/*845 or so integration formulas from CRC short table. [CRC number, integrand, answer, low, high, conditions] */ [7,x^n,x^(n+1)/(n+1),false,false,[]] [9,1/x,log(x),false,false,[]] [11,%e^x,%e^x,false,false,[]] [12,%e^(a*x),%e^(a*x)/a,false,false,[]] [13,b^(a*x),b^(a*x)/(a*log(b)),false,false,[b > 0]] [14,log(x),x*log(x)-x,false,false,[]] [15,a^x,a^x/log(a),false,false,[a > 0,a # 1]] [16,1/(x^2+a^2),atan(x/a)/a,false,false,[]] [17,1/(a^2-x^2),atanh(x/a)/a,false,false,[]] [18,1/(a^2-x^2),log((x+a)/(a-x))/(2*a),false,false,[a^2 > x^2]] [19,1/(x^2-a^2),-acoth(x/a)/a,false,false,[]] [20,1/(x^2-a^2),log((x-a)/(x+a))/(2*a),false,false,[x^2 > a^2]] [21,1/sqrt(a^2-x^2),asin(x/abs(a)),false,false,[]] [22,1/sqrt(a^2-x^2),-acos(x/abs(a)),false,false,[x^2 > a^2]] [23,1/sqrt(x^2+a^2*plus_minus),log(sqrt(x^2+a^2*plus_minus)+x),false,false,[]] [24,1/(x*sqrt(x^2-a^2)),asec(x/a)/abs(a),false,false,[]] [25,1/(x*sqrt(plus_minus*x^2+a^2)),-log((sqrt(plus_minus*x^2+a^2)+a)/x)/a,false,false,[]] [26,(b*x+a)^n,(b*x+a)^(n+1)/(b*(n+1)),false,false,[n # -1]] [27,x*(b*x+a)^n,(b*x+a)^(n+2)/(b^2*(n+2))-a*(b*x+a)^(n+1)/(b^2*(n+1)),false,false,[]] [28,x^2*(b*x+a)^n,((b*x+a)^(n+3)/(n+3)-2*a*(b*x+a)^(n+2)/(n+2)+a^2*(b*x+a)^(n+1)/(n+1))/b^3,false,false,[n # -1,n # -2,n # -3]] [29,x^m*(b*x+a)^n,a*n*('integrate(x^m*(b*x+a)^(n-1),x))/(n+m+1)+x^(m+1)*(b*x+a)^n/(n+m+1),false,false,[]] [30,x^m*(b*x+a)^n,((n+m+2)*'integrate(x^m*(b*x+a)^(n+1),x)-x^(m+1)*(b*x+a)^(n+1))/(a*(n+1)),false,false,[]] [31,x^m*(b*x+a)^n,(x^m*(b*x+a)^(n+1)-a*m*'integrate(x^(m-1)*(b*x+a)^n,x))/(b*(n+m+1)),false,false,[]] [32,1/(b*x+a),log(b*x+a)/b,false,false,[]] [33,1/(b*x+a)^2,-1/(b*(b*x+a)),false,false,[]] [34,1/(b*x+a)^3,-1/(2*b*(b*x+a)^2),false,false,[]] [35,x/(b*x+a),(-a*log(b*x+a)+b*x+a)/b^2,false,false,[]] [36,x/(b*x+a),x/b-a*log(b*x+a)/b^2,false,false,[]] [37,x/(b*x+a)^2,(log(b*x+a)+a/(b*x+a))/b^2,false,false,[]] [38,x/(b*x+a)^n,(a*(b*x+a)^(1-n)/(n-1)-(b*x+a)^(2-n)/(n-2))/b^2,false,false,[n # 1,n # 2]] [39,x^2/(b*x+a),(a^2*log(b*x+a)+(b*x+a)^2/2-2*a*(b*x+a))/b^3,false,false,[]] [40,x^2/(b*x+a)^2,(-2*a*log(b*x+a)-a^2/(b*x+a)+b*x+a)/b^3,false,false,[]] [41,x^2/(b*x+a)^3,(log(b*x+a)+2*a/(b*x+a)-a^2/(2*(b*x+a)^2))/b^3,false,false,[]] [42,x^2/(b*x+a)^n,(-(b*x+a)^(3-n)/(n-3)+2*a*(b*x+a)^(2-n)/(n-2)-a^2*(b*x+a)^(1-n)/(n-1))/b^3,false,false,[n # 1,n # 2,n # 3]] [43,1/(x*(b*x+a)),-log((b*x+a)/x)/a,false,false,[]] [44,1/(x*(b*x+a)^2),1/(a*(b*x+a))-log((b*x+a)/x)/a^2,false,false,[]] [45,1/(x*(b*x+a)^3),((b*x+2*a)^2/(2*(b*x+a)^2)-log((b*x+a)/x))/a^3,false,false,[]] [46,1/(x^2*(b*x+a)),b*log((b*x+a)/x)/a^2-1/(a*x),false,false,[]] [47,1/(x^3*(b*x+a)),b^2*log(x/(b*x+a))/a^3+(2*b*x-a)/(2*a^2*x^2),false,false,[]] [48,1/(x^2*(b*x+a)^2),2*b*log((b*x+a)/x)/a^3-(2*b*x+a)/(a^2*x*(b*x+a)),false,false,[]] [49,1/(x^2+c^2),atan(x/c)/c,false,false,[]] [50,1/(c^2-x^2),log((x+c)/(c-x))/(2*c),false,false,[c^2 > x^2]] [51,1/(x^2-c^2),log((x-c)/(x+c))/(2*c),false,false,[x^2 > c^2]] [52,x/(plus_minus*x^2+c^2),plus_minus*log(plus_minus*x^2+c^2)/2,false,false,[]] [53,x*(plus_minus*x^2+c^2)^(-n-1),-plus_minus/(2*n*(plus_minus*x^2+c^2)^n),false,false,[]] [54,1/(plus_minus*x^2+c^2)^n,((2*n-3)*'integrate((plus_minus*x^2+c^2)^(1-n),x)+x*(plus_minus*x^2+c^2)^(1-n))/(2*c^2*(n-1)),false,false,[]] [55,1/(x^2-c^2)^n,(-(2*n-3)*'integrate((x^2-c^2)^(1-n),x)-x*(x^2-c^2)^(1-n))/(2*c^2*(n-1)),false,false,[]] [56,x/(x^2-c^2),log(x^2-c^2)/2,false,false,[]] [57,x*(x^2-c^2)^(-n-1),-1/(2*n*(x^2-c^2)^n),false,false,[]] [58,1/((b*x+a)*(x*d_+c)),log((x*d_+c)/(b*x+a))/(a*d_-b*c),false,false,[]] [59,x/((b*x+a)*(x*d_+c)),(a*log(b*x+a)/b-c*log(x*d_+c)/d_)/(a*d_-b*c),false,false,[]] [60,1/((b*x+a)^2*(x*d_+c)),(d_*log((x*d_+c)/(b*x+a))/(a*d_-b*c)+1/(b*x+a))/(a*d_-b*c),false,false,[]] [61,x/((b*x+a)^2*(x*d_+c)),-c*log((x*d_+c)/(b*x+a))/(a*d_-b*c)^2-a/(b*(b*x+a)*(a*d_-b*c)),false,false,[]] [62,x^2/((b*x+a)^2*(x*d_+c)),(c^2*log(x*d_+c)/d_+a*log(b*x+a)*(a*d_-2*b*c)/b^2)/(a*d_-b*c)^2+a^2/(b^2*(b*x+a)*(a*d_-b*c)),false,false,[]] [63,1/((b*x+a)^n*(x*d_+c)^m),(-b*(n+m-2)*'integrate((x*d_+c)^(1-m)/(b*x+a)^n,x)-(b*x+a)^(1-n)*(x*d_+c)^(1-m))/((m-1)*(a*d_-b*c)),false,false,[]] [64,(b*x+a)/(x*d_+c),(a*d_-b*c)*log(x*d_+c)/d_^2+b*x/d_,false,false,[]] [65,(b*x+a)^m/(x*d_+c)^n,-(b*(n-m-2)*'integrate((b*x+a)^m*(x*d_+c)^(1-n),x)+(b*x+a)^(m+1)*(x*d_+c)^(1-n))/((n-1)*(a*d_-b*c)),false,false,[]] [66,(b*x+a)^m/(x*d_+c)^n,-(m*(a*d_-b*c)*'integrate((b*x+a)^(m-1)/(x*d_+c)^n,x)+(b*x+a)^m*(x*d_+c)^(1-n))/((n-m-1)*d_),false,false,[]] [67,(b*x+a)^m/(x*d_+c)^n,-((b*x+a)^m*(x*d_+c)^(1-n)-b*m*'integrate((b*x+a)^(m-1)*(x*d_+c)^(1-n),x))/((n-1)*d_),false,false,[]] [68,1/(b*x^2+a),atan(sqrt(a*b)*x/a)/sqrt(a*b),false,false,[a*b > 0]] [69,1/(b*x^2+a),log((sqrt(-a*b)*x+a)/(a-sqrt(-a*b)*x))/(2*sqrt(-a*b)),false,false,[a*b < 0]] [70,1/(b*x^2+a),atanh(sqrt(-a*b)*x/a)/sqrt(-a*b),false,false,[a*b < 0]] [71,1/(b^2*x^2+a^2),atan(b*x/a)/(a*b),false,false,[]] [72,x/(b*x^2+a),log(b*x^2+a)/(2*b),false,false,[]] [73,x^2/(b*x^2+a),x/b-a*('integrate(1/(b*x^2+a),x))/b,false,false,[]] [74,1/(b*x^2+a)^2,('integrate(1/(b*x^2+a),x))/(2*a)+x/(2*a*(b*x^2+a)),false,false,[]] [75,1/(a^2-b^2*x^2),log((b*x+a)/(a-b*x))/(2*a*b),false,false,[]] [76,(b*x^2+a)^(-m-1),(2*m-1)*('integrate(1/(b*x^2+a)^m,x))/(2*a*m)+x/(2*a*m*(b*x^2+a)^m),false,false,[]] [77,(b*x^2+a)^(-m-1),(2*m)!*(x*('sum(a^(r-m)*4^(r-m)*(r-1)!*r!/((2*r)!*(b*x^2+a)^r),r,1,m))/(2*a)+('integrate(1/(b*x^2+a),x))/(a^m*4^m))/m!^2,false,false,[]] [78,x*(b*x^2+a)^(-m-1),-1/(2*b*m*(b*x^2+a)^m),false,false,[]] [79,x^2*(b*x^2+a)^(-m-1),('integrate(1/(b*x^2+a)^m,x))/(2*b*m)-x/(2*b*m*(b*x^2+a)^m),false,false,[]] [80,1/(x*(b*x^2+a)),log(x^2/(b*x^2+a))/(2*a),false,false,[]] [81,1/(x^2*(b*x^2+a)),-b*('integrate(1/(b*x^2+a),x))/a-1/(a*x),false,false,[]] [82,(b*x^2+a)^(-m-1)/x,('integrate(1/(x*(b*x^2+a)^m),x))/a+1/(2*a*m*(b*x^2+a)^m),false,false,[]] [83,(b*x^2+a)^(-m-1)/x,a^(-m-1)*('sum(a^r/(r*(b*x^2+a)^r),r,1,m)+log(x^2/(b*x^2+a)))/2,false,false,[]] [84,(b*x^2+a)^(-m-1)/x^2,('integrate(1/(x^2*(b*x^2+a)^m),x))/a-b*('integrate((b*x^2+a)^(-m-1),x))/a,false,false,[]] [85,1/(b*x^3+a),(log((x+a^(1/3)/b^(1/3))^3/(b*x^3+a))/2+sqrt(3)*atan(b^(1/3)*(2*x-a^(1/3)/b^(1/3))/(sqrt(3)*a^(1/3))))/(3*a^(2/3)*b^(1/3)),false,false,[a^(1/3)/b^(1/3) >= a^(1/3)/b^(1/3),a^(1/3)/b^(1/3) <= a^(1/3)/b^(1/3)]] [86,x/(b*x^3+a),(log((b*x^3+a)/(x+a^(1/3)/b^(1/3))^3)/2+sqrt(3)*atan(b^(1/3)*(2*x-a^(1/3)/b^(1/3))/(sqrt(3)*a^(1/3))))/(3*a^(1/3)*b^(2/3)),false,false,[a^(1/3)/b^(1/3) >= a^(1/3)/b^(1/3),a^(1/3)/b^(1/3) <= a^(1/3)/b^(1/3)]] [87,x^2/(b*x^3+a),log(b*x^3+a)/(3*b),false,false,[]] [88,1/(b*x^4+a),(log((x^2+2*a^(1/3)*x/b^(1/3)+2*a^(2/3)/b^(2/3))/(x^2-2*a^(1/3)*x/b^(1/3)+2*a^(2/3)/b^(2/3)))/2+atan(2*a^(1/3)*x/(b^(1/3)*(2*a^(2/3)/b^(2/3)-x^2))))/(2*a^(2/3)*b^(1/3)),false,false,[a*b > 0,a^(1/3)/b^(1/3) >= (a/b)^(1/4)/sqrt(2),a^(1/3)/b^(1/3) <= (a/b)^(1/4)/sqrt(2)]] [89,1/(b*x^4+a),(log((x+a^(1/3)/b^(1/3))/(x-a^(1/3)/b^(1/3)))/2+atan(b^(1/3)*x/a^(1/3)))/(2*a^(2/3)*b^(1/3)),false,false,[a*b < 0,a^(1/3)/b^(1/3) >= (-a/b)^(1/4)/sqrt(2),a^(1/3)/b^(1/3) <= (-a/b)^(1/4)/sqrt(2)]] [90,1/(b*x^4+a),(log((x^2+2*a^(1/3)*x/b^(1/3)+2*a^(2/3)/b^(2/3))/(x^2-2*a^(1/3)*x/b^(1/3)+2*a^(2/3)/b^(2/3)))/2+atan(2*a^(1/3)*x/(b^(1/3)*(2*a^(2/3)/b^(2/3)-x^2))))/(2*a^(2/3)*b^(1/3)),false,false,[a*b > 0,a^(1/3)/b^(1/3) >= (a/b)^(1/4)/sqrt(2),a^(1/3)/b^(1/3) <= (a/b)^(1/4)/sqrt(2)]] [91,1/(b*x^4+a),(log((x+a^(1/3)/b^(1/3))/(x-a^(1/3)/b^(1/3)))/2+atan(b^(1/3)*x/a^(1/3)))/(2*a^(2/3)*b^(1/3)),false,false,[a*b < 0,a^(1/3)/b^(1/3) >= (-a/b)^(1/4)/sqrt(2),a^(1/3)/b^(1/3) <= (-a/b)^(1/4)/sqrt(2)]] [92,x/(b*x^4+a),atan(b^(1/3)*x^2/a^(1/3))/(2*a^(1/3)*b^(2/3)),false,false,[a*b > 0,a^(1/3)/b^(1/3) >= sqrt(a/b),a^(1/3)/b^(1/3) <= sqrt(a/b)]] [93,x/(b*x^4+a),log((x^2-a^(1/3)/b^(1/3))/(x^2+a^(1/3)/b^(1/3)))/(4*a^(1/3)*b^(2/3)),false,false,[a*b < 0,a^(1/3)/b^(1/3) >= sqrt(-a/b),a^(1/3)/b^(1/3) <= sqrt(-a/b)]] [94,x^2/(b*x^4+a),(log((x^2-2*a^(1/3)*x/b^(1/3)+2*a^(2/3)/b^(2/3))/(x^2+2*a^(1/3)*x/b^(1/3)+2*a^(2/3)/b^(2/3)))/2+atan(2*a^(1/3)*x/(b^(1/3)*(2*a^(2/3)/b^(2/3)-x^2))))/(4*a^(1/3)*b^(2/3)),false,false,[a*b > 0,a^(1/3)/b^(1/3) >= (a/b)^(1/4)/sqrt(2),a^(1/3)/b^(1/3) <= (a/b)^(1/4)/sqrt(2)]] [95,x^2/(b*x^4+a),(log((x-a^(1/3)/b^(1/3))/(x+a^(1/3)/b^(1/3)))+2*atan(b^(1/3)*x/a^(1/3)))/(4*a^(1/3)*b^(2/3)),false,false,[a*b < 