1,1,248,373,2.3490504,"\int (d \sin (e+f x))^n (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx","Integrate[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]),x]","\frac{a^3 \sin (e+f x) \cos (e+f x) (d \sin (e+f x))^n \left(\sin (e+f x) \left(\frac{(3 A+B) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right)}{n+2}+\sin (e+f x) \left(\frac{3 (A+B) \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(e+f x)\right)}{n+3}+\sin (e+f x) \left(\frac{(A+3 B) \, _2F_1\left(\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\sin ^2(e+f x)\right)}{n+4}+\frac{B \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n+5}{2};\frac{n+7}{2};\sin ^2(e+f x)\right)}{n+5}\right)\right)\right)+\frac{A \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{n+1}\right)}{f \sqrt{\cos ^2(e+f x)}}","\frac{a^3 (A (4 n+11)+B (4 n+9)) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right)}{d^2 f (n+2) (n+3) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 \left(A \left(4 n^2+21 n+20\right)+B \left(4 n^2+19 n+15\right)\right) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1) (n+2) (n+4) \sqrt{\cos ^2(e+f x)}}-\frac{a^3 \left(A \left(2 n^2+15 n+28\right)+B \left(2 n^2+14 n+27\right)\right) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (n+2) (n+3) (n+4)}-\frac{(A (n+4)+B (n+6)) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) (d \sin (e+f x))^{n+1}}{d f (n+3) (n+4)}-\frac{a B \cos (e+f x) (a \sin (e+f x)+a)^2 (d \sin (e+f x))^{n+1}}{d f (n+4)}",1,"(a^3*Cos[e + f*x]*Sin[e + f*x]*(d*Sin[e + f*x])^n*((A*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2])/(1 + n) + Sin[e + f*x]*(((3*A + B)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2])/(2 + n) + Sin[e + f*x]*((3*(A + B)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Sin[e + f*x]^2])/(3 + n) + Sin[e + f*x]*(((A + 3*B)*Hypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Sin[e + f*x]^2])/(4 + n) + (B*Hypergeometric2F1[1/2, (5 + n)/2, (7 + n)/2, Sin[e + f*x]^2]*Sin[e + f*x])/(5 + n))))))/(f*Sqrt[Cos[e + f*x]^2])","A",1
2,1,204,277,1.5483539,"\int (d \sin (e+f x))^n (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \, dx","Integrate[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]),x]","\frac{a^2 \sin (e+f x) \cos (e+f x) (d \sin (e+f x))^n \left(\sin (e+f x) \left(\frac{(2 A+B) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right)}{n+2}+\sin (e+f x) \left(\frac{(A+2 B) \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(e+f x)\right)}{n+3}+\frac{B \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\sin ^2(e+f x)\right)}{n+4}\right)\right)+\frac{A \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{n+1}\right)}{f \sqrt{\cos ^2(e+f x)}}","\frac{a^2 (2 A (n+3)+B (2 n+5)) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right)}{d^2 f (n+2) (n+3) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 (A (2 n+3)+2 B (n+1)) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1) (n+2) \sqrt{\cos ^2(e+f x)}}-\frac{a^2 (A (n+3)+B (n+4)) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (n+2) (n+3)}-\frac{B \cos (e+f x) \left(a^2 \sin (e+f x)+a^2\right) (d \sin (e+f x))^{n+1}}{d f (n+3)}",1,"(a^2*Cos[e + f*x]*Sin[e + f*x]*(d*Sin[e + f*x])^n*((A*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2])/(1 + n) + Sin[e + f*x]*(((2*A + B)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2])/(2 + n) + Sin[e + f*x]*(((A + 2*B)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Sin[e + f*x]^2])/(3 + n) + (B*Hypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Sin[e + f*x]^2]*Sin[e + f*x])/(4 + n)))))/(f*Sqrt[Cos[e + f*x]^2])","A",1
3,1,392,191,3.8457725,"\int (d \sin (e+f x))^n (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx","Integrate[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]),x]","-\frac{a 2^{-n-2} e^{i f n x} \left(1-e^{2 i (e+f x)}\right)^{-n} \left(-i e^{-i (e+f x)} \left(-1+e^{2 i (e+f x)}\right)\right)^n (\sin (e+f x)+1) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \left(\frac{2 (A+B) e^{-i (e+f (n+1) x)} \, _2F_1\left(\frac{1}{2} (-n-1),-n;\frac{1-n}{2};e^{2 i (e+f x)}\right)}{n+1}-\frac{2 (A+B) e^{i (e-f (n-1) x)} \, _2F_1\left(\frac{1-n}{2},-n;\frac{3-n}{2};e^{2 i (e+f x)}\right)}{n-1}+i \left(\frac{e^{-i f n x} \left(B n e^{2 i (e+f x)} \, _2F_1\left(1-\frac{n}{2},-n;2-\frac{n}{2};e^{2 i (e+f x)}\right)-2 (n-2) (2 A+B) \, _2F_1\left(-n,-\frac{n}{2};1-\frac{n}{2};e^{2 i (e+f x)}\right)\right)}{(n-2) n}+\frac{B e^{-i (2 e+f (n+2) x)} \, _2F_1\left(-\frac{n}{2}-1,-n;-\frac{n}{2};e^{2 i (e+f x)}\right)}{n+2}\right)\right)}{f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","\frac{a (A+B) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right)}{d^2 f (n+2) \sqrt{\cos ^2(e+f x)}}+\frac{a (A (n+2)+B (n+1)) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{d f (n+1) (n+2) \sqrt{\cos ^2(e+f x)}}-\frac{a B \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (n+2)}",1,"-((2^(-2 - n)*a*E^(I*f*n*x)*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^n*((2*(A + B)*Hypergeometric2F1[(-1 - n)/2, -n, (1 - n)/2, E^((2*I)*(e + f*x))])/(E^(I*(e + f*(1 + n)*x))*(1 + n)) - (2*(A + B)*E^(I*(e - f*(-1 + n)*x))*Hypergeometric2F1[(1 - n)/2, -n, (3 - n)/2, E^((2*I)*(e + f*x))])/(-1 + n) + I*((B*Hypergeometric2F1[-1 - n/2, -n, -1/2*n, E^((2*I)*(e + f*x))])/(E^(I*(2*e + f*(2 + n)*x))*(2 + n)) + (B*E^((2*I)*(e + f*x))*n*Hypergeometric2F1[1 - n/2, -n, 2 - n/2, E^((2*I)*(e + f*x))] - 2*(2*A + B)*(-2 + n)*Hypergeometric2F1[-n, -1/2*n, 1 - n/2, E^((2*I)*(e + f*x))])/(E^(I*f*n*x)*(-2 + n)*n)))*(d*Sin[e + f*x])^n*(1 + Sin[e + f*x]))/((1 - E^((2*I)*(e + f*x)))^n*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sin[e + f*x]^n))","C",1
4,1,157,202,0.8893384,"\int \frac{(d \sin (e+f x))^n (A+B \sin (e+f x))}{a+a \sin (e+f x)} \, dx","Integrate[((d*Sin[e + f*x])^n*(A + B*Sin[e + f*x]))/(a + a*Sin[e + f*x]),x]","\frac{\sin (e+f x) \cos (e+f x) (d \sin (e+f x))^n \left(\frac{(n+1) (A-B) \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right)}{(n+2) \sqrt{\cos ^2(e+f x)}}+\frac{(-A n+B n+B) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{(n+1) \sqrt{\cos ^2(e+f x)}}+\frac{A-B}{\sin (e+f x)+1}\right)}{a f}","\frac{(n+1) (A-B) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right)}{a d^2 f (n+2) \sqrt{\cos ^2(e+f x)}}+\frac{(-A n+B n+B) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{a d f (n+1) \sqrt{\cos ^2(e+f x)}}+\frac{(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (a \sin (e+f x)+a)}",1,"(Cos[e + f*x]*Sin[e + f*x]*(d*Sin[e + f*x])^n*(((B - A*n + B*n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2])/((1 + n)*Sqrt[Cos[e + f*x]^2]) + ((A - B)*(1 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]*Sin[e + f*x])/((2 + n)*Sqrt[Cos[e + f*x]^2]) + (A - B)/(1 + Sin[e + f*x])))/(a*f)","A",1
5,1,212,279,1.3268999,"\int \frac{(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^2} \, dx","Integrate[((d*Sin[e + f*x])^n*(A + B*Sin[e + f*x]))/(a + a*Sin[e + f*x])^2,x]","\frac{\sin (e+f x) \cos (e+f x) (d \sin (e+f x))^n \left(-\frac{n (-2 A n+A+2 B (n+1)) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{(n+1) \sqrt{\cos ^2(e+f x)}}+\frac{(n+1) (-2 A (n-1)+2 B n+B) \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right)}{(n+2) \sqrt{\cos ^2(e+f x)}}+\frac{n (B-A)}{\sin (e+f x)+1}+\frac{A (-n)+2 A+B n+B}{\sin (e+f x)+1}+\frac{A-B}{(\sin (e+f x)+1)^2}\right)}{3 a^2 f}","\frac{(n+1) (2 A (1-n)+2 B n+B) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right)}{3 a^2 d^2 f (n+2) \sqrt{\cos ^2(e+f x)}}-\frac{n (-2 A n+A+2 B (n+1)) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{3 a^2 d f (n+1) \sqrt{\cos ^2(e+f x)}}+\frac{(2 A (1-n)+2 B n+B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{3 a^2 d f (\sin (e+f x)+1)}+\frac{(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{3 d f (a \sin (e+f x)+a)^2}",1,"(Cos[e + f*x]*Sin[e + f*x]*(d*Sin[e + f*x])^n*(-((n*(A - 2*A*n + 2*B*(1 + n))*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2])/((1 + n)*Sqrt[Cos[e + f*x]^2])) + ((1 + n)*(B - 2*A*(-1 + n) + 2*B*n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]*Sin[e + f*x])/((2 + n)*Sqrt[Cos[e + f*x]^2]) + (A - B)/(1 + Sin[e + f*x])^2 + ((-A + B)*n)/(1 + Sin[e + f*x]) + (2*A + B - A*n + B*n)/(1 + Sin[e + f*x])))/(3*a^2*f)","A",1
6,1,260,362,4.5737881,"\int \frac{(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx","Integrate[((d*Sin[e + f*x])^n*(A + B*Sin[e + f*x]))/(a + a*Sin[e + f*x])^3,x]","\frac{(d \sin (e+f x))^n \left(\frac{2 \sin (e+f x) \cos (e+f x) \left(n \left(A \left(-4 n^2+9 n-2\right)+B \left(4 n^2+n-3\right)\right) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)+\frac{(n-1) (n+1)^2 (A (4 n-7)-B (4 n+3)) \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right)}{n+2}\right)}{(n+1) \sqrt{\cos ^2(e+f x)}}+\frac{(n-1) (A (4 n-7)-B (4 n+3)) \sin (2 (e+f x))}{\sin (e+f x)+1}+\frac{(A (5-2 n)+2 B n) \sin (2 (e+f x))}{(\sin (e+f x)+1)^2}+\frac{3 (A-B) \sin (2 (e+f x))}{(\sin (e+f x)+1)^3}\right)}{30 a^3 f}","\frac{(1-n) (n+1) (-4 A n+7 A+4 B n+3 B) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right)}{15 a^3 d^2 f (n+2) \sqrt{\cos ^2(e+f x)}}-\frac{n \left(A \left(4 n^2-9 n+2\right)+B \left(-4 n^2-n+3\right)\right) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right)}{15 a^3 d f (n+1) \sqrt{\cos ^2(e+f x)}}+\frac{(1-n) (-4 A n+7 A+4 B n+3 B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{15 d f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{(A (5-2 n)+2 B n) \cos (e+f x) (d \sin (e+f x))^{n+1}}{15 a d f (a \sin (e+f x)+a)^2}+\frac{(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{5 d f (a \sin (e+f x)+a)^3}",1,"((d*Sin[e + f*x])^n*((2*Cos[e + f*x]*Sin[e + f*x]*(n*(A*(-2 + 9*n - 4*n^2) + B*(-3 + n + 4*n^2))*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2] + ((-1 + n)*(1 + n)^2*(A*(-7 + 4*n) - B*(3 + 4*n))*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]*Sin[e + f*x])/(2 + n)))/((1 + n)*Sqrt[Cos[e + f*x]^2]) + (3*(A - B)*Sin[2*(e + f*x)])/(1 + Sin[e + f*x])^3 + ((A*(5 - 2*n) + 2*B*n)*Sin[2*(e + f*x)])/(1 + Sin[e + f*x])^2 + ((-1 + n)*(A*(-7 + 4*n) - B*(3 + 4*n))*Sin[2*(e + f*x)])/(1 + Sin[e + f*x])))/(30*a^3*f)","A",1
7,1,596,336,18.3825159,"\int (d \sin (e+f x))^n (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx","Integrate[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]),x]","\frac{2^{n+1} \tan \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) (a (\sin (e+f x)+1))^{5/2} \sin ^{-n}(e+f x) \left(\frac{\tan \left(\frac{1}{2} (e+f x)\right)}{\tan ^2\left(\frac{1}{2} (e+f x)\right)+1}\right)^n \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)^n (d \sin (e+f x))^n \left(\tan \left(\frac{1}{2} (e+f x)\right) \left(\frac{(5 A+2 B) \, _2F_1\left(\frac{n+2}{2},n+\frac{9}{2};\frac{n+4}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n+2}+\tan \left(\frac{1}{2} (e+f x)\right) \left(\frac{(11 A+10 B) \, _2F_1\left(\frac{n+3}{2},n+\frac{9}{2};\frac{n+5}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n+3}+\tan \left(\frac{1}{2} (e+f x)\right) \left(\frac{5 (3 A+4 B) \, _2F_1\left(\frac{n+4}{2},n+\frac{9}{2};\frac{n+6}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n+4}+\tan \left(\frac{1}{2} (e+f x)\right) \left(\frac{5 (3 A+4 B) \, _2F_1\left(n+\frac{9}{2},\frac{n+5}{2};\frac{n+7}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n+5}+\tan \left(\frac{1}{2} (e+f x)\right) \left(\frac{(11 A+10 B) \, _2F_1\left(n+\frac{9}{2},\frac{n+6}{2};\frac{n+8}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n+6}+\frac{(5 A+2 B) \tan \left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(n+\frac{9}{2},\frac{n+7}{2};\frac{n+9}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n+7}\right)\right)\right)\right)\right)+\frac{A \, _2F_1\left(\frac{n+1}{2},n+\frac{9}{2};\frac{n+3}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n+1}+\frac{A \tan ^7\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(\frac{n}{2}+4,n+\frac{9}{2};\frac{n}{2}+5;-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n+8}\right)}{f \sqrt{\sec ^2\left(\frac{1}{2} (e+f x)\right)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\frac{2 a^3 \left(A \left(32 n^3+224 n^2+478 n+301\right)+2 B \left(16 n^3+104 n^2+203 n+115\right)\right) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f (2 n+3) (2 n+5) (2 n+7) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^3 \left(A \left(8 n^2+50 n+77\right)+2 B \left(4 n^2+23 n+35\right)\right) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) (2 n+5) (2 n+7) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 (A (2 n+7)+2 B (n+5)) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (d \sin (e+f x))^{n+1}}{d f (2 n+5) (2 n+7)}-\frac{2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (d \sin (e+f x))^{n+1}}{d f (2 n+7)}",1,"(2^(1 + n)*Sec[(e + f*x)/2]*(d*Sin[e + f*x])^n*(a*(1 + Sin[e + f*x]))^(5/2)*Tan[(e + f*x)/2]*(Tan[(e + f*x)/2]/(1 + Tan[(e + f*x)/2]^2))^n*(1 + Tan[(e + f*x)/2]^2)^n*((A*Hypergeometric2F1[(1 + n)/2, 9/2 + n, (3 + n)/2, -Tan[(e + f*x)/2]^2])/(1 + n) + (A*Hypergeometric2F1[4 + n/2, 9/2 + n, 5 + n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^7)/(8 + n) + Tan[(e + f*x)/2]*(((5*A + 2*B)*Hypergeometric2F1[(2 + n)/2, 9/2 + n, (4 + n)/2, -Tan[(e + f*x)/2]^2])/(2 + n) + Tan[(e + f*x)/2]*(((11*A + 10*B)*Hypergeometric2F1[(3 + n)/2, 9/2 + n, (5 + n)/2, -Tan[(e + f*x)/2]^2])/(3 + n) + Tan[(e + f*x)/2]*((5*(3*A + 4*B)*Hypergeometric2F1[(4 + n)/2, 9/2 + n, (6 + n)/2, -Tan[(e + f*x)/2]^2])/(4 + n) + Tan[(e + f*x)/2]*((5*(3*A + 4*B)*Hypergeometric2F1[9/2 + n, (5 + n)/2, (7 + n)/2, -Tan[(e + f*x)/2]^2])/(5 + n) + Tan[(e + f*x)/2]*(((11*A + 10*B)*Hypergeometric2F1[9/2 + n, (6 + n)/2, (8 + n)/2, -Tan[(e + f*x)/2]^2])/(6 + n) + ((5*A + 2*B)*Hypergeometric2F1[9/2 + n, (7 + n)/2, (9 + n)/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(7 + n))))))))/(f*Sqrt[Sec[(e + f*x)/2]^2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sin[e + f*x]^n)","A",0
8,1,478,229,15.4040835,"\int (d \sin (e+f x))^n (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) \, dx","Integrate[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]),x]","\frac{2^{n+1} \tan \left(\frac{1}{2} (e+f x)\right) \sec \left(\frac{1}{2} (e+f x)\right) (a (\sin (e+f x)+1))^{3/2} \sin ^{-n}(e+f x) \left(\frac{\tan \left(\frac{1}{2} (e+f x)\right)}{\tan ^2\left(\frac{1}{2} (e+f x)\right)+1}\right)^n \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)^n (d \sin (e+f x))^n \left(\tan \left(\frac{1}{2} (e+f x)\right) \left(\frac{(3 A+2 B) \, _2F_1\left(\frac{n+2}{2},n+\frac{7}{2};\frac{n+4}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n+2}+\tan \left(\frac{1}{2} (e+f x)\right) \left(\frac{2 (2 A+3 B) \, _2F_1\left(\frac{n+3}{2},n+\frac{7}{2};\frac{n+5}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n+3}+\tan \left(\frac{1}{2} (e+f x)\right) \left(\frac{2 (2 A+3 B) \, _2F_1\left(n+\frac{7}{2},\frac{n+4}{2};\frac{n+6}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n+4}+\tan \left(\frac{1}{2} (e+f x)\right) \left(\frac{(3 A+2 B) \, _2F_1\left(n+\frac{7}{2},\frac{n+5}{2};\frac{n+7}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n+5}+\frac{A \tan \left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(n+\frac{7}{2},\frac{n+6}{2};\frac{n+8}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n+6}\right)\right)\right)\right)+\frac{A \, _2F_1\left(\frac{n+1}{2},n+\frac{7}{2};\frac{n+3}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n+1}\right)}{f \sqrt{\sec ^2\left(\frac{1}{2} (e+f x)\right)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\frac{2 a^2 \left(A \left(8 n^2+30 n+25\right)+2 B \left(4 n^2+13 n+9\right)\right) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f (2 n+3) (2 n+5) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 (A (2 n+5)+2 B (n+3)) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) (2 n+5) \sqrt{a \sin (e+f x)+a}}-\frac{2 a B \cos (e+f x) \sqrt{a \sin (e+f x)+a} (d \sin (e+f x))^{n+1}}{d f (2 n+5)}",1,"(2^(1 + n)*Sec[(e + f*x)/2]*(d*Sin[e + f*x])^n*(a*(1 + Sin[e + f*x]))^(3/2)*Tan[(e + f*x)/2]*(Tan[(e + f*x)/2]/(1 + Tan[(e + f*x)/2]^2))^n*(1 + Tan[(e + f*x)/2]^2)^n*((A*Hypergeometric2F1[(1 + n)/2, 7/2 + n, (3 + n)/2, -Tan[(e + f*x)/2]^2])/(1 + n) + Tan[(e + f*x)/2]*(((3*A + 2*B)*Hypergeometric2F1[(2 + n)/2, 7/2 + n, (4 + n)/2, -Tan[(e + f*x)/2]^2])/(2 + n) + Tan[(e + f*x)/2]*((2*(2*A + 3*B)*Hypergeometric2F1[(3 + n)/2, 7/2 + n, (5 + n)/2, -Tan[(e + f*x)/2]^2])/(3 + n) + Tan[(e + f*x)/2]*((2*(2*A + 3*B)*Hypergeometric2F1[7/2 + n, (4 + n)/2, (6 + n)/2, -Tan[(e + f*x)/2]^2])/(4 + n) + Tan[(e + f*x)/2]*(((3*A + 2*B)*Hypergeometric2F1[7/2 + n, (5 + n)/2, (7 + n)/2, -Tan[(e + f*x)/2]^2])/(5 + n) + (A*Hypergeometric2F1[7/2 + n, (6 + n)/2, (8 + n)/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(6 + n)))))))/(f*Sqrt[Sec[(e + f*x)/2]^2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sin[e + f*x]^n)","B",0
9,1,409,137,67.652985,"\int (d \sin (e+f x))^n \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) \, dx","Integrate[(d*Sin[e + f*x])^n*Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]),x]","-\frac{(1+i) 2^{-n-2} e^{i f n x-\frac{3 i e}{2}} \left(1-e^{2 i (e+f x)}\right)^{-n} \left(-i e^{-i (e+f x)} \left(-1+e^{2 i (e+f x)}\right)\right)^n \sqrt{a (\sin (e+f x)+1)} \sin ^{-n}(e+f x) (d \sin (e+f x))^n \left(2 e^{i e} \left(\frac{e^{\frac{1}{2} i (2 e+f (1-2 n) x)} \left(i B (2 n-1) e^{i (e+f x)} \, _2F_1\left(\frac{1}{4} (3-2 n),-n;\frac{1}{4} (7-2 n);e^{2 i (e+f x)}\right)-(2 n-3) (2 A+B) \, _2F_1\left(\frac{1}{4} (1-2 n),-n;\frac{1}{4} (5-2 n);e^{2 i (e+f x)}\right)\right)}{f (2 n-3) (2 n-1)}-\frac{i (2 A+B) e^{-\frac{1}{2} i f (2 n+1) x} \, _2F_1\left(\frac{1}{4} (-2 n-1),-n;\frac{1}{4} (3-2 n);e^{2 i (e+f x)}\right)}{2 f n+f}\right)+\frac{2 B e^{-\frac{1}{2} i f (2 n+3) x} \, _2F_1\left(\frac{1}{4} (-2 n-3),-n;\frac{1}{4} (1-2 n);e^{2 i (e+f x)}\right)}{f (2 n+3)}\right)}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}","-\frac{2 a (A (2 n+3)+2 B (n+1)) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a B \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}",1,"((-1 - I)*2^(-2 - n)*E^(((-3*I)/2)*e + I*f*n*x)*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^n*((2*B*Hypergeometric2F1[(-3 - 2*n)/4, -n, (1 - 2*n)/4, E^((2*I)*(e + f*x))])/(E^((I/2)*f*(3 + 2*n)*x)*f*(3 + 2*n)) + 2*E^(I*e)*(((-I)*(2*A + B)*Hypergeometric2F1[(-1 - 2*n)/4, -n, (3 - 2*n)/4, E^((2*I)*(e + f*x))])/(E^((I/2)*f*(1 + 2*n)*x)*(f + 2*f*n)) + (E^((I/2)*(2*e + f*(1 - 2*n)*x))*(-((2*A + B)*(-3 + 2*n)*Hypergeometric2F1[(1 - 2*n)/4, -n, (5 - 2*n)/4, E^((2*I)*(e + f*x))]) + I*B*E^(I*(e + f*x))*(-1 + 2*n)*Hypergeometric2F1[(3 - 2*n)/4, -n, (7 - 2*n)/4, E^((2*I)*(e + f*x))]))/(f*(-3 + 2*n)*(-1 + 2*n))))*(d*Sin[e + f*x])^n*Sqrt[a*(1 + Sin[e + f*x])])/((1 - E^((2*I)*(e + f*x)))^n*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[e + f*x]^n)","C",1
10,1,250,152,4.8566108,"\int \frac{(d \sin (e+f x))^n (A+B \sin (e+f x))}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[((d*Sin[e + f*x])^n*(A + B*Sin[e + f*x]))/Sqrt[a + a*Sin[e + f*x]],x]","\frac{\cos (e+f x) \sqrt{a (\sin (e+f x)+1)} \sin ^n(e+f x) \left(-\sin ^2(e+f x)\right)^{-n} \left(1-\frac{1}{\sin (e+f x)+1}\right)^{-n} (d \sin (e+f x))^n \left(4 (A-B) \sqrt{\frac{\sin (e+f x)-1}{\sin (e+f x)+1}} (-\sin (e+f x))^n F_1\left(-n-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{1}{\sin (e+f x)+1}\right)-(2 n+1) (A+B) \sqrt{2-2 \sin (e+f x)} \left(1-\frac{1}{\sin (e+f x)+1}\right)^n F_1\left(1;\frac{1}{2},-n;2;\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)\right)}{4 a f (2 n+1) (\sin (e+f x)-1)}","-\frac{(A-B) \cos (e+f x) \sin ^{-n}(e+f x) F_1\left(\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) (d \sin (e+f x))^n}{f \sqrt{a \sin (e+f x)+a}}-\frac{2 B \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f \sqrt{a \sin (e+f x)+a}}",1,"(Cos[e + f*x]*Sin[e + f*x]^n*(d*Sin[e + f*x])^n*Sqrt[a*(1 + Sin[e + f*x])]*(4*(A - B)*AppellF1[-1/2 - n, -1/2, -n, 1/2 - n, 2/(1 + Sin[e + f*x]), (1 + Sin[e + f*x])^(-1)]*(-Sin[e + f*x])^n*Sqrt[(-1 + Sin[e + f*x])/(1 + Sin[e + f*x])] - (A + B)*(1 + 2*n)*AppellF1[1, 1/2, -n, 2, (1 + Sin[e + f*x])/2, 1 + Sin[e + f*x]]*Sqrt[2 - 2*Sin[e + f*x]]*(1 - (1 + Sin[e + f*x])^(-1))^n))/(4*a*f*(1 + 2*n)*(-1 + Sin[e + f*x])*(-Sin[e + f*x]^2)^n*(1 - (1 + Sin[e + f*x])^(-1))^n)","A",0
11,1,523,226,21.9034275,"\int \frac{(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[((d*Sin[e + f*x])^n*(A + B*Sin[e + f*x]))/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\sec (e+f x) (d \sin (e+f x))^n \left(A \left(a^2 \sqrt{2-2 \sin (e+f x)} (\sin (e+f x)+1)^2 (-\sin (e+f x))^{-n} F_1\left(1;\frac{1}{2},-n;2;\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)-\frac{4 a \sqrt{\frac{\sin (e+f x)-1}{\sin (e+f x)+1}} (\sin (e+f x)+1) \left(1-\frac{1}{\sin (e+f x)+1}\right)^{-n} \left(2 a (2 n+1) F_1\left(\frac{1}{2}-n;-\frac{1}{2},-n;\frac{3}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{1}{\sin (e+f x)+1}\right)+a (2 n-1) (\sin (e+f x)+1) F_1\left(-n-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{1}{\sin (e+f x)+1}\right)\right)}{4 n^2-1}\right)+a B (\sin (e+f x)+1) \left(a \sqrt{2-2 \sin (e+f x)} (\sin (e+f x)+1) (-\sin (e+f x))^{-n} F_1\left(1;\frac{1}{2},-n;2;\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)-\frac{4 \sqrt{\frac{\sin (e+f x)-1}{\sin (e+f x)+1}} \left(1-\frac{1}{\sin (e+f x)+1}\right)^{-n} \left(a (2 n-1) (\sin (e+f x)+1) F_1\left(-n-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{1}{\sin (e+f x)+1}\right)-2 a (2 n+1) F_1\left(\frac{1}{2}-n;-\frac{1}{2},-n;\frac{3}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{1}{\sin (e+f x)+1}\right)\right)}{4 n^2-1}\right)\right)}{8 a^3 f \sqrt{a (\sin (e+f x)+1)}}","-\frac{(-4 A n+A+B (4 n+3)) \cos (e+f x) \sin ^{-n}(e+f x) F_1\left(\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) (d \sin (e+f x))^n}{4 a f \sqrt{a \sin (e+f x)+a}}-\frac{(2 n+1) (A-B) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{2 a f \sqrt{a \sin (e+f x)+a}}+\frac{(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{2 d f (a \sin (e+f x)+a)^{3/2}}",1,"(Sec[e + f*x]*(d*Sin[e + f*x])^n*(a*B*(1 + Sin[e + f*x])*((a*AppellF1[1, 1/2, -n, 2, (1 + Sin[e + f*x])/2, 1 + Sin[e + f*x]]*Sqrt[2 - 2*Sin[e + f*x]]*(1 + Sin[e + f*x]))/(-Sin[e + f*x])^n - (4*Sqrt[(-1 + Sin[e + f*x])/(1 + Sin[e + f*x])]*(-2*a*(1 + 2*n)*AppellF1[1/2 - n, -1/2, -n, 3/2 - n, 2/(1 + Sin[e + f*x]), (1 + Sin[e + f*x])^(-1)] + a*(-1 + 2*n)*AppellF1[-1/2 - n, -1/2, -n, 1/2 - n, 2/(1 + Sin[e + f*x]), (1 + Sin[e + f*x])^(-1)]*(1 + Sin[e + f*x])))/((-1 + 4*n^2)*(1 - (1 + Sin[e + f*x])^(-1))^n)) + A*((a^2*AppellF1[1, 1/2, -n, 2, (1 + Sin[e + f*x])/2, 1 + Sin[e + f*x]]*Sqrt[2 - 2*Sin[e + f*x]]*(1 + Sin[e + f*x])^2)/(-Sin[e + f*x])^n - (4*a*Sqrt[(-1 + Sin[e + f*x])/(1 + Sin[e + f*x])]*(1 + Sin[e + f*x])*(2*a*(1 + 2*n)*AppellF1[1/2 - n, -1/2, -n, 3/2 - n, 2/(1 + Sin[e + f*x]), (1 + Sin[e + f*x])^(-1)] + a*(-1 + 2*n)*AppellF1[-1/2 - n, -1/2, -n, 1/2 - n, 2/(1 + Sin[e + f*x]), (1 + Sin[e + f*x])^(-1)]*(1 + Sin[e + f*x])))/((-1 + 4*n^2)*(1 - (1 + Sin[e + f*x])^(-1))^n))))/(8*a^3*f*Sqrt[a*(1 + Sin[e + f*x])])","B",0
12,1,5918,221,22.1780455,"\int (d \sin (e+f x))^n (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\text{Result too large to show}","-\frac{2^{m+\frac{1}{2}} (A-B) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} \sin ^{-n}(e+f x) (a \sin (e+f x)+a)^m (d \sin (e+f x))^n F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f}-\frac{B 2^{m+\frac{3}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} \sin ^{-n}(e+f x) (a \sin (e+f x)+a)^m (d \sin (e+f x))^n F_1\left(\frac{1}{2};-n,-m-\frac{1}{2};\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f}",1,"Result too large to show","B",0
13,0,0,114,11.2524765,"\int (d \sin (e+f x))^n (a-a \sin (e+f x)) (a+a \sin (e+f x))^m \, dx","Integrate[(d*Sin[e + f*x])^n*(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^m,x]","\int (d \sin (e+f x))^n (a-a \sin (e+f x)) (a+a \sin (e+f x))^m \, dx","\frac{\sec (e+f x) (a-a \sin (e+f x)) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m (d \sin (e+f x))^{n+1} F_1\left(n+1;-\frac{1}{2},\frac{1}{2}-m;n+2;\sin (e+f x),-\sin (e+f x)\right)}{d f (n+1) \sqrt{1-\sin (e+f x)}}",1,"Integrate[(d*Sin[e + f*x])^n*(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^m, x]","F",-1
14,1,107,37,1.6347598,"\int \sin ^n(c+d x) (a+a \sin (c+d x))^{-2-n} (-1-n-(-2-n) \sin (c+d x)) \, dx","Integrate[Sin[c + d*x]^n*(a + a*Sin[c + d*x])^(-2 - n)*(-1 - n - (-2 - n)*Sin[c + d*x]),x]","-\frac{2^n \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (-\sin (c+d x)+\cos (c+d x)+1) (a (\sin (c+d x)+1))^{-n-2} \left(\left(\sin \left(\frac{3}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right) \cos \left(\frac{1}{4} (c+d x)\right)\right)^n}{d}","-\frac{\cos (c+d x) \sin ^{n+1}(c+d x) (a \sin (c+d x)+a)^{-n-2}}{d}",1,"-((2^n*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Cos[(c + d*x)/4]*(-Sin[(c + d*x)/4] + Sin[(3*(c + d*x))/4]))^n*(1 + Cos[c + d*x] - Sin[c + d*x])*(a*(1 + Sin[c + d*x]))^(-2 - n))/d)","B",1
15,1,35,35,0.3858777,"\int \sin ^{-2-m}(c+d x) (a+a \sin (c+d x))^m (1+m-m \sin (c+d x)) \, dx","Integrate[Sin[c + d*x]^(-2 - m)*(a + a*Sin[c + d*x])^m*(1 + m - m*Sin[c + d*x]),x]","-\frac{\cos (c+d x) \sin ^{-m-1}(c+d x) (a (\sin (c+d x)+1))^m}{d}","-\frac{\cos (c+d x) \sin ^{-m-1}(c+d x) (a \sin (c+d x)+a)^m}{d}",1,"-((Cos[c + d*x]*Sin[c + d*x]^(-1 - m)*(a*(1 + Sin[c + d*x]))^m)/d)","A",1
16,1,147,153,0.8789144,"\int \frac{\sin ^2(e+f x) (A+B \sin (e+f x))}{(a+b \sin (e+f x))^2} \, dx","Integrate[(Sin[e + f*x]^2*(A + B*Sin[e + f*x]))/(a + b*Sin[e + f*x])^2,x]","\frac{\frac{a^2 b (A b-a B) \cos (e+f x)}{(a-b) (a+b) (a+b \sin (e+f x))}+\frac{2 a \left(2 a^3 B-a^2 A b-3 a b^2 B+2 A b^3\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+(e+f x) (A b-2 a B)-b B \cos (e+f x)}{b^3 f}","\frac{a^2 (A b-a B) \cos (e+f x)}{b^2 f \left(a^2-b^2\right) (a+b \sin (e+f x))}-\frac{2 a \left(-2 a^3 B+a^2 A b+3 a b^2 B-2 A b^3\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 f \left(a^2-b^2\right)^{3/2}}+\frac{x (A b-2 a B)}{b^3}-\frac{B \cos (e+f x)}{b^2 f}",1,"((A*b - 2*a*B)*(e + f*x) + (2*a*(-(a^2*A*b) + 2*A*b^3 + 2*a^3*B - 3*a*b^2*B)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - b*B*Cos[e + f*x] + (a^2*b*(A*b - a*B)*Cos[e + f*x])/((a - b)*(a + b)*(a + b*Sin[e + f*x])))/(b^3*f)","A",1
17,1,131,182,0.9829716,"\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx","Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4,x]","\frac{a c^4 (120 (7 A-5 B) \cos (e+f x)+20 (13 A-7 B) \cos (3 (e+f x))+240 A \sin (2 (e+f x))-90 A \sin (4 (e+f x))-12 A \cos (5 (e+f x))+840 A f x+15 B \sin (2 (e+f x))+105 B \sin (4 (e+f x))-5 B \sin (6 (e+f x))+36 B \cos (5 (e+f x))-420 B f x)}{960 f}","\frac{7 a c^4 (2 A-B) \cos ^3(e+f x)}{24 f}+\frac{7 a (2 A-B) \cos ^3(e+f x) \left(c^4-c^4 \sin (e+f x)\right)}{40 f}+\frac{7 a c^4 (2 A-B) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{7}{16} a c^4 x (2 A-B)+\frac{a (2 A-B) \cos ^3(e+f x) \left(c^2-c^2 \sin (e+f x)\right)^2}{10 f}-\frac{a B c \cos ^3(e+f x) (c-c \sin (e+f x))^3}{6 f}",1,"(a*c^4*(840*A*f*x - 420*B*f*x + 120*(7*A - 5*B)*Cos[e + f*x] + 20*(13*A - 7*B)*Cos[3*(e + f*x)] - 12*A*Cos[5*(e + f*x)] + 36*B*Cos[5*(e + f*x)] + 240*A*Sin[2*(e + f*x)] + 15*B*Sin[2*(e + f*x)] - 90*A*Sin[4*(e + f*x)] + 105*B*Sin[4*(e + f*x)] - 5*B*Sin[6*(e + f*x)]))/(960*f)","A",1
18,1,95,142,0.8432521,"\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3,x]","\frac{a c^3 (15 (-(A-2 B) \sin (4 (e+f x))+4 f x (5 A-2 B)+8 A \sin (2 (e+f x)))+60 (4 A-3 B) \cos (e+f x)+10 (8 A-5 B) \cos (3 (e+f x))+6 B \cos (5 (e+f x)))}{480 f}","\frac{a c^3 (5 A-2 B) \cos ^3(e+f x)}{12 f}+\frac{a (5 A-2 B) \cos ^3(e+f x) \left(c^3-c^3 \sin (e+f x)\right)}{20 f}+\frac{a c^3 (5 A-2 B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a c^3 x (5 A-2 B)-\frac{a B c \cos ^3(e+f x) (c-c \sin (e+f x))^2}{5 f}",1,"(a*c^3*(60*(4*A - 3*B)*Cos[e + f*x] + 10*(8*A - 5*B)*Cos[3*(e + f*x)] + 6*B*Cos[5*(e + f*x)] + 15*(4*(5*A - 2*B)*f*x + 8*A*Sin[2*(e + f*x)] - (A - 2*B)*Sin[4*(e + f*x)])))/(480*f)","A",1
19,1,74,97,0.6745228,"\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2,x]","\frac{a c^2 (3 (8 A \sin (2 (e+f x))+16 A f x+B \sin (4 (e+f x))-4 B f x)+24 (A-B) \cos (e+f x)+8 (A-B) \cos (3 (e+f x)))}{96 f}","\frac{a c^2 (A-B) \cos ^3(e+f x)}{3 f}+\frac{a c^2 (4 A-B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a c^2 x (4 A-B)+\frac{a B c^2 \sin (e+f x) \cos ^3(e+f x)}{4 f}",1,"(a*c^2*(24*(A - B)*Cos[e + f*x] + 8*(A - B)*Cos[3*(e + f*x)] + 3*(16*A*f*x - 4*B*f*x + 8*A*Sin[2*(e + f*x)] + B*Sin[4*(e + f*x)])))/(96*f)","A",1
20,1,48,49,0.1564626,"\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x]),x]","-\frac{a c (-3 A (\sin (2 (e+f x))-2 e+2 f x)+3 B \cos (e+f x)+B \cos (3 (e+f x)))}{12 f}","\frac{a A c \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} a A c x-\frac{a B c \cos ^3(e+f x)}{3 f}",1,"-1/12*(a*c*(3*B*Cos[e + f*x] + B*Cos[3*(e + f*x)] - 3*A*(-2*e + 2*f*x + Sin[2*(e + f*x)])))/f","A",1
21,1,125,56,0.9281,"\int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx","Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x]),x]","\frac{a (\sin (e+f x)+1) \left(\frac{4 (A+B) \sin \left(\frac{f x}{2}\right)}{f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}-(x (A+2 B))-\frac{B \sin (e) \sin (f x)}{f}+\frac{B \cos (e) \cos (f x)}{f}\right)}{c \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","\frac{2 a (A+B) \cos (e+f x)}{f (c-c \sin (e+f x))}-\frac{a x (A+2 B)}{c}+\frac{a B \cos (e+f x)}{c f}",1,"(a*(-((A + 2*B)*x) + (B*Cos[e]*Cos[f*x])/f - (B*Sin[e]*Sin[f*x])/f + (4*(A + B)*Sin[(f*x)/2])/(f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(1 + Sin[e + f*x]))/(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)","B",1
22,1,160,72,0.6253255,"\int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx","Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^2,x]","-\frac{a \left(-6 (A+3 B) \cos \left(e+\frac{f x}{2}\right)+2 A \cos \left(e+\frac{3 f x}{2}\right)+9 B f x \sin \left(e+\frac{f x}{2}\right)+3 B f x \sin \left(e+\frac{3 f x}{2}\right)+14 B \cos \left(e+\frac{3 f x}{2}\right)+3 B f x \cos \left(2 e+\frac{3 f x}{2}\right)+24 B \sin \left(\frac{f x}{2}\right)-9 B f x \cos \left(\frac{f x}{2}\right)\right)}{6 c^2 f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\frac{a (A+7 B) \cos (e+f x)}{3 c^2 f (1-\sin (e+f x))}+\frac{2 a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^2}+\frac{a B x}{c^2}",1,"-1/6*(a*(-9*B*f*x*Cos[(f*x)/2] - 6*(A + 3*B)*Cos[e + (f*x)/2] + 2*A*Cos[e + (3*f*x)/2] + 14*B*Cos[e + (3*f*x)/2] + 3*B*f*x*Cos[2*e + (3*f*x)/2] + 24*B*Sin[(f*x)/2] + 9*B*f*x*Sin[e + (f*x)/2] + 3*B*f*x*Sin[e + (3*f*x)/2]))/(c^2*f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3)","B",1
23,1,147,104,0.7363166,"\int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx","Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^3,x]","\frac{a \left(15 (A-B) \cos \left(e+\frac{f x}{2}\right)-5 (A-B) \cos \left(e+\frac{3 f x}{2}\right)+A \sin \left(2 e+\frac{5 f x}{2}\right)+5 A \sin \left(\frac{f x}{2}\right)+15 B \sin \left(2 e+\frac{3 f x}{2}\right)-4 B \sin \left(2 e+\frac{5 f x}{2}\right)+25 B \sin \left(\frac{f x}{2}\right)\right)}{30 c^3 f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\frac{a (A-4 B) \cos (e+f x)}{15 f \left(c^3-c^3 \sin (e+f x)\right)}-\frac{a c (A+11 B) \cos (e+f x)}{15 f \left(c^2-c^2 \sin (e+f x)\right)^2}+\frac{2 a (A+B) \cos (e+f x)}{5 f (c-c \sin (e+f x))^3}",1,"(a*(15*(A - B)*Cos[e + (f*x)/2] - 5*(A - B)*Cos[e + (3*f*x)/2] + 5*A*Sin[(f*x)/2] + 25*B*Sin[(f*x)/2] + 15*B*Sin[2*e + (3*f*x)/2] + A*Sin[2*e + (5*f*x)/2] - 4*B*Sin[2*e + (5*f*x)/2]))/(30*c^3*f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5)","A",1
24,1,174,142,0.9022321,"\int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx","Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^4,x]","\frac{a \left(35 (4 A-B) \cos \left(e+\frac{f x}{2}\right)+14 A \sin \left(2 e+\frac{5 f x}{2}\right)-42 A \cos \left(e+\frac{3 f x}{2}\right)+2 A \cos \left(3 e+\frac{7 f x}{2}\right)+70 A \sin \left(\frac{f x}{2}\right)+105 B \sin \left(2 e+\frac{3 f x}{2}\right)-35 B \sin \left(2 e+\frac{5 f x}{2}\right)-5 B \cos \left(3 e+\frac{7 f x}{2}\right)+140 B \sin \left(\frac{f x}{2}\right)\right)}{420 c^4 f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\frac{a (2 A-5 B) \cos (e+f x)}{105 f \left(c^4-c^4 \sin (e+f x)\right)}-\frac{a (2 A-5 B) \cos (e+f x)}{105 f \left(c^2-c^2 \sin (e+f x)\right)^2}-\frac{a (A+15 B) \cos (e+f x)}{35 c f (c-c \sin (e+f x))^3}+\frac{2 a (A+B) \cos (e+f x)}{7 f (c-c \sin (e+f x))^4}",1,"(a*(35*(4*A - B)*Cos[e + (f*x)/2] - 42*A*Cos[e + (3*f*x)/2] + 2*A*Cos[3*e + (7*f*x)/2] - 5*B*Cos[3*e + (7*f*x)/2] + 70*A*Sin[(f*x)/2] + 140*B*Sin[(f*x)/2] + 105*B*Sin[2*e + (3*f*x)/2] + 14*A*Sin[2*e + (5*f*x)/2] - 35*B*Sin[2*e + (5*f*x)/2]))/(420*c^4*f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7)","A",1
25,1,200,176,0.888395,"\int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx","Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^5,x]","\frac{a \left(-42 (2 A+B) \cos \left(e+\frac{3 f x}{2}\right)+36 A \sin \left(2 e+\frac{5 f x}{2}\right)-A \sin \left(4 e+\frac{9 f x}{2}\right)+315 A \cos \left(e+\frac{f x}{2}\right)+9 A \cos \left(3 e+\frac{7 f x}{2}\right)+189 A \sin \left(\frac{f x}{2}\right)+210 B \sin \left(2 e+\frac{3 f x}{2}\right)-72 B \sin \left(2 e+\frac{5 f x}{2}\right)+2 B \sin \left(4 e+\frac{9 f x}{2}\right)-18 B \cos \left(3 e+\frac{7 f x}{2}\right)+252 B \sin \left(\frac{f x}{2}\right)\right)}{1260 c^5 f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}","-\frac{2 a (A-2 B) \cos (e+f x)}{315 f \left(c^5-c^5 \sin (e+f x)\right)}-\frac{2 a c (A-2 B) \cos (e+f x)}{315 f \left(c^3-c^3 \sin (e+f x)\right)^2}-\frac{a c (A-2 B) \cos (e+f x)}{105 f \left(c^2-c^2 \sin (e+f x)\right)^3}-\frac{a (A+19 B) \cos (e+f x)}{63 c f (c-c \sin (e+f x))^4}+\frac{2 a (A+B) \cos (e+f x)}{9 f (c-c \sin (e+f x))^5}",1,"(a*(315*A*Cos[e + (f*x)/2] - 42*(2*A + B)*Cos[e + (3*f*x)/2] + 9*A*Cos[3*e + (7*f*x)/2] - 18*B*Cos[3*e + (7*f*x)/2] + 189*A*Sin[(f*x)/2] + 252*B*Sin[(f*x)/2] + 210*B*Sin[2*e + (3*f*x)/2] + 36*A*Sin[2*e + (5*f*x)/2] - 72*B*Sin[2*e + (5*f*x)/2] - A*Sin[4*e + (9*f*x)/2] + 2*B*Sin[4*e + (9*f*x)/2]))/(1260*c^5*f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9)","A",1
26,1,219,229,2.0425254,"\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx","Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5,x]","\frac{(a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^5 (2520 (8 A-3 B) (e+f x)+560 (19 A-3 B) \sin (2 (e+f x))-280 (2 A-7 B) \sin (4 (e+f x))-560 (A-B) \sin (6 (e+f x))+560 (27 A-17 B) \cos (e+f x)+560 (13 A-7 B) \cos (3 (e+f x))+112 (11 A-B) \cos (5 (e+f x))-80 (A-3 B) \cos (7 (e+f x))-35 B \sin (8 (e+f x)))}{35840 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{10} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","\frac{3 a^2 c^5 (8 A-3 B) \cos ^5(e+f x)}{80 f}+\frac{3 a^2 (8 A-3 B) \cos ^5(e+f x) \left(c^5-c^5 \sin (e+f x)\right)}{112 f}+\frac{3 a^2 c^5 (8 A-3 B) \sin (e+f x) \cos ^3(e+f x)}{64 f}+\frac{9 a^2 c^5 (8 A-3 B) \sin (e+f x) \cos (e+f x)}{128 f}+\frac{9}{128} a^2 c^5 x (8 A-3 B)+\frac{a^2 c^3 (8 A-3 B) \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac{a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}",1,"((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5*(2520*(8*A - 3*B)*(e + f*x) + 560*(27*A - 17*B)*Cos[e + f*x] + 560*(13*A - 7*B)*Cos[3*(e + f*x)] + 112*(11*A - B)*Cos[5*(e + f*x)] - 80*(A - 3*B)*Cos[7*(e + f*x)] + 560*(19*A - 3*B)*Sin[2*(e + f*x)] - 280*(2*A - 7*B)*Sin[4*(e + f*x)] - 560*(A - B)*Sin[6*(e + f*x)] - 35*B*Sin[8*(e + f*x)]))/(35840*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)","A",1
27,1,163,189,1.5868447,"\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx","Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4,x]","\frac{a^2 c^4 (105 (16 A-11 B) \cos (e+f x)+105 (8 A-5 B) \cos (3 (e+f x))+1785 A \sin (2 (e+f x))+105 A \sin (4 (e+f x))-35 A \sin (6 (e+f x))+168 A \cos (5 (e+f x))+2940 A e+2940 A f x-210 B \sin (2 (e+f x))+210 B \sin (4 (e+f x))+70 B \sin (6 (e+f x))-63 B \cos (5 (e+f x))+15 B \cos (7 (e+f x))-840 B e-840 B f x)}{6720 f}","\frac{a^2 c^4 (7 A-2 B) \cos ^5(e+f x)}{30 f}+\frac{a^2 (7 A-2 B) \cos ^5(e+f x) \left(c^4-c^4 \sin (e+f x)\right)}{42 f}+\frac{a^2 c^4 (7 A-2 B) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{a^2 c^4 (7 A-2 B) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} a^2 c^4 x (7 A-2 B)-\frac{a^2 B \cos ^5(e+f x) \left(c^2-c^2 \sin (e+f x)\right)^2}{7 f}",1,"(a^2*c^4*(2940*A*e - 840*B*e + 2940*A*f*x - 840*B*f*x + 105*(16*A - 11*B)*Cos[e + f*x] + 105*(8*A - 5*B)*Cos[3*(e + f*x)] + 168*A*Cos[5*(e + f*x)] - 63*B*Cos[5*(e + f*x)] + 15*B*Cos[7*(e + f*x)] + 1785*A*Sin[2*(e + f*x)] - 210*B*Sin[2*(e + f*x)] + 105*A*Sin[4*(e + f*x)] + 210*B*Sin[4*(e + f*x)] - 35*A*Sin[6*(e + f*x)] + 70*B*Sin[6*(e + f*x)]))/(6720*f)","A",1
28,1,137,147,1.0852929,"\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3,x]","\frac{a^2 c^3 (120 (A-B) \cos (e+f x)+60 (A-B) \cos (3 (e+f x))+240 A \sin (2 (e+f x))+30 A \sin (4 (e+f x))+12 A \cos (5 (e+f x))+360 A e+360 A f x-15 B \sin (2 (e+f x))+15 B \sin (4 (e+f x))+5 B \sin (6 (e+f x))-12 B \cos (5 (e+f x))-60 B e-60 B f x)}{960 f}","\frac{a^2 c^3 (6 A-B) \cos ^5(e+f x)}{30 f}+\frac{a^2 c^3 (6 A-B) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{a^2 c^3 (6 A-B) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} a^2 c^3 x (6 A-B)-\frac{a^2 B \cos ^5(e+f x) \left(c^3-c^3 \sin (e+f x)\right)}{6 f}",1,"(a^2*c^3*(360*A*e - 60*B*e + 360*A*f*x - 60*B*f*x + 120*(A - B)*Cos[e + f*x] + 60*(A - B)*Cos[3*(e + f*x)] + 12*A*Cos[5*(e + f*x)] - 12*B*Cos[5*(e + f*x)] + 240*A*Sin[2*(e + f*x)] - 15*B*Sin[2*(e + f*x)] + 30*A*Sin[4*(e + f*x)] + 15*B*Sin[4*(e + f*x)] + 5*B*Sin[6*(e + f*x)]))/(960*f)","A",1
29,1,54,89,0.1493289,"\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2,x]","\frac{a^2 c^2 \left(5 A (12 (e+f x)+8 \sin (2 (e+f x))+\sin (4 (e+f x)))-32 B \cos ^5(e+f x)\right)}{160 f}","\frac{a^2 A c^2 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac{3 a^2 A c^2 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} a^2 A c^2 x-\frac{a^2 B c^2 \cos ^5(e+f x)}{5 f}",1,"(a^2*c^2*(-32*B*Cos[e + f*x]^5 + 5*A*(12*(e + f*x) + 8*Sin[2*(e + f*x)] + Sin[4*(e + f*x)])))/(160*f)","A",1
30,1,67,98,0.7957653,"\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x]),x]","-\frac{a^2 c (24 (A+B) \cos (e+f x)+8 (A+B) \cos (3 (e+f x))-12 f x (4 A+B)-24 A \sin (2 (e+f x))+3 B \sin (4 (e+f x)))}{96 f}","-\frac{a^2 c (4 A+B) \cos ^3(e+f x)}{12 f}+\frac{a^2 c (4 A+B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a^2 c x (4 A+B)-\frac{B c \cos ^3(e+f x) \left(a^2 \sin (e+f x)+a^2\right)}{4 f}",1,"-1/96*(a^2*c*(-12*(4*A + B)*f*x + 24*(A + B)*Cos[e + f*x] + 8*(A + B)*Cos[3*(e + f*x)] - 24*A*Sin[2*(e + f*x)] + 3*B*Sin[4*(e + f*x)]))/f","A",1
31,1,191,117,1.2992267,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x]),x]","\frac{a^2 (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right) (6 (2 A+3 B) (e+f x)-4 (A+3 B) \cos (e+f x)-B \sin (2 (e+f x)))-\sin \left(\frac{1}{2} (e+f x)\right) (-4 (A+3 B) \cos (e+f x)+4 A (3 e+3 f x+8)+2 B (9 e+9 f x+16)-B \sin (2 (e+f x)))\right)}{4 c f (\sin (e+f x)-1) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","\frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{f (c-c \sin (e+f x))^3}+\frac{3 a^2 (2 A+3 B) \cos (e+f x)}{2 c f}+\frac{a^2 (2 A+3 B) \cos ^3(e+f x)}{2 f (c-c \sin (e+f x))}-\frac{3 a^2 x (2 A+3 B)}{2 c}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2*(Cos[(e + f*x)/2]*(6*(2*A + 3*B)*(e + f*x) - 4*(A + 3*B)*Cos[e + f*x] - B*Sin[2*(e + f*x)]) - Sin[(e + f*x)/2]*(4*A*(8 + 3*e + 3*f*x) + 2*B*(16 + 9*e + 9*f*x) - 4*(A + 3*B)*Cos[e + f*x] - B*Sin[2*(e + f*x)])))/(4*c*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-1 + Sin[e + f*x]))","A",1
32,1,238,109,0.6309993,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^2,x]","\frac{a^2 (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(8 (A+B) \sin \left(\frac{1}{2} (e+f x)\right)+3 (A+4 B) (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-8 (2 A+5 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+4 (A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-3 B \cos (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{3 f (c-c \sin (e+f x))^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","-\frac{a^2 (A+4 B) \cos (e+f x)}{c^2 f}+\frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^4}+\frac{a^2 x (A+4 B)}{c^2}-\frac{2 a^2 (A+4 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^2}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(4*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) + 3*(A + 4*B)*(e + f*x)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 - 3*B*Cos[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + 8*(A + B)*Sin[(e + f*x)/2] - 8*(2*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2)/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^2)","B",1
33,1,278,112,0.736409,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^3,x]","\frac{a^2 (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(24 (A+B) \sin \left(\frac{1}{2} (e+f x)\right)+2 (3 A+43 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4-4 (3 A+8 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-8 (3 A+8 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+12 (A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-15 B (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5\right)}{15 f (c-c \sin (e+f x))^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","\frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}+\frac{2 a^2 B \cos (e+f x)}{f \left(c^3-c^3 \sin (e+f x)\right)}-\frac{a^2 B x}{c^3}-\frac{2 a^2 B \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(12*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 4*(3*A + 8*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 - 15*B*(e + f*x)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + 24*(A + B)*Sin[(e + f*x)/2] - 8*(3*A + 8*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + 2*(3*A + 43*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2)/(15*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^3)","B",1
34,1,191,75,0.9503716,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^4,x]","-\frac{a^2 \left(-35 (A+4 B) \cos \left(\frac{1}{2} (e+f x)\right)+7 (2 A+13 B) \cos \left(\frac{3}{2} (e+f x)\right)-70 A \sin \left(\frac{1}{2} (e+f x)\right)-35 A \sin \left(\frac{3}{2} (e+f x)\right)+7 A \sin \left(\frac{5}{2} (e+f x)\right)+A \cos \left(\frac{7}{2} (e+f x)\right)+70 B \sin \left(\frac{1}{2} (e+f x)\right)+35 B \sin \left(\frac{3}{2} (e+f x)\right)-7 B \sin \left(\frac{5}{2} (e+f x)\right)+35 B \cos \left(\frac{5}{2} (e+f x)\right)-6 B \cos \left(\frac{7}{2} (e+f x)\right)\right)}{140 c^4 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}","\frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}+\frac{a^2 c (A-6 B) \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^5}",1,"-1/140*(a^2*(-35*(A + 4*B)*Cos[(e + f*x)/2] + 7*(2*A + 13*B)*Cos[(3*(e + f*x))/2] + 35*B*Cos[(5*(e + f*x))/2] + A*Cos[(7*(e + f*x))/2] - 6*B*Cos[(7*(e + f*x))/2] - 70*A*Sin[(e + f*x)/2] + 70*B*Sin[(e + f*x)/2] - 35*A*Sin[(3*(e + f*x))/2] + 35*B*Sin[(3*(e + f*x))/2] + 7*A*Sin[(5*(e + f*x))/2] - 7*B*Sin[(5*(e + f*x))/2]))/(c^4*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7)","B",1
35,1,261,115,1.2422096,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^5,x]","-\frac{a^2 (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(315 (2 A+3 B) \cos \left(\frac{1}{2} (e+f x)\right)-63 (4 A+11 B) \cos \left(\frac{3}{2} (e+f x)\right)+882 A \sin \left(\frac{1}{2} (e+f x)\right)+420 A \sin \left(\frac{3}{2} (e+f x)\right)-72 A \sin \left(\frac{5}{2} (e+f x)\right)+2 A \sin \left(\frac{9}{2} (e+f x)\right)-18 A \cos \left(\frac{7}{2} (e+f x)\right)+63 B \sin \left(\frac{1}{2} (e+f x)\right)+105 B \sin \left(\frac{3}{2} (e+f x)\right)-63 B \sin \left(\frac{5}{2} (e+f x)\right)-7 B \sin \left(\frac{9}{2} (e+f x)\right)-315 B \cos \left(\frac{5}{2} (e+f x)\right)+63 B \cos \left(\frac{7}{2} (e+f x)\right)\right)}{2520 c^5 f (\sin (e+f x)-1)^5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","\frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac{a^2 (2 A-7 B) \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5}+\frac{a^2 c (2 A-7 B) \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6}",1,"-1/2520*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2*(315*(2*A + 3*B)*Cos[(e + f*x)/2] - 63*(4*A + 11*B)*Cos[(3*(e + f*x))/2] - 315*B*Cos[(5*(e + f*x))/2] - 18*A*Cos[(7*(e + f*x))/2] + 63*B*Cos[(7*(e + f*x))/2] + 882*A*Sin[(e + f*x)/2] + 63*B*Sin[(e + f*x)/2] + 420*A*Sin[(3*(e + f*x))/2] + 105*B*Sin[(3*(e + f*x))/2] - 72*A*Sin[(5*(e + f*x))/2] - 63*B*Sin[(5*(e + f*x))/2] + 2*A*Sin[(9*(e + f*x))/2] - 7*B*Sin[(9*(e + f*x))/2]))/(c^5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-1 + Sin[e + f*x])^5)","B",1
36,1,285,156,1.599098,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^6,x]","\frac{a^2 (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(231 (27 A+28 B) \cos \left(\frac{1}{2} (e+f x)\right)-2475 (A+2 B) \cos \left(\frac{3}{2} (e+f x)\right)+7623 A \sin \left(\frac{1}{2} (e+f x)\right)+3465 A \sin \left(\frac{3}{2} (e+f x)\right)-495 A \sin \left(\frac{5}{2} (e+f x)\right)+33 A \sin \left(\frac{9}{2} (e+f x)\right)-165 A \cos \left(\frac{7}{2} (e+f x)\right)+3 A \cos \left(\frac{11}{2} (e+f x)\right)+2772 B \sin \left(\frac{1}{2} (e+f x)\right)+2310 B \sin \left(\frac{3}{2} (e+f x)\right)-990 B \sin \left(\frac{5}{2} (e+f x)\right)-88 B \sin \left(\frac{9}{2} (e+f x)\right)-2310 B \cos \left(\frac{5}{2} (e+f x)\right)+440 B \cos \left(\frac{7}{2} (e+f x)\right)-8 B \cos \left(\frac{11}{2} (e+f x)\right)\right)}{27720 c^6 f (\sin (e+f x)-1)^6 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","\frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{2 a^2 (3 A-8 B) \cos ^5(e+f x)}{3465 c f (c-c \sin (e+f x))^5}+\frac{2 a^2 (3 A-8 B) \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^6}+\frac{a^2 c (3 A-8 B) \cos ^5(e+f x)}{99 f (c-c \sin (e+f x))^7}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2*(231*(27*A + 28*B)*Cos[(e + f*x)/2] - 2475*(A + 2*B)*Cos[(3*(e + f*x))/2] - 2310*B*Cos[(5*(e + f*x))/2] - 165*A*Cos[(7*(e + f*x))/2] + 440*B*Cos[(7*(e + f*x))/2] + 3*A*Cos[(11*(e + f*x))/2] - 8*B*Cos[(11*(e + f*x))/2] + 7623*A*Sin[(e + f*x)/2] + 2772*B*Sin[(e + f*x)/2] + 3465*A*Sin[(3*(e + f*x))/2] + 2310*B*Sin[(3*(e + f*x))/2] - 495*A*Sin[(5*(e + f*x))/2] - 990*B*Sin[(5*(e + f*x))/2] + 33*A*Sin[(9*(e + f*x))/2] - 88*B*Sin[(9*(e + f*x))/2]))/(27720*c^6*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-1 + Sin[e + f*x])^6)","A",1
37,1,313,197,3.6461797,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^7,x]","-\frac{a^2 (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(6006 (8 A+7 B) \cos \left(\frac{1}{2} (e+f x)\right)-1716 (11 A+19 B) \cos \left(\frac{3}{2} (e+f x)\right)+54912 A \sin \left(\frac{1}{2} (e+f x)\right)+24024 A \sin \left(\frac{3}{2} (e+f x)\right)-2860 A \sin \left(\frac{5}{2} (e+f x)\right)+312 A \sin \left(\frac{9}{2} (e+f x)\right)-4 A \sin \left(\frac{13}{2} (e+f x)\right)-1144 A \cos \left(\frac{7}{2} (e+f x)\right)+52 A \cos \left(\frac{11}{2} (e+f x)\right)+26598 B \sin \left(\frac{1}{2} (e+f x)\right)+21021 B \sin \left(\frac{3}{2} (e+f x)\right)-8580 B \sin \left(\frac{5}{2} (e+f x)\right)-702 B \sin \left(\frac{9}{2} (e+f x)\right)+9 B \sin \left(\frac{13}{2} (e+f x)\right)-15015 B \cos \left(\frac{5}{2} (e+f x)\right)+2574 B \cos \left(\frac{7}{2} (e+f x)\right)-117 B \cos \left(\frac{11}{2} (e+f x)\right)\right)}{240240 c^7 f (\sin (e+f x)-1)^7 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","\frac{2 a^2 (4 A-9 B) \cos ^5(e+f x)}{15015 c^2 f (c-c \sin (e+f x))^5}+\frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}+\frac{2 a^2 (4 A-9 B) \cos ^5(e+f x)}{3003 c f (c-c \sin (e+f x))^6}+\frac{a^2 (4 A-9 B) \cos ^5(e+f x)}{429 f (c-c \sin (e+f x))^7}+\frac{a^2 c (4 A-9 B) \cos ^5(e+f x)}{143 f (c-c \sin (e+f x))^8}",1,"-1/240240*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2*(6006*(8*A + 7*B)*Cos[(e + f*x)/2] - 1716*(11*A + 19*B)*Cos[(3*(e + f*x))/2] - 15015*B*Cos[(5*(e + f*x))/2] - 1144*A*Cos[(7*(e + f*x))/2] + 2574*B*Cos[(7*(e + f*x))/2] + 52*A*Cos[(11*(e + f*x))/2] - 117*B*Cos[(11*(e + f*x))/2] + 54912*A*Sin[(e + f*x)/2] + 26598*B*Sin[(e + f*x)/2] + 24024*A*Sin[(3*(e + f*x))/2] + 21021*B*Sin[(3*(e + f*x))/2] - 2860*A*Sin[(5*(e + f*x))/2] - 8580*B*Sin[(5*(e + f*x))/2] + 312*A*Sin[(9*(e + f*x))/2] - 702*B*Sin[(9*(e + f*x))/2] - 4*A*Sin[(13*(e + f*x))/2] + 9*B*Sin[(13*(e + f*x))/2]))/(c^7*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-1 + Sin[e + f*x])^7)","A",1
38,1,255,265,4.3076697,"\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^6 \, dx","Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^6,x]","\frac{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^6 (27720 (10 A-3 B) (e+f x)+1260 (144 A-25 B) \sin (2 (e+f x))+2520 (6 A+7 B) \sin (4 (e+f x))-210 (32 A-51 B) \sin (6 (e+f x))-315 (6 A-5 B) \sin (8 (e+f x))+5040 (33 A-19 B) \cos (e+f x)+3360 (29 A-15 B) \cos (3 (e+f x))+10080 (3 A-B) \cos (5 (e+f x))+360 (9 A+5 B) \cos (7 (e+f x))-280 (A-3 B) \cos (9 (e+f x))-126 B \sin (10 (e+f x)))}{645120 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{12} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{11 a^3 c^6 (10 A-3 B) \cos ^7(e+f x)}{560 f}+\frac{11 a^3 (10 A-3 B) \cos ^7(e+f x) \left(c^6-c^6 \sin (e+f x)\right)}{720 f}+\frac{11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos ^5(e+f x)}{480 f}+\frac{11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos ^3(e+f x)}{384 f}+\frac{11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos (e+f x)}{256 f}+\frac{11}{256} a^3 c^6 x (10 A-3 B)+\frac{a^3 (10 A-3 B) \cos ^7(e+f x) \left(c^3-c^3 \sin (e+f x)\right)^2}{90 f}-\frac{a^3 B \cos ^7(e+f x) \left(c^2-c^2 \sin (e+f x)\right)^3}{10 f}",1,"((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6*(27720*(10*A - 3*B)*(e + f*x) + 5040*(33*A - 19*B)*Cos[e + f*x] + 3360*(29*A - 15*B)*Cos[3*(e + f*x)] + 10080*(3*A - B)*Cos[5*(e + f*x)] + 360*(9*A + 5*B)*Cos[7*(e + f*x)] - 280*(A - 3*B)*Cos[9*(e + f*x)] + 1260*(144*A - 25*B)*Sin[2*(e + f*x)] + 2520*(6*A + 7*B)*Sin[4*(e + f*x)] - 210*(32*A - 51*B)*Sin[6*(e + f*x)] - 315*(6*A - 5*B)*Sin[8*(e + f*x)] - 126*B*Sin[10*(e + f*x)]))/(645120*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^12*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)","A",1
39,1,232,222,2.5709043,"\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx","Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5,x]","\frac{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^5 (2520 (9 A-2 B) (e+f x)+2016 (8 A-B) \sin (2 (e+f x))+504 (5 A+2 B) \sin (4 (e+f x))-63 (A-2 B) \sin (8 (e+f x))+504 (20 A-13 B) \cos (e+f x)+336 (18 A-11 B) \cos (3 (e+f x))+1008 (2 A-B) \cos (5 (e+f x))+36 (8 A-B) \cos (7 (e+f x))+672 B \sin (6 (e+f x))+28 B \cos (9 (e+f x)))}{64512 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{10} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{a^3 c^5 (9 A-2 B) \cos ^7(e+f x)}{56 f}+\frac{a^3 (9 A-2 B) \cos ^7(e+f x) \left(c^5-c^5 \sin (e+f x)\right)}{72 f}+\frac{a^3 c^5 (9 A-2 B) \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac{5 a^3 c^5 (9 A-2 B) \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{5 a^3 c^5 (9 A-2 B) \sin (e+f x) \cos (e+f x)}{128 f}+\frac{5}{128} a^3 c^5 x (9 A-2 B)-\frac{a^3 B c^3 \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}",1,"((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5*(2520*(9*A - 2*B)*(e + f*x) + 504*(20*A - 13*B)*Cos[e + f*x] + 336*(18*A - 11*B)*Cos[3*(e + f*x)] + 1008*(2*A - B)*Cos[5*(e + f*x)] + 36*(8*A - B)*Cos[7*(e + f*x)] + 28*B*Cos[9*(e + f*x)] + 2016*(8*A - B)*Sin[2*(e + f*x)] + 504*(5*A + 2*B)*Sin[4*(e + f*x)] + 672*B*Sin[6*(e + f*x)] - 63*(A - 2*B)*Sin[8*(e + f*x)]))/(64512*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)","A",1
40,1,209,181,1.9509649,"\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx","Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4,x]","\frac{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^4 (840 (8 A-B) (e+f x)+336 (15 A-B) \sin (2 (e+f x))+168 (6 A+B) \sin (4 (e+f x))+112 (A+B) \sin (6 (e+f x))+1680 (A-B) \cos (e+f x)+1008 (A-B) \cos (3 (e+f x))+336 (A-B) \cos (5 (e+f x))+48 (A-B) \cos (7 (e+f x))+21 B \sin (8 (e+f x)))}{21504 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^8 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{a^3 c^4 (8 A-B) \cos ^7(e+f x)}{56 f}+\frac{a^3 c^4 (8 A-B) \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac{5 a^3 c^4 (8 A-B) \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{5 a^3 c^4 (8 A-B) \sin (e+f x) \cos (e+f x)}{128 f}+\frac{5}{128} a^3 c^4 x (8 A-B)-\frac{a^3 B \cos ^7(e+f x) \left(c^4-c^4 \sin (e+f x)\right)}{8 f}",1,"((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4*(840*(8*A - B)*(e + f*x) + 1680*(A - B)*Cos[e + f*x] + 1008*(A - B)*Cos[3*(e + f*x)] + 336*(A - B)*Cos[5*(e + f*x)] + 48*(A - B)*Cos[7*(e + f*x)] + 336*(15*A - B)*Sin[2*(e + f*x)] + 168*(6*A + B)*Sin[4*(e + f*x)] + 112*(A + B)*Sin[6*(e + f*x)] + 21*B*Sin[8*(e + f*x)]))/(21504*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)","A",1
41,1,64,117,0.2289668,"\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3,x]","\frac{a^3 c^3 \left(7 A (45 \sin (2 (e+f x))+9 \sin (4 (e+f x))+\sin (6 (e+f x))+60 e+60 f x)-192 B \cos ^7(e+f x)\right)}{1344 f}","\frac{a^3 A c^3 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{5 a^3 A c^3 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{5 a^3 A c^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} a^3 A c^3 x-\frac{a^3 B c^3 \cos ^7(e+f x)}{7 f}",1,"(a^3*c^3*(-192*B*Cos[e + f*x]^7 + 7*A*(60*e + 60*f*x + 45*Sin[2*(e + f*x)] + 9*Sin[4*(e + f*x)] + Sin[6*(e + f*x)])))/(1344*f)","A",1
42,1,133,138,1.0863759,"\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2,x]","\frac{a^3 c^2 (-120 (A+B) \cos (e+f x)-60 (A+B) \cos (3 (e+f x))+240 A \sin (2 (e+f x))+30 A \sin (4 (e+f x))-12 A \cos (5 (e+f x))+360 A e+360 A f x+15 B \sin (2 (e+f x))-15 B \sin (4 (e+f x))-5 B \sin (6 (e+f x))-12 B \cos (5 (e+f x))+60 B e+60 B f x)}{960 f}","-\frac{a^3 c^2 (6 A+B) \cos ^5(e+f x)}{30 f}+\frac{a^3 c^2 (6 A+B) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{a^3 c^2 (6 A+B) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} a^3 c^2 x (6 A+B)-\frac{B c^2 \cos ^5(e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{6 f}",1,"(a^3*c^2*(360*A*e + 60*B*e + 360*A*f*x + 60*B*f*x - 120*(A + B)*Cos[e + f*x] - 60*(A + B)*Cos[3*(e + f*x)] - 12*A*Cos[5*(e + f*x)] - 12*B*Cos[5*(e + f*x)] + 240*A*Sin[2*(e + f*x)] + 15*B*Sin[2*(e + f*x)] + 30*A*Sin[4*(e + f*x)] - 15*B*Sin[4*(e + f*x)] - 5*B*Sin[6*(e + f*x)]))/(960*f)","A",1
43,1,95,140,0.854111,"\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x]),x]","\frac{a^3 c (15 (-(A+2 B) \sin (4 (e+f x))+4 f x (5 A+2 B)+8 A \sin (2 (e+f x)))-60 (4 A+3 B) \cos (e+f x)-10 (8 A+5 B) \cos (3 (e+f x))+6 B \cos (5 (e+f x)))}{480 f}","-\frac{a^3 c (5 A+2 B) \cos ^3(e+f x)}{12 f}-\frac{c (5 A+2 B) \cos ^3(e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{20 f}+\frac{a^3 c (5 A+2 B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a^3 c x (5 A+2 B)-\frac{a B c \cos ^3(e+f x) (a \sin (e+f x)+a)^2}{5 f}",1,"(a^3*c*(-60*(4*A + 3*B)*Cos[e + f*x] - 10*(8*A + 5*B)*Cos[3*(e + f*x)] + 6*B*Cos[5*(e + f*x)] + 15*(4*(5*A + 2*B)*f*x + 8*A*Sin[2*(e + f*x)] - (A + 2*B)*Sin[4*(e + f*x)])))/(480*f)","A",1
44,1,223,156,1.4235402,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x]),x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right) (30 (3 A+4 B) (e+f x)-3 (A+4 B) \sin (2 (e+f x))-3 (16 A+31 B) \cos (e+f x)+B \cos (3 (e+f x)))-\sin \left(\frac{1}{2} (e+f x)\right) (-3 (A+4 B) \sin (2 (e+f x))-3 (16 A+31 B) \cos (e+f x)+6 A (15 e+15 f x+32)+24 B (5 e+5 f x+8)+B \cos (3 (e+f x)))\right)}{12 c f (\sin (e+f x)-1) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{f (c-c \sin (e+f x))^4}+\frac{2 a^3 c^3 (3 A+4 B) \cos ^5(e+f x)}{f \left(c^2-c^2 \sin (e+f x)\right)^2}+\frac{5 a^3 (3 A+4 B) \cos ^3(e+f x)}{3 c f}-\frac{5 a^3 (3 A+4 B) \sin (e+f x) \cos (e+f x)}{2 c f}-\frac{5 a^3 x (3 A+4 B)}{2 c}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(Cos[(e + f*x)/2]*(30*(3*A + 4*B)*(e + f*x) - 3*(16*A + 31*B)*Cos[e + f*x] + B*Cos[3*(e + f*x)] - 3*(A + 4*B)*Sin[2*(e + f*x)]) - Sin[(e + f*x)/2]*(24*B*(8 + 5*e + 5*f*x) + 6*A*(32 + 15*e + 15*f*x) - 3*(16*A + 31*B)*Cos[e + f*x] + B*Cos[3*(e + f*x)] - 3*(A + 4*B)*Sin[2*(e + f*x)])))/(12*c*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-1 + Sin[e + f*x]))","A",1
45,1,280,163,0.8772631,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^2,x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(64 (A+B) \sin \left(\frac{1}{2} (e+f x)\right)+30 (2 A+5 B) (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-12 (A+5 B) \cos (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-32 (7 A+13 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+32 (A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-3 B \sin (2 (e+f x)) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{12 f (c-c \sin (e+f x))^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{3 f (c-c \sin (e+f x))^5}-\frac{5 a^3 (2 A+5 B) \cos (e+f x)}{2 c^2 f}-\frac{5 a^3 (2 A+5 B) \cos ^3(e+f x)}{6 f \left(c^2-c^2 \sin (e+f x)\right)}+\frac{5 a^3 x (2 A+5 B)}{2 c^2}-\frac{2 a^3 c (2 A+5 B) \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^3}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(32*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) + 30*(2*A + 5*B)*(e + f*x)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 - 12*(A + 5*B)*Cos[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + 64*(A + B)*Sin[(e + f*x)/2] - 32*(7*A + 13*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] - 3*B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sin[2*(e + f*x)]))/(12*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^2)","A",1
46,1,316,153,1.1045157,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^3,x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(48 (A+B) \sin \left(\frac{1}{2} (e+f x)\right)-15 (A+6 B) (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5+4 (23 A+93 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4-4 (11 A+21 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-8 (11 A+21 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+24 (A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+15 B \cos (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5\right)}{15 f (c-c \sin (e+f x))^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{a^3 (A+6 B) \cos (e+f x)}{c^3 f}+\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}+\frac{2 a^3 c^3 (A+6 B) \cos ^3(e+f x)}{3 f \left(c^3-c^3 \sin (e+f x)\right)^2}-\frac{a^3 x (A+6 B)}{c^3}-\frac{2 a^3 c (A+6 B) \cos ^5(e+f x)}{15 f (c-c \sin (e+f x))^4}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(24*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 4*(11*A + 21*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 - 15*(A + 6*B)*(e + f*x)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + 15*B*Cos[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + 48*(A + B)*Sin[(e + f*x)/2] - 8*(11*A + 21*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + 4*(23*A + 93*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3)/(15*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^3)","B",1
47,1,356,151,1.2124377,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^4,x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(240 (A+B) \sin \left(\frac{1}{2} (e+f x)\right)-2 (15 A+337 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6+2 (45 A+199 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5+4 (45 A+199 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4-12 (15 A+29 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-24 (15 A+29 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+120 (A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+105 B (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7\right)}{105 f (c-c \sin (e+f x))^4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac{2 a^3 B \cos (e+f x)}{f \left(c^4-c^4 \sin (e+f x)\right)}+\frac{a^3 B x}{c^4}+\frac{2 a^3 B c^2 \cos ^3(e+f x)}{3 f \left(c^2-c^2 \sin (e+f x)\right)^3}-\frac{2 a^3 B c \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(120*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 12*(15*A + 29*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + 2*(45*A + 199*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + 105*B*(e + f*x)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7 + 240*(A + B)*Sin[(e + f*x)/2] - 24*(15*A + 29*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + 4*(45*A + 199*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(e + f*x)/2] - 2*(15*A + 337*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6*Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3)/(105*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^4)","B",1
48,1,283,77,2.5158138,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^5,x]","-\frac{a^3 (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(315 (A-B) \cos \left(\frac{1}{2} (e+f x)\right)-189 (A-B) \cos \left(\frac{3}{2} (e+f x)\right)+189 A \sin \left(\frac{1}{2} (e+f x)\right)+105 A \sin \left(\frac{3}{2} (e+f x)\right)-27 A \sin \left(\frac{5}{2} (e+f x)\right)-A \sin \left(\frac{9}{2} (e+f x)\right)-63 A \cos \left(\frac{5}{2} (e+f x)\right)+9 A \cos \left(\frac{7}{2} (e+f x)\right)+693 B \sin \left(\frac{1}{2} (e+f x)\right)+483 B \sin \left(\frac{3}{2} (e+f x)\right)-225 B \sin \left(\frac{5}{2} (e+f x)\right)-63 B \sin \left(\frac{7}{2} (e+f x)\right)+8 B \sin \left(\frac{9}{2} (e+f x)\right)+63 B \cos \left(\frac{5}{2} (e+f x)\right)-9 B \cos \left(\frac{7}{2} (e+f x)\right)\right)}{504 c^5 f (\sin (e+f x)-1)^5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}+\frac{a^3 c^2 (A-8 B) \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^7}",1,"-1/504*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(315*(A - B)*Cos[(e + f*x)/2] - 189*(A - B)*Cos[(3*(e + f*x))/2] - 63*A*Cos[(5*(e + f*x))/2] + 63*B*Cos[(5*(e + f*x))/2] + 9*A*Cos[(7*(e + f*x))/2] - 9*B*Cos[(7*(e + f*x))/2] + 189*A*Sin[(e + f*x)/2] + 693*B*Sin[(e + f*x)/2] + 105*A*Sin[(3*(e + f*x))/2] + 483*B*Sin[(3*(e + f*x))/2] - 27*A*Sin[(5*(e + f*x))/2] - 225*B*Sin[(5*(e + f*x))/2] - 63*B*Sin[(7*(e + f*x))/2] - A*Sin[(9*(e + f*x))/2] + 8*B*Sin[(9*(e + f*x))/2]))/(c^5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-1 + Sin[e + f*x])^5)","B",1
49,1,313,118,2.9385529,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^6,x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(462 (11 A+3 B) \cos \left(\frac{1}{2} (e+f x)\right)-594 (5 A+2 B) \cos \left(\frac{3}{2} (e+f x)\right)+4158 A \sin \left(\frac{1}{2} (e+f x)\right)+2310 A \sin \left(\frac{3}{2} (e+f x)\right)-594 A \sin \left(\frac{5}{2} (e+f x)\right)-22 A \sin \left(\frac{9}{2} (e+f x)\right)-924 A \cos \left(\frac{5}{2} (e+f x)\right)+110 A \cos \left(\frac{7}{2} (e+f x)\right)-2 A \cos \left(\frac{11}{2} (e+f x)\right)+5544 B \sin \left(\frac{1}{2} (e+f x)\right)+4158 B \sin \left(\frac{3}{2} (e+f x)\right)-2178 B \sin \left(\frac{5}{2} (e+f x)\right)-693 B \sin \left(\frac{7}{2} (e+f x)\right)+99 B \sin \left(\frac{9}{2} (e+f x)\right)-693 B \cos \left(\frac{5}{2} (e+f x)\right)+198 B \cos \left(\frac{7}{2} (e+f x)\right)+9 B \cos \left(\frac{11}{2} (e+f x)\right)\right)}{11088 c^6 f (\sin (e+f x)-1)^6 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac{a^3 c^2 (2 A-9 B) \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac{a^3 c (2 A-9 B) \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^7}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(462*(11*A + 3*B)*Cos[(e + f*x)/2] - 594*(5*A + 2*B)*Cos[(3*(e + f*x))/2] - 924*A*Cos[(5*(e + f*x))/2] - 693*B*Cos[(5*(e + f*x))/2] + 110*A*Cos[(7*(e + f*x))/2] + 198*B*Cos[(7*(e + f*x))/2] - 2*A*Cos[(11*(e + f*x))/2] + 9*B*Cos[(11*(e + f*x))/2] + 4158*A*Sin[(e + f*x)/2] + 5544*B*Sin[(e + f*x)/2] + 2310*A*Sin[(3*(e + f*x))/2] + 4158*B*Sin[(3*(e + f*x))/2] - 594*A*Sin[(5*(e + f*x))/2] - 2178*B*Sin[(5*(e + f*x))/2] - 693*B*Sin[(7*(e + f*x))/2] - 22*A*Sin[(9*(e + f*x))/2] + 99*B*Sin[(9*(e + f*x))/2]))/(11088*c^6*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-1 + Sin[e + f*x])^6)","B",1
50,1,339,156,5.2852668,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^7,x]","-\frac{a^3 (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(6006 (9 A+5 B) \cos \left(\frac{1}{2} (e+f x)\right)-7722 (4 A+3 B) \cos \left(\frac{3}{2} (e+f x)\right)+48906 A \sin \left(\frac{1}{2} (e+f x)\right)+27027 A \sin \left(\frac{3}{2} (e+f x)\right)-6864 A \sin \left(\frac{5}{2} (e+f x)\right)-234 A \sin \left(\frac{9}{2} (e+f x)\right)+3 A \sin \left(\frac{13}{2} (e+f x)\right)-9009 A \cos \left(\frac{5}{2} (e+f x)\right)+858 A \cos \left(\frac{7}{2} (e+f x)\right)-39 A \cos \left(\frac{11}{2} (e+f x)\right)+47190 B \sin \left(\frac{1}{2} (e+f x)\right)+36036 B \sin \left(\frac{3}{2} (e+f x)\right)-19162 B \sin \left(\frac{5}{2} (e+f x)\right)-6006 B \sin \left(\frac{7}{2} (e+f x)\right)+780 B \sin \left(\frac{9}{2} (e+f x)\right)-10 B \sin \left(\frac{13}{2} (e+f x)\right)-12012 B \cos \left(\frac{5}{2} (e+f x)\right)+3146 B \cos \left(\frac{7}{2} (e+f x)\right)+130 B \cos \left(\frac{11}{2} (e+f x)\right)\right)}{144144 c^7 f (\sin (e+f x)-1)^7 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac{a^3 c^2 (3 A-10 B) \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac{2 a^3 (3 A-10 B) \cos ^7(e+f x)}{9009 f (c-c \sin (e+f x))^7}+\frac{2 a^3 c (3 A-10 B) \cos ^7(e+f x)}{1287 f (c-c \sin (e+f x))^8}",1,"-1/144144*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(6006*(9*A + 5*B)*Cos[(e + f*x)/2] - 7722*(4*A + 3*B)*Cos[(3*(e + f*x))/2] - 9009*A*Cos[(5*(e + f*x))/2] - 12012*B*Cos[(5*(e + f*x))/2] + 858*A*Cos[(7*(e + f*x))/2] + 3146*B*Cos[(7*(e + f*x))/2] - 39*A*Cos[(11*(e + f*x))/2] + 130*B*Cos[(11*(e + f*x))/2] + 48906*A*Sin[(e + f*x)/2] + 47190*B*Sin[(e + f*x)/2] + 27027*A*Sin[(3*(e + f*x))/2] + 36036*B*Sin[(3*(e + f*x))/2] - 6864*A*Sin[(5*(e + f*x))/2] - 19162*B*Sin[(5*(e + f*x))/2] - 6006*B*Sin[(7*(e + f*x))/2] - 234*A*Sin[(9*(e + f*x))/2] + 780*B*Sin[(9*(e + f*x))/2] + 3*A*Sin[(13*(e + f*x))/2] - 10*B*Sin[(13*(e + f*x))/2]))/(c^7*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-1 + Sin[e + f*x])^7)","B",1
51,1,378,197,6.6941532,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^8,x]","\frac{(a \sin (e+f x)+a)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(437580 A \sin \left(\frac{1}{2} (e+f x)\right)+240240 A \sin \left(\frac{3}{2} (e+f x)\right)-60060 A \sin \left(\frac{5}{2} (e+f x)\right)-1820 A \sin \left(\frac{9}{2} (e+f x)\right)+60 A \sin \left(\frac{13}{2} (e+f x)\right)+463320 A \cos \left(\frac{1}{2} (e+f x)\right)-260260 A \cos \left(\frac{3}{2} (e+f x)\right)-72072 A \cos \left(\frac{5}{2} (e+f x)\right)+5460 A \cos \left(\frac{7}{2} (e+f x)\right)-420 A \cos \left(\frac{11}{2} (e+f x)\right)+4 A \cos \left(\frac{15}{2} (e+f x)\right)+373230 B \sin \left(\frac{1}{2} (e+f x)\right)+285285 B \sin \left(\frac{3}{2} (e+f x)\right)-150150 B \sin \left(\frac{5}{2} (e+f x)\right)-45045 B \sin \left(\frac{7}{2} (e+f x)\right)+5005 B \sin \left(\frac{9}{2} (e+f x)\right)-165 B \sin \left(\frac{13}{2} (e+f x)\right)+302445 B \cos \left(\frac{1}{2} (e+f x)\right)-230230 B \cos \left(\frac{3}{2} (e+f x)\right)-117117 B \cos \left(\frac{5}{2} (e+f x)\right)+30030 B \cos \left(\frac{7}{2} (e+f x)\right)+1155 B \cos \left(\frac{11}{2} (e+f x)\right)-11 B \cos \left(\frac{15}{2} (e+f x)\right)\right)}{1441440 f (c-c \sin (e+f x))^8 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{a^3 c^2 (4 A-11 B) \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{2 a^3 (4 A-11 B) \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7}+\frac{2 a^3 (4 A-11 B) \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac{a^3 c (4 A-11 B) \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a + a*Sin[e + f*x])^3*(463320*A*Cos[(e + f*x)/2] + 302445*B*Cos[(e + f*x)/2] - 260260*A*Cos[(3*(e + f*x))/2] - 230230*B*Cos[(3*(e + f*x))/2] - 72072*A*Cos[(5*(e + f*x))/2] - 117117*B*Cos[(5*(e + f*x))/2] + 5460*A*Cos[(7*(e + f*x))/2] + 30030*B*Cos[(7*(e + f*x))/2] - 420*A*Cos[(11*(e + f*x))/2] + 1155*B*Cos[(11*(e + f*x))/2] + 4*A*Cos[(15*(e + f*x))/2] - 11*B*Cos[(15*(e + f*x))/2] + 437580*A*Sin[(e + f*x)/2] + 373230*B*Sin[(e + f*x)/2] + 240240*A*Sin[(3*(e + f*x))/2] + 285285*B*Sin[(3*(e + f*x))/2] - 60060*A*Sin[(5*(e + f*x))/2] - 150150*B*Sin[(5*(e + f*x))/2] - 45045*B*Sin[(7*(e + f*x))/2] - 1820*A*Sin[(9*(e + f*x))/2] + 5005*B*Sin[(9*(e + f*x))/2] + 60*A*Sin[(13*(e + f*x))/2] - 165*B*Sin[(13*(e + f*x))/2]))/(1441440*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^8)","A",1
