1,1,105,102,0.4553292,"\int \sin ^3(e+f x) (a+a \sin (e+f x))^2 \, dx","Integrate[Sin[e + f*x]^3*(a + a*Sin[e + f*x])^2,x]","-\frac{a^2 \cos (e+f x) \left(30 \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\left(4 \sin ^4(e+f x)+10 \sin ^3(e+f x)+12 \sin ^2(e+f x)+15 \sin (e+f x)+24\right) \sqrt{\cos ^2(e+f x)}\right)}{20 f \sqrt{\cos ^2(e+f x)}}","-\frac{a^2 \cos ^5(e+f x)}{5 f}+\frac{a^2 \cos ^3(e+f x)}{f}-\frac{2 a^2 \cos (e+f x)}{f}-\frac{a^2 \sin ^3(e+f x) \cos (e+f x)}{2 f}-\frac{3 a^2 \sin (e+f x) \cos (e+f x)}{4 f}+\frac{3 a^2 x}{4}",1,"-1/20*(a^2*Cos[e + f*x]*(30*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(24 + 15*Sin[e + f*x] + 12*Sin[e + f*x]^2 + 10*Sin[e + f*x]^3 + 4*Sin[e + f*x]^4)))/(f*Sqrt[Cos[e + f*x]^2])","A",1
2,1,115,129,0.5286168,"\int \sin ^3(e+f x) (a+a \sin (e+f x))^3 \, dx","Integrate[Sin[e + f*x]^3*(a + a*Sin[e + f*x])^3,x]","-\frac{a^3 \cos (e+f x) \left(690 \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\left(40 \sin ^5(e+f x)+144 \sin ^4(e+f x)+230 \sin ^3(e+f x)+272 \sin ^2(e+f x)+345 \sin (e+f x)+544\right) \sqrt{\cos ^2(e+f x)}\right)}{240 f \sqrt{\cos ^2(e+f x)}}","-\frac{3 a^3 \cos ^5(e+f x)}{5 f}+\frac{7 a^3 \cos ^3(e+f x)}{3 f}-\frac{4 a^3 \cos (e+f x)}{f}-\frac{a^3 \sin ^5(e+f x) \cos (e+f x)}{6 f}-\frac{23 a^3 \sin ^3(e+f x) \cos (e+f x)}{24 f}-\frac{23 a^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{23 a^3 x}{16}",1,"-1/240*(a^3*Cos[e + f*x]*(690*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(544 + 345*Sin[e + f*x] + 272*Sin[e + f*x]^2 + 230*Sin[e + f*x]^3 + 144*Sin[e + f*x]^4 + 40*Sin[e + f*x]^5)))/(f*Sqrt[Cos[e + f*x]^2])","A",1
3,1,101,53,0.1297697,"\int \frac{\sin ^4(x)}{a+a \sin (x)} \, dx","Integrate[Sin[x]^4/(a + a*Sin[x]),x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(-36 x \sin \left(\frac{x}{2}\right)+69 \sin \left(\frac{x}{2}\right)-18 \sin \left(\frac{3 x}{2}\right)+2 \sin \left(\frac{5 x}{2}\right)+\sin \left(\frac{7 x}{2}\right)-3 (12 x+7) \cos \left(\frac{x}{2}\right)-18 \cos \left(\frac{3 x}{2}\right)-2 \cos \left(\frac{5 x}{2}\right)+\cos \left(\frac{7 x}{2}\right)\right)}{24 a (\sin (x)+1)}","-\frac{3 x}{2 a}+\frac{4 \cos ^3(x)}{3 a}-\frac{4 \cos (x)}{a}+\frac{\sin ^3(x) \cos (x)}{a \sin (x)+a}+\frac{3 \sin (x) \cos (x)}{2 a}",1,"((Cos[x/2] + Sin[x/2])*(-3*(7 + 12*x)*Cos[x/2] - 18*Cos[(3*x)/2] - 2*Cos[(5*x)/2] + Cos[(7*x)/2] + 69*Sin[x/2] - 36*x*Sin[x/2] - 18*Sin[(3*x)/2] + 2*Sin[(5*x)/2] + Sin[(7*x)/2]))/(24*a*(1 + Sin[x]))","A",1
4,1,87,42,0.0853945,"\int \frac{\sin ^3(x)}{a+a \sin (x)} \, dx","Integrate[Sin[x]^3/(a + a*Sin[x]),x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(12 x \sin \left(\frac{x}{2}\right)-20 \sin \left(\frac{x}{2}\right)+3 \sin \left(\frac{3 x}{2}\right)-\sin \left(\frac{5 x}{2}\right)+4 (3 x+1) \cos \left(\frac{x}{2}\right)+3 \cos \left(\frac{3 x}{2}\right)+\cos \left(\frac{5 x}{2}\right)\right)}{8 a (\sin (x)+1)}","\frac{3 x}{2 a}+\frac{2 \cos (x)}{a}+\frac{\sin ^2(x) \cos (x)}{a \sin (x)+a}-\frac{3 \sin (x) \cos (x)}{2 a}",1,"((Cos[x/2] + Sin[x/2])*(4*(1 + 3*x)*Cos[x/2] + 3*Cos[(3*x)/2] + Cos[(5*x)/2] - 20*Sin[x/2] + 12*x*Sin[x/2] + 3*Sin[(3*x)/2] - Sin[(5*x)/2]))/(8*a*(1 + Sin[x]))","B",1
5,1,48,27,0.0650348,"\int \frac{\sin ^2(x)}{a+a \sin (x)} \, dx","Integrate[Sin[x]^2/(a + a*Sin[x]),x]","-\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(\cos \left(\frac{x}{2}\right) (x+\cos (x))+\sin \left(\frac{x}{2}\right) (x+\cos (x)-2)\right)}{a (\sin (x)+1)}","-\frac{x}{a}-\frac{\cos (x)}{a}-\frac{\cos (x)}{a (\sin (x)+1)}",1,"-(((Cos[x/2] + Sin[x/2])*(Cos[x/2]*(x + Cos[x]) + (-2 + x + Cos[x])*Sin[x/2]))/(a*(1 + Sin[x])))","A",1
6,1,42,17,0.0408867,"\int \frac{\sin (x)}{a+a \sin (x)} \, dx","Integrate[Sin[x]/(a + a*Sin[x]),x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left((x-2) \sin \left(\frac{x}{2}\right)+x \cos \left(\frac{x}{2}\right)\right)}{a (\sin (x)+1)}","\frac{x}{a}+\frac{\cos (x)}{a \sin (x)+a}",1,"((Cos[x/2] + Sin[x/2])*(x*Cos[x/2] + (-2 + x)*Sin[x/2]))/(a*(1 + Sin[x]))","B",1
7,1,29,12,0.025935,"\int \frac{1}{a+a \sin (x)} \, dx","Integrate[(a + a*Sin[x])^(-1),x]","\frac{2 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{a \sin (x)+a}","-\frac{\cos (x)}{a \sin (x)+a}",1,"(2*Sin[x/2]*(Cos[x/2] + Sin[x/2]))/(a + a*Sin[x])","B",1
8,1,74,20,0.0550697,"\int \frac{\csc (x)}{a+a \sin (x)} \, dx","Integrate[Csc[x]/(a + a*Sin[x]),x]","-\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(\cos \left(\frac{x}{2}\right) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)+\sin \left(\frac{x}{2}\right) \left(-\log \left(\sin \left(\frac{x}{2}\right)\right)+\log \left(\cos \left(\frac{x}{2}\right)\right)+2\right)\right)}{a (\sin (x)+1)}","\frac{\cos (x)}{a \sin (x)+a}-\frac{\tanh ^{-1}(\cos (x))}{a}",1,"-(((Cos[x/2] + Sin[x/2])*(Cos[x/2]*(Log[Cos[x/2]] - Log[Sin[x/2]]) + (2 + Log[Cos[x/2]] - Log[Sin[x/2]])*Sin[x/2]))/(a*(1 + Sin[x])))","B",1
9,1,63,26,0.1614348,"\int \frac{\csc ^2(x)}{a+a \sin (x)} \, dx","Integrate[Csc[x]^2/(a + a*Sin[x]),x]","\frac{\tan \left(\frac{x}{2}\right)-\cot \left(\frac{x}{2}\right)-2 \log \left(\sin \left(\frac{x}{2}\right)\right)+2 \log \left(\cos \left(\frac{x}{2}\right)\right)+\frac{4 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}}{2 a}","-\frac{2 \cot (x)}{a}+\frac{\tanh ^{-1}(\cos (x))}{a}+\frac{\cot (x)}{a \sin (x)+a}",1,"(-Cot[x/2] + 2*Log[Cos[x/2]] - 2*Log[Sin[x/2]] + (4*Sin[x/2])/(Cos[x/2] + Sin[x/2]) + Tan[x/2])/(2*a)","B",1
10,1,83,42,0.349583,"\int \frac{\csc ^3(x)}{a+a \sin (x)} \, dx","Integrate[Csc[x]^3/(a + a*Sin[x]),x]","\frac{-4 \tan \left(\frac{x}{2}\right)+4 \cot \left(\frac{x}{2}\right)-\csc ^2\left(\frac{x}{2}\right)+\sec ^2\left(\frac{x}{2}\right)+12 \log \left(\sin \left(\frac{x}{2}\right)\right)-12 \log \left(\cos \left(\frac{x}{2}\right)\right)-\frac{16 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}}{8 a}","\frac{2 \cot (x)}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{2 a}-\frac{3 \cot (x) \csc (x)}{2 a}+\frac{\cot (x) \csc (x)}{a \sin (x)+a}",1,"(4*Cot[x/2] - Csc[x/2]^2 - 12*Log[Cos[x/2]] + 12*Log[Sin[x/2]] + Sec[x/2]^2 - (16*Sin[x/2])/(Cos[x/2] + Sin[x/2]) - 4*Tan[x/2])/(8*a)","A",1
11,1,113,55,0.8131837,"\int \frac{\csc ^4(x)}{a+a \sin (x)} \, dx","Integrate[Csc[x]^4/(a + a*Sin[x]),x]","\frac{20 \tan \left(\frac{x}{2}\right)-20 \cot \left(\frac{x}{2}\right)+3 \csc ^2\left(\frac{x}{2}\right)-3 \sec ^2\left(\frac{x}{2}\right)-36 \log \left(\sin \left(\frac{x}{2}\right)\right)+36 \log \left(\cos \left(\frac{x}{2}\right)\right)+\frac{48 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}-\frac{1}{2} \sin (x) \csc ^4\left(\frac{x}{2}\right)+8 \sin ^4\left(\frac{x}{2}\right) \csc ^3(x)}{24 a}","-\frac{4 \cot ^3(x)}{3 a}-\frac{4 \cot (x)}{a}+\frac{3 \tanh ^{-1}(\cos (x))}{2 a}+\frac{3 \cot (x) \csc (x)}{2 a}+\frac{\cot (x) \csc ^2(x)}{a \sin (x)+a}",1,"(-20*Cot[x/2] + 3*Csc[x/2]^2 + 36*Log[Cos[x/2]] - 36*Log[Sin[x/2]] - 3*Sec[x/2]^2 + 8*Csc[x]^3*Sin[x/2]^4 + (48*Sin[x/2])/(Cos[x/2] + Sin[x/2]) - (Csc[x/2]^4*Sin[x])/2 + 20*Tan[x/2])/(24*a)","B",1
12,1,100,66,0.2527248,"\int \frac{\sin ^4(x)}{(a+a \sin (x))^2} \, dx","Integrate[Sin[x]^4/(a + a*Sin[x])^2,x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(21 (12 x-7) \cos \left(\frac{x}{2}\right)+(239-84 x) \cos \left(\frac{3 x}{2}\right)+3 \left(-5 \cos \left(\frac{5 x}{2}\right)+\cos \left(\frac{7 x}{2}\right)+2 \sin \left(\frac{x}{2}\right) (56 x+(28 x+27) \cos (x)+6 \cos (2 x)+\cos (3 x)-50)\right)\right)}{48 a^2 (\sin (x)+1)^2}","\frac{7 x}{2 a^2}+\frac{16 \cos (x)}{3 a^2}+\frac{8 \sin ^2(x) \cos (x)}{3 a^2 (\sin (x)+1)}-\frac{7 \sin (x) \cos (x)}{2 a^2}+\frac{\sin ^3(x) \cos (x)}{3 (a \sin (x)+a)^2}",1,"((Cos[x/2] + Sin[x/2])*(21*(-7 + 12*x)*Cos[x/2] + (239 - 84*x)*Cos[(3*x)/2] + 3*(-5*Cos[(5*x)/2] + Cos[(7*x)/2] + 2*(-50 + 56*x + (27 + 28*x)*Cos[x] + 6*Cos[2*x] + Cos[3*x])*Sin[x/2])))/(48*a^2*(1 + Sin[x])^2)","A",1
13,1,84,47,0.2369024,"\int \frac{\sin ^3(x)}{(a+a \sin (x))^2} \, dx","Integrate[Sin[x]^3/(a + a*Sin[x])^2,x]","-\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(6 (6 x-5) \cos \left(\frac{x}{2}\right)+(41-12 x) \cos \left(\frac{3 x}{2}\right)-3 \cos \left(\frac{5 x}{2}\right)+6 \sin \left(\frac{x}{2}\right) (8 x+4 (x+1) \cos (x)+\cos (2 x)-9)\right)}{12 a^2 (\sin (x)+1)^2}","-\frac{2 x}{a^2}-\frac{4 \cos (x)}{3 a^2}-\frac{2 \cos (x)}{a^2 (\sin (x)+1)}+\frac{\sin ^2(x) \cos (x)}{3 (a \sin (x)+a)^2}",1,"-1/12*((Cos[x/2] + Sin[x/2])*(6*(-5 + 6*x)*Cos[x/2] + (41 - 12*x)*Cos[(3*x)/2] - 3*Cos[(5*x)/2] + 6*(-9 + 8*x + 4*(1 + x)*Cos[x] + Cos[2*x])*Sin[x/2]))/(a^2*(1 + Sin[x])^2)","A",1
14,1,69,35,0.1290819,"\int \frac{\sin ^2(x)}{(a+a \sin (x))^2} \, dx","Integrate[Sin[x]^2/(a + a*Sin[x])^2,x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(3 (3 x-4) \cos \left(\frac{x}{2}\right)+(10-3 x) \cos \left(\frac{3 x}{2}\right)+6 \sin \left(\frac{x}{2}\right) (2 x+x \cos (x)-3)\right)}{6 a^2 (\sin (x)+1)^2}","\frac{x}{a^2}+\frac{5 \cos (x)}{3 a^2 (\sin (x)+1)}-\frac{\cos (x)}{3 (a \sin (x)+a)^2}",1,"((Cos[x/2] + Sin[x/2])*(3*(-4 + 3*x)*Cos[x/2] + (10 - 3*x)*Cos[(3*x)/2] + 6*(-3 + 2*x + x*Cos[x])*Sin[x/2]))/(6*a^2*(1 + Sin[x])^2)","A",1
15,1,29,33,0.0442472,"\int \frac{\sin (x)}{(a+a \sin (x))^2} \, dx","Integrate[Sin[x]/(a + a*Sin[x])^2,x]","-\frac{-4 \sin (x)+\sin (2 x)+\cos (x)+\cos (2 x)-3}{3 a^2 (\sin (x)+1)^2}","\frac{\cos (x)}{3 (a \sin (x)+a)^2}-\frac{2 \cos (x)}{3 \left(a^2 \sin (x)+a^2\right)}",1,"-1/3*(-3 + Cos[x] + Cos[2*x] - 4*Sin[x] + Sin[2*x])/(a^2*(1 + Sin[x])^2)","A",1
16,1,31,33,0.0277942,"\int \frac{1}{(a+a \sin (x))^2} \, dx","Integrate[(a + a*Sin[x])^(-2),x]","-\frac{-4 \sin (x)+\sin (2 x)+4 \cos (x)+\cos (2 x)-3}{6 a^2 (\sin (x)+1)^2}","-\frac{\cos (x)}{3 \left(a^2 \sin (x)+a^2\right)}-\frac{\cos (x)}{3 (a \sin (x)+a)^2}",1,"-1/6*(-3 + 4*Cos[x] + Cos[2*x] - 4*Sin[x] + Sin[2*x])/(a^2*(1 + Sin[x])^2)","A",1
17,1,129,38,0.1412957,"\int \frac{\csc (x)}{(a+a \sin (x))^2} \, dx","Integrate[Csc[x]/(a + a*Sin[x])^2,x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(\cos \left(\frac{3 x}{2}\right) \left(-3 \log \left(\sin \left(\frac{x}{2}\right)\right)+3 \log \left(\cos \left(\frac{x}{2}\right)\right)+8\right)+\cos \left(\frac{x}{2}\right) \left(9 \log \left(\sin \left(\frac{x}{2}\right)\right)-9 \log \left(\cos \left(\frac{x}{2}\right)\right)-6\right)-6 \sin \left(\frac{x}{2}\right) \left(-2 \log \left(\sin \left(\frac{x}{2}\right)\right)+2 \log \left(\cos \left(\frac{x}{2}\right)\right)+\cos (x) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)+3\right)\right)}{6 a^2 (\sin (x)+1)^2}","\frac{4 \cos (x)}{3 a^2 (\sin (x)+1)}-\frac{\tanh ^{-1}(\cos (x))}{a^2}+\frac{\cos (x)}{3 (a \sin (x)+a)^2}",1,"((Cos[x/2] + Sin[x/2])*(Cos[(3*x)/2]*(8 + 3*Log[Cos[x/2]] - 3*Log[Sin[x/2]]) + Cos[x/2]*(-6 - 9*Log[Cos[x/2]] + 9*Log[Sin[x/2]]) - 6*(3 + 2*Log[Cos[x/2]] + Cos[x]*(Log[Cos[x/2]] - Log[Sin[x/2]]) - 2*Log[Sin[x/2]])*Sin[x/2]))/(6*a^2*(1 + Sin[x])^2)","B",1
18,1,166,45,0.3748445,"\int \frac{\csc ^2(x)}{(a+a \sin (x))^2} \, dx","Integrate[Csc[x]^2/(a + a*Sin[x])^2,x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(4 \sin \left(\frac{x}{2}\right)+28 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2-2 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)+12 \log \left(\cos \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3-12 \log \left(\sin \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3+3 \tan \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3-3 \cot \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3\right)}{6 (a \sin (x)+a)^2}","-\frac{10 \cot (x)}{3 a^2}+\frac{2 \tanh ^{-1}(\cos (x))}{a^2}+\frac{2 \cot (x)}{a^2 (\sin (x)+1)}+\frac{\cot (x)}{3 (a \sin (x)+a)^2}",1,"((Cos[x/2] + Sin[x/2])*(4*Sin[x/2] - 2*(Cos[x/2] + Sin[x/2]) + 28*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 - 3*Cot[x/2]*(Cos[x/2] + Sin[x/2])^3 + 12*Log[Cos[x/2]]*(Cos[x/2] + Sin[x/2])^3 - 12*Log[Sin[x/2]]*(Cos[x/2] + Sin[x/2])^3 + 3*(Cos[x/2] + Sin[x/2])^3*Tan[x/2]))/(6*(a + a*Sin[x])^2)","B",1
19,1,203,64,0.663646,"\int \frac{\csc ^3(x)}{(a+a \sin (x))^2} \, dx","Integrate[Csc[x]^3/(a + a*Sin[x])^2,x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(-16 \sin \left(\frac{x}{2}\right)-160 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2+8 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)+3 \cos \left(\frac{x}{2}\right) \left(\tan \left(\frac{x}{2}\right)+1\right)^3-3 \sin \left(\frac{x}{2}\right) \left(\cot \left(\frac{x}{2}\right)+1\right)^3-84 \log \left(\cos \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3+84 \log \left(\sin \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3-24 \tan \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3+24 \cot \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3\right)}{24 a^2 (\sin (x)+1)^2}","\frac{16 \cot (x)}{3 a^2}-\frac{7 \tanh ^{-1}(\cos (x))}{2 a^2}-\frac{7 \cot (x) \csc (x)}{2 a^2}+\frac{8 \cot (x) \csc (x)}{3 a^2 (\sin (x)+1)}+\frac{\cot (x) \csc (x)}{3 (a \sin (x)+a)^2}",1,"((Cos[x/2] + Sin[x/2])*(-16*Sin[x/2] - 3*(1 + Cot[x/2])^3*Sin[x/2] + 8*(Cos[x/2] + Sin[x/2]) - 160*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 + 24*Cot[x/2]*(Cos[x/2] + Sin[x/2])^3 - 84*Log[Cos[x/2]]*(Cos[x/2] + Sin[x/2])^3 + 84*Log[Sin[x/2]]*(Cos[x/2] + Sin[x/2])^3 - 24*(Cos[x/2] + Sin[x/2])^3*Tan[x/2] + 3*Cos[x/2]*(1 + Tan[x/2])^3))/(24*a^2*(1 + Sin[x])^2)","B",1
20,1,238,65,3.5941354,"\int \frac{\csc ^4(x)}{(a+a \sin (x))^2} \, dx","Integrate[Csc[x]^4/(a + a*Sin[x])^2,x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(16 \sin \left(\frac{x}{2}\right)+\sin \left(\frac{x}{2}\right) \left(\tan \left(\frac{x}{2}\right)+1\right)^3+208 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2-8 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-6 \cos \left(\frac{x}{2}\right) \left(\tan \left(\frac{x}{2}\right)+1\right)^3-\cos \left(\frac{x}{2}\right) \left(\cot \left(\frac{x}{2}\right)+1\right)^3+6 \sin \left(\frac{x}{2}\right) \left(\cot \left(\frac{x}{2}\right)+1\right)^3+120 \log \left(\cos \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3-120 \log \left(\sin \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3+44 \tan \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3-44 \cot \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3\right)}{24 a^2 (\sin (x)+1)^2}","-\frac{\cot ^3(x)}{3 a^2}-\frac{4 \cot (x)}{a^2}-\frac{13 \cos (x)}{3 a^2 (\sin (x)+1)}-\frac{\cos (x)}{3 a^2 (\sin (x)+1)^2}+\frac{5 \tanh ^{-1}(\cos (x))}{a^2}+\frac{\cot (x) \csc (x)}{a^2}",1,"((Cos[x/2] + Sin[x/2])*(-(Cos[x/2]*(1 + Cot[x/2])^3) + 16*Sin[x/2] + 6*(1 + Cot[x/2])^3*Sin[x/2] - 8*(Cos[x/2] + Sin[x/2]) + 208*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 - 44*Cot[x/2]*(Cos[x/2] + Sin[x/2])^3 + 120*Log[Cos[x/2]]*(Cos[x/2] + Sin[x/2])^3 - 120*Log[Sin[x/2]]*(Cos[x/2] + Sin[x/2])^3 + 44*(Cos[x/2] + Sin[x/2])^3*Tan[x/2] - 6*Cos[x/2]*(1 + Tan[x/2])^3 + Sin[x/2]*(1 + Tan[x/2])^3))/(24*a^2*(1 + Sin[x])^2)","B",1
21,1,191,101,0.1048214,"\int \frac{\sin ^6(x)}{(a+a \sin (x))^3} \, dx","Integrate[Sin[x]^6/(a + a*Sin[x])^3,x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(24 \sin \left(\frac{x}{2}\right)-690 x \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5-405 \cos (x) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5+5 \cos (3 x) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5+45 \sin (2 x) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5+1576 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^4+112 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3-224 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2-12 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)}{60 (a \sin (x)+a)^3}","-\frac{23 x}{2 a^3}+\frac{136 \cos ^3(x)}{15 a^3}-\frac{136 \cos (x)}{5 a^3}+\frac{23 \sin ^3(x) \cos (x)}{3 \left(a^3 \sin (x)+a^3\right)}+\frac{23 \sin (x) \cos (x)}{2 a^3}+\frac{\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac{13 \sin ^4(x) \cos (x)}{15 a (a \sin (x)+a)^2}",1,"((Cos[x/2] + Sin[x/2])*(24*Sin[x/2] - 12*(Cos[x/2] + Sin[x/2]) - 224*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 + 112*(Cos[x/2] + Sin[x/2])^3 + 1576*Sin[x/2]*(Cos[x/2] + Sin[x/2])^4 - 690*x*(Cos[x/2] + Sin[x/2])^5 - 405*Cos[x]*(Cos[x/2] + Sin[x/2])^5 + 5*Cos[3*x]*(Cos[x/2] + Sin[x/2])^5 + 45*(Cos[x/2] + Sin[x/2])^5*Sin[2*x]))/(60*(a + a*Sin[x])^3)","A",1
22,1,170,90,0.0797915,"\int \frac{\sin ^5(x)}{(a+a \sin (x))^3} \, dx","Integrate[Sin[x]^5/(a + a*Sin[x])^3,x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(-24 \sin \left(\frac{x}{2}\right)+390 x \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5+180 \cos (x) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5-15 \sin (2 x) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5-1016 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^4-92 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3+184 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2+12 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)}{60 (a \sin (x)+a)^3}","\frac{13 x}{2 a^3}+\frac{152 \cos (x)}{15 a^3}+\frac{76 \sin ^2(x) \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}-\frac{13 \sin (x) \cos (x)}{2 a^3}+\frac{\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac{11 \sin ^3(x) \cos (x)}{15 a (a \sin (x)+a)^2}",1,"((Cos[x/2] + Sin[x/2])*(-24*Sin[x/2] + 12*(Cos[x/2] + Sin[x/2]) + 184*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 - 92*(Cos[x/2] + Sin[x/2])^3 - 1016*Sin[x/2]*(Cos[x/2] + Sin[x/2])^4 + 390*x*(Cos[x/2] + Sin[x/2])^5 + 180*Cos[x]*(Cos[x/2] + Sin[x/2])^5 - 15*(Cos[x/2] + Sin[x/2])^5*Sin[2*x]))/(60*(a + a*Sin[x])^3)","A",1
23,1,140,71,0.0800382,"\int \frac{\sin ^4(x)}{(a+a \sin (x))^3} \, dx","Integrate[Sin[x]^4/(a + a*Sin[x])^3,x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)-\cos \left(\frac{x}{2}\right)-15 x \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5-5 \cos (x) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5+48 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^4+6 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3-12 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2\right)}{5 (a \sin (x)+a)^3}","-\frac{3 x}{a^3}-\frac{9 \cos (x)}{5 a^3}-\frac{3 \cos (x)}{a^3 \sin (x)+a^3}+\frac{\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac{3 \sin ^2(x) \cos (x)}{5 a (a \sin (x)+a)^2}",1,"((Cos[x/2] + Sin[x/2])*(-Cos[x/2] + Sin[x/2] - 12*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 + 6*(Cos[x/2] + Sin[x/2])^3 + 48*Sin[x/2]*(Cos[x/2] + Sin[x/2])^4 - 15*x*(Cos[x/2] + Sin[x/2])^5 - 5*Cos[x]*(Cos[x/2] + Sin[x/2])^5))/(5*(a + a*Sin[x])^3)","A",1
24,1,112,59,0.1811535,"\int \frac{\sin ^3(x)}{(a+a \sin (x))^3} \, dx","Integrate[Sin[x]^3/(a + a*Sin[x])^3,x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(150 x \sin \left(\frac{x}{2}\right)-370 \sin \left(\frac{x}{2}\right)+75 x \sin \left(\frac{3 x}{2}\right)-90 \sin \left(\frac{3 x}{2}\right)-15 x \sin \left(\frac{5 x}{2}\right)+64 \sin \left(\frac{5 x}{2}\right)+30 (5 x-9) \cos \left(\frac{x}{2}\right)+(230-75 x) \cos \left(\frac{3 x}{2}\right)-15 x \cos \left(\frac{5 x}{2}\right)\right)}{60 a^3 (\sin (x)+1)^3}","\frac{x}{a^3}+\frac{29 \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}+\frac{\sin ^2(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac{7 \cos (x)}{15 a (a \sin (x)+a)^2}",1,"((Cos[x/2] + Sin[x/2])*(30*(-9 + 5*x)*Cos[x/2] + (230 - 75*x)*Cos[(3*x)/2] - 15*x*Cos[(5*x)/2] - 370*Sin[x/2] + 150*x*Sin[x/2] - 90*Sin[(3*x)/2] + 75*x*Sin[(3*x)/2] + 64*Sin[(5*x)/2] - 15*x*Sin[(5*x)/2]))/(60*a^3*(1 + Sin[x])^3)","A",1
25,1,47,50,0.0651813,"\int \frac{\sin ^2(x)}{(a+a \sin (x))^3} \, dx","Integrate[Sin[x]^2/(a + a*Sin[x])^3,x]","\frac{105 \sin (x)-12 \sin (2 x)-7 \sin (3 x)-15 \cos (x)-42 \cos (2 x)+7 \cos (3 x)+70}{60 a^3 (\sin (x)+1)^3}","-\frac{7 \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}+\frac{8 \cos (x)}{15 a (a \sin (x)+a)^2}-\frac{\cos (x)}{5 (a \sin (x)+a)^3}",1,"(70 - 15*Cos[x] - 42*Cos[2*x] + 7*Cos[3*x] + 105*Sin[x] - 12*Sin[2*x] - 7*Sin[3*x])/(60*a^3*(1 + Sin[x])^3)","A",1
26,1,41,50,0.0417198,"\int \frac{\sin (x)}{(a+a \sin (x))^3} \, dx","Integrate[Sin[x]/(a + a*Sin[x])^3,x]","\frac{\sin ^2\left(\frac{x}{2}\right) (8 \sin (x)+\sin (2 x)+4 \cos (x)-\cos (2 x)+7)}{5 a^3 (\sin (x)+1)^3}","-\frac{\cos (x)}{5 \left(a^3 \sin (x)+a^3\right)}-\frac{\cos (x)}{5 a (a \sin (x)+a)^2}+\frac{\cos (x)}{5 (a \sin (x)+a)^3}",1,"(Sin[x/2]^2*(7 + 4*Cos[x] - Cos[2*x] + 8*Sin[x] + Sin[2*x]))/(5*a^3*(1 + Sin[x])^3)","A",1
27,1,45,50,0.0605951,"\int \frac{1}{(a+a \sin (x))^3} \, dx","Integrate[(a + a*Sin[x])^(-3),x]","-\frac{-10 \sin \left(\frac{x}{2}\right)+\sin \left(\frac{5 x}{2}\right)+5 \cos \left(\frac{3 x}{2}\right)}{15 a^3 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5}","-\frac{2 \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}-\frac{2 \cos (x)}{15 a (a \sin (x)+a)^2}-\frac{\cos (x)}{5 (a \sin (x)+a)^3}",1,"-1/15*(5*Cos[(3*x)/2] - 10*Sin[x/2] + Sin[(5*x)/2])/(a^3*(Cos[x/2] + Sin[x/2])^5)","A",1
28,1,160,58,0.0723199,"\int \frac{\csc (x)}{(a+a \sin (x))^3} \, dx","Integrate[Csc[x]/(a + a*Sin[x])^3,x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(-6 \sin \left(\frac{x}{2}\right)-44 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^4+7 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3-14 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2+3 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-15 \log \left(\cos \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5+15 \log \left(\sin \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5\right)}{15 (a \sin (x)+a)^3}","\frac{22 \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}-\frac{\tanh ^{-1}(\cos (x))}{a^3}+\frac{7 \cos (x)}{15 a (a \sin (x)+a)^2}+\frac{\cos (x)}{5 (a \sin (x)+a)^3}",1,"((Cos[x/2] + Sin[x/2])*(-6*Sin[x/2] + 3*(Cos[x/2] + Sin[x/2]) - 14*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 + 7*(Cos[x/2] + Sin[x/2])^3 - 44*Sin[x/2]*(Cos[x/2] + Sin[x/2])^4 - 15*Log[Cos[x/2]]*(Cos[x/2] + Sin[x/2])^5 + 15*Log[Sin[x/2]]*(Cos[x/2] + Sin[x/2])^5))/(15*(a + a*Sin[x])^3)","B",1
29,1,206,65,0.15213,"\int \frac{\csc ^2(x)}{(a+a \sin (x))^3} \, dx","Integrate[Csc[x]^2/(a + a*Sin[x])^3,x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(4 \sin \left(\frac{x}{2}\right)+76 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^4-8 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3+16 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2-2 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)+30 \log \left(\cos \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5-30 \log \left(\sin \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5+5 \tan \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5-5 \cot \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5\right)}{10 (a \sin (x)+a)^3}","-\frac{24 \cot (x)}{5 a^3}+\frac{3 \tanh ^{-1}(\cos (x))}{a^3}+\frac{3 \cot (x)}{a^3 \sin (x)+a^3}+\frac{3 \cot (x)}{5 a (a \sin (x)+a)^2}+\frac{\cot (x)}{5 (a \sin (x)+a)^3}",1,"((Cos[x/2] + Sin[x/2])*(4*Sin[x/2] - 2*(Cos[x/2] + Sin[x/2]) + 16*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 - 8*(Cos[x/2] + Sin[x/2])^3 + 76*Sin[x/2]*(Cos[x/2] + Sin[x/2])^4 - 5*Cot[x/2]*(Cos[x/2] + Sin[x/2])^5 + 30*Log[Cos[x/2]]*(Cos[x/2] + Sin[x/2])^5 - 30*Log[Sin[x/2]]*(Cos[x/2] + Sin[x/2])^5 + 5*(Cos[x/2] + Sin[x/2])^5*Tan[x/2]))/(10*(a + a*Sin[x])^3)","B",1
30,1,247,86,0.4635291,"\int \frac{\csc ^3(x)}{(a+a \sin (x))^3} \, dx","Integrate[Csc[x]^3/(a + a*Sin[x])^3,x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(-48 \sin \left(\frac{x}{2}\right)+15 \cos ^3\left(\frac{x}{2}\right) \left(\tan \left(\frac{x}{2}\right)+1\right)^5-1712 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^4+136 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3-272 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2+24 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-15 \sin ^3\left(\frac{x}{2}\right) \left(\cot \left(\frac{x}{2}\right)+1\right)^5-780 \log \left(\cos \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5+780 \log \left(\sin \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5-180 \tan \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5+180 \cot \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5\right)}{120 a^3 (\sin (x)+1)^3}","\frac{152 \cot (x)}{15 a^3}-\frac{13 \tanh ^{-1}(\cos (x))}{2 a^3}-\frac{13 \cot (x) \csc (x)}{2 a^3}+\frac{76 \cot (x) \csc (x)}{15 \left(a^3 \sin (x)+a^3\right)}+\frac{11 \cot (x) \csc (x)}{15 a (a \sin (x)+a)^2}+\frac{\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}",1,"((Cos[x/2] + Sin[x/2])*(-48*Sin[x/2] - 15*(1 + Cot[x/2])^5*Sin[x/2]^3 + 24*(Cos[x/2] + Sin[x/2]) - 272*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 + 136*(Cos[x/2] + Sin[x/2])^3 - 1712*Sin[x/2]*(Cos[x/2] + Sin[x/2])^4 + 180*Cot[x/2]*(Cos[x/2] + Sin[x/2])^5 - 780*Log[Cos[x/2]]*(Cos[x/2] + Sin[x/2])^5 + 780*Log[Sin[x/2]]*(Cos[x/2] + Sin[x/2])^5 - 180*(Cos[x/2] + Sin[x/2])^5*Tan[x/2] + 15*Cos[x/2]^3*(1 + Tan[x/2])^5))/(120*a^3*(1 + Sin[x])^3)","B",1
31,1,299,103,0.9275309,"\int \frac{\csc ^4(x)}{(a+a \sin (x))^3} \, dx","Integrate[Csc[x]^4/(a + a*Sin[x])^3,x]","\frac{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(48 \sin \left(\frac{x}{2}\right)-45 \cos ^3\left(\frac{x}{2}\right) \left(\tan \left(\frac{x}{2}\right)+1\right)^5+2752 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^4-176 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3+352 \sin \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2-24 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)+45 \sin ^3\left(\frac{x}{2}\right) \left(\cot \left(\frac{x}{2}\right)+1\right)^5+5 \sin \left(\frac{x}{2}\right) \cos ^2\left(\frac{x}{2}\right) \left(\tan \left(\frac{x}{2}\right)+1\right)^5+1380 \log \left(\cos \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5-1380 \log \left(\sin \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5+400 \tan \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5-5 \sin ^2\left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right) \left(\cot \left(\frac{x}{2}\right)+1\right)^5-400 \cot \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5\right)}{120 a^3 (\sin (x)+1)^3}","-\frac{136 \cot ^3(x)}{15 a^3}-\frac{136 \cot (x)}{5 a^3}+\frac{23 \tanh ^{-1}(\cos (x))}{2 a^3}+\frac{23 \cot (x) \csc (x)}{2 a^3}+\frac{23 \cot (x) \csc ^2(x)}{3 \left(a^3 \sin (x)+a^3\right)}+\frac{13 \cot (x) \csc ^2(x)}{15 a (a \sin (x)+a)^2}+\frac{\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}",1,"((Cos[x/2] + Sin[x/2])*(48*Sin[x/2] - 5*Cos[x/2]*(1 + Cot[x/2])^5*Sin[x/2]^2 + 45*(1 + Cot[x/2])^5*Sin[x/2]^3 - 24*(Cos[x/2] + Sin[x/2]) + 352*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 - 176*(Cos[x/2] + Sin[x/2])^3 + 2752*Sin[x/2]*(Cos[x/2] + Sin[x/2])^4 - 400*Cot[x/2]*(Cos[x/2] + Sin[x/2])^5 + 1380*Log[Cos[x/2]]*(Cos[x/2] + Sin[x/2])^5 - 1380*Log[Sin[x/2]]*(Cos[x/2] + Sin[x/2])^5 + 400*(Cos[x/2] + Sin[x/2])^5*Tan[x/2] - 45*Cos[x/2]^3*(1 + Tan[x/2])^5 + 5*Cos[x/2]^2*Sin[x/2]*(1 + Tan[x/2])^5))/(120*a^3*(1 + Sin[x])^3)","B",1
32,1,165,158,0.4964325,"\int \sin ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(1890 \sin \left(\frac{1}{2} (c+d x)\right)-420 \sin \left(\frac{3}{2} (c+d x)\right)-252 \sin \left(\frac{5}{2} (c+d x)\right)+45 \sin \left(\frac{7}{2} (c+d x)\right)+35 \sin \left(\frac{9}{2} (c+d x)\right)-1890 \cos \left(\frac{1}{2} (c+d x)\right)-420 \cos \left(\frac{3}{2} (c+d x)\right)+252 \cos \left(\frac{5}{2} (c+d x)\right)+45 \cos \left(\frac{7}{2} (c+d x)\right)-35 \cos \left(\frac{9}{2} (c+d x)\right)\right)}{2520 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{2 a \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}-\frac{16 a \sin ^3(c+d x) \cos (c+d x)}{63 d \sqrt{a \sin (c+d x)+a}}-\frac{32 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 a d}+\frac{64 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 d}-\frac{32 a \cos (c+d x)}{45 d \sqrt{a \sin (c+d x)+a}}",1,"(Sqrt[a*(1 + Sin[c + d*x])]*(-1890*Cos[(c + d*x)/2] - 420*Cos[(3*(c + d*x))/2] + 252*Cos[(5*(c + d*x))/2] + 45*Cos[(7*(c + d*x))/2] - 35*Cos[(9*(c + d*x))/2] + 1890*Sin[(c + d*x)/2] - 420*Sin[(3*(c + d*x))/2] - 252*Sin[(5*(c + d*x))/2] + 45*Sin[(7*(c + d*x))/2] + 35*Sin[(9*(c + d*x))/2]))/(2520*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
33,1,141,122,0.2880315,"\int \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(105 \sin \left(\frac{1}{2} (c+d x)\right)-35 \sin \left(\frac{3}{2} (c+d x)\right)-7 \sin \left(\frac{5}{2} (c+d x)\right)+5 \sin \left(\frac{7}{2} (c+d x)\right)-105 \cos \left(\frac{1}{2} (c+d x)\right)-35 \cos \left(\frac{3}{2} (c+d x)\right)+7 \cos \left(\frac{5}{2} (c+d x)\right)+5 \cos \left(\frac{7}{2} (c+d x)\right)\right)}{140 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt{a \sin (c+d x)+a}}-\frac{12 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 a d}+\frac{8 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{35 d}-\frac{4 a \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}",1,"(Sqrt[a*(1 + Sin[c + d*x])]*(-105*Cos[(c + d*x)/2] - 35*Cos[(3*(c + d*x))/2] + 7*Cos[(5*(c + d*x))/2] + 5*Cos[(7*(c + d*x))/2] + 105*Sin[(c + d*x)/2] - 35*Sin[(3*(c + d*x))/2] - 7*Sin[(5*(c + d*x))/2] + 5*Sin[(7*(c + d*x))/2]))/(140*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
34,1,117,86,0.182396,"\int \sin ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Sin[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\sqrt{a (\sin (c+d x)+1)} \left(-30 \sin \left(\frac{1}{2} (c+d x)\right)+5 \sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)+30 \cos \left(\frac{1}{2} (c+d x)\right)+5 \cos \left(\frac{3}{2} (c+d x)\right)-3 \cos \left(\frac{5}{2} (c+d x)\right)\right)}{30 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 a d}+\frac{4 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 d}-\frac{14 a \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}",1,"-1/30*(Sqrt[a*(1 + Sin[c + d*x])]*(30*Cos[(c + d*x)/2] + 5*Cos[(3*(c + d*x))/2] - 3*Cos[(5*(c + d*x))/2] - 30*Sin[(c + d*x)/2] + 5*Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
35,1,81,56,0.1172389,"\int \sin (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Sin[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\sqrt{a (\sin (c+d x)+1)} \left(-4 \sin ^3\left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{3}{2} (c+d x)\right)\right)}{3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{2 a \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}",1,"-1/3*((3*Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2] - 4*Sin[(c + d*x)/2]^3)*Sqrt[a*(1 + Sin[c + d*x])])/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
36,1,65,26,0.0336538,"\int \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 \sqrt{a (\sin (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{2 a \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}",1,"(2*(-Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*Sqrt[a*(1 + Sin[c + d*x])])/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
37,1,94,37,0.0947128,"\int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}",1,"((-Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sqrt[a*(1 + Sin[c + d*x])])/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
38,1,178,64,0.7025765,"\int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(-2 \sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)+\sin (c+d x) \left(\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)\right)}{d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc \left(\frac{1}{4} (c+d x)\right)-\sec \left(\frac{1}{4} (c+d x)\right)\right) \left(\csc \left(\frac{1}{4} (c+d x)\right)+\sec \left(\frac{1}{4} (c+d x)\right)\right)}","-\frac{a \cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}",1,"-((Csc[(c + d*x)/2]^4*Sqrt[a*(1 + Sin[c + d*x])]*(2*Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2] + (Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sin[c + d*x]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4] - Sec[(c + d*x)/4])*(Csc[(c + d*x)/4] + Sec[(c + d*x)/4])))","B",1
39,1,249,102,0.7498452,"\int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\csc ^7\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right)-6 \sin \left(\frac{3}{2} (c+d x)\right)-2 \cos \left(\frac{1}{2} (c+d x)\right)-6 \cos \left(\frac{3}{2} (c+d x)\right)+3 \cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-3 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-3 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{4 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^2}","-\frac{3 a \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}",1,"(Csc[(c + d*x)/2]^7*Sqrt[a*(1 + Sin[c + d*x])]*(-2*Cos[(c + d*x)/2] - 6*Cos[(3*(c + d*x))/2] - 3*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 3*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2*Sin[(c + d*x)/2] - 6*Sin[(3*(c + d*x))/2]))/(4*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^2)","B",1
40,1,285,138,1.3404907,"\int \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\csc ^{10}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(84 \sin \left(\frac{1}{2} (c+d x)\right)-10 \sin \left(\frac{3}{2} (c+d x)\right)-30 \sin \left(\frac{5}{2} (c+d x)\right)-84 \cos \left(\frac{1}{2} (c+d x)\right)-10 \cos \left(\frac{3}{2} (c+d x)\right)+30 \cos \left(\frac{5}{2} (c+d x)\right)-45 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+45 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+15 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-15 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{24 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^3}","-\frac{5 a \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}-\frac{5 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}",1,"(Csc[(c + d*x)/2]^10*Sqrt[a*(1 + Sin[c + d*x])]*(-84*Cos[(c + d*x)/2] - 10*Cos[(3*(c + d*x))/2] + 30*Cos[(5*(c + d*x))/2] + 84*Sin[(c + d*x)/2] - 45*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 45*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 10*Sin[(3*(c + d*x))/2] - 30*Sin[(5*(c + d*x))/2] + 15*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 15*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)]))/(24*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3)","B",1
41,1,97,38,0.1007274,"\int \csc (c+d x) \sqrt{a-a \sin (c+d x)} \, dx","Integrate[Csc[c + d*x]*Sqrt[a - a*Sin[c + d*x]],x]","\frac{\sqrt{a-a \sin (c+d x)} \left(\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a-a \sin (c+d x)}}\right)}{d}",1,"((Log[1 - Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[1 + Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sqrt[a - a*Sin[c + d*x]])/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))","B",1
42,1,96,39,0.0783173,"\int \csc (c+d x) \sqrt{-a+a \sin (c+d x)} \, dx","Integrate[Csc[c + d*x]*Sqrt[-a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)-1)} \left(\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)-a}}\right)}{d}",1,"((Log[1 - Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[1 + Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sqrt[a*(-1 + Sin[c + d*x])])/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))","B",1
43,1,95,40,0.0841512,"\int \csc (c+d x) \sqrt{-a-a \sin (c+d x)} \, dx","Integrate[Csc[c + d*x]*Sqrt[-a - a*Sin[c + d*x]],x]","\frac{\sqrt{-a (\sin (c+d x)+1)} \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a (-\sin (c+d x))-a}}\right)}{d}",1,"((-Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sqrt[-(a*(1 + Sin[c + d*x]))])/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
44,1,165,162,0.5373203,"\int \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2),x]","\frac{(a (\sin (c+d x)+1))^{3/2} \left(3780 \sin \left(\frac{1}{2} (c+d x)\right)-1050 \sin \left(\frac{3}{2} (c+d x)\right)-378 \sin \left(\frac{5}{2} (c+d x)\right)+135 \sin \left(\frac{7}{2} (c+d x)\right)+35 \sin \left(\frac{9}{2} (c+d x)\right)-3780 \cos \left(\frac{1}{2} (c+d x)\right)-1050 \cos \left(\frac{3}{2} (c+d x)\right)+378 \cos \left(\frac{5}{2} (c+d x)\right)+135 \cos \left(\frac{7}{2} (c+d x)\right)-35 \cos \left(\frac{9}{2} (c+d x)\right)\right)}{2520 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\frac{2 a^2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}-\frac{34 a^2 \sin ^3(c+d x) \cos (c+d x)}{63 d \sqrt{a \sin (c+d x)+a}}-\frac{68 a^2 \cos (c+d x)}{45 d \sqrt{a \sin (c+d x)+a}}-\frac{68 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 d}+\frac{136 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 d}",1,"((a*(1 + Sin[c + d*x]))^(3/2)*(-3780*Cos[(c + d*x)/2] - 1050*Cos[(3*(c + d*x))/2] + 378*Cos[(5*(c + d*x))/2] + 135*Cos[(7*(c + d*x))/2] - 35*Cos[(9*(c + d*x))/2] + 3780*Sin[(c + d*x)/2] - 1050*Sin[(3*(c + d*x))/2] - 378*Sin[(5*(c + d*x))/2] + 135*Sin[(7*(c + d*x))/2] + 35*Sin[(9*(c + d*x))/2]))/(2520*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","A",1
45,1,141,116,0.3601484,"\int \sin ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","\frac{(a (\sin (c+d x)+1))^{3/2} \left(735 \sin \left(\frac{1}{2} (c+d x)\right)-175 \sin \left(\frac{3}{2} (c+d x)\right)-63 \sin \left(\frac{5}{2} (c+d x)\right)+15 \sin \left(\frac{7}{2} (c+d x)\right)-735 \cos \left(\frac{1}{2} (c+d x)\right)-175 \cos \left(\frac{3}{2} (c+d x)\right)+63 \cos \left(\frac{5}{2} (c+d x)\right)+15 \cos \left(\frac{7}{2} (c+d x)\right)\right)}{420 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\frac{152 a^2 \cos (c+d x)}{105 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}+\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}-\frac{38 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{105 d}",1,"((a*(1 + Sin[c + d*x]))^(3/2)*(-735*Cos[(c + d*x)/2] - 175*Cos[(3*(c + d*x))/2] + 63*Cos[(5*(c + d*x))/2] + 15*Cos[(7*(c + d*x))/2] + 735*Sin[(c + d*x)/2] - 175*Sin[(3*(c + d*x))/2] - 63*Sin[(5*(c + d*x))/2] + 15*Sin[(7*(c + d*x))/2]))/(420*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","A",1
46,1,115,86,0.1636553,"\int \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Sin[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{(a (\sin (c+d x)+1))^{3/2} \left(-20 \sin \left(\frac{1}{2} (c+d x)\right)+5 \sin \left(\frac{3}{2} (c+d x)\right)+\sin \left(\frac{5}{2} (c+d x)\right)+20 \cos \left(\frac{1}{2} (c+d x)\right)+5 \cos \left(\frac{3}{2} (c+d x)\right)-\cos \left(\frac{5}{2} (c+d x)\right)\right)}{10 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\frac{8 a^2 \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}",1,"-1/10*((a*(1 + Sin[c + d*x]))^(3/2)*(20*Cos[(c + d*x)/2] + 5*Cos[(3*(c + d*x))/2] - Cos[(5*(c + d*x))/2] - 20*Sin[(c + d*x)/2] + 5*Sin[(3*(c + d*x))/2] + Sin[(5*(c + d*x))/2]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","A",1
47,1,89,59,0.1377448,"\int (a+a \sin (c+d x))^{3/2} \, dx","Integrate[(a + a*Sin[c + d*x])^(3/2),x]","-\frac{(a (\sin (c+d x)+1))^{3/2} \left(-9 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)+9 \cos \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{3}{2} (c+d x)\right)\right)}{3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\frac{8 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}",1,"-1/3*((a*(1 + Sin[c + d*x]))^(3/2)*(9*Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2] - 9*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","A",1
48,1,118,66,0.1531902,"\int \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Csc[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","\frac{(a (\sin (c+d x)+1))^{3/2} \left(2 \sin \left(\frac{1}{2} (c+d x)\right)-2 \cos \left(\frac{1}{2} (c+d x)\right)-\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}-\frac{2 a^2 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}",1,"((-2*Cos[(c + d*x)/2] - Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2*Sin[(c + d*x)/2])*(a*(1 + Sin[c + d*x]))^(3/2))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","A",1
49,1,180,66,0.6544569,"\int \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(-2 \sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)+3 \sin (c+d x) \left(\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)\right)}{d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc \left(\frac{1}{4} (c+d x)\right)-\sec \left(\frac{1}{4} (c+d x)\right)\right) \left(\csc \left(\frac{1}{4} (c+d x)\right)+\sec \left(\frac{1}{4} (c+d x)\right)\right)}","-\frac{3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}-\frac{a^2 \cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}",1,"-((a*Csc[(c + d*x)/2]^4*Sqrt[a*(1 + Sin[c + d*x])]*(2*Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2] + 3*(Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sin[c + d*x]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4] - Sec[(c + d*x)/4])*(Csc[(c + d*x)/4] + Sec[(c + d*x)/4])))","B",1
50,1,250,106,0.6140177,"\int \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2),x]","\frac{a \csc ^7\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(-6 \sin \left(\frac{1}{2} (c+d x)\right)-14 \sin \left(\frac{3}{2} (c+d x)\right)+6 \cos \left(\frac{1}{2} (c+d x)\right)-14 \cos \left(\frac{3}{2} (c+d x)\right)+7 \cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-7 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-7 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+7 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{4 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^2}","-\frac{7 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{7 a^2 \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}",1,"(a*Csc[(c + d*x)/2]^7*Sqrt[a*(1 + Sin[c + d*x])]*(6*Cos[(c + d*x)/2] - 14*Cos[(3*(c + d*x))/2] - 7*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 7*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 7*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 7*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 6*Sin[(c + d*x)/2] - 14*Sin[(3*(c + d*x))/2]))/(4*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^2)","B",1
51,1,286,144,0.9537303,"\int \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]","\frac{a \csc ^{10}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(108 \sin \left(\frac{1}{2} (c+d x)\right)-22 \sin \left(\frac{3}{2} (c+d x)\right)-66 \sin \left(\frac{5}{2} (c+d x)\right)-108 \cos \left(\frac{1}{2} (c+d x)\right)-22 \cos \left(\frac{3}{2} (c+d x)\right)+66 \cos \left(\frac{5}{2} (c+d x)\right)-99 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+99 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+33 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-33 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{24 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^3}","-\frac{11 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{11 a^2 \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}-\frac{a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}",1,"(a*Csc[(c + d*x)/2]^10*Sqrt[a*(1 + Sin[c + d*x])]*(-108*Cos[(c + d*x)/2] - 22*Cos[(3*(c + d*x))/2] + 66*Cos[(5*(c + d*x))/2] + 108*Sin[(c + d*x)/2] - 99*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 99*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 22*Sin[(3*(c + d*x))/2] - 66*Sin[(5*(c + d*x))/2] + 33*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 33*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)]))/(24*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3)","A",1
52,1,189,203,1.256901,"\int \sin ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{(a (\sin (c+d x)+1))^{5/2} \left(-31878 \sin \left(\frac{1}{2} (c+d x)\right)+8778 \sin \left(\frac{3}{2} (c+d x)\right)+3465 \sin \left(\frac{5}{2} (c+d x)\right)-1287 \sin \left(\frac{7}{2} (c+d x)\right)-385 \sin \left(\frac{9}{2} (c+d x)\right)+63 \sin \left(\frac{11}{2} (c+d x)\right)+31878 \cos \left(\frac{1}{2} (c+d x)\right)+8778 \cos \left(\frac{3}{2} (c+d x)\right)-3465 \cos \left(\frac{5}{2} (c+d x)\right)-1287 \cos \left(\frac{7}{2} (c+d x)\right)+385 \cos \left(\frac{9}{2} (c+d x)\right)+63 \cos \left(\frac{11}{2} (c+d x)\right)\right)}{11088 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","-\frac{46 a^3 \sin ^4(c+d x) \cos (c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}-\frac{710 a^3 \sin ^3(c+d x) \cos (c+d x)}{693 d \sqrt{a \sin (c+d x)+a}}-\frac{284 a^3 \cos (c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{11 d}+\frac{568 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{693 d}-\frac{284 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{231 d}",1,"-1/11088*((a*(1 + Sin[c + d*x]))^(5/2)*(31878*Cos[(c + d*x)/2] + 8778*Cos[(3*(c + d*x))/2] - 3465*Cos[(5*(c + d*x))/2] - 1287*Cos[(7*(c + d*x))/2] + 385*Cos[(9*(c + d*x))/2] + 63*Cos[(11*(c + d*x))/2] - 31878*Sin[(c + d*x)/2] + 8778*Sin[(3*(c + d*x))/2] + 3465*Sin[(5*(c + d*x))/2] - 1287*Sin[(7*(c + d*x))/2] - 385*Sin[(9*(c + d*x))/2] + 63*Sin[(11*(c + d*x))/2]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","A",1
53,1,165,146,1.0198985,"\int \sin ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2),x]","\frac{(a (\sin (c+d x)+1))^{5/2} \left(8190 \sin \left(\frac{1}{2} (c+d x)\right)-2100 \sin \left(\frac{3}{2} (c+d x)\right)-756 \sin \left(\frac{5}{2} (c+d x)\right)+225 \sin \left(\frac{7}{2} (c+d x)\right)+35 \sin \left(\frac{9}{2} (c+d x)\right)-8190 \cos \left(\frac{1}{2} (c+d x)\right)-2100 \cos \left(\frac{3}{2} (c+d x)\right)+756 \cos \left(\frac{5}{2} (c+d x)\right)+225 \cos \left(\frac{7}{2} (c+d x)\right)-35 \cos \left(\frac{9}{2} (c+d x)\right)\right)}{2520 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","-\frac{832 a^3 \cos (c+d x)}{315 d \sqrt{a \sin (c+d x)+a}}-\frac{208 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{7/2}}{9 a d}+\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{63 d}-\frac{26 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 d}",1,"((a*(1 + Sin[c + d*x]))^(5/2)*(-8190*Cos[(c + d*x)/2] - 2100*Cos[(3*(c + d*x))/2] + 756*Cos[(5*(c + d*x))/2] + 225*Cos[(7*(c + d*x))/2] - 35*Cos[(9*(c + d*x))/2] + 8190*Sin[(c + d*x)/2] - 2100*Sin[(3*(c + d*x))/2] - 756*Sin[(5*(c + d*x))/2] + 225*Sin[(7*(c + d*x))/2] + 35*Sin[(9*(c + d*x))/2]))/(2520*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","A",1
54,1,141,116,0.6133445,"\int \sin (c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Sin[c + d*x]*(a + a*Sin[c + d*x])^(5/2),x]","\frac{(a (\sin (c+d x)+1))^{5/2} \left(315 \sin \left(\frac{1}{2} (c+d x)\right)-77 \sin \left(\frac{3}{2} (c+d x)\right)-21 \sin \left(\frac{5}{2} (c+d x)\right)+3 \sin \left(\frac{7}{2} (c+d x)\right)-315 \cos \left(\frac{1}{2} (c+d x)\right)-77 \cos \left(\frac{3}{2} (c+d x)\right)+21 \cos \left(\frac{5}{2} (c+d x)\right)+3 \cos \left(\frac{7}{2} (c+d x)\right)\right)}{84 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","-\frac{64 a^3 \cos (c+d x)}{21 d \sqrt{a \sin (c+d x)+a}}-\frac{16 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{21 d}-\frac{2 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{7 d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}",1,"((a*(1 + Sin[c + d*x]))^(5/2)*(-315*Cos[(c + d*x)/2] - 77*Cos[(3*(c + d*x))/2] + 21*Cos[(5*(c + d*x))/2] + 3*Cos[(7*(c + d*x))/2] + 315*Sin[(c + d*x)/2] - 77*Sin[(3*(c + d*x))/2] - 21*Sin[(5*(c + d*x))/2] + 3*Sin[(7*(c + d*x))/2]))/(84*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","A",1
55,1,117,89,0.318308,"\int (a+a \sin (c+d x))^{5/2} \, dx","Integrate[(a + a*Sin[c + d*x])^(5/2),x]","-\frac{(a (\sin (c+d x)+1))^{5/2} \left(-150 \sin \left(\frac{1}{2} (c+d x)\right)+25 \sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)+150 \cos \left(\frac{1}{2} (c+d x)\right)+25 \cos \left(\frac{3}{2} (c+d x)\right)-3 \cos \left(\frac{5}{2} (c+d x)\right)\right)}{30 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","-\frac{64 a^3 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}-\frac{16 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 d}-\frac{2 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}",1,"-1/30*((a*(1 + Sin[c + d*x]))^(5/2)*(150*Cos[(c + d*x)/2] + 25*Cos[(3*(c + d*x))/2] - 3*Cos[(5*(c + d*x))/2] - 150*Sin[(c + d*x)/2] + 25*Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","A",1
56,1,143,98,0.4123755,"\int \csc (c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Csc[c + d*x]*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{(a (\sin (c+d x)+1))^{5/2} \left(-15 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)+15 \cos \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{3}{2} (c+d x)\right)+3 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}-\frac{14 a^3 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}",1,"-1/3*((a*(1 + Sin[c + d*x]))^(5/2)*(15*Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2] + 3*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 3*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 15*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","A",1
57,1,182,94,0.8124619,"\int \csc ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{a^2 \csc ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(2 \sin \left(\frac{3}{2} (c+d x)\right)+2 \cos \left(\frac{3}{2} (c+d x)\right)+5 \sin (c+d x) \left(\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)\right)}{d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc \left(\frac{1}{4} (c+d x)\right)-\sec \left(\frac{1}{4} (c+d x)\right)\right) \left(\csc \left(\frac{1}{4} (c+d x)\right)+\sec \left(\frac{1}{4} (c+d x)\right)\right)}","-\frac{5 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}-\frac{a^3 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{a^2 \cot (c+d x) \sqrt{a \sin (c+d x)+a}}{d}",1,"-((a^2*Csc[(c + d*x)/2]^4*Sqrt[a*(1 + Sin[c + d*x])]*(2*Cos[(3*(c + d*x))/2] + 5*(Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sin[c + d*x] + 2*Sin[(3*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4] - Sec[(c + d*x)/4])*(Csc[(c + d*x)/4] + Sec[(c + d*x)/4])))","A",1
58,1,252,106,0.73858,"\int \csc ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2),x]","\frac{a^2 \csc ^7\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(-14 \sin \left(\frac{1}{2} (c+d x)\right)-22 \sin \left(\frac{3}{2} (c+d x)\right)+14 \cos \left(\frac{1}{2} (c+d x)\right)-22 \cos \left(\frac{3}{2} (c+d x)\right)+19 \cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-19 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-19 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+19 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{4 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^2}","-\frac{19 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{9 a^3 \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\frac{a^2 \cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{2 d}",1,"(a^2*Csc[(c + d*x)/2]^7*Sqrt[a*(1 + Sin[c + d*x])]*(14*Cos[(c + d*x)/2] - 22*Cos[(3*(c + d*x))/2] - 19*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 19*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 19*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 19*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 14*Sin[(c + d*x)/2] - 22*Sin[(3*(c + d*x))/2]))/(4*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^2)","B",1
59,1,288,144,1.1754072,"\int \csc ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2),x]","\frac{a^2 \csc ^{10}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(228 \sin \left(\frac{1}{2} (c+d x)\right)+14 \sin \left(\frac{3}{2} (c+d x)\right)-150 \sin \left(\frac{5}{2} (c+d x)\right)-228 \cos \left(\frac{1}{2} (c+d x)\right)+14 \cos \left(\frac{3}{2} (c+d x)\right)+150 \cos \left(\frac{5}{2} (c+d x)\right)-225 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+225 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+75 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-75 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{24 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^3}","-\frac{25 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{25 a^3 \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}-\frac{13 a^3 \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}-\frac{a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}",1,"(a^2*Csc[(c + d*x)/2]^10*Sqrt[a*(1 + Sin[c + d*x])]*(-228*Cos[(c + d*x)/2] + 14*Cos[(3*(c + d*x))/2] + 150*Cos[(5*(c + d*x))/2] + 228*Sin[(c + d*x)/2] - 225*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 225*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 14*Sin[(3*(c + d*x))/2] - 150*Sin[(5*(c + d*x))/2] + 75*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 75*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)]))/(24*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3)","A",1
60,1,370,182,1.7028994,"\int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{a^2 \csc ^{13}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(1030 \sin \left(\frac{1}{2} (c+d x)\right)+3102 \sin \left(\frac{3}{2} (c+d x)\right)+326 \sin \left(\frac{5}{2} (c+d x)\right)-978 \sin \left(\frac{7}{2} (c+d x)\right)-1030 \cos \left(\frac{1}{2} (c+d x)\right)+3102 \cos \left(\frac{3}{2} (c+d x)\right)-326 \cos \left(\frac{5}{2} (c+d x)\right)-978 \cos \left(\frac{7}{2} (c+d x)\right)-1956 \cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+489 \cos (4 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+1467 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+1956 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-489 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-1467 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{192 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^4}","-\frac{163 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{64 d}-\frac{163 a^3 \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}-\frac{163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}",1,"-1/192*(a^2*Csc[(c + d*x)/2]^13*Sqrt[a*(1 + Sin[c + d*x])]*(-1030*Cos[(c + d*x)/2] + 3102*Cos[(3*(c + d*x))/2] - 326*Cos[(5*(c + d*x))/2] - 978*Cos[(7*(c + d*x))/2] + 1467*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 1956*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 489*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 1467*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 1956*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 489*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 1030*Sin[(c + d*x)/2] + 3102*Sin[(3*(c + d*x))/2] + 326*Sin[(5*(c + d*x))/2] - 978*Sin[(7*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^4)","B",1
61,1,150,139,0.2233588,"\int \frac{\sin ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Sin[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(60 \sin \left(\frac{1}{2} (c+d x)\right)+5 \sin \left(\frac{3}{2} (c+d x)\right)-3 \sin \left(\frac{5}{2} (c+d x)\right)-60 \cos \left(\frac{1}{2} (c+d x)\right)+5 \cos \left(\frac{3}{2} (c+d x)\right)+3 \cos \left(\frac{5}{2} (c+d x)\right)+(-60-60 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{30 d \sqrt{a (\sin (c+d x)+1)}}","-\frac{2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 a d}-\frac{28 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*((-60 - 60*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] - 60*Cos[(c + d*x)/2] + 5*Cos[(3*(c + d*x))/2] + 3*Cos[(5*(c + d*x))/2] + 60*Sin[(c + d*x)/2] + 5*Sin[(3*(c + d*x))/2] - 3*Sin[(5*(c + d*x))/2]))/(30*d*Sqrt[a*(1 + Sin[c + d*x])])","C",1
62,1,105,105,0.2098614,"\int \frac{\sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Sin[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3-(6+6 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{3 d \sqrt{a (\sin (c+d x)+1)}}","-\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 a d}+\frac{4 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-1/3*(((-6 - 6*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] - 2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Sin[c + d*x])])","C",1
63,1,98,72,0.1014951,"\int \frac{\sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Sin[c + d*x]/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\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)+(1+i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{d \sqrt{a (\sin (c+d x)+1)}}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}",1,"(-2*((1 + I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] + Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Sin[c + d*x])])","C",1
64,1,73,47,0.0538186,"\int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[1/Sqrt[a + a*Sin[c + d*x]],x]","\frac{(2+2 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)}{d \sqrt{a (\sin (c+d x)+1)}}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((2 + 2*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Sin[c + d*x])])","C",1
65,1,128,84,0.0980887,"\int \frac{\csc (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Csc[c + d*x]/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left((2+2 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)+\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{d \sqrt{a (\sin (c+d x)+1)}}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-((((2 + 2*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] + Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Sin[c + d*x])]))","C",1
66,1,168,109,1.2632774,"\int \frac{\csc ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Csc[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-\tan \left(\frac{1}{4} (c+d x)\right)-\cot \left(\frac{1}{4} (c+d x)\right)+2 \sec \left(\frac{1}{2} (c+d x)\right)+(8+8 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)+2 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{4 d \sqrt{a (\sin (c+d x)+1)}}","-\frac{\cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*((8 + 8*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] - Cot[(c + d*x)/4] + 2*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2*Sec[(c + d*x)/2] - Tan[(c + d*x)/4]))/(4*d*Sqrt[a*(1 + Sin[c + d*x])])","C",1
67,1,307,146,3.4035681,"\int \frac{\csc ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Csc[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(4 \tan \left(\frac{1}{4} (c+d x)\right)+4 \cot \left(\frac{1}{4} (c+d x)\right)-\csc ^2\left(\frac{1}{4} (c+d x)\right)+\sec ^2\left(\frac{1}{4} (c+d x)\right)-\frac{8 \sin \left(\frac{1}{4} (c+d x)\right)}{\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}+\frac{8 \sin \left(\frac{1}{4} (c+d x)\right)}{\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)}+\frac{2}{\left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{2}{\left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}-(64+64 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)-28 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+28 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-8\right)}{32 d \sqrt{a (\sin (c+d x)+1)}}","\frac{\cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 \sqrt{a} d}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\cot (c+d x) \csc (c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-8 - (64 + 64*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] + 4*Cot[(c + d*x)/4] - Csc[(c + d*x)/4]^2 - 28*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 28*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sec[(c + d*x)/4]^2 + 2/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2 - (8*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]) - 2/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2 + (8*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4]) + 4*Tan[(c + d*x)/4]))/(32*d*Sqrt[a*(1 + Sin[c + d*x])])","C",1
68,1,178,183,0.4528598,"\int \frac{\sin ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Sin[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(55 \sin \left(\frac{1}{2} (c+d x)\right)-41 \sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)+\sin \left(\frac{7}{2} (c+d x)\right)-55 \cos \left(\frac{1}{2} (c+d x)\right)-41 \cos \left(\frac{3}{2} (c+d x)\right)-3 \cos \left(\frac{5}{2} (c+d x)\right)+\cos \left(\frac{7}{2} (c+d x)\right)-(150+150 i) (-1)^{3/4} (\sin (c+d x)+1) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{20 d (a (\sin (c+d x)+1))^{3/2}}","\frac{15 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{13 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{10 a^2 d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}-\frac{9 \sin ^2(c+d x) \cos (c+d x)}{10 a d \sqrt{a \sin (c+d x)+a}}-\frac{31 \cos (c+d x)}{5 a d \sqrt{a \sin (c+d x)+a}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-55*Cos[(c + d*x)/2] - 41*Cos[(3*(c + d*x))/2] - 3*Cos[(5*(c + d*x))/2] + Cos[(7*(c + d*x))/2] + 55*Sin[(c + d*x)/2] - (150 + 150*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(1 + Sin[c + d*x]) - 41*Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2] + Sin[(7*(c + d*x))/2]))/(20*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
69,1,156,145,0.262811,"\int \frac{\sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-11 \sin \left(\frac{1}{2} (c+d x)\right)+7 \sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{5}{2} (c+d x)\right)+11 \cos \left(\frac{1}{2} (c+d x)\right)+7 \cos \left(\frac{3}{2} (c+d x)\right)+\cos \left(\frac{5}{2} (c+d x)\right)+(33+33 i) (-1)^{3/4} (\sin (c+d x)+1) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{6 d (a (\sin (c+d x)+1))^{3/2}}","-\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{7 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{6 a^2 d}+\frac{\sin ^2(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}+\frac{13 \cos (c+d x)}{3 a d \sqrt{a \sin (c+d x)+a}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(11*Cos[(c + d*x)/2] + 7*Cos[(3*(c + d*x))/2] + Cos[(5*(c + d*x))/2] - 11*Sin[(c + d*x)/2] + (33 + 33*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(1 + Sin[c + d*x]) + 7*Sin[(3*(c + d*x))/2] - Sin[(5*(c + d*x))/2]))/(6*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
