(* ::Package:: *)

(* ::Title:: *)
(*Integration Problems for Products of Cosines and Cosines*)


(* ::Section::Closed:: *)
(*Cos[c+d x]^m (a+b Cos[c+d x])^n*)


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m*)
test[{grand_,var_,count_,ans_}]:=print[Simp[Int[grand,var]-ans]==0]

  p1={{a x,x,1,1/2*a*x^2},  (*simple trial *)
      {Cos[a + b*x], x, 1, Sin[a + b*x]/b},
      {Cos[a + b*x]^2, x, 1, x/2 + (Cos[a + b*x]*Sin[a + b*x])/(2*b)},
      {Cos[a + b*x]^3, x, 2, Sin[a + b*x]/b - Sin[a + b*x]^3/(3*b)},
      {Cos[a + b*x]^4, x, 2, (3*x)/8 + (3*Cos[a + b*x]*Sin[a + b*x])/(8*b) + (Cos[a + b*x]^3*Sin[a + b*x])/(4*b)},
      {Cos[a + b*x]^5, x, 2, Sin[a + b*x]/b - (2*Sin[a + b*x]^3)/(3*b) + Sin[a + b*x]^5/(5*b)}};

  Map[test,p1]


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2)*)
(**)

  p2={
    {Cos[a + b*x]^(5/2), x, 2, (6*EllipticE[(1/2)*(a + b*x), 2])/(5*b) + (2*Cos[a + b*x]^(3/2)*Sin[a + b*x])/(5*b)},
    {Cos[a + b*x]^(3/2), x, 2, (2*EllipticF[(1/2)*(a + b*x), 2])/(3*b) + (2*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(3*b)},
    {Cos[a + b*x]^(1/2), x, 1, (2*EllipticE[(1/2)*(a + b*x), 2])/b},
    {1/Cos[a + b*x]^(1/2), x, 1, (2*EllipticF[(1/2)*(a + b*x), 2])/b},
    {1/Cos[a + b*x]^(3/2), x, 2, -((2*EllipticE[(1/2)*(a + b*x), 2])/b) + (2*Sin[a + b*x])/(b*Sqrt[Cos[a + b*x]])},
{1/Cos[a + b*x]^(5/2), x, 2, (2*EllipticF[(1/2)*(a + b*x), 2])/(3*b) + (2*Sin[a + b*x])/(3*b*Cos[a + b*x]^(3/2))}
  }


(* ::Subsection::Closed:: *)
(*Integrands of the form (a+b Cos[c+d x])^n*)


{(a + b*Cos[c + d*x])^5, x, 4, (1/8)*a*(8*a^4 + 40*a^2*b^2 + 15*b^4)*x + (b*(107*a^4 + 192*a^2*b^2 + 16*b^4)*Sin[c + d*x])/(30*d) + (7*a*b^2*(22*a^2 + 23*b^2)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + (b*(47*a^2 + 16*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + (9*a*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (b*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)}
{(a + b*Cos[c + d*x])^4, x, 3, (1/8)*(8*a^4 + 24*a^2*b^2 + 3*b^4)*x + (a*b*(19*a^2 + 16*b^2)*Sin[c + d*x])/(6*d) + (b^2*(26*a^2 + 9*b^2)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (7*a*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)}
{(a + b*Cos[c + d*x])^3, x, 2, (1/2)*a*(2*a^2 + 3*b^2)*x + (2*b*(4*a^2 + b^2)*Sin[c + d*x])/(3*d) + (5*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)}
{(a + b*Cos[c + d*x])^2, x, 1, (1/2)*(2*a^2 + b^2)*x + (2*a*b*Sin[c + d*x])/d + (b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{(a + b*Cos[c + d*x]), x, 2, a*x + (b*Sin[c + d*x])/d}
{1/(a + b*Cos[c + d*x]), x, 1, (2*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*d)}
{1/(a + b*Cos[c + d*x])^2, x, 3, (2*a*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d) - (b*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{1/(a + b*Cos[c + d*x])^3, x, 4, ((2*a^2 + b^2)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - (b*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (3*a*b*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{1/(a + b*Cos[c + d*x])^4, x, 5, (a*(2*a^2 + 3*b^2)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) - (b*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (5*a*b*Sin[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (b*(11*a^2 + 4*b^2)*Sin[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}

{1/(3 + 5*Cos[c + d*x]), x, 1, ArcTanh[(1/2)*Tan[(1/2)*(c + d*x)]]/(2*d)}
{1/(3 + 5*Cos[c + d*x])^2, x, 3, -((3*ArcTanh[(1/2)*Tan[(1/2)*(c + d*x)]])/(32*d)) + (5*Sin[c + d*x])/(16*d*(3 + 5*Cos[c + d*x]))}
{1/(3 + 5*Cos[c + d*x])^3, x, 4, (43*ArcTanh[(1/2)*Tan[(1/2)*(c + d*x)]])/(1024*d) + (5*Sin[c + d*x])/(32*d*(3 + 5*Cos[c + d*x])^2) - (45*Sin[c + d*x])/(512*d*(3 + 5*Cos[c + d*x]))}
{1/(3 + 5*Cos[c + d*x])^4, x, 5, -((279*ArcTanh[(1/2)*Tan[(1/2)*(c + d*x)]])/(16384*d)) + (5*Sin[c + d*x])/(48*d*(3 + 5*Cos[c + d*x])^3) - (25*Sin[c + d*x])/(512*d*(3 + 5*Cos[c + d*x])^2) + (995*Sin[c + d*x])/(24576*d*(3 + 5*Cos[c + d*x]))}

{1/(5 + 3*Cos[c + d*x]), x, 1, x/4 - ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])]/(2*d)}
{1/(5 + 3*Cos[c + d*x])^2, x, 3, (5*x)/64 - (5*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(32*d) - (3*Sin[c + d*x])/(16*d*(5 + 3*Cos[c + d*x]))}
{1/(5 + 3*Cos[c + d*x])^3, x, 4, (59*x)/2048 - (59*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(1024*d) - (3*Sin[c + d*x])/(32*d*(5 + 3*Cos[c + d*x])^2) - (45*Sin[c + d*x])/(512*d*(5 + 3*Cos[c + d*x]))}
{1/(5 + 3*Cos[c + d*x])^4, x, 5, (385*x)/32768 - (385*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(16384*d) - Sin[c + d*x]/(16*d*(5 + 3*Cos[c + d*x])^3) - (25*Sin[c + d*x])/(512*d*(5 + 3*Cos[c + d*x])^2) - (311*Sin[c + d*x])/(8192*d*(5 + 3*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form (a+b Cos[c+d x])^(n/2)*)


{(a + b*Cos[c + d*x])^(7/2), x, 8, (32*a*(11*a^2 + 13*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(71*a^4 - 46*a^2*b^2 - 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(71*a^2 + 25*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (24*a*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*b*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)}
{(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(23*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (16*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (16*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}
{(a + b*Cos[c + d*x])^(3/2), x, 6, (8*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{(a + b*Cos[c + d*x])^(1/2), x, 2, (2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])}
{1/(a + b*Cos[c + d*x])^(1/2), x, 2, (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])}
{1/(a + b*Cos[c + d*x])^(3/2), x, 4, (2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/((a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*b*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])}
{1/(a + b*Cos[c + d*x])^(5/2), x, 7, (8*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (8*a*b*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])}
{1/(a + b*Cos[c + d*x])^(7/2), x, 8, (2*(23*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*(a^2 - b^2)^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (16*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*Sin[c + d*x])/(5*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(5/2)) - (16*a*b*Sin[c + d*x])/(15*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^(3/2)) - (2*b*(23*a^2 + 9*b^2)*Sin[c + d*x])/(15*(a^2 - b^2)^3*d*Sqrt[a + b*Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form (a+b Cos[c+d x])^(n/3)*)
(**)


{(a + b*Cos[c + d*x])^(5/3), x, 1, (3/5)*Int[((1/3)*(5*a^2 + 2*b^2) + (1/3)*(7*a*b)*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/3), x] + (3*b*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*d)}
{(a + b*Cos[c + d*x])^(4/3), x, 0, Int[(a + b*Cos[c + d*x])^(4/3), x]}
{(a + b*Cos[c + d*x])^(2/3), x, 0, Int[(a + b*Cos[c + d*x])^(2/3), x]}
{(a + b*Cos[c + d*x])^(1/3), x, 0, Int[(a + b*Cos[c + d*x])^(1/3), x]}
{1/(a + b*Cos[c + d*x])^(1/3), x, 0, Int[1/(a + b*Cos[c + d*x])^(1/3), x]}
{1/(a + b*Cos[c + d*x])^(2/3), x, 0, Int[1/(a + b*Cos[c + d*x])^(2/3), x]}
{1/(a + b*Cos[c + d*x])^(4/3), x, 1, -((3*Int[(-(a/3) - (1/3)*(2*b)*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/3), x])/(a^2 - b^2)) - (3*b*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x])^(1/3))}
{1/(a + b*Cos[c + d*x])^(5/3), x, 2, -((9*Int[((1/9)*(-2*a^2 + b^2) + (1/9)*(4*a*b)*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/3), x])/(2*(a^2 - b^2)^2)) - (3*b*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(2/3)) + (3*a*b*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(2*(a^2 - b^2)^2*d)}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (a+b Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]*(a + b*Cos[c + d*x]), x, 1, (b*x)/2 + (a*Sin[c + d*x])/d + (b*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{Cos[c + d*x]^2*(a + b*Cos[c + d*x]), x, 2, (a*x)/2 + (2*b*Sin[c + d*x])/(3*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (b*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^3*(a + b*Cos[c + d*x]), x, 3, (3*b*x)/8 + (2*a*Sin[c + d*x])/(3*d) + (3*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (b*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^4*(a + b*Cos[c + d*x]), x, 4, (3*a*x)/8 + (8*b*Sin[c + d*x])/(15*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (4*b*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (b*Cos[c + d*x]^4*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^5*(a + b*Cos[c + d*x]), x, 5, (5*b*x)/16 + (8*a*Sin[c + d*x])/(15*d) + (5*b*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (4*a*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (5*b*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) + (b*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^6*(a + b*Cos[c + d*x]), x, 6, (5*a*x)/16 + (16*b*Sin[c + d*x])/(35*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (8*b*Cos[c + d*x]^2*Sin[c + d*x])/(35*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (6*b*Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) + (b*Cos[c + d*x]^6*Sin[c + d*x])/(7*d)}


{Cos[c + d*x]*(a + b*Cos[c + d*x])^2, x, 2, a*b*x + ((3*a^2 + 2*b^2)*Sin[c + d*x])/(3*d) + (a*b*Cos[c + d*x]*Sin[c + d*x])/d + (b^2*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2, x, 3, (1/8)*(4*a^2 + 3*b^2)*x + (4*a*b*Sin[c + d*x])/(3*d) + ((4*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (2*a*b*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (b^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2, x, 4, (3*a*b*x)/4 + (2*(5*a^2 + 4*b^2)*Sin[c + d*x])/(15*d) + (3*a*b*Cos[c + d*x]*Sin[c + d*x])/(4*d) + ((5*a^2 + 4*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a*b*Cos[c + d*x]^3*Sin[c + d*x])/(2*d) + (b^2*Cos[c + d*x]^4*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^4*(a + b*Cos[c + d*x])^2, x, 5, (1/16)*(6*a^2 + 5*b^2)*x + (16*a*b*Sin[c + d*x])/(15*d) + ((6*a^2 + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (8*a*b*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + ((6*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (2*a*b*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) + (b^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^5*(a + b*Cos[c + d*x])^2, x, 6, (5*a*b*x)/8 + (8*(7*a^2 + 6*b^2)*Sin[c + d*x])/(105*d) + (5*a*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (4*(7*a^2 + 6*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + (5*a*b*Cos[c + d*x]^3*Sin[c + d*x])/(12*d) + ((7*a^2 + 6*b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + (a*b*Cos[c + d*x]^5*Sin[c + d*x])/(3*d) + (b^2*Cos[c + d*x]^6*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^6*(a + b*Cos[c + d*x])^2, x, 7, (5/128)*(8*a^2 + 7*b^2)*x + (32*a*b*Sin[c + d*x])/(35*d) + (5*(8*a^2 + 7*b^2)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (16*a*b*Cos[c + d*x]^2*Sin[c + d*x])/(35*d) + (5*(8*a^2 + 7*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (12*a*b*Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + ((8*a^2 + 7*b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (2*a*b*Cos[c + d*x]^6*Sin[c + d*x])/(7*d) + (b^2*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)}


{Cos[c + d*x]*(a + b*Cos[c + d*x])^3, x, 3, (3/8)*b*(4*a^2 + b^2)*x + (a*(a^2 + 4*b^2)*Sin[c + d*x])/(2*d) + (b*(2*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(4*d) + ((a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3, x, 4, (1/8)*a*(4*a^2 + 9*b^2)*x - ((3*a^4 - 52*a^2*b^2 - 16*b^4)*Sin[c + d*x])/(30*b*d) - (a*(6*a^2 - 71*b^2)*Cos[c + d*x]*Sin[c + d*x])/(120*d) - ((3*a^2 - 16*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) - (a*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + ((a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)}
{Cos[c + d*x]^3*(a + b*Cos[c + d*x])^3, x, 5, (1/16)*b*(18*a^2 + 5*b^2)*x + (2*a*(5*a^2 + 12*b^2)*Sin[c + d*x])/(15*d) + (b*(18*a^2 + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(5*a^2 + 12*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (b*(18*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (13*a*b^2*Cos[c + d*x]^4*Sin[c + d*x])/(30*d) + (b^2*Cos[c + d*x]^4*(a + b*Cos[c + d*x])*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^4*(a + b*Cos[c + d*x])^3, x, 6, (3/16)*a*(2*a^2 + 5*b^2)*x + (8*b*(7*a^2 + 2*b^2)*Sin[c + d*x])/(35*d) + (3*a*(2*a^2 + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (4*b*(7*a^2 + 2*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(35*d) + (a*(2*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) + (3*b*(7*a^2 + 2*b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + (5*a*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(14*d) + (b^2*Cos[c + d*x]^5*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^5*(a + b*Cos[c + d*x])^3, x, 7, (5/128)*b*(24*a^2 + 7*b^2)*x + (8*a*(7*a^2 + 18*b^2)*Sin[c + d*x])/(105*d) + (5*b*(24*a^2 + 7*b^2)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (4*a*(7*a^2 + 18*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + (5*b*(24*a^2 + 7*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (a*(7*a^2 + 18*b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + (b*(24*a^2 + 7*b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (17*a*b^2*Cos[c + d*x]^6*Sin[c + d*x])/(56*d) + (b^2*Cos[c + d*x]^6*(a + b*Cos[c + d*x])*Sin[c + d*x])/(8*d)}
{Cos[c + d*x]^6*(a + b*Cos[c + d*x])^3, x, 8, (5/128)*a*(8*a^2 + 21*b^2)*x + (16*b*(27*a^2 + 8*b^2)*Sin[c + d*x])/(315*d) + (5*a*(8*a^2 + 21*b^2)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (8*b*(27*a^2 + 8*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(315*d) + (5*a*(8*a^2 + 21*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (2*b*(27*a^2 + 8*b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(105*d) + (a*(8*a^2 + 21*b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (b*(27*a^2 + 8*b^2)*Cos[c + d*x]^6*Sin[c + d*x])/(63*d) + (19*a*b^2*Cos[c + d*x]^7*Sin[c + d*x])/(72*d) + (b^2*Cos[c + d*x]^7*(a + b*Cos[c + d*x])*Sin[c + d*x])/(9*d)}


{Cos[c + d*x]*(a + b*Cos[c + d*x])^4, x, 4, (1/2)*a*b*(4*a^2 + 3*b^2)*x + (2*(3*a^4 + 28*a^2*b^2 + 4*b^4)*Sin[c + d*x])/(15*d) + (a*b*(6*a^2 + 29*b^2)*Cos[c + d*x]*Sin[c + d*x])/(30*d) + ((3*a^2 + 4*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(15*d) + (a*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(5*d) + ((a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4, x, 5, (1/16)*(8*a^4 + 36*a^2*b^2 + 5*b^4)*x - (a*(4*a^4 - 121*a^2*b^2 - 128*b^4)*Sin[c + d*x])/(60*b*d) - ((8*a^4 - 178*a^2*b^2 - 75*b^4)*Cos[c + d*x]*Sin[c + d*x])/(240*d) - (a*(4*a^2 - 53*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) - ((4*a^2 - 25*b^2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*b*d) - (a*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + ((a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d)}
{Cos[c + d*x]^3*(a + b*Cos[c + d*x])^4, x, 6, (1/4)*a*b*(6*a^2 + 5*b^2)*x + (2*(35*a^4 + 168*a^2*b^2 + 24*b^4)*Sin[c + d*x])/(105*d) + (a*b*(6*a^2 + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(4*d) + ((35*a^4 + 168*a^2*b^2 + 24*b^4)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + (a*b*(6*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(6*d) + (b^2*(71*a^2 + 18*b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(105*d) + (8*a*b^2*Cos[c + d*x]^4*(a + b*Cos[c + d*x])*Sin[c + d*x])/(21*d) + (b^2*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^4*(a + b*Cos[c + d*x])^4, x, 7, (1/128)*(48*a^4 + 240*a^2*b^2 + 35*b^4)*x + (32*a*b*(7*a^2 + 6*b^2)*Sin[c + d*x])/(105*d) + ((48*a^4 + 240*a^2*b^2 + 35*b^4)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (16*a*b*(7*a^2 + 6*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + ((48*a^4 + 240*a^2*b^2 + 35*b^4)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (4*a*b*(7*a^2 + 6*b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + (b^2*(186*a^2 + 49*b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(336*d) + (9*a*b^2*Cos[c + d*x]^5*(a + b*Cos[c + d*x])*Sin[c + d*x])/(28*d) + (b^2*Cos[c + d*x]^5*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(8*d)}
{Cos[c + d*x]^5*(a + b*Cos[c + d*x])^4, x, 8, (5/32)*a*b*(8*a^2 + 7*b^2)*x + (8*(21*a^4 + 108*a^2*b^2 + 16*b^4)*Sin[c + d*x])/(315*d) + (5*a*b*(8*a^2 + 7*b^2)*Cos[c + d*x]*Sin[c + d*x])/(32*d) + (4*(21*a^4 + 108*a^2*b^2 + 16*b^4)*Cos[c + d*x]^2*Sin[c + d*x])/(315*d) + (5*a*b*(8*a^2 + 7*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(48*d) + ((21*a^4 + 108*a^2*b^2 + 16*b^4)*Cos[c + d*x]^4*Sin[c + d*x])/(105*d) + (a*b*(8*a^2 + 7*b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(12*d) + (b^2*(59*a^2 + 16*b^2)*Cos[c + d*x]^6*Sin[c + d*x])/(126*d) + (5*a*b^2*Cos[c + d*x]^6*(a + b*Cos[c + d*x])*Sin[c + d*x])/(18*d) + (b^2*Cos[c + d*x]^6*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^6*(a + b*Cos[c + d*x])^4, x, 9, (1/256)*(80*a^4 + 420*a^2*b^2 + 63*b^4)*x + (64*a*b*(9*a^2 + 8*b^2)*Sin[c + d*x])/(315*d) + ((80*a^4 + 420*a^2*b^2 + 63*b^4)*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (32*a*b*(9*a^2 + 8*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(315*d) + ((80*a^4 + 420*a^2*b^2 + 63*b^4)*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + (8*a*b*(9*a^2 + 8*b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(105*d) + ((80*a^4 + 420*a^2*b^2 + 63*b^4)*Cos[c + d*x]^5*Sin[c + d*x])/(480*d) + (4*a*b*(9*a^2 + 8*b^2)*Cos[c + d*x]^6*Sin[c + d*x])/(63*d) + (b^2*(292*a^2 + 81*b^2)*Cos[c + d*x]^7*Sin[c + d*x])/(720*d) + (11*a*b^2*Cos[c + d*x]^7*(a + b*Cos[c + d*x])*Sin[c + d*x])/(45*d) + (b^2*Cos[c + d*x]^7*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(10*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]/(a + b*Cos[c + d*x]), x, 2, x/b - (2*a*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2]*d)}
{Cos[c + d*x]^2/(a + b*Cos[c + d*x]), x, 4, -((a*x)/b^2) + (2*a^2*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]*d) + Sin[c + d*x]/(b*d)}
{Cos[c + d*x]^3/(a + b*Cos[c + d*x]), x, 4, ((2*a^2 + b^2)*x)/(2*b^3) - (2*a^3*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]*d) - (a*Sin[c + d*x])/(b^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)}
{Cos[c + d*x]^4/(a + b*Cos[c + d*x]), x, 5, -((a*(2*a^2 + b^2)*x)/(2*b^4)) + (2*a^4*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*d) + ((3*a^2 + 2*b^2)*Sin[c + d*x])/(3*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)}
{Cos[c + d*x]^5/(a + b*Cos[c + d*x]), x, 6, ((8*a^4 + 4*a^2*b^2 + 3*b^4)*x)/(8*b^5) - (2*a^5*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*Sqrt[a^2 - b^2]*d) - (a*(3*a^2 + 2*b^2)*Sin[c + d*x])/(3*b^4*d) + ((4*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) - (a*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)}
{Cos[c + d*x]^6/(a + b*Cos[c + d*x]), x, 7, -((a*(8*a^4 + 4*a^2*b^2 + 3*b^4)*x)/(8*b^6)) + (2*a^6*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*Sqrt[a^2 - b^2]*d) + ((15*a^4 + 10*a^2*b^2 + 8*b^4)*Sin[c + d*x])/(15*b^5*d) - (a*(4*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^4*d) + ((5*a^2 + 4*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(15*b^3*d) - (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*b^2*d) + (Cos[c + d*x]^4*Sin[c + d*x])/(5*b*d)}


{Cos[c + d*x]/(a + b*Cos[c + d*x])^2, x, 3, -((2*b*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d)) + (a*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^2/(a + b*Cos[c + d*x])^2, x, 3, x/b^2 - (2*a*(a^2 - 2*b^2)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(3/2)*d) - (a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^3/(a + b*Cos[c + d*x])^2, x, 4, -((2*a*x)/b^3) + (2*a^2*(2*a^2 - 3*b^2)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*d) + ((2*a^2 - b^2)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) - (a^2*Cos[c + d*x]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^4/(a + b*Cos[c + d*x])^2, x, 5, ((6*a^2 + b^2)*x)/(2*b^4) - (2*a^3*(3*a^2 - 4*b^2)*ArcTanh[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[-a^2 + b^2]])/(b^4*(-a^2 + b^2)^(3/2)*d) - (2*a*Sin[c + d*x])/(b^3*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) - (a^4*Sin[c + d*x])/(b^3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])), ((6*a^2 + b^2)*x)/(2*b^4) - (2*a^3*(3*a^2 - 4*b^2)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(3/2)*d) - (a*(3*a^2 - 2*b^2)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((3*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) - (a^2*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^5/(a + b*Cos[c + d*x])^2, x, 6, -((a*(4*a^2 + b^2)*x)/b^5) + (2*a^4*(4*a^2 - 5*b^2)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(3/2)*d) + ((12*a^4 - 7*a^2*b^2 - 2*b^4)*Sin[c + d*x])/(3*b^4*(a^2 - b^2)*d) - (a*(2*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((4*a^2 - b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^6/(a + b*Cos[c + d*x])^2, x, 7, ((40*a^4 + 12*a^2*b^2 + 3*b^4)*x)/(8*b^6) - (2*a^5*(5*a^2 - 6*b^2)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*(a^2 - b^2)^(3/2)*d) - (a*(15*a^4 - 8*a^2*b^2 - 4*b^4)*Sin[c + d*x])/(3*b^5*(a^2 - b^2)*d) + ((20*a^4 - 9*a^2*b^2 - 3*b^4)*Cos[c + d*x]*Sin[c + d*x])/(8*b^4*(a^2 - b^2)*d) - (a*(5*a^2 - 2*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d) + ((5*a^2 - b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(4*b^2*(a^2 - b^2)*d) - (a^2*Cos[c + d*x]^4*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}


{Cos[c + d*x]/(a + b*Cos[c + d*x])^3, x, 4, -((3*a*b*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d)) + (a*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2 + 2*b^2)*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^2/(a + b*Cos[c + d*x])^3, x, 4, ((a^2 + 2*b^2)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - (a^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(a^2 - 4*b^2)*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^3/(a + b*Cos[c + d*x])^3, x, 4, x/b^3 - (a*(2*a^4 - 5*a^2*b^2 + 6*b^4)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(5/2)*d) - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a^2*(2*a^2 - 5*b^2)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^4/(a + b*Cos[c + d*x])^3, x, 5, -((3*a*x)/b^4) + (3*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(5/2)*d) + ((6*a^4 - 11*a^2*b^2 + 2*b^4)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - (a^2*Cos[c + d*x]^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (3*a^2*(a^2 - 2*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^5/(a + b*Cos[c + d*x])^3, x, 6, ((12*a^2 + b^2)*x)/(2*b^5) - (a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(5/2)*d) - (3*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) + ((6*a^4 - 10*a^2*b^2 + b^4)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a^2*(4*a^2 - 7*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^6/(a + b*Cos[c + d*x])^3, x, 7, -((a*(20*a^2 + 3*b^2)*x)/(2*b^6)) + (a^4*(20*a^4 - 47*a^2*b^2 + 30*b^4)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*(a^2 - b^2)^(5/2)*d) + ((60*a^6 - 101*a^4*b^2 + 28*a^2*b^4 + 4*b^6)*Sin[c + d*x])/(6*b^5*(a^2 - b^2)^2*d) - (a*(10*a^4 - 16*a^2*b^2 + 3*b^4)*Cos[c + d*x]*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) + ((20*a^4 - 31*a^2*b^2 + 2*b^4)*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^2*d) - (a^2*Cos[c + d*x]^4*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a^2*(5*a^2 - 8*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}


{Cos[c + d*x]/(a + b*Cos[c + d*x])^4, x, 5, -((b*(4*a^2 + b^2)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d)) + (a*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((2*a^2 + 3*b^2)*Sin[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(2*a^2 + 13*b^2)*Sin[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^2/(a + b*Cos[c + d*x])^4, x, 5, (a*(a^2 + 4*b^2)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) - (a^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (a*(a^2 - 6*b^2)*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((a^4 - 10*a^2*b^2 - 6*b^4)*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^3/(a + b*Cos[c + d*x])^4, x, 5, -((b*(3*a^2 + 2*b^2)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d)) - (a^2*Cos[c + d*x]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a^2*(2*a^2 - 7*b^2)*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(2*a^4 - 5*a^2*b^2 + 18*b^4)*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^4/(a + b*Cos[c + d*x])^4, x, 5, x/b^4 - (a*(2*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 8*b^6)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(7/2)*d) - (a^2*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a^2*(3*a^2 - 8*b^2)*Cos[c + d*x]*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (a^2*(6*a^4 - 17*a^2*b^2 + 26*b^4)*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^5/(a + b*Cos[c + d*x])^4, x, 6, -((4*a*x)/b^5) + (a^2*(8*a^6 - 28*a^4*b^2 + 35*a^2*b^4 - 20*b^6)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(7/2)*d) + ((24*a^6 - 68*a^4*b^2 + 65*a^2*b^4 - 6*b^6)*Sin[c + d*x])/(6*b^4*(a^2 - b^2)^3*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a^2*(4*a^2 - 9*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (a^2*(4*a^4 - 11*a^2*b^2 + 12*b^4)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^6/(a + b*Cos[c + d*x])^4, x, 7, ((20*a^2 + b^2)*x)/(2*b^6) - (a^3*(20*a^6 - 69*a^4*b^2 + 84*a^2*b^4 - 40*b^6)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*(a^2 - b^2)^(7/2)*d) - (a*(60*a^6 - 167*a^4*b^2 + 146*a^2*b^4 - 24*b^6)*Sin[c + d*x])/(6*b^5*(a^2 - b^2)^3*d) + ((10*a^6 - 27*a^4*b^2 + 23*a^2*b^4 - b^6)*Cos[c + d*x]*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d) - (a^2*Cos[c + d*x]^4*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (5*a^2*(a^2 - 2*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (a^2*(20*a^4 - 53*a^2*b^2 + 48*b^4)*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (a+b Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]], x, 6, (2*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]], x, 7, -((2*(2*a^2 - 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (4*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)}
{Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]], x, 8, (2*a*(8*a^2 + 19*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(8*a^4 + 17*a^2*b^2 - 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(4*a^2 - 25*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^2*d) + (2*a*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b*d) + (2*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]], x, 9, -((2*(16*a^4 + 24*a^2*b^2 - 147*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (8*a*(4*a^4 + 5*a^2*b^2 - 9*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(8*a^2 + 13*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^3*d) - (2*(6*a^2 - 49*b^2)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) + (2*a*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*b*d) + (2*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*d)}


{Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2), x, 7, (2*(a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(5*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(5*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2), x, 8, -((4*a*(3*a^2 - 41*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(6*a^4 - 31*a^2*b^2 + 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*a^2 - 25*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) - (4*a*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)}
{Cos[c + d*x]^3*(a + b*Cos[c + d*x])^(3/2), x, 9, (2*(8*a^4 + 33*a^2*b^2 + 147*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(8*a^4 + 31*a^2*b^2 - 39*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (8*a*(a^2 - 22*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) + (2*(3*a^2 + 49*b^2)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) + (20*a*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*b*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^4*(a + b*Cos[c + d*x])^(3/2), x, 10, -((8*a*(4*a^4 + 9*a^2*b^2 - 174*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(1155*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(16*a^6 + 32*a^4*b^2 - 273*a^2*b^4 + 225*b^6)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(1155*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*a^4 + 19*a^2*b^2 + 225*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1155*b^3*d) - (4*a*(3*a^2 - 101*b^2)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1155*b^2*d) + (2*(a^2 + 27*b^2)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(231*b*d) + (8*a*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(33*d) + (2*b*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(11*d)}


{Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2), x, 8, (2*a*(3*a^2 + 29*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(21*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(3*a^4 + 2*a^2*b^2 - 5*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(21*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*a^2 + 5*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2), x, 9, -((2*(10*a^4 - 279*a^2*b^2 - 147*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (4*a*(5*a^4 - 62*a^2*b^2 + 57*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*(5*a^2 - 57*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) - (2*(10*a^2 - 49*b^2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) - (4*a*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)}
{Cos[c + d*x]^3*(a + b*Cos[c + d*x])^(5/2), x, 10, (2*a*(8*a^4 + 51*a^2*b^2 + 741*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(693*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(8*a^6 + 49*a^4*b^2 + 78*a^2*b^4 - 135*b^6)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(693*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(4*a^4 - 205*a^2*b^2 - 135*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*b^2*d) + (2*a*(3*a^2 + 229*b^2)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*b*d) + (2*(113*a^2 + 81*b^2)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (46*a*b*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(99*d) + (2*b^2*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^4*(a + b*Cos[c + d*x])^(5/2), x, 11, -((2*(240*a^6 + 760*a^4*b^2 - 30669*a^2*b^4 - 17787*b^6)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (8*a*(60*a^6 + 175*a^4*b^2 - 3496*a^2*b^4 + 3261*b^6)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(120*a^4 + 395*a^2*b^2 + 18973*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^3*d) - (2*(90*a^4 - 8145*a^2*b^2 - 5929*b^4)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^2*d) + (2*a*(15*a^2 + 2209*b^2)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9009*b*d) + (2*(159*a^2 + 121*b^2)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1287*d) + (54*a*b*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(143*d) + (2*b^2*Cos[c + d*x]^5*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(13*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]/Sqrt[a + b*Cos[c + d*x]], x, 5, (2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]])}
{Cos[c + d*x]^2/Sqrt[a + b*Cos[c + d*x]], x, 6, -((4*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(2*a^2 + b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)}
{Cos[c + d*x]^3/Sqrt[a + b*Cos[c + d*x]], x, 7, (2*(8*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(8*a^2 + 7*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (8*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d)}
{Cos[c + d*x]^4/Sqrt[a + b*Cos[c + d*x]], x, 8, -((8*a*(12*a^2 + 11*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(48*a^4 + 32*a^2*b^2 + 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(24*a^2 + 25*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^3*d) - (12*a*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b^2*d) + (2*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b*d)}


{Cos[c + d*x]/(a + b*Cos[c + d*x])^(3/2), x, 6, -((2*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])}
{Cos[c + d*x]^2/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*(2*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])}
{Cos[c + d*x]^3/(a + b*Cos[c + d*x])^(3/2), x, 7, -((2*a*(8*a^2 - 5*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(8*a^2 + b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Cos[c + d*x]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(4*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d)}
{Cos[c + d*x]^4/(a + b*Cos[c + d*x])^(3/2), x, 8, (2*(16*a^4 - 8*a^2*b^2 - 3*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(5*b^4*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (8*a*(4*a^2 + b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(5*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a*(8*a^2 - 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^3*(a^2 - b^2)*d) + (2*(6*a^2 - b^2)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d)}