52,1,274,190,2.3615213,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4)/(a + a*Sin[e + f*x]),x]","\frac{(c-c \sin (e+f x))^4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(3072 (A-B) \sin \left(\frac{1}{2} (e+f x)\right)-420 (4 A-5 B) (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-24 (47 A-75 B) \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+8 (A-5 B) \cos (3 (e+f x)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+24 (5 A-12 B) \sin (2 (e+f x)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+3 B \sin (4 (e+f x)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{96 a f (\sin (e+f x)+1) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^8}","-\frac{a^4 c^4 (A-B) \cos ^9(e+f x)}{f (a \sin (e+f x)+a)^5}-\frac{2 a^2 c^4 (4 A-5 B) \cos ^7(e+f x)}{f (a \sin (e+f x)+a)^3}-\frac{35 c^4 (4 A-5 B) \cos ^3(e+f x)}{12 a f}-\frac{7 c^4 (4 A-5 B) \cos ^5(e+f x)}{4 f (a \sin (e+f x)+a)}-\frac{35 c^4 (4 A-5 B) \sin (e+f x) \cos (e+f x)}{8 a f}-\frac{35 c^4 x (4 A-5 B)}{8 a}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^4*(3072*(A - B)*Sin[(e + f*x)/2] - 420*(4*A - 5*B)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 24*(47*A - 75*B)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 8*(A - 5*B)*Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 24*(5*A - 12*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[2*(e + f*x)] + 3*B*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[4*(e + f*x)]))/(96*a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8*(1 + Sin[e + f*x]))","A",1
53,1,220,157,1.3977676,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3)/(a + a*Sin[e + f*x]),x]","\frac{c^3 (\sin (e+f x)-1)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right) (30 (3 A-4 B) (e+f x)-3 (A-4 B) \sin (2 (e+f x))+(48 A-93 B) \cos (e+f x)+B \cos (3 (e+f x)))+\sin \left(\frac{1}{2} (e+f x)\right) (-3 (A-4 B) \sin (2 (e+f x))+(48 A-93 B) \cos (e+f x)+6 A (15 e+15 f x-32)-24 B (5 e+5 f x-8)+B \cos (3 (e+f x)))\right)}{12 a f (\sin (e+f x)+1) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}","-\frac{a^3 c^3 (A-B) \cos ^7(e+f x)}{f (a \sin (e+f x)+a)^4}-\frac{2 a^3 c^3 (3 A-4 B) \cos ^5(e+f x)}{f \left(a^2 \sin (e+f x)+a^2\right)^2}-\frac{5 c^3 (3 A-4 B) \cos ^3(e+f x)}{3 a f}-\frac{5 c^3 (3 A-4 B) \sin (e+f x) \cos (e+f x)}{2 a f}-\frac{5 c^3 x (3 A-4 B)}{2 a}",1,"(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^3*(Cos[(e + f*x)/2]*(30*(3*A - 4*B)*(e + f*x) + (48*A - 93*B)*Cos[e + f*x] + B*Cos[3*(e + f*x)] - 3*(A - 4*B)*Sin[2*(e + f*x)]) + Sin[(e + f*x)/2]*(-24*B*(-8 + 5*e + 5*f*x) + 6*A*(-32 + 15*e + 15*f*x) + (48*A - 93*B)*Cos[e + f*x] + B*Cos[3*(e + f*x)] - 3*(A - 4*B)*Sin[2*(e + f*x)])))/(12*a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6*(1 + Sin[e + f*x]))","A",1
54,1,188,118,1.3296961,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2)/(a + a*Sin[e + f*x]),x]","-\frac{c^2 (\sin (e+f x)-1)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right) (6 (2 A-3 B) (e+f x)+4 (A-3 B) \cos (e+f x)+B \sin (2 (e+f x)))+\sin \left(\frac{1}{2} (e+f x)\right) (4 (A-3 B) \cos (e+f x)+4 A (3 e+3 f x-8)-2 B (9 e+9 f x-16)+B \sin (2 (e+f x)))\right)}{4 a f (\sin (e+f x)+1) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}","-\frac{a^2 c^2 (A-B) \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^3}-\frac{3 c^2 (2 A-3 B) \cos (e+f x)}{2 a f}-\frac{c^2 (2 A-3 B) \cos ^3(e+f x)}{2 f (a \sin (e+f x)+a)}-\frac{3 c^2 x (2 A-3 B)}{2 a}",1,"-1/4*(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2*(Cos[(e + f*x)/2]*(6*(2*A - 3*B)*(e + f*x) + 4*(A - 3*B)*Cos[e + f*x] + B*Sin[2*(e + f*x)]) + Sin[(e + f*x)/2]*(4*A*(-8 + 3*e + 3*f*x) - 2*B*(-16 + 9*e + 9*f*x) + 4*(A - 3*B)*Cos[e + f*x] + B*Sin[2*(e + f*x)])))/(a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*(1 + Sin[e + f*x]))","A",1
55,1,127,57,0.5857506,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))}{a+a \sin (e+f x)} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x]))/(a + a*Sin[e + f*x]),x]","\frac{(c-c \sin (e+f x)) \left(\frac{4 (A-B) \sin \left(\frac{f x}{2}\right)}{f \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}-(x (A-2 B))-\frac{B \sin (e) \sin (f x)}{f}+\frac{B \cos (e) \cos (f x)}{f}\right)}{a \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}","-\frac{2 c (A-B) \cos (e+f x)}{f (a \sin (e+f x)+a)}-\frac{c x (A-2 B)}{a}+\frac{B c \cos (e+f x)}{a f}",1,"((-((A - 2*B)*x) + (B*Cos[e]*Cos[f*x])/f - (B*Sin[e]*Sin[f*x])/f + (4*(A - B)*Sin[(f*x)/2])/(f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))*(c - c*Sin[e + f*x]))/(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2)","B",1
56,1,35,35,0.0277489,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])),x]","\frac{A \tan (e+f x)}{a c f}+\frac{B \sec (e+f x)}{a c f}","\frac{A \tan (e+f x)}{a c f}+\frac{B \sec (e+f x)}{a c f}",1,"(B*Sec[e + f*x])/(a*c*f) + (A*Tan[e + f*x])/(a*c*f)","A",1
57,1,108,63,0.5919934,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^2),x]","\frac{\cos (e+f x) (-2 (A+B) \cos (e+f x)+(4 A-2 B) \cos (2 (e+f x))+8 A \sin (e+f x)+A \sin (2 (e+f x))-4 B \sin (e+f x)+B \sin (2 (e+f x))+6 B)}{12 a c^2 f (\sin (e+f x)-1)^2 (\sin (e+f x)+1)}","\frac{(2 A-B) \tan (e+f x)}{3 a c^2 f}+\frac{(A+B) \sec (e+f x)}{3 a f \left(c^2-c^2 \sin (e+f x)\right)}",1,"(Cos[e + f*x]*(6*B - 2*(A + B)*Cos[e + f*x] + (4*A - 2*B)*Cos[2*(e + f*x)] + 8*A*Sin[e + f*x] - 4*B*Sin[e + f*x] + A*Sin[2*(e + f*x)] + B*Sin[2*(e + f*x)]))/(12*a*c^2*f*(-1 + Sin[e + f*x])^2*(1 + Sin[e + f*x]))","A",1
58,1,157,102,0.8900201,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^3),x]","-\frac{\cos (e+f x) (5 (B-9 A) \cos (e+f x)+32 (3 A-2 B) \cos (2 (e+f x))+120 A \sin (e+f x)+36 A \sin (2 (e+f x))-24 A \sin (3 (e+f x))+9 A \cos (3 (e+f x))-80 B \sin (e+f x)-4 B \sin (2 (e+f x))+16 B \sin (3 (e+f x))+B (-\cos (3 (e+f x)))+80 B)}{240 a c^3 f (\sin (e+f x)-1)^3 (\sin (e+f x)+1)}","\frac{2 (3 A-2 B) \tan (e+f x)}{15 a c^3 f}+\frac{(3 A-2 B) \sec (e+f x)}{15 a f \left(c^3-c^3 \sin (e+f x)\right)}+\frac{(A+B) \sec (e+f x)}{5 a c f (c-c \sin (e+f x))^2}",1,"-1/240*(Cos[e + f*x]*(80*B + 5*(-9*A + B)*Cos[e + f*x] + 32*(3*A - 2*B)*Cos[2*(e + f*x)] + 9*A*Cos[3*(e + f*x)] - B*Cos[3*(e + f*x)] + 120*A*Sin[e + f*x] - 80*B*Sin[e + f*x] + 36*A*Sin[2*(e + f*x)] - 4*B*Sin[2*(e + f*x)] - 24*A*Sin[3*(e + f*x)] + 16*B*Sin[3*(e + f*x)]))/(a*c^3*f*(-1 + Sin[e + f*x])^3*(1 + Sin[e + f*x]))","A",1
59,1,240,142,1.1405174,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^4} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^4),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) ((182 B-406 A) \cos (e+f x)+224 (4 A-3 B) \cos (2 (e+f x))+896 A \sin (e+f x)+406 A \sin (2 (e+f x))-384 A \sin (3 (e+f x))-29 A \sin (4 (e+f x))+174 A \cos (3 (e+f x))-64 A \cos (4 (e+f x))-672 B \sin (e+f x)-182 B \sin (2 (e+f x))+288 B \sin (3 (e+f x))+13 B \sin (4 (e+f x))-78 B \cos (3 (e+f x))+48 B \cos (4 (e+f x))+560 B)}{2240 a c^4 f (\sin (e+f x)-1)^4 (\sin (e+f x)+1)}","\frac{2 (4 A-3 B) \tan (e+f x)}{35 a c^4 f}+\frac{(4 A-3 B) \sec (e+f x)}{35 a f \left(c^4-c^4 \sin (e+f x)\right)}+\frac{(4 A-3 B) \sec (e+f x)}{35 a f \left(c^2-c^2 \sin (e+f x)\right)^2}+\frac{(A+B) \sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(560*B + (-406*A + 182*B)*Cos[e + f*x] + 224*(4*A - 3*B)*Cos[2*(e + f*x)] + 174*A*Cos[3*(e + f*x)] - 78*B*Cos[3*(e + f*x)] - 64*A*Cos[4*(e + f*x)] + 48*B*Cos[4*(e + f*x)] + 896*A*Sin[e + f*x] - 672*B*Sin[e + f*x] + 406*A*Sin[2*(e + f*x)] - 182*B*Sin[2*(e + f*x)] - 384*A*Sin[3*(e + f*x)] + 288*B*Sin[3*(e + f*x)] - 29*A*Sin[4*(e + f*x)] + 13*B*Sin[4*(e + f*x)]))/(2240*a*c^4*f*(-1 + Sin[e + f*x])^4*(1 + Sin[e + f*x]))","A",1
60,1,354,240,2.06568,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5)/(a + a*Sin[e + f*x])^2,x]","\frac{(c-c \sin (e+f x))^5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(2048 (A-B) \sin \left(\frac{1}{2} (e+f x)\right)+1260 (4 A-7 B) (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+24 (95 A-217 B) \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-8 (A-7 B) \cos (3 (e+f x)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-24 (7 A-24 B) \sin (2 (e+f x)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-1024 (13 A-19 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-1024 (A-B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-3 B \sin (4 (e+f x)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{96 a^2 f (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{10}}","-\frac{a^5 c^5 (A-B) \cos ^{11}(e+f x)}{3 f (a \sin (e+f x)+a)^7}+\frac{2 a^3 c^5 (4 A-7 B) \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^5}+\frac{35 c^5 (4 A-7 B) \cos ^3(e+f x)}{4 a^2 f}+\frac{21 c^5 (4 A-7 B) \cos ^5(e+f x)}{4 f \left(a^2 \sin (e+f x)+a^2\right)}+\frac{105 c^5 (4 A-7 B) \sin (e+f x) \cos (e+f x)}{8 a^2 f}+\frac{105 c^5 x (4 A-7 B)}{8 a^2}+\frac{6 a^4 c^5 (4 A-7 B) \cos ^7(e+f x)}{f \left(a^2 \sin (e+f x)+a^2\right)^3}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^5*(2048*(A - B)*Sin[(e + f*x)/2] - 1024*(A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 1024*(13*A - 19*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 1260*(4*A - 7*B)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 24*(95*A - 217*B)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 8*(A - 7*B)*Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 24*(7*A - 24*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sin[2*(e + f*x)] - 3*B*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sin[4*(e + f*x)]))/(96*a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10*(1 + Sin[e + f*x])^2)","A",1
61,1,311,180,1.3185769,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4)/(a + a*Sin[e + f*x])^2,x]","\frac{(c-c \sin (e+f x))^4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(128 (A-B) \sin \left(\frac{1}{2} (e+f x)\right)+210 (A-2 B) (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+3 (24 A-71 B) \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-3 (A-6 B) \sin (2 (e+f x)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-128 (5 A-8 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-64 (A-B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+B \cos (3 (e+f x)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{12 a^2 f (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^8}","-\frac{a^4 c^4 (A-B) \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}+\frac{35 c^4 (A-2 B) \cos ^3(e+f x)}{3 a^2 f}+\frac{2 a^2 c^4 (A-2 B) \cos ^7(e+f x)}{f (a \sin (e+f x)+a)^4}+\frac{35 c^4 (A-2 B) \sin (e+f x) \cos (e+f x)}{2 a^2 f}+\frac{35 c^4 x (A-2 B)}{2 a^2}+\frac{14 c^4 (A-2 B) \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^4*(128*(A - B)*Sin[(e + f*x)/2] - 64*(A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 128*(5*A - 8*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 210*(A - 2*B)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 3*(24*A - 71*B)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + B*Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 3*(A - 6*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sin[2*(e + f*x)]))/(12*a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8*(1 + Sin[e + f*x])^2)","A",1
62,1,274,162,0.8826904,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3)/(a + a*Sin[e + f*x])^2,x]","\frac{(c-c \sin (e+f x))^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(64 (A-B) \sin \left(\frac{1}{2} (e+f x)\right)+30 (2 A-5 B) (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+12 (A-5 B) \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-32 (7 A-13 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-32 (A-B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+3 B \sin (2 (e+f x)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{12 a^2 f (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}","-\frac{a^3 c^3 (A-B) \cos ^7(e+f x)}{3 f (a \sin (e+f x)+a)^5}+\frac{5 c^3 (2 A-5 B) \cos (e+f x)}{2 a^2 f}+\frac{5 c^3 (2 A-5 B) \cos ^3(e+f x)}{6 f \left(a^2 \sin (e+f x)+a^2\right)}+\frac{5 c^3 x (2 A-5 B)}{2 a^2}+\frac{2 a c^3 (2 A-5 B) \cos ^5(e+f x)}{3 f (a \sin (e+f x)+a)^3}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^3*(64*(A - B)*Sin[(e + f*x)/2] - 32*(A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 32*(7*A - 13*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 30*(2*A - 5*B)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 12*(A - 5*B)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 3*B*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sin[2*(e + f*x)]))/(12*a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6*(1 + Sin[e + f*x])^2)","A",1
63,1,234,108,0.5955267,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^2,x]","\frac{(c-c \sin (e+f x))^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(8 (A-B) \sin \left(\frac{1}{2} (e+f x)\right)+3 (A-4 B) (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-8 (2 A-5 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-4 (A-B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-3 B \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{3 a^2 f (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}","\frac{c^2 (A-4 B) \cos (e+f x)}{a^2 f}-\frac{a^2 c^2 (A-B) \cos ^5(e+f x)}{3 f (a \sin (e+f x)+a)^4}+\frac{c^2 x (A-4 B)}{a^2}+\frac{2 c^2 (A-4 B) \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(8*(A - B)*Sin[(e + f*x)/2] - 4*(A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 8*(2*A - 5*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 3*(A - 4*B)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 3*B*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)*(c - c*Sin[e + f*x])^2)/(3*a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*(1 + Sin[e + f*x])^2)","B",1
64,1,156,72,0.5835935,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))}{(a+a \sin (e+f x))^2} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x]))/(a + a*Sin[e + f*x])^2,x]","\frac{c \left(-6 (A-3 B) \cos \left(e+\frac{f x}{2}\right)+2 A \cos \left(e+\frac{3 f x}{2}\right)-9 B f x \sin \left(e+\frac{f x}{2}\right)-3 B f x \sin \left(e+\frac{3 f x}{2}\right)-14 B \cos \left(e+\frac{3 f x}{2}\right)+3 B f x \cos \left(2 e+\frac{3 f x}{2}\right)+24 B \sin \left(\frac{f x}{2}\right)-9 B f x \cos \left(\frac{f x}{2}\right)\right)}{6 a^2 f \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","\frac{c (A-7 B) \cos (e+f x)}{3 a^2 f (\sin (e+f x)+1)}-\frac{B c x}{a^2}-\frac{2 c (A-B) \cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}",1,"(c*(-9*B*f*x*Cos[(f*x)/2] - 6*(A - 3*B)*Cos[e + (f*x)/2] + 2*A*Cos[e + (3*f*x)/2] - 14*B*Cos[e + (3*f*x)/2] + 3*B*f*x*Cos[2*e + (3*f*x)/2] + 24*B*Sin[(f*x)/2] - 9*B*f*x*Sin[e + (f*x)/2] - 3*B*f*x*Sin[e + (3*f*x)/2]))/(6*a^2*f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","B",1
65,1,110,62,0.5119306,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])),x]","\frac{\cos (e+f x) (-2 (A-B) \cos (e+f x)+2 (2 A+B) \cos (2 (e+f x))-8 A \sin (e+f x)-A \sin (2 (e+f x))-4 B \sin (e+f x)+B \sin (2 (e+f x))-6 B)}{12 a^2 c f (\sin (e+f x)-1) (\sin (e+f x)+1)^2}","\frac{(2 A+B) \tan (e+f x)}{3 a^2 c f}-\frac{(A-B) \sec (e+f x)}{3 c f \left(a^2 \sin (e+f x)+a^2\right)}",1,"(Cos[e + f*x]*(-6*B - 2*(A - B)*Cos[e + f*x] + 2*(2*A + B)*Cos[2*(e + f*x)] - 8*A*Sin[e + f*x] - 4*B*Sin[e + f*x] - A*Sin[2*(e + f*x)] + B*Sin[2*(e + f*x)]))/(12*a^2*c*f*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])^2)","A",1
66,1,53,62,0.1249633,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^2} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^2),x]","\frac{A \left(\frac{1}{3} \tan ^3(e+f x)+\tan (e+f x)\right)}{a^2 c^2 f}+\frac{B \sec ^3(e+f x)}{3 a^2 c^2 f}","\frac{A \tan ^3(e+f x)}{3 a^2 c^2 f}+\frac{A \tan (e+f x)}{a^2 c^2 f}+\frac{B \sec ^3(e+f x)}{3 a^2 c^2 f}",1,"(B*Sec[e + f*x]^3)/(3*a^2*c^2*f) + (A*(Tan[e + f*x] + Tan[e + f*x]^3/3))/(a^2*c^2*f)","A",1
67,1,237,93,1.0151216,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^3),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (54 (A+B) \cos (e+f x)-32 (4 A-B) \cos (2 (e+f x))-384 A \sin (e+f x)-18 A \sin (2 (e+f x))-128 A \sin (3 (e+f x))-9 A \sin (4 (e+f x))+18 A \cos (3 (e+f x))-64 A \cos (4 (e+f x))+96 B \sin (e+f x)-18 B \sin (2 (e+f x))+32 B \sin (3 (e+f x))-9 B \sin (4 (e+f x))+18 B \cos (3 (e+f x))+16 B \cos (4 (e+f x))-240 B)}{960 a^2 c^3 f (\sin (e+f x)-1)^3 (\sin (e+f x)+1)^2}","\frac{(4 A-B) \tan ^3(e+f x)}{15 a^2 c^3 f}+\frac{(4 A-B) \tan (e+f x)}{5 a^2 c^3 f}+\frac{(A+B) \sec ^3(e+f x)}{5 a^2 f \left(c^3-c^3 \sin (e+f x)\right)}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-240*B + 54*(A + B)*Cos[e + f*x] - 32*(4*A - B)*Cos[2*(e + f*x)] + 18*A*Cos[3*(e + f*x)] + 18*B*Cos[3*(e + f*x)] - 64*A*Cos[4*(e + f*x)] + 16*B*Cos[4*(e + f*x)] - 384*A*Sin[e + f*x] + 96*B*Sin[e + f*x] - 18*A*Sin[2*(e + f*x)] - 18*B*Sin[2*(e + f*x)] - 128*A*Sin[3*(e + f*x)] + 32*B*Sin[3*(e + f*x)] - 9*A*Sin[4*(e + f*x)] - 9*B*Sin[4*(e + f*x)]))/(960*a^2*c^3*f*(-1 + Sin[e + f*x])^3*(1 + Sin[e + f*x])^2)","B",1
68,1,285,135,0.963539,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^4),x]","-\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (42 (25 A+4 B) \cos (e+f x)-512 (5 A-2 B) \cos (2 (e+f x))-4480 A \sin (e+f x)-600 A \sin (2 (e+f x))-960 A \sin (3 (e+f x))-300 A \sin (4 (e+f x))+320 A \sin (5 (e+f x))+225 A \cos (3 (e+f x))-1280 A \cos (4 (e+f x))-75 A \cos (5 (e+f x))+1792 B \sin (e+f x)-96 B \sin (2 (e+f x))+384 B \sin (3 (e+f x))-48 B \sin (4 (e+f x))-128 B \sin (5 (e+f x))+36 B \cos (3 (e+f x))+512 B \cos (4 (e+f x))-12 B \cos (5 (e+f x))-2688 B)}{13440 a^2 c^4 f (\sin (e+f x)-1)^4 (\sin (e+f x)+1)^2}","\frac{4 (5 A-2 B) \tan ^3(e+f x)}{105 a^2 c^4 f}+\frac{4 (5 A-2 B) \tan (e+f x)}{35 a^2 c^4 f}+\frac{(5 A-2 B) \sec ^3(e+f x)}{35 a^2 f \left(c^4-c^4 \sin (e+f x)\right)}+\frac{(A+B) \sec ^3(e+f x)}{7 a^2 f \left(c^2-c^2 \sin (e+f x)\right)^2}",1,"-1/13440*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-2688*B + 42*(25*A + 4*B)*Cos[e + f*x] - 512*(5*A - 2*B)*Cos[2*(e + f*x)] + 225*A*Cos[3*(e + f*x)] + 36*B*Cos[3*(e + f*x)] - 1280*A*Cos[4*(e + f*x)] + 512*B*Cos[4*(e + f*x)] - 75*A*Cos[5*(e + f*x)] - 12*B*Cos[5*(e + f*x)] - 4480*A*Sin[e + f*x] + 1792*B*Sin[e + f*x] - 600*A*Sin[2*(e + f*x)] - 96*B*Sin[2*(e + f*x)] - 960*A*Sin[3*(e + f*x)] + 384*B*Sin[3*(e + f*x)] - 300*A*Sin[4*(e + f*x)] - 48*B*Sin[4*(e + f*x)] + 320*A*Sin[5*(e + f*x)] - 128*B*Sin[5*(e + f*x)]))/(a^2*c^4*f*(-1 + Sin[e + f*x])^4*(1 + Sin[e + f*x])^2)","B",1
69,1,329,175,1.1554245,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (180 (31 A-5 B) \cos (e+f x)-6912 (2 A-B) \cos (2 (e+f x))-18432 A \sin (e+f x)-4185 A \sin (2 (e+f x))-1024 A \sin (3 (e+f x))-1860 A \sin (4 (e+f x))+3072 A \sin (5 (e+f x))+155 A \sin (6 (e+f x))+310 A \cos (3 (e+f x))-6144 A \cos (4 (e+f x))-930 A \cos (5 (e+f x))+512 A \cos (6 (e+f x))+9216 B \sin (e+f x)+675 B \sin (2 (e+f x))+512 B \sin (3 (e+f x))+300 B \sin (4 (e+f x))-1536 B \sin (5 (e+f x))-25 B \sin (6 (e+f x))-50 B \cos (3 (e+f x))+3072 B \cos (4 (e+f x))+150 B \cos (5 (e+f x))-256 B \cos (6 (e+f x))-10752 B)}{64512 a^2 c^5 f (\sin (e+f x)-1)^5 (\sin (e+f x)+1)^2}","\frac{4 (2 A-B) \tan ^3(e+f x)}{63 a^2 c^5 f}+\frac{4 (2 A-B) \tan (e+f x)}{21 a^2 c^5 f}+\frac{(2 A-B) \sec ^3(e+f x)}{21 a^2 f \left(c^5-c^5 \sin (e+f x)\right)}+\frac{(2 A-B) \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac{(A+B) \sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-10752*B + 180*(31*A - 5*B)*Cos[e + f*x] - 6912*(2*A - B)*Cos[2*(e + f*x)] + 310*A*Cos[3*(e + f*x)] - 50*B*Cos[3*(e + f*x)] - 6144*A*Cos[4*(e + f*x)] + 3072*B*Cos[4*(e + f*x)] - 930*A*Cos[5*(e + f*x)] + 150*B*Cos[5*(e + f*x)] + 512*A*Cos[6*(e + f*x)] - 256*B*Cos[6*(e + f*x)] - 18432*A*Sin[e + f*x] + 9216*B*Sin[e + f*x] - 4185*A*Sin[2*(e + f*x)] + 675*B*Sin[2*(e + f*x)] - 1024*A*Sin[3*(e + f*x)] + 512*B*Sin[3*(e + f*x)] - 1860*A*Sin[4*(e + f*x)] + 300*B*Sin[4*(e + f*x)] + 3072*A*Sin[5*(e + f*x)] - 1536*B*Sin[5*(e + f*x)] + 155*A*Sin[6*(e + f*x)] - 25*B*Sin[6*(e + f*x)]))/(64512*a^2*c^5*f*(-1 + Sin[e + f*x])^5*(1 + Sin[e + f*x])^2)","A",1
70,1,388,243,2.5017022,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5)/(a + a*Sin[e + f*x])^3,x]","\frac{(c-c \sin (e+f x))^5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(768 (A-B) \sin \left(\frac{1}{2} (e+f x)\right)-630 (3 A-8 B) (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5-15 (32 A-127 B) \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5+15 (A-8 B) \sin (2 (e+f x)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5+128 (54 A-119 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4+64 (21 A-31 B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-128 (21 A-31 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-384 (A-B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-5 B \cos (3 (e+f x)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5\right)}{60 a^3 f (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{10}}","-\frac{a^5 c^5 (A-B) \cos ^{11}(e+f x)}{5 f (a \sin (e+f x)+a)^8}-\frac{7 c^5 (3 A-8 B) \cos ^3(e+f x)}{a^3 f}+\frac{2 a^3 c^5 (3 A-8 B) \cos ^9(e+f x)}{15 f (a \sin (e+f x)+a)^6}-\frac{21 c^5 (3 A-8 B) \sin (e+f x) \cos (e+f x)}{2 a^3 f}-\frac{21 c^5 x (3 A-8 B)}{2 a^3}-\frac{42 a^5 c^5 (3 A-8 B) \cos ^5(e+f x)}{5 f \left(a^4 \sin (e+f x)+a^4\right)^2}-\frac{6 a^5 c^5 (3 A-8 B) \cos ^7(e+f x)}{5 f \left(a^2 \sin (e+f x)+a^2\right)^4}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^5*(768*(A - B)*Sin[(e + f*x)/2] - 384*(A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 128*(21*A - 31*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 64*(21*A - 31*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 128*(54*A - 119*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 630*(3*A - 8*B)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 15*(32*A - 127*B)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 5*B*Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + 15*(A - 8*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sin[2*(e + f*x)]))/(60*a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10*(1 + Sin[e + f*x])^3)","A",1
71,1,348,201,1.6063535,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4)/(a + a*Sin[e + f*x])^3,x]","\frac{(c-c \sin (e+f x))^4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(384 (A-B) \sin \left(\frac{1}{2} (e+f x)\right)-210 (2 A-7 B) (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5-60 (A-7 B) \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5+64 (29 A-79 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4+64 (8 A-13 B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-128 (8 A-13 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-192 (A-B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-15 B \sin (2 (e+f x)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5\right)}{60 a^3 f (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^8}","-\frac{a^4 c^4 (A-B) \cos ^9(e+f x)}{5 f (a \sin (e+f x)+a)^7}-\frac{7 c^4 (2 A-7 B) \cos (e+f x)}{2 a^3 f}-\frac{7 c^4 (2 A-7 B) \cos ^3(e+f x)}{6 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{7 c^4 x (2 A-7 B)}{2 a^3}+\frac{2 a^2 c^4 (2 A-7 B) \cos ^7(e+f x)}{15 f (a \sin (e+f x)+a)^5}-\frac{14 c^4 (2 A-7 B) \cos ^5(e+f x)}{15 f (a \sin (e+f x)+a)^3}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^4*(384*(A - B)*Sin[(e + f*x)/2] - 192*(A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 128*(8*A - 13*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 64*(8*A - 13*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 64*(29*A - 79*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 210*(2*A - 7*B)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 60*(A - 7*B)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 15*B*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sin[2*(e + f*x)]))/(60*a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8*(1 + Sin[e + f*x])^3)","A",1
72,1,308,153,1.0921547,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3)/(a + a*Sin[e + f*x])^3,x]","\frac{(c-c \sin (e+f x))^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(48 (A-B) \sin \left(\frac{1}{2} (e+f x)\right)-15 (A-6 B) (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5+4 (23 A-93 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4+4 (11 A-21 B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-8 (11 A-21 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-24 (A-B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+15 B \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5\right)}{15 a^3 f (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}","-\frac{c^3 (A-6 B) \cos (e+f x)}{a^3 f}-\frac{a^3 c^3 (A-B) \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}-\frac{2 a^3 c^3 (A-6 B) \cos ^3(e+f x)}{3 f \left(a^3 \sin (e+f x)+a^3\right)^2}-\frac{c^3 x (A-6 B)}{a^3}+\frac{2 a c^3 (A-6 B) \cos ^5(e+f x)}{15 f (a \sin (e+f x)+a)^4}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(48*(A - B)*Sin[(e + f*x)/2] - 24*(A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 8*(11*A - 21*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 4*(11*A - 21*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 4*(23*A - 93*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 15*(A - 6*B)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + 15*B*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)*(c - c*Sin[e + f*x])^3)/(15*a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6*(1 + Sin[e + f*x])^3)","B",1
73,1,272,110,0.6978112,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^3,x]","\frac{(c-c \sin (e+f x))^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(24 (A-B) \sin \left(\frac{1}{2} (e+f x)\right)+2 (3 A-43 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4+4 (3 A-8 B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-8 (3 A-8 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-12 (A-B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+15 B (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5\right)}{15 a^3 f (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}","\frac{2 B c^2 \cos (e+f x)}{f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{B c^2 x}{a^3}-\frac{a^2 c^2 (A-B) \cos ^5(e+f x)}{5 f (a \sin (e+f x)+a)^5}-\frac{2 B c^2 \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^3}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(24*(A - B)*Sin[(e + f*x)/2] - 12*(A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 8*(3*A - 8*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 4*(3*A - 8*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 2*(3*A - 43*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + 15*B*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)*(c - c*Sin[e + f*x])^2)/(15*a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*(1 + Sin[e + f*x])^3)","B",1
74,1,139,103,0.8238272,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x]))/(a + a*Sin[e + f*x])^3,x]","\frac{c \left(-15 (A+B) \cos \left(e+\frac{f x}{2}\right)+5 (A+B) \cos \left(e+\frac{3 f x}{2}\right)+A \sin \left(2 e+\frac{5 f x}{2}\right)+5 A \sin \left(\frac{f x}{2}\right)-15 B \sin \left(2 e+\frac{3 f x}{2}\right)+4 B \sin \left(2 e+\frac{5 f x}{2}\right)-25 B \sin \left(\frac{f x}{2}\right)\right)}{30 a^3 f \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","\frac{c (A+4 B) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{a c (A-11 B) \cos (e+f x)}{15 f \left(a^2 \sin (e+f x)+a^2\right)^2}-\frac{2 c (A-B) \cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}",1,"(c*(-15*(A + B)*Cos[e + (f*x)/2] + 5*(A + B)*Cos[e + (3*f*x)/2] + 5*A*Sin[(f*x)/2] - 25*B*Sin[(f*x)/2] - 15*B*Sin[2*e + (3*f*x)/2] + A*Sin[2*e + (5*f*x)/2] + 4*B*Sin[2*e + (5*f*x)/2]))/(30*a^3*f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
75,1,156,102,0.8392535,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])),x]","\frac{\cos (e+f x) (-5 (9 A+B) \cos (e+f x)+32 (3 A+2 B) \cos (2 (e+f x))-120 A \sin (e+f x)-36 A \sin (2 (e+f x))+24 A \sin (3 (e+f x))+9 A \cos (3 (e+f x))-80 B \sin (e+f x)-4 B \sin (2 (e+f x))+16 B \sin (3 (e+f x))+B \cos (3 (e+f x))-80 B)}{240 a^3 c f (\sin (e+f x)-1) (\sin (e+f x)+1)^3}","\frac{2 (3 A+2 B) \tan (e+f x)}{15 a^3 c f}-\frac{(3 A+2 B) \sec (e+f x)}{15 c f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{(A-B) \sec (e+f x)}{5 a c f (a \sin (e+f x)+a)^2}",1,"(Cos[e + f*x]*(-80*B - 5*(9*A + B)*Cos[e + f*x] + 32*(3*A + 2*B)*Cos[2*(e + f*x)] + 9*A*Cos[3*(e + f*x)] + B*Cos[3*(e + f*x)] - 120*A*Sin[e + f*x] - 80*B*Sin[e + f*x] - 36*A*Sin[2*(e + f*x)] - 4*B*Sin[2*(e + f*x)] + 24*A*Sin[3*(e + f*x)] + 16*B*Sin[3*(e + f*x)]))/(240*a^3*c*f*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])^3)","A",1
76,1,237,90,1.0513533,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^2} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^2),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (54 (A-B) \cos (e+f x)-32 (4 A+B) \cos (2 (e+f x))+384 A \sin (e+f x)+18 A \sin (2 (e+f x))+128 A \sin (3 (e+f x))+9 A \sin (4 (e+f x))+18 A \cos (3 (e+f x))-64 A \cos (4 (e+f x))+96 B \sin (e+f x)-18 B \sin (2 (e+f x))+32 B \sin (3 (e+f x))-9 B \sin (4 (e+f x))-18 B \cos (3 (e+f x))-16 B \cos (4 (e+f x))+240 B)}{960 a^3 c^2 f (\sin (e+f x)-1)^2 (\sin (e+f x)+1)^3}","\frac{(4 A+B) \tan ^3(e+f x)}{15 a^3 c^2 f}+\frac{(4 A+B) \tan (e+f x)}{5 a^3 c^2 f}-\frac{(A-B) \sec ^3(e+f x)}{5 c^2 f \left(a^3 \sin (e+f x)+a^3\right)}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(240*B + 54*(A - B)*Cos[e + f*x] - 32*(4*A + B)*Cos[2*(e + f*x)] + 18*A*Cos[3*(e + f*x)] - 18*B*Cos[3*(e + f*x)] - 64*A*Cos[4*(e + f*x)] - 16*B*Cos[4*(e + f*x)] + 384*A*Sin[e + f*x] + 96*B*Sin[e + f*x] + 18*A*Sin[2*(e + f*x)] - 18*B*Sin[2*(e + f*x)] + 128*A*Sin[3*(e + f*x)] + 32*B*Sin[3*(e + f*x)] + 9*A*Sin[4*(e + f*x)] - 9*B*Sin[4*(e + f*x)]))/(960*a^3*c^2*f*(-1 + Sin[e + f*x])^2*(1 + Sin[e + f*x])^3)","B",1
77,1,65,84,0.2058823,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^3),x]","\frac{A \left(\frac{1}{5} \tan ^5(e+f x)+\frac{2}{3} \tan ^3(e+f x)+\tan (e+f x)\right)}{a^3 c^3 f}+\frac{B \sec ^5(e+f x)}{5 a^3 c^3 f}","\frac{A \tan ^5(e+f x)}{5 a^3 c^3 f}+\frac{2 A \tan ^3(e+f x)}{3 a^3 c^3 f}+\frac{A \tan (e+f x)}{a^3 c^3 f}+\frac{B \sec ^5(e+f x)}{5 a^3 c^3 f}",1,"(B*Sec[e + f*x]^5)/(5*a^3*c^3*f) + (A*(Tan[e + f*x] + (2*Tan[e + f*x]^3)/3 + Tan[e + f*x]^5/5))/(a^3*c^3*f)","A",1
78,1,325,121,1.1415235,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4),x]","-\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (1500 (A+B) \cos (e+f x)-640 (6 A-B) \cos (2 (e+f x))-15360 A \sin (e+f x)-375 A \sin (2 (e+f x))-7680 A \sin (3 (e+f x))-300 A \sin (4 (e+f x))-1536 A \sin (5 (e+f x))-75 A \sin (6 (e+f x))+750 A \cos (3 (e+f x))-3072 A \cos (4 (e+f x))+150 A \cos (5 (e+f x))-768 A \cos (6 (e+f x))+2560 B \sin (e+f x)-375 B \sin (2 (e+f x))+1280 B \sin (3 (e+f x))-300 B \sin (4 (e+f x))+256 B \sin (5 (e+f x))-75 B \sin (6 (e+f x))+750 B \cos (3 (e+f x))+512 B \cos (4 (e+f x))+150 B \cos (5 (e+f x))+128 B \cos (6 (e+f x))-8960 B)}{53760 a^3 c^4 f (\sin (e+f x)-1)^4 (\sin (e+f x)+1)^3}","\frac{(6 A-B) \tan ^5(e+f x)}{35 a^3 c^4 f}+\frac{2 (6 A-B) \tan ^3(e+f x)}{21 a^3 c^4 f}+\frac{(6 A-B) \tan (e+f x)}{7 a^3 c^4 f}+\frac{(A+B) \sec ^5(e+f x)}{7 a^3 f \left(c^4-c^4 \sin (e+f x)\right)}",1,"-1/53760*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-8960*B + 1500*(A + B)*Cos[e + f*x] - 640*(6*A - B)*Cos[2*(e + f*x)] + 750*A*Cos[3*(e + f*x)] + 750*B*Cos[3*(e + f*x)] - 3072*A*Cos[4*(e + f*x)] + 512*B*Cos[4*(e + f*x)] + 150*A*Cos[5*(e + f*x)] + 150*B*Cos[5*(e + f*x)] - 768*A*Cos[6*(e + f*x)] + 128*B*Cos[6*(e + f*x)] - 15360*A*Sin[e + f*x] + 2560*B*Sin[e + f*x] - 375*A*Sin[2*(e + f*x)] - 375*B*Sin[2*(e + f*x)] - 7680*A*Sin[3*(e + f*x)] + 1280*B*Sin[3*(e + f*x)] - 300*A*Sin[4*(e + f*x)] - 300*B*Sin[4*(e + f*x)] - 1536*A*Sin[5*(e + f*x)] + 256*B*Sin[5*(e + f*x)] - 75*A*Sin[6*(e + f*x)] - 75*B*Sin[6*(e + f*x)]))/(a^3*c^4*f*(-1 + Sin[e + f*x])^4*(1 + Sin[e + f*x])^3)","B",1
79,1,373,162,1.4144539,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (1125 (49 A+13 B) \cos (e+f x)-20480 (7 A-2 B) \cos (2 (e+f x))-322560 A \sin (e+f x)-24500 A \sin (2 (e+f x))-136192 A \sin (3 (e+f x))-19600 A \sin (4 (e+f x))-7168 A \sin (5 (e+f x))-4900 A \sin (6 (e+f x))+7168 A \sin (7 (e+f x))+23275 A \cos (3 (e+f x))-114688 A \cos (4 (e+f x))+1225 A \cos (5 (e+f x))-28672 A \cos (6 (e+f x))-1225 A \cos (7 (e+f x))+92160 B \sin (e+f x)-6500 B \sin (2 (e+f x))+38912 B \sin (3 (e+f x))-5200 B \sin (4 (e+f x))+2048 B \sin (5 (e+f x))-1300 B \sin (6 (e+f x))-2048 B \sin (7 (e+f x))+6175 B \cos (3 (e+f x))+32768 B \cos (4 (e+f x))+325 B \cos (5 (e+f x))+8192 B \cos (6 (e+f x))-325 B \cos (7 (e+f x))-184320 B)}{1290240 a^3 c^5 f (\sin (e+f x)-1)^5 (\sin (e+f x)+1)^3}","\frac{2 (7 A-2 B) \tan ^5(e+f x)}{105 a^3 c^5 f}+\frac{4 (7 A-2 B) \tan ^3(e+f x)}{63 a^3 c^5 f}+\frac{2 (7 A-2 B) \tan (e+f x)}{21 a^3 c^5 f}+\frac{(7 A-2 B) \sec ^5(e+f x)}{63 a^3 f \left(c^5-c^5 \sin (e+f x)\right)}+\frac{(A+B) \sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-184320*B + 1125*(49*A + 13*B)*Cos[e + f*x] - 20480*(7*A - 2*B)*Cos[2*(e + f*x)] + 23275*A*Cos[3*(e + f*x)] + 6175*B*Cos[3*(e + f*x)] - 114688*A*Cos[4*(e + f*x)] + 32768*B*Cos[4*(e + f*x)] + 1225*A*Cos[5*(e + f*x)] + 325*B*Cos[5*(e + f*x)] - 28672*A*Cos[6*(e + f*x)] + 8192*B*Cos[6*(e + f*x)] - 1225*A*Cos[7*(e + f*x)] - 325*B*Cos[7*(e + f*x)] - 322560*A*Sin[e + f*x] + 92160*B*Sin[e + f*x] - 24500*A*Sin[2*(e + f*x)] - 6500*B*Sin[2*(e + f*x)] - 136192*A*Sin[3*(e + f*x)] + 38912*B*Sin[3*(e + f*x)] - 19600*A*Sin[4*(e + f*x)] - 5200*B*Sin[4*(e + f*x)] - 7168*A*Sin[5*(e + f*x)] + 2048*B*Sin[5*(e + f*x)] - 4900*A*Sin[6*(e + f*x)] - 1300*B*Sin[6*(e + f*x)] + 7168*A*Sin[7*(e + f*x)] - 2048*B*Sin[7*(e + f*x)]))/(1290240*a^3*c^5*f*(-1 + Sin[e + f*x])^5*(1 + Sin[e + f*x])^3)","B",1
80,1,401,205,3.5476882,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6),x]","\frac{-3850 (107 A-3 B) \cos (e+f x)+135168 (8 A-3 B) \cos (2 (e+f x))+1802240 A \sin (e+f x)+247170 A \sin (2 (e+f x))+557056 A \sin (3 (e+f x))+187250 A \sin (4 (e+f x))-163840 A \sin (5 (e+f x))+37450 A \sin (6 (e+f x))-98304 A \sin (7 (e+f x))-3745 A \sin (8 (e+f x))-127330 A \cos (3 (e+f x))+819200 A \cos (4 (e+f x))+37450 A \cos (5 (e+f x))+163840 A \cos (6 (e+f x))+22470 A \cos (7 (e+f x))-16384 A \cos (8 (e+f x))-675840 B \sin (e+f x)-6930 B \sin (2 (e+f x))-208896 B \sin (3 (e+f x))-5250 B \sin (4 (e+f x))+61440 B \sin (5 (e+f x))-1050 B \sin (6 (e+f x))+36864 B \sin (7 (e+f x))+105 B \sin (8 (e+f x))+3570 B \cos (3 (e+f x))-307200 B \cos (4 (e+f x))-1050 B \cos (5 (e+f x))-61440 B \cos (6 (e+f x))-630 B \cos (7 (e+f x))+6144 B \cos (8 (e+f x))+1013760 B}{8110080 a^3 c^6 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","\frac{2 (8 A-3 B) \tan ^5(e+f x)}{165 a^3 c^6 f}+\frac{4 (8 A-3 B) \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac{2 (8 A-3 B) \tan (e+f x)}{33 a^3 c^6 f}+\frac{(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left(c^6-c^6 \sin (e+f x)\right)}+\frac{(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left(c^3-c^3 \sin (e+f x)\right)^2}+\frac{(A+B) \sec ^5(e+f x)}{11 a^3 f \left(c^2-c^2 \sin (e+f x)\right)^3}",1,"(1013760*B - 3850*(107*A - 3*B)*Cos[e + f*x] + 135168*(8*A - 3*B)*Cos[2*(e + f*x)] - 127330*A*Cos[3*(e + f*x)] + 3570*B*Cos[3*(e + f*x)] + 819200*A*Cos[4*(e + f*x)] - 307200*B*Cos[4*(e + f*x)] + 37450*A*Cos[5*(e + f*x)] - 1050*B*Cos[5*(e + f*x)] + 163840*A*Cos[6*(e + f*x)] - 61440*B*Cos[6*(e + f*x)] + 22470*A*Cos[7*(e + f*x)] - 630*B*Cos[7*(e + f*x)] - 16384*A*Cos[8*(e + f*x)] + 6144*B*Cos[8*(e + f*x)] + 1802240*A*Sin[e + f*x] - 675840*B*Sin[e + f*x] + 247170*A*Sin[2*(e + f*x)] - 6930*B*Sin[2*(e + f*x)] + 557056*A*Sin[3*(e + f*x)] - 208896*B*Sin[3*(e + f*x)] + 187250*A*Sin[4*(e + f*x)] - 5250*B*Sin[4*(e + f*x)] - 163840*A*Sin[5*(e + f*x)] + 61440*B*Sin[5*(e + f*x)] + 37450*A*Sin[6*(e + f*x)] - 1050*B*Sin[6*(e + f*x)] - 98304*A*Sin[7*(e + f*x)] + 36864*B*Sin[7*(e + f*x)] - 3745*A*Sin[8*(e + f*x)] + 105*B*Sin[8*(e + f*x)])/(8110080*a^3*c^6*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
81,1,149,198,2.9876115,"\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx","Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]","-\frac{a c^3 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 (60 (121 A-202 B) \cos (2 (e+f x))+30558 A \sin (e+f x)-770 A \sin (3 (e+f x))-35332 A-31530 B \sin (e+f x)+2870 B \sin (3 (e+f x))+315 B \cos (4 (e+f x))+27085 B)}{13860 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{256 a c^5 (11 A-5 B) \cos ^3(e+f x)}{3465 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a c^4 (11 A-5 B) \cos ^3(e+f x)}{1155 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a c^3 (11 A-5 B) \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{231 f}+\frac{2 a c^2 (11 A-5 B) \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{99 f}-\frac{2 a B c \cos ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{11 f}",1,"-1/13860*(a*c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c - c*Sin[e + f*x]]*(-35332*A + 27085*B + 60*(121*A - 202*B)*Cos[2*(e + f*x)] + 315*B*Cos[4*(e + f*x)] + 30558*A*Sin[e + f*x] - 31530*B*Sin[e + f*x] - 770*A*Sin[3*(e + f*x)] + 2870*B*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
82,1,123,157,1.5178577,"\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2),x]","-\frac{a c^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 ((648 A-741 B) \sin (e+f x)+30 (3 A-8 B) \cos (2 (e+f x))-942 A+35 B \sin (3 (e+f x))+664 B)}{630 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{64 a c^4 (3 A-B) \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac{16 a c^3 (3 A-B) \cos ^3(e+f x)}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a c^2 (3 A-B) \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{21 f}-\frac{2 a B c \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}",1,"-1/630*(a*c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c - c*Sin[e + f*x]]*(-942*A + 664*B + 30*(3*A - 8*B)*Cos[2*(e + f*x)] + (648*A - 741*B)*Sin[e + f*x] + 35*B*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
83,1,104,116,0.957969,"\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2),x]","\frac{a c \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 ((66 B-42 A) \sin (e+f x)+98 A+15 B \cos (2 (e+f x))-59 B)}{105 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{8 a c^3 (7 A-B) \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a c^2 (7 A-B) \cos ^3(e+f x)}{35 f \sqrt{c-c \sin (e+f x)}}-\frac{2 a B c \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{7 f}",1,"(a*c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(98*A - 59*B + 15*B*Cos[2*(e + f*x)] + (-42*A + 66*B)*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(105*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
84,1,87,73,0.4218293,"\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]],x]","\frac{2 a \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 (5 A+3 B \sin (e+f x)-2 B)}{15 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{2 a c^2 (5 A+B) \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}}-\frac{2 a B c \cos ^3(e+f x)}{5 f \sqrt{c-c \sin (e+f x)}}",1,"(2*a*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(5*A - 2*B + 3*B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(15*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
85,1,166,122,1.2640329,"\int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{a \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(6 \sqrt{2} (A+B) \sqrt{-c (\sin (e+f x)+1)} \tan ^{-1}\left(\frac{\sqrt{-c (\sin (e+f x)+1)}}{\sqrt{2} \sqrt{c}}\right)+\sqrt{c} (2 (3 A+5 B) \sin (e+f x)+6 A-B \cos (2 (e+f x))+9 B)\right)}{3 \sqrt{c} f \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{2 a (3 A+5 B) \cos (e+f x)}{3 f \sqrt{c-c \sin (e+f x)}}+\frac{2 \sqrt{2} a (A+B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{c} f}+\frac{2 a B \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{3 c f}",1,"-1/3*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(6*Sqrt[2]*(A + B)*ArcTan[Sqrt[-(c*(1 + Sin[e + f*x]))]/(Sqrt[2]*Sqrt[c])]*Sqrt[-(c*(1 + Sin[e + f*x]))] + Sqrt[c]*(6*A + 9*B - B*Cos[2*(e + f*x)] + 2*(3*A + 5*B)*Sin[e + f*x])))/(Sqrt[c]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]])","A",1
86,1,157,115,1.6055092,"\int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(3/2),x]","\frac{a \sec (e+f x) \left(2 \sqrt{c} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 (A-2 B \sin (e+f x)+3 B)+\sqrt{2} (A+5 B) \sqrt{-c (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \tan ^{-1}\left(\frac{\sqrt{-c (\sin (e+f x)+1)}}{\sqrt{2} \sqrt{c}}\right)\right)}{2 c^{3/2} f \sqrt{c-c \sin (e+f x)}}","-\frac{a (A+5 B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{2} c^{3/2} f}+\frac{a (A+B) \cos (e+f x)}{f (c-c \sin (e+f x))^{3/2}}+\frac{2 a B \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}",1,"(a*Sec[e + f*x]*(Sqrt[2]*(A + 5*B)*ArcTan[Sqrt[-(c*(1 + Sin[e + f*x]))]/(Sqrt[2]*Sqrt[c])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sqrt[-(c*(1 + Sin[e + f*x]))] + 2*Sqrt[c]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(A + 3*B - 2*B*Sin[e + f*x])))/(2*c^(3/2)*f*Sqrt[c - c*Sin[e + f*x]])","A",1
87,1,199,126,2.2396774,"\int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(5/2),x]","-\frac{a (\sin (e+f x)-1) (\sin (e+f x)+1) \left(\frac{2 \sqrt{c} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) ((A+9 B) \sin (e+f x)+3 A-5 B)}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\sqrt{2} (A-7 B) \sec (e+f x) \sqrt{-c (\sin (e+f x)+1)} \tan ^{-1}\left(\frac{\sqrt{-c (\sin (e+f x)+1)}}{\sqrt{2} \sqrt{c}}\right)\right)}{16 c^{5/2} f \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","-\frac{a (A-7 B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{8 \sqrt{2} c^{5/2} f}-\frac{a (A+9 B) \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{3/2}}+\frac{a (A+B) \cos (e+f x)}{2 f (c-c \sin (e+f x))^{5/2}}",1,"-1/16*(a*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])*(Sqrt[2]*(A - 7*B)*ArcTan[Sqrt[-(c*(1 + Sin[e + f*x]))]/(Sqrt[2]*Sqrt[c])]*Sec[e + f*x]*Sqrt[-(c*(1 + Sin[e + f*x]))] + (2*Sqrt[c]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(3*A - 5*B + (A + 9*B)*Sin[e + f*x]))/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5))/(c^(5/2)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sqrt[c - c*Sin[e + f*x]])","A",1
88,1,217,163,3.4512714,"\int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(7/2),x]","-\frac{a (\sin (e+f x)-1) (\sin (e+f x)+1) \left(\frac{\sqrt{c} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (4 (5 A+17 B) \sin (e+f x)+3 (A-3 B) \cos (2 (e+f x))+47 A-13 B)}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}+3 \sqrt{2} (A-3 B) \sec (e+f x) \sqrt{-c (\sin (e+f x)+1)} \tan ^{-1}\left(\frac{\sqrt{-c (\sin (e+f x)+1)}}{\sqrt{2} \sqrt{c}}\right)\right)}{192 c^{7/2} f \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","-\frac{a (A-3 B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{32 \sqrt{2} c^{7/2} f}-\frac{a (A-3 B) \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac{a (A+13 B) \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}+\frac{a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}",1,"-1/192*(a*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])*(3*Sqrt[2]*(A - 3*B)*ArcTan[Sqrt[-(c*(1 + Sin[e + f*x]))]/(Sqrt[2]*Sqrt[c])]*Sec[e + f*x]*Sqrt[-(c*(1 + Sin[e + f*x]))] + (Sqrt[c]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(47*A - 13*B + 3*(A - 3*B)*Cos[2*(e + f*x)] + 4*(5*A + 17*B)*Sin[e + f*x]))/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7))/(c^(7/2)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sqrt[c - c*Sin[e + f*x]])","A",0
89,1,1355,210,6.6417086,"\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx","Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]","\frac{(7 A-2 B) \sin \left(\frac{1}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{7/2}}{8 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{(7 A-2 B) \cos \left(\frac{1}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{7/2}}{8 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}-\frac{(4 A+B) \cos \left(\frac{3}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{7/2}}{32 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{(22 A-7 B) \cos \left(\frac{5}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{7/2}}{160 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{(A-4 B) \cos \left(\frac{7}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{7/2}}{112 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{A \cos \left(\frac{9}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{7/2}}{48 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{(2 A-3 B) \cos \left(\frac{11}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{7/2}}{352 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{B \cos \left(\frac{13}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{7/2}}{416 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{(4 A+B) (\sin (e+f x) a+a)^2 \sin \left(\frac{3}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{32 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{(22 A-7 B) (\sin (e+f x) a+a)^2 \sin \left(\frac{5}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{160 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}-\frac{(A-4 B) (\sin (e+f x) a+a)^2 \sin \left(\frac{7}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{112 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{A (\sin (e+f x) a+a)^2 \sin \left(\frac{9}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{48 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}-\frac{(2 A-3 B) (\sin (e+f x) a+a)^2 \sin \left(\frac{11}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{352 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{B (\sin (e+f x) a+a)^2 \sin \left(\frac{13}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{416 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}","\frac{256 a^2 c^6 (13 A-3 B) \cos ^5(e+f x)}{15015 f (c-c \sin (e+f x))^{5/2}}+\frac{64 a^2 c^5 (13 A-3 B) \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^2 c^4 (13 A-3 B) \cos ^5(e+f x)}{429 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^3 (13 A-3 B) \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{143 f}-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}",1,"((7*A - 2*B)*Cos[(e + f*x)/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - ((4*A + B)*Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((22*A - 7*B)*Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(160*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((A - 4*B)*Cos[(7*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (A*Cos[(9*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(48*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((2*A - 3*B)*Cos[(11*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(352*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (B*Cos[(13*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(416*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((7*A - 2*B)*Sin[(e + f*x)/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((4*A + B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(3*(e + f*x))/2])/(32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((22*A - 7*B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(5*(e + f*x))/2])/(160*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - ((A - 4*B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(7*(e + f*x))/2])/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (A*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(9*(e + f*x))/2])/(48*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - ((2*A - 3*B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(11*(e + f*x))/2])/(352*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (B*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(13*(e + f*x))/2])/(416*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)","B",1
90,1,1173,167,6.5146471,"\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2),x]","\frac{(6 A-B) \sin \left(\frac{1}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{5/2}}{8 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{(6 A-B) \cos \left(\frac{1}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{5/2}}{8 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}-\frac{(4 A+B) \cos \left(\frac{3}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{5/2}}{24 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{(8 A-3 B) \cos \left(\frac{5}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{5/2}}{80 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}-\frac{(2 A+3 B) \cos \left(\frac{7}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{5/2}}{112 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{(2 A-B) \cos \left(\frac{9}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{5/2}}{144 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}-\frac{B \cos \left(\frac{11}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{5/2}}{176 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{(4 A+B) (\sin (e+f x) a+a)^2 \sin \left(\frac{3}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2}}{24 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{(8 A-3 B) (\sin (e+f x) a+a)^2 \sin \left(\frac{5}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2}}{80 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{(2 A+3 B) (\sin (e+f x) a+a)^2 \sin \left(\frac{7}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2}}{112 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{(2 A-B) (\sin (e+f x) a+a)^2 \sin \left(\frac{9}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2}}{144 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{B (\sin (e+f x) a+a)^2 \sin \left(\frac{11}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2}}{176 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}","\frac{64 a^2 c^5 (11 A-B) \cos ^5(e+f x)}{3465 f (c-c \sin (e+f x))^{5/2}}+\frac{16 a^2 c^4 (11 A-B) \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^2 c^3 (11 A-B) \cos ^5(e+f x)}{99 f \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 B c^2 \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{11 f}",1,"((6*A - B)*Cos[(e + f*x)/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - ((4*A + B)*Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2))/(24*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((8*A - 3*B)*Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2))/(80*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - ((2*A + 3*B)*Cos[(7*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2))/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((2*A - B)*Cos[(9*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2))/(144*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - (B*Cos[(11*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2))/(176*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((6*A - B)*Sin[(e + f*x)/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((4*A + B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2)*Sin[(3*(e + f*x))/2])/(24*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((8*A - 3*B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2)*Sin[(5*(e + f*x))/2])/(80*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((2*A + 3*B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2)*Sin[(7*(e + f*x))/2])/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((2*A - B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2)*Sin[(9*(e + f*x))/2])/(144*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (B*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2)*Sin[(11*(e + f*x))/2])/(176*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)","B",1
91,1,106,120,4.594463,"\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2),x]","\frac{a^2 c \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5 ((130 B-90 A) \sin (e+f x)+162 A+35 B \cos (2 (e+f x))-87 B)}{315 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{8 a^2 c^4 (9 A+B) \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^{5/2}}+\frac{2 a^2 c^3 (9 A+B) \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^{3/2}}-\frac{2 a^2 B c^2 \cos ^5(e+f x)}{9 f \sqrt{c-c \sin (e+f x)}}",1,"(a^2*c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(162*A - 87*B + 35*B*Cos[2*(e + f*x)] + (-90*A + 130*B)*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(315*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
92,1,89,81,0.5824051,"\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]],x]","\frac{2 a^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5 (7 A+5 B \sin (e+f x)-2 B)}{35 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{2 a^2 c^3 (7 A+3 B) \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^{5/2}}-\frac{2 a^2 B c^2 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^{3/2}}",1,"(2*a^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(7*A - 2*B + 5*B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(35*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
93,1,175,161,1.1819081,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{a^2 (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (2 (5 A+11 B) \sin (e+f x)+70 A-3 B \cos (2 (e+f x))+79 B)+(120+120 i) \sqrt[4]{-1} (A+B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right)\right)}{15 f \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","-\frac{2 a^2 c (A+B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac{4 a^2 (A+B) \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}+\frac{4 \sqrt{2} a^2 (A+B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{c} f}-\frac{2 a^2 B c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}",1,"-1/15*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2*((120 + 120*I)*(-1)^(1/4)*(A + B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])] + (Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(70*A + 79*B - 3*B*Cos[2*(e + f*x)] + 2*(5*A + 11*B)*Sin[e + f*x])))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*Sqrt[c - c*Sin[e + f*x]])","C",1
94,1,355,176,0.8960332,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(3/2),x]","\frac{a^2 (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(12 (A+B) \sin \left(\frac{1}{2} (e+f x)\right)+3 (2 A+7 B) \cos \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+3 (2 A+7 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+6 (A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+(6+6 i) \sqrt[4]{-1} (3 A+7 B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2-B \cos \left(\frac{3}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+B \sin \left(\frac{3}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2\right)}{3 f (c-c \sin (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","-\frac{\sqrt{2} a^2 (3 A+7 B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{c^{3/2} f}+\frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{2 f (c-c \sin (e+f x))^{7/2}}+\frac{a^2 (3 A+7 B) \cos ^3(e+f x)}{6 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 (3 A+7 B) \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2*(6*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) + (6 + 6*I)*(-1)^(1/4)*(3*A + 7*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + 3*(2*A + 7*B)*Cos[(e + f*x)/2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 - B*Cos[(3*(e + f*x))/2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + 12*(A + B)*Sin[(e + f*x)/2] + 3*(2*A + 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(3*(e + f*x))/2]))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(3/2))","C",1
95,1,344,175,1.169733,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(5/2),x]","\frac{a^2 (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(8 (A+B) \sin \left(\frac{1}{2} (e+f x)\right)-(5 A+13 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-2 (5 A+13 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+4 (A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+(-3-3 i) \sqrt[4]{-1} (A+9 B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4-8 B \cos \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4-8 B \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4\right)}{4 f (c-c \sin (e+f x))^{5/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","\frac{3 a^2 (A+9 B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{4 \sqrt{2} c^{5/2} f}+\frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{4 f (c-c \sin (e+f x))^{9/2}}-\frac{3 a^2 (A+9 B) \cos (e+f x)}{8 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{a^2 (A+9 B) \cos ^3(e+f x)}{8 f (c-c \sin (e+f x))^{5/2}}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(4*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - (5*A + 13*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 - (3 + 3*I)*(-1)^(1/4)*(A + 9*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 - 8*B*Cos[(e + f*x)/2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + 8*(A + B)*Sin[(e + f*x)/2] - 2*(5*A + 13*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] - 8*B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2)/(4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(5/2))","C",1
96,1,342,175,1.7711877,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(7/2),x]","\frac{a^2 (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(64 (A+B) \sin \left(\frac{1}{2} (e+f x)\right)+3 (A+21 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5+6 (A+21 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4-4 (7 A+19 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-8 (7 A+19 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+32 (A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+(-3-3 i) \sqrt[4]{-1} (A-11 B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6\right)}{48 f (c-c \sin (e+f x))^{7/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","\frac{a^2 (A-11 B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{16 \sqrt{2} c^{7/2} f}+\frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{6 f (c-c \sin (e+f x))^{11/2}}-\frac{a^2 (A-11 B) \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 (A-11 B) \cos ^3(e+f x)}{24 f (c-c \sin (e+f x))^{7/2}}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(32*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 4*(7*A + 19*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + 3*(A + 21*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 - (3 + 3*I)*(-1)^(1/4)*(A - 11*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6 + 64*(A + B)*Sin[(e + f*x)/2] - 8*(7*A + 19*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + 6*(A + 21*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2)/(48*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(7/2))","C",1
97,1,357,222,2.6874003,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(9/2),x]","\frac{a^2 (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left((-24-24 i) \sqrt[4]{-1} (3 A-13 B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^8+2013 A \sin \left(\frac{1}{2} (e+f x)\right)+999 A \sin \left(\frac{3}{2} (e+f x)\right)-69 A \sin \left(\frac{5}{2} (e+f x)\right)+9 A \sin \left(\frac{7}{2} (e+f x)\right)+2013 A \cos \left(\frac{1}{2} (e+f x)\right)-999 A \cos \left(\frac{3}{2} (e+f x)\right)-69 A \cos \left(\frac{5}{2} (e+f x)\right)-9 A \cos \left(\frac{7}{2} (e+f x)\right)+1517 B \sin \left(\frac{1}{2} (e+f x)\right)+791 B \sin \left(\frac{3}{2} (e+f x)\right)-725 B \sin \left(\frac{5}{2} (e+f x)\right)-39 B \sin \left(\frac{7}{2} (e+f x)\right)+1517 B \cos \left(\frac{1}{2} (e+f x)\right)-791 B \cos \left(\frac{3}{2} (e+f x)\right)-725 B \cos \left(\frac{5}{2} (e+f x)\right)+39 B \cos \left(\frac{7}{2} (e+f x)\right)\right)}{6144 f (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","\frac{a^2 (3 A-13 B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{256 \sqrt{2} c^{9/2} f}+\frac{a^2 (3 A-13 B) \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}-\frac{a^2 (3 A-13 B) \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2*(2013*A*Cos[(e + f*x)/2] + 1517*B*Cos[(e + f*x)/2] - 999*A*Cos[(3*(e + f*x))/2] - 791*B*Cos[(3*(e + f*x))/2] - 69*A*Cos[(5*(e + f*x))/2] - 725*B*Cos[(5*(e + f*x))/2] - 9*A*Cos[(7*(e + f*x))/2] + 39*B*Cos[(7*(e + f*x))/2] - (24 + 24*I)*(-1)^(1/4)*(3*A - 13*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8 + 2013*A*Sin[(e + f*x)/2] + 1517*B*Sin[(e + f*x)/2] + 999*A*Sin[(3*(e + f*x))/2] + 791*B*Sin[(3*(e + f*x))/2] - 69*A*Sin[(5*(e + f*x))/2] - 725*B*Sin[(5*(e + f*x))/2] + 9*A*Sin[(7*(e + f*x))/2] - 39*B*Sin[(7*(e + f*x))/2]))/(6144*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2))","C",1
98,1,1569,210,6.8343315,"\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx","Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]","\frac{5 (8 A-B) \sin \left(\frac{1}{2} (e+f x)\right) (\sin (e+f x) a+a)^3 (c-c \sin (e+f x))^{7/2}}{64 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{5 (8 A-B) \cos \left(\frac{1}{2} (e+f x)\right) (\sin (e+f x) a+a)^3 (c-c \sin (e+f x))^{7/2}}{64 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{5 (6 A+B) \cos \left(\frac{3}{2} (e+f x)\right) (\sin (e+f x) a+a)^3 (c-c \sin (e+f x))^{7/2}}{192 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{3 (10 A-3 B) \cos \left(\frac{5}{2} (e+f x)\right) (\sin (e+f x) a+a)^3 (c-c \sin (e+f x))^{7/2}}{320 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{3 (4 A+3 B) \cos \left(\frac{7}{2} (e+f x)\right) (\sin (e+f x) a+a)^3 (c-c \sin (e+f x))^{7/2}}{448 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(12 A-5 B) \cos \left(\frac{9}{2} (e+f x)\right) (\sin (e+f x) a+a)^3 (c-c \sin (e+f x))^{7/2}}{576 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{(2 A+5 B) \cos \left(\frac{11}{2} (e+f x)\right) (\sin (e+f x) a+a)^3 (c-c \sin (e+f x))^{7/2}}{704 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(2 A-B) \cos \left(\frac{13}{2} (e+f x)\right) (\sin (e+f x) a+a)^3 (c-c \sin (e+f x))^{7/2}}{832 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{B \cos \left(\frac{15}{2} (e+f x)\right) (\sin (e+f x) a+a)^3 (c-c \sin (e+f x))^{7/2}}{960 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{5 (6 A+B) (\sin (e+f x) a+a)^3 \sin \left(\frac{3}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{192 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{3 (10 A-3 B) (\sin (e+f x) a+a)^3 \sin \left(\frac{5}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{320 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{3 (4 A+3 B) (\sin (e+f x) a+a)^3 \sin \left(\frac{7}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{448 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(12 A-5 B) (\sin (e+f x) a+a)^3 \sin \left(\frac{9}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{576 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(2 A+5 B) (\sin (e+f x) a+a)^3 \sin \left(\frac{11}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{704 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(2 A-B) (\sin (e+f x) a+a)^3 \sin \left(\frac{13}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{832 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{B (\sin (e+f x) a+a)^3 \sin \left(\frac{15}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{960 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{256 a^3 c^7 (15 A-B) \cos ^7(e+f x)}{45045 f (c-c \sin (e+f x))^{7/2}}+\frac{64 a^3 c^6 (15 A-B) \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^{5/2}}+\frac{8 a^3 c^5 (15 A-B) \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 c^4 (15 A-B) \cos ^7(e+f x)}{195 f \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 B c^3 \cos ^7(e+f x) \sqrt{c-c \sin (e+f x)}}{15 f}",1,"(5*(8*A - B)*Cos[(e + f*x)/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(64*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (5*(6*A + B)*Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(192*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (3*(10*A - 3*B)*Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(320*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (3*(4*A + 3*B)*Cos[(7*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(448*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((12*A - 5*B)*Cos[(9*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(576*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((2*A + 5*B)*Cos[(11*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(704*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((2*A - B)*Cos[(13*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(832*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (B*Cos[(15*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(960*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (5*(8*A - B)*Sin[(e + f*x)/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(64*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (5*(6*A + B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(3*(e + f*x))/2])/(192*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (3*(10*A - 3*B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(5*(e + f*x))/2])/(320*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (3*(4*A + 3*B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(7*(e + f*x))/2])/(448*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((12*A - 5*B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(9*(e + f*x))/2])/(576*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((2*A + 5*B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(11*(e + f*x))/2])/(704*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((2*A - B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(13*(e + f*x))/2])/(832*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (B*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(15*(e + f*x))/2])/(960*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)","B",1