0,a^(1/3)/b^(1/3) >= (-a/b)^(1/4),a^(1/3)/b^(1/3) <= (-a/b)^(1/4)]] [96,x^3/(b*x^4+a),log(b*x^4+a)/(4*b),false,false,[]] [97,1/(x*(b*x^n+a)),log(x^n/(b*x^n+a))/(a*n),false,false,[]] [98,(b*x^n+a)^(-m-1),('integrate(1/(b*x^n+a)^m,x))/a-b*('integrate(x^n*(b*x^n+a)^(-m-1),x))/a,false,false,[]] [99,x^m*(b*x^n+a)^(-p-1),('integrate(x^(m-n)/(b*x^n+a)^p,x))/b-a*('integrate(x^(m-n)*(b*x^n+a)^(-p-1),x))/b,false,false,[]] [100,(b*x^n+a)^(-p-1)/x^m,('integrate(1/(x^m*(b*x^n+a)^p),x))/a-b*('integrate(x^(n-m)*(b*x^n+a)^(-p-1),x))/a,false,false,[]] [101,x^m*(b*x^n+a)^p,(x^(-n+m+1)*(b*x^n+a)^(p+1)-a*(-n+m+1)*'integrate(x^(m-n)*(b*x^n+a)^p,x))/(b*(n*p+m+1)),false,false,[]] [102,x^m*(b*x^n+a)^p,(a*n*p*'integrate(x^m*(b*x^n+a)^(p-1),x)+x^(m+1)*(b*x^n+a)^p)/(n*p+m+1),false,false,[]] [103,x^m*(b*x^n+a)^p,(x^(m+1)*(b*x^n+a)^(p+1)-b*(n*p+n+m+1)*'integrate(x^(n+m)*(b*x^n+a)^p,x))/(a*(m+1)),false,false,[]] [104,x^m*(b*x^n+a)^p,((n*p+n+m+1)*'integrate(x^m*(b*x^n+a)^(p+1),x)-x^(m+1)*(b*x^n+a)^(p+1))/(a*n*(p+1)),false,false,[]] [105,1/(plus_minus*x^3+c^3),plus_minus*log((plus_minus*x+c)^3/(plus_minus*x^3+c^3))/(6*c^2)+atan((2*x-c*plus_minus)/(sqrt(3)*c))/(sqrt(3)*c^2),false,false,[]] [106,1/(plus_minus*x^3+c^3)^2,2*('integrate(1/(plus_minus*x^3+c^3),x))/(3*c^3)+x/(3*c^3*(plus_minus*x^3+c^3)),false,false,[]] [107,(plus_minus*x^3+c^3)^(-n-1),((3*n-1)*'integrate(1/(plus_minus*x^3+c^3)^n,x)+x/(plus_minus*x^3+c^3)^n)/(3*c^3*n),false,false,[]] [108,x/(plus_minus*x^3+c^3),log((plus_minus*x^3+c^3)/(plus_minus*x+c)^3)/(6*c)+plus_minus*atan((2*x-c*plus_minus)/(sqrt(3)*c))/(sqrt(3)*c),false,false,[]] [109,x/(plus_minus*x^3+c^3)^2,('integrate(x/(plus_minus*x^3+c^3),x))/(3*c^3)+x^2/(3*c^3*(plus_minus*x^3+c^3)),false,false,[]] [110,x*(plus_minus*x^3+c^3)^(-n-1),((3*n-2)*'integrate(x/(plus_minus*x^3+c^3)^n,x)+x^2/(plus_minus*x^3+c^3)^n)/(3*c^3*n),false,false,[]] [111,x^2/(plus_minus*x^3+c^3),plus_minus*log(plus_minus*x^3+c^3)/3,false,false,[]] [112,x^2*(plus_minus*x^3+c^3)^(-n-1),-plus_minus/(3*n*(plus_minus*x^3+c^3)^n),false,false,[]] [113,1/(x*(plus_minus*x^3+c^3)),log(x^3/(plus_minus*x^3+c^3))/(3*c^3),false,false,[]] [114,1/(x*(plus_minus*x^3+c^3)^2),log(x^3/(plus_minus*x^3+c^3))/(3*c^6)+1/(3*c^3*(plus_minus*x^3+c^3)),false,false,[]] [115,(plus_minus*x^3+c^3)^(-n-1)/x,('integrate(1/(x*(plus_minus*x^3+c^3)^n),x))/c^3+1/(3*c^3*n*(plus_minus*x^3+c^3)^n),false,false,[]] [116,1/(x^2*(plus_minus*x^3+c^3)),-plus_minus*('integrate(x/(plus_minus*x^3+c^3),x))/c^3-1/(c^3*x),false,false,[]] [117,(plus_minus*x^3+c^3)^(-n-1)/x^2,('integrate(1/(x^2*(plus_minus*x^3+c^3)^n),x))/c^3-plus_minus*('integrate(x*(plus_minus*x^3+c^3)^(-n-1),x))/c^3,false,false,[]] [118,1/(x^4+c^4),(log((x^2+sqrt(2)*c*x+c^2)/(x^2-sqrt(2)*c*x+c^2))/2+atan(sqrt(2)*c*x/(c^2-x^2)))/(2*sqrt(2)*c^3),false,false,[]] [119,1/(c^4-x^4),(log((x+c)/(c-x))/2+atan(x/c))/(2*c^3),false,false,[]] [120,x/(x^4+c^4),atan(x^2/c^2)/(2*c^2),false,false,[]] [121,x/(c^4-x^4),log((x^2+c^2)/(c^2-x^2))/(4*c^2),false,false,[]] [122,x^2/(x^4+c^4),(log((x^2-sqrt(2)*c*x+c^2)/(x^2+sqrt(2)*c*x+c^2))/2+atan(sqrt(2)*c*x/(c^2-x^2)))/(2*sqrt(2)*c),false,false,[]] [123,x^2/(c^4-x^4),(log((x+c)/(c-x))/2-atan(x/c))/(2*c),false,false,[]] [124,x^3/(plus_minus*x^4+c^4),plus_minus*log(plus_minus*x^4+c^4)/4,false,false,[]] [125,1/(c*x^2+b*x+a),2*atan((2*c*x+b)/sqrt(4*a*c-b^2))/sqrt(4*a*c-b^2),false,false,[4*a*c-b^2 > 0]] [126,1/(c*x^2+b*x+a),-2*atanh((2*c*x+b)/sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c),false,false,[]] [127,1/(c*x^2+b*x+a),log((2*c*x-sqrt(b^2-4*a*c)+b)/(2*c*x+sqrt(b^2-4*a*c)+b))/sqrt(b^2-4*a*c),false,false,[4*a*c-b^2 < 0]] [128,1/(c*x^2+b*x+a),2*atan((2*c*x+b)/sqrt(4*a*c-b^2))/sqrt(4*a*c-b^2),false,false,[4*a*c-b^2 > 0]] [129,1/(c*x^2+b*x+a),-2*atanh((2*c*x+b)/sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c),false,false,[4*a*c-b^2 < 0]] [130,1/(c*x^2+b*x+a),log((2*c*x-sqrt(b^2-4*a*c)+b)/(2*c*x+sqrt(b^2-4*a*c)+b))/sqrt(b^2-4*a*c),false,false,[4*a*c-b^2 < 0]] [131,1/(c*x^2+b*x+a)^2,2*c*('integrate(1/(c*x^2+b*x+a),x))/(4*a*c-b^2)+(2*c*x+b)/((4*a*c-b^2)*(c*x^2+b*x+a)),false,false,[]] [132,1/(c*x^2+b*x+a)^3,6*c^2*('integrate(1/(c*x^2+b*x+a),x))/(4*a*c-b^2)^2+(2*c*x+b)*(3*c/((4*a*c-b^2)*(c*x^2+b*x+a))+1/(2*(c*x^2+b*x+a)^2))/(4*a*c-b^2),false,false,[]] [133,(c*x^2+b*x+a)^(-n-1),2*c*(2*n-1)*('integrate(1/(c*x^2+b*x+a)^n,x))/((4*a*c-b^2)*n)+(2*c*x+b)/((4*a*c-b^2)*n*(c*x^2+b*x+a)^n),false,false,[]] [134,(c*x^2+b*x+a)^(-n-1),(c/(4*a*c-b^2))^n*(2*n)!*((2*c*x+b)*('sum((r-1)!*r!*((4*a*c-b^2)/(c*(c*x^2+b*x+a)))^r/(2*r)!,r,1,n))/(4*a*c-b^2)+'integrate(1/(c*x^2+b*x+a),x))/n!^2,false,false,[]] [135,x/(c*x^2+b*x+a),log(c*x^2+b*x+a)/(2*c)-b*('integrate(1/(c*x^2+b*x+a),x))/(2*c),false,false,[]] [136,x/(c*x^2+b*x+a)^2,-b*('integrate(1/(c*x^2+b*x+a),x))/(4*a*c-b^2)-(b*x+2*a)/((4*a*c-b^2)*(c*x^2+b*x+a)),false,false,[]] [137,x*(c*x^2+b*x+a)^(-n-1),-b*(2*n-1)*('integrate(1/(c*x^2+b*x+a)^n,x))/((4*a*c-b^2)*n)-(b*x+2*a)/((4*a*c-b^2)*n*(c*x^2+b*x+a)^n),false,false,[]] [138,x^2/(c*x^2+b*x+a),-b*log(c*x^2+b*x+a)/(2*c^2)+(b^2-2*a*c)*('integrate(1/(c*x^2+b*x+a),x))/(2*c^2)+x/c,false,false,[]] [139,x^2/(c*x^2+b*x+a)^2,2*a*('integrate(1/(c*x^2+b*x+a),x))/(4*a*c-b^2)+((b^2-2*a*c)*x+a*b)/(c*(4*a*c-b^2)*(c*x^2+b*x+a)),false,false,[]] [140,x^m*(c*x^2+b*x+a)^(-n-1),-b*(n-m+1)*('integrate(x^(m-1)*(c*x^2+b*x+a)^(-n-1),x))/(c*(2*n-m+1))+a*(m-1)*('integrate(x^(m-2)*(c*x^2+b*x+a)^(-n-1),x))/(c*(2*n-m+1))-x^(m-1)/(c*(2*n-m+1)*(c*x^2+b*x+a)^n),false,false,[]] [141,1/(x*(c*x^2+b*x+a)),log(x^2/(c*x^2+b*x+a))/(2*a)-b*('integrate(1/(c*x^2+b*x+a),x))/(2*a),false,false,[]] [142,1/(x^2*(c*x^2+b*x+a)),b*log((c*x^2+b*x+a)/x^2)/(2*a^2)+(b^2/(2*a^2)-c/a)*'integrate(1/(c*x^2+b*x+a),x)-1/(a*x),false,false,[]] [143,1/(x*(c*x^2+b*x+a)^n),-b*('integrate(1/(c*x^2+b*x+a)^n,x))/(2*a)+('integrate((c*x^2+b*x+a)^(1-n)/x,x))/a+(c*x^2+b*x+a)^(1-n)/(2*a*(n-1)),false,false,[]] [144,(c*x^2+b*x+a)^(-n-1)/x^m,-c*(2*n+m-1)*('integrate(x^(2-m)*(c*x^2+b*x+a)^(-n-1),x))/(a*(m-1))-b*(n+m-1)*('integrate(x^(1-m)*(c*x^2+b*x+a)^(-n-1),x))/(a*(m-1))-x^(1-m)/(a*(m-1)*(c*x^2+b*x+a)^n),false,false,[]] [145,sqrt(b*x+a),2*(b*x+a)^(3/2)/(3*b),false,false,[]] [146,x*sqrt(b*x+a),-2*(2*a-3*b*x)*(b*x+a)^(3/2)/(15*b^2),false,false,[]] [147,x^2*sqrt(b*x+a),2*(b*x+a)^(3/2)*(15*b^2*x^2-12*a*b*x+8*a^2)/(105*b^3),false,false,[]] [148,x^m*sqrt(b*x+a),2*(x^m*(b*x+a)^(3/2)-a*m*'integrate(x^(m-1)*sqrt(b*x+a),x))/(b*(2*m+3)),false,false,[]] [149,x^m*sqrt(b*x+a),2*b^(-m-1)*m!*sqrt(b*x+a)*'sum((-a)^(m-r)*(b*x+a)^(r+1)/((m-r)!*(2*r+3)*r!),r,0,m),false,false,[]] [150,sqrt(b*x+a)/x,a*'integrate(1/(x*sqrt(b*x+a)),x)+2*sqrt(b*x+a),false,false,[]] [151,sqrt(b*x+a)/x^m,-(b*(2*m-5)*('integrate(x^(1-m)*sqrt(b*x+a),x))/2+x^(1-m)*(b*x+a)^(3/2))/(a*(m-1)),false,false,[]] [152,1/sqrt(b*x+a),2*sqrt(b*x+a)/b,false,false,[]] [153,x/sqrt(b*x+a),-2*(2*a-b*x)*sqrt(b*x+a)/(3*b^2),false,false,[]] [154,x^2/sqrt(b*x+a),2*sqrt(b*x+a)*(3*b^2*x^2-4*a*b*x+8*a^2)/(15*b^3),false,false,[]] [155,x^m/sqrt(b*x+a),2*(x^m*sqrt(b*x+a)-a*m*'integrate(x^(m-1)/sqrt(b*x+a),x))/(b*(2*m+1)),false,false,[]] [156,x^m/sqrt(b*x+a),2*(-a)^m*b^(-m-1)*m!*sqrt(b*x+a)*'sum((-1)^r*(b*x+a)^r/(a^r*(m-r)!*(2*r+1)*r!),r,0,m),false,false,[]] [157,1/(x*sqrt(b*x+a)),log((sqrt(b*x+a)-sqrt(a))/(sqrt(b*x+a)+sqrt(a)))/sqrt(a),false,false,[a > 0]] [158,1/(x*sqrt(b*x+a)),2*atan(sqrt(-(b*x+a)/a))/sqrt(-a),false,false,[a < 0]] [159,1/(x*sqrt(b*x+a)),2*atan(sqrt(-(b*x+a)/a))/sqrt(-a),false,false,[a < 0]] [160,1/(x*sqrt(b*x+a)),log((sqrt(b*x+a)-sqrt(a))/(sqrt(b*x+a)+sqrt(a)))/sqrt(a),false,false,[a > 0]] [161,1/(x^2*sqrt(b*x+a)),-b*('integrate(1/(x*sqrt(b*x+a)),x))/(2*a)-sqrt(b*x+a)/(a*x),false,false,[]] [162,1/(x^n*sqrt(b*x+a)),-b*(2*n-3)*('integrate(x^(1-n)/sqrt(b*x+a),x))/(a*(2*n-2))-x^(1-n)*sqrt(b*x+a)/(a*(n-1)),false,false,[]] [163,1/(x^n*sqrt(b*x+a)),(2*n-2)!*((-b/a)^(n-1)*4^(1-n)*'integrate(1/(x*sqrt(b*x+a)),x)-sqrt(b*x+a)*('sum((-b/a)^(-r+n-1)*4^(r-n+1)*(r-1)!*r!/((2*r)!*x^r),r,1,n-1))/a)/(n-1)!^2,false,false,[]] [164,(b*x+a)^(n*plus_minus/2),2*(b*x+a)^((n*plus_minus+2)/2)/(b*(n*plus_minus+2)),false,false,[]] [165,x*(b*x+a)^(n*plus_minus/2),2*((b*x+a)^((n*plus_minus+4)/2)/(n*plus_minus+4)-a*(b*x+a)^((n*plus_minus+2)/2)/(n*plus_minus+2))/b^2,false,false,[]] [168,1/sqrt((b*x+a)*(x*d_+c)),2*atanh(sqrt(b*(b*x+a)*d_*(x*d_+c))/(b*(x*d_+c)))/sqrt(b*d_),false,false,[b*d_ > 0,a*d_-b*c < 0]] [169,1/sqrt((b*x+a)*(x*d_+c)),2*atanh(sqrt(b*(b*x+a)*d_*(x*d_+c))/((b*x+a)*d_))/sqrt(b*d_),false,false,[b*d_ > 0,a*d_-b*c > 0]] [170,1/sqrt((b*x+a)*(x*d_+c)),log((sqrt(b*(b*x+a)*d_*(x*d_+c))+b*(x*d_+c))^2/(x*d_+c))/sqrt(b*d_),false,false,[b*d_ > 0]] [171,1/sqrt((b*x+a)*(x*d_+c)),2*atan(sqrt(-b*(b*x+a)*d_*(x*d_+c))/(b*(x*d_+c)))/sqrt(-b*d_),false,false,[]] [172,1/sqrt((b*x+a)*(x*d_+c)),-asin((2*b*x*d_+a*d_+b*c)/abs(a*d_-b*c))/sqrt(-b*d_),false,false,[b*d_ < 0]] 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[187,(x*d_+c)^m/sqrt(b*x+a),2*m!^2*sqrt(b*x+a)*('sum(4^(m-r)*(2*r)!*(x*d_+c)^r*(-(a*d_-b*c)/b)^(m-r)/r!