{Cos[c + d*x]/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(a^2 + 3*b^2)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])}
{Cos[c + d*x]^2/(a + b*Cos[c + d*x])^(5/2), x, 7, -((4*a*(a^2 - 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(2*a^2 - 3*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (4*a*(a^2 - 3*b^2)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])}
{Cos[c + d*x]^3/(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(8*a^4 - 15*a^2*b^2 + 3*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(8*a^2 - 9*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Cos[c + d*x]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (8*a^2*(a^2 - 2*b^2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])}
{Cos[c + d*x]^4/(a + b*Cos[c + d*x])^(5/2), x, 8, -((8*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(16*a^4 - 16*a^2*b^2 - b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (4*a^2*(3*a^2 - 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*a^4 - 13*a^2*b^2 + b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d)}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (a+b Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x]), x, 6, (6*a*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (10*b*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]), x, 5, (6*b*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^(1/2)*(a + b*Cos[c + d*x]), x, 4, (2*a*EllipticE[(1/2)*(c + d*x), 2])/d + (2*b*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^(-1/2)*(a + b*Cos[c + d*x]), x, 3, (2*b*EllipticE[(1/2)*(c + d*x), 2])/d + (2*a*EllipticF[(1/2)*(c + d*x), 2])/d}
{Cos[c + d*x]^(-3/2)*(a + b*Cos[c + d*x]), x, 4, -((2*a*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*b*EllipticF[(1/2)*(c + d*x), 2])/d + (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(a + b*Cos[c + d*x]), x, 5, -((2*b*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*b*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)*(a + b*Cos[c + d*x]), x, 6, -((6*a*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*b*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*b*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*a*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])}


{Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2, x, 7, (2*(9*a^2 + 7*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (20*a*b*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (20*a*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*a^2 + 7*b^2)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (4*a*b*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*b^2*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2, x, 6, (12*a*b*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(7*a^2 + 5*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (4*a*b*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(1/2)*(a + b*Cos[c + d*x])^2, x, 5, (2*(5*a^2 + 3*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a*b*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (4*a*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^(-1/2)*(a + b*Cos[c + d*x])^2, x, 4, (4*a*b*EllipticE[(1/2)*(c + d*x), 2])/d + (2*(3*a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^(-3/2)*(a + b*Cos[c + d*x])^2, x, 4, -((2*(a^2 - b^2)*EllipticE[(1/2)*(c + d*x), 2])/d) + (4*a*b*EllipticF[(1/2)*(c + d*x), 2])/d + (2*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(a + b*Cos[c + d*x])^2, x, 5, -((4*a*b*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(a^2 + 3*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (4*a*b*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)*(a + b*Cos[c + d*x])^2, x, 6, -((2*(3*a^2 + 5*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a*b*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*a*b*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*a^2 + 5*b^2)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])}


{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3, x, 7, (2*b*(27*a^2 + 7*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*a*(7*a^2 + 15*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(7*a^2 + 15*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(27*a^2 + 7*b^2)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (40*a*b^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*b^2*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(1/2)*(a + b*Cos[c + d*x])^3, x, 6, (2*a*(5*a^2 + 9*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*b*(21*a^2 + 5*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b*(21*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (32*a*b^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b^2*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(-1/2)*(a + b*Cos[c + d*x])^3, x, 5, (6*b*(5*a^2 + b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/d + (8*a*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b^2*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^(-3/2)*(a + b*Cos[c + d*x])^3, x, 5, -((2*a*(a^2 - 3*b^2)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*b*(9*a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*b*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(a + b*Cos[c + d*x])^3, x, 5, -((2*b*(3*a^2 - b^2)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*(a^2 + 9*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (16*a^2*b*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(a + b*Cos[c + d*x])^3, x, 6, -((6*a*(a^2 + 5*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*b*(a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/d + (8*a^2*b*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (6*a*(a^2 + 5*b^2)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x]), x, 6, -((2*a*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d)) + (2*(3*a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^3*d) - (2*a^3*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^3*(a + b)*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)}
{Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x]), x, 5, (2*EllipticE[(1/2)*(c + d*x), 2])/(b*d) - (2*a*EllipticF[(1/2)*(c + d*x), 2])/(b^2*d) + (2*a^2*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^2*(a + b)*d)}
{Cos[c + d*x]^(1/2)/(a + b*Cos[c + d*x]), x, 3, (2*EllipticF[(1/2)*(c + d*x), 2])/(b*d) - (2*a*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b*(a + b)*d)}
{Cos[c + d*x]^(-1/2)/(a + b*Cos[c + d*x]), x, 1, (2*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a + b)*d)}
{Cos[c + d*x]^(-3/2)/(a + b*Cos[c + d*x]), x, 5, -((2*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) - (2*b*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*(a + b)*d) + (2*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)/(a + b*Cos[c + d*x]), x, 7, (2*b*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + (2*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (2*b^2*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a + b)*d) + (2*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*b*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]])}


{Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^2, x, 6, ((3*a^2 - 2*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d) - (a*(3*a^2 - 4*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d) + (a^2*(3*a^2 - 5*b^2)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^3*(a + b)^2*d) - (a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^2, x, 6, -((a*EllipticE[(1/2)*(c + d*x), 2])/(b*(a^2 - b^2)*d)) + ((a^2 - 2*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d) - (a*(a^2 - 3*b^2)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^2*(a + b)^2*d) + (a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^2, x, 6, EllipticE[(1/2)*(c + d*x), 2]/((a^2 - b^2)*d) + (a*EllipticF[(1/2)*(c + d*x), 2])/(b*(a^2 - b^2)*d) - ((a^2 + b^2)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b*(a + b)^2*d) - (b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-1/2)/(a + b*Cos[c + d*x])^2, x, 6, -((b*EllipticE[(1/2)*(c + d*x), 2])/(a*(a^2 - b^2)*d)) - EllipticF[(1/2)*(c + d*x), 2]/((a^2 - b^2)*d) + ((3*a^2 - b^2)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*(a - b)*(a + b)^2*d) + (b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)/(a + b*Cos[c + d*x])^2, x, 7, -(((2*a^2 - 3*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*(a^2 - b^2)*d)) + (b*EllipticF[(1/2)*(c + d*x), 2])/(a*(a^2 - b^2)*d) - (b*(5*a^2 - 3*b^2)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2 - 3*b^2)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)/(a + b*Cos[c + d*x])^2, x, 8, (b*(4*a^2 - 5*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(a^3*(a^2 - b^2)*d) + ((2*a^2 - 5*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*(a^2 - b^2)*d) + (b^2*(7*a^2 - 5*b^2)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^3*(a - b)*(a + b)^2*d) + ((2*a^2 - 5*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) - (b*(4*a^2 - 5*b^2)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))}


{Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^3, x, 7, -((3*a*(a^2 - 3*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^2*(a^2 - b^2)^2*d)) + ((3*a^4 - 5*a^2*b^2 + 8*b^4)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d) - (3*a*(a^4 - 2*a^2*b^2 + 5*b^4)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^3*(a + b)^3*d) - (a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (3*a*(a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^3, x, 7, -(((a^2 + 5*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(4*b*(a^2 - b^2)^2*d)) + (a*(a^2 - 7*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^2*(a^2 - b^2)^2*d) - ((a^4 - 10*a^2*b^2 - 3*b^4)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^2*(a + b)^3*d) + (a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^3, x, 7, ((5*a^2 + b^2)*EllipticE[(1/2)*(c + d*x), 2])/(4*a*(a^2 - b^2)^2*d) + (3*(a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/(4*b*(a^2 - b^2)^2*d) - ((3*a^4 + 10*a^2*b^2 - b^4)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a*(a - b)^2*b*(a + b)^3*d) - (b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (b*(5*a^2 + b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-1/2)/(a + b*Cos[c + d*x])^3, x, 7, -((3*b*(3*a^2 - b^2)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^2*(a^2 - b^2)^2*d)) - ((7*a^2 - b^2)*EllipticF[(1/2)*(c + d*x), 2])/(4*a*(a^2 - b^2)^2*d) + (3*(5*a^4 - 2*a^2*b^2 + b^4)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^2*(a - b)^2*(a + b)^3*d) + (b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (3*b^2*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)/(a + b*Cos[c + d*x])^3, x, 8, -(((8*a^4 - 29*a^2*b^2 + 15*b^4)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^3*(a^2 - b^2)^2*d)) + (b*(11*a^2 - 5*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(4*a^2*(a^2 - b^2)^2*d) - (b*(35*a^4 - 38*a^2*b^2 + 15*b^4)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 29*a^2*b^2 + 15*b^4)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2) + (b^2*(11*a^2 - 5*b^2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)/(a + b*Cos[c + d*x])^3, x, 9, (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*EllipticF[(1/2)*(c + d*x), 2])/(12*a^3*(a^2 - b^2)^2*d) + (b^2*(63*a^4 - 86*a^2*b^2 + 35*b^4)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2) + (b^2*(13*a^2 - 7*b^2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (a+b Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(1/2), x, 7, -(((3*a^2 - 16*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(24*b^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (a*(a - 14*b)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(12*b*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(a^2 + 4*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (a*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*b*d) + (Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) - ((3*a^2 - 16*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(24*b^2*d)}
{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(1/2), x, 6, (a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((a - 2*b)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) - ((a^2 - 4*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*b*d*Sqrt[a + b*Cos[c + d*x]]) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (a*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(4*b*d)}
{Cos[c + d*x]^(1/2)*(a + b*Cos[c + d*x])^(1/2), x, 5, (Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - (2*a*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/d}
{Cos[c + d*x]^(-1/2)*(a + b*Cos[c + d*x])^(1/2), x, 3, (2*(a - b)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (4*b*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]])}
{Cos[c + d*x]^(-3/2)*(a + b*Cos[c + d*x])^(1/2), x, 5, -((2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a + b)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/d}
{Cos[c + d*x]^(-5/2)*(a + b*Cos[c + d*x])^(1/2), x, 6, -((2*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a + b)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Sqrt[Cos[c + d*x]]) - (2*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a*d)}
{Cos[c + d*x]^(-7/2)*(a + b*Cos[c + d*x])^(1/2), x, 8, -((2*(9*a^2 - 2*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*a^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(9*a^2 + 7*a*b - 2*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2)) + (2*(9*a^2 - 2*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Sqrt[Cos[c + d*x]]) - (2*(9*a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(15*a^2*d)}
{Cos[c + d*x]^(-9/2)*(a + b*Cos[c + d*x])^(1/2), x, 8, -((2*b*(19*a^2 + 8*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*a^3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(25*a^3 + 19*a^2*b + 2*a*b^2 + 8*b^3)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2 - 4*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Cos[c + d*x]^(3/2)) + (2*b*(19*a^2 + 8*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^3*d*Sqrt[Cos[c + d*x]]) - (2*b*(19*a^2 + 8*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(105*a^3*d)}


{Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2), x, 8, -((a*(3*a^2 - 52*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(64*b^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + ((a^3 - 38*a^2*b + 12*a*b^2 - 24*b^3)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(32*b*d*Sqrt[a + b*Cos[c + d*x]]) + (3*(a^4 + 8*a^2*b^2 + 16*b^4)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(32*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((a^2 + 12*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d) + (3*a*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (b*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (a*(3*a^2 - 52*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(64*b^2*d)}
{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2), x, 7, ((3*a^2 + 16*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(24*b*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + (a*(7*a - 26*b)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(12*d*Sqrt[a + b*Cos[c + d*x]]) - (a*(a^2 - 12*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b*d*Sqrt[a + b*Cos[c + d*x]]) + (7*a*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (b*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + ((3*a^2 + 16*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(24*b*d)}
{Cos[c + d*x]^(1/2)*(a + b*Cos[c + d*x])^(3/2), x, 6, (5*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - ((4*a^2 - a*b + 2*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*a^2 + 4*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) + (b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (5*a*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(4*d)}
{Cos[c + d*x]^(-1/2)*(a + b*Cos[c + d*x])^(3/2), x, 7, (b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + (2*a*(a - 2*b)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (6*a*b*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/d}
{Cos[c + d*x]^(-3/2)*(a + b*Cos[c + d*x])^(3/2), x, 6, -((2*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a^2 + 2*a*b - b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (4*b^2*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*a*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/d}
{Cos[c + d*x]^(-5/2)*(a + b*Cos[c + d*x])^(3/2), x, 6, -((8*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a^2 + 4*a*b + 3*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (8*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) - (8*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*d)}
{Cos[c + d*x]^(-7/2)*(a + b*Cos[c + d*x])^(3/2), x, 7, -((2*(3*a^2 + b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(5*a*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(3*a^2 + 4*a*b + b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(5*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (2*(3*a^2 + b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]) - (2*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(5*a*d)}
{Cos[c + d*x]^(-9/2)*(a + b*Cos[c + d*x])^(3/2), x, 9, -((4*b*(41*a^2 - 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*a^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(25*a^3 + 82*a^2*b + 51*a*b^2 - 6*b^3)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (16*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Cos[c + d*x]^(3/2)) + (4*b*(41*a^2 - 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Sqrt[Cos[c + d*x]]) - (4*b*(41*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(105*a^2*d)}


{Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2), x, 9, -(((45*a^4 - 1692*a^2*b^2 - 1024*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(1920*b^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (a*(15*a^3 - 1146*a^2*b + 772*a*b^2 - 2312*b^3)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(960*b*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(3*a^4 + 40*a^2*b^2 + 240*b^4)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(64*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(15*a^2 + 772*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(960*b*d) + ((93*a^2 + 64*b^2)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(240*d) + (21*a*b*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(40*d) + (b^2*Cos[c + d*x]^(7/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d) - ((45*a^4 - 1692*a^2*b^2 - 1024*b^4)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(1920*b^2*d)}
{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2), x, 8, (a*(15*a^2 + 284*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(192*b*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((59*a^3 - 322*a^2*b + 36*a*b^2 - 72*b^3)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(96*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*a^4 - 120*a^2*b^2 - 48*b^4)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(32*b*d*Sqrt[a + b*Cos[c + d*x]]) + ((59*a^2 + 36*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(96*d) + (17*a*b*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d) + (b^2*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (a*(15*a^2 + 284*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(192*b*d)}
{Cos[c + d*x]^(1/2)*(a + b*Cos[c + d*x])^(5/2), x, 7, ((33*a^2 + 16*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(24*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - (a*(24*a^2 - 13*a*b + 38*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(12*d*Sqrt[a + b*Cos[c + d*x]]) + (5*a*(a^2 + 4*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + (13*a*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (b^2*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + ((33*a^2 + 16*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(24*d)}
{Cos[c + d*x]^(-1/2)*(a + b*Cos[c + d*x])^(5/2), x, 6, (9*a*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((4*a^3 - 12*a^2*b + a*b^2 - 2*b^3)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(15*a^2 + 4*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) + (b^2*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (9*a*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(4*d)}
{Cos[c + d*x]^(-3/2)*(a + b*Cos[c + d*x])^(5/2), x, 6, -(((2*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*a*(a^2 + 3*a*b - 3*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (10*a*b^2*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((2*a^2 - b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/d}
{Cos[c + d*x]^(-5/2)*(a + b*Cos[c + d*x])^(5/2), x, 7, -((14*a*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a^3 + 7*a^2*b + 9*a*b^2 - 3*b^3)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (4*b^3*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (14*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) - (14*a*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*d)}
{Cos[c + d*x]^(-7/2)*(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(9*a^2 + 23*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(9*a^3 + 17*a^2*b + 23*a*b^2 + 15*b^3)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (22*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*(9*a^2 + 23*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) - (2*(9*a^2 + 23*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(15*d)}
{Cos[c + d*x]^(-9/2)*(a + b*Cos[c + d*x])^(5/2), x, 8, -((2*b*(29*a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(21*a*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(5*a^3 + 29*a^2*b + 27*a*b^2 + 3*b^3)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(21*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (6*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(5/2)) + (2*(5*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*b*(29*a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*a*d*Sqrt[Cos[c + d*x]]) - (2*b*(29*a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(21*a*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{1/(Sqrt[Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]]), x, 1, (2*EllipticF[ArcSin[Tan[(c+d*x)/2]], 1/5])/(d*Sqrt[5])}
{1/(Sqrt[Cos[c + d*x]]*Sqrt[-2 + 3*Cos[c + d*x]]), x, 1, (2*EllipticF[ArcSin[Tan[(c+d*x)/2]], 5])/d}

{1/(Sqrt[Cos[c + d*x]]*Sqrt[2 - 3*Cos[c + d*x]]), x, 2, -(2*Sqrt[-Cos[c + d*x]]*EllipticF[ArcSin[Cot[(c+d*x)/2]], 1/5])/(d*Sqrt[5]*Sqrt[Cos[c + d*x]])}
{1/(Sqrt[Cos[c + d*x]]*Sqrt[-2 - 3*Cos[c + d*x]]), x, 2, -((2*Sqrt[-Cos[c + d*x]]*EllipticF[ArcSin[Cot[(c+d*x)/2]], 5])/(d*Sqrt[Cos[c + d*x]]))}

{1/(Sqrt[Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]), x, 1, (2*Sqrt[1 + Cos[c + d*x]]*Sqrt[(3 + 2*Cos[c + d*x])/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -(1/5)])/(Sqrt[5]*d*Sqrt[3 + 2*Cos[c + d*x]])}
{1/(Sqrt[Cos[c + d*x]]*Sqrt[3 - 2*Cos[c + d*x]]), x, 1, (2*Sqrt[1 + Cos[c + d*x]]*Sqrt[(3 - 2*Cos[c + d*x])/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -5])/(d*Sqrt[3 - 2*Cos[c + d*x]])}

{1/(Sqrt[Cos[c + d*x]]*Sqrt[-3 + 2*Cos[c + d*x]]), x, 1, (2*Sqrt[1 + Cos[c + d*x]]*Sqrt[(-3 + 2*Cos[c + d*x])/(-1 - Cos[c + d*x])]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -5])/(d*Sqrt[-3 + 2*Cos[c + d*x]])}
{1/(Sqrt[Cos[c + d*x]]*Sqrt[-3 - 2*Cos[c + d*x]]), x, 1, (2*Sqrt[-((-3 - 2*Cos[c + d*x])/(1 + Cos[c + d*x]))]*Sqrt[1 + Cos[c + d*x]]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -(1/5)])/(Sqrt[5]*d*Sqrt[-3 - 2*Cos[c + d*x]])}


{Sqrt[Cos[c + d*x]]/Sqrt[2 + 3*Cos[c + d*x]], x, 3, -((2*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], 1/5])/(Sqrt[5]*d)) + (4*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], 1/5])/(Sqrt[5]*d)}
{Sqrt[Cos[c + d*x]]/Sqrt[-2 + 3*Cos[c + d*x]], x, 3, -((2*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], 5])/d) + (4*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], 5])/d}

{Sqrt[Cos[c + d*x]]/Sqrt[2 - 3*Cos[c + d*x]], x, 4, (2*Sqrt[-Cos[c + d*x]]*EllipticF[ArcCsc[Tan[(1/2)*(c + d*x)]], 1/5])/(Sqrt[5]*d*Sqrt[Cos[c + d*x]]) + (4*Sqrt[(2 - 3*Cos[c + d*x])/(-1 - Cos[c + d*x])]*Sqrt[1 + Cos[c + d*x]]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], 5])/(d*Sqrt[2 - 3*Cos[c + d*x]])}
{Sqrt[Cos[c + d*x]]/Sqrt[-2 - 3*Cos[c + d*x]], x, 4, (2*Sqrt[-Cos[c + d*x]]*EllipticF[ArcCsc[Tan[(1/2)*(c + d*x)]], 5])/(d*Sqrt[Cos[c + d*x]]) + (4*Sqrt[-((-2 - 3*Cos[c + d*x])/(1 + Cos[c + d*x]))]*Sqrt[1 + Cos[c + d*x]]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], 1/5])/(Sqrt[5]*d*Sqrt[-2 - 3*Cos[c + d*x]])}

{Sqrt[Cos[c + d*x]]/Sqrt[3 + 2*Cos[c + d*x]], x, 3, -((2*Sqrt[1 + Cos[c + d*x]]*Sqrt[(3 + 2*Cos[c + d*x])/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -(1/5)])/(Sqrt[5]*d*Sqrt[3 + 2*Cos[c + d*x]])) + (4*Sqrt[1 + Cos[c + d*x]]*Sqrt[(3 + 2*Cos[c + d*x])/(1 + Cos[c + d*x])]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -(1/5)])/(Sqrt[5]*d*Sqrt[3 + 2*Cos[c + d*x]])}
{Sqrt[Cos[c + d*x]]/Sqrt[3 - 2*Cos[c + d*x]], x, 3, -((2*Sqrt[(3 - 2*Cos[c + d*x])/(1 + Cos[c + d*x])]*Sqrt[1 + Cos[c + d*x]]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -5])/(d*Sqrt[3 - 2*Cos[c + d*x]])) + (4*Sqrt[(3 - 2*Cos[c + d*x])/(1 + Cos[c + d*x])]*Sqrt[1 + Cos[c + d*x]]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -5])/(d*Sqrt[3 - 2*Cos[c + d*x]])}

{Sqrt[Cos[c + d*x]]/Sqrt[-3 + 2*Cos[c + d*x]], x, 3, -((2*Sqrt[1 + Cos[c + d*x]]*Sqrt[(-3 + 2*Cos[c + d*x])/(-1 - Cos[c + d*x])]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -5])/(d*Sqrt[-3 + 2*Cos[c + d*x]])) + (4*Sqrt[1 + Cos[c + d*x]]*Sqrt[(-3 + 2*Cos[c + d*x])/(-1 - Cos[c + d*x])]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -5])/(d*Sqrt[-3 + 2*Cos[c + d*x]])}
{Sqrt[Cos[c + d*x]]/Sqrt[-3 - 2*Cos[c + d*x]], x, 3, -((2*Sqrt[-((-3 - 2*Cos[c + d*x])/(1 + Cos[c + d*x]))]*Sqrt[1 + Cos[c + d*x]]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -(1/5)])/(Sqrt[5]*d*Sqrt[-3 - 2*Cos[c + d*x]])) + (4*Sqrt[-((-3 - 2*Cos[c + d*x])/(1 + Cos[c + d*x]))]*Sqrt[1 + Cos[c + d*x]]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -(1/5)])/(Sqrt[5]*d*Sqrt[-3 - 2*Cos[c + d*x]])}


{Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(1/2), x, 6, -((3*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + ((a - 2*b)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*b*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*a^2 + 4*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d) - (3*a*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(4*b^2*d)}
{Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(1/2), x, 3, (Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - (2*a*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(b*d)}
{Cos[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^(1/2), x, 3, -((2*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]])) + (4*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]])}
{Cos[c + d*x]^(-1/2)/(a + b*Cos[c + d*x])^(1/2), x, 1, (2*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]])}
{Cos[c + d*x]^(-3/2)/(a + b*Cos[c + d*x])^(1/2), x, 5, -((2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(a*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - (2*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(a*d)}
{Cos[c + d*x]^(-5/2)/(a + b*Cos[c + d*x])^(1/2), x, 6, (4*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + (2*(a - 2*b)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (4*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]) + (4*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^2*d)}


{Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(3/2), x, 6, ((3*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b^2*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - (2*a*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/((a - b)*b*d*Sqrt[a + b*Cos[c + d*x]]) - (6*a*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(b^2*(a^2 - b^2)*d)}
{Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(3/2), x, 6, -((2*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/((a - b)*d*Sqrt[a + b*Cos[c + d*x]]) + (4*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(b*(a^2 - b^2)*d)}
{Cos[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^(3/2), x, 5, (2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/((a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - (2*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/((a - b)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/((a^2 - b^2)*d)}
{Cos[c + d*x]^(-1/2)/(a + b*Cos[c + d*x])^(3/2), x, 5, -((2*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(a*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/((a - b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(a*(a^2 - b^2)*d)}
{Cos[c + d*x]^(-3/2)/(a + b*Cos[c + d*x])^(3/2), x, 7, -((2*(a^2 - 2*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(a^2*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a - 2*b)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(a*(a - b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2 - 2*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - (2*(a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(a^2*(a^2 - b^2)*d)}
{Cos[c + d*x]^(-5/2)/(a + b*Cos[c + d*x])^(3/2), x, 8, (2*b*(5*a^2 - 8*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^3*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + (2*(a^2 - 6*a*b + 8*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^2*(a - b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2 - 4*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) - (2*b*(5*a^2 - 8*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (2*b*(5*a^2 - 8*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^3*(a^2 - b^2)*d)}


{Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*a*(3*a^2 - 7*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(2*a^2 - 3*a*b - 3*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*(a - b)^2*b*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (4*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*a*(3*a^2 - 7*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a*(3*a^2 - 7*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*b^2*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(5/2), x, 6, -((8*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a + 3*b)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*(a - b)^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (8*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (8*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^(5/2), x, 6, (2*(3*a^2 + b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - (2*(3*a + b)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*(a - b)^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*b*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^(-1/2)/(a + b*Cos[c + d*x])^(5/2), x, 7, -((4*b*(3*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(3*a^2 + 3*a*b - 2*b^2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a*(a - b)^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (4*b^2*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (4*b*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^2*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^(-3/2)/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(3*a^3 - 9*a^2*b - 6*a*b^2 + 8*b^3)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^2*(a - b)^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)) + (8*b^2*(2*a^2 - b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) - (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^3*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^(-5/2)/(a + b*Cos[c + d*x])^(5/2), x, 8, (8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + (2*(a^4 - 9*a^3*b + 16*a^2*b^2 + 12*a*b^3 - 16*b^4)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^3*(a - b)^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - (8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^4*(a^2 - b^2)^2*d)}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/3) (a+b Cos[c+d x])^n*)


(* ::Subsubsection:: *)
(*n>0*)


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^(7/3)/(a + b*Cos[c + d*x]), x, 1, (3*Int[(a/3 + (1/3)*b*Cos[c + d*x] - (1/3)*(4*a)*Cos[c + d*x]^2)/(Cos[c + d*x]^(2/3)*(a + b*Cos[c + d*x])), x])/(4*b) + (3*Cos[c + d*x]^(1/3)*Sin[c + d*x])/(4*b*d)}
{Cos[c + d*x]^(5/3)/(a + b*Cos[c + d*x]), x, 0, Int[Cos[c + d*x]^(5/3)/(a + b*Cos[c + d*x]), x]}
{Cos[c + d*x]^(4/3)/(a + b*Cos[c + d*x]), x, 0, Int[Cos[c + d*x]^(4/3)/(a + b*Cos[c + d*x]), x]}
{Cos[c + d*x]^(2/3)/(a + b*Cos[c + d*x]), x, 0, Int[Cos[c + d*x]^(2/3)/(a + b*Cos[c + d*x]), x]}
{Cos[c + d*x]^(1/3)/(a + b*Cos[c + d*x]), x, 0, Int[Cos[c + d*x]^(1/3)/(a + b*Cos[c + d*x]), x]}
{Cos[c + d*x]^(-1/3)/(a + b*Cos[c + d*x]), x, 0, Int[1/(Cos[c + d*x]^(1/3)*(a + b*Cos[c + d*x])), x]}
{Cos[c + d*x]^(-2/3)/(a + b*Cos[c + d*x]), x, 0, Int[1/(Cos[c + d*x]^(2/3)*(a + b*Cos[c + d*x])), x]}
{Cos[c + d*x]^(-4/3)/(a + b*Cos[c + d*x]), x, 1, -((3*Int[(b/3 + (1/3)*(2*a)*Cos[c + d*x] + (1/3)*(2*b)*Cos[c + d*x]^2)/(Cos[c + d*x]^(1/3)*(a + b*Cos[c + d*x])), x])/a) + (3*Sin[c + d*x])/(a*d*Cos[c + d*x]^(1/3))}
{Cos[c + d*x]^(-5/3)/(a + b*Cos[c + d*x]), x, 1, -((3*Int[((2*b)/3 + (1/3)*a*Cos[c + d*x] + (1/3)*b*Cos[c + d*x]^2)/(Cos[c + d*x]^(2/3)*(a + b*Cos[c + d*x])), x])/(2*a)) + (3*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(2/3))}
{Cos[c + d*x]^(-7/3)/(a + b*Cos[c + d*x]), x, 2, (9*Int[((1/9)*(a^2 + 4*b^2) + (a*b)*Cos[c + d*x] + (1/9)*(8*b^2)*Cos[c + d*x]^2)/(Cos[c + d*x]^(1/3)*(a + b*Cos[c + d*x])), x])/(4*a^2) + (3*Sin[c + d*x])/(4*a*d*Cos[c + d*x]^(4/3)) - (3*b*Sin[c + d*x])/(a^2*d*Cos[c + d*x]^(1/3))}


{Cos[c + d*x]^(7/3)/Sqrt[a + b*Cos[c + d*x]], x, 1, (6*Int[(a/3 + (1/6)*(5*b)*Cos[c + d*x] - (1/3)*(4*a)*Cos[c + d*x]^2)/(Cos[c + d*x]^(2/3)*Sqrt[a + b*Cos[c + d*x]]), x])/(11*b) + (6*Cos[c + d*x]^(1/3)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(11*b*d)}
{Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Int[Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]], x]}
{Cos[c + d*x]^(4/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Int[Cos[c + d*x]^(4/3)/Sqrt[a + b*Cos[c + d*x]], x]}
{Cos[c + d*x]^(2/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Int[Cos[c + d*x]^(2/3)/Sqrt[a + b*Cos[c + d*x]], x]}
{Cos[c + d*x]^(1/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Int[Cos[c + d*x]^(1/3)/Sqrt[a + b*Cos[c + d*x]], x]}
{Cos[c + d*x]^(-1/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Int[1/(Cos[c + d*x]^(1/3)*Sqrt[a + b*Cos[c + d*x]]), x]}
{Cos[c + d*x]^(-2/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Int[1/(Cos[c + d*x]^(2/3)*Sqrt[a + b*Cos[c + d*x]]), x]}
{Cos[c + d*x]^(-4/3)/Sqrt[a + b*Cos[c + d*x]], x, 1, -((3*Int[(-(b/6) + (1/3)*(2*a)*Cos[c + d*x] + (1/6)*(7*b)*Cos[c + d*x]^2)/(Cos[c + d*x]^(1/3)*Sqrt[a + b*Cos[c + d*x]]), x])/a) + (3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(a*d*Cos[c + d*x]^(1/3))}
{Cos[c + d*x]^(-5/3)/Sqrt[a + b*Cos[c + d*x]], x, 1, -((3*Int[(b/6 + (1/3)*a*Cos[c + d*x] + (1/6)*(5*b)*Cos[c + d*x]^2)/(Cos[c + d*x]^(2/3)*Sqrt[a + b*Cos[c + d*x]]), x])/(2*a)) + (3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(2/3))}
{Cos[c + d*x]^(-7/3)/Sqrt[a + b*Cos[c + d*x]], x, 2, (9*Int[((1/36)*(4*a^2 - 5*b^2) + (1/2)*(a*b)*Cos[c + d*x] + (1/36)*(35*b^2)*Cos[c + d*x]^2)/(Cos[c + d*x]^(1/3)*Sqrt[a + b*Cos[c + d*x]]), x])/(4*a^2) + (3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*a*d*Cos[c + d*x]^(4/3)) - (15*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(8*a^2*d*Cos[c + d*x]^(1/3))}


(* ::Subsection::Closed:: *)
(*Integrands of the form (-Cos[c+d x])^(m/2) (a+b Cos[c+d x])^(n/2)*)


{1/(Sqrt[-Cos[c + d*x]]*Sqrt[2 - 3*Cos[c + d*x]]), x, 1, -((2*EllipticF[ArcSin[Cot[(c+d*x)/2]], 1/5])/(d*Sqrt[5]))}
{1/(Sqrt[-Cos[c + d*x]]*Sqrt[-2 - 3*Cos[c + d*x]]), x, 1, -((2*EllipticF[ArcSin[Cot[(c+d*x)/2]], 5])/d)}

{1/(Sqrt[-Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]]), x, 2, (2*Sqrt[Cos[c + d*x]]*EllipticF[ArcSin[Tan[(c+d*x)/2]], 1/5])/(d*Sqrt[5]*Sqrt[-Cos[c + d*x]])}
{1/(Sqrt[-Cos[c + d*x]]*Sqrt[-2 + 3*Cos[c + d*x]]), x, 2, (2*Sqrt[Cos[c + d*x]]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], 5])/(d*Sqrt[-Cos[c + d*x]])}