99,1,1351,161,6.7135478,"\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2),x]","\frac{5 A \sin \left(\frac{1}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^3}{8 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{5 A \cos \left(\frac{1}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^3}{8 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{5 (4 A+B) \cos \left(\frac{3}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^3}{96 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(2 A-B) \cos \left(\frac{5}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^3}{32 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{(5 A+2 B) \cos \left(\frac{7}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^3}{112 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(A-2 B) \cos \left(\frac{9}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^3}{144 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{(2 A+B) \cos \left(\frac{11}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^3}{352 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{B \cos \left(\frac{13}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^3}{416 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{5 (4 A+B) (c-c \sin (e+f x))^{5/2} \sin \left(\frac{3}{2} (e+f x)\right) (\sin (e+f x) a+a)^3}{96 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(2 A-B) (c-c \sin (e+f x))^{5/2} \sin \left(\frac{5}{2} (e+f x)\right) (\sin (e+f x) a+a)^3}{32 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(5 A+2 B) (c-c \sin (e+f x))^{5/2} \sin \left(\frac{7}{2} (e+f x)\right) (\sin (e+f x) a+a)^3}{112 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(A-2 B) (c-c \sin (e+f x))^{5/2} \sin \left(\frac{9}{2} (e+f x)\right) (\sin (e+f x) a+a)^3}{144 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(2 A+B) (c-c \sin (e+f x))^{5/2} \sin \left(\frac{11}{2} (e+f x)\right) (\sin (e+f x) a+a)^3}{352 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{B (c-c \sin (e+f x))^{5/2} \sin \left(\frac{13}{2} (e+f x)\right) (\sin (e+f x) a+a)^3}{416 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{64 a^3 c^6 (13 A+B) \cos ^7(e+f x)}{9009 f (c-c \sin (e+f x))^{7/2}}+\frac{16 a^3 c^5 (13 A+B) \cos ^7(e+f x)}{1287 f (c-c \sin (e+f x))^{5/2}}+\frac{2 a^3 c^4 (13 A+B) \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}-\frac{2 a^3 B c^3 \cos ^7(e+f x)}{13 f \sqrt{c-c \sin (e+f x)}}",1,"(5*A*Cos[(e + f*x)/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (5*(4*A + B)*Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/(96*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((2*A - B)*Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/(32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((5*A + 2*B)*Cos[(7*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((A - 2*B)*Cos[(9*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/(144*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((2*A + B)*Cos[(11*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/(352*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (B*Cos[(13*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/(416*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (5*A*Sin[(e + f*x)/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (5*(4*A + B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2)*Sin[(3*(e + f*x))/2])/(96*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((2*A - B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2)*Sin[(5*(e + f*x))/2])/(32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((5*A + 2*B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2)*Sin[(7*(e + f*x))/2])/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((A - 2*B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2)*Sin[(9*(e + f*x))/2])/(144*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((2*A + B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2)*Sin[(11*(e + f*x))/2])/(352*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (B*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2)*Sin[(13*(e + f*x))/2])/(416*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)","B",1
100,1,1157,124,6.5048784,"\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2),x]","\frac{(6 A+B) \sin \left(\frac{1}{2} (e+f x)\right) (c-c \sin (e+f x))^{3/2} (\sin (e+f x) a+a)^3}{8 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(6 A+B) \cos \left(\frac{1}{2} (e+f x)\right) (c-c \sin (e+f x))^{3/2} (\sin (e+f x) a+a)^3}{8 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{(8 A+3 B) \cos \left(\frac{3}{2} (e+f x)\right) (c-c \sin (e+f x))^{3/2} (\sin (e+f x) a+a)^3}{24 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{B \cos \left(\frac{5}{2} (e+f x)\right) (c-c \sin (e+f x))^{3/2} (\sin (e+f x) a+a)^3}{16 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{(6 A+B) \cos \left(\frac{7}{2} (e+f x)\right) (c-c \sin (e+f x))^{3/2} (\sin (e+f x) a+a)^3}{112 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{(2 A+3 B) \cos \left(\frac{9}{2} (e+f x)\right) (c-c \sin (e+f x))^{3/2} (\sin (e+f x) a+a)^3}{144 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{B \cos \left(\frac{11}{2} (e+f x)\right) (c-c \sin (e+f x))^{3/2} (\sin (e+f x) a+a)^3}{176 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(8 A+3 B) (c-c \sin (e+f x))^{3/2} \sin \left(\frac{3}{2} (e+f x)\right) (\sin (e+f x) a+a)^3}{24 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{B (c-c \sin (e+f x))^{3/2} \sin \left(\frac{5}{2} (e+f x)\right) (\sin (e+f x) a+a)^3}{16 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(6 A+B) (c-c \sin (e+f x))^{3/2} \sin \left(\frac{7}{2} (e+f x)\right) (\sin (e+f x) a+a)^3}{112 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{(2 A+3 B) (c-c \sin (e+f x))^{3/2} \sin \left(\frac{9}{2} (e+f x)\right) (\sin (e+f x) a+a)^3}{144 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{B (c-c \sin (e+f x))^{3/2} \sin \left(\frac{11}{2} (e+f x)\right) (\sin (e+f x) a+a)^3}{176 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{8 a^3 c^5 (11 A+3 B) \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^{7/2}}+\frac{2 a^3 c^4 (11 A+3 B) \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^{5/2}}-\frac{2 a^3 B c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}",1,"((6*A + B)*Cos[(e + f*x)/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((8*A + 3*B)*Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(24*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (B*Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(16*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((6*A + B)*Cos[(7*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((2*A + 3*B)*Cos[(9*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(144*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (B*Cos[(11*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(176*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((6*A + B)*Sin[(e + f*x)/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((8*A + 3*B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)*Sin[(3*(e + f*x))/2])/(24*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (B*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)*Sin[(5*(e + f*x))/2])/(16*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((6*A + B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)*Sin[(7*(e + f*x))/2])/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((2*A + 3*B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)*Sin[(9*(e + f*x))/2])/(144*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (B*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)*Sin[(11*(e + f*x))/2])/(176*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)","B",1
101,1,89,81,1.0136531,"\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]],x]","\frac{2 a^3 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7 (9 A+7 B \sin (e+f x)-2 B)}{63 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{2 a^3 c^4 (9 A+5 B) \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}-\frac{2 a^3 B c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}",1,"(2*a^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(9*A - 2*B + 7*B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(63*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
102,1,193,200,1.3475941,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{a^3 (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (-(448 A+673 B) \sin (e+f x)+6 (7 A+22 B) \cos (2 (e+f x))-2086 A+15 B \sin (3 (e+f x))-2236 B)+(6720+6720 i) \sqrt[4]{-1} (A+B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right)\right)}{420 f \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","-\frac{2 a^3 c^2 (A+B) \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac{4 a^3 c (A+B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac{8 a^3 (A+B) \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}+\frac{8 \sqrt{2} a^3 (A+B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{c} f}-\frac{2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}",1,"-1/420*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*((6720 + 6720*I)*(-1)^(1/4)*(A + B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])] - 2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-2086*A - 2236*B + 6*(7*A + 22*B)*Cos[2*(e + f*x)] - (448*A + 673*B)*Sin[e + f*x] + 15*B*Sin[3*(e + f*x)])))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c - c*Sin[e + f*x]])","C",1
103,1,444,218,1.6898261,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(3/2),x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(240 (A+B) \sin \left(\frac{1}{2} (e+f x)\right)+30 (9 A+20 B) \cos \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2-5 (2 A+9 B) \cos \left(\frac{3}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+30 (9 A+20 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+5 (2 A+9 B) \sin \left(\frac{3}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+120 (A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+(120+120 i) \sqrt[4]{-1} (5 A+9 B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2-3 B \cos \left(\frac{5}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2-3 B \sin \left(\frac{5}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2\right)}{30 f (c-c \sin (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","-\frac{2 \sqrt{2} a^3 (5 A+9 B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{c^{3/2} f}+\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}+\frac{a^3 c (5 A+9 B) \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}+\frac{a^3 (5 A+9 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 (5 A+9 B) \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(120*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) + (120 + 120*I)*(-1)^(1/4)*(5*A + 9*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + 30*(9*A + 20*B)*Cos[(e + f*x)/2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 - 5*(2*A + 9*B)*Cos[(3*(e + f*x))/2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 - 3*B*Cos[(5*(e + f*x))/2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + 240*(A + B)*Sin[(e + f*x)/2] + 30*(9*A + 20*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + 5*(2*A + 9*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(3*(e + f*x))/2] - 3*B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(5*(e + f*x))/2]))/(30*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^(3/2))","C",1
104,1,434,225,2.2092024,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(5/2),x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(24 (A+B) \sin \left(\frac{1}{2} (e+f x)\right)-6 (2 A+11 B) \cos \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4-6 (2 A+11 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4-3 (9 A+17 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-6 (9 A+17 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+12 (A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+(-15-15 i) \sqrt[4]{-1} (3 A+11 B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4+2 B \cos \left(\frac{3}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4-2 B \sin \left(\frac{3}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4\right)}{6 f (c-c \sin (e+f x))^{5/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{5 a^3 (3 A+11 B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{2 \sqrt{2} c^{5/2} f}+\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{4 f (c-c \sin (e+f x))^{11/2}}-\frac{5 a^3 (3 A+11 B) \cos (e+f x)}{4 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{a^3 c (3 A+11 B) \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{7/2}}-\frac{5 a^3 (3 A+11 B) \cos ^3(e+f x)}{24 c f (c-c \sin (e+f x))^{3/2}}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(12*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 3*(9*A + 17*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 - (15 + 15*I)*(-1)^(1/4)*(3*A + 11*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 - 6*(2*A + 11*B)*Cos[(e + f*x)/2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + 2*B*Cos[(3*(e + f*x))/2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + 24*(A + B)*Sin[(e + f*x)/2] - 6*(9*A + 17*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] - 6*(2*A + 11*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(e + f*x)/2] - 2*B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(3*(e + f*x))/2]))/(6*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^(5/2))","C",1
105,1,422,217,3.093306,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(7/2),x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(64 (A+B) \sin \left(\frac{1}{2} (e+f x)\right)+3 (11 A+47 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5+6 (11 A+47 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4-4 (13 A+25 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-8 (13 A+25 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+32 (A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+(15+15 i) \sqrt[4]{-1} (A+13 B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6+48 B \cos \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6+48 B \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6\right)}{24 f (c-c \sin (e+f x))^{7/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","-\frac{5 a^3 (A+13 B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{8 \sqrt{2} c^{7/2} f}+\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{6 f (c-c \sin (e+f x))^{13/2}}+\frac{5 a^3 (A+13 B) \cos (e+f x)}{16 c^3 f \sqrt{c-c \sin (e+f x)}}-\frac{a^3 c (A+13 B) \cos ^5(e+f x)}{24 f (c-c \sin (e+f x))^{9/2}}+\frac{5 a^3 (A+13 B) \cos ^3(e+f x)}{48 c f (c-c \sin (e+f x))^{5/2}}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(32*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 4*(13*A + 25*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + 3*(11*A + 47*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + (15 + 15*I)*(-1)^(1/4)*(A + 13*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6 + 48*B*Cos[(e + f*x)/2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6 + 64*(A + B)*Sin[(e + f*x)/2] - 8*(13*A + 25*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + 6*(11*A + 47*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(e + f*x)/2] + 48*B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6*Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3)/(24*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^(7/2))","C",1
106,1,355,217,4.3769704,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(9/2),x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left((120+120 i) \sqrt[4]{-1} (A-15 B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^8+1765 A \sin \left(\frac{1}{2} (e+f x)\right)+895 A \sin \left(\frac{3}{2} (e+f x)\right)-397 A \sin \left(\frac{5}{2} (e+f x)\right)-15 A \sin \left(\frac{7}{2} (e+f x)\right)+1765 A \cos \left(\frac{1}{2} (e+f x)\right)-895 A \cos \left(\frac{3}{2} (e+f x)\right)-397 A \cos \left(\frac{5}{2} (e+f x)\right)+15 A \cos \left(\frac{7}{2} (e+f x)\right)+405 B \sin \left(\frac{1}{2} (e+f x)\right)+2703 B \sin \left(\frac{3}{2} (e+f x)\right)+579 B \sin \left(\frac{5}{2} (e+f x)\right)-543 B \sin \left(\frac{7}{2} (e+f x)\right)+405 B \cos \left(\frac{1}{2} (e+f x)\right)-2703 B \cos \left(\frac{3}{2} (e+f x)\right)+579 B \cos \left(\frac{5}{2} (e+f x)\right)+543 B \cos \left(\frac{7}{2} (e+f x)\right)\right)}{3072 f (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","-\frac{5 a^3 (A-15 B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{128 \sqrt{2} c^{9/2} f}+\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac{5 a^3 (A-15 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{a^3 c (A-15 B) \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac{5 a^3 (A-15 B) \cos ^3(e+f x)}{192 c f (c-c \sin (e+f x))^{7/2}}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(1765*A*Cos[(e + f*x)/2] + 405*B*Cos[(e + f*x)/2] - 895*A*Cos[(3*(e + f*x))/2] - 2703*B*Cos[(3*(e + f*x))/2] - 397*A*Cos[(5*(e + f*x))/2] + 579*B*Cos[(5*(e + f*x))/2] + 15*A*Cos[(7*(e + f*x))/2] + 543*B*Cos[(7*(e + f*x))/2] + (120 + 120*I)*(-1)^(1/4)*(A - 15*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8 + 1765*A*Sin[(e + f*x)/2] + 405*B*Sin[(e + f*x)/2] + 895*A*Sin[(3*(e + f*x))/2] + 2703*B*Sin[(3*(e + f*x))/2] - 397*A*Sin[(5*(e + f*x))/2] + 579*B*Sin[(5*(e + f*x))/2] - 15*A*Sin[(7*(e + f*x))/2] - 543*B*Sin[(7*(e + f*x))/2]))/(3072*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^(9/2))","C",1
107,1,485,266,6.8327362,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(11/2),x]","\frac{(a \sin (e+f x)+a)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(56370 A \sin \left(\frac{1}{2} (e+f x)\right)+31140 A \sin \left(\frac{3}{2} (e+f x)\right)-10404 A \sin \left(\frac{5}{2} (e+f x)\right)-435 A \sin \left(\frac{7}{2} (e+f x)\right)-45 A \sin \left(\frac{9}{2} (e+f x)\right)+56370 A \cos \left(\frac{1}{2} (e+f x)\right)-31140 A \cos \left(\frac{3}{2} (e+f x)\right)-10404 A \cos \left(\frac{5}{2} (e+f x)\right)+435 A \cos \left(\frac{7}{2} (e+f x)\right)-45 A \cos \left(\frac{9}{2} (e+f x)\right)+38970 B \sin \left(\frac{1}{2} (e+f x)\right)+38580 B \sin \left(\frac{3}{2} (e+f x)\right)-12724 B \sin \left(\frac{5}{2} (e+f x)\right)-7775 B \sin \left(\frac{7}{2} (e+f x)\right)+255 B \sin \left(\frac{9}{2} (e+f x)\right)+38970 B \cos \left(\frac{1}{2} (e+f x)\right)-38580 B \cos \left(\frac{3}{2} (e+f x)\right)-12724 B \cos \left(\frac{5}{2} (e+f x)\right)+7775 B \cos \left(\frac{7}{2} (e+f x)\right)+255 B \cos \left(\frac{9}{2} (e+f x)\right)\right)}{122880 f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{\left(\frac{1}{512}+\frac{i}{512}\right) \sqrt[4]{-1} (3 A-17 B) (a \sin (e+f x)+a)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11} \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \sec \left(\frac{1}{4} (e+f x)\right) \left(\sin \left(\frac{1}{4} (e+f x)\right)+\cos \left(\frac{1}{4} (e+f x)\right)\right)\right)}{f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","-\frac{a^3 (3 A-17 B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{512 \sqrt{2} c^{11/2} f}-\frac{a^3 (3 A-17 B) \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}+\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac{a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}+\frac{a^3 c (3 A-17 B) \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac{a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}",1,"((1/512 + I/512)*(-1)^(1/4)*(3*A - 17*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*Sec[(e + f*x)/4]*(Cos[(e + f*x)/4] + Sin[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(a + a*Sin[e + f*x])^3)/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^(11/2)) + ((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a + a*Sin[e + f*x])^3*(56370*A*Cos[(e + f*x)/2] + 38970*B*Cos[(e + f*x)/2] - 31140*A*Cos[(3*(e + f*x))/2] - 38580*B*Cos[(3*(e + f*x))/2] - 10404*A*Cos[(5*(e + f*x))/2] - 12724*B*Cos[(5*(e + f*x))/2] + 435*A*Cos[(7*(e + f*x))/2] + 7775*B*Cos[(7*(e + f*x))/2] - 45*A*Cos[(9*(e + f*x))/2] + 255*B*Cos[(9*(e + f*x))/2] + 56370*A*Sin[(e + f*x)/2] + 38970*B*Sin[(e + f*x)/2] + 31140*A*Sin[(3*(e + f*x))/2] + 38580*B*Sin[(3*(e + f*x))/2] - 10404*A*Sin[(5*(e + f*x))/2] - 12724*B*Sin[(5*(e + f*x))/2] - 435*A*Sin[(7*(e + f*x))/2] - 7775*B*Sin[(7*(e + f*x))/2] - 45*A*Sin[(9*(e + f*x))/2] + 255*B*Sin[(9*(e + f*x))/2]))/(122880*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^(11/2))","C",1
108,1,157,200,5.5904828,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x]),x]","-\frac{c^3 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (196 (A-2 B) \cos (2 (e+f x))+2450 A \sin (e+f x)-14 A \sin (3 (e+f x))+4900 A-3430 B \sin (e+f x)+58 B \sin (3 (e+f x))+5 B \cos (4 (e+f x))-6125 B)}{140 a f (\sin (e+f x)+1) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{128 c^4 (7 A-9 B) \cos (e+f x)}{35 a f \sqrt{c-c \sin (e+f x)}}-\frac{32 c^3 (7 A-9 B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{35 a f}-\frac{12 c^2 (7 A-9 B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{35 a f}-\frac{c (7 A-9 B) \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 a f}-\frac{(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f}",1,"-1/140*(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(4900*A - 6125*B + 196*(A - 2*B)*Cos[2*(e + f*x)] + 5*B*Cos[4*(e + f*x)] + 2450*A*Sin[e + f*x] - 3430*B*Sin[e + f*x] - 14*A*Sin[3*(e + f*x)] + 58*B*Sin[3*(e + f*x)]))/(a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x]))","A",1
109,1,134,159,1.757782,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x]),x]","-\frac{c^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (25 (8 A-13 B) \sin (e+f x)+2 (5 A-16 B) \cos (2 (e+f x))+450 A+3 B \sin (3 (e+f x))-600 B)}{30 a f (\sin (e+f x)+1) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{32 c^3 (5 A-7 B) \cos (e+f x)}{15 a f \sqrt{c-c \sin (e+f x)}}-\frac{8 c^2 (5 A-7 B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{15 a f}-\frac{c (5 A-7 B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{5 a f}-\frac{(A-B) \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{a c f}",1,"-1/30*(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(450*A - 600*B + 2*(5*A - 16*B)*Cos[2*(e + f*x)] + 25*(8*A - 13*B)*Sin[e + f*x] + 3*B*Sin[3*(e + f*x)]))/(a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x]))","A",1
110,1,113,118,0.6007732,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2}}{a+a \sin (e+f x)} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x]),x]","\frac{c \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) ((14 B-6 A) \sin (e+f x)-18 A+B \cos (2 (e+f x))+27 B)}{3 a f (\sin (e+f x)+1) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{4 c^2 (3 A-5 B) \cos (e+f x)}{3 a f \sqrt{c-c \sin (e+f x)}}-\frac{c (3 A-5 B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{3 a f}-\frac{(A-B) \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{a c f}",1,"(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-18*A + 27*B + B*Cos[2*(e + f*x)] + (-6*A + 14*B)*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(3*a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x]))","A",1
111,1,44,73,0.2026362,"\int \frac{(A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)}}{a+a \sin (e+f x)} \, dx","Integrate[((A + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x]),x]","\frac{2 \sec (e+f x) \sqrt{c-c \sin (e+f x)} (-A+B \sin (e+f x)+2 B)}{a f}","-\frac{c (A-3 B) \cos (e+f x)}{a f \sqrt{c-c \sin (e+f x)}}-\frac{(A-B) \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{a c f}",1,"(2*Sec[e + f*x]*(-A + 2*B + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(a*f)","A",1
112,1,140,91,0.4501554,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(-(1+i) \sqrt[4]{-1} (A+B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-A+B\right)}{a f (\sin (e+f x)+1) \sqrt{c-c \sin (e+f x)}}","\frac{(A+B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{2} a \sqrt{c} f}-\frac{(A-B) \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{a c f}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-A + B - (1 + I)*(-1)^(1/4)*(A + B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))/(a*f*(1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])","C",1
113,1,284,136,0.5423655,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(2 (B-A) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+(A+B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+2 (A+B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-(1+i) \sqrt[4]{-1} (3 A-B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2\right)}{4 a f (\sin (e+f x)+1) (c-c \sin (e+f x))^{3/2}}","\frac{(3 A-B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{4 \sqrt{2} a c^{3/2} f}+\frac{(3 A-B) \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac{(A-B) \sec (e+f x)}{a c f \sqrt{c-c \sin (e+f x)}}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*(-A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + (A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - (1 + I)*(-1)^(1/4)*(3*A - B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(A + B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))/(4*a*f*(1 + Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2))","C",1
114,1,404,180,0.8526073,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(8 (B-A) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4+(7 A-B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3+2 (7 A-B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+4 (A+B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+8 (A+B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-(3+3 i) \sqrt[4]{-1} (5 A-3 B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4\right)}{32 a f (\sin (e+f x)+1) (c-c \sin (e+f x))^{5/2}}","\frac{3 (5 A-3 B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{32 \sqrt{2} a c^{5/2} f}-\frac{(5 A-3 B) \sec (e+f x)}{8 a c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{3 (5 A-3 B) \cos (e+f x)}{32 a c f (c-c \sin (e+f x))^{3/2}}+\frac{(A+B) \sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(8*(-A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + 4*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (7*A - B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - (3 + 3*I)*(-1)^(1/4)*(5*A - 3*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 8*(A + B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(7*A - B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))/(32*a*f*(1 + Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))","C",1
115,1,953,242,6.819311,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^2,x]","-\frac{(26 A-83 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 \sin \left(\frac{3}{2} (e+f x)\right) (c-c \sin (e+f x))^{9/2}}{12 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 (\sin (e+f x) a+a)^2}-\frac{(2 A-13 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 \sin \left(\frac{5}{2} (e+f x)\right) (c-c \sin (e+f x))^{9/2}}{20 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 (\sin (e+f x) a+a)^2}-\frac{B \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 \sin \left(\frac{7}{2} (e+f x)\right) (c-c \sin (e+f x))^{9/2}}{28 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 (\sin (e+f x) a+a)^2}+\frac{(164 A-351 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 (c-c \sin (e+f x))^{9/2}}{4 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 (\sin (e+f x) a+a)^2}+\frac{(164 A-351 B) \cos \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 (c-c \sin (e+f x))^{9/2}}{4 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 (\sin (e+f x) a+a)^2}+\frac{(26 A-83 B) \cos \left(\frac{3}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 (c-c \sin (e+f x))^{9/2}}{12 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 (\sin (e+f x) a+a)^2}-\frac{(2 A-13 B) \cos \left(\frac{5}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 (c-c \sin (e+f x))^{9/2}}{20 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 (\sin (e+f x) a+a)^2}+\frac{B \cos \left(\frac{7}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 (c-c \sin (e+f x))^{9/2}}{28 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 (\sin (e+f x) a+a)^2}+\frac{32 (2 A-3 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 (c-c \sin (e+f x))^{9/2}}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 (\sin (e+f x) a+a)^2}-\frac{32 (A-B) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) (c-c \sin (e+f x))^{9/2}}{3 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 (\sin (e+f x) a+a)^2}","\frac{2048 c^4 (7 A-13 B) \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{105 a^2 f}-\frac{512 c^3 (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{105 a^2 f}-\frac{(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac{64 c^2 (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac{(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac{16 c (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f}",1,"(-32*(A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^(9/2))/(3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) + (32*(2*A - 3*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(9/2))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) + ((164*A - 351*B)*Cos[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2))/(4*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) + ((26*A - 83*B)*Cos[(3*(e + f*x))/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2))/(12*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) - ((2*A - 13*B)*Cos[(5*(e + f*x))/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2))/(20*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) + (B*Cos[(7*(e + f*x))/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2))/(28*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) + ((164*A - 351*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2))/(4*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) - ((26*A - 83*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2)*Sin[(3*(e + f*x))/2])/(12*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) - ((2*A - 13*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2)*Sin[(5*(e + f*x))/2])/(20*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) - (B*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2)*Sin[(7*(e + f*x))/2])/(28*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2)","B",1
116,1,159,201,2.8270282,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^2} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^2,x]","-\frac{c^3 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (12 (25 A-62 B) \cos (2 (e+f x))-2730 A \sin (e+f x)-10 A \sin (3 (e+f x))-2100 A+5838 B \sin (e+f x)+46 B \sin (3 (e+f x))+3 B \cos (4 (e+f x))+4725 B)}{60 a^2 f (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{128 c^3 (5 A-11 B) \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{15 a^2 f}-\frac{(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{3 a^2 c^2 f}-\frac{32 c^2 (5 A-11 B) \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{15 a^2 f}-\frac{(5 A-11 B) \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{15 a^2 f}-\frac{4 c (5 A-11 B) \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^2 f}",1,"-1/60*(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(-2100*A + 4725*B + 12*(25*A - 62*B)*Cos[2*(e + f*x)] + 3*B*Cos[4*(e + f*x)] - 2730*A*Sin[e + f*x] + 5838*B*Sin[e + f*x] - 10*A*Sin[3*(e + f*x)] + 46*B*Sin[3*(e + f*x)]))/(a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2)","A",1
117,1,130,154,1.1620386,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^2} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^2,x]","-\frac{c^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) ((201 B-72 A) \sin (e+f x)+6 (A-4 B) \cos (2 (e+f x))-50 A+B \sin (3 (e+f x))+160 B)}{6 a^2 f (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{3 a^2 c^2 f}+\frac{32 c^2 (A-3 B) \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{3 a^2 f}-\frac{(A-3 B) \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 f}-\frac{8 c (A-3 B) \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 f}",1,"-1/6*(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(-50*A + 160*B + 6*(A - 4*B)*Cos[2*(e + f*x)] + (-72*A + 201*B)*Sin[e + f*x] + B*Sin[3*(e + f*x)]))/(a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2)","A",1
118,1,113,115,0.6573478,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^2} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^2,x]","\frac{c \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (6 (A-5 B) \sin (e+f x)+2 A+3 B \cos (2 (e+f x))-23 B)}{3 a^2 f (\sin (e+f x)+1)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^2 c^2 f}-\frac{(A-7 B) \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 f}+\frac{4 c (A-7 B) \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{3 a^2 f}",1,"(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*A - 23*B + 3*B*Cos[2*(e + f*x)] + 6*(A - 5*B)*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(3*a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2)","A",1
119,1,87,78,0.2816736,"\int \frac{(A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^2} \, dx","Integrate[((A + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^2,x]","-\frac{2 \sqrt{c-c \sin (e+f x)} (A+3 B \sin (e+f x)+2 B)}{3 a^2 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\frac{(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 c^2 f}-\frac{(A+5 B) \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{3 a^2 f}",1,"(-2*(A + 2*B + 3*B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(3*a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
120,1,176,135,0.5094438,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(-3 (A+B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+(-3-3 i) \sqrt[4]{-1} (A+B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+2 (B-A)\right)}{6 a^2 f (\sin (e+f x)+1)^2 \sqrt{c-c \sin (e+f x)}}","-\frac{(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 c^2 f}-\frac{(A+B) \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{2 a^2 c f}+\frac{(A+B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{2 \sqrt{2} a^2 \sqrt{c} f}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*(-A + B) - 3*(A + B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - (3 + 3*I)*(-1)^(1/4)*(A + B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3))/(6*a^2*f*(1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]])","C",1
121,1,300,175,0.8602614,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(3 (A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+6 (A+B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+4 (B-A) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+(-3-3 i) \sqrt[4]{-1} (5 A+B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-12 A \cos ^2(e+f x)\right)}{24 a^2 f (\sin (e+f x)+1)^2 (c-c \sin (e+f x))^{3/2}}","\frac{(5 A+B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{8 \sqrt{2} a^2 c^{3/2} f}-\frac{(A-B) \sec ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{3 a^2 c^2 f}+\frac{(5 A+B) \cos (e+f x)}{8 a^2 f (c-c \sin (e+f x))^{3/2}}-\frac{(5 A+B) \sec (e+f x)}{6 a^2 c f \sqrt{c-c \sin (e+f x)}}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-12*A*Cos[e + f*x]^2 + 4*(-A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + 3*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - (3 + 3*I)*(-1)^(1/4)*(5*A + B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 6*(A + B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3))/(24*a^2*f*(1 + Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(3/2))","C",1
122,1,430,225,1.4113033,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(3 (11 A+3 B) \cos ^3(e+f x)+24 (B-3 A) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4+16 (B-A) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4+6 (11 A+3 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+12 (A+B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+24 (A+B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+(-15-15 i) \sqrt[4]{-1} (7 A-B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4\right)}{192 a^2 f (\sin (e+f x)+1)^2 (c-c \sin (e+f x))^{5/2}}","\frac{5 (7 A-B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{64 \sqrt{2} a^2 c^{5/2} f}-\frac{(A-B) \sec ^3(e+f x)}{3 a^2 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{5 (7 A-B) \sec (e+f x)}{48 a^2 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{5 (7 A-B) \cos (e+f x)}{64 a^2 c f (c-c \sin (e+f x))^{3/2}}+\frac{(7 A-B) \sec (e+f x)}{24 a^2 c f (c-c \sin (e+f x))^{3/2}}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(3*(11*A + 3*B)*Cos[e + f*x]^3 + 16*(-A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + 24*(-3*A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 12*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - (15 + 15*I)*(-1)^(1/4)*(7*A - B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 24*(A + B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 6*(11*A + 3*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3))/(192*a^2*f*(1 + Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2))","C",1
123,1,176,242,4.186404,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^3} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^3,x]","-\frac{c^4 (\sin (e+f x)-1)^4 \sqrt{c-c \sin (e+f x)} (-40 (137 A-402 B) \cos (2 (e+f x))-10 (A-6 B) \cos (4 (e+f x))+15600 A \sin (e+f x)-400 A \sin (3 (e+f x))+11298 A-47430 B \sin (e+f x)+1335 B \sin (3 (e+f x))-3 B \sin (5 (e+f x))-33516 B)}{120 a^3 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\frac{(A-B) \sec ^5(e+f x) (c-c \sin (e+f x))^{15/2}}{5 a^3 c^3 f}-\frac{2048 c^3 (A-3 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{15 a^3 f}+\frac{512 c^2 (A-3 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 f}-\frac{(A-3 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 a^3 c f}-\frac{16 (A-3 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{15 a^3 f}-\frac{64 c (A-3 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{5 a^3 f}",1,"-1/120*(c^4*(-1 + Sin[e + f*x])^4*Sqrt[c - c*Sin[e + f*x]]*(11298*A - 33516*B - 40*(137*A - 402*B)*Cos[2*(e + f*x)] - 10*(A - 6*B)*Cos[4*(e + f*x)] + 15600*A*Sin[e + f*x] - 47430*B*Sin[e + f*x] - 400*A*Sin[3*(e + f*x)] + 1335*B*Sin[3*(e + f*x)] - 3*B*Sin[5*(e + f*x)]))/(a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
124,1,158,209,2.6323753,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^3} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^3,x]","-\frac{c^3 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) ((2200 B-540 A) \cos (2 (e+f x))+1410 A \sin (e+f x)-30 A \sin (3 (e+f x))+1092 A-6390 B \sin (e+f x)+170 B \sin (3 (e+f x))+5 B \cos (4 (e+f x))-4557 B)}{60 a^3 f (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{(A-B) \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{5 a^3 c^3 f}-\frac{128 c^2 (3 A-13 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{15 a^3 f}-\frac{(3 A-13 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{15 a^3 c f}-\frac{4 (3 A-13 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{5 a^3 f}+\frac{32 c (3 A-13 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 f}",1,"-1/60*(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(1092*A - 4557*B + (-540*A + 2200*B)*Cos[2*(e + f*x)] + 5*B*Cos[4*(e + f*x)] + 1410*A*Sin[e + f*x] - 6390*B*Sin[e + f*x] - 30*A*Sin[3*(e + f*x)] + 170*B*Sin[3*(e + f*x)]))/(a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3)","A",1
125,1,132,160,1.2145888,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^3,x]","-\frac{c^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (5 (8 A-133 B) \sin (e+f x)-30 (A-8 B) \cos (2 (e+f x))+58 A+15 B \sin (3 (e+f x))-488 B)}{30 a^3 f (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{(A-B) \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{5 a^3 c^3 f}-\frac{(A-11 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{5 a^3 c f}+\frac{8 (A-11 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 f}-\frac{32 c (A-11 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{15 a^3 f}",1,"-1/30*(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(58*A - 488*B - 30*(A - 8*B)*Cos[2*(e + f*x)] + 5*(8*A - 133*B)*Sin[e + f*x] + 15*B*Sin[3*(e + f*x)]))/(a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3)","A",1
126,1,113,121,0.6957482,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^3} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^3,x]","\frac{c \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (10 (A+3 B) \sin (e+f x)-2 A-15 B \cos (2 (e+f x))+27 B)}{15 a^3 f (\sin (e+f x)+1)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{(A-B) \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{5 a^3 c^3 f}-\frac{(A+9 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 c f}+\frac{4 (A+9 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{15 a^3 f}",1,"(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-2*A + 27*B - 15*B*Cos[2*(e + f*x)] + 10*(A + 3*B)*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(15*a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3)","A",1
127,1,89,85,0.3025527,"\int \frac{(A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx","Integrate[((A + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^3,x]","-\frac{2 \sqrt{c-c \sin (e+f x)} (3 A+5 B \sin (e+f x)+2 B)}{15 a^3 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\frac{(A-B) \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{5 a^3 c^3 f}-\frac{(3 A+7 B) \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{15 a^3 c f}",1,"(-2*(3*A + 2*B + 5*B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(15*a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
128,1,204,174,0.772018,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(-15 (A+B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4-10 (A+B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+(-15-15 i) \sqrt[4]{-1} (A+B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5+12 (B-A)\right)}{60 a^3 f (\sin (e+f x)+1)^3 \sqrt{c-c \sin (e+f x)}}","-\frac{(A-B) \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 c^3 f}-\frac{(A+B) \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{6 a^3 c^2 f}-\frac{(A+B) \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{4 a^3 c f}+\frac{(A+B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{4 \sqrt{2} a^3 \sqrt{c} f}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(12*(-A + B) - 10*(A + B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 15*(A + B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - (15 + 15*I)*(-1)^(1/4)*(A + B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5))/(60*a^3*f*(1 + Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]])","C",1
129,1,357,224,1.3619703,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(15 (A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5+30 (A+B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5-30 (3 A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4+24 (B-A) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+(-15-15 i) \sqrt[4]{-1} (7 A+3 B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5-40 A \cos ^2(e+f x)\right)}{240 a^3 f (\sin (e+f x)+1)^3 (c-c \sin (e+f x))^{3/2}}","\frac{(7 A+3 B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{16 \sqrt{2} a^3 c^{3/2} f}-\frac{(A-B) \sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}-\frac{(7 A+3 B) \sec ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{30 a^3 c^2 f}+\frac{(7 A+3 B) \cos (e+f x)}{16 a^3 f (c-c \sin (e+f x))^{3/2}}-\frac{(7 A+3 B) \sec (e+f x)}{12 a^3 c f \sqrt{c-c \sin (e+f x)}}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-40*A*Cos[e + f*x]^2 + 24*(-A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 - 30*(3*A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + 15*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - (15 + 15*I)*(-1)^(1/4)*(7*A + 3*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + 30*(A + B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5))/(240*a^3*f*(1 + Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))","C",1
130,1,479,258,2.2855158,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(15 (15 A+7 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5+60 (A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5+30 (15 A+7 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5+120 (A+B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5+80 (B-3 A) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+96 (B-A) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4+(-105-105 i) \sqrt[4]{-1} (9 A+B) \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \left(\tan \left(\frac{1}{4} (e+f x)\right)+1\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5-720 A \cos ^4(e+f x)\right)}{1920 a^3 f (\sin (e+f x)+1)^3 (c-c \sin (e+f x))^{5/2}}","\frac{7 (9 A+B) \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{128 \sqrt{2} a^3 c^{5/2} f}-\frac{(A-B) \sec ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{5 a^3 c^3 f}-\frac{(9 A+B) \sec ^3(e+f x)}{30 a^3 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{7 (9 A+B) \sec (e+f x)}{96 a^3 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{7 (9 A+B) \cos (e+f x)}{128 a^3 c f (c-c \sin (e+f x))^{3/2}}+\frac{7 (9 A+B) \sec (e+f x)}{240 a^3 c f (c-c \sin (e+f x))^{3/2}}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-720*A*Cos[e + f*x]^4 + 96*(-A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + 80*(-3*A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 60*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + 15*(15*A + 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - (105 + 105*I)*(-1)^(1/4)*(9*A + B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + 120*(A + B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + 30*(15*A + 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5))/(1920*a^3*f*(1 + Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))","C",1
131,1,118,94,1.0020432,"\int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]","-\frac{c^3 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (4 (23 B-60 A) \sin (e+f x)+4 \cos (2 (e+f x)) (4 (5 A-6 B) \sin (e+f x)-35 A+25 B)+\cos (4 (e+f x)) (5 A+4 B \sin (e+f x)-15 B))}{160 f}","\frac{a B \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{5 c f \sqrt{a \sin (e+f x)+a}}-\frac{a (A+B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt{a \sin (e+f x)+a}}",1,"-1/160*(c^3*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(4*(-60*A + 23*B)*Sin[e + f*x] + 4*Cos[2*(e + f*x)]*(-35*A + 25*B + 4*(5*A - 6*B)*Sin[e + f*x]) + Cos[4*(e + f*x)]*(5*A - 15*B + 4*B*Sin[e + f*x])))/f","A",1
132,1,102,94,0.8209175,"\int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2),x]","\frac{c^2 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (16 (7 A-2 B) \sin (e+f x)-4 \cos (2 (e+f x)) (4 (A-2 B) \sin (e+f x)-12 A+9 B)+3 B \cos (4 (e+f x)))}{96 f}","\frac{a B \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 c f \sqrt{a \sin (e+f x)+a}}-\frac{a (A+B) \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a \sin (e+f x)+a}}",1,"(c^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(3*B*Cos[4*(e + f*x)] + 16*(7*A - 2*B)*Sin[e + f*x] - 4*Cos[2*(e + f*x)]*(-12*A + 9*B + 4*(A - 2*B)*Sin[e + f*x])))/(96*f)","A",1
133,1,84,94,0.5558238,"\int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2),x]","\frac{c \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (2 (6 A-B) \sin (e+f x)+\cos (2 (e+f x)) (3 A+2 B \sin (e+f x)-3 B))}{12 f}","\frac{a B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt{a \sin (e+f x)+a}}-\frac{a (A+B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}",1,"(c*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(2*(6*A - B)*Sin[e + f*x] + Cos[2*(e + f*x)]*(3*A - 3*B + 2*B*Sin[e + f*x])))/(12*f)","A",1
134,1,63,92,0.1745021,"\int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]],x]","\frac{\sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (4 A \sin (e+f x)-B \cos (2 (e+f x)))}{4 f}","\frac{a B \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt{a \sin (e+f x)+a}}-\frac{a (A+B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}",1,"(Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*(-(B*Cos[2*(e + f*x)]) + 4*A*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(4*f)","A",1
135,1,120,100,1.1328773,"\int \frac{\sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x))}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{\sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(B \sin (e+f x)+(A+B) \left(2 \log \left(i-e^{i (e+f x)}\right)-i f x\right)\right)}{f \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{a B \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{c f \sqrt{a \sin (e+f x)+a}}-\frac{a (A+B) \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"-(((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*((A + B)*((-I)*f*x + 2*Log[I - E^(I*(e + f*x))]) + B*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]))","C",1
136,1,147,99,1.1569412,"\int \frac{\sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(3/2),x]","\frac{\sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(A+2 B \log \left(i-e^{i (e+f x)}\right)+i B \left(f x+2 i \log \left(i-e^{i (e+f x)}\right)\right) \sin (e+f x)-i B f x+B\right)}{f (c-c \sin (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{a (A+B) \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{a B \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(A + B - I*B*f*x + 2*B*Log[I - E^(I*(e + f*x))] + I*B*(f*x + (2*I)*Log[I - E^(I*(e + f*x))])*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^(3/2))","C",1
137,1,101,92,0.538958,"\int \frac{\sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(5/2),x]","\frac{\sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (A+2 B \sin (e+f x)-B)}{2 c^3 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{a (A+B) \cos (e+f x)}{2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{a B \cos (e+f x)}{c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}",1,"(Sqrt[a*(1 + Sin[e + f*x])]*(A - B + 2*B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(2*c^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
138,1,103,94,0.5962414,"\int \frac{\sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[(Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(7/2),x]","\frac{\sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (2 A+3 B \sin (e+f x)-B)}{6 c^4 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{a (A+B) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{a B \cos (e+f x)}{2 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}",1,"(Sqrt[a*(1 + Sin[e + f*x])]*(2*A - B + 3*B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(6*c^4*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
139,1,205,146,1.4855145,"\int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]","-\frac{c^3 (\sin (e+f x)-1)^3 (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)} (15 (16 A-11 B) \cos (2 (e+f x))+30 (2 A-B) \cos (4 (e+f x))+840 A \sin (e+f x)+60 A \sin (3 (e+f x))-12 A \sin (5 (e+f x))-240 B \sin (e+f x)+40 B \sin (3 (e+f x))+24 B \sin (5 (e+f x))+5 B \cos (6 (e+f x)))}{960 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\frac{a^2 (3 A-B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{30 f \sqrt{a \sin (e+f x)+a}}-\frac{a (3 A-B) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{15 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{6 f}",1,"-1/960*(c^3*(-1 + Sin[e + f*x])^3*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(15*(16*A - 11*B)*Cos[2*(e + f*x)] + 30*(2*A - B)*Cos[4*(e + f*x)] + 5*B*Cos[6*(e + f*x)] + 840*A*Sin[e + f*x] - 240*B*Sin[e + f*x] + 60*A*Sin[3*(e + f*x)] + 40*B*Sin[3*(e + f*x)] - 12*A*Sin[5*(e + f*x)] + 24*B*Sin[5*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
140,1,172,146,1.6119017,"\int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2),x]","\frac{c^2 (\sin (e+f x)-1)^2 (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)} (4 (100 A-11 B) \sin (e+f x)+3 \cos (4 (e+f x)) (5 A+4 B \sin (e+f x)-5 B)+4 \cos (2 (e+f x)) (4 (5 A+2 B) \sin (e+f x)+15 (A-B)))}{480 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\frac{a^2 (5 A-B) \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{30 f \sqrt{a \sin (e+f x)+a}}-\frac{a (5 A-B) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{20 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f}",1,"(c^2*(-1 + Sin[e + f*x])^2*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(4*(100*A - 11*B)*Sin[e + f*x] + 3*Cos[4*(e + f*x)]*(5*A - 5*B + 4*B*Sin[e + f*x]) + 4*Cos[2*(e + f*x)]*(15*(A - B) + 4*(5*A + 2*B)*Sin[e + f*x])))/(480*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
141,1,96,134,0.7479504,"\int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{c (\sin (e+f x)-1) \sec ^3(e+f x) (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)} (8 A (9 \sin (e+f x)+\sin (3 (e+f x)))-12 B \cos (2 (e+f x))-3 B \cos (4 (e+f x)))}{96 f}","-\frac{a^2 A \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{a A \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}{3 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}{4 f}",1,"-1/96*(c*Sec[e + f*x]^3*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(-12*B*Cos[2*(e + f*x)] - 3*B*Cos[4*(e + f*x)] + 8*A*(9*Sin[e + f*x] + Sin[3*(e + f*x)])))/f","A",1
142,1,81,96,0.5559005,"\int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]],x]","-\frac{a \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (\cos (2 (e+f x)) (3 (A+B)+2 B \sin (e+f x))-2 (6 A+B) \sin (e+f x))}{12 f}","\frac{c (A-B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}}+\frac{B c \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 a f \sqrt{c-c \sin (e+f x)}}",1,"-1/12*(a*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-2*(6*A + B)*Sin[e + f*x] + Cos[2*(e + f*x)]*(3*(A + B) + 2*B*Sin[e + f*x])))/f","A",1
143,1,136,145,0.6852246,"\int \frac{(a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x))}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[((a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{(a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(4 (A+2 B) \sin (e+f x)+16 (A+B) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-B \cos (2 (e+f x))\right)}{4 f \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\frac{2 a^2 (A+B) \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a (A+B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f \sqrt{c-c \sin (e+f x)}}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}}",1,"-1/4*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2)*(-(B*Cos[2*(e + f*x)]) + 16*(A + B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 4*(A + 2*B)*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c - c*Sin[e + f*x]])","A",1
144,1,210,158,0.8609103,"\int \frac{(a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(3/2),x]","-\frac{a \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-2 \sin (e+f x) \left(2 (A+3 B) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-B\right)+4 A \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+4 A+B \cos (2 (e+f x))+12 B \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+3 B\right)}{2 c f (\sin (e+f x)-1) \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{a^2 (A+3 B) \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{a (A+3 B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{2 c f \sqrt{c-c \sin (e+f x)}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{3/2}}",1,"-1/2*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(4*A + 3*B + B*Cos[2*(e + f*x)] + 4*A*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 12*B*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 2*(-B + 2*(A + 3*B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]])*Sin[e + f*x]))/(c*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])","A",1
145,1,198,149,0.9445828,"\int \frac{(a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(5/2),x]","\frac{a \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin (e+f x) \left(A+4 B \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+3 B\right)+B \cos (2 (e+f x)) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-B \left(3 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+2\right)\right)}{c^2 f (\sin (e+f x)-1)^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{a^2 B \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{4 f (c-c \sin (e+f x))^{5/2}}-\frac{a B \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f (c-c \sin (e+f x))^{3/2}}",1,"(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(B*Cos[2*(e + f*x)]*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - B*(2 + 3*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]) + (A + 3*B + 4*B*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]])*Sin[e + f*x]))/(c^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]])","A",1
146,1,125,96,0.9885848,"\int \frac{(a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(7/2),x]","-\frac{a \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (3 (A-B) \sin (e+f x)+A-3 B \cos (2 (e+f x))+4 B)}{6 c^3 f (\sin (e+f x)-1)^3 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{(A-5 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{24 c f (c-c \sin (e+f x))^{5/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{6 f (c-c \sin (e+f x))^{7/2}}",1,"-1/6*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(A + 4*B - 3*B*Cos[2*(e + f*x)] + 3*(A - B)*Sin[e + f*x]))/(c^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]])","A",1
147,1,123,146,1.3523274,"\int \frac{(a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(9/2),x]","\frac{a \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (4 A \sin (e+f x)+2 A-3 B \cos (2 (e+f x))+3 B)}{12 c^4 f (\sin (e+f x)-1)^4 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{96 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{24 c f (c-c \sin (e+f x))^{7/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{8 f (c-c \sin (e+f x))^{9/2}}",1,"(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(2*A + 3*B - 3*B*Cos[2*(e + f*x)] + 4*A*Sin[e + f*x]))/(12*c^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^4*Sqrt[c - c*Sin[e + f*x]])","A",1
148,1,126,154,1.9538905,"\int \frac{(a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(11/2),x]","-\frac{a \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (5 (3 A+B) \sin (e+f x)+9 (A+B)-10 B \cos (2 (e+f x)))}{60 c^5 f (\sin (e+f x)-1)^5 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{a^2 (3 A-7 B) \cos (e+f x)}{120 c^2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{a (3 A-7 B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{40 c f (c-c \sin (e+f x))^{9/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{10 f (c-c \sin (e+f x))^{11/2}}",1,"-1/60*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(9*(A + B) - 10*B*Cos[2*(e + f*x)] + 5*(3*A + B)*Sin[e + f*x]))/(c^5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^5*Sqrt[c - c*Sin[e + f*x]])","A",1
149,1,223,198,2.6494875,"\int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]","-\frac{c^3 (\sin (e+f x)-1)^3 (a (\sin (e+f x)+1))^{5/2} \sqrt{c-c \sin (e+f x)} (525 (A-B) \cos (2 (e+f x))+210 (A-B) \cos (4 (e+f x))+4200 A \sin (e+f x)+700 A \sin (3 (e+f x))+84 A \sin (5 (e+f x))+35 A \cos (6 (e+f x))-525 B \sin (e+f x)+35 B \sin (3 (e+f x))+63 B \sin (5 (e+f x))+15 B \sin (7 (e+f x))-35 B \cos (6 (e+f x)))}{6720 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\frac{a^3 (7 A-B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 (7 A-B) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{105 f}-\frac{a (7 A-B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{42 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}",1,"-1/6720*(c^3*(-1 + Sin[e + f*x])^3*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]]*(525*(A - B)*Cos[2*(e + f*x)] + 210*(A - B)*Cos[4*(e + f*x)] + 35*A*Cos[6*(e + f*x)] - 35*B*Cos[6*(e + f*x)] + 4200*A*Sin[e + f*x] - 525*B*Sin[e + f*x] + 700*A*Sin[3*(e + f*x)] + 35*B*Sin[3*(e + f*x)] + 84*A*Sin[5*(e + f*x)] + 63*B*Sin[5*(e + f*x)] + 15*B*Sin[7*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
150,1,113,180,0.798781,"\int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2),x]","\frac{a^2 c^2 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (600 A \sin (e+f x)+100 A \sin (3 (e+f x))+12 A \sin (5 (e+f x))-75 B \cos (2 (e+f x))-30 B \cos (4 (e+f x))-5 B \cos (6 (e+f x)))}{960 f}","-\frac{2 a^3 A \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 A \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{5 f}-\frac{a A \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}{6 f}",1,"(a^2*c^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-75*B*Cos[2*(e + f*x)] - 30*B*Cos[4*(e + f*x)] - 5*B*Cos[6*(e + f*x)] + 600*A*Sin[e + f*x] + 100*A*Sin[3*(e + f*x)] + 12*A*Sin[5*(e + f*x)]))/(960*f)","A",1
151,1,165,142,1.8016171,"\int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{c (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{5/2} \sqrt{c-c \sin (e+f x)} (4 (100 A+11 B) \sin (e+f x)+4 \cos (2 (e+f x)) (4 (5 A-2 B) \sin (e+f x)-15 (A+B))-3 \cos (4 (e+f x)) (5 (A+B)+4 B \sin (e+f x)))}{480 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","\frac{c^2 (5 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{30 f \sqrt{c-c \sin (e+f x)}}+\frac{c (5 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}{20 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{3/2}}{5 f}",1,"-1/480*(c*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]]*(4*(100*A + 11*B)*Sin[e + f*x] + 4*Cos[2*(e + f*x)]*(-15*(A + B) + 4*(5*A - 2*B)*Sin[e + f*x]) - 3*Cos[4*(e + f*x)]*(5*(A + B) + 4*B*Sin[e + f*x])))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
152,1,102,96,0.8324237,"\int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]],x]","\frac{a^2 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (16 (7 A+2 B) \sin (e+f x)-4 \cos (2 (e+f x)) (4 (A+2 B) \sin (e+f x)+12 A+9 B)+3 B \cos (4 (e+f x)))}{96 f}","\frac{c (A-B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sin (e+f x)}}+\frac{B c \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 a f \sqrt{c-c \sin (e+f x)}}",1,"(a^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(3*B*Cos[4*(e + f*x)] + 16*(7*A + 2*B)*Sin[e + f*x] - 4*Cos[2*(e + f*x)]*(12*A + 9*B + 4*(A + 2*B)*Sin[e + f*x])))/(96*f)","A",1
153,1,177,193,1.4983618,"\int \frac{(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{(a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left((36 A+51 B) \sin (e+f x)-3 (A+3 B) \cos (2 (e+f x))+96 A \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-B \sin (3 (e+f x))+96 B \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)}{12 f \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\frac{4 a^3 (A+B) \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 (A+B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f \sqrt{c-c \sin (e+f x)}}-\frac{a (A+B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sin (e+f x)}}",1,"-1/12*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2)*(-3*(A + 3*B)*Cos[2*(e + f*x)] + 96*A*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 96*B*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + (36*A + 51*B)*Sin[e + f*x] - B*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sqrt[c - c*Sin[e + f*x]])","A",1
154,1,231,210,1.6445384,"\int \frac{(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(3/2),x]","-\frac{a^2 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(2 (2 A+7 B) \cos (2 (e+f x))+\sin (e+f x) \left(-64 (A+2 B) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+8 A+31 B\right)+64 A \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+28 A+B \sin (3 (e+f x))+128 B \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+16 B\right)}{8 c f (\sin (e+f x)-1) \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{4 a^3 (A+2 B) \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 (A+2 B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}+\frac{a (A+2 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f \sqrt{c-c \sin (e+f x)}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{3/2}}",1,"-1/8*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(28*A + 16*B + 2*(2*A + 7*B)*Cos[2*(e + f*x)] + 64*A*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 128*B*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + (8*A + 31*B - 64*(A + 2*B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]])*Sin[e + f*x] + B*Sin[3*(e + f*x)]))/(c*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])","A",1
155,1,207,212,1.1648017,"\int \frac{(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(5/2),x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-4 (A+2 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2-2 (A+5 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+2 (A+B)-B \sin (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4\right)}{f (c-c \sin (e+f x))^{5/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\frac{a^3 (A+5 B) \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a^2 (A+5 B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{2 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{a (A+5 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{4 c f (c-c \sin (e+f x))^{3/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{4 f (c-c \sin (e+f x))^{5/2}}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2)*(2*(A + B) - 4*(A + 2*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 - 2*(A + 5*B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 - B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(5/2))","A",1
156,1,204,196,1.1905319,"\int \frac{(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(7/2),x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(3 (A+5 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4-6 (A+2 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+4 (A+B)+6 B \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)}{3 f (c-c \sin (e+f x))^{7/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","\frac{a^3 B \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{a^2 B \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f (c-c \sin (e+f x))^{3/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac{a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f (c-c \sin (e+f x))^{5/2}}",1,"((4*(A + B) - 6*(A + 2*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + 3*(A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + 6*B*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(7/2))","A",1
157,1,145,96,2.9642399,"\int \frac{(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(9/2),x]","\frac{a^2 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) ((4 A+17 B) \sin (e+f x)-3 (A-B) \cos (2 (e+f x))+5 A-3 B \sin (3 (e+f x))-5 B)}{12 c^4 f (\sin (e+f x)-1)^4 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{(A-7 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{48 c f (c-c \sin (e+f x))^{7/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{9/2}}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(5*A - 5*B - 3*(A - B)*Cos[2*(e + f*x)] + (4*A + 17*B)*Sin[e + f*x] - 3*B*Sin[3*(e + f*x)]))/(12*c^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^4*Sqrt[c - c*Sin[e + f*x]])","A",1
158,1,146,146,4.0798828,"\int \frac{(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(11/2),x]","\frac{a^2 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (-5 (8 A+13 B) \sin (e+f x)+10 (2 A+B) \cos (2 (e+f x))-36 A+15 B \sin (3 (e+f x))-6 B)}{120 c^5 f (\sin (e+f x)-1)^5 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{(A-4 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{240 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{(A-4 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{40 c f (c-c \sin (e+f x))^{9/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{10 f (c-c \sin (e+f x))^{11/2}}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(-36*A - 6*B + 10*(2*A + B)*Cos[2*(e + f*x)] - 5*(8*A + 13*B)*Sin[e + f*x] + 15*B*Sin[3*(e + f*x)]))/(120*c^5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^5*Sqrt[c - c*Sin[e + f*x]])","A",1
159,1,144,196,5.6160011,"\int \frac{(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(13/2),x]","\frac{a^2 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (6 (6 A+7 B) \sin (e+f x)-15 (A+B) \cos (2 (e+f x))+29 A-10 B \sin (3 (e+f x))+13 B)}{120 c^6 f (\sin (e+f x)-1)^6 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{960 c^3 f (c-c \sin (e+f x))^{7/2}}+\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{160 c^2 f (c-c \sin (e+f x))^{9/2}}+\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{40 c f (c-c \sin (e+f x))^{11/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(29*A + 13*B - 15*(A + B)*Cos[2*(e + f*x)] + 6*(6*A + 7*B)*Sin[e + f*x] - 10*B*Sin[3*(e + f*x)]))/(120*c^6*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^6*Sqrt[c - c*Sin[e + f*x]])","A",1