70,1,134,105,0.2743124,"\int \frac{\sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-3 \sin \left(\frac{1}{2} (c+d x)\right)+2 \sin \left(\frac{3}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{3}{2} (c+d x)\right)+(7+7 i) (-1)^{3/4} (\sin (c+d x)+1) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{2 d (a (\sin (c+d x)+1))^{3/2}}","\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{2 \cos (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}-\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}",1,"-1/2*((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(3*Cos[(c + d*x)/2] + 2*Cos[(3*(c + d*x))/2] - 3*Sin[(c + d*x)/2] + (7 + 7*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(1 + Sin[c + d*x]) + 2*Sin[(3*(c + d*x))/2]))/(d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
71,1,108,77,0.1956703,"\int \frac{\sin (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Sin[c + d*x]/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\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)+(3+3 i) (-1)^{3/4} (\sin (c+d x)+1) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{2 d (a (\sin (c+d x)+1))^{3/2}}","\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2] + (3 + 3*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(1 + Sin[c + d*x])))/(2*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
72,1,108,77,0.1482251,"\int \frac{1}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(-3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\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)+(1+i) (-1)^{3/4} (\sin (c+d x)+1) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{2 d (a (\sin (c+d x)+1))^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-Cos[(c + d*x)/2] + Sin[(c + d*x)/2] + (1 + I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(1 + Sin[c + d*x])))/(2*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
73,1,223,114,0.1992576,"\int \frac{\csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Csc[c + d*x]/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\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)-2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+(-5-5 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{2 d (a (\sin (c+d x)+1))^{3/2}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2] - (5 + 5*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 2*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 2*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2))/(2*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
74,1,449,144,0.6257081,"\int \frac{\csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(4 \sin \left(\frac{1}{2} (c+d x)\right)+\frac{2 \sin \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}-\frac{2 \sin \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)}+2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2-2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+6 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-6 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\tan \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2-\cot \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2+(18+18 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{4 d (a (\sin (c+d x)+1))^{3/2}}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 \cot (c+d x)}{2 a d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(4*Sin[(c + d*x)/2] - 2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (18 + 18*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - Cot[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 6*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 6*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (2*Sin[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]) - (2*Sin[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4]) - (Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*Tan[(c + d*x)/4]))/(4*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
75,1,620,186,4.7474598,"\int \frac{\csc ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-32 \sin \left(\frac{1}{2} (c+d x)\right)-\frac{24 \sin \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}+\frac{24 \sin \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)}+\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}-24 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2+16 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-76 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+76 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+12 \tan \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2+12 \cot \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2-\csc ^2\left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2+\sec ^2\left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2-(208+208 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{32 d (a (\sin (c+d x)+1))^{3/2}}","-\frac{19 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 a^{3/2} d}+\frac{13 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{7 \cot (c+d x)}{4 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-32*Sin[(c + d*x)/2] + 16*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 24*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (208 + 208*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 12*Cot[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - Csc[(c + d*x)/4]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 76*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 76*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + Sec[(c + d*x)/4]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2 - (24*Sin[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]) - (2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2 + (24*Sin[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4]) + 12*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*Tan[(c + d*x)/4]))/(32*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
76,1,221,221,0.5559819,"\int \frac{\sin ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Sin[c + d*x]^5/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-2547 \sin \left(\frac{1}{2} (c+d x)\right)+3603 \sin \left(\frac{3}{2} (c+d x)\right)+872 \sin \left(\frac{5}{2} (c+d x)\right)+52 \sin \left(\frac{7}{2} (c+d x)\right)-12 \sin \left(\frac{9}{2} (c+d x)\right)+2547 \cos \left(\frac{1}{2} (c+d x)\right)+3603 \cos \left(\frac{3}{2} (c+d x)\right)-872 \cos \left(\frac{5}{2} (c+d x)\right)+52 \cos \left(\frac{7}{2} (c+d x)\right)+12 \cos \left(\frac{9}{2} (c+d x)\right)+(8490+8490 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{480 d (a (\sin (c+d x)+1))^{5/2}}","\frac{283 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{787 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{240 a^3 d}-\frac{157 \sin ^2(c+d x) \cos (c+d x)}{80 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{1729 \cos (c+d x)}{120 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}+\frac{21 \sin ^3(c+d x) \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}",1,"-1/480*((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(2547*Cos[(c + d*x)/2] + 3603*Cos[(3*(c + d*x))/2] - 872*Cos[(5*(c + d*x))/2] + 52*Cos[(7*(c + d*x))/2] + 12*Cos[(9*(c + d*x))/2] - 2547*Sin[(c + d*x)/2] + (8490 + 8490*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + 3603*Sin[(3*(c + d*x))/2] + 872*Sin[(5*(c + d*x))/2] + 52*Sin[(7*(c + d*x))/2] - 12*Sin[(9*(c + d*x))/2]))/(d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
77,1,197,183,0.4775927,"\int \frac{\sin ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Sin[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-279 \sin \left(\frac{1}{2} (c+d x)\right)+399 \sin \left(\frac{3}{2} (c+d x)\right)+88 \sin \left(\frac{5}{2} (c+d x)\right)+8 \sin \left(\frac{7}{2} (c+d x)\right)+279 \cos \left(\frac{1}{2} (c+d x)\right)+399 \cos \left(\frac{3}{2} (c+d x)\right)-88 \cos \left(\frac{5}{2} (c+d x)\right)+8 \cos \left(\frac{7}{2} (c+d x)\right)+(978+978 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{96 d (a (\sin (c+d x)+1))^{5/2}}","-\frac{163 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{95 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{48 a^3 d}+\frac{197 \cos (c+d x)}{24 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}+\frac{17 \sin ^2(c+d x) \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(279*Cos[(c + d*x)/2] + 399*Cos[(3*(c + d*x))/2] - 88*Cos[(5*(c + d*x))/2] + 8*Cos[(7*(c + d*x))/2] - 279*Sin[(c + d*x)/2] + (978 + 978*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + 399*Sin[(3*(c + d*x))/2] + 88*Sin[(5*(c + d*x))/2] + 8*Sin[(7*(c + d*x))/2]))/(96*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
78,1,173,145,0.3221836,"\int \frac{\sin ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(45 \sin \left(\frac{1}{2} (c+d x)\right)-69 \sin \left(\frac{3}{2} (c+d x)\right)-16 \sin \left(\frac{5}{2} (c+d x)\right)-45 \cos \left(\frac{1}{2} (c+d x)\right)-69 \cos \left(\frac{3}{2} (c+d x)\right)+16 \cos \left(\frac{5}{2} (c+d x)\right)+(-150-150 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{32 d (a (\sin (c+d x)+1))^{5/2}}","\frac{75 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{9 \cos (c+d x)}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{\sin ^2(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac{13 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-45*Cos[(c + d*x)/2] - 69*Cos[(3*(c + d*x))/2] + 16*Cos[(5*(c + d*x))/2] + 45*Sin[(c + d*x)/2] - (150 + 150*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - 69*Sin[(3*(c + d*x))/2] - 16*Sin[(5*(c + d*x))/2]))/(32*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
79,1,196,107,0.2015645,"\int \frac{\sin ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(8 \sin \left(\frac{1}{2} (c+d x)\right)+13 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3-26 \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)^2-4 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+(19+19 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{16 d (a (\sin (c+d x)+1))^{5/2}}","-\frac{19 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{13 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(8*Sin[(c + d*x)/2] - 4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 26*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 13*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (19 + 19*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4))/(16*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
80,1,196,107,0.1893276,"\int \frac{\sin (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Sin[c + d*x]/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-8 \sin \left(\frac{1}{2} (c+d x)\right)-5 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3+10 \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)^2+4 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+(5+5 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{16 d (a (\sin (c+d x)+1))^{5/2}}","-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac{\cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-8*Sin[(c + d*x)/2] + 4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 10*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 5*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (5 + 5*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4))/(16*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
81,1,196,107,0.1617063,"\int \frac{1}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(-5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(8 \sin \left(\frac{1}{2} (c+d x)\right)-3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3+6 \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)^2-4 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+(3+3 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{16 d (a (\sin (c+d x)+1))^{5/2}}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{3 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(8*Sin[(c + d*x)/2] - 4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 6*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (3 + 3*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4))/(16*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
82,1,296,144,0.2680905,"\int \frac{\csc (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Csc[c + d*x]/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-8 \sin \left(\frac{1}{2} (c+d x)\right)+11 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3-22 \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)^2+4 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-16 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+16 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+(-43-43 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{16 d (a (\sin (c+d x)+1))^{5/2}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}+\frac{43 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{11 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac{\cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-8*Sin[(c + d*x)/2] + 4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 22*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 11*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - (43 + 43*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - 16*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + 16*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4))/(16*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
83,1,509,174,0.6261706,"\int \frac{\csc ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(8 \sin \left(\frac{1}{2} (c+d x)\right)+\frac{8 \sin \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}-\frac{8 \sin \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)}+8 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4-19 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3+38 \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)^2-4 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+40 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-40 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-4 \tan \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4-4 \cot \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4+(115+115 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{16 d (a (\sin (c+d x)+1))^{5/2}}","\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{115 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{35 \cot (c+d x)}{16 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{15 \cot (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac{\cot (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(8*Sin[(c + d*x)/2] - 4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 38*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 19*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 8*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + (115 + 115*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - 4*Cot[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + 40*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - 40*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + (8*Sin[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]) - (8*Sin[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4]) - 4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*Tan[(c + d*x)/4]))/(16*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
84,1,680,224,1.1340218,"\int \frac{\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-16 \sin \left(\frac{1}{2} (c+d x)\right)-\frac{40 \sin \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}+\frac{40 \sin \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)}+\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{\left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{\left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}-40 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4+54 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3-108 \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)^2+8 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-156 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+156 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+20 \tan \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4+20 \cot \left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4-\csc ^2\left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4+\sec ^2\left(\frac{1}{4} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4-(438+438 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{32 d (a (\sin (c+d x)+1))^{5/2}}","-\frac{39 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 a^{5/2} d}+\frac{219 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{63 \cot (c+d x)}{16 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{19 \cot (c+d x) \csc (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac{\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-16*Sin[(c + d*x)/2] + 8*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 108*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 54*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 40*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - (438 + 438*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + 20*Cot[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - Csc[(c + d*x)/4]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - 156*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + 156*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + Sec[(c + d*x)/4]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + (2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2 - (40*Sin[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]) - (2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2 + (40*Sin[(c + d*x)/4]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4]) + 20*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*Tan[(c + d*x)/4]))/(32*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
85,1,164,37,0.5121054,"\int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{\sin (e+f x)}} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]/Sqrt[Sin[e + f*x]],x]","\frac{(1+i) e^{\frac{1}{2} i (e+f x)} \sqrt{-i e^{-i (e+f x)} \left(-1+e^{2 i (e+f x)}\right)} \sqrt{a (\sin (e+f x)+1)} \left(\tan ^{-1}\left(\sqrt{-1+e^{2 i (e+f x)}}\right)-i \tanh ^{-1}\left(\frac{e^{i (e+f x)}}{\sqrt{-1+e^{2 i (e+f x)}}}\right)\right)}{\sqrt{2} f \sqrt{-1+e^{2 i (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 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{f}",1,"((1 + I)*E^((I/2)*(e + f*x))*Sqrt[((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x))]*(ArcTan[Sqrt[-1 + E^((2*I)*(e + f*x))]] - I*ArcTanh[E^(I*(e + f*x))/Sqrt[-1 + E^((2*I)*(e + f*x))]])*Sqrt[a*(1 + Sin[e + f*x])])/(Sqrt[2]*Sqrt[-1 + E^((2*I)*(e + f*x))]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","C",1
86,1,119,38,0.473459,"\int \frac{\sqrt{a-a \sin (e+f x)}}{\sqrt{-\sin (e+f x)}} \, dx","Integrate[Sqrt[a - a*Sin[e + f*x]]/Sqrt[-Sin[e + f*x]],x]","-\frac{\sqrt{-1+e^{2 i (e+f x)}} \sqrt{a-a \sin (e+f x)} \left(\tan ^{-1}\left(\sqrt{-1+e^{2 i (e+f x)}}\right)+i \tanh ^{-1}\left(\frac{e^{i (e+f x)}}{\sqrt{-1+e^{2 i (e+f x)}}}\right)\right)}{f \left(e^{i (e+f x)}-i\right) \sqrt{-\sin (e+f x)}}","\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-a \sin (e+f x)}}\right)}{f}",1,"-((Sqrt[-1 + E^((2*I)*(e + f*x))]*(ArcTan[Sqrt[-1 + E^((2*I)*(e + f*x))]] + I*ArcTanh[E^(I*(e + f*x))/Sqrt[-1 + E^((2*I)*(e + f*x))]])*Sqrt[a - a*Sin[e + f*x]])/((-I + E^(I*(e + f*x)))*f*Sqrt[-Sin[e + f*x]]))","C",1
87,1,123,17,2.5707248,"\int \frac{1}{\sqrt{\sin (x)} \sqrt{1+\sin (x)}} \, dx","Integrate[1/(Sqrt[Sin[x]]*Sqrt[1 + Sin[x]]),x]","\frac{2 \sqrt{\sin (x)} \sec ^2\left(\frac{x}{4}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(F\left(\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{4}\right)}}\right)\right|-1\right)-\Pi \left(1-\sqrt{2};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{4}\right)}}\right)\right|-1\right)-\Pi \left(1+\sqrt{2};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{4}\right)}}\right)\right|-1\right)\right)}{\sqrt{\sin (x)+1} \tan ^{\frac{3}{2}}\left(\frac{x}{4}\right) \sqrt{1-\cot ^2\left(\frac{x}{4}\right)}}","-\sqrt{2} \sin ^{-1}\left(\frac{\cos (x)}{\sin (x)+1}\right)",1,"(2*(EllipticF[ArcSin[1/Sqrt[Tan[x/4]]], -1] - EllipticPi[1 - Sqrt[2], ArcSin[1/Sqrt[Tan[x/4]]], -1] - EllipticPi[1 + Sqrt[2], ArcSin[1/Sqrt[Tan[x/4]]], -1])*Sec[x/4]^2*(Cos[x/2] + Sin[x/2])*Sqrt[Sin[x]])/(Sqrt[1 - Cot[x/4]^2]*Sqrt[1 + Sin[x]]*Tan[x/4]^(3/2))","C",1
88,1,125,42,0.0925608,"\int \frac{1}{\sqrt{\sin (x)} \sqrt{a+a \sin (x)}} \, dx","Integrate[1/(Sqrt[Sin[x]]*Sqrt[a + a*Sin[x]]),x]","\frac{2 \sqrt{\sin (x)} \sec ^2\left(\frac{x}{4}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(F\left(\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{4}\right)}}\right)\right|-1\right)-\Pi \left(1-\sqrt{2};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{4}\right)}}\right)\right|-1\right)-\Pi \left(1+\sqrt{2};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{4}\right)}}\right)\right|-1\right)\right)}{\tan ^{\frac{3}{2}}\left(\frac{x}{4}\right) \sqrt{1-\cot ^2\left(\frac{x}{4}\right)} \sqrt{a (\sin (x)+1)}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \cos (x)}{\sqrt{2} \sqrt{\sin (x)} \sqrt{a \sin (x)+a}}\right)}{\sqrt{a}}",1,"(2*(EllipticF[ArcSin[1/Sqrt[Tan[x/4]]], -1] - EllipticPi[1 - Sqrt[2], ArcSin[1/Sqrt[Tan[x/4]]], -1] - EllipticPi[1 + Sqrt[2], ArcSin[1/Sqrt[Tan[x/4]]], -1])*Sec[x/4]^2*(Cos[x/2] + Sin[x/2])*Sqrt[Sin[x]])/(Sqrt[1 - Cot[x/4]^2]*Sqrt[a*(1 + Sin[x])]*Tan[x/4]^(3/2))","C",1
89,1,125,31,2.5176776,"\int \frac{1}{\sqrt{1-\sin (x)} \sqrt{\sin (x)}} \, dx","Integrate[1/(Sqrt[1 - Sin[x]]*Sqrt[Sin[x]]),x]","\frac{2 \sin (x) \sec ^2\left(\frac{x}{4}\right) \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right) \left(F\left(\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{4}\right)}}\right)\right|-1\right)-\Pi \left(-1-\sqrt{2};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{4}\right)}}\right)\right|-1\right)-\Pi \left(-1+\sqrt{2};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{4}\right)}}\right)\right|-1\right)\right)}{\sqrt{-((\sin (x)-1) \sin (x))} \tan ^{\frac{3}{2}}\left(\frac{x}{4}\right) \sqrt{1-\cot ^2\left(\frac{x}{4}\right)}}","\sqrt{2} \tanh ^{-1}\left(\frac{\cos (x)}{\sqrt{2} \sqrt{1-\sin (x)} \sqrt{\sin (x)}}\right)",1,"(2*(EllipticF[ArcSin[1/Sqrt[Tan[x/4]]], -1] - EllipticPi[-1 - Sqrt[2], ArcSin[1/Sqrt[Tan[x/4]]], -1] - EllipticPi[-1 + Sqrt[2], ArcSin[1/Sqrt[Tan[x/4]]], -1])*Sec[x/4]^2*(Cos[x/2] - Sin[x/2])*Sin[x])/(Sqrt[1 - Cot[x/4]^2]*Sqrt[-((-1 + Sin[x])*Sin[x])]*Tan[x/4]^(3/2))","C",1
90,1,128,42,0.1001657,"\int \frac{1}{\sqrt{\sin (x)} \sqrt{a-a \sin (x)}} \, dx","Integrate[1/(Sqrt[Sin[x]]*Sqrt[a - a*Sin[x]]),x]","\frac{2 \sqrt{\sin (x)} \sec ^2\left(\frac{x}{4}\right) \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right) \left(F\left(\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{4}\right)}}\right)\right|-1\right)-\Pi \left(-1-\sqrt{2};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{4}\right)}}\right)\right|-1\right)-\Pi \left(-1+\sqrt{2};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{4}\right)}}\right)\right|-1\right)\right)}{\tan ^{\frac{3}{2}}\left(\frac{x}{4}\right) \sqrt{1-\cot ^2\left(\frac{x}{4}\right)} \sqrt{a-a \sin (x)}}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (x)}{\sqrt{2} \sqrt{\sin (x)} \sqrt{a-a \sin (x)}}\right)}{\sqrt{a}}",1,"(2*(EllipticF[ArcSin[1/Sqrt[Tan[x/4]]], -1] - EllipticPi[-1 - Sqrt[2], ArcSin[1/Sqrt[Tan[x/4]]], -1] - EllipticPi[-1 + Sqrt[2], ArcSin[1/Sqrt[Tan[x/4]]], -1])*Sec[x/4]^2*(Cos[x/2] - Sin[x/2])*Sqrt[Sin[x]])/(Sqrt[1 - Cot[x/4]^2]*Sqrt[a - a*Sin[x]]*Tan[x/4]^(3/2))","C",1
91,1,121,184,0.4015472,"\int \frac{\sqrt[3]{\sin (c+d x)}}{(a+a \sin (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^(1/3)/(a + a*Sin[c + d*x])^2,x]","\frac{\sqrt[3]{\sin (c+d x)} \sec ^3(c+d x) \left(80 \cos ^2(c+d x)^{3/2} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(c+d x)\right)+27 \sin (c+d x) \cos ^2(c+d x)^{3/2} \, _2F_1\left(\frac{2}{3},\frac{5}{2};\frac{5}{3};\sin ^2(c+d x)\right)+4 (27 \sin (c+d x)+5 \cos (2 (c+d x))-25)\right)}{180 a^2 d}","\frac{4 \sqrt[3]{\sin (c+d x)} \cos (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(c+d x)\right)}{9 a^2 d \sqrt{\cos ^2(c+d x)}}-\frac{\sin ^{\frac{4}{3}}(c+d x) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(c+d x)\right)}{36 a^2 d \sqrt{\cos ^2(c+d x)}}-\frac{\sqrt[3]{\sin (c+d x)} \cos (c+d x)}{9 a^2 d (\sin (c+d x)+1)}-\frac{\sqrt[3]{\sin (c+d x)} \cos (c+d x)}{3 d (a \sin (c+d x)+a)^2}",1,"(Sec[c + d*x]^3*Sin[c + d*x]^(1/3)*(80*(Cos[c + d*x]^2)^(3/2)*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[c + d*x]^2] + 27*(Cos[c + d*x]^2)^(3/2)*Hypergeometric2F1[2/3, 5/2, 5/3, Sin[c + d*x]^2]*Sin[c + d*x] + 4*(-25 + 5*Cos[2*(c + d*x)] + 27*Sin[c + d*x])))/(180*a^2*d)","A",1
92,1,160,161,0.6917215,"\int \sin ^3(c+d x) (a+a \sin (c+d x))^{2/3} \, dx","Integrate[Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(2/3),x]","\frac{3 (a (\sin (c+d x)+1))^{2/3} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(67 \sqrt{2} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)+\sqrt{1-\sin (c+d x)} (-92 \sin (c+d x)+10 \sin (3 (c+d x))+25 \cos (2 (c+d x))-144)\right)}{440 d \sqrt{1-\sin (c+d x)} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{67 \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{55\ 2^{5/6} d (\sin (c+d x)+1)^{7/6}}-\frac{3 \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{11 d}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{44 a d}-\frac{63 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{220 d}",1,"(3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*(1 + Sin[c + d*x]))^(2/3)*(67*Sqrt[2]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + Sqrt[1 - Sin[c + d*x]]*(-144 + 25*Cos[2*(c + d*x)] - 92*Sin[c + d*x] + 10*Sin[3*(c + d*x)])))/(440*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*Sqrt[1 - Sin[c + d*x]])","A",1
93,1,151,126,0.4391673,"\int \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx","Integrate[Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(2/3),x]","\frac{3 (a (\sin (c+d x)+1))^{2/3} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(19 \sqrt{2} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)+\sqrt{1-\sin (c+d x)} (5 \cos (2 (c+d x))-14 (\sin (c+d x)+2))\right)}{80 d \sqrt{1-\sin (c+d x)} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{19 \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{10\ 2^{5/6} d (\sin (c+d x)+1)^{7/6}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}+\frac{9 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{40 d}",1,"(3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*(1 + Sin[c + d*x]))^(2/3)*(19*Sqrt[2]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + Sqrt[1 - Sin[c + d*x]]*(5*Cos[2*(c + d*x)] - 14*(2 + Sin[c + d*x]))))/(80*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*Sqrt[1 - Sin[c + d*x]])","A",1
94,1,138,96,0.2330774,"\int \sin (c+d x) (a+a \sin (c+d x))^{2/3} \, dx","Integrate[Sin[c + d*x]*(a + a*Sin[c + d*x])^(2/3),x]","-\frac{3 (a (\sin (c+d x)+1))^{2/3} \left(\sqrt{1-\sin (c+d x)} (\sin (c+d x)+2)-\sqrt{2} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{5 d \sqrt{1-\sin (c+d x)} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{4 \sqrt[6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d (\sin (c+d x)+1)^{7/6}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 d}",1,"(-3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*(1 + Sin[c + d*x]))^(2/3)*(-(Sqrt[2]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2]) + Sqrt[1 - Sin[c + d*x]]*(2 + Sin[c + d*x])))/(5*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*Sqrt[1 - Sin[c + d*x]])","A",1
95,1,124,66,0.2045085,"\int (a+a \sin (c+d x))^{2/3} \, dx","Integrate[(a + a*Sin[c + d*x])^(2/3),x]","-\frac{3 (a (\sin (c+d x)+1))^{2/3} \left(\sqrt{2-2 \sin (c+d x)}-2 \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d \sqrt{2-2 \sin (c+d x)} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{2 \sqrt[6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{7/6}}",1,"(-3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(-2*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + Sqrt[2 - 2*Sin[c + d*x]])*(a*(1 + Sin[c + d*x]))^(2/3))/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*Sqrt[2 - 2*Sin[c + d*x]])","A",1
96,0,0,77,2.7905202,"\int \csc (c+d x) (a+a \sin (c+d x))^{2/3} \, dx","Integrate[Csc[c + d*x]*(a + a*Sin[c + d*x])^(2/3),x]","\int \csc (c+d x) (a+a \sin (c+d x))^{2/3} \, dx","-\frac{2 \sqrt[6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};1,-\frac{1}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{7/6}}",1,"Integrate[Csc[c + d*x]*(a + a*Sin[c + d*x])^(2/3), x]","F",-1
97,1,143,77,14.7494076,"\int \csc ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx","Integrate[Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(2/3),x]","-\frac{2 e^{i (c+d x)} \left(\left(1+i e^{-i (c+d x)}\right)^{2/3} \left(e^{i (c+d x)}-i\right) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-i e^{-i (c+d x)}\right)-e^{i (c+d x)}-i\right) (a (\sin (c+d x)+1))^{2/3}}{d \left(e^{i (c+d x)}-i\right) \left(e^{i (c+d x)}+i\right)^2}","-\frac{2 \sqrt[6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};2,-\frac{1}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{7/6}}",1,"(-2*E^(I*(c + d*x))*(-I - E^(I*(c + d*x)) + (1 + I/E^(I*(c + d*x)))^(2/3)*(-I + E^(I*(c + d*x)))*Hypergeometric2F1[1/3, 2/3, 4/3, (-I)/E^(I*(c + d*x))])*(a*(1 + Sin[c + d*x]))^(2/3))/(d*(-I + E^(I*(c + d*x)))*(I + E^(I*(c + d*x)))^2)","C",0
98,1,373,162,2.6446571,"\int \sin ^3(c+d x) (a+a \sin (c+d x))^{4/3} \, dx","Integrate[Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(4/3),x]","\frac{(a (\sin (c+d x)+1))^{4/3} \left(-\frac{3}{40} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (278 \sin (2 (c+d x))-35 \sin (4 (c+d x))+790 \cos (c+d x)-98 \cos (3 (c+d x))-1940)+\frac{291 (-1)^{3/4} e^{-\frac{3}{2} i (c+d x)} \left(e^{i (c+d x)}+i\right) \left(-2 \left(1+i e^{-i (c+d x)}\right)^{2/3} \left(1+e^{2 i (c+d x)}\right) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)+5 i \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-i e^{-i (c+d x)}\right) \sqrt{2-2 \sin (c+d x)}+20 e^{i (c+d x)} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-i e^{-i (c+d x)}\right) \sqrt{\cos ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)}\right)}{20 \sqrt{2} \left(1+i e^{-i (c+d x)}\right)^{2/3} \sqrt{i e^{-i (c+d x)} \left(e^{i (c+d x)}-i\right)^2}}\right)}{91 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\frac{388\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{455 d (\sin (c+d x)+1)^{5/6}}-\frac{3 \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{13 d}-\frac{6 \cos (c+d x) (a \sin (c+d x)+a)^{7/3}}{65 a d}-\frac{72 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{455 d}",1,"((a*(1 + Sin[c + d*x]))^(4/3)*((291*(-1)^(3/4)*(I + E^(I*(c + d*x)))*(20*E^(I*(c + d*x))*Sqrt[Cos[(2*c + Pi + 2*d*x)/4]^2]*Hypergeometric2F1[-1/3, 1/3, 2/3, (-I)/E^(I*(c + d*x))] - 2*(1 + I/E^(I*(c + d*x)))^(2/3)*(1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1/2, 5/6, 11/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + (5*I)*Hypergeometric2F1[1/3, 2/3, 5/3, (-I)/E^(I*(c + d*x))]*Sqrt[2 - 2*Sin[c + d*x]]))/(20*Sqrt[2]*E^(((3*I)/2)*(c + d*x))*(1 + I/E^(I*(c + d*x)))^(2/3)*Sqrt[(I*(-I + E^(I*(c + d*x)))^2)/E^(I*(c + d*x))]) - (3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-1940 + 790*Cos[c + d*x] - 98*Cos[3*(c + d*x)] + 278*Sin[2*(c + d*x)] - 35*Sin[4*(c + d*x)]))/40))/(91*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","C",1
99,1,363,127,2.6194518,"\int \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3} \, dx","Integrate[Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(4/3),x]","\frac{(a (\sin (c+d x)+1))^{4/3} \left(-\frac{3}{10} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (22 \sin (2 (c+d x))+60 \cos (c+d x)-7 \cos (3 (c+d x))-185)+\frac{111 (-1)^{3/4} e^{-\frac{3}{2} i (c+d x)} \left(e^{i (c+d x)}+i\right) \left(-2 \left(1+i e^{-i (c+d x)}\right)^{2/3} \left(1+e^{2 i (c+d x)}\right) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)+5 i \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-i e^{-i (c+d x)}\right) \sqrt{2-2 \sin (c+d x)}+20 e^{i (c+d x)} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-i e^{-i (c+d x)}\right) \sqrt{\cos ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)}\right)}{20 \sqrt{2} \left(1+i e^{-i (c+d x)}\right)^{2/3} \sqrt{i e^{-i (c+d x)} \left(e^{i (c+d x)}-i\right)^2}}\right)}{28 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\frac{37\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{35 d (\sin (c+d x)+1)^{5/6}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{7/3}}{10 a d}+\frac{9 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{70 d}",1,"((a*(1 + Sin[c + d*x]))^(4/3)*((111*(-1)^(3/4)*(I + E^(I*(c + d*x)))*(20*E^(I*(c + d*x))*Sqrt[Cos[(2*c + Pi + 2*d*x)/4]^2]*Hypergeometric2F1[-1/3, 1/3, 2/3, (-I)/E^(I*(c + d*x))] - 2*(1 + I/E^(I*(c + d*x)))^(2/3)*(1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1/2, 5/6, 11/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + (5*I)*Hypergeometric2F1[1/3, 2/3, 5/3, (-I)/E^(I*(c + d*x))]*Sqrt[2 - 2*Sin[c + d*x]]))/(20*Sqrt[2]*E^(((3*I)/2)*(c + d*x))*(1 + I/E^(I*(c + d*x)))^(2/3)*Sqrt[(I*(-I + E^(I*(c + d*x)))^2)/E^(I*(c + d*x))]) - (3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-185 + 60*Cos[c + d*x] - 7*Cos[3*(c + d*x)] + 22*Sin[2*(c + d*x)]))/10))/(28*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","C",1
100,1,351,97,1.9953983,"\int \sin (c+d x) (a+a \sin (c+d x))^{4/3} \, dx","Integrate[Sin[c + d*x]*(a + a*Sin[c + d*x])^(4/3),x]","\frac{(a (\sin (c+d x)+1))^{4/3} \left(-\frac{3}{2} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (\sin (2 (c+d x))+4 \cos (c+d x)-10)+\frac{3 (-1)^{3/4} e^{-\frac{3}{2} i (c+d x)} \left(e^{i (c+d x)}+i\right) \left(-2 \left(1+i e^{-i (c+d x)}\right)^{2/3} \left(1+e^{2 i (c+d x)}\right) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)+5 i \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-i e^{-i (c+d x)}\right) \sqrt{2-2 \sin (c+d x)}+20 e^{i (c+d x)} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-i e^{-i (c+d x)}\right) \sqrt{\cos ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)}\right)}{2 \sqrt{2} \left(1+i e^{-i (c+d x)}\right)^{2/3} \sqrt{i e^{-i (c+d x)} \left(e^{i (c+d x)}-i\right)^2}}\right)}{7 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\frac{8\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{7 d (\sin (c+d x)+1)^{5/6}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{7 d}",1,"((a*(1 + Sin[c + d*x]))^(4/3)*((3*(-1)^(3/4)*(I + E^(I*(c + d*x)))*(20*E^(I*(c + d*x))*Sqrt[Cos[(2*c + Pi + 2*d*x)/4]^2]*Hypergeometric2F1[-1/3, 1/3, 2/3, (-I)/E^(I*(c + d*x))] - 2*(1 + I/E^(I*(c + d*x)))^(2/3)*(1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1/2, 5/6, 11/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + (5*I)*Hypergeometric2F1[1/3, 2/3, 5/3, (-I)/E^(I*(c + d*x))]*Sqrt[2 - 2*Sin[c + d*x]]))/(2*Sqrt[2]*E^(((3*I)/2)*(c + d*x))*(1 + I/E^(I*(c + d*x)))^(2/3)*Sqrt[(I*(-I + E^(I*(c + d*x)))^2)/E^(I*(c + d*x))]) - (3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-10 + 4*Cos[c + d*x] + Sin[2*(c + d*x)]))/2))/(7*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","C",1
101,1,341,67,1.7217003,"\int (a+a \sin (c+d x))^{4/3} \, dx","Integrate[(a + a*Sin[c + d*x])^(4/3),x]","\frac{(a (\sin (c+d x)+1))^{4/3} \left(-\frac{3}{2} (\cos (c+d x)-5) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{3 (-1)^{3/4} e^{-\frac{3}{2} i (c+d x)} \left(e^{i (c+d x)}+i\right) \left(-2 \left(1+i e^{-i (c+d x)}\right)^{2/3} \left(1+e^{2 i (c+d x)}\right) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)+5 i \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-i e^{-i (c+d x)}\right) \sqrt{2-2 \sin (c+d x)}+20 e^{i (c+d x)} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-i e^{-i (c+d x)}\right) \sqrt{\cos ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)}\right)}{4 \sqrt{2} \left(1+i e^{-i (c+d x)}\right)^{2/3} \sqrt{i e^{-i (c+d x)} \left(e^{i (c+d x)}-i\right)^2}}\right)}{2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\frac{2\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{5/6}}",1,"(((-3*(-5 + Cos[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/2 + (3*(-1)^(3/4)*(I + E^(I*(c + d*x)))*(20*E^(I*(c + d*x))*Sqrt[Cos[(2*c + Pi + 2*d*x)/4]^2]*Hypergeometric2F1[-1/3, 1/3, 2/3, (-I)/E^(I*(c + d*x))] - 2*(1 + I/E^(I*(c + d*x)))^(2/3)*(1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1/2, 5/6, 11/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + (5*I)*Hypergeometric2F1[1/3, 2/3, 5/3, (-I)/E^(I*(c + d*x))]*Sqrt[2 - 2*Sin[c + d*x]]))/(4*Sqrt[2]*E^(((3*I)/2)*(c + d*x))*(1 + I/E^(I*(c + d*x)))^(2/3)*Sqrt[(I*(-I + E^(I*(c + d*x)))^2)/E^(I*(c + d*x))]))*(a*(1 + Sin[c + d*x]))^(4/3))/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","C",1
102,1,2791,78,9.6087236,"\int \csc (c+d x) (a+a \sin (c+d x))^{4/3} \, dx","Integrate[Csc[c + d*x]*(a + a*Sin[c + d*x])^(4/3),x]","\text{Result too large to show}","-\frac{2\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} F_1\left(\frac{1}{2};1,-\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{5/6}}",1,"(3*(a*(1 + Sin[c + d*x]))^(4/3))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - ((15 + 15*I)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*(a*(1 + Sin[c + d*x]))^(4/3)*(1 + Tan[(c + d*x)/2]))/(d*((5 + 5*I)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*Sec[(c + d*x)/2] + AppellF1[5/3, 1/3, 4/3, 8/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2]) + I*AppellF1[5/3, 4/3, 1/3, 8/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2]))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((15/2 + (15*I)/2)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Tan[(c + d*x)/2]), (1/2 - I/2)*(1 + Tan[(c + d*x)/2])]*(a*(1 + Sin[c + d*x]))^(4/3))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*((5 + 5*I)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Tan[(c + d*x)/2]), (1/2 - I/2)*(1 + Tan[(c + d*x)/2])] + (AppellF1[5/3, 1/3, 4/3, 8/3, (1/2 + I/2)*(1 + Tan[(c + d*x)/2]), (1/2 - I/2)*(1 + Tan[(c + d*x)/2])] + I*AppellF1[5/3, 4/3, 1/3, 8/3, (1/2 + I/2)*(1 + Tan[(c + d*x)/2]), (1/2 - I/2)*(1 + Tan[(c + d*x)/2])])*(1 + Tan[(c + d*x)/2]))) - (3*Cos[(3*(c + d*x))/2]*Csc[c + d*x]*(a*(1 + Sin[c + d*x]))^(4/3)*((1 + Tan[(c + d*x)/2])/Sqrt[Sec[(c + d*x)/2]^2])^(2/3)*(8 + (1 + I)*2^(2/3)*(((1 - I)*(I + Cot[(c + d*x)/2]))/(1 + Cot[(c + d*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/(2 + 2*Tan[(c + d*x)/2])]*(I + Tan[(c + d*x)/2]) - AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(c + d*x)/2])^(1/3)*((-1 - I)*(I + Cot[(c + d*x)/2]))^(1/3)*(1 + Tan[(c + d*x)/2])))/(4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(1 + Tan[(c + d*x)/2])*((-3*Sec[(c + d*x)/2]^2*((1 + Tan[(c + d*x)/2])/Sqrt[Sec[(c + d*x)/2]^2])^(2/3)*(8 + (1 + I)*2^(2/3)*(((1 - I)*(I + Cot[(c + d*x)/2]))/(1 + Cot[(c + d*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/(2 + 2*Tan[(c + d*x)/2])]*(I + Tan[(c + d*x)/2]) - AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(c + d*x)/2])^(1/3)*((-1 - I)*(I + Cot[(c + d*x)/2]))^(1/3)*(1 + Tan[(c + d*x)/2])))/(8*(1 + Tan[(c + d*x)/2])^2) + ((8 + (1 + I)*2^(2/3)*(((1 - I)*(I + Cot[(c + d*x)/2]))/(1 + Cot[(c + d*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/(2 + 2*Tan[(c + d*x)/2])]*(I + Tan[(c + d*x)/2]) - AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(c + d*x)/2])^(1/3)*((-1 - I)*(I + Cot[(c + d*x)/2]))^(1/3)*(1 + Tan[(c + d*x)/2]))*(Sqrt[Sec[(c + d*x)/2]^2]/2 - (Tan[(c + d*x)/2]*(1 + Tan[(c + d*x)/2]))/(2*Sqrt[Sec[(c + d*x)/2]^2])))/(2*(1 + Tan[(c + d*x)/2])*((1 + Tan[(c + d*x)/2])/Sqrt[Sec[(c + d*x)/2]^2])^(1/3)) + (3*((1 + Tan[(c + d*x)/2])/Sqrt[Sec[(c + d*x)/2]^2])^(2/3)*(-1/2*(AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(c + d*x)/2])^(1/3)*((-1 - I)*(I + Cot[(c + d*x)/2]))^(1/3)*Sec[(c + d*x)/2]^2) + ((1 + I)*(((1 - I)*(I + Cot[(c + d*x)/2]))/(1 + Cot[(c + d*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/(2 + 2*Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2)/2^(1/3) + ((1/3 + I/3)*2^(2/3)*(((1/2 - I/2)*(I + Cot[(c + d*x)/2])*Csc[(c + d*x)/2]^2)/(1 + Cot[(c + d*x)/2])^2 - ((1/2 - I/2)*Csc[(c + d*x)/2]^2)/(1 + Cot[(c + d*x)/2]))*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/(2 + 2*Tan[(c + d*x)/2])]*(I + Tan[(c + d*x)/2]))/(((1 - I)*(I + Cot[(c + d*x)/2]))/(1 + Cot[(c + d*x)/2]))^(2/3) - ((1/6 + I/6)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(c + d*x)/2])^(1/3)*Csc[(c + d*x)/2]^2*(1 + Tan[(c + d*x)/2]))/((-1 - I)*(I + Cot[(c + d*x)/2]))^(2/3) - ((1/3 - I/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*((-1 - I)*(I + Cot[(c + d*x)/2]))^(1/3)*Csc[(c + d*x)/2]^2*(1 + Tan[(c + d*x)/2]))/((2 + 2*I) - (2 - 2*I)*Cot[(c + d*x)/2])^(2/3) - ((2 + 2*I) - (2 - 2*I)*Cot[(c + d*x)/2])^(1/3)*((-1 - I)*(I + Cot[(c + d*x)/2]))^(1/3)*((-1/30 + I/30)*AppellF1[5/3, 1/3, 4/3, 8/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*Csc[(c + d*x)/2]^2 - (1/30 + I/30)*AppellF1[5/3, 4/3, 1/3, 8/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*Csc[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]) + ((2/3 + (2*I)/3)*2^(2/3)*(((1 - I)*(I + Cot[(c + d*x)/2]))/(1 + Cot[(c + d*x)/2]))^(1/3)*(I + Tan[(c + d*x)/2])*(2 + 2*Tan[(c + d*x)/2])*(-((Sec[(c + d*x)/2]^2*((1 + I) + (1 - I)*Tan[(c + d*x)/2]))/(2 + 2*Tan[(c + d*x)/2])^2) + ((1/2 - I/2)*Sec[(c + d*x)/2]^2)/(2 + 2*Tan[(c + d*x)/2]))*(-Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/(2 + 2*Tan[(c + d*x)/2])] + (1 - ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/(2 + 2*Tan[(c + d*x)/2]))^(-1/3)))/((1 + I) + (1 - I)*Tan[(c + d*x)/2])))/(4*(1 + Tan[(c + d*x)/2]))))","C",0
103,1,2800,78,10.5686066,"\int \csc ^2(c+d x) (a+a \sin (c+d x))^{4/3} \, dx","Integrate[Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(4/3),x]","\text{Result too large to show}","-\frac{2\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} F_1\left(\frac{1}{2};2,-\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{5/6}}",1,"((-1 - Cot[c + d*x])*(a*(1 + Sin[c + d*x]))^(4/3))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - ((15/2 + (15*I)/2)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*(a*(1 + Sin[c + d*x]))^(4/3)*(1 + Tan[(c + d*x)/2]))/(d*((5 + 5*I)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*Sec[(c + d*x)/2] + AppellF1[5/3, 1/3, 4/3, 8/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2]) + I*AppellF1[5/3, 4/3, 1/3, 8/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2]))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((10 + 10*I)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Tan[(c + d*x)/2]), (1/2 - I/2)*(1 + Tan[(c + d*x)/2])]*(a*(1 + Sin[c + d*x]))^(4/3))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*((5 + 5*I)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Tan[(c + d*x)/2]), (1/2 - I/2)*(1 + Tan[(c + d*x)/2])] + (AppellF1[5/3, 1/3, 4/3, 8/3, (1/2 + I/2)*(1 + Tan[(c + d*x)/2]), (1/2 - I/2)*(1 + Tan[(c + d*x)/2])] + I*AppellF1[5/3, 4/3, 1/3, 8/3, (1/2 + I/2)*(1 + Tan[(c + d*x)/2]), (1/2 - I/2)*(1 + Tan[(c + d*x)/2])])*(1 + Tan[(c + d*x)/2]))) + (Cos[(3*(c + d*x))/2]*Csc[c + d*x]*(a*(1 + Sin[c + d*x]))^(4/3)*((1 + Tan[(c + d*x)/2])/Sqrt[Sec[(c + d*x)/2]^2])^(2/3)*(8 + (1 + I)*2^(2/3)*(((1 - I)*(I + Cot[(c + d*x)/2]))/(1 + Cot[(c + d*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/(2 + 2*Tan[(c + d*x)/2])]*(I + Tan[(c + d*x)/2]) - AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(c + d*x)/2])^(1/3)*((-1 - I)*(I + Cot[(c + d*x)/2]))^(1/3)*(1 + Tan[(c + d*x)/2])))/(4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(1 + Tan[(c + d*x)/2])*((-3*Sec[(c + d*x)/2]^2*((1 + Tan[(c + d*x)/2])/Sqrt[Sec[(c + d*x)/2]^2])^(2/3)*(8 + (1 + I)*2^(2/3)*(((1 - I)*(I + Cot[(c + d*x)/2]))/(1 + Cot[(c + d*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/(2 + 2*Tan[(c + d*x)/2])]*(I + Tan[(c + d*x)/2]) - AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(c + d*x)/2])^(1/3)*((-1 - I)*(I + Cot[(c + d*x)/2]))^(1/3)*(1 + Tan[(c + d*x)/2])))/(8*(1 + Tan[(c + d*x)/2])^2) + ((8 + (1 + I)*2^(2/3)*(((1 - I)*(I + Cot[(c + d*x)/2]))/(1 + Cot[(c + d*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/(2 + 2*Tan[(c + d*x)/2])]*(I + Tan[(c + d*x)/2]) - AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(c + d*x)/2])^(1/3)*((-1 - I)*(I + Cot[(c + d*x)/2]))^(1/3)*(1 + Tan[(c + d*x)/2]))*(Sqrt[Sec[(c + d*x)/2]^2]/2 - (Tan[(c + d*x)/2]*(1 + Tan[(c + d*x)/2]))/(2*Sqrt[Sec[(c + d*x)/2]^2])))/(2*(1 + Tan[(c + d*x)/2])*((1 + Tan[(c + d*x)/2])/Sqrt[Sec[(c + d*x)/2]^2])^(1/3)) + (3*((1 + Tan[(c + d*x)/2])/Sqrt[Sec[(c + d*x)/2]^2])^(2/3)*(-1/2*(AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(c + d*x)/2])^(1/3)*((-1 - I)*(I + Cot[(c + d*x)/2]))^(1/3)*Sec[(c + d*x)/2]^2) + ((1 + I)*(((1 - I)*(I + Cot[(c + d*x)/2]))/(1 + Cot[(c + d*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/(2 + 2*Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2)/2^(1/3) + ((1/3 + I/3)*2^(2/3)*(((1/2 - I/2)*(I + Cot[(c + d*x)/2])*Csc[(c + d*x)/2]^2)/(1 + Cot[(c + d*x)/2])^2 - ((1/2 - I/2)*Csc[(c + d*x)/2]^2)/(1 + Cot[(c + d*x)/2]))*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/(2 + 2*Tan[(c + d*x)/2])]*(I + Tan[(c + d*x)/2]))/(((1 - I)*(I + Cot[(c + d*x)/2]))/(1 + Cot[(c + d*x)/2]))^(2/3) - ((1/6 + I/6)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(c + d*x)/2])^(1/3)*Csc[(c + d*x)/2]^2*(1 + Tan[(c + d*x)/2]))/((-1 - I)*(I + Cot[(c + d*x)/2]))^(2/3) - ((1/3 - I/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*((-1 - I)*(I + Cot[(c + d*x)/2]))^(1/3)*Csc[(c + d*x)/2]^2*(1 + Tan[(c + d*x)/2]))/((2 + 2*I) - (2 - 2*I)*Cot[(c + d*x)/2])^(2/3) - ((2 + 2*I) - (2 - 2*I)*Cot[(c + d*x)/2])^(1/3)*((-1 - I)*(I + Cot[(c + d*x)/2]))^(1/3)*((-1/30 + I/30)*AppellF1[5/3, 1/3, 4/3, 8/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*Csc[(c + d*x)/2]^2 - (1/30 + I/30)*AppellF1[5/3, 4/3, 1/3, 8/3, (1/2 + I/2)*(1 + Cot[(c + d*x)/2]), (1/2 - I/2)*(1 + Cot[(c + d*x)/2])]*Csc[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]) + ((2/3 + (2*I)/3)*2^(2/3)*(((1 - I)*(I + Cot[(c + d*x)/2]))/(1 + Cot[(c + d*x)/2]))^(1/3)*(I + Tan[(c + d*x)/2])*(2 + 2*Tan[(c + d*x)/2])*(-((Sec[(c + d*x)/2]^2*((1 + I) + (1 - I)*Tan[(c + d*x)/2]))/(2 + 2*Tan[(c + d*x)/2])^2) + ((1/2 - I/2)*Sec[(c + d*x)/2]^2)/(2 + 2*Tan[(c + d*x)/2]))*(-Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/(2 + 2*Tan[(c + d*x)/2])] + (1 - ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/(2 + 2*Tan[(c + d*x)/2]))^(-1/3)))/((1 + I) + (1 - I)*Tan[(c + d*x)/2])))/(4*(1 + Tan[(c + d*x)/2]))))","C",0
104,1,110,161,0.4611192,"\int \frac{\sin ^3(c+d x)}{\sqrt[3]{a+a \sin (c+d x)}} \, dx","Integrate[Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(1/3),x]","\frac{3 \cos (c+d x) \left(\sqrt{1-\sin (c+d x)} (2 \sin (c+d x)+5 \cos (2 (c+d x))-36)-37 \sqrt{2} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)\right)}{80 d \sqrt{1-\sin (c+d x)} \sqrt[3]{a (\sin (c+d x)+1)}}","\frac{37 \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{40\ 2^{5/6} d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \sin ^2(c+d x) \cos (c+d x)}{8 d \sqrt[3]{a \sin (c+d x)+a}}+\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{40 a d}-\frac{99 \cos (c+d x)}{80 d \sqrt[3]{a \sin (c+d x)+a}}",1,"(3*Cos[c + d*x]*(-37*Sqrt[2]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + Sqrt[1 - Sin[c + d*x]]*(-36 + 5*Cos[2*(c + d*x)] + 2*Sin[c + d*x])))/(80*d*Sqrt[1 - Sin[c + d*x]]*(a*(1 + Sin[c + d*x]))^(1/3))","A",0
105,1,95,126,0.2947238,"\int \frac{\sin ^2(c+d x)}{\sqrt[3]{a+a \sin (c+d x)}} \, dx","Integrate[Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(1/3),x]","-\frac{3 \cos (c+d x) \left(\sqrt{2-2 \sin (c+d x)} (2 \sin (c+d x)-1)-14 \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)\right)}{10 d \sqrt{2-2 \sin (c+d x)} \sqrt[3]{a (\sin (c+d x)+1)}}","-\frac{7 \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5\ 2^{5/6} d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 a d}+\frac{9 \cos (c+d x)}{10 d \sqrt[3]{a \sin (c+d x)+a}}",1,"(-3*Cos[c + d*x]*(-14*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + Sqrt[2 - 2*Sin[c + d*x]]*(-1 + 2*Sin[c + d*x])))/(10*d*Sqrt[2 - 2*Sin[c + d*x]]*(a*(1 + Sin[c + d*x]))^(1/3))","A",1
106,1,84,93,0.147214,"\int \frac{\sin (c+d x)}{\sqrt[3]{a+a \sin (c+d x)}} \, dx","Integrate[Sin[c + d*x]/(a + a*Sin[c + d*x])^(1/3),x]","-\frac{3 \cos (c+d x) \left(2 \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)+\sqrt{2-2 \sin (c+d x)}\right)}{2 d \sqrt{2-2 \sin (c+d x)} \sqrt[3]{a (\sin (c+d x)+1)}}","\frac{\cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{2^{5/6} d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \cos (c+d x)}{2 d \sqrt[3]{a \sin (c+d x)+a}}",1,"(-3*Cos[c + d*x]*(2*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + Sqrt[2 - 2*Sin[c + d*x]]))/(2*d*Sqrt[2 - 2*Sin[c + d*x]]*(a*(1 + Sin[c + d*x]))^(1/3))","A",1
107,1,70,66,0.0976667,"\int \frac{1}{\sqrt[3]{a+a \sin (c+d x)}} \, dx","Integrate[(a + a*Sin[c + d*x])^(-1/3),x]","\frac{3 \sqrt{2} \cos (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)}{d \sqrt{1-\sin (c+d x)} \sqrt[3]{a (\sin (c+d x)+1)}}","-\frac{\sqrt[6]{2} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"(3*Sqrt[2]*Cos[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2])/(d*Sqrt[1 - Sin[c + d*x]]*(a*(1 + Sin[c + d*x]))^(1/3))","A",1
108,0,0,77,3.4738796,"\int \frac{\csc (c+d x)}{\sqrt[3]{a+a \sin (c+d x)}} \, dx","Integrate[Csc[c + d*x]/(a + a*Sin[c + d*x])^(1/3),x]","\int \frac{\csc (c+d x)}{\sqrt[3]{a+a \sin (c+d x)}} \, dx","-\frac{\sqrt[6]{2} \cos (c+d x) F_1\left(\frac{1}{2};1,\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"Integrate[Csc[c + d*x]/(a + a*Sin[c + d*x])^(1/3), x]","F",-1
109,1,184,77,8.8385842,"\int \frac{\csc ^2(c+d x)}{\sqrt[3]{a+a \sin (c+d x)}} \, dx","Integrate[Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(1/3),x]","\frac{2\ 2^{2/3} \cos ^{\frac{2}{3}}\left(\frac{1}{4} (2 c+2 d x-\pi )\right) (\cos (c+d x)+i \sin (c+d x)) \left(4 i \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-i e^{-i (c+d x)}\right) \cos (c+d x) (\sin (c+d x)+i \cos (c+d x)+1)^{2/3}+4 \sin (c+d x)+1\right)}{5 d \left(-(-1)^{3/4} e^{-\frac{1}{2} i (c+d x)} \left(e^{i (c+d x)}+i\right)\right)^{2/3} \left(1+e^{2 i (c+d x)}\right) \sqrt[3]{a (\sin (c+d x)+1)}}","-\frac{\sqrt[6]{2} \cos (c+d x) F_1\left(\frac{1}{2};2,\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"(2*2^(2/3)*Cos[(2*c - Pi + 2*d*x)/4]^(2/3)*(Cos[c + d*x] + I*Sin[c + d*x])*(1 + 4*Sin[c + d*x] + (4*I)*Cos[c + d*x]*Hypergeometric2F1[1/3, 2/3, 4/3, (-I)/E^(I*(c + d*x))]*(1 + I*Cos[c + d*x] + Sin[c + d*x])^(2/3)))/(5*d*(-(((-1)^(3/4)*(I + E^(I*(c + d*x))))/E^((I/2)*(c + d*x))))^(2/3)*(1 + E^((2*I)*(c + d*x)))*(a*(1 + Sin[c + d*x]))^(1/3))","C",0
110,1,116,162,0.4960731,"\int \frac{\sin ^3(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx","Integrate[Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(4/3),x]","\frac{3 \cos (c+d x) \left(20 \sqrt{2} (\sin (c+d x)+1) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)+\sqrt{1-\sin (c+d x)} (4 \sin (c+d x)+\cos (2 (c+d x))+7)\right)}{10 d \sqrt{1-\sin (c+d x)} (a (\sin (c+d x)+1))^{4/3}}","-\frac{2 \sqrt[6]{2} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \sin ^2(c+d x) \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}+\frac{6 \cos (c+d x)}{5 a d \sqrt[3]{a \sin (c+d x)+a}}+\frac{6 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}",1,"(3*Cos[c + d*x]*(20*Sqrt[2]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2]*(1 + Sin[c + d*x]) + Sqrt[1 - Sin[c + d*x]]*(7 + Cos[2*(c + d*x)] + 4*Sin[c + d*x])))/(10*d*Sqrt[1 - Sin[c + d*x]]*(a*(1 + Sin[c + d*x]))^(4/3))","A",1
111,1,108,129,0.3326184,"\int \frac{\sin ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx","Integrate[Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(4/3),x]","-\frac{3 \cos (c+d x) \left(13 \sqrt{2} (\sin (c+d x)+1) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)+\sqrt{1-\sin (c+d x)} (5 \sin (c+d x)+7)\right)}{10 d \sqrt{1-\sin (c+d x)} (a (\sin (c+d x)+1))^{4/3}}","\frac{13 \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5\ 2^{5/6} a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \cos (c+d x)}{2 a d \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}",1,"(-3*Cos[c + d*x]*(13*Sqrt[2]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2]*(1 + Sin[c + d*x]) + Sqrt[1 - Sin[c + d*x]]*(7 + 5*Sin[c + d*x])))/(10*d*Sqrt[1 - Sin[c + d*x]]*(a*(1 + Sin[c + d*x]))^(4/3))","A",1
112,1,130,99,0.2896545,"\int \frac{\sin (c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx","Integrate[Sin[c + d*x]/(a + a*Sin[c + d*x])^(4/3),x]","\frac{3 \left(8 (\sin (c+d x)+1) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)+\sqrt{2-2 \sin (c+d x)}\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\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)}{5 d \sqrt{2-2 \sin (c+d x)} (a (\sin (c+d x)+1))^{4/3}}","\frac{3 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}-\frac{4 \sqrt[6]{2} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5 a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"(3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Sqrt[2 - 2*Sin[c + d*x]] + 8*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2]*(1 + Sin[c + d*x])))/(5*d*Sqrt[2 - 2*Sin[c + d*x]]*(a*(1 + Sin[c + d*x]))^(4/3))","A",1
113,1,130,69,0.2051083,"\int \frac{1}{(a+a \sin (c+d x))^{4/3}} \, dx","Integrate[(a + a*Sin[c + d*x])^(-4/3),x]","-\frac{3 \left(\sqrt{2-2 \sin (c+d x)}-2 (\sin (c+d x)+1) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\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)}{5 d \sqrt{2-2 \sin (c+d x)} (a (\sin (c+d x)+1))^{4/3}}","-\frac{\cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{2^{5/6} a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"(-3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Sqrt[2 - 2*Sin[c + d*x]] - 2*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2]*(1 + Sin[c + d*x])))/(5*d*Sqrt[2 - 2*Sin[c + d*x]]*(a*(1 + Sin[c + d*x]))^(4/3))","A",1
114,0,0,80,10.3158455,"\int \frac{\csc (c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx","Integrate[Csc[c + d*x]/(a + a*Sin[c + d*x])^(4/3),x]","\int \frac{\csc (c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx","-\frac{\cos (c+d x) F_1\left(\frac{1}{2};1,\frac{11}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{2^{5/6} a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"Integrate[Csc[c + d*x]/(a + a*Sin[c + d*x])^(4/3), x]","F",-1
115,1,230,80,14.1755736,"\int \frac{\csc ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx","Integrate[Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(4/3),x]","\frac{8\ 2^{2/3} \cos ^{\frac{8}{3}}\left(\frac{1}{4} (2 c+2 d x-\pi )\right) (\cos (2 (c+d x))+i \sin (2 (c+d x))) \left(14 i \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-i e^{-i (c+d x)}\right) (\sin (c+d x)+i \cos (c+d x)+1)^{2/3} (\sin (2 (c+d x))+2 \cos (c+d x))+35 \sin (c+d x)-14 \cos (2 (c+d x))+6\right)}{55 d \left(-1+i e^{i (c+d x)}\right)^3 \left(e^{i (c+d x)}-i\right) \left(-(-1)^{3/4} e^{-\frac{1}{2} i (c+d x)} \left(e^{i (c+d x)}+i\right)\right)^{2/3} (a (\sin (c+d x)+1))^{4/3}}","-\frac{\cos (c+d x) F_1\left(\frac{1}{2};2,\frac{11}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{2^{5/6} a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"(8*2^(2/3)*Cos[(2*c - Pi + 2*d*x)/4]^(8/3)*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])*(6 - 14*Cos[2*(c + d*x)] + 35*Sin[c + d*x] + (14*I)*Hypergeometric2F1[1/3, 2/3, 4/3, (-I)/E^(I*(c + d*x))]*(1 + I*Cos[c + d*x] + Sin[c + d*x])^(2/3)*(2*Cos[c + d*x] + Sin[2*(c + d*x)])))/(55*d*(-1 + I*E^(I*(c + d*x)))^3*(-I + E^(I*(c + d*x)))*(-(((-1)^(3/4)*(I + E^(I*(c + d*x))))/E^((I/2)*(c + d*x))))^(2/3)*(a*(1 + Sin[c + d*x]))^(4/3))","C",0
116,1,5109,96,22.5449541,"\int \sin ^n(e+f x) (1+\sin (e+f x))^{3/2} \, dx","Integrate[Sin[e + f*x]^n*(1 + Sin[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 (4 n+5) \cos (e+f x) \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f (2 n+3) \sqrt{\sin (e+f x)+1}}-\frac{2 \cos (e+f x) \sin ^{n+1}(e+f x)}{f (2 n+3) \sqrt{\sin (e+f x)+1}}",1,"Result too large to show","C",0
117,1,186,43,0.4445925,"\int \sin ^n(e+f x) \sqrt{1+\sin (e+f x)} \, dx","Integrate[Sin[e + f*x]^n*Sqrt[1 + Sin[e + f*x]],x]","\frac{2^{1-n} e^{i (e+f x)} \left(-i e^{-i (e+f x)} \left(-1+e^{2 i (e+f x)}\right)\right)^{n+1} \sqrt{\sin (e+f x)+1} \left(i (2 n-1) \, _2F_1\left(1,\frac{1}{4} (2 n+3);\frac{1}{4} (3-2 n);e^{2 i (e+f x)}\right)+(2 n+1) e^{i (e+f x)} \, _2F_1\left(1,\frac{1}{4} (2 n+5);\frac{1}{4} (5-2 n);e^{2 i (e+f x)}\right)\right)}{f (2 n-1) (2 n+1) \left(e^{i (e+f x)}+i\right)}","-\frac{2 \cos (e+f x) \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f \sqrt{\sin (e+f x)+1}}",1,"(2^(1 - n)*E^(I*(e + f*x))*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^(1 + n)*(I*(-1 + 2*n)*Hypergeometric2F1[1, (3 + 2*n)/4, (3 - 2*n)/4, E^((2*I)*(e + f*x))] + E^(I*(e + f*x))*(1 + 2*n)*Hypergeometric2F1[1, (5 + 2*n)/4, (5 - 2*n)/4, E^((2*I)*(e + f*x))])*Sqrt[1 + Sin[e + f*x]])/((I + E^(I*(e + f*x)))*f*(-1 + 2*n)*(1 + 2*n))","C",0
118,1,225,58,1.58174,"\int \frac{\sin ^n(e+f x)}{\sqrt{1+\sin (e+f x)}} \, dx","Integrate[Sin[e + f*x]^n/Sqrt[1 + Sin[e + f*x]],x]","\frac{\sqrt{\sin (e+f x)+1} \cos (e+f x) (-\sin (e+f x))^{-n} \sin ^n(e+f x) \left(1-\frac{1}{\sin (e+f x)+1}\right)^{-n} \left(4 \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) \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 f (2 n+1) (\sin (e+f x)-1)}","-\frac{\cos (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)}{f \sqrt{\sin (e+f x)+1}}",1,"(Cos[e + f*x]*Sin[e + f*x]^n*Sqrt[1 + Sin[e + f*x]]*(4*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])] - (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*f*(1 + 2*n)*(-1 + Sin[e + f*x])*(-Sin[e + f*x])^n*(1 - (1 + Sin[e + f*x])^(-1))^n)","B",0
119,1,263,60,3.7920211,"\int \frac{\sin ^n(e+f x)}{(1+\sin (e+f x))^{3/2}} \, dx","Integrate[Sin[e + f*x]^n/(1 + Sin[e + f*x])^(3/2),x]","\frac{\sec (e+f x) \sin ^n(e+f x) \left(\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 (\sin (e+f x)+1) \sqrt{1-\frac{2}{\sin (e+f x)+1}} \left(1-\frac{1}{\sin (e+f x)+1}\right)^{-n} \left(2 (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)+(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)}{8 f \sqrt{\sin (e+f x)+1}}","-\frac{\cos (e+f x) F_1\left(\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{2 f \sqrt{\sin (e+f x)+1}}",1,"(Sec[e + f*x]*Sin[e + f*x]^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 + Sin[e + f*x])^2)/(-Sin[e + f*x])^n - (4*(1 + Sin[e + f*x])*Sqrt[1 - 2/(1 + Sin[e + f*x])]*(2*(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)] + (-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*f*Sqrt[1 + Sin[e + f*x]])","B",0
120,1,5111,106,6.3346036,"\int \sin ^n(e+f x) (a+a \sin (e+f x))^{3/2} \, dx","Integrate[Sin[e + f*x]^n*(a + a*Sin[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 a^2 (4 n+5) \cos (e+f x) \, _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^2 \cos (e+f x) \sin ^{n+1}(e+f x)}{f (2 n+3) \sqrt{a \sin (e+f x)+a}}",1,"Result too large to show","C",0
121,1,264,46,4.2949163,"\int \sin ^n(e+f x) \sqrt{a+a \sin (e+f x)} \, dx","Integrate[Sin[e + f*x]^n*Sqrt[a + a*Sin[e + f*x]],x]","\frac{(1+i) e^{-\frac{1}{2} i f x} \sqrt{a (\sin (e+f x)+1)} \sin ^n(e+f x) \left(\sin ^2(e) e^{2 i f x}-i \sin (2 e) e^{2 i f x}+\cos ^2(e) \left(-e^{2 i f x}\right)+1\right)^{-n} \left((2 n+1) e^{i f x} \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right) \, _2F_1\left(\frac{1}{4} (1-2 n),-n;\frac{1}{4} (5-2 n);e^{2 i f x} (\cos (e)+i \sin (e))^2\right)+(2 n-1) \left(\sin \left(\frac{e}{2}\right)+i \cos \left(\frac{e}{2}\right)\right) \, _2F_1\left(\frac{1}{4} (-2 n-1),-n;\frac{1}{4} (3-2 n);e^{2 i f x} (\cos (e)+i \sin (e))^2\right)\right)}{f (2 n-1) (2 n+1) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{2 a \cos (e+f x) \, _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,"((1 + I)*(E^(I*f*x)*(1 + 2*n)*Hypergeometric2F1[(1 - 2*n)/4, -n, (5 - 2*n)/4, E^((2*I)*f*x)*(Cos[e] + I*Sin[e])^2]*(Cos[e/2] + I*Sin[e/2]) + (-1 + 2*n)*Hypergeometric2F1[(-1 - 2*n)/4, -n, (3 - 2*n)/4, E^((2*I)*f*x)*(Cos[e] + I*Sin[e])^2]*(I*Cos[e/2] + Sin[e/2]))*Sin[e + f*x]^n*Sqrt[a*(1 + Sin[e + f*x])])/(E^((I/2)*f*x)*f*(-1 + 2*n)*(1 + 2*n)*(1 - E^((2*I)*f*x)*Cos[e]^2 + E^((2*I)*f*x)*Sin[e]^2 - I*E^((2*I)*f*x)*Sin[2*e])^n*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","C",1
122,1,234,60,1.3383989,"\int \frac{\sin ^n(e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[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)} \sin ^{2 n}(e+f x) \left(-\sin ^2(e+f x)\right)^{-n} \left(1-\frac{1}{\sin (e+f x)+1}\right)^{-n} \left(4 \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) \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{\cos (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)}{f \sqrt{a \sin (e+f x)+a}}",1,"(Cos[e + f*x]*Sin[e + f*x]^(2*n)*Sqrt[a*(1 + Sin[e + f*x])]*(4*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])] - (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)","B",0
123,1,274,65,2.2697864,"\int \frac{\sin ^n(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[Sin[e + f*x]^n/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\sec (e+f x) \sin ^n(e+f x) \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 (\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)}{\left(4 n^2-1\right) \sqrt{1-\frac{2}{\sin (e+f x)+1}}}\right)}{8 a^3 f \sqrt{a (\sin (e+f x)+1)}}","-\frac{\cos (e+f x) F_1\left(\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{2 a f \sqrt{a \sin (e+f x)+a}}",1,"(Sec[e + f*x]*Sin[e + f*x]^n*((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*(-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)*Sqrt[1 - 2/(1 + Sin[e + f*x])]*(1 - (1 + Sin[e + f*x])^(-1))^n)))/(8*a^3*f*Sqrt[a*(1 + Sin[e + f*x])])","B",0