^2,r,0,m))/(b*(2*m+1)!),false,false,[]] [188,sqrt(x^2+a^2*plus_minus),(a^2*plus_minus*log(sqrt(x^2+a^2*plus_minus)+x)+x*sqrt(x^2+a^2*plus_minus))/2,false,false,[]] [189,1/sqrt(x^2+a^2*plus_minus),log(sqrt(x^2+a^2*plus_minus)+x),false,false,[]] [190,1/(x*sqrt(x^2-a^2)),asec(x/a)/abs(a),false,false,[]] [191,1/(x*sqrt(x^2+a^2)),-log((sqrt(x^2+a^2)+a)/x)/a,false,false,[]] [192,sqrt(x^2+a^2)/x,sqrt(x^2+a^2)-a*log((sqrt(x^2+a^2)+a)/x),false,false,[]] [193,sqrt(x^2-a^2)/x,sqrt(x^2-a^2)-abs(a)*asec(x/a),false,false,[]] [194,x/sqrt(x^2+a^2*plus_minus),sqrt(x^2+a^2*plus_minus),false,false,[]] [195,x*sqrt(x^2+a^2*plus_minus),(x^2+a^2*plus_minus)^(3/2)/3,false,false,[]] [196,(x^2+a^2*plus_minus)^(3/2),(3*a^4*log(sqrt(x^2+a^2*plus_minus)+x)/2+x*(x^2+a^2*plus_minus)^(3/2)+3*a^2*plus_minus*x*sqrt(x^2+a^2*plus_minus)/2)/4,false,false,[]] [197,1/(x^2+a^2*plus_minus)^(3/2),plus_minus*x/(a^2*sqrt(x^2+a^2*plus_minus)),false,false,[]] [198,x/(x^2+a^2*plus_minus)^(3/2),-1/sqrt(x^2+a^2*plus_minus),false,false,[]] [199,x*(x^2+a^2*plus_minus)^(3/2),(x^2+a^2*plus_minus)^(5/2)/5,false,false,[]] [200,x^2*sqrt(x^2+a^2*plus_minus),-a^4*log(sqrt(x^2+a^2*plus_minus)+x)/8+x*(x^2+a^2*plus_minus)^(3/2)/4-a^2*plus_minus*x*sqrt(x^2+a^2*plus_minus)/8,false,false,[]] [201,x^3*sqrt(x^2+a^2),(x^2+a^2)^(3/2)*(3*x^2-2*a^2)/15,false,false,[]] [202,x^3*sqrt(x^2-a^2),(x^2-a^2)^(5/2)/5+a^2*(x^2-a^2)^(3/2)/3,false,false,[]] [203,x^2/sqrt(x^2+a^2*plus_minus),x*sqrt(x^2+a^2*plus_minus)/2-a^2*plus_minus*log(sqrt(x^2+a^2*plus_minus)+x)/2,false,false,[]] [204,x^3/sqrt(x^2+a^2*plus_minus),(x^2+a^2*plus_minus)^(3/2)/3-a^2*plus_minus*sqrt(x^2+a^2*plus_minus),false,false,[]] [205,1/(x^2*sqrt(x^2+a^2*plus_minus)),-plus_minus*sqrt(x^2+a^2*plus_minus)/(a^2*x),false,false,[]] [206,1/(x^3*sqrt(x^2+a^2)),log((sqrt(x^2+a^2)+a)/x)/(2*a^3)-sqrt(x^2+a^2)/(2*a^2*x^2),false,false,[]] [207,1/(x^3*sqrt(x^2-a^2)),asec(x/a)/(2*abs(a)^3)+sqrt(x^2-a^2)/(2*a^2*x^2),false,false,[]] [208,x^2*(x^2+a^2*plus_minus)^(3/2),-a^6*plus_minus*log(sqrt(x^2+a^2*plus_minus)+x)/16+x*(x^2+a^2*plus_minus)^(5/2)/6-a^2*plus_minus*x*(x^2+a^2*plus_minus)^(3/2)/24-a^4*x*sqrt(x^2+a^2*plus_minus)/16,false,false,[]] [209,x^3*(x^2+a^2*plus_minus)^(3/2),(x^2+a^2*plus_minus)^(7/2)/7-a^2*plus_minus*(x^2+a^2*plus_minus)^(5/2)/5,false,false,[]] [210,sqrt(x^2+a^2*plus_minus)/x^2,log(sqrt(x^2+a^2*plus_minus)+x)-sqrt(x^2+a^2*plus_minus)/x,false,false,[]] [211,sqrt(x^2+a^2)/x^3,-log((sqrt(x^2+a^2)+a)/x)/(2*a)-sqrt(x^2+a^2)/(2*x^2),false,false,[]] [212,sqrt(x^2-a^2)/x^3,asec(x/a)/(2*abs(a))-sqrt(x^2-a^2)/(2*x^2),false,false,[]] [213,sqrt(x^2+a^2*plus_minus)/x^4,-plus_minus*(x^2+a^2*plus_minus)^(3/2)/(3*a^2*x^3),false,false,[]] [214,x^2/(x^2+a^2*plus_minus)^(3/2),log(sqrt(x^2+a^2*plus_minus)+x)-x/sqrt(x^2+a^2*plus_minus),false,false,[]] [215,x^3/(x^2+a^2*plus_minus)^(3/2),sqrt(x^2+a^2*plus_minus)+a^2*plus_minus/sqrt(x^2+a^2*plus_minus),false,false,[]] [216,1/(x*(x^2+a^2)^(3/2)),1/(a^2*sqrt(x^2+a^2))-log((sqrt(x^2+a^2)+a)/x)/a^3,false,false,[]] [217,1/(x*(x^2-a^2)^(3/2)),-asec(x/a)/abs(a^3)-1/(a^2*sqrt(x^2-a^2)),false,false,[]] [218,1/(x^2*(x^2+a^2*plus_minus)^(3/2)),-(sqrt(x^2+a^2*plus_minus)/x+x/sqrt(x^2+a^2*plus_minus))/a^4,false,false,[]] [219,1/(x^3*(x^2+a^2)^(3/2)),3*log((sqrt(x^2+a^2)+a)/x)/(2*a^5)-1/(2*a^2*x^2*sqrt(x^2+a^2))-3/(2*a^4*sqrt(x^2+a^2)),false,false,[]] [220,1/(x^3*(x^2-a^2)^(3/2)),-3*asec(x/a)/(2*abs(a^5))+1/(2*a^2*x^2*sqrt(x^2-a^2))-3/(2*a^4*sqrt(x^2-a^2)),false,false,[]] [221,x^m/sqrt(x^2+a^2*plus_minus),x^(m-1)*sqrt(x^2+a^2*plus_minus)/m-a^2*(m-1)*plus_minus*('integrate(x^(m-2)/sqrt(x^2+a^2*plus_minus),x))/m,false,false,[]] [222,x^(2*m)/sqrt(x^2+a^2*plus_minus),(2*m)!*(sqrt(x^2+a^2*plus_minus)*'sum(abs(a)^(2*(m-r))*(-plus_minus)^(m-r)*2^(2*r-1)*(r-1)!*r!*x^(2*r-1)/(2*r)!,r,1,m)+abs(a)^(2*m)*(-plus_minus)^m*log(sqrt(x^2+a^2*plus_minus)+x))/(2^(2*m)*m!^2),false,false,[]] [223,x^(2*m+1)/sqrt(x^2+a^2*plus_minus),m!^2*sqrt(x^2+a^2*plus_minus)*('sum(abs(a)^(2*(m-r))*(-plus_minus)^(m-r)*4^(m-r)*(2*r)!*x^(2*r)/r!^2,r,0,m))/(2*m+1)!,false,false,[]] [224,1/(x^m*sqrt(x^2+a^2*plus_minus)),-(m-2)*plus_minus*('integrate(x^(2-m)/sqrt(x^2+a^2*plus_minus),x))/(a^2*(m-1))-plus_minus*x^(1-m)*sqrt(x^2+a^2*plus_minus)/(a^2*(m-1)),false,false,[]] [225,1/(x^(2*m)*sqrt(x^2+a^2*plus_minus)),(m-1)!*m!*sqrt(x^2+a^2*plus_minus)*('sum(abs(a)^(2*(r-m))*(-plus_minus)^(r-m)*2^(-2*r+2*m-1)*(2*r)!*x^(-2*r-1)/r!^2,r,0,m-1))/(2*m)!,false,false,[]] 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[237,(a^2-x^2)^(3/2),(3*a^4*asin(x/abs(a))/2+x*(a^2-x^2)^(3/2)+3*a^2*x*sqrt(a^2-x^2)/2)/4,false,false,[]] [238,1/(a^2-x^2)^(3/2),x/(a^2*sqrt(a^2-x^2)),false,false,[]] [239,x/(a^2-x^2)^(3/2),1/sqrt(a^2-x^2),false,false,[]] [240,x*(a^2-x^2)^(3/2),-(a^2-x^2)^(5/2)/5,false,false,[]] [241,x^2*sqrt(a^2-x^2),a^2*(a^2*asin(x/abs(a))+x*sqrt(a^2-x^2))/8-x*(a^2-x^2)^(3/2)/4,false,false,[]] [242,x^3*sqrt(a^2-x^2),(a^2-x^2)^(3/2)*(-x^2/5-2*a^2/15),false,false,[]] [243,x^2*(a^2-x^2)^(3/2),a^6*asin(x/abs(a))/16-x*(a^2-x^2)^(5/2)/6+a^2*x*(a^2-x^2)^(3/2)/24+a^4*x*sqrt(a^2-x^2)/16,false,false,[]] [244,x^3*(a^2-x^2)^(3/2),(a^2-x^2)^(7/2)/7-a^2*(a^2-x^2)^(5/2)/5,false,false,[]] [245,x^2/sqrt(a^2-x^2),a^2*asin(x/abs(a))/2-x*sqrt(a^2-x^2)/2,false,false,[]] [246,1/(x^2*sqrt(a^2-x^2)),-sqrt(a^2-x^2)/(a^2*x),false,false,[]] [247,sqrt(a^2-x^2)/x^2,-asin(x/abs(a))-sqrt(a^2-x^2)/x,false,false,[]] [248,sqrt(a^2-x^2)/x^3,log((sqrt(a^2-x^2)+a)/x)/(2*a)-sqrt(a^2-x^2)/(2*x^2),false,false,[]] [249,sqrt(a^2-x^2)/x^4,-(a^2-x^2)^(3/2)/(3*a^2*x^3),false,false,[]] [250,x^2/(a^2-x^2)^(3/2),x/sqrt(a^2-x^2)-asin(x/abs(a)),false,false,[]] [251,x^3/sqrt(a^2-x^2),-2*(a^2-x^2)^(3/2)/3-x^2*sqrt(a^2-x^2),false,false,[]] [252,x^3/(a^2-x^2)^(3/2),2*sqrt(a^2-x^2)+x^2/sqrt(a^2-x^2),false,false,[]] [253,x^3/(a^2-x^2)^(3/2),sqrt(a^2-x^2)+a^2/sqrt(a^2-x^2),false,false,[]] [254,1/(x^3*sqrt(a^2-x^2)),-log((sqrt(a^2-x^2)+a)/x)/(2*a^3)-sqrt(a^2-x^2)/(2*a^2*x^2),false,false,[]] [255,1/(x*(a^2-x^2)^(3/2)),1/(a^2*sqrt(a^2-x^2))-log((sqrt(a^2-x^2)+a)/x)/a^3,false,false,[]] [256,1/(x^2*(a^2-x^2)^(3/2)),(x/sqrt(a^2-x^2)-sqrt(a^2-x^2)/x)/a^4,false,false,[]] [257,1/(x^3*(a^2-x^2)^(3/2)),-3*log((sqrt(a^2-x^2)+a)/x)/(2*a^5)-1/(2*a^2*x^2*sqrt(a^2-x^2))+3/(2*a^4*sqrt(a^2-x^2)),false,false,[]] [258,x^m/sqrt(a^2-x^2),a^2*(m-1)*('integrate(x^(m-2)/sqrt(a^2-x^2),x))/m-x^(m-1)*sqrt(a^2-x^2)/m,false,false,[]] [260,x^(2*m+1)/sqrt(a^2-x^2),-m!^2*sqrt(a^2-x^2)*('sum(abs(a)^(2*(m-r))*4^(m-r)*(2*r)!*x^(2*r)/r!^2,r,0,m))/(2*m+1)!,false,false,[]] [261,1/(x^m*sqrt(a^2-x^2)),(m-2)*('integrate(x^(2-m)/sqrt(a^2-x^2),x))/(a^2*(m-1))-x^(1-m)*sqrt(a^2-x^2)/(a^2*(m-1)),false,false,[]] [262,1/(x^(2*m)*sqrt(a^2-x^2)),-(m-1)!*m!*sqrt(a^2-x^2)*('sum(a^(2*r-2*m)*2^(-2*r+2*m-1)*(2*r)!*x^(-2*r-1)/r!^2,r,0,m-1))/(2*m)!,false,false,[]] [263,x^(-2*m-1)/sqrt(a^2-x^2),(2*m)!*(a^(-2*m-1)*log((a-sqrt(a^2-x^2))/x)/2^(2*m)-sqrt(a^2-x^2)*('sum(abs(a)^(2*(r-m))*4^(r-m)*(r-1)!*r!/((2*r)!*x^(2*r)),r,1,m))/(2*a^2))/m!^2,false,false,[]] [264,1/(sqrt(a^2-x^2)*(b^2-x^2)),log((b*sqrt(a^2-x^2)+sqrt(a^2-b^2)*x)^2/(b^2-x^2))/(2*b*sqrt(a^2-b^2)),false,false,[a^2 > b^2]] [265,1/(sqrt(a^2-x^2)*(b^2-x^2)),atan(sqrt(b^2-a^2)*x/(b*sqrt(a^2-x^2)))/(b*sqrt(b^2-a^2)),false,false,[b^2 > a^2]] [266,1/(sqrt(a^2-x^2)*(b^2-x^2)),log((b*sqrt(a^2-x^2)+sqrt(a^2-b^2)*x)^2/(b^2-x^2))/(2*b*sqrt(a^2-b^2)),false,false,[a^2 > b^2]] 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[276,1/(c*x^2+b*x+a)^(3/2),2*(2*c*x+b)/((4*a*c-b^2)*sqrt(c*x^2+b*x+a)),false,false,[]] [277,1/(c*x^2+b*x+a)^(5/2),2*(2*c*x+b)*(1/(c*x^2+b*x+a)+8*c/(4*a*c-b^2))/(3*(4*a*c-b^2)*sqrt(c*x^2+b*x+a)),false,false,[]] [278,(c*x^2+b*x+a)^(-n-1/2),8*c*(n-1)*('integrate((c*x^2+b*x+a)^(1/2-n),x))/((4*a*c-b^2)*(2*n-1))+2*(2*c*x+b)*(c*x^2+b*x+a)^(1/2-n)/((4*a*c-b^2)*(2*n-1)),false,false,[]] [279,(c*x^2+b*x+a)^(-n-1/2),(c/(4*a*c-b^2))^(n-1)*4^(2*n-1)*(n-1)!*n!*(2*c*x+b)*('sum((2*r)!/(16^r*r!^2*(c*(c*x^2+b*x+a)/(4*a*c-b^2))^r),r,0,n-1))/((4*a*c-b^2)*(2*n)!*sqrt(c*x^2+b*x+a)),false,false,[]] [280,sqrt(c*x^2+b*x+a),(4*a*c-b^2)*('integrate(1/sqrt(c*x^2+b*x+a),x))/(8*c)+(2*c*x+b)*sqrt(c*x^2+b*x+a)/(4*c),false,false,[]] [281,(c*x^2+b*x+a)^(3/2),3*(4*a*c-b^2)^2*('integrate(1/sqrt(c*x^2+b*x+a),x))/(128*c^2)+(2*c*x+b)*sqrt(c*x^2+b*x+a)*(c*x^2+b*x+3*(4*a*c-b^2)/(8*c)+a)/(8*c),false,false,[]] [282,(c*x^2+b*x+a)^(5/2),5*(4*a*c-b^2)^3*('integrate(1/sqrt(c*x^2+b*x+a),x))/(1024*c^3)+(2*c*x+b)*sqrt(c*x^2+b*x+a)*((c*x^2+b*x+a)^2+5*(4*a*c-b^2)*(c*x^2+b*x+a)/(16*c)+15*(4*a*c-b^2)^2/(128*c^2))/(12*c),false,false,[]] [283,(c*x^2+b*x+a)^(n+1/2),(4*a*c-b^2)*(2*n+1)*('integrate((c*x^2+b*x+a)^(n-1/2),x))/(8*c*(n+1))+(2*c*x+b)*(c*x^2+b*x+a)^(n+1/2)/(4*c*(n+1)),false,false,[]] [284,(c*x^2+b*x+a)^(n+1/2),(c/(4*a*c-b^2))^(-n-1)*16^(-n-1)*(2*n+2)!*(4*(2*c*x+b)*sqrt(c*x^2+b*x+a)*('sum(16^r*r!*(r+1)!*(c*(c*x^2+b*x+a)/(4*a*c-b^2))^r/(2*r+2)!,r,0,n))/(4*a*c-b^2)+'integrate(1/sqrt(c*x^2+b*x+a),x))/(n+1)!^2,false,false,[]] [285,x/sqrt(c*x^2+b*x+a),sqrt(c*x^2+b*x+a)/c-b*('integrate(1/sqrt(c*x^2+b*x+a),x))/(2*c),false,false,[]] [286,x/(c*x^2+b*x+a)^(3/2),-2*(b*x+2*a)/((4*a*c-b^2)*sqrt(c*x^2+b*x+a)),false,false,[]] [287,x*(c*x^2+b*x+a)^(-n-1/2),-b*('integrate((c*x^2+b*x+a)^(-n-1/2),x))/(2*c)-(c*x^2+b*x+a)^(1/2-n)/(c*(2*n-1)),false,false,[]] [288,x^2/sqrt(c*x^2+b*x+a),(3*b^2-4*a*c)*('integrate(1/sqrt(c*x^2+b*x+a),x))/(8*c^2)+(x/(2*c)-3*b/(4*c^2))*sqrt(c*x^2+b*x+a),false,false,[]] [289,x^2/(c*x^2+b*x+a)^(3/2),('integrate(1/sqrt(c*x^2+b*x+a),x))/c+((2*b^2-4*a*c)*x+2*a*b)/(c*(4*a*c-b^2)*sqrt(c*x^2+b*x+a)),false,false,[]] [290,x^2*(c*x^2+b*x+a)^(-n-1/2),(b^2*(2*n-3)+4*a*c)*('integrate((c*x^2+b*x+a)^(1/2-n),x))/(c*(4*a*c-b^2)*(2*n-1))+((2*b^2-4*a*c)*x+2*a*b)*(c*x^2+b*x+a)^(1/2-n)/(c*(4*a*c-b^2)*(2*n-1)),false,false,[]] [291,x^3/sqrt(c*x^2+b*x+a),(3*a*b/(4*c^2)-5*b^3/(16*c^3))*'integrate(1/sqrt(c*x^2+b*x+a),x)+(x^2/(3*c)-5*b*x/(12*c^2)-2*a/(3*c^2)+5*b^2/(8*c^3))*sqrt(c*x^2+b*x+a),false,false,[]] [292,x^n/sqrt(c*x^2+b*x+a),-b*(2*n-1)*('integrate(x^(n-1)/sqrt(c*x^2+b*x+a),x))/(2*c*n)-a*(n-1)*('integrate(x^(n-2)/sqrt(c*x^2+b*x+a),x))/(c*n)+x^(n-1)*sqrt(c*x^2+b*x+a)/(c*n),false,false,[]] 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[310,x^n*sqrt(2*a*x-x^2),sqrt(2*a*x-x^2)*(x^(n+1)/(n+2)-(2*n+1)!