160,1,269,250,6.8496047,"\int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(9/2),x]","\frac{a^3 c^4 (\sin (e+f x)-1)^4 (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (17640 (A-B) \cos (2 (e+f x))+8820 (A-B) \cos (4 (e+f x))+176400 A \sin (e+f x)+35280 A \sin (3 (e+f x))+7056 A \sin (5 (e+f x))+720 A \sin (7 (e+f x))+2520 A \cos (6 (e+f x))+315 A \cos (8 (e+f x))-17640 B \sin (e+f x)+2016 B \sin (5 (e+f x))+900 B \sin (7 (e+f x))+140 B \sin (9 (e+f x))-2520 B \cos (6 (e+f x))-315 B \cos (8 (e+f x)))}{322560 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\frac{a^4 (9 A-B) \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{315 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 (9 A-B) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{126 f}-\frac{a^2 (9 A-B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{84 f}-\frac{a (9 A-B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{9/2}}{72 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f}",1,"(a^3*c^4*(-1 + Sin[e + f*x])^4*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(17640*(A - B)*Cos[2*(e + f*x)] + 8820*(A - B)*Cos[4*(e + f*x)] + 2520*A*Cos[6*(e + f*x)] - 2520*B*Cos[6*(e + f*x)] + 315*A*Cos[8*(e + f*x)] - 315*B*Cos[8*(e + f*x)] + 176400*A*Sin[e + f*x] - 17640*B*Sin[e + f*x] + 35280*A*Sin[3*(e + f*x)] + 7056*A*Sin[5*(e + f*x)] + 2016*B*Sin[5*(e + f*x)] + 720*A*Sin[7*(e + f*x)] + 900*B*Sin[7*(e + f*x)] + 140*B*Sin[9*(e + f*x)]))/(322560*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7)","A",1
161,1,135,226,1.583275,"\int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]","\frac{a^3 c^3 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (19600 A \sin (e+f x)+3920 A \sin (3 (e+f x))+784 A \sin (5 (e+f x))+80 A \sin (7 (e+f x))-1960 B \cos (2 (e+f x))-980 B \cos (4 (e+f x))-280 B \cos (6 (e+f x))-35 B \cos (8 (e+f x)))}{35840 f}","-\frac{2 a^4 A \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{4 a^3 A \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac{a^2 A \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{a A \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{7/2}}{8 f}",1,"(a^3*c^3*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-1960*B*Cos[2*(e + f*x)] - 980*B*Cos[4*(e + f*x)] - 280*B*Cos[6*(e + f*x)] - 35*B*Cos[8*(e + f*x)] + 19600*A*Sin[e + f*x] + 3920*A*Sin[3*(e + f*x)] + 784*A*Sin[5*(e + f*x)] + 80*A*Sin[7*(e + f*x)]))/(35840*f)","A",1
162,1,232,192,2.0732215,"\int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2),x]","\frac{a^3 c^2 (\sin (e+f x)-1)^2 (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (-525 (A+B) \cos (2 (e+f x))-210 (A+B) \cos (4 (e+f x))+4200 A \sin (e+f x)+700 A \sin (3 (e+f x))+84 A \sin (5 (e+f x))-35 A \cos (6 (e+f x))+525 B \sin (e+f x)-35 B \sin (3 (e+f x))-63 B \sin (5 (e+f x))-15 B \sin (7 (e+f x))-35 B \cos (6 (e+f x)))}{6720 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\frac{c^3 (7 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{2 c^2 (7 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{105 f}+\frac{c (7 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{42 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{5/2}}{7 f}",1,"(a^3*c^2*(-1 + Sin[e + f*x])^2*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-525*(A + B)*Cos[2*(e + f*x)] - 210*(A + B)*Cos[4*(e + f*x)] - 35*A*Cos[6*(e + f*x)] - 35*B*Cos[6*(e + f*x)] + 4200*A*Sin[e + f*x] + 525*B*Sin[e + f*x] + 700*A*Sin[3*(e + f*x)] - 35*B*Sin[3*(e + f*x)] + 84*A*Sin[5*(e + f*x)] - 63*B*Sin[5*(e + f*x)] - 15*B*Sin[7*(e + f*x)]))/(6720*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7)","A",1
163,1,212,142,1.8516913,"\int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{a^3 c (\sin (e+f x)-1) (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (-15 (16 A+11 B) \cos (2 (e+f x))-30 (2 A+B) \cos (4 (e+f x))+840 A \sin (e+f x)+60 A \sin (3 (e+f x))-12 A \sin (5 (e+f x))+240 B \sin (e+f x)-40 B \sin (3 (e+f x))-24 B \sin (5 (e+f x))+5 B \cos (6 (e+f x)))}{960 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\frac{c^2 (3 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{30 f \sqrt{c-c \sin (e+f x)}}+\frac{c (3 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{15 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{6 f}",1,"-1/960*(a^3*c*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-15*(16*A + 11*B)*Cos[2*(e + f*x)] - 30*(2*A + B)*Cos[4*(e + f*x)] + 5*B*Cos[6*(e + f*x)] + 840*A*Sin[e + f*x] + 240*B*Sin[e + f*x] + 60*A*Sin[3*(e + f*x)] - 40*B*Sin[3*(e + f*x)] - 12*A*Sin[5*(e + f*x)] - 24*B*Sin[5*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7)","A",1
164,1,121,96,0.9653195,"\int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]],x]","\frac{a^3 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (4 (60 A+23 B) \sin (e+f x)+\cos (4 (e+f x)) (5 A+4 B \sin (e+f x)+15 B)-4 \cos (2 (e+f x)) (4 (5 A+6 B) \sin (e+f x)+5 (7 A+5 B)))}{160 f}","\frac{c (A-B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt{c-c \sin (e+f x)}}+\frac{B c \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{5 a f \sqrt{c-c \sin (e+f x)}}",1,"(a^3*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(4*(60*A + 23*B)*Sin[e + f*x] + Cos[4*(e + f*x)]*(5*A + 15*B + 4*B*Sin[e + f*x]) - 4*Cos[2*(e + f*x)]*(5*(7*A + 5*B) + 4*(5*A + 6*B)*Sin[e + f*x])))/(160*f)","A",1
165,1,183,239,2.8361539,"\int \frac{(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{a^3 (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(24 (29 A+36 B) \sin (e+f x)-8 (A+4 B) \sin (3 (e+f x))-12 (8 A+15 B) \cos (2 (e+f x))+1536 (A+B) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+3 B \cos (4 (e+f x))\right)}{96 f \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\frac{8 a^4 (A+B) \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 a^3 (A+B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f \sqrt{c-c \sin (e+f x)}}-\frac{a^2 (A+B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f \sqrt{c-c \sin (e+f x)}}-\frac{a (A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sin (e+f x)}}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt{c-c \sin (e+f x)}}",1,"-1/96*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*(-12*(8*A + 15*B)*Cos[2*(e + f*x)] + 3*B*Cos[4*(e + f*x)] + 1536*(A + B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 24*(29*A + 36*B)*Sin[e + f*x] - 8*(A + 4*B)*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*Sqrt[c - c*Sin[e + f*x]])","A",1
166,1,292,271,3.4958981,"\int \frac{(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(3/2),x]","\frac{a^3 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-2 (27 A+59 B) \cos (2 (e+f x))-117 A \sin (e+f x)-3 A \sin (3 (e+f x))-576 A \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+576 A \sin (e+f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-132 A-279 B \sin (e+f x)-13 B \sin (3 (e+f x))+B \cos (4 (e+f x))-960 B \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+960 B \sin (e+f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-45 B\right)}{24 c f (\sin (e+f x)-1) \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{4 a^4 (3 A+5 B) \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^3 (3 A+5 B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}+\frac{a^2 (3 A+5 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f \sqrt{c-c \sin (e+f x)}}+\frac{a (3 A+5 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 c f \sqrt{c-c \sin (e+f x)}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{3/2}}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(-132*A - 45*B - 2*(27*A + 59*B)*Cos[2*(e + f*x)] + B*Cos[4*(e + f*x)] - 576*A*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 960*B*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 117*A*Sin[e + f*x] - 279*B*Sin[e + f*x] + 576*A*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] + 960*B*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] - 3*A*Sin[3*(e + f*x)] - 13*B*Sin[3*(e + f*x)]))/(24*c*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])","A",1
167,1,251,263,2.5730426,"\int \frac{(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(5/2),x]","\frac{(a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-4 (A+6 B) \sin (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4-16 (3 A+5 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2-48 (A+3 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+16 (A+B)+B \cos (2 (e+f x)) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4\right)}{4 f (c-c \sin (e+f x))^{5/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\frac{6 a^4 (A+3 B) \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{3 a^3 (A+3 B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{3 a^2 (A+3 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{4 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{a (A+3 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 c f (c-c \sin (e+f x))^{3/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f (c-c \sin (e+f x))^{5/2}}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(7/2)*(16*(A + B) - 16*(3*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + B*Cos[2*(e + f*x)]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 - 48*(A + 3*B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 - 4*(A + 6*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[e + f*x]))/(4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(5/2))","A",1
168,1,244,264,2.9004296,"\int \frac{(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(7/2),x]","\frac{(a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(18 (A+3 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4-6 (3 A+5 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+6 (A+7 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+8 (A+B)+3 B \sin (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6\right)}{3 f (c-c \sin (e+f x))^{7/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\frac{a^4 (A+7 B) \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{a^3 (A+7 B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{2 c^3 f \sqrt{c-c \sin (e+f x)}}+\frac{a^2 (A+7 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac{a (A+7 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(7/2)*(8*(A + B) - 6*(3*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + 18*(A + 3*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + 6*(A + 7*B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6 + 3*B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6*Sin[e + f*x]))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(7/2))","A",1
169,1,238,247,2.6849657,"\int \frac{(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(9/2),x]","\frac{(a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-3 (A+7 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6+9 (A+3 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4-4 (3 A+5 B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+6 (A+B)-6 B \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^8 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)}{3 f (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\frac{a^4 B \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a^3 B \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 B \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}-\frac{a B \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c f (c-c \sin (e+f x))^{7/2}}",1,"((6*(A + B) - 4*(3*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + 9*(A + 3*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 - 3*(A + 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6 - 6*B*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(9/2))","A",1
170,1,434,96,6.9449903,"\int \frac{(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(11/2),x]","\frac{(-A-7 B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}{2 f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{2 (A+3 B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}{f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{(-3 A-5 B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{8 (A+B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{5 f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{B (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}{f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\frac{(A-9 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{80 c f (c-c \sin (e+f x))^{9/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}",1,"(8*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(7/2))/(5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2)) + ((-3*A - 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2)) + (2*(A + 3*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2)) + ((-A - 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2)) + (B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2))","B",1
171,1,442,146,6.9887686,"\int \frac{(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(13/2),x]","\frac{(-A-7 B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}{3 f (c-c \sin (e+f x))^{13/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{3 (A+3 B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}{2 f (c-c \sin (e+f x))^{13/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{4 (3 A+5 B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{5 f (c-c \sin (e+f x))^{13/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{4 (A+B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{3 f (c-c \sin (e+f x))^{13/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{B (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}{2 f (c-c \sin (e+f x))^{13/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\frac{(A-5 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{480 c^2 f (c-c \sin (e+f x))^{9/2}}+\frac{(A-5 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{60 c f (c-c \sin (e+f x))^{11/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}",1,"(4*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(13/2)) - (4*(3*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(13/2)) + (3*(A + 3*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2))/(2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(13/2)) + ((-A - 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(13/2)) + (B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(13/2))","B",1
172,1,442,202,7.1055034,"\int \frac{(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(15/2),x]","\frac{(-A-7 B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}{4 f (c-c \sin (e+f x))^{15/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{6 (A+3 B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}{5 f (c-c \sin (e+f x))^{15/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{2 (3 A+5 B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{3 f (c-c \sin (e+f x))^{15/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{8 (A+B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{7 f (c-c \sin (e+f x))^{15/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{B (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}{3 f (c-c \sin (e+f x))^{15/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\frac{(3 A-11 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6720 c^3 f (c-c \sin (e+f x))^{9/2}}+\frac{(3 A-11 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{840 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac{(3 A-11 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{168 c f (c-c \sin (e+f x))^{13/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}",1,"(8*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(7/2))/(7*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) - (2*(3*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) + (6*(A + 3*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2))/(5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) + ((-A - 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) + (B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2))","B",1
173,1,436,246,7.1427327,"\int \frac{(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(17/2),x]","\frac{(-A-7 B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}{5 f (c-c \sin (e+f x))^{17/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{(A+3 B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}{f (c-c \sin (e+f x))^{17/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{4 (3 A+5 B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{7 f (c-c \sin (e+f x))^{17/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{(A+B) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f (c-c \sin (e+f x))^{17/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{B (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}{4 f (c-c \sin (e+f x))^{17/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8960 c^4 f (c-c \sin (e+f x))^{9/2}}+\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{1120 c^3 f (c-c \sin (e+f x))^{11/2}}+\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{224 c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}",1,"((A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) - (4*(3*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(7*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) + ((A + 3*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) + ((-A - 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) + (B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2))","A",1
174,1,185,197,1.3056307,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{c^2 (\sin (e+f x)-1)^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left((36 A-51 B) \sin (e+f x)+3 (A-3 B) \cos (2 (e+f x))-96 A \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+B \sin (3 (e+f x))+96 B \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{12 f \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}","\frac{4 c^3 (A-B) \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c^2 (A-B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}+\frac{c (A-B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}-\frac{B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a \sin (e+f x)+a}}",1,"-1/12*(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]]*(3*(A - 3*B)*Cos[2*(e + f*x)] - 96*A*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 96*B*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (36*A - 51*B)*Sin[e + f*x] + B*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*Sqrt[a*(1 + Sin[e + f*x])])","A",1
175,1,146,146,0.6644181,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2))/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{c (\sin (e+f x)-1) \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(B \cos (2 (e+f x))-4 \left((A-2 B) \sin (e+f x)+4 (B-A) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}{4 f \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}","\frac{2 c^2 (A-B) \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{c (A-B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}-\frac{B \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}",1,"-1/4*(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]]*(B*Cos[2*(e + f*x)] - 4*(4*(-A + B)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (A - 2*B)*Sin[e + f*x])))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])])","A",1
176,1,119,96,1.1302966,"\int \frac{(A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[((A + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/Sqrt[a + a*Sin[e + f*x]],x]","\frac{\sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(B \sin (e+f x)+(A-B) \left(2 \log \left(e^{i (e+f x)}+i\right)-i f x\right)\right)}{f \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{c (A-B) \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{B \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((A - B)*((-I)*f*x + 2*Log[I + E^(I*(e + f*x))]) + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])])","C",1
177,1,97,113,0.3374462,"\int \frac{A+B \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(A + B*Sin[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]),x]","-\frac{\cos (e+f x) \left((A+B) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+(B-A) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{f \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}","\frac{(A-B) \cos (e+f x) \log (\sin (e+f x)+1)}{2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{(A+B) \cos (e+f x) \log (1-\sin (e+f x))}{2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"-((Cos[e + f*x]*((A + B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + (-A + B)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]))/(f*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]))","A",1
178,1,191,103,0.5430641,"\int \frac{A+B \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left((B-A) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+(A-B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+A+B\right)}{2 f \sqrt{a (\sin (e+f x)+1)} (c-c \sin (e+f x))^{3/2}}","\frac{(A+B) \cos (e+f x)}{2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{(A-B) \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"((A + B + (-A + B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + (A - B)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(2*f*Sqrt[a*(1 + Sin[e + f*x])]*(c - c*Sin[e + f*x])^(3/2))","A",1
179,1,222,153,0.6306083,"\int \frac{A+B \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left((A-B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2+(B-A) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+(A-B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+A+B\right)}{4 f \sqrt{a (\sin (e+f x)+1)} (c-c \sin (e+f x))^{5/2}}","\frac{(A-B) \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{4 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{(A-B) \cos (e+f x)}{4 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{(A+B) \cos (e+f x)}{4 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}",1,"((A + B + (A - B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + (-A + B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + (A - B)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(4*f*Sqrt[a*(1 + Sin[e + f*x])]*(c - c*Sin[e + f*x])^(5/2))","A",1
180,1,271,271,3.4750547,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{c^3 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(2 (27 A-59 B) \cos (2 (e+f x))-117 A \sin (e+f x)-3 A \sin (3 (e+f x))+576 A \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+576 A \sin (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+132 A+279 B \sin (e+f x)+13 B \sin (3 (e+f x))+B \cos (4 (e+f x))-960 B \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-960 B \sin (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-45 B\right)}{24 f (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{4 c^4 (3 A-5 B) \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 c^3 (3 A-5 B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}-\frac{c^2 (3 A-5 B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f \sqrt{a \sin (e+f x)+a}}-\frac{c (3 A-5 B) \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{6 a f \sqrt{a \sin (e+f x)+a}}-\frac{(A-B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{2 f (a \sin (e+f x)+a)^{3/2}}",1,"-1/24*(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(132*A - 45*B + 2*(27*A - 59*B)*Cos[2*(e + f*x)] + B*Cos[4*(e + f*x)] + 576*A*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 960*B*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 117*A*Sin[e + f*x] + 279*B*Sin[e + f*x] + 576*A*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] - 960*B*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] - 3*A*Sin[3*(e + f*x)] + 13*B*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2))","A",1
181,1,212,210,1.6040073,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{c^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(2 (2 A-7 B) \cos (2 (e+f x))+\sin (e+f x) \left(64 (A-2 B) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-8 A+31 B\right)+64 A \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+28 A+B \sin (3 (e+f x))-128 B \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-16 B\right)}{8 f (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{4 c^3 (A-2 B) \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 c^2 (A-2 B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}-\frac{c (A-2 B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f \sqrt{a \sin (e+f x)+a}}-\frac{(A-B) \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{2 f (a \sin (e+f x)+a)^{3/2}}",1,"-1/8*(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(28*A - 16*B + 2*(2*A - 7*B)*Cos[2*(e + f*x)] + 64*A*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 128*B*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (-8*A + 31*B + 64*(A - 2*B)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[e + f*x] + B*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2))","A",1
182,1,190,159,0.8637547,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{c \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(2 \sin (e+f x) \left(2 (A-3 B) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+B\right)+4 A \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+4 A-B \cos (2 (e+f x))-12 B \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-3 B\right)}{2 f (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{c^2 (A-3 B) \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{c (A-3 B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{2 a f \sqrt{a \sin (e+f x)+a}}-\frac{(A-B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f (a \sin (e+f x)+a)^{3/2}}",1,"-1/2*(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(4*A - 3*B - B*Cos[2*(e + f*x)] + 4*A*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 12*B*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 2*(B + 2*(A - 3*B)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2))","A",1
183,1,143,100,1.175561,"\int \frac{(A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(-A+2 B \log \left(e^{i (e+f x)}+i\right)+B \left(2 \log \left(e^{i (e+f x)}+i\right)-i f x\right) \sin (e+f x)-i B f x+B\right)}{f (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{B c \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{c (A-B) \cos (e+f x)}{f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(-A + B - I*B*f*x + 2*B*Log[I + E^(I*(e + f*x))] + B*((-I)*f*x + 2*Log[I + E^(I*(e + f*x))])*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
184,1,186,103,0.5624181,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(-\left((A+B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+(A+B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-A+B\right)}{2 f (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)}}","\frac{(A+B) \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{(A-B) \cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-A + B - (A + B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (A + B)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2))/(2*f*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
185,1,178,150,0.6712277,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)),x]","-\frac{\cos (e+f x) \left(2 A \sin (e+f x)-A \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+A \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+A \cos (2 (e+f x)) \left(\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+2 B\right)}{4 c f (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)}}","-\frac{(A-B) \cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac{A \cos (e+f x)}{2 a f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{A \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 a c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"-1/4*(Cos[e + f*x]*(2*B - A*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + A*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + A*Cos[2*(e + f*x)]*(-Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 2*A*Sin[e + f*x]))/(c*f*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
186,1,306,217,0.947333,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left((B-A) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4+(A+B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+(B-3 A) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+(3 A-B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+2 A \cos ^2(e+f x)\right)}{8 f (a (\sin (e+f x)+1))^{3/2} (c-c \sin (e+f x))^{5/2}}","\frac{(3 A-B) \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{8 a c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{(3 A-B) \cos (e+f x)}{8 a c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{(3 A-B) \cos (e+f x)}{8 a f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{(A-B) \cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*A*Cos[e + f*x]^2 + (-A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + (A + B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (-3*A + B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (3*A - B)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2))/(8*f*(a*(1 + Sin[e + f*x]))^(3/2)*(c - c*Sin[e + f*x])^(5/2))","A",1
187,1,573,323,7.0117444,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{(28 A-97 B) \sin (e+f x) (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}{4 f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}-\frac{(A-7 B) \cos (2 (e+f x)) (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}{4 f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}+\frac{16 (2 A-3 B) (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}{f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}-\frac{8 (A-B) (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}+\frac{16 (3 A-7 B) (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}-\frac{B \sin (3 (e+f x)) (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}{12 f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}","\frac{8 c^5 (3 A-7 B) \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{4 c^4 (3 A-7 B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{c^3 (3 A-7 B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{c^2 (3 A-7 B) \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{c (3 A-7 B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f (a \sin (e+f x)+a)^{3/2}}-\frac{(A-B) \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"(-8*(A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^(9/2))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2)) + (16*(2*A - 3*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(9/2))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2)) - ((A - 7*B)*Cos[2*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(9/2))/(4*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2)) + (16*(3*A - 7*B)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(9/2))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2)) - ((28*A - 97*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sin[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(4*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2)) - (B*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(9/2)*Sin[3*(e + f*x)])/(12*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2))","A",1
188,1,243,263,2.5397464,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(5/2),x]","\frac{(c-c \sin (e+f x))^{7/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(-4 (A-6 B) \sin (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4+16 (3 A-5 B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+48 (A-3 B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-16 (A-B)+B \cos (2 (e+f x)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4\right)}{4 f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}","\frac{6 c^4 (A-3 B) \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{3 c^3 (A-3 B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{3 c^2 (A-3 B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{4 a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{c (A-3 B) \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{2 a f (a \sin (e+f x)+a)^{3/2}}-\frac{(A-B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^(7/2)*(-16*(A - B) + 16*(3*A - 5*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + B*Cos[2*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + 48*(A - 3*B)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 4*(A - 6*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*Sin[e + f*x]))/(4*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(5/2))","A",1
189,1,199,211,1.1547947,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^(5/2),x]","\frac{(c-c \sin (e+f x))^{5/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(4 (A-2 B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+2 (A-5 B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-2 A+B \sin (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4+2 B\right)}{f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}","\frac{c^3 (A-5 B) \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{c^2 (A-5 B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{2 a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{c (A-5 B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{4 a f (a \sin (e+f x)+a)^{3/2}}-\frac{(A-B) \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^(5/2)*(-2*A + 2*B + 4*(A - 2*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 2*(A - 5*B)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + B*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(5/2))","A",1
190,1,179,149,0.9792886,"\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^(5/2),x]","\frac{c \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin (e+f x) \left(A-4 B \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-3 B\right)+B \cos (2 (e+f x)) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-B \left(3 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+2\right)\right)}{f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{B c^2 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{(A-B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}-\frac{B c \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f (a \sin (e+f x)+a)^{3/2}}",1,"(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(B*Cos[2*(e + f*x)]*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - B*(2 + 3*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + (A - 3*B - 4*B*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))","A",1
191,1,99,94,0.5088441,"\int \frac{(A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{\sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (A+2 B \sin (e+f x)+B)}{2 a^3 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\frac{c (A-B) \cos (e+f x)}{2 f (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}-\frac{B c \cos (e+f x)}{a f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}",1,"-1/2*(Sqrt[a*(1 + Sin[e + f*x])]*(A + B + 2*B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
192,1,214,151,0.6733288,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(-(A+B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-\left((A+B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+(A+B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-A+B\right)}{4 f (a (\sin (e+f x)+1))^{5/2} \sqrt{c-c \sin (e+f x)}}","\frac{(A+B) \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{4 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{(A+B) \cos (e+f x)}{4 a f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{(A-B) \cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-A + B - (A + B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - (A + B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + (A + B)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4))/(4*f*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
193,1,305,208,0.9613818,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left((A+B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4+(B-A) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2-\left((3 A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+(3 A+B) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-2 A \cos ^2(e+f x)\right)}{8 f (a (\sin (e+f x)+1))^{5/2} (c-c \sin (e+f x))^{3/2}}","\frac{(3 A+B) \cos (e+f x)}{8 a^2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{(3 A+B) \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{8 a^2 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{(3 A+B) \cos (e+f x)}{8 a f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}-\frac{(A-B) \cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{3/2}}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-2*A*Cos[e + f*x]^2 + (-A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + (A + B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - (3*A + B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + (3*A + B)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4))/(8*f*(a*(1 + Sin[e + f*x]))^(5/2)*(c - c*Sin[e + f*x])^(3/2))","A",1
194,1,246,245,0.9365238,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\sec ^3(e+f x) \left(22 A \sin (e+f x)+6 A \sin (3 (e+f x))-9 A \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-12 A \cos (2 (e+f x)) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-3 A \cos (4 (e+f x)) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+9 A \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+16 B\right)}{64 a^2 c^2 f \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}","\frac{3 A \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{8 a^2 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{3 A \cos (e+f x)}{8 a^2 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{3 A \cos (e+f x)}{8 a^2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{(A-B) \cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}-\frac{A \cos (e+f x)}{2 a f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}",1,"(Sec[e + f*x]^3*(16*B - 9*A*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 12*A*Cos[2*(e + f*x)]*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 3*A*Cos[4*(e + f*x)]*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 9*A*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 22*A*Sin[e + f*x] + 6*A*Sin[3*(e + f*x)]))/(64*a^2*c^2*f*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])","A",1
195,1,2903,174,14.2378541,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^n \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^n,x]","\text{Result too large to show}","\frac{c 2^{n+\frac{1}{2}} (A (m+n+1)+B (m-n)) \cos (e+f x) (1-\sin (e+f x))^{\frac{1}{2}-n} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2} (2 m+1),\frac{1}{2} (1-2 n);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1) (m+n+1)}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n}{f (m+n+1)}",1,"(4*(8*B*AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - (A + B)*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 8*B*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n*(A*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*n) + B*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*n)*Sin[e + f*x])*Tan[(-e + Pi/2 - f*x)/4])/(f*(1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)*((-4*m*(8*B*AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - (A + B)*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 8*B*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(2*n)*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(-1 - 2*m))/(1 + 2*n) - ((8*B*AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - (A + B)*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 8*B*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(2*n))/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) - (4*n*(8*B*AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - (A + B)*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 8*B*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(-1 + 2*n)*Tan[(-e + Pi/2 - f*x)/4])/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (4*m*(8*B*AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - (A + B)*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 8*B*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(1 + 2*n)*Tan[(-e + Pi/2 - f*x)/4])/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) - (4*(m + n)*(8*B*AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - (A + B)*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 8*B*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(2*n)*Tan[(-e + Pi/2 - f*x)/4]^2)/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) - (4*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(2*n)*Tan[(-e + Pi/2 - f*x)/4]*(-((A + B)*(-((m*(1/2 + n)*AppellF1[3/2 + n, 1 - 2*m, 1 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n)) - ((1/2 + n)*(1 + 2*(m + n))*AppellF1[3/2 + n, -2*m, 2 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 + n)))) - 8*B*(-((m*(1/2 + n)*AppellF1[3/2 + n, 1 - 2*m, 3 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n)) - ((1/2 + n)*(3 + 2*(m + n))*AppellF1[3/2 + n, -2*m, 4 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 + n))) + 8*B*(-((m*(1/2 + n)*AppellF1[3/2 + n, 1 - 2*m, 2*(1 + m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n)) - ((1/2 + n)*(1 + m + n)*AppellF1[3/2 + n, -2*m, 1 + 2*(1 + m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n))))/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))))","C",0
196,-1,0,145,180.0610483,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3,x]","\text{\$Aborted}","\frac{a^4 c^3 2^{m+\frac{1}{2}} (B (3-m)-A (m+4)) \cos ^7(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m-4} \, _2F_1\left(\frac{7}{2},\frac{1}{2}-m;\frac{9}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{7 f (m+4)}-\frac{a^3 B c^3 \cos ^7(e+f x) (a \sin (e+f x)+a)^{m-3}}{f (m+4)}",1,"$Aborted","F",-1
197,-1,0,145,180.0809493,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2,x]","\text{\$Aborted}","\frac{a^3 c^2 2^{m+\frac{1}{2}} (B (2-m)-A (m+3)) \cos ^5(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left(\frac{5}{2},\frac{1}{2}-m;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{5 f (m+3)}-\frac{a^2 B c^2 \cos ^5(e+f x) (a \sin (e+f x)+a)^{m-2}}{f (m+3)}",1,"$Aborted","F",-1
198,1,462,139,4.2203358,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x]),x]","\frac{i c 4^{-m-1} e^{i f m x} \left(1+i e^{-i (e+f x)}\right)^{-2 m} \left(-(-1)^{3/4} e^{-\frac{1}{2} i (e+f x)} \left(e^{i (e+f x)}+i\right)\right)^{2 m} (\sin (e+f x)-1) \sin ^{-2 m}\left(\frac{1}{4} (2 e+2 f x+\pi )\right) (a (\sin (e+f x)+1))^m \left(\frac{2 (B-i A) e^{-i (e+f (m+1) x)} \, _2F_1\left(-m-1,-2 m;-m;-i e^{-i (e+f x)}\right)}{m+1}+\frac{2 i A e^{i (e-f (m-1) x)} \, _2F_1\left(1-m,-2 m;2-m;-i e^{-i (e+f x)}\right)}{m-1}+\frac{4 A e^{-i f m x} \, _2F_1\left(-2 m,-m;1-m;-i e^{-i (e+f x)}\right)}{m}-\frac{i B e^{-i (2 e+f (m+2) x)} \, _2F_1\left(-m-2,-2 m;-m-1;-i e^{-i (e+f x)}\right)}{m+2}+\frac{2 B e^{i (e-f (m-1) x)} \, _2F_1\left(1-m,-2 m;2-m;-i e^{-i (e+f x)}\right)}{m-1}+\frac{i B e^{2 i e-i f (m-2) x} \, _2F_1\left(2-m,-2 m;3-m;-i e^{-i (e+f x)}\right)}{m-2}\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}","\frac{a^2 c 2^{m+\frac{1}{2}} (B (1-m)-A (m+2)) \cos ^3(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left(\frac{3}{2},\frac{1}{2}-m;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f (m+2)}-\frac{a B c \cos ^3(e+f x) (a \sin (e+f x)+a)^{m-1}}{f (m+2)}",1,"(I*4^(-1 - m)*c*E^(I*f*m*x)*(-(((-1)^(3/4)*(I + E^(I*(e + f*x))))/E^((I/2)*(e + f*x))))^(2*m)*(((-I)*B*Hypergeometric2F1[-2 - m, -2*m, -1 - m, (-I)/E^(I*(e + f*x))])/(E^(I*(2*e + f*(2 + m)*x))*(2 + m)) + (2*((-I)*A + B)*Hypergeometric2F1[-1 - m, -2*m, -m, (-I)/E^(I*(e + f*x))])/(E^(I*(e + f*(1 + m)*x))*(1 + m)) + ((2*I)*A*E^(I*(e - f*(-1 + m)*x))*Hypergeometric2F1[1 - m, -2*m, 2 - m, (-I)/E^(I*(e + f*x))])/(-1 + m) + (2*B*E^(I*(e - f*(-1 + m)*x))*Hypergeometric2F1[1 - m, -2*m, 2 - m, (-I)/E^(I*(e + f*x))])/(-1 + m) + (I*B*E^((2*I)*e - I*f*(-2 + m)*x)*Hypergeometric2F1[2 - m, -2*m, 3 - m, (-I)/E^(I*(e + f*x))])/(-2 + m) + (4*A*Hypergeometric2F1[-2*m, -m, 1 - m, (-I)/E^(I*(e + f*x))])/(E^(I*f*m*x)*m))*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^m)/((1 + I/E^(I*(e + f*x)))^(2*m)*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(2*e + Pi + 2*f*x)/4]^(2*m))","C",0
199,1,275,117,1.831743,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","-\frac{\sin ^{-2 m}\left(\frac{1}{4} (2 e+2 f x+\pi )\right) (a (\sin (e+f x)+1))^m \left(\frac{2 \sqrt{2} A \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right) \cos ^{2 m+1}\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \, _2F_1\left(\frac{1}{2},m+\frac{1}{2};m+\frac{3}{2};\sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)\right)}{(2 m+1) \sqrt{1-\sin (e+f x)}}+\frac{\sqrt[4]{-1} B 2^{-2 m-1} e^{-\frac{3}{2} i (e+f x)} \left(-(-1)^{3/4} e^{-\frac{1}{2} i (e+f x)} \left(e^{i (e+f x)}+i\right)\right)^{2 m+1} \left((m-1) e^{2 i (e+f x)} \, _2F_1\left(1,m;-m;-i e^{-i (e+f x)}\right)-(m+1) \, _2F_1\left(1,m+2;2-m;-i e^{-i (e+f x)}\right)\right)}{m^2-1}\right)}{f}","-\frac{2^{m+\frac{1}{2}} (A m+A+B m) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f (m+1)}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1)}",1,"-(((a*(1 + Sin[e + f*x]))^m*(((-1)^(1/4)*2^(-1 - 2*m)*B*(-(((-1)^(3/4)*(I + E^(I*(e + f*x))))/E^((I/2)*(e + f*x))))^(1 + 2*m)*(E^((2*I)*(e + f*x))*(-1 + m)*Hypergeometric2F1[1, m, -m, (-I)/E^(I*(e + f*x))] - (1 + m)*Hypergeometric2F1[1, 2 + m, 2 - m, (-I)/E^(I*(e + f*x))]))/(E^(((3*I)/2)*(e + f*x))*(-1 + m^2)) + (2*Sqrt[2]*A*Cos[(2*e - Pi + 2*f*x)/4]^(1 + 2*m)*Hypergeometric2F1[1/2, 1/2 + m, 3/2 + m, Sin[(2*e + Pi + 2*f*x)/4]^2]*Sin[(2*e - Pi + 2*f*x)/4])/((1 + 2*m)*Sqrt[1 - Sin[e + f*x]])))/(f*Sin[(2*e + Pi + 2*f*x)/4]^(2*m)))","C",0
200,1,7409,123,25.5681845,"\int \frac{(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx","Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x]),x]","\text{Result too large to show}","\frac{2^{m+\frac{1}{2}} (A m+B m+B) \sec (e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m \, _2F_1\left(-\frac{1}{2},\frac{1}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{c f m}-\frac{B \sec (e+f x) (a \sin (e+f x)+a)^{m+1}}{a c f m}",1,"Result too large to show","C",0
201,1,9240,148,6.934441,"\int \frac{(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx","Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^2,x]","\text{Result too large to show}","\frac{B \sec ^3(e+f x) (a \sin (e+f x)+a)^{m+2}}{a^2 c^2 f (1-m)}+\frac{2^{m+\frac{1}{2}} (A (1-m)-B (m+2)) \sec ^3(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(-\frac{3}{2},\frac{1}{2}-m;-\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 a c^2 f (1-m)}",1,"Result too large to show","C",0
202,1,12580,148,7.178262,"\int \frac{(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx","Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^3,x]","\text{Result too large to show}","\frac{B \sec ^5(e+f x) (a \sin (e+f x)+a)^{m+3}}{a^3 c^3 f (2-m)}+\frac{2^{m+\frac{1}{2}} (A (2-m)-B (m+3)) \sec ^5(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left(-\frac{5}{2},\frac{1}{2}-m;-\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{5 a^2 c^3 f (2-m)}",1,"Result too large to show","C",0
203,1,200,118,4.8698382,"\int \frac{(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin[e + f*x]],x]","\frac{2^{-2 m-\frac{3}{2}} \sin \left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (a (\sin (e+f x)+1))^m \left(2^{2 m+1} (A+B) \, _2F_1\left(1,2 m+1;2 (m+1);\sin \left(\frac{1}{4} (2 e+2 f x+\pi )\right)\right)+(A+B) \sec ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)^{2 m+1} \, _2F_1\left(2 m+1,2 m+1;2 (m+1);\frac{1}{2} \left(1-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)\right)-B 2^{2 m+3}\right)}{(2 f m+f) \sqrt{c-c \sin (e+f x)}}","\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left(1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1) \sqrt{c-c \sin (e+f x)}}-\frac{2 B \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt{c-c \sin (e+f x)}}",1,"(2^(-3/2 - 2*m)*(-(2^(3 + 2*m)*B) + 2^(1 + 2*m)*(A + B)*Hypergeometric2F1[1, 1 + 2*m, 2*(1 + m), Sin[(2*e + Pi + 2*f*x)/4]] + (A + B)*Hypergeometric2F1[1 + 2*m, 1 + 2*m, 2*(1 + m), (1 - Tan[(2*e - Pi + 2*f*x)/8]^2)/2]*(Sec[(2*e - Pi + 2*f*x)/8]^2)^(1 + 2*m))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^m*Sin[(2*e + Pi + 2*f*x)/4])/((f + 2*f*m)*Sqrt[c - c*Sin[e + f*x]])","A",1
204,1,200,118,2.3831001,"\int \frac{(A+B \sin (e+f x)) (c+c \sin (e+f x))^m}{\sqrt{a-a \sin (e+f x)}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + c*Sin[e + f*x])^m)/Sqrt[a - a*Sin[e + f*x]],x]","\frac{2^{-2 m-\frac{3}{2}} \sin \left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (c (\sin (e+f x)+1))^m \left(2^{2 m+1} (A+B) \, _2F_1\left(1,2 m+1;2 (m+1);\sin \left(\frac{1}{4} (2 e+2 f x+\pi )\right)\right)+(A+B) \sec ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)^{2 m+1} \, _2F_1\left(2 m+1,2 m+1;2 (m+1);\frac{1}{2} \left(1-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)\right)-B 2^{2 m+3}\right)}{(2 f m+f) \sqrt{a-a \sin (e+f x)}}","\frac{(A+B) \cos (e+f x) (c \sin (e+f x)+c)^m \, _2F_1\left(1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1) \sqrt{a-a \sin (e+f x)}}-\frac{2 B \cos (e+f x) (c \sin (e+f x)+c)^m}{f (2 m+1) \sqrt{a-a \sin (e+f x)}}",1,"(2^(-3/2 - 2*m)*(-(2^(3 + 2*m)*B) + 2^(1 + 2*m)*(A + B)*Hypergeometric2F1[1, 1 + 2*m, 2*(1 + m), Sin[(2*e + Pi + 2*f*x)/4]] + (A + B)*Hypergeometric2F1[1 + 2*m, 1 + 2*m, 2*(1 + m), (1 - Tan[(2*e - Pi + 2*f*x)/8]^2)/2]*(Sec[(2*e - Pi + 2*f*x)/8]^2)^(1 + 2*m))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c*(1 + Sin[e + f*x]))^m*Sin[(2*e + Pi + 2*f*x)/4])/((f + 2*f*m)*Sqrt[a - a*Sin[e + f*x]])","A",1
205,1,667,275,6.8194994,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2),x]","\frac{(c-c \sin (e+f x))^{5/2} (a (\sin (e+f x)+1))^m \left(\frac{\left(24 A m^2+184 A m+350 A-12 B m^2-104 B m-385 B\right) \left(\left(\frac{1}{8}-\frac{i}{8}\right) \cos \left(\frac{3}{2} (e+f x)\right)-\left(\frac{1}{8}+\frac{i}{8}\right) \sin \left(\frac{3}{2} (e+f x)\right)\right)}{(2 m+3) (2 m+5) (2 m+7)}+\frac{\left(24 A m^2+184 A m+350 A-12 B m^2-104 B m-385 B\right) \left(\left(\frac{1}{8}+\frac{i}{8}\right) \cos \left(\frac{3}{2} (e+f x)\right)-\left(\frac{1}{8}-\frac{i}{8}\right) \sin \left(\frac{3}{2} (e+f x)\right)\right)}{(2 m+3) (2 m+5) (2 m+7)}+\frac{\left(32 A m^3+304 A m^2+1272 A m+2100 A-8 B m^3-68 B m^2-110 B m-1575 B\right) \left(\left(\frac{1}{8}-\frac{i}{8}\right) \sin \left(\frac{1}{2} (e+f x)\right)+\left(\frac{1}{8}+\frac{i}{8}\right) \cos \left(\frac{1}{2} (e+f x)\right)\right)}{(2 m+1) (2 m+3) (2 m+5) (2 m+7)}+\frac{\left(32 A m^3+304 A m^2+1272 A m+2100 A-8 B m^3-68 B m^2-110 B m-1575 B\right) \left(\left(\frac{1}{8}+\frac{i}{8}\right) \sin \left(\frac{1}{2} (e+f x)\right)+\left(\frac{1}{8}-\frac{i}{8}\right) \cos \left(\frac{1}{2} (e+f x)\right)\right)}{(2 m+1) (2 m+3) (2 m+5) (2 m+7)}+\frac{(4 A m+14 A-6 B m-35 B) \left(\left(-\frac{1}{8}+\frac{i}{8}\right) \cos \left(\frac{5}{2} (e+f x)\right)-\left(\frac{1}{8}+\frac{i}{8}\right) \sin \left(\frac{5}{2} (e+f x)\right)\right)}{(2 m+5) (2 m+7)}+\frac{(4 A m+14 A-6 B m-35 B) \left(\left(-\frac{1}{8}-\frac{i}{8}\right) \cos \left(\frac{5}{2} (e+f x)\right)-\left(\frac{1}{8}-\frac{i}{8}\right) \sin \left(\frac{5}{2} (e+f x)\right)\right)}{(2 m+5) (2 m+7)}+\frac{\left(\frac{1}{8}-\frac{i}{8}\right) B \cos \left(\frac{7}{2} (e+f x)\right)-\left(\frac{1}{8}+\frac{i}{8}\right) B \sin \left(\frac{7}{2} (e+f x)\right)}{2 m+7}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) B \cos \left(\frac{7}{2} (e+f x)\right)-\left(\frac{1}{8}-\frac{i}{8}\right) B \sin \left(\frac{7}{2} (e+f x)\right)}{2 m+7}\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\frac{64 c^3 (B (5-2 m)-A (2 m+7)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}-\frac{16 c^2 (B (5-2 m)-A (2 m+7)) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) \left(4 m^2+16 m+15\right)}-\frac{2 c (B (5-2 m)-A (2 m+7)) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7)}-\frac{2 B \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}",1,"((a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^(5/2)*(((2100*A - 1575*B + 1272*A*m - 110*B*m + 304*A*m^2 - 68*B*m^2 + 32*A*m^3 - 8*B*m^3)*((1/8 + I/8)*Cos[(e + f*x)/2] + (1/8 - I/8)*Sin[(e + f*x)/2]))/((1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)) + ((2100*A - 1575*B + 1272*A*m - 110*B*m + 304*A*m^2 - 68*B*m^2 + 32*A*m^3 - 8*B*m^3)*((1/8 - I/8)*Cos[(e + f*x)/2] + (1/8 + I/8)*Sin[(e + f*x)/2]))/((1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)) + ((350*A - 385*B + 184*A*m - 104*B*m + 24*A*m^2 - 12*B*m^2)*((1/8 - I/8)*Cos[(3*(e + f*x))/2] - (1/8 + I/8)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)) + ((350*A - 385*B + 184*A*m - 104*B*m + 24*A*m^2 - 12*B*m^2)*((1/8 + I/8)*Cos[(3*(e + f*x))/2] - (1/8 - I/8)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)) + ((14*A - 35*B + 4*A*m - 6*B*m)*((-1/8 + I/8)*Cos[(5*(e + f*x))/2] - (1/8 + I/8)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)) + ((14*A - 35*B + 4*A*m - 6*B*m)*((-1/8 - I/8)*Cos[(5*(e + f*x))/2] - (1/8 - I/8)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)) + ((1/8 - I/8)*B*Cos[(7*(e + f*x))/2] - (1/8 + I/8)*B*Sin[(7*(e + f*x))/2])/(7 + 2*m) + ((1/8 + I/8)*B*Cos[(7*(e + f*x))/2] - (1/8 - I/8)*B*Sin[(7*(e + f*x))/2])/(7 + 2*m)))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5)","C",1
206,1,174,166,1.7655398,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2),x]","\frac{c \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (a (\sin (e+f x)+1))^m \left(-2 (2 m+1) (2 A m+5 A-2 B m-9 B) \sin (e+f x)+8 A m^2+40 A m+50 A+B \left(4 m^2+8 m+3\right) \cos (2 (e+f x))-4 B m^2-16 B m-39 B\right)}{f (2 m+1) (2 m+3) (2 m+5) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{2 B c^2 \cos (e+f x) (a \sin (e+f x)+a)^{m+2}}{a^2 f (2 m+5) \sqrt{c-c \sin (e+f x)}}+\frac{4 c^2 (A-B) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt{c-c \sin (e+f x)}}-\frac{2 c^2 (A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) \sqrt{c-c \sin (e+f x)}}",1,"(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^m*Sqrt[c - c*Sin[e + f*x]]*(50*A - 39*B + 40*A*m - 16*B*m + 8*A*m^2 - 4*B*m^2 + B*(3 + 8*m + 4*m^2)*Cos[2*(e + f*x)] - 2*(1 + 2*m)*(5*A - 9*B + 2*A*m - 2*B*m)*Sin[e + f*x]))/(f*(1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
207,1,116,104,0.4420166,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]],x]","\frac{2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (a (\sin (e+f x)+1))^m (A (2 m+3)+B (2 m+1) \sin (e+f x)-2 B)}{f (2 m+1) (2 m+3) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{2 c (A-B) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{2 B c \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) \sqrt{c-c \sin (e+f x)}}",1,"(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^m*Sqrt[c - c*Sin[e + f*x]]*(-2*B + A*(3 + 2*m) + B*(1 + 2*m)*Sin[e + f*x]))/(f*(1 + 2*m)*(3 + 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
208,1,200,118,2.2288028,"\int \frac{(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin[e + f*x]],x]","\frac{2^{-2 m-\frac{3}{2}} \sin \left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (a (\sin (e+f x)+1))^m \left(2^{2 m+1} (A+B) \, _2F_1\left(1,2 m+1;2 (m+1);\sin \left(\frac{1}{4} (2 e+2 f x+\pi )\right)\right)+(A+B) \sec ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)^{2 m+1} \, _2F_1\left(2 m+1,2 m+1;2 (m+1);\frac{1}{2} \left(1-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)\right)-B 2^{2 m+3}\right)}{(2 f m+f) \sqrt{c-c \sin (e+f x)}}","\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left(1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1) \sqrt{c-c \sin (e+f x)}}-\frac{2 B \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt{c-c \sin (e+f x)}}",1,"(2^(-3/2 - 2*m)*(-(2^(3 + 2*m)*B) + 2^(1 + 2*m)*(A + B)*Hypergeometric2F1[1, 1 + 2*m, 2*(1 + m), Sin[(2*e + Pi + 2*f*x)/4]] + (A + B)*Hypergeometric2F1[1 + 2*m, 1 + 2*m, 2*(1 + m), (1 - Tan[(2*e - Pi + 2*f*x)/8]^2)/2]*(Sec[(2*e - Pi + 2*f*x)/8]^2)^(1 + 2*m))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^m*Sin[(2*e + Pi + 2*f*x)/4])/((f + 2*f*m)*Sqrt[c - c*Sin[e + f*x]])","A",1
209,1,3178,134,6.7601804,"\int \frac{(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(3/2),x]","\text{Result too large to show}","\frac{(A (1-2 m)-B (2 m+3)) \cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left(1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{4 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^m}{2 f (c-c \sin (e+f x))^{3/2}}",1,"(2^(-3/2 - 2*m)*B*(-(4^m*Hypergeometric2F1[1, 2*m, 1 + 2*m, Cos[(-e + Pi/2 - f*x)/2]]) + Hypergeometric2F1[2*m, 2*m, 1 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a + a*Sin[e + f*x])^m)/(f*m*(c - c*Sin[e + f*x])^(3/2)) - ((A + B)*(Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a + a*Sin[e + f*x])^m*(AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^2 - (AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^2*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (2^(1 - 2*m)*AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(1 + 2*m)))/(8*Sqrt[2]*f*(c - c*Sin[e + f*x])^(3/2)*(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^3*(-1/8*(m*Cos[(-e + Pi/2 - f*x)/4]*(Cos[(-e + Pi/2 - f*x)/4]^2)^(-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/4]*(AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^2 - (AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^2*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (2^(1 - 2*m)*AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(1 + 2*m)))/Sqrt[2] + ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*((AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*Tan[(-e + Pi/2 - f*x)/4])/2 + m*AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^3 + (Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^2*(-1/2*(m*AppellF1[2, 1 - 2*m, 2*m, 3, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]) - (m*AppellF1[2, -2*m, 1 + 2*m, 3, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/2) + (m*AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^3*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + m*AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^3*(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(-1 - 2*m)*(Csc[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]*(Csc[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(2*(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m)) - (Cot[(-e + Pi/2 - f*x)/4]^2*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*((m*AppellF1[2, 1 - 2*m, 2*m, 3, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]*Csc[(-e + Pi/2 - f*x)/4]^2)/2 + (m*AppellF1[2, -2*m, 1 + 2*m, 3, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]*Csc[(-e + Pi/2 - f*x)/4]^2)/2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (m*AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Csc[(-e + Pi/2 - f*x)/4]*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Sec[(-e + Pi/2 - f*x)/4]*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(-1 + 2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(2^(2*m)*(1 + 2*m)) + (2^(1 - 2*m)*(-1/2*((1 + 2*m)*AppellF1[2 + 2*m, 2*m, 2, 3 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2 + 2*m) - (m*(1 + 2*m)*AppellF1[2 + 2*m, 1 + 2*m, 1, 3 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(2 + 2*m)))*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(1 + 2*m) - (2^(2 - 2*m)*m*AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]^3*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(-1 + 2*m))/(1 + 2*m)))/(8*Sqrt[2])))","C",0
210,1,8147,134,6.8716925,"\int \frac{(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(5/2),x]","\text{Result too large to show}","\frac{(A (3-2 m)-B (2 m+5)) \cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left(2,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{16 c^2 f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^m}{4 f (c-c \sin (e+f x))^{5/2}}",1,"Result too large to show","C",0
211,1,353,267,14.1311616,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-m} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-4 - m),x]","-\frac{2^{-m-18} \cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \csc ^{21}\left(\frac{1}{8} \left(-e-f x+\frac{\pi }{2}\right)\right) \sec ^7\left(\frac{1}{8} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^{-2 m}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-4} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{-2 (-m-4)} \left(\left(8 m^2+32 m+29\right) (3 A-2 B (m+2)) \sin (e+f x)+4 (m+2) (2 B (m+2)-3 A) \cos \left(2 \left(-e-f x+\frac{\pi }{2}\right)\right)+3 A \cos \left(3 \left(-e-f x+\frac{\pi }{2}\right)\right)-16 A m^3-96 A m^2-176 A m-96 A-2 B m \cos \left(3 \left(-e-f x+\frac{\pi }{2}\right)\right)-4 B \cos \left(3 \left(-e-f x+\frac{\pi }{2}\right)\right)+16 B m^2+64 B m+58 B\right)}{f (2 m+1) (2 m+3) (2 m+5) (2 m+7) \left(\cot ^2\left(\frac{1}{8} \left(-e-f x+\frac{\pi }{2}\right)\right)-1\right)^7}","\frac{2 (3 A-2 B (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c^3 f (2 m+5) (2 m+7) \left(4 m^2+8 m+3\right)}+\frac{2 (3 A-2 B (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{c^2 f (2 m+7) \left(4 m^2+16 m+15\right)}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-4}}{f (2 m+7)}+\frac{(3 A-2 B (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3}}{c f (2 m+5) (2 m+7)}",1,"-((2^(-18 - m)*Cos[(-e + Pi/2 - f*x)/2]*Csc[(-e + Pi/2 - f*x)/8]^21*Sec[(-e + Pi/2 - f*x)/8]^7*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-4 - m)*(-96*A + 58*B - 176*A*m + 64*B*m - 96*A*m^2 + 16*B*m^2 - 16*A*m^3 + 4*(2 + m)*(-3*A + 2*B*(2 + m))*Cos[2*(-e + Pi/2 - f*x)] + 3*A*Cos[3*(-e + Pi/2 - f*x)] - 4*B*Cos[3*(-e + Pi/2 - f*x)] - 2*B*m*Cos[3*(-e + Pi/2 - f*x)] + (29 + 32*m + 8*m^2)*(3*A - 2*B*(2 + m))*Sin[e + f*x]))/(f*(1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(-1 + Cot[(-e + Pi/2 - f*x)/8]^2)^7*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(-4 - m))))","A",0
212,1,269,191,10.73362,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-3-m} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-3 - m),x]","\frac{2^{-m-13} \cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \csc ^{15}\left(\frac{1}{8} \left(-e-f x+\frac{\pi }{2}\right)\right) \sec ^5\left(\frac{1}{8} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^{-2 m}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{-2 (-m-3)} \left(2 (2 m+3) (B (2 m+3)-2 A) \sin (e+f x)+(2 A-2 B m-3 B) \cos \left(2 \left(-e-f x+\frac{\pi }{2}\right)\right)+8 A m^2+24 A m+16 A-6 B m-9 B\right)}{f (2 m+1) (2 m+3) (2 m+5) \left(\cot ^2\left(\frac{1}{8} \left(-e-f x+\frac{\pi }{2}\right)\right)-1\right)^5}","\frac{(2 A-B (2 m+3)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c^2 f (2 m+5) \left(4 m^2+8 m+3\right)}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3}}{f (2 m+5)}+\frac{(2 A-B (2 m+3)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{c f (2 m+3) (2 m+5)}",1,"(2^(-13 - m)*Cos[(-e + Pi/2 - f*x)/2]*Csc[(-e + Pi/2 - f*x)/8]^15*Sec[(-e + Pi/2 - f*x)/8]^5*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m)*(16*A - 9*B + 24*A*m - 6*B*m + 8*A*m^2 + (2*A - 3*B - 2*B*m)*Cos[2*(-e + Pi/2 - f*x)] + 2*(3 + 2*m)*(-2*A + B*(3 + 2*m))*Sin[e + f*x]))/(f*(1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(-1 + Cot[(-e + Pi/2 - f*x)/8]^2)^5*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(-3 - m)))","A",0
213,1,211,114,8.7896959,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-2-m} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-2 - m),x]","-\frac{2^{-m-7} \cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \csc ^9\left(\frac{1}{8} \left(-e-f x+\frac{\pi }{2}\right)\right) \sec ^3\left(\frac{1}{8} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^{-2 m}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{-2 (-m-2)} ((A-2 B (m+1)) \sin (e+f x)-2 A (m+1)+B)}{f \left(4 m^2+8 m+3\right) \left(\cot ^2\left(\frac{1}{8} \left(-e-f x+\frac{\pi }{2}\right)\right)-1\right)^3}","\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{f (2 m+3)}+\frac{(A-2 B (m+1)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c f (2 m+1) (2 m+3)}",1,"-((2^(-7 - m)*Cos[(-e + Pi/2 - f*x)/2]*Csc[(-e + Pi/2 - f*x)/8]^9*Sec[(-e + Pi/2 - f*x)/8]^3*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m)*(B - 2*A*(1 + m) + (A - 2*B*(1 + m))*Sin[e + f*x]))/(f*(3 + 8*m + 4*m^2)*(-1 + Cot[(-e + Pi/2 - f*x)/8]^2)^3*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(-2 - m))))","A",0
214,1,675,163,11.6160272,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-m} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-1 - m),x]","-\frac{2^{-m} (2 m-3) \cos ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \cot \left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^{-2 m}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a \sin (e+f x)+a)^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-m-1} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{-2 (-m-1)} \left(8 B (2 m+1) \tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) F_1\left(\frac{1}{2}-m;-2 m,1;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)-(A+B) \left((2 m+1) \tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \, _2F_1\left(\frac{1}{2}-m,-2 m;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)+(2 m-1) \, _2F_1\left(-m-\frac{1}{2},-2 m;\frac{1}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)\right)\right)}{f \left(4 m^2-1\right) \left((2 m-3) \left((A+B) \left(\cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+1\right) \left(1-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{2 m}-4 B \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) F_1\left(\frac{1}{2}-m;-2 m,1;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)\right)-64 B m \sin ^4\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) F_1\left(\frac{3}{2}-m;1-2 m,1;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)-32 B \sin ^4\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) F_1\left(\frac{3}{2}-m;-2 m,2;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)\right)}","\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{f (2 m+1)}-\frac{B 2^{\frac{1}{2}-m} \cos (e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{2} (2 m+1),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1)}",1,"-(((-3 + 2*m)*Cos[(-e + Pi/2 - f*x)/4]^2*Cot[(-e + Pi/2 - f*x)/4]*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-1 - m)*(8*B*(1 + 2*m)*AppellF1[1/2 - m, -2*m, 1, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2 - (A + B)*((-1 + 2*m)*Hypergeometric2F1[-1/2 - m, -2*m, 1/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2] + (1 + 2*m)*Hypergeometric2F1[1/2 - m, -2*m, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2)))/(2^m*f*(-1 + 4*m^2)*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(-1 - m))*(-64*B*m*AppellF1[3/2 - m, 1 - 2*m, 1, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sin[(-e + Pi/2 - f*x)/4]^4 - 32*B*AppellF1[3/2 - m, -2*m, 2, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sin[(-e + Pi/2 - f*x)/4]^4 + (-3 + 2*m)*(-4*B*AppellF1[1/2 - m, -2*m, 1, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sin[(-e + Pi/2 - f*x)/2]^2 + (A + B)*(1 + Cos[(-e + Pi/2 - f*x)/2])*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)))))","C",0