124,1,5129,130,6.3066644,"\int (d \sin (e+f x))^n (1+\sin (e+f x))^{3/2} \, dx","Integrate[(d*Sin[e + f*x])^n*(1 + Sin[e + f*x])^(3/2),x]","\text{Result too large to show}","\frac{(4 n+5) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},n+1;n+2;\sin (e+f x)\right)}{d f (n+1) (2 n+3) \sqrt{1-\sin (e+f x)} \sqrt{\sin (e+f x)+1}}-\frac{2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{\sin (e+f x)+1}}",1,"Result too large to show","C",0
125,1,215,72,0.3537202,"\int (d \sin (e+f x))^n \sqrt{1+\sin (e+f x)} \, dx","Integrate[(d*Sin[e + f*x])^n*Sqrt[1 + Sin[e + f*x]],x]","\frac{(1-i) 2^{-n} e^{\frac{1}{2} i (e+f x)} \left(-i e^{-i (e+f x)} \left(-1+e^{2 i (e+f x)}\right)\right)^{n+1} \sqrt{\sin (e+f x)+1} \left(i (2 n-1) \, _2F_1\left(1,\frac{1}{4} (2 n+3);\frac{1}{4} (3-2 n);e^{2 i (e+f x)}\right)+(2 n+1) e^{i (e+f x)} \, _2F_1\left(1,\frac{1}{4} (2 n+5);\frac{1}{4} (5-2 n);e^{2 i (e+f x)}\right)\right) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f (2 n-1) (2 n+1) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{\cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},n+1;n+2;\sin (e+f x)\right)}{d f (n+1) \sqrt{1-\sin (e+f x)} \sqrt{\sin (e+f x)+1}}",1,"((1 - I)*E^((I/2)*(e + f*x))*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^(1 + n)*(I*(-1 + 2*n)*Hypergeometric2F1[1, (3 + 2*n)/4, (3 - 2*n)/4, E^((2*I)*(e + f*x))] + E^(I*(e + f*x))*(1 + 2*n)*Hypergeometric2F1[1, (5 + 2*n)/4, (5 - 2*n)/4, E^((2*I)*(e + f*x))])*(d*Sin[e + f*x])^n*Sqrt[1 + Sin[e + f*x]])/(2^n*f*(-1 + 2*n)*(1 + 2*n)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[e + f*x]^n)","C",0
126,1,227,78,0.4111198,"\int \frac{(d \sin (e+f x))^n}{\sqrt{1+\sin (e+f x)}} \, dx","Integrate[(d*Sin[e + f*x])^n/Sqrt[1 + Sin[e + f*x]],x]","\frac{\sqrt{\sin (e+f x)+1} \cos (e+f x) (-\sin (e+f x))^{-n} \left(1-\frac{1}{\sin (e+f x)+1}\right)^{-n} \left(4 \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) \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) (d \sin (e+f x))^n}{4 f (2 n+1) (\sin (e+f x)-1)}","-\frac{\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{\sin (e+f x)+1}}",1,"(Cos[e + f*x]*(d*Sin[e + f*x])^n*Sqrt[1 + Sin[e + f*x]]*(4*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])] - (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*f*(1 + 2*n)*(-1 + Sin[e + f*x])*(-Sin[e + f*x])^n*(1 - (1 + Sin[e + f*x])^(-1))^n)","B",0
127,1,265,80,0.9330168,"\int \frac{(d \sin (e+f x))^n}{(1+\sin (e+f x))^{3/2}} \, dx","Integrate[(d*Sin[e + f*x])^n/(1 + Sin[e + f*x])^(3/2),x]","\frac{\sec (e+f x) \left(\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 (\sin (e+f x)+1) \sqrt{1-\frac{2}{\sin (e+f x)+1}} \left(1-\frac{1}{\sin (e+f x)+1}\right)^{-n} \left(2 (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)+(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) (d \sin (e+f x))^n}{8 f \sqrt{\sin (e+f x)+1}}","-\frac{\cos (e+f x) \sin ^{-n}(e+f x) F_1\left(\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) (d \sin (e+f x))^n}{2 f \sqrt{\sin (e+f x)+1}}",1,"(Sec[e + f*x]*(d*Sin[e + f*x])^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 + Sin[e + f*x])^2)/(-Sin[e + f*x])^n - (4*(1 + Sin[e + f*x])*Sqrt[1 - 2/(1 + Sin[e + f*x])]*(2*(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)] + (-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*f*Sqrt[1 + Sin[e + f*x]])","B",0
128,1,5131,131,6.3369506,"\int (d \sin (e+f x))^n (a+a \sin (e+f x))^{3/2} \, dx","Integrate[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 a^2 (4 n+5) \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^2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}",1,"Result too large to show","C",0
129,1,266,66,0.4290846,"\int (d \sin (e+f x))^n \sqrt{a+a \sin (e+f x)} \, dx","Integrate[(d*Sin[e + f*x])^n*Sqrt[a + a*Sin[e + f*x]],x]","\frac{(1+i) e^{-\frac{1}{2} i f x} \sqrt{a (\sin (e+f x)+1)} (d \sin (e+f x))^n \left(\sin ^2(e) e^{2 i f x}-i \sin (2 e) e^{2 i f x}+\cos ^2(e) \left(-e^{2 i f x}\right)+1\right)^{-n} \left((2 n+1) e^{i f x} \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right) \, _2F_1\left(\frac{1}{4} (1-2 n),-n;\frac{1}{4} (5-2 n);e^{2 i f x} (\cos (e)+i \sin (e))^2\right)+(2 n-1) \left(\sin \left(\frac{e}{2}\right)+i \cos \left(\frac{e}{2}\right)\right) \, _2F_1\left(\frac{1}{4} (-2 n-1),-n;\frac{1}{4} (3-2 n);e^{2 i f x} (\cos (e)+i \sin (e))^2\right)\right)}{f (2 n-1) (2 n+1) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{2 a \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,"((1 + I)*(E^(I*f*x)*(1 + 2*n)*Hypergeometric2F1[(1 - 2*n)/4, -n, (5 - 2*n)/4, E^((2*I)*f*x)*(Cos[e] + I*Sin[e])^2]*(Cos[e/2] + I*Sin[e/2]) + (-1 + 2*n)*Hypergeometric2F1[(-1 - 2*n)/4, -n, (3 - 2*n)/4, E^((2*I)*f*x)*(Cos[e] + I*Sin[e])^2]*(I*Cos[e/2] + Sin[e/2]))*(d*Sin[e + f*x])^n*Sqrt[a*(1 + Sin[e + f*x])])/(E^((I/2)*f*x)*f*(-1 + 2*n)*(1 + 2*n)*(1 - E^((2*I)*f*x)*Cos[e]^2 + E^((2*I)*f*x)*Sin[e]^2 - I*E^((2*I)*f*x)*Sin[2*e])^n*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","C",1
130,1,242,80,0.7114587,"\int \frac{(d \sin (e+f x))^n}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(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)} \sin ^n(e+f x) \left(-\sin ^2(e+f x)\right)^{-n} \left(1-\frac{1}{\sin (e+f x)+1}\right)^{-n} \left(4 \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) \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) (d \sin (e+f x))^n}{4 a f (2 n+1) (\sin (e+f x)-1)}","-\frac{\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}}",1,"(Cos[e + f*x]*Sin[e + f*x]^n*(d*Sin[e + f*x])^n*Sqrt[a*(1 + Sin[e + f*x])]*(4*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])] - (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)","B",0
131,1,276,85,1.2642212,"\int \frac{(d \sin (e+f x))^n}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\sec (e+f x) (d \sin (e+f x))^n \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 (\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)}{\left(4 n^2-1\right) \sqrt{1-\frac{2}{\sin (e+f x)+1}}}\right)}{8 a^3 f \sqrt{a (\sin (e+f x)+1)}}","-\frac{\cos (e+f x) \sin ^{-n}(e+f x) F_1\left(\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) (d \sin (e+f x))^n}{2 a f \sqrt{a \sin (e+f x)+a}}",1,"(Sec[e + f*x]*(d*Sin[e + f*x])^n*((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*(-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)*Sqrt[1 - 2/(1 + Sin[e + f*x])]*(1 - (1 + Sin[e + f*x])^(-1))^n)))/(8*a^3*f*Sqrt[a*(1 + Sin[e + f*x])])","B",0
132,1,2805,71,14.9226665,"\int \sin ^n(e+f x) (1+\sin (e+f x))^m \, dx","Integrate[Sin[e + f*x]^n*(1 + Sin[e + f*x])^m,x]","\text{Result too large to show}","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) 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 \sqrt{\sin (e+f x)+1}}",1,"(-3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^(2*n)*(1 + Sin[e + f*x])^m)/(f*(Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)*((-3*n*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]^2*Sin[e + f*x]^(-1 + n))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) + (3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sin[e + f*x]^(1 + n))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) - (3*m*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^n*Tan[(-e + Pi/2 - f*x)/2])/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) + (3*Cos[e + f*x]*Sin[e + f*x]^n*(-1/3*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2]) - ((1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/3))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) - (3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^n*(-2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2] + 3*(-1/3*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2]) - ((1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/3) - 2*Tan[(-e + Pi/2 - f*x)/2]^2*(n*((-3*(1 + m + n)*AppellF1[5/2, 1 - n, 2 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5 + (3*(1 - n)*AppellF1[5/2, 2 - n, 1 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5) + (1 + m + n)*((-3*n*AppellF1[5/2, 1 - n, 2 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5 - (3*(2 + m + n)*AppellF1[5/2, -n, 3 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5))))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)^2)))","B",0
133,1,300,68,2.323968,"\int (1-\sin (e+f x))^m (-\sin (e+f x))^n \, dx","Integrate[(1 - Sin[e + f*x])^m*(-Sin[e + f*x])^n,x]","-\frac{(2 m+3) \cos (e+f x) (1-\sin (e+f x))^m (-\sin (e+f x))^n F_1\left(m+\frac{1}{2};-n,m+n+1;m+\frac{3}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f (2 m+1) \left((2 m+3) F_1\left(m+\frac{1}{2};-n,m+n+1;m+\frac{3}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)-2 \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \left(n F_1\left(m+\frac{3}{2};1-n,m+n+1;m+\frac{5}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+(m+n+1) F_1\left(m+\frac{3}{2};-n,m+n+2;m+\frac{5}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)\right)\right)}","\frac{2^{m+\frac{1}{2}} \cos (e+f x) F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};\sin (e+f x)+1,\frac{1}{2} (\sin (e+f x)+1)\right)}{f \sqrt{1-\sin (e+f x)}}",1,"-(((3 + 2*m)*AppellF1[1/2 + m, -n, 1 + m + n, 3/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Cos[e + f*x]*(1 - Sin[e + f*x])^m*(-Sin[e + f*x])^n)/(f*(1 + 2*m)*((3 + 2*m)*AppellF1[1/2 + m, -n, 1 + m + n, 3/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] - 2*(n*AppellF1[3/2 + m, 1 - n, 1 + m + n, 5/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] + (1 + m + n)*AppellF1[3/2 + m, -n, 2 + m + n, 5/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2])*Tan[(2*e - Pi + 2*f*x)/4]^2)))","B",0
134,1,2813,91,6.2230566,"\int (d \sin (e+f x))^n (1+\sin (e+f x))^m \, dx","Integrate[(d*Sin[e + f*x])^n*(1 + Sin[e + f*x])^m,x]","\text{Result too large to show}","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) \sin ^{-n}(e+f x) (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 \sqrt{\sin (e+f x)+1}}",1,"(-3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^n*(d*Sin[e + f*x])^n*(1 + Sin[e + f*x])^m)/(f*(Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)*((-3*n*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]^2*Sin[e + f*x]^(-1 + n))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) + (3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sin[e + f*x]^(1 + n))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) - (3*m*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^n*Tan[(-e + Pi/2 - f*x)/2])/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) + (3*Cos[e + f*x]*Sin[e + f*x]^n*(-1/3*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2]) - ((1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/3))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) - (3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^n*(-2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2] + 3*(-1/3*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2]) - ((1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/3) - 2*Tan[(-e + Pi/2 - f*x)/2]^2*(n*((-3*(1 + m + n)*AppellF1[5/2, 1 - n, 2 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5 + (3*(1 - n)*AppellF1[5/2, 2 - n, 1 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5) + (1 + m + n)*((-3*n*AppellF1[5/2, 1 - n, 2 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5 - (3*(2 + m + n)*AppellF1[5/2, -n, 3 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5))))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)^2)))","B",0
135,1,300,90,0.7246635,"\int (1-\sin (e+f x))^m (d \sin (e+f x))^n \, dx","Integrate[(1 - Sin[e + f*x])^m*(d*Sin[e + f*x])^n,x]","-\frac{(2 m+3) \cos (e+f x) (1-\sin (e+f x))^m (d \sin (e+f x))^n F_1\left(m+\frac{1}{2};-n,m+n+1;m+\frac{3}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f (2 m+1) \left((2 m+3) F_1\left(m+\frac{1}{2};-n,m+n+1;m+\frac{3}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)-2 \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \left(n F_1\left(m+\frac{3}{2};1-n,m+n+1;m+\frac{5}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+(m+n+1) F_1\left(m+\frac{3}{2};-n,m+n+2;m+\frac{5}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)\right)\right)}","\frac{2^{m+\frac{1}{2}} \cos (e+f x) (-\sin (e+f x))^{-n} (d \sin (e+f x))^n F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};\sin (e+f x)+1,\frac{1}{2} (\sin (e+f x)+1)\right)}{f \sqrt{1-\sin (e+f x)}}",1,"-(((3 + 2*m)*AppellF1[1/2 + m, -n, 1 + m + n, 3/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Cos[e + f*x]*(1 - Sin[e + f*x])^m*(d*Sin[e + f*x])^n)/(f*(1 + 2*m)*((3 + 2*m)*AppellF1[1/2 + m, -n, 1 + m + n, 3/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] - 2*(n*AppellF1[3/2 + m, 1 - n, 1 + m + n, 5/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] + (1 + m + n)*AppellF1[3/2 + m, -n, 2 + m + n, 5/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2])*Tan[(2*e - Pi + 2*f*x)/4]^2)))","B",0
136,1,2807,87,6.3065744,"\int \sin ^n(e+f x) (a+a \sin (e+f x))^m \, dx","Integrate[Sin[e + f*x]^n*(a + a*Sin[e + f*x])^m,x]","\text{Result too large to show}","-\frac{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 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}",1,"(-3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^(2*n)*(a + a*Sin[e + f*x])^m)/(f*(Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)*((-3*n*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]^2*Sin[e + f*x]^(-1 + n))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) + (3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sin[e + f*x]^(1 + n))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) - (3*m*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^n*Tan[(-e + Pi/2 - f*x)/2])/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) + (3*Cos[e + f*x]*Sin[e + f*x]^n*(-1/3*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2]) - ((1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/3))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) - (3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^n*(-2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2] + 3*(-1/3*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2]) - ((1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/3) - 2*Tan[(-e + Pi/2 - f*x)/2]^2*(n*((-3*(1 + m + n)*AppellF1[5/2, 1 - n, 2 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5 + (3*(1 - n)*AppellF1[5/2, 2 - n, 1 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5) + (1 + m + n)*((-3*n*AppellF1[5/2, 1 - n, 2 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5 - (3*(2 + m + n)*AppellF1[5/2, -n, 3 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5))))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)^2)))","B",0
137,1,301,85,0.5953121,"\int (-\sin (e+f x))^n (a-a \sin (e+f x))^m \, dx","Integrate[(-Sin[e + f*x])^n*(a - a*Sin[e + f*x])^m,x]","-\frac{(2 m+3) \cos (e+f x) (-\sin (e+f x))^n (a-a \sin (e+f x))^m F_1\left(m+\frac{1}{2};-n,m+n+1;m+\frac{3}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f (2 m+1) \left((2 m+3) F_1\left(m+\frac{1}{2};-n,m+n+1;m+\frac{3}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)-2 \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \left(n F_1\left(m+\frac{3}{2};1-n,m+n+1;m+\frac{5}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+(m+n+1) F_1\left(m+\frac{3}{2};-n,m+n+2;m+\frac{5}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)\right)\right)}","\frac{2^{m+\frac{1}{2}} \cos (e+f x) (1-\sin (e+f x))^{-m-\frac{1}{2}} (a-a \sin (e+f x))^m F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};\sin (e+f x)+1,\frac{1}{2} (\sin (e+f x)+1)\right)}{f}",1,"-(((3 + 2*m)*AppellF1[1/2 + m, -n, 1 + m + n, 3/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Cos[e + f*x]*(-Sin[e + f*x])^n*(a - a*Sin[e + f*x])^m)/(f*(1 + 2*m)*((3 + 2*m)*AppellF1[1/2 + m, -n, 1 + m + n, 3/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] - 2*(n*AppellF1[3/2 + m, 1 - n, 1 + m + n, 5/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] + (1 + m + n)*AppellF1[3/2 + m, -n, 2 + m + n, 5/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2])*Tan[(2*e - Pi + 2*f*x)/4]^2)))","B",0
138,1,2815,107,6.2389991,"\int (d \sin (e+f x))^n (a+a \sin (e+f x))^m \, dx","Integrate[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^m,x]","\text{Result too large to show}","-\frac{2^{m+\frac{1}{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,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f}",1,"(-3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^n*(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^m)/(f*(Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)*((-3*n*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]^2*Sin[e + f*x]^(-1 + n))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) + (3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sin[e + f*x]^(1 + n))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) - (3*m*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^n*Tan[(-e + Pi/2 - f*x)/2])/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) + (3*Cos[e + f*x]*Sin[e + f*x]^n*(-1/3*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2]) - ((1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/3))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) - (3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^n*(-2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2] + 3*(-1/3*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2]) - ((1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/3) - 2*Tan[(-e + Pi/2 - f*x)/2]^2*(n*((-3*(1 + m + n)*AppellF1[5/2, 1 - n, 2 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5 + (3*(1 - n)*AppellF1[5/2, 2 - n, 1 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5) + (1 + m + n)*((-3*n*AppellF1[5/2, 1 - n, 2 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5 - (3*(2 + m + n)*AppellF1[5/2, -n, 3 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5))))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)^2)))","B",0
139,1,301,107,0.5282274,"\int (d \sin (e+f x))^n (a-a \sin (e+f x))^m \, dx","Integrate[(d*Sin[e + f*x])^n*(a - a*Sin[e + f*x])^m,x]","-\frac{(2 m+3) \cos (e+f x) (a-a \sin (e+f x))^m (d \sin (e+f x))^n F_1\left(m+\frac{1}{2};-n,m+n+1;m+\frac{3}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f (2 m+1) \left((2 m+3) F_1\left(m+\frac{1}{2};-n,m+n+1;m+\frac{3}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)-2 \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \left(n F_1\left(m+\frac{3}{2};1-n,m+n+1;m+\frac{5}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+(m+n+1) F_1\left(m+\frac{3}{2};-n,m+n+2;m+\frac{5}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)\right)\right)}","\frac{2^{m+\frac{1}{2}} \cos (e+f x) (1-\sin (e+f x))^{-m-\frac{1}{2}} (-\sin (e+f x))^{-n} (a-a \sin (e+f x))^m (d \sin (e+f x))^n F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};\sin (e+f x)+1,\frac{1}{2} (\sin (e+f x)+1)\right)}{f}",1,"-(((3 + 2*m)*AppellF1[1/2 + m, -n, 1 + m + n, 3/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Cos[e + f*x]*(d*Sin[e + f*x])^n*(a - a*Sin[e + f*x])^m)/(f*(1 + 2*m)*((3 + 2*m)*AppellF1[1/2 + m, -n, 1 + m + n, 3/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] - 2*(n*AppellF1[3/2 + m, 1 - n, 1 + m + n, 5/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] + (1 + m + n)*AppellF1[3/2 + m, -n, 2 + m + n, 5/2 + m, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2])*Tan[(2*e - Pi + 2*f*x)/4]^2)))","B",0
140,-1,0,294,180.00457,"\int \sin ^4(c+d x) (a+a \sin (c+d x))^n \, dx","Integrate[Sin[c + d*x]^4*(a + a*Sin[c + d*x])^n,x]","\text{\$Aborted}","-\frac{2^{n+\frac{1}{2}} \left(n^4+6 n^3+17 n^2+12 n+9\right) \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (n+1) (n+2) (n+3) (n+4)}+\frac{\left(-n^2-n+9\right) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1) (n+2) (n+3) (n+4)}-\frac{\left(n^2+3 n+9\right) \cos (c+d x) (a \sin (c+d x)+a)^{n+1}}{a d (n+2) (n+3) (n+4)}-\frac{\sin ^3(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+4)}-\frac{n \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+3) (n+4)}",1,"$Aborted","F",-1
141,1,60244,215,128.810219,"\int \sin ^3(c+d x) (a+a \sin (c+d x))^n \, dx","Integrate[Sin[c + d*x]^3*(a + a*Sin[c + d*x])^n,x]","\text{Result too large to show}","-\frac{2^{n+\frac{1}{2}} n \left(n^2+3 n+5\right) \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (n+1) (n+2) (n+3)}-\frac{n \cos (c+d x) (a \sin (c+d x)+a)^{n+1}}{a d \left(n^2+5 n+6\right)}-\frac{\sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+3)}-\frac{(n+4) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1) (n+2) (n+3)}",1,"Result too large to show","C",0
142,1,28439,156,54.5783383,"\int \sin ^2(c+d x) (a+a \sin (c+d x))^n \, dx","Integrate[Sin[c + d*x]^2*(a + a*Sin[c + d*x])^n,x]","\text{Result too large to show}","-\frac{2^{n+\frac{1}{2}} \left(n^2+n+1\right) \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (n+1) (n+2)}+\frac{\cos (c+d x) (a \sin (c+d x)+a)^n}{d \left(n^2+3 n+2\right)}-\frac{\cos (c+d x) (a \sin (c+d x)+a)^{n+1}}{a d (n+2)}",1,"Result too large to show","C",0
143,1,178,109,0.4409141,"\int \sin (c+d x) (a+a \sin (c+d x))^n \, dx","Integrate[Sin[c + d*x]*(a + a*Sin[c + d*x])^n,x]","-\frac{\sqrt[4]{-1} 2^{-2 n-1} e^{-\frac{3}{2} i (c+d x)} \left(-(-1)^{3/4} e^{-\frac{1}{2} i (c+d x)} \left(e^{i (c+d x)}+i\right)\right)^{2 n+1} \left((n-1) e^{2 i (c+d x)} \, _2F_1\left(1,n;-n;-i e^{-i (c+d x)}\right)-(n+1) \, _2F_1\left(1,n+2;2-n;-i e^{-i (c+d x)}\right)\right) \sin ^{-2 n}\left(\frac{1}{4} (2 c+2 d x+\pi )\right) (a (\sin (c+d x)+1))^n}{d (n-1) (n+1)}","-\frac{2^{n+\frac{1}{2}} n \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (n+1)}-\frac{\cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1)}",1,"-(((-1)^(1/4)*2^(-1 - 2*n)*(-(((-1)^(3/4)*(I + E^(I*(c + d*x))))/E^((I/2)*(c + d*x))))^(1 + 2*n)*(E^((2*I)*(c + d*x))*(-1 + n)*Hypergeometric2F1[1, n, -n, (-I)/E^(I*(c + d*x))] - (1 + n)*Hypergeometric2F1[1, 2 + n, 2 - n, (-I)/E^(I*(c + d*x))])*(a*(1 + Sin[c + d*x]))^n)/(d*E^(((3*I)/2)*(c + d*x))*(-1 + n)*(1 + n)*Sin[(2*c + Pi + 2*d*x)/4]^(2*n)))","C",0
144,1,90,74,0.1659003,"\int (a+a \sin (c+d x))^n \, dx","Integrate[(a + a*Sin[c + d*x])^n,x]","\frac{\sqrt{2} \cos (c+d x) (a (\sin (c+d x)+1))^n \, _2F_1\left(\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};\frac{1}{4} \cos ^2(c+d x) \csc ^2\left(\frac{1}{4} (2 c+2 d x-\pi )\right)\right)}{(2 d n+d) \sqrt{1-\sin (c+d x)}}","-\frac{2^{n+\frac{1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d}",1,"(Sqrt[2]*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 + n, 3/2 + n, (Cos[c + d*x]^2*Csc[(2*c - Pi + 2*d*x)/4]^2)/4]*(a*(1 + Sin[c + d*x]))^n)/((d + 2*d*n)*Sqrt[1 - Sin[c + d*x]])","A",1
145,1,2560,85,16.1069561,"\int \csc (c+d x) (a+a \sin (c+d x))^n \, dx","Integrate[Csc[c + d*x]*(a + a*Sin[c + d*x])^n,x]","\text{Result too large to show}","-\frac{2^{n+\frac{1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n F_1\left(\frac{1}{2};1,\frac{1}{2}-n;\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d}",1,"-1/2*(Csc[c + d*x]*(a + a*Sin[c + d*x])^n*(AppellF1[2*n, n, n, 1 + 2*n, (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*((-I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n - AppellF1[2*n, n, n, 1 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*((-I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n))/(d*n*(Sec[(-c + Pi/2 - d*x)/2]^2)^n*(-1/2*(Tan[(-c + Pi/2 - d*x)/2]*(AppellF1[2*n, n, n, 1 + 2*n, (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*((-I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n - AppellF1[2*n, n, n, 1 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*((-I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n))/(Sec[(-c + Pi/2 - d*x)/2]^2)^n + ((((1 - I)*n^2*AppellF1[1 + 2*n, n, 1 + n, 2 + 2*n, (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*Sec[(-c + Pi/2 - d*x)/2]^2)/((1 + 2*n)*(-1 + Tan[(-c + Pi/2 - d*x)/2])^2) + ((1 + I)*n^2*AppellF1[1 + 2*n, 1 + n, n, 2 + 2*n, (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*Sec[(-c + Pi/2 - d*x)/2]^2)/((1 + 2*n)*(-1 + Tan[(-c + Pi/2 - d*x)/2])^2))*((-I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n + n*AppellF1[2*n, n, n, 1 + 2*n, (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*((-I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^(-1 + n)*((I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*(Sec[(-c + Pi/2 - d*x)/2]^2/(2*(-1 + Tan[(-c + Pi/2 - d*x)/2])) - (Sec[(-c + Pi/2 - d*x)/2]^2*(-I + Tan[(-c + Pi/2 - d*x)/2]))/(2*(-1 + Tan[(-c + Pi/2 - d*x)/2])^2)) + n*AppellF1[2*n, n, n, 1 + 2*n, (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*((-I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^(-1 + n)*(Sec[(-c + Pi/2 - d*x)/2]^2/(2*(-1 + Tan[(-c + Pi/2 - d*x)/2])) - (Sec[(-c + Pi/2 - d*x)/2]^2*(I + Tan[(-c + Pi/2 - d*x)/2]))/(2*(-1 + Tan[(-c + Pi/2 - d*x)/2])^2)) - ((-I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*(((-1 - I)*n^2*AppellF1[1 + 2*n, n, 1 + n, 2 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*Sec[(-c + Pi/2 - d*x)/2]^2)/((1 + 2*n)*(1 + Tan[(-c + Pi/2 - d*x)/2])^2) - ((1 - I)*n^2*AppellF1[1 + 2*n, 1 + n, n, 2 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*Sec[(-c + Pi/2 - d*x)/2]^2)/((1 + 2*n)*(1 + Tan[(-c + Pi/2 - d*x)/2])^2)) - n*AppellF1[2*n, n, n, 1 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*((-I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^(-1 + n)*((I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*(-1/2*(Sec[(-c + Pi/2 - d*x)/2]^2*(-I + Tan[(-c + Pi/2 - d*x)/2]))/(1 + Tan[(-c + Pi/2 - d*x)/2])^2 + Sec[(-c + Pi/2 - d*x)/2]^2/(2*(1 + Tan[(-c + Pi/2 - d*x)/2]))) - n*AppellF1[2*n, n, n, 1 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*((-I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^(-1 + n)*(-1/2*(Sec[(-c + Pi/2 - d*x)/2]^2*(I + Tan[(-c + Pi/2 - d*x)/2]))/(1 + Tan[(-c + Pi/2 - d*x)/2])^2 + Sec[(-c + Pi/2 - d*x)/2]^2/(2*(1 + Tan[(-c + Pi/2 - d*x)/2]))))/(2*n*(Sec[(-c + Pi/2 - d*x)/2]^2)^n)))","C",0
146,1,4206,85,26.563355,"\int \csc ^2(c+d x) (a+a \sin (c+d x))^n \, dx","Integrate[Csc[c + d*x]^2*(a + a*Sin[c + d*x])^n,x]","\text{Result too large to show}","-\frac{2^{n+\frac{1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n F_1\left(\frac{1}{2};2,\frac{1}{2}-n;\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d}",1,"-((Csc[c + d*x]^2*(a + a*Sin[c + d*x])^n*(-(AppellF1[1 + 2*n, n, n, 2 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*(-1 + Tan[(-c + Pi/2 - d*x)/2])*((-I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n) - AppellF1[1 + 2*n, n, n, 2*(1 + n), (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*((-I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*(1 + Tan[(-c + Pi/2 - d*x)/2])))/(d*(1 + 2*n)*(Sec[(-c + Pi/2 - d*x)/2]^2)^n*(-1 + Tan[(-c + Pi/2 - d*x)/2])*(1 + Tan[(-c + Pi/2 - d*x)/2])*(-1/2*((Sec[(-c + Pi/2 - d*x)/2]^2)^(1 - n)*(-(AppellF1[1 + 2*n, n, n, 2 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*(-1 + Tan[(-c + Pi/2 - d*x)/2])*((-I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n) - AppellF1[1 + 2*n, n, n, 2*(1 + n), (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*((-I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*(1 + Tan[(-c + Pi/2 - d*x)/2])))/((1 + 2*n)*(-1 + Tan[(-c + Pi/2 - d*x)/2])*(1 + Tan[(-c + Pi/2 - d*x)/2])^2) - ((Sec[(-c + Pi/2 - d*x)/2]^2)^(1 - n)*(-(AppellF1[1 + 2*n, n, n, 2 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*(-1 + Tan[(-c + Pi/2 - d*x)/2])*((-I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n) - AppellF1[1 + 2*n, n, n, 2*(1 + n), (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*((-I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*(1 + Tan[(-c + Pi/2 - d*x)/2])))/(2*(1 + 2*n)*(-1 + Tan[(-c + Pi/2 - d*x)/2])^2*(1 + Tan[(-c + Pi/2 - d*x)/2])) - (n*Tan[(-c + Pi/2 - d*x)/2]*(-(AppellF1[1 + 2*n, n, n, 2 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*(-1 + Tan[(-c + Pi/2 - d*x)/2])*((-I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n) - AppellF1[1 + 2*n, n, n, 2*(1 + n), (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*((-I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*(1 + Tan[(-c + Pi/2 - d*x)/2])))/((1 + 2*n)*(Sec[(-c + Pi/2 - d*x)/2]^2)^n*(-1 + Tan[(-c + Pi/2 - d*x)/2])*(1 + Tan[(-c + Pi/2 - d*x)/2])) + (-1/2*(AppellF1[1 + 2*n, n, n, 2*(1 + n), (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*Sec[(-c + Pi/2 - d*x)/2]^2*((-I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n) - (AppellF1[1 + 2*n, n, n, 2 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*Sec[(-c + Pi/2 - d*x)/2]^2*((-I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n)/2 - (((1/4 - I/4)*n*(1 + 2*n)*AppellF1[2 + 2*n, n, 1 + n, 1 + 2*(1 + n), (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*Sec[(-c + Pi/2 - d*x)/2]^2)/((1 + n)*(-1 + Tan[(-c + Pi/2 - d*x)/2])^2) + ((1/4 + I/4)*n*(1 + 2*n)*AppellF1[2 + 2*n, 1 + n, n, 1 + 2*(1 + n), (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*Sec[(-c + Pi/2 - d*x)/2]^2)/((1 + n)*(-1 + Tan[(-c + Pi/2 - d*x)/2])^2))*((-I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*(1 + Tan[(-c + Pi/2 - d*x)/2]) - n*AppellF1[1 + 2*n, n, n, 2*(1 + n), (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*((-I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^(-1 + n)*((I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*(1 + Tan[(-c + Pi/2 - d*x)/2])*(Sec[(-c + Pi/2 - d*x)/2]^2/(2*(-1 + Tan[(-c + Pi/2 - d*x)/2])) - (Sec[(-c + Pi/2 - d*x)/2]^2*(-I + Tan[(-c + Pi/2 - d*x)/2]))/(2*(-1 + Tan[(-c + Pi/2 - d*x)/2])^2)) - n*AppellF1[1 + 2*n, n, n, 2*(1 + n), (-1 - I)/(-1 + Tan[(-c + Pi/2 - d*x)/2]), (-1 + I)/(-1 + Tan[(-c + Pi/2 - d*x)/2])]*((-I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(-1 + Tan[(-c + Pi/2 - d*x)/2]))^(-1 + n)*(1 + Tan[(-c + Pi/2 - d*x)/2])*(Sec[(-c + Pi/2 - d*x)/2]^2/(2*(-1 + Tan[(-c + Pi/2 - d*x)/2])) - (Sec[(-c + Pi/2 - d*x)/2]^2*(I + Tan[(-c + Pi/2 - d*x)/2]))/(2*(-1 + Tan[(-c + Pi/2 - d*x)/2])^2)) - (-1 + Tan[(-c + Pi/2 - d*x)/2])*((-I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*(((-1/2 - I/2)*n*(1 + 2*n)*AppellF1[2 + 2*n, n, 1 + n, 3 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*Sec[(-c + Pi/2 - d*x)/2]^2)/((2 + 2*n)*(1 + Tan[(-c + Pi/2 - d*x)/2])^2) - ((1/2 - I/2)*n*(1 + 2*n)*AppellF1[2 + 2*n, 1 + n, n, 3 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*Sec[(-c + Pi/2 - d*x)/2]^2)/((2 + 2*n)*(1 + Tan[(-c + Pi/2 - d*x)/2])^2)) - n*AppellF1[1 + 2*n, n, n, 2 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*(-1 + Tan[(-c + Pi/2 - d*x)/2])*((-I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^(-1 + n)*((I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*(-1/2*(Sec[(-c + Pi/2 - d*x)/2]^2*(-I + Tan[(-c + Pi/2 - d*x)/2]))/(1 + Tan[(-c + Pi/2 - d*x)/2])^2 + Sec[(-c + Pi/2 - d*x)/2]^2/(2*(1 + Tan[(-c + Pi/2 - d*x)/2]))) - n*AppellF1[1 + 2*n, n, n, 2 + 2*n, (1 - I)/(1 + Tan[(-c + Pi/2 - d*x)/2]), (1 + I)/(1 + Tan[(-c + Pi/2 - d*x)/2])]*(-1 + Tan[(-c + Pi/2 - d*x)/2])*((-I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^n*((I + Tan[(-c + Pi/2 - d*x)/2])/(1 + Tan[(-c + Pi/2 - d*x)/2]))^(-1 + n)*(-1/2*(Sec[(-c + Pi/2 - d*x)/2]^2*(I + Tan[(-c + Pi/2 - d*x)/2]))/(1 + Tan[(-c + Pi/2 - d*x)/2])^2 + Sec[(-c + Pi/2 - d*x)/2]^2/(2*(1 + Tan[(-c + Pi/2 - d*x)/2]))))/((1 + 2*n)*(Sec[(-c + Pi/2 - d*x)/2]^2)^n*(-1 + Tan[(-c + Pi/2 - d*x)/2])*(1 + Tan[(-c + Pi/2 - d*x)/2])))))","C",0
147,1,88,58,0.151475,"\int (1+\sin (c+d x))^n \, dx","Integrate[(1 + Sin[c + d*x])^n,x]","\frac{\sqrt{2} \cos (c+d x) (\sin (c+d x)+1)^n \, _2F_1\left(\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};\frac{1}{4} \cos ^2(c+d x) \csc ^2\left(\frac{1}{4} (2 c+2 d x-\pi )\right)\right)}{(2 d n+d) \sqrt{1-\sin (c+d x)}}","-\frac{2^{n+\frac{1}{2}} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d \sqrt{\sin (c+d x)+1}}",1,"(Sqrt[2]*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 + n, 3/2 + n, (Cos[c + d*x]^2*Csc[(2*c - Pi + 2*d*x)/4]^2)/4]*(1 + Sin[c + d*x])^n)/((d + 2*d*n)*Sqrt[1 - Sin[c + d*x]])","A",1
148,1,90,57,0.1104711,"\int (1-\sin (c+d x))^n \, dx","Integrate[(1 - Sin[c + d*x])^n,x]","\frac{\cos (c+d x) (1-\sin (c+d x))^n \cos ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)^{-n-\frac{1}{2}} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{4} \cos ^2(c+d x) \csc ^2\left(\frac{1}{4} (2 c+2 d x-\pi )\right)\right)}{d}","\frac{2^{n+\frac{1}{2}} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d \sqrt{1-\sin (c+d x)}}",1,"(Cos[c + d*x]*(Cos[(2*c + Pi + 2*d*x)/4]^2)^(-1/2 - n)*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (Cos[c + d*x]^2*Csc[(2*c - Pi + 2*d*x)/4]^2)/4]*(1 - Sin[c + d*x])^n)/d","A",0
149,1,76,77,0.1652458,"\int \sin ^3(e+f x) (a+b \sin (e+f x)) \, dx","Integrate[Sin[e + f*x]^3*(a + b*Sin[e + f*x]),x]","-\frac{3 a \cos (e+f x)}{4 f}+\frac{a \cos (3 (e+f x))}{12 f}+\frac{3 b (e+f x)}{8 f}-\frac{b \sin (2 (e+f x))}{4 f}+\frac{b \sin (4 (e+f x))}{32 f}","\frac{a \cos ^3(e+f x)}{3 f}-\frac{a \cos (e+f x)}{f}-\frac{b \sin ^3(e+f x) \cos (e+f x)}{4 f}-\frac{3 b \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3 b x}{8}",1,"(3*b*(e + f*x))/(8*f) - (3*a*Cos[e + f*x])/(4*f) + (a*Cos[3*(e + f*x)])/(12*f) - (b*Sin[2*(e + f*x)])/(4*f) + (b*Sin[4*(e + f*x)])/(32*f)","A",1
150,1,60,55,0.0603854,"\int \sin ^2(e+f x) (a+b \sin (e+f x)) \, dx","Integrate[Sin[e + f*x]^2*(a + b*Sin[e + f*x]),x]","\frac{a (e+f x)}{2 f}-\frac{a \sin (2 (e+f x))}{4 f}-\frac{3 b \cos (e+f x)}{4 f}+\frac{b \cos (3 (e+f x))}{12 f}","-\frac{a \sin (e+f x) \cos (e+f x)}{2 f}+\frac{a x}{2}+\frac{b \cos ^3(e+f x)}{3 f}-\frac{b \cos (e+f x)}{f}",1,"(a*(e + f*x))/(2*f) - (3*b*Cos[e + f*x])/(4*f) + (b*Cos[3*(e + f*x)])/(12*f) - (a*Sin[2*(e + f*x)])/(4*f)","A",1
151,1,35,39,0.0963895,"\int \sin (e+f x) (a+b \sin (e+f x)) \, dx","Integrate[Sin[e + f*x]*(a + b*Sin[e + f*x]),x]","-\frac{4 a \cos (e+f x)+b (\sin (2 (e+f x))-2 (e+f x))}{4 f}","-\frac{a \cos (e+f x)}{f}-\frac{b \sin (e+f x) \cos (e+f x)}{2 f}+\frac{b x}{2}",1,"-1/4*(4*a*Cos[e + f*x] + b*(-2*(e + f*x) + Sin[2*(e + f*x)]))/f","A",1
152,1,27,16,0.0063282,"\int (a+b \sin (e+f x)) \, dx","Integrate[a + b*Sin[e + f*x],x]","a x+\frac{b \sin (e) \sin (f x)}{f}-\frac{b \cos (e) \cos (f x)}{f}","a x-\frac{b \cos (e+f x)}{f}",1,"a*x - (b*Cos[e]*Cos[f*x])/f + (b*Sin[e]*Sin[f*x])/f","A",1
153,1,43,17,0.0151591,"\int \csc (e+f x) (a+b \sin (e+f x)) \, dx","Integrate[Csc[e + f*x]*(a + b*Sin[e + f*x]),x]","\frac{a \log \left(\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{f}-\frac{a \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{f}+b x","b x-\frac{a \tanh ^{-1}(\cos (e+f x))}{f}",1,"b*x - (a*Log[Cos[e/2 + (f*x)/2]])/f + (a*Log[Sin[e/2 + (f*x)/2]])/f","B",1
154,1,52,26,0.024779,"\int \csc ^2(e+f x) (a+b \sin (e+f x)) \, dx","Integrate[Csc[e + f*x]^2*(a + b*Sin[e + f*x]),x]","-\frac{a \cot (e+f x)}{f}+\frac{b \log \left(\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{f}-\frac{b \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{f}","-\frac{a \cot (e+f x)}{f}-\frac{b \tanh ^{-1}(\cos (e+f x))}{f}",1,"-((a*Cot[e + f*x])/f) - (b*Log[Cos[e/2 + (f*x)/2]])/f + (b*Log[Sin[e/2 + (f*x)/2]])/f","A",1
155,1,91,48,0.0316063,"\int \csc ^3(e+f x) (a+b \sin (e+f x)) \, dx","Integrate[Csc[e + f*x]^3*(a + b*Sin[e + f*x]),x]","-\frac{a \csc ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{a \sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{a \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}-\frac{a \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}-\frac{b \cot (e+f x)}{f}","-\frac{a \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a \cot (e+f x) \csc (e+f x)}{2 f}-\frac{b \cot (e+f x)}{f}",1,"-((b*Cot[e + f*x])/f) - (a*Csc[(e + f*x)/2]^2)/(8*f) - (a*Log[Cos[(e + f*x)/2]])/(2*f) + (a*Log[Sin[(e + f*x)/2]])/(2*f) + (a*Sec[(e + f*x)/2]^2)/(8*f)","A",1
156,1,115,64,0.0291932,"\int \csc ^4(e+f x) (a+b \sin (e+f x)) \, dx","Integrate[Csc[e + f*x]^4*(a + b*Sin[e + f*x]),x]","-\frac{2 a \cot (e+f x)}{3 f}-\frac{a \cot (e+f x) \csc ^2(e+f x)}{3 f}-\frac{b \csc ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{b \sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{b \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}-\frac{b \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}","-\frac{a \cot ^3(e+f x)}{3 f}-\frac{a \cot (e+f x)}{f}-\frac{b \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{b \cot (e+f x) \csc (e+f x)}{2 f}",1,"(-2*a*Cot[e + f*x])/(3*f) - (b*Csc[(e + f*x)/2]^2)/(8*f) - (a*Cot[e + f*x]*Csc[e + f*x]^2)/(3*f) - (b*Log[Cos[(e + f*x)/2]])/(2*f) + (b*Log[Sin[(e + f*x)/2]])/(2*f) + (b*Sec[(e + f*x)/2]^2)/(8*f)","A",1
157,1,91,112,0.3504959,"\int \sin ^3(e+f x) (a+b \sin (e+f x))^2 \, dx","Integrate[Sin[e + f*x]^3*(a + b*Sin[e + f*x])^2,x]","\frac{-30 \left(6 a^2+5 b^2\right) \cos (e+f x)+5 \left(4 a^2+5 b^2\right) \cos (3 (e+f x))-3 b (b \cos (5 (e+f x))-5 a (12 (e+f x)-8 \sin (2 (e+f x))+\sin (4 (e+f x))))}{240 f}","\frac{\left(a^2+2 b^2\right) \cos ^3(e+f x)}{3 f}-\frac{\left(a^2+b^2\right) \cos (e+f x)}{f}-\frac{a b \sin ^3(e+f x) \cos (e+f x)}{2 f}-\frac{3 a b \sin (e+f x) \cos (e+f x)}{4 f}+\frac{3 a b x}{4}-\frac{b^2 \cos ^5(e+f x)}{5 f}",1,"(-30*(6*a^2 + 5*b^2)*Cos[e + f*x] + 5*(4*a^2 + 5*b^2)*Cos[3*(e + f*x)] - 3*b*(b*Cos[5*(e + f*x)] - 5*a*(12*(e + f*x) - 8*Sin[2*(e + f*x)] + Sin[4*(e + f*x)])))/(240*f)","A",1
158,1,117,101,0.1620429,"\int \sin ^2(e+f x) (a+b \sin (e+f x))^2 \, dx","Integrate[Sin[e + f*x]^2*(a + b*Sin[e + f*x])^2,x]","\frac{a^2 (e+f x)}{2 f}-\frac{a^2 \sin (2 (e+f x))}{4 f}-\frac{3 a b \cos (e+f x)}{2 f}+\frac{a b \cos (3 (e+f x))}{6 f}+\frac{3 b^2 (e+f x)}{8 f}-\frac{b^2 \sin (2 (e+f x))}{4 f}+\frac{b^2 \sin (4 (e+f x))}{32 f}","-\frac{\left(4 a^2+3 b^2\right) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} x \left(4 a^2+3 b^2\right)+\frac{2 a b \cos ^3(e+f x)}{3 f}-\frac{2 a b \cos (e+f x)}{f}-\frac{b^2 \sin ^3(e+f x) \cos (e+f x)}{4 f}",1,"(a^2*(e + f*x))/(2*f) + (3*b^2*(e + f*x))/(8*f) - (3*a*b*Cos[e + f*x])/(2*f) + (a*b*Cos[3*(e + f*x)])/(6*f) - (a^2*Sin[2*(e + f*x)])/(4*f) - (b^2*Sin[2*(e + f*x)])/(4*f) + (b^2*Sin[4*(e + f*x)])/(32*f)","A",1
159,1,59,71,0.2203496,"\int \sin (e+f x) (a+b \sin (e+f x))^2 \, dx","Integrate[Sin[e + f*x]*(a + b*Sin[e + f*x])^2,x]","\frac{b (12 a (e+f x)-6 a \sin (2 (e+f x))+b \cos (3 (e+f x)))-3 \left(4 a^2+3 b^2\right) \cos (e+f x)}{12 f}","-\frac{2 \left(a^2+b^2\right) \cos (e+f x)}{3 f}-\frac{\cos (e+f x) (a+b \sin (e+f x))^2}{3 f}-\frac{a b \sin (e+f x) \cos (e+f x)}{3 f}+a b x",1,"(-3*(4*a^2 + 3*b^2)*Cos[e + f*x] + b*(12*a*(e + f*x) + b*Cos[3*(e + f*x)] - 6*a*Sin[2*(e + f*x)]))/(12*f)","A",1
160,1,46,50,0.1040802,"\int (a+b \sin (e+f x))^2 \, dx","Integrate[(a + b*Sin[e + f*x])^2,x]","-\frac{-2 \left(2 a^2+b^2\right) (e+f x)+8 a b \cos (e+f x)+b^2 \sin (2 (e+f x))}{4 f}","\frac{1}{2} x \left(2 a^2+b^2\right)-\frac{2 a b \cos (e+f x)}{f}-\frac{b^2 \sin (e+f x) \cos (e+f x)}{2 f}",1,"-1/4*(-2*(2*a^2 + b^2)*(e + f*x) + 8*a*b*Cos[e + f*x] + b^2*Sin[2*(e + f*x)])/f","A",1
161,1,76,35,0.0239881,"\int \csc (e+f x) (a+b \sin (e+f x))^2 \, dx","Integrate[Csc[e + f*x]*(a + b*Sin[e + f*x])^2,x]","\frac{a^2 \log \left(\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{f}-\frac{a^2 \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{f}+2 a b x+\frac{b^2 \sin (e) \sin (f x)}{f}-\frac{b^2 \cos (e) \cos (f x)}{f}","-\frac{a^2 \tanh ^{-1}(\cos (e+f x))}{f}+2 a b x-\frac{b^2 \cos (e+f x)}{f}",1,"2*a*b*x - (b^2*Cos[e]*Cos[f*x])/f - (a^2*Log[Cos[e/2 + (f*x)/2]])/f + (a^2*Log[Sin[e/2 + (f*x)/2]])/f + (b^2*Sin[e]*Sin[f*x])/f","B",1
162,1,76,34,0.2385403,"\int \csc ^2(e+f x) (a+b \sin (e+f x))^2 \, dx","Integrate[Csc[e + f*x]^2*(a + b*Sin[e + f*x])^2,x]","\frac{a^2 \tan \left(\frac{1}{2} (e+f x)\right)+a^2 \left(-\cot \left(\frac{1}{2} (e+f x)\right)\right)+2 b \left(2 a \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-2 a \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)+b e+b f x\right)}{2 f}","-\frac{a^2 \cot (e+f x)}{f}-\frac{2 a b \tanh ^{-1}(\cos (e+f x))}{f}+b^2 x",1,"(-(a^2*Cot[(e + f*x)/2]) + 2*b*(b*e + b*f*x - 2*a*Log[Cos[(e + f*x)/2]] + 2*a*Log[Sin[(e + f*x)/2]]) + a^2*Tan[(e + f*x)/2])/(2*f)","B",1
163,1,133,59,0.4736023,"\int \csc ^3(e+f x) (a+b \sin (e+f x))^2 \, dx","Integrate[Csc[e + f*x]^3*(a + b*Sin[e + f*x])^2,x]","\frac{a^2 \left(-\csc ^2\left(\frac{1}{2} (e+f x)\right)\right)+a^2 \sec ^2\left(\frac{1}{2} (e+f x)\right)+4 a^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-4 a^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)+8 a b \tan \left(\frac{1}{2} (e+f x)\right)-8 a b \cot \left(\frac{1}{2} (e+f x)\right)+8 b^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-8 b^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{8 f}","-\frac{\left(a^2+2 b^2\right) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a^2 \cot (e+f x) \csc (e+f x)}{2 f}-\frac{2 a b \cot (e+f x)}{f}",1,"(-8*a*b*Cot[(e + f*x)/2] - a^2*Csc[(e + f*x)/2]^2 - 4*a^2*Log[Cos[(e + f*x)/2]] - 8*b^2*Log[Cos[(e + f*x)/2]] + 4*a^2*Log[Sin[(e + f*x)/2]] + 8*b^2*Log[Sin[(e + f*x)/2]] + a^2*Sec[(e + f*x)/2]^2 + 8*a*b*Tan[(e + f*x)/2])/(8*f)","B",1
164,1,132,82,0.0417658,"\int \csc ^4(e+f x) (a+b \sin (e+f x))^2 \, dx","Integrate[Csc[e + f*x]^4*(a + b*Sin[e + f*x])^2,x]","-\frac{2 a^2 \cot (e+f x)}{3 f}-\frac{a^2 \cot (e+f x) \csc ^2(e+f x)}{3 f}-\frac{a b \csc ^2\left(\frac{1}{2} (e+f x)\right)}{4 f}+\frac{a b \sec ^2\left(\frac{1}{2} (e+f x)\right)}{4 f}+\frac{a b \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f}-\frac{a b \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f}-\frac{b^2 \cot (e+f x)}{f}","-\frac{\left(2 a^2+3 b^2\right) \cot (e+f x)}{3 f}-\frac{a^2 \cot (e+f x) \csc ^2(e+f x)}{3 f}-\frac{a b \tanh ^{-1}(\cos (e+f x))}{f}-\frac{a b \cot (e+f x) \csc (e+f x)}{f}",1,"(-2*a^2*Cot[e + f*x])/(3*f) - (b^2*Cot[e + f*x])/f - (a*b*Csc[(e + f*x)/2]^2)/(4*f) - (a^2*Cot[e + f*x]*Csc[e + f*x]^2)/(3*f) - (a*b*Log[Cos[(e + f*x)/2]])/f + (a*b*Log[Sin[(e + f*x)/2]])/f + (a*b*Sec[(e + f*x)/2]^2)/(4*f)","A",1
165,1,255,110,0.0422247,"\int \csc ^5(e+f x) (a+b \sin (e+f x))^2 \, dx","Integrate[Csc[e + f*x]^5*(a + b*Sin[e + f*x])^2,x]","-\frac{a^2 \csc ^4\left(\frac{1}{2} (e+f x)\right)}{64 f}-\frac{3 a^2 \csc ^2\left(\frac{1}{2} (e+f x)\right)}{32 f}+\frac{a^2 \sec ^4\left(\frac{1}{2} (e+f x)\right)}{64 f}+\frac{3 a^2 \sec ^2\left(\frac{1}{2} (e+f x)\right)}{32 f}+\frac{3 a^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{8 f}-\frac{3 a^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{8 f}-\frac{4 a b \cot (e+f x)}{3 f}-\frac{2 a b \cot (e+f x) \csc ^2(e+f x)}{3 f}-\frac{b^2 \csc ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{b^2 \sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{b^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}-\frac{b^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}","-\frac{\left(3 a^2+4 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{\left(3 a^2+4 b^2\right) \cot (e+f x) \csc (e+f x)}{8 f}-\frac{a^2 \cot (e+f x) \csc ^3(e+f x)}{4 f}-\frac{2 a b \cot ^3(e+f x)}{3 f}-\frac{2 a b \cot (e+f x)}{f}",1,"(-4*a*b*Cot[e + f*x])/(3*f) - (3*a^2*Csc[(e + f*x)/2]^2)/(32*f) - (b^2*Csc[(e + f*x)/2]^2)/(8*f) - (a^2*Csc[(e + f*x)/2]^4)/(64*f) - (2*a*b*Cot[e + f*x]*Csc[e + f*x]^2)/(3*f) - (3*a^2*Log[Cos[(e + f*x)/2]])/(8*f) - (b^2*Log[Cos[(e + f*x)/2]])/(2*f) + (3*a^2*Log[Sin[(e + f*x)/2]])/(8*f) + (b^2*Log[Sin[(e + f*x)/2]])/(2*f) + (3*a^2*Sec[(e + f*x)/2]^2)/(32*f) + (b^2*Sec[(e + f*x)/2]^2)/(8*f) + (a^2*Sec[(e + f*x)/2]^4)/(64*f)","B",0
166,1,147,171,0.7358685,"\int \sin ^3(e+f x) (a+b \sin (e+f x))^3 \, dx","Integrate[Sin[e + f*x]^3*(a + b*Sin[e + f*x])^3,x]","\frac{20 \left(4 a^3+15 a b^2\right) \cos (3 (e+f x))-360 a \left(2 a^2+5 b^2\right) \cos (e+f x)+b \left(5 \left(-9 \left(16 a^2+5 b^2\right) \sin (2 (e+f x))+9 \left(2 a^2+b^2\right) \sin (4 (e+f x))+216 a^2 e+216 a^2 f x-b^2 \sin (6 (e+f x))+60 b^2 e+60 b^2 f x\right)-36 a b \cos (5 (e+f x))\right)}{960 f}","\frac{a \left(a^2+6 b^2\right) \cos ^3(e+f x)}{3 f}-\frac{a \left(a^2+3 b^2\right) \cos (e+f x)}{f}-\frac{b \left(18 a^2+5 b^2\right) \sin ^3(e+f x) \cos (e+f x)}{24 f}-\frac{b \left(18 a^2+5 b^2\right) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} b x \left(18 a^2+5 b^2\right)-\frac{3 a b^2 \cos ^5(e+f x)}{5 f}-\frac{b^3 \sin ^5(e+f x) \cos (e+f x)}{6 f}",1,"(-360*a*(2*a^2 + 5*b^2)*Cos[e + f*x] + 20*(4*a^3 + 15*a*b^2)*Cos[3*(e + f*x)] + b*(-36*a*b*Cos[5*(e + f*x)] + 5*(216*a^2*e + 60*b^2*e + 216*a^2*f*x + 60*b^2*f*x - 9*(16*a^2 + 5*b^2)*Sin[2*(e + f*x)] + 9*(2*a^2 + b^2)*Sin[4*(e + f*x)] - b^2*Sin[6*(e + f*x)])))/(960*f)","A",1
167,1,117,160,0.6506173,"\int \sin ^2(e+f x) (a+b \sin (e+f x))^3 \, dx","Integrate[Sin[e + f*x]^2*(a + b*Sin[e + f*x])^3,x]","\frac{10 \left(12 a^2 b+5 b^3\right) \cos (3 (e+f x))+15 a \left(4 \left(4 a^2+9 b^2\right) (e+f x)-8 \left(a^2+3 b^2\right) \sin (2 (e+f x))+3 b^2 \sin (4 (e+f x))\right)-60 b \left(18 a^2+5 b^2\right) \cos (e+f x)-6 b^3 \cos (5 (e+f x))}{480 f}","\frac{b \left(15 a^2+4 b^2\right) \cos ^3(e+f x)}{15 f}-\frac{b \left(15 a^2+4 b^2\right) \cos (e+f x)}{5 f}-\frac{a \left(4 a^2+9 b^2\right) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a x \left(4 a^2+9 b^2\right)-\frac{11 a b^2 \sin ^3(e+f x) \cos (e+f x)}{20 f}-\frac{b^2 \sin ^3(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{5 f}",1,"(-60*b*(18*a^2 + 5*b^2)*Cos[e + f*x] + 10*(12*a^2*b + 5*b^3)*Cos[3*(e + f*x)] - 6*b^3*Cos[5*(e + f*x)] + 15*a*(4*(4*a^2 + 9*b^2)*(e + f*x) - 8*(a^2 + 3*b^2)*Sin[2*(e + f*x)] + 3*b^2*Sin[4*(e + f*x)]))/(480*f)","A",1
168,1,100,121,0.3532293,"\int \sin (e+f x) (a+b \sin (e+f x))^3 \, dx","Integrate[Sin[e + f*x]*(a + b*Sin[e + f*x])^3,x]","\frac{b \left(-8 \left(3 a^2+b^2\right) \sin (2 (e+f x))+48 a^2 e+48 a^2 f x+8 a b \cos (3 (e+f x))+b^2 \sin (4 (e+f x))+12 b^2 e+12 b^2 f x\right)-8 a \left(4 a^2+9 b^2\right) \cos (e+f x)}{32 f}","-\frac{a \left(a^2+4 b^2\right) \cos (e+f x)}{2 f}-\frac{b \left(2 a^2+3 b^2\right) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} b x \left(4 a^2+b^2\right)-\frac{\cos (e+f x) (a+b \sin (e+f x))^3}{4 f}-\frac{a \cos (e+f x) (a+b \sin (e+f x))^2}{4 f}",1,"(-8*a*(4*a^2 + 9*b^2)*Cos[e + f*x] + b*(48*a^2*e + 12*b^2*e + 48*a^2*f*x + 12*b^2*f*x + 8*a*b*Cos[3*(e + f*x)] - 8*(3*a^2 + b^2)*Sin[2*(e + f*x)] + b^2*Sin[4*(e + f*x)]))/(32*f)","A",1
169,1,71,90,0.1698112,"\int (a+b \sin (e+f x))^3 \, dx","Integrate[(a + b*Sin[e + f*x])^3,x]","\frac{6 a \left(2 a^2+3 b^2\right) (e+f x)-9 b \left(4 a^2+b^2\right) \cos (e+f x)-9 a b^2 \sin (2 (e+f x))+b^3 \cos (3 (e+f x))}{12 f}","-\frac{2 b \left(4 a^2+b^2\right) \cos (e+f x)}{3 f}+\frac{1}{2} a x \left(2 a^2+3 b^2\right)-\frac{5 a b^2 \sin (e+f x) \cos (e+f x)}{6 f}-\frac{b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}",1,"(6*a*(2*a^2 + 3*b^2)*(e + f*x) - 9*b*(4*a^2 + b^2)*Cos[e + f*x] + b^3*Cos[3*(e + f*x)] - 9*a*b^2*Sin[2*(e + f*x)])/(12*f)","A",1
170,1,81,74,0.1661592,"\int \csc (e+f x) (a+b \sin (e+f x))^3 \, dx","Integrate[Csc[e + f*x]*(a + b*Sin[e + f*x])^3,x]","-\frac{-4 a^3 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)+4 a^3 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)-2 b \left(6 a^2+b^2\right) (e+f x)+12 a b^2 \cos (e+f x)+b^3 \sin (2 (e+f x))}{4 f}","-\frac{a^3 \tanh ^{-1}(\cos (e+f x))}{f}+\frac{1}{2} b x \left(6 a^2+b^2\right)-\frac{5 a b^2 \cos (e+f x)}{2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 f}",1,"-1/4*(-2*b*(6*a^2 + b^2)*(e + f*x) + 12*a*b^2*Cos[e + f*x] + 4*a^3*Log[Cos[(e + f*x)/2]] - 4*a^3*Log[Sin[(e + f*x)/2]] + b^3*Sin[2*(e + f*x)])/f","A",1
171,1,87,68,0.5424573,"\int \csc ^2(e+f x) (a+b \sin (e+f x))^3 \, dx","Integrate[Csc[e + f*x]^2*(a + b*Sin[e + f*x])^3,x]","\frac{a^3 \tan \left(\frac{1}{2} (e+f x)\right)+a^3 \left(-\cot \left(\frac{1}{2} (e+f x)\right)\right)+6 a b \left(a \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-a \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)+b (e+f x)\right)-2 b^3 \cos (e+f x)}{2 f}","\frac{b \left(a^2-b^2\right) \cos (e+f x)}{f}-\frac{3 a^2 b \tanh ^{-1}(\cos (e+f x))}{f}-\frac{a^2 \cot (e+f x) (a+b \sin (e+f x))}{f}+3 a b^2 x",1,"(-2*b^3*Cos[e + f*x] - a^3*Cot[(e + f*x)/2] + 6*a*b*(b*(e + f*x) - a*Log[Cos[(e + f*x)/2]] + a*Log[Sin[(e + f*x)/2]]) + a^3*Tan[(e + f*x)/2])/(2*f)","A",1
172,1,152,79,0.6751676,"\int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx","Integrate[Csc[e + f*x]^3*(a + b*Sin[e + f*x])^3,x]","\frac{a^3 \left(-\csc ^2\left(\frac{1}{2} (e+f x)\right)\right)+a^3 \sec ^2\left(\frac{1}{2} (e+f x)\right)+4 a^3 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-4 a^3 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)+12 a^2 b \tan \left(\frac{1}{2} (e+f x)\right)-12 a^2 b \cot \left(\frac{1}{2} (e+f x)\right)+24 a b^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-24 a b^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)+8 b^3 e+8 b^3 f x}{8 f}","-\frac{a \left(a^2+6 b^2\right) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{5 a^2 b \cot (e+f x)}{2 f}-\frac{a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f}+b^3 x",1,"(8*b^3*e + 8*b^3*f*x - 12*a^2*b*Cot[(e + f*x)/2] - a^3*Csc[(e + f*x)/2]^2 - 4*a^3*Log[Cos[(e + f*x)/2]] - 24*a*b^2*Log[Cos[(e + f*x)/2]] + 4*a^3*Log[Sin[(e + f*x)/2]] + 24*a*b^2*Log[Sin[(e + f*x)/2]] + a^3*Sec[(e + f*x)/2]^2 + 12*a^2*b*Tan[(e + f*x)/2])/(8*f)","A",1
173,1,525,109,6.1984693,"\int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx","Integrate[Csc[e + f*x]^4*(a + b*Sin[e + f*x])^3,x]","\frac{\sin ^3(e+f x) \csc \left(\frac{1}{2} (e+f x)\right) \left(-2 a^3 \cos \left(\frac{1}{2} (e+f x)\right)-9 a b^2 \cos \left(\frac{1}{2} (e+f x)\right)\right) (a \csc (e+f x)+b)^3}{6 f (a+b \sin (e+f x))^3}+\frac{\sin ^3(e+f x) \sec \left(\frac{1}{2} (e+f x)\right) \left(2 a^3 \sin \left(\frac{1}{2} (e+f x)\right)+9 a b^2 \sin \left(\frac{1}{2} (e+f x)\right)\right) (a \csc (e+f x)+b)^3}{6 f (a+b \sin (e+f x))^3}-\frac{a^3 \sin ^3(e+f x) \cot \left(\frac{1}{2} (e+f x)\right) \csc ^2\left(\frac{1}{2} (e+f x)\right) (a \csc (e+f x)+b)^3}{24 f (a+b \sin (e+f x))^3}+\frac{a^3 \sin ^3(e+f x) \tan \left(\frac{1}{2} (e+f x)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right) (a \csc (e+f x)+b)^3}{24 f (a+b \sin (e+f x))^3}+\frac{\left(3 a^2 b+2 b^3\right) \sin ^3(e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right) (a \csc (e+f x)+b)^3}{2 f (a+b \sin (e+f x))^3}+\frac{\left(-3 a^2 b-2 b^3\right) \sin ^3(e+f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right) (a \csc (e+f x)+b)^3}{2 f (a+b \sin (e+f x))^3}-\frac{3 a^2 b \sin ^3(e+f x) \csc ^2\left(\frac{1}{2} (e+f x)\right) (a \csc (e+f x)+b)^3}{8 f (a+b \sin (e+f x))^3}+\frac{3 a^2 b \sin ^3(e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) (a \csc (e+f x)+b)^3}{8 f (a+b \sin (e+f x))^3}","-\frac{a \left(2 a^2+9 b^2\right) \cot (e+f x)}{3 f}-\frac{b \left(3 a^2+2 b^2\right) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac{a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}",1,"((-2*a^3*Cos[(e + f*x)/2] - 9*a*b^2*Cos[(e + f*x)/2])*Csc[(e + f*x)/2]*(b + a*Csc[e + f*x])^3*Sin[e + f*x]^3)/(6*f*(a + b*Sin[e + f*x])^3) - (3*a^2*b*Csc[(e + f*x)/2]^2*(b + a*Csc[e + f*x])^3*Sin[e + f*x]^3)/(8*f*(a + b*Sin[e + f*x])^3) - (a^3*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2*(b + a*Csc[e + f*x])^3*Sin[e + f*x]^3)/(24*f*(a + b*Sin[e + f*x])^3) + ((-3*a^2*b - 2*b^3)*(b + a*Csc[e + f*x])^3*Log[Cos[(e + f*x)/2]]*Sin[e + f*x]^3)/(2*f*(a + b*Sin[e + f*x])^3) + ((3*a^2*b + 2*b^3)*(b + a*Csc[e + f*x])^3*Log[Sin[(e + f*x)/2]]*Sin[e + f*x]^3)/(2*f*(a + b*Sin[e + f*x])^3) + (3*a^2*b*(b + a*Csc[e + f*x])^3*Sec[(e + f*x)/2]^2*Sin[e + f*x]^3)/(8*f*(a + b*Sin[e + f*x])^3) + ((b + a*Csc[e + f*x])^3*Sec[(e + f*x)/2]*(2*a^3*Sin[(e + f*x)/2] + 9*a*b^2*Sin[(e + f*x)/2])*Sin[e + f*x]^3)/(6*f*(a + b*Sin[e + f*x])^3) + (a^3*(b + a*Csc[e + f*x])^3*Sec[(e + f*x)/2]^2*Sin[e + f*x]^3*Tan[(e + f*x)/2])/(24*f*(a + b*Sin[e + f*x])^3)","B",1
174,1,322,134,6.1809411,"\int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx","Integrate[Csc[e + f*x]^5*(a + b*Sin[e + f*x])^3,x]","-\frac{3 \left(a^3+4 a b^2\right) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{32 f}+\frac{3 \left(a^3+4 a b^2\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{32 f}+\frac{3 \left(a^3+4 a b^2\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{8 f}-\frac{3 \left(a^3+4 a b^2\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{8 f}-\frac{a^3 \csc ^4\left(\frac{1}{2} (e+f x)\right)}{64 f}+\frac{a^3 \sec ^4\left(\frac{1}{2} (e+f x)\right)}{64 f}+\frac{\csc \left(\frac{1}{2} (e+f x)\right) \left(b^3 \left(-\cos \left(\frac{1}{2} (e+f x)\right)\right)-2 a^2 b \cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}+\frac{\sec \left(\frac{1}{2} (e+f x)\right) \left(2 a^2 b \sin \left(\frac{1}{2} (e+f x)\right)+b^3 \sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}-\frac{a^2 b \cot \left(\frac{1}{2} (e+f x)\right) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{a^2 b \tan \left(\frac{1}{2} (e+f x)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}","-\frac{b \left(2 a^2+b^2\right) \cot (e+f x)}{f}-\frac{3 a \left(a^2+4 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{3 a \left(a^2+4 b^2\right) \cot (e+f x) \csc (e+f x)}{8 f}-\frac{3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{4 f}-\frac{a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}",1,"((-2*a^2*b*Cos[(e + f*x)/2] - b^3*Cos[(e + f*x)/2])*Csc[(e + f*x)/2])/(2*f) - (3*(a^3 + 4*a*b^2)*Csc[(e + f*x)/2]^2)/(32*f) - (a^2*b*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2)/(8*f) - (a^3*Csc[(e + f*x)/2]^4)/(64*f) - (3*(a^3 + 4*a*b^2)*Log[Cos[(e + f*x)/2]])/(8*f) + (3*(a^3 + 4*a*b^2)*Log[Sin[(e + f*x)/2]])/(8*f) + (3*(a^3 + 4*a*b^2)*Sec[(e + f*x)/2]^2)/(32*f) + (a^3*Sec[(e + f*x)/2]^4)/(64*f) + (Sec[(e + f*x)/2]*(2*a^2*b*Sin[(e + f*x)/2] + b^3*Sin[(e + f*x)/2]))/(2*f) + (a^2*b*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(8*f)","B",0
175,1,106,137,0.3674379,"\int (a+b \sin (e+f x))^4 \, dx","Integrate[(a + b*Sin[e + f*x])^4,x]","\frac{-96 a b \left(4 a^2+3 b^2\right) \cos (e+f x)+3 \left(-8 \left(6 a^2 b^2+b^4\right) \sin (2 (e+f x))+4 \left(8 a^4+24 a^2 b^2+3 b^4\right) (e+f x)+b^4 \sin (4 (e+f x))\right)+32 a b^3 \cos (3 (e+f x))}{96 f}","-\frac{a b \left(19 a^2+16 b^2\right) \cos (e+f x)}{6 f}-\frac{b^2 \left(26 a^2+9 b^2\right) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} x \left(8 a^4+24 a^2 b^2+3 b^4\right)-\frac{b \cos (e+f x) (a+b \sin (e+f x))^3}{4 f}-\frac{7 a b \cos (e+f x) (a+b \sin (e+f x))^2}{12 f}",1,"(-96*a*b*(4*a^2 + 3*b^2)*Cos[e + f*x] + 32*a*b^3*Cos[3*(e + f*x)] + 3*(4*(8*a^4 + 24*a^2*b^2 + 3*b^4)*(e + f*x) - 8*(6*a^2*b^2 + b^4)*Sin[2*(e + f*x)] + b^4*Sin[4*(e + f*x)]))/(96*f)","A",1
176,1,98,110,0.2597849,"\int \frac{\sin ^4(x)}{a+b \sin (x)} \, dx","Integrate[Sin[x]^4/(a + b*Sin[x]),x]","\frac{-6 a x \left(2 a^2+b^2\right)-3 b \left(4 a^2+3 b^2\right) \cos (x)+\frac{24 a^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+3 a b^2 \sin (2 x)+b^3 \cos (3 x)}{12 b^4}","-\frac{a x \left(2 a^2+b^2\right)}{2 b^4}-\frac{\left(3 a^2+2 b^2\right) \cos (x)}{3 b^3}+\frac{2 a^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 \sqrt{a^2-b^2}}+\frac{a \sin (x) \cos (x)}{2 b^2}-\frac{\sin ^2(x) \cos (x)}{3 b}",1,"(-6*a*(2*a^2 + b^2)*x + (24*a^4*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 3*b*(4*a^2 + 3*b^2)*Cos[x] + b^3*Cos[3*x] + 3*a*b^2*Sin[2*x])/(12*b^4)","A",1
177,1,78,82,0.1107764,"\int \frac{\sin ^3(x)}{a+b \sin (x)} \, dx","Integrate[Sin[x]^3/(a + b*Sin[x]),x]","\frac{4 a^2 x-\frac{8 a^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+4 a b \cos (x)+2 b^2 x-b^2 \sin (2 x)}{4 b^3}","\frac{x \left(2 a^2+b^2\right)}{2 b^3}-\frac{2 a^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 \sqrt{a^2-b^2}}+\frac{a \cos (x)}{b^2}-\frac{\sin (x) \cos (x)}{2 b}",1,"(4*a^2*x + 2*b^2*x - (8*a^3*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 4*a*b*Cos[x] - b^2*Sin[2*x])/(4*b^3)","A",1
178,1,56,61,0.0910295,"\int \frac{\sin ^2(x)}{a+b \sin (x)} \, dx","Integrate[Sin[x]^2/(a + b*Sin[x]),x]","-\frac{-\frac{2 a^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+a x+b \cos (x)}{b^2}","\frac{2 a^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 \sqrt{a^2-b^2}}-\frac{a x}{b^2}-\frac{\cos (x)}{b}",1,"-((a*x - (2*a^2*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + b*Cos[x])/b^2)","A",1
179,1,47,50,0.0416744,"\int \frac{\sin (x)}{a+b \sin (x)} \, dx","Integrate[Sin[x]/(a + b*Sin[x]),x]","\frac{x-\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}}{b}","\frac{x}{b}-\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b \sqrt{a^2-b^2}}",1,"(x - (2*a*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2])/b","A",1
180,1,40,40,0.0217563,"\int \frac{1}{a+b \sin (x)} \, dx","Integrate[(a + b*Sin[x])^(-1),x]","\frac{2 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}","\frac{2 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}",1,"(2*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2]","A",1
181,1,62,53,0.0552263,"\int \frac{\csc (x)}{a+b \sin (x)} \, dx","Integrate[Csc[x]/(a + b*Sin[x]),x]","\frac{-\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)}{a}","-\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a \sqrt{a^2-b^2}}-\frac{\tanh ^{-1}(\cos (x))}{a}",1,"((-2*b*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - Log[Cos[x/2]] + Log[Sin[x/2]])/a","A",1
182,1,91,62,0.2480788,"\int \frac{\csc ^2(x)}{a+b \sin (x)} \, dx","Integrate[Csc[x]^2/(a + b*Sin[x]),x]","\frac{\csc \left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) \left(\frac{2 b^2 \sin (x) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-a \cos (x)+b \sin (x) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)\right)}{2 a^2}","\frac{2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 \sqrt{a^2-b^2}}+\frac{b \tanh ^{-1}(\cos (x))}{a^2}-\frac{\cot (x)}{a}",1,"(Csc[x/2]*Sec[x/2]*(-(a*Cos[x]) + (2*b^2*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]]*Sin[x])/Sqrt[a^2 - b^2] + b*(Log[Cos[x/2]] - Log[Sin[x/2]])*Sin[x]))/(2*a^2)","A",1
183,1,144,84,0.5181627,"\int \frac{\csc ^3(x)}{a+b \sin (x)} \, dx","Integrate[Csc[x]^3/(a + b*Sin[x]),x]","\frac{-\frac{16 b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-a^2 \csc ^2\left(\frac{x}{2}\right)+a^2 \sec ^2\left(\frac{x}{2}\right)+4 a^2 \log \left(\sin \left(\frac{x}{2}\right)\right)-4 a^2 \log \left(\cos \left(\frac{x}{2}\right)\right)-4 a b \tan \left(\frac{x}{2}\right)+4 a b \cot \left(\frac{x}{2}\right)+8 b^2 \log \left(\sin \left(\frac{x}{2}\right)\right)-8 b^2 \log \left(\cos \left(\frac{x}{2}\right)\right)}{8 a^3}","\frac{b \cot (x)}{a^2}-\frac{\left(a^2+2 b^2\right) \tanh ^{-1}(\cos (x))}{2 a^3}-\frac{2 b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 \sqrt{a^2-b^2}}-\frac{\cot (x) \csc (x)}{2 a}",1,"((-16*b^3*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 4*a*b*Cot[x/2] - a^2*Csc[x/2]^2 - 4*a^2*Log[Cos[x/2]] - 8*b^2*Log[Cos[x/2]] + 4*a^2*Log[Sin[x/2]] + 8*b^2*Log[Sin[x/2]] + a^2*Sec[x/2]^2 - 4*a*b*Tan[x/2])/(8*a^3)","A",1
184,1,125,112,1.6968184,"\int \frac{\csc ^4(x)}{a+b \sin (x)} \, dx","Integrate[Csc[x]^4/(a + b*Sin[x]),x]","\frac{a \left(2 a^2+3 b^2\right) \cos (3 x) \csc ^3(x)-3 a \cot (x) \csc (x) \left(\left(2 a^2+b^2\right) \csc (x)-2 a b\right)+6 b \left(a^2+2 b^2\right) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)+\frac{24 b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}}{12 a^4}","\frac{b \cot (x) \csc (x)}{2 a^2}+\frac{b \left(a^2+2 b^2\right) \tanh ^{-1}(\cos (x))}{2 a^4}+\frac{2 b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 \sqrt{a^2-b^2}}-\frac{\left(2 a^2+3 b^2\right) \cot (x)}{3 a^3}-\frac{\cot (x) \csc ^2(x)}{3 a}",1,"((24*b^4*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + a*(2*a^2 + 3*b^2)*Cos[3*x]*Csc[x]^3 - 3*a*Cot[x]*Csc[x]*(-2*a*b + (2*a^2 + b^2)*Csc[x]) + 6*b*(a^2 + 2*b^2)*(Log[Cos[x/2]] - Log[Sin[x/2]]))/(12*a^4)","A",1
185,1,115,169,0.6024613,"\int \frac{\sin ^4(x)}{(a+b \sin (x))^2} \, dx","Integrate[Sin[x]^4/(a + b*Sin[x])^2,x]","\frac{4 a b \cos (x) \left(\frac{a^3}{(a-b) (a+b) (a+b \sin (x))}+2\right)+12 a^2 x-\frac{8 a^3 \left(3 a^2-4 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+2 b^2 x-b^2 \sin (2 x)}{4 b^4}","\frac{a^2 \sin ^2(x) \cos (x)}{b \left(a^2-b^2\right) (a+b \sin (x))}-\frac{\left(3 a^2-b^2\right) \sin (x) \cos (x)}{2 b^2 \left(a^2-b^2\right)}+\frac{x \left(6 a^2+b^2\right)}{2 b^4}+\frac{a \left(3 a^2-2 b^2\right) \cos (x)}{b^3 \left(a^2-b^2\right)}-\frac{2 a^3 \left(3 a^2-4 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 \left(a^2-b^2\right)^{3/2}}",1,"(12*a^2*x + 2*b^2*x - (8*a^3*(3*a^2 - 4*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 4*a*b*Cos[x]*(2 + a^3/((a - b)*(a + b)*(a + b*Sin[x]))) - b^2*Sin[2*x])/(4*b^4)","A",1
186,1,94,124,0.4269831,"\int \frac{\sin ^3(x)}{(a+b \sin (x))^2} \, dx","Integrate[Sin[x]^3/(a + b*Sin[x])^2,x]","\frac{b \cos (x) \left(-\frac{a^3}{(a-b) (a+b) (a+b \sin (x))}-1\right)+\frac{2 a^2 \left(2 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-2 a x}{b^3}","-\frac{\left(2 a^2-b^2\right) \cos (x)}{b^2 \left(a^2-b^2\right)}+\frac{a^2 \sin (x) \cos (x)}{b \left(a^2-b^2\right) (a+b \sin (x))}+\frac{2 a^2 \left(2 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 \left(a^2-b^2\right)^{3/2}}-\frac{2 a x}{b^3}",1,"(-2*a*x + (2*a^2*(2*a^2 - 3*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + b*Cos[x]*(-1 - a^3/((a - b)*(a + b)*(a + b*Sin[x]))))/b^3","A",1
187,1,83,87,0.2272935,"\int \frac{\sin ^2(x)}{(a+b \sin (x))^2} \, dx","Integrate[Sin[x]^2/(a + b*Sin[x])^2,x]","\frac{-\frac{2 a \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{a^2 b \cos (x)}{(a-b) (a+b) (a+b \sin (x))}+x}{b^2}","-\frac{2 a \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 \left(a^2-b^2\right)^{3/2}}+\frac{a^2 \cos (x)}{b \left(a^2-b^2\right) (a+b \sin (x))}+\frac{x}{b^2}",1,"(x - (2*a*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (a^2*b*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])))/b^2","A",1