{1/(Sqrt[-Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]), x, 1, -((2*Sqrt[1 - Cos[c + d*x]]*Sqrt[(3 + 2*Cos[c + d*x])/(1 - Cos[c + d*x])]*EllipticF[ArcCsc[Tan[(1/2)*(c + d*x)]], -5])/(d*Sqrt[3 + 2*Cos[c + d*x]]))}
{1/(Sqrt[-Cos[c + d*x]]*Sqrt[3 - 2*Cos[c + d*x]]), x, 1, -((2*Sqrt[(3 - 2*Cos[c + d*x])/(1 - Cos[c + d*x])]*Sqrt[1 - Cos[c + d*x]]*EllipticF[ArcCsc[Tan[(1/2)*(c + d*x)]], -(1/5)])/(Sqrt[5]*d*Sqrt[3 - 2*Cos[c + d*x]]))}

{1/(Sqrt[-Cos[c + d*x]]*Sqrt[-3 + 2*Cos[c + d*x]]), x, 1, -((2*Sqrt[1 - Cos[c + d*x]]*Sqrt[-((-3 + 2*Cos[c + d*x])/(1 - Cos[c + d*x]))]*EllipticF[ArcCsc[Tan[(1/2)*(c + d*x)]], -(1/5)])/(Sqrt[5]*d*Sqrt[-3 + 2*Cos[c + d*x]]))}
{1/(Sqrt[-Cos[c + d*x]]*Sqrt[-3 - 2*Cos[c + d*x]]), x, 1, -((2*Sqrt[1 - Cos[c + d*x]]*Sqrt[(-3 - 2*Cos[c + d*x])/(-1 + Cos[c + d*x])]*EllipticF[ArcCsc[Tan[(1/2)*(c + d*x)]], -5])/(d*Sqrt[-3 - 2*Cos[c + d*x]]))}

{1/(Sqrt[-Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]), x, 1, -((2*Sqrt[1 - Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a - b)*(1 - Cos[c + d*x]))]*EllipticF[ArcCsc[Tan[(1/2)*(c + d*x)]], -((a + b)/(a - b))])/(d*Sqrt[a + b*Cos[c + d*x]]))}


{Sqrt[-Cos[c + d*x]]/Sqrt[2 - 3*Cos[c + d*x]], x, 3, (2*EllipticF[ArcCsc[Tan[(1/2)*(c + d*x)]], 1/5])/(Sqrt[5]*d) - (4*EllipticPi[-1, ArcCsc[Tan[(1/2)*(c + d*x)]], 1/5])/(Sqrt[5]*d)}
{Sqrt[-Cos[c + d*x]]/Sqrt[-2 - 3*Cos[c + d*x]], x, 3, (2*EllipticF[ArcCsc[Tan[(1/2)*(c + d*x)]], 5])/d - (4*EllipticPi[-1, ArcCsc[Tan[(1/2)*(c + d*x)]], 5])/d}

{Sqrt[-Cos[c + d*x]]/Sqrt[2 + 3*Cos[c + d*x]], x, 4, -((2*Sqrt[Cos[c + d*x]]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], 1/5])/(Sqrt[5]*d*Sqrt[-Cos[c + d*x]])) - (4*Sqrt[1 - Cos[c + d*x]]*Sqrt[(2 + 3*Cos[c + d*x])/(-1 + Cos[c + d*x])]*EllipticPi[-1, ArcCsc[Tan[(1/2)*(c + d*x)]], 5])/(d*Sqrt[2 + 3*Cos[c + d*x]])}
{Sqrt[-Cos[c + d*x]]/Sqrt[-2 + 3*Cos[c + d*x]], x, 4, -((2*Sqrt[Cos[c + d*x]]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], 5])/(d*Sqrt[-Cos[c + d*x]])) - (4*Sqrt[1 - Cos[c + d*x]]*Sqrt[-((-2 + 3*Cos[c + d*x])/(1 - Cos[c + d*x]))]*EllipticPi[-1, ArcCsc[Tan[(1/2)*(c + d*x)]], 1/5])/(Sqrt[5]*d*Sqrt[-2 + 3*Cos[c + d*x]])}

{Sqrt[-Cos[c + d*x]]/Sqrt[3 + 2*Cos[c + d*x]], x, 3, (2*Sqrt[1 - Cos[c + d*x]]*Sqrt[(3 + 2*Cos[c + d*x])/(1 - Cos[c + d*x])]*EllipticF[ArcCsc[Tan[(1/2)*(c + d*x)]], -5])/(d*Sqrt[3 + 2*Cos[c + d*x]]) - (4*Sqrt[1 - Cos[c + d*x]]*Sqrt[(3 + 2*Cos[c + d*x])/(1 - Cos[c + d*x])]*EllipticPi[-1, ArcCsc[Tan[(1/2)*(c + d*x)]], -5])/(d*Sqrt[3 + 2*Cos[c + d*x]])}
{Sqrt[-Cos[c + d*x]]/Sqrt[3 - 2*Cos[c + d*x]], x, 3, (2*Sqrt[(3 - 2*Cos[c + d*x])/(1 - Cos[c + d*x])]*Sqrt[1 - Cos[c + d*x]]*EllipticF[ArcCsc[Tan[(1/2)*(c + d*x)]], -(1/5)])/(Sqrt[5]*d*Sqrt[3 - 2*Cos[c + d*x]]) - (4*Sqrt[(3 - 2*Cos[c + d*x])/(1 - Cos[c + d*x])]*Sqrt[1 - Cos[c + d*x]]*EllipticPi[-1, ArcCsc[Tan[(1/2)*(c + d*x)]], -(1/5)])/(Sqrt[5]*d*Sqrt[3 - 2*Cos[c + d*x]])}

{Sqrt[-Cos[c + d*x]]/Sqrt[-3 + 2*Cos[c + d*x]], x, 3, (2*Sqrt[1 - Cos[c + d*x]]*Sqrt[-((-3 + 2*Cos[c + d*x])/(1 - Cos[c + d*x]))]*EllipticF[ArcCsc[Tan[(1/2)*(c + d*x)]], -(1/5)])/(Sqrt[5]*d*Sqrt[-3 + 2*Cos[c + d*x]]) - (4*Sqrt[1 - Cos[c + d*x]]*Sqrt[-((-3 + 2*Cos[c + d*x])/(1 - Cos[c + d*x]))]*EllipticPi[-1, ArcCsc[Tan[(1/2)*(c + d*x)]], -(1/5)])/(Sqrt[5]*d*Sqrt[-3 + 2*Cos[c + d*x]])}
{Sqrt[-Cos[c + d*x]]/Sqrt[-3 - 2*Cos[c + d*x]], x, 3, (2*Sqrt[1 - Cos[c + d*x]]*Sqrt[(-3 - 2*Cos[c + d*x])/(-1 + Cos[c + d*x])]*EllipticF[ArcCsc[Tan[(1/2)*(c + d*x)]], -5])/(d*Sqrt[-3 - 2*Cos[c + d*x]]) - (4*Sqrt[1 - Cos[c + d*x]]*Sqrt[(-3 - 2*Cos[c + d*x])/(-1 + Cos[c + d*x])]*EllipticPi[-1, ArcCsc[Tan[(1/2)*(c + d*x)]], -5])/(d*Sqrt[-3 - 2*Cos[c + d*x]])}

{Sqrt[-Cos[c + d*x]]/Sqrt[a + b*Cos[c + d*x]], x, 3, (2*Sqrt[1 - Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a - b)*(1 - Cos[c + d*x]))]*EllipticF[ArcCsc[Tan[(1/2)*(c + d*x)]], -((a + b)/(a - b))])/(d*Sqrt[a + b*Cos[c + d*x]]) - (4*Sqrt[1 - Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a - b)*(1 - Cos[c + d*x]))]*EllipticPi[-1, ArcCsc[Tan[(1/2)*(c + d*x)]], -((a + b)/(a - b))])/(d*Sqrt[a + b*Cos[c + d*x]])}


(* ::Section::Closed:: *)
(*Cos[c+d x]^m (A+B Cos[c+d x]) (a+b Cos[c+d x])^n*)


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x])*)


{(A + B*Cos[c + d*x]), x, 2, A*x + (B*Sin[c + d*x])/d}
{Cos[c + d*x]*(A + B*Cos[c + d*x]), x, 1, (B*x)/2 + (A*Sin[c + d*x])/d + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x]), x, 2, (A*x)/2 + (2*B*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x]), x, 3, (3*B*x)/8 + (2*A*Sin[c + d*x])/(3*d) + (3*B*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x]), x, 4, (3*A*x)/8 + (8*B*Sin[c + d*x])/(15*d) + (3*A*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (4*B*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (B*Cos[c + d*x]^4*Sin[c + d*x])/(5*d)}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x])*)
(**)


{(A + B*Cos[c + d*x])*Cos[c + d*x]^(5/2), x, 6, (6*A*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (10*B*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)}
{(A + B*Cos[c + d*x])*Cos[c + d*x]^(3/2), x, 5, (6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*A*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)}
{(A + B*Cos[c + d*x])*Cos[c + d*x]^(1/2), x, 4, (2*A*EllipticE[(1/2)*(c + d*x), 2])/d + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{(A + B*Cos[c + d*x])/Cos[c + d*x]^(1/2), x, 3, (2*B*EllipticE[(1/2)*(c + d*x), 2])/d + (2*A*EllipticF[(1/2)*(c + d*x), 2])/d}
{(A + B*Cos[c + d*x])/Cos[c + d*x]^(3/2), x, 4, -((2*A*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/d + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{(A + B*Cos[c + d*x])/Cos[c + d*x]^(5/2), x, 5, -((2*B*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*A*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form (A+B Cos[c+d x]) (a+b Cos[c+d x])^n*)


{(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 4, (1/8)*(8*a^4*A + 24*a^2*A*b^2 + 3*A*b^4 + 16*a^3*b*B + 12*a*b^3*B)*x + ((95*a^3*A*b + 80*a*A*b^3 + 12*a^4*B + 112*a^2*b^2*B + 16*b^4*B)*Sin[c + d*x])/(30*d) + (b*(130*a^2*A*b + 45*A*b^3 + 24*a^3*B + 116*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((35*a*A*b + 12*a^2*B + 16*b^2*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + ((5*A*b + 4*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (B*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)}
{(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 3, (1/8)*(8*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 3*b^3*B)*x + ((16*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*Sin[c + d*x])/(6*d) + (b*(20*a*A*b + 6*a^2*B + 9*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*A*b + 3*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (B*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)}
{(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 2, (1/2)*(2*a^2*A + A*b^2 + 2*a*b*B)*x + (2*(3*a*A*b + a^2*B + b^2*B)*Sin[c + d*x])/(3*d) + (b*(3*A*b + 2*a*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)}
{(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 1, (1/2)*(2*a*A + b*B)*x + ((A*b + a*B)*Sin[c + d*x])/d + (b*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 2, (B*x)/b + (2*(A*b - a*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2]*d)}
{(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 3, (2*(a*A - b*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d) - ((A*b - a*B)*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 4, ((2*a^2*A + A*b^2 - 3*a*b*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - ((A*b - a*B)*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*a*A*b - a^2*B - 2*b^2*B)*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4, x, 5, ((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) - ((A*b - a*B)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*Sin[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Sin[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}


{(b*B/a + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 2, (B*x)/b - (2*Sqrt[a^2 - b^2]*B*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a*b*d)}
{(a*B/b + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 2, (B*x)/b}

{(a + b*Cos[c + d*x])/(b + a*Cos[c + d*x])^2, x, 2, Sin[c + d*x]/(d*(b + a*Cos[c + d*x]))}
{(3 + Cos[c + d*x])/(2 - Cos[c + d*x]), x, 2, -x + (5*x)/Sqrt[3] + (10*ArcTan[Sin[c + d*x]/(2 + Sqrt[3] - Cos[c + d*x])])/(Sqrt[3]*d)}


(* ::Subsection::Closed:: *)
(*Integrands of the form (A+B Cos[c+d x]) (a+b Cos[c+d x])^(n/2)*)


{(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 8, (2*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(56*a^3*A*b - 56*a*A*b^3 + 15*a^4*B + 10*a^2*b^2*B - 25*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(56*a*A*b + 15*a^2*B + 25*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*A*b + 5*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*B*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)}
{(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 7, (2*(20*a*A*b + 3*a^2*B + 9*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(5*a^2*A*b - 5*A*b^3 + 3*a^3*B - 3*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}
{(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 6, (2*(3*A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 5, (2*B*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(A*b - a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]])}
{(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*(A*b - a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b - a*B)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])}
{(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(4*a*A*b - a^2*B - 3*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(A*b - a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b - a*B)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(4*a*A*b - a^2*B - 3*b^2*B)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form (A+B Cos[c+d x]) (a+b Cos[c+d x])^(n/3)*)
(**)


{(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(2/3), x, 1, (3/5)*Int[((1/3)*(5*a*A + 2*b*B) + (1/3)*(5*A*b + 2*a*B)*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/3), x] + (3*B*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*d)}
{(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/3), x, 0, Int[(a + b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]), x]}
{(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/3), x, 0, Int[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/3), x]}
{(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(2/3), x, 1, (3*Int[(a + b*Cos[c + d*x])^(1/3)*((1/3)*(a*A - b*B) - (4/3)*(A*b - a*B)*Cos[c + d*x]), x])/(a^2 - b^2) + (3*(A*b - a*B)*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/((a^2 - b^2)*d)}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]) (a+b Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 2, (1/2)*(A*b + a*B)*x + ((3*a*A + 2*b*B)*Sin[c + d*x])/(3*d) + ((A*b + a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (b*B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 3, (1/8)*(4*a*A + 3*b*B)*x + (2*(A*b + a*B)*Sin[c + d*x])/(3*d) + ((4*a*A + 3*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((A*b + a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (b*B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 4, (3/8)*(A*b + a*B)*x + (2*(5*a*A + 4*b*B)*Sin[c + d*x])/(15*d) + (3*(A*b + a*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((5*a*A + 4*b*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + ((A*b + a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (b*B*Cos[c + d*x]^4*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 5, (1/16)*(6*a*A + 5*b*B)*x + (8*(A*b + a*B)*Sin[c + d*x])/(15*d) + ((6*a*A + 5*b*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (4*(A*b + a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + ((6*a*A + 5*b*B)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + ((A*b + a*B)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) + (b*B*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 3, (1/8)*(8*a*A*b + 4*a^2*B + 3*b^2*B)*x + ((4*a^2*A*b + 4*A*b^3 - a^3*B + 8*a*b^2*B)*Sin[c + d*x])/(6*b*d) + ((8*a*A*b - 2*a^2*B + 9*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*A*b - a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*b*d) + (B*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*b*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 4, (1/8)*(4*a^2*A + 3*A*b^2 + 6*a*b*B)*x + (2*(10*a*A*b + 5*a^2*B + 4*b^2*B)*Sin[c + d*x])/(15*d) + ((4*a^2*A + 3*A*b^2 + 6*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((10*a*A*b + 5*a^2*B + 4*b^2*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (b*(5*A*b + 6*a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (b*B*Cos[c + d*x]^3*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 5, (1/16)*(12*a*A*b + 6*a^2*B + 5*b^2*B)*x + (2*(5*a^2*A + 4*A*b^2 + 8*a*b*B)*Sin[c + d*x])/(15*d) + ((12*a*A*b + 6*a^2*B + 5*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((5*a^2*A + 4*A*b^2 + 8*a*b*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + ((12*a*A*b + 6*a^2*B + 5*b^2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (b*(6*A*b + 7*a*B)*Cos[c + d*x]^4*Sin[c + d*x])/(30*d) + (b*B*Cos[c + d*x]^4*(a + b*Cos[c + d*x])*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 6, (1/16)*(6*a^2*A + 5*A*b^2 + 10*a*b*B)*x + (8*(14*a*A*b + 7*a^2*B + 6*b^2*B)*Sin[c + d*x])/(105*d) + ((6*a^2*A + 5*A*b^2 + 10*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (4*(14*a*A*b + 7*a^2*B + 6*b^2*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + ((6*a^2*A + 5*A*b^2 + 10*a*b*B)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + ((14*a*A*b + 7*a^2*B + 6*b^2*B)*Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + (b*(7*A*b + 8*a*B)*Cos[c + d*x]^5*Sin[c + d*x])/(42*d) + (b*B*Cos[c + d*x]^5*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 4, (1/8)*(12*a^2*A*b + 3*A*b^3 + 4*a^3*B + 9*a*b^2*B)*x + ((15*a^3*A*b + 60*a*A*b^3 - 3*a^4*B + 52*a^2*b^2*B + 16*b^4*B)*Sin[c + d*x])/(30*b*d) + ((30*a^2*A*b + 45*A*b^3 - 6*a^3*B + 71*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((15*a*A*b - 3*a^2*B + 16*b^2*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) + ((5*A*b - a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + (B*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 5, (1/16)*(8*a^3*A + 18*a*A*b^2 + 18*a^2*b*B + 5*b^3*B)*x + (2*(15*a^2*A*b + 4*A*b^3 + 5*a^3*B + 12*a*b^2*B)*Sin[c + d*x])/(15*d) + ((8*a^3*A + 18*a*A*b^2 + 18*a^2*b*B + 5*b^3*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((15*a^2*A*b + 4*A*b^3 + 5*a^3*B + 12*a*b^2*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (b*(66*a*A*b + 38*a^2*B + 25*b^2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(120*d) + (b*(3*A*b + 4*a*B)*Cos[c + d*x]^3*(a + b*Cos[c + d*x])*Sin[c + d*x])/(15*d) + (b*B*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 6, (1/16)*(18*a^2*A*b + 5*A*b^3 + 6*a^3*B + 15*a*b^2*B)*x + (2*(35*a^3*A + 84*a*A*b^2 + 84*a^2*b*B + 24*b^3*B)*Sin[c + d*x])/(105*d) + ((18*a^2*A*b + 5*A*b^3 + 6*a^3*B + 15*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((35*a^3*A + 84*a*A*b^2 + 84*a^2*b*B + 24*b^3*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + ((18*a^2*A*b + 5*A*b^3 + 6*a^3*B + 15*a*b^2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (b*(91*a*A*b + 51*a^2*B + 36*b^2*B)*Cos[c + d*x]^4*Sin[c + d*x])/(210*d) + (b*(7*A*b + 9*a*B)*Cos[c + d*x]^4*(a + b*Cos[c + d*x])*Sin[c + d*x])/(42*d) + (b*B*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 7, (1/128)*(48*a^3*A + 120*a*A*b^2 + 120*a^2*b*B + 35*b^3*B)*x + (8*(21*a^2*A*b + 6*A*b^3 + 7*a^3*B + 18*a*b^2*B)*Sin[c + d*x])/(105*d) + ((48*a^3*A + 120*a*A*b^2 + 120*a^2*b*B + 35*b^3*B)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (4*(21*a^2*A*b + 6*A*b^3 + 7*a^3*B + 18*a*b^2*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + ((48*a^3*A + 120*a*A*b^2 + 120*a^2*b*B + 35*b^3*B)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + ((21*a^2*A*b + 6*A*b^3 + 7*a^3*B + 18*a*b^2*B)*Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + (b*(120*a*A*b + 66*a^2*B + 49*b^2*B)*Cos[c + d*x]^5*Sin[c + d*x])/(336*d) + (b*(4*A*b + 5*a*B)*Cos[c + d*x]^5*(a + b*Cos[c + d*x])*Sin[c + d*x])/(28*d) + (b*B*Cos[c + d*x]^5*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(8*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 5, (1/16)*(32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B)*x + ((24*a^4*A*b + 224*a^2*A*b^3 + 32*A*b^5 - 4*a^5*B + 121*a^3*b^2*B + 128*a*b^4*B)*Sin[c + d*x])/(60*b*d) + ((48*a^3*A*b + 232*a*A*b^3 - 8*a^4*B + 178*a^2*b^2*B + 75*b^4*B)*Cos[c + d*x]*Sin[c + d*x])/(240*d) + ((24*a^2*A*b + 32*A*b^3 - 4*a^3*B + 53*a*b^2*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) + ((24*a*A*b - 4*a^2*B + 25*b^2*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*b*d) + ((6*A*b - a*B)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + (B*(a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 6, (1/16)*(8*a^4*A + 36*a^2*A*b^2 + 5*A*b^4 + 24*a^3*b*B + 20*a*b^3*B)*x + (2*(140*a^3*A*b + 112*a*A*b^3 + 35*a^4*B + 168*a^2*b^2*B + 24*b^4*B)*Sin[c + d*x])/(105*d) + ((8*a^4*A + 36*a^2*A*b^2 + 5*A*b^4 + 24*a^3*b*B + 20*a*b^3*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((140*a^3*A*b + 112*a*A*b^3 + 35*a^4*B + 168*a^2*b^2*B + 24*b^4*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + (b*(728*a^2*A*b + 175*A*b^3 + 272*a^3*B + 556*a*b^2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(840*d) + (b*(49*a*A*b + 31*a^2*B + 18*b^2*B)*Cos[c + d*x]^3*(a + b*Cos[c + d*x])*Sin[c + d*x])/(105*d) + (b*(7*A*b + 10*a*B)*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(42*d) + (b*B*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 7, (1/128)*(192*a^3*A*b + 160*a*A*b^3 + 48*a^4*B + 240*a^2*b^2*B + 35*b^4*B)*x + (2*(35*a^4*A + 168*a^2*A*b^2 + 24*A*b^4 + 112*a^3*b*B + 96*a*b^3*B)*Sin[c + d*x])/(105*d) + ((192*a^3*A*b + 160*a*A*b^3 + 48*a^4*B + 240*a^2*b^2*B + 35*b^4*B)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + ((35*a^4*A + 168*a^2*A*b^2 + 24*A*b^4 + 112*a^3*b*B + 96*a*b^3*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + ((192*a^3*A*b + 160*a*A*b^3 + 48*a^4*B + 240*a^2*b^2*B + 35*b^4*B)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (b*(1136*a^2*A*b + 288*A*b^3 + 414*a^3*B + 907*a*b^2*B)*Cos[c + d*x]^4*Sin[c + d*x])/(1680*d) + (b*(128*a*A*b + 78*a^2*B + 49*b^2*B)*Cos[c + d*x]^4*(a + b*Cos[c + d*x])*Sin[c + d*x])/(336*d) + (b*(8*A*b + 11*a*B)*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(56*d) + (b*B*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(8*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 8, (1/128)*(48*a^4*A + 240*a^2*A*b^2 + 35*A*b^4 + 160*a^3*b*B + 140*a*b^3*B)*x + (8*(84*a^3*A*b + 72*a*A*b^3 + 21*a^4*B + 108*a^2*b^2*B + 16*b^4*B)*Sin[c + d*x])/(315*d) + ((48*a^4*A + 240*a^2*A*b^2 + 35*A*b^4 + 160*a^3*b*B + 140*a*b^3*B)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (4*(84*a^3*A*b + 72*a*A*b^3 + 21*a^4*B + 108*a^2*b^2*B + 16*b^4*B)*Cos[c + d*x]^2*Sin[c + d*x])/(315*d) + ((48*a^4*A + 240*a^2*A*b^2 + 35*A*b^4 + 160*a^3*b*B + 140*a*b^3*B)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + ((84*a^3*A*b + 72*a*A*b^3 + 21*a^4*B + 108*a^2*b^2*B + 16*b^4*B)*Cos[c + d*x]^4*Sin[c + d*x])/(105*d) + (b*(558*a^2*A*b + 147*A*b^3 + 200*a^3*B + 460*a*b^2*B)*Cos[c + d*x]^5*Sin[c + d*x])/(1008*d) + (b*(81*a*A*b + 48*a^2*B + 32*b^2*B)*Cos[c + d*x]^5*(a + b*Cos[c + d*x])*Sin[c + d*x])/(252*d) + (b*(3*A*b + 4*a*B)*Cos[c + d*x]^5*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(24*d) + (b*B*Cos[c + d*x]^5*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 4, ((A*b - a*B)*x)/b^2 - (2*a*(A*b - a*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]*d) + (B*Sin[c + d*x])/(b*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 4, -(((2*a*A*b - 2*a^2*B - b^2*B)*x)/(2*b^3)) + (2*a^2*(A*b - a*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]*d) + ((A*b - a*B)*Sin[c + d*x])/(b^2*d) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 5, ((2*a^2*A*b + A*b^3 - 2*a^3*B - a*b^2*B)*x)/(2*b^4) - (2*a^3*(A*b - a*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*d) - ((3*a*A*b - 3*a^2*B - 2*b^2*B)*Sin[c + d*x])/(3*b^3*d) + ((A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (B*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 6, -(((8*a^3*A*b + 4*a*A*b^3 - 8*a^4*B - 4*a^2*b^2*B - 3*b^4*B)*x)/(8*b^5)) + (2*a^4*(A*b - a*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*Sqrt[a^2 - b^2]*d) + ((3*a^2*A*b + 2*A*b^3 - 3*a^3*B - 2*a*b^2*B)*Sin[c + d*x])/(3*b^4*d) - ((4*a*A*b - 4*a^2*B - 3*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) + ((A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (B*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 3, (B*x)/b^2 - (2*(A*b^3 + a^3*B - 2*a*b^2*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(3/2)*d) + (a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 4, ((A*b - 2*a*B)*x)/b^3 - (2*a*(a^2*A*b - 2*A*b^3 - 2*a^3*B + 3*a*b^2*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*d) - ((a*A*b - 2*a^2*B + b^2*B)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 5, -(((4*a*A*b - 6*a^2*B - b^2*B)*x)/(2*b^4)) + (2*a^2*(2*a^2*A*b - 3*A*b^3 - 3*a^3*B + 4*a*b^2*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(3/2)*d) + ((2*a^2*A*b - A*b^3 - 3*a^3*B + 2*a*b^2*B)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) - ((2*a*A*b - 3*a^2*B + b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 6, ((6*a^2*A*b + A*b^3 - 8*a^3*B - 2*a*b^2*B)*x)/(2*b^5) - (2*a^3*(3*a^2*A*b - 4*A*b^3 - 4*a^3*B + 5*a*b^2*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(3/2)*d) - ((9*a^3*A*b - 6*a*A*b^3 - 12*a^4*B + 7*a^2*b^2*B + 2*b^4*B)*Sin[c + d*x])/(3*b^4*(a^2 - b^2)*d) + ((3*a^2*A*b - A*b^3 - 4*a^3*B + 2*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) - ((3*a*A*b - 4*a^2*B + b^2*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}


{Cos[c + d*x]*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 4, -(((3*a*A*b - a^2*B - 2*b^2*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d)) + (a*(A*b - a*B)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*A*b + 2*A*b^3 + a^3*B - 4*a*b^2*B)*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 4, (B*x)/b^3 + ((a^2*A*b^3 + 2*A*b^5 - 2*a^5*B + 5*a^3*b^2*B - 6*a*b^4*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(5/2)*d) + (a*(A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a*(3*A*b^3 + 2*a^3*B - 5*a*b^2*B)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 5, ((A*b - 3*a*B)*x)/b^4 - (a*(2*a^4*A*b - 5*a^2*A*b^3 + 6*A*b^5 - 6*a^5*B + 15*a^3*b^2*B - 12*a*b^4*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(5/2)*d) - ((2*a^3*A*b - 5*a*A*b^3 - 6*a^4*B + 11*a^2*b^2*B - 2*b^4*B)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(a^2*A*b - 4*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 6, -(((6*a*A*b - 12*a^2*B - b^2*B)*x)/(2*b^5)) + (a^2*(6*a^4*A*b - 15*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 29*a^3*b^2*B - 20*a*b^4*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(5/2)*d) + ((6*a^4*A*b - 11*a^2*A*b^3 + 2*A*b^5 - 12*a^5*B + 21*a^3*b^2*B - 6*a*b^4*B)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*A*b - 6*a*A*b^3 - 6*a^4*B + 10*a^2*b^2*B - b^4*B)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(2*a^2*A*b - 5*A*b^3 - 4*a^3*B + 7*a*b^2*B)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}