215,1,2552,158,17.2041957,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-m} \, dx","Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^m,x]","\text{Result too large to show}","\frac{c 2^{\frac{1}{2}-m} (A+2 B m) \cos (e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{2} (2 m+1),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1)}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m}}{f}",1,"(2^(2 - m)*((A + B)*AppellF1[1/2 - m, -2*m, 1, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*B*(-AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*m)*(a + a*Sin[e + f*x])^m*((A*Cos[(-e + Pi/2 - f*x)/2]^(2*m))/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m) + (B*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sin[e + f*x])/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m))*Tan[(-e + Pi/2 - f*x)/4])/(f*(-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(c - c*Sin[e + f*x])^m*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(-((2^(2 - m)*m*((A + B)*AppellF1[1/2 - m, -2*m, 1, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*B*(-AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]))*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(-1 - 2*m))/((-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*m))) - (((A + B)*AppellF1[1/2 - m, -2*m, 1, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*B*(-AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]))*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sec[(-e + Pi/2 - f*x)/4]^2)/(2^m*(-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (2^(2 - m)*m*((A + B)*AppellF1[1/2 - m, -2*m, 1, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*B*(-AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]))*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(-1 - 2*m)*Tan[(-e + Pi/2 - f*x)/4])/((-1 + 2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (2^(2 - m)*m*((A + B)*AppellF1[1/2 - m, -2*m, 1, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*B*(-AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]))*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(1 - 2*m)*Tan[(-e + Pi/2 - f*x)/4])/((-1 + 2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) - (2^(2 - m)*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Tan[(-e + Pi/2 - f*x)/4]*((A + B)*(-(((1/2 - m)*m*AppellF1[3/2 - m, 1 - 2*m, 1, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m)) - ((1/2 - m)*AppellF1[3/2 - m, -2*m, 2, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 - m))) + 8*B*(((1/2 - m)*m*AppellF1[3/2 - m, 1 - 2*m, 2, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m) - ((1/2 - m)*m*AppellF1[3/2 - m, 1 - 2*m, 3, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m) + ((1/2 - m)*AppellF1[3/2 - m, -2*m, 3, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m) - (3*(1/2 - m)*AppellF1[3/2 - m, -2*m, 4, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 - m)))))/((-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))))","C",0
216,1,3601,170,92.4708524,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{1-m} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1 - m),x]","\text{Result too large to show}","\frac{c^2 2^{\frac{1}{2}-m} (2 A-B (1-2 m)) \cos (e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{2} (2 m-1),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1)}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{1-m}}{2 f}",1,"(2^(5 - m)*((A + B)*AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - (A + 9*B)*AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*B*(2*AppellF1[1/2 - m, -2*m, 4, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - AppellF1[1/2 - m, -2*m, 5, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]))*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 - m)*(Cos[Pi/4 + (e - Pi/2 + f*x)/2]^2*((A*Cos[(-e + Pi/2 - f*x)/2]^(2*m))/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m) + (B*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sin[e + f*x])/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m)) + (A*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sin[Pi/4 + (e - Pi/2 + f*x)/2]^2)/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m) + (B*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sin[e + f*x]*Sin[Pi/4 + (e - Pi/2 + f*x)/2]^2)/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m) + Cos[Pi/4 + (e - Pi/2 + f*x)/2]*((-2*A*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sin[Pi/4 + (e - Pi/2 + f*x)/2])/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m) - (2*B*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sin[e + f*x]*Sin[Pi/4 + (e - Pi/2 + f*x)/2])/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m)))*Tan[(-e + Pi/2 - f*x)/4])/(f*(-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(1 - m))*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(-((2^(5 - m)*m*((A + B)*AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - (A + 9*B)*AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*B*(2*AppellF1[1/2 - m, -2*m, 4, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - AppellF1[1/2 - m, -2*m, 5, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]))*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(-1 - 2*m))/((-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*m))) - (2^(3 - m)*((A + B)*AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - (A + 9*B)*AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*B*(2*AppellF1[1/2 - m, -2*m, 4, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - AppellF1[1/2 - m, -2*m, 5, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]))*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sec[(-e + Pi/2 - f*x)/4]^2)/((-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (2^(5 - m)*m*((A + B)*AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - (A + 9*B)*AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*B*(2*AppellF1[1/2 - m, -2*m, 4, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - AppellF1[1/2 - m, -2*m, 5, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]))*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(-1 - 2*m)*Tan[(-e + Pi/2 - f*x)/4])/((-1 + 2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (2^(5 - m)*m*((A + B)*AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - (A + 9*B)*AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*B*(2*AppellF1[1/2 - m, -2*m, 4, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - AppellF1[1/2 - m, -2*m, 5, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]))*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(1 - 2*m)*Tan[(-e + Pi/2 - f*x)/4])/((-1 + 2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) - (2^(5 - m)*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Tan[(-e + Pi/2 - f*x)/4]*((A + B)*(-(((1/2 - m)*m*AppellF1[3/2 - m, 1 - 2*m, 2, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m)) - ((1/2 - m)*AppellF1[3/2 - m, -2*m, 3, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m)) - (A + 9*B)*(-(((1/2 - m)*m*AppellF1[3/2 - m, 1 - 2*m, 3, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m)) - (3*(1/2 - m)*AppellF1[3/2 - m, -2*m, 4, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 - m))) + 8*B*(((1/2 - m)*m*AppellF1[3/2 - m, 1 - 2*m, 5, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m) + (5*(1/2 - m)*AppellF1[3/2 - m, -2*m, 6, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 - m)) + 2*(-(((1/2 - m)*m*AppellF1[3/2 - m, 1 - 2*m, 4, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m)) - (2*(1/2 - m)*AppellF1[3/2 - m, -2*m, 5, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m)))))/((-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))))","C",0
217,1,5163,173,49.2153431,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{2-m} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(2 - m),x]","\text{Result too large to show}","\frac{c^3 2^{\frac{5}{2}-m} (3 A-2 B (1-m)) \cos (e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{2} (2 m-3),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{3 f (2 m+1)}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{2-m}}{3 f}",1,"Result too large to show","C",0
218,1,63,34,0.5395077,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^n (B (3-n)-B (4+n) \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^n*(B*(3 - n) - B*(4 + n)*Sin[e + f*x]),x]","\frac{a^3 B (14 \sin (2 (e+f x))-\sin (4 (e+f x))+14 \cos (e+f x)-6 \cos (3 (e+f x))) (c-c \sin (e+f x))^n}{8 f}","\frac{a^3 B c^3 \cos ^7(e+f x) (c-c \sin (e+f x))^{n-3}}{f}",1,"(a^3*B*(c - c*Sin[e + f*x])^n*(14*Cos[e + f*x] - 6*Cos[3*(e + f*x)] + 14*Sin[2*(e + f*x)] - Sin[4*(e + f*x)]))/(8*f)","A",1
219,1,67,34,1.1419585,"\int (a-a \sin (e+f x))^3 (c+c \sin (e+f x))^n (B (3-n)+B (4+n) \sin (e+f x)) \, dx","Integrate[(a - a*Sin[e + f*x])^3*(c + c*Sin[e + f*x])^n*(B*(3 - n) + B*(4 + n)*Sin[e + f*x]),x]","-\frac{a^3 B \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (c (\sin (e+f x)+1))^n}{f}","-\frac{a^3 B c^3 \cos ^7(e+f x) (c \sin (e+f x)+c)^{n-3}}{f}",1,"-((a^3*B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c*(1 + Sin[e + f*x]))^n)/f)","A",1
220,1,66,33,1.1090877,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 (B (-3+m)-B (4+m) \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^3*(B*(-3 + m) - B*(4 + m)*Sin[e + f*x]),x]","\frac{B c^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (a (\sin (e+f x)+1))^m}{f}","\frac{a^3 B c^3 \cos ^7(e+f x) (a \sin (e+f x)+a)^{m-3}}{f}",1,"(B*c^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^m)/f","A",1
221,1,61,35,0.5400512,"\int (a-a \sin (e+f x))^m (c+c \sin (e+f x))^3 (B (-3+m)+B (4+m) \sin (e+f x)) \, dx","Integrate[(a - a*Sin[e + f*x])^m*(c + c*Sin[e + f*x])^3*(B*(-3 + m) + B*(4 + m)*Sin[e + f*x]),x]","\frac{B c^3 (-14 \sin (2 (e+f x))+\sin (4 (e+f x))-14 \cos (e+f x)+6 \cos (3 (e+f x))) (a-a \sin (e+f x))^m}{8 f}","-\frac{a^3 B c^3 \cos ^7(e+f x) (a-a \sin (e+f x))^{m-3}}{f}",1,"(B*c^3*(a - a*Sin[e + f*x])^m*(-14*Cos[e + f*x] + 6*Cos[3*(e + f*x)] - 14*Sin[2*(e + f*x)] + Sin[4*(e + f*x)]))/(8*f)","A",1
222,1,36,36,0.4871376,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n (B (m-n)-B (1+m+n) \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n*(B*(m - n) - B*(1 + m + n)*Sin[e + f*x]),x]","\frac{B \cos (e+f x) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n}{f}","\frac{B \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n}{f}",1,"(B*Cos[e + f*x]*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^n)/f","A",1
223,1,37,37,0.4748399,"\int (a-a \sin (e+f x))^m (c+c \sin (e+f x))^n (B (m-n)+B (1+m+n) \sin (e+f x)) \, dx","Integrate[(a - a*Sin[e + f*x])^m*(c + c*Sin[e + f*x])^n*(B*(m - n) + B*(1 + m + n)*Sin[e + f*x]),x]","-\frac{B \cos (e+f x) (a-a \sin (e+f x))^m (c (\sin (e+f x)+1))^n}{f}","-\frac{B \cos (e+f x) (a-a \sin (e+f x))^m (c \sin (e+f x)+c)^n}{f}",1,"-((B*Cos[e + f*x]*(c*(1 + Sin[e + f*x]))^n*(a - a*Sin[e + f*x])^m)/f)","A",1
224,1,87,140,0.1398412,"\int \sin ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx","Integrate[Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]","\frac{a^3 A (-210 \sin (2 (c+d x))-210 \sin (4 (c+d x))+70 \sin (6 (c+d x))-1365 \cos (c+d x)-175 \cos (3 (c+d x))+147 \cos (5 (c+d x))-15 \cos (7 (c+d x))+840 c+840 d x)}{6720 d}","-\frac{a^3 A \cos ^7(c+d x)}{7 d}+\frac{3 a^3 A \cos ^5(c+d x)}{5 d}-\frac{2 a^3 A \cos ^3(c+d x)}{3 d}+\frac{a^3 A \sin ^5(c+d x) \cos (c+d x)}{3 d}-\frac{a^3 A \sin ^3(c+d x) \cos (c+d x)}{12 d}-\frac{a^3 A \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^3 A x",1,"(a^3*A*(840*c + 840*d*x - 1365*Cos[c + d*x] - 175*Cos[3*(c + d*x)] + 147*Cos[5*(c + d*x)] - 15*Cos[7*(c + d*x)] - 210*Sin[2*(c + d*x)] - 210*Sin[4*(c + d*x)] + 70*Sin[6*(c + d*x)]))/(6720*d)","A",1
225,1,77,121,0.1022211,"\int \sin ^2(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx","Integrate[Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]","\frac{a^3 A (-15 \sin (2 (c+d x))-45 \sin (4 (c+d x))+5 \sin (6 (c+d x))-240 \cos (c+d x)-40 \cos (3 (c+d x))+24 \cos (5 (c+d x))+180 c+180 d x)}{960 d}","\frac{2 a^3 A \cos ^5(c+d x)}{5 d}-\frac{2 a^3 A \cos ^3(c+d x)}{3 d}+\frac{a^3 A \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac{5 a^3 A \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{3 a^3 A \sin (c+d x) \cos (c+d x)}{16 d}+\frac{3}{16} a^3 A x",1,"(a^3*A*(180*c + 180*d*x - 240*Cos[c + d*x] - 40*Cos[3*(c + d*x)] + 24*Cos[5*(c + d*x)] - 15*Sin[2*(c + d*x)] - 45*Sin[4*(c + d*x)] + 5*Sin[6*(c + d*x)]))/(960*d)","A",1
226,1,55,96,0.4762119,"\int \sin (c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx","Integrate[Sin[c + d*x]*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]","\frac{a^3 A (-90 \cos (c+d x)-25 \cos (3 (c+d x))+3 (-5 \sin (4 (c+d x))+\cos (5 (c+d x))+20 d x))}{240 d}","\frac{a^3 A \cos ^5(c+d x)}{5 d}-\frac{2 a^3 A \cos ^3(c+d x)}{3 d}+\frac{a^3 A \sin ^3(c+d x) \cos (c+d x)}{2 d}-\frac{a^3 A \sin (c+d x) \cos (c+d x)}{4 d}+\frac{1}{4} a^3 A x",1,"(a^3*A*(-90*Cos[c + d*x] - 25*Cos[3*(c + d*x)] + 3*(20*d*x + Cos[5*(c + d*x)] - 5*Sin[4*(c + d*x)])))/(240*d)","A",1
227,1,54,82,0.3466177,"\int (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx","Integrate[(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]","\frac{a^3 A (24 \sin (2 (c+d x))-3 \sin (4 (c+d x))-48 \cos (c+d x)-16 \cos (3 (c+d x))+60 d x)}{96 d}","-\frac{5 a^3 A \cos ^3(c+d x)}{12 d}-\frac{A \cos ^3(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{4 d}+\frac{5 a^3 A \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5}{8} a^3 A x",1,"(a^3*A*(60*d*x - 48*Cos[c + d*x] - 16*Cos[3*(c + d*x)] + 24*Sin[2*(c + d*x)] - 3*Sin[4*(c + d*x)]))/(96*d)","A",1
228,1,74,76,0.1467426,"\int \csc (c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]","\frac{a^3 A \left(9 \cos (c+d x)-\cos (3 (c+d x))+6 \left(\sin (2 (c+d x))+2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2 c+2 d x\right)\right)}{12 d}","-\frac{a^3 A \cos ^3(c+d x)}{3 d}+\frac{a^3 A \cos (c+d x)}{d}+\frac{a^3 A \sin (c+d x) \cos (c+d x)}{d}-\frac{a^3 A \tanh ^{-1}(\cos (c+d x))}{d}+a^3 A x",1,"(a^3*A*(9*Cos[c + d*x] - Cos[3*(c + d*x)] + 6*(-2*c + 2*d*x - 2*Log[Cos[(c + d*x)/2]] + 2*Log[Sin[(c + d*x)/2]] + Sin[2*(c + d*x)])))/(12*d)","A",1
229,1,77,79,0.1896197,"\int \csc ^2(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]","\frac{a^3 A \left(-8 \sin (c) \sin (d x)+\sin (2 (c+d x))+8 \cos (c) \cos (d x)-4 \cot (c+d x)+8 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-8 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2 c-2 d x\right)}{4 d}","\frac{2 a^3 A \cos (c+d x)}{d}-\frac{a^3 A \cot (c+d x)}{d}+\frac{a^3 A \sin (c+d x) \cos (c+d x)}{2 d}-\frac{2 a^3 A \tanh ^{-1}(\cos (c+d x))}{d}-\frac{1}{2} a^3 A x",1,"(a^3*A*(-2*c - 2*d*x + 8*Cos[c]*Cos[d*x] - 4*Cot[c + d*x] - 8*Log[Cos[(c + d*x)/2]] + 8*Log[Sin[(c + d*x)/2]] - 8*Sin[c]*Sin[d*x] + Sin[2*(c + d*x)]))/(4*d)","A",1
230,1,142,78,0.038027,"\int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]","-\frac{a^3 A \sin (c) \sin (d x)}{d}+\frac{a^3 A \cos (c) \cos (d x)}{d}-\frac{2 a^3 A \cot (c+d x)}{d}-\frac{a^3 A \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a^3 A \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a^3 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{a^3 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-2 a^3 A x","\frac{a^3 A \cos (c+d x)}{d}-\frac{2 a^3 A \cot (c+d x)}{d}-\frac{a^3 A \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 A \cot (c+d x) \csc (c+d x)}{2 d}-2 a^3 A x",1,"-2*a^3*A*x + (a^3*A*Cos[c]*Cos[d*x])/d - (2*a^3*A*Cot[c + d*x])/d - (a^3*A*Csc[(c + d*x)/2]^2)/(8*d) - (a^3*A*Log[Cos[(c + d*x)/2]])/(2*d) + (a^3*A*Log[Sin[(c + d*x)/2]])/(2*d) + (a^3*A*Sec[(c + d*x)/2]^2)/(8*d) - (a^3*A*Sin[c]*Sin[d*x])/d","A",1
231,1,141,78,0.4899032,"\int \csc ^4(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]","-\frac{a^3 A \left(-8 \tan \left(\frac{1}{2} (c+d x)\right)+8 \cot \left(\frac{1}{2} (c+d x)\right)+6 \csc ^2\left(\frac{1}{2} (c+d x)\right)-6 \sec ^2\left(\frac{1}{2} (c+d x)\right)+24 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-24 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{1}{2} \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-8 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+24 c+24 d x\right)}{24 d}","-\frac{a^3 A \cot ^3(c+d x)}{3 d}-\frac{a^3 A \cot (c+d x)}{d}+\frac{a^3 A \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^3 A \cot (c+d x) \csc (c+d x)}{d}-a^3 A x",1,"-1/24*(a^3*A*(24*c + 24*d*x + 8*Cot[(c + d*x)/2] + 6*Csc[(c + d*x)/2]^2 - 24*Log[Cos[(c + d*x)/2]] + 24*Log[Sin[(c + d*x)/2]] - 6*Sec[(c + d*x)/2]^2 - 8*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + (Csc[(c + d*x)/2]^4*Sin[c + d*x])/2 - 8*Tan[(c + d*x)/2]))/d","A",1
232,1,210,86,0.0689729,"\int \csc ^5(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]","a^3 A \left(-\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{3 d}+\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{3 d}-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{3 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{3 \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}-\frac{5 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}+\frac{5 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{12 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{12 d}\right)","-\frac{2 a^3 A \cot ^3(c+d x)}{3 d}+\frac{5 a^3 A \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 A \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^3 A \cot (c+d x) \csc (c+d x)}{8 d}",1,"a^3*A*(Cot[(c + d*x)/2]/(3*d) - (3*Csc[(c + d*x)/2]^2)/(32*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(12*d) - Csc[(c + d*x)/2]^4/(64*d) + (5*Log[Cos[(c + d*x)/2]])/(8*d) - (5*Log[Sin[(c + d*x)/2]])/(8*d) + (3*Sec[(c + d*x)/2]^2)/(32*d) + Sec[(c + d*x)/2]^4/(64*d) - Tan[(c + d*x)/2]/(3*d) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(12*d))","B",1
233,1,268,105,0.0736676,"\int \csc ^6(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]","a^3 A \left(-\frac{7 \tan \left(\frac{1}{2} (c+d x)\right)}{30 d}+\frac{7 \cot \left(\frac{1}{2} (c+d x)\right)}{30 d}-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{16 d}+\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right)}{32 d}-\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{16 d}-\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}+\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^4\left(\frac{1}{2} (c+d x)\right)}{160 d}-\frac{19 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{480 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)}{160 d}+\frac{19 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{480 d}\right)","-\frac{a^3 A \cot ^5(c+d x)}{5 d}-\frac{2 a^3 A \cot ^3(c+d x)}{3 d}+\frac{a^3 A \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{a^3 A \cot (c+d x) \csc ^3(c+d x)}{2 d}+\frac{a^3 A \cot (c+d x) \csc (c+d x)}{4 d}",1,"a^3*A*((7*Cot[(c + d*x)/2])/(30*d) + Csc[(c + d*x)/2]^2/(16*d) - (19*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(480*d) - Csc[(c + d*x)/2]^4/(32*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(160*d) + Log[Cos[(c + d*x)/2]]/(4*d) - Log[Sin[(c + d*x)/2]]/(4*d) - Sec[(c + d*x)/2]^2/(16*d) + Sec[(c + d*x)/2]^4/(32*d) - (7*Tan[(c + d*x)/2])/(30*d) + (19*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(480*d) + (Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(160*d))","B",1
234,1,306,130,0.0799491,"\int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]","a^3 A \left(-\frac{2 \tan \left(\frac{1}{2} (c+d x)\right)}{15 d}+\frac{2 \cot \left(\frac{1}{2} (c+d x)\right)}{15 d}-\frac{\csc ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{3 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{\sec ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}+\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{3 \sec ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}+\frac{3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^4\left(\frac{1}{2} (c+d x)\right)}{80 d}+\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{240 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)}{80 d}-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{240 d}\right)","-\frac{2 a^3 A \cot ^5(c+d x)}{5 d}-\frac{2 a^3 A \cot ^3(c+d x)}{3 d}+\frac{3 a^3 A \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a^3 A \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{3 a^3 A \cot (c+d x) \csc (c+d x)}{16 d}",1,"a^3*A*((2*Cot[(c + d*x)/2])/(15*d) + (3*Csc[(c + d*x)/2]^2)/(64*d) + (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(240*d) - Csc[(c + d*x)/2]^4/(64*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(80*d) - Csc[(c + d*x)/2]^6/(384*d) + (3*Log[Cos[(c + d*x)/2]])/(16*d) - (3*Log[Sin[(c + d*x)/2]])/(16*d) - (3*Sec[(c + d*x)/2]^2)/(64*d) + Sec[(c + d*x)/2]^4/(64*d) + Sec[(c + d*x)/2]^6/(384*d) - (2*Tan[(c + d*x)/2])/(15*d) - (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(240*d) + (Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(80*d))","B",1
235,1,254,129,0.9447012,"\int \frac{\sin ^4(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sin[c + d*x]^4*(A - A*Sin[c + d*x]))/(a + a*Sin[c + d*x])^3,x]","\frac{A \left(-11400 d x \sin \left(c+\frac{d x}{2}\right)-5700 d x \sin \left(c+\frac{3 d x}{2}\right)+1830 \sin \left(2 c+\frac{3 d x}{2}\right)-4234 \sin \left(2 c+\frac{5 d x}{2}\right)+1140 d x \sin \left(3 c+\frac{5 d x}{2}\right)+165 \sin \left(4 c+\frac{7 d x}{2}\right)-15 \sin \left(4 c+\frac{9 d x}{2}\right)+12060 \cos \left(c+\frac{d x}{2}\right)-14090 \cos \left(c+\frac{3 d x}{2}\right)+5700 d x \cos \left(2 c+\frac{3 d x}{2}\right)+1140 d x \cos \left(2 c+\frac{5 d x}{2}\right)+1050 \cos \left(3 c+\frac{5 d x}{2}\right)+165 \cos \left(3 c+\frac{7 d x}{2}\right)+15 \cos \left(5 c+\frac{9 d x}{2}\right)+19780 \sin \left(\frac{d x}{2}\right)-11400 d x \cos \left(\frac{d x}{2}\right)\right)}{480 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","-\frac{4 A \cos (c+d x)}{a^3 d}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{199 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)}+\frac{41 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)^2}-\frac{2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}-\frac{19 A x}{2 a^3}",1,"(A*(-11400*d*x*Cos[(d*x)/2] + 12060*Cos[c + (d*x)/2] - 14090*Cos[c + (3*d*x)/2] + 5700*d*x*Cos[2*c + (3*d*x)/2] + 1140*d*x*Cos[2*c + (5*d*x)/2] + 1050*Cos[3*c + (5*d*x)/2] + 165*Cos[3*c + (7*d*x)/2] + 15*Cos[5*c + (9*d*x)/2] + 19780*Sin[(d*x)/2] - 11400*d*x*Sin[c + (d*x)/2] - 5700*d*x*Sin[c + (3*d*x)/2] + 1830*Sin[2*c + (3*d*x)/2] - 4234*Sin[2*c + (5*d*x)/2] + 1140*d*x*Sin[3*c + (5*d*x)/2] + 165*Sin[4*c + (7*d*x)/2] - 15*Sin[4*c + (9*d*x)/2]))/(480*a^3*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","A",1
236,1,228,103,0.8007897,"\int \frac{\sin ^3(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sin[c + d*x]^3*(A - A*Sin[c + d*x]))/(a + a*Sin[c + d*x])^3,x]","-\frac{A \left(-1200 d x \sin \left(c+\frac{d x}{2}\right)-600 d x \sin \left(c+\frac{3 d x}{2}\right)+405 \sin \left(2 c+\frac{3 d x}{2}\right)-491 \sin \left(2 c+\frac{5 d x}{2}\right)+120 d x \sin \left(3 c+\frac{5 d x}{2}\right)+15 \sin \left(4 c+\frac{7 d x}{2}\right)+1665 \cos \left(c+\frac{d x}{2}\right)-1675 \cos \left(c+\frac{3 d x}{2}\right)+600 d x \cos \left(2 c+\frac{3 d x}{2}\right)+120 d x \cos \left(2 c+\frac{5 d x}{2}\right)+75 \cos \left(3 c+\frac{5 d x}{2}\right)+15 \cos \left(3 c+\frac{7 d x}{2}\right)+2495 \sin \left(\frac{d x}{2}\right)-1200 d x \cos \left(\frac{d x}{2}\right)\right)}{120 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","\frac{A \cos (c+d x)}{a^3 d}+\frac{104 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)}-\frac{31 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)^2}+\frac{2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}+\frac{4 A x}{a^3}",1,"-1/120*(A*(-1200*d*x*Cos[(d*x)/2] + 1665*Cos[c + (d*x)/2] - 1675*Cos[c + (3*d*x)/2] + 600*d*x*Cos[2*c + (3*d*x)/2] + 120*d*x*Cos[2*c + (5*d*x)/2] + 75*Cos[3*c + (5*d*x)/2] + 15*Cos[3*c + (7*d*x)/2] + 2495*Sin[(d*x)/2] - 1200*d*x*Sin[c + (d*x)/2] - 600*d*x*Sin[c + (3*d*x)/2] + 405*Sin[2*c + (3*d*x)/2] - 491*Sin[2*c + (5*d*x)/2] + 120*d*x*Sin[3*c + (5*d*x)/2] + 15*Sin[4*c + (7*d*x)/2]))/(a^3*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","B",1
237,1,189,89,0.7555579,"\int \frac{\sin ^2(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sin[c + d*x]^2*(A - A*Sin[c + d*x]))/(a + a*Sin[c + d*x])^3,x]","\frac{A \left(-50 d x \sin \left(c+\frac{d x}{2}\right)-25 d x \sin \left(c+\frac{3 d x}{2}\right)+40 \sin \left(2 c+\frac{3 d x}{2}\right)-26 \sin \left(2 c+\frac{5 d x}{2}\right)+5 d x \sin \left(3 c+\frac{5 d x}{2}\right)+110 \cos \left(c+\frac{d x}{2}\right)-90 \cos \left(c+\frac{3 d x}{2}\right)+25 d x \cos \left(2 c+\frac{3 d x}{2}\right)+5 d x \cos \left(2 c+\frac{5 d x}{2}\right)+150 \sin \left(\frac{d x}{2}\right)-50 d x \cos \left(\frac{d x}{2}\right)\right)}{20 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","-\frac{13 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)}+\frac{7 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^2}-\frac{2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}-\frac{A x}{a^3}",1,"(A*(-50*d*x*Cos[(d*x)/2] + 110*Cos[c + (d*x)/2] - 90*Cos[c + (3*d*x)/2] + 25*d*x*Cos[2*c + (3*d*x)/2] + 5*d*x*Cos[2*c + (5*d*x)/2] + 150*Sin[(d*x)/2] - 50*d*x*Sin[c + (d*x)/2] - 25*d*x*Sin[c + (3*d*x)/2] + 40*Sin[2*c + (3*d*x)/2] - 26*Sin[2*c + (5*d*x)/2] + 5*d*x*Sin[3*c + (5*d*x)/2]))/(20*a^3*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","B",1
238,1,107,82,0.4620478,"\int \frac{\sin (c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sin[c + d*x]*(A - A*Sin[c + d*x]))/(a + a*Sin[c + d*x])^3,x]","-\frac{A \left(15 \sin \left(2 c+\frac{3 d x}{2}\right)-4 \sin \left(2 c+\frac{5 d x}{2}\right)+15 \cos \left(c+\frac{d x}{2}\right)-5 \cos \left(c+\frac{3 d x}{2}\right)+25 \sin \left(\frac{d x}{2}\right)\right)}{30 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","\frac{4 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)}-\frac{11 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)^2}+\frac{2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}",1,"-1/30*(A*(15*Cos[c + (d*x)/2] - 5*Cos[c + (3*d*x)/2] + 25*Sin[(d*x)/2] + 15*Sin[2*c + (3*d*x)/2] - 4*Sin[2*c + (5*d*x)/2]))/(a^3*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","A",1
239,1,92,58,0.2387193,"\int \frac{A-A \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(A - A*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{A \left(\sin \left(2 c+\frac{5 d x}{2}\right)-15 \cos \left(c+\frac{d x}{2}\right)+5 \cos \left(c+\frac{3 d x}{2}\right)+5 \sin \left(\frac{d x}{2}\right)\right)}{30 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","-\frac{A \cos ^3(c+d x)}{15 d (a \sin (c+d x)+a)^3}-\frac{a A \cos ^3(c+d x)}{5 d (a \sin (c+d x)+a)^4}",1,"(A*(-15*Cos[c + (d*x)/2] + 5*Cos[c + (3*d*x)/2] + 5*Sin[(d*x)/2] + Sin[2*c + (5*d*x)/2]))/(30*a^3*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","A",1
240,1,313,98,1.0064763,"\int \frac{\csc (c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Csc[c + d*x]*(A - A*Sin[c + d*x]))/(a + a*Sin[c + d*x])^3,x]","\frac{(A-A \sin (c+d x)) \left(2 \sin \left(\frac{d x}{2}\right) (-19 \sin (c+d x)+4 \cos (2 (c+d x))-17)+\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(3 \cos \left(\frac{c}{2}\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2-3 \sin \left(\frac{c}{2}\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2-5 \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4+5 \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4-2 \sin \left(\frac{c}{2}\right)+2 \cos \left(\frac{c}{2}\right)\right)\right)}{5 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","\frac{8 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)}+\frac{3 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^2}+\frac{2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}-\frac{A \tanh ^{-1}(\cos (c+d x))}{a^3 d}",1,"(((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(2*Cos[c/2] - 2*Sin[c/2] + 3*Cos[c/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 3*Sin[c/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 5*Log[Cos[(c + d*x)/2]]*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + 5*Log[Sin[(c + d*x)/2]]*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + 2*Sin[(d*x)/2]*(-17 + 4*Cos[2*(c + d*x)] - 19*Sin[c + d*x]))*(A - A*Sin[c + d*x]))/(5*a^3*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","B",1
241,1,167,113,2.973677,"\int \frac{\csc ^2(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Csc[c + d*x]^2*(A - A*Sin[c + d*x]))/(a + a*Sin[c + d*x])^3,x]","-\frac{A \left(-15 \tan \left(\frac{1}{2} (c+d x)\right)+15 \cot \left(\frac{1}{2} (c+d x)\right)+120 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-120 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right) (-354 \sin (c+d x)+79 \cos (2 (c+d x))-287)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}+\frac{38}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{12}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}\right)}{30 a^3 d}","-\frac{A \cot (c+d x)}{a^3 d}+\frac{4 A \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{104 A \cot (c+d x)}{15 a^3 d (\csc (c+d x)+1)}+\frac{31 A \cot (c+d x)}{15 a^3 d (\csc (c+d x)+1)^2}-\frac{2 A \cot (c+d x)}{5 a^3 d (\csc (c+d x)+1)^3}",1,"-1/30*(A*(15*Cot[(c + d*x)/2] - 120*Log[Cos[(c + d*x)/2]] + 120*Log[Sin[(c + d*x)/2]] + 12/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + 38/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (2*Sin[(c + d*x)/2]*(-287 + 79*Cos[2*(c + d*x)] - 354*Sin[c + d*x]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 - 15*Tan[(c + d*x)/2]))/(a^3*d)","A",1
242,1,245,138,3.8875693,"\int \frac{\csc ^3(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Csc[c + d*x]^3*(A - A*Sin[c + d*x]))/(a + a*Sin[c + d*x])^3,x]","\frac{A \left(-240 \tan \left(\frac{1}{2} (c+d x)\right)+240 \cot \left(\frac{1}{2} (c+d x)\right)-15 \csc ^2\left(\frac{1}{2} (c+d x)\right)+15 \sec ^2\left(\frac{1}{2} (c+d x)\right)+1140 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-1140 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{2624 \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{232}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{464 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{48}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{96 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}\right)}{120 a^3 d}","\frac{4 A \cot (c+d x)}{a^3 d}+\frac{164 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)}+\frac{29 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)^2}+\frac{2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}-\frac{19 A \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{A \cot (c+d x) \csc (c+d x)}{2 a^3 d}",1,"(A*(240*Cot[(c + d*x)/2] - 15*Csc[(c + d*x)/2]^2 - 1140*Log[Cos[(c + d*x)/2]] + 1140*Log[Sin[(c + d*x)/2]] + 15*Sec[(c + d*x)/2]^2 - (96*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 + 48/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - (464*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 232/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (2624*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 240*Tan[(c + d*x)/2]))/(120*a^3*d)","A",1
243,1,348,153,6.2289147,"\int \frac{\csc ^4(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Csc[c + d*x]^4*(A - A*Sin[c + d*x]))/(a + a*Sin[c + d*x])^3,x]","\frac{A \left(\frac{29 \tan \left(\frac{1}{2} (c+d x)\right)}{6 d}-\frac{29 \cot \left(\frac{1}{2} (c+d x)\right)}{6 d}+\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{2 d}-\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{2 d}-\frac{18 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{18 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{186 \sin \left(\frac{1}{2} (c+d x)\right)}{5 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{13}{5 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{26 \sin \left(\frac{1}{2} (c+d x)\right)}{5 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}-\frac{2}{5 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{4 \sin \left(\frac{1}{2} (c+d x)\right)}{5 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}\right)}{a^3}","-\frac{A \cot ^3(c+d x)}{3 a^3 d}-\frac{10 A \cot (c+d x)}{a^3 d}-\frac{93 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)}-\frac{13 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^2}-\frac{2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}+\frac{18 A \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{2 A \cot (c+d x) \csc (c+d x)}{a^3 d}",1,"(A*((-29*Cot[(c + d*x)/2])/(6*d) + Csc[(c + d*x)/2]^2/(2*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*d) + (18*Log[Cos[(c + d*x)/2]])/d - (18*Log[Sin[(c + d*x)/2]])/d - Sec[(c + d*x)/2]^2/(2*d) + (4*Sin[(c + d*x)/2])/(5*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5) - 2/(5*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + (26*Sin[(c + d*x)/2])/(5*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - 13/(5*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (186*Sin[(c + d*x)/2])/(5*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (29*Tan[(c + d*x)/2])/(6*d) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*d)))/a^3","B",0
244,1,267,327,1.999559,"\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3,x]","\frac{a (\sin (e+f x)+1) \left(10 d \left(4 A d (3 c+d)+B \left(12 c^2+12 c d+5 d^2\right)\right) \cos (3 (e+f x))+15 \left(-8 \left(A d \left(3 c^2+3 c d+d^2\right)+B (c+d)^3\right) \sin (2 (e+f x))+4 f x \left(A \left(8 c^3+12 c^2 d+12 c d^2+3 d^3\right)+B \left(4 c^3+12 c^2 d+9 c d^2+3 d^3\right)\right)+d^2 (A d+B (3 c+d)) \sin (4 (e+f x))\right)-60 \left(2 A \left(4 c^3+12 c^2 d+9 c d^2+3 d^3\right)+B \left(8 c^3+18 c^2 d+18 c d^2+5 d^3\right)\right) \cos (e+f x)-6 B d^3 \cos (5 (e+f x))\right)}{480 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","-\frac{a \left(5 A d \left(6 c^2+20 c d+9 d^2\right)-B \left(6 c^3-30 c^2 d-71 c d^2-45 d^3\right)\right) \sin (e+f x) \cos (e+f x)}{120 f}+\frac{1}{8} a x \left(A \left(8 c^3+12 c^2 d+12 c d^2+3 d^3\right)+B \left(4 c^3+12 c^2 d+9 c d^2+3 d^3\right)\right)-\frac{a \left(5 A d \left(3 c^3+16 c^2 d+12 c d^2+4 d^3\right)-B \left(3 c^4-15 c^3 d-52 c^2 d^2-60 c d^3-16 d^4\right)\right) \cos (e+f x)}{30 d f}-\frac{a \left(4 d^2 (5 A+4 B)-3 c (B c-5 d (A+B))\right) \cos (e+f x) (c+d \sin (e+f x))^2}{60 d f}+\frac{a (B c-5 d (A+B)) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac{a B \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}",1,"(a*(1 + Sin[e + f*x])*(-60*(2*A*(4*c^3 + 12*c^2*d + 9*c*d^2 + 3*d^3) + B*(8*c^3 + 18*c^2*d + 18*c*d^2 + 5*d^3))*Cos[e + f*x] + 10*d*(4*A*d*(3*c + d) + B*(12*c^2 + 12*c*d + 5*d^2))*Cos[3*(e + f*x)] - 6*B*d^3*Cos[5*(e + f*x)] + 15*(4*(B*(4*c^3 + 12*c^2*d + 9*c*d^2 + 3*d^3) + A*(8*c^3 + 12*c^2*d + 12*c*d^2 + 3*d^3))*f*x - 8*(B*(c + d)^3 + A*d*(3*c^2 + 3*c*d + d^2))*Sin[2*(e + f*x)] + d^2*(A*d + B*(3*c + d))*Sin[4*(e + f*x)])))/(480*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)","A",1
245,1,185,213,1.1227125,"\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]","\frac{a (\sin (e+f x)+1) \left(3 \left(4 f x \left(4 A \left(2 c^2+2 c d+d^2\right)+B \left(4 c^2+8 c d+3 d^2\right)\right)-8 \left(A d (2 c+d)+B (c+d)^2\right) \sin (2 (e+f x))+B d^2 \sin (4 (e+f x))\right)-24 \left(A \left(4 c^2+8 c d+3 d^2\right)+B \left(4 c^2+6 c d+3 d^2\right)\right) \cos (e+f x)+8 d (A d+B (2 c+d)) \cos (3 (e+f x))\right)}{96 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","\frac{1}{8} a x \left(4 A \left(2 c^2+2 c d+d^2\right)+B \left(4 c^2+8 c d+3 d^2\right)\right)-\frac{a \left(4 A d \left(c^2+3 c d+d^2\right)-B \left(c^3-4 c^2 d-8 c d^2-4 d^3\right)\right) \cos (e+f x)}{6 d f}-\frac{a \left(3 d^2 (4 A+3 B)-2 c (B c-4 d (A+B))\right) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{a (B c-4 d (A+B)) \cos (e+f x) (c+d \sin (e+f x))^2}{12 d f}-\frac{a B \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}",1,"(a*(1 + Sin[e + f*x])*(-24*(B*(4*c^2 + 6*c*d + 3*d^2) + A*(4*c^2 + 8*c*d + 3*d^2))*Cos[e + f*x] + 8*d*(A*d + B*(2*c + d))*Cos[3*(e + f*x)] + 3*(4*(4*A*(2*c^2 + 2*c*d + d^2) + B*(4*c^2 + 8*c*d + 3*d^2))*f*x - 8*(B*(c + d)^2 + A*d*(2*c + d))*Sin[2*(e + f*x)] + B*d^2*Sin[4*(e + f*x)])))/(96*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)","A",1
246,1,104,111,0.4386217,"\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]","\frac{a (-3 (4 A (c+d)+B (4 c+3 d)) \cos (e+f x)+12 A c f x-3 A d \sin (2 (e+f x))+6 A d f x-3 B c \sin (2 (e+f x))+6 B c f x-3 B d \sin (2 (e+f x))+B d \cos (3 (e+f x))+6 B d f x)}{12 f}","-\frac{a (3 A (c+d)+B (3 c+d)) \cos (e+f x)}{3 f}-\frac{a (3 A d+3 B c-B d) \sin (e+f x) \cos (e+f x)}{6 f}+\frac{1}{2} a x (A (2 c+d)+B (c+d))-\frac{B d \cos (e+f x) (a \sin (e+f x)+a)^2}{3 a f}",1,"(a*(12*A*c*f*x + 6*B*c*f*x + 6*A*d*f*x + 6*B*d*f*x - 3*(4*A*(c + d) + B*(4*c + 3*d))*Cos[e + f*x] + B*d*Cos[3*(e + f*x)] - 3*B*c*Sin[2*(e + f*x)] - 3*A*d*Sin[2*(e + f*x)] - 3*B*d*Sin[2*(e + f*x)]))/(12*f)","A",1
247,1,45,48,0.0984129,"\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]),x]","\frac{a (-4 (A+B) \cos (e+f x)+4 A f x-B \sin (2 (e+f x))+2 B e+2 B f x)}{4 f}","-\frac{a (A+B) \cos (e+f x)}{f}+\frac{1}{2} a x (2 A+B)-\frac{a B \sin (e+f x) \cos (e+f x)}{2 f}",1,"(a*(2*B*e + 4*A*f*x + 2*B*f*x - 4*(A + B)*Cos[e + f*x] - B*Sin[2*(e + f*x)]))/(4*f)","A",1
248,1,196,98,0.6534233,"\int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{c+d \sin (e+f x)} \, dx","Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x]),x]","\frac{a (\sin (e+f x)+1) \left(\frac{2 (c-d) (\cos (e)-i \sin (e)) (B c-A d) \tan ^{-1}\left(\frac{(\cos (e)-i \sin (e)) \sec \left(\frac{f x}{2}\right) \left(c \sin \left(\frac{f x}{2}\right)+d \cos \left(e+\frac{f x}{2}\right)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{f \sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}+A d x+B x (d-c)+\frac{B d \sin (e) \sin (f x)}{f}-\frac{B d \cos (e) \cos (f x)}{f}\right)}{d^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","\frac{2 a (c-d) (B c-A d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^2 f \sqrt{c^2-d^2}}-\frac{a x (B c-d (A+B))}{d^2}-\frac{a B \cos (e+f x)}{d f}",1,"(a*(A*d*x + B*(-c + d)*x - (B*d*Cos[e]*Cos[f*x])/f + (2*(c - d)*(B*c - A*d)*ArcTan[(Sec[(f*x)/2]*(Cos[e] - I*Sin[e])*(d*Cos[e + (f*x)/2] + c*Sin[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*(Cos[e] - I*Sin[e]))/(Sqrt[c^2 - d^2]*f*Sqrt[(Cos[e] - I*Sin[e])^2]) + (B*d*Sin[e]*Sin[f*x])/f)*(1 + Sin[e + f*x]))/(d^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)","C",1
249,1,217,124,1.2707277,"\int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx","Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2,x]","\frac{a (\sin (e+f x)+1) \left(\frac{2 (\cos (e)-i \sin (e)) \left(A d^2-B \left(c^2+c d-d^2\right)\right) \tan ^{-1}\left(\frac{(\cos (e)-i \sin (e)) \sec \left(\frac{f x}{2}\right) \left(c \sin \left(\frac{f x}{2}\right)+d \cos \left(e+\frac{f x}{2}\right)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{f (c+d) \sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}+\frac{\csc (e) (A d-B c) (c \cos (e)+d \sin (f x))}{f (c+d) (c+d \sin (e+f x))}+B x\right)}{d^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","\frac{2 a \left(d^2 (A+B) (c-d)-B c \left(c^2-d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^2 f \left(c^2-d^2\right)^{3/2}}+\frac{a (B c-A d) \cos (e+f x)}{d f (c+d) (c+d \sin (e+f x))}+\frac{a B x}{d^2}",1,"(a*(1 + Sin[e + f*x])*(B*x + (2*(A*d^2 - B*(c^2 + c*d - d^2))*ArcTan[(Sec[(f*x)/2]*(Cos[e] - I*Sin[e])*(d*Cos[e + (f*x)/2] + c*Sin[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*(Cos[e] - I*Sin[e]))/((c + d)*Sqrt[c^2 - d^2]*f*Sqrt[(Cos[e] - I*Sin[e])^2]) + ((-(B*c) + A*d)*Csc[e]*(c*Cos[e] + d*Sin[f*x]))/((c + d)*f*(c + d*Sin[e + f*x]))))/(d^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)","C",1
250,1,345,176,2.7112883,"\int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c+d \sin (e+f x))^3} \, dx","Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3,x]","\frac{a (\sin (e+f x)+1) \left(\frac{d \csc (e) \left(\left(A d^2 (d-2 c)+B c \left(2 c^2+2 c d-3 d^2\right)\right) \sin (2 e+f x)-d \left(A d (c-2 d)+B \left(c^2+2 c d-2 d^2\right)\right) \cos (e+2 f x)+\sin (f x) \left(B c \left(2 c^2+6 c d-5 d^2\right)-A d \left(-4 c^2+6 c d+d^2\right)\right)\right)+\left(2 c^2+d^2\right) \cot (e) \left(A d (c-2 d)+B \left(c^2+2 c d-2 d^2\right)\right)}{d^2 (c+d \sin (e+f x))^2}+\frac{4 (\cos (e)-i \sin (e)) (2 A c-A d+B c-2 B d) \tan ^{-1}\left(\frac{(\cos (e)-i \sin (e)) \sec \left(\frac{f x}{2}\right) \left(c \sin \left(\frac{f x}{2}\right)+d \cos \left(e+\frac{f x}{2}\right)\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right)}{4 f (c-d) (c+d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","\frac{a (2 A c-A d+B c-2 B d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c+d) \left(c^2-d^2\right)^{3/2}}-\frac{a \left(A d (c-2 d)+B \left(c^2+2 c d-2 d^2\right)\right) \cos (e+f x)}{2 d f (c-d) (c+d)^2 (c+d \sin (e+f x))}+\frac{a (B c-A d) \cos (e+f x)}{2 d f (c+d) (c+d \sin (e+f x))^2}",1,"(a*(1 + Sin[e + f*x])*((4*(2*A*c + B*c - A*d - 2*B*d)*ArcTan[(Sec[(f*x)/2]*(Cos[e] - I*Sin[e])*(d*Cos[e + (f*x)/2] + c*Sin[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*(Cos[e] - I*Sin[e]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + ((2*c^2 + d^2)*(A*(c - 2*d)*d + B*(c^2 + 2*c*d - 2*d^2))*Cot[e] + d*Csc[e]*(-(d*(A*(c - 2*d)*d + B*(c^2 + 2*c*d - 2*d^2))*Cos[e + 2*f*x]) + (B*c*(2*c^2 + 6*c*d - 5*d^2) - A*d*(-4*c^2 + 6*c*d + d^2))*Sin[f*x] + (A*d^2*(-2*c + d) + B*c*(2*c^2 + 2*c*d - 3*d^2))*Sin[2*e + f*x]))/(d^2*(c + d*Sin[e + f*x])^2)))/(4*(c - d)*(c + d)^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)","C",1
251,1,437,464,3.1277214,"\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3,x]","-\frac{a^2 \cos (e+f x) \left(60 \left(6 A \left(4 c^3+8 c^2 d+7 c d^2+2 d^3\right)+B \left(16 c^3+42 c^2 d+36 c d^2+11 d^3\right)\right) \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\sqrt{\cos ^2(e+f x)} \left(-16 \left(3 A d \left(5 c^2+10 c d+4 d^2\right)+B \left(5 c^3+30 c^2 d+36 c d^2+14 d^3\right)\right) \cos (2 (e+f x))+12 d^2 (A d+3 B c+2 B d) \cos (4 (e+f x))+240 A c^3 \sin (e+f x)+960 A c^3+1440 A c^2 d \sin (e+f x)+2640 A c^2 d+1530 A c d^2 \sin (e+f x)-90 A c d^2 \sin (3 (e+f x))+2400 A c d^2+540 A d^3 \sin (e+f x)-60 A d^3 \sin (3 (e+f x))+756 A d^3+480 B c^3 \sin (e+f x)+880 B c^3+1530 B c^2 d \sin (e+f x)-90 B c^2 d \sin (3 (e+f x))+2400 B c^2 d+1620 B c d^2 \sin (e+f x)-180 B c d^2 \sin (3 (e+f x))+2268 B c d^2+545 B d^3 \sin (e+f x)-80 B d^3 \sin (3 (e+f x))+5 B d^3 \sin (5 (e+f x))+712 B d^3\right)\right)}{480 f \sqrt{\cos ^2(e+f x)}}","\frac{a^2 \left(6 A d (c-10 d)-B \left(2 c^2-12 c d+55 d^2\right)\right) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac{a^2 \left(6 A d \left(c^2-10 c d-12 d^2\right)-B \left(2 c^3-12 c^2 d+51 c d^2+64 d^3\right)\right) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}+\frac{1}{16} a^2 x \left(6 A \left(4 c^3+8 c^2 d+7 c d^2+2 d^3\right)+B \left(16 c^3+42 c^2 d+36 c d^2+11 d^3\right)\right)+\frac{a^2 \left(6 A d \left(2 c^3-20 c^2 d-57 c d^2-30 d^3\right)-B \left(4 c^4-24 c^3 d+96 c^2 d^2+284 c d^3+165 d^4\right)\right) \sin (e+f x) \cos (e+f x)}{240 d f}+\frac{a^2 \left(6 A d \left(c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right)-B \left(2 c^5-12 c^4 d+47 c^3 d^2+208 c^2 d^3+216 c d^4+64 d^5\right)\right) \cos (e+f x)}{60 d^2 f}+\frac{a^2 (-6 A d+2 B c-7 B d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac{B \cos (e+f x) \left(a^2 \sin (e+f x)+a^2\right) (c+d \sin (e+f x))^4}{6 d f}",1,"-1/480*(a^2*Cos[e + f*x]*(60*(6*A*(4*c^3 + 8*c^2*d + 7*c*d^2 + 2*d^3) + B*(16*c^3 + 42*c^2*d + 36*c*d^2 + 11*d^3))*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(960*A*c^3 + 880*B*c^3 + 2640*A*c^2*d + 2400*B*c^2*d + 2400*A*c*d^2 + 2268*B*c*d^2 + 756*A*d^3 + 712*B*d^3 - 16*(3*A*d*(5*c^2 + 10*c*d + 4*d^2) + B*(5*c^3 + 30*c^2*d + 36*c*d^2 + 14*d^3))*Cos[2*(e + f*x)] + 12*d^2*(3*B*c + A*d + 2*B*d)*Cos[4*(e + f*x)] + 240*A*c^3*Sin[e + f*x] + 480*B*c^3*Sin[e + f*x] + 1440*A*c^2*d*Sin[e + f*x] + 1530*B*c^2*d*Sin[e + f*x] + 1530*A*c*d^2*Sin[e + f*x] + 1620*B*c*d^2*Sin[e + f*x] + 540*A*d^3*Sin[e + f*x] + 545*B*d^3*Sin[e + f*x] - 90*B*c^2*d*Sin[3*(e + f*x)] - 90*A*c*d^2*Sin[3*(e + f*x)] - 180*B*c*d^2*Sin[3*(e + f*x)] - 60*A*d^3*Sin[3*(e + f*x)] - 80*B*d^3*Sin[3*(e + f*x)] + 5*B*d^3*Sin[5*(e + f*x)])))/(f*Sqrt[Cos[e + f*x]^2])","A",1
252,1,296,336,1.5673987,"\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]","-\frac{a^2 \cos (e+f x) \left(60 \left(A \left(12 c^2+16 c d+7 d^2\right)+2 B \left(4 c^2+7 c d+3 d^2\right)\right) \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\sqrt{\cos ^2(e+f x)} \left(-8 \left(10 A d (c+d)+B \left(5 c^2+20 c d+12 d^2\right)\right) \cos (2 (e+f x))+120 A c^2 \sin (e+f x)+480 A c^2+480 A c d \sin (e+f x)+880 A c d+255 A d^2 \sin (e+f x)-15 A d^2 \sin (3 (e+f x))+400 A d^2+240 B c^2 \sin (e+f x)+440 B c^2+510 B c d \sin (e+f x)-30 B c d \sin (3 (e+f x))+800 B c d+270 B d^2 \sin (e+f x)-30 B d^2 \sin (3 (e+f x))+6 B d^2 \cos (4 (e+f x))+378 B d^2\right)\right)}{240 f \sqrt{\cos ^2(e+f x)}}","\frac{a^2 \left(5 A d (c-8 d)-2 B \left(c^2-5 c d+18 d^2\right)\right) \cos (e+f x) (c+d \sin (e+f x))^2}{60 d^2 f}+\frac{1}{8} a^2 x \left(12 A c^2+16 A c d+7 A d^2+8 B c^2+14 B c d+6 B d^2\right)+\frac{a^2 \left(5 A d \left(2 c^2-16 c d-21 d^2\right)-B \left(4 c^3-20 c^2 d+66 c d^2+90 d^3\right)\right) \sin (e+f x) \cos (e+f x)}{120 d f}+\frac{a^2 \left(5 A d \left(c^3-8 c^2 d-20 c d^2-8 d^3\right)-2 B \left(c^4-5 c^3 d+16 c^2 d^2+40 c d^3+18 d^4\right)\right) \cos (e+f x)}{30 d^2 f}+\frac{a^2 (2 B (c-3 d)-5 A d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d^2 f}-\frac{B \cos (e+f x) \left(a^2 \sin (e+f x)+a^2\right) (c+d \sin (e+f x))^3}{5 d f}",1,"-1/240*(a^2*Cos[e + f*x]*(60*(2*B*(4*c^2 + 7*c*d + 3*d^2) + A*(12*c^2 + 16*c*d + 7*d^2))*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(480*A*c^2 + 440*B*c^2 + 880*A*c*d + 800*B*c*d + 400*A*d^2 + 378*B*d^2 - 8*(10*A*d*(c + d) + B*(5*c^2 + 20*c*d + 12*d^2))*Cos[2*(e + f*x)] + 6*B*d^2*Cos[4*(e + f*x)] + 120*A*c^2*Sin[e + f*x] + 240*B*c^2*Sin[e + f*x] + 480*A*c*d*Sin[e + f*x] + 510*B*c*d*Sin[e + f*x] + 255*A*d^2*Sin[e + f*x] + 270*B*d^2*Sin[e + f*x] - 30*B*c*d*Sin[3*(e + f*x)] - 15*A*d^2*Sin[3*(e + f*x)] - 30*B*d^2*Sin[3*(e + f*x)])))/(f*Sqrt[Cos[e + f*x]^2])","A",1
253,1,160,166,0.755302,"\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]","-\frac{a^2 \cos (e+f x) \left(6 (12 A c+8 A d+8 B c+7 B d) \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\sqrt{\cos ^2(e+f x)} \left(8 (A d+B (c+2 d)) \sin ^2(e+f x)+3 (4 A c+8 A d+8 B c+7 B d) \sin (e+f x)+8 (6 A c+5 A d+5 B c+4 B d)+6 B d \sin ^3(e+f x)\right)\right)}{24 f \sqrt{\cos ^2(e+f x)}}","-\frac{a^2 (12 A c+8 A d+8 B c+7 B d) \cos (e+f x)}{6 f}-\frac{a^2 (12 A c+8 A d+8 B c+7 B d) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} a^2 x (12 A c+8 A d+8 B c+7 B d)-\frac{(4 A d+4 B c-B d) \cos (e+f x) (a \sin (e+f x)+a)^2}{12 f}-\frac{B d \cos (e+f x) (a \sin (e+f x)+a)^3}{4 a f}",1,"-1/24*(a^2*Cos[e + f*x]*(6*(12*A*c + 8*B*c + 8*A*d + 7*B*d)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(8*(6*A*c + 5*B*c + 5*A*d + 4*B*d) + 3*(4*A*c + 8*B*c + 8*A*d + 7*B*d)*Sin[e + f*x] + 8*(A*d + B*(c + 2*d))*Sin[e + f*x]^2 + 6*B*d*Sin[e + f*x]^3)))/(f*Sqrt[Cos[e + f*x]^2])","A",1
254,1,106,94,0.324986,"\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]),x]","-\frac{a^2 \cos (e+f x) \left(6 (3 A+2 B) \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\sqrt{\cos ^2(e+f x)} \left(3 (A+2 B) \sin (e+f x)+2 (6 A+5 B)+2 B \sin ^2(e+f x)\right)\right)}{6 f \sqrt{\cos ^2(e+f x)}}","-\frac{2 a^2 (3 A+2 B) \cos (e+f x)}{3 f}-\frac{a^2 (3 A+2 B) \sin (e+f x) \cos (e+f x)}{6 f}+\frac{1}{2} a^2 x (3 A+2 B)-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^2}{3 f}",1,"-1/6*(a^2*Cos[e + f*x]*(6*(3*A + 2*B)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(2*(6*A + 5*B) + 3*(A + 2*B)*Sin[e + f*x] + 2*B*Sin[e + f*x]^2)))/(f*Sqrt[Cos[e + f*x]^2])","A",1
255,1,177,171,0.6228289,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{c+d \sin (e+f x)} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x]),x]","\frac{a^2 (\sin (e+f x)+1)^2 \left(2 (e+f x) \left(2 A d (2 d-c)+B \left(2 c^2-4 c d+3 d^2\right)\right)-\frac{8 (c-d)^2 (B c-A d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}-4 d (A d-B c+2 B d) \cos (e+f x)-B d^2 \sin (2 (e+f x))\right)}{4 d^3 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","-\frac{2 a^2 (c-d)^2 (B c-A d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f \sqrt{c^2-d^2}}-\frac{a^2 x \left(2 A d (c-2 d)-B \left(2 c^2-4 c d+3 d^2\right)\right)}{2 d^3}+\frac{a^2 (-2 A d+2 B c-3 B d) \cos (e+f x)}{2 d^2 f}-\frac{B \cos (e+f x) \left(a^2 \sin (e+f x)+a^2\right)}{2 d f}",1,"(a^2*(1 + Sin[e + f*x])^2*(2*(2*A*d*(-c + 2*d) + B*(2*c^2 - 4*c*d + 3*d^2))*(e + f*x) - (8*(c - d)^2*(B*c - A*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] - 4*d*(-(B*c) + A*d + 2*B*d)*Cos[e + f*x] - B*d^2*Sin[2*(e + f*x)]))/(4*d^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)","A",1
256,1,192,198,0.9957345,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2,x]","\frac{a^2 (\sin (e+f x)+1)^2 \left(\frac{2 (c-d) \left(B \left(2 c^2+2 c d-d^2\right)-A d (c+2 d)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{(c+d) \sqrt{c^2-d^2}}+(e+f x) (A d-2 B c+2 B d)-\frac{d (d-c) (A d-B c) \cos (e+f x)}{(c+d) (c+d \sin (e+f x))}-B d \cos (e+f x)\right)}{d^3 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","-\frac{2 a^2 (c-d) \left(A d (c+2 d)-B \left(2 c^2+2 c d-d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f (c+d) \sqrt{c^2-d^2}}-\frac{a^2 x (-A d+2 B c-2 B d)}{d^3}+\frac{a^2 (A d-B (2 c+d)) \cos (e+f x)}{d^2 f (c+d)}+\frac{(B c-A d) \cos (e+f x) \left(a^2 \sin (e+f x)+a^2\right)}{d f (c+d) (c+d \sin (e+f x))}",1,"(a^2*(1 + Sin[e + f*x])^2*((-2*B*c + A*d + 2*B*d)*(e + f*x) + (2*(c - d)*(-(A*d*(c + 2*d)) + B*(2*c^2 + 2*c*d - d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)*Sqrt[c^2 - d^2]) - B*d*Cos[e + f*x] - (d*(-c + d)*(-(B*c) + A*d)*Cos[e + f*x])/((c + d)*(c + d*Sin[e + f*x]))))/(d^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)","A",1
257,1,226,215,1.3661451,"\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^3} \, dx","Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3,x]","\frac{a^2 (\sin (e+f x)+1)^2 \left(-\frac{d \left(A d (c+4 d)+B \left(-3 c^2-4 c d+2 d^2\right)\right) \cos (e+f x)}{(c+d)^2 (c+d \sin (e+f x))}-\frac{2 \left(B \left(2 c^3+4 c^2 d+c d^2-4 d^3\right)-3 A d^3\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{(c+d)^2 \sqrt{c^2-d^2}}-\frac{d (d-c) (A d-B c) \cos (e+f x)}{(c+d) (c+d \sin (e+f x))^2}+2 B (e+f x)\right)}{2 d^3 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}","-\frac{a^2 \left(3 A d^2-B \left(2 c^2+3 c d-2 d^2\right)\right) \cos (e+f x)}{2 d^2 f (c+d)^2 (c+d \sin (e+f x))}+\frac{a^2 \left(3 A d^3-B \left(2 c^3+4 c^2 d+c d^2-4 d^3\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f (c+d)^2 \sqrt{c^2-d^2}}+\frac{(B c-A d) \cos (e+f x) \left(a^2 \sin (e+f x)+a^2\right)}{2 d f (c+d) (c+d \sin (e+f x))^2}+\frac{a^2 B x}{d^3}",1,"(a^2*(1 + Sin[e + f*x])^2*(2*B*(e + f*x) - (2*(-3*A*d^3 + B*(2*c^3 + 4*c^2*d + c*d^2 - 4*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)^2*Sqrt[c^2 - d^2]) - (d*(-c + d)*(-(B*c) + A*d)*Cos[e + f*x])/((c + d)*(c + d*Sin[e + f*x])^2) - (d*(A*d*(c + 4*d) + B*(-3*c^2 - 4*c*d + 2*d^2))*Cos[e + f*x])/((c + d)^2*(c + d*Sin[e + f*x]))))/(2*d^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)","A",1
258,1,528,604,4.7626786,"\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3,x]","-\frac{a^3 \cos (e+f x) \left(420 \left(A \left(40 c^3+90 c^2 d+78 c d^2+23 d^3\right)+3 B \left(10 c^3+26 c^2 d+23 c d^2+7 d^3\right)\right) \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\sqrt{\cos ^2(e+f x)} \left(18 d \left(14 A d (c+d)+B \left(14 c^2+42 c d+23 d^2\right)\right) \cos (4 (e+f x))-\left(112 A \left(5 c^3+45 c^2 d+66 c d^2+26 d^3\right)+3 B \left(560 c^3+2464 c^2 d+2912 c d^2+1083 d^3\right)\right) \cos (2 (e+f x))+5040 A c^3 \sin (e+f x)+12880 A c^3+20790 A c^2 d \sin (e+f x)-630 A c^2 d \sin (3 (e+f x))+35280 A c^2 d+22050 A c d^2 \sin (e+f x)-1890 A c d^2 \sin (3 (e+f x))+32676 A c d^2+7595 A d^3 \sin (e+f x)-980 A d^3 \sin (3 (e+f x))+35 A d^3 \sin (5 (e+f x))+10276 A d^3+6930 B c^3 \sin (e+f x)-210 B c^3 \sin (3 (e+f x))+11760 B c^3+22050 B c^2 d \sin (e+f x)-1890 B c^2 d \sin (3 (e+f x))+32676 B c^2 d+22785 B c d^2 \sin (e+f x)-2940 B c d^2 \sin (3 (e+f x))+105 B c d^2 \sin (5 (e+f x))+30828 B c d^2+7665 B d^3 \sin (e+f x)-1260 B d^3 \sin (3 (e+f x))+105 B d^3 \sin (5 (e+f x))-15 B d^3 \cos (6 (e+f x))+9762 B d^3\right)\right)}{3360 f \sqrt{\cos ^2(e+f x)}}","-\frac{a^3 \left(-14 A c d+91 A d^2+6 B c^2-27 B c d+87 B d^2\right) \cos (e+f x) (c+d \sin (e+f x))^4}{210 d^3 f}-\frac{a^3 \left(7 A d \left(2 c^2-18 c d+115 d^2\right)-B \left(6 c^3-42 c^2 d+177 c d^2-735 d^3\right)\right) \cos (e+f x) (c+d \sin (e+f x))^3}{840 d^3 f}+\frac{1}{16} a^3 x \left(A \left(40 c^3+90 c^2 d+78 c d^2+23 d^3\right)+3 B \left(10 c^3+26 c^2 d+23 c d^2+7 d^3\right)\right)-\frac{a^3 \left(7 A d \left(2 c^3-18 c^2 d+111 c d^2+136 d^3\right)-B \left(6 c^4-42 c^3 d+165 c^2 d^2-651 c d^3-864 d^4\right)\right) \cos (e+f x) (c+d \sin (e+f x))^2}{840 d^3 f}-\frac{a^3 \left(7 A d \left(4 c^4-36 c^3 d+216 c^2 d^2+626 c d^3+345 d^4\right)-3 B \left(4 c^5-28 c^4 d+104 c^3 d^2-392 c^2 d^3-1263 c d^4-735 d^5\right)\right) \sin (e+f x) \cos (e+f x)}{1680 d^2 f}-\frac{a^3 \left(7 A d \left(2 c^5-18 c^4 d+107 c^3 d^2+472 c^2 d^3+456 c d^4+136 d^5\right)-3 B \left(2 c^6-14 c^5 d+51 c^4 d^2-189 c^3 d^3-920 c^2 d^4-952 c d^5-288 d^6\right)\right) \cos (e+f x)}{420 d^3 f}+\frac{(3 B (c-3 d)-7 A d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^4}{42 d^2 f}-\frac{a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f}",1,"-1/3360*(a^3*Cos[e + f*x]*(420*(3*B*(10*c^3 + 26*c^2*d + 23*c*d^2 + 7*d^3) + A*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3))*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(12880*A*c^3 + 11760*B*c^3 + 35280*A*c^2*d + 32676*B*c^2*d + 32676*A*c*d^2 + 30828*B*c*d^2 + 10276*A*d^3 + 9762*B*d^3 - (112*A*(5*c^3 + 45*c^2*d + 66*c*d^2 + 26*d^3) + 3*B*(560*c^3 + 2464*c^2*d + 2912*c*d^2 + 1083*d^3))*Cos[2*(e + f*x)] + 18*d*(14*A*d*(c + d) + B*(14*c^2 + 42*c*d + 23*d^2))*Cos[4*(e + f*x)] - 15*B*d^3*Cos[6*(e + f*x)] + 5040*A*c^3*Sin[e + f*x] + 6930*B*c^3*Sin[e + f*x] + 20790*A*c^2*d*Sin[e + f*x] + 22050*B*c^2*d*Sin[e + f*x] + 22050*A*c*d^2*Sin[e + f*x] + 22785*B*c*d^2*Sin[e + f*x] + 7595*A*d^3*Sin[e + f*x] + 7665*B*d^3*Sin[e + f*x] - 210*B*c^3*Sin[3*(e + f*x)] - 630*A*c^2*d*Sin[3*(e + f*x)] - 1890*B*c^2*d*Sin[3*(e + f*x)] - 1890*A*c*d^2*Sin[3*(e + f*x)] - 2940*B*c*d^2*Sin[3*(e + f*x)] - 980*A*d^3*Sin[3*(e + f*x)] - 1260*B*d^3*Sin[3*(e + f*x)] + 105*B*c*d^2*Sin[5*(e + f*x)] + 35*A*d^3*Sin[5*(e + f*x)] + 105*B*d^3*Sin[5*(e + f*x)])))/(f*Sqrt[Cos[e + f*x]^2])","A",1
259,1,355,463,2.4249697,"\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]","-\frac{a^3 \cos (e+f x) \left(60 \left(A \left(40 c^2+60 c d+26 d^2\right)+B \left(30 c^2+52 c d+23 d^2\right)\right) \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\sqrt{\cos ^2(e+f x)} \left(-16 \left(A \left(5 c^2+30 c d+22 d^2\right)+B \left(15 c^2+44 c d+26 d^2\right)\right) \cos (2 (e+f x))+12 d (A d+2 B c+3 B d) \cos (4 (e+f x))+720 A c^2 \sin (e+f x)+1840 A c^2+1980 A c d \sin (e+f x)-60 A c d \sin (3 (e+f x))+3360 A c d+1050 A d^2 \sin (e+f x)-90 A d^2 \sin (3 (e+f x))+1556 A d^2+990 B c^2 \sin (e+f x)-30 B c^2 \sin (3 (e+f x))+1680 B c^2+2100 B c d \sin (e+f x)-180 B c d \sin (3 (e+f x))+3112 B c d+1085 B d^2 \sin (e+f x)-140 B d^2 \sin (3 (e+f x))+5 B d^2 \sin (5 (e+f x))+1468 B d^2\right)\right)}{480 f \sqrt{\cos ^2(e+f x)}}","\frac{1}{16} a^3 x \left(A \left(40 c^2+60 c d+26 d^2\right)+B \left(30 c^2+52 c d+23 d^2\right)\right)+\frac{a^3 \left(2 A d (2 c-11 d)-B \left(2 c^2-8 c d+21 d^2\right)\right) \cos (e+f x) (c+d \sin (e+f x))^3}{40 d^3 f}-\frac{a^3 \left(2 A d \left(2 c^2-15 c d+76 d^2\right)-B \left(2 c^3-12 c^2 d+41 c d^2-136 d^3\right)\right) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^3 f}-\frac{a^3 \left(2 A d \left(4 c^3-30 c^2 d+146 c d^2+195 d^3\right)-B \left(4 c^4-24 c^3 d+76 c^2 d^2-236 c d^3-345 d^4\right)\right) \sin (e+f x) \cos (e+f x)}{240 d^2 f}-\frac{a^3 \left(2 A d \left(2 c^4-15 c^3 d+72 c^2 d^2+180 c d^3+76 d^4\right)-B \left(2 c^5-12 c^4 d+37 c^3 d^2-112 c^2 d^3-304 c d^4-136 d^5\right)\right) \cos (e+f x)}{60 d^3 f}+\frac{(-6 A d+3 B c-8 B d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^3}{30 d^2 f}-\frac{a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^3}{6 d f}",1,"-1/480*(a^3*Cos[e + f*x]*(60*(B*(30*c^2 + 52*c*d + 23*d^2) + A*(40*c^2 + 60*c*d + 26*d^2))*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(1840*A*c^2 + 1680*B*c^2 + 3360*A*c*d + 3112*B*c*d + 1556*A*d^2 + 1468*B*d^2 - 16*(A*(5*c^2 + 30*c*d + 22*d^2) + B*(15*c^2 + 44*c*d + 26*d^2))*Cos[2*(e + f*x)] + 12*d*(2*B*c + A*d + 3*B*d)*Cos[4*(e + f*x)] + 720*A*c^2*Sin[e + f*x] + 990*B*c^2*Sin[e + f*x] + 1980*A*c*d*Sin[e + f*x] + 2100*B*c*d*Sin[e + f*x] + 1050*A*d^2*Sin[e + f*x] + 1085*B*d^2*Sin[e + f*x] - 30*B*c^2*Sin[3*(e + f*x)] - 60*A*c*d*Sin[3*(e + f*x)] - 180*B*c*d*Sin[3*(e + f*x)] - 90*A*d^2*Sin[3*(e + f*x)] - 140*B*d^2*Sin[3*(e + f*x)] + 5*B*d^2*Sin[5*(e + f*x)])))/(f*Sqrt[Cos[e + f*x]^2])","A",1
260,1,156,201,0.9128322,"\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]","\frac{\cos (e+f x) \left(-\frac{1}{4} a^4 (5 A d+5 B c-B d) (\sin (e+f x)+1)^3-\frac{a^4 (20 A c+15 A d+15 B c+13 B d) \left(30 \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\left(2 \sin ^2(e+f x)+9 \sin (e+f x)+22\right) \sqrt{\cos ^2(e+f x)}\right)}{24 \sqrt{\cos ^2(e+f x)}}-B d (a \sin (e+f x)+a)^4\right)}{5 a f}","\frac{a^3 (20 A c+15 A d+15 B c+13 B d) \cos ^3(e+f x)}{60 f}-\frac{a^3 (20 A c+15 A d+15 B c+13 B d) \cos (e+f x)}{5 f}-\frac{3 a^3 (20 A c+15 A d+15 B c+13 B d) \sin (e+f x) \cos (e+f x)}{40 f}+\frac{1}{8} a^3 x (20 A c+15 A d+15 B c+13 B d)-\frac{(5 A d+5 B c-B d) \cos (e+f x) (a \sin (e+f x)+a)^3}{20 f}-\frac{B d \cos (e+f x) (a \sin (e+f x)+a)^4}{5 a f}",1,"(Cos[e + f*x]*(-1/4*(a^4*(5*B*c + 5*A*d - B*d)*(1 + Sin[e + f*x])^3) - B*d*(a + a*Sin[e + f*x])^4 - (a^4*(20*A*c + 15*B*c + 15*A*d + 13*B*d)*(30*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(22 + 9*Sin[e + f*x] + 2*Sin[e + f*x]^2)))/(24*Sqrt[Cos[e + f*x]^2])))/(5*a*f)","A",1
261,1,120,127,0.4968649,"\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]),x]","-\frac{a^3 \cos (e+f x) \left(30 (4 A+3 B) \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\sqrt{\cos ^2(e+f x)} \left(8 (A+3 B) \sin ^2(e+f x)+9 (4 A+5 B) \sin (e+f x)+88 A+6 B \sin ^3(e+f x)+72 B\right)\right)}{24 f \sqrt{\cos ^2(e+f x)}}","-\frac{5 a^3 (4 A+3 B) \cos (e+f x)}{6 f}-\frac{5 a^3 (4 A+3 B) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{5}{8} a^3 x (4 A+3 B)-\frac{a (4 A+3 B) \cos (e+f x) (a \sin (e+f x)+a)^2}{12 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^3}{4 f}",1,"-1/24*(a^3*Cos[e + f*x]*(30*(4*A + 3*B)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(88*A + 72*B + 9*(4*A + 5*B)*Sin[e + f*x] + 8*(A + 3*B)*Sin[e + f*x]^2 + 6*B*Sin[e + f*x]^3)))/(f*Sqrt[Cos[e + f*x]^2])","A",1
262,1,233,246,0.9811942,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{c+d \sin (e+f x)} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x]),x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(-3 d \left(4 A d (3 d-c)+B \left(4 c^2-12 c d+15 d^2\right)\right) \cos (e+f x)+\frac{24 (c-d)^3 (B c-A d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}+6 (e+f x) \left(A d \left(2 c^2-6 c d+7 d^2\right)+B \left(-2 c^3+6 c^2 d-7 c d^2+5 d^3\right)\right)-3 d^2 (A d-B c+3 B d) \sin (2 (e+f x))+B d^3 \cos (3 (e+f x))\right)}{12 d^4 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{2 a^3 (c-d)^3 (B c-A d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^4 f \sqrt{c^2-d^2}}+\frac{a^3 \left(A d (2 c-5 d)-B \left(2 c^2-5 c d+5 d^2\right)\right) \cos (e+f x)}{2 d^3 f}+\frac{a^3 x \left(A d \left(2 c^2-6 c d+7 d^2\right)-B \left(2 c^3-6 c^2 d+7 c d^2-5 d^3\right)\right)}{2 d^4}+\frac{(-3 A d+3 B c-5 B d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{6 d^2 f}-\frac{a B \cos (e+f x) (a \sin (e+f x)+a)^2}{3 d f}",1,"(a^3*(1 + Sin[e + f*x])^3*(6*(A*d*(2*c^2 - 6*c*d + 7*d^2) + B*(-2*c^3 + 6*c^2*d - 7*c*d^2 + 5*d^3))*(e + f*x) + (24*(c - d)^3*(B*c - A*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] - 3*d*(4*A*d*(-c + 3*d) + B*(4*c^2 - 12*c*d + 15*d^2))*Cos[e + f*x] + B*d^3*Cos[3*(e + f*x)] - 3*d^2*(-(B*c) + A*d + 3*B*d)*Sin[2*(e + f*x)]))/(12*d^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)","A",1
263,1,244,283,1.4971281,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2,x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(2 (e+f x) \left(2 A d (3 d-2 c)+B \left(6 c^2-12 c d+7 d^2\right)\right)-\frac{8 (c-d)^2 \left(B \left(3 c^2+3 c d-d^2\right)-A d (2 c+3 d)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{(c+d) \sqrt{c^2-d^2}}-4 d (A d-2 B c+3 B d) \cos (e+f x)+\frac{4 d (c-d)^2 (B c-A d) \cos (e+f x)}{(c+d) (c+d \sin (e+f x))}-B d^2 \sin (2 (e+f x))\right)}{4 d^4 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{2 a^3 (c-d)^2 \left(A d (2 c+3 d)-B \left(3 c^2+3 c d-d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^4 f (c+d) \sqrt{c^2-d^2}}-\frac{a^3 x \left(2 A d (2 c-3 d)-B \left(6 c^2-12 c d+7 d^2\right)\right)}{2 d^4}-\frac{a^3 \left(4 A c d-B \left(6 c^2-3 c d-5 d^2\right)\right) \cos (e+f x)}{2 d^3 f (c+d)}+\frac{(2 A d-B (3 c+d)) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{2 d^2 f (c+d)}+\frac{a (B c-A d) \cos (e+f x) (a \sin (e+f x)+a)^2}{d f (c+d) (c+d \sin (e+f x))}",1,"(a^3*(1 + Sin[e + f*x])^3*(2*(2*A*d*(-2*c + 3*d) + B*(6*c^2 - 12*c*d + 7*d^2))*(e + f*x) - (8*(c - d)^2*(-(A*d*(2*c + 3*d)) + B*(3*c^2 + 3*c*d - d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)*Sqrt[c^2 - d^2]) - 4*d*(-2*B*c + A*d + 3*B*d)*Cos[e + f*x] + (4*(c - d)^2*d*(B*c - A*d)*Cos[e + f*x])/((c + d)*(c + d*Sin[e + f*x])) - B*d^2*Sin[2*(e + f*x)]))/(4*d^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)","A",1
264,1,830,305,3.1448089,"\int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^3} \, dx","Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3,x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(\frac{4 (c-d) \left(3 B \left(2 c^3+4 d c^2+d^2 c-2 d^3\right)-A d \left(2 c^2+6 d c+7 d^2\right)\right) \tan ^{-1}\left(\frac{d+c \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}+\frac{-12 B e c^5-12 B f x c^5+4 A d e c^4-12 B d e c^4+4 A d f x c^4-12 B d f x c^4-24 B d e \sin (e+f x) c^4-24 B d f x \sin (e+f x) c^4+8 A d^2 e c^3+6 B d^2 e c^3+8 A d^2 f x c^3+6 B d^2 f x c^3+8 A d^2 e \sin (e+f x) c^3-24 B d^2 e \sin (e+f x) c^3+8 A d^2 f x \sin (e+f x) c^3-24 B d^2 f x \sin (e+f x) c^3-9 B d^2 \sin (2 (e+f x)) c^3+6 A d^3 e c^2+6 B d^3 e c^2+6 A d^3 f x c^2+6 B d^3 f x c^2+B d^3 \cos (3 (e+f x)) c^2+16 A d^3 e \sin (e+f x) c^2+24 B d^3 e \sin (e+f x) c^2+16 A d^3 f x \sin (e+f x) c^2+24 B d^3 f x \sin (e+f x) c^2+3 A d^3 \sin (2 (e+f x)) c^2-9 B d^3 \sin (2 (e+f x)) c^2+4 A d^4 e c+6 B d^4 e c+4 A d^4 f x c+6 B d^4 f x c+2 B d^4 \cos (3 (e+f x)) c+8 A d^4 e \sin (e+f x) c+24 B d^4 e \sin (e+f x) c+8 A d^4 f x \sin (e+f x) c+24 B d^4 f x \sin (e+f x) c+3 A d^4 \sin (2 (e+f x)) c+4 B d^4 \sin (2 (e+f x)) c+2 A d^5 e+6 B d^5 e+2 A d^5 f x+6 B d^5 f x-d \left(2 A d \left(-2 c^3-4 d c^2+5 d^2 c+d^3\right)+B \left(12 c^4+12 d c^3-9 d^2 c^2+4 d^3 c+d^4\right)\right) \cos (e+f x)-2 d^2 (c+d)^2 (-3 B c+A d+3 B d) (e+f x) \cos (2 (e+f x))+B d^5 \cos (3 (e+f x))-6 A d^5 \sin (2 (e+f x))-2 B d^5 \sin (2 (e+f x))}{(c+d \sin (e+f x))^2}\right)}{4 d^4 (c+d)^2 f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}","-\frac{\left(A d (c+4 d)-B \left(3 c^2+4 c d-2 d^2\right)\right) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{2 d^2 f (c+d)^2 (c+d \sin (e+f x))}-\frac{a^3 (c-d) \left(A d \left(2 c^2+6 c d+7 d^2\right)-3 B \left(2 c^3+4 c^2 d+c d^2-2 d^3\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^4 f (c+d)^2 \sqrt{c^2-d^2}}-\frac{a^3 x (-A d+3 B c-3 B d)}{d^4}-\frac{a^3 (3 B c (2 c+3 d)-A d (2 c+5 d)) \cos (e+f x)}{2 d^3 f (c+d)^2}+\frac{a (B c-A d) \cos (e+f x) (a \sin (e+f x)+a)^2}{2 d f (c+d) (c+d \sin (e+f x))^2}",1,"(a^3*(1 + Sin[e + f*x])^3*((4*(c - d)*(-(A*d*(2*c^2 + 6*c*d + 7*d^2)) + 3*B*(2*c^3 + 4*c^2*d + c*d^2 - 2*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] + (-12*B*c^5*e + 4*A*c^4*d*e - 12*B*c^4*d*e + 8*A*c^3*d^2*e + 6*B*c^3*d^2*e + 6*A*c^2*d^3*e + 6*B*c^2*d^3*e + 4*A*c*d^4*e + 6*B*c*d^4*e + 2*A*d^5*e + 6*B*d^5*e - 12*B*c^5*f*x + 4*A*c^4*d*f*x - 12*B*c^4*d*f*x + 8*A*c^3*d^2*f*x + 6*B*c^3*d^2*f*x + 6*A*c^2*d^3*f*x + 6*B*c^2*d^3*f*x + 4*A*c*d^4*f*x + 6*B*c*d^4*f*x + 2*A*d^5*f*x + 6*B*d^5*f*x - d*(2*A*d*(-2*c^3 - 4*c^2*d + 5*c*d^2 + d^3) + B*(12*c^4 + 12*c^3*d - 9*c^2*d^2 + 4*c*d^3 + d^4))*Cos[e + f*x] - 2*d^2*(c + d)^2*(-3*B*c + A*d + 3*B*d)*(e + f*x)*Cos[2*(e + f*x)] + B*c^2*d^3*Cos[3*(e + f*x)] + 2*B*c*d^4*Cos[3*(e + f*x)] + B*d^5*Cos[3*(e + f*x)] - 24*B*c^4*d*e*Sin[e + f*x] + 8*A*c^3*d^2*e*Sin[e + f*x] - 24*B*c^3*d^2*e*Sin[e + f*x] + 16*A*c^2*d^3*e*Sin[e + f*x] + 24*B*c^2*d^3*e*Sin[e + f*x] + 8*A*c*d^4*e*Sin[e + f*x] + 24*B*c*d^4*e*Sin[e + f*x] - 24*B*c^4*d*f*x*Sin[e + f*x] + 8*A*c^3*d^2*f*x*Sin[e + f*x] - 24*B*c^3*d^2*f*x*Sin[e + f*x] + 16*A*c^2*d^3*f*x*Sin[e + f*x] + 24*B*c^2*d^3*f*x*Sin[e + f*x] + 8*A*c*d^4*f*x*Sin[e + f*x] + 24*B*c*d^4*f*x*Sin[e + f*x] - 9*B*c^3*d^2*Sin[2*(e + f*x)] + 3*A*c^2*d^3*Sin[2*(e + f*x)] - 9*B*c^2*d^3*Sin[2*(e + f*x)] + 3*A*c*d^4*Sin[2*(e + f*x)] + 4*B*c*d^4*Sin[2*(e + f*x)] - 6*A*d^5*Sin[2*(e + f*x)] - 2*B*d^5*Sin[2*(e + f*x)])/(c + d*Sin[e + f*x])^2))/(4*d^4*(c + d)^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)","B",1
265,1,788,220,1.2721109,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(a + a*Sin[e + f*x]),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(9 d \left(A d (d-4 c)+B \left(-4 c^2+3 c d-2 d^2\right)\right) \cos \left(\frac{3}{2} (e+f x)\right)+3 \cos \left(\frac{1}{2} (e+f x)\right) \left(4 A d \left(6 c^2 (e+f x)-3 c d (2 e+2 f x+1)+d^2 (3 e+3 f x+1)\right)+B \left(8 c^3 (e+f x)-12 c^2 d (2 e+2 f x+1)+12 c d^2 (3 e+3 f x+1)-d^3 (12 e+12 f x+7)\right)\right)+48 A c^3 \sin \left(\frac{1}{2} (e+f x)\right)-144 A c^2 d \sin \left(\frac{1}{2} (e+f x)\right)+72 A c^2 d e \sin \left(\frac{1}{2} (e+f x)\right)+72 A c^2 d f x \sin \left(\frac{1}{2} (e+f x)\right)+180 A c d^2 \sin \left(\frac{1}{2} (e+f x)\right)-72 A c d^2 e \sin \left(\frac{1}{2} (e+f x)\right)-72 A c d^2 f x \sin \left(\frac{1}{2} (e+f x)\right)-36 A c d^2 \sin \left(\frac{3}{2} (e+f x)\right)-60 A d^3 \sin \left(\frac{1}{2} (e+f x)\right)+36 A d^3 e \sin \left(\frac{1}{2} (e+f x)\right)+36 A d^3 f x \sin \left(\frac{1}{2} (e+f x)\right)+9 A d^3 \sin \left(\frac{3}{2} (e+f x)\right)-3 A d^3 \sin \left(\frac{5}{2} (e+f x)\right)+3 A d^3 \cos \left(\frac{5}{2} (e+f x)\right)-48 B c^3 \sin \left(\frac{1}{2} (e+f x)\right)+24 B c^3 e \sin \left(\frac{1}{2} (e+f x)\right)+24 B c^3 f x \sin \left(\frac{1}{2} (e+f x)\right)+180 B c^2 d \sin \left(\frac{1}{2} (e+f x)\right)-72 B c^2 d e \sin \left(\frac{1}{2} (e+f x)\right)-72 B c^2 d f x \sin \left(\frac{1}{2} (e+f x)\right)-36 B c^2 d \sin \left(\frac{3}{2} (e+f x)\right)-180 B c d^2 \sin \left(\frac{1}{2} (e+f x)\right)+108 B c d^2 e \sin \left(\frac{1}{2} (e+f x)\right)+108 B c d^2 f x \sin \left(\frac{1}{2} (e+f x)\right)+27 B c d^2 \sin \left(\frac{3}{2} (e+f x)\right)-9 B c d^2 \sin \left(\frac{5}{2} (e+f x)\right)+9 B c d^2 \cos \left(\frac{5}{2} (e+f x)\right)+69 B d^3 \sin \left(\frac{1}{2} (e+f x)\right)-36 B d^3 e \sin \left(\frac{1}{2} (e+f x)\right)-36 B d^3 f x \sin \left(\frac{1}{2} (e+f x)\right)-18 B d^3 \sin \left(\frac{3}{2} (e+f x)\right)+2 B d^3 \sin \left(\frac{5}{2} (e+f x)\right)+B d^3 \sin \left(\frac{7}{2} (e+f x)\right)-2 B d^3 \cos \left(\frac{5}{2} (e+f x)\right)+B d^3 \cos \left(\frac{7}{2} (e+f x)\right)\right)}{24 a f (\sin (e+f x)+1)}","\frac{2 d \left(3 A \left(c^2-3 c d+d^2\right)-B \left(7 c^2-9 c d+4 d^2\right)\right) \cos (e+f x)}{3 a f}+\frac{x \left(3 A d \left(2 c^2-2 c d+d^2\right)+B \left(2 c^3-6 c^2 d+9 c d^2-3 d^3\right)\right)}{2 a}+\frac{d^2 (6 A c-9 A d-11 B c+9 B d) \sin (e+f x) \cos (e+f x)}{6 a f}-\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}+\frac{d (3 A-4 B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(3*(4*A*d*(6*c^2*(e + f*x) - 3*c*d*(1 + 2*e + 2*f*x) + d^2*(1 + 3*e + 3*f*x)) + B*(8*c^3*(e + f*x) - 12*c^2*d*(1 + 2*e + 2*f*x) + 12*c*d^2*(1 + 3*e + 3*f*x) - d^3*(7 + 12*e + 12*f*x)))*Cos[(e + f*x)/2] + 9*d*(A*d*(-4*c + d) + B*(-4*c^2 + 3*c*d - 2*d^2))*Cos[(3*(e + f*x))/2] + 9*B*c*d^2*Cos[(5*(e + f*x))/2] + 3*A*d^3*Cos[(5*(e + f*x))/2] - 2*B*d^3*Cos[(5*(e + f*x))/2] + B*d^3*Cos[(7*(e + f*x))/2] + 48*A*c^3*Sin[(e + f*x)/2] - 48*B*c^3*Sin[(e + f*x)/2] - 144*A*c^2*d*Sin[(e + f*x)/2] + 180*B*c^2*d*Sin[(e + f*x)/2] + 180*A*c*d^2*Sin[(e + f*x)/2] - 180*B*c*d^2*Sin[(e + f*x)/2] - 60*A*d^3*Sin[(e + f*x)/2] + 69*B*d^3*Sin[(e + f*x)/2] + 24*B*c^3*e*Sin[(e + f*x)/2] + 72*A*c^2*d*e*Sin[(e + f*x)/2] - 72*B*c^2*d*e*Sin[(e + f*x)/2] - 72*A*c*d^2*e*Sin[(e + f*x)/2] + 108*B*c*d^2*e*Sin[(e + f*x)/2] + 36*A*d^3*e*Sin[(e + f*x)/2] - 36*B*d^3*e*Sin[(e + f*x)/2] + 24*B*c^3*f*x*Sin[(e + f*x)/2] + 72*A*c^2*d*f*x*Sin[(e + f*x)/2] - 72*B*c^2*d*f*x*Sin[(e + f*x)/2] - 72*A*c*d^2*f*x*Sin[(e + f*x)/2] + 108*B*c*d^2*f*x*Sin[(e + f*x)/2] + 36*A*d^3*f*x*Sin[(e + f*x)/2] - 36*B*d^3*f*x*Sin[(e + f*x)/2] - 36*B*c^2*d*Sin[(3*(e + f*x))/2] - 36*A*c*d^2*Sin[(3*(e + f*x))/2] + 27*B*c*d^2*Sin[(3*(e + f*x))/2] + 9*A*d^3*Sin[(3*(e + f*x))/2] - 18*B*d^3*Sin[(3*(e + f*x))/2] - 9*B*c*d^2*Sin[(5*(e + f*x))/2] - 3*A*d^3*Sin[(5*(e + f*x))/2] + 2*B*d^3*Sin[(5*(e + f*x))/2] + B*d^3*Sin[(7*(e + f*x))/2]))/(24*a*f*(1 + Sin[e + f*x]))","B",1