188,1,67,66,0.1092851,"\int \frac{\sin (x)}{(a+b \sin (x))^2} \, dx","Integrate[Sin[x]/(a + b*Sin[x])^2,x]","-\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-\frac{a \cos (x)}{(a-b) (a+b) (a+b \sin (x))}","-\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-\frac{a \cos (x)}{\left(a^2-b^2\right) (a+b \sin (x))}",1,"(-2*b*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - (a*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x]))","A",1
189,1,66,65,0.0982136,"\int \frac{1}{(a+b \sin (x))^2} \, dx","Integrate[(a + b*Sin[x])^(-2),x]","\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{b \cos (x)}{(a-b) (a+b) (a+b \sin (x))}","\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{b \cos (x)}{\left(a^2-b^2\right) (a+b \sin (x))}",1,"(2*a*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (b*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x]))","A",1
190,1,99,93,0.2500998,"\int \frac{\csc (x)}{(a+b \sin (x))^2} \, dx","Integrate[Csc[x]/(a + b*Sin[x])^2,x]","\frac{\frac{2 b \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-\frac{a b^2 \cos (x)}{(a-b) (a+b) (a+b \sin (x))}+\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)}{a^2}","-\frac{2 b \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right)^{3/2}}-\frac{b^2 \cos (x)}{a \left(a^2-b^2\right) (a+b \sin (x))}-\frac{\tanh ^{-1}(\cos (x))}{a^2}",1,"((2*b*(-2*a^2 + b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - Log[Cos[x/2]] + Log[Sin[x/2]] - (a*b^2*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])))/a^2","A",1
191,1,127,123,0.7245632,"\int \frac{\csc ^2(x)}{(a+b \sin (x))^2} \, dx","Integrate[Csc[x]^2/(a + b*Sin[x])^2,x]","\frac{\frac{4 b^2 \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{2 a b^3 \cos (x)}{(a-b) (a+b) (a+b \sin (x))}+a \tan \left(\frac{x}{2}\right)-a \cot \left(\frac{x}{2}\right)-4 b \log \left(\sin \left(\frac{x}{2}\right)\right)+4 b \log \left(\cos \left(\frac{x}{2}\right)\right)}{2 a^3}","\frac{2 b \tanh ^{-1}(\cos (x))}{a^3}-\frac{\left(a^2-2 b^2\right) \cot (x)}{a^2 \left(a^2-b^2\right)}-\frac{b^2 \cot (x)}{a \left(a^2-b^2\right) (a+b \sin (x))}+\frac{2 b^2 \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 \left(a^2-b^2\right)^{3/2}}",1,"((4*b^2*(3*a^2 - 2*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - a*Cot[x/2] + 4*b*Log[Cos[x/2]] - 4*b*Log[Sin[x/2]] + (2*a*b^3*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])) + a*Tan[x/2])/(2*a^3)","A",1
192,1,171,168,1.0093284,"\int \frac{\csc ^3(x)}{(a+b \sin (x))^2} \, dx","Integrate[Csc[x]^3/(a + b*Sin[x])^2,x]","\frac{4 \left(a^2+6 b^2\right) \log \left(\sin \left(\frac{x}{2}\right)\right)-4 \left(a^2+6 b^2\right) \log \left(\cos \left(\frac{x}{2}\right)\right)+\frac{16 b^3 \left(3 b^2-4 a^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-a^2 \csc ^2\left(\frac{x}{2}\right)+a^2 \sec ^2\left(\frac{x}{2}\right)-\frac{8 a b^4 \cos (x)}{(a-b) (a+b) (a+b \sin (x))}-8 a b \tan \left(\frac{x}{2}\right)+8 a b \cot \left(\frac{x}{2}\right)}{8 a^4}","-\frac{\left(a^2-3 b^2\right) \cot (x) \csc (x)}{2 a^2 \left(a^2-b^2\right)}-\frac{b^2 \cot (x) \csc (x)}{a \left(a^2-b^2\right) (a+b \sin (x))}-\frac{\left(a^2+6 b^2\right) \tanh ^{-1}(\cos (x))}{2 a^4}-\frac{2 b^3 \left(4 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 \left(a^2-b^2\right)^{3/2}}+\frac{b \left(2 a^2-3 b^2\right) \cot (x)}{a^3 \left(a^2-b^2\right)}",1,"((16*b^3*(-4*a^2 + 3*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 8*a*b*Cot[x/2] - a^2*Csc[x/2]^2 - 4*(a^2 + 6*b^2)*Log[Cos[x/2]] + 4*(a^2 + 6*b^2)*Log[Sin[x/2]] + a^2*Sec[x/2]^2 - (8*a*b^4*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])) - 8*a*b*Tan[x/2])/(8*a^4)","A",1
193,1,164,243,1.0387639,"\int \frac{\sin ^5(x)}{(a+b \sin (x))^3} \, dx","Integrate[Sin[x]^5/(a + b*Sin[x])^3,x]","\frac{-\frac{2 a^5 b \cos (x)}{(a-b) (a+b) (a+b \sin (x))^2}+2 x \left(12 a^2+b^2\right)+\frac{2 a^4 b \left(7 a^2-10 b^2\right) \cos (x)}{(a-b)^2 (a+b)^2 (a+b \sin (x))}-\frac{4 a^3 \left(12 a^4-29 a^2 b^2+20 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+12 a b \cos (x)-b^2 \sin (2 x)}{4 b^5}","\frac{a^2 \sin ^3(x) \cos (x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^2}+\frac{a^2 \left(4 a^2-7 b^2\right) \sin ^2(x) \cos (x)}{2 b^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}+\frac{x \left(12 a^2+b^2\right)}{2 b^5}+\frac{3 a \left(4 a^4-7 a^2 b^2+2 b^4\right) \cos (x)}{2 b^4 \left(a^2-b^2\right)^2}-\frac{\left(6 a^4-10 a^2 b^2+b^4\right) \sin (x) \cos (x)}{2 b^3 \left(a^2-b^2\right)^2}-\frac{a^3 \left(12 a^4-29 a^2 b^2+20 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 \left(a^2-b^2\right)^{5/2}}",1,"(2*(12*a^2 + b^2)*x - (4*a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 12*a*b*Cos[x] - (2*a^5*b*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])^2) + (2*a^4*b*(7*a^2 - 10*b^2)*Cos[x])/((a - b)^2*(a + b)^2*(a + b*Sin[x])) - b^2*Sin[2*x])/(4*b^5)","A",1
194,1,144,179,0.8388989,"\int \frac{\sin ^4(x)}{(a+b \sin (x))^3} \, dx","Integrate[Sin[x]^4/(a + b*Sin[x])^3,x]","\frac{\frac{a^4 b \cos (x)}{(a-b) (a+b) (a+b \sin (x))^2}+\frac{6 a^2 \left(2 a^4-5 a^2 b^2+4 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a^3 b \left(8 b^2-5 a^2\right) \cos (x)}{(a-b)^2 (a+b)^2 (a+b \sin (x))}-6 a x-2 b \cos (x)}{2 b^4}","\frac{a^2 \sin ^2(x) \cos (x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^2}-\frac{\left(3 a^2-2 b^2\right) \cos (x)}{2 b^3 \left(a^2-b^2\right)}+\frac{3 a^2 \left(2 a^4-5 a^2 b^2+4 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 \left(a^2-b^2\right)^{5/2}}-\frac{3 a^3 \left(a^2-2 b^2\right) \cos (x)}{2 b^3 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{3 a x}{b^4}",1,"(-6*a*x + (6*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 2*b*Cos[x] + (a^4*b*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])^2) + (a^3*b*(-5*a^2 + 8*b^2)*Cos[x])/((a - b)^2*(a + b)^2*(a + b*Sin[x])))/(2*b^4)","A",1
195,1,136,144,0.571871,"\int \frac{\sin ^3(x)}{(a+b \sin (x))^3} \, dx","Integrate[Sin[x]^3/(a + b*Sin[x])^3,x]","\frac{-\frac{a^3 b \cos (x)}{(a-b) (a+b) (a+b \sin (x))^2}+\frac{3 a^2 b \left(a^2-2 b^2\right) \cos (x)}{(a-b)^2 (a+b)^2 (a+b \sin (x))}-\frac{2 a \left(2 a^4-5 a^2 b^2+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+2 x}{2 b^3}","\frac{a^2 \left(2 a^2-5 b^2\right) \cos (x)}{2 b^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}+\frac{a^2 \sin (x) \cos (x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^2}-\frac{a \left(2 a^4-5 a^2 b^2+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 \left(a^2-b^2\right)^{5/2}}+\frac{x}{b^3}",1,"(2*x - (2*a*(2*a^4 - 5*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - (a^3*b*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])^2) + (3*a^2*b*(a^2 - 2*b^2)*Cos[x])/((a - b)^2*(a + b)^2*(a + b*Sin[x])))/(2*b^3)","A",1
196,1,94,118,0.3767443,"\int \frac{\sin ^2(x)}{(a+b \sin (x))^3} \, dx","Integrate[Sin[x]^2/(a + b*Sin[x])^3,x]","\frac{\left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a \cos (x) \left(3 a b-\left(a^2-4 b^2\right) \sin (x)\right)}{2 (a-b)^2 (a+b)^2 (a+b \sin (x))^2}","\frac{\left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a^2 \cos (x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^2}-\frac{a \left(a^2-4 b^2\right) \cos (x)}{2 b \left(a^2-b^2\right)^2 (a+b \sin (x))}",1,"((a^2 + 2*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (a*Cos[x]*(3*a*b - (a^2 - 4*b^2)*Sin[x]))/(2*(a - b)^2*(a + b)^2*(a + b*Sin[x])^2)","A",1
197,1,94,103,0.298677,"\int \frac{\sin (x)}{(a+b \sin (x))^3} \, dx","Integrate[Sin[x]/(a + b*Sin[x])^3,x]","-\frac{3 a b \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-\frac{\cos (x) \left(b \left(a^2+2 b^2\right) \sin (x)+a \left(2 a^2+b^2\right)\right)}{2 (a-b)^2 (a+b)^2 (a+b \sin (x))^2}","-\frac{3 a b \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-\frac{\left(a^2+2 b^2\right) \cos (x)}{2 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{a \cos (x)}{2 \left(a^2-b^2\right) (a+b \sin (x))^2}",1,"(-3*a*b*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - (Cos[x]*(a*(2*a^2 + b^2) + b*(a^2 + 2*b^2)*Sin[x]))/(2*(a - b)^2*(a + b)^2*(a + b*Sin[x])^2)","A",1
198,1,93,102,0.2027287,"\int \frac{1}{(a+b \sin (x))^3} \, dx","Integrate[(a + b*Sin[x])^(-3),x]","\frac{\left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{b \cos (x) \left(4 a^2+3 a b \sin (x)-b^2\right)}{2 (a-b)^2 (a+b)^2 (a+b \sin (x))^2}","\frac{\left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{3 a b \cos (x)}{2 \left(a^2-b^2\right)^2 (a+b \sin (x))}+\frac{b \cos (x)}{2 \left(a^2-b^2\right) (a+b \sin (x))^2}",1,"((2*a^2 + b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (b*Cos[x]*(4*a^2 - b^2 + 3*a*b*Sin[x]))/(2*(a - b)^2*(a + b)^2*(a + b*Sin[x])^2)","A",1
199,1,140,145,0.9050956,"\int \frac{\csc (x)}{(a+b \sin (x))^3} \, dx","Integrate[Csc[x]/(a + b*Sin[x])^3,x]","-\frac{\frac{2 b \left(6 a^4-5 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a b^2 \cos (x) \left(6 a^3+b \left(5 a^2-2 b^2\right) \sin (x)-3 a b^2\right)}{(a-b)^2 (a+b)^2 (a+b \sin (x))^2}-2 \log \left(\sin \left(\frac{x}{2}\right)\right)+2 \log \left(\cos \left(\frac{x}{2}\right)\right)}{2 a^3}","-\frac{\tanh ^{-1}(\cos (x))}{a^3}-\frac{b^2 \left(5 a^2-2 b^2\right) \cos (x)}{2 a^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{b^2 \cos (x)}{2 a \left(a^2-b^2\right) (a+b \sin (x))^2}-\frac{b \left(6 a^4-5 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 \left(a^2-b^2\right)^{5/2}}",1,"-1/2*((2*b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 2*Log[Cos[x/2]] - 2*Log[Sin[x/2]] + (a*b^2*Cos[x]*(6*a^3 - 3*a*b^2 + b*(5*a^2 - 2*b^2)*Sin[x]))/((a - b)^2*(a + b)^2*(a + b*Sin[x])^2))/a^3","A",1
200,1,174,187,1.3818272,"\int \frac{\csc ^2(x)}{(a+b \sin (x))^3} \, dx","Integrate[Csc[x]^2/(a + b*Sin[x])^3,x]","\frac{\frac{a^2 b^3 \cos (x)}{(a-b) (a+b) (a+b \sin (x))^2}+\frac{a b^3 \left(7 a^2-4 b^2\right) \cos (x)}{(a-b)^2 (a+b)^2 (a+b \sin (x))}+\frac{6 b^2 \left(4 a^4-5 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+a \tan \left(\frac{x}{2}\right)-a \cot \left(\frac{x}{2}\right)-6 b \log \left(\sin \left(\frac{x}{2}\right)\right)+6 b \log \left(\cos \left(\frac{x}{2}\right)\right)}{2 a^4}","\frac{3 b \tanh ^{-1}(\cos (x))}{a^4}-\frac{3 b^2 \left(2 a^2-b^2\right) \cot (x)}{2 a^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{b^2 \cot (x)}{2 a \left(a^2-b^2\right) (a+b \sin (x))^2}+\frac{3 b^2 \left(4 a^4-5 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 \left(a^2-b^2\right)^{5/2}}-\frac{\left(2 a^4-11 a^2 b^2+6 b^4\right) \cot (x)}{2 a^3 \left(a^2-b^2\right)^2}",1,"((6*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - a*Cot[x/2] + 6*b*Log[Cos[x/2]] - 6*b*Log[Sin[x/2]] + (a^2*b^3*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])^2) + (a*b^3*(7*a^2 - 4*b^2)*Cos[x])/((a - b)^2*(a + b)^2*(a + b*Sin[x])) + a*Tan[x/2])/(2*a^4)","A",1
201,1,220,241,2.0848695,"\int \frac{\csc ^3(x)}{(a+b \sin (x))^3} \, dx","Integrate[Csc[x]^3/(a + b*Sin[x])^3,x]","\frac{-\frac{4 a^2 b^4 \cos (x)}{(a-b) (a+b) (a+b \sin (x))^2}+4 \left(a^2+12 b^2\right) \log \left(\sin \left(\frac{x}{2}\right)\right)-4 \left(a^2+12 b^2\right) \log \left(\cos \left(\frac{x}{2}\right)\right)+\frac{12 a b^4 \left(2 b^2-3 a^2\right) \cos (x)}{(a-b)^2 (a+b)^2 (a+b \sin (x))}-a^2 \csc ^2\left(\frac{x}{2}\right)+a^2 \sec ^2\left(\frac{x}{2}\right)-\frac{8 b^3 \left(20 a^4-29 a^2 b^2+12 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-12 a b \tan \left(\frac{x}{2}\right)+12 a b \cot \left(\frac{x}{2}\right)}{8 a^5}","-\frac{b^2 \left(7 a^2-4 b^2\right) \cot (x) \csc (x)}{2 a^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{b^2 \cot (x) \csc (x)}{2 a \left(a^2-b^2\right) (a+b \sin (x))^2}-\frac{\left(a^2+12 b^2\right) \tanh ^{-1}(\cos (x))}{2 a^5}+\frac{3 b \left(2 a^4-7 a^2 b^2+4 b^4\right) \cot (x)}{2 a^4 \left(a^2-b^2\right)^2}-\frac{b^3 \left(20 a^4-29 a^2 b^2+12 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 \left(a^2-b^2\right)^{5/2}}-\frac{\left(a^4-10 a^2 b^2+6 b^4\right) \cot (x) \csc (x)}{2 a^3 \left(a^2-b^2\right)^2}",1,"((-8*b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 12*a*b*Cot[x/2] - a^2*Csc[x/2]^2 - 4*(a^2 + 12*b^2)*Log[Cos[x/2]] + 4*(a^2 + 12*b^2)*Log[Sin[x/2]] + a^2*Sec[x/2]^2 - (4*a^2*b^4*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])^2) + (12*a*b^4*(-3*a^2 + 2*b^2)*Cos[x])/((a - b)^2*(a + b)^2*(a + b*Sin[x])) - 12*a*b*Tan[x/2])/(8*a^5)","A",1
202,1,157,182,1.0249218,"\int \frac{1}{(a+b \sin (c+d x))^4} \, dx","Integrate[(a + b*Sin[c + d*x])^(-4),x]","\frac{\frac{6 a \left(2 a^2+3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}+\frac{b \cos (c+d x) \left(18 a^4+b^2 \left(11 a^2+4 b^2\right) \sin ^2(c+d x)+3 a b \left(9 a^2+b^2\right) \sin (c+d x)-5 a^2 b^2+2 b^4\right)}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))^3}}{6 d}","\frac{a \left(2 a^2+3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{b \left(11 a^2+4 b^2\right) \cos (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{5 a b \cos (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{b \cos (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^3}",1,"((6*a*(2*a^2 + 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (b*Cos[c + d*x]*(18*a^4 - 5*a^2*b^2 + 2*b^4 + 3*a*b*(9*a^2 + b^2)*Sin[c + d*x] + b^2*(11*a^2 + 4*b^2)*Sin[c + d*x]^2))/((a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])^3))/(6*d)","A",1
203,1,143,172,3.140387,"\int \sin (e+f x) \sqrt{a+b \sin (e+f x)} \, dx","Integrate[Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]],x]","-\frac{2 \left(-\left(a^2-b^2\right) \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right)+b \cos (e+f x) (a+b \sin (e+f x))+a (a+b) \sqrt{\frac{a+b \sin (e+f x)}{a+b}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right)\right)}{3 b f \sqrt{a+b \sin (e+f x)}}","-\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b f \sqrt{a+b \sin (e+f x)}}-\frac{2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f}+\frac{2 a \sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}",1,"(-2*(b*Cos[e + f*x]*(a + b*Sin[e + f*x]) + a*(a + b)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)] - (a^2 - b^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)]))/(3*b*f*Sqrt[a + b*Sin[e + f*x]])","A",1
204,1,61,62,0.0717831,"\int \sqrt{a+b \sin (e+f x)} \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]],x]","-\frac{2 \sqrt{a+b \sin (e+f x)} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right)}{f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}","\frac{2 \sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}",1,"(-2*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(f*Sqrt[(a + b*Sin[e + f*x])/(a + b)])","A",1
205,1,89,128,16.1794766,"\int \csc (e+f x) \sqrt{a+b \sin (e+f x)} \, dx","Integrate[Csc[e + f*x]*Sqrt[a + b*Sin[e + f*x]],x]","-\frac{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \left(b F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right)+a \Pi \left(2;\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right)\right)}{f \sqrt{a+b \sin (e+f x)}}","\frac{2 b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}+\frac{2 a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}",1,"(-2*(b*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)] + a*EllipticPi[2, (-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)])*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])","A",1
206,1,312,213,8.8716349,"\int \csc ^2(e+f x) \sqrt{a+b \sin (e+f x)} \, dx","Integrate[Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]],x]","\frac{-4 \cot (e+f x) \sqrt{a+b \sin (e+f x)}-\frac{2 b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (e+f x)}}+\frac{2 i \sec (e+f x) \sqrt{-\frac{b (\sin (e+f x)-1)}{a+b}} \sqrt{-\frac{b (\sin (e+f x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}}{4 f}","-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{f}+\frac{a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}+\frac{b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}",1,"(((2*I)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b)/(a - b)]))*Sec[e + f*x]*Sqrt[-((b*(-1 + Sin[e + f*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[e + f*x]))/(a - b))])/(a*b*Sqrt[-(a + b)^(-1)]) - 4*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]] - (2*b*EllipticPi[2, (-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/Sqrt[a + b*Sin[e + f*x]])/(4*f)","C",1
207,1,94,132,2.3955076,"\int \frac{\sin (e+f x)}{\sqrt{a+b \sin (e+f x)}} \, dx","Integrate[Sin[e + f*x]/Sqrt[a + b*Sin[e + f*x]],x]","-\frac{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \left((a+b) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right)\right)}{b f \sqrt{a+b \sin (e+f x)}}","\frac{2 \sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{b f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}-\frac{2 a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{b f \sqrt{a+b \sin (e+f x)}}",1,"(-2*((a + b)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)] - a*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)])*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(b*f*Sqrt[a + b*Sin[e + f*x]])","A",1
208,1,61,62,0.0544017,"\int \frac{1}{\sqrt{a+b \sin (e+f x)}} \, dx","Integrate[1/Sqrt[a + b*Sin[e + f*x]],x]","-\frac{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}","\frac{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}",1,"(-2*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])","A",1
209,1,62,63,0.0811154,"\int \frac{\csc (e+f x)}{\sqrt{a+b \sin (e+f x)}} \, dx","Integrate[Csc[e + f*x]/Sqrt[a + b*Sin[e + f*x]],x]","-\frac{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}","\frac{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}",1,"(-2*EllipticPi[2, (-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])","A",1
210,1,315,222,10.0487244,"\int \frac{\csc ^2(e+f x)}{\sqrt{a+b \sin (e+f x)}} \, dx","Integrate[Csc[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]],x]","\frac{-4 \cot (e+f x) \sqrt{a+b \sin (e+f x)}+\frac{6 b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (e+f x)}}+\frac{2 i \sec (e+f x) \sqrt{-\frac{b (\sin (e+f x)-1)}{a+b}} \sqrt{-\frac{b (\sin (e+f x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}}{4 a f}","-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{a f}+\frac{\sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}-\frac{b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a f \sqrt{a+b \sin (e+f x)}}",1,"(((2*I)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b)/(a - b)]))*Sec[e + f*x]*Sqrt[-((b*(-1 + Sin[e + f*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[e + f*x]))/(a - b))])/(a*b*Sqrt[-(a + b)^(-1)]) - 4*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]] + (6*b*EllipticPi[2, (-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/Sqrt[a + b*Sin[e + f*x]])/(4*a*f)","C",1
211,1,10847,371,26.8300846,"\int \sqrt{\sin (c+d x)} \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Sqrt[Sin[c + d*x]]*Sqrt[a + b*Sin[c + d*x]],x]","\text{Result too large to show}","-\frac{\cos (c+d x) \sqrt{a+b \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}-\frac{\sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{(a-b) \sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{a \sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}",1,"Result too large to show","C",0
212,1,172,109,3.3431724,"\int \frac{1}{\sqrt{\sin (c+d x)} \sqrt{a+b \sin (c+d x)}} \, dx","Integrate[1/(Sqrt[Sin[c + d*x]]*Sqrt[a + b*Sin[c + d*x]]),x]","\frac{8 a \sin ^4\left(\frac{1}{4} (2 c+2 d x-\pi )\right) \sec (c+d x) \sqrt{-\frac{(a+b) \sin (c+d x) (a+b \sin (c+d x))}{a^2 (\sin (c+d x)-1)^2}} \sqrt{-\frac{(a+b) \cot ^2\left(\frac{1}{4} (2 c+2 d x-\pi )\right)}{a-b}} F\left(\sin ^{-1}\left(\sqrt{-\frac{a+b \sin (c+d x)}{a (\sin (c+d x)-1)}}\right)|\frac{2 a}{a-b}\right)}{d (a+b) \sqrt{\sin (c+d x)} \sqrt{a+b \sin (c+d x)}}","-\frac{2 \sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}",1,"(8*a*Sqrt[-(((a + b)*Cot[(2*c - Pi + 2*d*x)/4]^2)/(a - b))]*EllipticF[ArcSin[Sqrt[-((a + b*Sin[c + d*x])/(a*(-1 + Sin[c + d*x])))]], (2*a)/(a - b)]*Sec[c + d*x]*Sqrt[-(((a + b)*Sin[c + d*x]*(a + b*Sin[c + d*x]))/(a^2*(-1 + Sin[c + d*x])^2))]*Sin[(2*c - Pi + 2*d*x)/4]^4)/((a + b)*d*Sqrt[Sin[c + d*x]]*Sqrt[a + b*Sin[c + d*x]])","A",1
213,1,199,270,0.8048886,"\int (d \sin (e+f x))^m (a+b \sin (e+f x))^3 \, dx","Integrate[(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x])^3,x]","\frac{\sin (e+f x) \cos (e+f x) (d \sin (e+f x))^m \left(\frac{b \left(3 a^2 (m+3)+b^2 (m+2)\right) \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right)}{(m+2) \sqrt{\cos ^2(e+f x)}}+\frac{a (m+3) \left(a^2 (m+2)+3 b^2 (m+1)\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)}{(m+1) (m+2) \sqrt{\cos ^2(e+f x)}}-b^2 (a+b \sin (e+f x))-\frac{a b^2 (2 m+7)}{m+2}\right)}{f (m+3)}","\frac{b \left(3 a^2 (m+3)+b^2 (m+2)\right) \cos (e+f x) (d \sin (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right)}{d^2 f (m+2) (m+3) \sqrt{\cos ^2(e+f x)}}+\frac{a \left(a^2 (m+2)+3 b^2 (m+1)\right) \cos (e+f x) (d \sin (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)}{d f (m+1) (m+2) \sqrt{\cos ^2(e+f x)}}-\frac{a b^2 (2 m+7) \cos (e+f x) (d \sin (e+f x))^{m+1}}{d f (m+2) (m+3)}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x)) (d \sin (e+f x))^{m+1}}{d f (m+3)}",1,"(Cos[e + f*x]*Sin[e + f*x]*(d*Sin[e + f*x])^m*(-((a*b^2*(7 + 2*m))/(2 + m)) + (a*(3 + m)*(3*b^2*(1 + m) + a^2*(2 + m))*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[e + f*x]^2])/((1 + m)*(2 + m)*Sqrt[Cos[e + f*x]^2]) + (b*(b^2*(2 + m) + 3*a^2*(3 + m))*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Sin[e + f*x]^2]*Sin[e + f*x])/((2 + m)*Sqrt[Cos[e + f*x]^2]) - b^2*(a + b*Sin[e + f*x])))/(f*(3 + m))","A",1
214,1,144,194,0.328671,"\int (d \sin (e+f x))^m (a+b \sin (e+f x))^2 \, dx","Integrate[(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x])^2,x]","-\frac{\cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-m-1)} (d \sin (e+f x))^m \left(a \left(a \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3}{2};\cos ^2(e+f x)\right)+2 b \sqrt{\sin ^2(e+f x)} \, _2F_1\left(\frac{1}{2},-\frac{m}{2};\frac{3}{2};\cos ^2(e+f x)\right)\right)+b^2 \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-m-1);\frac{3}{2};\cos ^2(e+f x)\right)\right)}{f}","\frac{\left(a^2 (m+2)+b^2 (m+1)\right) \cos (e+f x) (d \sin (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)}{d f (m+1) (m+2) \sqrt{\cos ^2(e+f x)}}+\frac{2 a b \cos (e+f x) (d \sin (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right)}{d^2 f (m+2) \sqrt{\cos ^2(e+f x)}}-\frac{b^2 \cos (e+f x) (d \sin (e+f x))^{m+1}}{d f (m+2)}",1,"-((Cos[e + f*x]*(d*Sin[e + f*x])^m*(Sin[e + f*x]^2)^((-1 - m)/2)*(b^2*Hypergeometric2F1[1/2, (-1 - m)/2, 3/2, Cos[e + f*x]^2]*Sin[e + f*x] + a*(a*Hypergeometric2F1[1/2, (1 - m)/2, 3/2, Cos[e + f*x]^2]*Sin[e + f*x] + 2*b*Hypergeometric2F1[1/2, -1/2*m, 3/2, Cos[e + f*x]^2]*Sqrt[Sin[e + f*x]^2])))/f)","A",1
215,1,111,139,0.1598787,"\int (d \sin (e+f x))^m (a+b \sin (e+f x)) \, dx","Integrate[(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x]),x]","\frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) (d \sin (e+f x))^m \left(a (m+2) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)+b (m+1) \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right)\right)}{f (m+1) (m+2)}","\frac{a \cos (e+f x) (d \sin (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)}{d f (m+1) \sqrt{\cos ^2(e+f x)}}+\frac{b \cos (e+f x) (d \sin (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right)}{d^2 f (m+2) \sqrt{\cos ^2(e+f x)}}",1,"(Sqrt[Cos[e + f*x]^2]*(d*Sin[e + f*x])^m*(a*(2 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[e + f*x]^2] + b*(1 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Sin[e + f*x]^2]*Sin[e + f*x])*Tan[e + f*x])/(f*(1 + m)*(2 + m))","A",1
216,1,1590,195,17.6870663,"\int \frac{(d \sin (e+f x))^m}{a+b \sin (e+f x)} \, dx","Integrate[(d*Sin[e + f*x])^m/(a + b*Sin[e + f*x]),x]","\frac{\sec ^2(e+f x)^{m/2} (d \sin (e+f x))^m \tan (e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^m \left(a b (m+2) F_1\left(\frac{m+1}{2};\frac{m}{2},1;\frac{m+3}{2};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)+(m+1) \left(\left(a^2-b^2\right) F_1\left(\frac{m+2}{2};\frac{m-1}{2},1;\frac{m+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-a^2 \, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(e+f x)\right)\right) \tan (e+f x)\right)}{a^2 b f (m+1) (m+2) (a+b \sin (e+f x)) \left(\frac{\left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^m \left(a b (m+2) F_1\left(\frac{m+1}{2};\frac{m}{2},1;\frac{m+3}{2};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)+(m+1) \left(\left(a^2-b^2\right) F_1\left(\frac{m+2}{2};\frac{m-1}{2},1;\frac{m+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-a^2 \, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(e+f x)\right)\right) \tan (e+f x)\right) \sec ^2(e+f x)^{\frac{m}{2}+1}}{a^2 b (m+1) (m+2)}+\frac{m \tan ^2(e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^m \left(a b (m+2) F_1\left(\frac{m+1}{2};\frac{m}{2},1;\frac{m+3}{2};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)+(m+1) \left(\left(a^2-b^2\right) F_1\left(\frac{m+2}{2};\frac{m-1}{2},1;\frac{m+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-a^2 \, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(e+f x)\right)\right) \tan (e+f x)\right) \sec ^2(e+f x)^{m/2}}{a^2 b (m+1) (m+2)}+\frac{m \tan (e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^{m-1} \left(a b (m+2) F_1\left(\frac{m+1}{2};\frac{m}{2},1;\frac{m+3}{2};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)+(m+1) \left(\left(a^2-b^2\right) F_1\left(\frac{m+2}{2};\frac{m-1}{2},1;\frac{m+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-a^2 \, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(e+f x)\right)\right) \tan (e+f x)\right) \left(\sqrt{\sec ^2(e+f x)}-\frac{\tan ^2(e+f x)}{\sqrt{\sec ^2(e+f x)}}\right) \sec ^2(e+f x)^{m/2}}{a^2 b (m+1) (m+2)}+\frac{\tan (e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^m \left((m+1) \left(\left(a^2-b^2\right) F_1\left(\frac{m+2}{2};\frac{m-1}{2},1;\frac{m+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-a^2 \, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(e+f x)\right)\right) \sec ^2(e+f x)+a b (m+2) \left(\frac{2 \left(b^2-a^2\right) (m+1) F_1\left(\frac{m+1}{2}+1;\frac{m}{2},2;\frac{m+3}{2}+1;-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \sec ^2(e+f x) \tan (e+f x)}{a^2 (m+3)}-\frac{m (m+1) F_1\left(\frac{m+1}{2}+1;\frac{m}{2}+1,1;\frac{m+3}{2}+1;-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \sec ^2(e+f x) \tan (e+f x)}{m+3}\right)+(m+1) \tan (e+f x) \left(\left(a^2-b^2\right) \left(\frac{2 \left(\frac{b^2}{a^2}-1\right) (m+2) F_1\left(\frac{m+2}{2}+1;\frac{m-1}{2},2;\frac{m+4}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{m+4}-\frac{(m-1) (m+2) F_1\left(\frac{m+2}{2}+1;\frac{m-1}{2}+1,1;\frac{m+4}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{m+4}\right)-a^2 (m+2) \csc (e+f x) \sec (e+f x) \left(\left(\tan ^2(e+f x)+1\right)^{\frac{1}{2} (-m-1)}-\, _2F_1\left(\frac{m+1}{2},\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(e+f x)\right)\right)\right)\right) \sec ^2(e+f x)^{m/2}}{a^2 b (m+1) (m+2)}\right)}","\frac{b \cos (e+f x) \sin ^2(e+f x)^{-m/2} (d \sin (e+f x))^m F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{a d \cos (e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (d \sin (e+f x))^{m-1} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}",1,"((Sec[e + f*x]^2)^(m/2)*(d*Sin[e + f*x])^m*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^m*(a*b*(2 + m)*AppellF1[(1 + m)/2, m/2, 1, (3 + m)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] + (1 + m)*((a^2 - b^2)*AppellF1[(2 + m)/2, (-1 + m)/2, 1, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - a^2*Hypergeometric2F1[(1 + m)/2, (2 + m)/2, (4 + m)/2, -Tan[e + f*x]^2])*Tan[e + f*x]))/(a^2*b*f*(1 + m)*(2 + m)*(a + b*Sin[e + f*x])*(((Sec[e + f*x]^2)^(1 + m/2)*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^m*(a*b*(2 + m)*AppellF1[(1 + m)/2, m/2, 1, (3 + m)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] + (1 + m)*((a^2 - b^2)*AppellF1[(2 + m)/2, (-1 + m)/2, 1, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - a^2*Hypergeometric2F1[(1 + m)/2, (2 + m)/2, (4 + m)/2, -Tan[e + f*x]^2])*Tan[e + f*x]))/(a^2*b*(1 + m)*(2 + m)) + (m*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]^2*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^m*(a*b*(2 + m)*AppellF1[(1 + m)/2, m/2, 1, (3 + m)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] + (1 + m)*((a^2 - b^2)*AppellF1[(2 + m)/2, (-1 + m)/2, 1, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - a^2*Hypergeometric2F1[(1 + m)/2, (2 + m)/2, (4 + m)/2, -Tan[e + f*x]^2])*Tan[e + f*x]))/(a^2*b*(1 + m)*(2 + m)) + (m*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(-1 + m)*(a*b*(2 + m)*AppellF1[(1 + m)/2, m/2, 1, (3 + m)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] + (1 + m)*((a^2 - b^2)*AppellF1[(2 + m)/2, (-1 + m)/2, 1, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - a^2*Hypergeometric2F1[(1 + m)/2, (2 + m)/2, (4 + m)/2, -Tan[e + f*x]^2])*Tan[e + f*x])*(Sqrt[Sec[e + f*x]^2] - Tan[e + f*x]^2/Sqrt[Sec[e + f*x]^2]))/(a^2*b*(1 + m)*(2 + m)) + ((Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^m*((1 + m)*((a^2 - b^2)*AppellF1[(2 + m)/2, (-1 + m)/2, 1, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - a^2*Hypergeometric2F1[(1 + m)/2, (2 + m)/2, (4 + m)/2, -Tan[e + f*x]^2])*Sec[e + f*x]^2 + a*b*(2 + m)*(-((m*(1 + m)*AppellF1[1 + (1 + m)/2, 1 + m/2, 1, 1 + (3 + m)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + m)) + (2*(-a^2 + b^2)*(1 + m)*AppellF1[1 + (1 + m)/2, m/2, 2, 1 + (3 + m)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Sec[e + f*x]^2*Tan[e + f*x])/(a^2*(3 + m))) + (1 + m)*Tan[e + f*x]*((a^2 - b^2)*(-(((-1 + m)*(2 + m)*AppellF1[1 + (2 + m)/2, 1 + (-1 + m)/2, 1, 1 + (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(4 + m)) + (2*(-1 + b^2/a^2)*(2 + m)*AppellF1[1 + (2 + m)/2, (-1 + m)/2, 2, 1 + (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(4 + m)) - a^2*(2 + m)*Csc[e + f*x]*Sec[e + f*x]*(-Hypergeometric2F1[(1 + m)/2, (2 + m)/2, (4 + m)/2, -Tan[e + f*x]^2] + (1 + Tan[e + f*x]^2)^((-1 - m)/2)))))/(a^2*b*(1 + m)*(2 + m))))","B",0
217,1,1790,306,18.8762368,"\int \frac{(d \sin (e+f x))^m}{(a+b \sin (e+f x))^2} \, dx","Integrate[(d*Sin[e + f*x])^m/(a + b*Sin[e + f*x])^2,x]","-\frac{\sec ^2(e+f x)^{m/2} (d \sin (e+f x))^m \tan (e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^m \left(2 b \left(a b (m+2) F_1\left(\frac{m+1}{2};\frac{m}{2},2;\frac{m+3}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)+\left(a^2-b^2\right) (m+1) F_1\left(\frac{m+2}{2};\frac{m-1}{2},2;\frac{m+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)\right)-a \left(a^2+b^2\right) (m+2) F_1\left(\frac{m+1}{2};\frac{m}{2},1;\frac{m+3}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right)}{a^3 \left(a^2-b^2\right) f (m+1) (m+2) (a+b \sin (e+f x))^2 \left(-\frac{\left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^m \left(2 b \left(a b (m+2) F_1\left(\frac{m+1}{2};\frac{m}{2},2;\frac{m+3}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)+\left(a^2-b^2\right) (m+1) F_1\left(\frac{m+2}{2};\frac{m-1}{2},2;\frac{m+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)\right)-a \left(a^2+b^2\right) (m+2) F_1\left(\frac{m+1}{2};\frac{m}{2},1;\frac{m+3}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right) \sec ^2(e+f x)^{\frac{m}{2}+1}}{a^3 \left(a^2-b^2\right) (m+1) (m+2)}-\frac{m \tan ^2(e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^m \left(2 b \left(a b (m+2) F_1\left(\frac{m+1}{2};\frac{m}{2},2;\frac{m+3}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)+\left(a^2-b^2\right) (m+1) F_1\left(\frac{m+2}{2};\frac{m-1}{2},2;\frac{m+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)\right)-a \left(a^2+b^2\right) (m+2) F_1\left(\frac{m+1}{2};\frac{m}{2},1;\frac{m+3}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right) \sec ^2(e+f x)^{m/2}}{a^3 \left(a^2-b^2\right) (m+1) (m+2)}-\frac{m \tan (e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^{m-1} \left(\sqrt{\sec ^2(e+f x)}-\frac{\tan ^2(e+f x)}{\sqrt{\sec ^2(e+f x)}}\right) \left(2 b \left(a b (m+2) F_1\left(\frac{m+1}{2};\frac{m}{2},2;\frac{m+3}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)+\left(a^2-b^2\right) (m+1) F_1\left(\frac{m+2}{2};\frac{m-1}{2},2;\frac{m+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)\right)-a \left(a^2+b^2\right) (m+2) F_1\left(\frac{m+1}{2};\frac{m}{2},1;\frac{m+3}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right) \sec ^2(e+f x)^{m/2}}{a^3 \left(a^2-b^2\right) (m+1) (m+2)}-\frac{\tan (e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^m \left(2 b \left(\left(a^2-b^2\right) (m+1) F_1\left(\frac{m+2}{2};\frac{m-1}{2},2;\frac{m+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x)+a b (m+2) \left(\frac{4 \left(\frac{b^2}{a^2}-1\right) (m+1) F_1\left(\frac{m+1}{2}+1;\frac{m}{2},3;\frac{m+3}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{m+3}-\frac{m (m+1) F_1\left(\frac{m+1}{2}+1;\frac{m}{2}+1,2;\frac{m+3}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{m+3}\right)+\left(a^2-b^2\right) (m+1) \tan (e+f x) \left(\frac{4 \left(\frac{b^2}{a^2}-1\right) (m+2) F_1\left(\frac{m+2}{2}+1;\frac{m-1}{2},3;\frac{m+4}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{m+4}-\frac{(m-1) (m+2) F_1\left(\frac{m+2}{2}+1;\frac{m-1}{2}+1,2;\frac{m+4}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{m+4}\right)\right)-a \left(a^2+b^2\right) (m+2) \left(\frac{2 \left(\frac{b^2}{a^2}-1\right) (m+1) F_1\left(\frac{m+1}{2}+1;\frac{m}{2},2;\frac{m+3}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{m+3}-\frac{m (m+1) F_1\left(\frac{m+1}{2}+1;\frac{m}{2}+1,1;\frac{m+3}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{m+3}\right)\right) \sec ^2(e+f x)^{m/2}}{a^3 \left(a^2-b^2\right) (m+1) (m+2)}\right)}","-\frac{b^2 \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-m-1)} (d \sin (e+f x))^{m+1} F_1\left(\frac{1}{2};\frac{1}{2} (-m-1),2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d f \left(a^2-b^2\right)^2}-\frac{a^2 d \cos (e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (d \sin (e+f x))^{m-1} F_1\left(\frac{1}{2};\frac{1-m}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{2 a b \cos (e+f x) \sin ^2(e+f x)^{-m/2} (d \sin (e+f x))^m F_1\left(\frac{1}{2};-\frac{m}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}",1,"-(((Sec[e + f*x]^2)^(m/2)*(d*Sin[e + f*x])^m*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^m*(-(a*(a^2 + b^2)*(2 + m)*AppellF1[(1 + m)/2, m/2, 1, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + 2*b*(a*b*(2 + m)*AppellF1[(1 + m)/2, m/2, 2, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (a^2 - b^2)*(1 + m)*AppellF1[(2 + m)/2, (-1 + m)/2, 2, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x])))/(a^3*(a^2 - b^2)*f*(1 + m)*(2 + m)*(a + b*Sin[e + f*x])^2*(-(((Sec[e + f*x]^2)^(1 + m/2)*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^m*(-(a*(a^2 + b^2)*(2 + m)*AppellF1[(1 + m)/2, m/2, 1, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + 2*b*(a*b*(2 + m)*AppellF1[(1 + m)/2, m/2, 2, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (a^2 - b^2)*(1 + m)*AppellF1[(2 + m)/2, (-1 + m)/2, 2, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x])))/(a^3*(a^2 - b^2)*(1 + m)*(2 + m))) - (m*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]^2*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^m*(-(a*(a^2 + b^2)*(2 + m)*AppellF1[(1 + m)/2, m/2, 1, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + 2*b*(a*b*(2 + m)*AppellF1[(1 + m)/2, m/2, 2, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (a^2 - b^2)*(1 + m)*AppellF1[(2 + m)/2, (-1 + m)/2, 2, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x])))/(a^3*(a^2 - b^2)*(1 + m)*(2 + m)) - (m*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(-1 + m)*(Sqrt[Sec[e + f*x]^2] - Tan[e + f*x]^2/Sqrt[Sec[e + f*x]^2])*(-(a*(a^2 + b^2)*(2 + m)*AppellF1[(1 + m)/2, m/2, 1, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + 2*b*(a*b*(2 + m)*AppellF1[(1 + m)/2, m/2, 2, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (a^2 - b^2)*(1 + m)*AppellF1[(2 + m)/2, (-1 + m)/2, 2, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x])))/(a^3*(a^2 - b^2)*(1 + m)*(2 + m)) - ((Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^m*(-(a*(a^2 + b^2)*(2 + m)*(-((m*(1 + m)*AppellF1[1 + (1 + m)/2, 1 + m/2, 1, 1 + (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + m)) + (2*(-1 + b^2/a^2)*(1 + m)*AppellF1[1 + (1 + m)/2, m/2, 2, 1 + (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + m))) + 2*b*((a^2 - b^2)*(1 + m)*AppellF1[(2 + m)/2, (-1 + m)/2, 2, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2 + a*b*(2 + m)*(-((m*(1 + m)*AppellF1[1 + (1 + m)/2, 1 + m/2, 2, 1 + (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + m)) + (4*(-1 + b^2/a^2)*(1 + m)*AppellF1[1 + (1 + m)/2, m/2, 3, 1 + (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + m)) + (a^2 - b^2)*(1 + m)*Tan[e + f*x]*(-(((-1 + m)*(2 + m)*AppellF1[1 + (2 + m)/2, 1 + (-1 + m)/2, 2, 1 + (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(4 + m)) + (4*(-1 + b^2/a^2)*(2 + m)*AppellF1[1 + (2 + m)/2, (-1 + m)/2, 3, 1 + (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(4 + m)))))/(a^3*(a^2 - b^2)*(1 + m)*(2 + m)))))","B",0
218,1,2298,406,18.6869164,"\int \frac{(d \sin (e+f x))^m}{(a+b \sin (e+f x))^3} \, dx","Integrate[(d*Sin[e + f*x])^m/(a + b*Sin[e + f*x])^3,x]","\text{Result too large to show}","-\frac{3 a b^2 \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-m-1)} (d \sin (e+f x))^{m+1} F_1\left(\frac{1}{2};\frac{1}{2} (-m-1),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d f \left(a^2-b^2\right)^3}+\frac{3 a^2 b \cos (e+f x) \sin ^2(e+f x)^{-m/2} (d \sin (e+f x))^m F_1\left(\frac{1}{2};-\frac{m}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}+\frac{b^3 \cos (e+f x) \sin ^2(e+f x)^{-m/2} (d \sin (e+f x))^m F_1\left(\frac{1}{2};\frac{1}{2} (-m-2),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}-\frac{a^3 d \cos (e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (d \sin (e+f x))^{m-1} F_1\left(\frac{1}{2};\frac{1-m}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}",1,"-(((Sec[e + f*x]^2)^(m/2)*(d*Sin[e + f*x])^m*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^m*(-(a*(a^2 + 3*b^2)*(2 + m)*AppellF1[(1 + m)/2, (-2 + m)/2, 2, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + b*(4*a*b*(2 + m)*AppellF1[(1 + m)/2, (-2 + m)/2, 3, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (1 + m)*((3*a^2 + b^2)*AppellF1[(2 + m)/2, (-1 + m)/2, 2, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 4*b^2*AppellF1[(2 + m)/2, (-1 + m)/2, 3, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])*Tan[e + f*x])))/(a^4*(a^2 - b^2)*f*(1 + m)*(2 + m)*(a + b*Sin[e + f*x])^3*(-(((Sec[e + f*x]^2)^(1 + m/2)*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^m*(-(a*(a^2 + 3*b^2)*(2 + m)*AppellF1[(1 + m)/2, (-2 + m)/2, 2, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + b*(4*a*b*(2 + m)*AppellF1[(1 + m)/2, (-2 + m)/2, 3, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (1 + m)*((3*a^2 + b^2)*AppellF1[(2 + m)/2, (-1 + m)/2, 2, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 4*b^2*AppellF1[(2 + m)/2, (-1 + m)/2, 3, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])*Tan[e + f*x])))/(a^4*(a^2 - b^2)*(1 + m)*(2 + m))) - (m*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]^2*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^m*(-(a*(a^2 + 3*b^2)*(2 + m)*AppellF1[(1 + m)/2, (-2 + m)/2, 2, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + b*(4*a*b*(2 + m)*AppellF1[(1 + m)/2, (-2 + m)/2, 3, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (1 + m)*((3*a^2 + b^2)*AppellF1[(2 + m)/2, (-1 + m)/2, 2, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 4*b^2*AppellF1[(2 + m)/2, (-1 + m)/2, 3, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])*Tan[e + f*x])))/(a^4*(a^2 - b^2)*(1 + m)*(2 + m)) - (m*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(-1 + m)*(Sqrt[Sec[e + f*x]^2] - Tan[e + f*x]^2/Sqrt[Sec[e + f*x]^2])*(-(a*(a^2 + 3*b^2)*(2 + m)*AppellF1[(1 + m)/2, (-2 + m)/2, 2, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + b*(4*a*b*(2 + m)*AppellF1[(1 + m)/2, (-2 + m)/2, 3, (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (1 + m)*((3*a^2 + b^2)*AppellF1[(2 + m)/2, (-1 + m)/2, 2, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 4*b^2*AppellF1[(2 + m)/2, (-1 + m)/2, 3, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])*Tan[e + f*x])))/(a^4*(a^2 - b^2)*(1 + m)*(2 + m)) - ((Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^m*(-(a*(a^2 + 3*b^2)*(2 + m)*(-(((-2 + m)*(1 + m)*AppellF1[1 + (1 + m)/2, 1 + (-2 + m)/2, 2, 1 + (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + m)) + (4*(-1 + b^2/a^2)*(1 + m)*AppellF1[1 + (1 + m)/2, (-2 + m)/2, 3, 1 + (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + m))) + b*((1 + m)*((3*a^2 + b^2)*AppellF1[(2 + m)/2, (-1 + m)/2, 2, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 4*b^2*AppellF1[(2 + m)/2, (-1 + m)/2, 3, (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])*Sec[e + f*x]^2 + 4*a*b*(2 + m)*(-(((-2 + m)*(1 + m)*AppellF1[1 + (1 + m)/2, 1 + (-2 + m)/2, 3, 1 + (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + m)) + (6*(-1 + b^2/a^2)*(1 + m)*AppellF1[1 + (1 + m)/2, (-2 + m)/2, 4, 1 + (3 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + m)) + (1 + m)*Tan[e + f*x]*((3*a^2 + b^2)*(-(((-1 + m)*(2 + m)*AppellF1[1 + (2 + m)/2, 1 + (-1 + m)/2, 2, 1 + (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(4 + m)) + (4*(-1 + b^2/a^2)*(2 + m)*AppellF1[1 + (2 + m)/2, (-1 + m)/2, 3, 1 + (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(4 + m)) - 4*b^2*(-(((-1 + m)*(2 + m)*AppellF1[1 + (2 + m)/2, 1 + (-1 + m)/2, 3, 1 + (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(4 + m)) + (6*(-1 + b^2/a^2)*(2 + m)*AppellF1[1 + (2 + m)/2, (-1 + m)/2, 4, 1 + (4 + m)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(4 + m))))))/(a^4*(a^2 - b^2)*(1 + m)*(2 + m)))))","B",0
219,1,188,142,0.3030199,"\int \sin ^{-1-\frac{a^2}{a^2+b^2}}(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Sin[c + d*x]^(-1 - a^2/(a^2 + b^2))*(a + b*Sin[c + d*x])^2,x]","-\frac{\cos (c+d x) \sin ^{-\frac{a^2}{a^2+b^2}}(c+d x) \sin ^2(c+d x)^{-\frac{b^2}{2 \left(a^2+b^2\right)}} \left(\sqrt{\sin ^2(c+d x)} \left(a^2 \, _2F_1\left(\frac{1}{2},\frac{a^2}{2 \left(a^2+b^2\right)}+1;\frac{3}{2};\cos ^2(c+d x)\right)+b^2 \, _2F_1\left(\frac{1}{2},\frac{a^2}{2 \left(a^2+b^2\right)};\frac{3}{2};\cos ^2(c+d x)\right)\right)+2 a b \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} \left(\frac{a^2}{a^2+b^2}+1\right);\frac{3}{2};\cos ^2(c+d x)\right)\right)}{d}","\frac{2 a \left(a^2+b^2\right) \cos (c+d x) \sin ^{\frac{b^2}{a^2+b^2}}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{b^2}{2 \left(a^2+b^2\right)};\frac{1}{2} \left(3-\frac{a^2}{a^2+b^2}\right);\sin ^2(c+d x)\right)}{b d \sqrt{\cos ^2(c+d x)}}-\frac{\left(a^2+b^2\right) \cos (c+d x) \sin ^{-\frac{a^2}{a^2+b^2}}(c+d x)}{d}",1,"-((Cos[c + d*x]*(2*a*b*Hypergeometric2F1[1/2, (1 + a^2/(a^2 + b^2))/2, 3/2, Cos[c + d*x]^2]*Sin[c + d*x] + (b^2*Hypergeometric2F1[1/2, a^2/(2*(a^2 + b^2)), 3/2, Cos[c + d*x]^2] + a^2*Hypergeometric2F1[1/2, 1 + a^2/(2*(a^2 + b^2)), 3/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2]))/(d*Sin[c + d*x]^(a^2/(a^2 + b^2))*(Sin[c + d*x]^2)^(b^2/(2*(a^2 + b^2)))))","A",1
220,1,73,73,0.1009653,"\int \frac{(1+2 \sin (c+d x))^2}{\sin ^{\frac{6}{5}}(c+d x)} \, dx","Integrate[(1 + 2*Sin[c + d*x])^2/Sin[c + d*x]^(6/5),x]","-\frac{4 \sin ^{\frac{4}{5}}(c+d x) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{3}{5};\frac{3}{2};\cos ^2(c+d x)\right)}{d \sin ^2(c+d x)^{2/5}}-\frac{5 \cos (c+d x)}{d \sqrt[5]{\sin (c+d x)}}","\frac{5 \sin ^{\frac{4}{5}}(c+d x) \cos (c+d x) \, _2F_1\left(\frac{2}{5},\frac{1}{2};\frac{7}{5};\sin ^2(c+d x)\right)}{d \sqrt{\cos ^2(c+d x)}}-\frac{5 \cos (c+d x)}{d \sqrt[5]{\sin (c+d x)}}",1,"(-5*Cos[c + d*x])/(d*Sin[c + d*x]^(1/5)) - (4*Cos[c + d*x]*Hypergeometric2F1[1/2, 3/5, 3/2, Cos[c + d*x]^2]*Sin[c + d*x]^(4/5))/(d*(Sin[c + d*x]^2)^(2/5))","A",1
221,0,0,24,2.3741305,"\int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx","Integrate[Sin[c + d*x]^m*(a + b*Sin[c + d*x])^n,x]","\int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx","\text{Int}\left(\sin ^m(c+d x) (a+b \sin (c+d x))^n,x\right)",0,"Integrate[Sin[c + d*x]^m*(a + b*Sin[c + d*x])^n, x]","A",-1
222,0,0,351,4.143534,"\int \sin ^3(c+d x) (a+b \sin (c+d x))^n \, dx","Integrate[Sin[c + d*x]^3*(a + b*Sin[c + d*x])^n,x]","\int \sin ^3(c+d x) (a+b \sin (c+d x))^n \, dx","\frac{\sqrt{2} a \left(2 a^2+b^2 \left(n^2+5 n+4\right)\right) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b^3 d (n+2) (n+3) \sqrt{\sin (c+d x)+1}}-\frac{\sqrt{2} (a+b) \left(2 a^2+b^2 (n+2)^2\right) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b^3 d (n+2) (n+3) \sqrt{\sin (c+d x)+1}}+\frac{2 a \cos (c+d x) (a+b \sin (c+d x))^{n+1}}{b^2 d (n+2) (n+3)}-\frac{\sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{n+1}}{b d (n+3)}",1,"Integrate[Sin[c + d*x]^3*(a + b*Sin[c + d*x])^n, x]","F",-1
223,0,0,274,6.8137932,"\int \sin ^2(c+d x) (a+b \sin (c+d x))^n \, dx","Integrate[Sin[c + d*x]^2*(a + b*Sin[c + d*x])^n,x]","\int \sin ^2(c+d x) (a+b \sin (c+d x))^n \, dx","-\frac{\sqrt{2} \left(a^2+b^2 (n+1)\right) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b^2 d (n+2) \sqrt{\sin (c+d x)+1}}+\frac{\sqrt{2} a (a+b) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b^2 d (n+2) \sqrt{\sin (c+d x)+1}}-\frac{\cos (c+d x) (a+b \sin (c+d x))^{n+1}}{b d (n+2)}",1,"Integrate[Sin[c + d*x]^2*(a + b*Sin[c + d*x])^n, x]","F",-1
224,1,193,220,0.4885002,"\int \sin (c+d x) (a+b \sin (c+d x))^n \, dx","Integrate[Sin[c + d*x]*(a + b*Sin[c + d*x])^n,x]","\frac{\sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} (a+b \sin (c+d x))^{n+1} \left((n+1) (a+b \sin (c+d x)) F_1\left(n+2;\frac{1}{2},\frac{1}{2};n+3;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)-a (n+2) F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)\right)}{b^2 d (n+1) (n+2)}","\frac{\sqrt{2} a \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b d \sqrt{\sin (c+d x)+1}}-\frac{\sqrt{2} (a+b) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b d \sqrt{\sin (c+d x)+1}}",1,"(Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]*(a + b*Sin[c + d*x])^(1 + n)*(-(a*(2 + n)*AppellF1[1 + n, 1/2, 1/2, 2 + n, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]) + (1 + n)*AppellF1[2 + n, 1/2, 1/2, 3 + n, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])))/(b^2*d*(1 + n)*(2 + n))","A",0
225,1,120,104,0.2332627,"\int (a+b \sin (c+d x))^n \, dx","Integrate[(a + b*Sin[c + d*x])^n,x]","\frac{\sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} (a+b \sin (c+d x))^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (n+1)}","-\frac{\sqrt{2} \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{d \sqrt{\sin (c+d x)+1}}",1,"(AppellF1[1 + n, 1/2, 1/2, 2 + n, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]*(a + b*Sin[c + d*x])^(1 + n))/(b*d*(1 + n))","A",0
226,0,0,22,2.4442202,"\int \csc (c+d x) (a+b \sin (c+d x))^n \, dx","Integrate[Csc[c + d*x]*(a + b*Sin[c + d*x])^n,x]","\int \csc (c+d x) (a+b \sin (c+d x))^n \, dx","\text{Int}\left(\csc (c+d x) (a+b \sin (c+d x))^n,x\right)",0,"Integrate[Csc[c + d*x]*(a + b*Sin[c + d*x])^n, x]","A",-1
227,1,64,116,0.5609501,"\int (a+a \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx","Integrate[(a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^4,x]","\frac{a c^4 (120 \sin (2 (e+f x))-45 \sin (4 (e+f x))+420 \cos (e+f x)+130 \cos (3 (e+f x))-6 \cos (5 (e+f x))+420 f x)}{480 f}","\frac{7 a c^4 \cos ^3(e+f x)}{12 f}+\frac{7 a \cos ^3(e+f x) \left(c^4-c^4 \sin (e+f x)\right)}{20 f}+\frac{7 a c^4 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{7}{8} a c^4 x+\frac{a \cos ^3(e+f x) \left(c^2-c^2 \sin (e+f x)\right)^2}{5 f}",1,"(a*c^4*(420*f*x + 420*Cos[e + f*x] + 130*Cos[3*(e + f*x)] - 6*Cos[5*(e + f*x)] + 120*Sin[2*(e + f*x)] - 45*Sin[4*(e + f*x)]))/(480*f)","A",1
228,1,54,83,0.3675026,"\int (a+a \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^3,x]","\frac{a c^3 (24 \sin (2 (e+f x))-3 \sin (4 (e+f x))+48 \cos (e+f x)+16 \cos (3 (e+f x))+60 f x)}{96 f}","\frac{5 a c^3 \cos ^3(e+f x)}{12 f}+\frac{a \cos ^3(e+f x) \left(c^3-c^3 \sin (e+f x)\right)}{4 f}+\frac{5 a c^3 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{5}{8} a c^3 x",1,"(a*c^3*(60*f*x + 48*Cos[e + f*x] + 16*Cos[3*(e + f*x)] + 24*Sin[2*(e + f*x)] - 3*Sin[4*(e + f*x)]))/(96*f)","A",1
229,1,42,52,0.277003,"\int (a+a \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^2,x]","\frac{a c^2 (3 \sin (2 (e+f x))+3 \cos (e+f x)+\cos (3 (e+f x))+6 f x)}{12 f}","\frac{a c^2 \cos ^3(e+f x)}{3 f}+\frac{a c^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} a c^2 x",1,"(a*c^2*(6*f*x + 3*Cos[e + f*x] + Cos[3*(e + f*x)] + 3*Sin[2*(e + f*x)]))/(12*f)","A",1
230,1,25,29,0.0227735,"\int (a+a \sin (e+f x)) (c-c \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])*(c - c*Sin[e + f*x]),x]","\frac{a c (2 (e+f x)+\sin (2 (e+f x)))}{4 f}","\frac{a c \sin (e+f x) \cos (e+f x)}{2 f}+\frac{a c x}{2}",1,"(a*c*(2*(e + f*x) + Sin[2*(e + f*x)]))/(4*f)","A",1
231,1,83,33,0.1914092,"\int \frac{a+a \sin (e+f x)}{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x]),x]","\frac{a \left(f x \sin \left(e+\frac{f x}{2}\right)+4 \sin \left(\frac{f x}{2}\right)-f x \cos \left(\frac{f x}{2}\right)\right)}{c 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)}","\frac{2 a \cos (e+f x)}{f (c-c \sin (e+f x))}-\frac{a x}{c}",1,"(a*(-(f*x*Cos[(f*x)/2]) + 4*Sin[(f*x)/2] + f*x*Sin[e + (f*x)/2]))/(c*f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","B",1
232,1,74,30,0.2790555,"\int \frac{a+a \sin (e+f x)}{(c-c \sin (e+f x))^2} \, dx","Integrate[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^2,x]","-\frac{a \left(\cos \left(e+\frac{3 f x}{2}\right)-3 \cos \left(e+\frac{f x}{2}\right)\right)}{3 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 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}",1,"-1/3*(a*(-3*Cos[e + (f*x)/2] + Cos[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
233,1,96,60,0.3384375,"\int \frac{a+a \sin (e+f x)}{(c-c \sin (e+f x))^3} \, dx","Integrate[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^3,x]","\frac{a \left(\sin \left(2 e+\frac{5 f x}{2}\right)+15 \cos \left(e+\frac{f x}{2}\right)-5 \cos \left(e+\frac{3 f x}{2}\right)+5 \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 \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^3}+\frac{a c \cos ^3(e+f x)}{5 f (c-c \sin (e+f x))^4}",1,"(a*(15*Cos[e + (f*x)/2] - 5*Cos[e + (3*f*x)/2] + 5*Sin[(f*x)/2] + 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
234,1,109,92,0.4822461,"\int \frac{a+a \sin (e+f x)}{(c-c \sin (e+f x))^4} \, dx","Integrate[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^4,x]","\frac{a \left(7 \sin \left(2 e+\frac{5 f x}{2}\right)+70 \cos \left(e+\frac{f x}{2}\right)-21 \cos \left(e+\frac{3 f x}{2}\right)+\cos \left(3 e+\frac{7 f x}{2}\right)+35 \sin \left(\frac{f x}{2}\right)\right)}{210 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{2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^3}+\frac{2 a \cos ^3(e+f x)}{35 f (c-c \sin (e+f x))^4}+\frac{a c \cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}",1,"(a*(70*Cos[e + (f*x)/2] - 21*Cos[e + (3*f*x)/2] + Cos[3*e + (7*f*x)/2] + 35*Sin[(f*x)/2] + 7*Sin[2*e + (5*f*x)/2]))/(210*c^4*f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7)","A",1
235,1,124,126,0.6191855,"\int \frac{a+a \sin (e+f x)}{(c-c \sin (e+f x))^5} \, dx","Integrate[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^5,x]","\frac{a \left(36 \sin \left(2 e+\frac{5 f x}{2}\right)-\sin \left(4 e+\frac{9 f x}{2}\right)+315 \cos \left(e+\frac{f x}{2}\right)-84 \cos \left(e+\frac{3 f x}{2}\right)+9 \cos \left(3 e+\frac{7 f x}{2}\right)+189 \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 c \cos ^3(e+f x)}{315 f \left(c^2-c^2 \sin (e+f x)\right)^3}+\frac{2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^4}+\frac{a \cos ^3(e+f x)}{21 f (c-c \sin (e+f x))^5}+\frac{a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}",1,"(a*(315*Cos[e + (f*x)/2] - 84*Cos[e + (3*f*x)/2] + 9*Cos[3*e + (7*f*x)/2] + 189*Sin[(f*x)/2] + 36*Sin[2*e + (5*f*x)/2] - 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
236,1,89,152,1.1109869,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5 \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5,x]","\frac{a^2 c^5 (665 \sin (2 (e+f x))-35 \sin (4 (e+f x))-35 \sin (6 (e+f x))+945 \cos (e+f x)+455 \cos (3 (e+f x))+77 \cos (5 (e+f x))-5 \cos (7 (e+f x))+1260 e+1260 f x)}{2240 f}","\frac{3 a^2 c^5 \cos ^5(e+f x)}{10 f}+\frac{3 a^2 \cos ^5(e+f x) \left(c^5-c^5 \sin (e+f x)\right)}{14 f}+\frac{3 a^2 c^5 \sin (e+f x) \cos ^3(e+f x)}{8 f}+\frac{9 a^2 c^5 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{9}{16} a^2 c^5 x+\frac{a^2 c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{7 f}",1,"(a^2*c^5*(1260*e + 1260*f*x + 945*Cos[e + f*x] + 455*Cos[3*(e + f*x)] + 77*Cos[5*(e + f*x)] - 5*Cos[7*(e + f*x)] + 665*Sin[2*(e + f*x)] - 35*Sin[4*(e + f*x)] - 35*Sin[6*(e + f*x)]))/(2240*f)","A",1
237,1,79,118,0.7265106,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4 \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^4,x]","\frac{a^2 c^4 (255 \sin (2 (e+f x))+15 \sin (4 (e+f x))-5 \sin (6 (e+f x))+240 \cos (e+f x)+120 \cos (3 (e+f x))+24 \cos (5 (e+f x))+420 e+420 f x)}{960 f}","\frac{7 a^2 c^4 \cos ^5(e+f x)}{30 f}+\frac{a^2 \cos ^5(e+f x) \left(c^4-c^4 \sin (e+f x)\right)}{6 f}+\frac{7 a^2 c^4 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{7 a^2 c^4 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{7}{16} a^2 c^4 x",1,"(a^2*c^4*(420*e + 420*f*x + 240*Cos[e + f*x] + 120*Cos[3*(e + f*x)] + 24*Cos[5*(e + f*x)] + 255*Sin[2*(e + f*x)] + 15*Sin[4*(e + f*x)] - 5*Sin[6*(e + f*x)]))/(960*f)","A",1
238,1,69,85,1.6426154,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^3,x]","\frac{a^2 c^3 (40 \sin (2 (e+f x))+5 \sin (4 (e+f x))+20 \cos (e+f x)+10 \cos (3 (e+f x))+2 \cos (5 (e+f x))+60 e+60 f x)}{160 f}","\frac{a^2 c^3 \cos ^5(e+f x)}{5 f}+\frac{a^2 c^3 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac{3 a^2 c^3 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} a^2 c^3 x",1,"(a^2*c^3*(60*e + 60*f*x + 20*Cos[e + f*x] + 10*Cos[3*(e + f*x)] + 2*Cos[5*(e + f*x)] + 40*Sin[2*(e + f*x)] + 5*Sin[4*(e + f*x)]))/(160*f)","A",1
239,1,39,64,0.043687,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^2,x]","\frac{a^2 c^2 (12 (e+f x)+8 \sin (2 (e+f x))+\sin (4 (e+f x)))}{32 f}","\frac{a^2 c^2 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac{3 a^2 c^2 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} a^2 c^2 x",1,"(a^2*c^2*(12*(e + f*x) + 8*Sin[2*(e + f*x)] + Sin[4*(e + f*x)]))/(32*f)","A",1
240,1,43,52,0.3292927,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","-\frac{a^2 c (-3 (\sin (2 (e+f x))+2 f x)+3 \cos (e+f x)+\cos (3 (e+f x)))}{12 f}","-\frac{a^2 c \cos ^3(e+f x)}{3 f}+\frac{a^2 c \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} a^2 c x",1,"-1/12*(a^2*c*(3*Cos[e + f*x] + Cos[3*(e + f*x)] - 3*(2*f*x + Sin[2*(e + f*x)])))/f","A",1
241,1,130,57,0.3802139,"\int \frac{(a+a \sin (e+f x))^2}{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(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) (3 (e+f x)-\cos (e+f x))+\sin \left(\frac{1}{2} (e+f x)\right) (\cos (e+f x)-3 e-3 f x-8)\right)}{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{3 a^2 \cos (e+f x)}{c f}+\frac{2 a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^2}-\frac{3 a^2 x}{c}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2]*(3*(e + f*x) - Cos[e + f*x]) + (-8 - 3*e - 3*f*x + Cos[e + f*x])*Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2)/(c*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-1 + Sin[e + f*x]))","B",1
242,1,121,72,0.6194877,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^2} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^2,x]","-\frac{a^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-3 (3 e+3 f x+8) \cos \left(\frac{1}{2} (e+f x)\right)+(3 e+3 f x+16) \cos \left(\frac{3}{2} (e+f x)\right)+6 \sin \left(\frac{1}{2} (e+f x)\right) (2 (e+f x+2)+(e+f x) \cos (e+f x))\right)}{6 c^2 f (\sin (e+f x)-1)^2}","-\frac{2 a^2 \cos (e+f x)}{f \left(c^2-c^2 \sin (e+f x)\right)}+\frac{a^2 x}{c^2}+\frac{2 a^2 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}",1,"-1/6*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(-3*(8 + 3*e + 3*f*x)*Cos[(e + f*x)/2] + (16 + 3*e + 3*f*x)*Cos[(3*(e + f*x))/2] + 6*(2*(2 + e + f*x) + (e + f*x)*Cos[e + f*x])*Sin[(e + f*x)/2]))/(c^2*f*(-1 + Sin[e + f*x])^2)","A",1
243,1,81,34,0.4009511,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^3} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^3,x]","\frac{a^2 \left(-10 \sin \left(\frac{1}{2} (e+f x)\right)-5 \sin \left(\frac{3}{2} (e+f x)\right)+\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)}{10 c^3 f (\sin (e+f x)-1)^3}","\frac{a^2 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(-10*Sin[(e + f*x)/2] - 5*Sin[(3*(e + f*x))/2] + Sin[(5*(e + f*x))/2]))/(10*c^3*f*(-1 + Sin[e + f*x])^3)","B",1
244,1,117,67,0.6275866,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^4} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^4,x]","-\frac{a^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-70 \sin \left(\frac{1}{2} (e+f x)\right)-35 \sin \left(\frac{3}{2} (e+f x)\right)+7 \sin \left(\frac{5}{2} (e+f x)\right)-35 \cos \left(\frac{1}{2} (e+f x)\right)+14 \cos \left(\frac{3}{2} (e+f x)\right)+\cos \left(\frac{7}{2} (e+f x)\right)\right)}{140 c^4 f (\sin (e+f x)-1)^4}","\frac{a^2 c^2 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}+\frac{a^2 c \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^5}",1,"-1/140*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(-35*Cos[(e + f*x)/2] + 14*Cos[(3*(e + f*x))/2] + Cos[(7*(e + f*x))/2] - 70*Sin[(e + f*x)/2] - 35*Sin[(3*(e + f*x))/2] + 7*Sin[(5*(e + f*x))/2]))/(c^4*f*(-1 + Sin[e + f*x])^4)","A",1
245,1,121,98,0.5798005,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^5} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^5,x]","\frac{a^2 \left(441 \sin \left(\frac{1}{2} (e+f x)\right)+210 \sin \left(\frac{3}{2} (e+f x)\right)-36 \sin \left(\frac{5}{2} (e+f x)\right)+\sin \left(\frac{9}{2} (e+f x)\right)+315 \cos \left(\frac{1}{2} (e+f x)\right)-126 \cos \left(\frac{3}{2} (e+f x)\right)-9 \cos \left(\frac{7}{2} (e+f x)\right)\right)}{1260 c^5 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}","\frac{a^2 c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac{2 a^2 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5}+\frac{2 a^2 c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6}",1,"(a^2*(315*Cos[(e + f*x)/2] - 126*Cos[(3*(e + f*x))/2] - 9*Cos[(7*(e + f*x))/2] + 441*Sin[(e + f*x)/2] + 210*Sin[(3*(e + f*x))/2] - 36*Sin[(5*(e + f*x))/2] + Sin[(9*(e + f*x))/2]))/(1260*c^5*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9)","A",1
246,1,133,132,0.7426446,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^6} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^6,x]","\frac{a^2 \left(2541 \sin \left(\frac{1}{2} (e+f x)\right)+1155 \sin \left(\frac{3}{2} (e+f x)\right)-165 \sin \left(\frac{5}{2} (e+f x)\right)+11 \sin \left(\frac{9}{2} (e+f x)\right)+2079 \cos \left(\frac{1}{2} (e+f x)\right)-825 \cos \left(\frac{3}{2} (e+f x)\right)-55 \cos \left(\frac{7}{2} (e+f x)\right)+\cos \left(\frac{11}{2} (e+f x)\right)\right)}{9240 c^6 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11}}","\frac{a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{2 a^2 \cos ^5(e+f x)}{1155 c f (c-c \sin (e+f x))^5}+\frac{2 a^2 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^6}+\frac{a^2 c \cos ^5(e+f x)}{33 f (c-c \sin (e+f x))^7}",1,"(a^2*(2079*Cos[(e + f*x)/2] - 825*Cos[(3*(e + f*x))/2] - 55*Cos[(7*(e + f*x))/2] + Cos[(11*(e + f*x))/2] + 2541*Sin[(e + f*x)/2] + 1155*Sin[(3*(e + f*x))/2] - 165*Sin[(5*(e + f*x))/2] + 11*Sin[(9*(e + f*x))/2]))/(9240*c^6*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11)","A",1
247,1,109,180,2.1353416,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6 \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6,x]","\frac{a^3 c^6 (18144 \sin (2 (e+f x))+1512 \sin (4 (e+f x))-672 \sin (6 (e+f x))-189 \sin (8 (e+f x))+16632 \cos (e+f x)+9744 \cos (3 (e+f x))+3024 \cos (5 (e+f x))+324 \cos (7 (e+f x))-28 \cos (9 (e+f x))+27720 e+27720 f x)}{64512 f}","\frac{11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac{11 a^3 \cos ^7(e+f x) \left(c^6-c^6 \sin (e+f x)\right)}{72 f}+\frac{11 a^3 c^6 \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac{55 a^3 c^6 \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{55 a^3 c^6 \sin (e+f x) \cos (e+f x)}{128 f}+\frac{55}{128} a^3 c^6 x+\frac{a^3 \cos ^7(e+f x) \left(c^3-c^3 \sin (e+f x)\right)^2}{9 f}",1,"(a^3*c^6*(27720*e + 27720*f*x + 16632*Cos[e + f*x] + 9744*Cos[3*(e + f*x)] + 3024*Cos[5*(e + f*x)] + 324*Cos[7*(e + f*x)] - 28*Cos[9*(e + f*x)] + 18144*Sin[2*(e + f*x)] + 1512*Sin[4*(e + f*x)] - 672*Sin[6*(e + f*x)] - 189*Sin[8*(e + f*x)]))/(64512*f)","A",1
248,1,89,145,1.2118136,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5,x]","\frac{a^3 c^5 (1792 \sin (2 (e+f x))+280 \sin (4 (e+f x))-7 \sin (8 (e+f x))+1120 \cos (e+f x)+672 \cos (3 (e+f x))+224 \cos (5 (e+f x))+32 \cos (7 (e+f x))+2520 e+2520 f x)}{7168 f}","\frac{9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac{a^3 \cos ^7(e+f x) \left(c^5-c^5 \sin (e+f x)\right)}{8 f}+\frac{3 a^3 c^5 \sin (e+f x) \cos ^5(e+f x)}{16 f}+\frac{15 a^3 c^5 \sin (e+f x) \cos ^3(e+f x)}{64 f}+\frac{45 a^3 c^5 \sin (e+f x) \cos (e+f x)}{128 f}+\frac{45}{128} a^3 c^5 x",1,"(a^3*c^5*(2520*e + 2520*f*x + 1120*Cos[e + f*x] + 672*Cos[3*(e + f*x)] + 224*Cos[5*(e + f*x)] + 32*Cos[7*(e + f*x)] + 1792*Sin[2*(e + f*x)] + 280*Sin[4*(e + f*x)] - 7*Sin[8*(e + f*x)]))/(7168*f)","A",1