*('sum(a^(-r+n+1)*2^(r-n)*r!^2*x^r/(2*r+1)!,r,0,n))/(n!*(n+2)!))+a^(n+2)*(2*n+1)!*asin((x-a)/abs(a))/(2^n*n!*(n+2)!),false,false,[]] [311,sqrt(2*a*x-x^2)/x^n,(n-3)*('integrate(x^(1-n)*sqrt(2*a*x-x^2),x))/(a*(2*n-3))+(2*a*x-x^2)^(3/2)/(a*(3-2*n)*x^n),false,false,[]] [312,x^n/sqrt(2*a*x-x^2),a*(2*n-1)*('integrate(x^(n-1)/sqrt(2*a*x-x^2),x))/n-x^(n-1)*sqrt(2*a*x-x^2)/n,false,false,[]] [313,x^n/sqrt(2*a*x-x^2),a^n*(2*n)!*asin((x-a)/abs(a))/(2^n*n!^2)-(2*n)!*sqrt(2*a*x-x^2)*('sum(a^(n-r)*2^(r-n)*(r-1)!*r!*x^(r-1)/(2*r)!,r,1,n))/n!^2,false,false,[]] [314,1/(x^n*sqrt(2*a*x-x^2)),(n-1)*('integrate(x^(1-n)/sqrt(2*a*x-x^2),x))/(a*(2*n-1))+sqrt(2*a*x-x^2)/(a*(1-2*n)*x^n),false,false,[]] [315,1/(x^n*sqrt(2*a*x-x^2)),-(n-1)!*n!*sqrt(2*a*x-x^2)*('sum(a^(r-n)*2^(n-r)*(2*r)!*x^(-r-1)/r!^2,r,0,n-1))/(2*n)!,false,false,[]] [316,1/(2*a*x-x^2)^(3/2),(x-a)/(a^2*sqrt(2*a*x-x^2)),false,false,[]] [317,x/(2*a*x-x^2)^(3/2),x/(a*sqrt(2*a*x-x^2)),false,false,[]] [318,1/sqrt(x^2+2*a*x),log(sqrt(x^2+2*a*x)+x+a),false,false,[]] [319,sqrt(a*x^2+c),c*log(sqrt(a*x^2+c)+sqrt(a)*x)/(2*sqrt(a))+x*sqrt(a*x^2+c)/2,false,false,[a > 0]] [320,sqrt(a*x^2+c),c*asin(sqrt(-a/c)*x)/(2*sqrt(-a))+x*sqrt(a*x^2+c)/2,false,false,[a < 0]] [321,sqrt(a*x^2+c),c*asin(sqrt(-a/c)*x)/(2*sqrt(-a))+x*sqrt(a*x^2+c)/2,false,false,[a < 0]] [322,sqrt(a*x^2+c),c*log(sqrt(a*x^2+c)+sqrt(a)*x)/(2*sqrt(a))+x*sqrt(a*x^2+c)/2,false,false,[a > 0]] [323,sqrt((x+1)/(1-x)),asin(x)-sqrt(1-x^2),false,false,[]] [324,1/(x*sqrt(a*x^n+c)),log((sqrt(a*x^n+c)-sqrt(c))/(sqrt(a*x^n+c)+sqrt(c)))/(sqrt(c)*n),false,false,[]] [325,1/(x*sqrt(a*x^n+c)),2*log((sqrt(a*x^n+c)-sqrt(c))/sqrt(x^n))/(sqrt(c)*n),false,false,[c > 0]] [326,1/(x*sqrt(a*x^n+c)),2*asec(sqrt(-a*x^n/c))/(sqrt(-c)*n),false,false,[c < 0]] [328,1/sqrt(a*x^2+c),asin(sqrt(-a/c)*x)/sqrt(-a),false,false,[a < 0]] [329,1/sqrt(a*x^2+c),asin(sqrt(-a/c)*x)/sqrt(-a),false,false,[a < 0]] [331,(a*x^2+c)^(m+1/2),c*(2*m+1)*('integrate((a*x^2+c)^(m-1/2),x))/(2*(m+1))+x*(a*x^2+c)^(m+1/2)/(2*(m+1)),false,false,[]] [332,(a*x^2+c)^(m+1/2),(2*m+1)!*x*sqrt(a*x^2+c)*('sum(c^(m-r)*2^(2*r-2*m-1)*r!^2*(a*x^2+c)^r/(2*r+1)!,r,0,m))/(m!*(m+1)!)+c^(m+1)*2^(-2*m-1)*(2*m+1)!*('integrate(1/sqrt(a*x^2+c),x))/(m!*(m+1)!),false,false,[]] [333,x*(a*x^2+c)^(m+1/2),(a*x^2+c)^(m+3/2)/(a*(2*m+3)),false,false,[]] [334,(a*x^2+c)^(m+1/2)/x,c*'integrate((a*x^2+c)^(m-1/2)/x,x)+(a*x^2+c)^(m+1/2)/(2*m+1),false,false,[]] [335,(a*x^2+c)^(m+1/2)/x,sqrt(a*x^2+c)*'sum(c^(m-r)*(a*x^2+c)^r/(2*r+1),r,0,m)+c^(m+1)*'integrate(1/(x*sqrt(a*x^2+c)),x),false,false,[]] [336,(a*x^2+c)^(-m-1/2),(2*m-2)*('integrate((a*x^2+c)^(1/2-m),x))/(c*(2*m-1))+x*(a*x^2+c)^(1/2-m)/(c*(2*m-1)),false,false,[]] [337,(a*x^2+c)^(-m-1/2),(m-1)!*m!*x*('sum(c^(r-m)*2^(-2*r+2*m-1)*(2*r)!/(r!^2*(a*x^2+c)^r),r,0,m-1))/((2*m)!*sqrt(a*x^2+c)),false,false,[]] [338,1/(x^m*sqrt(a*x^2+c)),-a*(m-2)*('integrate(x^(2-m)/sqrt(a*x^2+c),x))/(c*(m-1))-x^(1-m)*sqrt(a*x^2+c)/(c*(m-1)),false,false,[]] [339,(x^2+1)/((1-x^2)*sqrt(x^4+1)),log((sqrt(x^4+1)+sqrt(2)*x)/(1-x^2))/sqrt(2),false,false,[]] [340,(1-x^2)/((x^2+1)*sqrt(x^4+1)),atan(sqrt(2)*x/sqrt(x^4+1))/sqrt(2),false,false,[]] [341,1/(x*sqrt(x^n+a^2)),-2*log((sqrt(x^n+a^2)+a)/sqrt(x^n))/(a*n),false,false,[]] [342,1/(x*sqrt(x^n-a^2)),-2*asin(a/sqrt(x^n))/(a*n),false,false,[]] [343,sqrt(x/(a^3-x^3)),2*asin((x/a)^(3/2))/3,false,false,[]] [344,sin(a*x),-cos(a*x)/a,false,false,[]] [345,cos(a*x),sin(a*x)/a,false,false,[]] [346,tan(a*x),-log(cos(a*x))/a,false,false,[]] [347,tan(a*x),log(sec(a*x))/a,false,false,[]] [348,cot(a*x),log(sin(a*x))/a,false,false,[]] [349,cot(a*x),-log(csc(a*x))/a,false,false,[]] [350,sec(a*x),log(tan(a*x)+sec(a*x))/a,false,false,[]] [351,sec(a*x),log(tan(a*x/2+%pi/4))/a,false,false,[]] [352,csc(a*x),log(csc(a*x)-cot(a*x))/a,false,false,[]] [353,csc(a*x),log(tan(a*x/2))/a,false,false,[]] [354,sin(a*x)^2,x/2-cos(a*x)*sin(a*x)/(2*a),false,false,[]] [355,sin(a*x)^2,x/2-sin(2*a*x)/(4*a),false,false,[]] [356,sin(a*x)^3,-cos(a*x)*(sin(a*x)^2+2)/(3*a),false,false,[]] [357,sin(a*x)^4,sin(4*a*x)/(32*a)-sin(2*a*x)/(4*a)+3*x/8,false,false,[]] [358,sin(a*x)^n,(n-1)*('integrate(sin(a*x)^(n-2),x))/n-cos(a*x)*sin(a*x)^(n-1)/(a*n),false,false,[]] [359,sin(a*x)^(2*m),(2*m)!*x/(2^(2*m)*m!^2)-(2*m)!*cos(a*x)*('sum(2^(2*r-2*m)*r!^2*sin(a*x)^(2*r+1)/(2*r+1)!,r,0,m-1))/(a*m!^2),false,false,[]] [360,sin(a*x)^(2*m+1),-m!^2*cos(a*x)*('sum(2^(2*m-2*r)*(2*r)!*sin(a*x)^(2*r)/r!^2,r,0,m-1))/(a*(2*m+1)!),false,false,[]] [361,cos(a*x)^2,cos(a*x)*sin(a*x)/(2*a)+x/2,false,false,[]] [362,cos(a*x)^2,sin(2*a*x)/(4*a)+x/2,false,false,[]] [363,cos(a*x)^3,(cos(a*x)^2+2)*sin(a*x)/(3*a),false,false,[]] [364,cos(a*x)^4,sin(4*a*x)/(32*a)+sin(2*a*x)/(4*a)+3*x/8,false,false,[]] [365,cos(a*x)^n,(n-1)*('integrate(cos(a*x)^(n-2),x))/n+cos(a*x)^(n-1)*sin(a*x)/(a*n),false,false,[]] [366,cos(a*x)^(2*m),(2*m)!*sin(a*x)*('sum(2^(2*r-2*m)*r!^2*cos(a*x)^(2*r+1)/(2*r+1)!,r,0,m-1))/(a*m!^2)+(2*m)!*x/(2^(2*m)*m!^2),false,false,[]] [367,cos(a*x)^(2*m+1),m!^2*sin(a*x)*('sum(2^(2*m-2*r)*(2*r)!*cos(a*x)^(2*r)/r!^2,r,0,m))/(a*(2*m+1)!),false,false,[]] [368,csc(a*x)^2,-cot(a*x)/a,false,false,[]] [369,csc(a*x)^2,-cot(a*x)/a,false,false,[]] [370,1/sin(a*x)^m,(m-2)*('integrate(sin(a*x)^(2-m),x))/(m-1)-cos(a*x)*sin(a*x)^(1-m)/(a*(m-1)),false,false,[]] [371,csc(a*x)^m,(m-2)*('integrate(sin(a*x)^(2-m),x))/(m-1)-cos(a*x)*sin(a*x)^(1-m)/(a*(m-1)),false,false,[]] [372,1/sin(a*x)^(2*m),-(m-1)!*m!*cos(a*x)*('sum(2^(-2*r+2*m-1)*(2*r)!*sin(a*x)^(-2*r-1)/r!^2,r,0,m-1))/(a*(2*m)!),false,false,[]] [373,csc(a*x)^(2*m),-(m-1)!*m!*cos(a*x)*('sum(2^(-2*r+2*m-1)*(2*r)!*sin(a*x)^(-2*r-1)/r!^2,r,0,m-1))/(a*(2*m)!),false,false,[]] [374,sin(a*x)^(-2*m-1),(2*m)!*log(tan(a*x/2))/(a*2^(2*m)*m!^2)-(2*m)!*cos(a*x)*('sum(2^(2*r-2*m)*r!^2*sin(a*x)^(-2*r-2)/(2*r+1)!,r,0,m-1))/(a*m!^2),false,false,[]] [375,csc(a*x)^(2*m+1),(2*m)!*log(tan(a*x/2))/(a*2^(2*m)*m!^2)-(2*m)!*cos(a*x)*('sum(2^(2*r-2*m)*r!^2*sin(a*x)^(-2*r-2)/(2*r+1)!,r,0,m-1))/(a*m!^2),false,false,[]] [376,sec(a*x)^2,tan(a*x)/a,false,false,[]] [377,sec(a*x)^2,tan(a*x)/a,false,false,[]] [378,1/cos(a*x)^m,(m-2)*('integrate(cos(a*x)^(2-m),x))/(m-1)+cos(a*x)^(1-m)*sin(a*x)/(a*(m-1)),false,false,[]] [379,sec(a*x)^m,(m-2)*('integrate(cos(a*x)^(2-m),x))/(m-1)+cos(a*x)^(1-m)*sin(a*x)/(a*(m-1)),false,false,[]] [380,1/cos(a*x)^(2*m),(m-1)!*m!*sin(a*x)*('sum(2^(-2*r+2*m-1)*(2*r)!*cos(a*x)^(-2*r-1)/r!^2,r,0,m-1))/(a*(2*m)!),false,false,[]] [381,sec(a*x)^(2*m),(m-1)!*m!*sin(a*x)*('sum(2^(-2*r+2*m-1)*(2*r)!*cos(a*x)^(-2*r-1)/r!^2,r,0,m-1))/(a*(2*m)!),false,false,[]] [382,cos(a*x)^(-2*m-1),(2*m)!*log(tan(a*x)+sec(a*x))/(a*2^(2*m)*m!^2)+(2*m)!*sin(a*x)*('sum(2^(2*r-2*m)*r!^2*cos(a*x)^(-2*r-2)/(2*r+1)!,r,0,m-1))/(a*m!^2),false,false,[]] [383,sec(a*x)^(2*m+1),(2*m)!*log(tan(a*x)+sec(a*x))/(a*2^(2*m)*m!^2)+(2*m)!*sin(a*x)*('sum(2^(2*r-2*m)*r!^2*cos(a*x)^(-2*r-2)/(2*r+1)!,r,0,m-1))/(a*m!^2),false,false,[]] [384,sin(m*x)*sin(n*x),sin((m-n)*x)/(2*(m-n))-sin((n+m)*x)/(2*(n+m)),false,false,[m^2 # n^2]] [385,cos(m*x)*cos(n*x),sin((n+m)*x)/(2*(n+m))+sin((m-n)*x)/(2*(m-n)),false,false,[m^2 # n^2]] [386,cos(a*x)*sin(a*x),sin(a*x)^2/(2*a),false,false,[]] [387,sin(m*x)*cos(n*x),-cos((n+m)*x)/(2*(n+m))-cos((m-n)*x)/(2*(m-n)),false,false,[m^2 # n^2]] [388,cos(a*x)^2*sin(a*x)^2,x/8-sin(4*a*x)/(32*a),false,false,[]] [389,cos(a*x)^m*sin(a*x),-cos(a*x)^(m+1)/(a*(m+1)),false,false,[]] [390,cos(a*x)*sin(a*x)^m,sin(a*x)^(m+1)/(a*(m+1)),false,false,[]] [391,cos(a*x)^m*sin(a*x)^n,(m-1)*('integrate(cos(a*x)^(m-2)*sin(a*x)^n,x))/(n+m)+cos(a*x)^(m-1)*sin(a*x)^(n+1)/(a*(n+m)),false,false,[]] [392,cos(a*x)^m*sin(a*x)^n,(n-1)*('integrate(cos(a*x)^m*sin(a*x)^(n-2),x))/(n+m)-cos(a*x)^(m+1)*sin(a*x)^(n-1)/(a*(n+m)),false,false,[]] [393,cos(a*x)^m/sin(a*x)^n,-(-n+m+2)*('integrate(cos(a*x)^m*sin(a*x)^(2-n),x))/(n-1)-cos(a*x)^(m+1)*sin(a*x)^(1-n)/(a*(n-1)),false,false,[]] [394,cos(a*x)^m/sin(a*x)^n,(m-1)*('integrate(cos(a*x)^(m-2)/sin(a*x)^n,x))/(m-n)+cos(a*x)^(m-1)*sin(a*x)^(1-n)/(a*(m-n)),false,false,[]] [395,sin(a*x)^m/cos(a*x)^n,cos(a*x)^(1-n)*sin(a*x)^(m+1)/(a*(n-1))-(-n+m+2)*('integrate(cos(a*x)^(2-n)*sin(a*x)^m,x))/(n-1),false,false,[]] [396,sin(a*x)^m/cos(a*x)^n,(m-1)*('integrate(sin(a*x)^(m-2)/cos(a*x)^n,x))/(m-n)-cos(a*x)^(1-n)*sin(a*x)^(m-1)/(a*(m-n)),false,false,[]] [397,sec(a*x)*tan(a*x),sec(a*x)/a,false,false,[]] [398,sec(a*x)*tan(a*x),sec(a*x)/a,false,false,[]] [399,sin(a*x)*tan(a*x),log(tan(a*x/2+%pi/4))/a-sin(a*x)/a,false,false,[]] [400,cot(a*x)*csc(a*x),-csc(a*x)/a,false,false,[]] [401,cot(a*x)*csc(a*x),-csc(a*x)/a,false,false,[]] [402,csc(a*x)*sec(a*x),log(tan(a*x))/a,false,false,[]] [403,csc(a*x)*sec(a*x)^2,(log(tan(a*x/2))+sec(a*x))/a,false,false,[]] [404,csc(a*x)/cos(a*x)^n,'integrate(cos(a*x)^(2-n)*csc(a*x),x)+cos(a*x)^(1-n)/(a*(n-1)),false,false,[]] [405,csc(a*x)^2*sec(a*x),log(tan(a*x/2+%pi/4))/a-csc(a*x)/a,false,false,[]] [406,csc(a*x)^2*sec(a*x)^2,-2*cot(2*a*x)/a,false,false,[]] [407,1/(cos(a*x)^n*sin(a*x)^m),(n+m-2)*('integrate(sin(a*x)^(2-m)/cos(a*x)^n,x))/(m-1)-cos(a*x)^(1-n)*sin(a*x)^(1-m)/(a*(m-1)),false,false,[]] [408,1/(cos(a*x)^n*sin(a*x)^m),(n+m-2)*('integrate(cos(a*x)^(2-n)/sin(a*x)^m,x))/(n-1)+cos(a*x)^(1-n)*sin(a*x)^(1-m)/(a*(n-1)),false,false,[]] [409,sin(b*x+a),-cos(b*x+a)/b,false,false,[]] [410,cos(b*x+a),sin(b*x+a)/b,false,false,[]] [411,1/(plus_minus*sin(a*x)+1),plus_minus*tan(a*plus_minus*x/2-%pi/4)/a,false,false,[]] [412,1/(cos(a*x)+1),tan(a*x/2)/a,false,false,[]] [413,1/(1-cos(a*x)),-cot(a*x/2)/a,false,false,[]] [414,1/(b*sin(x)+a),2*atan((a*tan(x/2)+b)/sqrt(a^2-b^2))/sqrt(a^2-b^2),false,false,[a^2 > b^2]] [416,1/(b*cos(x)+a),2*atan(sqrt(a^2-b^2)*tan(x/2)/(b+a))/sqrt(a^2-b^2),false,false,[a^2 > b^2]] [417,1/(b*cos(x)+a),log((sqrt(b^2-a^2)*tan(x/2)+b+a)/(sqrt(b^2-a^2)*tan(x/2)-b-a))/sqrt(b^2-a^2),false,false,[b^2 > a^2]] [418,1/(b*sin(x)+c*cos(x)+a),log(((a-c)*tan(x/2)-sqrt(c^2+b^2-a^2)+b)/((a-c)*tan(x/2)+sqrt(c^2+b^2-a^2)+b))/sqrt(c^2+b^2-a^2),false,false,[a # c,a^2 < c^2+b^2]] [419,1/(b*sin(x)+c*cos(x)+a),2*atan(((a-c)*tan(x/2)+b)/sqrt(-c^2-b^2+a^2))/sqrt(-c^2-b^2+a^2),false,false,[a^2 > c^2+b^2]] [420,1/(b*sin(x)+c*cos(x)+a),(-(c+b)*sin(x)-(b-c)*sin(x)+a)/(a*(-(c+b)*sin(x)+(b-c)*sin(x)+a)),false,false,[a^2 >= c^2+b^2,a^2 <= c^2+b^2]] [421,sin(x)^2/(b*cos(x)^2+a),sqrt((b+a)/a)*atan(sqrt(a/(b+a))*tan(x))/b-x/b,false,false,[a*b > 0,a > abs(b),a > -abs(b)]] [422,1/(b^2*sin(x)^2+a^2*cos(x)^2),atan(b*tan(x)/a)/(a*b),false,false,[]] [423,cos(c*x)^2/(b^2*sin(c*x)^2+a^2),sqrt(b^2+a^2)*atan(sqrt(b^2+a^2)*tan(c*x)/a)/(a*b^2*c)-x/b^2,false,false,[]] [424,cos(c*x)*sin(c*x)/(b*sin(c*x)^2+a*cos(c*x)^2),log(b*sin(c*x)^2+a*cos(c*x)^2)/(2*(b-a)*c),false,false,[]] [425,cos(c*x)/(b*sin(c*x)+a*cos(c*x)),'integrate(1/(b*tan(c*x)+a),x),false,false,[]] [426,cos(c*x)/(b*sin(c*x)+a*cos(c*x)),(b*log(b*sin(c*x)+a*cos(c*x))+a*c*x)/((b^2+a^2)*c),false,false,[]] [427,sin(c*x)/(b*sin(c*x)+a*cos(c*x)),'integrate(1/(a*cot(c*x)+b),x),false,false,[]] [428,sin(c*x)/(b*sin(c*x)+a*cos(c*x)),(b*c*x-a*log(b*sin(c*x)+a*cos(c*x)))/((b^2+a^2)*c),false,false,[]] [429,1/(c*sin(x)^2+2*b*cos(x)*sin(x)+a*cos(x)^2),log((c*tan(x)-sqrt(b^2-a*c)+b)/(c*tan(x)+sqrt(b^2-a*c)+b))/(2*sqrt(b^2-a*c)),false,false,[b^2 > a*c]] [430,1/(c*sin(x)^2+2*b*cos(x)*sin(x)+a*cos(x)^2),atan((c*tan(x)+b)/sqrt(a*c-b^2))/sqrt(a*c-b^2),false,false,[b^2 < a*c]] [431,1/(c*sin(x)^2+2*b*cos(x)*sin(x)+a*cos(x)^2),-1/(c*tan(x)+b),false,false,[b^2 >= a*c,b^2 <= a*c]] [432,sin(a*x)/(plus_minus*sin(a*x)+1),plus_minus*x-tan(a*plus_minus*x/2-%pi/4)/a,false,false,[]] [433,csc(a*x)/(plus_minus*sin(a*x)+1),log(tan(a*x/2))/a-tan(a*plus_minus*x/2-%pi/4)/a,false,false,[]] [434,1/(sin(a*x)+1)^2,tan(a*x/2-%pi/4)^3/(6*a)+tan(a*x/2-%pi/4)/(2*a),false,false,[]] [435,1/(1-sin(a*x))^2,-cot(a*x/2-%pi/4)^3/(6*a)-cot(a*x/2-%pi/4)/(2*a),false,false,[]] [436,sin(a*x)/(sin(a*x)+1)^2,tan(a*x/2-%pi/4)/(2*a)-tan(a*x/2-%pi/4)^3/(6*a),false,false,[]] [437,sin(a*x)/(1-sin(a*x))^2,cot(a*x/2-%pi/4)/(2*a)-cot(a*x/2-%pi/4)^3/(6*a),false,false,[]] [438,sin(x)/(b*sin(x)+a),x/b-a*('integrate(1/(b*sin(x)+a),x))/b,false,false,[]] [439,csc(x)/(b*sin(x)+a),log(tan(x/2))/a-b*('integrate(1/(b*sin(x)+a),x))/a,false,false,[]] [440,1/(b*sin(x)+a)^2,a*('integrate(1/(b*sin(x)+a),x))/(a^2-b^2)+b*cos(x)/((a^2-b^2)*(b*sin(x)+a)),false,false,[]] [441,sin(x)/(b*sin(x)+a)^2,b*('integrate(1/(b*sin(x)+a),x))/(b^2-a^2)+a*cos(x)/((b^2-a^2)*(b*sin(x)+a)),false,false,[]] [442,1/(b*sin(x)+a)^2,a*('integrate(1/(b*sin(x)+a),x))/(a^2-b^2)+b*cos(x)/((a^2-b^2)*(b*sin(x)+a)),false,false,[]] [443,1/(b*sin(x)+a)^2,b*('integrate(1/(b*sin(x)+a),x))/(b^2-a^2)+a*cos(x)/((b^2-a^2)*(b*sin(x)+a)),false,false,[]] [444,1/(b^2*sin(c*x)^2+a^2),atan(sqrt(b^2+a^2)*tan(c*x)/a)/(a*sqrt(b^2+a^2)*c),false,false,[]] [445,1/(a^2-b^2*sin(c*x)^2),atan(sqrt(a^2-b^2)*tan(c*x)/a)/(a*sqrt(a^2-b^2)*c),false,false,[a^2 > b^2]] [446,1/(a^2-b^2*sin(c*x)^2),log((sqrt(b^2-a^2)*tan(c*x)+a)/(sqrt(b^2-a^2)*tan(c*x)-a))/(2*a*sqrt(b^2-a^2)*c),false,false,[a^2 < b^2]] [447,cos(a*x)/(cos(a*x)+1),x-tan(a*x/2)/a,false,false,[]] [448,cos(a*x)/(1-cos(a*x)),-cot(a*x/2)/a-x,false,false,[]] [449,sec(a*x)/(cos(a*x)+1),log(tan(a*x/2+%pi/4))/a-tan(a*x/2)/a,false,false,[]] [450,sec(a*x)/(1-cos(a*x)),log(tan(a*x/2+%pi/4))/a-cot(a*x/2)/a,false,false,[]] [451,1/(cos(a*x)+1)^2,tan(a*x/2)^3/(6*a)+tan(a*x/2)/(2*a),false,false,[]] [452,1/(1-cos(a*x))^2,-cot(a*x/2)^3/(6*a)-cot(a*x/2)/(2*a),false,false,[]] [453,cos(a*x)/(cos(a*x)+1)^2,tan(a*x/2)/(2*a)-tan(a*x/2)^3/(6*a),false,false,[]] [454,cos(a*x)/(1-cos(a*x))^2,cot(a*x/2)/(2*a)-cot(a*x/2)^3/(6*a),false,false,[]] [455,cos(x)/(b*cos(x)+a),x/b-a*('integrate(1/(b*cos(x)+a),x))/b,false,false,[]] [456,sec(x)/(b*cos(x)+a),log(tan(x/2+%pi/4))/a-b*('integrate(1/(b*cos(x)+a),x))/a,false,false,[]] [457,1/(b*cos(x)+a)^2,b*sin(x)/((b^2-a^2)*(b*cos(x)+a))-a*('integrate(1/(b*cos(x)+a),x))/(b^2-a^2),false,false,[]] [458,cos(x)/(b*cos(x)+a)^2,a*sin(x)/((a^2-b^2)*(b*cos(x)+a))-b*('integrate(1/(b*cos(x)+a),x))/(a^2-b^2),false,false,[]] [459,1/(-2*a*b*cos(c*x)+b^2+a^2),2*atan((b+a)*tan(c*x/2)/(a-b))/((a^2-b^2)*c),false,false,[]] [460,1/(b^2*cos(c*x)^2+a^2),atan(a*tan(c*x)/sqrt(b^2+a^2))/(a*sqrt(b^2+a^2)*c),false,false,[]] [461,1/(a^2-b^2*cos(c*x)^2),atan(a*tan(c*x)/sqrt(a^2-b^2))/(a*sqrt(a^2-b^2)*c),false,false,[a^2 > b^2]] [462,1/(a^2-b^2*cos(c*x)^2),log((a*tan(c*x)-sqrt(b^2-a^2))/(a*tan(c*x)+sqrt(b^2-a^2)))/(2*a*sqrt(b^2-a^2)*c),false,false,[b^2 > a^2]] [463,sin(a*x)/(plus_minus*cos(a*x)+1),-plus_minus*log(plus_minus*cos(a*x)+1)/a,false,false,[]] [464,cos(a*x)/(plus_minus*sin(a*x)+1),plus_minus*log(plus_minus*sin(a*x)+1)/a,false,false,[]] [465,csc(a*x)/(plus_minus*cos(a*x)+1),log(tan(a*x/2))/(2*a)+plus_minus/(2*a*(plus_minus*cos(a*x)+1)),false,false,[]] [466,sec(a*x)/(plus_minus*sin(a*x)+1),log(tan(a*x/2+%pi/4))/(2*a)-plus_minus/(2*a*(plus_minus*sin(a*x)+1)),false,false,[]] [467,tan(a*x)/(plus_minus*cos(a*x)+1),log(sec(a*x)+plus_minus)/a,false,false,[]] [468,cot(a*x)/(plus_minus*sin(a*x)+1),-log(csc(a*x)+plus_minus)/a,false,false,[]] [469,tan(a*x)/(plus_minus*sin(a*x)+1),plus_minus*log(tan(a*x/2+%pi/4))/(2*a)+1/(2*a*(plus_minus*sin(a*x)+1)),false,false,[]] [470,cot(a*x)/(plus_minus*cos(a*x)+1),plus_minus*log(tan(a*x/2))/(2*a)-1/(2*a*(plus_minus*cos(a*x)+1)),false,false,[]] [471,1/(sin(a*x)+plus_minus*cos(a*x)),log(tan(a*x/2+%pi*plus_minus/8))/(sqrt(2)*a),false,false,[]] [472,1/(sin(a*x)+plus_minus*cos(a*x))^2,tan(a*x-%pi*plus_minus/4)/(2*a),false,false,[]] [473,1/(plus_minus*sin(a*x)+cos(a*x)+1),plus_minus*log(plus_minus*tan(a*x/2)+1)/a,false,false,[]] [474,1/(a^2*cos(c*x)^2-b^2*sin(c*x)^2),log((b*tan(c*x)+a)/(b*tan(c*x)-a))/(2*a*b*c),false,false,[]] [475,x*sin(a*x),sin(a*x)/a^2-x*cos(a*x)/a,false,false,[]] [476,x^2*sin(a*x),2*x*sin(a*x)/a^2+(2-a^2*x^2)*cos(a*x)/a^3,false,false,[]] [477,x^3*sin(a*x),(3*a^2*x^2-6)*sin(a*x)/a^4+(6*x-a^2*x^3)*cos(a*x)/a^3,false,false,[]] [478,x^m*sin(a*x),m*('integrate(x^(m-1)*cos(a*x),x))/a-x^m*cos(a*x)/a,false,false,[]] [479,x^m*sin(a*x),m!*cos(a*x)*'sum(a^(-2*r-1)*(-1)^(r+1)*x^(m-2*r)/(m-2*r)!,r,0,floor(m/2))+m!*sin(a*x)*'sum(a^(-2*r-2)*(-1)^r*x^(-2*r+m-1)/(-2*r+m-1)!,r,0,floor((m-1)/2)),false,false,[]] [480,x*cos(a*x),x*sin(a*x)/a+cos(a*x)/a^2,false,false,[]] [481,x^2*cos(a*x),(a^2*x^2-2)*sin(a*x)/a^3+2*x*cos(a*x)/a^2,false,false,[]] [482,x^3*cos(a*x),(a^2*x^3-6*x)*sin(a*x)/a^3+(3*a^2*x^2-6)*cos(a*x)/a^4,false,false,[]] [483,x^m*cos(a*x),x^m*sin(a*x)/a-m*('integrate(x^(m-1)*sin(a*x),x))/a,false,false,[]] [484,x^m*cos(a*x),m!*sin(a*x)*'sum(a^(-2*r-1)*(-1)^r*x^(m-2*r)/(m-2*r)!,r,0,floor(m/2))+m!