266,1,200,143,0.4648332,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/(a + a*Sin[e + f*x]),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(2 (e+f x) \left(2 A d (2 c-d)+B \left(2 c^2-4 c d+3 d^2\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+8 (A-B) (c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right)+4 d (B (d-2 c)-A d) \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-B d^2 \sin (2 (e+f x)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{4 a f (\sin (e+f x)+1)}","\frac{x \left(2 A d (2 c-d)+B \left(2 c^2-4 c d+3 d^2\right)\right)}{2 a}+\frac{2 d (A (c-d)-B (2 c-d)) \cos (e+f x)}{a f}-\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{f (a \sin (e+f x)+a)}+\frac{d^2 (2 A-3 B) \sin (e+f x) \cos (e+f x)}{2 a f}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(8*(A - B)*(c - d)^2*Sin[(e + f*x)/2] + 2*(2*A*(2*c - d)*d + B*(2*c^2 - 4*c*d + 3*d^2))*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 4*d*(-(A*d) + B*(-2*c + d))*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - B*d^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[2*(e + f*x)]))/(4*a*f*(1 + Sin[e + f*x]))","A",1
267,1,126,67,0.4688739,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))}{a+a \sin (e+f x)} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]))/(a + a*Sin[e + f*x]),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right) ((e+f x) (A d+B (c-d))-B d \cos (e+f x))+\sin \left(\frac{1}{2} (e+f x)\right) (2 A c+A d (e+f x-2)+B (c-d) (e+f x-2)-B d \cos (e+f x))\right)}{a f (\sin (e+f x)+1)}","-\frac{(A-B) (c-d) \cos (e+f x)}{a f (\sin (e+f x)+1)}+\frac{x (A d+B (c-d))}{a}-\frac{B d \cos (e+f x)}{a f}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(Cos[(e + f*x)/2]*((B*(c - d) + A*d)*(e + f*x) - B*d*Cos[e + f*x]) + (2*A*c + B*(c - d)*(-2 + e + f*x) + A*d*(-2 + e + f*x) - B*d*Cos[e + f*x])*Sin[(e + f*x)/2]))/(a*f*(1 + Sin[e + f*x]))","A",1
268,1,79,35,0.1593669,"\int \frac{A+B \sin (e+f x)}{a+a \sin (e+f x)} \, dx","Integrate[(A + B*Sin[e + f*x])/(a + a*Sin[e + f*x]),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right) (2 A+B (e+f x-2))+B (e+f x) \cos \left(\frac{1}{2} (e+f x)\right)\right)}{a f (\sin (e+f x)+1)}","\frac{B x}{a}-\frac{(A-B) \cos (e+f x)}{f (a \sin (e+f x)+a)}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(B*(e + f*x)*Cos[(e + f*x)/2] + (2*A + B*(-2 + e + f*x))*Sin[(e + f*x)/2]))/(a*f*(1 + Sin[e + f*x]))","B",1
269,1,148,101,0.3269695,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])),x]","\frac{2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left((A-B) \sqrt{c^2-d^2} \sin \left(\frac{1}{2} (e+f x)\right)+(B c-A d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)\right)}{a f (c-d) \sqrt{c^2-d^2} (\sin (e+f x)+1)}","\frac{2 (B c-A d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a f (c-d) \sqrt{c^2-d^2}}-\frac{(A-B) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a)}",1,"(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((A - B)*Sqrt[c^2 - d^2]*Sin[(e + f*x)/2] + (B*c - A*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))/(a*(c - d)*Sqrt[c^2 - d^2]*f*(1 + Sin[e + f*x]))","A",1
270,1,209,181,1.2222897,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^2} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\frac{2 \left(B \left(c^2+c d+d^2\right)-A d (2 c+d)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{(c+d) \sqrt{c^2-d^2}}+\frac{d (B c-A d) \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{(c+d) (c+d \sin (e+f x))}+2 (A-B) \sin \left(\frac{1}{2} (e+f x)\right)\right)}{a f (c-d)^2 (\sin (e+f x)+1)}","-\frac{2 \left(A d (2 c+d)-B \left(c^2+c d+d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a f (c-d) \left(c^2-d^2\right)^{3/2}}+\frac{d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{a f (c-d)^2 (c+d) (c+d \sin (e+f x))}-\frac{(A-B) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*(A - B)*Sin[(e + f*x)/2] + (2*(-(A*d*(2*c + d)) + B*(c^2 + c*d + d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/((c + d)*Sqrt[c^2 - d^2]) + (d*(B*c - A*d)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/((c + d)*(c + d*Sin[e + f*x]))))/(a*(c - d)^2*f*(1 + Sin[e + f*x]))","A",1
271,1,313,283,1.6062165,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^3),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\frac{d \left(B \left(3 c^2+2 c d+2 d^2\right)-A d (5 c+2 d)\right) \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{(c+d)^2 (c+d \sin (e+f x))}+\frac{2 \left(B \left(2 c^3+4 c^2 d+7 c d^2+2 d^3\right)-3 A d \left(2 c^2+2 c d+d^2\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{(c+d)^2 \sqrt{c^2-d^2}}+\frac{d (c-d) (B c-A d) \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{(c+d) (c+d \sin (e+f x))^2}+4 (A-B) \sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 a f (c-d)^3 (\sin (e+f x)+1)}","-\frac{d \left(2 A c^2+9 A c d+4 A d^2-5 B c^2-6 B c d-4 B d^2\right) \cos (e+f x)}{2 a f (c-d)^3 (c+d)^2 (c+d \sin (e+f x))}-\frac{\left(3 A d \left(2 c^2+2 c d+d^2\right)-B \left(2 c^3+4 c^2 d+7 c d^2+2 d^3\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a f (c-d) \left(c^2-d^2\right)^{5/2}}-\frac{d (2 A c+3 A d-3 B c-2 B d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(4*(A - B)*Sin[(e + f*x)/2] + (2*(-3*A*d*(2*c^2 + 2*c*d + d^2) + B*(2*c^3 + 4*c^2*d + 7*c*d^2 + 2*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/((c + d)^2*Sqrt[c^2 - d^2]) + ((c - d)*d*(B*c - A*d)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/((c + d)*(c + d*Sin[e + f*x])^2) + (d*(-(A*d*(5*c + 2*d)) + B*(3*c^2 + 2*c*d + 2*d^2))*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/((c + d)^2*(c + d*Sin[e + f*x]))))/(2*a*(c - d)^3*f*(1 + Sin[e + f*x]))","A",1
272,1,547,228,3.571263,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(a + a*Sin[e + f*x])^2,x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(3 \cos \left(\frac{1}{2} (e+f x)\right) \left(8 A d \left(6 c^2+3 c d (3 e+3 f x-4)+d^2 (-6 e-6 f x+5)\right)+B \left(16 c^3+24 c^2 d (3 e+3 f x-4)-24 c d^2 (6 e+6 f x-5)+7 d^3 (12 e+12 f x-7)\right)\right)-\cos \left(\frac{3}{2} (e+f x)\right) \left(4 A \left(4 c^3+24 c^2 d+6 c d^2 (3 e+3 f x-10)+d^3 (-12 e-12 f x+41)\right)+B \left(32 c^3+24 c^2 d (3 e+3 f x-10)-12 c d^2 (12 e+12 f x-41)+d^3 (84 e+84 f x-239)\right)\right)+3 \left(2 \sin \left(\frac{1}{2} (e+f x)\right) \left(d \cos (e+f x) \left(8 A d (3 c (e+f x)-2 d (e+f x+1))+B \left(24 c^2 (e+f x)-48 c d (e+f x+1)+d^2 (28 e+28 f x+27)\right)\right)+2 d^2 (-2 A d-6 B c+3 B d) \cos (2 (e+f x))+8 A c^3+24 A c^2 d+48 A c d^2 e+48 A c d^2 f x-72 A c d^2-32 A d^3 e-32 A d^3 f x+36 A d^3+8 B c^3+48 B c^2 d e+48 B c^2 d f x-72 B c^2 d-96 B c d^2 e-96 B c d^2 f x+108 B c d^2+B d^3 \cos (3 (e+f x))+56 B d^3 e+56 B d^3 f x-50 B d^3\right)+d^2 (4 A d+12 B c-5 B d) \cos \left(\frac{5}{2} (e+f x)\right)+B d^3 \cos \left(\frac{7}{2} (e+f x)\right)\right)\right)}{48 a^2 f (\sin (e+f x)+1)^2}","\frac{2 d \left(A \left(c^2+6 c d-5 d^2\right)+B \left(2 c^2-15 c d+8 d^2\right)\right) \cos (e+f x)}{3 a^2 f}+\frac{d x \left(2 A d (3 c-2 d)+B \left(6 c^2-12 c d+7 d^2\right)\right)}{2 a^2}+\frac{d^2 (2 A (c+6 d)+B (4 c-21 d)) \sin (e+f x) \cos (e+f x)}{6 a^2 f}-\frac{(A (c+5 d)+2 B (c-4 d)) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f (\sin (e+f x)+1)}-\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(3*(8*A*d*(6*c^2 + d^2*(5 - 6*e - 6*f*x) + 3*c*d*(-4 + 3*e + 3*f*x)) + B*(16*c^3 + 24*c^2*d*(-4 + 3*e + 3*f*x) - 24*c*d^2*(-5 + 6*e + 6*f*x) + 7*d^3*(-7 + 12*e + 12*f*x)))*Cos[(e + f*x)/2] - (4*A*(4*c^3 + 24*c^2*d + d^3*(41 - 12*e - 12*f*x) + 6*c*d^2*(-10 + 3*e + 3*f*x)) + B*(32*c^3 + 24*c^2*d*(-10 + 3*e + 3*f*x) - 12*c*d^2*(-41 + 12*e + 12*f*x) + d^3*(-239 + 84*e + 84*f*x)))*Cos[(3*(e + f*x))/2] + 3*(d^2*(12*B*c + 4*A*d - 5*B*d)*Cos[(5*(e + f*x))/2] + B*d^3*Cos[(7*(e + f*x))/2] + 2*(8*A*c^3 + 8*B*c^3 + 24*A*c^2*d - 72*B*c^2*d - 72*A*c*d^2 + 108*B*c*d^2 + 36*A*d^3 - 50*B*d^3 + 48*B*c^2*d*e + 48*A*c*d^2*e - 96*B*c*d^2*e - 32*A*d^3*e + 56*B*d^3*e + 48*B*c^2*d*f*x + 48*A*c*d^2*f*x - 96*B*c*d^2*f*x - 32*A*d^3*f*x + 56*B*d^3*f*x + d*(8*A*d*(3*c*(e + f*x) - 2*d*(1 + e + f*x)) + B*(24*c^2*(e + f*x) - 48*c*d*(1 + e + f*x) + d^2*(27 + 28*e + 28*f*x)))*Cos[e + f*x] + 2*d^2*(-6*B*c - 2*A*d + 3*B*d)*Cos[2*(e + f*x)] + B*d^3*Cos[3*(e + f*x)])*Sin[(e + f*x)/2])))/(48*a^2*f*(1 + Sin[e + f*x])^2)","B",1
273,1,338,132,1.6745967,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^2,x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(6 \cos \left(\frac{1}{2} (e+f x)\right) \left(A d (4 c+d (3 e+3 f x-4))+B \left(2 c^2+2 c d (3 e+3 f x-4)+d^2 (-6 e-6 f x+5)\right)\right)-\cos \left(\frac{3}{2} (e+f x)\right) \left(2 A \left(2 c^2+8 c d+d^2 (3 e+3 f x-10)\right)+B \left(8 c^2+4 c d (3 e+3 f x-10)+d^2 (-12 e-12 f x+41)\right)\right)+6 \sin \left(\frac{1}{2} (e+f x)\right) \left(-2 d \cos (e+f x) (-A d (e+f x)-2 B c (e+f x)+2 B d (e+f x+1))+2 A c^2+4 A c d+4 A d^2 e+4 A d^2 f x-6 A d^2+2 B c^2+8 B c d e+8 B c d f x-12 B c d-B d^2 \cos (2 (e+f x))-8 B d^2 e-8 B d^2 f x+9 B d^2\right)+3 B d^2 \cos \left(\frac{5}{2} (e+f x)\right)\right)}{12 a^2 f (\sin (e+f x)+1)^2}","-\frac{(c-d) (A (c+3 d)+2 B (c-3 d)) \cos (e+f x)}{3 a^2 f (\sin (e+f x)+1)}+\frac{d x (A d+2 B (c-d))}{a^2}+\frac{d^2 (A-4 B) \cos (e+f x)}{3 a^2 f}-\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(6*(A*d*(4*c + d*(-4 + 3*e + 3*f*x)) + B*(2*c^2 + d^2*(5 - 6*e - 6*f*x) + 2*c*d*(-4 + 3*e + 3*f*x)))*Cos[(e + f*x)/2] - (B*(8*c^2 + d^2*(41 - 12*e - 12*f*x) + 4*c*d*(-10 + 3*e + 3*f*x)) + 2*A*(2*c^2 + 8*c*d + d^2*(-10 + 3*e + 3*f*x)))*Cos[(3*(e + f*x))/2] + 3*B*d^2*Cos[(5*(e + f*x))/2] + 6*(2*A*c^2 + 2*B*c^2 + 4*A*c*d - 12*B*c*d - 6*A*d^2 + 9*B*d^2 + 8*B*c*d*e + 4*A*d^2*e - 8*B*d^2*e + 8*B*c*d*f*x + 4*A*d^2*f*x - 8*B*d^2*f*x - 2*d*(-2*B*c*(e + f*x) - A*d*(e + f*x) + 2*B*d*(1 + e + f*x))*Cos[e + f*x] - B*d^2*Cos[2*(e + f*x)])*Sin[(e + f*x)/2]))/(12*a^2*f*(1 + Sin[e + f*x])^2)","B",1
274,1,180,85,0.3394728,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^2} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]))/(a + a*Sin[e + f*x])^2,x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(2 (A-B) (c-d) \sin \left(\frac{1}{2} (e+f x)\right)+2 (A c+2 A d+2 B c-5 B d) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-(A-B) (c-d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+3 B d (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{3 a^2 f (\sin (e+f x)+1)^2}","-\frac{(A c+2 A d+2 B c-5 B d) \cos (e+f x)}{3 a^2 f (\sin (e+f x)+1)}+\frac{B d x}{a^2}-\frac{(A-B) (c-d) \cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*(A - B)*(c - d)*Sin[(e + f*x)/2] - (A - B)*(c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(A*c + 2*B*c + 2*A*d - 5*B*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 3*B*d*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3))/(3*a^2*f*(1 + Sin[e + f*x])^2)","B",1
275,1,43,65,0.0565477,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx","Integrate[(A + B*Sin[e + f*x])/(a + a*Sin[e + f*x])^2,x]","-\frac{\cos (e+f x) ((A+2 B) \sin (e+f x)+2 A+B)}{3 a^2 f (\sin (e+f x)+1)^2}","-\frac{(A+2 B) \cos (e+f x)}{3 f \left(a^2 \sin (e+f x)+a^2\right)}-\frac{(A-B) \cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}",1,"-1/3*(Cos[e + f*x]*(2*A + B + (A + 2*B)*Sin[e + f*x]))/(a^2*f*(1 + Sin[e + f*x])^2)","A",1
276,1,229,152,0.6278536,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\frac{6 d (A d-B c) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}+2 (A-B) (c-d) \sin \left(\frac{1}{2} (e+f x)\right)+2 (A (c-4 d)+B (2 c+d)) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+(B-A) (c-d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1)^2}","-\frac{2 d (B c-A d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^2 f (c-d)^2 \sqrt{c^2-d^2}}-\frac{(A (c-4 d)+B (2 c+d)) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1)}-\frac{(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*(A - B)*(c - d)*Sin[(e + f*x)/2] + (-A + B)*(c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(A*(c - 4*d) + B*(2*c + d))*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (6*d*(-(B*c) + A*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/Sqrt[c^2 - d^2]))/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])^2)","A",1
277,1,313,275,2.8297041,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(-\frac{6 d \left(B \left(2 c^2+2 c d+d^2\right)-A d (3 c+2 d)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{(c+d) \sqrt{c^2-d^2}}+\frac{3 d^2 (A d-B c) \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}{(c+d) (c+d \sin (e+f x))}+2 (A-B) (c-d) \sin \left(\frac{1}{2} (e+f x)\right)+2 (A (c-7 d)+2 B (c+2 d)) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+(B-A) (c-d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{3 a^2 f (c-d)^3 (\sin (e+f x)+1)^2}","\frac{2 d \left(A d (3 c+2 d)-B \left(2 c^2+2 c d+d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^2 f (c-d)^3 (c+d) \sqrt{c^2-d^2}}-\frac{d \left(A \left(c^2-6 c d-10 d^2\right)+B \left(2 c^2+9 c d+4 d^2\right)\right) \cos (e+f x)}{3 a^2 f (c-d)^3 (c+d) (c+d \sin (e+f x))}-\frac{(A c-6 A d+2 B c+3 B d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) (c+d \sin (e+f x))}-\frac{(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*(A - B)*(c - d)*Sin[(e + f*x)/2] + (-A + B)*(c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(A*(c - 7*d) + 2*B*(c + 2*d))*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - (6*d*(-(A*d*(3*c + 2*d)) + B*(2*c^2 + 2*c*d + d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/((c + d)*Sqrt[c^2 - d^2]) + (3*d^2*(-(B*c) + A*d)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/((c + d)*(c + d*Sin[e + f*x]))))/(3*a^2*(c - d)^3*f*(1 + Sin[e + f*x])^2)","A",1
278,1,1522,386,6.3655981,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(48 B \cos \left(\frac{1}{2} (e+f x)\right) c^5-16 A \cos \left(\frac{3}{2} (e+f x)\right) c^5-32 B \cos \left(\frac{3}{2} (e+f x)\right) c^5+48 A \sin \left(\frac{1}{2} (e+f x)\right) c^5+48 B \sin \left(\frac{1}{2} (e+f x)\right) c^5-96 A d \cos \left(\frac{1}{2} (e+f x)\right) c^4+240 B d \cos \left(\frac{1}{2} (e+f x)\right) c^4+80 A d \cos \left(\frac{3}{2} (e+f x)\right) c^4-224 B d \cos \left(\frac{3}{2} (e+f x)\right) c^4-224 A d \sin \left(\frac{1}{2} (e+f x)\right) c^4+416 B d \sin \left(\frac{1}{2} (e+f x)\right) c^4+48 B d \sin \left(\frac{3}{2} (e+f x)\right) c^4-16 A d \sin \left(\frac{5}{2} (e+f x)\right) c^4-32 B d \sin \left(\frac{5}{2} (e+f x)\right) c^4-524 A d^2 \cos \left(\frac{1}{2} (e+f x)\right) c^3+536 B d^2 \cos \left(\frac{1}{2} (e+f x)\right) c^3+536 A d^2 \cos \left(\frac{3}{2} (e+f x)\right) c^3-728 B d^2 \cos \left(\frac{3}{2} (e+f x)\right) c^3+24 B d^2 \cos \left(\frac{5}{2} (e+f x)\right) c^3+4 A d^2 \cos \left(\frac{7}{2} (e+f x)\right) c^3+8 B d^2 \cos \left(\frac{7}{2} (e+f x)\right) c^3-872 A d^2 \sin \left(\frac{1}{2} (e+f x)\right) c^3+992 B d^2 \sin \left(\frac{1}{2} (e+f x)\right) c^3-132 A d^2 \sin \left(\frac{3}{2} (e+f x)\right) c^3+96 B d^2 \sin \left(\frac{3}{2} (e+f x)\right) c^3+116 A d^2 \sin \left(\frac{5}{2} (e+f x)\right) c^3-224 B d^2 \sin \left(\frac{5}{2} (e+f x)\right) c^3-776 A d^3 \cos \left(\frac{1}{2} (e+f x)\right) c^2+701 B d^3 \cos \left(\frac{1}{2} (e+f x)\right) c^2+1028 A d^3 \cos \left(\frac{3}{2} (e+f x)\right) c^2-893 B d^3 \cos \left(\frac{3}{2} (e+f x)\right) c^2-12 A d^3 \cos \left(\frac{5}{2} (e+f x)\right) c^2+21 B d^3 \cos \left(\frac{5}{2} (e+f x)\right) c^2-32 A d^3 \cos \left(\frac{7}{2} (e+f x)\right) c^2+59 B d^3 \cos \left(\frac{7}{2} (e+f x)\right) c^2-1144 A d^3 \sin \left(\frac{1}{2} (e+f x)\right) c^2+967 B d^3 \sin \left(\frac{1}{2} (e+f x)\right) c^2-204 A d^3 \sin \left(\frac{3}{2} (e+f x)\right) c^2+207 B d^3 \sin \left(\frac{3}{2} (e+f x)\right) c^2+412 A d^3 \sin \left(\frac{5}{2} (e+f x)\right) c^2-409 B d^3 \sin \left(\frac{5}{2} (e+f x)\right) c^2+15 B d^3 \sin \left(\frac{7}{2} (e+f x)\right) c^2-487 A d^4 \cos \left(\frac{1}{2} (e+f x)\right) c+400 B d^4 \cos \left(\frac{1}{2} (e+f x)\right) c+695 A d^4 \cos \left(\frac{3}{2} (e+f x)\right) c-482 B d^4 \cos \left(\frac{3}{2} (e+f x)\right) c-15 A d^4 \cos \left(\frac{5}{2} (e+f x)\right) c-18 B d^4 \cos \left(\frac{5}{2} (e+f x)\right) c-97 A d^4 \cos \left(\frac{7}{2} (e+f x)\right) c+76 B d^4 \cos \left(\frac{7}{2} (e+f x)\right) c-685 A d^4 \sin \left(\frac{1}{2} (e+f x)\right) c+496 B d^4 \sin \left(\frac{1}{2} (e+f x)\right) c-165 A d^4 \sin \left(\frac{3}{2} (e+f x)\right) c+174 B d^4 \sin \left(\frac{3}{2} (e+f x)\right) c+403 A d^4 \sin \left(\frac{5}{2} (e+f x)\right) c-286 B d^4 \sin \left(\frac{5}{2} (e+f x)\right) c-21 A d^4 \sin \left(\frac{7}{2} (e+f x)\right) c+12 B d^4 \sin \left(\frac{7}{2} (e+f x)\right) c-112 A d^5 \cos \left(\frac{1}{2} (e+f x)\right)+70 B d^5 \cos \left(\frac{1}{2} (e+f x)\right)+134 A d^5 \cos \left(\frac{3}{2} (e+f x)\right)-98 B d^5 \cos \left(\frac{3}{2} (e+f x)\right)+6 A d^5 \cos \left(\frac{5}{2} (e+f x)\right)-6 B d^5 \cos \left(\frac{5}{2} (e+f x)\right)-52 A d^5 \cos \left(\frac{7}{2} (e+f x)\right)+34 B d^5 \cos \left(\frac{7}{2} (e+f x)\right)-168 A d^5 \sin \left(\frac{1}{2} (e+f x)\right)+126 B d^5 \sin \left(\frac{1}{2} (e+f x)\right)-66 A d^5 \sin \left(\frac{3}{2} (e+f x)\right)+42 B d^5 \sin \left(\frac{3}{2} (e+f x)\right)+114 A d^5 \sin \left(\frac{5}{2} (e+f x)\right)-78 B d^5 \sin \left(\frac{5}{2} (e+f x)\right)-12 A d^5 \sin \left(\frac{7}{2} (e+f x)\right)+6 B d^5 \sin \left(\frac{7}{2} (e+f x)\right)\right)}{48 (c-d)^4 (c+d)^2 f (\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^2}-\frac{d \left(6 B c^3-12 A d c^2+12 B d c^2-16 A d^2 c+13 B d^2 c-7 A d^3+4 B d^3\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (e+f x)\right) \left(d \cos \left(\frac{1}{2} (e+f x)\right)+c \sin \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{c^2-d^2}}\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}{(c-d)^4 (c+d)^2 \sqrt{c^2-d^2} f (\sin (e+f x) a+a)^2}","-\frac{d \left(A \left(2 c^2-16 c d-21 d^2\right)+B \left(4 c^2+19 c d+12 d^2\right)\right) \cos (e+f x)}{6 a^2 f (c-d)^3 (c+d) (c+d \sin (e+f x))^2}+\frac{d \left(A d \left(12 c^2+16 c d+7 d^2\right)-B \left(6 c^3+12 c^2 d+13 c d^2+4 d^3\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^2 f (c-d)^4 (c+d)^2 \sqrt{c^2-d^2}}-\frac{d \left(A \left(2 c^3-16 c^2 d-59 c d^2-32 d^3\right)+B \left(4 c^3+37 c^2 d+44 c d^2+20 d^3\right)\right) \cos (e+f x)}{6 a^2 f (c-d)^4 (c+d)^2 (c+d \sin (e+f x))}-\frac{(A c-8 A d+2 B c+5 B d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}",1,"-((d*(6*B*c^3 - 12*A*c^2*d + 12*B*c^2*d - 16*A*c*d^2 + 13*B*c*d^2 - 7*A*d^3 + 4*B*d^3)*ArcTan[(Sec[(e + f*x)/2]*(d*Cos[(e + f*x)/2] + c*Sin[(e + f*x)/2]))/Sqrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/((c - d)^4*(c + d)^2*Sqrt[c^2 - d^2]*f*(a + a*Sin[e + f*x])^2)) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(48*B*c^5*Cos[(e + f*x)/2] - 96*A*c^4*d*Cos[(e + f*x)/2] + 240*B*c^4*d*Cos[(e + f*x)/2] - 524*A*c^3*d^2*Cos[(e + f*x)/2] + 536*B*c^3*d^2*Cos[(e + f*x)/2] - 776*A*c^2*d^3*Cos[(e + f*x)/2] + 701*B*c^2*d^3*Cos[(e + f*x)/2] - 487*A*c*d^4*Cos[(e + f*x)/2] + 400*B*c*d^4*Cos[(e + f*x)/2] - 112*A*d^5*Cos[(e + f*x)/2] + 70*B*d^5*Cos[(e + f*x)/2] - 16*A*c^5*Cos[(3*(e + f*x))/2] - 32*B*c^5*Cos[(3*(e + f*x))/2] + 80*A*c^4*d*Cos[(3*(e + f*x))/2] - 224*B*c^4*d*Cos[(3*(e + f*x))/2] + 536*A*c^3*d^2*Cos[(3*(e + f*x))/2] - 728*B*c^3*d^2*Cos[(3*(e + f*x))/2] + 1028*A*c^2*d^3*Cos[(3*(e + f*x))/2] - 893*B*c^2*d^3*Cos[(3*(e + f*x))/2] + 695*A*c*d^4*Cos[(3*(e + f*x))/2] - 482*B*c*d^4*Cos[(3*(e + f*x))/2] + 134*A*d^5*Cos[(3*(e + f*x))/2] - 98*B*d^5*Cos[(3*(e + f*x))/2] + 24*B*c^3*d^2*Cos[(5*(e + f*x))/2] - 12*A*c^2*d^3*Cos[(5*(e + f*x))/2] + 21*B*c^2*d^3*Cos[(5*(e + f*x))/2] - 15*A*c*d^4*Cos[(5*(e + f*x))/2] - 18*B*c*d^4*Cos[(5*(e + f*x))/2] + 6*A*d^5*Cos[(5*(e + f*x))/2] - 6*B*d^5*Cos[(5*(e + f*x))/2] + 4*A*c^3*d^2*Cos[(7*(e + f*x))/2] + 8*B*c^3*d^2*Cos[(7*(e + f*x))/2] - 32*A*c^2*d^3*Cos[(7*(e + f*x))/2] + 59*B*c^2*d^3*Cos[(7*(e + f*x))/2] - 97*A*c*d^4*Cos[(7*(e + f*x))/2] + 76*B*c*d^4*Cos[(7*(e + f*x))/2] - 52*A*d^5*Cos[(7*(e + f*x))/2] + 34*B*d^5*Cos[(7*(e + f*x))/2] + 48*A*c^5*Sin[(e + f*x)/2] + 48*B*c^5*Sin[(e + f*x)/2] - 224*A*c^4*d*Sin[(e + f*x)/2] + 416*B*c^4*d*Sin[(e + f*x)/2] - 872*A*c^3*d^2*Sin[(e + f*x)/2] + 992*B*c^3*d^2*Sin[(e + f*x)/2] - 1144*A*c^2*d^3*Sin[(e + f*x)/2] + 967*B*c^2*d^3*Sin[(e + f*x)/2] - 685*A*c*d^4*Sin[(e + f*x)/2] + 496*B*c*d^4*Sin[(e + f*x)/2] - 168*A*d^5*Sin[(e + f*x)/2] + 126*B*d^5*Sin[(e + f*x)/2] + 48*B*c^4*d*Sin[(3*(e + f*x))/2] - 132*A*c^3*d^2*Sin[(3*(e + f*x))/2] + 96*B*c^3*d^2*Sin[(3*(e + f*x))/2] - 204*A*c^2*d^3*Sin[(3*(e + f*x))/2] + 207*B*c^2*d^3*Sin[(3*(e + f*x))/2] - 165*A*c*d^4*Sin[(3*(e + f*x))/2] + 174*B*c*d^4*Sin[(3*(e + f*x))/2] - 66*A*d^5*Sin[(3*(e + f*x))/2] + 42*B*d^5*Sin[(3*(e + f*x))/2] - 16*A*c^4*d*Sin[(5*(e + f*x))/2] - 32*B*c^4*d*Sin[(5*(e + f*x))/2] + 116*A*c^3*d^2*Sin[(5*(e + f*x))/2] - 224*B*c^3*d^2*Sin[(5*(e + f*x))/2] + 412*A*c^2*d^3*Sin[(5*(e + f*x))/2] - 409*B*c^2*d^3*Sin[(5*(e + f*x))/2] + 403*A*c*d^4*Sin[(5*(e + f*x))/2] - 286*B*c*d^4*Sin[(5*(e + f*x))/2] + 114*A*d^5*Sin[(5*(e + f*x))/2] - 78*B*d^5*Sin[(5*(e + f*x))/2] + 15*B*c^2*d^3*Sin[(7*(e + f*x))/2] - 21*A*c*d^4*Sin[(7*(e + f*x))/2] + 12*B*c*d^4*Sin[(7*(e + f*x))/2] - 12*A*d^5*Sin[(7*(e + f*x))/2] + 6*B*d^5*Sin[(7*(e + f*x))/2]))/(48*(c - d)^4*(c + d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2)","B",1
279,1,366,225,2.9708954,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(a + a*Sin[e + f*x])^3,x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(2 (c-d) \left(A \left(2 c^2+11 c d+32 d^2\right)+3 B \left(c^2+8 c d-24 d^2\right)\right) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4-15 d^2 (e+f x) (-A d-3 B c+3 B d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5+6 (A-B) (c-d)^3 \sin \left(\frac{1}{2} (e+f x)\right)-(c-d)^2 (A (2 c+13 d)+3 B (c-6 d)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+2 (c-d)^2 (A (2 c+13 d)+3 B (c-6 d)) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-3 (A-B) (c-d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-15 B d^3 \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5\right)}{15 a^3 f (\sin (e+f x)+1)^3}","-\frac{(c-d) \left(A \left(2 c^2+7 c d+15 d^2\right)+3 B \left(c^2+6 c d-15 d^2\right)\right) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{d^2 (A (2 c+7 d)+3 B (c-9 d)) \cos (e+f x)}{15 a^3 f}+\frac{d^2 x (A d+3 B (c-d))}{a^3}-\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}-\frac{(2 A (c+2 d)+3 B (c-3 d)) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(6*(A - B)*(c - d)^3*Sin[(e + f*x)/2] - 3*(A - B)*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(c - d)^2*(3*B*(c - 6*d) + A*(2*c + 13*d))*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - (c - d)^2*(3*B*(c - 6*d) + A*(2*c + 13*d))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 2*(c - d)*(3*B*(c^2 + 8*c*d - 24*d^2) + A*(2*c^2 + 11*c*d + 32*d^2))*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 15*d^2*(-3*B*c - A*d + 3*B*d)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 15*B*d^3*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5))/(15*a^3*f*(1 + Sin[e + f*x])^3)","A",1
280,1,514,164,0.8984046,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^3,x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(30 \cos \left(\frac{1}{2} (e+f x)\right) \left(2 A d (c+d)+B \left(c^2+4 c d+d^2 (5 e+5 f x-9)\right)\right)-5 \cos \left(\frac{3}{2} (e+f x)\right) \left(4 A \left(c^2+3 c d+2 d^2\right)+B \left(6 c^2+16 c d+d^2 (15 e+15 f x-46)\right)\right)+40 A c^2 \sin \left(\frac{1}{2} (e+f x)\right)-4 A c^2 \sin \left(\frac{5}{2} (e+f x)\right)+60 A c d \sin \left(\frac{1}{2} (e+f x)\right)-12 A c d \sin \left(\frac{5}{2} (e+f x)\right)+80 A d^2 \sin \left(\frac{1}{2} (e+f x)\right)+30 A d^2 \sin \left(\frac{3}{2} (e+f x)\right)-14 A d^2 \sin \left(\frac{5}{2} (e+f x)\right)+30 B c^2 \sin \left(\frac{1}{2} (e+f x)\right)-6 B c^2 \sin \left(\frac{5}{2} (e+f x)\right)+160 B c d \sin \left(\frac{1}{2} (e+f x)\right)+60 B c d \sin \left(\frac{3}{2} (e+f x)\right)-28 B c d \sin \left(\frac{5}{2} (e+f x)\right)-370 B d^2 \sin \left(\frac{1}{2} (e+f x)\right)+150 B d^2 e \sin \left(\frac{1}{2} (e+f x)\right)+150 B d^2 f x \sin \left(\frac{1}{2} (e+f x)\right)-90 B d^2 \sin \left(\frac{3}{2} (e+f x)\right)+75 B d^2 e \sin \left(\frac{3}{2} (e+f x)\right)+75 B d^2 f x \sin \left(\frac{3}{2} (e+f x)\right)+64 B d^2 \sin \left(\frac{5}{2} (e+f x)\right)-15 B d^2 e \sin \left(\frac{5}{2} (e+f x)\right)-15 B d^2 f x \sin \left(\frac{5}{2} (e+f x)\right)-15 B d^2 e \cos \left(\frac{5}{2} (e+f x)\right)-15 B d^2 f x \cos \left(\frac{5}{2} (e+f x)\right)\right)}{60 a^3 f (\sin (e+f x)+1)^3}","-\frac{\left(2 A \left(c^2+3 c d+2 d^2\right)+B \left(3 c^2+14 c d-29 d^2\right)\right) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{B d^2 x}{a^3}-\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 A (c+d)+B (3 c-7 d)) \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(30*(2*A*d*(c + d) + B*(c^2 + 4*c*d + d^2*(-9 + 5*e + 5*f*x)))*Cos[(e + f*x)/2] - 5*(4*A*(c^2 + 3*c*d + 2*d^2) + B*(6*c^2 + 16*c*d + d^2*(-46 + 15*e + 15*f*x)))*Cos[(3*(e + f*x))/2] - 15*B*d^2*e*Cos[(5*(e + f*x))/2] - 15*B*d^2*f*x*Cos[(5*(e + f*x))/2] + 40*A*c^2*Sin[(e + f*x)/2] + 30*B*c^2*Sin[(e + f*x)/2] + 60*A*c*d*Sin[(e + f*x)/2] + 160*B*c*d*Sin[(e + f*x)/2] + 80*A*d^2*Sin[(e + f*x)/2] - 370*B*d^2*Sin[(e + f*x)/2] + 150*B*d^2*e*Sin[(e + f*x)/2] + 150*B*d^2*f*x*Sin[(e + f*x)/2] + 60*B*c*d*Sin[(3*(e + f*x))/2] + 30*A*d^2*Sin[(3*(e + f*x))/2] - 90*B*d^2*Sin[(3*(e + f*x))/2] + 75*B*d^2*e*Sin[(3*(e + f*x))/2] + 75*B*d^2*f*x*Sin[(3*(e + f*x))/2] - 4*A*c^2*Sin[(5*(e + f*x))/2] - 6*B*c^2*Sin[(5*(e + f*x))/2] - 12*A*c*d*Sin[(5*(e + f*x))/2] - 28*B*c*d*Sin[(5*(e + f*x))/2] - 14*A*d^2*Sin[(5*(e + f*x))/2] + 64*B*d^2*Sin[(5*(e + f*x))/2] - 15*B*d^2*e*Sin[(5*(e + f*x))/2] - 15*B*d^2*f*x*Sin[(5*(e + f*x))/2]))/(60*a^3*f*(1 + Sin[e + f*x])^3)","B",1
281,1,176,127,0.6888261,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]))/(a + a*Sin[e + f*x])^3,x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(15 (A d+B (c+2 d)) \cos \left(\frac{1}{2} (e+f x)\right)-5 (2 A c+3 A d+3 B c+4 B d) \cos \left(\frac{3}{2} (e+f x)\right)-2 \sin \left(\frac{1}{2} (e+f x)\right) ((2 A c+3 A d+3 B c-8 B d) \cos (e+f x)+(2 A c+3 A d+3 B c+7 B d) \cos (2 (e+f x))-3 (3 A c+2 A d+2 B c+8 B d))\right)}{30 a^3 f (\sin (e+f x)+1)^3}","-\frac{(2 A c+3 A d+3 B c+7 B d) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{(2 A c+3 A d+3 B c-8 B d) \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}-\frac{(A-B) (c-d) \cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(15*(A*d + B*(c + 2*d))*Cos[(e + f*x)/2] - 5*(2*A*c + 3*B*c + 3*A*d + 4*B*d)*Cos[(3*(e + f*x))/2] - 2*(-3*(3*A*c + 2*B*c + 2*A*d + 8*B*d) + (2*A*c + 3*B*c + 3*A*d - 8*B*d)*Cos[e + f*x] + (2*A*c + 3*B*c + 3*A*d + 7*B*d)*Cos[2*(e + f*x)])*Sin[(e + f*x)/2]))/(30*a^3*f*(1 + Sin[e + f*x])^3)","A",1
282,1,63,102,0.0925967,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx","Integrate[(A + B*Sin[e + f*x])/(a + a*Sin[e + f*x])^3,x]","-\frac{\cos (e+f x) \left((2 A+3 B) \sin ^2(e+f x)+(6 A+9 B) \sin (e+f x)+7 A+3 B\right)}{15 a^3 f (\sin (e+f x)+1)^3}","-\frac{(2 A+3 B) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{(2 A+3 B) \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}-\frac{(A-B) \cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}",1,"-1/15*(Cos[e + f*x]*(7*A + 3*B + (6*A + 9*B)*Sin[e + f*x] + (2*A + 3*B)*Sin[e + f*x]^2))/(a^3*f*(1 + Sin[e + f*x])^3)","A",1
283,1,502,229,1.2505295,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(-\frac{60 d^2 (A d-B c) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}+20 A c^2 \sin \left(\frac{1}{2} (e+f x)\right)-2 A c^2 \sin \left(\frac{5}{2} (e+f x)\right)-10 A c^2 \cos \left(\frac{3}{2} (e+f x)\right)-75 A c d \sin \left(\frac{1}{2} (e+f x)\right)+9 A c d \sin \left(\frac{5}{2} (e+f x)\right)-15 A c d \cos \left(\frac{1}{2} (e+f x)\right)+45 A c d \cos \left(\frac{3}{2} (e+f x)\right)+145 A d^2 \sin \left(\frac{1}{2} (e+f x)\right)+15 A d^2 \sin \left(\frac{3}{2} (e+f x)\right)-22 A d^2 \sin \left(\frac{5}{2} (e+f x)\right)+75 A d^2 \cos \left(\frac{1}{2} (e+f x)\right)-95 A d^2 \cos \left(\frac{3}{2} (e+f x)\right)+15 B c^2 \sin \left(\frac{1}{2} (e+f x)\right)-3 B c^2 \sin \left(\frac{5}{2} (e+f x)\right)+15 B c^2 \cos \left(\frac{1}{2} (e+f x)\right)-15 B c^2 \cos \left(\frac{3}{2} (e+f x)\right)-85 B c d \sin \left(\frac{1}{2} (e+f x)\right)-15 B c d \sin \left(\frac{3}{2} (e+f x)\right)+16 B c d \sin \left(\frac{5}{2} (e+f x)\right)-75 B c d \cos \left(\frac{1}{2} (e+f x)\right)+65 B c d \cos \left(\frac{3}{2} (e+f x)\right)-20 B d^2 \sin \left(\frac{1}{2} (e+f x)\right)+2 B d^2 \sin \left(\frac{5}{2} (e+f x)\right)+10 B d^2 \cos \left(\frac{3}{2} (e+f x)\right)\right)}{30 a^3 f (c-d)^3 (\sin (e+f x)+1)^3}","\frac{2 d^2 (B c-A d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^3 f (c-d)^3 \sqrt{c^2-d^2}}-\frac{\left(A \left(2 c^2-9 c d+22 d^2\right)+B \left(3 c^2-16 c d-2 d^2\right)\right) \cos (e+f x)}{15 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right)}-\frac{(2 A c-7 A d+3 B c+2 B d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2}-\frac{(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(15*B*c^2*Cos[(e + f*x)/2] - 15*A*c*d*Cos[(e + f*x)/2] - 75*B*c*d*Cos[(e + f*x)/2] + 75*A*d^2*Cos[(e + f*x)/2] - 10*A*c^2*Cos[(3*(e + f*x))/2] - 15*B*c^2*Cos[(3*(e + f*x))/2] + 45*A*c*d*Cos[(3*(e + f*x))/2] + 65*B*c*d*Cos[(3*(e + f*x))/2] - 95*A*d^2*Cos[(3*(e + f*x))/2] + 10*B*d^2*Cos[(3*(e + f*x))/2] + 20*A*c^2*Sin[(e + f*x)/2] + 15*B*c^2*Sin[(e + f*x)/2] - 75*A*c*d*Sin[(e + f*x)/2] - 85*B*c*d*Sin[(e + f*x)/2] + 145*A*d^2*Sin[(e + f*x)/2] - 20*B*d^2*Sin[(e + f*x)/2] - (60*d^2*(-(B*c) + A*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/Sqrt[c^2 - d^2] - 15*B*c*d*Sin[(3*(e + f*x))/2] + 15*A*d^2*Sin[(3*(e + f*x))/2] - 2*A*c^2*Sin[(5*(e + f*x))/2] - 3*B*c^2*Sin[(5*(e + f*x))/2] + 9*A*c*d*Sin[(5*(e + f*x))/2] + 16*B*c*d*Sin[(5*(e + f*x))/2] - 22*A*d^2*Sin[(5*(e + f*x))/2] + 2*B*d^2*Sin[(5*(e + f*x))/2]))/(30*a^3*(c - d)^3*f*(1 + Sin[e + f*x])^3)","B",1
284,1,1253,381,6.3836837,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2),x]","\frac{2 d^2 \left(3 B c^2-4 A d c+3 B d c-3 A d^2+B d^2\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (e+f x)\right) \left(d \cos \left(\frac{1}{2} (e+f x)\right)+c \sin \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{c^2-d^2}}\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}{(c-d)^4 (c+d) \sqrt{c^2-d^2} f (\sin (e+f x) a+a)^3}+\frac{\left(60 B \cos \left(\frac{1}{2} (e+f x)\right) c^4-40 A \cos \left(\frac{3}{2} (e+f x)\right) c^4-60 B \cos \left(\frac{3}{2} (e+f x)\right) c^4+80 A \sin \left(\frac{1}{2} (e+f x)\right) c^4+60 B \sin \left(\frac{1}{2} (e+f x)\right) c^4-8 A \sin \left(\frac{5}{2} (e+f x)\right) c^4-12 B \sin \left(\frac{5}{2} (e+f x)\right) c^4-80 A d \cos \left(\frac{1}{2} (e+f x)\right) c^3-390 B d \cos \left(\frac{1}{2} (e+f x)\right) c^3+196 A d \cos \left(\frac{3}{2} (e+f x)\right) c^3+304 B d \cos \left(\frac{3}{2} (e+f x)\right) c^3+4 A d \cos \left(\frac{7}{2} (e+f x)\right) c^3+6 B d \cos \left(\frac{7}{2} (e+f x)\right) c^3-340 A d \sin \left(\frac{1}{2} (e+f x)\right) c^3-440 B d \sin \left(\frac{1}{2} (e+f x)\right) c^3-90 B d \sin \left(\frac{3}{2} (e+f x)\right) c^3+28 A d \sin \left(\frac{5}{2} (e+f x)\right) c^3+62 B d \sin \left(\frac{5}{2} (e+f x)\right) c^3+540 A d^2 \cos \left(\frac{1}{2} (e+f x)\right) c^2-1090 B d^2 \cos \left(\frac{1}{2} (e+f x)\right) c^2-476 A d^2 \cos \left(\frac{3}{2} (e+f x)\right) c^2+1076 B d^2 \cos \left(\frac{3}{2} (e+f x)\right) c^2+60 B d^2 \cos \left(\frac{5}{2} (e+f x)\right) c^2-24 A d^2 \cos \left(\frac{7}{2} (e+f x)\right) c^2-46 B d^2 \cos \left(\frac{7}{2} (e+f x)\right) c^2+820 A d^2 \sin \left(\frac{1}{2} (e+f x)\right) c^2-1520 B d^2 \sin \left(\frac{1}{2} (e+f x)\right) c^2+120 A d^2 \sin \left(\frac{3}{2} (e+f x)\right) c^2-390 B d^2 \sin \left(\frac{3}{2} (e+f x)\right) c^2-52 A d^2 \sin \left(\frac{5}{2} (e+f x)\right) c^2+362 B d^2 \sin \left(\frac{5}{2} (e+f x)\right) c^2+1430 A d^3 \cos \left(\frac{1}{2} (e+f x)\right) c-885 B d^3 \cos \left(\frac{1}{2} (e+f x)\right) c-1546 A d^3 \cos \left(\frac{3}{2} (e+f x)\right) c+1181 B d^3 \cos \left(\frac{3}{2} (e+f x)\right) c-90 A d^3 \cos \left(\frac{5}{2} (e+f x)\right) c+15 B d^3 \cos \left(\frac{5}{2} (e+f x)\right) c+86 A d^3 \cos \left(\frac{7}{2} (e+f x)\right) c-111 B d^3 \cos \left(\frac{7}{2} (e+f x)\right) c+2140 A d^3 \sin \left(\frac{1}{2} (e+f x)\right) c-1435 B d^3 \sin \left(\frac{1}{2} (e+f x)\right) c+540 A d^3 \sin \left(\frac{3}{2} (e+f x)\right) c-315 B d^3 \sin \left(\frac{3}{2} (e+f x)\right) c-568 A d^3 \sin \left(\frac{5}{2} (e+f x)\right) c+553 B d^3 \sin \left(\frac{5}{2} (e+f x)\right) c-15 B d^3 \sin \left(\frac{7}{2} (e+f x)\right) c+735 A d^4 \cos \left(\frac{1}{2} (e+f x)\right)-320 B d^4 \cos \left(\frac{1}{2} (e+f x)\right)-969 A d^4 \cos \left(\frac{3}{2} (e+f x)\right)+334 B d^4 \cos \left(\frac{3}{2} (e+f x)\right)-15 A d^4 \cos \left(\frac{5}{2} (e+f x)\right)+30 B d^4 \cos \left(\frac{5}{2} (e+f x)\right)+129 A d^4 \cos \left(\frac{7}{2} (e+f x)\right)-44 B d^4 \cos \left(\frac{7}{2} (e+f x)\right)+975 A d^4 \sin \left(\frac{1}{2} (e+f x)\right)-340 B d^4 \sin \left(\frac{1}{2} (e+f x)\right)+285 A d^4 \sin \left(\frac{3}{2} (e+f x)\right)-150 B d^4 \sin \left(\frac{3}{2} (e+f x)\right)-555 A d^4 \sin \left(\frac{5}{2} (e+f x)\right)+190 B d^4 \sin \left(\frac{5}{2} (e+f x)\right)+15 A d^4 \sin \left(\frac{7}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}{120 (c-d)^4 (c+d) f (\sin (e+f x) a+a)^3 (c+d \sin (e+f x))}","-\frac{2 d^2 \left(A d (4 c+3 d)-B \left(3 c^2+3 c d+d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^3 f (c-d)^4 (c+d) \sqrt{c^2-d^2}}-\frac{\left(A \left(2 c^2-12 c d+45 d^2\right)+B \left(3 c^2-23 c d-15 d^2\right)\right) \cos (e+f x)}{15 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))}-\frac{d \left(A \left(2 c^3-12 c^2 d+43 c d^2+72 d^3\right)+B \left(3 c^3-23 c^2 d-63 c d^2-22 d^3\right)\right) \cos (e+f x)}{15 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))}-\frac{(2 A c-9 A d+3 B c+4 B d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))}-\frac{(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))}",1,"(2*d^2*(3*B*c^2 - 4*A*c*d + 3*B*c*d - 3*A*d^2 + B*d^2)*ArcTan[(Sec[(e + f*x)/2]*(d*Cos[(e + f*x)/2] + c*Sin[(e + f*x)/2]))/Sqrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)/((c - d)^4*(c + d)*Sqrt[c^2 - d^2]*f*(a + a*Sin[e + f*x])^3) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(60*B*c^4*Cos[(e + f*x)/2] - 80*A*c^3*d*Cos[(e + f*x)/2] - 390*B*c^3*d*Cos[(e + f*x)/2] + 540*A*c^2*d^2*Cos[(e + f*x)/2] - 1090*B*c^2*d^2*Cos[(e + f*x)/2] + 1430*A*c*d^3*Cos[(e + f*x)/2] - 885*B*c*d^3*Cos[(e + f*x)/2] + 735*A*d^4*Cos[(e + f*x)/2] - 320*B*d^4*Cos[(e + f*x)/2] - 40*A*c^4*Cos[(3*(e + f*x))/2] - 60*B*c^4*Cos[(3*(e + f*x))/2] + 196*A*c^3*d*Cos[(3*(e + f*x))/2] + 304*B*c^3*d*Cos[(3*(e + f*x))/2] - 476*A*c^2*d^2*Cos[(3*(e + f*x))/2] + 1076*B*c^2*d^2*Cos[(3*(e + f*x))/2] - 1546*A*c*d^3*Cos[(3*(e + f*x))/2] + 1181*B*c*d^3*Cos[(3*(e + f*x))/2] - 969*A*d^4*Cos[(3*(e + f*x))/2] + 334*B*d^4*Cos[(3*(e + f*x))/2] + 60*B*c^2*d^2*Cos[(5*(e + f*x))/2] - 90*A*c*d^3*Cos[(5*(e + f*x))/2] + 15*B*c*d^3*Cos[(5*(e + f*x))/2] - 15*A*d^4*Cos[(5*(e + f*x))/2] + 30*B*d^4*Cos[(5*(e + f*x))/2] + 4*A*c^3*d*Cos[(7*(e + f*x))/2] + 6*B*c^3*d*Cos[(7*(e + f*x))/2] - 24*A*c^2*d^2*Cos[(7*(e + f*x))/2] - 46*B*c^2*d^2*Cos[(7*(e + f*x))/2] + 86*A*c*d^3*Cos[(7*(e + f*x))/2] - 111*B*c*d^3*Cos[(7*(e + f*x))/2] + 129*A*d^4*Cos[(7*(e + f*x))/2] - 44*B*d^4*Cos[(7*(e + f*x))/2] + 80*A*c^4*Sin[(e + f*x)/2] + 60*B*c^4*Sin[(e + f*x)/2] - 340*A*c^3*d*Sin[(e + f*x)/2] - 440*B*c^3*d*Sin[(e + f*x)/2] + 820*A*c^2*d^2*Sin[(e + f*x)/2] - 1520*B*c^2*d^2*Sin[(e + f*x)/2] + 2140*A*c*d^3*Sin[(e + f*x)/2] - 1435*B*c*d^3*Sin[(e + f*x)/2] + 975*A*d^4*Sin[(e + f*x)/2] - 340*B*d^4*Sin[(e + f*x)/2] - 90*B*c^3*d*Sin[(3*(e + f*x))/2] + 120*A*c^2*d^2*Sin[(3*(e + f*x))/2] - 390*B*c^2*d^2*Sin[(3*(e + f*x))/2] + 540*A*c*d^3*Sin[(3*(e + f*x))/2] - 315*B*c*d^3*Sin[(3*(e + f*x))/2] + 285*A*d^4*Sin[(3*(e + f*x))/2] - 150*B*d^4*Sin[(3*(e + f*x))/2] - 8*A*c^4*Sin[(5*(e + f*x))/2] - 12*B*c^4*Sin[(5*(e + f*x))/2] + 28*A*c^3*d*Sin[(5*(e + f*x))/2] + 62*B*c^3*d*Sin[(5*(e + f*x))/2] - 52*A*c^2*d^2*Sin[(5*(e + f*x))/2] + 362*B*c^2*d^2*Sin[(5*(e + f*x))/2] - 568*A*c*d^3*Sin[(5*(e + f*x))/2] + 553*B*c*d^3*Sin[(5*(e + f*x))/2] - 555*A*d^4*Sin[(5*(e + f*x))/2] + 190*B*d^4*Sin[(5*(e + f*x))/2] - 15*B*c*d^3*Sin[(7*(e + f*x))/2] + 15*A*d^4*Sin[(7*(e + f*x))/2]))/(120*(c - d)^4*(c + d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x]))","B",1
285,1,548,508,4.9602816,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(4 \left(A \left(2 c^2-19 c d+107 d^2\right)+3 B \left(c^2-12 c d-19 d^2\right)\right) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4+\frac{15 d^3 \left(B \left(7 c^2+6 c d+2 d^2\right)-3 A d (3 c+2 d)\right) \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}{(c+d)^2 (c+d \sin (e+f x))}+\frac{30 d^2 \left(3 B \left(4 c^3+8 c^2 d+7 c d^2+2 d^3\right)-A d \left(20 c^2+30 c d+13 d^2\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{(c+d)^2 \sqrt{c^2-d^2}}+\frac{15 d^3 (c-d) (B c-A d) \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}{(c+d) (c+d \sin (e+f x))^2}+12 (A-B) (c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right)-2 (c-d) (A (2 c-17 d)+3 B (c+4 d)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+4 (c-d) (A (2 c-17 d)+3 B (c+4 d)) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+6 (B-A) (c-d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{30 a^3 f (c-d)^5 (\sin (e+f x)+1)^3}","-\frac{\left(A \left(2 c^2-15 c d+76 d^2\right)+3 B \left(c^2-10 c d-12 d^2\right)\right) \cos (e+f x)}{15 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^2}-\frac{d^2 \left(A d \left(20 c^2+30 c d+13 d^2\right)-3 B \left(4 c^3+8 c^2 d+7 c d^2+2 d^3\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^3 f (c-d)^5 (c+d)^2 \sqrt{c^2-d^2}}-\frac{d \left(A \left(4 c^3-30 c^2 d+146 c d^2+195 d^3\right)+3 B \left(2 c^3-20 c^2 d-57 c d^2-30 d^3\right)\right) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^2}-\frac{d \left(A \left(4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right)+3 B \left(2 c^4-20 c^3 d-119 c^2 d^2-130 c d^3-48 d^4\right)\right) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 (c+d \sin (e+f x))}-\frac{(2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(12*(A - B)*(c - d)^2*Sin[(e + f*x)/2] + 6*(-A + B)*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 4*(c - d)*(A*(2*c - 17*d) + 3*B*(c + 4*d))*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 2*(c - d)*(A*(2*c - 17*d) + 3*B*(c + 4*d))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 4*(3*B*(c^2 - 12*c*d - 19*d^2) + A*(2*c^2 - 19*c*d + 107*d^2))*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + (30*d^2*(-(A*d*(20*c^2 + 30*c*d + 13*d^2)) + 3*B*(4*c^3 + 8*c^2*d + 7*c*d^2 + 2*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/((c + d)^2*Sqrt[c^2 - d^2]) + (15*(c - d)*d^3*(B*c - A*d)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/((c + d)*(c + d*Sin[e + f*x])^2) + (15*d^3*(-3*A*d*(3*c + 2*d) + B*(7*c^2 + 6*c*d + 2*d^2))*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/((c + d)^2*(c + d*Sin[e + f*x]))))/(30*a^3*(c - d)^5*f*(1 + Sin[e + f*x])^3)","A",1
286,1,305,256,1.2478728,"\int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3,x]","-\frac{\sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-4 d \left(27 A d (7 c+2 d)+B \left(189 c^2+162 c d+83 d^2\right)\right) \cos (2 (e+f x))+2520 A c^3+2520 A c^2 d \sin (e+f x)+5040 A c^2 d+2016 A c d^2 \sin (e+f x)+4788 A c d^2+846 A d^3 \sin (e+f x)-90 A d^3 \sin (3 (e+f x))+1368 A d^3+840 B c^3 \sin (e+f x)+1680 B c^3+2016 B c^2 d \sin (e+f x)+4788 B c^2 d+2538 B c d^2 \sin (e+f x)-270 B c d^2 \sin (3 (e+f x))+4104 B c d^2+752 B d^3 \sin (e+f x)-80 B d^3 \sin (3 (e+f x))+35 B d^3 \cos (4 (e+f x))+1321 B d^3\right)}{1260 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{4 a (c+d) \left(15 c^2+10 c d+7 d^2\right) (-9 A d+B c-8 B d) \cos (e+f x)}{315 d f \sqrt{a \sin (e+f x)+a}}+\frac{2 a (-9 A d+B c-8 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt{a \sin (e+f x)+a}}+\frac{4 d (c+d) (-9 A d+B c-8 B d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 a f}+\frac{8 (5 c-d) (c+d) (-9 A d+B c-8 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 f}-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt{a \sin (e+f x)+a}}",1,"-1/1260*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(2520*A*c^3 + 1680*B*c^3 + 5040*A*c^2*d + 4788*B*c^2*d + 4788*A*c*d^2 + 4104*B*c*d^2 + 1368*A*d^3 + 1321*B*d^3 - 4*d*(27*A*d*(7*c + 2*d) + B*(189*c^2 + 162*c*d + 83*d^2))*Cos[2*(e + f*x)] + 35*B*d^3*Cos[4*(e + f*x)] + 840*B*c^3*Sin[e + f*x] + 2520*A*c^2*d*Sin[e + f*x] + 2016*B*c^2*d*Sin[e + f*x] + 2016*A*c*d^2*Sin[e + f*x] + 2538*B*c*d^2*Sin[e + f*x] + 846*A*d^3*Sin[e + f*x] + 752*B*d^3*Sin[e + f*x] - 270*B*c*d^2*Sin[3*(e + f*x)] - 90*A*d^3*Sin[3*(e + f*x)] - 80*B*d^3*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
287,1,176,192,0.7430275,"\int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]","-\frac{\sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\left(56 A d (5 c+2 d)+B \left(140 c^2+224 c d+141 d^2\right)\right) \sin (e+f x)-6 d (7 A d+14 B c+6 B d) \cos (2 (e+f x))+420 A c^2+560 A c d+266 A d^2+280 B c^2+532 B c d-15 B d^2 \sin (3 (e+f x))+228 B d^2\right)}{210 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{2 a \left(15 c^2+10 c d+7 d^2\right) (-7 A d+B c-6 B d) \cos (e+f x)}{105 d f \sqrt{a \sin (e+f x)+a}}+\frac{2 d (-7 A d+B c-6 B d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 a f}+\frac{4 (5 c-d) (-7 A d+B c-6 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt{a \sin (e+f x)+a}}",1,"-1/210*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(420*A*c^2 + 280*B*c^2 + 560*A*c*d + 532*B*c*d + 266*A*d^2 + 228*B*d^2 - 6*d*(14*B*c + 7*A*d + 6*B*d)*Cos[2*(e + f*x)] + (56*A*d*(5*c + 2*d) + B*(140*c^2 + 224*c*d + 141*d^2))*Sin[e + f*x] - 15*B*d^2*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
288,1,117,118,0.3585466,"\int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]","-\frac{\sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (2 (5 A d+5 B c+4 B d) \sin (e+f x)+30 A c+20 A d+20 B c-3 B d \cos (2 (e+f x))+19 B d)}{15 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{2 (5 A d+5 B c-2 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{2 a (15 A c+5 A d+5 B c+7 B d) \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{2 B d \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}",1,"-1/15*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(30*A*c + 20*B*c + 20*A*d + 19*B*d - 3*B*d*Cos[2*(e + f*x)] + 2*(5*B*c + 5*A*d + 4*B*d)*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
289,1,82,62,0.1232083,"\int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]),x]","-\frac{2 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (3 A+B \sin (e+f x)+2 B)}{3 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{2 a (3 A+B) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 B \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}",1,"(-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(3*A + 2*B + B*Sin[e + f*x]))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
290,1,903,100,9.0345393,"\int \frac{\sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x))}{c+d \sin (e+f x)} \, dx","Integrate[(Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x]),x]","\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \left(-\frac{(2-2 i) B \sqrt{d} \cos \left(\frac{f x}{2}\right) \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right)}{f}+\frac{(A d-B c) \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right) \left((-1+i) x \cos (e)+(1+i) x \sin (e)+\frac{\text{RootSum}\left[d e^{2 i e} \text{$\#$1}^4+2 i c e^{i e} \text{$\#$1}^2-d\&,\frac{-\sqrt{d} \sqrt{c+d} e^{i e} f x \text{$\#$1}^3-2 i \sqrt{d} \sqrt{c+d} e^{i e} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^3+\frac{(1-i) c f x \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\frac{(2+2 i) c \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^2}{\sqrt{e^{-i e}}}-i \sqrt{d} \sqrt{c+d} f x \text{$\#$1}+2 \sqrt{d} \sqrt{c+d} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}+(1+i) d \sqrt{e^{-i e}} f x-(2-2 i) d \sqrt{e^{-i e}} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right)}{d-i c e^{i e} \text{$\#$1}^2}\&\right] (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}{4 f}\right)}{\sqrt{c+d} (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}+\frac{(A d-B c) \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right) \left((1-i) x \cos (e)-(1+i) x \sin (e)+\frac{\text{RootSum}\left[d e^{2 i e} \text{$\#$1}^4+2 i c e^{i e} \text{$\#$1}^2-d\&,\frac{-i \sqrt{d} \sqrt{c+d} e^{i e} f x \text{$\#$1}^3+2 \sqrt{d} \sqrt{c+d} e^{i e} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^3-\frac{(1+i) c f x \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\frac{(2-2 i) c \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\sqrt{d} \sqrt{c+d} f x \text{$\#$1}+2 i \sqrt{d} \sqrt{c+d} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}+(1-i) d \sqrt{e^{-i e}} f x+(2+2 i) d \sqrt{e^{-i e}} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right)}{d-i c e^{i e} \text{$\#$1}^2}\&\right] \sqrt{\cos (e)-i \sin (e)} (-i \cos (e)+\sin (e)-1)}{4 f}\right)}{\sqrt{c+d} (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}+\frac{(2-2 i) B \sqrt{d} \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) \sin \left(\frac{f x}{2}\right)}{f}\right) \sqrt{a (\sin (e+f x)+1)}}{d^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{2 \sqrt{a} (B c-A d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{d^{3/2} f \sqrt{c+d}}-\frac{2 a B \cos (e+f x)}{d f \sqrt{a \sin (e+f x)+a}}",1,"((1/2 + I/2)*(((-2 + 2*I)*B*Sqrt[d]*Cos[(f*x)/2]*(Cos[e/2] - Sin[e/2]))/f + ((-(B*c) + A*d)*(Cos[e/2] + I*Sin[e/2])*((-1 + I)*x*Cos[e] + (RootSum[-d + (2*I)*c*E^(I*e)*#1^2 + d*E^((2*I)*e)*#1^4 & , ((1 + I)*d*Sqrt[E^((-I)*e)]*f*x - (2 - 2*I)*d*Sqrt[E^((-I)*e)]*Log[E^((I/2)*f*x) - #1] - I*Sqrt[d]*Sqrt[c + d]*f*x*#1 + 2*Sqrt[d]*Sqrt[c + d]*Log[E^((I/2)*f*x) - #1]*#1 + ((1 - I)*c*f*x*#1^2)/Sqrt[E^((-I)*e)] + ((2 + 2*I)*c*Log[E^((I/2)*f*x) - #1]*#1^2)/Sqrt[E^((-I)*e)] - Sqrt[d]*Sqrt[c + d]*E^(I*e)*f*x*#1^3 - (2*I)*Sqrt[d]*Sqrt[c + d]*E^(I*e)*Log[E^((I/2)*f*x) - #1]*#1^3)/(d - I*c*E^(I*e)*#1^2) & ]*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]])/(4*f) + (1 + I)*x*Sin[e]))/(Sqrt[c + d]*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]]) + ((-(B*c) + A*d)*(Cos[e/2] + I*Sin[e/2])*((1 - I)*x*Cos[e] - (1 + I)*x*Sin[e] + (RootSum[-d + (2*I)*c*E^(I*e)*#1^2 + d*E^((2*I)*e)*#1^4 & , ((1 - I)*d*Sqrt[E^((-I)*e)]*f*x + (2 + 2*I)*d*Sqrt[E^((-I)*e)]*Log[E^((I/2)*f*x) - #1] + Sqrt[d]*Sqrt[c + d]*f*x*#1 + (2*I)*Sqrt[d]*Sqrt[c + d]*Log[E^((I/2)*f*x) - #1]*#1 - ((1 + I)*c*f*x*#1^2)/Sqrt[E^((-I)*e)] + ((2 - 2*I)*c*Log[E^((I/2)*f*x) - #1]*#1^2)/Sqrt[E^((-I)*e)] - I*Sqrt[d]*Sqrt[c + d]*E^(I*e)*f*x*#1^3 + 2*Sqrt[d]*Sqrt[c + d]*E^(I*e)*Log[E^((I/2)*f*x) - #1]*#1^3)/(d - I*c*E^(I*e)*#1^2) & ]*Sqrt[Cos[e] - I*Sin[e]]*(-1 - I*Cos[e] + Sin[e]))/(4*f)))/(Sqrt[c + d]*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]]) + ((2 - 2*I)*B*Sqrt[d]*(Cos[e/2] + Sin[e/2])*Sin[(f*x)/2])/f)*Sqrt[a*(1 + Sin[e + f*x])])/(d^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","C",1
291,1,901,126,8.7965297,"\int \frac{\sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx","Integrate[(Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2,x]","\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{a (\sin (e+f x)+1)} \left(\frac{(A d+B (c+2 d)) \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right) \left((-1+i) x \cos (e)+(1+i) x \sin (e)+\frac{\text{RootSum}\left[d e^{2 i e} \text{$\#$1}^4+2 i c e^{i e} \text{$\#$1}^2-d\&,\frac{-\sqrt{d} \sqrt{c+d} e^{i e} f x \text{$\#$1}^3-2 i \sqrt{d} \sqrt{c+d} e^{i e} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^3+\frac{(1-i) c f x \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\frac{(2+2 i) c \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^2}{\sqrt{e^{-i e}}}-i \sqrt{d} \sqrt{c+d} f x \text{$\#$1}+2 \sqrt{d} \sqrt{c+d} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}+(1+i) d \sqrt{e^{-i e}} f x-(2-2 i) d \sqrt{e^{-i e}} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right)}{d-i c e^{i e} \text{$\#$1}^2}\&\right] (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}{4 f}\right)}{(c+d)^{3/2} (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}+\frac{(A d+B (c+2 d)) \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right) \left((1-i) x \cos (e)-(1+i) x \sin (e)+\frac{\text{RootSum}\left[d e^{2 i e} \text{$\#$1}^4+2 i c e^{i e} \text{$\#$1}^2-d\&,\frac{-i \sqrt{d} \sqrt{c+d} e^{i e} f x \text{$\#$1}^3+2 \sqrt{d} \sqrt{c+d} e^{i e} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^3-\frac{(1+i) c f x \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\frac{(2-2 i) c \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\sqrt{d} \sqrt{c+d} f x \text{$\#$1}+2 i \sqrt{d} \sqrt{c+d} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}+(1-i) d \sqrt{e^{-i e}} f x+(2+2 i) d \sqrt{e^{-i e}} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right)}{d-i c e^{i e} \text{$\#$1}^2}\&\right] \sqrt{\cos (e)-i \sin (e)} (-i \cos (e)+\sin (e)-1)}{4 f}\right)}{(c+d)^{3/2} (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}-\frac{(2-2 i) \sqrt{d} (A d-B c) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{(c+d) f (c+d \sin (e+f x))}\right)}{d^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{a (B c-A d) \cos (e+f x)}{d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{\sqrt{a} (A d+B (c+2 d)) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{d^{3/2} f (c+d)^{3/2}}",1,"((1/4 + I/4)*Sqrt[a*(1 + Sin[e + f*x])]*(((A*d + B*(c + 2*d))*(Cos[e/2] + I*Sin[e/2])*((-1 + I)*x*Cos[e] + (RootSum[-d + (2*I)*c*E^(I*e)*#1^2 + d*E^((2*I)*e)*#1^4 & , ((1 + I)*d*Sqrt[E^((-I)*e)]*f*x - (2 - 2*I)*d*Sqrt[E^((-I)*e)]*Log[E^((I/2)*f*x) - #1] - I*Sqrt[d]*Sqrt[c + d]*f*x*#1 + 2*Sqrt[d]*Sqrt[c + d]*Log[E^((I/2)*f*x) - #1]*#1 + ((1 - I)*c*f*x*#1^2)/Sqrt[E^((-I)*e)] + ((2 + 2*I)*c*Log[E^((I/2)*f*x) - #1]*#1^2)/Sqrt[E^((-I)*e)] - Sqrt[d]*Sqrt[c + d]*E^(I*e)*f*x*#1^3 - (2*I)*Sqrt[d]*Sqrt[c + d]*E^(I*e)*Log[E^((I/2)*f*x) - #1]*#1^3)/(d - I*c*E^(I*e)*#1^2) & ]*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]])/(4*f) + (1 + I)*x*Sin[e]))/((c + d)^(3/2)*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]]) + ((A*d + B*(c + 2*d))*(Cos[e/2] + I*Sin[e/2])*((1 - I)*x*Cos[e] - (1 + I)*x*Sin[e] + (RootSum[-d + (2*I)*c*E^(I*e)*#1^2 + d*E^((2*I)*e)*#1^4 & , ((1 - I)*d*Sqrt[E^((-I)*e)]*f*x + (2 + 2*I)*d*Sqrt[E^((-I)*e)]*Log[E^((I/2)*f*x) - #1] + Sqrt[d]*Sqrt[c + d]*f*x*#1 + (2*I)*Sqrt[d]*Sqrt[c + d]*Log[E^((I/2)*f*x) - #1]*#1 - ((1 + I)*c*f*x*#1^2)/Sqrt[E^((-I)*e)] + ((2 - 2*I)*c*Log[E^((I/2)*f*x) - #1]*#1^2)/Sqrt[E^((-I)*e)] - I*Sqrt[d]*Sqrt[c + d]*E^(I*e)*f*x*#1^3 + 2*Sqrt[d]*Sqrt[c + d]*E^(I*e)*Log[E^((I/2)*f*x) - #1]*#1^3)/(d - I*c*E^(I*e)*#1^2) & ]*Sqrt[Cos[e] - I*Sin[e]]*(-1 - I*Cos[e] + Sin[e]))/(4*f)))/((c + d)^(3/2)*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]]) - ((2 - 2*I)*Sqrt[d]*(-(B*c) + A*d)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c + d)*f*(c + d*Sin[e + f*x]))))/(d^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","C",1
292,1,967,192,10.266511,"\int \frac{\sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^3} \, dx","Integrate[(Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3,x]","\frac{\left(\frac{1}{16}+\frac{i}{16}\right) \sqrt{a (\sin (e+f x)+1)} \left(\frac{(3 A d+B (c+4 d)) \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right) \left((-1+i) x \cos (e)+(1+i) x \sin (e)+\frac{\text{RootSum}\left[d e^{2 i e} \text{$\#$1}^4+2 i c e^{i e} \text{$\#$1}^2-d\&,\frac{-\sqrt{d} \sqrt{c+d} e^{i e} f x \text{$\#$1}^3-2 i \sqrt{d} \sqrt{c+d} e^{i e} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^3+\frac{(1-i) c f x \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\frac{(2+2 i) c \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^2}{\sqrt{e^{-i e}}}-i \sqrt{d} \sqrt{c+d} f x \text{$\#$1}+2 \sqrt{d} \sqrt{c+d} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}+(1+i) d \sqrt{e^{-i e}} f x-(2-2 i) d \sqrt{e^{-i e}} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right)}{d-i c e^{i e} \text{$\#$1}^2}\&\right] (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}{4 f}\right)}{(c+d)^{5/2} (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}+\frac{(3 A d+B (c+4 d)) \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right) \left((1-i) x \cos (e)-(1+i) x \sin (e)+\frac{\text{RootSum}\left[d e^{2 i e} \text{$\#$1}^4+2 i c e^{i e} \text{$\#$1}^2-d\&,\frac{-i \sqrt{d} \sqrt{c+d} e^{i e} f x \text{$\#$1}^3+2 \sqrt{d} \sqrt{c+d} e^{i e} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^3-\frac{(1+i) c f x \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\frac{(2-2 i) c \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\sqrt{d} \sqrt{c+d} f x \text{$\#$1}+2 i \sqrt{d} \sqrt{c+d} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}+(1-i) d \sqrt{e^{-i e}} f x+(2+2 i) d \sqrt{e^{-i e}} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right)}{d-i c e^{i e} \text{$\#$1}^2}\&\right] \sqrt{\cos (e)-i \sin (e)} (-i \cos (e)+\sin (e)-1)}{4 f}\right)}{(c+d)^{5/2} (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}-\frac{(2-2 i) \sqrt{d} (3 A d+B (c+4 d)) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{(c+d)^2 f (c+d \sin (e+f x))}-\frac{(4-4 i) \sqrt{d} (A d-B c) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{(c+d) f (c+d \sin (e+f x))^2}\right)}{d^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{\sqrt{a} (3 A d+B (c+4 d)) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 d^{3/2} f (c+d)^{5/2}}-\frac{a (3 A d+B (c+4 d)) \cos (e+f x)}{4 d f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac{a (B c-A d) \cos (e+f x)}{2 d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}",1,"((1/16 + I/16)*Sqrt[a*(1 + Sin[e + f*x])]*(((3*A*d + B*(c + 4*d))*(Cos[e/2] + I*Sin[e/2])*((-1 + I)*x*Cos[e] + (RootSum[-d + (2*I)*c*E^(I*e)*#1^2 + d*E^((2*I)*e)*#1^4 & , ((1 + I)*d*Sqrt[E^((-I)*e)]*f*x - (2 - 2*I)*d*Sqrt[E^((-I)*e)]*Log[E^((I/2)*f*x) - #1] - I*Sqrt[d]*Sqrt[c + d]*f*x*#1 + 2*Sqrt[d]*Sqrt[c + d]*Log[E^((I/2)*f*x) - #1]*#1 + ((1 - I)*c*f*x*#1^2)/Sqrt[E^((-I)*e)] + ((2 + 2*I)*c*Log[E^((I/2)*f*x) - #1]*#1^2)/Sqrt[E^((-I)*e)] - Sqrt[d]*Sqrt[c + d]*E^(I*e)*f*x*#1^3 - (2*I)*Sqrt[d]*Sqrt[c + d]*E^(I*e)*Log[E^((I/2)*f*x) - #1]*#1^3)/(d - I*c*E^(I*e)*#1^2) & ]*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]])/(4*f) + (1 + I)*x*Sin[e]))/((c + d)^(5/2)*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]]) + ((3*A*d + B*(c + 4*d))*(Cos[e/2] + I*Sin[e/2])*((1 - I)*x*Cos[e] - (1 + I)*x*Sin[e] + (RootSum[-d + (2*I)*c*E^(I*e)*#1^2 + d*E^((2*I)*e)*#1^4 & , ((1 - I)*d*Sqrt[E^((-I)*e)]*f*x + (2 + 2*I)*d*Sqrt[E^((-I)*e)]*Log[E^((I/2)*f*x) - #1] + Sqrt[d]*Sqrt[c + d]*f*x*#1 + (2*I)*Sqrt[d]*Sqrt[c + d]*Log[E^((I/2)*f*x) - #1]*#1 - ((1 + I)*c*f*x*#1^2)/Sqrt[E^((-I)*e)] + ((2 - 2*I)*c*Log[E^((I/2)*f*x) - #1]*#1^2)/Sqrt[E^((-I)*e)] - I*Sqrt[d]*Sqrt[c + d]*E^(I*e)*f*x*#1^3 + 2*Sqrt[d]*Sqrt[c + d]*E^(I*e)*Log[E^((I/2)*f*x) - #1]*#1^3)/(d - I*c*E^(I*e)*#1^2) & ]*Sqrt[Cos[e] - I*Sin[e]]*(-1 - I*Cos[e] + Sin[e]))/(4*f)))/((c + d)^(5/2)*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]]) - ((4 - 4*I)*Sqrt[d]*(-(B*c) + A*d)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c + d)*f*(c + d*Sin[e + f*x])^2) - ((2 - 2*I)*Sqrt[d]*(3*A*d + B*(c + 4*d))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c + d)^2*f*(c + d*Sin[e + f*x]))))/(d^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","C",1
293,1,390,374,4.6068587,"\int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3,x]","-\frac{a \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-8 \left(11 A d \left(189 c^2+351 c d+137 d^2\right)+3 B \left(231 c^3+1287 c^2 d+1507 c d^2+581 d^3\right)\right) \cos (2 (e+f x))+70 d^2 (11 A d+33 B c+21 B d) \cos (4 (e+f x))+18480 A c^3 \sin (e+f x)+92400 A c^3+99792 A c^2 d \sin (e+f x)+216216 A c^2 d+100188 A c d^2 \sin (e+f x)-5940 A c d^2 \sin (3 (e+f x))+195624 A c d^2+35156 A d^3 \sin (e+f x)-3740 A d^3 \sin (3 (e+f x))+59158 A d^3+33264 B c^3 \sin (e+f x)+72072 B c^3+100188 B c^2 d \sin (e+f x)-5940 B c^2 d \sin (3 (e+f x))+195624 B c^2 d+105468 B c d^2 \sin (e+f x)-11220 B c d^2 \sin (3 (e+f x))+177474 B c d^2+34734 B d^3 \sin (e+f x)-4935 B d^3 \sin (3 (e+f x))+315 B d^3 \sin (5 (e+f x))+55482 B d^3\right)}{27720 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{2 a^2 \left(11 A d (c-17 d)-3 B \left(c^2-9 c d+56 d^2\right)\right) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{4 a^2 (c+d) \left(15 c^2+10 c d+7 d^2\right) \left(11 A d (c-17 d)-3 B \left(c^2-9 c d+56 d^2\right)\right) \cos (e+f x)}{3465 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{4 (c+d) \left(11 A d (c-17 d)-3 B \left(c^2-9 c d+56 d^2\right)\right) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 f}+\frac{8 a (5 c-d) (c+d) \left(11 A d (c-17 d)-3 B \left(c^2-9 c d+56 d^2\right)\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3465 d f}-\frac{2 a B \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}",1,"-1/27720*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(92400*A*c^3 + 72072*B*c^3 + 216216*A*c^2*d + 195624*B*c^2*d + 195624*A*c*d^2 + 177474*B*c*d^2 + 59158*A*d^3 + 55482*B*d^3 - 8*(11*A*d*(189*c^2 + 351*c*d + 137*d^2) + 3*B*(231*c^3 + 1287*c^2*d + 1507*c*d^2 + 581*d^3))*Cos[2*(e + f*x)] + 70*d^2*(33*B*c + 11*A*d + 21*B*d)*Cos[4*(e + f*x)] + 18480*A*c^3*Sin[e + f*x] + 33264*B*c^3*Sin[e + f*x] + 99792*A*c^2*d*Sin[e + f*x] + 100188*B*c^2*d*Sin[e + f*x] + 100188*A*c*d^2*Sin[e + f*x] + 105468*B*c*d^2*Sin[e + f*x] + 35156*A*d^3*Sin[e + f*x] + 34734*B*d^3*Sin[e + f*x] - 5940*B*c^2*d*Sin[3*(e + f*x)] - 5940*A*c*d^2*Sin[3*(e + f*x)] - 11220*B*c*d^2*Sin[3*(e + f*x)] - 3740*A*d^3*Sin[3*(e + f*x)] - 4935*B*d^3*Sin[3*(e + f*x)] + 315*B*d^3*Sin[5*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
294,1,267,294,2.2649905,"\int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]","-\frac{a \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-4 \left(9 A d (14 c+13 d)+B \left(63 c^2+234 c d+137 d^2\right)\right) \cos (2 (e+f x))+840 A c^2 \sin (e+f x)+4200 A c^2+3024 A c d \sin (e+f x)+6552 A c d+1518 A d^2 \sin (e+f x)-90 A d^2 \sin (3 (e+f x))+2964 A d^2+1512 B c^2 \sin (e+f x)+3276 B c^2+3036 B c d \sin (e+f x)-180 B c d \sin (3 (e+f x))+5928 B c d+1598 B d^2 \sin (e+f x)-170 B d^2 \sin (3 (e+f x))+35 B d^2 \cos (4 (e+f x))+2689 B d^2\right)}{1260 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{2 a^2 \left(15 c^2+10 c d+7 d^2\right) \left(3 A d (c-13 d)-B \left(c^2-7 c d+34 d^2\right)\right) \cos (e+f x)}{315 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (-9 A d+3 B c-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 \left(3 A d (c-13 d)-B \left(c^2-7 c d+34 d^2\right)\right) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}+\frac{4 a (5 c-d) \left(3 A d (c-13 d)-B \left(c^2-7 c d+34 d^2\right)\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 d f}-\frac{2 a B \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}",1,"-1/1260*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(4200*A*c^2 + 3276*B*c^2 + 6552*A*c*d + 5928*B*c*d + 2964*A*d^2 + 2689*B*d^2 - 4*(9*A*d*(14*c + 13*d) + B*(63*c^2 + 234*c*d + 137*d^2))*Cos[2*(e + f*x)] + 35*B*d^2*Cos[4*(e + f*x)] + 840*A*c^2*Sin[e + f*x] + 1512*B*c^2*Sin[e + f*x] + 3024*A*c*d*Sin[e + f*x] + 3036*B*c*d*Sin[e + f*x] + 1518*A*d^2*Sin[e + f*x] + 1598*B*d^2*Sin[e + f*x] - 180*B*c*d*Sin[3*(e + f*x)] - 90*A*d^2*Sin[3*(e + f*x)] - 170*B*d^2*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
295,1,144,165,1.0748608,"\int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]","-\frac{a \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) ((140 A c+252 A d+252 B c+253 B d) \sin (e+f x)-6 (7 A d+7 B c+13 B d) \cos (2 (e+f x))+700 A c+546 A d+546 B c-15 B d \sin (3 (e+f x))+494 B d)}{210 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{8 a^2 (35 A c+21 A d+21 B c+19 B d) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{2 (7 A d+7 B c-2 B d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac{2 a (35 A c+21 A d+21 B c+19 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{2 B d \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 a f}",1,"-1/210*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(700*A*c + 546*B*c + 546*A*d + 494*B*d - 6*(7*B*c + 7*A*d + 13*B*d)*Cos[2*(e + f*x)] + (140*A*c + 252*B*c + 252*A*d + 253*B*d)*Sin[e + f*x] - 15*B*d*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
296,1,101,101,0.3968694,"\int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]),x]","-\frac{a \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (2 (5 A+9 B) \sin (e+f x)+50 A-3 B \cos (2 (e+f x))+39 B)}{15 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{8 a^2 (5 A+3 B) \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a (5 A+3 B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{2 B \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}",1,"-1/15*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(50*A + 39*B - 3*B*Cos[2*(e + f*x)] + 2*(5*A + 9*B)*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
297,1,356,153,3.5350469,"\int \frac{(a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x))}{c+d \sin (e+f x)} \, dx","Integrate[((a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x]),x]","\frac{(a (\sin (e+f x)+1))^{3/2} \left(6 \sqrt{d} (2 A d-2 B c+3 B d) \sin \left(\frac{1}{2} (e+f x)\right)-6 \sqrt{d} (2 A d-2 B c+3 B d) \cos \left(\frac{1}{2} (e+f x)\right)+\frac{3 (c-d) (B c-A d) \left(2 \log \left(\sqrt{d} \sqrt{c+d} \left(\tan ^2\left(\frac{1}{4} (e+f x)\right)+2 \tan \left(\frac{1}{4} (e+f x)\right)-1\right)+(c+d) \sec ^2\left(\frac{1}{4} (e+f x)\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{\sqrt{c+d}}-\frac{3 (c-d) (B c-A d) \left(2 \log \left(-\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(-\sqrt{d} \sqrt{c+d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \sqrt{c+d} \cos \left(\frac{1}{2} (e+f x)\right)+c+d\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{\sqrt{c+d}}-2 B d^{3/2} \sin \left(\frac{3}{2} (e+f x)\right)-2 B d^{3/2} \cos \left(\frac{3}{2} (e+f x)\right)\right)}{6 d^{5/2} f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\frac{2 a^{3/2} (c-d) (B c-A d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{d^{5/2} f \sqrt{c+d}}+\frac{2 a^2 (-3 A d+3 B c-4 B d) \cos (e+f x)}{3 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a B \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 d f}",1,"((a*(1 + Sin[e + f*x]))^(3/2)*(-6*Sqrt[d]*(-2*B*c + 2*A*d + 3*B*d)*Cos[(e + f*x)/2] - 2*B*d^(3/2)*Cos[(3*(e + f*x))/2] - (3*(c - d)*(B*c - A*d)*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[-(Sec[(e + f*x)/4]^2*(c + d + Sqrt[d]*Sqrt[c + d]*Cos[(e + f*x)/2] - Sqrt[d]*Sqrt[c + d]*Sin[(e + f*x)/2]))]))/Sqrt[c + d] + (3*(c - d)*(B*c - A*d)*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[(c + d)*Sec[(e + f*x)/4]^2 + Sqrt[d]*Sqrt[c + d]*(-1 + 2*Tan[(e + f*x)/4] + Tan[(e + f*x)/4]^2)]))/Sqrt[c + d] + 6*Sqrt[d]*(-2*B*c + 2*A*d + 3*B*d)*Sin[(e + f*x)/2] - 2*B*d^(3/2)*Sin[(3*(e + f*x))/2]))/(6*d^(5/2)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","B",1
298,1,381,191,5.0134608,"\int \frac{(a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx","Integrate[((a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2,x]","\frac{(a (\sin (e+f x)+1))^{3/2} \left(\frac{\left(A d (c+3 d)+B \left(-3 c^2-3 c d+2 d^2\right)\right) \left(2 \log \left(\sqrt{d} \sqrt{c+d} \left(\tan ^2\left(\frac{1}{4} (e+f x)\right)+2 \tan \left(\frac{1}{4} (e+f x)\right)-1\right)+(c+d) \sec ^2\left(\frac{1}{4} (e+f x)\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{(c+d)^{3/2}}+\frac{\left(B \left(3 c^2+3 c d-2 d^2\right)-A d (c+3 d)\right) \left(2 \log \left(-\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(-\sqrt{d} \sqrt{c+d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \sqrt{c+d} \cos \left(\frac{1}{2} (e+f x)\right)+c+d\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{(c+d)^{3/2}}-\frac{4 \sqrt{d} (d-c) (A d-B c) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{(c+d) (c+d \sin (e+f x))}+8 B \sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)-8 B \sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)\right)}{4 d^{5/2} f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\frac{a^{3/2} \left(A d (c+3 d)-B \left(3 c^2+3 c d-2 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{d^{5/2} f (c+d)^{3/2}}-\frac{a^2 (-A d+3 B c+2 B d) \cos (e+f x)}{d^2 f (c+d) \sqrt{a \sin (e+f x)+a}}+\frac{a (B c-A d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{d f (c+d) (c+d \sin (e+f x))}",1,"((a*(1 + Sin[e + f*x]))^(3/2)*(-8*B*Sqrt[d]*Cos[(e + f*x)/2] + ((-(A*d*(c + 3*d)) + B*(3*c^2 + 3*c*d - 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[-(Sec[(e + f*x)/4]^2*(c + d + Sqrt[d]*Sqrt[c + d]*Cos[(e + f*x)/2] - Sqrt[d]*Sqrt[c + d]*Sin[(e + f*x)/2]))]))/(c + d)^(3/2) + ((A*d*(c + 3*d) + B*(-3*c^2 - 3*c*d + 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[(c + d)*Sec[(e + f*x)/4]^2 + Sqrt[d]*Sqrt[c + d]*(-1 + 2*Tan[(e + f*x)/4] + Tan[(e + f*x)/4]^2)]))/(c + d)^(3/2) + 8*B*Sqrt[d]*Sin[(e + f*x)/2] - (4*Sqrt[d]*(-c + d)*(-(B*c) + A*d)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c + d)*(c + d*Sin[e + f*x]))))/(4*d^(5/2)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