{Cos[c + d*x]*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4, x, 5, -(((4*a^2*A*b + A*b^3 - a^3*B - 4*a*b^2*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d)) + (a*(A*b - a*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((2*a^2*A*b + 3*A*b^3 + a^3*B - 6*a*b^2*B)*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((2*a^3*A*b + 13*a*A*b^3 + a^4*B - 10*a^2*b^2*B - 6*b^4*B)*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4, x, 5, ((a^3*A + 4*a*A*b^2 - 3*a^2*b*B - 2*b^3*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + (a*(A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a*(a^2*A*b + 4*A*b^3 + 2*a^3*B - 7*a*b^2*B)*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((a^4*A*b - 10*a^2*A*b^3 - 6*A*b^5 + 2*a^5*B - 5*a^3*b^2*B + 18*a*b^4*B)*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4, x, 5, (B*x)/b^4 - ((3*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B - 7*a^5*b^2*B + 8*a^3*b^4*B - 8*a*b^6*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(7/2)*d) + (a*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a*(5*A*b^3 + 3*a^3*B - 8*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(4*a^2*A*b^3 + 11*A*b^5 - 6*a^5*B + 17*a^3*b^2*B - 26*a*b^4*B)*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4, x, 6, ((A*b - 4*a*B)*x)/b^5 - (a*(2*a^6*A*b - 7*a^4*A*b^3 + 8*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 28*a^5*b^2*B - 35*a^3*b^4*B + 20*a*b^6*B)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(7/2)*d) - ((6*a^5*A*b - 17*a^3*A*b^3 + 26*a*A*b^5 - 24*a^6*B + 68*a^4*b^2*B - 65*a^2*b^4*B + 6*b^6*B)*Sin[c + d*x])/(6*b^4*(a^2 - b^2)^3*d) + (a*(A*b - a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (a*(a^2*A*b - 6*A*b^3 - 4*a^3*B + 9*a*b^2*B)*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(a^4*A*b - 2*a^2*A*b^3 + 6*A*b^5 - 4*a^5*B + 11*a^3*b^2*B - 12*a*b^4*B)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]) (a+b Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(5*a*A*b - 2*a^2*B + 9*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(5*a^2*A*b - 5*A*b^3 - 2*a^3*B + 2*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*A*b - 2*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 8, -((2*(14*a^2*A*b - 63*A*b^3 - 8*a^3*B - 19*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(14*a^3*A*b - 14*a*A*b^3 - 8*a^4*B - 17*a^2*b^2*B + 25*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(7*a*A*b - 4*a^2*B + 25*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^2*d) + (2*(7*A*b + a*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b*d) + (2*B*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 9, (2*(24*a^3*A*b + 57*a*A*b^3 - 16*a^4*B - 24*a^2*b^2*B + 147*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(24*a^4*A*b + 51*a^2*A*b^3 - 75*A*b^5 - 16*a^5*B - 20*a^3*b^2*B + 36*a*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(12*a^2*A*b - 75*A*b^3 - 8*a^3*B - 13*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^3*d) + (2*(9*a*A*b - 6*a^2*B + 49*b^2*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) + (2*(9*A*b + a*B)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*b*d) + (2*B*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 10, -((2*(176*a^4*A*b + 264*a^2*A*b^3 - 1617*A*b^5 - 128*a^5*B - 156*a^3*b^2*B - 471*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^5*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(176*a^5*A*b + 220*a^3*A*b^3 - 396*a*A*b^5 - 128*a^6*B - 124*a^4*b^2*B - 423*a^2*b^4*B + 675*b^6*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^5*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(88*a^3*A*b + 143*a*A*b^3 - 64*a^4*B - 86*a^2*b^2*B + 675*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^4*d) - (2*(66*a^2*A*b - 539*A*b^3 - 48*a^3*B - 67*a*b^2*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^3*d) + (2*(11*a*A*b - 8*a^2*B + 81*b^2*B)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*b^2*d) + (2*(11*A*b + a*B)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(99*b*d) + (2*B*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(11*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 8, (2*(21*a^2*A*b + 63*A*b^3 - 6*a^3*B + 82*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(21*a^3*A*b - 21*a*A*b^3 - 6*a^4*B + 31*a^2*b^2*B - 25*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(21*a*A*b - 6*a^2*B + 25*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) + (2*(7*A*b - 2*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*B*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 9, -((2*(18*a^3*A*b - 246*a*A*b^3 - 8*a^4*B - 33*a^2*b^2*B - 147*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(18*a^4*A*b - 93*a^2*A*b^3 + 75*A*b^5 - 8*a^5*B - 31*a^3*b^2*B + 39*a*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(9*a^2*A*b + 75*A*b^3 - 4*a^3*B + 88*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) + (2*(72*a*A*b + 3*a^2*B + 49*b^2*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) + (2*(9*A*b + 10*a*B)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*b*B*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 10, (2*(88*a^4*A*b + 363*a^2*A*b^3 + 1617*A*b^5 - 48*a^5*B - 108*a^3*b^2*B + 2088*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(88*a^5*A*b + 341*a^3*A*b^3 - 429*a*A*b^5 - 48*a^6*B - 96*a^4*b^2*B + 819*a^2*b^4*B - 675*b^6*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(44*a^3*A*b - 968*a*A*b^3 - 24*a^4*B - 57*a^2*b^2*B - 675*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^3*d) + (2*(33*a^2*A*b + 539*A*b^3 - 18*a^3*B + 606*a*b^2*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^2*d) + (2*(110*a*A*b + 3*a^2*B + 81*b^2*B)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*b*d) + (2*(11*A*b + 12*a*B)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(99*d) + (2*b*B*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 11, -((2*(624*a^5*A*b + 1404*a^3*A*b^3 - 27144*a*A*b^5 - 384*a^6*B - 644*a^4*b^2*B - 3525*a^2*b^4*B - 17787*b^6*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^5*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(624*a^6*A*b + 1248*a^4*A*b^3 - 10647*a^2*A*b^5 + 8775*A*b^7 - 384*a^7*B - 548*a^5*b^2*B - 3337*a^3*b^4*B + 4269*a*b^6*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^5*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(312*a^4*A*b + 741*a^2*A*b^3 + 8775*A*b^5 - 192*a^5*B - 346*a^3*b^2*B + 10198*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^4*d) - (2*(234*a^3*A*b - 7878*a*A*b^3 - 144*a^4*B - 267*a^2*b^2*B - 5929*b^4*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^3*d) + (2*(39*a^2*A*b + 1053*A*b^3 - 24*a^3*B + 1156*a*b^2*B)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9009*b^2*d) + (2*(156*a*A*b + 3*a^2*B + 121*b^2*B)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1287*b*d) + (2*(13*A*b + 14*a*B)*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(143*d) + (2*b*B*Cos[c + d*x]^5*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(13*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(45*a^3*A*b + 435*a*A*b^3 - 10*a^4*B + 279*a^2*b^2*B + 147*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(45*a^4*A*b + 30*a^2*A*b^3 - 75*A*b^5 - 10*a^5*B + 124*a^3*b^2*B - 114*a*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(45*a^2*A*b + 75*A*b^3 - 10*a^3*B + 114*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) + (2*(45*a*A*b - 10*a^2*B + 49*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) + (2*(9*A*b - 2*a*B)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*B*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 10, -((2*(110*a^4*A*b - 3069*a^2*A*b^3 - 1617*A*b^5 - 40*a^5*B - 255*a^3*b^2*B - 3705*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(110*a^5*A*b - 1364*a^3*A*b^3 + 1254*a*A*b^5 - 40*a^6*B - 245*a^4*b^2*B - 390*a^2*b^4*B + 675*b^6*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(55*a^3*A*b + 1793*a*A*b^3 - 20*a^4*B + 1025*a^2*b^2*B + 675*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^2*d) + (2*(825*a^2*A*b + 539*A*b^3 + 15*a^3*B + 1145*a*b^2*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b*d) + (2*(209*a*A*b + 113*a^2*B + 81*b^2*B)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (2*b*(11*A*b + 14*a*B)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(99*d) + (2*b*B*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 11, (2*(520*a^5*A*b + 3315*a^3*A*b^3 + 48165*a*A*b^5 - 240*a^6*B - 760*a^4*b^2*B + 30669*a^2*b^4*B + 17787*b^6*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(520*a^6*A*b + 3185*a^4*A*b^3 + 5070*a^2*A*b^5 - 8775*A*b^7 - 240*a^7*B - 700*a^5*b^2*B + 13984*a^3*b^4*B - 13044*a*b^6*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(260*a^4*A*b - 13325*a^2*A*b^3 - 8775*A*b^5 - 120*a^5*B - 395*a^3*b^2*B - 18973*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^3*d) + (2*(195*a^3*A*b + 14885*a*A*b^3 - 90*a^4*B + 8145*a^2*b^2*B + 5929*b^4*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^2*d) + (2*(1469*a^2*A*b + 1053*A*b^3 + 15*a^3*B + 2209*a*b^2*B)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9009*b*d) + (2*(299*a*A*b + 159*a^2*B + 121*b^2*B)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1287*d) + (2*b*(13*A*b + 16*a*B)*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(143*d) + (2*b*B*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(13*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 12, -((2*(240*a^6*A*b + 760*a^4*A*b^3 - 30669*a^2*A*b^5 - 17787*A*b^7 - 128*a^7*B - 284*a^5*b^2*B - 2475*a^3*b^4*B - 40665*a*b^6*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^5*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(240*a^7*A*b + 700*a^5*A*b^3 - 13984*a^3*A*b^5 + 13044*a*A*b^7 - 128*a^8*B - 252*a^6*b^2*B - 2395*a^4*b^4*B - 4830*a^2*b^6*B + 7605*b^8*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^5*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(120*a^5*A*b + 395*a^3*A*b^3 + 18973*a*A*b^5 - 64*a^6*B - 150*a^4*b^2*B + 10753*a^2*b^4*B + 7605*b^6*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^4*d) - (2*(90*a^4*A*b - 8145*a^2*A*b^3 - 5929*A*b^5 - 48*a^5*B - 115*a^3*b^2*B - 12541*a*b^4*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^3*d) + (2*(75*a^3*A*b + 11045*a*A*b^3 - 40*a^4*B + 5909*a^2*b^2*B + 4563*b^4*B)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^2*d) + (2*(795*a^2*A*b + 605*A*b^3 + 5*a^3*B + 1259*a*b^2*B)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(6435*b*d) + (2*(405*a*A*b + 213*a^2*B + 169*b^2*B)*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2145*d) + (2*b*(5*A*b + 6*a*B)*Cos[c + d*x]^5*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(65*d) + (2*b*B*Cos[c + d*x]^5*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 6, (2*(3*A*b - 2*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(3*a*A*b - 2*a^2*B - b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 7, -((2*(10*a*A*b - 8*a^2*B - 9*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(10*a^2*A*b + 5*A*b^3 - 8*a^3*B - 7*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*A*b - 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*B*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 8, (2*(56*a^2*A*b + 63*A*b^3 - 48*a^3*B - 44*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(56*a^3*A*b + 49*a*A*b^3 - 48*a^4*B - 32*a^2*b^2*B - 25*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(28*a*A*b - 24*a^2*B - 25*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^3*d) + (2*(7*A*b - 6*a*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b^2*d) + (2*B*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 9, -((2*(144*a^3*A*b + 132*a*A*b^3 - 128*a^4*B - 108*a^2*b^2*B - 147*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^5*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(144*a^4*A*b + 96*a^2*A*b^3 + 75*A*b^5 - 128*a^5*B - 76*a^3*b^2*B - 111*a*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^5*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(72*a^2*A*b + 75*A*b^3 - 64*a^3*B - 62*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^4*d) - (2*(54*a*A*b - 48*a^2*B - 49*b^2*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^3*d) + (2*(9*A*b - 8*a*B)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*b^2*d) + (2*B*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*b*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 6, -((2*(a*A*b - 2*a^2*B + b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(A*b - 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 7, (2*(6*a^2*A*b - 3*A*b^3 - 8*a^3*B + 5*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(6*a*A*b - 8*a^2*B - b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(3*a*A*b - 4*a^2*B + b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 8, -((2*(40*a^3*A*b - 25*a*A*b^3 - 48*a^4*B + 24*a^2*b^2*B + 9*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^4*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(40*a^2*A*b + 5*A*b^3 - 48*a^3*B - 12*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(20*a^2*A*b - 5*A*b^3 - 24*a^3*B + 9*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)*d) - (2*(5*a*A*b - 6*a^2*B + b^2*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 9, (2*(336*a^4*A*b - 168*a^2*A*b^3 - 63*A*b^5 - 384*a^5*B + 172*a^3*b^2*B + 107*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^5*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(336*a^3*A*b + 84*a*A*b^3 - 384*a^4*B - 116*a^2*b^2*B - 25*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^5*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(168*a^3*A*b - 63*a*A*b^3 - 192*a^4*B + 62*a^2*b^2*B + 25*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^4*(a^2 - b^2)*d) + (2*(42*a^2*A*b - 7*A*b^3 - 48*a^3*B + 13*a*b^2*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b^3*(a^2 - b^2)*d) - (2*(7*a*A*b - 8*a^2*B + b^2*B)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b^2*(a^2 - b^2)*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(a*A*b + 2*a^2*B - 3*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(2*a^3*A*b - 6*a*A*b^3 - 8*a^4*B + 15*a^2*b^2*B - 3*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(2*a^2*A*b - 3*A*b^3 - 8*a^3*B + 9*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*a*(a^2*A*b - 5*A*b^3 - 4*a^3*B + 8*a*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 8, (2*(8*a^4*A*b - 15*a^2*A*b^3 + 3*A*b^5 - 16*a^5*B + 28*a^3*b^2*B - 8*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(8*a^3*A*b - 9*a*A*b^3 - 16*a^4*B + 16*a^2*b^2*B + b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*a*(3*a^2*A*b - 7*A*b^3 - 6*a^3*B + 10*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(4*a^3*A*b - 8*a*A*b^3 - 8*a^4*B + 13*a^2*b^2*B - b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 9, -((2*(80*a^5*A*b - 140*a^3*A*b^3 + 40*a*A*b^5 - 128*a^6*B + 212*a^4*b^2*B - 55*a^2*b^4*B - 9*b^6*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(80*a^4*A*b - 80*a^2*A*b^3 - 5*A*b^5 - 128*a^5*B + 116*a^3*b^2*B + 17*a*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*a*(5*a^2*A*b - 9*A*b^3 - 8*a^3*B + 12*a*b^2*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(40*a^4*A*b - 65*a^2*A*b^3 + 5*A*b^5 - 64*a^5*B + 98*a^3*b^2*B - 14*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^4*(a^2 - b^2)^2*d) - (2*(30*a^3*A*b - 50*a*A*b^3 - 48*a^4*B + 71*a^2*b^2*B - 3*b^4*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)^2*d)}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x]) (a+b Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 7, (2*(9*a*A + 7*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*(A*b + a*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*(A*b + a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*a*A + 7*b*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(A*b + a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*b*B*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 6, (6*(A*b + a*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(7*a*A + 5*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*a*A + 5*b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(A*b + a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 5, (2*(5*a*A + 3*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(A*b + a*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 4, (2*(A*b + a*B)*EllipticE[(1/2)*(c + d*x), 2])/d + (2*(3*a*A + b*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 4, -((2*(a*A - b*B)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(A*b + a*B)*EllipticF[(1/2)*(c + d*x), 2])/d + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 5, -((2*(A*b + a*B)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(a*A + 3*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 6, -((2*(3*a*A + 5*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(A*b + a*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*a*A + 5*b*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 8, (2*(9*a^2*A + 7*A*b^2 + 14*a*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*(22*a*A*b + 11*a^2*B + 9*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (10*(22*a*A*b + 11*a^2*B + 9*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(9*a^2*A + 7*A*b^2 + 14*a*b*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(22*a*A*b + 11*a^2*B + 9*b^2*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*b*(11*A*b + 13*a*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*b*B*Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 7, (2*(18*a*A*b + 9*a^2*B + 7*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(7*a^2*A + 5*A*b^2 + 10*a*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*a^2*A + 5*A*b^2 + 10*a*b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(18*a*A*b + 9*a^2*B + 7*b^2*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(9*A*b + 11*a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*b*B*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 6, (2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(7*A*b + 9*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b*B*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 5, (2*(10*a*A*b + 5*a^2*B + 3*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(3*a^2*A + A*b^2 + 2*a*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*(5*A*b + 7*a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 5, -((2*(a^2*A - A*b^2 - 2*a*b*B)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(6*a*A*b + 3*a^2*B + b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*b*(3*a*A - b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + b*Cos[c + d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 5, -((2*(2*a*A*b + a^2*B - b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*(5*A*b + 3*a*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 6, -((2*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(2*a*A*b + a^2*B + 3*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*(7*A*b + 5*a*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}


{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 8, (2*(27*a^2*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45*b^3*B)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (2*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45*b^3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(27*a^2*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(220*a*A*b + 129*a^2*B + 81*b^2*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*b*(11*A*b + 15*a*B)*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(99*d) + (2*b*B*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 7, (2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(144*a*A*b + 89*a^2*B + 49*b^2*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*b*(9*A*b + 13*a*B)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(63*d) + (2*b*B*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 6, (2*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(21*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b*(84*a*A*b + 57*a^2*B + 25*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*b*(7*A*b + 11*a*B)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(35*d) + (2*b*B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 6, -((2*(5*a^3*A - 15*a*A*b^2 - 15*a^2*b*B - 3*b^3*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(9*a^2*A*b + A*b^3 + 3*a^3*B + 3*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*b*(15*a^2*A - 5*A*b^2 - 12*a*b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b*(5*a*A - b*B)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d) + (2*a*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 6, -((2*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*b*(8*a*A*b + 3*a^2*B - b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*(7*A*b + 3*a*B)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 6, -((2*(3*a^3*A + 15*a*A*b^2 + 15*a^2*b*B - 5*b^3*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(3*a^2*A*b + 3*A*b^3 + a^3*B + 9*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*(9*a^2*A + 33*A*b^2 + 40*a*b*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*(9*A*b + 5*a*B)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 7, -((2*(5*a*A*b - 5*a^2*B - 3*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d)) + (2*(3*a^2*A*b + A*b^3 - 3*a^3*B - a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^4*d) - (2*a^3*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^4*(a + b)*d) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 6, (2*(A*b - a*B)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d) - (2*(3*a*A*b - 3*a^2*B - b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^3*d) + (2*a^2*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^3*(a + b)*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 5, (2*B*EllipticE[(1/2)*(c + d*x), 2])/(b*d) + (2*(A*b - a*B)*EllipticF[(1/2)*(c + d*x), 2])/(b^2*d) - (2*a*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^2*(a + b)*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 3, (2*B*EllipticF[(1/2)*(c + d*x), 2])/(b*d) + (2*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b*(a + b)*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 5, -((2*A*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) - (2*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*(a + b)*d) + (2*A*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 7, (2*(A*b - a*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + (2*A*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (2*b*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a + b)*d) + (2*A*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]])}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 7, ((3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d) - ((9*a^3*A*b - 12*a*A*b^3 - 15*a^4*B + 16*a^2*b^2*B + 2*b^4*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^4*(a^2 - b^2)*d) + (a^2*(3*a^2*A*b - 5*A*b^3 - 5*a^3*B + 7*a*b^2*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^4*(a + b)^2*d) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 6, -(((a*A*b - 3*a^2*B + 2*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d)) + ((a^2*A*b - 2*A*b^3 - 3*a^3*B + 4*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d) - (a*(a^2*A*b - 3*A*b^3 - 3*a^3*B + 5*a*b^2*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^3*(a + b)^2*d) + (a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 6, ((A*b - a*B)*EllipticE[(1/2)*(c + d*x), 2])/(b*(a^2 - b^2)*d) + ((a*A*b + a^2*B - 2*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d) - ((a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^2*(a + b)^2*d) - ((A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 6, -(((A*b - a*B)*EllipticE[(1/2)*(c + d*x), 2])/(a*(a^2 - b^2)*d)) - ((A*b - a*B)*EllipticF[(1/2)*(c + d*x), 2])/(b*(a^2 - b^2)*d) + ((3*a^2*A*b - A*b^3 - a^3*B - a*b^2*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*(a - b)*b*(a + b)^2*d) + (b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 7, -(((2*a^2*A - 3*A*b^2 + a*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*(a^2 - b^2)*d)) + ((A*b - a*B)*EllipticF[(1/2)*(c + d*x), 2])/(a*(a^2 - b^2)*d) - ((5*a^2*A*b - 3*A*b^3 - 3*a^3*B + a*b^2*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2*A - 3*A*b^2 + a*b*B)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 8, ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*(a^2 - b^2)*d) + (b*(7*a^2*A*b - 5*A*b^3 - 5*a^3*B + 3*a*b^2*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^3*(a - b)*(a + b)^2*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) - ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 7, -(((3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d)) + ((3*a^4*A*b - 5*a^2*A*b^3 + 8*A*b^5 - 15*a^5*B + 33*a^3*b^2*B - 24*a*b^4*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^4*(a^2 - b^2)^2*d) - (a*(3*a^4*A*b - 6*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 38*a^3*b^2*B - 35*a*b^4*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^4*(a + b)^3*d) + (a*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 7, -(((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^2*(a^2 - b^2)^2*d)) + ((a^3*A*b - 7*a*A*b^3 + 3*a^4*B - 5*a^2*b^2*B + 8*b^4*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d) - ((a^4*A*b - 10*a^2*A*b^3 - 3*A*b^5 + 3*a^5*B - 6*a^3*b^2*B + 15*a*b^4*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^3*(a + b)^3*d) + (a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 7, ((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*a*b*(a^2 - b^2)^2*d) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^2*(a^2 - b^2)^2*d) - ((3*a^4*A*b + 10*a^2*A*b^3 - A*b^5 + a^5*B - 10*a^3*b^2*B - 3*a*b^4*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a*(a - b)^2*b^2*(a + b)^3*d) - ((A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 7, -(((9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^2*(a^2 - b^2)^2*d)) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*a*b*(a^2 - b^2)^2*d) + ((15*a^4*A*b - 6*a^2*A*b^3 + 3*A*b^5 - 3*a^5*B - 10*a^3*b^2*B + a*b^4*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^2*(a - b)^2*b*(a + b)^3*d) + (b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 8, -(((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^3*(a^2 - b^2)^2*d)) + ((11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*a^2*(a^2 - b^2)^2*d) - ((35*a^4*A*b - 38*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 6*a^3*b^2*B - 3*a*b^4*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2) + (b*(11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 9, ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*EllipticF[(1/2)*(c + d*x), 2])/(12*a^3*(a^2 - b^2)^2*d) + (b*(63*a^4*A*b - 86*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 38*a^3*b^2*B - 15*a*b^4*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2) + (b*(13*a^2*A*b - 7*A*b^3 - 9*a^3*B + 3*a*b^2*B)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x]) (a+b Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 8, -(((24*a^2*A*b - 128*A*b^3 - 15*a^3*B - 28*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(192*b^3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + ((8*a^2*A*b - 112*a*A*b^2 - 5*a^3*B - 2*a^2*b*B + 36*a*b^2*B - 72*b^3*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(96*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((8*a^3*A*b + 32*a*A*b^3 - 5*a^4*B - 8*a^2*b^2*B + 48*b^4*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(32*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + ((8*a*A*b - 5*a^2*B + 36*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(96*b^2*d) + ((8*A*b + a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b*d) + (B*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - ((24*a^2*A*b - 128*A*b^3 - 15*a^3*B - 28*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(192*b^3*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 7, ((6*a*A*b - 3*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(24*b^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((6*a*A*b - 12*A*b^2 + a^2*B - 14*a*b*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(12*b*d*Sqrt[a + b*Cos[c + d*x]]) - ((2*a^2*A*b - 8*A*b^3 - a^3*B - 4*a*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((6*A*b + a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*b*d) + (B*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + ((6*a*A*b - 3*a^2*B + 16*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(24*b^2*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 6, ((4*A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - ((4*a*A - a*B + 2*b*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a*A*b - a^2*B + 4*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*b*d*Sqrt[a + b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + ((4*A*b + a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(4*b*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 7, (B*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + (2*(a*A - A*b - a*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*(2*A*b + a*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/d}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 6, -((2*A*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a*A + A*b + a*B - b*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (4*b*B*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/d}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 6, -((2*(A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a*A + A*b + 3*a*B + 3*b*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Sqrt[Cos[c + d*x]]) - (2*(A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a*d)}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 8, -((2*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*a^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(9*a^2*A + 7*a*A*b - 2*A*b^2 + 5*a^2*B + 5*a*b*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2)) + (2*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Sqrt[Cos[c + d*x]]) - (2*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(15*a^2*d)}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 8, -((2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*a^3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(25*a^3*A + 19*a^2*A*b + 2*a*A*b^2 + 8*A*b^3 + 63*a^3*B + 49*a^2*b*B - 14*a*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2*A - 4*A*b^2 + 7*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Cos[c + d*x]^(3/2)) + (2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^3*d*Sqrt[Cos[c + d*x]]) - (2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(105*a^3*d)}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 9, -(((90*a^3*A*b - 1560*a*A*b^3 - 45*a^4*B - 132*a^2*b^2*B - 1024*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(1920*b^3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + ((30*a^3*A*b - 1140*a^2*A*b^2 + 360*a*A*b^3 - 720*A*b^4 - 15*a^4*B - 6*a^3*b*B + 412*a^2*b^2*B - 1592*a*b^3*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(960*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((6*a^4*A*b + 48*a^2*A*b^3 + 96*A*b^5 - 3*a^5*B - 8*a^3*b^2*B + 144*a*b^4*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(64*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + ((30*a^2*A*b + 360*A*b^3 - 15*a^3*B + 412*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(960*b^2*d) + ((90*a*A*b + 3*a^2*B + 64*b^2*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(240*b*d) + ((10*A*b + 11*a*B)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(40*d) + (b*B*Cos[c + d*x]^(7/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d) - ((90*a^3*A*b - 1560*a*A*b^3 - 45*a^4*B - 132*a^2*b^2*B - 1024*b^4*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(1920*b^3*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 8, ((24*a^2*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(192*b^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((56*a^2*A*b - 208*a*A*b^2 + 3*a^3*B - 114*a^2*b*B + 36*a*b^2*B - 72*b^3*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(96*b*d*Sqrt[a + b*Cos[c + d*x]]) - ((8*a^3*A*b - 96*a*A*b^3 - 3*a^4*B - 24*a^2*b^2*B - 48*b^4*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(32*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((56*a*A*b + 3*a^2*B + 36*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(96*b*d) + ((8*A*b + 9*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d) + (b*B*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + ((24*a^2*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(192*b^2*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 7, ((30*a*A*b + 3*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(24*b*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - ((24*a^2*A - 6*a*A*b + 12*A*b^2 - 7*a^2*B + 26*a*b*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(12*d*Sqrt[a + b*Cos[c + d*x]]) + ((6*a^2*A*b + 8*A*b^3 - a^3*B + 12*a*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b*d*Sqrt[a + b*Cos[c + d*x]]) + ((6*A*b + 7*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (b*B*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + ((30*a*A*b + 3*a^2*B + 16*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(24*b*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 6, ((4*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((4*a^2*A - 8*a*A*b - 4*a^2*B + a*b*B - 2*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) + ((12*a*A*b + 3*a^2*B + 4*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) + (b*B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + ((4*A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(4*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 6, -(((2*a*A - b*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a^2*A + 2*a*A*b - A*b^2 + a^2*B - 2*a*b*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(2*A*b + 3*a*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((2*a*A - b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/d}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 7, -((2*(4*A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a^2*A + 4*a*A*b + 3*A*b^2 + 3*a^2*B + 6*a*b*B - 3*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (4*b^2*B*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(4*A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) - (2*(4*A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*d)}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 7, -((2*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*a*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(9*a^2*A + 12*a*A*b + 3*A*b^2 + 5*a^2*B + 20*a*b*B + 15*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(6*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]) - (2*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(15*a*d)}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 9, -((2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*a^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(25*a^3*A + 82*a^2*A*b + 51*a*A*b^2 - 6*A*b^3 + 63*a^3*B + 84*a^2*b*B + 21*a*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(8*A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Cos[c + d*x]^(3/2)) + (2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Sqrt[Cos[c + d*x]]) - (2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(105*a^2*d)}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 10, -(((180*a^4*A*b - 6768*a^2*A*b^3 - 4096*A*b^5 - 75*a^5*B - 320*a^3*b^2*B - 9040*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(7680*b^3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + ((60*a^4*A*b - 4584*a^3*A*b^2 + 3088*a^2*A*b^3 - 9248*a*A*b^4 - 25*a^5*B - 10*a^4*b*B + 1720*a^3*b^2*B - 9920*a^2*b^3*B + 1200*a*b^4*B - 2400*b^5*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3840*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((12*a^5*A*b + 160*a^3*A*b^3 + 960*a*A*b^5 - 5*a^6*B - 20*a^4*b^2*B + 720*a^2*b^4*B + 320*b^6*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(256*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + ((60*a^3*A*b + 3088*a*A*b^3 - 25*a^4*B + 1720*a^2*b^2*B + 1200*b^4*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3840*b^2*d) + ((372*a^2*A*b + 256*A*b^3 + 5*a^3*B + 540*a*b^2*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(960*b*d) + ((252*a*A*b + 135*a^2*B + 100*b^2*B)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(480*d) + (b*(4*A*b + 5*a*B)*Cos[c + d*x]^(7/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(20*d) + (b*B*Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(6*d) - ((180*a^4*A*b - 6768*a^2*A*b^3 - 4096*A*b^5 - 75*a^5*B - 320*a^3*b^2*B - 9040*a*b^4*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(7680*b^3*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 9, ((150*a^3*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(1920*b^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((590*a^3*A*b - 3220*a^2*A*b^2 + 360*a*A*b^3 - 720*A*b^4 + 15*a^4*B - 1146*a^3*b*B + 772*a^2*b^2*B - 2312*a*b^3*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(960*b*d*Sqrt[a + b*Cos[c + d*x]]) - ((10*a^4*A*b - 240*a^2*A*b^3 - 96*A*b^5 - 3*a^5*B - 40*a^3*b^2*B - 240*a*b^4*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(64*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((590*a^2*A*b + 360*A*b^3 + 15*a^3*B + 772*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(960*b*d) + ((170*a*A*b + 93*a^2*B + 64*b^2*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(240*d) + (b*(10*A*b + 13*a*B)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(40*d) + (b*B*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + ((150*a^3*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(1920*b^2*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 8, ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(192*b*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - ((192*a^3*A - 104*a^2*A*b + 304*a*A*b^2 - 59*a^3*B + 322*a^2*b*B - 36*a*b^2*B + 72*b^3*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(96*d*Sqrt[a + b*Cos[c + d*x]]) + ((40*a^3*A*b + 160*a*A*b^3 - 5*a^4*B + 120*a^2*b^2*B + 48*b^4*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(32*b*d*Sqrt[a + b*Cos[c + d*x]]) + ((104*a*A*b + 59*a^2*B + 36*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(96*d) + (b*(8*A*b + 11*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d) + (b*B*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d) + ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(192*b*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 7, ((54*a*A*b + 33*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(24*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((24*a^3*A - 72*a^2*A*b + 6*a*A*b^2 - 12*A*b^3 - 24*a^3*B + 13*a^2*b*B - 38*a*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(12*d*Sqrt[a + b*Cos[c + d*x]]) + ((30*a^2*A*b + 8*A*b^3 + 5*a^3*B + 20*a*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (b*B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + ((54*a*A*b + 33*a^2*B + 16*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(24*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 7, -(((8*a^2*A - 4*A*b^2 - 9*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + ((4*a^3*A + 12*a^2*A*b - 12*a*A*b^2 + 4*a^3*B - 12*a^2*b*B + a*b^2*B - 2*b^3*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(20*a*A*b + 15*a^2*B + 4*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(4*a*A - b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((8*a^2*A - 4*A*b^2 - 9*a*b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(4*d)}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 7, -(((14*a*A*b + 6*a^2*B - 3*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a^3*A + 7*a^2*A*b + 9*a*A*b^2 - 3*A*b^3 + 3*a^3*B + 9*a^2*b*B - 9*a*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*(2*A*b + 5*a*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(2*A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) - ((14*a*A*b + 6*a^2*B - 3*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*d)}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 8, -((2*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(9*a^3*A + 17*a^2*A*b + 23*a*A*b^2 + 15*A*b^3 + 5*a^3*B + 35*a^2*b*B + 45*a*b^2*B - 15*b^3*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (4*b^3*B*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(8*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) - (2*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(15*d)}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 8, -((2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*a*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(25*a^3*A + 145*a^2*A*b + 135*a*A*b^2 + 15*A*b^3 + 63*a^3*B + 119*a^2*b*B + 161*a*b^2*B + 105*b^3*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(10*A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2*A + 45*A*b^2 + 77*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) - (2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(105*a*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 7, -(((18*a*A*b - 15*a^2*B - 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(24*b^3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + ((6*a*A*b - 12*A*b^2 - 5*a^2*B - 2*a*b*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(12*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((6*a^2*A*b + 8*A*b^3 - 5*a^3*B - 4*a*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + ((6*A*b - 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*b^2*d) + (B*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d) - ((18*a*A*b - 15*a^2*B - 16*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(24*b^3*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 6, ((4*A*b - 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((a - 2*b)*B*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*b*d*Sqrt[a + b*Cos[c + d*x]]) - ((4*a*A*b - 3*a^2*B - 4*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d) + ((4*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(4*b^2*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 6, (B*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - (2*A*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*(2*A*b - a*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(b*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 3, (2*(A - B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (4*B*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]])}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 5, -((2*A*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(a*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(A + B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(a*d)}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(2*A*b - 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + (2*(a*A - 2*A*b + 3*a*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*(2*A*b - 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]) + (2*(2*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^2*d)}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 7, ((12*a^2*A*b - 4*A*b^3 - 15*a^3*B + 7*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b^3*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - ((4*a*A*b - 5*a^2*B + 3*a*b*B - 2*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*(a - b)*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((12*a*A*b - 15*a^2*B - 4*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((4*a*A*b - 5*a^2*B + b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) + ((12*a^2*A*b - 4*A*b^3 - 15*a^3*B + 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(4*b^3*(a^2 - b^2)*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 6, -(((2*a*A*b - 3*a^2*B + b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b^2*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(A*b - a*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/((a - b)*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(2*A*b - 3*a*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((2*a*A*b - 3*a^2*B + b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(b^2*(a^2 - b^2)*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*(A*b - a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - (2*(A - B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/((a - b)*d*Sqrt[a + b*Cos[c + d*x]]) + (4*B*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(b*(a^2 - b^2)*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 6, -((2*(A*b - a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(a*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(A - B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/((a - b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(a*(a^2 - b^2)*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 7, -((2*(a^2*A - 2*A*b^2 + a*b*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(a^2*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a*A - 2*A*b + a*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(a*(a - b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2*A - 2*A*b^2 + a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - (2*(a^2*A - 2*A*b^2 + a*b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(a^2*(a^2 - b^2)*d)}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 7, (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^3*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + (2*(a^2*A - 6*a*A*b + 8*A*b^2 + 3*a^2*B - 6*a*b*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^2*(a - b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2*A - 4*A*b^2 + 3*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) - (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^3*(a^2 - b^2)*d)}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 7, -(((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(2*a^2*A*b - 3*a*A*b^2 - 3*A*b^3 - 5*a^3*B + 3*a^2*b*B + 6*a*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*(a - b)^2*b^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(2*A*b - 5*a*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*a*(2*a^2*A*b - 6*A*b^3 - 5*a^3*B + 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*b^3*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a*A*b + 3*A*b^2 + 2*a^2*B - 3*a*b*B - 3*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*(a - b)^2*b*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (4*B*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*b^2*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 6, (2*(3*a^2*A + A*b^2 - 4*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - (2*(3*a*A + A*b - a*B - 3*b*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*(a - b)^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*b*(3*a^2*A + A*b^2 - 4*a*b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*a^2*A + A*b^2 - 4*a*b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 6, -((2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(3*a^2*A + 3*a*A*b - 2*A*b^2 - 3*a^2*B - a*b*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a*(a - b)^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^2*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(3*a^3*A - 9*a^2*A*b - 6*a*A*b^2 + 8*A*b^3 + 3*a^3*B + 3*a^2*b*B - 2*a*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^2*(a - b)^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(8*a^2*A*b - 4*A*b^3 - 5*a^3*B + a*b^2*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) - (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^3*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + (2*(a^4*A - 9*a^3*A*b + 16*a^2*A*b^2 + 12*a*A*b^3 - 16*A*b^4 + 3*a^4*B - 9*a^3*b*B - 6*a^2*b^2*B + 8*a*b^3*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^3*(a - b)^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(10*a^2*A*b - 6*A*b^3 - 7*a^3*B + 3*a*b^2*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - (2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^4*(a^2 - b^2)^2*d)}


(* ::Section::Closed:: *)
(*Cos[c+d x]^m (A+C Cos[c+d x]^2) (a+b Cos[c+d x])^n*)


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+C Cos[c+d x]^2)*)


{(A + C*Cos[c + d*x]^2), x, 2, A*x + (C*x)/2 + (C*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{Cos[c + d*x]*(A + C*Cos[c + d*x]^2), x, 2, ((3*A + 2*C)*Sin[c + d*x])/(3*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2), x, 2, (1/8)*(4*A + 3*C)*x + ((4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2), x, 3, ((5*A + 4*C)*Sin[c + d*x])/(5*d) + (C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - ((5*A + 4*C)*Sin[c + d*x]^3)/(15*d)}
{Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2), x, 3, (1/16)*(6*A + 5*C)*x + ((6*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((6*A + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (C*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)}