*cos(a*x)*'sum(a^(-2*r-2)*(-1)^r*x^(-2*r+m-1)/(-2*r+m-1)!,r,0,floor((m-1)/2)),false,false,[]] [485,sin(a*x)/x,'sum((-1)^n*(a*x)^(2*n+1)/((2*n+1)*(2*n+1)!),n,0,inf),false,false,[]] [486,cos(a*x)/x,('sum((-1)^n*(a*x)^(2*n)/(n*(2*n)!),n,0,inf))/2,false,false,[]] [487,x*sin(a*x)^2,-x*sin(2*a*x)/(4*a)-cos(2*a*x)/(8*a^2)+x^2/4,false,false,[]] [488,x^2*sin(a*x)^2,-(x^2/(4*a)-1/(8*a^3))*sin(2*a*x)-x*cos(2*a*x)/(4*a^2)+x^3/6,false,false,[]] [489,x*sin(a*x)^3,-sin(3*a*x)/(36*a^2)+x*cos(3*a*x)/(12*a)+3*sin(a*x)/(4*a^2)-3*x*cos(a*x)/(4*a),false,false,[]] [490,x*cos(a*x)^2,x*sin(2*a*x)/(4*a)+cos(2*a*x)/(8*a^2)+x^2/4,false,false,[]] [491,x^2*cos(a*x)^2,(x^2/(4*a)-1/(8*a^3))*sin(2*a*x)+x*cos(2*a*x)/(4*a^2)+x^3/6,false,false,[]] [492,x*cos(a*x)^3,x*sin(3*a*x)/(12*a)+cos(3*a*x)/(36*a^2)+3*x*sin(a*x)/(4*a)+3*cos(a*x)/(4*a^2),false,false,[]] [493,sin(a*x)/x^m,a*('integrate(x^(1-m)*cos(a*x),x))/(m-1)+x^(1-m)*sin(a*x)/(1-m),false,false,[]] [494,cos(a*x)/x^m,a*('integrate(x^(1-m)*sin(a*x),x))/(1-m)+x^(1-m)*cos(a*x)/(1-m),false,false,[]] [495,x/(plus_minus*sin(a*x)+1),log(plus_minus*sin(a*x)+1)/a^2-plus_minus*x*cos(a*x)/(a*(plus_minus*sin(a*x)+1)),false,false,[]] [496,x/(cos(a*x)+1),2*log(cos(a*x/2))/a^2+x*tan(a*x/2)/a,false,false,[]] [497,x/(1-cos(a*x)),2*log(sin(a*x/2))/a^2-x*cot(a*x/2)/a,false,false,[]] [498,(sin(x)+x)/(cos(x)+1),x*tan(x/2),false,false,[]] [499,(x-sin(x))/(1-cos(x)),-x*cot(x/2),false,false,[]] [500,sqrt(1-cos(a*x)),-2*sin(a*x)/(a*sqrt(1-cos(a*x))),false,false,[]] [501,sqrt(1-cos(a*x)),-2*sqrt(2)*cos(a*x/2)/a,false,false,[]] [502,sqrt(cos(a*x)+1),2*sin(a*x)/(a*sqrt(cos(a*x)+1)),false,false,[]] [503,sqrt(cos(a*x)+1),2*sqrt(2)*sin(a*x/2)/a,false,false,[]] [504,sqrt(sin(x)+1),2*(sin(x/2)-cos(x/2)),false,false,[\<_\<\=(%pi*(8*k-1)/2,x,%pi*(8*k+3)/2)]] [505,sqrt(sin(x)+1),-2*(sin(x/2)-cos(x/2)),false,false,[\<_\<\=(%pi*(8*k+3)/2,x,%pi*(8*k+7)/2)]] [506,sqrt(1-sin(x)),2*(sin(x/2)+cos(x/2)),false,false,[\<_\<\=(%pi*(8*k-3)/2,x,%pi*(8*k+1)/2)]] [507,sqrt(1-sin(x)),-2*(sin(x/2)+cos(x/2)),false,false,[\<_\<\=(%pi*(8*k+1)/2,x,%pi*(8*k+5)/2)]] [508,1/sqrt(1-cos(x)),sqrt(2)*log(tan(x/4)),false,false,[4*%pi*k < x,x <= %pi*(4*k+2)]] [509,1/sqrt(1-cos(x)),-sqrt(2)*log(tan(x/4)),false,false,[%pi*(4*k-2) < x,x <= 4*%pi*k]] [510,1/sqrt(cos(x)+1),sqrt(2)*log(tan((x+%pi)/4)),false,false,[%pi*(4*k-1) < x,x <= %pi*(4*k+1)]] [511,1/sqrt(cos(x)+1),-sqrt(2)*log(tan((x+%pi)/4)),false,false,[%pi*(4*k+1) < x,x <= %pi*(4*k+3)]] [512,1/sqrt(1-sin(x)),sqrt(2)*log(tan(x/4-%pi/8)),false,false,[\<_\<\=(%pi*(8*k+1)/2,x,%pi*(8*k+5)/2)]] [513,1/sqrt(1-sin(x)),-sqrt(2)*log(tan(x/4-%pi/8)),false,false,[\<_\<\=(%pi*(8*k-3)/2,x,%pi*(8*k+1)/2)]] [514,1/sqrt(sin(x)+1),sqrt(2)*log(tan(x/4+%pi/8)),false,false,[\<_\<\=(%pi*(8*k-1)/2,x,%pi*(8*k+3)/2)]] [515,1/sqrt(sin(x)+1),-sqrt(2)*log(tan(x/4+%pi/8)),false,false,[\<_\<\=(%pi*(8*k+3)/2,x,%pi*(8*k+7)/2)]] [516,tan(a*x)^2,tan(a*x)/a-x,false,false,[]] [517,tan(a*x)^3,log(cos(a*x))/a+tan(a*x)^2/(2*a),false,false,[]] [518,tan(a*x)^4,tan(a*x)^3/(3*a)-tan(a*x)/a+x,false,false,[]] [519,tan(a*x)^n,tan(a*x)^(n-1)/(a*(n-1))-'integrate(tan(a*x)^(n-2),x),false,false,[]] [520,cot(a*x)^2,-cot(a*x)/a-x,false,false,[]] [521,cot(a*x)^3,-log(sin(a*x))/a-cot(a*x)^2/(2*a),false,false,[]] [522,cot(a*x)^4,-cot(a*x)^3/(3*a)+cot(a*x)/a+x,false,false,[]] [523,cot(a*x)^n,-'integrate(cot(a*x)^(n-2),x)-cot(a*x)^(n-1)/(a*(n-1)),false,false,[]] [524,x*csc(a*x)^2,'integrate(x*csc(a*x)^2,x),false,false,[]] [525,x*csc(a*x)^2,log(sin(a*x))/a^2-x*cot(a*x)/a,false,false,[]] [526,x/sin(a*x)^n,'integrate(x*csc(a*x)^n,x),false,false,[]] [527,x/sin(a*x)^n,(n-2)*('integrate(x*sin(a*x)^(2-n),x))/(n-1)-sin(a*x)^(2-n)/(a^2*(n-2)*(n-1))-x*cos(a*x)*sin(a*x)^(1-n)/(a*(n-1)),false,false,[]] [528,x*sec(a*x)^2,'integrate(x*sec(a*x)^2,x),false,false,[]] [529,x*sec(a*x)^2,log(cos(a*x))/a^2+x*tan(a*x)/a,false,false,[]] [530,x/cos(a*x)^n,'integrate(x*sec(a*x)^n,x),false,false,[]] [531,x/cos(a*x)^n,(n-2)*('integrate(x*cos(a*x)^(2-n),x))/(n-1)+x*cos(a*x)^(1-n)*sin(a*x)/(a*(n-1))-cos(a*x)^(2-n)/(a^2*(n-2)*(n-1)),false,false,[]] [532,sin(a*x)/sqrt(b^2*sin(a*x)^2+1),-asin(b*cos(a*x)/sqrt(b^2+1))/(a*b),false,false,[]] [533,sin(a*x)/sqrt(1-b^2*sin(a*x)^2),-log(sqrt(1-b^2*sin(a*x)^2)+b*cos(a*x))/(a*b),false,false,[]] [534,sin(a*x)*sqrt(b^2*sin(a*x)^2+1),-(b^2+1)*asin(b*cos(a*x)/sqrt(b^2+1))/(2*a*b)-cos(a*x)*sqrt(b^2*sin(a*x)^2+1)/(2*a),false,false,[]] [535,sin(a*x)*sqrt(1-b^2*sin(a*x)^2),-(1-b^2)*log(sqrt(1-b^2*sin(a*x)^2)+b*cos(a*x))/(2*a*b)-cos(a*x)*sqrt(1-b^2*sin(a*x)^2)/(2*a),false,false,[]] [536,cos(a*x)/sqrt(b^2*sin(a*x)^2+1),log(sqrt(b^2*sin(a*x)^2+1)+b*sin(a*x))/(a*b),false,false,[]] [537,cos(a*x)/sqrt(1-b^2*sin(a*x)^2),asin(b*sin(a*x))/(a*b),false,false,[]] [538,cos(a*x)*sqrt(b^2*sin(a*x)^2+1),log(sqrt(b^2*sin(a*x)^2+1)+b*sin(a*x))/(2*a*b)+sin(a*x)*sqrt(b^2*sin(a*x)^2+1)/(2*a),false,false,[]] [539,cos(a*x)*sqrt(1-b^2*sin(a*x)^2),asin(b*sin(a*x))/(2*a*b)+sin(a*x)*sqrt(1-b^2*sin(a*x)^2)/(2*a),false,false,[]] [540,1/sqrt(b*tan(c*x)^2+a),asin(sqrt((a-b)*sin(c*x)/a))/((a-b)^(1/4)*sqrt(c)),false,false,[\<_\<\=(%pi*(2*k-1)/2,x,%pi*(2*k+1)/2)]] [541,1/sqrt(b*tan(c*x)^2+a),-asin(sqrt(a-b)*sqrt(sin(c*x)/a))/((a-b)^(1/4)*sqrt(c)),false,false,[\<_\<\=(%pi*(2*k+1)/2,x,%pi*(2*k+3)/2)]] [542,cos(x)^(2*n),2^(1-2*n)*('sum(binomial(2*n,k)*sin(2*(n-k)*x)/(n-k),k,0,n-1))/2+binomial(2*n,n)*x/4^n,false,false,[]] [543,sin(x)^(2*n),binomial(2*n,n)*x/4^n+2^(1-2*n)*('sum(binomial(2*n,k)*sin(2*(n-k)*(%pi/2-x))/(k-n),k,0,n-1))/2,false,false,[]] [544,cos(x)^(2*n+1),('sum(binomial(2*n+1,k)*sin((2*n-2*k+1)*x)/(2*n-2*k+1),k,0,n))/4^n,false,false,[]] [545,cos(x)^(2*n+1),('sum(binomial(2*n+1,k)*sin((2*n-2*k+1)*(%pi/2-x))/(2*n-2*k+1),k,0,n))/4^n,false,false,[]] [546,asin(a*x),x*asin(a*x)+sqrt(1-a^2*x^2)/a,false,false,[]] [547,acos(a*x),x*acos(a*x)-sqrt(1-a^2*x^2)/a,false,false,[]] [548,atan(a*x),x*atan(a*x)-log(a^2*x^2+1)/(2*a),false,false,[]] [549,acot(a*x),log(a^2*x^2+1)/(2*a)+x*acot(a*x),false,false,[]] [550,asec(a*x),x*asec(a*x)-log(sqrt(a^2*x^2-1)+a*x)/a,false,false,[]] [551,acsc(a*x),log(sqrt(a^2*x^2-1)+a*x)/a+x*acsc(a*x),false,false,[]] [552,asin(x/a),x*asin(x/a)+sqrt(a^2-x^2),false,false,[a > 0]] [553,acos(x/a),x*acos(x/a)-sqrt(a^2-x^2),false,false,[a > 0]] [554,atan(x/a),x*atan(x/a)-a*log(x^2+a^2)/2,false,false,[]] [555,acot(x/a),a*log(x^2+a^2)/2+x*acot(x/a),false,false,[]] [556,x*asin(a*x),((2*a^2*x^2-1)*asin(a*x)+a*x*sqrt(1-a^2*x^2))/(4*a^2),false,false,[]] [557,x*acos(a*x),((2*a^2*x^2-1)*acos(a*x)-a*x*sqrt(1-a^2*x^2))/(4*a^2),false,false,[]] [558,x^n*asin(a*x),x^(n+1)*asin(a*x)/(n+1)-a*('integrate(x^(n+1)/sqrt(1-a^2*x^2),x))/(n+1),false,false,[n # -1]] [559,x^n*acos(a*x),a*('integrate(x^(n+1)/sqrt(1-a^2*x^2),x))/(n+1)+x^(n+1)*acos(a*x)/(n+1),false,false,[n # -1]] [560,x*atan(a*x),(a^2*x^2+1)*atan(a*x)/(2*a^2)-x/(2*a),false,false,[]] [561,x^n*atan(a*x),x^(n+1)*atan(a*x)/(n+1)-a*('integrate(x^(n+1)/(a^2*x^2+1),x))/(n+1),false,false,[]] [562,x*acot(a*x),(a^2*x^2+1)*acot(a*x)/(2*a^2)+x/(2*a),false,false,[]] [563,x^n*acot(a*x),a*('integrate(x^(n+1)/(a^2*x^2+1),x))/(n+1)+x^(n+1)*acot(a*x)/(n+1),false,false,[]] [564,asin(a*x)/x^2,a*log((1-sqrt(1-a^2*x^2))/x)-asin(a*x)/x,false,false,[]] [565,acos(a*x)/x^2,a*log((sqrt(1-a^2*x^2)+1)/x)-acos(a*x)/x,false,false,[]] [566,atan(a*x)/x^2,-a*log((a^2*x^2+1)/x^2)/2-atan(a*x)/x,false,false,[]] [567,acot(a*x)/x^2,-a*log(x^2/(a^2*x^2+1))/2-acot(a*x)/x,false,false,[]] [568,asin(a*x)^2,x*asin(a*x)^2+2*sqrt(1-a^2*x^2)*asin(a*x)/a-2*x,false,false,[]] [569,acos(a*x)^2,x*acos(a*x)^2-2*sqrt(1-a^2*x^2)*acos(a*x)/a-2*x,false,false,[]] [571,asin(a*x)^n,n!*x*'sum((-1)^r*asin(a*x)^(n-2*r)/(n-2*r)!,r,0,floor(n/2))+n!*sqrt(1-a^2*x^2)*('sum((-1)^r*asin(a*x)^(-2*r+n-1)/(-2*r+n-1)!,r,0,floor((n-1)/2)))/a,false,false,[]] [572,acos(a*x)^n,-(n-1)*n*'integrate(acos(a*x)^(n-2),x)+x*acos(a*x)^n-n*sqrt(1-a^2*x^2)*acos(a*x)^(n-1)/a,false,false,[]] [573,acos(a*x)^n,n!*x*'sum((-1)^r*acos(a*x)^(n-2*r)/(n-2*r)!,r,0,floor(n/2))-n!*sqrt(1-a^2*x^2)*('sum((-1)^r*acos(a*x)^(-2*r+n-1)/(-2*r+n-1)!,r,0,floor((n-1)/2)))/a,false,false,[]] [574,asin(a*x)/sqrt(1-a^2*x^2),asin(a*x)^2/(2*a),false,false,[]] [575,x^n*asin(a*x)/sqrt(1-a^2*x^2),(n-1)*('integrate(x^(n-2)*asin(a*x)/sqrt(1-a^2*x^2),x))/(a^2*n)-x^(n-1)*sqrt(1-a^2*x^2)*asin(a*x)/(a^2*n)+x^n/(a*n^2),false,false,[]] [576,acos(a*x)/sqrt(1-a^2*x^2),-acos(a*x)^2/(2*a),false,false,[]] [577,x^n*acos(a*x)/sqrt(1-a^2*x^2),(n-1)*('integrate(x^(n-2)*acos(a*x)/sqrt(1-a^2*x^2),x))/(a^2*n)-x^(n-1)*sqrt(1-a^2*x^2)*acos(a*x)/(a^2*n)-x^n/(a*n^2),false,false,[]] [578,atan(a*x)/(a^2*x^2+1),atan(a*x)^2/(2*a),false,false,[]] [579,acot(a*x)/(a^2*x^2+1),-acot(a*x)^2/(2*a),false,false,[]] [580,x*asec(a*x),x^2*asec(a*x)/2-sqrt(a^2*x^2-1)/(2*a^2),false,false,[]] [581,x^n*asec(a*x),x^(n+1)*asec(a*x)/(n+1)-('integrate(x^n/sqrt(a^2*x^2-1),x))/(n+1),false,false,[]] [582,asec(a*x)/x^2,sqrt(a^2*x^2-1)/x-asec(a*x)/x,false,false,[]] [583,x*acsc(a*x),x^2*acsc(a*x)/2+sqrt(a^2*x^2-1)/(2*a^2),false,false,[]] [584,x^n*acsc(a*x),('integrate(x^n/sqrt(a^2*x^2-1),x))/(n+1)+x^(n+1)*acsc(a*x)/(n+1),false,false,[]] [585,acsc(a*x)/x^2,-acsc(a*x)/x-sqrt(a^2*x^2-1)/x,false,false,[]] [586,log(x),x*log(x)-x,false,false,[]] [587,x*log(x),x^2*log(x)/2-x^2/4,false,false,[]] [588,x^2*log(x),x^3*log(x)/3-x^3/9,false,false,[]] [589,x^n*log(x),x^(n+1)*log(x)/(n+1)-x^(n+1)/(n+1)^2,false,false,[]] [590,log(x)^2,x*log(x)^2-2*x*log(x)+2*x,false,false,[]] [591,log(x)^n,x*log(x)^n-n*'integrate(log(x)^(n-1),x),false,false,[n # -1]] [592,log(x)^n,(-1)^n*n!