299,1,416,221,5.3406764,"\int \frac{(a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^3} \, dx","Integrate[((a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3,x]","\frac{(a (\sin (e+f x)+1))^{3/2} \left(-\frac{4 \sqrt{d} \left(A d (c+7 d)+B \left(-5 c^2-7 c d+4 d^2\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{(c+d)^2 (c+d \sin (e+f x))}+\frac{\left(A d (c+7 d)+3 B \left(c^2+3 c d+4 d^2\right)\right) \left(2 \log \left(\sqrt{d} \sqrt{c+d} \left(\tan ^2\left(\frac{1}{4} (e+f x)\right)+2 \tan \left(\frac{1}{4} (e+f x)\right)-1\right)+(c+d) \sec ^2\left(\frac{1}{4} (e+f x)\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{(c+d)^{5/2}}-\frac{\left(A d (c+7 d)+3 B \left(c^2+3 c d+4 d^2\right)\right) \left(2 \log \left(-\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(-\sqrt{d} \sqrt{c+d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \sqrt{c+d} \cos \left(\frac{1}{2} (e+f x)\right)+c+d\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{(c+d)^{5/2}}-\frac{8 \sqrt{d} (d-c) (A d-B c) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{(c+d) (c+d \sin (e+f x))^2}\right)}{16 d^{5/2} f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\frac{a^{3/2} \left(A d (c+7 d)+3 B \left(c^2+3 c d+4 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 d^{5/2} f (c+d)^{5/2}}+\frac{a^2 \left(A d (c-5 d)+B \left(3 c^2+5 c d-4 d^2\right)\right) \cos (e+f x)}{4 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac{a (B c-A d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{2 d f (c+d) (c+d \sin (e+f x))^2}",1,"((a*(1 + Sin[e + f*x]))^(3/2)*(-(((A*d*(c + 7*d) + 3*B*(c^2 + 3*c*d + 4*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[-(Sec[(e + f*x)/4]^2*(c + d + Sqrt[d]*Sqrt[c + d]*Cos[(e + f*x)/2] - Sqrt[d]*Sqrt[c + d]*Sin[(e + f*x)/2]))]))/(c + d)^(5/2)) + ((A*d*(c + 7*d) + 3*B*(c^2 + 3*c*d + 4*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[(c + d)*Sec[(e + f*x)/4]^2 + Sqrt[d]*Sqrt[c + d]*(-1 + 2*Tan[(e + f*x)/4] + Tan[(e + f*x)/4]^2)]))/(c + d)^(5/2) - (8*Sqrt[d]*(-c + d)*(-(B*c) + A*d)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c + d)*(c + d*Sin[e + f*x])^2) - (4*Sqrt[d]*(A*d*(c + 7*d) + B*(-5*c^2 - 7*c*d + 4*d^2))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c + d)^2*(c + d*Sin[e + f*x]))))/(16*d^(5/2)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
300,1,1565,534,6.8686345,"\int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3,x]","\frac{B \cos \left(\frac{13}{2} (e+f x)\right) (a (\sin (e+f x)+1))^{5/2} d^3}{416 f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}-\frac{B (a (\sin (e+f x)+1))^{5/2} \sin \left(\frac{13}{2} (e+f x)\right) d^3}{416 f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{\left(40 A c^3+30 B c^3+90 A d c^2+78 B d c^2+78 A d^2 c+69 B d^2 c+23 A d^3+21 B d^3\right) \left(\left(\frac{1}{16}-\frac{i}{16}\right) \sin \left(\frac{1}{2} (e+f x)\right)-\left(\frac{1}{16}+\frac{i}{16}\right) \cos \left(\frac{1}{2} (e+f x)\right)\right) (a (\sin (e+f x)+1))^{5/2}}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{\left(40 A c^3+30 B c^3+90 A d c^2+78 B d c^2+78 A d^2 c+69 B d^2 c+23 A d^3+21 B d^3\right) \left(\left(\frac{1}{16}+\frac{i}{16}\right) \sin \left(\frac{1}{2} (e+f x)\right)-\left(\frac{1}{16}-\frac{i}{16}\right) \cos \left(\frac{1}{2} (e+f x)\right)\right) (a (\sin (e+f x)+1))^{5/2}}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{\left(80 A c^3+88 B c^3+264 A d c^2+240 B d c^2+240 A d^2 c+228 B d^2 c+76 A d^3+71 B d^3\right) (a (\sin (e+f x)+1))^{5/2} \left(\left(-\frac{1}{192}+\frac{i}{192}\right) \cos \left(\frac{3}{2} (e+f x)\right)-\left(\frac{1}{192}+\frac{i}{192}\right) \sin \left(\frac{3}{2} (e+f x)\right)\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{\left(80 A c^3+88 B c^3+264 A d c^2+240 B d c^2+240 A d^2 c+228 B d^2 c+76 A d^3+71 B d^3\right) (a (\sin (e+f x)+1))^{5/2} \left(\left(-\frac{1}{192}-\frac{i}{192}\right) \cos \left(\frac{3}{2} (e+f x)\right)-\left(\frac{1}{192}-\frac{i}{192}\right) \sin \left(\frac{3}{2} (e+f x)\right)\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{\left(16 A c^3+40 B c^3+120 A d c^2+144 B d c^2+144 A d^2 c+150 B d^2 c+50 A d^3+51 B d^3\right) (a (\sin (e+f x)+1))^{5/2} \left(\left(\frac{1}{320}-\frac{i}{320}\right) \cos \left(\frac{5}{2} (e+f x)\right)-\left(\frac{1}{320}+\frac{i}{320}\right) \sin \left(\frac{5}{2} (e+f x)\right)\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{\left(16 A c^3+40 B c^3+120 A d c^2+144 B d c^2+144 A d^2 c+150 B d^2 c+50 A d^3+51 B d^3\right) (a (\sin (e+f x)+1))^{5/2} \left(\left(\frac{1}{320}+\frac{i}{320}\right) \cos \left(\frac{5}{2} (e+f x)\right)-\left(\frac{1}{320}-\frac{i}{320}\right) \sin \left(\frac{5}{2} (e+f x)\right)\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{\left(4 B c^3+12 A d c^2+30 B d c^2+30 A d^2 c+39 B d^2 c+13 A d^3+15 B d^3\right) (a (\sin (e+f x)+1))^{5/2} \left(\left(\frac{1}{224}+\frac{i}{224}\right) \cos \left(\frac{7}{2} (e+f x)\right)+\left(\frac{1}{224}-\frac{i}{224}\right) \sin \left(\frac{7}{2} (e+f x)\right)\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{\left(4 B c^3+12 A d c^2+30 B d c^2+30 A d^2 c+39 B d^2 c+13 A d^3+15 B d^3\right) (a (\sin (e+f x)+1))^{5/2} \left(\left(\frac{1}{224}-\frac{i}{224}\right) \cos \left(\frac{7}{2} (e+f x)\right)+\left(\frac{1}{224}+\frac{i}{224}\right) \sin \left(\frac{7}{2} (e+f x)\right)\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{\left(6 B c^2+6 A d c+15 B d c+5 A d^2+7 B d^2\right) (a (\sin (e+f x)+1))^{5/2} \left(\left(\frac{1}{288}-\frac{i}{288}\right) d \sin \left(\frac{9}{2} (e+f x)\right)-\left(\frac{1}{288}+\frac{i}{288}\right) d \cos \left(\frac{9}{2} (e+f x)\right)\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{\left(6 B c^2+6 A d c+15 B d c+5 A d^2+7 B d^2\right) (a (\sin (e+f x)+1))^{5/2} \left(\left(\frac{1}{288}+\frac{i}{288}\right) d \sin \left(\frac{9}{2} (e+f x)\right)-\left(\frac{1}{288}-\frac{i}{288}\right) d \cos \left(\frac{9}{2} (e+f x)\right)\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{(6 B c+2 A d+5 B d) (a (\sin (e+f x)+1))^{5/2} \left(\left(-\frac{1}{704}+\frac{i}{704}\right) \cos \left(\frac{11}{2} (e+f x)\right) d^2-\left(\frac{1}{704}+\frac{i}{704}\right) \sin \left(\frac{11}{2} (e+f x)\right) d^2\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{(6 B c+2 A d+5 B d) (a (\sin (e+f x)+1))^{5/2} \left(\left(-\frac{1}{704}-\frac{i}{704}\right) \cos \left(\frac{11}{2} (e+f x)\right) d^2-\left(\frac{1}{704}-\frac{i}{704}\right) \sin \left(\frac{11}{2} (e+f x)\right) d^2\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\frac{2 a^3 \left(-39 A c d+299 A d^2+15 B c^2-75 B c d+280 B d^2\right) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^3 \left(13 A d \left(3 c^2-38 c d+355 d^2\right)-B \left(15 c^3-150 c^2 d+799 c d^2-4184 d^3\right)\right) \cos (e+f x) (c+d \sin (e+f x))^3}{9009 d^3 f \sqrt{a \sin (e+f x)+a}}-\frac{4 a^3 (c+d) \left(15 c^2+10 c d+7 d^2\right) \left(13 A d \left(3 c^2-38 c d+355 d^2\right)-B \left(15 c^3-150 c^2 d+799 c d^2-4184 d^3\right)\right) \cos (e+f x)}{45045 d^3 f \sqrt{a \sin (e+f x)+a}}-\frac{8 a^2 (5 c-d) (c+d) \left(13 A d \left(3 c^2-38 c d+355 d^2\right)-B \left(15 c^3-150 c^2 d+799 c d^2-4184 d^3\right)\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{45045 d^2 f}+\frac{2 a^2 (-13 A d+5 B c-16 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac{4 a (c+d) \left(13 A d \left(3 c^2-38 c d+355 d^2\right)-B \left(15 c^3-150 c^2 d+799 c d^2-4184 d^3\right)\right) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{15015 d f}-\frac{2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^4}{13 d f}",1,"(B*d^3*Cos[(13*(e + f*x))/2]*(a*(1 + Sin[e + f*x]))^(5/2))/(416*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((40*A*c^3 + 30*B*c^3 + 90*A*c^2*d + 78*B*c^2*d + 78*A*c*d^2 + 69*B*c*d^2 + 23*A*d^3 + 21*B*d^3)*((-1/16 - I/16)*Cos[(e + f*x)/2] + (1/16 - I/16)*Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((40*A*c^3 + 30*B*c^3 + 90*A*c^2*d + 78*B*c^2*d + 78*A*c*d^2 + 69*B*c*d^2 + 23*A*d^3 + 21*B*d^3)*((-1/16 + I/16)*Cos[(e + f*x)/2] + (1/16 + I/16)*Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((80*A*c^3 + 88*B*c^3 + 264*A*c^2*d + 240*B*c^2*d + 240*A*c*d^2 + 228*B*c*d^2 + 76*A*d^3 + 71*B*d^3)*(a*(1 + Sin[e + f*x]))^(5/2)*((-1/192 + I/192)*Cos[(3*(e + f*x))/2] - (1/192 + I/192)*Sin[(3*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((80*A*c^3 + 88*B*c^3 + 264*A*c^2*d + 240*B*c^2*d + 240*A*c*d^2 + 228*B*c*d^2 + 76*A*d^3 + 71*B*d^3)*(a*(1 + Sin[e + f*x]))^(5/2)*((-1/192 - I/192)*Cos[(3*(e + f*x))/2] - (1/192 - I/192)*Sin[(3*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((16*A*c^3 + 40*B*c^3 + 120*A*c^2*d + 144*B*c^2*d + 144*A*c*d^2 + 150*B*c*d^2 + 50*A*d^3 + 51*B*d^3)*(a*(1 + Sin[e + f*x]))^(5/2)*((1/320 - I/320)*Cos[(5*(e + f*x))/2] - (1/320 + I/320)*Sin[(5*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((16*A*c^3 + 40*B*c^3 + 120*A*c^2*d + 144*B*c^2*d + 144*A*c*d^2 + 150*B*c*d^2 + 50*A*d^3 + 51*B*d^3)*(a*(1 + Sin[e + f*x]))^(5/2)*((1/320 + I/320)*Cos[(5*(e + f*x))/2] - (1/320 - I/320)*Sin[(5*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((4*B*c^3 + 12*A*c^2*d + 30*B*c^2*d + 30*A*c*d^2 + 39*B*c*d^2 + 13*A*d^3 + 15*B*d^3)*(a*(1 + Sin[e + f*x]))^(5/2)*((1/224 + I/224)*Cos[(7*(e + f*x))/2] + (1/224 - I/224)*Sin[(7*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((4*B*c^3 + 12*A*c^2*d + 30*B*c^2*d + 30*A*c*d^2 + 39*B*c*d^2 + 13*A*d^3 + 15*B*d^3)*(a*(1 + Sin[e + f*x]))^(5/2)*((1/224 - I/224)*Cos[(7*(e + f*x))/2] + (1/224 + I/224)*Sin[(7*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((6*B*c^2 + 6*A*c*d + 15*B*c*d + 5*A*d^2 + 7*B*d^2)*(a*(1 + Sin[e + f*x]))^(5/2)*((-1/288 - I/288)*d*Cos[(9*(e + f*x))/2] + (1/288 - I/288)*d*Sin[(9*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((6*B*c^2 + 6*A*c*d + 15*B*c*d + 5*A*d^2 + 7*B*d^2)*(a*(1 + Sin[e + f*x]))^(5/2)*((-1/288 + I/288)*d*Cos[(9*(e + f*x))/2] + (1/288 + I/288)*d*Sin[(9*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((6*B*c + 2*A*d + 5*B*d)*(a*(1 + Sin[e + f*x]))^(5/2)*((-1/704 + I/704)*d^2*Cos[(11*(e + f*x))/2] - (1/704 + I/704)*d^2*Sin[(11*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + ((6*B*c + 2*A*d + 5*B*d)*(a*(1 + Sin[e + f*x]))^(5/2)*((-1/704 - I/704)*d^2*Cos[(11*(e + f*x))/2] - (1/704 - I/704)*d^2*Sin[(11*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) - (B*d^3*(a*(1 + Sin[e + f*x]))^(5/2)*Sin[(13*(e + f*x))/2])/(416*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","C",1
301,1,891,429,6.6081481,"\int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(-277200 A \cos \left(\frac{1}{2} (e+f x)\right) c^2-207900 B \cos \left(\frac{1}{2} (e+f x)\right) c^2-46200 A \cos \left(\frac{3}{2} (e+f x)\right) c^2-50820 B \cos \left(\frac{3}{2} (e+f x)\right) c^2+5544 A \cos \left(\frac{5}{2} (e+f x)\right) c^2+13860 B \cos \left(\frac{5}{2} (e+f x)\right) c^2+1980 B \cos \left(\frac{7}{2} (e+f x)\right) c^2+277200 A \sin \left(\frac{1}{2} (e+f x)\right) c^2+207900 B \sin \left(\frac{1}{2} (e+f x)\right) c^2-46200 A \sin \left(\frac{3}{2} (e+f x)\right) c^2-50820 B \sin \left(\frac{3}{2} (e+f x)\right) c^2-5544 A \sin \left(\frac{5}{2} (e+f x)\right) c^2-13860 B \sin \left(\frac{5}{2} (e+f x)\right) c^2+1980 B \sin \left(\frac{7}{2} (e+f x)\right) c^2-415800 A d \cos \left(\frac{1}{2} (e+f x)\right) c-360360 B d \cos \left(\frac{1}{2} (e+f x)\right) c-101640 A d \cos \left(\frac{3}{2} (e+f x)\right) c-92400 B d \cos \left(\frac{3}{2} (e+f x)\right) c+27720 A d \cos \left(\frac{5}{2} (e+f x)\right) c+33264 B d \cos \left(\frac{5}{2} (e+f x)\right) c+3960 A d \cos \left(\frac{7}{2} (e+f x)\right) c+9900 B d \cos \left(\frac{7}{2} (e+f x)\right) c-1540 B d \cos \left(\frac{9}{2} (e+f x)\right) c+415800 A d \sin \left(\frac{1}{2} (e+f x)\right) c+360360 B d \sin \left(\frac{1}{2} (e+f x)\right) c-101640 A d \sin \left(\frac{3}{2} (e+f x)\right) c-92400 B d \sin \left(\frac{3}{2} (e+f x)\right) c-27720 A d \sin \left(\frac{5}{2} (e+f x)\right) c-33264 B d \sin \left(\frac{5}{2} (e+f x)\right) c+3960 A d \sin \left(\frac{7}{2} (e+f x)\right) c+9900 B d \sin \left(\frac{7}{2} (e+f x)\right) c+1540 B d \sin \left(\frac{9}{2} (e+f x)\right) c-180180 A d^2 \cos \left(\frac{1}{2} (e+f x)\right)-159390 B d^2 \cos \left(\frac{1}{2} (e+f x)\right)-46200 A d^2 \cos \left(\frac{3}{2} (e+f x)\right)-43890 B d^2 \cos \left(\frac{3}{2} (e+f x)\right)+16632 A d^2 \cos \left(\frac{5}{2} (e+f x)\right)+17325 B d^2 \cos \left(\frac{5}{2} (e+f x)\right)+4950 A d^2 \cos \left(\frac{7}{2} (e+f x)\right)+6435 B d^2 \cos \left(\frac{7}{2} (e+f x)\right)-770 A d^2 \cos \left(\frac{9}{2} (e+f x)\right)-1925 B d^2 \cos \left(\frac{9}{2} (e+f x)\right)-315 B d^2 \cos \left(\frac{11}{2} (e+f x)\right)+180180 A d^2 \sin \left(\frac{1}{2} (e+f x)\right)+159390 B d^2 \sin \left(\frac{1}{2} (e+f x)\right)-46200 A d^2 \sin \left(\frac{3}{2} (e+f x)\right)-43890 B d^2 \sin \left(\frac{3}{2} (e+f x)\right)-16632 A d^2 \sin \left(\frac{5}{2} (e+f x)\right)-17325 B d^2 \sin \left(\frac{5}{2} (e+f x)\right)+4950 A d^2 \sin \left(\frac{7}{2} (e+f x)\right)+6435 B d^2 \sin \left(\frac{7}{2} (e+f x)\right)+770 A d^2 \sin \left(\frac{9}{2} (e+f x)\right)+1925 B d^2 \sin \left(\frac{9}{2} (e+f x)\right)-315 B d^2 \sin \left(\frac{11}{2} (e+f x)\right)\right)}{55440 f \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}","\frac{2 a^3 \left(11 A d (3 c-19 d)-B \left(15 c^2-65 c d+194 d^2\right)\right) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^3 \left(15 c^2+10 c d+7 d^2\right) \left(11 A d \left(c^2-10 c d+73 d^2\right)-B \left(5 c^3-40 c^2 d+169 c d^2-710 d^3\right)\right) \cos (e+f x)}{3465 d^3 f \sqrt{a \sin (e+f x)+a}}-\frac{4 a^2 (5 c-d) \left(11 A d \left(c^2-10 c d+73 d^2\right)-B \left(5 c^3-40 c^2 d+169 c d^2-710 d^3\right)\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3465 d^2 f}+\frac{2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac{2 a \left(11 A d \left(c^2-10 c d+73 d^2\right)-B \left(5 c^3-40 c^2 d+169 c d^2-710 d^3\right)\right) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 d f}-\frac{2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}",1,"((a*(1 + Sin[e + f*x]))^(5/2)*(-277200*A*c^2*Cos[(e + f*x)/2] - 207900*B*c^2*Cos[(e + f*x)/2] - 415800*A*c*d*Cos[(e + f*x)/2] - 360360*B*c*d*Cos[(e + f*x)/2] - 180180*A*d^2*Cos[(e + f*x)/2] - 159390*B*d^2*Cos[(e + f*x)/2] - 46200*A*c^2*Cos[(3*(e + f*x))/2] - 50820*B*c^2*Cos[(3*(e + f*x))/2] - 101640*A*c*d*Cos[(3*(e + f*x))/2] - 92400*B*c*d*Cos[(3*(e + f*x))/2] - 46200*A*d^2*Cos[(3*(e + f*x))/2] - 43890*B*d^2*Cos[(3*(e + f*x))/2] + 5544*A*c^2*Cos[(5*(e + f*x))/2] + 13860*B*c^2*Cos[(5*(e + f*x))/2] + 27720*A*c*d*Cos[(5*(e + f*x))/2] + 33264*B*c*d*Cos[(5*(e + f*x))/2] + 16632*A*d^2*Cos[(5*(e + f*x))/2] + 17325*B*d^2*Cos[(5*(e + f*x))/2] + 1980*B*c^2*Cos[(7*(e + f*x))/2] + 3960*A*c*d*Cos[(7*(e + f*x))/2] + 9900*B*c*d*Cos[(7*(e + f*x))/2] + 4950*A*d^2*Cos[(7*(e + f*x))/2] + 6435*B*d^2*Cos[(7*(e + f*x))/2] - 1540*B*c*d*Cos[(9*(e + f*x))/2] - 770*A*d^2*Cos[(9*(e + f*x))/2] - 1925*B*d^2*Cos[(9*(e + f*x))/2] - 315*B*d^2*Cos[(11*(e + f*x))/2] + 277200*A*c^2*Sin[(e + f*x)/2] + 207900*B*c^2*Sin[(e + f*x)/2] + 415800*A*c*d*Sin[(e + f*x)/2] + 360360*B*c*d*Sin[(e + f*x)/2] + 180180*A*d^2*Sin[(e + f*x)/2] + 159390*B*d^2*Sin[(e + f*x)/2] - 46200*A*c^2*Sin[(3*(e + f*x))/2] - 50820*B*c^2*Sin[(3*(e + f*x))/2] - 101640*A*c*d*Sin[(3*(e + f*x))/2] - 92400*B*c*d*Sin[(3*(e + f*x))/2] - 46200*A*d^2*Sin[(3*(e + f*x))/2] - 43890*B*d^2*Sin[(3*(e + f*x))/2] - 5544*A*c^2*Sin[(5*(e + f*x))/2] - 13860*B*c^2*Sin[(5*(e + f*x))/2] - 27720*A*c*d*Sin[(5*(e + f*x))/2] - 33264*B*c*d*Sin[(5*(e + f*x))/2] - 16632*A*d^2*Sin[(5*(e + f*x))/2] - 17325*B*d^2*Sin[(5*(e + f*x))/2] + 1980*B*c^2*Sin[(7*(e + f*x))/2] + 3960*A*c*d*Sin[(7*(e + f*x))/2] + 9900*B*c*d*Sin[(7*(e + f*x))/2] + 4950*A*d^2*Sin[(7*(e + f*x))/2] + 6435*B*d^2*Sin[(7*(e + f*x))/2] + 1540*B*c*d*Sin[(9*(e + f*x))/2] + 770*A*d^2*Sin[(9*(e + f*x))/2] + 1925*B*d^2*Sin[(9*(e + f*x))/2] - 315*B*d^2*Sin[(11*(e + f*x))/2]))/(55440*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","B",1
302,1,202,212,4.2425833,"\int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]","-\frac{a^2 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (-4 (63 A c+180 A d+180 B c+254 B d) \cos (2 (e+f x))+2352 A c \sin (e+f x)+7476 A c+3030 A d \sin (e+f x)-90 A d \sin (3 (e+f x))+6240 A d+3030 B c \sin (e+f x)-90 B c \sin (3 (e+f x))+6240 B c+3116 B d \sin (e+f x)-260 B d \sin (3 (e+f x))+35 B d \cos (4 (e+f x))+5653 B d)}{1260 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{64 a^3 (21 A c+15 A d+15 B c+13 B d) \cos (e+f x)}{315 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 (21 A c+15 A d+15 B c+13 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 f}-\frac{2 (9 A d+9 B c-2 B d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{63 f}-\frac{2 a (21 A c+15 A d+15 B c+13 B d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}-\frac{2 B d \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f}",1,"-1/1260*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(7476*A*c + 6240*B*c + 6240*A*d + 5653*B*d - 4*(63*A*c + 180*B*c + 180*A*d + 254*B*d)*Cos[2*(e + f*x)] + 35*B*d*Cos[4*(e + f*x)] + 2352*A*c*Sin[e + f*x] + 3030*B*c*Sin[e + f*x] + 3030*A*d*Sin[e + f*x] + 3116*B*d*Sin[e + f*x] - 90*B*c*Sin[3*(e + f*x)] - 90*A*d*Sin[3*(e + f*x)] - 260*B*d*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
303,1,119,138,1.5118857,"\int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]),x]","-\frac{a^2 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) ((392 A+505 B) \sin (e+f x)-6 (7 A+20 B) \cos (2 (e+f x))+1246 A-15 B \sin (3 (e+f x))+1040 B)}{210 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{64 a^3 (7 A+5 B) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 (7 A+5 B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{2 a (7 A+5 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac{2 B \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f}",1,"-1/210*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(1246*A + 1040*B - 6*(7*A + 20*B)*Cos[2*(e + f*x)] + (392*A + 505*B)*Sin[e + f*x] - 15*B*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
304,1,450,218,5.8224277,"\int \frac{(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{c+d \sin (e+f x)} \, dx","Integrate[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x]),x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(30 \sqrt{d} \left(A d (5 d-2 c)+B \left(2 c^2-5 c d+5 d^2\right)\right) \sin \left(\frac{1}{2} (e+f x)\right)-30 \sqrt{d} \left(A d (5 d-2 c)+B \left(2 c^2-5 c d+5 d^2\right)\right) \cos \left(\frac{1}{2} (e+f x)\right)-5 d^{3/2} (2 A d-2 B c+5 B d) \sin \left(\frac{3}{2} (e+f x)\right)-5 d^{3/2} (2 A d-2 B c+5 B d) \cos \left(\frac{3}{2} (e+f x)\right)-\frac{15 (c-d)^2 (B c-A d) \left(2 \log \left(\sqrt{d} \sqrt{c+d} \left(\tan ^2\left(\frac{1}{4} (e+f x)\right)+2 \tan \left(\frac{1}{4} (e+f x)\right)-1\right)+(c+d) \sec ^2\left(\frac{1}{4} (e+f x)\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{\sqrt{c+d}}+\frac{15 (c-d)^2 (B c-A d) \left(2 \log \left(-\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(-\sqrt{d} \sqrt{c+d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \sqrt{c+d} \cos \left(\frac{1}{2} (e+f x)\right)+c+d\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{\sqrt{c+d}}-3 B d^{5/2} \sin \left(\frac{5}{2} (e+f x)\right)+3 B d^{5/2} \cos \left(\frac{5}{2} (e+f x)\right)\right)}{30 d^{7/2} f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","\frac{2 a^{5/2} (c-d)^2 (B c-A d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{d^{7/2} f \sqrt{c+d}}+\frac{2 a^3 \left(5 A d (3 c-7 d)-B \left(15 c^2-35 c d+32 d^2\right)\right) \cos (e+f x)}{15 d^3 f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (-5 A d+5 B c-8 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 d^2 f}-\frac{2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 d f}",1,"((a*(1 + Sin[e + f*x]))^(5/2)*(-30*Sqrt[d]*(A*d*(-2*c + 5*d) + B*(2*c^2 - 5*c*d + 5*d^2))*Cos[(e + f*x)/2] - 5*d^(3/2)*(-2*B*c + 2*A*d + 5*B*d)*Cos[(3*(e + f*x))/2] + 3*B*d^(5/2)*Cos[(5*(e + f*x))/2] + (15*(c - d)^2*(B*c - A*d)*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[-(Sec[(e + f*x)/4]^2*(c + d + Sqrt[d]*Sqrt[c + d]*Cos[(e + f*x)/2] - Sqrt[d]*Sqrt[c + d]*Sin[(e + f*x)/2]))]))/Sqrt[c + d] - (15*(c - d)^2*(B*c - A*d)*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[(c + d)*Sec[(e + f*x)/4]^2 + Sqrt[d]*Sqrt[c + d]*(-1 + 2*Tan[(e + f*x)/4] + Tan[(e + f*x)/4]^2)]))/Sqrt[c + d] + 30*Sqrt[d]*(A*d*(-2*c + 5*d) + B*(2*c^2 - 5*c*d + 5*d^2))*Sin[(e + f*x)/2] - 5*d^(3/2)*(-2*B*c + 2*A*d + 5*B*d)*Sin[(3*(e + f*x))/2] - 3*B*d^(5/2)*Sin[(5*(e + f*x))/2]))/(30*d^(7/2)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","B",0
305,1,460,265,5.966176,"\int \frac{(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx","Integrate[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2,x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(\frac{3 (c-d) \left(B \left(5 c^2+5 c d-2 d^2\right)-A d (3 c+5 d)\right) \left(2 \log \left(\sqrt{d} \sqrt{c+d} \left(\tan ^2\left(\frac{1}{4} (e+f x)\right)+2 \tan \left(\frac{1}{4} (e+f x)\right)-1\right)+(c+d) \sec ^2\left(\frac{1}{4} (e+f x)\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{(c+d)^{3/2}}-\frac{3 (c-d) \left(B \left(5 c^2+5 c d-2 d^2\right)-A d (3 c+5 d)\right) \left(2 \log \left(-\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(-\sqrt{d} \sqrt{c+d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \sqrt{c+d} \cos \left(\frac{1}{2} (e+f x)\right)+c+d\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{(c+d)^{3/2}}+12 \sqrt{d} (2 A d-4 B c+5 B d) \sin \left(\frac{1}{2} (e+f x)\right)-12 \sqrt{d} (2 A d-4 B c+5 B d) \cos \left(\frac{1}{2} (e+f x)\right)-\frac{12 \sqrt{d} (c-d)^2 (A d-B c) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{(c+d) (c+d \sin (e+f x))}-4 B d^{3/2} \sin \left(\frac{3}{2} (e+f x)\right)-4 B d^{3/2} \cos \left(\frac{3}{2} (e+f x)\right)\right)}{12 d^{7/2} f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","\frac{a^{5/2} (c-d) \left(A d (3 c+5 d)-B \left(5 c^2+5 c d-2 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{d^{7/2} f (c+d)^{3/2}}-\frac{a^3 \left(3 A d (3 c+d)-B \left(15 c^2-5 c d-14 d^2\right)\right) \cos (e+f x)}{3 d^3 f (c+d) \sqrt{a \sin (e+f x)+a}}-\frac{a^2 (-3 A d+5 B c+2 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 d^2 f (c+d)}+\frac{a (B c-A d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{d f (c+d) (c+d \sin (e+f x))}",1,"((a*(1 + Sin[e + f*x]))^(5/2)*(-12*Sqrt[d]*(-4*B*c + 2*A*d + 5*B*d)*Cos[(e + f*x)/2] - 4*B*d^(3/2)*Cos[(3*(e + f*x))/2] - (3*(c - d)*(-(A*d*(3*c + 5*d)) + B*(5*c^2 + 5*c*d - 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[-(Sec[(e + f*x)/4]^2*(c + d + Sqrt[d]*Sqrt[c + d]*Cos[(e + f*x)/2] - Sqrt[d]*Sqrt[c + d]*Sin[(e + f*x)/2]))]))/(c + d)^(3/2) + (3*(c - d)*(-(A*d*(3*c + 5*d)) + B*(5*c^2 + 5*c*d - 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[(c + d)*Sec[(e + f*x)/4]^2 + Sqrt[d]*Sqrt[c + d]*(-1 + 2*Tan[(e + f*x)/4] + Tan[(e + f*x)/4]^2)]))/(c + d)^(3/2) + 12*Sqrt[d]*(-4*B*c + 2*A*d + 5*B*d)*Sin[(e + f*x)/2] - (12*(c - d)^2*Sqrt[d]*(-(B*c) + A*d)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c + d)*(c + d*Sin[e + f*x])) - 4*B*d^(3/2)*Sin[(3*(e + f*x))/2]))/(12*d^(7/2)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",0
306,1,504,308,8.2399054,"\int \frac{(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^3} \, dx","Integrate[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3,x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(\frac{\left(A d \left(3 c^2+10 c d+19 d^2\right)-B \left(15 c^3+30 c^2 d+7 c d^2-20 d^3\right)\right) \left(2 \log \left(\sqrt{d} \sqrt{c+d} \left(\tan ^2\left(\frac{1}{4} (e+f x)\right)+2 \tan \left(\frac{1}{4} (e+f x)\right)-1\right)+(c+d) \sec ^2\left(\frac{1}{4} (e+f x)\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{(c+d)^{5/2}}+\frac{\left(B \left(15 c^3+30 c^2 d+7 c d^2-20 d^3\right)-A d \left(3 c^2+10 c d+19 d^2\right)\right) \left(2 \log \left(-\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(-\sqrt{d} \sqrt{c+d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \sqrt{c+d} \cos \left(\frac{1}{2} (e+f x)\right)+c+d\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{(c+d)^{5/2}}-\frac{4 \sqrt{d} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(d \left(A d \left(-5 c^2-6 c d+11 d^2\right)+B \left(25 c^3+34 c^2 d+c d^2+4 d^3\right)\right) \sin (e+f x)-3 A c^3 d-8 A c^2 d^2+9 A c d^3+2 A d^4+15 B c^4+20 B c^3 d-B c^2 d^2+10 B c d^3-4 B d^2 (c+d)^2 \cos (2 (e+f x))+4 B d^4\right)}{(c+d)^2 (c+d \sin (e+f x))^2}\right)}{16 d^{7/2} f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\frac{a^{5/2} \left(A d \left(3 c^2+10 c d+19 d^2\right)-B \left(15 c^3+30 c^2 d+7 c d^2-20 d^3\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 d^{7/2} f (c+d)^{5/2}}+\frac{a^3 \left(3 A d (c+3 d)-B \left(15 c^2+25 c d+4 d^2\right)\right) \cos (e+f x)}{4 d^3 f (c+d)^2 \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \left(A d (c+7 d)-B \left(5 c^2+7 c d-4 d^2\right)\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{4 d^2 f (c+d)^2 (c+d \sin (e+f x))}+\frac{a (B c-A d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 d f (c+d) (c+d \sin (e+f x))^2}",1,"((a*(1 + Sin[e + f*x]))^(5/2)*(((-(A*d*(3*c^2 + 10*c*d + 19*d^2)) + B*(15*c^3 + 30*c^2*d + 7*c*d^2 - 20*d^3))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[-(Sec[(e + f*x)/4]^2*(c + d + Sqrt[d]*Sqrt[c + d]*Cos[(e + f*x)/2] - Sqrt[d]*Sqrt[c + d]*Sin[(e + f*x)/2]))]))/(c + d)^(5/2) + ((A*d*(3*c^2 + 10*c*d + 19*d^2) - B*(15*c^3 + 30*c^2*d + 7*c*d^2 - 20*d^3))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[(c + d)*Sec[(e + f*x)/4]^2 + Sqrt[d]*Sqrt[c + d]*(-1 + 2*Tan[(e + f*x)/4] + Tan[(e + f*x)/4]^2)]))/(c + d)^(5/2) - (4*Sqrt[d]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(15*B*c^4 - 3*A*c^3*d + 20*B*c^3*d - 8*A*c^2*d^2 - B*c^2*d^2 + 9*A*c*d^3 + 10*B*c*d^3 + 2*A*d^4 + 4*B*d^4 - 4*B*d^2*(c + d)^2*Cos[2*(e + f*x)] + d*(A*d*(-5*c^2 - 6*c*d + 11*d^2) + B*(25*c^3 + 34*c^2*d + c*d^2 + 4*d^3))*Sin[e + f*x]))/((c + d)^2*(c + d*Sin[e + f*x])^2)))/(16*d^(7/2)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",0
307,1,375,284,0.8634874,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/Sqrt[a + a*Sin[e + f*x]],x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(-35 d \left(2 A d (6 c-d)+B \left(12 c^2-6 c d+5 d^2\right)\right) \sin \left(\frac{3}{2} (e+f x)\right)-35 d \left(2 A d (6 c-d)+B \left(12 c^2-6 c d+5 d^2\right)\right) \cos \left(\frac{3}{2} (e+f x)\right)+105 \left(4 A d \left(6 c^2-3 c d+2 d^2\right)+B \left(8 c^3-12 c^2 d+24 c d^2-5 d^3\right)\right) \sin \left(\frac{1}{2} (e+f x)\right)-105 \left(4 A d \left(6 c^2-3 c d+2 d^2\right)+B \left(8 c^3-12 c^2 d+24 c d^2-5 d^3\right)\right) \cos \left(\frac{1}{2} (e+f x)\right)+21 d^2 (B (d-6 c)-2 A d) \sin \left(\frac{5}{2} (e+f x)\right)+21 d^2 (2 A d+6 B c-B d) \cos \left(\frac{5}{2} (e+f x)\right)+(840+840 i) (-1)^{3/4} (A-B) (c-d)^3 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)+15 B d^3 \sin \left(\frac{7}{2} (e+f x)\right)+15 B d^3 \cos \left(\frac{7}{2} (e+f x)\right)\right)}{420 f \sqrt{a (\sin (e+f x)+1)}}","-\frac{2 d \left(7 A d (9 c-d)+B \left(24 c^2-15 c d+31 d^2\right)\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 a f}-\frac{4 \left(7 A d \left(21 c^2-12 c d+7 d^2\right)+B \left(36 c^3-63 c^2 d+144 c d^2-37 d^3\right)\right) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{2 (7 A d+6 B c-B d) \cos (e+f x) (c+d \sin (e+f x))^2}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{2} (A-B) (c-d)^3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{2 B \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt{a \sin (e+f x)+a}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((840 + 840*I)*(-1)^(3/4)*(A - B)*(c - d)^3*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] - 105*(4*A*d*(6*c^2 - 3*c*d + 2*d^2) + B*(8*c^3 - 12*c^2*d + 24*c*d^2 - 5*d^3))*Cos[(e + f*x)/2] - 35*d*(2*A*(6*c - d)*d + B*(12*c^2 - 6*c*d + 5*d^2))*Cos[(3*(e + f*x))/2] + 21*d^2*(6*B*c + 2*A*d - B*d)*Cos[(5*(e + f*x))/2] + 15*B*d^3*Cos[(7*(e + f*x))/2] + 105*(4*A*d*(6*c^2 - 3*c*d + 2*d^2) + B*(8*c^3 - 12*c^2*d + 24*c*d^2 - 5*d^3))*Sin[(e + f*x)/2] - 35*d*(2*A*(6*c - d)*d + B*(12*c^2 - 6*c*d + 5*d^2))*Sin[(3*(e + f*x))/2] + 21*d^2*(-2*A*d + B*(-6*c + d))*Sin[(5*(e + f*x))/2] + 15*B*d^3*Sin[(7*(e + f*x))/2]))/(420*f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
308,1,246,200,0.5381131,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/Sqrt[a + a*Sin[e + f*x]],x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(30 \left(A d (4 c-d)+2 B \left(c^2-c d+d^2\right)\right) \sin \left(\frac{1}{2} (e+f x)\right)-30 \left(A d (4 c-d)+2 B \left(c^2-c d+d^2\right)\right) \cos \left(\frac{1}{2} (e+f x)\right)+5 d (B (d-4 c)-2 A d) \sin \left(\frac{3}{2} (e+f x)\right)+5 d (B (d-4 c)-2 A d) \cos \left(\frac{3}{2} (e+f x)\right)+(60+60 i) (-1)^{3/4} (A-B) (c-d)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)-3 B d^2 \sin \left(\frac{5}{2} (e+f x)\right)+3 B d^2 \cos \left(\frac{5}{2} (e+f x)\right)\right)}{30 f \sqrt{a (\sin (e+f x)+1)}}","-\frac{4 \left(5 A d (3 c-d)+B \left(6 c^2-7 c d+7 d^2\right)\right) \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{2 d (5 A d+4 B c-B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 a f}-\frac{\sqrt{2} (A-B) (c-d)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{2 B \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt{a \sin (e+f x)+a}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((60 + 60*I)*(-1)^(3/4)*(A - B)*(c - d)^2*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] - 30*(A*(4*c - d)*d + 2*B*(c^2 - c*d + d^2))*Cos[(e + f*x)/2] + 5*d*(-2*A*d + B*(-4*c + d))*Cos[(3*(e + f*x))/2] + 3*B*d^2*Cos[(5*(e + f*x))/2] + 30*(A*(4*c - d)*d + 2*B*(c^2 - c*d + d^2))*Sin[(e + f*x)/2] + 5*d*(-2*A*d + B*(-4*c + d))*Sin[(3*(e + f*x))/2] - 3*B*d^2*Sin[(5*(e + f*x))/2]))/(30*f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
309,1,135,130,0.4796394,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]))/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (3 A d+3 B c+B d \sin (e+f x)-B d)-(6+6 i) (-1)^{3/4} (A-B) (c-d) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)\right)}{3 f \sqrt{a (\sin (e+f x)+1)}}","-\frac{2 (3 A d+3 B c-2 B d) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{2} (A-B) (c-d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{2 B d \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 a f}",1,"-1/3*((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((-6 - 6*I)*(-1)^(3/4)*(A - B)*(c - d)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] + 2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(3*B*c + 3*A*d - B*d + B*d*Sin[e + f*x])))/(f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
310,1,106,79,0.217873,"\int \frac{A+B \sin (e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(A + B*Sin[e + f*x])/Sqrt[a + a*Sin[e + f*x]],x]","\frac{2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(B \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)\right)+(1+i) (-1)^{3/4} (A-B) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)\right)}{f \sqrt{a (\sin (e+f x)+1)}}","-\frac{\sqrt{2} (A-B) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{2 B \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}",1,"(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((1 + I)*(-1)^(3/4)*(A - B)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] + B*(-Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))/(f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
311,1,238,136,3.3434215,"\int \frac{A+B \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Integrate[(A + B*Sin[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","\frac{(-1)^{3/4} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\sqrt[4]{-1} (B c-A d) \left(\log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{c+d}-\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-\log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{c+d}+\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)-\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)+(2+2 i) \sqrt{d} (A-B) \sqrt{c+d} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)\right)}{\sqrt{d} f (c-d) \sqrt{c+d} \sqrt{a (\sin (e+f x)+1)}}","-\frac{\sqrt{2} (A-B) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f (c-d)}-\frac{2 (B c-A d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} \sqrt{d} f (c-d) \sqrt{c+d}}",1,"((-1)^(3/4)*((2 + 2*I)*(A - B)*Sqrt[d]*Sqrt[c + d]*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] + (-1)^(1/4)*(B*c - A*d)*(Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] + Sqrt[d]*Cos[(e + f*x)/2] - Sqrt[d]*Sin[(e + f*x)/2])] - Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] - Sqrt[d]*Cos[(e + f*x)/2] + Sqrt[d]*Sin[(e + f*x)/2])]))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/((c - d)*Sqrt[d]*Sqrt[c + d]*f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
312,1,374,207,6.9695938,"\int \frac{A+B \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2} \, dx","Integrate[(A + B*Sin[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(-\frac{\left(B \left(c^2+c d+2 d^2\right)-A d (3 c+d)\right) \left(2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{c+d}-\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{\sqrt{d} (c+d)^{3/2}}+\frac{\left(B \left(c^2+c d+2 d^2\right)-A d (3 c+d)\right) \left(2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{c+d}+\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)-\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{\sqrt{d} (c+d)^{3/2}}-\frac{4 (c-d) (B c-A d) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{(c+d) (c+d \sin (e+f x))}+(8+8 i) (-1)^{3/4} (A-B) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)\right)}{4 f (c-d)^2 \sqrt{a (\sin (e+f x)+1)}}","-\frac{(B c-A d) \cos (e+f x)}{f \left(c^2-d^2\right) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac{\left(A d (3 c+d)-B \left(c^2+c d+2 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} \sqrt{d} f (c-d)^2 (c+d)^{3/2}}-\frac{\sqrt{2} (A-B) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f (c-d)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((8 + 8*I)*(-1)^(3/4)*(A - B)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] - ((-(A*d*(3*c + d)) + B*(c^2 + c*d + 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] + Sqrt[d]*Cos[(e + f*x)/2] - Sqrt[d]*Sin[(e + f*x)/2])]))/(Sqrt[d]*(c + d)^(3/2)) + ((-(A*d*(3*c + d)) + B*(c^2 + c*d + 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] - Sqrt[d]*Cos[(e + f*x)/2] + Sqrt[d]*Sin[(e + f*x)/2])]))/(Sqrt[d]*(c + d)^(3/2)) - (4*(c - d)*(B*c - A*d)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c + d)*(c + d*Sin[e + f*x]))))/(4*(c - d)^2*f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
313,1,847,309,10.7780903,"\int \frac{A+B \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3} \, dx","Integrate[(A + B*Sin[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^3),x]","\frac{(2+2 i) (A-B) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{1}{4} (e+f x)\right) \left(\cos \left(\frac{1}{4} (e+f x)\right)-\sin \left(\frac{1}{4} (e+f x)\right)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}{\left(\sqrt[4]{-1} c^3-3 \sqrt[4]{-1} d c^2+3 \sqrt[4]{-1} d^2 c-\sqrt[4]{-1} d^3\right) f \sqrt{a (\sin (e+f x)+1)}}-\frac{\left(B \left(3 c^3+6 d c^2+19 d^2 c+4 d^3\right)-A d \left(15 c^2+10 d c+7 d^2\right)\right) \left(e+f x-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)-\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{c+d}\right)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}{16 (c-d)^3 \sqrt{d} (c+d)^{5/2} f \sqrt{a (\sin (e+f x)+1)}}+\frac{\left(B \left(3 c^3+6 d c^2+19 d^2 c+4 d^3\right)-A d \left(15 c^2+10 d c+7 d^2\right)\right) \left(e+f x-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(-\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{c+d}\right)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}{16 (c-d)^3 \sqrt{d} (c+d)^{5/2} f \sqrt{a (\sin (e+f x)+1)}}+\frac{\left(-3 B \cos \left(\frac{1}{2} (e+f x)\right) c^2+3 B \sin \left(\frac{1}{2} (e+f x)\right) c^2+7 A d \cos \left(\frac{1}{2} (e+f x)\right) c-B d \cos \left(\frac{1}{2} (e+f x)\right) c-7 A d \sin \left(\frac{1}{2} (e+f x)\right) c+B d \sin \left(\frac{1}{2} (e+f x)\right) c+A d^2 \cos \left(\frac{1}{2} (e+f x)\right)-4 B d^2 \cos \left(\frac{1}{2} (e+f x)\right)-A d^2 \sin \left(\frac{1}{2} (e+f x)\right)+4 B d^2 \sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}{4 (c-d)^2 (c+d)^2 f \sqrt{a (\sin (e+f x)+1)} (c+d \sin (e+f x))}+\frac{\left(-B c \cos \left(\frac{1}{2} (e+f x)\right)+A d \cos \left(\frac{1}{2} (e+f x)\right)+B c \sin \left(\frac{1}{2} (e+f x)\right)-A d \sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 (c-d) (c+d) f \sqrt{a (\sin (e+f x)+1)} (c+d \sin (e+f x))^2}","\frac{\left(A d (7 c+d)-B \left(3 c^2+c d+4 d^2\right)\right) \cos (e+f x)}{4 f \left(c^2-d^2\right)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{(B c-A d) \cos (e+f x)}{2 f \left(c^2-d^2\right) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}+\frac{\left(A d \left(15 c^2+10 c d+7 d^2\right)-B \left(3 c^3+6 c^2 d+19 c d^2+4 d^3\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 \sqrt{a} \sqrt{d} f (c-d)^3 (c+d)^{5/2}}-\frac{\sqrt{2} (A-B) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f (c-d)^3}",1,"((2 + 2*I)*(A - B)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(e + f*x)/4]*(Cos[(e + f*x)/4] - Sin[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(((-1)^(1/4)*c^3 - 3*(-1)^(1/4)*c^2*d + 3*(-1)^(1/4)*c*d^2 - (-1)^(1/4)*d^3)*f*Sqrt[a*(1 + Sin[e + f*x])]) - ((-(A*d*(15*c^2 + 10*c*d + 7*d^2)) + B*(3*c^3 + 6*c^2*d + 19*c*d^2 + 4*d^3))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] + Sqrt[d]*Cos[(e + f*x)/2] - Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(16*(c - d)^3*Sqrt[d]*(c + d)^(5/2)*f*Sqrt[a*(1 + Sin[e + f*x])]) + ((-(A*d*(15*c^2 + 10*c*d + 7*d^2)) + B*(3*c^3 + 6*c^2*d + 19*c*d^2 + 4*d^3))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] - Sqrt[d]*Cos[(e + f*x)/2] + Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(16*(c - d)^3*Sqrt[d]*(c + d)^(5/2)*f*Sqrt[a*(1 + Sin[e + f*x])]) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-(B*c*Cos[(e + f*x)/2]) + A*d*Cos[(e + f*x)/2] + B*c*Sin[(e + f*x)/2] - A*d*Sin[(e + f*x)/2]))/(2*(c - d)*(c + d)*f*Sqrt[a*(1 + Sin[e + f*x])]*(c + d*Sin[e + f*x])^2) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-3*B*c^2*Cos[(e + f*x)/2] + 7*A*c*d*Cos[(e + f*x)/2] - B*c*d*Cos[(e + f*x)/2] + A*d^2*Cos[(e + f*x)/2] - 4*B*d^2*Cos[(e + f*x)/2] + 3*B*c^2*Sin[(e + f*x)/2] - 7*A*c*d*Sin[(e + f*x)/2] + B*c*d*Sin[(e + f*x)/2] - A*d^2*Sin[(e + f*x)/2] + 4*B*d^2*Sin[(e + f*x)/2]))/(4*(c - d)^2*(c + d)^2*f*Sqrt[a*(1 + Sin[e + f*x])]*(c + d*Sin[e + f*x]))","C",0
314,1,684,283,1.0665824,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left((30+30 i) (-1)^{3/4} (c-d)^2 (A (c+11 d)+3 B (c-5 d)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)+30 A c^3 \sin \left(\frac{1}{2} (e+f x)\right)-30 A c^3 \cos \left(\frac{1}{2} (e+f x)\right)-90 A c^2 d \sin \left(\frac{1}{2} (e+f x)\right)+90 A c^2 d \cos \left(\frac{1}{2} (e+f x)\right)+270 A c d^2 \sin \left(\frac{1}{2} (e+f x)\right)-180 A c d^2 \sin \left(\frac{3}{2} (e+f x)\right)-270 A c d^2 \cos \left(\frac{1}{2} (e+f x)\right)-180 A c d^2 \cos \left(\frac{3}{2} (e+f x)\right)-110 A d^3 \sin \left(\frac{1}{2} (e+f x)\right)+70 A d^3 \sin \left(\frac{3}{2} (e+f x)\right)-10 A d^3 \sin \left(\frac{5}{2} (e+f x)\right)+110 A d^3 \cos \left(\frac{1}{2} (e+f x)\right)+70 A d^3 \cos \left(\frac{3}{2} (e+f x)\right)+10 A d^3 \cos \left(\frac{5}{2} (e+f x)\right)-30 B c^3 \sin \left(\frac{1}{2} (e+f x)\right)+30 B c^3 \cos \left(\frac{1}{2} (e+f x)\right)+270 B c^2 d \sin \left(\frac{1}{2} (e+f x)\right)-180 B c^2 d \sin \left(\frac{3}{2} (e+f x)\right)-270 B c^2 d \cos \left(\frac{1}{2} (e+f x)\right)-180 B c^2 d \cos \left(\frac{3}{2} (e+f x)\right)-330 B c d^2 \sin \left(\frac{1}{2} (e+f x)\right)+210 B c d^2 \sin \left(\frac{3}{2} (e+f x)\right)-30 B c d^2 \sin \left(\frac{5}{2} (e+f x)\right)+330 B c d^2 \cos \left(\frac{1}{2} (e+f x)\right)+210 B c d^2 \cos \left(\frac{3}{2} (e+f x)\right)+30 B c d^2 \cos \left(\frac{5}{2} (e+f x)\right)+165 B d^3 \sin \left(\frac{1}{2} (e+f x)\right)-123 B d^3 \sin \left(\frac{3}{2} (e+f x)\right)+9 B d^3 \sin \left(\frac{5}{2} (e+f x)\right)+3 B d^3 \sin \left(\frac{7}{2} (e+f x)\right)-165 B d^3 \cos \left(\frac{1}{2} (e+f x)\right)-123 B d^3 \cos \left(\frac{3}{2} (e+f x)\right)-9 B d^3 \cos \left(\frac{5}{2} (e+f x)\right)+3 B d^3 \cos \left(\frac{7}{2} (e+f x)\right)\right)}{60 f (a (\sin (e+f x)+1))^{3/2}}","-\frac{(c-d)^2 (A (c+11 d)+3 B (c-5 d)) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}+\frac{d^2 (15 A c-35 A d-51 B c+39 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{30 a^2 f}+\frac{d \left(15 A c^2-120 A c d+65 A d^2-99 B c^2+168 B c d-93 B d^2\right) \cos (e+f x)}{15 a f \sqrt{a \sin (e+f x)+a}}-\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{2 f (a \sin (e+f x)+a)^{3/2}}+\frac{d (5 A-9 B) \cos (e+f x) (c+d \sin (e+f x))^2}{10 a f \sqrt{a \sin (e+f x)+a}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-30*A*c^3*Cos[(e + f*x)/2] + 30*B*c^3*Cos[(e + f*x)/2] + 90*A*c^2*d*Cos[(e + f*x)/2] - 270*B*c^2*d*Cos[(e + f*x)/2] - 270*A*c*d^2*Cos[(e + f*x)/2] + 330*B*c*d^2*Cos[(e + f*x)/2] + 110*A*d^3*Cos[(e + f*x)/2] - 165*B*d^3*Cos[(e + f*x)/2] - 180*B*c^2*d*Cos[(3*(e + f*x))/2] - 180*A*c*d^2*Cos[(3*(e + f*x))/2] + 210*B*c*d^2*Cos[(3*(e + f*x))/2] + 70*A*d^3*Cos[(3*(e + f*x))/2] - 123*B*d^3*Cos[(3*(e + f*x))/2] + 30*B*c*d^2*Cos[(5*(e + f*x))/2] + 10*A*d^3*Cos[(5*(e + f*x))/2] - 9*B*d^3*Cos[(5*(e + f*x))/2] + 3*B*d^3*Cos[(7*(e + f*x))/2] + 30*A*c^3*Sin[(e + f*x)/2] - 30*B*c^3*Sin[(e + f*x)/2] - 90*A*c^2*d*Sin[(e + f*x)/2] + 270*B*c^2*d*Sin[(e + f*x)/2] + 270*A*c*d^2*Sin[(e + f*x)/2] - 330*B*c*d^2*Sin[(e + f*x)/2] - 110*A*d^3*Sin[(e + f*x)/2] + 165*B*d^3*Sin[(e + f*x)/2] + (30 + 30*I)*(-1)^(3/4)*(c - d)^2*(3*B*(c - 5*d) + A*(c + 11*d))*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 180*B*c^2*d*Sin[(3*(e + f*x))/2] - 180*A*c*d^2*Sin[(3*(e + f*x))/2] + 210*B*c*d^2*Sin[(3*(e + f*x))/2] + 70*A*d^3*Sin[(3*(e + f*x))/2] - 123*B*d^3*Sin[(3*(e + f*x))/2] - 30*B*c*d^2*Sin[(5*(e + f*x))/2] - 10*A*d^3*Sin[(5*(e + f*x))/2] + 9*B*d^3*Sin[(5*(e + f*x))/2] + 3*B*d^3*Sin[(7*(e + f*x))/2]))/(60*f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
315,1,357,203,0.7423399,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(6 (A-B) (c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right)-3 (A-B) (c-d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+6 d (-2 A d-4 B c+3 B d) \cos \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-6 d (-2 A d-4 B c+3 B d) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+(3+3 i) (-1)^{3/4} (c-d) (A c+7 A d+3 B c-11 B d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)-2 B d^2 \cos \left(\frac{3}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-2 B d^2 \sin \left(\frac{3}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2\right)}{6 f (a (\sin (e+f x)+1))^{3/2}}","-\frac{(c-d) (A c+7 A d+3 B c-11 B d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}+\frac{d^2 (3 A-7 B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{6 a^2 f}+\frac{d (3 A c-9 A d-15 B c+13 B d) \cos (e+f x)}{3 a f \sqrt{a \sin (e+f x)+a}}-\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(6*(A - B)*(c - d)^2*Sin[(e + f*x)/2] - 3*(A - B)*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (3 + 3*I)*(-1)^(3/4)*(c - d)*(A*c + 3*B*c + 7*A*d - 11*B*d)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 6*d*(-4*B*c - 2*A*d + 3*B*d)*Cos[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 2*B*d^2*Cos[(3*(e + f*x))/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 6*d*(-4*B*c - 2*A*d + 3*B*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 2*B*d^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sin[(3*(e + f*x))/2]))/(6*f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
316,1,246,133,0.456095,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]))/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(2 (A-B) (c-d) \sin \left(\frac{1}{2} (e+f x)\right)-(A-B) (c-d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+(1+i) (-1)^{3/4} (A c+3 A d+3 B c-7 B d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)-4 B d \cos \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+4 B d \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2\right)}{2 f (a (\sin (e+f x)+1))^{3/2}}","-\frac{(A c+3 A d+3 B c-7 B d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{(A-B) (c-d) \cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2}}-\frac{2 B d \cos (e+f x)}{a f \sqrt{a \sin (e+f x)+a}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*(A - B)*(c - d)*Sin[(e + f*x)/2] - (A - B)*(c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (1 + I)*(-1)^(3/4)*(A*c + 3*B*c + 3*A*d - 7*B*d)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 4*B*d*Cos[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 4*B*d*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2))/(2*f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
317,1,150,87,0.2048089,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(2 (A-B) \sin \left(\frac{1}{2} (e+f x)\right)+(B-A) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+(1+i) (-1)^{3/4} (A+3 B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)\right)}{2 f (a (\sin (e+f x)+1))^{3/2}}","-\frac{(A+3 B) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{(A-B) \cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*(A - B)*Sin[(e + f*x)/2] + (-A + B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (1 + I)*(-1)^(3/4)*(A + 3*B)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2))/(2*f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
318,1,419,187,3.136518,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(2 (A-B) (c-d) \sin \left(\frac{1}{2} (e+f x)\right)+(B-A) (c-d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\frac{\sqrt{d} (B c-A d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{c+d}-\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{\sqrt{c+d}}+\frac{\sqrt{d} (A d-B c) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{c+d}+\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)-\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{\sqrt{c+d}}+(1+i) (-1)^{3/4} (A (c-5 d)+B (3 c+d)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)\right)}{2 f (c-d)^2 (a (\sin (e+f x)+1))^{3/2}}","-\frac{(A (c-5 d)+B (3 c+d)) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^2}+\frac{2 \sqrt{d} (B c-A d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{a^{3/2} f (c-d)^2 \sqrt{c+d}}-\frac{(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*(A - B)*(c - d)*Sin[(e + f*x)/2] + (-A + B)*(c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (1 + I)*(-1)^(3/4)*(A*(c - 5*d) + B*(3*c + d))*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (Sqrt[d]*(B*c - A*d)*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] + Sqrt[d]*Cos[(e + f*x)/2] - Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/Sqrt[c + d] + (Sqrt[d]*(-(B*c) + A*d)*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] - Sqrt[d]*Cos[(e + f*x)/2] + Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/Sqrt[c + d]))/(2*(c - d)^2*f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
319,1,542,292,9.5425824,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\frac{\sqrt{d} \left(B \left(3 c^2+3 c d+2 d^2\right)-A d (5 c+3 d)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{c+d}-\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{(c+d)^{3/2}}+\frac{\sqrt{d} \left(A d (5 c+3 d)-B \left(3 c^2+3 c d+2 d^2\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{c+d}+\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)-\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{(c+d)^{3/2}}+4 (A-B) (c-d) \sin \left(\frac{1}{2} (e+f x)\right)+\frac{4 d (c-d) (B c-A d) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}{(c+d) (c+d \sin (e+f x))}+2 (B-A) (c-d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+(2+2 i) (-1)^{3/4} (A c-9 A d+3 B c+5 B d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)\right)}{4 f (c-d)^3 (a (\sin (e+f x)+1))^{3/2}}","-\frac{\sqrt{d} \left(A d (5 c+3 d)-B \left(3 c^2+3 c d+2 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{a^{3/2} f (c-d)^3 (c+d)^{3/2}}-\frac{(A c-9 A d+3 B c+5 B d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^3}+\frac{d (B (3 c+d)-A (c+3 d)) \cos (e+f x)}{2 a f (c-d)^2 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(4*(A - B)*(c - d)*Sin[(e + f*x)/2] + 2*(-A + B)*(c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (2 + 2*I)*(-1)^(3/4)*(A*c + 3*B*c - 9*A*d + 5*B*d)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (Sqrt[d]*(-(A*d*(5*c + 3*d)) + B*(3*c^2 + 3*c*d + 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] + Sqrt[d]*Cos[(e + f*x)/2] - Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/(c + d)^(3/2) + (Sqrt[d]*(A*d*(5*c + 3*d) - B*(3*c^2 + 3*c*d + 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] - Sqrt[d]*Cos[(e + f*x)/2] + Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/(c + d)^(3/2) + (4*(c - d)*d*(B*c - A*d)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/((c + d)*(c + d*Sin[e + f*x]))))/(4*(c - d)^3*f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
320,1,1395,402,13.4844226,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3),x]","\frac{(1+i) (A c+3 B c-13 A d+9 B d) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{1}{4} (e+f x)\right) \left(\cos \left(\frac{1}{4} (e+f x)\right)-\sin \left(\frac{1}{4} (e+f x)\right)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{\left(2 \sqrt[4]{-1} c^4-8 \sqrt[4]{-1} d c^3+12 \sqrt[4]{-1} d^2 c^2-8 \sqrt[4]{-1} d^3 c+2 \sqrt[4]{-1} d^4\right) f (a (\sin (e+f x)+1))^{3/2}}+\frac{\sqrt{d} \left(3 B \left(5 c^3+10 d c^2+13 d^2 c+4 d^3\right)-A d \left(35 c^2+42 d c+19 d^2\right)\right) \left(e+f x-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)-\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{c+d}\right)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{16 (c-d)^4 (c+d)^{5/2} f (a (\sin (e+f x)+1))^{3/2}}-\frac{\sqrt{d} \left(3 B \left(5 c^3+10 d c^2+13 d^2 c+4 d^3\right)-A d \left(35 c^2+42 d c+19 d^2\right)\right) \left(e+f x-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(-\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{c+d}\right)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{16 (c-d)^4 (c+d)^{5/2} f (a (\sin (e+f x)+1))^{3/2}}+\frac{\left(-8 A \cos \left(\frac{1}{2} (e+f x)\right) c^4+8 B \cos \left(\frac{1}{2} (e+f x)\right) c^4+8 A \sin \left(\frac{1}{2} (e+f x)\right) c^4-8 B \sin \left(\frac{1}{2} (e+f x)\right) c^4-8 A d \cos \left(\frac{1}{2} (e+f x)\right) c^3+26 B d \cos \left(\frac{1}{2} (e+f x)\right) c^3-8 A d \cos \left(\frac{3}{2} (e+f x)\right) c^3+26 B d \cos \left(\frac{3}{2} (e+f x)\right) c^3+8 A d \sin \left(\frac{1}{2} (e+f x)\right) c^3-26 B d \sin \left(\frac{1}{2} (e+f x)\right) c^3-8 A d \sin \left(\frac{3}{2} (e+f x)\right) c^3+26 B d \sin \left(\frac{3}{2} (e+f x)\right) c^3-22 A d^2 \cos \left(\frac{1}{2} (e+f x)\right) c^2+6 B d^2 \cos \left(\frac{1}{2} (e+f x)\right) c^2-40 A d^2 \cos \left(\frac{3}{2} (e+f x)\right) c^2+31 B d^2 \cos \left(\frac{3}{2} (e+f x)\right) c^2+2 A d^2 \cos \left(\frac{5}{2} (e+f x)\right) c^2-9 B d^2 \cos \left(\frac{5}{2} (e+f x)\right) c^2+22 A d^2 \sin \left(\frac{1}{2} (e+f x)\right) c^2-6 B d^2 \sin \left(\frac{1}{2} (e+f x)\right) c^2-40 A d^2 \sin \left(\frac{3}{2} (e+f x)\right) c^2+31 B d^2 \sin \left(\frac{3}{2} (e+f x)\right) c^2-2 A d^2 \sin \left(\frac{5}{2} (e+f x)\right) c^2+9 B d^2 \sin \left(\frac{5}{2} (e+f x)\right) c^2-10 A d^3 \cos \left(\frac{1}{2} (e+f x)\right) c+4 B d^3 \cos \left(\frac{1}{2} (e+f x)\right) c-25 A d^3 \cos \left(\frac{3}{2} (e+f x)\right) c+13 B d^3 \cos \left(\frac{3}{2} (e+f x)\right) c+15 A d^3 \cos \left(\frac{5}{2} (e+f x)\right) c-9 B d^3 \cos \left(\frac{5}{2} (e+f x)\right) c+10 A d^3 \sin \left(\frac{1}{2} (e+f x)\right) c-4 B d^3 \sin \left(\frac{1}{2} (e+f x)\right) c-25 A d^3 \sin \left(\frac{3}{2} (e+f x)\right) c+13 B d^3 \sin \left(\frac{3}{2} (e+f x)\right) c-15 A d^3 \sin \left(\frac{5}{2} (e+f x)\right) c+9 B d^3 \sin \left(\frac{5}{2} (e+f x)\right) c+4 B d^4 \cos \left(\frac{1}{2} (e+f x)\right)+A d^4 \cos \left(\frac{3}{2} (e+f x)\right)+2 B d^4 \cos \left(\frac{3}{2} (e+f x)\right)+7 A d^4 \cos \left(\frac{5}{2} (e+f x)\right)-6 B d^4 \cos \left(\frac{5}{2} (e+f x)\right)-4 B d^4 \sin \left(\frac{1}{2} (e+f x)\right)+A d^4 \sin \left(\frac{3}{2} (e+f x)\right)+2 B d^4 \sin \left(\frac{3}{2} (e+f x)\right)-7 A d^4 \sin \left(\frac{5}{2} (e+f x)\right)+6 B d^4 \sin \left(\frac{5}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}{16 (c-d)^3 (c+d)^2 f (a (\sin (e+f x)+1))^{3/2} (c+d \sin (e+f x))^2}","-\frac{\sqrt{d} \left(A d \left(35 c^2+42 c d+19 d^2\right)-3 B \left(5 c^3+10 c^2 d+13 c d^2+4 d^3\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 a^{3/2} f (c-d)^4 (c+d)^{5/2}}-\frac{(A (c-13 d)+3 B (c+3 d)) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^4}+\frac{d \left(3 B \left(3 c^2+3 c d+2 d^2\right)-A \left(2 c^2+15 c d+7 d^2\right)\right) \cos (e+f x)}{4 a f (c-d)^3 (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac{d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{2 a f (c-d)^2 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}",1,"((1 + I)*(A*c + 3*B*c - 13*A*d + 9*B*d)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(e + f*x)/4]*(Cos[(e + f*x)/4] - Sin[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/((2*(-1)^(1/4)*c^4 - 8*(-1)^(1/4)*c^3*d + 12*(-1)^(1/4)*c^2*d^2 - 8*(-1)^(1/4)*c*d^3 + 2*(-1)^(1/4)*d^4)*f*(a*(1 + Sin[e + f*x]))^(3/2)) + (Sqrt[d]*(-(A*d*(35*c^2 + 42*c*d + 19*d^2)) + 3*B*(5*c^3 + 10*c^2*d + 13*c*d^2 + 4*d^3))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] + Sqrt[d]*Cos[(e + f*x)/2] - Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/(16*(c - d)^4*(c + d)^(5/2)*f*(a*(1 + Sin[e + f*x]))^(3/2)) - (Sqrt[d]*(-(A*d*(35*c^2 + 42*c*d + 19*d^2)) + 3*B*(5*c^3 + 10*c^2*d + 13*c*d^2 + 4*d^3))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] - Sqrt[d]*Cos[(e + f*x)/2] + Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/(16*(c - d)^4*(c + d)^(5/2)*f*(a*(1 + Sin[e + f*x]))^(3/2)) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-8*A*c^4*Cos[(e + f*x)/2] + 8*B*c^4*Cos[(e + f*x)/2] - 8*A*c^3*d*Cos[(e + f*x)/2] + 26*B*c^3*d*Cos[(e + f*x)/2] - 22*A*c^2*d^2*Cos[(e + f*x)/2] + 6*B*c^2*d^2*Cos[(e + f*x)/2] - 10*A*c*d^3*Cos[(e + f*x)/2] + 4*B*c*d^3*Cos[(e + f*x)/2] + 4*B*d^4*Cos[(e + f*x)/2] - 8*A*c^3*d*Cos[(3*(e + f*x))/2] + 26*B*c^3*d*Cos[(3*(e + f*x))/2] - 40*A*c^2*d^2*Cos[(3*(e + f*x))/2] + 31*B*c^2*d^2*Cos[(3*(e + f*x))/2] - 25*A*c*d^3*Cos[(3*(e + f*x))/2] + 13*B*c*d^3*Cos[(3*(e + f*x))/2] + A*d^4*Cos[(3*(e + f*x))/2] + 2*B*d^4*Cos[(3*(e + f*x))/2] + 2*A*c^2*d^2*Cos[(5*(e + f*x))/2] - 9*B*c^2*d^2*Cos[(5*(e + f*x))/2] + 15*A*c*d^3*Cos[(5*(e + f*x))/2] - 9*B*c*d^3*Cos[(5*(e + f*x))/2] + 7*A*d^4*Cos[(5*(e + f*x))/2] - 6*B*d^4*Cos[(5*(e + f*x))/2] + 8*A*c^4*Sin[(e + f*x)/2] - 8*B*c^4*Sin[(e + f*x)/2] + 8*A*c^3*d*Sin[(e + f*x)/2] - 26*B*c^3*d*Sin[(e + f*x)/2] + 22*A*c^2*d^2*Sin[(e + f*x)/2] - 6*B*c^2*d^2*Sin[(e + f*x)/2] + 10*A*c*d^3*Sin[(e + f*x)/2] - 4*B*c*d^3*Sin[(e + f*x)/2] - 4*B*d^4*Sin[(e + f*x)/2] - 8*A*c^3*d*Sin[(3*(e + f*x))/2] + 26*B*c^3*d*Sin[(3*(e + f*x))/2] - 40*A*c^2*d^2*Sin[(3*(e + f*x))/2] + 31*B*c^2*d^2*Sin[(3*(e + f*x))/2] - 25*A*c*d^3*Sin[(3*(e + f*x))/2] + 13*B*c*d^3*Sin[(3*(e + f*x))/2] + A*d^4*Sin[(3*(e + f*x))/2] + 2*B*d^4*Sin[(3*(e + f*x))/2] - 2*A*c^2*d^2*Sin[(5*(e + f*x))/2] + 9*B*c^2*d^2*Sin[(5*(e + f*x))/2] - 15*A*c*d^3*Sin[(5*(e + f*x))/2] + 9*B*c*d^3*Sin[(5*(e + f*x))/2] - 7*A*d^4*Sin[(5*(e + f*x))/2] + 6*B*d^4*Sin[(5*(e + f*x))/2]))/(16*(c - d)^3*(c + d)^2*f*(a*(1 + Sin[e + f*x]))^(3/2)*(c + d*Sin[e + f*x])^2)","C",0
321,1,523,308,1.791438,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(a + a*Sin[e + f*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left((3+3 i) (-1)^{3/4} (c-d) \left(3 A \left(c^2+6 c d+25 d^2\right)+B \left(5 c^2+62 c d-163 d^2\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)+(24+24 i) d^2 (-2 A d-6 B c+5 B d) \left(\cos \left(\frac{1}{2} (e+f x)\right)+i \sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4+(24+24 i) d^2 (2 A d+6 B c-5 B d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+i \cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4+24 (A-B) (c-d)^3 \sin \left(\frac{1}{2} (e+f x)\right)-3 (c-d)^2 (3 A (c+7 d)+B (5 c-29 d)) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+6 (c-d)^2 (3 A (c+7 d)+B (5 c-29 d)) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-12 (A-B) (c-d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-16 B d^3 \cos \left(\frac{3}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4-16 B d^3 \sin \left(\frac{3}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4\right)}{48 f (a (\sin (e+f x)+1))^{5/2}}","-\frac{(c-d) \left(3 A \left(c^2+6 c d+25 d^2\right)+B \left(5 c^2+62 c d-163 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}+\frac{d^2 (9 A c+39 A d+15 B c-95 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{48 a^3 f}+\frac{d \left(A \left(9 c^2+36 c d-93 d^2\right)+B \left(15 c^2-228 c d+197 d^2\right)\right) \cos (e+f x)}{24 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}-\frac{(3 A c+9 A d+5 B c-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{16 a f (a \sin (e+f x)+a)^{3/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(24*(A - B)*(c - d)^3*Sin[(e + f*x)/2] - 12*(A - B)*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 6*(c - d)^2*(B*(5*c - 29*d) + 3*A*(c + 7*d))*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 3*(c - d)^2*(B*(5*c - 29*d) + 3*A*(c + 7*d))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (3 + 3*I)*(-1)^(3/4)*(c - d)*(B*(5*c^2 + 62*c*d - 163*d^2) + 3*A*(c^2 + 6*c*d + 25*d^2))*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 16*B*d^3*Cos[(3*(e + f*x))/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + (24 + 24*I)*d^2*(-6*B*c - 2*A*d + 5*B*d)*(Cos[(e + f*x)/2] + I*Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + (24 + 24*I)*d^2*(6*B*c + 2*A*d - 5*B*d)*(I*Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 16*B*d^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*Sin[(3*(e + f*x))/2]))/(48*f*(a*(1 + Sin[e + f*x]))^(5/2))","C",1
322,1,544,219,1.140424,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left((2+2 i) (-1)^{3/4} \left(A \left(3 c^2+10 c d+19 d^2\right)+B \left(5 c^2+38 c d-75 d^2\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)+11 A c^2 \sin \left(\frac{1}{2} (e+f x)\right)-3 A c^2 \sin \left(\frac{3}{2} (e+f x)\right)-11 A c^2 \cos \left(\frac{1}{2} (e+f x)\right)-3 A c^2 \cos \left(\frac{3}{2} (e+f x)\right)-6 A c d \sin \left(\frac{1}{2} (e+f x)\right)-10 A c d \sin \left(\frac{3}{2} (e+f x)\right)+6 A c d \cos \left(\frac{1}{2} (e+f x)\right)-10 A c d \cos \left(\frac{3}{2} (e+f x)\right)-5 A d^2 \sin \left(\frac{1}{2} (e+f x)\right)+13 A d^2 \sin \left(\frac{3}{2} (e+f x)\right)+5 A d^2 \cos \left(\frac{1}{2} (e+f x)\right)+13 A d^2 \cos \left(\frac{3}{2} (e+f x)\right)-3 B c^2 \sin \left(\frac{1}{2} (e+f x)\right)-5 B c^2 \sin \left(\frac{3}{2} (e+f x)\right)+3 B c^2 \cos \left(\frac{1}{2} (e+f x)\right)-5 B c^2 \cos \left(\frac{3}{2} (e+f x)\right)-10 B c d \sin \left(\frac{1}{2} (e+f x)\right)+26 B c d \sin \left(\frac{3}{2} (e+f x)\right)+10 B c d \cos \left(\frac{1}{2} (e+f x)\right)+26 B c d \cos \left(\frac{3}{2} (e+f x)\right)+45 B d^2 \sin \left(\frac{1}{2} (e+f x)\right)-69 B d^2 \sin \left(\frac{3}{2} (e+f x)\right)-16 B d^2 \sin \left(\frac{5}{2} (e+f x)\right)-45 B d^2 \cos \left(\frac{1}{2} (e+f x)\right)-69 B d^2 \cos \left(\frac{3}{2} (e+f x)\right)+16 B d^2 \cos \left(\frac{5}{2} (e+f x)\right)\right)}{32 f (a (\sin (e+f x)+1))^{5/2}}","-\frac{\left(A \left(3 c^2+10 c d+19 d^2\right)+B \left(5 c^2+38 c d-75 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}+\frac{d^2 (A-9 B) \cos (e+f x)}{4 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{4 f (a \sin (e+f x)+a)^{5/2}}-\frac{(c-d) (3 A c+5 A d+5 B c-13 B d) \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-11*A*c^2*Cos[(e + f*x)/2] + 3*B*c^2*Cos[(e + f*x)/2] + 6*A*c*d*Cos[(e + f*x)/2] + 10*B*c*d*Cos[(e + f*x)/2] + 5*A*d^2*Cos[(e + f*x)/2] - 45*B*d^2*Cos[(e + f*x)/2] - 3*A*c^2*Cos[(3*(e + f*x))/2] - 5*B*c^2*Cos[(3*(e + f*x))/2] - 10*A*c*d*Cos[(3*(e + f*x))/2] + 26*B*c*d*Cos[(3*(e + f*x))/2] + 13*A*d^2*Cos[(3*(e + f*x))/2] - 69*B*d^2*Cos[(3*(e + f*x))/2] + 16*B*d^2*Cos[(5*(e + f*x))/2] + 11*A*c^2*Sin[(e + f*x)/2] - 3*B*c^2*Sin[(e + f*x)/2] - 6*A*c*d*Sin[(e + f*x)/2] - 10*B*c*d*Sin[(e + f*x)/2] - 5*A*d^2*Sin[(e + f*x)/2] + 45*B*d^2*Sin[(e + f*x)/2] + (2 + 2*I)*(-1)^(3/4)*(B*(5*c^2 + 38*c*d - 75*d^2) + A*(3*c^2 + 10*c*d + 19*d^2))*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 3*A*c^2*Sin[(3*(e + f*x))/2] - 5*B*c^2*Sin[(3*(e + f*x))/2] - 10*A*c*d*Sin[(3*(e + f*x))/2] + 26*B*c*d*Sin[(3*(e + f*x))/2] + 13*A*d^2*Sin[(3*(e + f*x))/2] - 69*B*d^2*Sin[(3*(e + f*x))/2] - 16*B*d^2*Sin[(5*(e + f*x))/2]))/(32*f*(a*(1 + Sin[e + f*x]))^(5/2))","C",1
323,1,267,151,0.7702628,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]))/(a + a*Sin[e + f*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(8 (A-B) (c-d) \sin \left(\frac{1}{2} (e+f x)\right)-(3 A c+5 A d+5 B c-13 B d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+2 (3 A c+5 A d+5 B c-13 B d) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-4 (A-B) (c-d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+(1+i) (-1)^{3/4} (3 A c+5 A d+5 B c+19 B d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)\right)}{16 f (a (\sin (e+f x)+1))^{5/2}}","-\frac{(3 A c+5 A d+5 B c+19 B d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}-\frac{(3 A c+5 A d+5 B c-13 B d) \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac{(A-B) (c-d) \cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(8*(A - B)*(c - d)*Sin[(e + f*x)/2] - 4*(A - B)*(c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(3*A*c + 5*B*c + 5*A*d - 13*B*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - (3*A*c + 5*B*c + 5*A*d - 13*B*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (1 + I)*(-1)^(3/4)*(3*A*c + 5*B*c + 5*A*d + 19*B*d)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4))/(16*f*(a*(1 + Sin[e + f*x]))^(5/2))","C",1
324,1,227,126,0.3723947,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/(a + a*Sin[e + f*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(8 (A-B) \sin \left(\frac{1}{2} (e+f x)\right)-(3 A+5 B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+2 (3 A+5 B) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+4 (B-A) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+(1+i) (-1)^{3/4} (3 A+5 B) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)\right)}{16 f (a (\sin (e+f x)+1))^{5/2}}","-\frac{(3 A+5 B) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}-\frac{(3 A+5 B) \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac{(A-B) \cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(8*(A - B)*Sin[(e + f*x)/2] + 4*(-A + B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(3*A + 5*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - (3*A + 5*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (1 + I)*(-1)^(3/4)*(3*A + 5*B)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4))/(16*f*(a*(1 + Sin[e + f*x]))^(5/2))","C",1
325,1,550,261,5.6103958,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left((1+i) (-1)^{3/4} \left(A \left(3 c^2-14 c d+43 d^2\right)+B \left(5 c^2-34 c d-3 d^2\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)+\frac{8 d^{3/2} (A d-B c) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \left(2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{c+d}-\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{\sqrt{c+d}}+\frac{8 d^{3/2} (B c-A d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \left(2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{c+d}+\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)-\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+e+f x\right)}{\sqrt{c+d}}+8 (A-B) (c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right)-(c-d) (3 A c-11 A d+5 B c+3 B d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+2 (c-d) (3 A c-11 A d+5 B c+3 B d) \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+4 (B-A) (c-d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{16 f (c-d)^3 (a (\sin (e+f x)+1))^{5/2}}","-\frac{\left(A \left(3 c^2-14 c d+43 d^2\right)+B \left(5 c^2-34 c d-3 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^3}-\frac{2 d^{3/2} (B c-A d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{a^{5/2} f (c-d)^3 \sqrt{c+d}}-\frac{(3 A c-11 A d+5 B c+3 B d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2}}-\frac{(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(8*(A - B)*(c - d)^2*Sin[(e + f*x)/2] + 4*(-A + B)*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(c - d)*(3*A*c + 5*B*c - 11*A*d + 3*B*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - (c - d)*(3*A*c + 5*B*c - 11*A*d + 3*B*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (1 + I)*(-1)^(3/4)*(B*(5*c^2 - 34*c*d - 3*d^2) + A*(3*c^2 - 14*c*d + 43*d^2))*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + (8*d^(3/2)*(-(B*c) + A*d)*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] + Sqrt[d]*Cos[(e + f*x)/2] - Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/Sqrt[c + d] + (8*d^(3/2)*(B*c - A*d)*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] - Sqrt[d]*Cos[(e + f*x)/2] + Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/Sqrt[c + d]))/(16*(c - d)^3*f*(a*(1 + Sin[e + f*x]))^(5/2))","C",1