{(-1 + 3*Cos[c + d*x]^2)/Sqrt[Cos[c + d*x]], x, 1, (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d}
{Cos[c + d*x]^m*((-C)*((m + 1)/(m + 2)) + C*Cos[c + d*x]^2), x, 1, (C*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m))}
{Cos[c + d*x]^m*(A + (-A)*((m + 2)/(m + 1))*Cos[c + d*x]^2), x, 1, -((A*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m)))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (A+C Cos[c+d x]^2)*)
(**)


{(A + C*Cos[c + d*x]^2)*Cos[c + d*x]^(5/2), x, 3, (2*(9*A + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)}
{(A + C*Cos[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 3, (2*(7*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)}
{(A + C*Cos[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 2, (2*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)}
{(A + C*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 2, (2*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{(A + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 2, -((2*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 2, (2*(A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}


(* ::Subsection::Closed:: *)
(*Integrands of the form (A+C Cos[c+d x]^2) (a+b Cos[c+d x])^n*)


{(A + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 5, (1/16)*(16*a^4*A + 48*a^2*A*b^2 + 6*A*b^4 + 8*a^4*C + 36*a^2*b^2*C + 5*b^4*C)*x + (a*(190*a^2*A*b^2 + 160*A*b^4 - 4*a^4*C + 121*a^2*b^2*C + 128*b^4*C)*Sin[c + d*x])/(60*b*d) + ((260*a^2*A*b^2 + 90*A*b^4 - 8*a^4*C + 178*a^2*b^2*C + 75*b^4*C)*Cos[c + d*x]*Sin[c + d*x])/(240*d) + (a*(70*A*b^2 - 4*a^2*C + 53*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) + ((30*A*b^2 - 4*a^2*C + 25*b^2*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*b*d) - (a*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + (C*(a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d)}
{(A + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 4, (1/8)*a*(8*a^2*A + 12*A*b^2 + 4*a^2*C + 9*b^2*C)*x + ((80*a^2*A*b^2 + 20*A*b^4 - 3*a^4*C + 52*a^2*b^2*C + 16*b^4*C)*Sin[c + d*x])/(30*b*d) + (a*(100*A*b^2 - 6*a^2*C + 71*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((20*A*b^2 - 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) - (a*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)}
{(A + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 3, (1/8)*(8*a^2*A + 4*A*b^2 + 4*a^2*C + 3*b^2*C)*x + (a*(12*A*b^2 - a^2*C + 8*b^2*C)*Sin[c + d*x])/(6*b*d) + ((12*A*b^2 - 2*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) - (a*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*b*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*b*d)}
{(A + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 2, (1/2)*a*(2*A + C)*x + ((3*A*b^2 - a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*b*d) - (a*C*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*b*d)}
{(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 3, -((a*C*x)/b^2) + (2*(A*b^2 + a^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]*d) + (C*Sin[c + d*x])/(b*d)}
{(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 3, (C*x)/b^2 + (2*a*(A*b^2 - a^2*C + 2*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(3/2)*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 4, ((2*a^2*A + A*b^2 + a^2*C + 2*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a*(3*A*b^2 - a^2*C + 4*b^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 5, (a*(2*a^2*A + 3*A*b^2 + a^2*C + 4*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a*(5*A*b^2 - a^2*C + 6*b^2*C)*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((11*a^2*A*b^2 + 4*A*b^4 - a^4*C + 10*a^2*b^2*C + 6*b^4*C)*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}


{(a^2 - b^2*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 5, (1/16)*(16*a^6 + 40*a^4*b^2 - 30*a^2*b^4 - 5*b^6)*x + (a*b*(194*a^4 + 39*a^2*b^2 - 128*b^4)*Sin[c + d*x])/(60*d) + (b^2*(268*a^4 - 88*a^2*b^2 - 75*b^4)*Cos[c + d*x]*Sin[c + d*x])/(240*d) + (a*b*(74*a^2 - 53*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*d) + (b*(34*a^2 - 25*b^2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*d) + (a*b*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*d) - (b*(a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*d)}
{(a^2 - b^2*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 4, (1/8)*a*(8*a^4 + 8*a^2*b^2 - 9*b^4)*x + (b*(83*a^4 - 32*a^2*b^2 - 16*b^4)*Sin[c + d*x])/(30*d) + (a*b^2*(106*a^2 - 71*b^2)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + (b*(23*a^2 - 16*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + (a*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) - (b*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)}
{(a^2 - b^2*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 3, (1/8)*(8*a^4 - 3*b^4)*x + (a*b*(13*a^2 - 8*b^2)*Sin[c + d*x])/(6*d) + (b^2*(14*a^2 - 9*b^2)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (a*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) - (b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)}
{(a^2 - b^2*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 2, (1/2)*a*(2*a^2 - b^2)*x + (2*b*(2*a^2 - b^2)*Sin[c + d*x])/(3*d) + (a*b^2*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)}
{(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 3, a*x - (b*Sin[c + d*x])/d}
{(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 3, -x + (4*a*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*d)}
{(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 4, (2*(a^2 + b^2)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d) - (2*a*b*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 5, (2*a*(a^2 + 2*b^2)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - (a*b*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (b*(2*a^2 + b^2)*Sin[c + d*x])/((a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form (A+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/2)*)


{(A + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(483*a^2*A*b^2 + 189*A*b^4 - 10*a^4*C + 279*a^2*b^2*C + 147*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*(84*a^2*A*b^2 - 84*A*b^4 - 5*a^4*C + 62*a^2*b^2*C - 57*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (4*a*(84*A*b^2 - 5*a^2*C + 57*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) + (2*(63*A*b^2 - 10*a^2*C + 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) - (4*a*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)}
{(A + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 8, (4*a*(70*A*b^2 - 3*a^2*C + 41*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(35*a^2*A*b^2 - 35*A*b^4 - 6*a^4*C + 31*a^2*b^2*C - 25*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(35*A*b^2 - 6*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) - (4*a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)}
{(A + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(15*A*b^2 - 2*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (4*a*(a^2 - b^2)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)}
{(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 6, -((4*a*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(3*A*b^2 + 2*a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)}
{(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*(A*b^2 + 2*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])}
{(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, (4*a*(2*A*b^2 - a^2*C + 3*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - 2*a^2*C + 3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (4*a*(2*A*b^2 - a^2*C + 3*b^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])}


{(a^2 - b^2*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(493*a^4 - 90*a^2*b^2 - 147*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*(89*a^4 - 146*a^2*b^2 + 57*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*d*Sqrt[a + b*Cos[c + d*x]]) + (4*a*b*(89*a^2 - 57*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*b*(73*a^2 - 49*b^2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*d) + (4*a*b*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) - (2*b*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*d)}
{(a^2 - b^2*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 8, (4*a*(73*a^2 - 41*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(41*a^4 - 66*a^2*b^2 + 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(41*a^2 - 25*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (4*a*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) - (2*b*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)}
{(a^2 - b^2*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(17*a^2 - 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (4*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}
{(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 6, (4*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 6, -((2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (4*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])}
{(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, (4*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/((a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*b*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form (A+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/3)*)
(**)


{(A + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(2/3), x, 2, (9*Int[((1/9)*(40*a*A*b + 19*a*b*C) + (1/9)*(40*A*b^2 - 6*a^2*C + 25*b^2*C)*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/3), x])/(40*b) - (9*a*C*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(40*b*d) + (3*C*(a + b*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*b*d)}
{(A + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/3), x, 1, (3*Int[(a + b*Cos[c + d*x])^(1/3)*((1/3)*b*(7*A + 4*C) - a*C*Cos[c + d*x]), x])/(7*b) + (3*C*(a + b*Cos[c + d*x])^(4/3)*Sin[c + d*x])/(7*b*d)}
{(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/3), x, 1, (3*Int[((1/3)*b*(5*A + 2*C) - a*C*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/3), x])/(5*b) + (3*C*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*b*d)}
{(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(2/3), x, 2, (9*Int[(a + b*Cos[c + d*x])^(1/3)*((4/9)*(a*A*b + a*b*C) - (4/9)*(4*A*b^2 + 3*a^2*C + b^2*C)*Cos[c + d*x]), x])/(4*b*(a^2 - b^2)) + (3*C*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*b*d) + (3*(4*A*b^2 + 3*a^2*C + b^2*C)*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*b*(a^2 - b^2)*d)}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+C Cos[c+d x]^2) (a+b Cos[c+d x])^n*)


(* ::Subsubsection:: *)
(*n>0*)


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 4, ((2*A*b^2 + 2*a^2*C + b^2*C)*x)/(2*b^3) - (2*a*(A*b^2 + a^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]*d) - (a*C*Sin[c + d*x])/(b^2*d) + (C*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)}
{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 5, -((a*(2*A*b^2 + 2*a^2*C + b^2*C)*x)/(2*b^4)) + (2*a^2*(A*b^2 + a^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*d) + ((3*A*b^2 + 3*a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*b^3*d) - (a*C*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)}
{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, ((8*a^2*A*b^2 + 4*A*b^4 + 8*a^4*C + 4*a^2*b^2*C + 3*b^4*C)*x)/(8*b^5) - (2*a^3*(A*b^2 + a^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*Sqrt[a^2 - b^2]*d) - (a*(3*A*b^2 + 3*a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*b^4*d) + ((4*A*b^2 + 4*a^2*C + 3*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) - (a*C*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)}


{Cos[c + d*x]*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 4, -((2*a*C*x)/b^3) - (2*(A*b^4 - 2*a^4*C + 3*a^2*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*d) + ((A*b^2 + 2*a^2*C - b^2*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 5, ((2*A*b^2 + 6*a^2*C + b^2*C)*x)/(2*b^4) - (2*a*(a^2*A*b^2 - 2*A*b^4 + 3*a^4*C - 4*a^2*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(3/2)*d) - (a*(A*b^2 + 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((2*A*b^2 + 3*a^2*C - b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, -((a*(2*A*b^2 + 4*a^2*C + b^2*C)*x)/b^5) + (2*a^2*(2*a^2*A*b^2 - 3*A*b^4 + 4*a^4*C - 5*a^2*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(3/2)*d) + ((6*a^2*A*b^2 - 3*A*b^4 + 12*a^4*C - 7*a^2*b^2*C - 2*b^4*C)*Sin[c + d*x])/(3*b^4*(a^2 - b^2)*d) - (a*(A*b^2 + 2*a^2*C - b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((3*A*b^2 + 4*a^2*C - b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}


(* ::Section::Closed:: *)
(*Cos[c+d x]^m (B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n*)


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (B Cos[c+d x]+C Cos[c+d x]^2)*)


{(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, (C*x)/2 + (B*Sin[c + d*x])/d + (C*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 2, (B*x)/2 + (2*C*Sin[c + d*x])/(3*d) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, (3*C*x)/8 + (2*B*Sin[c + d*x])/(3*d) + (3*C*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 4, (3*B*x)/8 + (8*C*Sin[c + d*x])/(15*d) + (3*B*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (4*C*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, (5*C*x)/16 + (8*B*Sin[c + d*x])/(15*d) + (5*C*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (4*B*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (5*C*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (B*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) + (C*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (B Cos[c+d x]+C Cos[c+d x]^2)*)
(**)


{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(5/2), x, 7, (14*C*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*B*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (14*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 6, (6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (10*C*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 5, (6*C*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 4, (2*B*EllipticE[(1/2)*(c + d*x), 2])/d + (2*C*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 4, (2*C*EllipticE[(1/2)*(c + d*x), 2])/d + (2*B*EllipticF[(1/2)*(c + d*x), 2])/d}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 5, -((2*B*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*C*EllipticF[(1/2)*(c + d*x), 2])/d + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form (B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n*)


{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 5, (1/16)*(32*a^3*b*B + 24*a*b^3*B + 8*a^4*C + 36*a^2*b^2*C + 5*b^4*C)*x + ((24*a^4*b*B + 224*a^2*b^3*B + 32*b^5*B - 4*a^5*C + 121*a^3*b^2*C + 128*a*b^4*C)*Sin[c + d*x])/(60*b*d) + ((48*a^3*b*B + 232*a*b^3*B - 8*a^4*C + 178*a^2*b^2*C + 75*b^4*C)*Cos[c + d*x]*Sin[c + d*x])/(240*d) + ((24*a^2*b*B + 32*b^3*B - 4*a^3*C + 53*a*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) + ((24*a*b*B - 4*a^2*C + 25*b^2*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*b*d) + ((6*b*B - a*C)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + (C*(a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 4, (1/8)*(12*a^2*b*B + 3*b^3*B + 4*a^3*C + 9*a*b^2*C)*x + ((15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 52*a^2*b^2*C + 16*b^4*C)*Sin[c + d*x])/(30*b*d) + ((30*a^2*b*B + 45*b^3*B - 6*a^3*C + 71*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((15*a*b*B - 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) + ((5*b*B - a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 3, (1/8)*(8*a*b*B + 4*a^2*C + 3*b^2*C)*x + ((4*a^2*b*B + 4*b^3*B - a^3*C + 8*a*b^2*C)*Sin[c + d*x])/(6*b*d) + ((8*a*b*B - 2*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*b*B - a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*b*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*b*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 2, (1/2)*(b*B + a*C)*x + ((3*a*b*B - a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*b*d) + ((3*b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*b*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 4, ((b*B - a*C)*x)/b^2 - (2*a*(b*B - a*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]*d) + (C*Sin[c + d*x])/(b*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 3, (C*x)/b^2 - (2*(b^3*B + a^3*C - 2*a*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(3/2)*d) + (a*(b*B - a*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 4, -(((3*a*b*B - a^2*C - 2*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d)) + (a*(b*B - a*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*b*B + 2*b^3*B + a^3*C - 4*a*b^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 5, -(((4*a^2*b*B + b^3*B - a^3*C - 4*a*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d)) + (a*(b*B - a*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((2*a^2*b*B + 3*b^3*B + a^3*C - 6*a*b^2*C)*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((2*a^3*b*B + 13*a*b^3*B + a^4*C - 10*a^2*b^2*C - 6*b^4*C)*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form (B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/2)*)
(**)


{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(45*a^3*b*B + 435*a*b^3*B - 10*a^4*C + 279*a^2*b^2*C + 147*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(45*a^4*b*B + 30*a^2*b^3*B - 75*b^5*B - 10*a^5*C + 124*a^3*b^2*C - 114*a*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(45*a^2*b*B + 75*b^3*B - 10*a^3*C + 114*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) + (2*(45*a*b*B - 10*a^2*C + 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) + (2*(9*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 8, (2*(21*a^2*b*B + 63*b^3*B - 6*a^3*C + 82*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(21*a^3*b*B - 21*a*b^3*B - 6*a^4*C + 31*a^2*b^2*C - 25*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) + (2*(7*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(5*a*b*B - 2*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(5*a^2*b*B - 5*b^3*B - 2*a^3*C + 2*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 6, (2*(3*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(3*a*b*B - 2*a^2*C - b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 6, -((2*(a*b*B - 2*a^2*C + b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(b*B - a*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])}
{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(a*b*B + 2*a^2*C - 3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(b*B - a*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n*)


(* ::Subsubsection:: *)
(*n>0*)


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 4, -(((2*a*b*B - 2*a^2*C - b^2*C)*x)/(2*b^3)) + (2*a^2*(b*B - a*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]*d) + ((b*B - a*C)*Sin[c + d*x])/(b^2*d) + (C*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)}
{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 5, ((2*a^2*b*B + b^3*B - 2*a^3*C - a*b^2*C)*x)/(2*b^4) - (2*a^3*(b*B - a*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*d) - ((3*a*b*B - 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/(3*b^3*d) + ((b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)}
{Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, -(((8*a^3*b*B + 4*a*b^3*B - 8*a^4*C - 4*a^2*b^2*C - 3*b^4*C)*x)/(8*b^5)) + (2*a^4*(b*B - a*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*Sqrt[a^2 - b^2]*d) + ((3*a^2*b*B + 2*b^3*B - 3*a^3*C - 2*a*b^2*C)*Sin[c + d*x])/(3*b^4*d) - ((4*a*b*B - 4*a^2*C - 3*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) + ((b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)}


{Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 4, ((b*B - 2*a*C)*x)/b^3 - (2*a*(a^2*b*B - 2*b^3*B - 2*a^3*C + 3*a*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*d) - ((a*b*B - 2*a^2*C + b^2*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 5, -(((4*a*b*B - 6*a^2*C - b^2*C)*x)/(2*b^4)) + (2*a^2*(2*a^2*b*B - 3*b^3*B - 3*a^3*C + 4*a*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(3/2)*d) + ((2*a^2*b*B - b^3*B - 3*a^3*C + 2*a*b^2*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) - ((2*a*b*B - 3*a^2*C + b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, ((6*a^2*b*B + b^3*B - 8*a^3*C - 2*a*b^2*C)*x)/(2*b^5) - (2*a^3*(3*a^2*b*B - 4*b^3*B - 4*a^3*C + 5*a*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(3/2)*d) - ((9*a^3*b*B - 6*a*b^3*B - 12*a^4*C + 7*a^2*b^2*C + 2*b^4*C)*Sin[c + d*x])/(3*b^4*(a^2 - b^2)*d) + ((3*a^2*b*B - b^3*B - 4*a^3*C + 2*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) - ((3*a*b*B - 4*a^2*C + b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}


(* ::Section::Closed:: *)
(*Cos[c+d x]^m (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n*)


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]+C Cos[c+d x]^2)*)


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, A*x + (C*x)/2 + (B*Sin[c + d*x])/d + (C*Cos[c + d*x]*Sin[c + d*x])/(2*d)}
{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 2, (B*x)/2 + ((3*A + 2*C)*Sin[c + d*x])/(3*d) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, (1/8)*(4*A + 3*C)*x + (2*B*Sin[c + d*x])/(3*d) + ((4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 4, (3*B*x)/8 + (2*(5*A + 4*C)*Sin[c + d*x])/(15*d) + (3*B*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((5*A + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, (1/16)*(6*A + 5*C)*x + (8*B*Sin[c + d*x])/(15*d) + ((6*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (4*B*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + ((6*A + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (B*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) + (C*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x]+C Cos[c+d x]^2)*)
(**)


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(5/2), x, 7, (2*(9*A + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*B*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 6, (6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(7*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 5, (2*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 4, (2*B*EllipticE[(1/2)*(c + d*x), 2])/d + (2*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 4, -((2*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/d + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 5, -((2*B*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n*)


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 5, (1/16)*(16*a^4*A + 48*a^2*A*b^2 + 6*A*b^4 + 32*a^3*b*B + 24*a*b^3*B + 8*a^4*C + 36*a^2*b^2*C + 5*b^4*C)*x + ((190*a^3*A*b^2 + 160*a*A*b^4 + 24*a^4*b*B + 224*a^2*b^3*B + 32*b^5*B - 4*a^5*C + 121*a^3*b^2*C + 128*a*b^4*C)*Sin[c + d*x])/(60*b*d) + ((260*a^2*A*b^2 + 90*A*b^4 + 48*a^3*b*B + 232*a*b^3*B - 8*a^4*C + 178*a^2*b^2*C + 75*b^4*C)*Cos[c + d*x]*Sin[c + d*x])/(240*d) + ((70*a*A*b^2 + 24*a^2*b*B + 32*b^3*B - 4*a^3*C + 53*a*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) + ((30*A*b^2 + 24*a*b*B - 4*a^2*C + 25*b^2*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*b*d) + ((6*b*B - a*C)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + (C*(a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 4, (1/8)*(8*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 3*b^3*B + 4*a^3*C + 9*a*b^2*C)*x + ((80*a^2*A*b^2 + 20*A*b^4 + 15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 52*a^2*b^2*C + 16*b^4*C)*Sin[c + d*x])/(30*b*d) + ((100*a*A*b^2 + 30*a^2*b*B + 45*b^3*B - 6*a^3*C + 71*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((20*A*b^2 + 15*a*b*B - 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) + ((5*b*B - a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 3, (1/8)*(8*a^2*A + 4*A*b^2 + 8*a*b*B + 4*a^2*C + 3*b^2*C)*x + ((12*a*A*b^2 + 4*a^2*b*B + 4*b^3*B - a^3*C + 8*a*b^2*C)*Sin[c + d*x])/(6*b*d) + ((12*A*b^2 + 8*a*b*B - 2*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*b*B - a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*b*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*b*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 2, (1/2)*(2*a*A + b*B + a*C)*x + ((3*A*b^2 + 3*a*b*B - a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*b*d) + ((3*b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*b*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 3, ((b*B - a*C)*x)/b^2 + (2*(A*b^2 - a*b*B + a^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]*d) + (C*Sin[c + d*x])/(b*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 3, (C*x)/b^2 + (2*(a*A*b^2 - b^3*B - a^3*C + 2*a*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(3/2)*d) - ((A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 4, ((2*a^2*A + A*b^2 - 3*a*b*B + a^2*C + 2*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - ((A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*a*A*b^2 - a^2*b*B - 2*b^3*B - a^3*C + 4*a*b^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 5, ((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B + a^3*C + 4*a*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) - ((A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((5*a*A*b^2 - 2*a^2*b*B - 3*b^3*B - a^3*C + 6*a*b^2*C)*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((11*a^2*A*b^2 + 4*A*b^4 - 2*a^3*b*B - 13*a*b^3*B - a^4*C + 10*a^2*b^2*C + 6*b^4*C)*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}


{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 5, (1/16)*(16*a^5*b*B + 80*a^3*b^3*B + 30*a*b^5*B - 16*a^6*C - 40*a^4*b^2*C + 30*a^2*b^4*C + 5*b^6*C)*x + (b*(214*a^4*b*B + 384*a^2*b^3*B + 32*b^5*B - 194*a^5*C - 39*a^3*b^2*C + 128*a*b^4*C)*Sin[c + d*x])/(60*d) + (b^2*(308*a^3*b*B + 322*a*b^3*B - 268*a^4*C + 88*a^2*b^2*C + 75*b^4*C)*Cos[c + d*x]*Sin[c + d*x])/(240*d) + (b*(94*a^2*b*B + 32*b^3*B - 74*a^3*C + 53*a*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*d) + (b*(54*a*b*B - 34*a^2*C + 25*b^2*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*d) + (b*(6*b*B - a*C)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*d) + (b*C*(a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*d)}
{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 4, (1/8)*(8*a^4*b*B + 24*a^2*b^3*B + 3*b^5*B - 8*a^5*C - 8*a^3*b^2*C + 9*a*b^4*C)*x + (b*(95*a^3*b*B + 80*a*b^3*B - 83*a^4*C + 32*a^2*b^2*C + 16*b^4*C)*Sin[c + d*x])/(30*d) + (b^2*(130*a^2*b*B + 45*b^3*B - 106*a^3*C + 71*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + (b*(35*a*b*B - 23*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + (b*(5*b*B - a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (b*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)}
{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 3, (1/8)*(8*a^3*b*B + 12*a*b^3*B - 8*a^4*C + 3*b^4*C)*x + (b*(16*a^2*b*B + 4*b^3*B - 13*a^3*C + 8*a*b^2*C)*Sin[c + d*x])/(6*d) + (b^2*(20*a*b*B - 14*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (b*(4*b*B - a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (b*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)}
{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 2, (1/2)*(2*a^2*b*B + b^3*B - 2*a^3*C + a*b^2*C)*x + (2*b*(3*a*b*B - 2*a^2*C + b^2*C)*Sin[c + d*x])/(3*d) + (b^2*(3*b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (b*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)}
{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 3, (b*B - a*C)*x + (b*C*Sin[c + d*x])/d}
{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 3, C*x + (2*(b*B - 2*a*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*d)}
{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 4, (2*(a*b*B - a^2*C - b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d) - (b*(b*B - 2*a*C)*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 5, ((2*a^2*b*B + b^3*B - 2*a^3*C - 4*a*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - (b*(b*B - 2*a*C)*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (b*(3*a*b*B - 4*a^2*C - 2*b^2*C)*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/2)*)


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(483*a^2*A*b^2 + 189*A*b^4 + 45*a^3*b*B + 435*a*b^3*B - 10*a^4*C + 279*a^2*b^2*C + 147*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(168*a^3*A*b^2 - 168*a*A*b^4 + 45*a^4*b*B + 30*a^2*b^3*B - 75*b^5*B - 10*a^5*C + 124*a^3*b^2*C - 114*a*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(168*a*A*b^2 + 45*a^2*b*B + 75*b^3*B - 10*a^3*C + 114*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) + (2*(63*A*b^2 + 45*a*b*B - 10*a^2*C + 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) + (2*(9*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 8, (2*(140*a*A*b^2 + 21*a^2*b*B + 63*b^3*B - 6*a^3*C + 82*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(35*a^2*A*b^2 - 35*A*b^4 + 21*a^3*b*B - 21*a*b^3*B - 6*a^4*C + 31*a^2*b^2*C - 25*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(35*A*b^2 + 21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) + (2*(7*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(15*A*b^2 + 5*a*b*B - 2*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(5*a^2*b*B - 5*b^3*B - 2*a^3*C + 2*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 6, (2*(3*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(3*A*b^2 - 3*a*b*B + 2*a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(4*a*A*b^2 - a^2*b*B - 3*b^3*B - 2*a^3*C + 6*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - a*b*B - 2*a^2*C + 3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(4*a*A*b^2 - a^2*b*B - 3*b^3*B - 2*a^3*C + 6*a*b^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])}


{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(528*a^3*b*B + 624*a*b^3*B - 493*a^4*C + 90*a^2*b^2*C + 147*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(213*a^4*b*B - 138*a^2*b^3*B - 75*b^5*B - 178*a^5*C + 292*a^3*b^2*C - 114*a*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(213*a^2*b*B + 75*b^3*B - 178*a^3*C + 114*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*b*(108*a*b*B - 73*a^2*C + 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*d) + (2*b*(9*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*b*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*d)}
{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 8, (2*(161*a^2*b*B + 63*b^3*B - 146*a^3*C + 82*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(56*a^3*b*B - 56*a*b^3*B - 41*a^4*C + 66*a^2*b^2*C - 25*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(56*a*b*B - 41*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*b*(7*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*b*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)}
{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(20*a*b*B - 17*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(5*a^2*b*B - 5*b^3*B - 2*a^3*C + 2*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(5*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)}
{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 6, (2*(3*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)}
{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])}
{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/((a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*(b*B - 2*a*C)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])}


(* ::Subsection::Closed:: *)
(*Integrands of the form (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/3)*)
(**)