249,1,89,112,1.0670846,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4 \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4,x]","\frac{a^3 c^4 (315 \sin (2 (e+f x))+63 \sin (4 (e+f x))+7 \sin (6 (e+f x))+105 \cos (e+f x)+63 \cos (3 (e+f x))+21 \cos (5 (e+f x))+3 \cos (7 (e+f x))+420 e+420 f x)}{1344 f}","\frac{a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac{a^3 c^4 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{5 a^3 c^4 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{5 a^3 c^4 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} a^3 c^4 x",1,"(a^3*c^4*(420*e + 420*f*x + 105*Cos[e + f*x] + 63*Cos[3*(e + f*x)] + 21*Cos[5*(e + f*x)] + 3*Cos[7*(e + f*x)] + 315*Sin[2*(e + f*x)] + 63*Sin[4*(e + f*x)] + 7*Sin[6*(e + f*x)]))/(1344*f)","A",1
250,1,49,91,0.0485232,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^3,x]","\frac{a^3 c^3 (45 \sin (2 (e+f x))+9 \sin (4 (e+f x))+\sin (6 (e+f x))+60 e+60 f x)}{192 f}","\frac{a^3 c^3 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{5 a^3 c^3 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{5 a^3 c^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} a^3 c^3 x",1,"(a^3*c^3*(60*e + 60*f*x + 45*Sin[2*(e + f*x)] + 9*Sin[4*(e + f*x)] + Sin[6*(e + f*x)]))/(192*f)","A",1
251,1,69,85,1.6500585,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^2,x]","\frac{a^3 c^2 (40 \sin (2 (e+f x))+5 \sin (4 (e+f x))-20 \cos (e+f x)-10 \cos (3 (e+f x))-2 \cos (5 (e+f x))+60 e+60 f x)}{160 f}","-\frac{a^3 c^2 \cos ^5(e+f x)}{5 f}+\frac{a^3 c^2 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac{3 a^3 c^2 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} a^3 c^2 x",1,"(a^3*c^2*(60*e + 60*f*x - 20*Cos[e + f*x] - 10*Cos[3*(e + f*x)] - 2*Cos[5*(e + f*x)] + 40*Sin[2*(e + f*x)] + 5*Sin[4*(e + f*x)]))/(160*f)","A",1
252,1,54,82,0.3829959,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x]),x]","\frac{a^3 c (24 \sin (2 (e+f x))-3 \sin (4 (e+f x))-48 \cos (e+f x)-16 \cos (3 (e+f x))+60 f x)}{96 f}","-\frac{5 a^3 c \cos ^3(e+f x)}{12 f}-\frac{c \cos ^3(e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{4 f}+\frac{5 a^3 c \sin (e+f x) \cos (e+f x)}{8 f}+\frac{5}{8} a^3 c x",1,"(a^3*c*(60*f*x - 48*Cos[e + f*x] - 16*Cos[3*(e + f*x)] + 24*Sin[2*(e + f*x)] - 3*Sin[4*(e + f*x)]))/(96*f)","A",1
253,1,153,94,0.5314499,"\int \frac{(a+a \sin (e+f x))^3}{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(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 (e+f x)-\sin (2 (e+f x))-16 \cos (e+f x))+\sin \left(\frac{1}{2} (e+f x)\right) (\sin (2 (e+f x))+16 \cos (e+f x)-30 e-30 f x-64)\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)^6}","\frac{2 a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^3}+\frac{15 a^3 \cos (e+f x)}{2 c f}+\frac{5 a^3 \cos ^3(e+f x)}{2 f (c-c \sin (e+f x))}-\frac{15 a^3 x}{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*(e + f*x) - 16*Cos[e + f*x] - Sin[2*(e + f*x)]) + Sin[(e + f*x)/2]*(-64 - 30*e - 30*f*x + 16*Cos[e + f*x] + Sin[2*(e + f*x)])))/(4*c*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-1 + Sin[e + f*x]))","A",1
254,1,149,92,1.0019389,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^2} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^2,x]","\frac{a^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(6 (15 e+15 f x+23) \cos \left(\frac{1}{2} (e+f x)\right)-(30 e+30 f x+121) \cos \left(\frac{3}{2} (e+f x)\right)+3 \cos \left(\frac{5}{2} (e+f x)\right)-6 \sin \left(\frac{1}{2} (e+f x)\right) (2 (5 e+5 f x-2) \cos (e+f x)-\cos (2 (e+f x))+20 e+20 f x+31)\right)}{12 c^2 f (\sin (e+f x)-1)^2}","-\frac{5 a^3 \cos (e+f x)}{c^2 f}+\frac{2 a^3 c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^4}+\frac{5 a^3 x}{c^2}-\frac{10 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^2}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(6*(23 + 15*e + 15*f*x)*Cos[(e + f*x)/2] - (121 + 30*e + 30*f*x)*Cos[(3*(e + f*x))/2] + 3*Cos[(5*(e + f*x))/2] - 6*(31 + 20*e + 20*f*x + 2*(-2 + 5*e + 5*f*x)*Cos[e + f*x] - Cos[2*(e + f*x)])*Sin[(e + f*x)/2]))/(12*c^2*f*(-1 + Sin[e + f*x])^2)","A",1
255,1,249,106,0.4583181,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^3} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^3,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(48 \sin \left(\frac{1}{2} (e+f x)\right)-15 (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5+92 \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)^4-44 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-88 \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 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\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{2 a^3 \cos (e+f x)}{f \left(c^3-c^3 \sin (e+f x)\right)}-\frac{a^3 x}{c^3}+\frac{2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}-\frac{2 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(24*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 44*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 - 15*(e + f*x)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + 48*Sin[(e + f*x)/2] - 88*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + 92*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(e + f*x)/2])*(a + a*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
256,1,93,34,0.8026663,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^4,x]","\frac{a^3 \left(35 \cos \left(\frac{1}{2} (e+f x)\right)-21 \cos \left(\frac{3}{2} (e+f x)\right)-7 \cos \left(\frac{5}{2} (e+f x)\right)+\cos \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)}{28 c^4 f (\sin (e+f x)-1)^4}","\frac{a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}",1,"(a^3*(35*Cos[(e + f*x)/2] - 21*Cos[(3*(e + f*x))/2] - 7*Cos[(5*(e + f*x))/2] + Cos[(7*(e + f*x))/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/(28*c^4*f*(-1 + Sin[e + f*x])^4)","B",1
257,1,135,69,0.7388709,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^5,x]","\frac{a^3 \left(189 \sin \left(\frac{1}{2} (e+f x)\right)+105 \sin \left(\frac{3}{2} (e+f x)\right)-27 \sin \left(\frac{5}{2} (e+f x)\right)-\sin \left(\frac{9}{2} (e+f x)\right)+315 \cos \left(\frac{1}{2} (e+f x)\right)-189 \cos \left(\frac{3}{2} (e+f x)\right)-63 \cos \left(\frac{5}{2} (e+f x)\right)+9 \cos \left(\frac{7}{2} (e+f x)\right)\right)}{504 c^5 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}","\frac{a^3 c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}+\frac{a^3 c^2 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^7}",1,"(a^3*(315*Cos[(e + f*x)/2] - 189*Cos[(3*(e + f*x))/2] - 63*Cos[(5*(e + f*x))/2] + 9*Cos[(7*(e + f*x))/2] + 189*Sin[(e + f*x)/2] + 105*Sin[(3*(e + f*x))/2] - 27*Sin[(5*(e + f*x))/2] - Sin[(9*(e + f*x))/2]))/(504*c^5*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9)","A",1
258,1,145,101,0.9012456,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^6} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^6,x]","-\frac{a^3 \left(-2079 \sin \left(\frac{1}{2} (e+f x)\right)-1155 \sin \left(\frac{3}{2} (e+f x)\right)+297 \sin \left(\frac{5}{2} (e+f x)\right)+11 \sin \left(\frac{9}{2} (e+f x)\right)-2541 \cos \left(\frac{1}{2} (e+f x)\right)+1485 \cos \left(\frac{3}{2} (e+f x)\right)+462 \cos \left(\frac{5}{2} (e+f x)\right)-55 \cos \left(\frac{7}{2} (e+f x)\right)+\cos \left(\frac{11}{2} (e+f x)\right)\right)}{5544 c^6 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11}}","\frac{a^3 c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac{2 a^3 c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac{2 a^3 c \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^7}",1,"-1/5544*(a^3*(-2541*Cos[(e + f*x)/2] + 1485*Cos[(3*(e + f*x))/2] + 462*Cos[(5*(e + f*x))/2] - 55*Cos[(7*(e + f*x))/2] + Cos[(11*(e + f*x))/2] - 2079*Sin[(e + f*x)/2] - 1155*Sin[(3*(e + f*x))/2] + 297*Sin[(5*(e + f*x))/2] + 11*Sin[(9*(e + f*x))/2]))/(c^6*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11)","A",1
259,1,157,132,1.9847717,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^7} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^7,x]","\frac{a^3 \left(16302 \sin \left(\frac{1}{2} (e+f x)\right)+9009 \sin \left(\frac{3}{2} (e+f x)\right)-2288 \sin \left(\frac{5}{2} (e+f x)\right)-78 \sin \left(\frac{9}{2} (e+f x)\right)+\sin \left(\frac{13}{2} (e+f x)\right)+18018 \cos \left(\frac{1}{2} (e+f x)\right)-10296 \cos \left(\frac{3}{2} (e+f x)\right)-3003 \cos \left(\frac{5}{2} (e+f x)\right)+286 \cos \left(\frac{7}{2} (e+f x)\right)-13 \cos \left(\frac{11}{2} (e+f x)\right)\right)}{48048 c^7 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{13}}","\frac{a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac{3 a^3 c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac{2 a^3 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^7}+\frac{2 a^3 c \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^8}",1,"(a^3*(18018*Cos[(e + f*x)/2] - 10296*Cos[(3*(e + f*x))/2] - 3003*Cos[(5*(e + f*x))/2] + 286*Cos[(7*(e + f*x))/2] - 13*Cos[(11*(e + f*x))/2] + 16302*Sin[(e + f*x)/2] + 9009*Sin[(3*(e + f*x))/2] - 2288*Sin[(5*(e + f*x))/2] - 78*Sin[(9*(e + f*x))/2] + Sin[(13*(e + f*x))/2]))/(48048*c^7*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^13)","A",1
260,1,209,166,1.9245113,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^8} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(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(109395 \sin \left(\frac{1}{2} (e+f x)\right)+60060 \sin \left(\frac{3}{2} (e+f x)\right)-15015 \sin \left(\frac{5}{2} (e+f x)\right)-455 \sin \left(\frac{9}{2} (e+f x)\right)+15 \sin \left(\frac{13}{2} (e+f x)\right)+115830 \cos \left(\frac{1}{2} (e+f x)\right)-65065 \cos \left(\frac{3}{2} (e+f x)\right)-18018 \cos \left(\frac{5}{2} (e+f x)\right)+1365 \cos \left(\frac{7}{2} (e+f x)\right)-105 \cos \left(\frac{11}{2} (e+f x)\right)+\cos \left(\frac{15}{2} (e+f x)\right)\right)}{360360 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 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{8 a^3 \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7}+\frac{8 a^3 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac{4 a^3 c \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*(115830*Cos[(e + f*x)/2] - 65065*Cos[(3*(e + f*x))/2] - 18018*Cos[(5*(e + f*x))/2] + 1365*Cos[(7*(e + f*x))/2] - 105*Cos[(11*(e + f*x))/2] + Cos[(15*(e + f*x))/2] + 109395*Sin[(e + f*x)/2] + 60060*Sin[(3*(e + f*x))/2] - 15015*Sin[(5*(e + f*x))/2] - 455*Sin[(9*(e + f*x))/2] + 15*Sin[(13*(e + f*x))/2]))/(360360*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^8)","A",1
261,1,175,118,1.4362673,"\int \frac{(c-c \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx","Integrate[(c - c*Sin[e + f*x])^4/(a + a*Sin[e + f*x]),x]","-\frac{c^4 (\sin (e+f x)-1)^4 \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) (-15 \sin (2 (e+f x))+141 \cos (e+f x)-\cos (3 (e+f x))+210 e+210 f x-384)+\cos \left(\frac{1}{2} (e+f x)\right) (-15 \sin (2 (e+f x))+141 \cos (e+f x)-\cos (3 (e+f x))+210 e+210 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)^8}","-\frac{2 a^3 c^4 \cos ^7(e+f x)}{f (a \sin (e+f x)+a)^4}-\frac{35 c^4 \cos ^3(e+f x)}{3 a f}-\frac{14 a c^4 \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^2}-\frac{35 c^4 \sin (e+f x) \cos (e+f x)}{2 a f}-\frac{35 c^4 x}{2 a}",1,"-1/12*(c^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^4*(Sin[(e + f*x)/2]*(-384 + 210*e + 210*f*x + 141*Cos[e + f*x] - Cos[3*(e + f*x)] - 15*Sin[2*(e + f*x)]) + Cos[(e + f*x)/2]*(210*e + 210*f*x + 141*Cos[e + f*x] - Cos[3*(e + f*x)] - 15*Sin[2*(e + f*x)])))/(a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8*(1 + Sin[e + f*x]))","A",1
262,1,155,92,0.5106082,"\int \frac{(c-c \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx","Integrate[(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(\sin \left(\frac{1}{2} (e+f x)\right) (-\sin (2 (e+f x))+16 \cos (e+f x)+30 e+30 f x-64)+\cos \left(\frac{1}{2} (e+f x)\right) (30 (e+f x)-\sin (2 (e+f x))+16 \cos (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)^6}","-\frac{2 a^2 c^3 \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^3}-\frac{15 c^3 \cos (e+f x)}{2 a f}-\frac{5 c^3 \cos ^3(e+f x)}{2 f (a \sin (e+f x)+a)}-\frac{15 c^3 x}{2 a}",1,"(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^3*(Sin[(e + f*x)/2]*(-64 + 30*e + 30*f*x + 16*Cos[e + f*x] - Sin[2*(e + f*x)]) + Cos[(e + f*x)/2]*(30*(e + f*x) + 16*Cos[e + f*x] - Sin[2*(e + f*x)])))/(4*a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6*(1 + Sin[e + f*x]))","A",1
263,1,129,56,0.3667403,"\int \frac{(c-c \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx","Integrate[(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) (3 (e+f x)+\cos (e+f x))+\sin \left(\frac{1}{2} (e+f x)\right) (\cos (e+f x)+3 e+3 f x-8)\right)}{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{3 c^2 \cos (e+f x)}{a f}-\frac{2 a c^2 \cos ^3(e+f x)}{f (a \sin (e+f x)+a)^2}-\frac{3 c^2 x}{a}",1,"-((c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(Cos[(e + f*x)/2]*(3*(e + f*x) + Cos[e + f*x]) + (-8 + 3*e + 3*f*x + Cos[e + f*x])*Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2)/(a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*(1 + Sin[e + f*x])))","B",1
264,1,79,32,0.1862497,"\int \frac{c-c \sin (e+f x)}{a+a \sin (e+f x)} \, dx","Integrate[(c - c*Sin[e + f*x])/(a + a*Sin[e + f*x]),x]","-\frac{c \left(f x \sin \left(e+\frac{f x}{2}\right)-4 \sin \left(\frac{f x}{2}\right)+f x \cos \left(\frac{f x}{2}\right)\right)}{a 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)}","-\frac{2 c \cos (e+f x)}{f (a \sin (e+f x)+a)}-\frac{c x}{a}",1,"-((c*(f*x*Cos[(f*x)/2] - 4*Sin[(f*x)/2] + f*x*Sin[e + (f*x)/2]))/(a*f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))","B",1
265,1,16,16,0.0115889,"\int \frac{1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))} \, dx","Integrate[1/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])),x]","\frac{\tan (e+f x)}{a c f}","\frac{\tan (e+f x)}{a c f}",1,"Tan[e + f*x]/(a*c*f)","A",1
266,1,87,53,0.4214513,"\int \frac{1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx","Integrate[1/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^2),x]","\frac{\sin (e+f x)+8 \sin (2 (e+f x))+\sin (3 (e+f x))+4 \cos (e+f x)-2 \cos (2 (e+f x))+4 \cos (3 (e+f x))-2}{24 a c^2 f (\sin (e+f x)-1)^2 (\sin (e+f x)+1)}","\frac{2 \tan (e+f x)}{3 a c^2 f}+\frac{\sec (e+f x)}{3 a f \left(c^2-c^2 \sin (e+f x)\right)}",1,"(-2 + 4*Cos[e + f*x] - 2*Cos[2*(e + f*x)] + 4*Cos[3*(e + f*x)] + Sin[e + f*x] + 8*Sin[2*(e + f*x)] + Sin[3*(e + f*x)])/(24*a*c^2*f*(-1 + Sin[e + f*x])^2*(1 + Sin[e + f*x]))","A",1
267,1,111,85,0.667166,"\int \frac{1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx","Integrate[1/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^3),x]","-\frac{12 \sin (e+f x)+32 \sin (2 (e+f x))+12 \sin (3 (e+f x))-8 \sin (4 (e+f x))+32 \cos (e+f x)-12 \cos (2 (e+f x))+32 \cos (3 (e+f x))+3 \cos (4 (e+f x))-15}{160 a c^3 f (\sin (e+f x)-1)^3 (\sin (e+f x)+1)}","\frac{2 \tan (e+f x)}{5 a c^3 f}+\frac{\sec (e+f x)}{5 a f \left(c^3-c^3 \sin (e+f x)\right)}+\frac{\sec (e+f x)}{5 a c f (c-c \sin (e+f x))^2}",1,"-1/160*(-15 + 32*Cos[e + f*x] - 12*Cos[2*(e + f*x)] + 32*Cos[3*(e + f*x)] + 3*Cos[4*(e + f*x)] + 12*Sin[e + f*x] + 32*Sin[2*(e + f*x)] + 12*Sin[3*(e + f*x)] - 8*Sin[4*(e + f*x)])/(a*c^3*f*(-1 + Sin[e + f*x])^3*(1 + Sin[e + f*x]))","A",1
268,1,131,118,0.740997,"\int \frac{1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^4} \, dx","Integrate[1/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^4),x]","\frac{406 \sin (e+f x)+512 \sin (2 (e+f x))+377 \sin (3 (e+f x))-384 \sin (4 (e+f x))-29 \sin (5 (e+f x))+896 \cos (e+f x)-232 \cos (2 (e+f x))+832 \cos (3 (e+f x))+174 \cos (4 (e+f x))-64 \cos (5 (e+f x))-406}{4480 a c^4 f (\sin (e+f x)-1)^4 (\sin (e+f x)+1)}","\frac{8 \tan (e+f x)}{35 a c^4 f}+\frac{4 \sec (e+f x)}{35 a f \left(c^4-c^4 \sin (e+f x)\right)}+\frac{4 \sec (e+f x)}{35 a f \left(c^2-c^2 \sin (e+f x)\right)^2}+\frac{\sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}",1,"(-406 + 896*Cos[e + f*x] - 232*Cos[2*(e + f*x)] + 832*Cos[3*(e + f*x)] + 174*Cos[4*(e + f*x)] - 64*Cos[5*(e + f*x)] + 406*Sin[e + f*x] + 512*Sin[2*(e + f*x)] + 377*Sin[3*(e + f*x)] - 384*Sin[4*(e + f*x)] - 29*Sin[5*(e + f*x)])/(4480*a*c^4*f*(-1 + Sin[e + f*x])^4*(1 + Sin[e + f*x]))","A",1
269,1,276,148,0.7235981,"\int \frac{(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx","Integrate[(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(256 \sin \left(\frac{1}{2} (e+f x)\right)+630 (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+285 \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-\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)^3-21 \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)^3-1664 \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)^2-128 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{12 f (a \sin (e+f x)+a)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{10}}","-\frac{2 a^4 c^5 \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}+\frac{35 c^5 \cos ^3(e+f x)}{a^2 f}+\frac{6 a^2 c^5 \cos ^7(e+f x)}{f (a \sin (e+f x)+a)^4}+\frac{105 c^5 \sin (e+f x) \cos (e+f x)}{2 a^2 f}+\frac{105 c^5 x}{2 a^2}+\frac{42 c^5 \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])^5*(256*Sin[(e + f*x)/2] - 128*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 1664*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 630*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 285*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 21*(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])^10*(a + a*Sin[e + f*x])^2)","A",1
270,1,243,135,0.49426,"\int \frac{(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx","Integrate[(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 \sin \left(\frac{1}{2} (e+f x)\right)+210 (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+72 \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 \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)^3-640 \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 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{12 f (a \sin (e+f x)+a)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^8}","-\frac{2 a^3 c^4 \cos ^7(e+f x)}{3 f (a \sin (e+f x)+a)^5}+\frac{35 c^4 \cos (e+f x)}{2 a^2 f}+\frac{35 c^4 \cos ^3(e+f x)}{6 f \left(a^2 \sin (e+f x)+a^2\right)}+\frac{35 c^4 x}{2 a^2}+\frac{14 a^4 c^4 \cos ^5(e+f x)}{3 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])^4*(128*Sin[(e + f*x)/2] - 64*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 640*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 210*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 72*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 3*(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])^8*(a + a*Sin[e + f*x])^2)","A",1
271,1,210,90,0.3791203,"\int \frac{(c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx","Integrate[(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(16 \sin \left(\frac{1}{2} (e+f x)\right)+15 (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 \cos (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-56 \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)^2-8 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{3 f (a \sin (e+f x)+a)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{5 c^3 \cos (e+f x)}{a^2 f}-\frac{2 a^2 c^3 \cos ^5(e+f x)}{3 f (a \sin (e+f x)+a)^4}+\frac{5 c^3 x}{a^2}+\frac{10 c^3 \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])*(16*Sin[(e + f*x)/2] - 8*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 56*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 15*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 3*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)*(c - c*Sin[e + f*x])^3)/(3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6*(a + a*Sin[e + f*x])^2)","B",1
272,1,119,70,0.6186986,"\int \frac{(c-c \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c - c*Sin[e + f*x])^2/(a + a*Sin[e + f*x])^2,x]","\frac{c^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(3 (3 e+3 f x-8) \cos \left(\frac{1}{2} (e+f x)\right)+(-3 e-3 f x+16) \cos \left(\frac{3}{2} (e+f x)\right)+6 \sin \left(\frac{1}{2} (e+f x)\right) (2 (e+f x-2)+(e+f x) \cos (e+f x))\right)}{6 a^2 f (\sin (e+f x)+1)^2}","\frac{2 c^2 \cos (e+f x)}{f \left(a^2 \sin (e+f x)+a^2\right)}+\frac{c^2 x}{a^2}-\frac{2 a c^2 \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^3}",1,"(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(3*(-8 + 3*e + 3*f*x)*Cos[(e + f*x)/2] + (16 - 3*e - 3*f*x)*Cos[(3*(e + f*x))/2] + 6*(2*(-2 + e + f*x) + (e + f*x)*Cos[e + f*x])*Sin[(e + f*x)/2]))/(6*a^2*f*(1 + Sin[e + f*x])^2)","A",1
273,1,70,29,0.2712752,"\int \frac{c-c \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c - c*Sin[e + f*x])/(a + a*Sin[e + f*x])^2,x]","\frac{c \left(\cos \left(e+\frac{3 f x}{2}\right)-3 \cos \left(e+\frac{f x}{2}\right)\right)}{3 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{a c \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^3}",1,"(c*(-3*Cos[e + (f*x)/2] + Cos[e + (3*f*x)/2]))/(3*a^2*f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","B",1
274,1,87,52,0.479195,"\int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx","Integrate[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])),x]","-\frac{\sin (e+f x)+8 \sin (2 (e+f x))+\sin (3 (e+f x))-4 \cos (e+f x)+2 \cos (2 (e+f x))-4 \cos (3 (e+f x))+2}{24 a^2 c f (\sin (e+f x)-1) (\sin (e+f x)+1)^2}","\frac{2 \tan (e+f x)}{3 a^2 c f}-\frac{\sec (e+f x)}{3 c f \left(a^2 \sin (e+f x)+a^2\right)}",1,"-1/24*(2 - 4*Cos[e + f*x] + 2*Cos[2*(e + f*x)] - 4*Cos[3*(e + f*x)] + Sin[e + f*x] + 8*Sin[2*(e + f*x)] + Sin[3*(e + f*x)])/(a^2*c*f*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])^2)","A",1
275,1,29,38,0.0523771,"\int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^2} \, dx","Integrate[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^2),x]","\frac{\frac{1}{3} \tan ^3(e+f x)+\tan (e+f x)}{a^2 c^2 f}","\frac{\tan ^3(e+f x)}{3 a^2 c^2 f}+\frac{\tan (e+f x)}{a^2 c^2 f}",1,"(Tan[e + f*x] + Tan[e + f*x]^3/3)/(a^2*c^2*f)","A",1
276,1,131,76,0.8944542,"\int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx","Integrate[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^3),x]","-\frac{18 \sin (e+f x)+512 \sin (2 (e+f x))+27 \sin (3 (e+f x))+128 \sin (4 (e+f x))+9 \sin (5 (e+f x))+128 \cos (e+f x)-72 \cos (2 (e+f x))+192 \cos (3 (e+f x))-18 \cos (4 (e+f x))+64 \cos (5 (e+f x))-54}{1920 a^2 c^3 f (\sin (e+f x)-1)^3 (\sin (e+f x)+1)^2}","\frac{4 \tan ^3(e+f x)}{15 a^2 c^3 f}+\frac{4 \tan (e+f x)}{5 a^2 c^3 f}+\frac{\sec ^3(e+f x)}{5 a^2 f \left(c^3-c^3 \sin (e+f x)\right)}",1,"-1/1920*(-54 + 128*Cos[e + f*x] - 72*Cos[2*(e + f*x)] + 192*Cos[3*(e + f*x)] - 18*Cos[4*(e + f*x)] + 64*Cos[5*(e + f*x)] + 18*Sin[e + f*x] + 512*Sin[2*(e + f*x)] + 27*Sin[3*(e + f*x)] + 128*Sin[4*(e + f*x)] + 9*Sin[5*(e + f*x)])/(a^2*c^3*f*(-1 + Sin[e + f*x])^3*(1 + Sin[e + f*x])^2)","A",1
277,1,151,111,0.9504029,"\int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4} \, dx","Integrate[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^4),x]","\frac{120 \sin (e+f x)+1088 \sin (2 (e+f x))+180 \sin (3 (e+f x))+128 \sin (4 (e+f x))+60 \sin (5 (e+f x))-64 \sin (6 (e+f x))+512 \cos (e+f x)-255 \cos (2 (e+f x))+768 \cos (3 (e+f x))-30 \cos (4 (e+f x))+256 \cos (5 (e+f x))+15 \cos (6 (e+f x))-210}{5376 a^2 c^4 f (\sin (e+f x)-1)^4 (\sin (e+f x)+1)^2}","\frac{4 \tan ^3(e+f x)}{21 a^2 c^4 f}+\frac{4 \tan (e+f x)}{7 a^2 c^4 f}+\frac{\sec ^3(e+f x)}{7 a^2 f \left(c^4-c^4 \sin (e+f x)\right)}+\frac{\sec ^3(e+f x)}{7 a^2 f \left(c^2-c^2 \sin (e+f x)\right)^2}",1,"(-210 + 512*Cos[e + f*x] - 255*Cos[2*(e + f*x)] + 768*Cos[3*(e + f*x)] - 30*Cos[4*(e + f*x)] + 256*Cos[5*(e + f*x)] + 15*Cos[6*(e + f*x)] + 120*Sin[e + f*x] + 1088*Sin[2*(e + f*x)] + 180*Sin[3*(e + f*x)] + 128*Sin[4*(e + f*x)] + 60*Sin[5*(e + f*x)] - 64*Sin[6*(e + f*x)])/(5376*a^2*c^4*f*(-1 + Sin[e + f*x])^4*(1 + Sin[e + f*x])^2)","A",1
278,1,193,144,1.1616521,"\int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx","Integrate[1/((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) (18432 \sin (e+f x)+4185 \sin (2 (e+f x))+1024 \sin (3 (e+f x))+1860 \sin (4 (e+f x))-3072 \sin (5 (e+f x))-155 \sin (6 (e+f x))-5580 \cos (e+f x)+13824 \cos (2 (e+f x))-310 \cos (3 (e+f x))+6144 \cos (4 (e+f x))+930 \cos (5 (e+f x))-512 \cos (6 (e+f x)))}{64512 f (a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^5}","\frac{8 \tan ^3(e+f x)}{63 a^2 c^5 f}+\frac{8 \tan (e+f x)}{21 a^2 c^5 f}+\frac{2 \sec ^3(e+f x)}{21 a^2 f \left(c^5-c^5 \sin (e+f x)\right)}+\frac{2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac{\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])*(-5580*Cos[e + f*x] + 13824*Cos[2*(e + f*x)] - 310*Cos[3*(e + f*x)] + 6144*Cos[4*(e + f*x)] + 930*Cos[5*(e + f*x)] - 512*Cos[6*(e + f*x)] + 18432*Sin[e + f*x] + 4185*Sin[2*(e + f*x)] + 1024*Sin[3*(e + f*x)] + 1860*Sin[4*(e + f*x)] - 3072*Sin[5*(e + f*x)] - 155*Sin[6*(e + f*x)]))/(64512*f*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5)","A",1
279,1,303,161,0.8572263,"\int \frac{(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx","Integrate[(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(256 \sin \left(\frac{1}{2} (e+f x)\right)-630 (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5-160 \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+5 \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+2304 \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+448 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-896 \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)^2-128 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{20 f (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)^{10}}","-\frac{2 a^4 c^5 \cos ^9(e+f x)}{5 f (a \sin (e+f x)+a)^7}-\frac{63 c^5 \cos (e+f x)}{2 a^3 f}-\frac{21 c^5 \cos ^3(e+f x)}{2 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{63 c^5 x}{2 a^3}+\frac{6 a^2 c^5 \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^5}-\frac{42 c^5 \cos ^5(e+f x)}{5 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])^5*(256*Sin[(e + f*x)/2] - 128*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 896*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 448*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 2304*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 630*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 160*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + 5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sin[2*(e + f*x)]))/(20*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10*(a + a*Sin[e + f*x])^3)","A",1
280,1,270,124,0.6181418,"\int \frac{(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx","Integrate[(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(96 \sin \left(\frac{1}{2} (e+f x)\right)-105 (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 \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+464 \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+128 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-256 \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)^2-48 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{15 f (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)^8}","-\frac{7 c^4 \cos (e+f x)}{a^3 f}-\frac{2 a^3 c^4 \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}-\frac{7 c^4 x}{a^3}+\frac{14 a c^4 \cos ^5(e+f x)}{15 f (a \sin (e+f x)+a)^4}-\frac{14 c^4 \cos ^3(e+f x)}{3 a f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(96*Sin[(e + f*x)/2] - 48*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 256*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 128*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 464*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 105*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 15*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)*(c - c*Sin[e + f*x])^4)/(15*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8*(a + a*Sin[e + f*x])^3)","B",1
281,1,239,103,0.4413222,"\int \frac{(c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx","Integrate[(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 \sin \left(\frac{1}{2} (e+f x)\right)-15 (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5+92 \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+44 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-88 \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 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{15 f (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)^6}","-\frac{2 c^3 \cos (e+f x)}{f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{c^3 x}{a^3}-\frac{2 a^2 c^3 \cos ^5(e+f x)}{5 f (a \sin (e+f x)+a)^5}+\frac{2 c^3 \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])*(48*Sin[(e + f*x)/2] - 24*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 88*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 44*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 92*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 15*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)*(c - c*Sin[e + f*x])^3)/(15*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6*(a + a*Sin[e + f*x])^3)","B",1
282,1,81,33,0.4086254,"\int \frac{(c-c \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx","Integrate[(c - c*Sin[e + f*x])^2/(a + a*Sin[e + f*x])^3,x]","\frac{c^2 \left(10 \sin \left(\frac{1}{2} (e+f x)\right)+5 \sin \left(\frac{3}{2} (e+f x)\right)-\sin \left(\frac{5}{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)}{10 a^3 f (\sin (e+f x)+1)^3}","-\frac{a^2 c^2 \cos ^5(e+f x)}{5 f (a \sin (e+f x)+a)^5}",1,"(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(10*Sin[(e + f*x)/2] + 5*Sin[(3*(e + f*x))/2] - Sin[(5*(e + f*x))/2]))/(10*a^3*f*(1 + Sin[e + f*x])^3)","B",1
283,1,92,58,0.3435091,"\int \frac{c-c \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx","Integrate[(c - c*Sin[e + f*x])/(a + a*Sin[e + f*x])^3,x]","\frac{c \left(\sin \left(2 e+\frac{5 f x}{2}\right)-15 \cos \left(e+\frac{f x}{2}\right)+5 \cos \left(e+\frac{3 f x}{2}\right)+5 \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 \cos ^3(e+f x)}{15 f (a \sin (e+f x)+a)^3}-\frac{a c \cos ^3(e+f x)}{5 f (a \sin (e+f x)+a)^4}",1,"(c*(-15*Cos[e + (f*x)/2] + 5*Cos[e + (3*f*x)/2] + 5*Sin[(f*x)/2] + 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
284,1,111,83,0.6704186,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))} \, dx","Integrate[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])),x]","\frac{-12 \sin (e+f x)-32 \sin (2 (e+f x))-12 \sin (3 (e+f x))+8 \sin (4 (e+f x))+32 \cos (e+f x)-12 \cos (2 (e+f x))+32 \cos (3 (e+f x))+3 \cos (4 (e+f x))-15}{160 a^3 c f (\sin (e+f x)-1) (\sin (e+f x)+1)^3}","\frac{2 \tan (e+f x)}{5 a^3 c f}-\frac{\sec (e+f x)}{5 c f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{\sec (e+f x)}{5 a c f (a \sin (e+f x)+a)^2}",1,"(-15 + 32*Cos[e + f*x] - 12*Cos[2*(e + f*x)] + 32*Cos[3*(e + f*x)] + 3*Cos[4*(e + f*x)] - 12*Sin[e + f*x] - 32*Sin[2*(e + f*x)] - 12*Sin[3*(e + f*x)] + 8*Sin[4*(e + f*x)])/(160*a^3*c*f*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])^3)","A",1
285,1,131,75,0.7945099,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^2} \, dx","Integrate[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^2),x]","\frac{18 \sin (e+f x)+512 \sin (2 (e+f x))+27 \sin (3 (e+f x))+128 \sin (4 (e+f x))+9 \sin (5 (e+f x))-128 \cos (e+f x)+72 \cos (2 (e+f x))-192 \cos (3 (e+f x))+18 \cos (4 (e+f x))-64 \cos (5 (e+f x))+54}{1920 a^3 c^2 f (\sin (e+f x)-1)^2 (\sin (e+f x)+1)^3}","\frac{4 \tan ^3(e+f x)}{15 a^3 c^2 f}+\frac{4 \tan (e+f x)}{5 a^3 c^2 f}-\frac{\sec ^3(e+f x)}{5 c^2 f \left(a^3 \sin (e+f x)+a^3\right)}",1,"(54 - 128*Cos[e + f*x] + 72*Cos[2*(e + f*x)] - 192*Cos[3*(e + f*x)] + 18*Cos[4*(e + f*x)] - 64*Cos[5*(e + f*x)] + 18*Sin[e + f*x] + 512*Sin[2*(e + f*x)] + 27*Sin[3*(e + f*x)] + 128*Sin[4*(e + f*x)] + 9*Sin[5*(e + f*x)])/(1920*a^3*c^2*f*(-1 + Sin[e + f*x])^2*(1 + Sin[e + f*x])^3)","A",1
286,1,41,59,0.1292692,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3} \, dx","Integrate[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^3),x]","\frac{\frac{1}{5} \tan ^5(e+f x)+\frac{2}{3} \tan ^3(e+f x)+\tan (e+f x)}{a^3 c^3 f}","\frac{\tan ^5(e+f x)}{5 a^3 c^3 f}+\frac{2 \tan ^3(e+f x)}{3 a^3 c^3 f}+\frac{\tan (e+f x)}{a^3 c^3 f}",1,"(Tan[e + f*x] + (2*Tan[e + f*x]^3)/3 + Tan[e + f*x]^5/5)/(a^3*c^3*f)","A",1
287,1,193,97,1.1901968,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx","Integrate[1/((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) (5120 \sin (e+f x)+125 \sin (2 (e+f x))+2560 \sin (3 (e+f x))+100 \sin (4 (e+f x))+512 \sin (5 (e+f x))+25 \sin (6 (e+f x))-500 \cos (e+f x)+1280 \cos (2 (e+f x))-250 \cos (3 (e+f x))+1024 \cos (4 (e+f x))-50 \cos (5 (e+f x))+256 \cos (6 (e+f x)))}{17920 f (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^4}","\frac{6 \tan ^5(e+f x)}{35 a^3 c^4 f}+\frac{4 \tan ^3(e+f x)}{7 a^3 c^4 f}+\frac{6 \tan (e+f x)}{7 a^3 c^4 f}+\frac{\sec ^5(e+f x)}{7 a^3 f \left(c^4-c^4 \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])*(-500*Cos[e + f*x] + 1280*Cos[2*(e + f*x)] - 250*Cos[3*(e + f*x)] + 1024*Cos[4*(e + f*x)] - 50*Cos[5*(e + f*x)] + 256*Cos[6*(e + f*x)] + 5120*Sin[e + f*x] + 125*Sin[2*(e + f*x)] + 2560*Sin[3*(e + f*x)] + 100*Sin[4*(e + f*x)] + 512*Sin[5*(e + f*x)] + 25*Sin[6*(e + f*x)]))/(17920*f*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4)","A",1
288,1,213,131,1.5437016,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx","Integrate[1/((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) (46080 \sin (e+f x)+3500 \sin (2 (e+f x))+19456 \sin (3 (e+f x))+2800 \sin (4 (e+f x))+1024 \sin (5 (e+f x))+700 \sin (6 (e+f x))-1024 \sin (7 (e+f x))-7875 \cos (e+f x)+20480 \cos (2 (e+f x))-3325 \cos (3 (e+f x))+16384 \cos (4 (e+f x))-175 \cos (5 (e+f x))+4096 \cos (6 (e+f x))+175 \cos (7 (e+f x)))}{184320 f (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^5}","\frac{2 \tan ^5(e+f x)}{15 a^3 c^5 f}+\frac{4 \tan ^3(e+f x)}{9 a^3 c^5 f}+\frac{2 \tan (e+f x)}{3 a^3 c^5 f}+\frac{\sec ^5(e+f x)}{9 a^3 f \left(c^5-c^5 \sin (e+f x)\right)}+\frac{\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])*(-7875*Cos[e + f*x] + 20480*Cos[2*(e + f*x)] - 3325*Cos[3*(e + f*x)] + 16384*Cos[4*(e + f*x)] - 175*Cos[5*(e + f*x)] + 4096*Cos[6*(e + f*x)] + 175*Cos[7*(e + f*x)] + 46080*Sin[e + f*x] + 3500*Sin[2*(e + f*x)] + 19456*Sin[3*(e + f*x)] + 2800*Sin[4*(e + f*x)] + 1024*Sin[5*(e + f*x)] + 700*Sin[6*(e + f*x)] - 1024*Sin[7*(e + f*x)]))/(184320*f*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5)","A",1
289,1,233,167,1.6253797,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx","Integrate[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6),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) (1802240 \sin (e+f x)+247170 \sin (2 (e+f x))+557056 \sin (3 (e+f x))+187250 \sin (4 (e+f x))-163840 \sin (5 (e+f x))+37450 \sin (6 (e+f x))-98304 \sin (7 (e+f x))-3745 \sin (8 (e+f x))-411950 \cos (e+f x)+1081344 \cos (2 (e+f x))-127330 \cos (3 (e+f x))+819200 \cos (4 (e+f x))+37450 \cos (5 (e+f x))+163840 \cos (6 (e+f x))+22470 \cos (7 (e+f x))-16384 \cos (8 (e+f x)))}{8110080 f (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^6}","\frac{16 \tan ^5(e+f x)}{165 a^3 c^6 f}+\frac{32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac{16 \tan (e+f x)}{33 a^3 c^6 f}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left(c^6-c^6 \sin (e+f x)\right)}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left(c^3-c^3 \sin (e+f x)\right)^2}+\frac{\sec ^5(e+f x)}{11 a^3 f \left(c^2-c^2 \sin (e+f x)\right)^3}",1,"((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-411950*Cos[e + f*x] + 1081344*Cos[2*(e + f*x)] - 127330*Cos[3*(e + f*x)] + 819200*Cos[4*(e + f*x)] + 37450*Cos[5*(e + f*x)] + 163840*Cos[6*(e + f*x)] + 22470*Cos[7*(e + f*x)] - 16384*Cos[8*(e + f*x)] + 1802240*Sin[e + f*x] + 247170*Sin[2*(e + f*x)] + 557056*Sin[3*(e + f*x)] + 187250*Sin[4*(e + f*x)] - 163840*Sin[5*(e + f*x)] + 37450*Sin[6*(e + f*x)] - 98304*Sin[7*(e + f*x)] - 3745*Sin[8*(e + f*x)]))/(8110080*f*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6)","A",1
290,1,104,137,0.8597884,"\int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx","Integrate[(a + a*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 (-1389 \sin (e+f x)+35 \sin (3 (e+f x))-330 \cos (2 (e+f x))+1606)}{630 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 \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a c^4 \cos ^3(e+f x)}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a c^3 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{21 f}+\frac{2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}",1,"(a*c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c - c*Sin[e + f*x]]*(1606 - 330*Cos[2*(e + f*x)] - 1389*Sin[e + f*x] + 35*Sin[3*(e + f*x)]))/(630*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
291,1,94,103,0.571705,"\int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*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 (108 \sin (e+f x)+15 \cos (2 (e+f x))-157)}{105 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 \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}+\frac{16 a c^3 \cos ^3(e+f x)}{35 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a c^2 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{7 f}",1,"-1/105*(a*c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(-157 + 15*Cos[2*(e + f*x)] + 108*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
292,1,82,69,0.3650581,"\int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{2 a c (3 \sin (e+f x)-7) \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}{15 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 \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a c^2 \cos ^3(e+f x)}{5 f \sqrt{c-c \sin (e+f x)}}",1,"(-2*a*c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(-7 + 3*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
293,1,71,34,0.1250909,"\int (a+a \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(a + a*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}{3 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 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}",1,"(2*a*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c - c*Sin[e + f*x]])/(3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","B",1
294,1,135,77,0.653496,"\int \frac{a+a \sin (e+f x)}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{2 a \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sqrt{c} (\sin (e+f x)+1)+\sqrt{2} \sqrt{-c (\sin (e+f x)+1)} \tan ^{-1}\left(\frac{\sqrt{-c (\sin (e+f x)+1)}}{\sqrt{2} \sqrt{c}}\right)\right)}{\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 \sqrt{2} a \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 \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}",1,"(-2*a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Sqrt[c]*(1 + Sin[e + f*x]) + Sqrt[2]*ArcTan[Sqrt[-(c*(1 + Sin[e + f*x]))]/(Sqrt[2]*Sqrt[c])]*Sqrt[-(c*(1 + 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
295,1,107,76,0.7076939,"\int \frac{a+a \sin (e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^(3/2),x]","\frac{a \sec (e+f x) \left(2 \sqrt{c} (\sin (e+f x)+1)-\sqrt{2} (\sin (e+f x)-1) \sqrt{-c (\sin (e+f x)+1)} \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 \cos (e+f x)}{f (c-c \sin (e+f x))^{3/2}}-\frac{a \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}",1,"(a*Sec[e + f*x]*(2*Sqrt[c]*(1 + Sin[e + f*x]) - Sqrt[2]*ArcTan[Sqrt[-(c*(1 + Sin[e + f*x]))]/(Sqrt[2]*Sqrt[c])]*(-1 + Sin[e + f*x])*Sqrt[-(c*(1 + Sin[e + f*x]))]))/(2*c^(3/2)*f*Sqrt[c - c*Sin[e + f*x]])","A",1
296,1,176,113,0.9678884,"\int \frac{a+a \sin (e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^(5/2),x]","\frac{a \left(2 \sqrt{2} \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)^4 \tan ^{-1}\left(\frac{\sqrt{-c (\sin (e+f x)+1)}}{\sqrt{2} \sqrt{c}}\right)-2 \sqrt{c} (-8 \sin (e+f x)+\cos (2 (e+f x))-7)\right)}{32 c^{5/2} f \sqrt{c-c \sin (e+f x)} \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)}","-\frac{a \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 \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{3/2}}+\frac{a \cos (e+f x)}{2 f (c-c \sin (e+f x))^{5/2}}",1,"(a*(-2*Sqrt[c]*(-7 + Cos[2*(e + f*x)] - 8*Sin[e + f*x]) + 2*Sqrt[2]*ArcTan[Sqrt[-(c*(1 + Sin[e + f*x]))]/(Sqrt[2]*Sqrt[c])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sqrt[-(c*(1 + Sin[e + f*x]))]))/(32*c^(5/2)*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]])","A",1
297,1,189,145,1.1611572,"\int \frac{a+a \sin (e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^(7/2),x]","\frac{a \left(2 \sqrt{c} (131 \sin (e+f x)+3 (\sin (3 (e+f x))+38)-14 \cos (2 (e+f x)))+12 \sqrt{2} \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)^6 \tan ^{-1}\left(\frac{\sqrt{-c (\sin (e+f x)+1)}}{\sqrt{2} \sqrt{c}}\right)\right)}{768 c^{7/2} f \sqrt{c-c \sin (e+f x)} \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 \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 \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac{a \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}+\frac{a \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}",1,"(a*(12*Sqrt[2]*ArcTan[Sqrt[-(c*(1 + Sin[e + f*x]))]/(Sqrt[2]*Sqrt[c])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6*Sqrt[-(c*(1 + Sin[e + f*x]))] + 2*Sqrt[c]*(-14*Cos[2*(e + f*x)] + 131*Sin[e + f*x] + 3*(38 + Sin[3*(e + f*x)]))))/(768*c^(7/2)*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]])","A",1
298,1,1105,145,6.4356684,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2),x]","\frac{7 \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 \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{\cos \left(\frac{3}{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{11 \cos \left(\frac{5}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{7/2}}{80 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{\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{\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{\cos \left(\frac{11}{2} (e+f x)\right) (\sin (e+f x) a+a)^2 (c-c \sin (e+f x))^{7/2}}{176 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{(\sin (e+f x) a+a)^2 \sin \left(\frac{3}{2} (e+f x)\right) (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{11 (\sin (e+f x) a+a)^2 \sin \left(\frac{5}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{80 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{(\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{(\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{(\sin (e+f x) a+a)^2 \sin \left(\frac{11}{2} (e+f x)\right) (c-c \sin (e+f x))^{7/2}}{176 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 \cos ^5(e+f x)}{1155 f (c-c \sin (e+f x))^{5/2}}+\frac{64 a^2 c^5 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^2 c^4 \cos ^5(e+f x)}{33 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^3 \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{11 f}",1,"(7*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) - (Cos[(3*(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) + (11*Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(80*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (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) + (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) + (Cos[(11*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(176*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (7*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) + ((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(3*(e + f*x))/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) + (11*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(5*(e + f*x))/2])/(80*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*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*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) - ((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(11*(e + f*x))/2])/(176*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
299,1,96,109,5.6934553,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2),x]","-\frac{a^2 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 (220 \sin (e+f x)+35 \cos (2 (e+f x))-249)}{315 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{64 a^2 c^5 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^{5/2}}+\frac{16 a^2 c^4 \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^2 c^3 \cos ^5(e+f x)}{9 f \sqrt{c-c \sin (e+f x)}}",1,"-1/315*(a^2*c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(-249 + 35*Cos[2*(e + f*x)] + 220*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
300,1,84,73,1.3882319,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{2 a^2 c (5 \sin (e+f x)-9) \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}{35 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 \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^{5/2}}+\frac{2 a^2 c^3 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^{3/2}}",1,"(-2*a^2*c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(-9 + 5*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
301,1,73,36,0.2327113,"\int (a+a \sin (e+f x))^2 \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^2*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}{5 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 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}",1,"(2*a^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sqrt[c - c*Sin[e + f*x]])/(5*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","B",1
302,1,130,115,0.4729359,"\int \frac{(a+a \sin (e+f x))^2}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])^2/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{a^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(15 \sin \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{3}{2} (e+f x)\right)+15 \cos \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{3}{2} (e+f x)\right)+(24+24 i) \sqrt[4]{-1} \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)}{3 f \sqrt{c-c \sin (e+f x)}}","-\frac{2 a^2 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac{4 a^2 \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}+\frac{4 \sqrt{2} a^2 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{c} f}",1,"-1/3*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*((24 + 24*I)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])] + 15*Cos[(e + f*x)/2] - Cos[(3*(e + f*x))/2] + 15*Sin[(e + f*x)/2] + Sin[(3*(e + f*x))/2]))/(f*Sqrt[c - c*Sin[e + f*x]])","C",1
303,1,149,115,0.6854516,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^(3/2),x]","-\frac{a^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(3 \sin \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{3}{2} (e+f x)\right)+3 \cos \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{3}{2} (e+f x)\right)-(6+6 i) \sqrt[4]{-1} (\sin (e+f x)-1) \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)}{c f (\sin (e+f x)-1) \sqrt{c-c \sin (e+f x)}}","-\frac{3 \sqrt{2} a^2 \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 \cos ^3(e+f x)}{f (c-c \sin (e+f x))^{5/2}}+\frac{3 a^2 \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])*(3*Cos[(e + f*x)/2] + Cos[(3*(e + f*x))/2] + 3*Sin[(e + f*x)/2] - (6 + 6*I)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(-1 + Sin[e + f*x]) - Sin[(3*(e + f*x))/2]))/(c*f*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]]))","C",1
304,1,163,122,0.9737258,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^(5/2),x]","\frac{a^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(3 \sin \left(\frac{1}{2} (e+f x)\right)+5 \sin \left(\frac{3}{2} (e+f x)\right)+3 \cos \left(\frac{1}{2} (e+f x)\right)-5 \cos \left(\frac{3}{2} (e+f x)\right)+(3+3 i) \sqrt[4]{-1} \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) (4 \sin (e+f x)+\cos (2 (e+f x))-3)\right)}{8 c^2 f (\sin (e+f x)-1)^2 \sqrt{c-c \sin (e+f x)}}","\frac{3 a^2 \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 \cos ^3(e+f x)}{2 f (c-c \sin (e+f x))^{7/2}}-\frac{3 a^2 \cos (e+f x)}{4 c f (c-c \sin (e+f x))^{3/2}}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(3*Cos[(e + f*x)/2] - 5*Cos[(3*(e + f*x))/2] + 3*Sin[(e + f*x)/2] + (3 + 3*I)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(-3 + Cos[2*(e + f*x)] + 4*Sin[e + f*x]) + 5*Sin[(3*(e + f*x))/2]))/(8*c^2*f*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]])","C",1
305,1,307,156,0.9830597,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(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 \sin \left(\frac{1}{2} (e+f x)\right)+3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5+6 \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)^4-28 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-56 \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 \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} \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 \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 \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{9/2}}-\frac{a^2 \cos (e+f x)}{4 c f (c-c \sin (e+f x))^{5/2}}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(32*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 28*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + 3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 - (3 + 3*I)*(-1)^(1/4)*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*Sin[(e + f*x)/2] - 56*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + 6*(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
306,1,371,190,1.4558997,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^{9/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(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(256 \sin \left(\frac{1}{2} (e+f x)\right)+3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7+6 \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+4 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5+8 \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)^4-96 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-192 \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+128 \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} \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\right)}{256 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{3 a^2 \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{3 a^2 \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{a^2 c \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{11/2}}-\frac{a^2 \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{7/2}}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(128*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 96*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + 4*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + 3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7 - (3 + 3*I)*(-1)^(1/4)*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 + 256*Sin[(e + f*x)/2] - 192*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + 8*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(e + f*x)/2] + 6*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6*Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2)/(256*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2))","C",1
307,1,112,145,6.2672197,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2),x]","\frac{a^3 c^3 \cos ^6(e+f x) \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) (-6377 \sin (e+f x)+231 \sin (3 (e+f x))-1890 \cos (2 (e+f x))+5230)}{6006 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}","\frac{256 a^3 c^7 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^{7/2}}+\frac{64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac{24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt{c-c \sin (e+f x)}}",1,"(a^3*c^3*Cos[e + f*x]^6*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(5230 - 1890*Cos[2*(e + f*x)] - 6377*Sin[e + f*x] + 231*Sin[3*(e + f*x)]))/(6006*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7)","A",1
308,1,1105,109,6.4770951,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2),x]","\frac{5 \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 \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 \cos \left(\frac{3}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2} (\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)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{\cos \left(\frac{5}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2} (\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)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{5 \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{\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{\cos \left(\frac{11}{2} (e+f x)\right) (c-c \sin (e+f x))^{5/2} (\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)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{5 (c-c \sin (e+f x))^{5/2} \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)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{(c-c \sin (e+f x))^{5/2} \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)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{5 (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{(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{(c-c \sin (e+f x))^{5/2} \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)^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 \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^{7/2}}+\frac{16 a^3 c^5 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^{5/2}}+\frac{2 a^3 c^4 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}",1,"(5*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*Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(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])^6) + (Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/(16*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (5*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) + (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) - (Cos[(11*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(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])^6) + (5*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*(a + a*Sin[e + f*x])^3*(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])^6) + ((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2)*Sin[(5*(e + f*x))/2])/(16*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 + 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 + 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) + ((a + a*Sin[e + f*x])^3*(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])^6)","B",1
309,1,84,73,3.0942927,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{2 a^3 c (7 \sin (e+f x)-11) \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}{63 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{8 a^3 c^5 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}+\frac{2 a^3 c^4 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}",1,"(-2*a^3*c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(-11 + 7*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
310,1,73,36,0.3758487,"\int (a+a \sin (e+f x))^3 \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^3*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}{7 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 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}",1,"(2*a^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*Sqrt[c - c*Sin[e + f*x]])/(7*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","B",1
311,1,156,151,0.7531435,"\int \frac{(a+a \sin (e+f x))^3}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])^3/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{a^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(330 \sin \left(\frac{1}{2} (e+f x)\right)+35 \sin \left(\frac{3}{2} (e+f x)\right)-3 \sin \left(\frac{5}{2} (e+f x)\right)+330 \cos \left(\frac{1}{2} (e+f x)\right)-35 \cos \left(\frac{3}{2} (e+f x)\right)-3 \cos \left(\frac{5}{2} (e+f x)\right)+(480+480 i) \sqrt[4]{-1} \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)}{30 f \sqrt{c-c \sin (e+f x)}}","-\frac{2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac{4 a^3 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac{8 a^3 \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}+\frac{8 \sqrt{2} a^3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{c} f}",1,"-1/30*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*((480 + 480*I)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])] + 330*Cos[(e + f*x)/2] - 35*Cos[(3*(e + f*x))/2] - 3*Cos[(5*(e + f*x))/2] + 330*Sin[(e + f*x)/2] + 35*Sin[(3*(e + f*x))/2] - 3*Sin[(5*(e + f*x))/2]))/(f*Sqrt[c - c*Sin[e + f*x]])","C",1
312,1,173,150,0.8413914,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(3/2),x]","-\frac{a^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(50 \sin \left(\frac{1}{2} (e+f x)\right)-25 \sin \left(\frac{3}{2} (e+f x)\right)+\sin \left(\frac{5}{2} (e+f x)\right)+50 \cos \left(\frac{1}{2} (e+f x)\right)+25 \cos \left(\frac{3}{2} (e+f x)\right)+\cos \left(\frac{5}{2} (e+f x)\right)-(120+120 i) \sqrt[4]{-1} (\sin (e+f x)-1) \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)}{6 c f (\sin (e+f x)-1) \sqrt{c-c \sin (e+f x)}}","-\frac{10 \sqrt{2} a^3 \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^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}+\frac{5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac{10 a^3 \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}",1,"-1/6*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(50*Cos[(e + f*x)/2] + 25*Cos[(3*(e + f*x))/2] + Cos[(5*(e + f*x))/2] + 50*Sin[(e + f*x)/2] - (120 + 120*I)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(-1 + Sin[e + f*x]) - 25*Sin[(3*(e + f*x))/2] + Sin[(5*(e + f*x))/2]))/(c*f*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])","C",1
313,1,187,157,1.0505789,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(5/2),x]","\frac{a^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-5 \sin \left(\frac{1}{2} (e+f x)\right)+15 \sin \left(\frac{3}{2} (e+f x)\right)+2 \sin \left(\frac{5}{2} (e+f x)\right)-5 \cos \left(\frac{1}{2} (e+f x)\right)-15 \cos \left(\frac{3}{2} (e+f x)\right)+2 \cos \left(\frac{5}{2} (e+f x)\right)+(15+15 i) \sqrt[4]{-1} \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) (4 \sin (e+f x)+\cos (2 (e+f x))-3)\right)}{4 c^2 f (\sin (e+f x)-1)^2 \sqrt{c-c \sin (e+f x)}}","\frac{15 a^3 \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^2 \cos ^5(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}-\frac{15 a^3 \cos (e+f x)}{4 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{5 a^3 \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(-5*Cos[(e + f*x)/2] - 15*Cos[(3*(e + f*x))/2] + 2*Cos[(5*(e + f*x))/2] - 5*Sin[(e + f*x)/2] + (15 + 15*I)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(-3 + Cos[2*(e + f*x)] + 4*Sin[e + f*x]) + 15*Sin[(3*(e + f*x))/2] + 2*Sin[(5*(e + f*x))/2]))/(4*c^2*f*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]])","C",1
314,1,307,157,1.5742695,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(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 \sin \left(\frac{1}{2} (e+f x)\right)+33 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5+66 \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)^4-52 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-104 \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 \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} \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)}{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 \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^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^{11/2}}+\frac{5 a^3 \cos (e+f x)}{8 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac{5 a^3 \cos ^3(e+f x)}{12 f (c-c \sin (e+f x))^{7/2}}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(32*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 52*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + 33*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + (15 + 15*I)*(-1)^(1/4)*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*Sin[(e + f*x)/2] - 104*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + 66*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*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
315,1,371,191,2.6548022,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^{9/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(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(768 \sin \left(\frac{1}{2} (e+f x)\right)-15 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7-30 \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+236 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5+472 \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)^4-544 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-1088 \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+384 \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} \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\right)}{384 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 \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{5 a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{a^3 c^2 \cos ^5(e+f x)}{4 f (c-c \sin (e+f x))^{13/2}}+\frac{5 a^3 \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{5/2}}-\frac{5 a^3 \cos ^3(e+f x)}{24 f (c-c \sin (e+f x))^{9/2}}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(384*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 544*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + 236*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 - 15*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7 + (15 + 15*I)*(-1)^(1/4)*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 + 768*Sin[(e + f*x)/2] - 1088*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + 472*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(e + f*x)/2] - 30*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6*Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3)/(384*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^(9/2))","C",1