326,1,1318,395,12.3748959,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^2),x]","\frac{(1+i) \left(3 A c^2+5 B c^2-22 A d c-58 B d c+115 A d^2-43 B d^2\right) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{1}{4} (e+f x)\right) \left(\cos \left(\frac{1}{4} (e+f x)\right)-\sin \left(\frac{1}{4} (e+f x)\right)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}{\left(16 \sqrt[4]{-1} c^4-64 \sqrt[4]{-1} d c^3+96 \sqrt[4]{-1} d^2 c^2-64 \sqrt[4]{-1} d^3 c+16 \sqrt[4]{-1} d^4\right) f (a (\sin (e+f x)+1))^{5/2}}+\frac{d^{3/2} \left(A d (7 c+5 d)-B \left(5 c^2+5 d c+2 d^2\right)\right) \left(e+f x-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)-\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{c+d}\right)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}{4 (c-d)^4 (c+d)^{3/2} f (a (\sin (e+f x)+1))^{5/2}}+\frac{d^{3/2} \left(B \left(5 c^2+5 d c+2 d^2\right)-A d (7 c+5 d)\right) \left(e+f x-2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)+2 \log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(-\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{c+d}\right)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}{4 (c-d)^4 (c+d)^{3/2} f (a (\sin (e+f x)+1))^{5/2}}+\frac{\left(-22 A \cos \left(\frac{1}{2} (e+f x)\right) c^3+6 B \cos \left(\frac{1}{2} (e+f x)\right) c^3-6 A \cos \left(\frac{3}{2} (e+f x)\right) c^3-10 B \cos \left(\frac{3}{2} (e+f x)\right) c^3+22 A \sin \left(\frac{1}{2} (e+f x)\right) c^3-6 B \sin \left(\frac{1}{2} (e+f x)\right) c^3-6 A \sin \left(\frac{3}{2} (e+f x)\right) c^3-10 B \sin \left(\frac{3}{2} (e+f x)\right) c^3+40 A d \cos \left(\frac{1}{2} (e+f x)\right) c^2-40 B d \cos \left(\frac{1}{2} (e+f x)\right) c^2+21 A d \cos \left(\frac{3}{2} (e+f x)\right) c^2-29 B d \cos \left(\frac{3}{2} (e+f x)\right) c^2+3 A d \cos \left(\frac{5}{2} (e+f x)\right) c^2+5 B d \cos \left(\frac{5}{2} (e+f x)\right) c^2-40 A d \sin \left(\frac{1}{2} (e+f x)\right) c^2+40 B d \sin \left(\frac{1}{2} (e+f x)\right) c^2+21 A d \sin \left(\frac{3}{2} (e+f x)\right) c^2-29 B d \sin \left(\frac{3}{2} (e+f x)\right) c^2-3 A d \sin \left(\frac{5}{2} (e+f x)\right) c^2-5 B d \sin \left(\frac{5}{2} (e+f x)\right) c^2+54 A d^2 \cos \left(\frac{1}{2} (e+f x)\right) c-70 B d^2 \cos \left(\frac{1}{2} (e+f x)\right) c+54 A d^2 \cos \left(\frac{3}{2} (e+f x)\right) c-86 B d^2 \cos \left(\frac{3}{2} (e+f x)\right) c-16 A d^2 \cos \left(\frac{5}{2} (e+f x)\right) c+32 B d^2 \cos \left(\frac{5}{2} (e+f x)\right) c-54 A d^2 \sin \left(\frac{1}{2} (e+f x)\right) c+70 B d^2 \sin \left(\frac{1}{2} (e+f x)\right) c+54 A d^2 \sin \left(\frac{3}{2} (e+f x)\right) c-86 B d^2 \sin \left(\frac{3}{2} (e+f x)\right) c+16 A d^2 \sin \left(\frac{5}{2} (e+f x)\right) c-32 B d^2 \sin \left(\frac{5}{2} (e+f x)\right) c+24 A d^3 \cos \left(\frac{1}{2} (e+f x)\right)+8 B d^3 \cos \left(\frac{1}{2} (e+f x)\right)+75 A d^3 \cos \left(\frac{3}{2} (e+f x)\right)-19 B d^3 \cos \left(\frac{3}{2} (e+f x)\right)-35 A d^3 \cos \left(\frac{5}{2} (e+f x)\right)+11 B d^3 \cos \left(\frac{5}{2} (e+f x)\right)-24 A d^3 \sin \left(\frac{1}{2} (e+f x)\right)-8 B d^3 \sin \left(\frac{1}{2} (e+f x)\right)+75 A d^3 \sin \left(\frac{3}{2} (e+f x)\right)-19 B d^3 \sin \left(\frac{3}{2} (e+f x)\right)+35 A d^3 \sin \left(\frac{5}{2} (e+f x)\right)-11 B d^3 \sin \left(\frac{5}{2} (e+f x)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}{64 (c-d)^3 (c+d) f (a (\sin (e+f x)+1))^{5/2} (c+d \sin (e+f x))}","-\frac{\left(A \left(3 c^2-22 c d+115 d^2\right)+B \left(5 c^2-58 c d-43 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^4}+\frac{d^{3/2} \left(A d (7 c+5 d)-B \left(5 c^2+5 c d+2 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{a^{5/2} f (c-d)^4 (c+d)^{3/2}}-\frac{d \left(A \left(3 c^2-16 c d-35 d^2\right)+B \left(5 c^2+32 c d+11 d^2\right)\right) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{(3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}-\frac{(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}",1,"((1 + I)*(3*A*c^2 + 5*B*c^2 - 22*A*c*d - 58*B*c*d + 115*A*d^2 - 43*B*d^2)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(e + f*x)/4]*(Cos[(e + f*x)/4] - Sin[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/((16*(-1)^(1/4)*c^4 - 64*(-1)^(1/4)*c^3*d + 96*(-1)^(1/4)*c^2*d^2 - 64*(-1)^(1/4)*c*d^3 + 16*(-1)^(1/4)*d^4)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + (d^(3/2)*(A*d*(7*c + 5*d) - B*(5*c^2 + 5*c*d + 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] + Sqrt[d]*Cos[(e + f*x)/2] - Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/(4*(c - d)^4*(c + d)^(3/2)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + (d^(3/2)*(-(A*d*(7*c + 5*d)) + B*(5*c^2 + 5*c*d + 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] - Sqrt[d]*Cos[(e + f*x)/2] + Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/(4*(c - d)^4*(c + d)^(3/2)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-22*A*c^3*Cos[(e + f*x)/2] + 6*B*c^3*Cos[(e + f*x)/2] + 40*A*c^2*d*Cos[(e + f*x)/2] - 40*B*c^2*d*Cos[(e + f*x)/2] + 54*A*c*d^2*Cos[(e + f*x)/2] - 70*B*c*d^2*Cos[(e + f*x)/2] + 24*A*d^3*Cos[(e + f*x)/2] + 8*B*d^3*Cos[(e + f*x)/2] - 6*A*c^3*Cos[(3*(e + f*x))/2] - 10*B*c^3*Cos[(3*(e + f*x))/2] + 21*A*c^2*d*Cos[(3*(e + f*x))/2] - 29*B*c^2*d*Cos[(3*(e + f*x))/2] + 54*A*c*d^2*Cos[(3*(e + f*x))/2] - 86*B*c*d^2*Cos[(3*(e + f*x))/2] + 75*A*d^3*Cos[(3*(e + f*x))/2] - 19*B*d^3*Cos[(3*(e + f*x))/2] + 3*A*c^2*d*Cos[(5*(e + f*x))/2] + 5*B*c^2*d*Cos[(5*(e + f*x))/2] - 16*A*c*d^2*Cos[(5*(e + f*x))/2] + 32*B*c*d^2*Cos[(5*(e + f*x))/2] - 35*A*d^3*Cos[(5*(e + f*x))/2] + 11*B*d^3*Cos[(5*(e + f*x))/2] + 22*A*c^3*Sin[(e + f*x)/2] - 6*B*c^3*Sin[(e + f*x)/2] - 40*A*c^2*d*Sin[(e + f*x)/2] + 40*B*c^2*d*Sin[(e + f*x)/2] - 54*A*c*d^2*Sin[(e + f*x)/2] + 70*B*c*d^2*Sin[(e + f*x)/2] - 24*A*d^3*Sin[(e + f*x)/2] - 8*B*d^3*Sin[(e + f*x)/2] - 6*A*c^3*Sin[(3*(e + f*x))/2] - 10*B*c^3*Sin[(3*(e + f*x))/2] + 21*A*c^2*d*Sin[(3*(e + f*x))/2] - 29*B*c^2*d*Sin[(3*(e + f*x))/2] + 54*A*c*d^2*Sin[(3*(e + f*x))/2] - 86*B*c*d^2*Sin[(3*(e + f*x))/2] + 75*A*d^3*Sin[(3*(e + f*x))/2] - 19*B*d^3*Sin[(3*(e + f*x))/2] - 3*A*c^2*d*Sin[(5*(e + f*x))/2] - 5*B*c^2*d*Sin[(5*(e + f*x))/2] + 16*A*c*d^2*Sin[(5*(e + f*x))/2] - 32*B*c*d^2*Sin[(5*(e + f*x))/2] + 35*A*d^3*Sin[(5*(e + f*x))/2] - 11*B*d^3*Sin[(5*(e + f*x))/2]))/(64*(c - d)^3*(c + d)*f*(a*(1 + Sin[e + f*x]))^(5/2)*(c + d*Sin[e + f*x]))","C",0
327,1,2103,519,13.8569732,"\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3),x]","\text{Result too large to show}","-\frac{\left(3 A \left(c^2-10 c d+73 d^2\right)+B \left(5 c^2-82 c d-115 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^5}+\frac{d^{3/2} \left(3 A d \left(21 c^2+30 c d+13 d^2\right)-B \left(35 c^3+70 c^2 d+67 c d^2+20 d^3\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 a^{5/2} f (c-d)^5 (c+d)^{5/2}}-\frac{d \left(A \left(3 c^2-20 c d-31 d^2\right)+B \left(5 c^2+28 c d+15 d^2\right)\right) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}-\frac{d \left(3 A \left(c^3-7 c^2 d-37 c d^2-21 d^3\right)+B \left(5 c^3+73 c^2 d+79 c d^2+35 d^3\right)\right) \cos (e+f x)}{16 a^2 f (c-d)^4 (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{(3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}",1,"((1 + I)*(3*A*c^2 + 5*B*c^2 - 30*A*c*d - 82*B*c*d + 219*A*d^2 - 115*B*d^2)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(e + f*x)/4]*(Cos[(e + f*x)/4] - Sin[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/((16*(-1)^(1/4)*c^5 - 80*(-1)^(1/4)*c^4*d + 160*(-1)^(1/4)*c^3*d^2 - 160*(-1)^(1/4)*c^2*d^3 + 80*(-1)^(1/4)*c*d^4 - 16*(-1)^(1/4)*d^5)*f*(a*(1 + Sin[e + f*x]))^(5/2)) - (d^(3/2)*(-3*A*d*(21*c^2 + 30*c*d + 13*d^2) + B*(35*c^3 + 70*c^2*d + 67*c*d^2 + 20*d^3))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] + Sqrt[d]*Cos[(e + f*x)/2] - Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/(16*(c - d)^5*(c + d)^(5/2)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + (d^(3/2)*(-3*A*d*(21*c^2 + 30*c*d + 13*d^2) + B*(35*c^3 + 70*c^2*d + 67*c*d^2 + 20*d^3))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] - Sqrt[d]*Cos[(e + f*x)/2] + Sqrt[d]*Sin[(e + f*x)/2])])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/(16*(c - d)^5*(c + d)^(5/2)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-44*A*c^5*Cos[(e + f*x)/2] + 12*B*c^5*Cos[(e + f*x)/2] + 84*A*c^4*d*Cos[(e + f*x)/2] - 116*B*c^4*d*Cos[(e + f*x)/2] + 249*A*c^3*d^2*Cos[(e + f*x)/2] - 433*B*c^3*d^2*Cos[(e + f*x)/2] + 385*A*c^2*d^3*Cos[(e + f*x)/2] - 277*B*c^2*d^3*Cos[(e + f*x)/2] + 239*A*c*d^4*Cos[(e + f*x)/2] - 95*B*c*d^4*Cos[(e + f*x)/2] + 47*A*d^5*Cos[(e + f*x)/2] - 51*B*d^5*Cos[(e + f*x)/2] - 12*A*c^5*Cos[(3*(e + f*x))/2] - 20*B*c^5*Cos[(3*(e + f*x))/2] + 40*A*c^4*d*Cos[(3*(e + f*x))/2] - 104*B*c^4*d*Cos[(3*(e + f*x))/2] + 261*A*c^3*d^2*Cos[(3*(e + f*x))/2] - 581*B*c^3*d^2*Cos[(3*(e + f*x))/2] + 781*A*c^2*d^3*Cos[(3*(e + f*x))/2] - 665*B*c^2*d^3*Cos[(3*(e + f*x))/2] + 579*A*c*d^4*Cos[(3*(e + f*x))/2] - 299*B*c*d^4*Cos[(3*(e + f*x))/2] + 79*A*d^5*Cos[(3*(e + f*x))/2] - 59*B*d^5*Cos[(3*(e + f*x))/2] + 12*A*c^4*d*Cos[(5*(e + f*x))/2] + 20*B*c^4*d*Cos[(5*(e + f*x))/2] - 73*A*c^3*d^2*Cos[(5*(e + f*x))/2] + 217*B*c^3*d^2*Cos[(5*(e + f*x))/2] - 353*A*c^2*d^3*Cos[(5*(e + f*x))/2] + 397*B*c^2*d^3*Cos[(5*(e + f*x))/2] - 419*A*c*d^4*Cos[(5*(e + f*x))/2] + 251*B*c*d^4*Cos[(5*(e + f*x))/2] - 127*A*d^5*Cos[(5*(e + f*x))/2] + 75*B*d^5*Cos[(5*(e + f*x))/2] + 3*A*c^3*d^2*Cos[(7*(e + f*x))/2] + 5*B*c^3*d^2*Cos[(7*(e + f*x))/2] - 21*A*c^2*d^3*Cos[(7*(e + f*x))/2] + 73*B*c^2*d^3*Cos[(7*(e + f*x))/2] - 111*A*c*d^4*Cos[(7*(e + f*x))/2] + 79*B*c*d^4*Cos[(7*(e + f*x))/2] - 63*A*d^5*Cos[(7*(e + f*x))/2] + 35*B*d^5*Cos[(7*(e + f*x))/2] + 44*A*c^5*Sin[(e + f*x)/2] - 12*B*c^5*Sin[(e + f*x)/2] - 84*A*c^4*d*Sin[(e + f*x)/2] + 116*B*c^4*d*Sin[(e + f*x)/2] - 249*A*c^3*d^2*Sin[(e + f*x)/2] + 433*B*c^3*d^2*Sin[(e + f*x)/2] - 385*A*c^2*d^3*Sin[(e + f*x)/2] + 277*B*c^2*d^3*Sin[(e + f*x)/2] - 239*A*c*d^4*Sin[(e + f*x)/2] + 95*B*c*d^4*Sin[(e + f*x)/2] - 47*A*d^5*Sin[(e + f*x)/2] + 51*B*d^5*Sin[(e + f*x)/2] - 12*A*c^5*Sin[(3*(e + f*x))/2] - 20*B*c^5*Sin[(3*(e + f*x))/2] + 40*A*c^4*d*Sin[(3*(e + f*x))/2] - 104*B*c^4*d*Sin[(3*(e + f*x))/2] + 261*A*c^3*d^2*Sin[(3*(e + f*x))/2] - 581*B*c^3*d^2*Sin[(3*(e + f*x))/2] + 781*A*c^2*d^3*Sin[(3*(e + f*x))/2] - 665*B*c^2*d^3*Sin[(3*(e + f*x))/2] + 579*A*c*d^4*Sin[(3*(e + f*x))/2] - 299*B*c*d^4*Sin[(3*(e + f*x))/2] + 79*A*d^5*Sin[(3*(e + f*x))/2] - 59*B*d^5*Sin[(3*(e + f*x))/2] - 12*A*c^4*d*Sin[(5*(e + f*x))/2] - 20*B*c^4*d*Sin[(5*(e + f*x))/2] + 73*A*c^3*d^2*Sin[(5*(e + f*x))/2] - 217*B*c^3*d^2*Sin[(5*(e + f*x))/2] + 353*A*c^2*d^3*Sin[(5*(e + f*x))/2] - 397*B*c^2*d^3*Sin[(5*(e + f*x))/2] + 419*A*c*d^4*Sin[(5*(e + f*x))/2] - 251*B*c*d^4*Sin[(5*(e + f*x))/2] + 127*A*d^5*Sin[(5*(e + f*x))/2] - 75*B*d^5*Sin[(5*(e + f*x))/2] + 3*A*c^3*d^2*Sin[(7*(e + f*x))/2] + 5*B*c^3*d^2*Sin[(7*(e + f*x))/2] - 21*A*c^2*d^3*Sin[(7*(e + f*x))/2] + 73*B*c^2*d^3*Sin[(7*(e + f*x))/2] - 111*A*c*d^4*Sin[(7*(e + f*x))/2] + 79*B*c*d^4*Sin[(7*(e + f*x))/2] - 63*A*d^5*Sin[(7*(e + f*x))/2] + 35*B*d^5*Sin[(7*(e + f*x))/2]))/(128*(c - d)^4*(c + d)^2*f*(a*(1 + Sin[e + f*x]))^(5/2)*(c + d*Sin[e + f*x])^2)","C",0
328,0,0,221,26.0822974,"\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","-\frac{4 \sqrt{2} a^2 (A-B) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};-\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1}}-\frac{8 \sqrt{2} a^2 B \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};-\frac{5}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1}}",1,"Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n, x]","F",-1
329,0,0,217,9.0733972,"\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","-\frac{2 \sqrt{2} a (A-B) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1}}-\frac{4 \sqrt{2} a B \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};-\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1}}",1,"Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n, x]","F",-1
330,0,0,221,4.6516595,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x]),x]","\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","-\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{\sqrt{2} a f \sqrt{\sin (e+f x)+1}}-\frac{\sqrt{2} B \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{a f \sqrt{\sin (e+f x)+1}}",1,"Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x]), x]","F",-1
331,0,0,223,28.1409524,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^2,x]","\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","-\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{5}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{2 \sqrt{2} a^2 f \sqrt{\sin (e+f x)+1}}-\frac{B \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{\sqrt{2} a^2 f \sqrt{\sin (e+f x)+1}}",1,"Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^2, x]","F",-1
332,1,245,427,26.6504299,"\int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} \left(-30 (A+B) (c-d (4 n+5)) \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};-\frac{d (\sin (e+f x)-1)}{c+d}\right)+30 (A+B) (c+d) \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{n+1}+20 B d (2 n+3) (\sin (e+f x)-1) \, _2F_1\left(\frac{3}{2},-n;\frac{5}{2};-\frac{d (\sin (e+f x)-1)}{c+d}\right)+6 B d (2 n+3) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 \, _2F_1\left(\frac{5}{2},-n;\frac{7}{2};-\frac{d (\sin (e+f x)-1)}{c+d}\right)\right)}{15 d f (2 n+3) \sqrt{a (\sin (e+f x)+1)}}","\frac{2 a^2 (A-B) (c-d (4 n+5)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right)}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 (A-B) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 B \left(3 c^2-2 c d (4 n+7)+d^2 \left(16 n^2+56 n+43\right)\right) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right)}{d^2 f (2 n+3) (2 n+5) \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 B (3 c-d (4 n+11)) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d^2 f (2 n+3) (2 n+5) \sqrt{a \sin (e+f x)+a}}-\frac{2 a B \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}",1,"-1/15*(a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^n*(-30*(A + B)*(c - d*(5 + 4*n))*Hypergeometric2F1[1/2, -n, 3/2, -((d*(-1 + Sin[e + f*x]))/(c + d))] + 6*B*d*(3 + 2*n)*Hypergeometric2F1[5/2, -n, 7/2, -((d*(-1 + Sin[e + f*x]))/(c + d))]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + 20*B*d*(3 + 2*n)*Hypergeometric2F1[3/2, -n, 5/2, -((d*(-1 + Sin[e + f*x]))/(c + d))]*(-1 + Sin[e + f*x]) + 30*(A + B)*(c + d)*((c + d*Sin[e + f*x])/(c + d))^(1 + n)))/(d*f*(3 + 2*n)*Sqrt[a*(1 + Sin[e + f*x])]*((c + d*Sin[e + f*x])/(c + d))^n)","A",1
333,0,0,167,8.2548109,"\int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","\int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","-\frac{2 a \cos (e+f x) (A d (2 n+3)-B (c-2 d (n+1))) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right)}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}",1,"Integrate[Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n, x]","F",-1
334,1,244,220,5.5680599,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/Sqrt[a + a*Sin[e + f*x]],x]","\frac{\cos (e+f x) \sqrt{a (\sin (e+f x)+1)} (c+d \sin (e+f x))^n \left(\frac{4 (A-B) \sqrt{\frac{\sin (e+f x)-1}{\sin (e+f x)+1}} \left(\frac{c-d}{d \sin (e+f x)+d}+1\right)^{-n} F_1\left(-n-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{d-c}{\sin (e+f x) d+d}\right)}{2 n+1}-(A+B) \sqrt{2-2 \sin (e+f x)} \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(1;\frac{1}{2},-n;2;\frac{1}{2} (\sin (e+f x)+1),\frac{d (\sin (e+f x)+1)}{d-c}\right)\right)}{4 a f (\sin (e+f x)-1)}","-\frac{(A-B) \cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left(n+1;\frac{1}{2},1;n+2;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right)}{f (n+1) (c-d) (1-\sin (e+f x)) \sqrt{a \sin (e+f x)+a}}-\frac{2 B \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{a \sin (e+f x)+a}}",1,"(Cos[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*(c + d*Sin[e + f*x])^n*(-(((A + B)*AppellF1[1, 1/2, -n, 2, (1 + Sin[e + f*x])/2, (d*(1 + Sin[e + f*x]))/(-c + d)]*Sqrt[2 - 2*Sin[e + f*x]])/((c + d*Sin[e + f*x])/(c - d))^n) + (4*(A - B)*AppellF1[-1/2 - n, -1/2, -n, 1/2 - n, 2/(1 + Sin[e + f*x]), (-c + d)/(d + d*Sin[e + f*x])]*Sqrt[(-1 + Sin[e + f*x])/(1 + Sin[e + f*x])])/((1 + 2*n)*(1 + (c - d)/(d + d*Sin[e + f*x]))^n)))/(4*a*f*(-1 + Sin[e + f*x]))","A",0
335,1,603,269,17.2269176,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\sec (e+f x) (c+d \sin (e+f x))^n \left(a A (\sin (e+f x)+1) \left(a \sqrt{2-2 \sin (e+f x)} (\sin (e+f x)+1) \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(1;\frac{1}{2},-n;2;\frac{1}{2} (\sin (e+f x)+1),\frac{d (\sin (e+f x)+1)}{d-c}\right)-\frac{4 \sqrt{\frac{\sin (e+f x)-1}{\sin (e+f x)+1}} \left(\frac{c-d}{d \sin (e+f x)+d}+1\right)^{-n} \left(2 a (2 n+1) F_1\left(\frac{1}{2}-n;-\frac{1}{2},-n;\frac{3}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{d-c}{\sin (e+f x) d+d}\right)+a (2 n-1) (\sin (e+f x)+1) F_1\left(-n-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{d-c}{\sin (e+f x) d+d}\right)\right)}{4 n^2-1}\right)+a B (\sin (e+f x)+1) \left(a \sqrt{2-2 \sin (e+f x)} (\sin (e+f x)+1) \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(1;\frac{1}{2},-n;2;\frac{1}{2} (\sin (e+f x)+1),\frac{d (\sin (e+f x)+1)}{d-c}\right)-\frac{4 \sqrt{\frac{\sin (e+f x)-1}{\sin (e+f x)+1}} \left(\frac{c-d}{d \sin (e+f x)+d}+1\right)^{-n} \left(a (2 n-1) (\sin (e+f x)+1) F_1\left(-n-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{d-c}{\sin (e+f x) d+d}\right)-2 a (2 n+1) F_1\left(\frac{1}{2}-n;-\frac{1}{2},-n;\frac{3}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{d-c}{\sin (e+f x) d+d}\right)\right)}{4 n^2-1}\right)\right)}{8 a^3 f \sqrt{a (\sin (e+f x)+1)}}","\frac{d (A-B) \cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left(n+1;\frac{1}{2},2;n+2;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right)}{f (n+1) (c-d)^2 (a-a \sin (e+f x)) \sqrt{a \sin (e+f x)+a}}-\frac{B \cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left(n+1;\frac{1}{2},1;n+2;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right)}{a f (n+1) (c-d) (1-\sin (e+f x)) \sqrt{a \sin (e+f x)+a}}",1,"(Sec[e + f*x]*(c + d*Sin[e + f*x])^n*(a*B*(1 + Sin[e + f*x])*((a*AppellF1[1, 1/2, -n, 2, (1 + Sin[e + f*x])/2, (d*(1 + Sin[e + f*x]))/(-c + d)]*Sqrt[2 - 2*Sin[e + f*x]]*(1 + Sin[e + f*x]))/((c + d*Sin[e + f*x])/(c - d))^n - (4*Sqrt[(-1 + Sin[e + f*x])/(1 + Sin[e + f*x])]*(-2*a*(1 + 2*n)*AppellF1[1/2 - n, -1/2, -n, 3/2 - n, 2/(1 + Sin[e + f*x]), (-c + d)/(d + d*Sin[e + f*x])] + a*(-1 + 2*n)*AppellF1[-1/2 - n, -1/2, -n, 1/2 - n, 2/(1 + Sin[e + f*x]), (-c + d)/(d + d*Sin[e + f*x])]*(1 + Sin[e + f*x])))/((-1 + 4*n^2)*(1 + (c - d)/(d + d*Sin[e + f*x]))^n)) + a*A*(1 + Sin[e + f*x])*((a*AppellF1[1, 1/2, -n, 2, (1 + Sin[e + f*x])/2, (d*(1 + Sin[e + f*x]))/(-c + d)]*Sqrt[2 - 2*Sin[e + f*x]]*(1 + Sin[e + f*x]))/((c + d*Sin[e + f*x])/(c - d))^n - (4*Sqrt[(-1 + Sin[e + f*x])/(1 + Sin[e + f*x])]*(2*a*(1 + 2*n)*AppellF1[1/2 - n, -1/2, -n, 3/2 - n, 2/(1 + Sin[e + f*x]), (-c + d)/(d + d*Sin[e + f*x])] + a*(-1 + 2*n)*AppellF1[-1/2 - n, -1/2, -n, 1/2 - n, 2/(1 + Sin[e + f*x]), (-c + d)/(d + d*Sin[e + f*x])]*(1 + Sin[e + f*x])))/((-1 + 4*n^2)*(1 + (c - d)/(d + d*Sin[e + f*x]))^n))))/(8*a^3*f*Sqrt[a*(1 + Sin[e + f*x])])","B",0
336,1,300,351,7.7260017,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]","-\frac{\csc ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^m (a (\sin (e+f x)+1))^m \left(-2 (A+B) (c+d)^2 \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right) \, _2F_1\left(\frac{1}{2},m+4;\frac{3}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)-\frac{2}{7} (A-B) (c-d)^2 \tan ^7\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \, _2F_1\left(\frac{7}{2},m+4;\frac{9}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)-\frac{2}{5} (c-d) (A (3 c+d)-B (c+3 d)) \tan ^5\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \, _2F_1\left(\frac{5}{2},m+4;\frac{7}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)-\frac{2}{3} (c+d) (3 A c-A d+B c-3 B d) \tan ^3\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \, _2F_1\left(\frac{3}{2},m+4;\frac{5}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)\right)}{f}","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) \left(A (m+3) \left(c^2 \left(m^2+3 m+2\right)+2 c d m (m+2)+d^2 \left(m^2+m+1\right)\right)+B \left(c^2 m \left(m^2+5 m+6\right)+2 c d \left(m^3+4 m^2+4 m+3\right)+d^2 m \left(m^2+3 m+5\right)\right)\right) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f (m+1) (m+2) (m+3)}+\frac{\cos (e+f x) \left(d (A d (m+3)+B (2 c+d m))-2 (m+2) \left(A c d (m+3)+B \left(c^2+c d m+d^2\right)\right)\right) (a \sin (e+f x)+a)^m}{f (m+1) (m+2) (m+3)}-\frac{d \cos (e+f x) (A d (m+3)+B (2 c+d m)) (a \sin (e+f x)+a)^{m+1}}{a f (m+2) (m+3)}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^2}{f (m+3)}",1,"-(((Csc[(2*e + Pi + 2*f*x)/4]^2)^m*(a*(1 + Sin[e + f*x]))^m*(-2*(A + B)*(c + d)^2*Hypergeometric2F1[1/2, 4 + m, 3/2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Tan[(2*e - Pi + 2*f*x)/4] - (2*(c + d)*(3*A*c + B*c - A*d - 3*B*d)*Hypergeometric2F1[3/2, 4 + m, 5/2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Tan[(2*e - Pi + 2*f*x)/4]^3)/3 - (2*(c - d)*(A*(3*c + d) - B*(c + 3*d))*Hypergeometric2F1[5/2, 4 + m, 7/2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Tan[(2*e - Pi + 2*f*x)/4]^5)/5 - (2*(A - B)*(c - d)^2*Hypergeometric2F1[7/2, 4 + m, 9/2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Tan[(2*e - Pi + 2*f*x)/4]^7)/7))/f)","A",0
337,1,212,199,3.5889014,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]","-\frac{\csc ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^m (a (\sin (e+f x)+1))^m \left(-2 (A+B) (c+d) \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right) \, _2F_1\left(\frac{1}{2},m+3;\frac{3}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)-\frac{2}{5} (A-B) (c-d) \tan ^5\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \, _2F_1\left(\frac{5}{2},m+3;\frac{7}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)-\frac{4}{3} (A c-B d) \tan ^3\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \, _2F_1\left(\frac{3}{2},m+3;\frac{5}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)\right)}{f}","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) \left(A (m+2) (c m+c+d m)+B \left(c m (m+2)+d \left(m^2+m+1\right)\right)\right) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f (m+1) (m+2)}+\frac{\cos (e+f x) (B d-(m+2) (A d+B c)) (a \sin (e+f x)+a)^m}{f (m+1) (m+2)}-\frac{B d \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (m+2)}",1,"-(((Csc[(2*e + Pi + 2*f*x)/4]^2)^m*(a*(1 + Sin[e + f*x]))^m*(-2*(A + B)*(c + d)*Hypergeometric2F1[1/2, 3 + m, 3/2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Tan[(2*e - Pi + 2*f*x)/4] - (4*(A*c - B*d)*Hypergeometric2F1[3/2, 3 + m, 5/2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Tan[(2*e - Pi + 2*f*x)/4]^3)/3 - (2*(A - B)*(c - d)*Hypergeometric2F1[5/2, 3 + m, 7/2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Tan[(2*e - Pi + 2*f*x)/4]^5)/5))/f)","A",1
338,1,275,117,1.8158703,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","-\frac{\sin ^{-2 m}\left(\frac{1}{4} (2 e+2 f x+\pi )\right) (a (\sin (e+f x)+1))^m \left(\frac{2 \sqrt{2} A \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right) \cos ^{2 m+1}\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \, _2F_1\left(\frac{1}{2},m+\frac{1}{2};m+\frac{3}{2};\sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)\right)}{(2 m+1) \sqrt{1-\sin (e+f x)}}+\frac{\sqrt[4]{-1} B 2^{-2 m-1} e^{-\frac{3}{2} i (e+f x)} \left(-(-1)^{3/4} e^{-\frac{1}{2} i (e+f x)} \left(e^{i (e+f x)}+i\right)\right)^{2 m+1} \left((m-1) e^{2 i (e+f x)} \, _2F_1\left(1,m;-m;-i e^{-i (e+f x)}\right)-(m+1) \, _2F_1\left(1,m+2;2-m;-i e^{-i (e+f x)}\right)\right)}{m^2-1}\right)}{f}","-\frac{2^{m+\frac{1}{2}} (A m+A+B m) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f (m+1)}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1)}",1,"-(((a*(1 + Sin[e + f*x]))^m*(((-1)^(1/4)*2^(-1 - 2*m)*B*(-(((-1)^(3/4)*(I + E^(I*(e + f*x))))/E^((I/2)*(e + f*x))))^(1 + 2*m)*(E^((2*I)*(e + f*x))*(-1 + m)*Hypergeometric2F1[1, m, -m, (-I)/E^(I*(e + f*x))] - (1 + m)*Hypergeometric2F1[1, 2 + m, 2 - m, (-I)/E^(I*(e + f*x))]))/(E^(((3*I)/2)*(e + f*x))*(-1 + m^2)) + (2*Sqrt[2]*A*Cos[(2*e - Pi + 2*f*x)/4]^(1 + 2*m)*Hypergeometric2F1[1/2, 1/2 + m, 3/2 + m, Sin[(2*e + Pi + 2*f*x)/4]^2]*Sin[(2*e - Pi + 2*f*x)/4])/((1 + 2*m)*Sqrt[1 - Sin[e + f*x]])))/(f*Sin[(2*e + Pi + 2*f*x)/4]^(2*m)))","C",0
339,1,473,191,7.0782618,"\int \frac{(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{c+d \sin (e+f x)} \, dx","Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x]),x]","\frac{(a (\sin (e+f x)+1))^m \left(\frac{6 (c+d) (B c-A d) \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^{\frac{1}{2}-m} \cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)^{m+\frac{1}{2}} F_1\left(\frac{1}{2};\frac{1}{2}-m,1;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{d (c+d \sin (e+f x)) \left(\sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \left(4 d F_1\left(\frac{3}{2};\frac{1}{2}-m,2;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-(2 m-1) (c+d) F_1\left(\frac{3}{2};\frac{3}{2}-m,1;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)+3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,1;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)}+\frac{\sqrt{2} B \cos (e+f x) \, _2F_1\left(\frac{1}{2},m+\frac{1}{2};m+\frac{3}{2};\frac{1}{4} \cos ^2(e+f x) \csc ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{(2 d m+d) \sqrt{1-\sin (e+f x)}}\right)}{f}","-\frac{\sqrt{2} (B c-A d) \cos (e+f x) (a \sin (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{d f (2 m+1) (c-d) \sqrt{1-\sin (e+f x)}}-\frac{B 2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{d f}",1,"((a*(1 + Sin[e + f*x]))^m*((Sqrt[2]*B*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 + m, 3/2 + m, (Cos[e + f*x]^2*Csc[(2*e - Pi + 2*f*x)/4]^2)/4])/((d + 2*d*m)*Sqrt[1 - Sin[e + f*x]]) + (6*(c + d)*(B*c - A*d)*AppellF1[1/2, 1/2 - m, 1, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)]*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*Sec[(2*e - Pi + 2*f*x)/4]^2*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(d*(c + d*Sin[e + f*x])*(3*(c + d)*AppellF1[1/2, 1/2 - m, 1, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (4*d*AppellF1[3/2, 1/2 - m, 2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] - (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, 1, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])*Sin[(2*e - Pi + 2*f*x)/4]^2))))/f","B",0
340,1,651,293,2.4960585,"\int \frac{(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx","Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2,x]","\frac{6 (c+d) \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) \sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^{\frac{1}{2}-m} \cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)^{m-\frac{1}{2}} (a (\sin (e+f x)+1))^m \left(\frac{(A d-B c) F_1\left(\frac{1}{2};\frac{1}{2}-m,2;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left((2 m-1) (c+d) F_1\left(\frac{3}{2};\frac{3}{2}-m,2;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-8 d F_1\left(\frac{3}{2};\frac{1}{2}-m,3;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)-3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,2;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}+\frac{B (c+d \sin (e+f x)) F_1\left(\frac{1}{2};\frac{1}{2}-m,1;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left((2 m-1) (c+d) F_1\left(\frac{3}{2};\frac{3}{2}-m,1;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-4 d F_1\left(\frac{3}{2};\frac{1}{2}-m,2;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)-3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,1;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}\right)}{d f (c+d \sin (e+f x))^2}","\frac{\sqrt{2} \cos (e+f x) \left(A d (c (1-m)-d m)-B \left(c^2 (-m)-c d m+d^2\right)\right) (a \sin (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{d f (2 m+1) (c-d)^2 (c+d) \sqrt{1-\sin (e+f x)}}+\frac{2^{m+\frac{1}{2}} m (B c-A d) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{d f \left(c^2-d^2\right)}-\frac{(B c-A d) \cos (e+f x) (a \sin (e+f x)+a)^m}{f \left(c^2-d^2\right) (c+d \sin (e+f x))}",1,"(6*(c + d)*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*(a*(1 + Sin[e + f*x]))^m*(((-(B*c) + A*d)*AppellF1[1/2, 1/2 - m, 2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])/(-3*(c + d)*AppellF1[1/2, 1/2 - m, 2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (-8*d*AppellF1[3/2, 1/2 - m, 3, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, 2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])*Cos[(2*e + Pi + 2*f*x)/4]^2) + (B*AppellF1[1/2, 1/2 - m, 1, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)]*(c + d*Sin[e + f*x]))/(-3*(c + d)*AppellF1[1/2, 1/2 - m, 1, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (-4*d*AppellF1[3/2, 1/2 - m, 2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, 1, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])*Cos[(2*e + Pi + 2*f*x)/4]^2))*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(d*f*(c + d*Sin[e + f*x])^2)","B",0
341,1,651,467,2.5325245,"\int \frac{(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c+d \sin (e+f x))^3} \, dx","Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3,x]","\frac{6 (c+d) \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) \sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^{\frac{1}{2}-m} \cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)^{m-\frac{1}{2}} (a (\sin (e+f x)+1))^m \left(\frac{(A d-B c) F_1\left(\frac{1}{2};\frac{1}{2}-m,3;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left((2 m-1) (c+d) F_1\left(\frac{3}{2};\frac{3}{2}-m,3;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-12 d F_1\left(\frac{3}{2};\frac{1}{2}-m,4;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)-3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,3;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}+\frac{B (c+d \sin (e+f x)) F_1\left(\frac{1}{2};\frac{1}{2}-m,2;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left((2 m-1) (c+d) F_1\left(\frac{3}{2};\frac{3}{2}-m,2;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-8 d F_1\left(\frac{3}{2};\frac{1}{2}-m,3;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)-3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,2;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}\right)}{d f (c+d \sin (e+f x))^3}","\frac{\cos (e+f x) \left(B \left(c^3 (1-m) m+2 c^2 d (1-m) m-c d^2 \left(m^2-3 m+3\right)+2 d^3 m\right)-A d \left(-\left(c^2 \left(m^2-3 m+2\right)\right)+2 c d (2-m) m-d^2 \left(m^2-m+1\right)\right)\right) (a \sin (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{\sqrt{2} d f (2 m+1) (c-d)^3 (c+d)^2 \sqrt{1-\sin (e+f x)}}-\frac{2^{m-\frac{1}{2}} m \cos (e+f x) \left(A d (c (3-m)-d m)-B \left(c^2 (1-m)-c d m+2 d^2\right)\right) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{d f \left(c^2-d^2\right)^2}+\frac{\cos (e+f x) \left(A d (c (3-m)-d m)-B \left(c^2 (1-m)-c d m+2 d^2\right)\right) (a \sin (e+f x)+a)^m}{2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))}-\frac{(B c-A d) \cos (e+f x) (a \sin (e+f x)+a)^m}{2 f \left(c^2-d^2\right) (c+d \sin (e+f x))^2}",1,"(6*(c + d)*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*(a*(1 + Sin[e + f*x]))^m*(((-(B*c) + A*d)*AppellF1[1/2, 1/2 - m, 3, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])/(-3*(c + d)*AppellF1[1/2, 1/2 - m, 3, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (-12*d*AppellF1[3/2, 1/2 - m, 4, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, 3, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])*Cos[(2*e + Pi + 2*f*x)/4]^2) + (B*AppellF1[1/2, 1/2 - m, 2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)]*(c + d*Sin[e + f*x]))/(-3*(c + d)*AppellF1[1/2, 1/2 - m, 2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (-8*d*AppellF1[3/2, 1/2 - m, 3, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, 2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])*Cos[(2*e + Pi + 2*f*x)/4]^2))*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(d*f*(c + d*Sin[e + f*x])^3)","A",0
342,1,3281,284,8.1355817,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2),x]","\text{Result too large to show}","\frac{\sqrt{2} (A-B) (c-d) \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{1}{2};\frac{1}{2},-\frac{3}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}+\frac{\sqrt{2} B (c-d) \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{3}{2};\frac{1}{2},-\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (2 m+3) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}",1,"-((((-2*B*c*AppellF1[3/2, (1 - 2*m)/2, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^3*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2 + (-1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(3*Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]) - (2*A*d*AppellF1[3/2, (1 - 2*m)/2, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^3*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2 + (-1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(3*Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]) + (B*d*AppellF1[5/2, (1 - 2*m)/2, -1/2, 7/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^5*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2 + (-1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(5*Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]) - (2*B*d*AppellF1[3/2, (-1 - 2*m)/2, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((-1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^3*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((-1 - 2*m)/2 + (1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (3*B*d*(c + d)*AppellF1[1/2, -3/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(3 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 + (-4 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(3/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(-3*(c + d)*AppellF1[1/2, -3/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (2*d*AppellF1[3/2, -3/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(3 + 2*m)*AppellF1[3/2, -1/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) - (6*B*c*(c + d)*AppellF1[1/2, -1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 + (-2 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(1/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(-3*(c + d)*AppellF1[1/2, -1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (2*d*AppellF1[3/2, -1/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(1 + 2*m)*AppellF1[3/2, 1/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) - (6*A*d*(c + d)*AppellF1[1/2, -1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 + (-2 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(1/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(-3*(c + d)*AppellF1[1/2, -1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (2*d*AppellF1[3/2, -1/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(1 + 2*m)*AppellF1[3/2, 1/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) + (6*A*c*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 - m)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(-1/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(3*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (2*d*AppellF1[3/2, 1/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) + (3*B*d*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 - m)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(-1/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(3*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (2*d*AppellF1[3/2, 1/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2))*(a + a*Sin[e + f*x])^m)/(f*Cos[(-e + Pi/2 - f*x)/2]^(2*m)))","B",0
343,1,672,274,11.893628,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \sqrt{c+d \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]],x]","\frac{6 (c+d) \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) \sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^{\frac{1}{2}-m} \cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)^{m-\frac{1}{2}} (a (\sin (e+f x)+1))^m \sqrt{c+d \sin (e+f x)} \left(\frac{(B c-A d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-\frac{1}{2};\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left(-2 d F_1\left(\frac{3}{2};\frac{1}{2}-m,\frac{1}{2};\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-(2 m-1) (c+d) F_1\left(\frac{3}{2};\frac{3}{2}-m,-\frac{1}{2};\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)+3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-\frac{1}{2};\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}-\frac{B (c+d \sin (e+f x)) F_1\left(\frac{1}{2};\frac{1}{2}-m,-\frac{3}{2};\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left(-6 d F_1\left(\frac{3}{2};\frac{1}{2}-m,-\frac{1}{2};\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-(2 m-1) (c+d) F_1\left(\frac{3}{2};\frac{3}{2}-m,-\frac{3}{2};\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)+3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-\frac{3}{2};\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}\right)}{d f}","\frac{\sqrt{2} (A-B) \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{1}{2};\frac{1}{2},-\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}+\frac{\sqrt{2} B \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{3}{2};\frac{1}{2},-\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (2 m+3) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}",1,"(6*(c + d)*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*(a*(1 + Sin[e + f*x]))^m*Sqrt[c + d*Sin[e + f*x]]*(((B*c - A*d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])/(3*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (-2*d*AppellF1[3/2, 1/2 - m, 1/2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] - (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -1/2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])*Cos[(2*e + Pi + 2*f*x)/4]^2) - (B*AppellF1[1/2, 1/2 - m, -3/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)]*(c + d*Sin[e + f*x]))/(3*(c + d)*AppellF1[1/2, 1/2 - m, -3/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (-6*d*AppellF1[3/2, 1/2 - m, -1/2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] - (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -3/2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])*Cos[(2*e + Pi + 2*f*x)/4]^2))*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(d*f)","B",0
344,1,672,274,5.713515,"\int \frac{(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]],x]","\frac{6 (c+d) \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) \sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^{\frac{1}{2}-m} \cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)^{m-\frac{1}{2}} (a (\sin (e+f x)+1))^m \left(\frac{(B c-A d) F_1\left(\frac{1}{2};\frac{1}{2}-m,\frac{1}{2};\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left(2 d F_1\left(\frac{3}{2};\frac{1}{2}-m,\frac{3}{2};\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-(2 m-1) (c+d) F_1\left(\frac{3}{2};\frac{3}{2}-m,\frac{1}{2};\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)+3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,\frac{1}{2};\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}-\frac{B (c+d \sin (e+f x)) F_1\left(\frac{1}{2};\frac{1}{2}-m,-\frac{1}{2};\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left(-2 d F_1\left(\frac{3}{2};\frac{1}{2}-m,\frac{1}{2};\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-(2 m-1) (c+d) F_1\left(\frac{3}{2};\frac{3}{2}-m,-\frac{1}{2};\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)+3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-\frac{1}{2};\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}\right)}{d f \sqrt{c+d \sin (e+f x)}}","\frac{\sqrt{2} (A-B) \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{1}{2};\frac{1}{2},\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{2} B \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{3}{2};\frac{1}{2},\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (2 m+3) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}",1,"(6*(c + d)*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*(a*(1 + Sin[e + f*x]))^m*(((B*c - A*d)*AppellF1[1/2, 1/2 - m, 1/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])/(3*(c + d)*AppellF1[1/2, 1/2 - m, 1/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (2*d*AppellF1[3/2, 1/2 - m, 3/2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] - (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, 1/2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])*Cos[(2*e + Pi + 2*f*x)/4]^2) - (B*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)]*(c + d*Sin[e + f*x]))/(3*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (-2*d*AppellF1[3/2, 1/2 - m, 1/2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] - (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -1/2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])*Cos[(2*e + Pi + 2*f*x)/4]^2))*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(d*f*Sqrt[c + d*Sin[e + f*x]])","B",0
345,1,672,288,6.2236602,"\int \frac{(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^(3/2),x]","\frac{6 (c+d) \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) \sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^{\frac{1}{2}-m} \cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)^{m-\frac{1}{2}} (a (\sin (e+f x)+1))^m \left(\frac{(B c-A d) F_1\left(\frac{1}{2};\frac{1}{2}-m,\frac{3}{2};\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left(6 d F_1\left(\frac{3}{2};\frac{1}{2}-m,\frac{5}{2};\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-(2 m-1) (c+d) F_1\left(\frac{3}{2};\frac{3}{2}-m,\frac{3}{2};\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)+3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,\frac{3}{2};\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}-\frac{B (c+d \sin (e+f x)) F_1\left(\frac{1}{2};\frac{1}{2}-m,\frac{1}{2};\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left(2 d F_1\left(\frac{3}{2};\frac{1}{2}-m,\frac{3}{2};\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-(2 m-1) (c+d) F_1\left(\frac{3}{2};\frac{3}{2}-m,\frac{1}{2};\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)+3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,\frac{1}{2};\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}\right)}{d f (c+d \sin (e+f x))^{3/2}}","\frac{\sqrt{2} (A-B) \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{1}{2};\frac{1}{2},\frac{3}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) (c-d) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{2} B \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{3}{2};\frac{1}{2},\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (2 m+3) (c-d) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}",1,"(6*(c + d)*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*(a*(1 + Sin[e + f*x]))^m*(((B*c - A*d)*AppellF1[1/2, 1/2 - m, 3/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])/(3*(c + d)*AppellF1[1/2, 1/2 - m, 3/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (6*d*AppellF1[3/2, 1/2 - m, 5/2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] - (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, 3/2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])*Cos[(2*e + Pi + 2*f*x)/4]^2) - (B*AppellF1[1/2, 1/2 - m, 1/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)]*(c + d*Sin[e + f*x]))/(3*(c + d)*AppellF1[1/2, 1/2 - m, 1/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (2*d*AppellF1[3/2, 1/2 - m, 3/2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] - (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, 1/2, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])*Cos[(2*e + Pi + 2*f*x)/4]^2))*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(d*f*(c + d*Sin[e + f*x])^(3/2))","B",0
346,1,682,270,6.0823063,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","\frac{6 (c+d) \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) \sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^{\frac{1}{2}-m} \cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)^{m-\frac{1}{2}} (a (\sin (e+f x)+1))^m (c+d \sin (e+f x))^n \left(\frac{(B c-A d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-n;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left(-4 d n F_1\left(\frac{3}{2};\frac{1}{2}-m,1-n;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-(2 m-1) (c+d) F_1\left(\frac{3}{2};\frac{3}{2}-m,-n;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)+3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-n;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}-\frac{B (c+d \sin (e+f x)) F_1\left(\frac{1}{2};\frac{1}{2}-m,-n-1;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left(-4 d (n+1) F_1\left(\frac{3}{2};\frac{1}{2}-m,-n;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-(2 m-1) (c+d) F_1\left(\frac{3}{2};\frac{3}{2}-m,-n-1;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)+3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-n-1;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}\right)}{d f}","\frac{\sqrt{2} (A-B) \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(m+\frac{1}{2};\frac{1}{2},-n;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) \sqrt{1-\sin (e+f x)}}+\frac{\sqrt{2} B \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(m+\frac{3}{2};\frac{1}{2},-n;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (2 m+3) \sqrt{1-\sin (e+f x)}}",1,"(6*(c + d)*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*(a*(1 + Sin[e + f*x]))^m*(c + d*Sin[e + f*x])^n*(((B*c - A*d)*AppellF1[1/2, 1/2 - m, -n, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])/(3*(c + d)*AppellF1[1/2, 1/2 - m, -n, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (-4*d*n*AppellF1[3/2, 1/2 - m, 1 - n, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] - (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -n, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])*Cos[(2*e + Pi + 2*f*x)/4]^2) - (B*AppellF1[1/2, 1/2 - m, -1 - n, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)]*(c + d*Sin[e + f*x]))/(3*(c + d)*AppellF1[1/2, 1/2 - m, -1 - n, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (-4*d*(1 + n)*AppellF1[3/2, 1/2 - m, -n, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] - (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -1 - n, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])*Cos[(2*e + Pi + 2*f*x)/4]^2))*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(d*f)","B",0
347,1,573,277,6.9359819,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^{-1-m} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^(-1 - m),x]","\frac{2 \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) \sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^{\frac{1}{2}-m} \cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)^{m-\frac{1}{2}} (a (\sin (e+f x)+1))^m (c+d \sin (e+f x))^{-m} \left(-A \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{(c-d) \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d \sin (e+f x)}\right) \left(\frac{(c+d) \cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d \sin (e+f x)}\right)^{\frac{1}{2}-m}-\frac{3 B (c+d)^2 F_1\left(\frac{1}{2};\frac{1}{2}-m,m;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{d \left(3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,m;\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left((2 m-1) (c+d) F_1\left(\frac{3}{2};\frac{3}{2}-m,m;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-4 d m F_1\left(\frac{3}{2};\frac{1}{2}-m,m+1;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right)\right)}+\frac{B c \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{(c-d) \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d \sin (e+f x)}\right) \left(\frac{(c+d) \cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d \sin (e+f x)}\right)^{\frac{1}{2}-m}}{d}\right)}{f (c+d)}","\frac{\sqrt{2} B \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^{-m} \left(\frac{c+d \sin (e+f x)}{c-d}\right)^m F_1\left(m+\frac{3}{2};\frac{1}{2},m+1;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (2 m+3) (c-d) \sqrt{1-\sin (e+f x)}}-\frac{a 2^{m+\frac{1}{2}} (A-B) \cos (e+f x) (a \sin (e+f x)+a)^{m-1} \left(\frac{(c+d) (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right)^{\frac{1}{2}-m} (c+d \sin (e+f x))^{-m} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right)}{f (c+d)}",1,"(2*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*(a*(1 + Sin[e + f*x]))^m*((-3*B*(c + d)^2*AppellF1[1/2, 1/2 - m, m, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])/(d*(3*(c + d)*AppellF1[1/2, 1/2 - m, m, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] - (-4*d*m*AppellF1[3/2, 1/2 - m, 1 + m, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, m, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)])*Cos[(2*e + Pi + 2*f*x)/4]^2)) - A*Hypergeometric2F1[1/2, 1/2 - m, 3/2, ((c - d)*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d*Sin[e + f*x])]*(((c + d)*Cos[(2*e - Pi + 2*f*x)/4]^2)/(c + d*Sin[e + f*x]))^(1/2 - m) + (B*c*Hypergeometric2F1[1/2, 1/2 - m, 3/2, ((c - d)*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d*Sin[e + f*x])]*(((c + d)*Cos[(2*e - Pi + 2*f*x)/4]^2)/(c + d*Sin[e + f*x]))^(1/2 - m))/d)*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/((c + d)*f*(c + d*Sin[e + f*x])^m)","B",0
348,0,0,132,10.1145643,"\int (a-a \sin (e+f x)) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Integrate[(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]","\int (a-a \sin (e+f x)) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","\frac{2 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(m+\frac{1}{2};-\frac{1}{2},-n;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1)}",1,"Integrate[(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x]","F",-1
349,0,0,139,4.8741141,"\int (a-a \sin (e+f x)) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx","Integrate[(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-1 - m),x]","\int (a-a \sin (e+f x)) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx","\frac{2 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^{-m} \left(\frac{c+d \sin (e+f x)}{c-d}\right)^m F_1\left(m+\frac{1}{2};-\frac{1}{2},m+1;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) (c-d)}",1,"Integrate[(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-1 - m), x]","F",-1
350,1,39,39,0.6784455,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} (d-(c-d) m+(c+(c-d) m) \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-2 - m)*(d - (c - d)*m + (c + (c - d)*m)*Sin[e + f*x]),x]","-\frac{\cos (e+f x) (a (\sin (e+f x)+1))^m (c+d \sin (e+f x))^{-m-1}}{f}","-\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{-m-1}}{f}",1,"-((Cos[e + f*x]*(a*(1 + Sin[e + f*x]))^m*(c + d*Sin[e + f*x])^(-1 - m))/f)","A",1
351,1,40,40,0.7434534,"\int (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} (d+(c+d) m+(c+(c+d) m) \sin (e+f x)) \, dx","Integrate[(a - a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-2 - m)*(d + (c + d)*m + (c + (c + d)*m)*Sin[e + f*x]),x]","-\frac{\cos (e+f x) (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-m-1}}{f}","-\frac{\cos (e+f x) (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-m-1}}{f}",1,"-((Cos[e + f*x]*(a - a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-1 - m))/f)","A",1
352,1,188,199,1.5956773,"\int \frac{(a+b \sin (e+f x))^2 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx","Integrate[((a + b*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2,x]","\frac{\frac{2 (b c-a d) \left(a d^2 (B d-A c)+b \left(-A c^2 d+2 A d^3+2 B c^3-3 B c d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{\left(c^2-d^2\right)^{3/2}}+b (e+f x) (2 a B d+A b d-2 b B c)+\frac{d (b c-a d)^2 (A d-B c) \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))}+b^2 (-B) d \cos (e+f x)}{d^3 f}","-\frac{(b c-a d)^2 (B c-A d) \cos (e+f x)}{d^2 f \left(c^2-d^2\right) (c+d \sin (e+f x))}-\frac{2 (b c-a d) \left(a d^2 (A c-B d)-b \left(-A c^2 d+2 A d^3+2 B c^3-3 B c d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f \left(c^2-d^2\right)^{3/2}}-\frac{b x (-2 a B d-A b d+2 b B c)}{d^3}-\frac{b^2 B \cos (e+f x)}{d^2 f}",1,"(b*(-2*b*B*c + A*b*d + 2*a*B*d)*(e + f*x) + (2*(b*c - a*d)*(a*d^2*(-(A*c) + B*d) + b*(2*B*c^3 - A*c^2*d - 3*B*c*d^2 + 2*A*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(3/2) - b^2*B*d*Cos[e + f*x] + (d*(b*c - a*d)^2*(-(B*c) + A*d)*Cos[e + f*x])/((c - d)*(c + d)*(c + d*Sin[e + f*x])))/(d^3*f)","A",1
353,1,2042,840,6.7503528,"\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2))/(a + b*Sin[e + f*x])^(3/2),x]","\text{Result too large to show}","\frac{2 (A b-a B) (b c-a d) \sqrt{c+d \sin (e+f x)} \cos (e+f x)}{b \left(a^2-b^2\right) f \sqrt{a+b \sin (e+f x)}}-\frac{\left(2 A b (b c-a d)-B \left(-3 d a^2+2 b c a+b^2 d\right)\right) \sqrt{c+d \sin (e+f x)} \cos (e+f x)}{b \left(a^2-b^2\right) f \sqrt{a+b \sin (e+f x)}}+\frac{(c-d) \sqrt{c+d} \left(3 B d a^2-2 b B c a-2 A b d a+2 A b^2 c-b^2 B d\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{(a-b) b^2 \sqrt{a+b} (b c-a d) f}+\frac{\sqrt{c+d} (3 b B c+2 A b d-3 a B d) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^3 \sqrt{a+b} f}+\frac{\sqrt{a+b} \left(2 A b (b (c-2 d)+a d)-B \left(3 d a^2-6 b d a+b^2 (2 c+d)\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(a-b) b^3 \sqrt{c+d} f}",1,"(-2*(A*b^2*c*Cos[e + f*x] - a*b*B*c*Cos[e + f*x] - a*A*b*d*Cos[e + f*x] + a^2*B*d*Cos[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/(b*(-a^2 + b^2)*f*Sqrt[a + b*Sin[e + f*x]]) + ((-4*(-(b*c) + a*d)*(2*a*A*b*c^2 - 2*b^2*B*c^2 - 2*A*b^2*c*d + 2*a*b*B*c*d + a^2*B*d^2 - b^2*B*d^2)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + a*d)*(2*A*b^2*c^2 - 2*a*b*B*c^2 + 4*a^2*B*c*d - 4*b^2*B*c*d - 2*A*b^2*d^2 + 2*a*b*B*d^2)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/((a + b)*d), ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) + 2*(-2*A*b^2*c*d + 2*a*b*B*c*d + 2*a*A*b*d^2 - 3*a^2*B*d^2 + b^2*B*d^2)*((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*Sqrt[a + b*Sin[e + f*x]]) + (Sqrt[(a - b)/(a + b)]*(a + b)*Cos[(-e + Pi/2 - f*x)/2]*EllipticE[ArcSin[(Sqrt[(a - b)/(a + b)]*Sin[(-e + Pi/2 - f*x)/2])/Sqrt[(a + b*Sin[e + f*x])/(a + b)]], (2*(-(b*c) + a*d))/((a - b)*(c + d))]*Sqrt[c + d*Sin[e + f*x]])/(b*d*Sqrt[((a + b)*Cos[(-e + Pi/2 - f*x)/2]^2)/(a + b*Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/(a + b)]*Sqrt[((a + b)*(c + d*Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x]))]) - (2*(-(b*c) + a*d)*((((a + b)*c + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/((a + b)*d), ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])))/(b*d)))/(2*(a - b)*b*(a + b)*f)","B",0
354,1,1901,630,10.185714,"\int \frac{(A+B \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}{(a+b \sin (e+f x))^{3/2}} \, dx","Integrate[((A + B*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/(a + b*Sin[e + f*x])^(3/2),x]","\frac{-\frac{4 (a A c-b B c) (a d-b c) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{d-c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{\sqrt{2}}\right)|\frac{2 (a d-b c)}{(a+b) (d-c)}\right) \sec (e+f x) \sqrt{\frac{(c+d) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a d-b c}} \sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}} \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{(a+b) (c+d) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-4 (a d-b c) (A b c-a B c+a A d-b B d) \left(\frac{\sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{d-c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{\sqrt{2}}\right)|\frac{2 (a d-b c)}{(a+b) (d-c)}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{\frac{(c+d) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a d-b c}} \sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{(a+b) (c+d) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{d-c}} \Pi \left(\frac{a d-b c}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{\sqrt{2}}\right)|\frac{2 (a d-b c)}{(a+b) (d-c)}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{\frac{(c+d) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a d-b c}} \sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{(a+b) d \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\right)+2 (a B d-A b d) \left(\frac{\sqrt{c+d \sin (e+f x)} \cos (e+f x)}{d \sqrt{a+b \sin (e+f x)}}-\frac{2 (a d-b c) \left(\frac{((a+b) c+a d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{d-c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{\sqrt{2}}\right)|\frac{2 (a d-b c)}{(a+b) (d-c)}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{\frac{(c+d) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a d-b c}} \sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{(a+b) (c+d) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-\frac{(b c+a d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{d-c}} \Pi \left(\frac{a d-b c}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{\sqrt{2}}\right)|\frac{2 (a d-b c)}{(a+b) (d-c)}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{\frac{(c+d) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a d-b c}} \sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{(a+b) d \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\right)}{b d}+\frac{\sqrt{\frac{a-b}{a+b}} (a+b) \cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{a-b}{a+b}} \sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{\sqrt{\frac{a+b \sin (e+f x)}{a+b}}}\right)|\frac{2 (a d-b c)}{(a-b) (c+d)}\right) \sqrt{c+d \sin (e+f x)}}{b d \sqrt{\frac{(a+b) \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{a+b \sin (e+f x)}} \sqrt{a+b \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \sqrt{\frac{(a+b) (c+d \sin (e+f x))}{(c+d) (a+b \sin (e+f x))}}}\right)}{(a-b) (a+b) f}-\frac{2 (a B \cos (e+f x)-A b \cos (e+f x)) \sqrt{c+d \sin (e+f x)}}{\left(a^2-b^2\right) f \sqrt{a+b \sin (e+f x)}}","\frac{2 \sqrt{a+b} (c-d) (A b-a B) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{b f (a-b) \sqrt{c+d} (b c-a d)}+\frac{2 (c-d) \sqrt{c+d} (A b-a B) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b f (a-b) \sqrt{a+b} (b c-a d)}+\frac{2 B \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left(\frac{(a+b) d}{b (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{b^2 f \sqrt{c+d}}",1,"(-2*(-(A*b*Cos[e + f*x]) + a*B*Cos[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*f*Sqrt[a + b*Sin[e + f*x]]) + ((-4*(a*A*c - b*B*c)*(-(b*c) + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + a*d)*(A*b*c - a*B*c + a*A*d - b*B*d)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/((a + b)*d), ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) + 2*(-(A*b*d) + a*B*d)*((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*Sqrt[a + b*Sin[e + f*x]]) + (Sqrt[(a - b)/(a + b)]*(a + b)*Cos[(-e + Pi/2 - f*x)/2]*EllipticE[ArcSin[(Sqrt[(a - b)/(a + b)]*Sin[(-e + Pi/2 - f*x)/2])/Sqrt[(a + b*Sin[e + f*x])/(a + b)]], (2*(-(b*c) + a*d))/((a - b)*(c + d))]*Sqrt[c + d*Sin[e + f*x]])/(b*d*Sqrt[((a + b)*Cos[(-e + Pi/2 - f*x)/2]^2)/(a + b*Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/(a + b)]*Sqrt[((a + b)*(c + d*Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x]))]) - (2*(-(b*c) + a*d)*((((a + b)*c + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/((a + b)*d), ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])))/(b*d)))/((a - b)*(a + b)*f)","B",0
355,1,1949,417,6.5633652,"\int \frac{A+B \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]),x]","\frac{-\frac{4 (a d-b c) \left(A d a^2-A b c a+b^2 B c-A b^2 d\right) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{d-c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{\sqrt{2}}\right)|\frac{2 (a d-b c)}{(a+b) (d-c)}\right) \sec (e+f x) \sqrt{\frac{(c+d) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a d-b c}} \sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}} \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{(a+b) (c+d) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-4 (a d-b c) \left(B d a^2+b B c a-A b d a-A b^2 c\right) \left(\frac{\sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{d-c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{\sqrt{2}}\right)|\frac{2 (a d-b c)}{(a+b) (d-c)}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{\frac{(c+d) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a d-b c}} \sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{(a+b) (c+d) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{d-c}} \Pi \left(\frac{a d-b c}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{\sqrt{2}}\right)|\frac{2 (a d-b c)}{(a+b) (d-c)}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{\frac{(c+d) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a d-b c}} \sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{(a+b) d \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\right)+2 \left(A b^2 d-a b B d\right) \left(\frac{\sqrt{c+d \sin (e+f x)} \cos (e+f x)}{d \sqrt{a+b \sin (e+f x)}}-\frac{2 (a d-b c) \left(\frac{((a+b) c+a d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{d-c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{\sqrt{2}}\right)|\frac{2 (a d-b c)}{(a+b) (d-c)}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{\frac{(c+d) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a d-b c}} \sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{(a+b) (c+d) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-\frac{(b c+a d) \sqrt{\frac{(c+d) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{d-c}} \Pi \left(\frac{a d-b c}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{\sqrt{2}}\right)|\frac{2 (a d-b c)}{(a+b) (d-c)}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{\frac{(c+d) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a d-b c}} \sqrt{\frac{(-a-b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (c+d \sin (e+f x))}{a d-b c}}}{(a+b) d \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\right)}{b d}+\frac{\sqrt{\frac{a-b}{a+b}} (a+b) \cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{a-b}{a+b}} \sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{\sqrt{\frac{a+b \sin (e+f x)}{a+b}}}\right)|\frac{2 (a d-b c)}{(a-b) (c+d)}\right) \sqrt{c+d \sin (e+f x)}}{b d \sqrt{\frac{(a+b) \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{a+b \sin (e+f x)}} \sqrt{a+b \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \sqrt{\frac{(a+b) (c+d \sin (e+f x))}{(c+d) (a+b \sin (e+f x))}}}\right)}{(a-b) (a+b) (a d-b c) f}-\frac{2 \left(A b^2 \cos (e+f x)-a b B \cos (e+f x)\right) \sqrt{c+d \sin (e+f x)}}{\left(a^2-b^2\right) (a d-b c) f \sqrt{a+b \sin (e+f x)}}","\frac{2 \sqrt{a+b} (A-B) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f (a-b) \sqrt{c+d} (b c-a d)}+\frac{2 (c-d) \sqrt{c+d} (A b-a B) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{f (a-b) \sqrt{a+b} (b c-a d)^2}",1,"(-2*(A*b^2*Cos[e + f*x] - a*b*B*Cos[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*(-(b*c) + a*d)*f*Sqrt[a + b*Sin[e + f*x]]) + ((-4*(-(b*c) + a*d)*(-(a*A*b*c) + b^2*B*c + a^2*A*d - A*b^2*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + a*d)*(-(A*b^2*c) + a*b*B*c - a*A*b*d + a^2*B*d)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/((a + b)*d), ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) + 2*(A*b^2*d - a*b*B*d)*((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*Sqrt[a + b*Sin[e + f*x]]) + (Sqrt[(a - b)/(a + b)]*(a + b)*Cos[(-e + Pi/2 - f*x)/2]*EllipticE[ArcSin[(Sqrt[(a - b)/(a + b)]*Sin[(-e + Pi/2 - f*x)/2])/Sqrt[(a + b*Sin[e + f*x])/(a + b)]], (2*(-(b*c) + a*d))/((a - b)*(c + d))]*Sqrt[c + d*Sin[e + f*x]])/(b*d*Sqrt[((a + b)*Cos[(-e + Pi/2 - f*x)/2]^2)/(a + b*Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/(a + b)]*Sqrt[((a + b)*(c + d*Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x]))]) - (2*(-(b*c) + a*d)*((((a + b)*c + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/((a + b)*d), ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])))/(b*d)))/((a - b)*(a + b)*(-(b*c) + a*d)*f)","B",0
356,1,2266,544,7.4255684,"\int \frac{A+B \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2)),x]","\text{Result too large to show}","-\frac{2 \sec (e+f x) \left(A \left(a^2 d^2+b^2 \left(c^2-2 d^2\right)\right)-B \left(a^2 c d+a b \left(c^2-d^2\right)-b^2 c d\right)\right) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f \sqrt{a+b} (c-d) \sqrt{c+d} (b c-a d)^3}+\frac{2 b (A b-a B) \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 \sec (e+f x) (-a A d+a B d+A b c-2 A b d+b B c) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f \sqrt{a+b} (c-d) \sqrt{c+d} (b c-a d)^2}",1,"(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((2*(A*b^3*Cos[e + f*x] - a*b^2*B*Cos[e + f*x]))/((a^2 - b^2)*(-(b*c) + a*d)^2*(a + b*Sin[e + f*x])) - (2*(B*c*d^2*Cos[e + f*x] - A*d^3*Cos[e + f*x]))/((b*c - a*d)^2*(c^2 - d^2)*(c + d*Sin[e + f*x]))))/f + ((-4*(-(b*c) + a*d)*(a*A*b^2*c^3 - b^3*B*c^3 - 2*a^2*A*b*c^2*d + 2*A*b^3*c^2*d + a^3*A*c*d^2 - 2*a*A*b^2*c*d^2 + b^3*B*c*d^2 + 2*a^2*A*b*d^3 - 2*A*b^3*d^3 - a^3*B*d^3 + a*b^2*B*d^3)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + a*d)*(A*b^3*c^3 - a*b^2*B*c^3 + a*A*b^2*c^2*d - 2*a^2*b*B*c^2*d + b^3*B*c^2*d + a^2*A*b*c*d^2 - 2*A*b^3*c*d^2 - a^3*B*c*d^2 + 2*a*b^2*B*c*d^2 + a^3*A*d^3 - 2*a*A*b^2*d^3 + a^2*b*B*d^3)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/((a + b)*d), ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) + 2*(-(A*b^3*c^2*d) + a*b^2*B*c^2*d + a^2*b*B*c*d^2 - b^3*B*c*d^2 - a^2*A*b*d^3 + 2*A*b^3*d^3 - a*b^2*B*d^3)*((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*Sqrt[a + b*Sin[e + f*x]]) + (Sqrt[(a - b)/(a + b)]*(a + b)*Cos[(-e + Pi/2 - f*x)/2]*EllipticE[ArcSin[(Sqrt[(a - b)/(a + b)]*Sin[(-e + Pi/2 - f*x)/2])/Sqrt[(a + b*Sin[e + f*x])/(a + b)]], (2*(-(b*c) + a*d))/((a - b)*(c + d))]*Sqrt[c + d*Sin[e + f*x]])/(b*d*Sqrt[((a + b)*Cos[(-e + Pi/2 - f*x)/2]^2)/(a + b*Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/(a + b)]*Sqrt[((a + b)*(c + d*Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x]))]) - (2*(-(b*c) + a*d)*((((a + b)*c + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/((a + b)*d), ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])))/(b*d)))/((a - b)*(a + b)*(c - d)*(c + d)*(-(b*c) + a*d)^2*f)","B",0
357,1,2837,858,8.7699861,"\int \frac{A+B \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}} \, dx","Integrate[(A + B*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(5/2)),x]","\text{Result too large to show}","\frac{2 d \left(A \left(\left(3 c^2-4 d^2\right) b^2+a^2 d^2\right)-B \left(c d a^2+3 b \left(c^2-d^2\right) a-b^2 c d\right)\right) \sqrt{a+b \sin (e+f x)} \cos (e+f x)}{3 \left(a^2-b^2\right) (b c-a d)^2 \left(c^2-d^2\right) f (c+d \sin (e+f x))^{3/2}}+\frac{2 b (A b-a B) \cos (e+f x)}{\left(a^2-b^2\right) (b c-a d) f \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac{2 \left(B \left(-d^2 \left(c^2+3 d^2\right) a^3+2 b c d \left(3 c^2-d^2\right) a^2+b^2 \left(3 c^4-5 d^2 c^2+6 d^4\right) a-2 b^3 c d \left(3 c^2-d^2\right)\right)+A \left(-\left(\left(3 c^4-15 d^2 c^2+8 d^4\right) b^3\right)-4 a c d^3 b^2-a^2 d^2 \left(9 c^2-5 d^2\right) b+4 a^3 c d^3\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt{a+b} (c-d)^2 (c+d)^{3/2} (b c-a d)^4 f}-\frac{2 \left(B \left(-c \left(3 c^2+3 d c-2 d^2\right) b^2-6 a d \left(c^2-d^2\right) b+a^2 d^2 (c+3 d)\right)-A \left(\left(3 c^3-9 d c^2-6 d^2 c+8 d^3\right) b^2-6 a d \left(c^2-d^2\right) b+a^2 d^2 (3 c+d)\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt{a+b} (c-d)^2 (c+d)^{3/2} (b c-a d)^3 f}",1,"(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*(A*b^4*Cos[e + f*x] - a*b^3*B*Cos[e + f*x]))/((a^2 - b^2)*(-(b*c) + a*d)^3*(a + b*Sin[e + f*x])) + (2*(-(B*c*d^2*Cos[e + f*x]) + A*d^3*Cos[e + f*x]))/(3*(b*c - a*d)^2*(c^2 - d^2)*(c + d*Sin[e + f*x])^2) - (2*(6*b*B*c^3*d^2*Cos[e + f*x] - 9*A*b*c^2*d^3*Cos[e + f*x] - a*B*c^2*d^3*Cos[e + f*x] + 4*a*A*c*d^4*Cos[e + f*x] - 2*b*B*c*d^4*Cos[e + f*x] + 5*A*b*d^5*Cos[e + f*x] - 3*a*B*d^5*Cos[e + f*x]))/(3*(b*c - a*d)^3*(c^2 - d^2)^2*(c + d*Sin[e + f*x]))))/f + ((-4*(-(b*c) + a*d)*(-3*a*A*b^3*c^5 + 3*b^4*B*c^5 + 9*a^2*A*b^2*c^4*d - 9*A*b^4*c^4*d - 9*a^3*A*b*c^3*d^2 + 15*a*A*b^3*c^3*d^2 - a^2*b^2*B*c^3*d^2 - 5*b^4*B*c^3*d^2 + 3*a^4*A*c^2*d^3 - 20*a^2*A*b^2*c^2*d^3 + 17*A*b^4*c^2*d^3 + 10*a^3*b*B*c^2*d^3 - 10*a*b^3*B*c^2*d^3 + 5*a^3*A*b*c*d^4 - 8*a*A*b^3*c*d^4 - 4*a^4*B*c*d^4 + 5*a^2*b^2*B*c*d^4 + 2*b^4*B*c*d^4 + a^4*A*d^5 + 7*a^2*A*b^2*d^5 - 8*A*b^4*d^5 - 6*a^3*b*B*d^5 + 6*a*b^3*B*d^5)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + a*d)*(-3*A*b^4*c^5 + 3*a*b^3*B*c^5 - 3*a*A*b^3*c^4*d + 9*a^2*b^2*B*c^4*d - 6*b^4*B*c^4*d - 9*a^2*A*b^2*c^3*d^2 + 15*A*b^4*c^3*d^2 + 5*a^3*b*B*c^3*d^2 - 11*a*b^3*B*c^3*d^2 - 5*a^3*A*b*c^2*d^3 + 11*a*A*b^3*c^2*d^3 - a^4*B*c^2*d^3 - 7*a^2*b^2*B*c^2*d^3 + 2*b^4*B*c^2*d^3 + 4*a^4*A*c*d^4 + a^2*A*b^2*c*d^4 - 8*A*b^4*c*d^4 - 5*a^3*b*B*c*d^4 + 8*a*b^3*B*c*d^4 + 5*a^3*A*b*d^5 - 8*a*A*b^3*d^5 - 3*a^4*B*d^5 + 6*a^2*b^2*B*d^5)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/((a + b)*d), ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) + 2*(3*A*b^4*c^4*d - 3*a*b^3*B*c^4*d - 6*a^2*b^2*B*c^3*d^2 + 6*b^4*B*c^3*d^2 + 9*a^2*A*b^2*c^2*d^3 - 15*A*b^4*c^2*d^3 + a^3*b*B*c^2*d^3 + 5*a*b^3*B*c^2*d^3 - 4*a^3*A*b*c*d^4 + 4*a*A*b^3*c*d^4 + 2*a^2*b^2*B*c*d^4 - 2*b^4*B*c*d^4 - 5*a^2*A*b^2*d^5 + 8*A*b^4*d^5 + 3*a^3*b*B*d^5 - 6*a*b^3*B*d^5)*((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*Sqrt[a + b*Sin[e + f*x]]) + (Sqrt[(a - b)/(a + b)]*(a + b)*Cos[(-e + Pi/2 - f*x)/2]*EllipticE[ArcSin[(Sqrt[(a - b)/(a + b)]*Sin[(-e + Pi/2 - f*x)/2])/Sqrt[(a + b*Sin[e + f*x])/(a + b)]], (2*(-(b*c) + a*d))/((a - b)*(c + d))]*Sqrt[c + d*Sin[e + f*x]])/(b*d*Sqrt[((a + b)*Cos[(-e + Pi/2 - f*x)/2]^2)/(a + b*Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/(a + b)]*Sqrt[((a + b)*(c + d*Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x]))]) - (2*(-(b*c) + a*d)*((((a + b)*c + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/((a + b)*d), ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])))/(b*d)))/(3*(a - b)*(a + b)*(c - d)^2*(c + d)^2*(-(b*c) + a*d)^3*f)","B",0
358,0,0,38,17.6941458,"\int (a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Integrate[(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","\int (a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","\text{Int}\left((A+B \sin (e+f x)) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n,x\right)",0,"Integrate[(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n, x]","A",-1