{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(2/3), x, 2, (9*Int[((1/9)*(40*a*A*b + 16*b^2*B + 19*a*b*C) + (1/9)*(40*A*b^2 + 16*a*b*B - 6*a^2*C + 25*b^2*C)*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/3), x])/(40*b) + (3*(8*b*B - 3*a*C)*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(40*b*d) + (3*C*(a + b*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*b*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/3), x, 1, (3*Int[(a + b*Cos[c + d*x])^(1/3)*((1/3)*b*(7*A + 4*C) + (1/3)*(7*b*B - 3*a*C)*Cos[c + d*x]), x])/(7*b) + (3*C*(a + b*Cos[c + d*x])^(4/3)*Sin[c + d*x])/(7*b*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/3), x, 1, (3*Int[((1/3)*b*(5*A + 2*C) + (1/3)*(5*b*B - 3*a*C)*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/3), x])/(5*b) + (3*C*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*b*d)}
{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(2/3), x, 2, (9*Int[(a + b*Cos[c + d*x])^(1/3)*((4/9)*(a*A*b - b^2*B + a*b*C) - (4/9)*(4*A*b^2 - 4*a*b*B + 3*a^2*C + b^2*C)*Cos[c + d*x]), x])/(4*b*(a^2 - b^2)) + (3*C*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*b*d) + (3*(4*A*b^2 - 4*a*b*B + 3*a^2*C + b^2*C)*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*b*(a^2 - b^2)*d)}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 3, (1/8)*(4*A*b + 4*a*B + 3*b*C)*x + ((3*a*A + 2*b*B + 2*a*C)*Sin[c + d*x])/(3*d) + ((4*A*b + 4*a*B + 3*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((4*b*B + a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(12*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(4*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 4, (1/8)*(4*a*A + 3*b*B + 3*a*C)*x + (2*(5*A*b + 5*a*B + 4*b*C)*Sin[c + d*x])/(15*d) + ((4*a*A + 3*b*B + 3*a*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((5*A*b + 5*a*B + 4*b*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + ((5*b*B + a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (C*Cos[c + d*x]^3*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 5, (1/16)*(6*A*b + 6*a*B + 5*b*C)*x + (2*(5*a*A + 4*b*B + 4*a*C)*Sin[c + d*x])/(15*d) + ((6*A*b + 6*a*B + 5*b*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((5*a*A + 4*b*B + 4*a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + ((6*A*b + 6*a*B + 5*b*C)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + ((6*b*B + a*C)*Cos[c + d*x]^4*Sin[c + d*x])/(30*d) + (C*Cos[c + d*x]^4*(a + b*Cos[c + d*x])*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 6, (1/16)*(6*a*A + 5*b*B + 5*a*C)*x + (8*(7*A*b + 7*a*B + 6*b*C)*Sin[c + d*x])/(105*d) + ((6*a*A + 5*b*B + 5*a*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (4*(7*A*b + 7*a*B + 6*b*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + ((6*a*A + 5*b*B + 5*a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + ((7*A*b + 7*a*B + 6*b*C)*Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + ((7*b*B + a*C)*Cos[c + d*x]^5*Sin[c + d*x])/(42*d) + (C*Cos[c + d*x]^5*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 4, (1/8)*(8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*x + ((15*a^2*A + 10*A*b^2 + 20*a*b*B + 10*a^2*C + 8*b^2*C)*Sin[c + d*x])/(15*d) + ((8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((20*A*b^2 + 25*a*b*B + 2*a^2*C + 16*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(60*d) + ((5*b*B + 2*a*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(20*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 5, (1/16)*(8*a^2*A + 6*A*b^2 + 12*a*b*B + 6*a^2*C + 5*b^2*C)*x + (2*(10*a*A*b + 5*a^2*B + 4*b^2*B + 8*a*b*C)*Sin[c + d*x])/(15*d) + ((8*a^2*A + 6*A*b^2 + 12*a*b*B + 6*a^2*C + 5*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((10*a*A*b + 5*a^2*B + 4*b^2*B + 8*a*b*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + ((30*A*b^2 + 36*a*b*B + 2*a^2*C + 25*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(120*d) + ((3*b*B + a*C)*Cos[c + d*x]^3*(a + b*Cos[c + d*x])*Sin[c + d*x])/(15*d) + (C*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 6, (1/16)*(12*a*A*b + 6*a^2*B + 5*b^2*B + 10*a*b*C)*x + (2*(35*a^2*A + 28*A*b^2 + 56*a*b*B + 28*a^2*C + 24*b^2*C)*Sin[c + d*x])/(105*d) + ((12*a*A*b + 6*a^2*B + 5*b^2*B + 10*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((35*a^2*A + 28*A*b^2 + 56*a*b*B + 28*a^2*C + 24*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + ((12*a*A*b + 6*a^2*B + 5*b^2*B + 10*a*b*C)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + ((42*A*b^2 + 49*a*b*B + 2*a^2*C + 36*b^2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(210*d) + ((7*b*B + 2*a*C)*Cos[c + d*x]^4*(a + b*Cos[c + d*x])*Sin[c + d*x])/(42*d) + (C*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 7, (1/128)*(48*a^2*A + 40*A*b^2 + 80*a*b*B + 40*a^2*C + 35*b^2*C)*x + (8*(14*a*A*b + 7*a^2*B + 6*b^2*B + 12*a*b*C)*Sin[c + d*x])/(105*d) + ((48*a^2*A + 40*A*b^2 + 80*a*b*B + 40*a^2*C + 35*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (4*(14*a*A*b + 7*a^2*B + 6*b^2*B + 12*a*b*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + ((48*a^2*A + 40*A*b^2 + 80*a*b*B + 40*a^2*C + 35*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + ((14*a*A*b + 7*a^2*B + 6*b^2*B + 12*a*b*C)*Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + ((56*A*b^2 + 64*a*b*B + 2*a^2*C + 49*b^2*C)*Cos[c + d*x]^5*Sin[c + d*x])/(336*d) + ((4*b*B + a*C)*Cos[c + d*x]^5*(a + b*Cos[c + d*x])*Sin[c + d*x])/(28*d) + (C*Cos[c + d*x]^5*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(8*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 5, (1/16)*(24*a^2*A*b + 6*A*b^3 + 8*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 5*b^3*C)*x + ((15*a^3*A + 30*a*A*b^2 + 30*a^2*b*B + 8*b^3*B + 10*a^3*C + 24*a*b^2*C)*Sin[c + d*x])/(15*d) + ((24*a^2*A*b + 6*A*b^3 + 8*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 5*b^3*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((90*a*A*b^2 + 54*a^2*b*B + 32*b^3*B + 2*a^3*C + 71*a*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(120*d) + ((30*A*b^2 + 42*a*b*B + 6*a^2*C + 25*b^2*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(120*d) + ((2*b*B + a*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(10*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(6*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 6, (1/16)*(8*a^3*A + 18*a*A*b^2 + 18*a^2*b*B + 5*b^3*B + 6*a^3*C + 15*a*b^2*C)*x + (2*(105*a^2*A*b + 28*A*b^3 + 35*a^3*B + 84*a*b^2*B + 84*a^2*b*C + 24*b^3*C)*Sin[c + d*x])/(105*d) + ((8*a^3*A + 18*a*A*b^2 + 18*a^2*b*B + 5*b^3*B + 6*a^3*C + 15*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((105*a^2*A*b + 28*A*b^3 + 35*a^3*B + 84*a*b^2*B + 84*a^2*b*C + 24*b^3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + ((462*a*A*b^2 + 266*a^2*b*B + 175*b^3*B + 6*a^3*C + 381*a*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(840*d) + ((21*A*b^2 + 28*a*b*B + 3*a^2*C + 18*b^2*C)*Cos[c + d*x]^3*(a + b*Cos[c + d*x])*Sin[c + d*x])/(105*d) + ((7*b*B + 3*a*C)*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(42*d) + (C*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 7, (1/128)*(144*a^2*A*b + 40*A*b^3 + 48*a^3*B + 120*a*b^2*B + 120*a^2*b*C + 35*b^3*C)*x + (2*(35*a^3*A + 84*a*A*b^2 + 84*a^2*b*B + 24*b^3*B + 28*a^3*C + 72*a*b^2*C)*Sin[c + d*x])/(105*d) + ((144*a^2*A*b + 40*A*b^3 + 48*a^3*B + 120*a*b^2*B + 120*a^2*b*C + 35*b^3*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + ((35*a^3*A + 84*a*A*b^2 + 84*a^2*b*B + 24*b^3*B + 28*a^3*C + 72*a*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + ((144*a^2*A*b + 40*A*b^3 + 48*a^3*B + 120*a*b^2*B + 120*a^2*b*C + 35*b^3*C)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + ((728*a*A*b^2 + 408*a^2*b*B + 288*b^3*B + 6*a^3*C + 619*a*b^2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(1680*d) + ((56*A*b^2 + 72*a*b*B + 6*a^2*C + 49*b^2*C)*Cos[c + d*x]^4*(a + b*Cos[c + d*x])*Sin[c + d*x])/(336*d) + ((8*b*B + 3*a*C)*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(56*d) + (C*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(8*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 8, (1/128)*(48*a^3*A + 120*a*A*b^2 + 120*a^2*b*B + 35*b^3*B + 40*a^3*C + 105*a*b^2*C)*x + (8*(63*a^2*A*b + 18*A*b^3 + 21*a^3*B + 54*a*b^2*B + 54*a^2*b*C + 16*b^3*C)*Sin[c + d*x])/(315*d) + ((48*a^3*A + 120*a*A*b^2 + 120*a^2*b*B + 35*b^3*B + 40*a^3*C + 105*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (4*(63*a^2*A*b + 18*A*b^3 + 21*a^3*B + 54*a*b^2*B + 54*a^2*b*C + 16*b^3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(315*d) + ((48*a^3*A + 120*a*A*b^2 + 120*a^2*b*B + 35*b^3*B + 40*a^3*C + 105*a*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + ((63*a^2*A*b + 18*A*b^3 + 21*a^3*B + 54*a*b^2*B + 54*a^2*b*C + 16*b^3*C)*Cos[c + d*x]^4*Sin[c + d*x])/(105*d) + ((360*a*A*b^2 + 198*a^2*b*B + 147*b^3*B + 2*a^3*C + 313*a*b^2*C)*Cos[c + d*x]^5*Sin[c + d*x])/(1008*d) + ((36*A*b^2 + 45*a*b*B + 3*a^2*C + 32*b^2*C)*Cos[c + d*x]^5*(a + b*Cos[c + d*x])*Sin[c + d*x])/(252*d) + ((3*b*B + a*C)*Cos[c + d*x]^5*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(24*d) + (C*Cos[c + d*x]^5*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 6, (1/16)*(32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 24*a^3*b*C + 20*a*b^3*C)*x + ((105*a^4*A + 420*a^2*A*b^2 + 56*A*b^4 + 280*a^3*b*B + 224*a*b^3*B + 70*a^4*C + 336*a^2*b^2*C + 48*b^4*C)*Sin[c + d*x])/(105*d) + ((32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 24*a^3*b*C + 20*a*b^3*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((1008*a^2*A*b^2 + 224*A*b^4 + 392*a^3*b*B + 721*a*b^3*B + 8*a^4*C + 788*a^2*b^2*C + 192*b^4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(840*d) + ((504*a*A*b^2 + 336*a^2*b*B + 175*b^3*B + 24*a^3*C + 412*a*b^2*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(840*d) + ((14*A*b^2 + 21*a*b*B + 4*a^2*C + 12*b^2*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(70*d) + ((7*b*B + 4*a*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(42*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 7, (1/128)*(64*a^4*A + 288*a^2*A*b^2 + 40*A*b^4 + 192*a^3*b*B + 160*a*b^3*B + 48*a^4*C + 240*a^2*b^2*C + 35*b^4*C)*x + (2*(140*a^3*A*b + 112*a*A*b^3 + 35*a^4*B + 168*a^2*b^2*B + 24*b^4*B + 112*a^3*b*C + 96*a*b^3*C)*Sin[c + d*x])/(105*d) + ((64*a^4*A + 288*a^2*A*b^2 + 40*A*b^4 + 192*a^3*b*B + 160*a*b^3*B + 48*a^4*C + 240*a^2*b^2*C + 35*b^4*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + ((140*a^3*A*b + 112*a*A*b^3 + 35*a^4*B + 168*a^2*b^2*B + 24*b^4*B + 112*a^3*b*C + 96*a*b^3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + ((5824*a^2*A*b^2 + 1400*A*b^4 + 2176*a^3*b*B + 4448*a*b^3*B + 24*a^4*C + 4772*a^2*b^2*C + 1225*b^4*C)*Cos[c + d*x]^3*Sin[c + d*x])/(6720*d) + ((392*a*A*b^2 + 248*a^2*b*B + 144*b^3*B + 12*a^3*C + 331*a*b^2*C)*Cos[c + d*x]^3*(a + b*Cos[c + d*x])*Sin[c + d*x])/(840*d) + ((56*A*b^2 + 80*a*b*B + 12*a^2*C + 49*b^2*C)*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(336*d) + ((2*b*B + a*C)*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(14*d) + (C*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(8*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 8, (1/128)*(192*a^3*A*b + 160*a*A*b^3 + 48*a^4*B + 240*a^2*b^2*B + 35*b^4*B + 160*a^3*b*C + 140*a*b^3*C)*x + (2*(105*a^4*A + 504*a^2*A*b^2 + 72*A*b^4 + 336*a^3*b*B + 288*a*b^3*B + 84*a^4*C + 432*a^2*b^2*C + 64*b^4*C)*Sin[c + d*x])/(315*d) + ((192*a^3*A*b + 160*a*A*b^3 + 48*a^4*B + 240*a^2*b^2*B + 35*b^4*B + 160*a^3*b*C + 140*a*b^3*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + ((105*a^4*A + 504*a^2*A*b^2 + 72*A*b^4 + 336*a^3*b*B + 288*a*b^3*B + 84*a^4*C + 432*a^2*b^2*C + 64*b^4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(315*d) + ((192*a^3*A*b + 160*a*A*b^3 + 48*a^4*B + 240*a^2*b^2*B + 35*b^4*B + 160*a^3*b*C + 140*a*b^3*C)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + ((3408*a^2*A*b^2 + 864*A*b^4 + 1242*a^3*b*B + 2721*a*b^3*B + 8*a^4*C + 2884*a^2*b^2*C + 768*b^4*C)*Cos[c + d*x]^4*Sin[c + d*x])/(5040*d) + ((384*a*A*b^2 + 234*a^2*b*B + 147*b^3*B + 8*a^3*C + 332*a*b^2*C)*Cos[c + d*x]^4*(a + b*Cos[c + d*x])*Sin[c + d*x])/(1008*d) + ((72*A*b^2 + 99*a*b*B + 12*a^2*C + 64*b^2*C)*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(504*d) + ((9*b*B + 4*a*C)*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(72*d) + (C*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 9, (1/256)*(96*a^4*A + 480*a^2*A*b^2 + 70*A*b^4 + 320*a^3*b*B + 280*a*b^3*B + 80*a^4*C + 420*a^2*b^2*C + 63*b^4*C)*x + (8*(84*a^3*A*b + 72*a*A*b^3 + 21*a^4*B + 108*a^2*b^2*B + 16*b^4*B + 72*a^3*b*C + 64*a*b^3*C)*Sin[c + d*x])/(315*d) + ((96*a^4*A + 480*a^2*A*b^2 + 70*A*b^4 + 320*a^3*b*B + 280*a*b^3*B + 80*a^4*C + 420*a^2*b^2*C + 63*b^4*C)*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (4*(84*a^3*A*b + 72*a*A*b^3 + 21*a^4*B + 108*a^2*b^2*B + 16*b^4*B + 72*a^3*b*C + 64*a*b^3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(315*d) + ((96*a^4*A + 480*a^2*A*b^2 + 70*A*b^4 + 320*a^3*b*B + 280*a*b^3*B + 80*a^4*C + 420*a^2*b^2*C + 63*b^4*C)*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + ((84*a^3*A*b + 72*a*A*b^3 + 21*a^4*B + 108*a^2*b^2*B + 16*b^4*B + 72*a^3*b*C + 64*a*b^3*C)*Cos[c + d*x]^4*Sin[c + d*x])/(105*d) + ((5580*a^2*A*b^2 + 1470*A*b^4 + 2000*a^3*b*B + 4600*a*b^3*B + 8*a^4*C + 4834*a^2*b^2*C + 1323*b^4*C)*Cos[c + d*x]^5*Sin[c + d*x])/(10080*d) + ((810*a*A*b^2 + 480*a^2*b*B + 320*b^3*B + 12*a^3*C + 713*a*b^2*C)*Cos[c + d*x]^5*(a + b*Cos[c + d*x])*Sin[c + d*x])/(2520*d) + ((30*A*b^2 + 40*a*b*B + 4*a^2*C + 27*b^2*C)*Cos[c + d*x]^5*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(240*d) + ((5*b*B + 2*a*C)*Cos[c + d*x]^5*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(45*d) + (C*Cos[c + d*x]^5*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(10*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 4, ((2*A*b^2 - 2*a*b*B + 2*a^2*C + b^2*C)*x)/(2*b^3) - (2*a*(A*b^2 - a*b*B + a^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]*d) + ((b*B - a*C)*Sin[c + d*x])/(b^2*d) + (C*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 5, -(((2*a*A*b^2 - 2*a^2*b*B - b^3*B + 2*a^3*C + a*b^2*C)*x)/(2*b^4)) + (2*a^2*(A*b^2 - a*b*B + a^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*d) + ((3*A*b^2 - 3*a*b*B + 3*a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*b^3*d) + ((b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, ((8*a^2*A*b^2 + 4*A*b^4 - 8*a^3*b*B - 4*a*b^3*B + 8*a^4*C + 4*a^2*b^2*C + 3*b^4*C)*x)/(8*b^5) - (2*a^3*(A*b^2 - a*b*B + a^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*Sqrt[a^2 - b^2]*d) - ((3*a*A*b^2 - 3*a^2*b*B - 2*b^3*B + 3*a^3*C + 2*a*b^2*C)*Sin[c + d*x])/(3*b^4*d) + ((4*A*b^2 - 4*a*b*B + 4*a^2*C + 3*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) + ((b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, -(((8*a^3*A*b^2 + 4*a*A*b^4 - 8*a^4*b*B - 4*a^2*b^3*B - 3*b^5*B + 8*a^5*C + 4*a^3*b^2*C + 3*a*b^4*C)*x)/(8*b^6)) + (2*a^4*(A*b^2 - a*b*B + a^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*Sqrt[a^2 - b^2]*d) + ((15*a^2*A*b^2 + 10*A*b^4 - 15*a^3*b*B - 10*a*b^3*B + 15*a^4*C + 10*a^2*b^2*C + 8*b^4*C)*Sin[c + d*x])/(15*b^5*d) - ((4*a*A*b^2 - 4*a^2*b*B - 3*b^3*B + 4*a^3*C + 3*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(8*b^4*d) + ((5*A*b^2 - 5*a*b*B + 5*a^2*C + 4*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*b^3*d) + ((b*B - a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*b^2*d) + (C*Cos[c + d*x]^4*Sin[c + d*x])/(5*b*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 4, ((b*B - 2*a*C)*x)/b^3 - (2*(A*b^4 + a^3*b*B - 2*a*b^3*B - 2*a^4*C + 3*a^2*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*d) + ((A*b^2 - a*b*B + 2*a^2*C - b^2*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 5, ((2*A*b^2 - 4*a*b*B + 6*a^2*C + b^2*C)*x)/(2*b^4) - (2*a*(a^2*A*b^2 - 2*A*b^4 - 2*a^3*b*B + 3*a*b^3*B + 3*a^4*C - 4*a^2*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(3/2)*d) - ((a*A*b^2 - 2*a^2*b*B + b^3*B + 3*a^3*C - 2*a*b^2*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, -(((4*a*A*b^2 - 6*a^2*b*B - b^3*B + 8*a^3*C + 2*a*b^2*C)*x)/(2*b^5)) + (2*a^2*(2*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 4*a*b^3*B + 4*a^4*C - 5*a^2*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(3/2)*d) + ((6*a^2*A*b^2 - 3*A*b^4 - 9*a^3*b*B + 6*a*b^3*B + 12*a^4*C - 7*a^2*b^2*C - 2*b^4*C)*Sin[c + d*x])/(3*b^4*(a^2 - b^2)*d) - ((2*a*A*b^2 - 3*a^2*b*B + b^3*B + 4*a^3*C - 2*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) + ((3*A*b^2 - 3*a*b*B + 4*a^2*C - b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 7, ((24*a^2*A*b^2 + 4*A*b^4 - 32*a^3*b*B - 8*a*b^3*B + 40*a^4*C + 12*a^2*b^2*C + 3*b^4*C)*x)/(8*b^6) - (2*a^3*(3*a^2*A*b^2 - 4*A*b^4 - 4*a^3*b*B + 5*a*b^3*B + 5*a^4*C - 6*a^2*b^2*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*(a^2 - b^2)^(3/2)*d) - ((9*a^3*A*b^2 - 6*a*A*b^4 - 12*a^4*b*B + 7*a^2*b^3*B + 2*b^5*B + 15*a^5*C - 8*a^3*b^2*C - 4*a*b^4*C)*Sin[c + d*x])/(3*b^5*(a^2 - b^2)*d) + ((12*a^2*A*b^2 - 4*A*b^4 - 16*a^3*b*B + 8*a*b^3*B + 20*a^4*C - 9*a^2*b^2*C - 3*b^4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*b^4*(a^2 - b^2)*d) - ((3*a*A*b^2 - 4*a^2*b*B + b^3*B + 5*a^3*C - 2*a*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d) + ((4*A*b^2 - 4*a*b*B + 5*a^2*C - b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}


{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 4, (C*x)/b^3 - ((3*a*A*b^4 - a^2*b^3*B - 2*b^5*B + 2*a^5*C - 5*a^3*b^2*C + 6*a*b^4*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(5/2)*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((2*a^2*A*b^2 + A*b^4 - 3*a*b^3*B - 2*a^4*C + 5*a^2*b^2*C)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 5, ((b*B - 3*a*C)*x)/b^4 + ((a^2*A*b^4 + 2*A*b^6 - 2*a^5*b*B + 5*a^3*b^3*B - 6*a*b^5*B + 6*a^6*C - 15*a^4*b^2*C + 12*a^2*b^4*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(5/2)*d) - ((3*A*b^4 + 2*a^3*b*B - 5*a*b^3*B - 6*a^4*C + 11*a^2*b^2*C - 2*b^4*C)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*A*b^2 + 2*A*b^4 + a^3*b*B - 4*a*b^3*B - 3*a^4*C + 6*a^2*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 6, ((2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*x)/(2*b^5) - (a*(2*a^4*A*b^2 - 5*a^2*A*b^4 + 6*A*b^6 - 6*a^5*b*B + 15*a^3*b^3*B - 12*a*b^5*B + 12*a^6*C - 29*a^4*b^2*C + 20*a^2*b^4*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(5/2)*d) - ((2*a^3*A*b^2 - 5*a*A*b^4 - 6*a^4*b*B + 11*a^2*b^3*B - 2*b^5*B + 12*a^5*C - 21*a^3*b^2*C + 6*a*b^4*C)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) + ((a^2*A*b^2 - 4*A*b^4 - 3*a^3*b*B + 6*a*b^3*B + 6*a^4*C - 10*a^2*b^2*C + b^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((3*A*b^4 + 2*a^3*b*B - 5*a*b^3*B - 4*a^4*C + 7*a^2*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, -(((6*a*A*b^2 - 12*a^2*b*B - b^3*B + 20*a^3*C + 3*a*b^2*C)*x)/(2*b^6)) + (a^2*(6*a^4*A*b^2 - 15*a^2*A*b^4 + 12*A*b^6 - 12*a^5*b*B + 29*a^3*b^3*B - 20*a*b^5*B + 20*a^6*C - 47*a^4*b^2*C + 30*a^2*b^4*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*(a^2 - b^2)^(5/2)*d) + ((18*a^4*A*b^2 - 33*a^2*A*b^4 + 6*A*b^6 - 36*a^5*b*B + 63*a^3*b^3*B - 18*a*b^5*B + 60*a^6*C - 101*a^4*b^2*C + 28*a^2*b^4*C + 4*b^6*C)*Sin[c + d*x])/(6*b^5*(a^2 - b^2)^2*d) - ((3*a^3*A*b^2 - 6*a*A*b^4 - 6*a^4*b*B + 10*a^2*b^3*B - b^5*B + 10*a^5*C - 16*a^3*b^2*C + 3*a*b^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) + ((6*a^2*A*b^2 - 15*A*b^4 - 12*a^3*b*B + 21*a*b^3*B + 20*a^4*C - 31*a^2*b^2*C + 2*b^4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((a^2*A*b^2 - 4*A*b^4 - 3*a^3*b*B + 6*a*b^3*B + 5*a^4*C - 8*a^2*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}


{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 5, -(((4*a^2*A*b + A*b^3 - a^3*B - 4*a*b^2*B + 3*a^2*b*C + 2*b^3*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d)) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((4*a^2*A*b^2 + A*b^4 - a^3*b*B - 4*a*b^3*B - 2*a^4*C + 7*a^2*b^2*C)*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((2*a^3*A*b^2 + 13*a*A*b^4 + a^4*b*B - 10*a^2*b^3*B - 6*b^5*B + 2*a^5*C - 5*a^3*b^2*C + 18*a*b^4*C)*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 5, (C*x)/b^4 + ((a^3*A*b^4 + 4*a*A*b^6 - 3*a^2*b^5*B - 2*b^7*B - 2*a^7*C + 7*a^5*b^2*C - 8*a^3*b^4*C + 8*a*b^6*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(7/2)*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((3*a^2*A*b^2 + 2*A*b^4 - 5*a*b^3*B - 3*a^4*C + 8*a^2*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((13*a^2*A*b^4 + 2*A*b^6 - 4*a^3*b^3*B - 11*a*b^5*B + 6*a^6*C - 17*a^4*b^2*C + 26*a^2*b^4*C)*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 6, ((b*B - 4*a*C)*x)/b^5 - ((3*a^2*A*b^6 + 2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + 20*a^2*b^6*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(7/2)*d) + ((4*a^2*A*b^4 + 11*A*b^6 - 6*a^5*b*B + 17*a^3*b^3*B - 26*a*b^5*B + 24*a^6*C - 68*a^4*b^2*C + 65*a^2*b^4*C - 6*b^6*C)*Sin[c + d*x])/(6*b^4*(a^2 - b^2)^3*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((2*a^2*A*b^2 + 3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + 9*a^2*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((3*a^2*A*b^4 + 2*A*b^6 - a^5*b*B + 2*a^3*b^3*B - 6*a*b^5*B + 4*a^6*C - 11*a^4*b^2*C + 12*a^2*b^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))}
(* {Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 7, ((2*A*b^2 - 8*a*b*B + 20*a^2*C + b^2*C)*x)/(2*b^6) - (a*(2*a^6*A*b^2 - 7*a^4*A*b^4 + 8*a^2*A*b^6 - 8*A*b^8 - 8*a^7*b*B + 28*a^5*b^3*B - 35*a^3*b^5*B + 20*a*b^7*B + 20*a^8*C - 69*a^6*b^2*C + 84*a^4*b^4*C - 40*a^2*b^6*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*(a^2 - b^2)^(7/2)*d) - ((6*a^5*A*b^2 - 17*a^3*A*b^4 + 26*a*A*b^6 - 24*a^6*b*B + 68*a^4*b^3*B - 65*a^2*b^5*B + 6*b^7*B + 60*a^7*C - 167*a^5*b^2*C + 146*a^3*b^4*C - 24*a*b^6*C)*Sin[c + d*x])/(6*b^5*(a^2 - b^2)^3*d) + ((a^4*A*b^2 - 2*a^2*A*b^4 + 6*A*b^6 - 4*a^5*b*B + 11*a^3*b^3*B - 12*a*b^5*B + 10*a^6*C - 27*a^4*b^2*C + 23*a^2*b^4*C - b^6*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((a^2*A*b^2 + 4*A*b^4 + 2*a^3*b*B - 7*a*b^3*B - 5*a^4*C + 10*a^2*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((2*a^4*A*b^2 + a^2*A*b^4 + 12*A*b^6 - 8*a^5*b*B + 20*a^3*b^3*B - 27*a*b^5*B + 20*a^6*C - 53*a^4*b^2*C + 48*a^2*b^4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} *)


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 8, (2*(35*a*A*b^2 - 14*a^2*b*B + 63*b^3*B + 8*a^3*C + 19*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(35*a^2*A*b^2 - 35*A*b^4 - 14*a^3*b*B + 14*a*b^3*B + 8*a^4*C + 17*a^2*b^2*C - 25*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(35*A*b^2 + 7*a*b*B - 4*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^2*d) + (2*(7*b*B + a*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b*d) + (2*C*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 9, -((2*(42*a^2*A*b^2 - 189*A*b^4 - 24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(42*a^3*A*b^2 - 42*a*A*b^4 - 24*a^4*b*B - 51*a^2*b^3*B + 75*b^5*B + 16*a^5*C + 20*a^3*b^2*C - 36*a*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(21*a*A*b^2 - 12*a^2*b*B + 75*b^3*B + 8*a^3*C + 13*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^3*d) + (2*(63*A*b^2 + 9*a*b*B - 6*a^2*C + 49*b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) + (2*(9*b*B + a*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*b*d) + (2*C*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 10, (2*(264*a^3*A*b^2 + 627*a*A*b^4 - 176*a^4*b*B - 264*a^2*b^3*B + 1617*b^5*B + 128*a^5*C + 156*a^3*b^2*C + 471*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^5*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(264*a^4*A*b^2 + 561*a^2*A*b^4 - 825*A*b^6 - 176*a^5*b*B - 220*a^3*b^3*B + 396*a*b^5*B + 128*a^6*C + 124*a^4*b^2*C + 423*a^2*b^4*C - 675*b^6*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^5*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(132*a^2*A*b^2 - 825*A*b^4 - 88*a^3*b*B - 143*a*b^3*B + 64*a^4*C + 86*a^2*b^2*C - 675*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^4*d) + (2*(99*a*A*b^2 - 66*a^2*b*B + 539*b^3*B + 48*a^3*C + 67*a*b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^3*d) + (2*(99*A*b^2 + 11*a*b*B - 8*a^2*C + 81*b^2*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*b^2*d) + (2*(11*b*B + a*C)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(99*b*d) + (2*C*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 11, -((2*(2288*a^4*A*b^2 + 3432*a^2*A*b^4 - 21021*A*b^6 - 1664*a^5*b*B - 2028*a^3*b^3*B - 6123*a*b^5*B + 1280*a^6*C + 1384*a^4*b^2*C + 2598*a^2*b^4*C - 17787*b^6*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^6*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(2288*a^5*A*b^2 + 2860*a^3*A*b^4 - 5148*a*A*b^6 - 1664*a^6*b*B - 1612*a^4*b^3*B - 5499*a^2*b^5*B + 8775*b^7*B + 1280*a^7*C + 1064*a^5*b^2*C + 2162*a^3*b^4*C - 4506*a*b^6*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^6*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(1144*a^3*A*b^2 + 1859*a*A*b^4 - 832*a^4*b*B - 1118*a^2*b^3*B + 8775*b^5*B + 640*a^5*C + 772*a^3*b^2*C + 1423*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^5*d) - (2*(858*a^2*A*b^2 - 7007*A*b^4 - 624*a^3*b*B - 871*a*b^3*B + 480*a^4*C + 604*a^2*b^2*C - 5929*b^4*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^4*d) + (2*(143*a*A*b^2 - 104*a^2*b*B + 1053*b^3*B + 80*a^3*C + 103*a*b^2*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9009*b^3*d) + (2*(143*A*b^2 + 13*a*b*B - 10*a^2*C + 121*b^2*C)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1287*b^2*d) + (2*(13*b*B + a*C)*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(143*b*d) + (2*C*Cos[c + d*x]^5*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(13*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 9, (2*(63*a^2*A*b^2 + 189*A*b^4 - 18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(63*a^3*A*b^2 - 63*a*A*b^4 - 18*a^4*b*B + 93*a^2*b^3*B - 75*b^5*B + 8*a^5*C + 31*a^3*b^2*C - 39*a*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(126*a*A*b^2 + 9*a^2*b*B + 75*b^3*B - 4*a^3*C + 88*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) + (2*(63*A*b^2 + 72*a*b*B + 3*a^2*C + 49*b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) + (2*(3*b*B + a*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 10, -((2*(198*a^3*A*b^2 - 2706*a*A*b^4 - 88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(198*a^4*A*b^2 - 1023*a^2*A*b^4 + 825*A*b^6 - 88*a^5*b*B - 341*a^3*b^3*B + 429*a*b^5*B + 48*a^6*C + 96*a^4*b^2*C - 819*a^2*b^4*C + 675*b^6*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(99*a^2*A*b^2 + 825*A*b^4 - 44*a^3*b*B + 968*a*b^3*B + 24*a^4*C + 57*a^2*b^2*C + 675*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^3*d) + (2*(792*a*A*b^2 + 33*a^2*b*B + 539*b^3*B - 18*a^3*C + 606*a*b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^2*d) + (2*(99*A*b^2 + 110*a*b*B + 3*a^2*C + 81*b^2*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*b*d) + (2*(11*b*B + 3*a*C)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 11, (2*(1144*a^4*A*b^2 + 4719*a^2*A*b^4 + 21021*A*b^6 - 624*a^5*b*B - 1404*a^3*b^3*B + 27144*a*b^5*B + 384*a^6*C + 644*a^4*b^2*C + 3525*a^2*b^4*C + 17787*b^6*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^5*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(1144*a^5*A*b^2 + 4433*a^3*A*b^4 - 5577*a*A*b^6 - 624*a^6*b*B - 1248*a^4*b^3*B + 10647*a^2*b^5*B - 8775*b^7*B + 384*a^7*C + 548*a^5*b^2*C + 3337*a^3*b^4*C - 4269*a*b^6*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^5*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(572*a^3*A*b^2 - 12584*a*A*b^4 - 312*a^4*b*B - 741*a^2*b^3*B - 8775*b^5*B + 192*a^5*C + 346*a^3*b^2*C - 10198*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^4*d) + (2*(429*a^2*A*b^2 + 7007*A*b^4 - 234*a^3*b*B + 7878*a*b^3*B + 144*a^4*C + 267*a^2*b^2*C + 5929*b^4*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^3*d) + (2*(1430*a*A*b^2 + 39*a^2*b*B + 1053*b^3*B - 24*a^3*C + 1156*a*b^2*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9009*b^2*d) + (2*(143*A*b^2 + 156*a*b*B + 3*a^2*C + 121*b^2*C)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1287*b*d) + (2*(13*b*B + 3*a*C)*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(13*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 12, -((2*(624*a^5*A*b^2 + 1404*a^3*A*b^4 - 27144*a*A*b^6 - 384*a^6*b*B - 644*a^4*b^3*B - 3525*a^2*b^5*B - 17787*b^7*B + 256*a^7*C + 360*a^5*b^2*C + 1050*a^3*b^4*C - 22878*a*b^6*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^6*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (1/(45045*b^6*d*Sqrt[a + b*Cos[c + d*x]]))*(2*(624*a^6*A*b^2 + 1248*a^4*A*b^4 - 10647*a^2*A*b^6 + 8775*A*b^8 - 384*a^7*b*B - 548*a^5*b^3*B - 3337*a^3*b^5*B + 4269*a*b^7*B + 256*a^8*C + 296*a^6*b^2*C + 942*a^4*b^4*C - 9099*a^2*b^6*C + 7605*b^8*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)]) + (2*(312*a^4*A*b^2 + 741*a^2*A*b^4 + 8775*A*b^6 - 192*a^5*b*B - 346*a^3*b^3*B + 10198*a*b^5*B + 128*a^6*C + 196*a^4*b^2*C + 555*a^2*b^4*C + 7605*b^6*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^5*d) - (2*(234*a^3*A*b^2 - 7878*a*A*b^4 - 144*a^4*b*B - 267*a^2*b^3*B - 5929*b^5*B + 96*a^5*C + 152*a^3*b^2*C - 6612*a*b^4*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^4*d) + (2*(195*a^2*A*b^2 + 5265*A*b^4 - 120*a^3*b*B + 5780*a*b^3*B + 80*a^4*C + 129*a^2*b^2*C + 4563*b^4*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^3*d) + (2*(780*a*A*b^2 + 15*a^2*b*B + 605*b^3*B - 10*a^3*C + 654*a*b^2*C)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(6435*b^2*d) + (2*(195*A*b^2 + 210*a*b*B + 3*a^2*C + 169*b^2*C)*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2145*b*d) + (2*(5*b*B + a*C)*Cos[c + d*x]^5*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(65*d) + (2*C*Cos[c + d*x]^5*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 10, (2*(495*a^3*A*b^2 + 4785*a*A*b^4 - 110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(495*a^4*A*b^2 + 330*a^2*A*b^4 - 825*A*b^6 - 110*a^5*b*B + 1364*a^3*b^3*B - 1254*a*b^5*B + 40*a^6*C + 245*a^4*b^2*C + 390*a^2*b^4*C - 675*b^6*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(1485*a^2*A*b^2 + 825*A*b^4 + 55*a^3*b*B + 1793*a*b^3*B - 20*a^4*C + 1025*a^2*b^2*C + 675*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^2*d) + (2*(1485*a*A*b^2 + 825*a^2*b*B + 539*b^3*B + 15*a^3*C + 1145*a*b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b*d) + (2*(33*A*b^2 + 44*a*b*B + 5*a^2*C + 27*b^2*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(11*b*B + 5*a*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 11, -((2*(1430*a^4*A*b^2 - 39897*a^2*A*b^4 - 21021*A*b^6 - 520*a^5*b*B - 3315*a^3*b^3*B - 48165*a*b^5*B + 240*a^6*C + 760*a^4*b^2*C - 30669*a^2*b^4*C - 17787*b^6*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(1430*a^5*A*b^2 - 17732*a^3*A*b^4 + 16302*a*A*b^6 - 520*a^6*b*B - 3185*a^4*b^3*B - 5070*a^2*b^5*B + 8775*b^7*B + 240*a^7*C + 700*a^5*b^2*C - 13984*a^3*b^4*C + 13044*a*b^6*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(715*a^3*A*b^2 + 23309*a*A*b^4 - 260*a^4*b*B + 13325*a^2*b^3*B + 8775*b^5*B + 120*a^5*C + 395*a^3*b^2*C + 18973*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^3*d) + (2*(10725*a^2*A*b^2 + 7007*A*b^4 + 195*a^3*b*B + 14885*a*b^3*B - 90*a^4*C + 8145*a^2*b^2*C + 5929*b^4*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^2*d) + (2*(2717*a*A*b^2 + 1469*a^2*b*B + 1053*b^3*B + 15*a^3*C + 2209*a*b^2*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9009*b*d) + (2*(143*A*b^2 + 182*a*b*B + 15*a^2*C + 121*b^2*C)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1287*d) + (2*(13*b*B + 5*a*C)*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 12, (2*(520*a^5*A*b^2 + 3315*a^3*A*b^4 + 48165*a*A*b^6 - 240*a^6*b*B - 760*a^4*b^3*B + 30669*a^2*b^5*B + 17787*b^7*B + 128*a^7*C + 284*a^5*b^2*C + 2475*a^3*b^4*C + 40665*a*b^6*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^5*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(520*a^6*A*b^2 + 3185*a^4*A*b^4 + 5070*a^2*A*b^6 - 8775*A*b^8 - 240*a^7*b*B - 700*a^5*b^3*B + 13984*a^3*b^5*B - 13044*a*b^7*B + 128*a^8*C + 252*a^6*b^2*C + 2395*a^4*b^4*C + 4830*a^2*b^6*C - 7605*b^8*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^5*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(260*a^4*A*b^2 - 13325*a^2*A*b^4 - 8775*A*b^6 - 120*a^5*b*B - 395*a^3*b^3*B - 18973*a*b^5*B + 64*a^6*C + 150*a^4*b^2*C - 10753*a^2*b^4*C - 7605*b^6*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^4*d) + (2*(195*a^3*A*b^2 + 14885*a*A*b^4 - 90*a^4*b*B + 8145*a^2*b^3*B + 5929*b^5*B + 48*a^5*C + 115*a^3*b^2*C + 12541*a*b^4*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^3*d) + (2*(7345*a^2*A*b^2 + 5265*A*b^4 + 75*a^3*b*B + 11045*a*b^3*B - 40*a^4*C + 5909*a^2*b^2*C + 4563*b^4*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^2*d) + (2*(1495*a*A*b^2 + 795*a^2*b*B + 605*b^3*B + 5*a^3*C + 1259*a*b^2*C)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(6435*b*d) + (2*(195*A*b^2 + 240*a*b*B + 15*a^2*C + 169*b^2*C)*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2145*d) + (2*(3*b*B + a*C)*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(39*d) + (2*C*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(15*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 13, -((2*(4080*a^6*A*b^2 + 12920*a^4*A*b^4 - 521373*a^2*A*b^6 - 302379*A*b^8 - 2176*a^7*b*B - 4828*a^5*b^3*B - 42075*a^3*b^5*B - 691305*a*b^7*B + 1280*a^8*C + 2280*a^6*b^2*C + 9630*a^4*b^4*C - 438705*a^2*b^6*C - 266805*b^8*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(765765*b^6*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(4080*a^7*A*b^2 + 11900*a^5*A*b^4 - 237728*a^3*A*b^6 + 221748*a*A*b^8 - 2176*a^8*b*B - 4284*a^6*b^3*B - 40715*a^4*b^5*B - 82110*a^2*b^7*B + 129285*b^9*B + 1280*a^9*C + 1960*a^7*b^2*C + 8970*a^5*b^4*C - 202020*a^3*b^6*C + 189810*a*b^8*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(765765*b^6*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(2040*a^5*A*b^2 + 6715*a^3*A*b^4 + 322541*a*A*b^6 - 1088*a^6*b*B - 2550*a^4*b^3*B + 182801*a^2*b^5*B + 129285*b^7*B + 640*a^7*C + 1220*a^5*b^2*C + 4995*a^3*b^4*C + 278745*a*b^6*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(765765*b^5*d) - (2*(1530*a^4*A*b^2 - 138465*a^2*A*b^4 - 100793*A*b^6 - 816*a^5*b*B - 1955*a^3*b^3*B - 213197*a*b^5*B + 480*a^6*C + 940*a^4*b^2*C - 115845*a^2*b^4*C - 88935*b^6*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(765765*b^4*d) + (2*(1275*a^3*A*b^2 + 187765*a*A*b^4 - 680*a^4*b*B + 100453*a^2*b^3*B + 77571*b^5*B + 400*a^5*C + 795*a^3*b^2*C + 162165*a*b^4*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(765765*b^3*d) + (2*(13515*a^2*A*b^2 + 10285*A*b^4 + 85*a^3*b*B + 21403*a*b^3*B - 50*a^4*C + 11295*a^2*b^2*C + 9075*b^4*C)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(109395*b^2*d) + (2*(6885*a*A*b^2 + 3621*a^2*b*B + 2873*b^3*B + 15*a^3*C + 5945*a*b^2*C)*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(36465*b*d) + (2*(85*A*b^2 + 102*a*b*B + 5*a^2*C + 75*b^2*C)*Cos[c + d*x]^5*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1105*d) + (2*(17*b*B + 5*a*C)*Cos[c + d*x]^5*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(255*d) + (2*C*Cos[c + d*x]^5*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(17*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(15*a*A*b^2 - 10*a^2*b*B - 5*b^3*B + 8*a^3*C + 7*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B - 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*C*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 8, -((2*(70*a*A*b^2 - 56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(70*a^2*A*b^2 + 35*A*b^4 - 56*a^3*b*B - 49*a*b^3*B + 48*a^4*C + 32*a^2*b^2*C + 25*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(35*A*b^2 - 28*a*b*B + 24*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^3*d) + (2*(7*b*B - 6*a*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b^2*d) + (2*C*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 9, (2*(168*a^2*A*b^2 + 189*A*b^4 - 144*a^3*b*B - 132*a*b^3*B + 128*a^4*C + 108*a^2*b^2*C + 147*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^5*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(168*a^3*A*b^2 + 147*a*A*b^4 - 144*a^4*b*B - 96*a^2*b^3*B - 75*b^5*B + 128*a^5*C + 76*a^3*b^2*C + 111*a*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^5*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(84*a*A*b^2 - 72*a^2*b*B - 75*b^3*B + 64*a^3*C + 62*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^4*d) + (2*(63*A*b^2 - 54*a*b*B + 48*a^2*C + 49*b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^3*d) + (2*(9*b*B - 8*a*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*b^2*d) + (2*C*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*b*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 10, -((2*(1584*a^3*A*b^2 + 1452*a*A*b^4 - 1408*a^4*b*B - 1188*a^2*b^3*B - 1617*b^5*B + 1280*a^5*C + 1032*a^3*b^2*C + 1146*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^6*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(1584*a^4*A*b^2 + 1056*a^2*A*b^4 + 825*A*b^6 - 1408*a^5*b*B - 836*a^3*b^3*B - 1221*a*b^5*B + 1280*a^6*C + 712*a^4*b^2*C + 798*a^2*b^4*C + 675*b^6*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^6*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(792*a^2*A*b^2 + 825*A*b^4 - 704*a^3*b*B - 682*a*b^3*B + 640*a^4*C + 596*a^2*b^2*C + 675*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^5*d) - (2*(594*a*A*b^2 - 528*a^2*b*B - 539*b^3*B + 480*a^3*C + 472*a*b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^4*d) + (2*(99*A*b^2 - 88*a*b*B + 80*a^2*C + 81*b^2*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*b^3*d) + (2*(11*b*B - 10*a*C)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(99*b^2*d) + (2*C*Cos[c + d*x]^4*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(11*b*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 7, -((2*(3*a*A*b^2 - 6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(3*A*b^2 - 6*a*b*B + 8*a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*A*b^2 - 3*a*b*B + 4*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d)}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 8, (2*(30*a^2*A*b^2 - 15*A*b^4 - 40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^4*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(30*a*A*b^2 - 40*a^2*b*B - 5*b^3*B + 48*a^3*C + 12*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(15*a*A*b^2 - 20*a^2*b*B + 5*b^3*B + 24*a^3*C - 9*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)*d) + (2*(5*A*b^2 - 5*a*b*B + 6*a^2*C - b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 9, -((2*(280*a^3*A*b^2 - 175*a*A*b^4 - 336*a^4*b*B + 168*a^2*b^3*B + 63*b^5*B + 384*a^5*C - 172*a^3*b^2*C - 107*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^5*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(280*a^2*A*b^2 + 35*A*b^4 - 336*a^3*b*B - 84*a*b^3*B + 384*a^4*C + 116*a^2*b^2*C + 25*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^5*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(140*a^2*A*b^2 - 35*A*b^4 - 168*a^3*b*B + 63*a*b^3*B + 192*a^4*C - 62*a^2*b^2*C - 25*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^4*(a^2 - b^2)*d) - (2*(35*a*A*b^2 - 42*a^2*b*B + 7*b^3*B + 48*a^3*C - 13*a*b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b^3*(a^2 - b^2)*d) + (2*(7*A*b^2 - 7*a*b*B + 8*a^2*C - b^2*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b^2*(a^2 - b^2)*d)}
{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 10, (2*(1008*a^4*A*b^2 - 504*a^2*A*b^4 - 189*A*b^6 - 1152*a^5*b*B + 516*a^3*b^3*B + 321*a*b^5*B + 1280*a^6*C - 536*a^4*b^2*C - 282*a^2*b^4*C - 147*b^6*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^6*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(1008*a^3*A*b^2 + 252*a*A*b^4 - 1152*a^4*b*B - 348*a^2*b^3*B - 75*b^5*B + 1280*a^5*C + 424*a^3*b^2*C + 186*a*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^6*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(504*a^3*A*b^2 - 189*a*A*b^4 - 576*a^4*b*B + 186*a^2*b^3*B + 75*b^5*B + 640*a^5*C - 188*a^3*b^2*C - 137*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^5*(a^2 - b^2)*d) + (2*(378*a^2*A*b^2 - 63*A*b^4 - 432*a^3*b*B + 117*a*b^3*B + 480*a^4*C - 116*a^2*b^2*C - 49*b^4*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^4*(a^2 - b^2)*d) - (2*(63*a*A*b^2 - 72*a^2*b*B + 9*b^3*B + 80*a^3*C - 17*a*b^2*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*b^3*(a^2 - b^2)*d) + (2*(9*A*b^2 - 9*a*b*B + 10*a^2*C - b^2*C)*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*b^2*(a^2 - b^2)*d)}


{Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(a^2*A*b^2 + 3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 8*a^4*C + 15*a^2*b^2*C - 3*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(a*A*b^2 + 2*a^2*b*B - 3*b^3*B - 8*a^3*C + 9*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(2*a^2*A*b^2 + 2*A*b^4 + a^3*b*B - 5*a*b^3*B - 4*a^4*C + 8*a^2*b^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])}
{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 8, -((2*(2*a^3*A*b^2 - 6*a*A*b^4 - 8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(2*a^2*A*b^2 - 3*A*b^4 - 8*a^3*b*B + 9*a*b^3*B + 16*a^4*C - 16*a^2*b^2*C - b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(4*A*b^4 + 3*a^3*b*B - 7*a*b^3*B - 6*a^4*C + 10*a^2*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2*A*b^2 - 5*A*b^4 - 4*a^3*b*B + 8*a*b^3*B + 8*a^4*C - 13*a^2*b^2*C + b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(40*a^4*A*b^2 - 75*a^2*A*b^4 + 15*A*b^6 - 80*a^5*b*B + 140*a^3*b^3*B - 40*a*b^5*B + 128*a^6*C - 212*a^4*b^2*C + 55*a^2*b^4*C + 9*b^6*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(40*a^3*A*b^2 - 45*a*A*b^4 - 80*a^4*b*B + 80*a^2*b^3*B + 5*b^5*B + 128*a^5*C - 116*a^3*b^2*C - 17*a*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(2*a^2*A*b^2 - 6*A*b^4 - 5*a^3*b*B + 9*a*b^3*B + 8*a^4*C - 12*a^2*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(20*a^3*A*b^2 - 40*a*A*b^4 - 40*a^4*b*B + 65*a^2*b^3*B - 5*b^5*B + 64*a^5*C - 98*a^3*b^2*C + 14*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^4*(a^2 - b^2)^2*d) + (2*(15*a^2*A*b^2 - 35*A*b^4 - 30*a^3*b*B + 50*a*b^3*B + 48*a^4*C - 71*a^2*b^2*C + 3*b^4*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)^2*d)}
(* {Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 10, -((2*(560*a^5*A*b^2 - 980*a^3*A*b^4 + 280*a*A*b^6 - 896*a^6*b*B + 1484*a^4*b^3*B - 385*a^2*b^5*B - 63*b^7*B + 1280*a^7*C - 2040*a^5*b^2*C + 450*a^3*b^4*C + 170*a*b^6*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^6*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(560*a^4*A*b^2 - 560*a^2*A*b^4 - 35*A*b^6 - 896*a^5*b*B + 812*a^3*b^3*B + 119*a*b^5*B + 1280*a^6*C - 1080*a^4*b^2*C - 210*a^2*b^4*C - 25*b^6*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^6*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(4*a^2*A*b^2 - 8*A*b^4 - 7*a^3*b*B + 11*a*b^3*B + 10*a^4*C - 14*a^2*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(280*a^4*A*b^2 - 455*a^2*A*b^4 + 35*A*b^6 - 448*a^5*b*B + 686*a^3*b^3*B - 98*a*b^5*B + 640*a^6*C - 940*a^4*b^2*C + 135*a^2*b^4*C + 25*b^6*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^5*(a^2 - b^2)^2*d) - (2*(210*a^3*A*b^2 - 350*a*A*b^4 - 336*a^4*b*B + 497*a^2*b^3*B - 21*b^5*B + 480*a^5*C - 680*a^3*b^2*C + 60*a*b^4*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^4*(a^2 - b^2)^2*d) + (2*(35*a^2*A*b^2 - 63*A*b^4 - 56*a^3*b*B + 84*a*b^3*B + 80*a^4*C - 111*a^2*b^2*C + 3*b^4*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*b^3*(a^2 - b^2)^2*d)} *)


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 8, (2*(9*a*A + 7*b*B + 7*a*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*(11*A*b + 11*a*B + 9*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (10*(11*A*b + 11*a*B + 9*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(9*a*A + 7*b*B + 7*a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(11*A*b + 11*a*B + 9*b*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*(11*b*B + 2*a*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 7, (2*(9*A*b + 9*a*B + 7*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(7*a*A + 5*b*B + 5*a*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*a*A + 5*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*A*b + 9*a*B + 7*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(9*b*B + 2*a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*C*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 6, (2*(5*a*A + 3*b*B + 3*a*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(7*A*b + 7*a*B + 5*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*A*b + 7*a*B + 5*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(7*b*B + 2*a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 5, (2*(5*A*b + 5*a*B + 3*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(3*a*A + b*B + a*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*(5*b*B + 2*a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 5, -((2*(a*A - b*B - a*C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(3*A*b + 3*a*B + b*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*b*(3*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 5, -((2*(A*b + a*B - b*C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(a*A + 3*b*B + 3*a*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*(2*A*b + 3*a*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 6, -((2*(3*a*A + 5*b*B + 5*a*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(A*b + a*B + 3*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*(2*A*b + 5*a*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*(3*a*A + 5*b*B + 5*a*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 7, -((2*(3*A*b + 3*a*B + 5*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(5*a*A + 7*b*B + 7*a*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(2*A*b + 7*a*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(5*a*A + 7*b*B + 7*a*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(3*A*b + 3*a*B + 5*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}


{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 8, (2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(77*a^2*A + 55*A*b^2 + 110*a*b*B + 55*a^2*C + 45*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (2*(77*a^2*A + 55*A*b^2 + 110*a*b*B + 55*a^2*C + 45*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(99*A*b^2 + 121*a*b*B + 8*a^2*C + 81*b^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*(11*b*B + 4*a*C)*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 7, (2*(15*a^2*A + 9*A*b^2 + 18*a*b*B + 9*a^2*C + 7*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(63*A*b^2 + 81*a*b*B + 8*a^2*C + 49*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(9*b*B + 4*a*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(63*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 6, (2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(21*a^2*A + 7*A*b^2 + 14*a*b*B + 7*a^2*C + 5*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(35*A*b^2 + 49*a*b*B + 8*a^2*C + 25*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*b*B + 4*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(35*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 6, -((2*(5*a^2*A - 5*A*b^2 - 10*a*b*B - 5*a^2*C - 3*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(6*a*A*b + 3*a^2*B + b^2*B + 2*a*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*b*(15*a*A - 5*b*B - 7*a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b*(5*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 6, -((2*(2*a*A*b + a^2*B - b^2*B - 2*a*b*C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(a^2*A + 3*A*b^2 + 6*a*b*B + 3*a^2*C + b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*b*(5*A*b + 3*a*B - b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*(4*A*b + 3*a*B)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 6, -((2*(3*a^2*A + 5*A*b^2 + 10*a*b*B + 5*a^2*C - 5*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(2*a*A*b + a^2*B + 3*b^2*B + 6*a*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*(9*a^2*A + 8*A*b^2 + 25*a*b*B + 15*a^2*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(4*A*b + 5*a*B)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 7, -((2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(5*a^2*A + 7*A*b^2 + 14*a*b*B + 7*a^2*C + 21*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(25*a^2*A + 8*A*b^2 + 49*a*b*B + 35*a^2*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*(4*A*b + 7*a*B)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}


{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 8, (2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B + 9*a^3*C + 21*a*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(231*a^2*A*b + 55*A*b^3 + 77*a^3*B + 165*a*b^2*B + 165*a^2*b*C + 45*b^3*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (2*(231*a^2*A*b + 55*A*b^3 + 77*a^3*B + 165*a*b^2*B + 165*a^2*b*C + 45*b^3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(1584*a*A*b^2 + 979*a^2*b*B + 539*b^3*B + 48*a^3*C + 1212*a*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d) + (2*(99*A*b^2 + 143*a*b*B + 24*a^2*C + 81*b^2*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(693*d) + (2*(11*b*B + 6*a*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 7, (2*(45*a^2*A*b + 9*A*b^3 + 15*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 7*b^3*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(21*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B + 7*a^3*C + 15*a*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(252*a*A*b^2 + 171*a^2*b*B + 75*b^3*B + 16*a^3*C + 176*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*(63*A*b^2 + 99*a*b*B + 24*a^2*C + 49*b^2*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(315*d) + (2*(3*b*B + 2*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 7, -((1/(5*d))*(2*(5*a^3*A - 15*a*A*b^2 - 15*a^2*b*B - 3*b^3*B - 5*a^3*C - 9*a*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])) + (1/(21*d))*(2*(63*a^2*A*b + 7*A*b^3 + 21*a^3*B + 21*a*b^2*B + 21*a^2*b*C + 5*b^3*C)*EllipticF[(1/2)*(c + d*x), 2]) - (2*b*(105*a^2*A - 35*A*b^2 - 84*a*b*B - 57*a^2*C - 25*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) - (2*b*(35*a*A - 7*b*B - 11*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(35*d) - (2*b*(7*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 7, -((1/(5*d))*(2*(15*a^2*A*b - 5*A*b^3 + 5*a^3*B - 15*a*b^2*B - 15*a^2*b*C - 3*b^3*C)*EllipticE[(1/2)*(c + d*x), 2])) + (2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B + 3*a^3*C + 3*a*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*b*(40*a*A*b + 15*a^2*B - 5*b^2*B - 12*a*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b*(35*A*b + 15*a*B - 3*b*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(15*d) + (2*(2*A*b + a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 7, -((2*(3*a^3*A + 15*a*A*b^2 + 15*a^2*b*B - 5*b^3*B + 5*a^3*C - 15*a*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(3*a^2*A*b + 3*A*b^3 + a^3*B + 9*a*b^2*B + 9*a^2*b*C + b^3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*b*(9*a^2*A + 33*A*b^2 + 40*a*b*B + 15*a^2*C - 5*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(9*a^2*A + 24*A*b^2 + 35*a*b*B + 15*a^2*C)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(6*A*b + 5*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 7, -((2*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B + 15*a^2*b*C - 5*b^3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(5*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 21*b^3*B + 7*a^3*C + 63*a*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(164*a^2*A*b + 48*A*b^3 + 63*a^3*B + 231*a*b^2*B + 280*a^2*b*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*(25*a^2*A + 24*A*b^2 + 63*a*b*B + 35*a^2*C)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(6*A*b + 7*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}
{Cos[c + d*x]^(-11/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 8, -((2*(7*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 15*b^3*B + 9*a^3*C + 45*a*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (2*(15*a^2*A*b + 7*A*b^3 + 5*a^3*B + 21*a*b^2*B + 21*a^2*b*C + 21*b^3*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(176*a^2*A*b + 16*A*b^3 + 75*a^3*B + 171*a*b^2*B + 252*a^2*b*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)) + (2*(7*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 15*b^3*B + 9*a^3*C + 45*a*b^2*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(49*a^2*A + 24*A*b^2 + 99*a*b*B + 63*a^2*C)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(2*A*b + 3*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))}


{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 9, (2*(195*a^4*A + 702*a^2*A*b^2 + 91*A*b^4 + 468*a^3*b*B + 364*a*b^3*B + 117*a^4*C + 546*a^2*b^2*C + 77*b^4*C)*EllipticE[(1/2)*(c + d*x), 2])/(195*d) + (2*(308*a^3*A*b + 220*a*A*b^3 + 77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 220*a^3*b*C + 180*a*b^3*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (2*(308*a^3*A*b + 220*a*A*b^3 + 77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 220*a^3*b*C + 180*a*b^3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(33319*a^2*A*b^2 + 7007*A*b^4 + 13351*a^3*b*B + 22763*a*b^3*B + 384*a^4*C + 25217*a^2*b^2*C + 5929*b^4*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45045*d) + (2*(3146*a*A*b^2 + 2171*a^2*b*B + 1053*b^3*B + 192*a^3*C + 2518*a*b^2*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(9009*d) + (2*(143*A*b^2 + 221*a*b*B + 48*a^2*C + 121*b^2*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(1287*d) + (2*(13*b*B + 8*a*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(13*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 8, (2*(60*a^3*A*b + 36*a*A*b^3 + 15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 36*a^3*b*C + 28*a*b^3*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(231*a^4*A + 462*a^2*A*b^2 + 55*A*b^4 + 308*a^3*b*B + 220*a*b^3*B + 77*a^4*C + 330*a^2*b^2*C + 45*b^4*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (2*(4653*a^2*A*b^2 + 825*A*b^4 + 2057*a^3*b*B + 2761*a*b^3*B + 128*a^4*C + 3199*a^2*b^2*C + 675*b^4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*(1782*a*A*b^2 + 1353*a^2*b*B + 539*b^3*B + 192*a^3*C + 1346*a*b^2*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3465*d) + (2*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d) + (2*(11*b*B + 8*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 8, -((2*(15*a^4*A - 90*a^2*A*b^2 - 9*A*b^4 - 60*a^3*b*B - 36*a*b^3*B - 15*a^4*C - 54*a^2*b^2*C - 7*b^4*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (2*(84*a^3*A*b + 28*a*A*b^3 + 21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*C + 20*a*b^3*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) - (2*b*(315*a^3*A - 357*a*A*b^2 - 423*a^2*b*B - 75*b^3*B - 187*a^3*C - 251*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d) - (2*b*(315*a^2*A - 63*A*b^2 - 162*a*b*B - 123*a^2*C - 49*b^2*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(315*d) - (2*b*(21*a*A - 3*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) - (2*b*(9*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 8, -((2*(20*a^3*A*b - 20*a*A*b^3 + 5*a^4*B - 30*a^2*b^2*B - 3*b^4*B - 20*a^3*b*C - 12*a*b^3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(7*a^4*A + 126*a^2*A*b^2 + 7*A*b^4 + 84*a^3*b*B + 28*a*b^3*B + 21*a^4*C + 42*a^2*b^2*C + 5*b^4*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) - (2*b*(385*a^2*A*b - 35*A*b^3 + 105*a^3*B - 119*a*b^2*B - 141*a^2*b*C - 25*b^3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) - (2*b*(350*a*A*b + 105*a^2*B - 21*b^2*B - 54*a*b*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(105*d) - (2*b*(21*A*b + 7*a*B - b*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(8*A*b + 3*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 8, -((2*(3*a^4*A + 30*a^2*A*b^2 - 5*A*b^4 + 20*a^3*b*B - 20*a*b^3*B + 5*a^4*C - 30*a^2*b^2*C - 3*b^4*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(4*a^3*A*b + 12*a*A*b^3 + a^4*B + 18*a^2*b^2*B + b^4*B + 12*a^3*b*C + 4*a*b^3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*b*(9*a^3*A + 73*a*A*b^2 + 55*a^2*b*B - 5*b^3*B + 15*a^3*C - 17*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b*(9*a^2*A + 59*A*b^2 + 50*a*b*B + 15*a^2*C - 3*b^2*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(15*d) + (2*(3*a^2*A + 16*A*b^2 + 15*a*b*B + 5*a^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*(8*A*b + 5*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 8, -((2*(12*a^3*A*b + 20*a*A*b^3 + 3*a^4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a^3*b*C - 20*a*b^3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(5*a^4*A + 42*a^2*A*b^2 + 21*A*b^4 + 28*a^3*b*B + 84*a*b^3*B + 7*a^4*C + 126*a^2*b^2*C + 7*b^4*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) - (2*b*(227*a^2*A*b + 279*A*b^3 + 63*a^3*B + 511*a*b^2*B + 385*a^2*b*C - 35*b^3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(202*a^2*A*b + 192*A*b^3 + 63*a^3*B + 413*a*b^2*B + 350*a^2*b*C)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*(25*a^2*A + 48*A*b^2 + 77*a*b*B + 35*a^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(8*A*b + 7*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))}
{Cos[c + d*x]^(-11/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 8, -((2*(7*a^4*A + 54*a^2*A*b^2 + 15*A*b^4 + 36*a^3*b*B + 60*a*b^3*B + 9*a^4*C + 90*a^2*b^2*C - 15*b^4*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (2*(20*a^3*A*b + 28*a*A*b^3 + 5*a^4*B + 42*a^2*b^2*B + 21*b^4*B + 28*a^3*b*C + 84*a*b^3*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(147*a^4*A + 883*a^2*A*b^2 + 128*A*b^4 + 681*a^3*b*B + 837*a*b^3*B + 189*a^4*C + 1533*a^2*b^2*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]) + (2*(202*a^2*A*b + 64*A*b^3 + 75*a^3*B + 261*a*b^2*B + 294*a^2*b*C)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)) + (2*(49*a^2*A + 48*A*b^2 + 117*a*b*B + 63*a^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(8*A*b + 9*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))}
{Cos[c + d*x]^(-13/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 9, -((2*(28*a^3*A*b + 36*a*A*b^3 + 7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 36*a^3*b*C + 60*a*b^3*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (2*(45*a^4*A + 330*a^2*A*b^2 + 77*A*b^4 + 220*a^3*b*B + 308*a*b^3*B + 55*a^4*C + 462*a^2*b^2*C + 231*b^4*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (2*(675*a^4*A + 3199*a^2*A*b^2 + 128*A*b^4 + 2761*a^3*b*B + 2057*a*b^3*B + 825*a^4*C + 4653*a^2*b^2*C)*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(3/2)) + (2*(28*a^3*A*b + 36*a*A*b^3 + 7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 36*a^3*b*C + 60*a*b^3*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(1346*a^2*A*b + 192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 1782*a^2*b*C)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(5/2)) + (2*(27*a^2*A + 16*A*b^2 + 55*a*b*B + 33*a^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (2*(8*A*b + 11*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 9, (2*(15*a^2*A*b^2 + 9*A*b^4 - 15*a^3*b*B - 9*a*b^3*B + 15*a^4*C + 9*a^2*b^2*C + 7*b^4*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*b^5*d) - (2*(21*a^3*A*b^2 + 7*a*A*b^4 - 21*a^4*b*B - 7*a^2*b^3*B - 5*b^5*B + 21*a^5*C + 7*a^3*b^2*C + 5*a*b^4*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*b^6*d) + (2*a^4*(A*b^2 - a*b*B + a^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^6*(a + b)*d) - (2*(7*a*A*b^2 - 7*a^2*b*B - 5*b^3*B + 7*a^3*C + 5*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*b^4*d) + (2*(9*A*b^2 - 9*a*b*B + 9*a^2*C + 7*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*b^3*d) + (2*(b*B - a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*b^2*d) + (2*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*b*d)}
{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 8, -((2*(5*a*A*b^2 - 5*a^2*b*B - 3*b^3*B + 5*a^3*C + 3*a*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*b^4*d)) + (2*(21*a^2*A*b^2 + 7*A*b^4 - 21*a^3*b*B - 7*a*b^3*B + 21*a^4*C + 7*a^2*b^2*C + 5*b^4*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*b^5*d) - (2*a^3*(A*b^2 - a*b*B + a^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^5*(a + b)*d) + (2*(7*A*b^2 - 7*a*b*B + 7*a^2*C + 5*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*b^3*d) + (2*(b*B - a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b^2*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*b*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, (2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d) - (2*(3*a*A*b^2 - 3*a^2*b*B - b^3*B + 3*a^3*C + a*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^4*d) + (2*a^2*(A*b^2 - a*b*B + a^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^4*(a + b)*d) + (2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, (2*(b*B - a*C)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d) + (2*(3*A*b^2 - 3*a*b*B + 3*a^2*C + b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^3*d) - (2*a*(A*b^2 - a*b*B + a^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^3*(a + b)*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 5, (2*C*EllipticE[(1/2)*(c + d*x), 2])/(b*d) + (2*(b*B - a*C)*EllipticF[(1/2)*(c + d*x), 2])/(b^2*d) + (2*(A*b^2 - a*b*B + a^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^2*(a + b)*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, -((2*A*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) + (2*C*EllipticF[(1/2)*(c + d*x), 2])/(b*d) - (2*(A*b^2 - a*b*B + a^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*b*(a + b)*d) + (2*A*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, (2*(A*b - a*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + (2*A*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (2*(A*b^2 - a*b*B + a^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a + b)*d) + (2*A*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 8, -((2*(3*a^2*A + 5*A*b^2 - 5*a*b*B + 5*a^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*a^3*d)) - (2*(A*b - a*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (2*b*(A*b^2 - a*b*B + a^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^3*(a + b)*d) + (2*A*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) + (2*(3*a^2*A + 5*A*b^2 - 5*a*b*B + 5*a^2*C)*Sin[c + d*x])/(5*a^3*d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 9, (2*(3*a^2*A*b + 5*A*b^3 - 3*a^3*B - 5*a*b^2*B + 5*a^2*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*a^4*d) + (2*(5*a^2*A + 7*A*b^2 - 7*a*b*B + 7*a^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*a^3*d) + (2*b^2*(A*b^2 - a*b*B + a^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^4*(a + b)*d) + (2*A*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(5*a^2*d*Cos[c + d*x]^(5/2)) + (2*(5*a^2*A + 7*A*b^2 - 7*a*b*B + 7*a^2*C)*Sin[c + d*x])/(21*a^3*d*Cos[c + d*x]^(3/2)) - (2*(3*a^2*A*b + 5*A*b^3 - 3*a^3*B - 5*a*b^2*B + 5*a^2*b*C)*Sin[c + d*x])/(5*a^4*d*Sqrt[Cos[c + d*x]])}
{Cos[c + d*x]^(-11/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 10, -((2*(7*a^4*A + 9*a^2*A*b^2 + 15*A*b^4 - 9*a^3*b*B - 15*a*b^3*B + 9*a^4*C + 15*a^2*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*a^5*d)) - (2*(5*a^2*A*b + 7*A*b^3 - 5*a^3*B - 7*a*b^2*B + 7*a^2*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*a^4*d) - (2*b^3*(A*b^2 - a*b*B + a^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^5*(a + b)*d) + (2*A*Sin[c + d*x])/(9*a*d*Cos[c + d*x]^(9/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(7*a^2*d*Cos[c + d*x]^(7/2)) + (2*(7*a^2*A + 9*A*b^2 - 9*a*b*B + 9*a^2*C)*Sin[c + d*x])/(45*a^3*d*Cos[c + d*x]^(5/2)) - (2*(5*a^2*A*b + 7*A*b^3 - 5*a^3*B - 7*a*b^2*B + 7*a^2*b*C)*Sin[c + d*x])/(21*a^4*d*Cos[c + d*x]^(3/2)) + (2*(7*a^4*A + 9*a^2*A*b^2 + 15*A*b^4 - 9*a^3*b*B - 15*a*b^3*B + 9*a^4*C + 15*a^2*b^2*C)*Sin[c + d*x])/(15*a^5*d*Sqrt[Cos[c + d*x]])}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 8, ((15*a^2*A*b^2 - 10*A*b^4 - 25*a^3*b*B + 20*a*b^3*B + 35*a^4*C - 24*a^2*b^2*C - 6*b^4*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*b^4*(a^2 - b^2)*d) - ((9*a^3*A*b^2 - 12*a*A*b^4 - 15*a^4*b*B + 16*a^2*b^3*B + 2*b^5*B + 21*a^5*C - 20*a^3*b^2*C - 4*a*b^4*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^5*(a^2 - b^2)*d) + (a^2*(3*a^2*A*b^2 - 5*A*b^4 - 5*a^3*b*B + 7*a*b^3*B + 7*a^4*C - 9*a^2*b^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^5*(a + b)^2*d) - ((3*a*A*b^2 - 5*a^2*b*B + 2*b^3*B + 7*a^3*C - 4*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d) + ((5*A*b^2 - 5*a*b*B + 7*a^2*C - 2*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 7, -(((a*A*b^2 - 3*a^2*b*B + 2*b^3*B + 5*a^3*C - 4*a*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d)) + ((3*a^2*A*b^2 - 6*A*b^4 - 9*a^3*b*B + 12*a*b^3*B + 15*a^4*C - 16*a^2*b^2*C - 2*b^4*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^4*(a^2 - b^2)*d) - (a*(a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 5*a*b^3*B + 5*a^4*C - 7*a^2*b^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^4*(a + b)^2*d) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, ((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d) + ((a*A*b^2 + a^2*b*B - 2*b^3*B - 3*a^3*C + 4*a*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d) - ((a^2*A*b^2 + A*b^4 + a^3*b*B - 3*a*b^3*B - 3*a^4*C + 5*a^2*b^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^3*(a + b)^2*d) - ((A*b^2 - a*b*B + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, -(((A*b^2 - a*b*B + a^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(a*b*(a^2 - b^2)*d)) - ((A*b^2 - a*b*B - a^2*C + 2*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d) + ((3*a^2*A*b^2 - A*b^4 - a^3*b*B - a*b^3*B - a^4*C + 3*a^2*b^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 - a*b*B + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 7, -(((2*a^2*A - 3*A*b^2 + a*b*B - a^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*(a^2 - b^2)*d)) + ((A*b^2 - a*b*B + a^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(a*b*(a^2 - b^2)*d) - ((5*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + a*b^3*B + a^4*C + a^2*b^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a - b)*b*(a + b)^2*d) + ((2*a^2*A - 3*A*b^2 + a*b*B - a^2*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 8, ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B - a^2*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B - 3*a^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*(a^2 - b^2)*d) + ((7*a^2*A*b^2 - 5*A*b^4 - 5*a^3*b*B + 3*a*b^3*B + 3*a^4*C - a^2*b^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^3*(a - b)*(a + b)^2*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B - 3*a^2*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) - ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B - a^2*b*C)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 9, -(((6*a^4*A + 24*a^2*A*b^2 - 35*A*b^4 - 20*a^3*b*B + 25*a*b^3*B + 10*a^4*C - 15*a^2*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*a^4*(a^2 - b^2)*d)) - ((4*a^2*A*b - 7*A*b^3 - 2*a^3*B + 5*a*b^2*B - 3*a^2*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^3*(a^2 - b^2)*d) - (b*(9*a^2*A*b^2 - 7*A*b^4 - 7*a^3*b*B + 5*a*b^3*B + 5*a^4*C - 3*a^2*b^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^4*(a - b)*(a + b)^2*d) + ((2*a^2*A - 7*A*b^2 + 5*a*b*B - 5*a^2*C)*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)) - ((4*a^2*A*b - 7*A*b^3 - 2*a^3*B + 5*a*b^2*B - 3*a^2*b*C)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) + ((6*a^4*A + 24*a^2*A*b^2 - 35*A*b^4 - 20*a^3*b*B + 25*a*b^3*B + 10*a^4*C - 15*a^2*b^2*C)*Sin[c + d*x])/(5*a^4*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x]))}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 8, -(((3*a^3*A*b^2 - 9*a*A*b^4 - 15*a^4*b*B + 29*a^2*b^3*B - 8*b^5*B + 35*a^5*C - 65*a^3*b^2*C + 24*a*b^4*C)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^4*(a^2 - b^2)^2*d)) + ((9*a^4*A*b^2 - 15*a^2*A*b^4 + 24*A*b^6 - 45*a^5*b*B + 99*a^3*b^3*B - 72*a*b^5*B + 105*a^6*C - 223*a^4*b^2*C + 128*a^2*b^4*C + 8*b^6*C)*EllipticF[(1/2)*(c + d*x), 2])/(12*b^5*(a^2 - b^2)^2*d) - (a*(3*a^4*A*b^2 - 6*a^2*A*b^4 + 15*A*b^6 - 15*a^5*b*B + 38*a^3*b^3*B - 35*a*b^5*B + 35*a^6*C - 86*a^4*b^2*C + 63*a^2*b^4*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^5*(a + b)^3*d) + ((3*a^2*A*b^2 - 21*A*b^4 - 15*a^3*b*B + 33*a*b^3*B + 35*a^4*C - 61*a^2*b^2*C + 8*b^4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*A*b^2 + 5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 7*a^4*C + 13*a^2*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, -(((a^2*A*b^2 + 5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 15*a^4*C + 29*a^2*b^2*C - 8*b^4*C)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d)) + ((a^3*A*b^2 - 7*a*A*b^4 + 3*a^4*b*B - 5*a^2*b^3*B + 8*b^5*B - 15*a^5*C + 33*a^3*b^2*C - 24*a*b^4*C)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^4*(a^2 - b^2)^2*d) - ((a^4*A*b^2 - 10*a^2*A*b^4 - 3*A*b^6 + 3*a^5*b*B - 6*a^3*b^3*B + 15*a*b^5*B - 15*a^6*C + 38*a^4*b^2*C - 35*a^2*b^4*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^4*(a + b)^3*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((3*a^2*A*b^2 + 3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4*C + 11*a^2*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, ((5*a^2*A*b^2 + A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + 9*a^2*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(4*a*b^2*(a^2 - b^2)^2*d) + ((3*a^2*A*b^2 + 3*A*b^4 + a^3*b*B - 7*a*b^3*B + 3*a^4*C - 5*a^2*b^2*C + 8*b^4*C)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d) - ((3*a^4*A*b^2 + 10*a^2*A*b^4 - A*b^6 + a^5*b*B - 10*a^3*b^3*B - 3*a*b^5*B + 3*a^6*C - 6*a^4*b^2*C + 15*a^2*b^4*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((A*b^2 - a*b*B + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((5*a^2*A*b^2 + A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + 9*a^2*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, -(((9*a^2*A*b^2 - 3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + 5*a^2*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^2*b*(a^2 - b^2)^2*d)) - ((7*a^2*A*b^2 - A*b^4 - 3*a^3*b*B - 3*a*b^3*B - a^4*C + 7*a^2*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(4*a*b^2*(a^2 - b^2)^2*d) + ((15*a^4*A*b^2 - 6*a^2*A*b^4 + 3*A*b^6 - 3*a^5*b*B - 10*a^3*b^3*B + a*b^5*B - a^6*C + 10*a^4*b^2*C + 3*a^2*b^4*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) + ((A*b^2 - a*b*B + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((9*a^2*A*b^2 - 3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + 5*a^2*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 8, -(((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^3*(a^2 - b^2)^2*d)) + ((11*a^2*A*b^2 - 5*A*b^4 - 7*a^3*b*B + a*b^3*B + 3*a^4*C + 3*a^2*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(4*a^2*b*(a^2 - b^2)^2*d) - ((35*a^4*A*b^2 - 38*a^2*A*b^4 + 15*A*b^6 - 15*a^5*b*B + 6*a^3*b^3*B - 3*a*b^5*B + 3*a^6*C + 10*a^4*b^2*C - a^2*b^4*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*C)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2) + ((11*a^2*A*b^2 - 5*A*b^4 - 7*a^3*b*B + a*b^3*B + 3*a^4*C + 3*a^2*b^2*C)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 9, ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B - 9*a^4*b*C + 3*a^2*b^3*C)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B - 21*a^4*C + 3*a^2*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(12*a^3*(a^2 - b^2)^2*d) + ((63*a^4*A*b^2 - 86*a^2*A*b^4 + 35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 15*a*b^5*B + 15*a^6*C - 6*a^4*b^2*C + 3*a^2*b^4*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B - 21*a^4*C + 3*a^2*b^2*C)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B - 9*a^4*b*C + 3*a^2*b^3*C)*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2) + ((13*a^2*A*b^2 - 7*A*b^4 - 9*a^3*b*B + 3*a*b^3*B + 5*a^4*C + a^2*b^2*C)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))}