316,1,435,225,4.1104536,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^{11/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(11/2),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(4096 \sin \left(\frac{1}{2} (e+f x)\right)-15 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9-30 \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)^8-20 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7-40 \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+992 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5+1984 \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)^4-2688 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3-5376 \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+2048 \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} \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)^{10}\right)}{2560 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{3 a^3 \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{3 a^3 \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}-\frac{a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}+\frac{a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{15/2}}+\frac{a^3 \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{7/2}}-\frac{a^3 \cos ^3(e+f x)}{8 f (c-c \sin (e+f x))^{11/2}}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(2048*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 2688*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + 992*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 - 20*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7 - 15*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9 + (15 + 15*I)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10 + 4096*Sin[(e + f*x)/2] - 5376*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + 1984*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(e + f*x)/2] - 40*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6*Sin[(e + f*x)/2] - 30*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8*Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3)/(2560*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^(11/2))","C",1
317,1,112,132,2.1853671,"\int \frac{(c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx","Integrate[(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) (-175 \sin (e+f x)+\sin (3 (e+f x))-14 \cos (2 (e+f x))-350)}{10 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{256 c^3 \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{5 a f}+\frac{64 c^2 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{5 a f}+\frac{2 \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 a f}+\frac{8 c \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{5 a f}",1,"(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(-350 - 14*Cos[2*(e + f*x)] - 175*Sin[e + f*x] + Sin[3*(e + f*x)]))/(10*a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x]))","A",1
318,1,102,98,0.7528659,"\int \frac{(c-c \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx","Integrate[(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) (20 \sin (e+f x)+\cos (2 (e+f x))+45)}{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{64 c^2 \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{3 a f}+\frac{2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}+\frac{16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f}",1,"-1/3*(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(45 + Cos[2*(e + f*x)] + 20*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x]))","A",1
319,1,88,60,0.3065187,"\int \frac{(c-c \sin (e+f x))^{3/2}}{a+a \sin (e+f x)} \, dx","Integrate[(c - c*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x]),x]","-\frac{2 c (\sin (e+f x)+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)}{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{2 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{a f}-\frac{8 c \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{a f}",1,"(-2*c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(3 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x]))","A",1
320,1,29,29,0.0969955,"\int \frac{\sqrt{c-c \sin (e+f x)}}{a+a \sin (e+f x)} \, dx","Integrate[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 f}","-\frac{2 \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{a f}",1,"(-2*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f)","A",1
321,1,97,83,0.3320117,"\int \frac{1}{(a+a \sin (e+f x)) \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[1/((a + a*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]]),x]","-\frac{\cos (e+f x) \left(1+(1+i) \sqrt[4]{-1} \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)\right)}{a f (\sin (e+f x)+1) \sqrt{c-c \sin (e+f x)}}","\frac{\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{\sec (e+f x) \sqrt{c-c \sin (e+f x)}}{a c f}",1,"-((Cos[e + f*x]*(1 + (1 + I)*(-1)^(1/4)*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
322,1,125,117,0.6248124,"\int \frac{1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2)),x]","-\frac{\sec (e+f x) \left(-3 \sin (e+f x)+(3+3 i) \sqrt[4]{-1} \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+1\right)}{4 a c f \sqrt{c-c \sin (e+f x)}}","\frac{3 \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 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac{\sec (e+f x)}{a c f \sqrt{c-c \sin (e+f x)}}",1,"-1/4*(Sec[e + f*x]*(1 + (3 + 3*I)*(-1)^(1/4)*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*Sin[e + f*x]))/(a*c*f*Sqrt[c - c*Sin[e + f*x]])","C",1
323,1,162,156,0.8391042,"\int \frac{1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\left(\frac{1}{128}+\frac{i}{128}\right) \cos (e+f x) \left((1-i) (40 \sin (e+f x)+15 \cos (2 (e+f x))-9)-60 \sqrt[4]{-1} \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)\right)}{a c^2 f (\sin (e+f x)-1)^2 (\sin (e+f x)+1) \sqrt{c-c \sin (e+f x)}}","\frac{15 \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 \sec (e+f x)}{8 a c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{15 \cos (e+f x)}{32 a c f (c-c \sin (e+f x))^{3/2}}+\frac{\sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}",1,"((1/128 + I/128)*Cos[e + f*x]*(-60*(-1)^(1/4)*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]) + (1 - I)*(-9 + 15*Cos[2*(e + f*x)] + 40*Sin[e + f*x])))/(a*c^2*f*(-1 + Sin[e + f*x])^2*(1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])","C",1
324,1,124,176,3.1231522,"\int \frac{(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c - c*Sin[e + f*x])^(9/2)/(a + a*Sin[e + f*x])^2,x]","\frac{c^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) (8568 \sin (e+f x)+56 \sin (3 (e+f x))-1044 \cos (2 (e+f x))+3 \cos (4 (e+f x))+6825)}{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{4096 c^3 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{15 a^2 f}-\frac{1024 c^2 \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^2 f}+\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 a^2 c f}+\frac{32 \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{15 a^2 f}+\frac{128 c \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{5 a^2 f}",1,"(c^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(6825 - 1044*Cos[2*(e + f*x)] + 3*Cos[4*(e + f*x)] + 8568*Sin[e + f*x] + 56*Sin[3*(e + f*x)]))/(60*a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2)","A",1
325,1,112,136,1.2472113,"\int \frac{(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^2} \, dx","Integrate[(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) (273 \sin (e+f x)+\sin (3 (e+f x))-30 \cos (2 (e+f x))+210)}{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{256 c^2 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 f}+\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{3 a^2 c f}+\frac{8 \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{a^2 f}-\frac{64 c \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{a^2 f}",1,"(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(210 - 30*Cos[2*(e + f*x)] + 273*Sin[e + f*x] + Sin[3*(e + f*x)]))/(6*a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2)","A",1
326,1,104,100,0.8011106,"\int \frac{(c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^2} \, dx","Integrate[(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) (36 \sin (e+f x)-3 \cos (2 (e+f x))+25)}{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{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{a^2 c f}-\frac{16 \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{a^2 f}+\frac{64 c \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 f}",1,"(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(25 - 3*Cos[2*(e + f*x)] + 36*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
327,1,92,68,0.3660387,"\int \frac{(c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c - c*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^2,x]","\frac{2 c (3 \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)}{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{8 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 f}-\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{a^2 c f}",1,"(2*c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(1 + 3*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
328,1,73,36,0.1332118,"\int \frac{\sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^2} \, dx","Integrate[Sqrt[c - c*Sin[e + f*x]]/(a + a*Sin[e + f*x])^2,x]","-\frac{2 \sqrt{c-c \sin (e+f x)}}{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{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 c f}",1,"(-2*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)","B",1
329,1,109,124,0.5137309,"\int \frac{1}{(a+a \sin (e+f x))^2 \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{\cos (e+f x) \left(-3 \sin (e+f x)+(-3-3 i) \sqrt[4]{-1} \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)^3-5\right)}{6 a^2 f (\sin (e+f x)+1)^2 \sqrt{c-c \sin (e+f x)}}","-\frac{\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 c^2 f}-\frac{\sec (e+f x) \sqrt{c-c \sin (e+f x)}}{2 a^2 c f}+\frac{\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]*(-5 - (3 + 3*I)*(-1)^(1/4)*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 - 3*Sin[e + f*x]))/(6*a^2*f*(1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]])","C",1
330,1,164,155,0.7889502,"\int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\left(\frac{1}{96}+\frac{i}{96}\right) \cos (e+f x) \left((1-i) (-20 \sin (e+f x)+15 \cos (2 (e+f x))+11)+60 \sqrt[4]{-1} \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\right)}{a^2 c f (\sin (e+f x)-1) (\sin (e+f x)+1)^2 \sqrt{c-c \sin (e+f x)}}","\frac{5 \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{\sec ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{3 a^2 c^2 f}+\frac{5 \cos (e+f x)}{8 a^2 f (c-c \sin (e+f x))^{3/2}}-\frac{5 \sec (e+f x)}{6 a^2 c f \sqrt{c-c \sin (e+f x)}}",1,"((1/96 + I/96)*Cos[e + f*x]*(60*(-1)^(1/4)*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 + (1 - I)*(11 + 15*Cos[2*(e + f*x)] - 20*Sin[e + f*x])))/(a^2*c*f*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]])","C",1
331,1,156,192,1.1733968,"\int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2)),x]","-\frac{\left(\frac{1}{1536}+\frac{i}{1536}\right) \sec ^3(e+f x) \left((1-i) (-329 \sin (e+f x)-105 \sin (3 (e+f x))+70 \cos (2 (e+f x))+102)+840 \sqrt[4]{-1} \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)}{a^2 c^2 f \sqrt{c-c \sin (e+f x)}}","\frac{35 \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{\sec ^3(e+f x)}{3 a^2 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{35 \sec (e+f x)}{48 a^2 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{35 \cos (e+f x)}{64 a^2 c f (c-c \sin (e+f x))^{3/2}}+\frac{7 \sec (e+f x)}{24 a^2 c f (c-c \sin (e+f x))^{3/2}}",1,"((-1/1536 - I/1536)*Sec[e + f*x]^3*(840*(-1)^(1/4)*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 + (1 - I)*(102 + 70*Cos[2*(e + f*x)] - 329*Sin[e + f*x] - 105*Sin[3*(e + f*x)])))/(a^2*c^2*f*Sqrt[c - c*Sin[e + f*x]])","C",1
332,1,124,174,3.1514888,"\int \frac{(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^3} \, dx","Integrate[(c - c*Sin[e + f*x])^(9/2)/(a + a*Sin[e + f*x])^3,x]","\frac{c^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) (-7800 \sin (e+f x)+200 \sin (3 (e+f x))+2740 \cos (2 (e+f x))+5 \cos (4 (e+f x))-5649)}{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{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^3 c^2 f}-\frac{4096 c^2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^3 f}+\frac{32 \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{3 a^3 c f}-\frac{128 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 f}+\frac{1024 c \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 f}",1,"(c^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(-5649 + 2740*Cos[2*(e + f*x)] + 5*Cos[4*(e + f*x)] - 7800*Sin[e + f*x] + 200*Sin[3*(e + f*x)]))/(60*a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3)","A",1
333,1,114,134,1.2750617,"\int \frac{(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^3} \, dx","Integrate[(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) (-235 \sin (e+f x)+5 \sin (3 (e+f x))+90 \cos (2 (e+f x))-182)}{10 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{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{a^3 c^2 f}-\frac{24 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 c f}+\frac{64 \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{a^3 f}-\frac{256 c \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 f}",1,"(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(-182 + 90*Cos[2*(e + f*x)] - 235*Sin[e + f*x] + 5*Sin[3*(e + f*x)]))/(10*a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3)","A",1
334,1,104,104,0.8678039,"\int \frac{(c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx","Integrate[(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) (-20 \sin (e+f x)+15 \cos (2 (e+f x))-29)}{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{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 c^2 f}+\frac{16 \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 c f}-\frac{64 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^3 f}",1,"(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-29 + 15*Cos[2*(e + f*x)] - 20*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
335,1,92,73,0.3922448,"\int \frac{(c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^3} \, dx","Integrate[(c - c*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^3,x]","\frac{2 c (5 \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)}{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{8 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^3 c f}-\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 c^2 f}",1,"(2*c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + 5*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
336,1,73,36,0.1515129,"\int \frac{\sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx","Integrate[Sqrt[c - c*Sin[e + f*x]]/(a + a*Sin[e + f*x])^3,x]","-\frac{2 \sqrt{c-c \sin (e+f x)}}{5 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{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 c^2 f}",1,"(-2*Sqrt[c - c*Sin[e + f*x]])/(5*a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","B",1
337,1,189,160,0.6147259,"\int \frac{1}{(a+a \sin (e+f x))^3 \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[1/((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 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4-10 \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} \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\right)}{60 a^3 f (\sin (e+f x)+1)^3 \sqrt{c-c \sin (e+f x)}}","-\frac{\sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 c^3 f}-\frac{\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{6 a^3 c^2 f}-\frac{\sec (e+f x) \sqrt{c-c \sin (e+f x)}}{4 a^3 c f}+\frac{\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 - 10*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 15*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - (15 + 15*I)*(-1)^(1/4)*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
338,1,174,191,1.1916029,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\left(\frac{1}{1920}+\frac{i}{1920}\right) \cos (e+f x) \left((1-i) (-231 \sin (e+f x)+105 \sin (3 (e+f x))+350 \cos (2 (e+f x))+206)+840 \sqrt[4]{-1} \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\right)}{a^3 c f (\sin (e+f x)-1) (\sin (e+f x)+1)^3 \sqrt{c-c \sin (e+f x)}}","\frac{7 \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{\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}-\frac{7 \sec ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{30 a^3 c^2 f}+\frac{7 \cos (e+f x)}{16 a^3 f (c-c \sin (e+f x))^{3/2}}-\frac{7 \sec (e+f x)}{12 a^3 c f \sqrt{c-c \sin (e+f x)}}",1,"((1/1920 + I/1920)*Cos[e + f*x]*(840*(-1)^(1/4)*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 + (1 - I)*(206 + 350*Cos[2*(e + f*x)] - 231*Sin[e + f*x] + 105*Sin[3*(e + f*x)])))/(a^3*c*f*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]])","C",1
339,1,443,228,1.5366389,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[1/((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(-240 \cos ^4(e+f x)+75 \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+20 \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+150 \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+40 \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 \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)^2-32 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4+(-315-315 i) \sqrt[4]{-1} \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\right)}{640 a^3 f (\sin (e+f x)+1)^3 (c-c \sin (e+f x))^{5/2}}","\frac{63 \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{\sec ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{5 a^3 c^3 f}-\frac{3 \sec ^3(e+f x)}{10 a^3 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{21 \sec (e+f x)}{32 a^3 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{63 \cos (e+f x)}{128 a^3 c f (c-c \sin (e+f x))^{3/2}}+\frac{21 \sec (e+f x)}{80 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])*(-240*Cos[e + f*x]^4 - 32*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 - 80*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 20*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + 75*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - (315 + 315*I)*(-1)^(1/4)*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 + 40*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + 150*(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))/(640*a^3*f*(1 + Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))","C",1
340,1,83,43,0.3973838,"\int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx","Integrate[Sqrt[a + a*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)} (8 (\sin (3 (e+f x))-7 \sin (e+f x))-28 \cos (2 (e+f x))+\cos (4 (e+f x)))}{32 f}","-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt{a \sin (e+f x)+a}}",1,"-1/32*(c^3*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-28*Cos[2*(e + f*x)] + Cos[4*(e + f*x)] + 8*(-7*Sin[e + f*x] + Sin[3*(e + f*x)])))/f","A",1
341,1,74,43,0.3035247,"\int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx","Integrate[Sqrt[a + a*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)} (15 \sin (e+f x)-\sin (3 (e+f x))+6 \cos (2 (e+f x)))}{12 f}","-\frac{a \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]]*(6*Cos[2*(e + f*x)] + 15*Sin[e + f*x] - Sin[3*(e + f*x)]))/(12*f)","A",1
342,1,60,43,0.2165028,"\int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx","Integrate[Sqrt[a + a*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)} (4 \sin (e+f x)+\cos (2 (e+f x)))}{4 f}","-\frac{a \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])]*(Cos[2*(e + f*x)] + 4*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(4*f)","A",1
343,1,39,41,0.0876281,"\int \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]],x]","\frac{\tan (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}{f}","-\frac{a \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}",1,"(Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*Tan[e + f*x])/f","A",1
344,1,119,52,0.9287894,"\int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{\sqrt{2} \left(e^{i (e+f x)}-i\right) \left(f x+2 i \log \left(i-e^{i (e+f x)}\right)\right) \sqrt{a (\sin (e+f x)+1)}}{f \left(e^{i (e+f x)}+i\right) \sqrt{i c e^{-i (e+f x)} \left(e^{i (e+f x)}-i\right)^2}}","-\frac{a \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,"-((Sqrt[2]*(-I + E^(I*(e + f*x)))*(f*x + (2*I)*Log[I - E^(I*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(Sqrt[(I*c*(-I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*(I + E^(I*(e + f*x)))*f))","C",1
345,1,84,40,0.2070653,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]/(c - c*Sin[e + f*x])^(3/2),x]","\frac{\sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}{c^2 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)}","\frac{a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}",1,"(Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])/(c^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","B",1
346,1,87,43,0.2246196,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[Sqrt[a + a*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)}}{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 \cos (e+f x)}{2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}",1,"(Sqrt[a*(1 + 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]))","B",1
347,1,87,43,0.2832487,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[Sqrt[a + a*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)}}{3 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 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}",1,"(Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])/(3*c^4*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","B",1
348,1,146,89,1.0444576,"\int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(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)} (70 \sin (e+f x)+5 \sin (3 (e+f x))-\sin (5 (e+f x))+20 \cos (2 (e+f x))+5 \cos (4 (e+f x)))}{80 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 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{10 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{5 f}",1,"-1/80*(c^3*(-1 + Sin[e + f*x])^3*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(20*Cos[2*(e + f*x)] + 5*Cos[4*(e + f*x)] + 70*Sin[e + f*x] + 5*Sin[3*(e + f*x)] - 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
349,1,137,89,0.6457746,"\int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(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)} (8 (9 \sin (e+f x)+\sin (3 (e+f x)))+12 \cos (2 (e+f x))+3 \cos (4 (e+f x)))}{96 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 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{6 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{4 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]]*(12*Cos[2*(e + f*x)] + 3*Cos[4*(e + f*x)] + 8*(9*Sin[e + f*x] + Sin[3*(e + f*x)])))/(96*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
350,1,70,89,0.4310917,"\int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{c (\sin (e+f x)-1) (9 \sin (e+f x)+\sin (3 (e+f x))) \sec ^3(e+f x) (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)}}{12 f}","-\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}{3 f}",1,"-1/12*(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]]*(9*Sin[e + f*x] + Sin[3*(e + f*x)]))/f","A",1
351,1,60,43,0.2123776,"\int (a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*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))-4 \sin (e+f x))}{4 f}","\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}}",1,"-1/4*(a*Sec[e + f*x]*(Cos[2*(e + f*x)] - 4*Sin[e + f*x])*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])/f","A",1
352,1,113,96,0.3356711,"\int \frac{(a+a \sin (e+f x))^{3/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/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(\sin (e+f x)+4 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\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)^3}","-\frac{2 a^2 \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 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f \sqrt{c-c \sin (e+f x)}}",1,"-(((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2)*(4*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])^3*Sqrt[c - c*Sin[e + f*x]]))","A",1
353,1,153,97,0.46988,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(3/2),x]","\frac{2 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) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \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)-1\right)}{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 \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 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}",1,"(2*a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(-1 - Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 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
354,1,99,42,0.4808409,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(5/2),x]","\frac{a \sin (e+f x) \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)}{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{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{4 f (c-c \sin (e+f x))^{5/2}}",1,"(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sin[e + f*x]*Sqrt[a*(1 + 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]])","B",1
355,1,106,88,0.5840485,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(7/2),x]","-\frac{a (3 \sin (e+f x)+1) \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 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{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{24 c f (c-c \sin (e+f x))^{5/2}}+\frac{\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])]*(1 + 3*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
356,1,106,92,1.0712465,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(9/2),x]","\frac{a (2 \sin (e+f x)+1) \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 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 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{4 f (c-c \sin (e+f x))^{9/2}}-\frac{a^2 \cos (e+f x)}{12 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}",1,"(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(1 + 2*Sin[e + f*x]))/(6*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
357,1,106,92,1.4945202,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(11/2),x]","-\frac{a (5 \sin (e+f x)+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)}{20 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 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{5 f (c-c \sin (e+f x))^{11/2}}-\frac{a^2 \cos (e+f x)}{20 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}",1,"-1/20*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(3 + 5*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
358,1,156,134,1.3775168,"\int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(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)} (600 \sin (e+f x)+100 \sin (3 (e+f x))+12 \sin (5 (e+f x))+75 \cos (2 (e+f x))+30 \cos (4 (e+f x))+5 \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)^5}","-\frac{a^3 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{15 f}-\frac{a \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]))^(5/2)*Sqrt[c - c*Sin[e + f*x]]*(75*Cos[2*(e + f*x)] + 30*Cos[4*(e + f*x)] + 5*Cos[6*(e + f*x)] + 600*Sin[e + f*x] + 100*Sin[3*(e + f*x)] + 12*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])^5)","A",1
359,1,77,134,0.542548,"\int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2),x]","\frac{a^2 c^2 (150 \sin (e+f x)+25 \sin (3 (e+f x))+3 \sin (5 (e+f x))) \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}{240 f}","-\frac{2 a^3 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{5 f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f}",1,"(a^2*c^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(150*Sin[e + f*x] + 25*Sin[3*(e + f*x)] + 3*Sin[5*(e + f*x)]))/(240*f)","A",1
360,1,133,89,0.6888704,"\int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(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)} (8 (9 \sin (e+f x)+\sin (3 (e+f x)))-12 \cos (2 (e+f x))-3 \cos (4 (e+f x)))}{96 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 \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 f \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}{4 f}",1,"-1/96*(c*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]]*(-12*Cos[2*(e + f*x)] - 3*Cos[4*(e + f*x)] + 8*(9*Sin[e + f*x] + Sin[3*(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
361,1,72,43,0.2706147,"\int (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*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)} (-15 \sin (e+f x)+\sin (3 (e+f x))+6 \cos (2 (e+f x)))}{12 f}","\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sin (e+f x)}}",1,"-1/12*(a^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(6*Cos[2*(e + f*x)] - 15*Sin[e + f*x] + Sin[3*(e + f*x)]))/f","A",1
362,1,127,141,0.5934658,"\int \frac{(a+a \sin (e+f x))^{5/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/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(12 \sin (e+f x)-\cos (2 (e+f x))+32 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\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)^5}","-\frac{4 a^3 \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 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f \sqrt{c-c \sin (e+f x)}}-\frac{a \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]))^(5/2)*(-Cos[2*(e + f*x)] + 32*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 12*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sqrt[c - c*Sin[e + f*x]])","A",1
363,1,169,144,0.8075905,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/(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(\cos (2 (e+f x))+16 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+\sin (e+f x) \left(2-16 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+7\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{4 a^3 \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 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}",1,"-1/2*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(7 + Cos[2*(e + f*x)] + 16*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + (2 - 16*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
364,1,190,147,1.1624584,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(5/2),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(\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)-3 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+4 \sin (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)+1\right)-2\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^3 \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 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f (c-c \sin (e+f x))^{3/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(-2 - 3*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Cos[2*(e + f*x)]*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 4*(1 + 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
365,1,110,42,0.9834977,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(7/2),x]","\frac{a^2 (3 \cos (2 (e+f x))-5) \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 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{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 f (c-c \sin (e+f x))^{7/2}}",1,"(a^2*(-5 + 3*Cos[2*(e + f*x)])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])])/(6*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]])","B",1
366,1,118,88,2.2507931,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/(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 \sin (e+f x)-3 \cos (2 (e+f x))+5)}{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{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{48 c f (c-c \sin (e+f x))^{7/2}}+\frac{\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 - 3*Cos[2*(e + f*x)] + 4*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
367,1,118,133,3.4241322,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/(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) (10 \sin (e+f x)-5 \cos (2 (e+f x))+9)}{30 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{\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{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{40 c f (c-c \sin (e+f x))^{9/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{10 f (c-c \sin (e+f x))^{11/2}}",1,"-1/30*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(9 - 5*Cos[2*(e + f*x)] + 10*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
368,1,118,140,4.9093007,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/(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) (36 \sin (e+f x)-15 \cos (2 (e+f x))+29)}{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 \cos (e+f x)}{60 c^2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 c f (c-c \sin (e+f x))^{11/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{6 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 - 15*Cos[2*(e + f*x)] + 36*Sin[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
369,1,127,179,5.2894973,"\int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2),x]","\frac{a^3 c^4 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (19600 \sin (e+f x)+3920 \sin (3 (e+f x))+784 \sin (5 (e+f x))+80 \sin (7 (e+f x))+1960 \cos (2 (e+f x))+980 \cos (4 (e+f x))+280 \cos (6 (e+f x))+35 \cos (8 (e+f x)))}{35840 f}","-\frac{a^4 \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{14 f}-\frac{3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{28 f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{9/2}}{8 f}",1,"(a^3*c^4*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(1960*Cos[2*(e + f*x)] + 980*Cos[4*(e + f*x)] + 280*Cos[6*(e + f*x)] + 35*Cos[8*(e + f*x)] + 19600*Sin[e + f*x] + 3920*Sin[3*(e + f*x)] + 784*Sin[5*(e + f*x)] + 80*Sin[7*(e + f*x)]))/(35840*f)","A",1
370,1,87,179,0.9874755,"\int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(7/2),x]","\frac{a^3 c^3 (1225 \sin (e+f x)+245 \sin (3 (e+f x))+49 \sin (5 (e+f x))+5 \sin (7 (e+f x))) \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}{2240 f}","-\frac{2 a^4 \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 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}",1,"(a^3*c^3*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(1225*Sin[e + f*x] + 245*Sin[3*(e + f*x)] + 49*Sin[5*(e + f*x)] + 5*Sin[7*(e + f*x)]))/(2240*f)","A",1
371,1,107,134,1.261903,"\int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(5/2),x]","\frac{a^3 c^2 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (600 \sin (e+f x)+100 \sin (3 (e+f x))+12 \sin (5 (e+f x))-75 \cos (2 (e+f x))-30 \cos (4 (e+f x))-5 \cos (6 (e+f x)))}{960 f}","\frac{c^3 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{15 f \sqrt{c-c \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{15 f}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{6 f}",1,"(a^3*c^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-75*Cos[2*(e + f*x)] - 30*Cos[4*(e + f*x)] - 5*Cos[6*(e + f*x)] + 600*Sin[e + f*x] + 100*Sin[3*(e + f*x)] + 12*Sin[5*(e + f*x)]))/(960*f)","A",1
372,1,93,89,0.9631885,"\int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{a^3 c \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (-70 \sin (e+f x)-5 \sin (3 (e+f x))+\sin (5 (e+f x))+20 \cos (2 (e+f x))+5 \cos (4 (e+f x)))}{80 f}","\frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{5 f}",1,"-1/80*(a^3*c*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(20*Cos[2*(e + f*x)] + 5*Cos[4*(e + f*x)] - 70*Sin[e + f*x] - 5*Sin[3*(e + f*x)] + Sin[5*(e + f*x)]))/f","A",1
373,1,82,43,0.3354811,"\int (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)*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)} (56 \sin (e+f x)-8 \sin (3 (e+f x))-28 \cos (2 (e+f x))+\cos (4 (e+f x)))}{32 f}","\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 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]]*(-28*Cos[2*(e + f*x)] + Cos[4*(e + f*x)] + 56*Sin[e + f*x] - 8*Sin[3*(e + f*x)]))/(32*f)","A",1
374,1,150,184,1.0350253,"\int \frac{(a+a \sin (e+f x))^{7/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)/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(87 \sin (e+f x)-\sin (3 (e+f x))-12 \cos (2 (e+f x))+192 \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)^7}","-\frac{8 a^4 \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 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f \sqrt{c-c \sin (e+f x)}}-\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sin (e+f x)}}",1,"-1/12*(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*Cos[2*(e + f*x)] + 192*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 87*Sin[e + f*x] - 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
375,1,179,192,1.6325696,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)/(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(\sin (3 (e+f x))+18 \cos (2 (e+f x))+192 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+\sin (e+f x) \left(39-192 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+44\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{12 a^4 \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{6 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}+\frac{3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f \sqrt{c-c \sin (e+f x)}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{f (c-c \sin (e+f x))^{3/2}}",1,"-1/8*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(44 + 18*Cos[2*(e + f*x)] + 192*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + (39 - 192*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]])*Sin[e + f*x] + 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
376,1,207,195,1.9373142,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(5/2),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(\sin (3 (e+f x))-72 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+4 \cos (2 (e+f x)) \left(6 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-1\right)+\sin (e+f x) \left(96 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+41\right)-28\right)}{4 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{6 a^4 \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 \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 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f (c-c \sin (e+f x))^{3/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(-28 - 72*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 4*Cos[2*(e + f*x)]*(-1 + 6*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]) + (41 + 96*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]])*Sin[e + f*x] + Sin[3*(e + f*x)]))/(4*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
377,1,232,193,2.3175819,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(7/2),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(-3 \sin (3 (e+f x)) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-30 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+18 \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)+1\right)+9 \sin (e+f x) \left(5 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+4\right)-34\right)}{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^4 \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 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f (c-c \sin (e+f x))^{3/2}}-\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f (c-c \sin (e+f x))^{5/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(-34 - 30*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 18*Cos[2*(e + f*x)]*(1 + Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]) + 9*(4 + 5*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]])*Sin[e + f*x] - 3*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(e + f*x)]))/(6*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
378,1,115,42,4.5005301,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(9/2),x]","-\frac{a^3 (\sin (3 (e+f x))-7 \sin (e+f x)) \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 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{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}",1,"-1/4*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(-7*Sin[e + f*x] + Sin[3*(e + f*x)]))/(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]])","B",1
379,1,331,88,6.58665,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(11/2),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)^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 (\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 (\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 (\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{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{80 c f (c-c \sin (e+f x))^{9/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}",1,"(8*(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*(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*(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)) - ((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",1
380,1,335,133,6.6580783,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(13/2),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)^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 (\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{12 (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 (\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{\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{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{60 c f (c-c \sin (e+f x))^{11/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}",1,"(4*(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)) - (12*(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*(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)) - ((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",1
381,1,333,178,6.6905983,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(15/2),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)^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 (\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 (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))^{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 (\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{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2240 c^3 f (c-c \sin (e+f x))^{9/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}",1,"(8*(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*(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])^(15/2)) + (6*(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)) - ((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))","A",1
382,1,329,188,6.7578902,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(17/2),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)^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 (\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{12 (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 (\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{a^4 \cos (e+f x)}{280 c^3 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{11/2}}+\frac{a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{56 c^2 f (c-c \sin (e+f x))^{13/2}}-\frac{3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}",1,"((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)) - (12*(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)) + ((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)) - ((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))","A",1
383,1,136,139,0.5526319,"\int \frac{(c-c \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(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(12 \sin (e+f x)+\cos (2 (e+f x))-32 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\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)^5}","\frac{4 c^3 \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 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}+\frac{c \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^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2*(Cos[2*(e + f*x)] - 32*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 12*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*Sqrt[a*(1 + Sin[e + f*x])])","A",1
384,1,119,93,0.3090379,"\int \frac{(c-c \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(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(\sin (e+f x)-4 \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)} \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 \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 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}",1,"(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])*(-4*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + Sin[e + f*x])*Sqrt[c - c*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
385,1,118,49,0.9840536,"\int \frac{\sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[Sqrt[c - c*Sin[e + f*x]]/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{\sqrt{2} \left(e^{i (e+f x)}+i\right) \left(f x+2 i \log \left(e^{i (e+f x)}+i\right)\right) \sqrt{c-c \sin (e+f x)}}{f \left(e^{i (e+f x)}-i\right) \sqrt{-i a e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2}}","\frac{c \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)}}",1,"-((Sqrt[2]*(I + E^(I*(e + f*x)))*(f*x + (2*I)*Log[I + E^(I*(e + f*x))])*Sqrt[c - c*Sin[e + f*x]])/((-I + E^(I*(e + f*x)))*Sqrt[((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*f))","C",1
386,1,89,46,0.233494,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[1/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]),x]","-\frac{\cos (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)}{f \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) \tanh ^{-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]*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 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
387,1,161,95,0.4011913,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[1/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)),x]","-\frac{\cos (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)+\sin (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)+1\right)}{2 c f (\sin (e+f x)-1) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x)}{2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{\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,"-1/2*(Cos[e + f*x]*(1 - Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[e + f*x]))/(c*f*(-1 + Sin[e + f*x])*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])","A",1
388,1,224,140,0.6327324,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[1/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\cos (e+f x) \left(-3 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+\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 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\sin (e+f x) \left(4 \log \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\right)+4\right)}{8 c^2 f (\sin (e+f x)-1)^2 \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}","\frac{\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{\cos (e+f x)}{4 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{\cos (e+f x)}{4 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}",1,"(Cos[e + f*x]*(4 - 3*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 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*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (-2 + 4*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 4*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[e + f*x]))/(8*c^2*f*(-1 + Sin[e + f*x])^2*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])","A",1
389,1,162,191,1.6085603,"\int \frac{(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(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(\sin (3 (e+f x))-18 \cos (2 (e+f x))-192 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\sin (e+f x) \left(39-192 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-44\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{12 c^4 \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{6 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}-\frac{3 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f \sqrt{a \sin (e+f x)+a}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{f (a \sin (e+f x)+a)^{3/2}}",1,"(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(-44 - 18*Cos[2*(e + f*x)] - 192*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (39 - 192*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[e + f*x] + Sin[3*(e + f*x)]))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2))","A",1
390,1,153,143,0.7660633,"\int \frac{(c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(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(\cos (2 (e+f x))+16 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+2 \sin (e+f x) \left(8 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-1\right)+7\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{4 c^3 \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 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)^{3/2}}",1,"-1/2*(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(7 + Cos[2*(e + f*x)] + 16*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 2*(-1 + 8*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
391,1,134,97,0.4399971,"\int \frac{(c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(c - c*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{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) \left(\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)+\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+1\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{c^2 \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 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f (a \sin (e+f x)+a)^{3/2}}",1,"(-2*c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(1 + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 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
392,1,85,41,0.1915139,"\int \frac{\sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[Sqrt[c - c*Sin[e + f*x]]/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{\sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}{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{c \cos (e+f x)}{f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}",1,"-((Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3))","B",1
393,1,148,95,0.40523,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]),x]","-\frac{\cos (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)+\sin (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)+1\right)}{2 f (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)}}","\frac{\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{\cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}",1,"-1/2*(Cos[e + f*x]*(1 + Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[e + f*x]))/(f*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
394,1,170,143,0.6174734,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\cos (e+f x) \left(-2 \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)+\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)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\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{\cos (e+f x)}{2 a f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac{\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,"(Cos[e + f*x]*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 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]]) - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 2*Sin[e + f*x]))/(4*c*f*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
395,1,287,191,0.7673017,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[1/((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(2 \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+\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-3 \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 \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)\right)}{8 f (a (\sin (e+f x)+1))^{3/2} (c-c \sin (e+f x))^{5/2}}","\frac{3 \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 \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 \cos (e+f x)}{8 a f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{\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*Cos[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*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*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
396,1,202,237,5.1197298,"\int \frac{(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(c - c*Sin[e + f*x])^(9/2)/(a + a*Sin[e + f*x])^(5/2),x]","\frac{c^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) \left(320 \sin (e+f x)+24 \sin (3 (e+f x))+\cos (4 (e+f x))+\cos (2 (e+f x)) \left(106-384 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+1152 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+1536 \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)+273\right)}{16 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{24 c^5 \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{12 c^4 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{3 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{a f (a \sin (e+f x)+a)^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{2 f (a \sin (e+f x)+a)^{5/2}}",1,"(c^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(273 + Cos[4*(e + f*x)] + Cos[2*(e + f*x)]*(106 - 384*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 1152*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 320*Sin[e + f*x] + 1536*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] + 24*Sin[3*(e + f*x)]))/(16*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))","A",1
397,1,187,193,1.9981307,"\int \frac{(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(c - c*Sin[e + f*x])^(7/2)/(a + a*Sin[e + f*x])^(5/2),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(\sin (3 (e+f x))+\cos (2 (e+f x)) \left(4-24 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+72 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\sin (e+f x) \left(96 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+41\right)+28\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)}","\frac{6 c^4 \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 \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 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f (a \sin (e+f x)+a)^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{2 f (a \sin (e+f x)+a)^{5/2}}",1,"(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(28 + Cos[2*(e + f*x)]*(4 - 24*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 72*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (41 + 96*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[e + f*x] + Sin[3*(e + f*x)]))/(4*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))","A",1
398,1,172,143,1.1877387,"\int \frac{(c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(c - c*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^(5/2),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(-\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)+3 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+4 \sin (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)+1\right)+2\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{c^3 \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 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f (a \sin (e+f x)+a)^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f (a \sin (e+f x)+a)^{5/2}}",1,"(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(2 + 3*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - Cos[2*(e + f*x)]*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 4*(1 + 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
399,1,86,42,0.4580831,"\int \frac{(c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(c - c*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^(5/2),x]","\frac{c \sin (e+f x) \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)}{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{\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))","B",1
400,1,87,43,0.2198178,"\int \frac{\sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[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)}}{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 \cos (e+f x)}{2 f (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}",1,"-1/2*(Sqrt[a*(1 + 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)","B",1
401,1,211,140,0.6484763,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{\cos (e+f x) \left(-3 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+\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 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\sin (e+f x) \left(-4 \log \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\right)-4\right)}{8 f (a (\sin (e+f x)+1))^{5/2} \sqrt{c-c \sin (e+f x)}}","\frac{\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{\cos (e+f x)}{4 a f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{\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]*(-4 - 3*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 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*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (-2 - 4*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 4*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[e + f*x]))/(8*f*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
402,1,287,188,0.7462305,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[1/((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(-2 \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-\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2-3 \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)+3 \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)\right)}{8 f (a (\sin (e+f x)+1))^{5/2} (c-c \sin (e+f x))^{3/2}}","\frac{3 \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 \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 \cos (e+f x)}{8 a f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}-\frac{\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*Cos[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*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*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
403,1,237,236,0.9712163,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[1/((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 \sin (e+f x)+6 \sin (3 (e+f x))-9 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-12 \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 \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 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{64 a^2 c^2 f \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}","\frac{3 \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 \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 \cos (e+f x)}{8 a^2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{\cos (e+f x)}{2 a f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}-\frac{\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}",1,"(Sec[e + f*x]^3*(-9*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 12*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*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*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 22*Sin[e + f*x] + 6*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
404,1,365,110,2.9568924,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{4 (2 n+3) \sin \left(\frac{1}{8} (2 e+2 f x-\pi )\right) \cos ^3\left(\frac{1}{8} (2 e+2 f x-\pi )\right) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n F_1\left(n+\frac{1}{2};-2 m,2 (m+n)+1;n+\frac{3}{2};\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)}{f (2 n+1) \left((2 n+3) \cos ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right) F_1\left(n+\frac{1}{2};-2 m,2 (m+n)+1;n+\frac{3}{2};\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)-2 \sin ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right) \left(2 m F_1\left(n+\frac{3}{2};1-2 m,2 (m+n)+1;n+\frac{5}{2};\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)+(2 m+2 n+1) F_1\left(n+\frac{3}{2};-2 m,2 (m+n+1);n+\frac{5}{2};\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)\right)\right)}","\frac{c 2^{n+\frac{1}{2}} \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)}",1,"(4*(3 + 2*n)*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2]*Cos[(2*e - Pi + 2*f*x)/8]^3*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^n*Sin[(2*e - Pi + 2*f*x)/8])/(f*(1 + 2*n)*((3 + 2*n)*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2]*Cos[(2*e - Pi + 2*f*x)/8]^2 - 2*(2*m*AppellF1[3/2 + n, 1 - 2*m, 1 + 2*(m + n), 5/2 + n, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2] + (1 + 2*m + 2*n)*AppellF1[3/2 + n, -2*m, 2*(1 + m + n), 5/2 + n, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2])*Sin[(2*e - Pi + 2*f*x)/8]^2))","C",0
405,-1,0,86,180.0111481,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^3,x]","\text{\$Aborted}","-\frac{a^4 c^3 2^{m+\frac{1}{2}} \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}",1,"$Aborted","F",-1
406,1,88512,86,153.5142706,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^2,x]","\text{Result too large to show}","-\frac{a^3 c^2 2^{m+\frac{1}{2}} \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}",1,"Result too large to show","C",0