*x*'sum((-log(x))^r/r!,r,0,n),false,false,[n # -1]] [593,log(x)^n/x,log(x)^(n+1)/(n+1),false,false,[]] [594,1/log(x),?etc+log(log(x))+log(x)^3/18+log(x)^2/4+log(x)+1,false,false,[series(?truncated)]] [595,1/(x*log(x)),log(log(x)),false,false,[]] [596,1/(x*log(x)^n),log(x)^(1-n)/(1-n),false,false,[]] [597,x^m/log(x)^n,(m+1)*('integrate(x^m*log(x)^(1-n),x))/(n-1)+x^(m+1)*log(x)^(1-n)/(1-n),false,false,[]] [598,x^m*log(x)^n,x^(m+1)*log(x)^n/(m+1)-n*('integrate(x^m*log(x)^(n-1),x))/(m+1),false,false,[]] [600,x^p*cos(b*log(x)),x^(p+1)*(b*sin(b*log(x))+(p+1)*cos(b*log(x)))/((p+1)^2+b^2)+c,false,false,[]] [601,x^p*sin(b*log(x)),x^(p+1)*((p+1)*sin(b*log(x))-b*cos(b*log(x)))/((p+1)^2+b^2)+c,false,false,[]] [602,log(a*x+b),(a*x+b)*log(a*x+b)/a-x,false,false,[]] [603,log(a*x+b)/x^2,a*log(x)/b-(a*x+b)*log(a*x+b)/(b*x),false,false,[]] [604,x^m*log(a*x+b),(x^(m+1)-(-b/a)^(m+1))*log(a*x+b)/(m+1)-(-b/a)^(m+1)*('sum((-a*x/b)^r/r,r,1,m+1))/(m+1),false,false,[]] [605,log(a*x+b)/x^m,(-a/b)^(m-1)*log((a*x+b)/x)/(m-1)-x^(1-m)*log(a*x+b)/(m-1)+(-a/b)^(m-1)*('sum((-b/(a*x))^r/r,r,1,m-2))/(m-1),false,false,[m > 2]] [606,log((x+a)/(x-a)),(x+a)*log(x+a)-(x-a)*log(x-a),false,false,[]] [607,x^m*log((x+a)/(x-a)),2*a^(m+1)*('sum((x/a)^(-2*r+m+2)/(-2*r+m+2),r,1,floor((m+1)/2)))/(m+1)+(x^(m+1)-(-a)^(m+1))*log(x+a)/(m+1)-(x^(m+1)-a^(m+1))*log(x-a)/(m+1),false,false,[]] [608,log((x+a)/(x-a))/x^2,log((x-a)/(x+a))/x-log((x^2-a^2)/x^2)/a,false,false,[]] [609,log(c*x^2+b*x+a),result_for(609),false,false,[b^2-4*a*c < 0]] [610,log(c*x^2+b*x+a),result_for(610),false,false,[b^2-4*a*c > 0]] [611,x^n*log(c*x^2+b*x+a),-2*c*('integrate(x^(n+2)/(c*x^2+b*x+a),x))/(n+1)-b*('integrate(x^(n+1)/(c*x^2+b*x+a),x))/(n+1)+x^(n+1)*log(c*x^2+b*x+a)/(n+1),false,false,[]] [612,log(x^2+a^2),x*log(x^2+a^2)+2*a*atan(x/a)-2*x,false,false,[]] [613,log(x^2-a^2),x*log(x^2-a^2)+a*log((x+a)/(x-a))-2*x,false,false,[]] [614,x*log(x^2+a^2),(x^2+a^2)*log(x^2+a^2)/2-x^2/2,false,false,[]] [615,log(sqrt(x^2+a^2*plus_minus)+x),x*log(sqrt(x^2+a^2*plus_minus)+x)-sqrt(x^2+a^2*plus_minus),false,false,[]] [616,x*log(sqrt(x^2+a^2*plus_minus)+x),(x^2/2+a^2*plus_minus/4)*log(sqrt(x^2+a^2*plus_minus)+x)-x*sqrt(x^2+a^2*plus_minus)/4,false,false,[]] [617,x^m*log(sqrt(x^2+a^2*plus_minus)+x),x^(m+1)*log(sqrt(x^2+a^2*plus_minus)+x)/(m+1)-('integrate(x^(m+1)/sqrt(x^2+a^2*plus_minus),x))/(m+1),false,false,[]] [618,log(sqrt(x^2+a^2)+x)/x^2,-log((sqrt(x^2+a^2)+a)/x)/a-log(sqrt(x^2+a^2)+x)/x,false,false,[]] [619,log(sqrt(x^2-a^2)+x)/x^2,asec(x/a)/abs(a)-log(sqrt(x^2-a^2)+x)/x,false,false,[]] [620,x^n*log(x^2-a^2),(-2*'sum(a^(2*r)*x^(-2*r+n+1)/(-2*r+n+1),r,0,floor(n/2))+x^(n+1)*log(x^2-a^2)-(-a)^(n+1)*log(x+a)-a^(n+1)*log(x-a))/(n+1),false,false,[]] [621,%e^x,%e^x,false,false,[]] [622,%e^-x,-%e^-x,false,false,[]] [623,%e^(a*x),%e^(a*x)/a,false,false,[]] [624,x*%e^(a*x),(a*x-1)*%e^(a*x)/a^2,false,false,[]] [625,x^m*%e^(a*x),x^m*%e^(a*x)/a-m*('integrate(x^(m-1)*%e^(a*x),x))/a,false,false,[]] [626,x^m*%e^(a*x),m!*%e^(a*x)*'sum(a^(-r-1)*(-1)^r*x^(m-r)/(m-r)!,r,0,m),false,false,[]] [627,%e^(a*x)/x,?etc+log(x)+a^3*x^3/18+a^2*x^2/4+a*x+1,false,false,[series(?truncated)]] [628,%e^(a*x)/x^m,a*('integrate(x^(1-m)*%e^(a*x),x))/(m-1)+x^(1-m)*%e^(a*x)/(1-m),false,false,[]] [629,%e^(a*x)*log(x),%e^(a*x)*log(x)/a-('integrate(%e^(a*x)/x,x))/a,false,false,[]] [630,1/(%e^x+1),x-log(%e^x+1),false,false,[]] [631,1/(%e^x+1),log(%e^x/(%e^x+1)),false,false,[]] [632,1/(b*%e^(p*x)+a),x/a-log(b*%e^(p*x)+a)/(a*p),false,false,[]] [633,1/(a*%e^(m*x)+b*%e^-(m*x)),atan(sqrt(a/b)*%e^(m*x))/(sqrt(a*b)*m),false,false,[a > 0,b > 0]] [634,1/(a*%e^(m*x)-b*%e^-(m*x)),log((sqrt(a)*%e^(m*x)-sqrt(b))/(sqrt(a)*%e^(m*x)+sqrt(b)))/(2*sqrt(a*b)*m),false,false,[]] [635,1/(a*%e^(m*x)-b*%e^-(m*x)),-atanh(sqrt(a/b)*%e^(m*x))/(sqrt(a*b)*m),false,false,[a > 0,b > 0]] [636,a^x-1/a^x,(a^x+1/a^x)/log(a),false,false,[]] [637,%e^(a*x)/(c*%e^(a*x)+b),log(c*%e^(a*x)+b)/(a*c),false,false,[]] [638,x*%e^(a*x)/(a*x+1)^2,%e^(a*x)/(a^2*(a*x+1)),false,false,[]] [639,x*%e^-x^2,-%e^-x^2/2,false,false,[]] [640,%e^(a*x)*sin(b*x),%e^(a*x)*(a*sin(b*x)-b*cos(b*x))/(b^2+a^2),false,false,[]] [641,%e^(a*x)*sin(b*x)*sin(c*x),%e^(a*x)*((b-c)*sin((b-c)*x)+a*cos((b-c)*x))/(2*((b-c)^2+a^2))-%e^(a*x)*((c+b)*sin((c+b)*x)+a*cos((c+b)*x))/(2*((c+b)^2+a^2)),false,false,[]] [642,%e^(a*x)*sin(b*x)*cos(c*x),%e^(a*x)*(a*sin((c+b)*x)-(c+b)*cos((c+b)*x))/(2*((c+b)^2+a^2))+%e^(a*x)*(a*sin((b-c)*x)-(b-c)*cos((b-c)*x))/(2*((b-c)^2+a^2)),false,false,[]] [643,%e^(a*x)*sin(b*x)*sin(b*x+c),cos(c)*%e^(a*x)/(2*a)-%e^(a*x)*(2*b*sin(2*b*x+c)+a*cos(2*b*x+c))/(2*(4*b^2+a^2)),false,false,[]] [644,%e^(a*x)*sin(b*x)*cos(b*x+c),%e^(a*x)*(a*sin(2*b*x+c)-2*b*cos(2*b*x+c))/(2*(4*b^2+a^2))-sin(c)*%e^(a*x)/(2*a),false,false,[]] [645,%e^(a*x)*cos(b*x),%e^(a*x)*(b*sin(b*x)+a*cos(b*x))/(b^2+a^2),false,false,[]] [646,%e^(a*x)*cos(b*x)*cos(c*x),%e^(a*x)*((c+b)*sin((c+b)*x)+a*cos((c+b)*x))/(2*((c+b)^2+a^2))+%e^(a*x)*((b-c)*sin((b-c)*x)+a*cos((b-c)*x))/(2*((b-c)^2+a^2)),false,false,[]] [647,%e^(a*x)*cos(b*x)*cos(b*x+c),%e^(a*x)*(2*b*sin(2*b*x+c)+a*cos(2*b*x+c))/(2*(4*b^2+a^2))+cos(c)*%e^(a*x)/(2*a),false,false,[]] [648,%e^(a*x)*cos(b*x)*sin(b*x+c),%e^(a*x)*(a*sin(2*b*x+c)-2*b*cos(2*b*x+c))/(2*(4*b^2+a^2))+sin(c)*%e^(a*x)/(2*a),false,false,[]] [649,%e^(a*x)*sin(b*x)^n,(b^2*(n-1)*n*'integrate(%e^(a*x)*sin(b*x)^(n-2),x)+%e^(a*x)*sin(b*x)^(n-1)*(a*sin(b*x)-b*n*cos(b*x)))/(b^2*n^2+a^2),false,false,[]] [650,%e^(a*x)*cos(b*x)^n,(b^2*(n-1)*n*'integrate(%e^(a*x)*cos(b*x)^(n-2),x)+%e^(a*x)*cos(b*x)^(n-1)*(b*n*sin(b*x)+a*cos(b*x)))/(b^2*n^2+a^2),false,false,[]] [651,x^m*%e^x*sin(x),-m*('integrate(x^(m-1)*%e^x*sin(x),x))/2+m*('integrate(x^(m-1)*%e^x*cos(x),x))/2+x^m*%e^x*(sin(x)-cos(x))/2,false,false,[]] [652,x^m*%e^(a*x)*sin(b*x),x^m*%e^(a*x)*(a*sin(b*x)-b*cos(b*x))/(b^2+a^2)-m*('integrate(x^(m-1)*%e^(a*x)*(a*sin(b*x)-b*cos(b*x)),x))/(b^2+a^2),false,false,[]] [653,x^m*%e^x*cos(x),-m*('integrate(x^(m-1)*%e^x*sin(x),x))/2-m*('integrate(x^(m-1)*%e^x*cos(x),x))/2+x^m*%e^x*(sin(x)+cos(x))/2,false,false,[]] [654,x^m*%e^(a*x)*cos(b*x),x^m*%e^(a*x)*(b*sin(b*x)+a*cos(b*x))/(b^2+a^2)-m*('integrate(x^(m-1)*%e^(a*x)*(b*sin(b*x)+a*cos(b*x)),x))/(b^2+a^2),false,false,[]] [655,%e^(a*x)*cos(x)^m*sin(x)^n,(m-1)*(n+m)*('integrate(%e^(a*x)*cos(x)^(m-2)*sin(x)^n,x))/((n+m)^2+a^2)-a*n*('integrate(%e^(a*x)*cos(x)^(m-1)*sin(x)^(n-1),x))/((n+m)^2+a^2)+%e^(a*x)*cos(x)^(m-1)*sin(x)^n*((n+m)*sin(x)+a*cos(x))/((n+m)^2+a^2),false,false,[]] [656,%e^(a*x)*cos(x)^m*sin(x)^n,a*m*('integrate(%e^(a*x)*cos(x)^(m-1)*sin(x)^(n-1),x))/((n+m)^2+a^2)+(n-1)*(n+m)*('integrate(%e^(a*x)*cos(x)^m*sin(x)^(n-2),x))/((n+m)^2+a^2)+%e^(a*x)*cos(x)^m*sin(x)^(n-1)*(a*sin(x)-(n+m)*cos(x))/((n+m)^2+a^2),false,false,[]] [657,%e^(a*x)*cos(x)^m*sin(x)^n,(m-1)*m*('integrate(%e^(a*x)*cos(x)^(m-2)*sin(x)^n,x))/((n+m)^2+a^2)+(n-1)*n*('integrate(%e^(a*x)*cos(x)^m*sin(x)^(n-2),x))/((n+m)^2+a^2)+%e^(a*x)*cos(x)^(m-1)*sin(x)^(n-1)*(m*sin(x)^2+a*cos(x)*sin(x)-n*cos(x)^2)/((n+m)^2+a^2),false,false,[]] [658,%e^(a*x)*cos(x)^m*sin(x)^n,(n-m)*(n+m-1)*('integrate(%e^(a*x)*cos(x)^m*sin(x)^(n-2),x))/((n+m)^2+a^2)+(m-1)*m*('integrate(%e^(a*x)*cos(x)^(m-2)*sin(x)^(n-2),x))/((n+m)^2+a^2)+%e^(a*x)*cos(x)^(m-1)*sin(x)^(n-1)*(m*sin(x)^2+a*cos(x)*sin(x)-n*cos(x)^2)/((n+m)^2+a^2),false,false,[]] [659,x*%e^(a*x)*sin(b*x),x*%e^(a*x)*(a*sin(b*x)-b*cos(b*x))/(b^2+a^2)-%e^(a*x)*((a^2-b^2)*sin(b*x)-2*a*b*cos(b*x))/(b^2+a^2)^2,false,false,[]] [660,x*%e^(a*x)*cos(b*x),x*%e^(a*x)*(b*sin(b*x)+a*cos(b*x))/(b^2+a^2)-%e^(a*x)*(2*a*b*sin(b*x)+(a^2-b^2)*cos(b*x))/(b^2+a^2)^2,false,false,[]] [661,%e^(a*x)/sin(x)^n,((n-2)^2+a^2)*('integrate(%e^(a*x)*sin(x)^(2-n),x))/((n-2)*(n-1))-%e^(a*x)*sin(x)^(1-n)*(a*sin(x)+(n-2)*cos(x))/((n-2)*(n-1)),false,false,[]] [662,%e^(a*x)/cos(x)^n,((n-2)^2+a^2)*('integrate(%e^(a*x)*cos(x)^(2-n),x))/((n-2)*(n-1))-%e^(a*x)*cos(x)^(1-n)*(a*cos(x)-(n-2)*sin(x))/((n-2)*(n-1)),false,false,[]] [663,%e^(a*x)*tan(x)^n,-a*('integrate(%e^(a*x)*tan(x)^(n-1),x))/(n-1)-'integrate(%e^(a*x)*tan(x)^(n-2),x)+%e^(a*x)*tan(x)^(n-1)/(n-1),false,false,[]] [664,sinh(x),cosh(x),false,false,[]] [665,cosh(x),sinh(x),false,false,[]] [666,tanh(x),log(cosh(x)),false,false,[]] [667,coth(x),log(sinh(x)),false,false,[]] [668,sech(x),atan(sinh(x)),false,false,[]] [669,csch(x),log(tanh(x/2)),false,false,[]] [670,x*sinh(x),x*cosh(x)-sinh(x),false,false,[]] [671,x^n*sinh(x),x^n*cosh(x)-n*'integrate(x^(n-1)*cosh(x),x),false,false,[]] [672,x*cosh(x),x*sinh(x)-cosh(x),false,false,[]] [673,x^n*cosh(x),x^n*sinh(x)-n*'integrate(x^(n-1)*sinh(x),x),false,false,[]] [674,sech(x)*tanh(x),-sech(x),false,false,[]] [675,coth(x)*csch(x),-csch(x),false,false,[]] [676,sinh(x)^2,sinh(2*x)/4-x/2,false,false,[]] [677,cosh(x)^n*sinh(x)^m,(n-1)*('integrate(cosh(x)^(n-2)*sinh(x)^m,x))/(n+m)+cosh(x)^(n-1)*sinh(x)^(m+1)/(n+m),false,false,[n+m # 0]] [678,cosh(x)^n*sinh(x)^m,cosh(x)^(n+1)*sinh(x)^(m-1)/(n+m)-(m-1)*('integrate(cosh(x)^n*sinh(x)^(m-2),x))/(n+m),false,false,[n+m # 0]] [679,1/(cosh(x)^n*sinh(x)^m),-(n+m-2)*('integrate(sinh(x)^(2-m)/cosh(x)^n,x))/(m-1)-cosh(x)^(1-n)*sinh(x)^(1-m)/(m-1),false,false,[m # 1]] [680,1/(cosh(x)^n*sinh(x)^m),(n+m-2)*('integrate(cosh(x)^(2-n)/sinh(x)^m,x))/(n-1)+cosh(x)^(1-n)*sinh(x)^(1-m)/(n-1),false,false,[n # 1]] [681,tanh(x)^2,x-tanh(x),false,false,[]] [682,tanh(x)^n,'integrate(tanh(x)^(n-2),x)-tanh(x)^(n-1)/(n-1),false,false,[n # 