(* ::Subsection::Closed:: *)
(*Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/2)*)


(* ::Subsubsection::Closed:: *)
(*n>0*)


{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 8, ((48*a*A*b^2 - 24*a^2*b*B + 128*b^3*B + 15*a^3*C + 28*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(192*b^3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((48*a*A*b^2 - 96*A*b^3 + 8*a^2*b*B - 112*a*b^2*B - 5*a^3*C - 2*a^2*b*C + 36*a*b^2*C - 72*b^3*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(96*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((16*a^2*A*b^2 - 64*A*b^4 - 8*a^3*b*B - 32*a*b^3*B + 5*a^4*C + 8*a^2*b^2*C - 48*b^4*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(32*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + ((48*A*b^2 + 8*a*b*B - 5*a^2*C + 36*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(96*b^2*d) + ((8*b*B + a*C)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b*d) + (C*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + ((48*a*A*b^2 - 24*a^2*b*B + 128*b^3*B + 15*a^3*C + 28*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(192*b^3*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 7, ((24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(24*b^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - ((24*a*A*b - 6*a*b*B + 12*b^2*B - a^2*C + 14*a*b*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(12*b*d*Sqrt[a + b*Cos[c + d*x]]) + ((8*a*A*b^2 - 2*a^2*b*B + 8*b^3*B + a^3*C + 4*a*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((6*b*B + a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*b*d) + (C*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + ((24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(24*b^2*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 6, ((4*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((4*a*A - 4*A*b - 4*a*B + a*C - 2*b*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) + ((8*A*b^2 + 4*a*b*B - a^2*C + 4*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*b*d*Sqrt[a + b*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + ((4*b*B + a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(4*b*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 6, -(((2*A - C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a*A + A*b + a*B - b*B - a*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*(2*b*B + a*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((2*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/d}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 7, -((2*(A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a*A + A*b + 3*a*B + 3*b*B + 3*a*C - 3*b*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (4*b*C*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Sqrt[Cos[c + d*x]]) - (2*(A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a*d)}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 8, -((2*(9*a^2*A - 2*A*b^2 + 5*a*b*B + 15*a^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*a^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(9*a^2*A + 7*a*A*b - 2*A*b^2 + 5*a^2*B + 5*a*b*B + 15*a^2*C + 15*a*b*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2)) + (2*(9*a^2*A - 2*A*b^2 + 5*a*b*B + 15*a^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Sqrt[Cos[c + d*x]]) - (2*(9*a^2*A - 2*A*b^2 + 5*a*b*B + 15*a^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(15*a^2*d)}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 8, -((2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B + 35*a^2*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*a^3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(25*a^3*A + 19*a^2*A*b + 2*a*A*b^2 + 8*A*b^3 + 63*a^3*B + 49*a^2*b*B - 14*a*b^2*B + 35*a^3*C + 35*a^2*b*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2*A - 4*A*b^2 + 7*a*b*B + 35*a^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Cos[c + d*x]^(3/2)) + (2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B + 35*a^2*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^3*d*Sqrt[Cos[c + d*x]]) - (2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B + 35*a^2*b*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(105*a^3*d)}


{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 9, ((240*a^2*A*b^2 + 1280*A*b^4 - 90*a^3*b*B + 1560*a*b^3*B + 45*a^4*C + 132*a^2*b^2*C + 1024*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(1920*b^3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((560*a^2*A*b^2 - 2080*a*A*b^3 + 30*a^3*b*B - 1140*a^2*b^2*B + 360*a*b^3*B - 720*b^4*B - 15*a^4*C - 6*a^3*b*C + 412*a^2*b^2*C - 1592*a*b^3*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(960*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((16*a^3*A*b^2 - 192*a*A*b^4 - 6*a^4*b*B - 48*a^2*b^3*B - 96*b^5*B + 3*a^5*C + 8*a^3*b^2*C - 144*a*b^4*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(64*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + ((560*a*A*b^2 + 30*a^2*b*B + 360*b^3*B - 15*a^3*C + 412*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(960*b^2*d) + ((80*A*b^2 + 90*a*b*B + 3*a^2*C + 64*b^2*C)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(240*b*d) + ((10*b*B + 3*a*C)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(40*d) + (C*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + ((240*a^2*A*b^2 + 1280*A*b^4 - 90*a^3*b*B + 1560*a*b^3*B + 45*a^4*C + 132*a^2*b^2*C + 1024*b^4*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(1920*b^3*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 8, ((240*a*A*b^2 + 24*a^2*b*B + 128*b^3*B - 9*a^3*C + 156*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(192*b^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - ((192*a^2*A*b - 48*a*A*b^2 + 96*A*b^3 - 56*a^2*b*B + 208*a*b^2*B - 3*a^3*C + 114*a^2*b*C - 36*a*b^2*C + 72*b^3*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(96*b*d*Sqrt[a + b*Cos[c + d*x]]) + ((48*a^2*A*b^2 + 64*A*b^4 - 8*a^3*b*B + 96*a*b^3*B + 3*a^4*C + 24*a^2*b^2*C + 48*b^4*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(32*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((48*A*b^2 + 56*a*b*B + 3*a^2*C + 36*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(96*b*d) + ((8*b*B + 3*a*C)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d) + (C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d) + ((240*a*A*b^2 + 24*a^2*b*B + 128*b^3*B - 9*a^3*C + 156*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(192*b^2*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 7, ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(24*b*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((24*a^2*A - 48*a*A*b - 24*a^2*B + 6*a*b*B - 12*b^2*B + 7*a^2*C - 26*a*b*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(12*d*Sqrt[a + b*Cos[c + d*x]]) + ((24*a*A*b^2 + 6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b*d*Sqrt[a + b*Cos[c + d*x]]) + ((2*b*B + a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(24*b*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 7, -(((8*a*A - 4*b*B - 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + ((4*a^2*A + 8*a*A*b - 4*A*b^2 + 4*a^2*B - 8*a*b*B - 4*a^2*C + a*b*C - 2*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) + ((8*A*b^2 + 12*a*b*B + 3*a^2*C + 4*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(4*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((8*a*A - 4*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(4*d)}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 7, -(((8*A*b + 6*a*B - 3*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a^2*A + 4*a*A*b + 3*A*b^2 + 3*a^2*B + 6*a*b*B - 3*b^2*B + 3*a^2*C - 6*a*b*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(2*b*B + 3*a*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) - ((8*A*b + 6*a*B - 3*b*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*d)}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 8, -((2*(9*a^2*A + 3*A*b^2 + 20*a*b*B + 15*a^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*a*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(9*a^2*A + 12*a*A*b + 3*A*b^2 + 5*a^2*B + 20*a*b*B + 15*b^2*B + 15*a^2*C + 30*a*b*C - 15*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (4*b^2*C*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*(9*a^2*A + 3*A*b^2 + 20*a*b*B + 15*a^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) - (2*(9*a^2*A + 3*A*b^2 + 20*a*b*B + 15*a^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(15*a*d)}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 9, -((2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 140*a^2*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*a^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(25*a^3*A + 82*a^2*A*b + 51*a*A*b^2 - 6*A*b^3 + 63*a^3*B + 84*a^2*b*B + 21*a*b^2*B + 35*a^3*C + 140*a^2*b*C + 105*a*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B + 35*a^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Cos[c + d*x]^(3/2)) + (2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 140*a^2*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) - (2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 140*a^2*b*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(105*a^2*d)}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 11, -(((2520*a^4*A*b^2 - 94752*a^2*A*b^4 - 57344*A*b^6 - 1050*a^5*b*B - 4480*a^3*b^3*B - 126560*a*b^5*B + 525*a^6*C + 1400*a^4*b^2*C - 74736*a^2*b^4*C - 49152*b^6*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(107520*b^4*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (1/(53760*b^3*d*Sqrt[a + b*Cos[c + d*x]]))*((840*a^4*A*b^2 - 64176*a^3*A*b^3 + 43232*a^2*A*b^4 - 129472*a*A*b^5 - 350*a^5*b*B - 140*a^4*b^2*B + 24080*a^3*b^3*B - 138880*a^2*b^4*B + 16800*a*b^5*B - 33600*b^6*B + 175*a^6*C + 70*a^5*b*C + 480*a^4*b^2*C - 49968*a^3*b^3*C + 35856*a^2*b^4*C - 108576*a*b^5*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))]) + (1/(512*b^4*d*Sqrt[a + b*Cos[c + d*x]]))*((24*a^5*A*b^2 + 320*a^3*A*b^4 + 1920*a*A*b^6 - 10*a^6*b*B - 40*a^4*b^3*B + 1440*a^2*b^5*B + 640*b^7*B + 5*a^7*C + 12*a^5*b^2*C + 240*a^3*b^4*C + 1600*a*b^6*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))]) + ((840*a^3*A*b^2 + 43232*a*A*b^4 - 350*a^4*b*B + 24080*a^2*b^3*B + 16800*b^5*B + 175*a^5*C + 480*a^3*b^2*C + 35856*a*b^4*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(53760*b^3*d) + ((5208*a^2*A*b^2 + 3584*A*b^4 + 70*a^3*b*B + 7560*a*b^3*B - 35*a^4*C + 4084*a^2*b^2*C + 3072*b^4*C)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(13440*b^2*d) + ((3528*a*A*b^2 + 1890*a^2*b*B + 1400*b^3*B + 15*a^3*C + 2924*a*b^2*C)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(6720*b*d) + ((56*A*b^2 + 70*a*b*B + 5*a^2*C + 48*b^2*C)*Cos[c + d*x]^(7/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(280*d) + ((14*b*B + 5*a*C)*Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(84*d) + (C*Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) - ((2520*a^4*A*b^2 - 94752*a^2*A*b^4 - 57344*A*b^6 - 1050*a^5*b*B - 4480*a^3*b^3*B - 126560*a*b^5*B + 525*a^6*C + 1400*a^4*b^2*C - 74736*a^2*b^4*C - 49152*b^6*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(107520*b^4*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 10, ((600*a^3*A*b^2 + 11360*a*A*b^4 - 180*a^4*b*B + 6768*a^2*b^3*B + 4096*b^5*B + 75*a^5*C + 320*a^3*b^2*C + 9040*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(7680*b^3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + (1/(3840*b^2*d*Sqrt[a + b*Cos[c + d*x]]))*((2360*a^3*A*b^2 - 12880*a^2*A*b^3 + 1440*a*A*b^4 - 2880*A*b^5 + 60*a^4*b*B - 4584*a^3*b^2*B + 3088*a^2*b^3*B - 9248*a*b^4*B - 25*a^5*C - 10*a^4*b*C + 1720*a^3*b^2*C - 9920*a^2*b^3*C + 1200*a*b^4*C - 2400*b^5*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))]) - ((40*a^4*A*b^2 - 960*a^2*A*b^4 - 384*A*b^6 - 12*a^5*b*B - 160*a^3*b^3*B - 960*a*b^5*B + 5*a^6*C + 20*a^4*b^2*C - 720*a^2*b^4*C - 320*b^6*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(256*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + ((2360*a^2*A*b^2 + 1440*A*b^4 + 60*a^3*b*B + 3088*a*b^3*B - 25*a^4*C + 1720*a^2*b^2*C + 1200*b^4*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3840*b^2*d) + ((680*a*A*b^2 + 372*a^2*b*B + 256*b^3*B + 5*a^3*C + 540*a*b^2*C)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(960*b*d) + ((120*A*b^2 + 156*a*b*B + 15*a^2*C + 100*b^2*C)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(480*d) + ((12*b*B + 5*a*C)*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(60*d) + (C*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d) + ((600*a^3*A*b^2 + 11360*a*A*b^4 - 180*a^4*b*B + 6768*a^2*b^3*B + 4096*b^5*B + 75*a^5*C + 320*a^3*b^2*C + 9040*a*b^4*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(7680*b^3*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 9, ((2640*a^2*A*b^2 + 1280*A*b^4 + 150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(1920*b^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - (1/(960*b*d*Sqrt[a + b*Cos[c + d*x]]))*((1920*a^3*A*b - 1040*a^2*A*b^2 + 3040*a*A*b^3 - 590*a^3*b*B + 3220*a^2*b^2*B - 360*a*b^3*B + 720*b^4*B - 15*a^4*C + 1146*a^3*b*C - 772*a^2*b^2*C + 2312*a*b^3*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))]) + ((80*a^3*A*b^2 + 320*a*A*b^4 - 10*a^4*b*B + 240*a^2*b^3*B + 96*b^5*B + 3*a^5*C + 40*a^3*b^2*C + 240*a*b^4*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(64*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((1040*a*A*b^2 + 590*a^2*b*B + 360*b^3*B + 15*a^3*C + 772*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(960*b*d) + ((80*A*b^2 + 110*a*b*B + 15*a^2*C + 64*b^2*C)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(240*d) + ((2*b*B + a*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d) + ((2640*a^2*A*b^2 + 1280*A*b^4 + 150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(1920*b^2*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 8, ((432*a*A*b^2 + 264*a^2*b*B + 128*b^3*B + 15*a^3*C + 284*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(192*b*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((192*a^3*A - 576*a^2*A*b + 48*a*A*b^2 - 96*A*b^3 - 192*a^3*B + 104*a^2*b*B - 304*a*b^2*B + 59*a^3*C - 322*a^2*b*C + 36*a*b^2*C - 72*b^3*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(96*d*Sqrt[a + b*Cos[c + d*x]]) + ((240*a^2*A*b^2 + 64*A*b^4 + 40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2*b^2*C + 48*b^4*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(32*b*d*Sqrt[a + b*Cos[c + d*x]]) + ((16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + ((8*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d) + ((432*a*A*b^2 + 264*a^2*b*B + 128*b^3*B + 15*a^3*C + 284*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(192*b*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 8, -(((48*a^2*A - 24*A*b^2 - 54*a*b*B - 33*a^2*C - 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(24*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + ((24*a^3*A + 72*a^2*A*b - 72*a*A*b^2 + 24*a^3*B - 72*a^2*b*B + 6*a*b^2*B - 12*b^3*B - 24*a^3*C + 13*a^2*b*C - 38*a*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(12*d*Sqrt[a + b*Cos[c + d*x]]) + ((40*a*A*b^2 + 30*a^2*b*B + 8*b^3*B + 5*a^3*C + 20*a*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(8*a*A - 2*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (b*(6*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((48*a^2*A - 24*A*b^2 - 54*a*b*B - 33*a^2*C - 16*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(24*d)}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 8, -(((56*a*A*b + 24*a^2*B - 12*b^2*B - 27*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(12*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + ((4*a^3*A + 28*a^2*A*b + 36*a*A*b^2 - 12*A*b^3 + 12*a^3*B + 36*a^2*b*B - 36*a*b^2*B + 12*a^3*C - 36*a^2*b*C + 3*a*b^2*C - 6*b^3*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(6*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(8*A*b^2 + 20*a*b*B + 15*a^2*C + 4*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(8*A*b + 4*a*B - b*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*(5*A*b + 3*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) - ((56*a*A*b + 24*a^2*B - 12*b^2*B - 27*a*b*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(12*d)}
{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 8, -(((18*a^2*A + 46*A*b^2 + 70*a*b*B + 30*a^2*C - 15*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(9*a^3*A + 17*a^2*A*b + 23*a*A*b^2 + 15*A*b^3 + 5*a^3*B + 35*a^2*b*B + 45*a*b^2*B - 15*b^3*B + 15*a^3*C + 45*a^2*b*C - 45*a*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*(2*b*B + 5*a*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*a^2*A + 5*A*b^2 + 10*a*b*B + 5*a^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*(A*b + a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) - ((18*a^2*A + 46*A*b^2 + 70*a*b*B + 30*a^2*C - 15*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(15*d)}
{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 9, -((2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 245*a^2*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(105*a*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (1/(105*d*Sqrt[a + b*Cos[c + d*x]]))*(2*(25*a^3*A + 145*a^2*A*b + 135*a*A*b^2 + 15*A*b^3 + 63*a^3*B + 119*a^2*b*B + 161*a*b^2*B + 105*b^3*B + 35*a^3*C + 245*a^2*b*C + 315*a*b^2*C - 105*b^3*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))]) + (4*b^3*C*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*(25*a^2*A + 15*A*b^2 + 56*a*b*B + 35*a^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 245*a^2*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Sqrt[Cos[c + d*x]]) + (2*(5*A*b + 7*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) - (2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 245*a^2*b*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(105*a*d)}


(* ::Subsubsection::Closed:: *)
(*n<0*)


{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 7, ((24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(24*b^3*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + ((6*a*b*B - 12*b^2*B - 5*a^2*C - 2*a*b*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(12*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((8*a*A*b^2 - 6*a^2*b*B - 8*b^3*B + 5*a^3*C + 4*a*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + ((6*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*b^2*d) + (C*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d) + ((24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(24*b^3*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 6, ((4*b*B - 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - ((4*A*b - a*C + 2*b*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*b*d*Sqrt[a + b*Cos[c + d*x]]) + ((8*A*b^2 - 4*a*b*B + 3*a^2*C + 4*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d) + ((4*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(4*b^2*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 5, (C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + (2*(A - B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*(2*b*B - a*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(b*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 6, -((2*A*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(a*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(A + B - C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (4*C*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(a*d)}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(2*A*b - 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + (2*(a*A - 2*A*b + 3*a*B + 3*a*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*(2*A*b - 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]) + (2*(2*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^2*d)}

{Cos[c + d*x]^(1/2)*(a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 6, ((4*A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - ((4*a*A - a*B + 2*b*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a*A*b - a^2*B + 4*b^2*B)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*b*d*Sqrt[a + b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + ((4*A*b + a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(4*b*d)}


{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 8, ((72*a^2*A*b^2 - 24*A*b^4 - 90*a^3*b*B + 42*a*b^3*B + 105*a^4*C - 41*a^2*b^2*C - 16*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(24*b^4*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - ((24*a*A*b^2 - 30*a^2*b*B + 18*a*b^2*B - 12*b^3*B + 35*a^3*C - 21*a^2*b*C + 10*a*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(12*(a - b)*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - ((24*a*A*b^2 - 30*a^2*b*B - 8*b^3*B + 35*a^3*C + 12*a*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((24*a*A*b^2 - 30*a^2*b*B + 6*b^3*B + 35*a^3*C - 11*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*b^3*(a^2 - b^2)*d) + ((6*A*b^2 - 6*a*b*B + 7*a^2*C - b^2*C)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) + ((72*a^2*A*b^2 - 24*A*b^4 - 90*a^3*b*B + 42*a*b^3*B + 105*a^4*C - 41*a^2*b^2*C - 16*b^4*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(24*b^4*(a^2 - b^2)*d)}
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 7, -(((8*a*A*b^2 - 12*a^2*b*B + 4*b^3*B + 15*a^3*C - 7*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(4*b^3*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + ((4*A*b^2 - 4*a*b*B + 5*a^2*C - 3*a*b*C + 2*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*(a - b)*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((8*A*b^2 - 12*a*b*B + 15*a^2*C + 4*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*A*b^2 - 4*a*b*B + 5*a^2*C - b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) - ((8*a*A*b^2 - 12*a^2*b*B + 4*b^3*B + 15*a^3*C - 7*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(4*b^3*(a^2 - b^2)*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 6, ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b^2*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - (2*(A*b - b*B + a*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/((a - b)*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(2*b*B - 3*a*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(b^2*(a^2 - b^2)*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 6, -((2*(A*b^2 - a*b*B + a^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(a*b*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(A - B + C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/((a - b)*d*Sqrt[a + b*Cos[c + d*x]]) + (4*C*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 - a*b*B + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(a*b*(a^2 - b^2)*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 7, -((2*(a^2*A - 2*A*b^2 + a*b*B - a^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(a^2*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a*A - 2*A*b + a*B - a*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(a*(a - b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2*A - 2*A*b^2 + a*b*B - a^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - (2*(a^2*A - 2*A*b^2 + a*b*B - a^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(a^2*(a^2 - b^2)*d)}
{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 7, (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B - 3*a^2*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^3*(a^2 - b^2)*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + (2*(a^2*A - 6*a*A*b + 8*A*b^2 + 3*a^2*B - 6*a*b*B + 3*a^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^2*(a - b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2*A - 4*A*b^2 + 3*a*b*B - 3*a^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) - (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B - 3*a^2*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B - 3*a^2*b*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^3*(a^2 - b^2)*d)}


(* {Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 8, -(((24*a^3*A*b^2 - 56*a*A*b^4 - 60*a^4*b*B + 104*a^2*b^3*B - 12*b^5*B + 105*a^5*C - 170*a^3*b^2*C + 33*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(12*b^4*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + ((8*a^2*A*b^2 - 12*a*A*b^3 - 12*A*b^4 - 20*a^3*b*B + 12*a^2*b^2*B + 24*a*b^3*B + 35*a^4*C - 21*a^3*b*C - 33*a^2*b^2*C + 9*a*b^3*C - 6*b^4*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(6*(a - b)^2*b^3*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + ((8*A*b^2 - 20*a*b*B + 35*a^2*C + 4*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(2*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(a^2*A*b^2 - 5*A*b^4 - 4*a^3*b*B + 8*a*b^3*B + 7*a^4*C - 11*a^2*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((8*a^2*A*b^2 - 24*A*b^4 - 20*a^3*b*B + 36*a*b^3*B + 35*a^4*C - 54*a^2*b^2*C + 3*b^4*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^2*d) - ((24*a^3*A*b^2 - 56*a*A*b^4 - 60*a^4*b*B + 104*a^2*b^3*B - 12*b^5*B + 105*a^5*C - 170*a^3*b^2*C + 33*a*b^4*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(12*b^4*(a^2 - b^2)^2*d)} *)
{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, -(((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(a*A*b^2 + 3*A*b^3 + 2*a^2*b*B - 3*a*b^2*B - 3*b^3*B - 5*a^3*C + 3*a^2*b*C + 6*a*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*(a - b)^2*b^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(2*b*B - 5*a*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(a^2*A*b^2 + 3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + 9*a^2*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*b^3*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(3*a^2*A*b^2 + A*b^4 - 4*a*b^3*B - 3*a^4*C + 7*a^2*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a*b^2*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - (2*(3*a*A*b + A*b^2 - a*b*B - 3*b^2*B - 2*a^2*C + 3*a*b*C + 3*b^2*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*(a - b)^2*b*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (4*C*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(3*a^2*A*b^2 + A*b^4 - 4*a*b^3*B - 3*a^4*C + 7*a^2*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*a^2*A*b^2 + A*b^4 - 4*a*b^3*B - 3*a^4*C + 7*a^2*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a*b^2*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 6, -((2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B + 4*a^2*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(3*a^2*A + 3*a*A*b - 2*A*b^2 - 3*a^2*B - a*b*B + a^2*C + 3*a*b*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a*(a - b)^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 - a*b*B + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B + 4*a^2*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B + 4*a^2*b*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^2*(a^2 - b^2)^2*d)}
{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B - 3*a^4*C - a^2*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])) + (2*(3*a^3*A - 9*a^2*A*b - 6*a*A*b^2 + 8*A*b^3 + 3*a^3*B + 3*a^2*b*B - 2*a*b^2*B - 3*a^3*C - a^2*b*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^2*(a - b)^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)) + (2*(8*a^2*A*b^2 - 4*A*b^4 - 5*a^3*b*B + a*b^3*B + 2*a^4*C + 2*a^2*b^2*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B - 3*a^4*C - a^2*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) - (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B - 3*a^4*C - a^2*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^3*(a^2 - b^2)^2*d)}
(* {Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 6*a^4*b*C + 2*a^2*b^3*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) + (1/(9*a^4*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]))*(16*((3/8)*a^2*b*(a^4*A + 7*a^2*A*b^2 - 4*A*b^4 - 6*a^3*b*B + 2*a*b^3*B + 3*a^4*C + a^2*b^2*C) + (3/8)*a*b*(-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B + 6*a^4*b*C - 2*a^2*b^3*C))*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))]) + (1/(9*a^4*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]))*(32*((-(3/8))*a*b*(-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B + 6*a^4*b*C - 2*a^2*b^3*C) - (3/8)*a*b*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 6*a^4*b*C + 2*a^2*b^3*C))*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))]) + (2*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)) - (4*((-(3/2))*b^2*(a^2*A - 2*A*b^2 + a*b*B - a^2*C) - 2*a^2*(A*b^2 - a*b*B + a^2*C) + (3/2)*a^2*b*((-A)*b + a*B - b*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B - 5*a^4*C + a^2*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) + (2*(-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B + 6*a^4*b*C - 2*a^2*b^3*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 6*a^4*b*C + 2*a^2*b^3*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(3*a^4*(a^2 - b^2)^2*d)} *)