407,1,261,84,1.6636002,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x]),x]","-\frac{(-1)^{3/4} c 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} (\sin (e+f x)-1) \left((m-1) m e^{2 i (e+f x)} \, _2F_1\left(1,m;-m;-i e^{-i (e+f x)}\right)-(m+1) \left(2 (m-1) e^{i (e+f x)} \, _2F_1\left(1,m+1;1-m;-i e^{-i (e+f x)}\right)-m \, _2F_1\left(1,m+2;2-m;-i e^{-i (e+f x)}\right)\right)\right) \sin ^{-2 m}\left(\frac{1}{4} (2 e+2 f x+\pi )\right) (a (\sin (e+f x)+1))^m}{f (m-1) m (m+1) \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}} \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}",1,"-(((-1)^(3/4)*2^(-1 - 2*m)*c*(-(((-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)*m*Hypergeometric2F1[1, m, -m, (-I)/E^(I*(e + f*x))] - (1 + m)*(2*E^(I*(e + f*x))*(-1 + m)*Hypergeometric2F1[1, 1 + m, 1 - m, (-I)/E^(I*(e + f*x))] - m*Hypergeometric2F1[1, 2 + m, 2 - m, (-I)/E^(I*(e + f*x))]))*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^m)/(E^(((3*I)/2)*(e + f*x))*f*(-1 + m)*m*(1 + m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(2*e + Pi + 2*f*x)/4]^(2*m)))","C",0
408,1,3844,76,6.3915036,"\int \frac{(a+a \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x]),x]","\text{Result too large to show}","\frac{2^{m+\frac{1}{2}} \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}",1,"-1/2*((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Cot[(-e + Pi/2 - f*x)/4]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(a + a*Sin[e + f*x])^m*(-(AppellF1[-1/2, -2*m, 2*m, 1/2, 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)) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2)))/(f*(c - c*Sin[e + f*x])*(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^2*(-1/2*(m*(Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(-(AppellF1[-1/2, -2*m, 2*m, 1/2, 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)) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2))) - ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Csc[(-e + Pi/2 - f*x)/4]^2*(-(AppellF1[-1/2, -2*m, 2*m, 1/2, 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)) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2)))/8 + ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Cot[(-e + Pi/2 - f*x)/4]*(-(m*AppellF1[-1/2, -2*m, 2*m, 1/2, 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]) - (Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(m*AppellF1[1/2, 1 - 2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4] + m*AppellF1[1/2, -2*m, 1 + 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(2*(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2)) + (3*Tan[(-e + Pi/2 - f*x)/4]^2*(-1/3*(m*AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]) - (m*AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/3)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2) - (3*m*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]^3*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(-1 + 2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2) - (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(-2*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4] + 3*(-1/3*(m*AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]) - (m*AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/3) - 4*m*Tan[(-e + Pi/2 - f*x)/4]^2*((-6*m*AppellF1[5/2, 1 - 2*m, 1 + 2*m, 7/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/5 + (3*(1 - 2*m)*AppellF1[5/2, 2 - 2*m, 2*m, 7/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/10 - (3*(1 + 2*m)*AppellF1[5/2, -2*m, 2 + 2*m, 7/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/10)))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2)^2))/2))","C",0
409,1,5391,86,6.423568,"\int \frac{(a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^2} \, dx","Integrate[(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^2,x]","\text{Result too large to show}","\frac{2^{m+\frac{1}{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,"Result too large to show","C",0
410,1,7184,86,22.7761535,"\int \frac{(a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx","Integrate[(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^3,x]","\text{Result too large to show}","\frac{2^{m+\frac{1}{2}} \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}",1,"Result too large to show","C",0
411,1,149,160,2.5078441,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2),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) (a (\sin (e+f x)+1))^m \left(4 \left(4 m^2+16 m+7\right) \sin (e+f x)+\left(4 m^2+8 m+3\right) \cos (2 (e+f x))-12 m^2-56 m-89\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{64 c^3 \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}+\frac{16 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f \left(4 m^2+16 m+15\right)}+\frac{2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}",1,"-((c^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]]*(-89 - 56*m - 12*m^2 + (3 + 8*m + 4*m^2)*Cos[2*(e + f*x)] + 4*(7 + 16*m + 4*m^2)*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
412,1,110,100,0.4778603,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{2 c \sqrt{c-c \sin (e+f x)} ((2 m+1) \sin (e+f x)-2 m-5) \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 (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{8 c^2 \cos (e+f x) (a \sin (e+f x)+a)^m}{f \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}+\frac{2 c \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}",1,"(-2*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]]*(-5 - 2*m + (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
413,1,85,46,0.1660999,"\int (a+a \sin (e+f x))^m \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^m*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}{f (2 m+1) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{2 c \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \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]])/(f*(1 + 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
414,1,157,68,1.4560036,"\int \frac{(a+a \sin (e+f x))^m}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])^m/Sqrt[c - c*Sin[e + f*x]],x]","\frac{2^{-2 m-\frac{3}{2}} \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(4^m \, _2F_1\left(1,2 m;2 m+1;\sin \left(\frac{1}{4} (2 e+2 f x+\pi )\right)\right)-\sec ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)^{2 m} \, _2F_1\left(2 m,2 m;2 m+1;\frac{1}{2} \left(1-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)\right)\right)}{f m \sqrt{c-c \sin (e+f x)}}","\frac{\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)}}",1,"(2^(-3/2 - 2*m)*(4^m*Hypergeometric2F1[1, 2*m, 1 + 2*m, Sin[(2*e + Pi + 2*f*x)/4]] - Hypergeometric2F1[2*m, 2*m, 1 + 2*m, (1 - Tan[(2*e - Pi + 2*f*x)/8]^2)/2]*(Sec[(2*e - Pi + 2*f*x)/8]^2)^(2*m))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^m)/(f*m*Sqrt[c - c*Sin[e + f*x]])","B",1
415,1,3006,74,17.2336139,"\int \frac{(a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(3/2),x]","\text{Result too large to show}","\frac{\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)}{2 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}",1,"-1/8*((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)))/(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
416,1,5136,74,21.981983,"\int \frac{(a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(5/2),x]","\text{Result too large to show}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left(3,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{4 c^2 f (2 m+1) \sqrt{c-c \sin (e+f x)}}",1,"Result too large to show","C",0
417,1,157,68,0.5266323,"\int \frac{(a+a \sin (e+f x))^m}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])^m/Sqrt[c - c*Sin[e + f*x]],x]","\frac{2^{-2 m-\frac{3}{2}} \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(4^m \, _2F_1\left(1,2 m;2 m+1;\sin \left(\frac{1}{4} (2 e+2 f x+\pi )\right)\right)-\sec ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)^{2 m} \, _2F_1\left(2 m,2 m;2 m+1;\frac{1}{2} \left(1-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)\right)\right)}{f m \sqrt{c-c \sin (e+f x)}}","\frac{\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)}}",1,"(2^(-3/2 - 2*m)*(4^m*Hypergeometric2F1[1, 2*m, 1 + 2*m, Sin[(2*e + Pi + 2*f*x)/4]] - Hypergeometric2F1[2*m, 2*m, 1 + 2*m, (1 - Tan[(2*e - Pi + 2*f*x)/8]^2)/2]*(Sec[(2*e - Pi + 2*f*x)/8]^2)^(2*m))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^m)/(f*m*Sqrt[c - c*Sin[e + f*x]])","B",1
418,1,157,68,0.4384616,"\int \frac{(c+c \sin (e+f x))^m}{\sqrt{a-a \sin (e+f x)}} \, dx","Integrate[(c + c*Sin[e + f*x])^m/Sqrt[a - a*Sin[e + f*x]],x]","\frac{2^{-2 m-\frac{3}{2}} \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(4^m \, _2F_1\left(1,2 m;2 m+1;\sin \left(\frac{1}{4} (2 e+2 f x+\pi )\right)\right)-\sec ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)^{2 m} \, _2F_1\left(2 m,2 m;2 m+1;\frac{1}{2} \left(1-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)\right)\right)}{f m \sqrt{a-a \sin (e+f x)}}","\frac{\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)}}",1,"(2^(-3/2 - 2*m)*(4^m*Hypergeometric2F1[1, 2*m, 1 + 2*m, Sin[(2*e + Pi + 2*f*x)/4]] - Hypergeometric2F1[2*m, 2*m, 1 + 2*m, (1 - Tan[(2*e - Pi + 2*f*x)/8]^2)/2]*(Sec[(2*e - Pi + 2*f*x)/8]^2)^(2*m))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c*(1 + Sin[e + f*x]))^m)/(f*m*Sqrt[a - a*Sin[e + f*x]])","B",1
419,1,174,164,8.5326503,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m),x]","\frac{2^{-m-2} \cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^{-2 m-5}\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) \sin (e+f x)+\cos \left(2 \left(-e-f x+\frac{\pi }{2}\right)\right)+4 \left(m^2+3 m+2\right)\right)}{f (2 m+1) (2 m+3) (2 m+5)}","\frac{2 \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{2 \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{c f \left(4 m^2+16 m+15\right)}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3}}{f (2 m+5)}",1,"(2^(-2 - m)*Cos[(-e + Pi/2 - f*x)/2]*Sin[(-e + Pi/2 - f*x)/2]^(-5 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m)*(4*(2 + 3*m + m^2) + Cos[2*(-e + Pi/2 - f*x)] - 2*(3 + 2*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])^(2*(-3 - m)))","A",1
420,1,136,101,3.044368,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m),x]","-\frac{2^{-m} \cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^{-2 m-3}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (\sin (e+f x)-2 (m+1)) (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)}}{f \left(8 m^2+16 m+6\right)}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c f \left(4 m^2+8 m+3\right)}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{f (2 m+3)}",1,"-((Cos[(-e + Pi/2 - f*x)/2]*Sin[(-e + Pi/2 - f*x)/2]^(-3 - 2*m)*(-2*(1 + m) + Sin[e + f*x])*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(2^m*f*(6 + 16*m + 8*m^2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(-2 - m))))","A",1
421,1,107,46,1.5269951,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m),x]","\frac{2^{-m} \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) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^{-m} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{2 m}}{c f (2 m+1)}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{f (2 m+1)}",1,"(Cos[(2*e + Pi + 2*f*x)/4]^(-1 - 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*m)*(a*(1 + Sin[e + f*x]))^m*Sin[(2*e + Pi + 2*f*x)/4])/(2^m*c*f*(1 + 2*m)*(c - c*Sin[e + f*x])^m)","B",1
422,1,388,112,2.9154081,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-m} \, dx","Integrate[(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^m,x]","\frac{2^{1-m} (2 m-3) \sin ^2\left(\frac{1}{8} (2 e+2 f x+3 \pi )\right) \cos ^{1-2 m}\left(\frac{1}{4} (2 e+2 f x+\pi )\right) (a (\sin (e+f x)+1))^m F_1\left(\frac{1}{2}-m;-2 m,1;\frac{3}{2}-m;\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right) (c-c \sin (e+f x))^{-m} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{2 m}}{f (2 m-1) \left(2 \sin ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right) \left(2 m F_1\left(\frac{3}{2}-m;1-2 m,1;\frac{5}{2}-m;\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)+F_1\left(\frac{3}{2}-m;-2 m,2;\frac{5}{2}-m;\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)\right)+(2 m-3) \cos ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right) F_1\left(\frac{1}{2}-m;-2 m,1;\frac{3}{2}-m;\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)\right)}","\frac{c 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,"(2^(1 - m)*(-3 + 2*m)*AppellF1[1/2 - m, -2*m, 1, 3/2 - m, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2]*Cos[(2*e + Pi + 2*f*x)/4]^(1 - 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*m)*(a*(1 + Sin[e + f*x]))^m*Sin[(2*e + 3*Pi + 2*f*x)/8]^2)/(f*(-1 + 2*m)*(c - c*Sin[e + f*x])^m*((-3 + 2*m)*AppellF1[1/2 - m, -2*m, 1, 3/2 - m, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2]*Cos[(2*e - Pi + 2*f*x)/8]^2 + 2*(2*m*AppellF1[3/2 - m, 1 - 2*m, 1, 5/2 - m, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2] + AppellF1[3/2 - m, -2*m, 2, 5/2 - m, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2])*Sin[(2*e - Pi + 2*f*x)/8]^2))","C",0
423,1,602,114,7.1800538,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 - m),x]","-\frac{c 2^{2-m} (2 m-3) (\sin (e+f x)-1) \cos ^{3-2 m}\left(\frac{1}{4} (2 e+2 f x+\pi )\right) (a (\sin (e+f x)+1))^m \left(F_1\left(\frac{1}{2}-m;-2 m,2;\frac{3}{2}-m;\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)-F_1\left(\frac{1}{2}-m;-2 m,3;\frac{3}{2}-m;\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)\right) (c-c \sin (e+f x))^{-m} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{2 (m-1)}}{f (2 m-1) \left(2 \tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right) \left(2 m F_1\left(\frac{3}{2}-m;1-2 m,2;\frac{5}{2}-m;\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)-2 m F_1\left(\frac{3}{2}-m;1-2 m,3;\frac{5}{2}-m;\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)+2 F_1\left(\frac{3}{2}-m;-2 m,3;\frac{5}{2}-m;\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)-3 F_1\left(\frac{3}{2}-m;-2 m,4;\frac{5}{2}-m;\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)\right)+(2 m-3) F_1\left(\frac{1}{2}-m;-2 m,2;\frac{3}{2}-m;\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)+(3-2 m) F_1\left(\frac{1}{2}-m;-2 m,3;\frac{3}{2}-m;\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right),-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)\right)}","\frac{c^2 2^{\frac{3}{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,"-((2^(2 - m)*c*(-3 + 2*m)*(AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2] - AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2])*Cos[(2*e + Pi + 2*f*x)/4]^(3 - 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(-1 + m))*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^m)/(f*(-1 + 2*m)*(c - c*Sin[e + f*x])^m*((-3 + 2*m)*AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2] + (3 - 2*m)*AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2] + 2*(2*m*AppellF1[3/2 - m, 1 - 2*m, 2, 5/2 - m, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2] - 2*m*AppellF1[3/2 - m, 1 - 2*m, 3, 5/2 - m, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2] + 2*AppellF1[3/2 - m, -2*m, 3, 5/2 - m, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2] - 3*AppellF1[3/2 - m, -2*m, 4, 5/2 - m, Tan[(-2*e + Pi - 2*f*x)/8]^2, -Tan[(2*e - Pi + 2*f*x)/8]^2])*Tan[(2*e - Pi + 2*f*x)/8]^2)))","C",0
424,1,1201,114,12.7111028,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{2-m} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(2 - m),x]","\frac{2^{4-m} (2 m-3) \left(F_1\left(\frac{1}{2}-m;-2 m,3;\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)-2 F_1\left(\frac{1}{2}-m;-2 m,4;\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)+F_1\left(\frac{1}{2}-m;-2 m,5;\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) \cos \left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin \left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^{4-2 m}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{-2 (2-m)} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{2-m}}{f (2 m-1) \left(-8 F_1\left(\frac{3}{2}-m;-2 m,5;\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) \sec ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+8 F_1\left(\frac{3}{2}-m;-2 m,5;\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) \cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sec ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+4 m F_1\left(\frac{3}{2}-m;1-2 m,3;\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) \tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)-8 m F_1\left(\frac{3}{2}-m;1-2 m,4;\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) \tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+4 m F_1\left(\frac{3}{2}-m;1-2 m,5;\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) \tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+6 F_1\left(\frac{3}{2}-m;-2 m,4;\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) \tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+10 F_1\left(\frac{3}{2}-m;-2 m,6;\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) \tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+(2 m-3) F_1\left(\frac{1}{2}-m;-2 m,3;\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)+(6-4 m) F_1\left(\frac{1}{2}-m;-2 m,4;\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)+2 m F_1\left(\frac{1}{2}-m;-2 m,5;\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)-3 F_1\left(\frac{1}{2}-m;-2 m,5;\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)}","\frac{c^3 2^{\frac{5}{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-3),\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,"(2^(4 - m)*(-3 + 2*m)*(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] - 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)/4]*Sin[(-e + Pi/2 - f*x)/4]*Sin[(-e + Pi/2 - f*x)/2]^(4 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(2 - m))/(f*(-1 + 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(2 - m))*((-3 + 2*m)*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] + (6 - 4*m)*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] - 3*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] + 2*m*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] - 8*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 + 8*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]*Cos[(-e + Pi/2 - f*x)/2]*Sec[(-e + Pi/2 - f*x)/4]^2 + 4*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]*Tan[(-e + Pi/2 - f*x)/4]^2 - 8*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]*Tan[(-e + Pi/2 - f*x)/4]^2 + 4*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]*Tan[(-e + Pi/2 - f*x)/4]^2 + 6*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]*Tan[(-e + Pi/2 - f*x)/4]^2 + 10*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]*Tan[(-e + Pi/2 - f*x)/4]^2))","C",0
425,1,207,227,1.3623527,"\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^4 \, dx","Integrate[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^4,x]","\frac{a (\sin (e+f x)+1) \left(10 d^2 \left(24 c^2+16 c d+5 d^2\right) \cos (3 (e+f x))+15 \left(-8 d \left(4 c^3+6 c^2 d+4 c d^2+d^3\right) \sin (2 (e+f x))+4 f x \left(8 c^4+16 c^3 d+24 c^2 d^2+12 c d^3+3 d^4\right)+d^3 (4 c+d) \sin (4 (e+f x))\right)-60 \left(8 c^4+32 c^3 d+36 c^2 d^2+24 c d^3+5 d^4\right) \cos (e+f x)-6 d^4 \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(12 c^2+35 c d+16 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^2}{60 f}-\frac{a d \left(24 c^3+130 c^2 d+116 c d^2+45 d^3\right) \sin (e+f x) \cos (e+f x)}{120 f}-\frac{a \left(12 c^4+95 c^3 d+112 c^2 d^2+80 c d^3+16 d^4\right) \cos (e+f x)}{30 f}+\frac{1}{8} a x \left(8 c^4+16 c^3 d+24 c^2 d^2+12 c d^3+3 d^4\right)-\frac{a \cos (e+f x) (c+d \sin (e+f x))^4}{5 f}-\frac{a (4 c+5 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 f}",1,"(a*(1 + Sin[e + f*x])*(-60*(8*c^4 + 32*c^3*d + 36*c^2*d^2 + 24*c*d^3 + 5*d^4)*Cos[e + f*x] + 10*d^2*(24*c^2 + 16*c*d + 5*d^2)*Cos[3*(e + f*x)] - 6*d^4*Cos[5*(e + f*x)] + 15*(4*(8*c^4 + 16*c^3*d + 24*c^2*d^2 + 12*c*d^3 + 3*d^4)*f*x - 8*d*(4*c^3 + 6*c^2*d + 4*c*d^2 + d^3)*Sin[2*(e + f*x)] + d^3*(4*c + d)*Sin[4*(e + f*x)])))/(480*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)","A",1
426,1,124,162,0.8012435,"\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^3,x]","\frac{a \left(3 \left(-8 d \left(3 c^2+3 c d+d^2\right) \sin (2 (e+f x))+4 f x \left(8 c^3+12 c^2 d+12 c d^2+3 d^3\right)+d^3 \sin (4 (e+f x))\right)-24 \left(4 c^3+12 c^2 d+9 c d^2+3 d^3\right) \cos (e+f x)+8 d^2 (3 c+d) \cos (3 (e+f x))\right)}{96 f}","-\frac{a d \left(6 c^2+20 c d+9 d^2\right) \sin (e+f x) \cos (e+f x)}{24 f}-\frac{a \left(3 c^3+16 c^2 d+12 c d^2+4 d^3\right) \cos (e+f x)}{6 f}+\frac{1}{8} a x \left(8 c^3+12 c^2 d+12 c d^2+3 d^3\right)-\frac{a \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}-\frac{a (3 c+4 d) \cos (e+f x) (c+d \sin (e+f x))^2}{12 f}",1,"(a*(-24*(4*c^3 + 12*c^2*d + 9*c*d^2 + 3*d^3)*Cos[e + f*x] + 8*d^2*(3*c + d)*Cos[3*(e + f*x)] + 3*(4*(8*c^3 + 12*c^2*d + 12*c*d^2 + 3*d^3)*f*x - 8*d*(3*c^2 + 3*c*d + d^2)*Sin[2*(e + f*x)] + d^3*Sin[4*(e + f*x)])))/(96*f)","A",1
427,1,89,99,0.3960991,"\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]","\frac{a \left(-3 \left(4 c^2+8 c d+3 d^2\right) \cos (e+f x)+12 c^2 f x-6 c d \sin (2 (e+f x))+12 c d f x-3 d^2 \sin (2 (e+f x))+d^2 \cos (3 (e+f x))+6 d^2 f x\right)}{12 f}","-\frac{2 a \left(c^2+3 c d+d^2\right) \cos (e+f x)}{3 f}+\frac{1}{2} a x \left(2 c^2+2 c d+d^2\right)-\frac{a \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}-\frac{a d (2 c+3 d) \sin (e+f x) \cos (e+f x)}{6 f}",1,"(a*(12*c^2*f*x + 12*c*d*f*x + 6*d^2*f*x - 3*(4*c^2 + 8*c*d + 3*d^2)*Cos[e + f*x] + d^2*Cos[3*(e + f*x)] - 6*c*d*Sin[2*(e + f*x)] - 3*d^2*Sin[2*(e + f*x)]))/(12*f)","A",1
428,1,45,48,0.108291,"\int (a+a \sin (e+f x)) (c+d \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]","\frac{a (-4 (c+d) \cos (e+f x)+4 c f x-d \sin (2 (e+f x))+2 d e+2 d f x)}{4 f}","-\frac{a (c+d) \cos (e+f x)}{f}+\frac{1}{2} a x (2 c+d)-\frac{a d \sin (e+f x) \cos (e+f x)}{2 f}",1,"(a*(2*d*e + 4*c*f*x + 2*d*f*x - 4*(c + d)*Cos[e + f*x] - d*Sin[2*(e + f*x)]))/(4*f)","A",1
429,1,27,16,0.0068918,"\int (a+a \sin (e+f x)) \, dx","Integrate[a + a*Sin[e + f*x],x]","\frac{a \sin (e) \sin (f x)}{f}-\frac{a \cos (e) \cos (f x)}{f}+a x","a x-\frac{a \cos (e+f x)}{f}",1,"a*x - (a*Cos[e]*Cos[f*x])/f + (a*Sin[e]*Sin[f*x])/f","A",1
430,1,182,63,0.3231264,"\int \frac{a+a \sin (e+f x)}{c+d \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x]),x]","\frac{a (\sin (e+f x)+1) \left(f x \sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}-2 (c-d) (\cos (e)-i \sin (e)) \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)\right)}{d f \sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","\frac{a x}{d}-\frac{2 a (c-d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d f \sqrt{c^2-d^2}}",1,"(a*(-2*(c - 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*x*Sqrt[(Cos[e] - I*Sin[e])^2])*(1 + Sin[e + f*x]))/(d*Sqrt[c^2 - d^2]*f*Sqrt[(Cos[e] - I*Sin[e])^2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)","C",1
431,1,220,83,0.5891598,"\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx","Integrate[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^2,x]","\frac{a (\sin (e+f x)+1) \left(2 \sqrt{c^2-d^2} \csc (e) \sqrt{(\cos (e)-i \sin (e))^2} (c \cos (e)+d \sin (f x))+4 d (\cos (e)-i \sin (e)) (c+d \sin (e+f x)) \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)\right)}{2 d f (c+d) \sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 (c+d \sin (e+f x))}","\frac{2 a \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c+d) \sqrt{c^2-d^2}}-\frac{a \cos (e+f x)}{f (c+d) (c+d \sin (e+f x))}",1,"(a*(1 + Sin[e + f*x])*(2*Sqrt[c^2 - d^2]*Csc[e]*Sqrt[(Cos[e] - I*Sin[e])^2]*(c*Cos[e] + d*Sin[f*x]) + 4*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])*(c + d*Sin[e + f*x])))/(2*d*(c + d)*Sqrt[c^2 - d^2]*f*Sqrt[(Cos[e] - I*Sin[e])^2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(c + d*Sin[e + f*x]))","C",1
432,1,242,134,1.2006242,"\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx","Integrate[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^3,x]","\frac{a (\sin (e+f x)+1) \left(\frac{4 (2 c-d) (\cos (e)-i \sin (e)) \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)}{(c-d) \sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}+\frac{2 (c+d) \csc (e) (c \cos (e)+d \sin (f x))}{d (c+d \sin (e+f x))^2}+\frac{2 (c-2 d) \csc (e) \sin (f x)+(2 d-4 c) \cot (e)}{(c-d) (c+d \sin (e+f x))}\right)}{4 f (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 c-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 (c-2 d) \cos (e+f x)}{2 f (c-d) (c+d)^2 (c+d \sin (e+f x))}-\frac{a \cos (e+f x)}{2 f (c+d) (c+d \sin (e+f x))^2}",1,"(a*(1 + Sin[e + f*x])*((4*(2*c - 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]))/((c - d)*Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + (2*(c + d)*Csc[e]*(c*Cos[e] + d*Sin[f*x]))/(d*(c + d*Sin[e + f*x])^2) + ((-4*c + 2*d)*Cot[e] + 2*(c - 2*d)*Csc[e]*Sin[f*x])/((c - d)*(c + d*Sin[e + f*x]))))/(4*(c + d)^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)","C",1
433,1,428,192,2.7056463,"\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^4} \, dx","Integrate[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^4,x]","\frac{a (\sin (e+f x)+1) \left(\frac{2 c \left(4 c^4-18 c^3 d+14 c^2 d^2-27 c d^3+12 d^4\right) \cot (e)-d \csc (e) \left(-24 c^4 \sin (f x)+30 c^3 d \sin (2 e+f x)+78 c^3 d \sin (f x)-30 c^2 d^2 \sin (2 e+f x)+2 c^2 d^2 \sin (2 e+3 f x)-3 d^2 \left(2 c^2-2 c d+d^2\right) \cos (3 e+2 f x)-24 c^2 d^2 \sin (f x)+3 d \left(4 c^3-16 c^2 d+6 c d^2+d^3\right) \cos (e+2 f x)+15 c d^3 \sin (2 e+f x)-9 c d^3 \sin (2 e+3 f x)+12 c d^3 \sin (f x)+4 d^4 \sin (2 e+3 f x)-12 d^4 \sin (f x)\right)}{d (c+d \sin (e+f x))^3}+\frac{24 \left(2 c^2-2 c d+d^2\right) (\cos (e)-i \sin (e)) \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)}{24 f (c-d)^2 (c+d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","\frac{a \left(2 c^2-2 c d+d^2\right) \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)^{5/2}}-\frac{a (c-4 d) (2 c-d) \cos (e+f x)}{6 f (c-d)^2 (c+d)^3 (c+d \sin (e+f x))}-\frac{a (2 c-3 d) \cos (e+f x)}{6 f (c-d) (c+d)^2 (c+d \sin (e+f x))^2}-\frac{a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^3}",1,"(a*(1 + Sin[e + f*x])*((24*(2*c^2 - 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]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + (2*c*(4*c^4 - 18*c^3*d + 14*c^2*d^2 - 27*c*d^3 + 12*d^4)*Cot[e] - d*Csc[e]*(3*d*(4*c^3 - 16*c^2*d + 6*c*d^2 + d^3)*Cos[e + 2*f*x] - 3*d^2*(2*c^2 - 2*c*d + d^2)*Cos[3*e + 2*f*x] - 24*c^4*Sin[f*x] + 78*c^3*d*Sin[f*x] - 24*c^2*d^2*Sin[f*x] + 12*c*d^3*Sin[f*x] - 12*d^4*Sin[f*x] + 30*c^3*d*Sin[2*e + f*x] - 30*c^2*d^2*Sin[2*e + f*x] + 15*c*d^3*Sin[2*e + f*x] + 2*c^2*d^2*Sin[2*e + 3*f*x] - 9*c*d^3*Sin[2*e + 3*f*x] + 4*d^4*Sin[2*e + 3*f*x]))/(d*(c + d*Sin[e + f*x])^3)))/(24*(c - d)^2*(c + d)^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)","C",1
434,1,262,318,1.3988818,"\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^4,x]","-\frac{a^2 \cos (e+f x) \left(30 \left(24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right) \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\sqrt{\cos ^2(e+f x)} \left(10 d^2 \left(36 c^2+48 c d+11 d^2\right) \sin ^3(e+f x)+64 d \left(5 c^3+15 c^2 d+9 c d^2+2 d^3\right) \sin ^2(e+f x)+15 \left(8 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right) \sin (e+f x)+32 \left(15 c^4+50 c^3 d+60 c^2 d^2+36 c d^3+8 d^4\right)+96 d^3 (2 c+d) \sin ^4(e+f x)+40 d^4 \sin ^5(e+f x)\right)\right)}{240 f \sqrt{\cos ^2(e+f x)}}","\frac{a^2 \left(4 c^2-48 c d-55 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac{a^2 \left(4 c^3-48 c^2 d-123 c d^2-64 d^3\right) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac{a^2 \left(8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right) \sin (e+f x) \cos (e+f x)}{240 f}+\frac{1}{16} a^2 x \left(24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right)+\frac{a^2 \left(4 c^5-48 c^4 d-311 c^3 d^2-448 c^2 d^3-288 c d^4-64 d^5\right) \cos (e+f x)}{60 d f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac{a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}",1,"-1/240*(a^2*Cos[e + f*x]*(30*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(32*(15*c^4 + 50*c^3*d + 60*c^2*d^2 + 36*c*d^3 + 8*d^4) + 15*(8*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*Sin[e + f*x] + 64*d*(5*c^3 + 15*c^2*d + 9*c*d^2 + 2*d^3)*Sin[e + f*x]^2 + 10*d^2*(36*c^2 + 48*c*d + 11*d^2)*Sin[e + f*x]^3 + 96*d^3*(2*c + d)*Sin[e + f*x]^4 + 40*d^4*Sin[e + f*x]^5)))/(f*Sqrt[Cos[e + f*x]^2])","A",1
435,1,204,233,0.9404466,"\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3,x]","-\frac{a^2 \cos (e+f x) \left(30 \left(4 c^3+8 c^2 d+7 c d^2+2 d^3\right) \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\sqrt{\cos ^2(e+f x)} \left(8 d \left(5 c^2+10 c d+3 d^2\right) \sin ^2(e+f x)+5 \left(4 c^3+24 c^2 d+21 c d^2+6 d^3\right) \sin (e+f x)+8 \left(10 c^3+25 c^2 d+20 c d^2+6 d^3\right)+10 d^2 (3 c+2 d) \sin ^3(e+f x)+8 d^3 \sin ^4(e+f x)\right)\right)}{40 f \sqrt{\cos ^2(e+f x)}}","\frac{a^2 \left(c^2-10 c d-12 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac{3}{8} a^2 x (2 c+d) \left(2 c^2+3 c d+2 d^2\right)+\frac{a^2 \left(2 c^3-20 c^2 d-57 c d^2-30 d^3\right) \sin (e+f x) \cos (e+f x)}{40 f}+\frac{a^2 \left(c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right) \cos (e+f x)}{10 d f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac{a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}",1,"-1/40*(a^2*Cos[e + f*x]*(30*(4*c^3 + 8*c^2*d + 7*c*d^2 + 2*d^3)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(8*(10*c^3 + 25*c^2*d + 20*c*d^2 + 6*d^3) + 5*(4*c^3 + 24*c^2*d + 21*c*d^2 + 6*d^3)*Sin[e + f*x] + 8*d*(5*c^2 + 10*c*d + 3*d^2)*Sin[e + f*x]^2 + 10*d^2*(3*c + 2*d)*Sin[e + f*x]^3 + 8*d^3*Sin[e + f*x]^4)))/(f*Sqrt[Cos[e + f*x]^2])","A",1
436,1,148,156,0.54477,"\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2,x]","-\frac{a^2 \cos (e+f x) \left(6 \left(12 c^2+16 c d+7 d^2\right) \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\sqrt{\cos ^2(e+f x)} \left(3 \left(4 c^2+16 c d+7 d^2\right) \sin (e+f x)+16 \left(3 c^2+5 c d+2 d^2\right)+16 d (c+d) \sin ^2(e+f x)+6 d^2 \sin ^3(e+f x)\right)\right)}{24 f \sqrt{\cos ^2(e+f x)}}","-\frac{a^2 \left(12 c^2+16 c d+7 d^2\right) \cos (e+f x)}{6 f}-\frac{a^2 \left(12 c^2+16 c d+7 d^2\right) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} a^2 x \left(12 c^2+16 c d+7 d^2\right)-\frac{d (8 c-d) \cos (e+f x) (a \sin (e+f x)+a)^2}{12 f}-\frac{d^2 \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*c^2 + 16*c*d + 7*d^2)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(16*(3*c^2 + 5*c*d + 2*d^2) + 3*(4*c^2 + 16*c*d + 7*d^2)*Sin[e + f*x] + 16*d*(c + d)*Sin[e + f*x]^2 + 6*d^2*Sin[e + f*x]^3)))/(f*Sqrt[Cos[e + f*x]^2])","A",1
437,1,106,94,0.3313255,"\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x]),x]","-\frac{a^2 \cos (e+f x) \left(6 (3 c+2 d) \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\sqrt{\cos ^2(e+f x)} \left(3 (c+2 d) \sin (e+f x)+2 (6 c+5 d)+2 d \sin ^2(e+f x)\right)\right)}{6 f \sqrt{\cos ^2(e+f x)}}","-\frac{2 a^2 (3 c+2 d) \cos (e+f x)}{3 f}-\frac{a^2 (3 c+2 d) \sin (e+f x) \cos (e+f x)}{6 f}+\frac{1}{2} a^2 x (3 c+2 d)-\frac{d \cos (e+f x) (a \sin (e+f x)+a)^2}{3 f}",1,"-1/6*(a^2*Cos[e + f*x]*(6*(3*c + 2*d)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(2*(6*c + 5*d) + 3*(c + 2*d)*Sin[e + f*x] + 2*d*Sin[e + f*x]^2)))/(f*Sqrt[Cos[e + f*x]^2])","A",1
438,1,34,45,0.1856162,"\int (a+a \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^2,x]","-\frac{a^2 (-6 (e+f x)+\sin (2 (e+f x))+8 \cos (e+f x))}{4 f}","-\frac{2 a^2 \cos (e+f x)}{f}-\frac{a^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{3 a^2 x}{2}",1,"-1/4*(a^2*(-6*(e + f*x) + 8*Cos[e + f*x] + Sin[2*(e + f*x)]))/f","A",1
439,1,130,92,0.4116128,"\int \frac{(a+a \sin (e+f x))^2}{c+d \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x]),x]","-\frac{a^2 (\sin (e+f x)+1)^2 \left(\sqrt{c^2-d^2} ((c-2 d) (e+f x)+d \cos (e+f x))-2 (c-d)^2 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)\right)}{d^2 f \sqrt{c^2-d^2} \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 \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^2 x (c-2 d)}{d^2}-\frac{a^2 \cos (e+f x)}{d f}",1,"-((a^2*(-2*(c - d)^2*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]] + Sqrt[c^2 - d^2]*((c - 2*d)*(e + f*x) + d*Cos[e + f*x]))*(1 + Sin[e + f*x])^2)/(d^2*Sqrt[c^2 - d^2]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4))","A",1
440,1,139,112,0.4851285,"\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^2} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^2,x]","\frac{a^2 (\sin (e+f x)+1)^2 \left(-\frac{2 \left(c^2+c d-2 d^2\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 (c-d) \cos (e+f x)}{(c+d) (c+d \sin (e+f x))}+e+f x\right)}{d^2 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 (c+2 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 \left(c^2-d^2\right)^{3/2}}+\frac{a^2 (c-d) \cos (e+f x)}{d f (c+d) (c+d \sin (e+f x))}+\frac{a^2 x}{d^2}",1,"(a^2*(1 + Sin[e + f*x])^2*(e + f*x - (2*(c^2 + c*d - 2*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)*Sqrt[c^2 - d^2]) + ((c - d)*d*Cos[e + f*x])/((c + d)*(c + d*Sin[e + f*x]))))/(d^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)","A",1
441,1,140,138,0.6204636,"\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^3} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^3,x]","\frac{a^2 \cos (e+f x) \left(-\frac{(c+4 d) \sin (e+f x)+4 c+d}{(c+d) (c+d \sin (e+f x))^2}-\frac{6 \tanh ^{-1}\left(\frac{\sqrt{c-d} \sqrt{1-\sin (e+f x)}}{\sqrt{-c-d} \sqrt{\sin (e+f x)+1}}\right)}{(-c-d)^{3/2} \sqrt{c-d} \sqrt{\cos ^2(e+f x)}}\right)}{2 f (c+d)}","\frac{3 a^2 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c+d)^2 \sqrt{c^2-d^2}}-\frac{a^2 (c+4 d) \cos (e+f x)}{2 d f (c+d)^2 (c+d \sin (e+f x))}+\frac{a^2 (c-d) \cos (e+f x)}{2 d f (c+d) (c+d \sin (e+f x))^2}",1,"(a^2*Cos[e + f*x]*((-6*ArcTanh[(Sqrt[c - d]*Sqrt[1 - Sin[e + f*x]])/(Sqrt[-c - d]*Sqrt[1 + Sin[e + f*x]])])/((-c - d)^(3/2)*Sqrt[c - d]*Sqrt[Cos[e + f*x]^2]) - (4*c + d + (c + 4*d)*Sin[e + f*x])/((c + d)*(c + d*Sin[e + f*x])^2)))/(2*(c + d)*f)","A",1
442,1,196,207,2.5608885,"\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^4,x]","\frac{a^2 \cos (e+f x) \left(-\frac{d (\sin (e+f x)+1)^2}{(c+d \sin (e+f x))^3}-\frac{(3 c-2 d) \left(\frac{6 \tanh ^{-1}\left(\frac{\sqrt{c-d} \sqrt{1-\sin (e+f x)}}{\sqrt{-c-d} \sqrt{\sin (e+f x)+1}}\right)}{\sqrt{-c-d} \sqrt{c-d}}-\frac{\sqrt{\cos ^2(e+f x)} ((c+4 d) \sin (e+f x)+4 c+d)}{(c+d \sin (e+f x))^2}\right)}{2 (c+d)^2 \sqrt{\cos ^2(e+f x)}}\right)}{3 f (d-c) (c+d)}","\frac{a^2 (3 c-2 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) (c+d)^3 \sqrt{c^2-d^2}}-\frac{a^2 \left(c^2+6 c d-10 d^2\right) \cos (e+f x)}{6 d f (c-d) (c+d)^3 (c+d \sin (e+f x))}-\frac{a^2 (c+6 d) \cos (e+f x)}{6 d f (c+d)^2 (c+d \sin (e+f x))^2}+\frac{a^2 (c-d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3}",1,"(a^2*Cos[e + f*x]*(-((d*(1 + Sin[e + f*x])^2)/(c + d*Sin[e + f*x])^3) - ((3*c - 2*d)*((6*ArcTanh[(Sqrt[c - d]*Sqrt[1 - Sin[e + f*x]])/(Sqrt[-c - d]*Sqrt[1 + Sin[e + f*x]])])/(Sqrt[-c - d]*Sqrt[c - d]) - (Sqrt[Cos[e + f*x]^2]*(4*c + d + (c + 4*d)*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2))/(2*(c + d)^2*Sqrt[Cos[e + f*x]^2])))/(3*(-c + d)*(c + d)*f)","A",1
443,1,269,286,4.8882401,"\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^5} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^5,x]","-\frac{a^2 \cos (e+f x) \left(\frac{\left(12 c^2-16 c d+7 d^2\right) (c+d \sin (e+f x))^2 \left(\sqrt{-c-d} \sqrt{c-d} \sqrt{\cos ^2(e+f x)} ((c+4 d) \sin (e+f x)+4 c+d)-6 (c+d \sin (e+f x))^2 \tanh ^{-1}\left(\frac{\sqrt{c-d} \sqrt{1-\sin (e+f x)}}{\sqrt{-c-d} \sqrt{\sin (e+f x)+1}}\right)\right)}{(-c-d)^{7/2} (c-d)^{3/2} \sqrt{\cos ^2(e+f x)}}+\frac{2 d (5 c-2 d) (\sin (e+f x)+1)^2 (c+d \sin (e+f x))}{(c-d) (c+d)}+6 d (\sin (e+f x)+1)^2\right)}{24 f (d-c) (c+d) (c+d \sin (e+f x))^4}","\frac{a^2 \left(12 c^2-16 c d+7 d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{4 f (c-d)^2 (c+d)^4 \sqrt{c^2-d^2}}-\frac{a^2 \left(2 c^2+16 c d-21 d^2\right) \cos (e+f x)}{24 d f (c-d) (c+d)^3 (c+d \sin (e+f x))^2}-\frac{a^2 \left(2 c^3+16 c^2 d-59 c d^2+32 d^3\right) \cos (e+f x)}{24 d f (c-d)^2 (c+d)^4 (c+d \sin (e+f x))}-\frac{a^2 (c+8 d) \cos (e+f x)}{12 d f (c+d)^2 (c+d \sin (e+f x))^3}+\frac{a^2 (c-d) \cos (e+f x)}{4 d f (c+d) (c+d \sin (e+f x))^4}",1,"-1/24*(a^2*Cos[e + f*x]*(6*d*(1 + Sin[e + f*x])^2 + (2*(5*c - 2*d)*d*(1 + Sin[e + f*x])^2*(c + d*Sin[e + f*x]))/((c - d)*(c + d)) + ((12*c^2 - 16*c*d + 7*d^2)*(c + d*Sin[e + f*x])^2*(-6*ArcTanh[(Sqrt[c - d]*Sqrt[1 - Sin[e + f*x]])/(Sqrt[-c - d]*Sqrt[1 + Sin[e + f*x]])]*(c + d*Sin[e + f*x])^2 + Sqrt[-c - d]*Sqrt[c - d]*Sqrt[Cos[e + f*x]^2]*(4*c + d + (c + 4*d)*Sin[e + f*x])))/((-c - d)^(7/2)*(c - d)^(3/2)*Sqrt[Cos[e + f*x]^2])))/((-c + d)*(c + d)*f*(c + d*Sin[e + f*x])^4)","A",1
444,1,233,215,1.3925169,"\int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3,x]","-\frac{a^3 \cos (e+f x) \left(30 \left(40 c^3+90 c^2 d+78 c d^2+23 d^3\right) \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\sqrt{\cos ^2(e+f x)} \left(10 d \left(18 c^2+54 c d+23 d^2\right) \sin ^3(e+f x)+16 \left(5 c^3+45 c^2 d+57 c d^2+17 d^3\right) \sin ^2(e+f x)+15 \left(24 c^3+90 c^2 d+78 c d^2+23 d^3\right) \sin (e+f x)+16 \left(55 c^3+135 c^2 d+114 c d^2+34 d^3\right)+144 d^2 (c+d) \sin ^4(e+f x)+40 d^3 \sin ^5(e+f x)\right)\right)}{240 f \sqrt{\cos ^2(e+f x)}}","-\frac{a^3 d \left(18 c^2+54 c d+23 d^2\right) \sin ^3(e+f x) \cos (e+f x)}{24 f}-\frac{a^3 \left(24 c^3+90 c^2 d+78 c d^2+23 d^3\right) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} a^3 x \left(40 c^3+90 c^2 d+78 c d^2+23 d^3\right)-\frac{3 a^3 d^2 (c+d) \cos ^5(e+f x)}{5 f}+\frac{a^3 (c+d)^2 (c+7 d) \cos ^3(e+f x)}{3 f}-\frac{4 a^3 (c+d)^3 \cos (e+f x)}{f}-\frac{a^3 d^3 \sin ^5(e+f x) \cos (e+f x)}{6 f}",1,"-1/240*(a^3*Cos[e + f*x]*(30*(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]*(16*(55*c^3 + 135*c^2*d + 114*c*d^2 + 34*d^3) + 15*(24*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*Sin[e + f*x] + 16*(5*c^3 + 45*c^2*d + 57*c*d^2 + 17*d^3)*Sin[e + f*x]^2 + 10*d*(18*c^2 + 54*c*d + 23*d^2)*Sin[e + f*x]^3 + 144*d^2*(c + d)*Sin[e + f*x]^4 + 40*d^3*Sin[e + f*x]^5)))/(f*Sqrt[Cos[e + f*x]^2])","A",1
445,1,177,164,0.7404186,"\int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2,x]","-\frac{a^3 \cos (e+f x) \left(30 \left(20 c^2+30 c d+13 d^2\right) \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\sqrt{\cos ^2(e+f x)} \left(8 \left(5 c^2+30 c d+19 d^2\right) \sin ^2(e+f x)+15 \left(12 c^2+30 c d+13 d^2\right) \sin (e+f x)+8 \left(55 c^2+90 c d+38 d^2\right)+30 d (2 c+3 d) \sin ^3(e+f x)+24 d^2 \sin ^4(e+f x)\right)\right)}{120 f \sqrt{\cos ^2(e+f x)}}","\frac{a^3 \left(c^2+6 c d+5 d^2\right) \cos ^3(e+f x)}{3 f}-\frac{a^3 \left(12 c^2+30 c d+13 d^2\right) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a^3 x \left(20 c^2+30 c d+13 d^2\right)-\frac{4 a^3 (c+d)^2 \cos (e+f x)}{f}-\frac{a^3 d (2 c+3 d) \sin ^3(e+f x) \cos (e+f x)}{4 f}-\frac{a^3 d^2 \cos ^5(e+f x)}{5 f}",1,"-1/120*(a^3*Cos[e + f*x]*(30*(20*c^2 + 30*c*d + 13*d^2)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(8*(55*c^2 + 90*c*d + 38*d^2) + 15*(12*c^2 + 30*c*d + 13*d^2)*Sin[e + f*x] + 8*(5*c^2 + 30*c*d + 19*d^2)*Sin[e + f*x]^2 + 30*d*(2*c + 3*d)*Sin[e + f*x]^3 + 24*d^2*Sin[e + f*x]^4)))/(f*Sqrt[Cos[e + f*x]^2])","A",1
446,1,120,110,0.5183255,"\int (a+a \sin (e+f x))^3 (c+d \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x]),x]","-\frac{a^3 \cos (e+f x) \left(30 (4 c+3 d) \sin ^{-1}\left(\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right)+\sqrt{\cos ^2(e+f x)} \left(8 (c+3 d) \sin ^2(e+f x)+9 (4 c+5 d) \sin (e+f x)+88 c+6 d \sin ^3(e+f x)+72 d\right)\right)}{24 f \sqrt{\cos ^2(e+f x)}}","\frac{a^3 (c+3 d) \cos ^3(e+f x)}{3 f}-\frac{4 a^3 (c+d) \cos (e+f x)}{f}-\frac{3 a^3 (4 c+5 d) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{5}{8} a^3 x (4 c+3 d)-\frac{a^3 d \sin ^3(e+f x) \cos (e+f x)}{4 f}",1,"-1/24*(a^3*Cos[e + f*x]*(30*(4*c + 3*d)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(88*c + 72*d + 9*(4*c + 5*d)*Sin[e + f*x] + 8*(c + 3*d)*Sin[e + f*x]^2 + 6*d*Sin[e + f*x]^3)))/(f*Sqrt[Cos[e + f*x]^2])","A",1
447,1,44,63,0.3535985,"\int (a+a \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^3,x]","\frac{a^3 (-9 \sin (2 (e+f x))-45 \cos (e+f x)+\cos (3 (e+f x))+30 e+30 f x)}{12 f}","\frac{a^3 \cos ^3(e+f x)}{3 f}-\frac{4 a^3 \cos (e+f x)}{f}-\frac{3 a^3 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{5 a^3 x}{2}",1,"(a^3*(30*e + 30*f*x - 45*Cos[e + f*x] + Cos[3*(e + f*x)] - 9*Sin[2*(e + f*x)]))/(12*f)","A",1
448,1,162,143,0.6582942,"\int \frac{(a+a \sin (e+f x))^3}{c+d \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x]),x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(\sqrt{c^2-d^2} \left(2 \left(2 c^2-6 c d+7 d^2\right) (e+f x)+4 d (c-3 d) \cos (e+f x)+d^2 (-\sin (2 (e+f x)))\right)-8 (c-d)^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)\right)}{4 d^3 f \sqrt{c^2-d^2} \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 \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^3 x \left(2 c^2-6 c d+7 d^2\right)}{2 d^3}+\frac{a^3 (2 c-5 d) \cos (e+f x)}{2 d^2 f}-\frac{\cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{2 d f}",1,"(a^3*(1 + Sin[e + f*x])^3*(-8*(c - d)^3*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]] + Sqrt[c^2 - d^2]*(2*(2*c^2 - 6*c*d + 7*d^2)*(e + f*x) + 4*(c - 3*d)*d*Cos[e + f*x] - d^2*Sin[2*(e + f*x)])))/(4*d^3*Sqrt[c^2 - d^2]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)","A",1
449,1,162,161,0.6763438,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^2} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^2,x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(\frac{2 (2 c+3 d) (c-d)^2 \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}}+(3 d-2 c) (e+f x)-\frac{d (c-d)^2 \cos (e+f x)}{(c+d) (c+d \sin (e+f x))}-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)^6}","\frac{2 a^3 (c-d)^2 (2 c+3 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 (c+d) \sqrt{c^2-d^2}}-\frac{a^3 x (2 c-3 d)}{d^3}-\frac{2 a^3 c \cos (e+f x)}{d^2 f (c+d)}+\frac{(c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{d f (c+d) (c+d \sin (e+f x))}",1,"(a^3*(1 + Sin[e + f*x])^3*((-2*c + 3*d)*(e + f*x) + (2*(c - d)^2*(2*c + 3*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)*Sqrt[c^2 - d^2]) - d*Cos[e + f*x] - ((c - d)^2*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])^6)","A",1
450,1,196,187,1.0081208,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^3,x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(\frac{3 d \left(c^2+c d-2 d^2\right) \cos (e+f x)}{(c+d)^2 (c+d \sin (e+f x))}-\frac{2 \left(2 c^3+4 c^2 d+c d^2-7 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 (c-d)^2 \cos (e+f x)}{(c+d) (c+d \sin (e+f x))^2}+2 (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)^6}","-\frac{a^3 (c-d) \left(2 c^2+6 c d+7 d^2\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{a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 f (c+d)^2 (c+d \sin (e+f x))}+\frac{(c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{2 d f (c+d) (c+d \sin (e+f x))^2}+\frac{a^3 x}{d^3}",1,"(a^3*(1 + Sin[e + f*x])^3*(2*(e + f*x) - (2*(2*c^3 + 4*c^2*d + c*d^2 - 7*d^3)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)^2*Sqrt[c^2 - d^2]) - ((c - d)^2*d*Cos[e + f*x])/((c + d)*(c + d*Sin[e + f*x])^2) + (3*d*(c^2 + 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])^6)","A",1
451,1,178,207,2.3530919,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^4,x]","\frac{a^3 \cos (e+f x) \left(-\frac{(\sin (e+f x)+1)^2}{(c+d \sin (e+f x))^3}-\frac{5 (\sin (e+f x)+1)}{2 (c+d) (c+d \sin (e+f x))^2}-\frac{15}{2 (c+d)^2 (c+d \sin (e+f x))}+\frac{15 \tanh ^{-1}\left(\frac{\sqrt{c-d} \sqrt{1-\sin (e+f x)}}{\sqrt{-c-d} \sqrt{\sin (e+f x)+1}}\right)}{(-c-d)^{5/2} \sqrt{c-d} \sqrt{\cos ^2(e+f x)}}\right)}{3 f (c+d)}","\frac{5 a^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c+d)^3 \sqrt{c^2-d^2}}-\frac{a^3 \left(2 c^2+9 c d+22 d^2\right) \cos (e+f x)}{6 d^2 f (c+d)^3 (c+d \sin (e+f x))}+\frac{a^3 (c-d) (2 c+7 d) \cos (e+f x)}{6 d^2 f (c+d)^2 (c+d \sin (e+f x))^2}+\frac{(c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{3 d f (c+d) (c+d \sin (e+f x))^3}",1,"(a^3*Cos[e + f*x]*((15*ArcTanh[(Sqrt[c - d]*Sqrt[1 - Sin[e + f*x]])/(Sqrt[-c - d]*Sqrt[1 + Sin[e + f*x]])])/((-c - d)^(5/2)*Sqrt[c - d]*Sqrt[Cos[e + f*x]^2]) - (1 + Sin[e + f*x])^2/(c + d*Sin[e + f*x])^3 - (5*(1 + Sin[e + f*x]))/(2*(c + d)*(c + d*Sin[e + f*x])^2) - 15/(2*(c + d)^2*(c + d*Sin[e + f*x]))))/(3*(c + d)*f)","A",1
452,1,240,289,3.2063937,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^5} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^5,x]","\frac{a^3 \cos (e+f x) \left(-\frac{(4 c-3 d) \left(-\frac{\sqrt{\cos ^2(e+f x)} \left(\left(2 c^2+9 c d+22 d^2\right) \sin ^2(e+f x)+\left(9 c^2+48 c d+9 d^2\right) \sin (e+f x)+22 c^2+9 c d+2 d^2\right)}{6 (c+d)^3 (c+d \sin (e+f x))^3}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{c-d} \sqrt{1-\sin (e+f x)}}{\sqrt{-c-d} \sqrt{\sin (e+f x)+1}}\right)}{(-c-d)^{7/2} \sqrt{c-d}}\right)}{\sqrt{\cos ^2(e+f x)}}-\frac{d (\sin (e+f x)+1)^3}{(c+d \sin (e+f x))^4}\right)}{4 f (d-c) (c+d)}","\frac{5 a^3 (4 c-3 d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{4 f (c-d) (c+d)^4 \sqrt{c^2-d^2}}-\frac{a^3 \left(2 c^2+12 c d+45 d^2\right) \cos (e+f x)}{24 d^2 f (c+d)^3 (c+d \sin (e+f x))^2}-\frac{a^3 \left(2 c^3+12 c^2 d+43 c d^2-72 d^3\right) \cos (e+f x)}{24 d^2 f (c-d) (c+d)^4 (c+d \sin (e+f x))}+\frac{a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 f (c+d)^2 (c+d \sin (e+f x))^3}+\frac{(c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{4 d f (c+d) (c+d \sin (e+f x))^4}",1,"(a^3*Cos[e + f*x]*(-((d*(1 + Sin[e + f*x])^3)/(c + d*Sin[e + f*x])^4) - ((4*c - 3*d)*((-5*ArcTanh[(Sqrt[c - d]*Sqrt[1 - Sin[e + f*x]])/(Sqrt[-c - d]*Sqrt[1 + Sin[e + f*x]])])/((-c - d)^(7/2)*Sqrt[c - d]) - (Sqrt[Cos[e + f*x]^2]*(22*c^2 + 9*c*d + 2*d^2 + (9*c^2 + 48*c*d + 9*d^2)*Sin[e + f*x] + (2*c^2 + 9*c*d + 22*d^2)*Sin[e + f*x]^2))/(6*(c + d)^3*(c + d*Sin[e + f*x])^3)))/Sqrt[Cos[e + f*x]^2]))/(4*(-c + d)*(c + d)*f)","A",1
453,1,234,189,0.41212,"\int \frac{(c+d \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx","Integrate[(c + d*Sin[e + f*x])^4/(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(-3 d^2 \left(24 c^2-16 c d+7 d^2\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)-6 d \left(-8 c^3+12 c^2 d-12 c d^2+3 d^3\right) (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-3 d^3 (4 c-d) \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)+24 (c-d)^4 \sin \left(\frac{1}{2} (e+f x)\right)+d^4 \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)\right)}{12 a f (\sin (e+f x)+1)}","\frac{d^2 \left(6 c^2-20 c d+9 d^2\right) \sin (e+f x) \cos (e+f x)}{6 a f}+\frac{2 d \left(3 c^3-16 c^2 d+12 c d^2-4 d^3\right) \cos (e+f x)}{3 a f}+\frac{d x \left(8 c^3-12 c^2 d+12 c d^2-3 d^3\right)}{2 a}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}+\frac{d (3 c-4 d) \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])*(24*(c - d)^4*Sin[(e + f*x)/2] - 6*d*(-8*c^3 + 12*c^2*d - 12*c*d^2 + 3*d^3)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 3*d^2*(24*c^2 - 16*c*d + 7*d^2)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + d^4*Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 3*(4*c - d)*d^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[2*(e + f*x)]))/(12*a*f*(1 + Sin[e + f*x]))","A",1
454,1,192,121,0.5857468,"\int \frac{(c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx","Integrate[(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(d \cos \left(\frac{1}{2} (e+f x)\right) \left(6 \left(2 c^2-2 c d+d^2\right) (e+f x)-4 d (3 c-d) \cos (e+f x)+d^2 (-\sin (2 (e+f x)))\right)+\sin \left(\frac{1}{2} (e+f x)\right) \left(2 \left(4 c^3+6 c^2 d (e+f x-2)-6 c d^2 (e+f x-2)+d^3 (3 e+3 f x-4)\right)-4 d^2 (3 c-d) \cos (e+f x)+d^3 (-\sin (2 (e+f x)))\right)\right)}{4 a f (\sin (e+f x)+1)}","\frac{2 d \left(c^2-3 c d+d^2\right) \cos (e+f x)}{a f}+\frac{3 d x \left(2 c^2-2 c d+d^2\right)}{2 a}+\frac{d^2 (2 c-3 d) \sin (e+f x) \cos (e+f x)}{2 a f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{f (a \sin (e+f x)+a)}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(d*Cos[(e + f*x)/2]*(6*(2*c^2 - 2*c*d + d^2)*(e + f*x) - 4*(3*c - d)*d*Cos[e + f*x] - d^2*Sin[2*(e + f*x)]) + Sin[(e + f*x)/2]*(2*(4*c^3 + 6*c^2*d*(-2 + e + f*x) - 6*c*d^2*(-2 + e + f*x) + d^3*(-4 + 3*e + 3*f*x)) - 4*(3*c - d)*d^2*Cos[e + f*x] - d^3*Sin[2*(e + f*x)])))/(4*a*f*(1 + Sin[e + f*x]))","A",1
455,1,122,62,0.4641375,"\int \frac{(c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx","Integrate[(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(\sin \left(\frac{1}{2} (e+f x)\right) \left(-2 c^2-2 c d (e+f x-2)+d^2 (e+f x-2)+d^2 \cos (e+f x)\right)+d \cos \left(\frac{1}{2} (e+f x)\right) (d \cos (e+f x)-(2 c-d) (e+f x))\right)}{a f (\sin (e+f x)+1)}","-\frac{(c-d)^2 \cos (e+f x)}{a f (\sin (e+f x)+1)}+\frac{d x (2 c-d)}{a}-\frac{d^2 \cos (e+f x)}{a f}",1,"-(((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(d*Cos[(e + f*x)/2]*(-((2*c - d)*(e + f*x)) + d*Cos[e + f*x]) + (-2*c^2 - 2*c*d*(-2 + e + f*x) + d^2*(-2 + e + f*x) + d^2*Cos[e + f*x])*Sin[(e + f*x)/2]))/(a*f*(1 + Sin[e + f*x])))","A",1
456,1,79,35,0.1654737,"\int \frac{c+d \sin (e+f x)}{a+a \sin (e+f x)} \, dx","Integrate[(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(\sin \left(\frac{1}{2} (e+f x)\right) (2 c+d (e+f x-2))+d (e+f x) \cos \left(\frac{1}{2} (e+f x)\right)\right)}{a f (\sin (e+f x)+1)}","\frac{d x}{a}-\frac{(c-d) \cos (e+f x)}{f (a \sin (e+f x)+a)}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(d*(e + f*x)*Cos[(e + f*x)/2] + (2*c + d*(-2 + e + f*x))*Sin[(e + f*x)/2]))/(a*f*(1 + Sin[e + f*x]))","B",1
457,1,48,23,0.0421319,"\int \frac{1}{a+a \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^(-1),x]","\frac{2 \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)}{f (a \sin (e+f x)+a)}","-\frac{\cos (e+f x)}{f (a \sin (e+f x)+a)}",1,"(2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(f*(a + a*Sin[e + f*x]))","B",1
458,1,114,89,0.3572107,"\int \frac{1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))} \, dx","Integrate[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])),x]","\frac{\cos (e+f x) \left(\frac{2 d \tanh ^{-1}\left(\frac{\sqrt{c-d} \sqrt{1-\sin (e+f x)}}{\sqrt{-c-d} \sqrt{\sin (e+f x)+1}}\right)}{\sqrt{-c-d} \sqrt{c-d} \sqrt{\cos ^2(e+f x)}}+\frac{1}{\sin (e+f x)+1}\right)}{a f (d-c)}","-\frac{2 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{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a)}",1,"(Cos[e + f*x]*((2*d*ArcTanh[(Sqrt[c - d]*Sqrt[1 - Sin[e + f*x]])/(Sqrt[-c - d]*Sqrt[1 + Sin[e + f*x]])])/(Sqrt[-c - d]*Sqrt[c - d]*Sqrt[Cos[e + f*x]^2]) + (1 + Sin[e + f*x])^(-1)))/(a*(-c + d)*f)","A",1
459,1,162,150,0.646693,"\int \frac{1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^2} \, dx","Integrate[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^2),x]","\frac{\cos (e+f x) \left(-\frac{d}{(\sin (e+f x)+1) (c+d \sin (e+f x))}+\frac{c+2 d}{(c-d) (\sin (e+f x)+1)}+\frac{2 d (2 c+d) \tanh ^{-1}\left(\frac{\sqrt{c-d} \sqrt{1-\sin (e+f x)}}{\sqrt{-c-d} \sqrt{\sin (e+f x)+1}}\right)}{\sqrt{-c-d} (c-d)^{3/2} \sqrt{\cos ^2(e+f x)}}\right)}{a f (d-c) (c+d)}","-\frac{2 d (2 c+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) \left(c^2-d^2\right)^{3/2}}-\frac{d (c+2 d) \cos (e+f x)}{a f (c-d)^2 (c+d) (c+d \sin (e+f x))}-\frac{\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*d*(2*c + d)*ArcTanh[(Sqrt[c - d]*Sqrt[1 - Sin[e + f*x]])/(Sqrt[-c - d]*Sqrt[1 + Sin[e + f*x]])])/(Sqrt[-c - d]*(c - d)^(3/2)*Sqrt[Cos[e + f*x]^2]) + (c + 2*d)/((c - d)*(1 + Sin[e + f*x])) - d/((1 + Sin[e + f*x])*(c + d*Sin[e + f*x]))))/(a*(-c + d)*(c + d)*f)","A",1
460,1,230,213,1.983082,"\int \frac{1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx","Integrate[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^3),x]","\frac{\cos (e+f x) \left(\frac{2 c^2+9 c d+4 d^2}{(c-d)^2 (c+d) (\sin (e+f x)+1)}-\frac{6 d \left(2 c^2+2 c d+d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \sqrt{1-\sin (e+f x)}}{\sqrt{-c-d} \sqrt{\sin (e+f x)+1}}\right)}{(-c-d)^{3/2} (c-d)^{5/2} \sqrt{\cos ^2(e+f x)}}-\frac{d (4 c+d)}{(c-d) (c+d) (\sin (e+f x)+1) (c+d \sin (e+f x))}-\frac{d}{(\sin (e+f x)+1) (c+d \sin (e+f x))^2}\right)}{2 a f (d-c) (c+d)}","-\frac{3 d \left(2 c^2+2 c d+d^2\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 c+d) (c+4 d) \cos (e+f x)}{2 a f (c-d)^3 (c+d)^2 (c+d \sin (e+f x))}-\frac{d (2 c+3 d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^2}-\frac{\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]*((-6*d*(2*c^2 + 2*c*d + d^2)*ArcTanh[(Sqrt[c - d]*Sqrt[1 - Sin[e + f*x]])/(Sqrt[-c - d]*Sqrt[1 + Sin[e + f*x]])])/((-c - d)^(3/2)*(c - d)^(5/2)*Sqrt[Cos[e + f*x]^2]) + (2*c^2 + 9*c*d + 4*d^2)/((c - d)^2*(c + d)*(1 + Sin[e + f*x])) - d/((1 + Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - (d*(4*c + d))/((c - d)*(c + d)*(1 + Sin[e + f*x])*(c + d*Sin[e + f*x]))))/(2*a*(-c + d)*(c + d)*f)","A",1
461,1,837,260,1.6971038,"\int \frac{(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c + d*Sin[e + f*x])^5/(a + a*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(48 \sin \left(\frac{1}{2} (e+f x)\right) c^5+240 d \sin \left(\frac{1}{2} (e+f x)\right) c^4-1440 d^2 \sin \left(\frac{1}{2} (e+f x)\right) c^3+720 d^2 e \sin \left(\frac{1}{2} (e+f x)\right) c^3+720 d^2 f x \sin \left(\frac{1}{2} (e+f x)\right) c^3+240 d^2 e \sin \left(\frac{3}{2} (e+f x)\right) c^3+240 d^2 f x \sin \left(\frac{3}{2} (e+f x)\right) c^3+120 d^3 \cos \left(\frac{5}{2} (e+f x)\right) c^2+2640 d^3 \sin \left(\frac{1}{2} (e+f x)\right) c^2-1440 d^3 e \sin \left(\frac{1}{2} (e+f x)\right) c^2-1440 d^3 f x \sin \left(\frac{1}{2} (e+f x)\right) c^2-360 d^3 \sin \left(\frac{3}{2} (e+f x)\right) c^2-480 d^3 e \sin \left(\frac{3}{2} (e+f x)\right) c^2-480 d^3 f x \sin \left(\frac{3}{2} (e+f x)\right) c^2-120 d^3 \sin \left(\frac{5}{2} (e+f x)\right) c^2-75 d^4 \cos \left(\frac{5}{2} (e+f x)\right) c+15 d^4 \cos \left(\frac{7}{2} (e+f x)\right) c-1905 d^4 \sin \left(\frac{1}{2} (e+f x)\right) c+1260 d^4 e \sin \left(\frac{1}{2} (e+f x)\right) c+1260 d^4 f x \sin \left(\frac{1}{2} (e+f x)\right) c+315 d^4 \sin \left(\frac{3}{2} (e+f x)\right) c+420 d^4 e \sin \left(\frac{3}{2} (e+f x)\right) c+420 d^4 f x \sin \left(\frac{3}{2} (e+f x)\right) c+75 d^4 \sin \left(\frac{5}{2} (e+f x)\right) c+15 d^4 \sin \left(\frac{7}{2} (e+f x)\right) c+3 d \left(80 c^4+80 d (3 e+3 f x-4) c^3-80 d^2 (6 e+6 f x-5) c^2+35 d^3 (12 e+12 f x-7) c-4 d^4 (30 e+30 f x-13)\right) \cos \left(\frac{1}{2} (e+f x)\right)-\left(16 c^5+160 d c^4+80 d^2 (3 e+3 f x-10) c^3-40 d^3 (12 e+12 f x-41) c^2+5 d^4 (84 e+84 f x-239) c-6 d^5 (20 e+20 f x-57)\right) \cos \left(\frac{3}{2} (e+f x)\right)+30 d^5 \cos \left(\frac{5}{2} (e+f x)\right)-3 d^5 \cos \left(\frac{7}{2} (e+f x)\right)-d^5 \cos \left(\frac{9}{2} (e+f x)\right)+516 d^5 \sin \left(\frac{1}{2} (e+f x)\right)-360 d^5 e \sin \left(\frac{1}{2} (e+f x)\right)-360 d^5 f x \sin \left(\frac{1}{2} (e+f x)\right)-118 d^5 \sin \left(\frac{3}{2} (e+f x)\right)-120 d^5 e \sin \left(\frac{3}{2} (e+f x)\right)-120 d^5 f x \sin \left(\frac{3}{2} (e+f x)\right)-30 d^5 \sin \left(\frac{5}{2} (e+f x)\right)-3 d^5 \sin \left(\frac{7}{2} (e+f x)\right)+d^5 \sin \left(\frac{9}{2} (e+f x)\right)\right)}{48 a^2 f (\sin (e+f x)+1)^2}","\frac{d \left(c^2+10 c d-12 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f}+\frac{5 d^2 x (2 c-d) \left(2 c^2-3 c d+2 d^2\right)}{2 a^2}+\frac{d^2 \left(2 c^3+20 c^2 d-57 c d^2+30 d^3\right) \sin (e+f x) \cos (e+f x)}{6 a^2 f}+\frac{2 d \left(c^4+10 c^3 d-44 c^2 d^2+40 c d^3-12 d^4\right) \cos (e+f x)}{3 a^2 f}-\frac{(c-d) (c+10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(3*d*(80*c^4 + 80*c^3*d*(-4 + 3*e + 3*f*x) - 80*c^2*d^2*(-5 + 6*e + 6*f*x) + 35*c*d^3*(-7 + 12*e + 12*f*x) - 4*d^4*(-13 + 30*e + 30*f*x))*Cos[(e + f*x)/2] - (16*c^5 + 160*c^4*d + 80*c^3*d^2*(-10 + 3*e + 3*f*x) - 40*c^2*d^3*(-41 + 12*e + 12*f*x) - 6*d^5*(-57 + 20*e + 20*f*x) + 5*c*d^4*(-239 + 84*e + 84*f*x))*Cos[(3*(e + f*x))/2] + 120*c^2*d^3*Cos[(5*(e + f*x))/2] - 75*c*d^4*Cos[(5*(e + f*x))/2] + 30*d^5*Cos[(5*(e + f*x))/2] + 15*c*d^4*Cos[(7*(e + f*x))/2] - 3*d^5*Cos[(7*(e + f*x))/2] - d^5*Cos[(9*(e + f*x))/2] + 48*c^5*Sin[(e + f*x)/2] + 240*c^4*d*Sin[(e + f*x)/2] - 1440*c^3*d^2*Sin[(e + f*x)/2] + 2640*c^2*d^3*Sin[(e + f*x)/2] - 1905*c*d^4*Sin[(e + f*x)/2] + 516*d^5*Sin[(e + f*x)/2] + 720*c^3*d^2*e*Sin[(e + f*x)/2] - 1440*c^2*d^3*e*Sin[(e + f*x)/2] + 1260*c*d^4*e*Sin[(e + f*x)/2] - 360*d^5*e*Sin[(e + f*x)/2] + 720*c^3*d^2*f*x*Sin[(e + f*x)/2] - 1440*c^2*d^3*f*x*Sin[(e + f*x)/2] + 1260*c*d^4*f*x*Sin[(e + f*x)/2] - 360*d^5*f*x*Sin[(e + f*x)/2] - 360*c^2*d^3*Sin[(3*(e + f*x))/2] + 315*c*d^4*Sin[(3*(e + f*x))/2] - 118*d^5*Sin[(3*(e + f*x))/2] + 240*c^3*d^2*e*Sin[(3*(e + f*x))/2] - 480*c^2*d^3*e*Sin[(3*(e + f*x))/2] + 420*c*d^4*e*Sin[(3*(e + f*x))/2] - 120*d^5*e*Sin[(3*(e + f*x))/2] + 240*c^3*d^2*f*x*Sin[(3*(e + f*x))/2] - 480*c^2*d^3*f*x*Sin[(3*(e + f*x))/2] + 420*c*d^4*f*x*Sin[(3*(e + f*x))/2] - 120*d^5*f*x*Sin[(3*(e + f*x))/2] - 120*c^2*d^3*Sin[(5*(e + f*x))/2] + 75*c*d^4*Sin[(5*(e + f*x))/2] - 30*d^5*Sin[(5*(e + f*x))/2] + 15*c*d^4*Sin[(7*(e + f*x))/2] - 3*d^5*Sin[(7*(e + f*x))/2] + d^5*Sin[(9*(e + f*x))/2]))/(48*a^2*f*(1 + Sin[e + f*x])^2)","B",1
462,1,378,195,1.9474757,"\int \frac{(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c + d*Sin[e + f*x])^4/(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 d \cos \left(\frac{1}{2} (e+f x)\right) \left(64 c^3+48 c^2 d (3 e+3 f x-4)-32 c d^2 (6 e+6 f x-5)+7 d^3 (12 e+12 f x-7)\right)-\cos \left(\frac{3}{2} (e+f x)\right) \left(16 c^4+128 c^3 d+48 c^2 d^2 (3 e+3 f x-10)-16 c d^3 (12 e+12 f x-41)+d^4 (84 e+84 f x-239)\right)+3 \left(2 \sin \left(\frac{1}{2} (e+f x)\right) \left(8 c^4+32 c^3 d+d^2 \cos (e+f x) \left(48 c^2 (e+f x)-64 c d (e+f x+1)+d^2 (28 e+28 f x+27)\right)+96 c^2 d^2 e+96 c^2 d^2 f x-144 c^2 d^2-2 d^3 (8 c-3 d) \cos (2 (e+f x))-128 c d^3 e-128 c d^3 f x+144 c d^3+d^4 \cos (3 (e+f x))+56 d^4 e+56 d^4 f x-50 d^4\right)+d^3 (16 c-5 d) \cos \left(\frac{5}{2} (e+f x)\right)+d^4 \cos \left(\frac{7}{2} (e+f x)\right)\right)\right)}{48 a^2 f (\sin (e+f x)+1)^2}","\frac{d^2 \left(2 c^2+16 c d-21 d^2\right) \sin (e+f x) \cos (e+f x)}{6 a^2 f}+\frac{d^2 x \left(12 c^2-16 c d+7 d^2\right)}{2 a^2}+\frac{2 d \left(c^3+8 c^2 d-20 c d^2+8 d^3\right) \cos (e+f x)}{3 a^2 f}-\frac{(c-d) (c+8 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \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*d*(64*c^3 + 48*c^2*d*(-4 + 3*e + 3*f*x) - 32*c*d^2*(-5 + 6*e + 6*f*x) + 7*d^3*(-7 + 12*e + 12*f*x))*Cos[(e + f*x)/2] - (16*c^4 + 128*c^3*d + 48*c^2*d^2*(-10 + 3*e + 3*f*x) - 16*c*d^3*(-41 + 12*e + 12*f*x) + d^4*(-239 + 84*e + 84*f*x))*Cos[(3*(e + f*x))/2] + 3*((16*c - 5*d)*d^3*Cos[(5*(e + f*x))/2] + d^4*Cos[(7*(e + f*x))/2] + 2*(8*c^4 + 32*c^3*d - 144*c^2*d^2 + 144*c*d^3 - 50*d^4 + 96*c^2*d^2*e - 128*c*d^3*e + 56*d^4*e + 96*c^2*d^2*f*x - 128*c*d^3*f*x + 56*d^4*f*x + d^2*(48*c^2*(e + f*x) - 64*c*d*(1 + e + f*x) + d^2*(27 + 28*e + 28*f*x))*Cos[e + f*x] - 2*(8*c - 3*d)*d^3*Cos[2*(e + f*x)] + d^4*Cos[3*(e + f*x)])*Sin[(e + f*x)/2])))/(48*a^2*f*(1 + Sin[e + f*x])^2)","A",1
463,1,212,120,0.3544226,"\int \frac{(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx","Integrate[(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 d^2 (3 c-2 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+2 (c-d)^3 \sin \left(\frac{1}{2} (e+f x)\right)-(c-d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+2 (c+8 d) (c-d)^2 \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 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)^3\right)}{3 a^2 f (\sin (e+f x)+1)^2}","\frac{d^2 (c-4 d) \cos (e+f x)}{3 a^2 f}+\frac{d^2 x (3 c-2 d)}{a^2}-\frac{(c+6 d) (c-d)^2 \cos (e+f x)}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \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])*(2*(c - d)^3*Sin[(e + f*x)/2] - (c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(c - d)^2*(c + 8*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 3*(3*c - 2*d)*d^2*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 3*d^3*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3))/(3*a^2*f*(1 + Sin[e + f*x])^2)","A",1
464,1,172,85,0.2806586,"\int \frac{(c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx","Integrate[(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(2 \left(c^2+4 c d-5 d^2\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)^2+2 (c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right)-(c-d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+3 d^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)}{3 a^2 f (\sin (e+f x)+1)^2}","-\frac{(c-d) (c+4 d) \cos (e+f x)}{3 a^2 f (\sin (e+f x)+1)}+\frac{d^2 x}{a^2}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))}{3 f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*(c - d)^2*Sin[(e + f*x)/2] - (c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(c^2 + 4*c*d - 5*d^2)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 3*d^2*(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
465,1,43,65,0.0602774,"\int \frac{c+d \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c + d*Sin[e + f*x])/(a + a*Sin[e + f*x])^2,x]","-\frac{\cos (e+f x) ((c+2 d) \sin (e+f x)+2 c+d)}{3 a^2 f (\sin (e+f x)+1)^2}","-\frac{(c+2 d) \cos (e+f x)}{3 f \left(a^2 \sin (e+f x)+a^2\right)}-\frac{(c-d) \cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}",1,"-1/3*(Cos[e + f*x]*(2*c + d + (c + 2*d)*Sin[e + f*x]))/(a^2*f*(1 + Sin[e + f*x])^2)","A",1