1]] [683,sech(x)^2,tanh(x),false,false,[]] [684,cosh(x)^2,sinh(2*x)/4+x/2,false,false,[]] [685,coth(x)^2,x-coth(x),false,false,[]] [686,coth(x)^n,'integrate(coth(x)^(n-2),x)-coth(x)^(n-1)/(n-1),false,false,[n # 1]] [687,csch(x)^2,-coth(x),false,false,[]] [688,sinh(m*x)*sinh(n*x),sinh((n+m)*x)/(2*(n+m))-sinh((m-n)*x)/(2*(m-n)),false,false,[m^2 # n^2]] [689,cosh(m*x)*cosh(n*x),sinh((n+m)*x)/(2*(n+m))+sinh((m-n)*x)/(2*(m-n)),false,false,[m^2 # n^2]] [690,sinh(m*x)*cosh(n*x),cosh((n+m)*x)/(2*(n+m))+cosh((m-n)*x)/(2*(m-n)),false,false,[m^2 # n^2]] [691,asinh(x/a),x*asinh(x/a)-sqrt(x^2+a^2),false,false,[a > 0]] [692,x*asinh(x/a),(x^2/2+a^2/4)*asinh(x/a)-x*sqrt(x^2+a^2)/4,false,false,[a > 0]] [693,x^n*asinh(x),x^(n+1)*asinh(x)/(n+1)-('integrate(x^(n+1)/sqrt(x^2+1),x))/(n+1),false,false,[n # -1]] [696,x*%acosh(x/a),(x^2/2-a^2/4)*%acosh(x/a)-x*sqrt(x^2-a^2)/4,false,false,[]] [697,x^n*%acosh(x),x^(n+1)*%acosh(x)/(n+1)-('integrate(x^(n+1)/sqrt(x^2-1),x))/(n+1),false,false,[n # -1]] 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[710,x^(p+1)*bessel_j(x,p),bessel_j(p+1,x)*x^(p+1),false,false,[]] [711,x^(1-p)*bessel_j(x,p),-bessel_j(p-1,x)*x^(1-p),false,false,[]] [712,x*bessel_j(a*x,p)^2,(bessel_j(p,a*x)^2-bessel_j(p-1,a*x)*bessel_j(p+1,a*x))*x^2/2,false,false,[]] [713,bessel_j(x,1),-bessel_j(0,x),false,false,[]] [714,x*bessel_j(x,0),bessel_j(1,x)*x,false,false,[]] [715,x^(n-1)*%e^-x,gamma(n),0,inf,[]] [716,x^n/p^x,n!*log(p)^(-n-1),0,inf,[p > 0]] [717,x^(n-1)*%e^-((a+1)*x),gamma(n)/(a+1)^n,0,inf,[n > 0,a > -1]] [718,x^m*(-log(x))^n,(m+1)^(-n-1)*gamma(n+1),0,1,[m > -1,n > -1]] [719,(1-x)^(n-1)*x^(m-1),gamma(m)*gamma(n)/gamma(n+m),0,1,[n > 0,m > 0]] [720,x^(m-1)*(x+1)^(-n-m),gamma(m)*gamma(n)/gamma(n+m),0,inf,[n > 0,m > 0]] [722,1/x^m,1/(m-1),1,inf,[m > 1]] [723,1/(x^p*(x+1)),%pi*csc(%pi*p),0,inf,[0 < p,p < 1]] [724,1/((1-x)*x^p),-%pi*cot(%pi*p),0,inf,[0 < p,p < 1]] [725,x^p/(1-x)^p,%pi*p*csc(%pi*p),0,1,[p < 1,-p < 1]] [726,(1-x)^(-p-1)*x^p,-%pi*csc(%pi*p),0,1,[-1 < p,p < 0]] [727,(1-x)^p*x^(-p-1),-%pi*csc(%pi*p),0,1,[-1 < p,p < 0]] [728,x^(p-1)/(x+1),%pi*csc(%pi*p),0,inf,[0 < p,p < 1]] [729,x^(m-1)/(x^n+1),%pi*csc(%pi*m/n)/n,0,inf,[0 < m,m < n]] [730,x^a/(x^b+m)^c,gamma((a+1)/b)*gamma(c-(a+1)/b)*m^((-b*c+a+1)/b)/(b*gamma(c)),0,inf,[a > -1,b > 0,m > 0,c > (a+1)/b]] [731,1/(sqrt(x)*(x+1)),%pi,0,inf,[]] [734,1/(x^2+a^2),-%pi/(2*abs(a)),0,inf,[a # 0]] [736,(1-x^2)^(n/2)/2,%pi*factorial_double(n)/(2*factorial_double(n+1)),0,1,[]] [737,x^m*(1-x^2)^(n/2),gamma((m+1)/2)*gamma((n+2)/2)/(2*gamma((n+m+3)/2)),0,1,[m > -1,n > -2]] [738,sin(x)^n,sqrt(%pi)*gamma((n+1)/2)/(2*gamma((n+2)/2)),0,%pi/2,[n > -1]] [739,cos(x)^n,sqrt(%pi)*gamma((n+1)/2)/(2*gamma((n+2)/2)),0,%pi/2,[n > -1]] [740,sin(x)^n,%pi*factorial_double(n-1)/(2*factorial_double(n)),0,%pi/2,[n # 0]] [741,cos(x)^n,%pi*factorial_double(n-1)/(2*factorial_double(n)),0,%pi/2,[n # 0]] [742,sin(x)^n,factorial_double(n-1)/factorial_double(n),0,%pi/2,[n # 1]] 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[766,sin(p*x)*cos(q*x)/x,%pi/2,0,inf,[p > q,q > 0]] [767,cos(p*x)*sin(p*x)/x,%pi/4,0,inf,[p > 0]] [768,cos(m*x)/(x^2+a^2),%pi*%e^-abs(a*m)/(2*abs(a)),0,inf,[]] [769,cos(x^2),2^-(3/2)*sqrt(%pi),0,inf,[]] [770,sin(x^2),2^-(3/2)*sqrt(%pi),0,inf,[]] [771,sin(a*x^n),gamma(1/n)*sin(%pi/(2*n))/(a^(1/n)*n),0,inf,[n > 1]] [772,cos(a*x^n),gamma(1/n)*cos(%pi/(2*n))/(a^(1/n)*n),0,inf,[n > 1]] [773,sin(x)/sqrt(x),sqrt(%pi)/sqrt(2),0,inf,[]] [774,cos(x)/sqrt(x),sqrt(%pi)/sqrt(2),0,inf,[]] [775,sin(x)^3/x,%pi/4,0,inf,[]] [776,sin(x)^3/x^2,3*log(3)/4,0,inf,[]] [777,sin(x)^3/x^3,3*%pi/8,0,inf,[]] [778,sin(x)^4/x^4,%pi/3,0,inf,[]] [779,1/(a*cos(x)+1),acos(a)/sqrt(1-a^2),0,%pi/2,[a < 1,-a < 1]] [780,1/(b*cos(x)+a),%pi/sqrt(a^2-b^2),0,%pi,[a > b,b >= 0]] [781,1/(a*cos(x)+1),2*%pi/sqrt(1-a^2),0,2*%pi,[a < 1,-a < 1]] [782,(cos(a*x)-cos(b*x))/x,log(abs(b/a)),0,inf,[]] [783,1/(a^2*sin(x)^2+b^2*cos(x)^2),%pi/(2*abs(a*b)),0,%pi/2,[]] [784,1/(a^2*sin(x)^2+b^2*cos(x)^2)^2,%pi*(b^2+a^2)/(4*a^3*b^3),0,%pi/2,[a > 0,b > 0]] [785,cos(x)^(m-1)*sin(x)^(n-1),beta(n/2,m/2)/2,0,%pi/2,[]] [786,sin(x)^(2*n+1),factorial_double(2*n)/factorial_double(2*n+1),0,%pi/2,[]] [787,sin(x)^(2*n),%pi*factorial_double(2*n-1)/(2*factorial_double(2*n)),0,%pi/2,[]] [788,x*csc(x),42082/11025,0,%pi/2,[series(?truncated)]] [789,1/(tan(x)^m+1),%pi/4,0,%pi/2,[]] [790,sqrt(cos(x)),2*sqrt(2)*%pi^(3/2)/gamma(1/4)^2,0,%pi/2,[]] [791,tan(x)^h,%pi*sec(%pi*h/2)/2,0,%pi/2,[0 < h,h < 1]] [792,(atan(a*x)-atan(b*x))/x,%pi*log(a/b)/2,0,%pi/2,[a > 0,b > 0]] [793,%e^-(a*x),1/a,0,inf,[a > 0]] [794,(%e^-(a*x)-%e^-(b*x))/x,log(b/a),0,inf,[a > 0,b > 0]] [795,x^n*%e^-(a*x),a^(-n-1)*gamma(n+1),0,inf,[a > 0,n > -1]] [796,x^n*%e^-(a*x),a^(-n-1)*n!,0,inf,[a > 0]] [797,x^n*%e^-(a*x^p),gamma((n+1)/p)/(a^((n+1)/p)*p),0,inf,[a > 0,p > 0,n > -1]] [798,%e^-(a^2*x^2),sqrt(%pi)/(2*a),0,inf,[a > 0]] [799,%e^-(a*x^2),sqrt(%pi)*erf(sqrt(a)*x)/(2*sqrt(a)),false,false,[a > 0]] [800,%e^(a*x^2),sqrt(%pi)*erfi(sqrt(a)*x)/(2*sqrt(a)),false,false,[a > 0]] [800,%e^x^2,sqrt(%pi)*erfi(x)/2,false,false,[]] [801,x*%e^-x^2,1/2,0,inf,[a > 0]] [802,x^2*%e^-x^2,sqrt(%pi)/4,0,inf,[]] [803,x^n*%e^-(a*x^2),sqrt(%pi)*a^(-n/2-1/2)*2^(-n/2-1)*factorial_double(n-1),0,inf,[a > 0,n > 0]] [804,x^(2*n+1)*%e^-(a*x^2),a^(-n-1)*n!/2,0,inf,[a > 0,n > -1]] [805,x^m*%e^-(a*x),a^(-m-1)*m!*(1-%e^-a*'sum(a^r/r!,r,0,m)),0,1,[]] [806,%e^(-x^2-a^2/x^2),sqrt(%pi)*%e^-(2*abs(a))/2,0,inf,[]] [807,%e^(-a*x^2-b/x^2),sqrt(%pi)*%e^-(2*sqrt(a*b))/(2*sqrt(a)),0,inf,[a > 0,b > 0]] [808,sqrt(x)*%e^-(a*x),sqrt(%pi)/(2*a^(3/2)),0,inf,[a > 0]] [809,%e^-(a*x)/sqrt(x),sqrt(%pi)/sqrt(a),0,inf,[a > 0]] [810,%e^-(a*x)*cos(m*x),a/(m^2+a^2),0,inf,[a > 0]] [811,%e^-(a*x)*cos(b*x+c),(a*cos(c)-b*sin(c))/(b^2+a^2),0,inf,[a > 0]] [812,%e^-(a*x)*sin(m*x),m/(m^2+a^2),0,inf,[a > 0]] [813,%e^-(a*x)*sin(b*x+c),(a*sin(c)+b*cos(c))/(b^2+a^2),0,inf,[a > 0]] [814,x*%e^-(a*x)*sin(b*x),2*a*b/(b^2+a^2)^2,0,inf,[a > 0]] [815,x*%e^-(a*x)*cos(b*x),(a^2-b^2)/(b^2+a^2)^2,0,inf,[a > 0]] [816,x^n*%e^-(a*x)*sin(b*x),-(b^2+a^2)^(-n-1)*i*((b*i+a)^(n+1)-(a-b*i)^(n+1))*n!/2,0,inf,[a > 0]] [817,x^n*%e^-(a*x)*cos(b*x),(b^2+a^2)^(-n-1)*((b*i+a)^(n+1)+(a-b*i)^(n+1))*n!/2,0,inf,[a > 0,n > -1]] [818,%e^-(a*x)*sin(x)/x,acot(a),0,inf,[a > 0]] [819,%e^-(a^2*x^2)*cos(b*x),sqrt(%pi)*%e^-(b^2/(4*a^2))/(2*abs(a)),0,inf,[a*b > 0]] [820,x^(b-1)*%e^-(cos(phi)*x)*sin(sin(phi)*x),gamma(b)*sin(b*phi),0,inf,[b > 0,\<_\<(-%pi/2,phi,%pi/2)]] [821,x^(b-1)*%e^-(cos(phi)*x)*cos(sin(phi)*x),gamma(b)*cos(b*phi),0,inf,[b > 0,\<_\<(-%pi/2,phi,%pi/2)]] [822,x^(b-1)*cos(x),gamma(b)*cos(%pi*b/2),0,inf,[0 < b,b < 1]] [823,x^(b-1)*sin(x),gamma(b)*sin(%pi*b/2),0,inf,[0 < b,b < 1]] [824,log(x)^n,(-1)^n*n!,0,1,[n < -1]] [825,sqrt(-log(x)),sqrt(%pi)/2,0,1,[]] [826,1/sqrt(-log(x)),sqrt(%pi),0,1,[]] [827,(-log(x))^n,n!,0,1,[]] [828,log(1-x)*x,-3/4,0,1,[]] [829,x*log(x+1),1/4,0,1,[]] [830,x^m*log(x)^n,(m+1)^(-m-1)*(-1)^n*gamma(n+1),0,1,[m > -1]] [831,log(x)/(x+1),-%pi^2/12,0,1,[]] [832,log(x)/(1-x),-%pi^2/6,0,1,[]] [833,log(x+1)/x,%pi^2/12,0,1,[]] [834,log(1-x)/x,-%pi^2/6,0,1,[]] [835,log(x)*log(x+1),-2*log(2)-%pi^2/12+2,0,1,[]] [836,log(1-x)*log(x),2-%pi^2/6,0,1,[]] [837,log(x)/(1-x^2),-%pi^2/8,0,1,[]] [838,log((x+1)/(1-x))/x,%pi^2/4,0,1,[]] [839,log(x)/sqrt(1-x^2),-%pi*log(2)/2,0,1,[]] [840,x^m*(-log(x))^n,(m+1)^(-n-1)*gamma(n+1),0,1,[m > -1,n > -1]] [841,(x^p-x^q)/log(x),log((p+1)/(q+1)),0,1,[p > -1,q > -1]] [842,1/sqrt(-log(x)),sqrt(%pi),0,1,[]] [843,log((%e^x+1)/(%e^x-1)),%pi^2/4,0,inf,[]] [844,log(sin(x)),-%pi*log(2)/2,0,%pi/2,[]] [845,log(cos(x)),-%pi*log(2)/2,0,%pi/2,[]] [846,log(sec(x)),%pi*log(2)/2,0,%pi/2,[]] [847,log(csc(x)),%pi*log(2)/2,0,%pi/2,[]] [848,x*log(sin(x)),-%pi^2*log(2)/2,0,%pi,[]] [849,sin(x)*log(sin(x)),log(2)-1,0,%pi/2,[]] [850,log(tan(x)),0,0,%pi/2,[]] [851,log(b*plus_minus*cos(x)+a),%pi*log((sqrt(a^2-b^2)+a)/2),0,%pi,[a >= b]] [852,log(-2*a*b*cos(x)+b^2+a^2),2*%pi*log(a),0,%pi,[a >= b,b > 0]] [853,log(-2*a*b*cos(x)+b^2+a^2),2*%pi*log(b),0,%pi,[b >= a,a > 0]] [854,sin(a*x)*csch(b*x),%pi*tanh(%pi*a/(2*abs(b)))/(2*b),0,inf,[]] [855,cos(a*x)*sech(b*x),%pi*sech(%pi*a/(2*b))/(2*b),0,inf,[]] [856,sech(a*x),%pi/(2*abs(a)),0,inf,[]] [857,x*csch(a*x),%pi^2/(4*a^2),0,inf,[a >= 0]] [858,%e^-(a*x)*cosh(b*x),a/(a^2-b^2),0,inf,[b < a,-b < a]] [859,%e^-(a*x)*sinh(b*x),b/(a^2-b^2),0,inf,[b < a,-b < a]] [860,sinh(a*x)/(%e^(b*x)+1),%pi*csc(%pi*a/b)/(2*b)-1/(2*a),0,inf,[b >= 0]] [861,sinh(a*x)/(%e^(b*x)-1),1/(2*a)-%pi*cot(%pi*a/b)/(2*b),0,inf,[b >= 0]] [862,1/sqrt(1-k^2*sin(x)^2),?etc+%pi*('sum(k^(2*r)*(2*r)!^2/(2^(4*r)*r!^2),r,0,inf))/2,0,%pi/2,[k^2 < 1,series(?truncated)]] [863,1/(1-k^2*sin(x)^2)^(3/2),?etc+%pi*('sum(k^(2*r)*(2*r+1)*(2*r)!^2/(2^(4*r)*r!^2),r,0,inf))/4,0,%pi/2,[k^2 < 1,series(?truncated)]] [864,sqrt(1-k^2*sin(x)^2),?etc+%pi*('sum(k^(2*r)*(2*r)!^2/((2*r+1)*2^(4*r)*r!^2),r,0,inf))/4,0,%pi/2,[k^2 < 1,series(?truncated)]] [865,%e^-x*log(x),-euler_gamma,0,inf,[]] [866,%e^-x^2*log(x),-sqrt(%pi)*(euler_gamma+2*log(2))/4,0,inf,[]] [867,(1/(1-%e^-x)-1/x)*%e^-x,euler_gamma,0,inf,[]]