466,1,54,55,0.1001285,"\int \frac{1}{(a+a \sin (e+f x))^2} \, dx","Integrate[(a + a*Sin[e + f*x])^(-2),x]","-\frac{-4 \sin (e+f x)+\sin (2 (e+f x))+4 \cos (e+f x)+\cos (2 (e+f x))-3}{6 a^2 f (\sin (e+f x)+1)^2}","-\frac{\cos (e+f x)}{3 f \left(a^2 \sin (e+f x)+a^2\right)}-\frac{\cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}",1,"-1/6*(-3 + 4*Cos[e + f*x] + Cos[2*(e + f*x)] - 4*Sin[e + f*x] + Sin[2*(e + f*x)])/(a^2*f*(1 + Sin[e + f*x])^2)","A",1
467,1,204,131,0.3505357,"\int \frac{1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx","Integrate[1/((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^2 \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 (c-d) \sin \left(\frac{1}{2} (e+f x)\right)+2 (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-(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^2 \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{(c-4 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1)}-\frac{\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*(c - d)*Sin[(e + f*x)/2] - (c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(c - 4*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (6*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)/Sqrt[c^2 - d^2]))/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])^2)","A",1
468,1,267,221,1.3720513,"\int \frac{1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2} \, dx","Integrate[1/((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^2 (3 c+2 d) \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^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)^3}{(c+d) (c+d \sin (e+f x))}+2 (c-d) \sin \left(\frac{1}{2} (e+f x)\right)+2 (c-7 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-(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^2 (3 c+2 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)^3 (c+d) \sqrt{c^2-d^2}}-\frac{d \left(c^2-6 c d-10 d^2\right) \cos (e+f x)}{3 a^2 f (c-d)^3 (c+d) (c+d \sin (e+f x))}-\frac{(c-6 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) (c+d \sin (e+f x))}-\frac{\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*(c - d)*Sin[(e + f*x)/2] - (c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(c - 7*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (6*d^2*(3*c + 2*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)/((c + d)*Sqrt[c^2 - d^2]) + (3*d^3*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
469,1,338,294,1.2028753,"\int \frac{1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx","Integrate[1/((a + a*Sin[e + f*x])^2*(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{6 d^2 \left(12 c^2+16 c d+7 d^2\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)^2 \sqrt{c^2-d^2}}+\frac{3 d^3 (7 c+4 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)^3}{(c+d)^2 (c+d \sin (e+f x))}+\frac{3 d^3 (c-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)^3}{(c+d) (c+d \sin (e+f x))^2}+4 (c-d) \sin \left(\frac{1}{2} (e+f x)\right)+4 (c-10 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)^2-2 (c-d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{6 a^2 f (c-d)^4 (\sin (e+f x)+1)^2}","\frac{d^2 \left(12 c^2+16 c d+7 d^2\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(2 c^2-16 c d-21 d^2\right) \cos (e+f x)}{6 a^2 f (c-d)^3 (c+d) (c+d \sin (e+f x))^2}-\frac{d \left(2 c^3-16 c^2 d-59 c d^2-32 d^3\right) \cos (e+f x)}{6 a^2 f (c-d)^4 (c+d)^2 (c+d \sin (e+f x))}-\frac{(c-8 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{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(4*(c - d)*Sin[(e + f*x)/2] - 2*(c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 4*(c - 10*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (6*d^2*(12*c^2 + 16*c*d + 7*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)^2*Sqrt[c^2 - d^2]) + (3*(c - d)*d^3*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/((c + d)*(c + d*Sin[e + f*x])^2) + (3*d^3*(7*c + 4*d)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/((c + d)^2*(c + d*Sin[e + f*x]))))/(6*a^2*(c - d)^4*f*(1 + Sin[e + f*x])^2)","A",1
470,1,560,354,2.8627622,"\int \frac{(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^6/(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(16 \left(2 c^2+26 c d+197 d^2\right) (c-d)^4 \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)^4-45 d^4 \left(20 c^2-24 c d+9 d^2\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5 (\cos (e+f x)-i \sin (e+f x))-45 d^4 \left(20 c^2-24 c d+9 d^2\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5 (\cos (e+f x)+i \sin (e+f x))-60 d^3 \left(-40 c^3+90 c^2 d-78 c d^2+23 d^3\right) (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5-45 i d^5 (2 c-d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5 (\cos (2 (e+f x))-i \sin (2 (e+f x)))+45 i d^5 (2 c-d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5 (\cos (2 (e+f x))+i \sin (2 (e+f x)))+48 (c-d)^6 \sin \left(\frac{1}{2} (e+f x)\right)-24 (c-d)^6 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-16 (c+14 d) (c-d)^5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+32 (c+14 d) (c-d)^5 \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+10 d^6 \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)}{120 a^3 f (\sin (e+f x)+1)^3}","-\frac{(c-d) \left(2 c^2+18 c d+115 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^3}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{d \left(2 c^3+18 c^2 d+111 c d^2-136 d^3\right) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a^3 f}+\frac{d^3 x \left(40 c^3-90 c^2 d+78 c d^2-23 d^3\right)}{2 a^3}+\frac{d^2 \left(4 c^4+36 c^3 d+216 c^2 d^2-626 c d^3+345 d^4\right) \sin (e+f x) \cos (e+f x)}{30 a^3 f}+\frac{2 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) \cos (e+f x)}{15 a^3 f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(48*(c - d)^6*Sin[(e + f*x)/2] - 24*(c - d)^6*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 32*(c - d)^5*(c + 14*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 16*(c - d)^5*(c + 14*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 16*(c - d)^4*(2*c^2 + 26*c*d + 197*d^2)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 60*d^3*(-40*c^3 + 90*c^2*d - 78*c*d^2 + 23*d^3)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + 10*d^6*Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 45*d^4*(20*c^2 - 24*c*d + 9*d^2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(Cos[e + f*x] - I*Sin[e + f*x]) - 45*d^4*(20*c^2 - 24*c*d + 9*d^2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(Cos[e + f*x] + I*Sin[e + f*x]) - (45*I)*(2*c - d)*d^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)]) + (45*I)*(2*c - d)*d^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])))/(120*a^3*f*(1 + Sin[e + f*x])^3)","C",1
471,1,992,278,7.8993131,"\int \frac{(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^5/(a + a*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(-160 \cos \left(\frac{3}{2} (e+f x)\right) c^5+320 \sin \left(\frac{1}{2} (e+f x)\right) c^5-32 \sin \left(\frac{5}{2} (e+f x)\right) c^5+1200 d \cos \left(\frac{1}{2} (e+f x)\right) c^4-1200 d \cos \left(\frac{3}{2} (e+f x)\right) c^4+1200 d \sin \left(\frac{1}{2} (e+f x)\right) c^4-240 d \sin \left(\frac{5}{2} (e+f x)\right) c^4+4800 d^2 \cos \left(\frac{1}{2} (e+f x)\right) c^3-3200 d^2 \cos \left(\frac{3}{2} (e+f x)\right) c^3+6400 d^2 \sin \left(\frac{1}{2} (e+f x)\right) c^3+2400 d^2 \sin \left(\frac{3}{2} (e+f x)\right) c^3-1120 d^2 \sin \left(\frac{5}{2} (e+f x)\right) c^3-21600 d^3 \cos \left(\frac{1}{2} (e+f x)\right) c^2+12000 d^3 (e+f x) \cos \left(\frac{1}{2} (e+f x)\right) c^2+18400 d^3 \cos \left(\frac{3}{2} (e+f x)\right) c^2-6000 d^3 (e+f x) \cos \left(\frac{3}{2} (e+f x)\right) c^2-1200 d^3 (e+f x) \cos \left(\frac{5}{2} (e+f x)\right) c^2-29600 d^3 \sin \left(\frac{1}{2} (e+f x)\right) c^2+12000 d^3 (e+f x) \sin \left(\frac{1}{2} (e+f x)\right) c^2-7200 d^3 \sin \left(\frac{3}{2} (e+f x)\right) c^2+6000 d^3 (e+f x) \sin \left(\frac{3}{2} (e+f x)\right) c^2+5120 d^3 \sin \left(\frac{5}{2} (e+f x)\right) c^2-1200 d^3 (e+f x) \sin \left(\frac{5}{2} (e+f x)\right) c^2+22500 d^4 \cos \left(\frac{1}{2} (e+f x)\right) c-18000 d^4 (e+f x) \cos \left(\frac{1}{2} (e+f x)\right) c-24300 d^4 \cos \left(\frac{3}{2} (e+f x)\right) c+9000 d^4 (e+f x) \cos \left(\frac{3}{2} (e+f x)\right) c+1500 d^4 \cos \left(\frac{5}{2} (e+f x)\right) c+1800 d^4 (e+f x) \cos \left(\frac{5}{2} (e+f x)\right) c+300 d^4 \cos \left(\frac{7}{2} (e+f x)\right) c+35100 d^4 \sin \left(\frac{1}{2} (e+f x)\right) c-18000 d^4 (e+f x) \sin \left(\frac{1}{2} (e+f x)\right) c+4500 d^4 \sin \left(\frac{3}{2} (e+f x)\right) c-9000 d^4 (e+f x) \sin \left(\frac{3}{2} (e+f x)\right) c-7260 d^4 \sin \left(\frac{5}{2} (e+f x)\right) c+1800 d^4 (e+f x) \sin \left(\frac{5}{2} (e+f x)\right) c+300 d^4 \sin \left(\frac{7}{2} (e+f x)\right) c-7560 d^5 \cos \left(\frac{1}{2} (e+f x)\right)+7800 d^5 (e+f x) \cos \left(\frac{1}{2} (e+f x)\right)+9230 d^5 \cos \left(\frac{3}{2} (e+f x)\right)-3900 d^5 (e+f x) \cos \left(\frac{3}{2} (e+f x)\right)-750 d^5 \cos \left(\frac{5}{2} (e+f x)\right)-780 d^5 (e+f x) \cos \left(\frac{5}{2} (e+f x)\right)-105 d^5 \cos \left(\frac{7}{2} (e+f x)\right)-15 d^5 \cos \left(\frac{9}{2} (e+f x)\right)-12760 d^5 \sin \left(\frac{1}{2} (e+f x)\right)+7800 d^5 (e+f x) \sin \left(\frac{1}{2} (e+f x)\right)-930 d^5 \sin \left(\frac{3}{2} (e+f x)\right)+3900 d^5 (e+f x) \sin \left(\frac{3}{2} (e+f x)\right)+2782 d^5 \sin \left(\frac{5}{2} (e+f x)\right)-780 d^5 (e+f x) \sin \left(\frac{5}{2} (e+f x)\right)-105 d^5 \sin \left(\frac{7}{2} (e+f x)\right)+15 d^5 \sin \left(\frac{9}{2} (e+f x)\right)\right)}{480 f (\sin (e+f x) a+a)^3}","-\frac{(c-d) \left(2 c^2+15 c d+76 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^2}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{d^3 x \left(20 c^2-30 c d+13 d^2\right)}{2 a^3}+\frac{d^2 \left(4 c^3+30 c^2 d+146 c d^2-195 d^3\right) \sin (e+f x) \cos (e+f x)}{30 a^3 f}+\frac{2 d \left(2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right) \cos (e+f x)}{15 a^3 f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+11 d) \cos (e+f x) (c+d \sin (e+f x))^3}{15 a f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(1200*c^4*d*Cos[(e + f*x)/2] + 4800*c^3*d^2*Cos[(e + f*x)/2] - 21600*c^2*d^3*Cos[(e + f*x)/2] + 22500*c*d^4*Cos[(e + f*x)/2] - 7560*d^5*Cos[(e + f*x)/2] + 12000*c^2*d^3*(e + f*x)*Cos[(e + f*x)/2] - 18000*c*d^4*(e + f*x)*Cos[(e + f*x)/2] + 7800*d^5*(e + f*x)*Cos[(e + f*x)/2] - 160*c^5*Cos[(3*(e + f*x))/2] - 1200*c^4*d*Cos[(3*(e + f*x))/2] - 3200*c^3*d^2*Cos[(3*(e + f*x))/2] + 18400*c^2*d^3*Cos[(3*(e + f*x))/2] - 24300*c*d^4*Cos[(3*(e + f*x))/2] + 9230*d^5*Cos[(3*(e + f*x))/2] - 6000*c^2*d^3*(e + f*x)*Cos[(3*(e + f*x))/2] + 9000*c*d^4*(e + f*x)*Cos[(3*(e + f*x))/2] - 3900*d^5*(e + f*x)*Cos[(3*(e + f*x))/2] + 1500*c*d^4*Cos[(5*(e + f*x))/2] - 750*d^5*Cos[(5*(e + f*x))/2] - 1200*c^2*d^3*(e + f*x)*Cos[(5*(e + f*x))/2] + 1800*c*d^4*(e + f*x)*Cos[(5*(e + f*x))/2] - 780*d^5*(e + f*x)*Cos[(5*(e + f*x))/2] + 300*c*d^4*Cos[(7*(e + f*x))/2] - 105*d^5*Cos[(7*(e + f*x))/2] - 15*d^5*Cos[(9*(e + f*x))/2] + 320*c^5*Sin[(e + f*x)/2] + 1200*c^4*d*Sin[(e + f*x)/2] + 6400*c^3*d^2*Sin[(e + f*x)/2] - 29600*c^2*d^3*Sin[(e + f*x)/2] + 35100*c*d^4*Sin[(e + f*x)/2] - 12760*d^5*Sin[(e + f*x)/2] + 12000*c^2*d^3*(e + f*x)*Sin[(e + f*x)/2] - 18000*c*d^4*(e + f*x)*Sin[(e + f*x)/2] + 7800*d^5*(e + f*x)*Sin[(e + f*x)/2] + 2400*c^3*d^2*Sin[(3*(e + f*x))/2] - 7200*c^2*d^3*Sin[(3*(e + f*x))/2] + 4500*c*d^4*Sin[(3*(e + f*x))/2] - 930*d^5*Sin[(3*(e + f*x))/2] + 6000*c^2*d^3*(e + f*x)*Sin[(3*(e + f*x))/2] - 9000*c*d^4*(e + f*x)*Sin[(3*(e + f*x))/2] + 3900*d^5*(e + f*x)*Sin[(3*(e + f*x))/2] - 32*c^5*Sin[(5*(e + f*x))/2] - 240*c^4*d*Sin[(5*(e + f*x))/2] - 1120*c^3*d^2*Sin[(5*(e + f*x))/2] + 5120*c^2*d^3*Sin[(5*(e + f*x))/2] - 7260*c*d^4*Sin[(5*(e + f*x))/2] + 2782*d^5*Sin[(5*(e + f*x))/2] - 1200*c^2*d^3*(e + f*x)*Sin[(5*(e + f*x))/2] + 1800*c*d^4*(e + f*x)*Sin[(5*(e + f*x))/2] - 780*d^5*(e + f*x)*Sin[(5*(e + f*x))/2] + 300*c*d^4*Sin[(7*(e + f*x))/2] - 105*d^5*Sin[(7*(e + f*x))/2] + 15*d^5*Sin[(9*(e + f*x))/2]))/(480*f*(a + a*Sin[e + f*x])^3)","B",1
472,1,683,195,1.454884,"\int \frac{(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^4/(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(80 c^4 \sin \left(\frac{1}{2} (e+f x)\right)-8 c^4 \sin \left(\frac{5}{2} (e+f x)\right)+240 c^3 d \sin \left(\frac{1}{2} (e+f x)\right)-48 c^3 d \sin \left(\frac{5}{2} (e+f x)\right)+960 c^2 d^2 \sin \left(\frac{1}{2} (e+f x)\right)+360 c^2 d^2 \sin \left(\frac{3}{2} (e+f x)\right)-168 c^2 d^2 \sin \left(\frac{5}{2} (e+f x)\right)+15 d \cos \left(\frac{1}{2} (e+f x)\right) \left(16 c^3+48 c^2 d+16 c d^2 (5 e+5 f x-9)-15 d^3 (4 e+4 f x-5)\right)-5 \cos \left(\frac{3}{2} (e+f x)\right) \left(8 c^4+48 c^3 d+96 c^2 d^2+8 c d^3 (15 e+15 f x-46)-9 d^4 (10 e+10 f x-27)\right)-2960 c d^3 \sin \left(\frac{1}{2} (e+f x)\right)+1200 c d^3 e \sin \left(\frac{1}{2} (e+f x)\right)+1200 c d^3 f x \sin \left(\frac{1}{2} (e+f x)\right)-720 c d^3 \sin \left(\frac{3}{2} (e+f x)\right)+600 c d^3 e \sin \left(\frac{3}{2} (e+f x)\right)+600 c d^3 f x \sin \left(\frac{3}{2} (e+f x)\right)+512 c d^3 \sin \left(\frac{5}{2} (e+f x)\right)-120 c d^3 e \sin \left(\frac{5}{2} (e+f x)\right)-120 c d^3 f x \sin \left(\frac{5}{2} (e+f x)\right)-120 c d^3 e \cos \left(\frac{5}{2} (e+f x)\right)-120 c d^3 f x \cos \left(\frac{5}{2} (e+f x)\right)+1755 d^4 \sin \left(\frac{1}{2} (e+f x)\right)-900 d^4 e \sin \left(\frac{1}{2} (e+f x)\right)-900 d^4 f x \sin \left(\frac{1}{2} (e+f x)\right)+225 d^4 \sin \left(\frac{3}{2} (e+f x)\right)-450 d^4 e \sin \left(\frac{3}{2} (e+f x)\right)-450 d^4 f x \sin \left(\frac{3}{2} (e+f x)\right)-363 d^4 \sin \left(\frac{5}{2} (e+f x)\right)+90 d^4 e \sin \left(\frac{5}{2} (e+f x)\right)+90 d^4 f x \sin \left(\frac{5}{2} (e+f x)\right)+15 d^4 \sin \left(\frac{7}{2} (e+f x)\right)+75 d^4 \cos \left(\frac{5}{2} (e+f x)\right)+90 d^4 e \cos \left(\frac{5}{2} (e+f x)\right)+90 d^4 f x \cos \left(\frac{5}{2} (e+f x)\right)+15 d^4 \cos \left(\frac{7}{2} (e+f x)\right)\right)}{120 a^3 f (\sin (e+f x)+1)^3}","\frac{d^2 \left(2 c^2+10 c d-27 d^2\right) \cos (e+f x)}{15 a^3 f}-\frac{(c-d)^2 \left(2 c^2+12 c d+45 d^2\right) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{d^3 x (4 c-3 d)}{a^3}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+9 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])*(15*d*(16*c^3 + 48*c^2*d - 15*d^3*(-5 + 4*e + 4*f*x) + 16*c*d^2*(-9 + 5*e + 5*f*x))*Cos[(e + f*x)/2] - 5*(8*c^4 + 48*c^3*d + 96*c^2*d^2 - 9*d^4*(-27 + 10*e + 10*f*x) + 8*c*d^3*(-46 + 15*e + 15*f*x))*Cos[(3*(e + f*x))/2] + 75*d^4*Cos[(5*(e + f*x))/2] - 120*c*d^3*e*Cos[(5*(e + f*x))/2] + 90*d^4*e*Cos[(5*(e + f*x))/2] - 120*c*d^3*f*x*Cos[(5*(e + f*x))/2] + 90*d^4*f*x*Cos[(5*(e + f*x))/2] + 15*d^4*Cos[(7*(e + f*x))/2] + 80*c^4*Sin[(e + f*x)/2] + 240*c^3*d*Sin[(e + f*x)/2] + 960*c^2*d^2*Sin[(e + f*x)/2] - 2960*c*d^3*Sin[(e + f*x)/2] + 1755*d^4*Sin[(e + f*x)/2] + 1200*c*d^3*e*Sin[(e + f*x)/2] - 900*d^4*e*Sin[(e + f*x)/2] + 1200*c*d^3*f*x*Sin[(e + f*x)/2] - 900*d^4*f*x*Sin[(e + f*x)/2] + 360*c^2*d^2*Sin[(3*(e + f*x))/2] - 720*c*d^3*Sin[(3*(e + f*x))/2] + 225*d^4*Sin[(3*(e + f*x))/2] + 600*c*d^3*e*Sin[(3*(e + f*x))/2] - 450*d^4*e*Sin[(3*(e + f*x))/2] + 600*c*d^3*f*x*Sin[(3*(e + f*x))/2] - 450*d^4*f*x*Sin[(3*(e + f*x))/2] - 8*c^4*Sin[(5*(e + f*x))/2] - 48*c^3*d*Sin[(5*(e + f*x))/2] - 168*c^2*d^2*Sin[(5*(e + f*x))/2] + 512*c*d^3*Sin[(5*(e + f*x))/2] - 363*d^4*Sin[(5*(e + f*x))/2] - 120*c*d^3*e*Sin[(5*(e + f*x))/2] + 90*d^4*e*Sin[(5*(e + f*x))/2] - 120*c*d^3*f*x*Sin[(5*(e + f*x))/2] + 90*d^4*f*x*Sin[(5*(e + f*x))/2] + 15*d^4*Sin[(7*(e + f*x))/2]))/(120*a^3*f*(1 + Sin[e + f*x])^3)","B",1
473,1,408,142,5.6096288,"\int \frac{(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx","Integrate[(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(40 c^3 \sin \left(\frac{1}{2} (e+f x)\right)-4 c^3 \sin \left(\frac{5}{2} (e+f x)\right)+30 d \cos \left(\frac{1}{2} (e+f x)\right) \left(3 c^2+6 c d+d^2 (5 e+5 f x-9)\right)+90 c^2 d \sin \left(\frac{1}{2} (e+f x)\right)-18 c^2 d \sin \left(\frac{5}{2} (e+f x)\right)-5 \cos \left(\frac{3}{2} (e+f x)\right) \left(4 c^3+18 c^2 d+24 c d^2+d^3 (15 e+15 f x-46)\right)+240 c d^2 \sin \left(\frac{1}{2} (e+f x)\right)+90 c d^2 \sin \left(\frac{3}{2} (e+f x)\right)-42 c d^2 \sin \left(\frac{5}{2} (e+f x)\right)-370 d^3 \sin \left(\frac{1}{2} (e+f x)\right)+150 d^3 e \sin \left(\frac{1}{2} (e+f x)\right)+150 d^3 f x \sin \left(\frac{1}{2} (e+f x)\right)-90 d^3 \sin \left(\frac{3}{2} (e+f x)\right)+75 d^3 e \sin \left(\frac{3}{2} (e+f x)\right)+75 d^3 f x \sin \left(\frac{3}{2} (e+f x)\right)+64 d^3 \sin \left(\frac{5}{2} (e+f x)\right)-15 d^3 e \sin \left(\frac{5}{2} (e+f x)\right)-15 d^3 f x \sin \left(\frac{5}{2} (e+f x)\right)-15 d^3 e \cos \left(\frac{5}{2} (e+f x)\right)-15 d^3 f x \cos \left(\frac{5}{2} (e+f x)\right)\right)}{60 a^3 f (\sin (e+f x)+1)^3}","-\frac{(c-d) \left(2 c^2+11 c d+29 d^2\right) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{d^3 x}{a^3}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d)^2 (2 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*d*(3*c^2 + 6*c*d + d^2*(-9 + 5*e + 5*f*x))*Cos[(e + f*x)/2] - 5*(4*c^3 + 18*c^2*d + 24*c*d^2 + d^3*(-46 + 15*e + 15*f*x))*Cos[(3*(e + f*x))/2] - 15*d^3*e*Cos[(5*(e + f*x))/2] - 15*d^3*f*x*Cos[(5*(e + f*x))/2] + 40*c^3*Sin[(e + f*x)/2] + 90*c^2*d*Sin[(e + f*x)/2] + 240*c*d^2*Sin[(e + f*x)/2] - 370*d^3*Sin[(e + f*x)/2] + 150*d^3*e*Sin[(e + f*x)/2] + 150*d^3*f*x*Sin[(e + f*x)/2] + 90*c*d^2*Sin[(3*(e + f*x))/2] - 90*d^3*Sin[(3*(e + f*x))/2] + 75*d^3*e*Sin[(3*(e + f*x))/2] + 75*d^3*f*x*Sin[(3*(e + f*x))/2] - 4*c^3*Sin[(5*(e + f*x))/2] - 18*c^2*d*Sin[(5*(e + f*x))/2] - 42*c*d^2*Sin[(5*(e + f*x))/2] + 64*d^3*Sin[(5*(e + f*x))/2] - 15*d^3*e*Sin[(5*(e + f*x))/2] - 15*d^3*f*x*Sin[(5*(e + f*x))/2]))/(60*a^3*f*(1 + Sin[e + f*x])^3)","B",1
474,1,84,125,0.1262021,"\int \frac{(c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^2/(a + a*Sin[e + f*x])^3,x]","-\frac{\cos (e+f x) \left(\left(2 c^2+6 c d+7 d^2\right) \sin ^2(e+f x)+6 \left(c^2+3 c d+d^2\right) \sin (e+f x)+7 c^2+6 c d+2 d^2\right)}{15 a^3 f (\sin (e+f x)+1)^3}","-\frac{\left(2 c^2+6 c d+7 d^2\right) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+5 d) \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}",1,"-1/15*(Cos[e + f*x]*(7*c^2 + 6*c*d + 2*d^2 + 6*(c^2 + 3*c*d + d^2)*Sin[e + f*x] + (2*c^2 + 6*c*d + 7*d^2)*Sin[e + f*x]^2))/(a^3*f*(1 + Sin[e + f*x])^3)","A",1
475,1,63,102,0.0829945,"\int \frac{c+d \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])/(a + a*Sin[e + f*x])^3,x]","-\frac{\cos (e+f x) \left((2 c+3 d) \sin ^2(e+f x)+(6 c+9 d) \sin (e+f x)+7 c+3 d\right)}{15 a^3 f (\sin (e+f x)+1)^3}","-\frac{(2 c+3 d) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{(2 c+3 d) \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}-\frac{(c-d) \cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}",1,"-1/15*(Cos[e + f*x]*(7*c + 3*d + (6*c + 9*d)*Sin[e + f*x] + (2*c + 3*d)*Sin[e + f*x]^2))/(a^3*f*(1 + Sin[e + f*x])^3)","A",1
476,1,76,83,0.1226427,"\int \frac{1}{(a+a \sin (e+f x))^3} \, dx","Integrate[(a + a*Sin[e + f*x])^(-3),x]","\frac{15 \sin (e+f x)-6 \sin (2 (e+f x))-\sin (3 (e+f x))-15 \cos (e+f x)-6 \cos (2 (e+f x))+\cos (3 (e+f x))+10}{30 a^3 f (\sin (e+f x)+1)^3}","-\frac{2 \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{2 \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}",1,"(10 - 15*Cos[e + f*x] - 6*Cos[2*(e + f*x)] + Cos[3*(e + f*x)] + 15*Sin[e + f*x] - 6*Sin[2*(e + f*x)] - Sin[3*(e + f*x)])/(30*a^3*f*(1 + Sin[e + f*x])^3)","A",1
477,1,301,186,0.7059956,"\int \frac{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx","Integrate[1/((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(2 \left(2 c^2-9 c d+22 d^2\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{30 d^3 \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}}+6 (c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right)+(c-d) (7 d-2 c) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+2 (2 c-7 d) (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)^2-3 (c-d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{15 a^3 f (c-d)^3 (\sin (e+f x)+1)^3}","-\frac{\left(2 c^2-9 c d+22 d^2\right) \cos (e+f x)}{15 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right)}-\frac{2 d^3 \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{(2 c-7 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2}-\frac{\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])*(6*(c - d)^2*Sin[(e + f*x)/2] - 3*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(2*c - 7*d)*(c - d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (c - d)*(-2*c + 7*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 2*(2*c^2 - 9*c*d + 22*d^2)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - (30*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)/Sqrt[c^2 - d^2]))/(15*a^3*(c - d)^3*f*(1 + Sin[e + f*x])^3)","A",1
478,1,361,298,2.5526443,"\int \frac{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2} \, dx","Integrate[1/((a + a*Sin[e + f*x])^3*(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(2 \left(2 c^2-14 c d+57 d^2\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{30 d^3 (4 c+3 d) \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) \sqrt{c^2-d^2}}-\frac{15 d^4 \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))}+6 (c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right)-2 (c-6 d) (c-d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+4 (c-6 d) (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)^2-3 (c-d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{15 a^3 f (c-d)^4 (\sin (e+f x)+1)^3}","-\frac{\left(2 c^2-12 c d+45 d^2\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{2 d^3 (4 c+3 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)^4 (c+d) \sqrt{c^2-d^2}}-\frac{d \left(2 c^3-12 c^2 d+43 c d^2+72 d^3\right) \cos (e+f x)}{15 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))}-\frac{(2 c-9 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{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(6*(c - d)^2*Sin[(e + f*x)/2] - 3*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 4*(c - 6*d)*(c - d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 2*(c - 6*d)*(c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 2*(2*c^2 - 14*c*d + 57*d^2)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - (30*d^3*(4*c + 3*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)/((c + d)*Sqrt[c^2 - d^2]) - (15*d^4*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/((c + d)*(c + d*Sin[e + f*x]))))/(15*a^3*(c - d)^4*f*(1 + Sin[e + f*x])^3)","A",1
479,1,1066,378,6.3243269,"\int \frac{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx","Integrate[1/((a + a*Sin[e + f*x])^3*(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(-160 \cos \left(\frac{3}{2} (e+f x)\right) c^6+320 \sin \left(\frac{1}{2} (e+f x)\right) c^6-32 \sin \left(\frac{5}{2} (e+f x)\right) c^6-400 d \cos \left(\frac{1}{2} (e+f x)\right) c^5+848 d \cos \left(\frac{3}{2} (e+f x)\right) c^5+32 d \cos \left(\frac{7}{2} (e+f x)\right) c^5-1520 d \sin \left(\frac{1}{2} (e+f x)\right) c^5+80 d \sin \left(\frac{5}{2} (e+f x)\right) c^5+3400 d^2 \cos \left(\frac{1}{2} (e+f x)\right) c^4-2400 d^2 \cos \left(\frac{3}{2} (e+f x)\right) c^4-200 d^2 \cos \left(\frac{7}{2} (e+f x)\right) c^4+4568 d^2 \sin \left(\frac{1}{2} (e+f x)\right) c^4+800 d^2 \sin \left(\frac{3}{2} (e+f x)\right) c^4-32 d^2 \sin \left(\frac{5}{2} (e+f x)\right) c^4+8 d^2 \sin \left(\frac{9}{2} (e+f x)\right) c^4+19340 d^3 \cos \left(\frac{1}{2} (e+f x)\right) c^3-19396 d^3 \cos \left(\frac{3}{2} (e+f x)\right) c^3-1260 d^3 \cos \left(\frac{5}{2} (e+f x)\right) c^3+836 d^3 \cos \left(\frac{7}{2} (e+f x)\right) c^3+27340 d^3 \sin \left(\frac{1}{2} (e+f x)\right) c^3+7500 d^3 \sin \left(\frac{3}{2} (e+f x)\right) c^3-6820 d^3 \sin \left(\frac{5}{2} (e+f x)\right) c^3-60 d^3 \sin \left(\frac{9}{2} (e+f x)\right) c^3+30400 d^4 \cos \left(\frac{1}{2} (e+f x)\right) c^2-35280 d^4 \cos \left(\frac{3}{2} (e+f x)\right) c^2-2640 d^4 \cos \left(\frac{5}{2} (e+f x)\right) c^2+4480 d^4 \cos \left(\frac{7}{2} (e+f x)\right) c^2+40904 d^4 \sin \left(\frac{1}{2} (e+f x)\right) c^2+13280 d^4 \sin \left(\frac{3}{2} (e+f x)\right) c^2-18080 d^4 \sin \left(\frac{5}{2} (e+f x)\right) c^2-60 d^4 \sin \left(\frac{7}{2} (e+f x)\right) c^2+284 d^4 \sin \left(\frac{9}{2} (e+f x)\right) c^2+19940 d^5 \cos \left(\frac{1}{2} (e+f x)\right) c-24742 d^5 \cos \left(\frac{3}{2} (e+f x)\right) c-2250 d^5 \cos \left(\frac{5}{2} (e+f x)\right) c+5747 d^5 \cos \left(\frac{7}{2} (e+f x)\right) c-135 d^5 \cos \left(\frac{9}{2} (e+f x)\right) c+26020 d^5 \sin \left(\frac{1}{2} (e+f x)\right) c+9690 d^5 \sin \left(\frac{3}{2} (e+f x)\right) c-15670 d^5 \sin \left(\frac{5}{2} (e+f x)\right) c+135 d^5 \sin \left(\frac{7}{2} (e+f x)\right) c+915 d^5 \sin \left(\frac{9}{2} (e+f x)\right) c+4810 d^6 \cos \left(\frac{1}{2} (e+f x)\right)-5810 d^6 \cos \left(\frac{3}{2} (e+f x)\right)-870 d^6 \cos \left(\frac{5}{2} (e+f x)\right)+2200 d^6 \cos \left(\frac{7}{2} (e+f x)\right)-90 d^6 \cos \left(\frac{9}{2} (e+f x)\right)+6318 d^6 \sin \left(\frac{1}{2} (e+f x)\right)+2750 d^6 \sin \left(\frac{3}{2} (e+f x)\right)-4266 d^6 \sin \left(\frac{5}{2} (e+f x)\right)+60 d^6 \sin \left(\frac{7}{2} (e+f x)\right)+518 d^6 \sin \left(\frac{9}{2} (e+f x)\right)\right)}{480 (c-d)^5 (c+d)^2 f (\sin (e+f x) a+a)^3 (c+d \sin (e+f x))^2}-\frac{d^3 \left(20 c^2+30 d c+13 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)^5 (c+d)^2 \sqrt{c^2-d^2} f (\sin (e+f x) a+a)^3}","-\frac{\left(2 c^2-15 c d+76 d^2\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^3 \left(20 c^2+30 c d+13 d^2\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(4 c^3-30 c^2 d+146 c d^2+195 d^3\right) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^2}-\frac{d \left(4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 (c+d \sin (e+f x))}-\frac{(2 c-11 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{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}",1,"-((d^3*(20*c^2 + 30*c*d + 13*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)^5*(c + d)^2*Sqrt[c^2 - d^2]*f*(a + a*Sin[e + f*x])^3)) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-400*c^5*d*Cos[(e + f*x)/2] + 3400*c^4*d^2*Cos[(e + f*x)/2] + 19340*c^3*d^3*Cos[(e + f*x)/2] + 30400*c^2*d^4*Cos[(e + f*x)/2] + 19940*c*d^5*Cos[(e + f*x)/2] + 4810*d^6*Cos[(e + f*x)/2] - 160*c^6*Cos[(3*(e + f*x))/2] + 848*c^5*d*Cos[(3*(e + f*x))/2] - 2400*c^4*d^2*Cos[(3*(e + f*x))/2] - 19396*c^3*d^3*Cos[(3*(e + f*x))/2] - 35280*c^2*d^4*Cos[(3*(e + f*x))/2] - 24742*c*d^5*Cos[(3*(e + f*x))/2] - 5810*d^6*Cos[(3*(e + f*x))/2] - 1260*c^3*d^3*Cos[(5*(e + f*x))/2] - 2640*c^2*d^4*Cos[(5*(e + f*x))/2] - 2250*c*d^5*Cos[(5*(e + f*x))/2] - 870*d^6*Cos[(5*(e + f*x))/2] + 32*c^5*d*Cos[(7*(e + f*x))/2] - 200*c^4*d^2*Cos[(7*(e + f*x))/2] + 836*c^3*d^3*Cos[(7*(e + f*x))/2] + 4480*c^2*d^4*Cos[(7*(e + f*x))/2] + 5747*c*d^5*Cos[(7*(e + f*x))/2] + 2200*d^6*Cos[(7*(e + f*x))/2] - 135*c*d^5*Cos[(9*(e + f*x))/2] - 90*d^6*Cos[(9*(e + f*x))/2] + 320*c^6*Sin[(e + f*x)/2] - 1520*c^5*d*Sin[(e + f*x)/2] + 4568*c^4*d^2*Sin[(e + f*x)/2] + 27340*c^3*d^3*Sin[(e + f*x)/2] + 40904*c^2*d^4*Sin[(e + f*x)/2] + 26020*c*d^5*Sin[(e + f*x)/2] + 6318*d^6*Sin[(e + f*x)/2] + 800*c^4*d^2*Sin[(3*(e + f*x))/2] + 7500*c^3*d^3*Sin[(3*(e + f*x))/2] + 13280*c^2*d^4*Sin[(3*(e + f*x))/2] + 9690*c*d^5*Sin[(3*(e + f*x))/2] + 2750*d^6*Sin[(3*(e + f*x))/2] - 32*c^6*Sin[(5*(e + f*x))/2] + 80*c^5*d*Sin[(5*(e + f*x))/2] - 32*c^4*d^2*Sin[(5*(e + f*x))/2] - 6820*c^3*d^3*Sin[(5*(e + f*x))/2] - 18080*c^2*d^4*Sin[(5*(e + f*x))/2] - 15670*c*d^5*Sin[(5*(e + f*x))/2] - 4266*d^6*Sin[(5*(e + f*x))/2] - 60*c^2*d^4*Sin[(7*(e + f*x))/2] + 135*c*d^5*Sin[(7*(e + f*x))/2] + 60*d^6*Sin[(7*(e + f*x))/2] + 8*c^4*d^2*Sin[(9*(e + f*x))/2] - 60*c^3*d^3*Sin[(9*(e + f*x))/2] + 284*c^2*d^4*Sin[(9*(e + f*x))/2] + 915*c*d^5*Sin[(9*(e + f*x))/2] + 518*d^6*Sin[(9*(e + f*x))/2]))/(480*(c - d)^5*(c + d)^2*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2)","B",1
480,1,55,75,0.0648344,"\int \frac{A+B \sin (x)}{(1+\sin (x))^4} \, dx","Integrate[(A + B*Sin[x])/(1 + Sin[x])^4,x]","-\frac{\cos (x) \left((6 A+8 B) \sin ^3(x)+8 (3 A+4 B) \sin ^2(x)+13 (3 A+4 B) \sin (x)+36 A+13 B\right)}{105 (\sin (x)+1)^4}","-\frac{2 (3 A+4 B) \cos (x)}{105 (\sin (x)+1)}-\frac{2 (3 A+4 B) \cos (x)}{105 (\sin (x)+1)^2}-\frac{(3 A+4 B) \cos (x)}{35 (\sin (x)+1)^3}-\frac{(A-B) \cos (x)}{7 (\sin (x)+1)^4}",1,"-1/105*(Cos[x]*(36*A + 13*B + 13*(3*A + 4*B)*Sin[x] + 8*(3*A + 4*B)*Sin[x]^2 + (6*A + 8*B)*Sin[x]^3))/(1 + Sin[x])^4","A",1
481,1,54,81,0.0694356,"\int \frac{A+B \sin (x)}{(1-\sin (x))^4} \, dx","Integrate[(A + B*Sin[x])/(1 - Sin[x])^4,x]","\frac{\cos (x) \left((8 B-6 A) \sin ^3(x)+8 (3 A-4 B) \sin ^2(x)+(52 B-39 A) \sin (x)+36 A-13 B\right)}{105 (\sin (x)-1)^4}","\frac{2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))}+\frac{2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))^2}+\frac{(3 A-4 B) \cos (x)}{35 (1-\sin (x))^3}+\frac{(A+B) \cos (x)}{7 (1-\sin (x))^4}",1,"(Cos[x]*(36*A - 13*B + (-39*A + 52*B)*Sin[x] + 8*(3*A - 4*B)*Sin[x]^2 + (-6*A + 8*B)*Sin[x]^3))/(105*(-1 + Sin[x])^4)","A",1
482,1,3531,290,6.7638898,"\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 a \left(15 c^2+56 c d+25 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 f}-\frac{2 a \left(c^2-d^2\right) \left(15 c^2+56 c d+25 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d f \sqrt{c+d \sin (e+f x)}}+\frac{2 a \left(15 c^3+161 c^2 d+145 c d^2+63 d^3\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}-\frac{2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 f}",1,"a*((c^3*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(7*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (23*c^2*d*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(15*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (29*c*d^2*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(21*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (3*d^3*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(5*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + ((1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*(-1/210*((180*c^2 + 308*c*d + 115*d^2)*Cos[e]*Cos[f*x])/f + (d^2*Cos[3*e]*Cos[3*f*x])/(14*f) - (d*(15*c + 7*d)*Cos[2*f*x]*Sin[2*e])/(35*f) + ((180*c^2 + 308*c*d + 115*d^2)*Sin[e]*Sin[f*x])/(210*f) - (d*(15*c + 7*d)*Cos[2*e]*Sin[2*f*x])/(35*f) - (d^2*Sin[3*e]*Sin[3*f*x])/(14*f) + (2*(15*c^3 + 161*c^2*d + 145*c*d^2 + 63*d^3)*Tan[e])/(105*d*f)))/(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2 + (18*c^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(7*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]) + (2*c^3*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(d*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]) + (34*c*d*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(15*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]) + (10*d^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(21*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]))","C",0
483,1,2625,231,6.397211,"\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 a (3 c+5 d) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \sqrt{c+d \sin (e+f x)}}+\frac{2 a \left(3 c^2+20 c d+9 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac{2 a (3 c+5 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 f}",1,"a*((c^2*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(5*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (4*c*d*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(3*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (3*d^2*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(5*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + ((1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*((-2*(6*c + 5*d)*Cos[e]*Cos[f*x])/(15*f) - (d*Cos[2*f*x]*Sin[2*e])/(5*f) + (2*(6*c + 5*d)*Sin[e]*Sin[f*x])/(15*f) - (d*Cos[2*e]*Sin[2*f*x])/(5*f) + (2*(3*c^2 + 20*c*d + 9*d^2)*Tan[e])/(15*d*f)))/(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2 + (8*c*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(5*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]) + (2*c^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(d*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]) + (2*d*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(3*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]))","C",0
484,1,1736,179,6.2769569,"\int (a+a \sin (e+f x)) \sqrt{c+d \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]],x]","a \left(\frac{c \sec (e) \left(-\frac{F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(1-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}\right)},-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}-1\right)}\right) \cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1} \sqrt{\frac{\cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} d+\sqrt{\cot ^2(e)+1} d}{d \sqrt{\cot ^2(e)+1}-c \csc (e)}} \sqrt{\frac{d \sqrt{\cot ^2(e)+1}-d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1}}{\sqrt{\cot ^2(e)+1} d+c \csc (e)}} \sqrt{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}-\frac{\frac{2 d \sin (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac{\cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1}}}{\sqrt{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}\right) (\sin (e+f x)+1)}{3 f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{d \sec (e) \left(-\frac{F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(1-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}\right)},-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}-1\right)}\right) \cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1} \sqrt{\frac{\cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} d+\sqrt{\cot ^2(e)+1} d}{d \sqrt{\cot ^2(e)+1}-c \csc (e)}} \sqrt{\frac{d \sqrt{\cot ^2(e)+1}-d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1}}{\sqrt{\cot ^2(e)+1} d+c \csc (e)}} \sqrt{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}-\frac{\frac{2 d \sin (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac{\cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1}}}{\sqrt{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}\right) (\sin (e+f x)+1)}{f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{\sqrt{c+d \sin (e+f x)} \left(-\frac{2 \cos (e) \cos (f x)}{3 f}+\frac{2 \sin (e) \sin (f x)}{3 f}+\frac{2 (c+3 d) \tan (e)}{3 d f}\right) (\sin (e+f x)+1)}{\left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{2 c F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(1-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}\right)},-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}-1\right)}\right) \sec (e) \sec \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\frac{d \sqrt{\tan ^2(e)+1}-d \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} d+c \sec (e)}} \sqrt{\frac{\sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1} d+\sqrt{\tan ^2(e)+1} d}{d \sqrt{\tan ^2(e)+1}-c \sec (e)}} \sqrt{c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}} (\sin (e+f x)+1)}{d f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2 \sqrt{\tan ^2(e)+1}}+\frac{2 F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(1-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}\right)},-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}-1\right)}\right) \sec (e) \sec \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\frac{d \sqrt{\tan ^2(e)+1}-d \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} d+c \sec (e)}} \sqrt{\frac{\sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1} d+\sqrt{\tan ^2(e)+1} d}{d \sqrt{\tan ^2(e)+1}-c \sec (e)}} \sqrt{c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}} (\sin (e+f x)+1)}{3 f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2 \sqrt{\tan ^2(e)+1}}\right)","-\frac{2 a \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \sqrt{c+d \sin (e+f x)}}-\frac{2 a \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f}+\frac{2 a (c+3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"a*((c*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(3*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (d*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + ((1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*((-2*Cos[e]*Cos[f*x])/(3*f) + (2*Sin[e]*Sin[f*x])/(3*f) + (2*(c + 3*d)*Tan[e])/(3*d*f)))/(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2 + (2*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(3*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]) + (2*c*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(d*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]))","C",0
485,1,880,138,6.2465606,"\int \frac{a+a \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])/Sqrt[c + d*Sin[e + f*x]],x]","a \left(\frac{\sec (e) \left(-\frac{F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(1-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}\right)},-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}-1\right)}\right) \cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1} \sqrt{\frac{\cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} d+\sqrt{\cot ^2(e)+1} d}{d \sqrt{\cot ^2(e)+1}-c \csc (e)}} \sqrt{\frac{d \sqrt{\cot ^2(e)+1}-d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1}}{\sqrt{\cot ^2(e)+1} d+c \csc (e)}} \sqrt{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}-\frac{\frac{2 d \sin (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac{\cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1}}}{\sqrt{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}\right) (\sin (e+f x)+1)}{f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{2 \sqrt{c+d \sin (e+f x)} \tan (e) (\sin (e+f x)+1)}{d f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{2 F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(1-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}\right)},-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}-1\right)}\right) \sec (e) \sec \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\frac{d \sqrt{\tan ^2(e)+1}-d \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} d+c \sec (e)}} \sqrt{\frac{\sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1} d+\sqrt{\tan ^2(e)+1} d}{d \sqrt{\tan ^2(e)+1}-c \sec (e)}} \sqrt{c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}} (\sin (e+f x)+1)}{d f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2 \sqrt{\tan ^2(e)+1}}\right)","\frac{2 a \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{c+d \sin (e+f x)}}",1,"a*((Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (2*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*Tan[e])/(d*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (2*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(d*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]))","C",0
486,1,938,169,6.3879046,"\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(3/2),x]","a \left(\frac{\sqrt{c+d \sin (e+f x)} \left(\frac{2 \csc (e) (c \cos (e)+d \sin (f x))}{d (c+d) f (c+d \sin (e+f x))}-\frac{2 \csc (e) \sec (e)}{d (c+d) f}\right) (\sin (e+f x)+1)}{\left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}-\frac{\sec (e) \left(-\frac{F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(1-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}\right)},-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}-1\right)}\right) \cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1} \sqrt{\frac{\cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} d+\sqrt{\cot ^2(e)+1} d}{d \sqrt{\cot ^2(e)+1}-c \csc (e)}} \sqrt{\frac{d \sqrt{\cot ^2(e)+1}-d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1}}{\sqrt{\cot ^2(e)+1} d+c \csc (e)}} \sqrt{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}-\frac{\frac{2 d \sin (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac{\cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1}}}{\sqrt{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}\right) (\sin (e+f x)+1)}{(c+d) f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{2 F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(1-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}\right)},-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}-1\right)}\right) \sec (e) \sec \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\frac{d \sqrt{\tan ^2(e)+1}-d \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} d+c \sec (e)}} \sqrt{\frac{\sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1} d+\sqrt{\tan ^2(e)+1} d}{d \sqrt{\tan ^2(e)+1}-c \sec (e)}} \sqrt{c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}} (\sin (e+f x)+1)}{d (c+d) f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2 \sqrt{\tan ^2(e)+1}}\right)","-\frac{2 a \cos (e+f x)}{f (c+d) \sqrt{c+d \sin (e+f x)}}+\frac{2 a \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{c+d \sin (e+f x)}}-\frac{2 a \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"a*(((1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*((-2*Csc[e]*Sec[e])/(d*(c + d)*f) + (2*Csc[e]*(c*Cos[e] + d*Sin[f*x]))/(d*(c + d)*f*(c + d*Sin[e + f*x]))))/(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2 - (Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/((c + d)*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (2*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(d*(c + d)*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]))","C",0
487,1,1870,237,6.7738764,"\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(5/2),x]","a \left(\frac{\sqrt{c+d \sin (e+f x)} \left(-\frac{2 (c-3 d) \csc (e) \sec (e)}{3 (c-d) d (c+d)^2 f}-\frac{2 \csc (e) (3 c \cos (e)-d \cos (e)-c \sin (f x)+3 d \sin (f x))}{3 (c-d) (c+d)^2 f (c+d \sin (e+f x))}+\frac{2 \csc (e) (c \cos (e)+d \sin (f x))}{3 d (c+d) f (c+d \sin (e+f x))^2}\right) (\sin (e+f x)+1)}{\left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{d \sec (e) \left(-\frac{F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(1-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}\right)},-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}-1\right)}\right) \cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1} \sqrt{\frac{\cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} d+\sqrt{\cot ^2(e)+1} d}{d \sqrt{\cot ^2(e)+1}-c \csc (e)}} \sqrt{\frac{d \sqrt{\cot ^2(e)+1}-d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1}}{\sqrt{\cot ^2(e)+1} d+c \csc (e)}} \sqrt{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}-\frac{\frac{2 d \sin (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac{\cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1}}}{\sqrt{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}\right) (\sin (e+f x)+1)}{(c-d) (c+d)^2 f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}-\frac{c \sec (e) \left(-\frac{F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(1-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}\right)},-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}-1\right)}\right) \cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1} \sqrt{\frac{\cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} d+\sqrt{\cot ^2(e)+1} d}{d \sqrt{\cot ^2(e)+1}-c \csc (e)}} \sqrt{\frac{d \sqrt{\cot ^2(e)+1}-d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1}}{\sqrt{\cot ^2(e)+1} d+c \csc (e)}} \sqrt{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}-\frac{\frac{2 d \sin (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac{\cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1}}}{\sqrt{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}\right) (\sin (e+f x)+1)}{3 (c-d) (c+d)^2 f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}-\frac{2 F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(1-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}\right)},-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}-1\right)}\right) \sec (e) \sec \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\frac{d \sqrt{\tan ^2(e)+1}-d \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} d+c \sec (e)}} \sqrt{\frac{\sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1} d+\sqrt{\tan ^2(e)+1} d}{d \sqrt{\tan ^2(e)+1}-c \sec (e)}} \sqrt{c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}} (\sin (e+f x)+1)}{3 (c-d) (c+d)^2 f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2 \sqrt{\tan ^2(e)+1}}+\frac{2 c F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(1-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}\right)},-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}-1\right)}\right) \sec (e) \sec \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\frac{d \sqrt{\tan ^2(e)+1}-d \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} d+c \sec (e)}} \sqrt{\frac{\sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1} d+\sqrt{\tan ^2(e)+1} d}{d \sqrt{\tan ^2(e)+1}-c \sec (e)}} \sqrt{c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}} (\sin (e+f x)+1)}{(c-d) d (c+d)^2 f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2 \sqrt{\tan ^2(e)+1}}\right)","-\frac{2 a (c-3 d) \cos (e+f x)}{3 f (c-d) (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}+\frac{2 a \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{2 a (c-3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f (c-d) (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"a*(((1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*((-2*(c - 3*d)*Csc[e]*Sec[e])/(3*(c - d)*d*(c + d)^2*f) + (2*Csc[e]*(c*Cos[e] + d*Sin[f*x]))/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^2) - (2*Csc[e]*(3*c*Cos[e] - d*Cos[e] - c*Sin[f*x] + 3*d*Sin[f*x]))/(3*(c - d)*(c + d)^2*f*(c + d*Sin[e + f*x]))))/(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2 - (c*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(3*(c - d)*(c + d)^2*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (d*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/((c - d)*(c + d)^2*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) - (2*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(3*(c - d)*(c + d)^2*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]) + (2*c*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/((c - d)*d*(c + d)^2*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]))","C",0
488,1,2815,318,7.1538907,"\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx","Integrate[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(7/2),x]","\text{Result too large to show}","-\frac{2 a \left(3 c^2-20 c d+9 d^2\right) \cos (e+f x)}{15 f (c-d)^2 (c+d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 a \left(3 c^2-20 c d+9 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f (c-d)^2 (c+d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a (3 c-5 d) \cos (e+f x)}{15 f (c-d) (c+d)^2 (c+d \sin (e+f x))^{3/2}}-\frac{2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}+\frac{2 a (3 c-5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f (c-d) (c+d)^2 \sqrt{c+d \sin (e+f x)}}",1,"a*(((1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*((-2*(3*c^2 - 20*c*d + 9*d^2)*Csc[e]*Sec[e])/(15*(c - d)^2*d*(c + d)^3*f) + (2*Csc[e]*(c*Cos[e] + d*Sin[f*x]))/(5*d*(c + d)*f*(c + d*Sin[e + f*x])^3) - (2*Csc[e]*(5*c*Cos[e] - 3*d*Cos[e] - 3*c*Sin[f*x] + 5*d*Sin[f*x]))/(15*(c - d)*(c + d)^2*f*(c + d*Sin[e + f*x])^2) - (2*Csc[e]*(15*c^2*Cos[e] - 12*c*d*Cos[e] + 5*d^2*Cos[e] - 3*c^2*Sin[f*x] + 20*c*d*Sin[f*x] - 9*d^2*Sin[f*x]))/(15*(c - d)^2*(c + d)^3*f*(c + d*Sin[e + f*x]))))/(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2 - (c^2*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(5*(c - d)^2*(c + d)^3*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (4*c*d*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(3*(c - d)^2*(c + d)^3*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) - (3*d^2*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(5*(c - d)^2*(c + d)^3*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) - (8*c*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(5*(c - d)^2*(c + d)^3*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]) + (2*c^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/((c - d)^2*d*(c + d)^3*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]) + (2*d*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(3*(c - d)^2*(c + d)^3*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]))","C",0
489,1,322,378,1.9778771,"\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2),x]","\frac{a^2 (\sin (e+f x)+1)^2 \left(16 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(\left(5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)-d^2 \left(235 c^3+405 c^2 d+309 c d^2+75 d^3\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)-d (c+d \sin (e+f x)) \left(2 d \left(\sin (2 (e+f x)) \left(150 c^2+540 c d-35 d^2 \cos (2 (e+f x))+259 d^2\right)-5 d (19 c+18 d) \cos (3 (e+f x))\right)+2 \left(20 c^3+1080 c^2 d+1671 c d^2+690 d^3\right) \cos (e+f x)\right)\right)}{1260 d^2 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \sqrt{c+d \sin (e+f x)}}","\frac{4 a^2 \left(5 c^3-45 c^2 d-141 c d^2-75 d^3\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d f}+\frac{4 a^2 \left(c^2-d^2\right) \left(5 c^3-45 c^2 d-141 c d^2-75 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 \left(5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{4 a^2 \left(5 c (c-9 d)-56 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac{4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}",1,"(a^2*(1 + Sin[e + f*x])^2*(16*(-(d^2*(235*c^3 + 405*c^2*d + 309*c*d^2 + 75*d^3)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]) + (5*c^4 - 45*c^3*d - 381*c^2*d^2 - 435*c*d^3 - 168*d^4)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - d*(c + d*Sin[e + f*x])*(2*(20*c^3 + 1080*c^2*d + 1671*c*d^2 + 690*d^3)*Cos[e + f*x] + 2*d*(-5*d*(19*c + 18*d)*Cos[3*(e + f*x)] + (150*c^2 + 540*c*d + 259*d^2 - 35*d^2*Cos[2*(e + f*x)])*Sin[2*(e + f*x)]))))/(1260*d^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*Sqrt[c + d*Sin[e + f*x]])","A",1
490,1,262,298,2.1111519,"\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2),x]","\frac{a^2 \left(d \cos (e+f x) \left(-4 c^3-d \left(36 c^2+168 c d+95 d^2\right) \sin (e+f x)-112 c^2 d+2 d^2 (13 c+14 d) \cos (2 (e+f x))-106 c d^2+5 d^3 \sin (3 (e+f x))-28 d^3\right)-8 \left(c^4-7 c^3 d-11 c^2 d^2+7 c d^3+10 d^4\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+8 \left(c^4-6 c^3 d-44 c^2 d^2-58 c d^3-21 d^4\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{70 d^2 f \sqrt{c+d \sin (e+f x)}}","\frac{4 a^2 \left(c^2-7 c d-10 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{35 d f}+\frac{4 a^2 \left(c^2-7 c d-10 d^2\right) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{35 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 (c+3 d) \left(c^2-10 c d-7 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{35 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac{4 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}",1,"(a^2*(8*(c^4 - 6*c^3*d - 44*c^2*d^2 - 58*c*d^3 - 21*d^4)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - 8*(c^4 - 7*c^3*d - 11*c^2*d^2 + 7*c*d^3 + 10*d^4)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d*Cos[e + f*x]*(-4*c^3 - 112*c^2*d - 106*c*d^2 - 28*d^3 + 2*d^2*(13*c + 14*d)*Cos[2*(e + f*x)] - d*(36*c^2 + 168*c*d + 95*d^2)*Sin[e + f*x] + 5*d^3*Sin[3*(e + f*x)])))/(70*d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",1
491,1,244,239,1.4144915,"\int (a+a \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]],x]","-\frac{a^2 (\sin (e+f x)+1)^2 \left(-d \cos (e+f x) \left(-2 c^2-4 d (2 c+5 d) \sin (e+f x)-20 c d+3 d^2 \cos (2 (e+f x))-3 d^2\right)+4 \left(c^3-5 c^2 d-c d^2+5 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-4 \left(c^3-4 c^2 d-17 c d^2-12 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{15 d^2 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \sqrt{c+d \sin (e+f x)}}","\frac{4 a^2 (c-5 d) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 \left(c^2-5 c d-12 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}+\frac{4 a^2 (c-5 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d f}",1,"-1/15*(a^2*(1 + Sin[e + f*x])^2*(-4*(c^3 - 4*c^2*d - 17*c*d^2 - 12*d^3)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + 4*(c^3 - 5*c^2*d - c*d^2 + 5*d^3)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - d*Cos[e + f*x]*(-2*c^2 - 20*c*d - 3*d^2 + 3*d^2*Cos[2*(e + f*x)] - 4*d*(2*c + 5*d)*Sin[e + f*x])))/(d^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*Sqrt[c + d*Sin[e + f*x]])","A",1
492,1,193,189,1.0820487,"\int \frac{(a+a \sin (e+f x))^2}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])^2/Sqrt[c + d*Sin[e + f*x]],x]","-\frac{2 a^2 (\sin (e+f x)+1)^2 \left(2 \left(c^2-3 c d+2 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-2 \left(c^2-2 c d-3 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+d \cos (e+f x) (c+d \sin (e+f x))\right)}{3 d^2 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \sqrt{c+d \sin (e+f x)}}","\frac{4 a^2 (c-2 d) (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 (c-3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d f}",1,"(-2*a^2*(1 + Sin[e + f*x])^2*(d*Cos[e + f*x]*(c + d*Sin[e + f*x]) - 2*(c^2 - 2*c*d - 3*d^2)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + 2*(c^2 - 3*c*d + 2*d^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(3*d^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*Sqrt[c + d*Sin[e + f*x]])","A",1
493,1,175,189,0.9117332,"\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(3/2),x]","-\frac{2 a^2 (\sin (e+f x)+1)^2 \left(2 c (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-(c-d) \left(2 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+d \cos (e+f x)\right)\right)}{d^2 f (c+d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \sqrt{c+d \sin (e+f x)}}","-\frac{4 a^2 (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^2 c \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d^2 f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt{c+d \sin (e+f x)}}",1,"(-2*a^2*(1 + Sin[e + f*x])^2*(2*c*(c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - (c - d)*(d*Cos[e + f*x] + 2*(c + d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])))/(d^2*(c + d)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*Sqrt[c + d*Sin[e + f*x]])","A",1
494,1,207,247,1.7687047,"\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(5/2),x]","-\frac{2 a^2 (\sin (e+f x)+1)^2 \left(d \cos (e+f x) \left(c^2+2 d (c+3 d) \sin (e+f x)+6 c d+d^2\right)+2 (c+2 d) (c+d)^2 \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{3/2} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-2 (c+3 d) (c+d)^2 \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{3/2} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{3 d^2 f (c+d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 (c+d \sin (e+f x))^{3/2}}","\frac{4 a^2 (c+2 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 (c+3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{4 a^2 (c+3 d) \cos (e+f x)}{3 d f (c+d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{2 a^2 (c-d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}",1,"(-2*a^2*(1 + Sin[e + f*x])^2*(-2*(c + d)^2*(c + 3*d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*((c + d*Sin[e + f*x])/(c + d))^(3/2) + 2*(c + d)^2*(c + 2*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*((c + d*Sin[e + f*x])/(c + d))^(3/2) + d*Cos[e + f*x]*(c^2 + 6*c*d + d^2 + 2*d*(c + 3*d)*Sin[e + f*x])))/(3*d^2*(c + d)^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c + d*Sin[e + f*x])^(3/2))","A",1
495,1,283,320,2.026273,"\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(7/2),x]","\frac{2 a^2 (\sin (e+f x)+1)^2 \left(d \cos (e+f x) \left(-2 \left(c^2+5 c d-12 d^2\right) (c+d \sin (e+f x))^2-2 (c-d) (c+5 d) (c+d) (c+d \sin (e+f x))+3 (c-d)^2 (c+d)^2\right)-2 (c+d \sin (e+f x))^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(d^2 (11 c-5 d) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-\left(c^2+5 c d-12 d^2\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)\right)}{15 d^2 f (c-d) (c+d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 (c+d \sin (e+f x))^{5/2}}","-\frac{4 a^2 \left(c^2+5 c d-12 d^2\right) \cos (e+f x)}{15 d f (c-d) (c+d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 \left(c^2+5 c d-12 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f (c-d) (c+d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{4 a^2 (c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 (c+5 d) \cos (e+f x)}{15 d f (c+d)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 a^2 (c-d) \cos (e+f x)}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}",1,"(2*a^2*(1 + Sin[e + f*x])^2*(-2*((11*c - 5*d)*d^2*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - (c^2 + 5*c*d - 12*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*(c + d*Sin[e + f*x])^2*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d*Cos[e + f*x]*(3*(c - d)^2*(c + d)^2 - 2*(c - d)*(c + d)*(c + 5*d)*(c + d*Sin[e + f*x]) - 2*(c^2 + 5*c*d - 12*d^2)*(c + d*Sin[e + f*x])^2)))/(15*(c - d)*d^2*(c + d)^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c + d*Sin[e + f*x])^(5/2))","A",1
496,1,377,467,1.9028924,"\int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2),x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(d (c+d \sin (e+f x)) \left(d^2 \left(452 c^2+2508 c d+1701 d^2\right) \cos (3 (e+f x))-4 d \left(6 c^3+990 c^2 d+2401 c d^2+1155 d^3\right) \sin (2 (e+f x))+2 \left(32 c^4-264 c^3 d-8994 c^2 d^2-13926 c d^3-5859 d^4\right) \cos (e+f x)+14 d^3 (23 c+33 d) \sin (4 (e+f x))-63 d^4 \cos (5 (e+f x))\right)-32 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(d^2 \left(c^4+858 c^3 d+1668 c^2 d^2+1254 c d^3+315 d^4\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+\left(4 c^5-33 c^4 d+174 c^3 d^2+1452 c^2 d^3+1806 c d^4+693 d^5\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)\right)}{5544 d^3 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6 \sqrt{c+d \sin (e+f x)}}","-\frac{4 a^3 \left(4 c^2-33 c d+189 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac{4 a^3 \left(4 c^3-33 c^2 d+182 c d^2+231 d^3\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac{4 a^3 \left(4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{693 d^2 f}-\frac{4 a^3 \left(c^2-d^2\right) \left(4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{693 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 (c+3 d) \left(4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{693 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^{7/2}}{11 d f}",1,"(a^3*(1 + Sin[e + f*x])^3*(-32*(d^2*(c^4 + 858*c^3*d + 1668*c^2*d^2 + 1254*c*d^3 + 315*d^4)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (4*c^5 - 33*c^4*d + 174*c^3*d^2 + 1452*c^2*d^3 + 1806*c*d^4 + 693*d^5)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d*(c + d*Sin[e + f*x])*(2*(32*c^4 - 264*c^3*d - 8994*c^2*d^2 - 13926*c*d^3 - 5859*d^4)*Cos[e + f*x] + d^2*(452*c^2 + 2508*c*d + 1701*d^2)*Cos[3*(e + f*x)] - 63*d^4*Cos[5*(e + f*x)] - 4*d*(6*c^3 + 990*c^2*d + 2401*c*d^2 + 1155*d^3)*Sin[2*(e + f*x)] + 14*d^3*(23*c + 33*d)*Sin[4*(e + f*x)])))/(5544*d^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Sin[e + f*x]])","A",1
497,1,318,390,2.2741696,"\int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2),x]","\frac{a^3 (\sin (e+f x)+1)^3 \left(d (c+d \sin (e+f x)) \left(2 d \left(5 d (10 c+27 d) \cos (3 (e+f x))-\sin (2 (e+f x)) \left(6 c^2+432 c d-35 d^2 \cos (2 (e+f x))+511 d^2\right)\right)+\left(32 c^3-216 c^2 d-3828 c d^2-2910 d^3\right) \cos (e+f x)\right)-16 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(d^2 \left(c^3+387 c^2 d+471 c d^2+165 d^3\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+\left(4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)\right)}{1260 d^3 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6 \sqrt{c+d \sin (e+f x)}}","-\frac{4 a^3 \left(4 c^2-27 c d+119 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}-\frac{4 a^3 \left(4 c^3-27 c^2 d+114 c d^2+165 d^3\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}-\frac{4 a^3 \left(c^2-d^2\right) \left(4 c^3-27 c^2 d+114 c d^2+165 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left(4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^{5/2}}{9 d f}",1,"(a^3*(1 + Sin[e + f*x])^3*(-16*(d^2*(c^3 + 387*c^2*d + 471*c*d^2 + 165*d^3)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d*(c + d*Sin[e + f*x])*((32*c^3 - 216*c^2*d - 3828*c*d^2 - 2910*d^3)*Cos[e + f*x] + 2*d*(5*d*(10*c + 27*d)*Cos[3*(e + f*x)] - (6*c^2 + 432*c*d + 511*d^2 - 35*d^2*Cos[2*(e + f*x)])*Sin[2*(e + f*x)]))))/(1260*d^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Sin[e + f*x]])","A",1
498,1,266,318,2.7225294,"\int (a+a \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]],x]","-\frac{a^3 \left(-2 d \cos (e+f x) \left(16 c^3+d \left(4 c^2-336 c d-565 d^2\right) \sin (e+f x)-84 c^2 d+18 d^2 (2 c+7 d) \cos (2 (e+f x))-556 c d^2+15 d^3 \sin (3 (e+f x))-126 d^3\right)-16 \left(4 c^4-21 c^3 d+61 c^2 d^2+21 c d^3-65 d^4\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+16 \left(4 c^4-17 c^3 d+41 c^2 d^2+209 c d^3+147 d^4\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{420 d^3 f \sqrt{c+d \sin (e+f x)}}","-\frac{4 a^3 \left(4 c^2-21 c d+65 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d^2 f}-\frac{4 a^3 \left(c^2-d^2\right) \left(4 c^2-21 c d+65 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left(4 c^3-21 c^2 d+62 c d^2+147 d^3\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac{2 \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^{3/2}}{7 d f}",1,"-1/420*(a^3*(16*(4*c^4 - 17*c^3*d + 41*c^2*d^2 + 209*c*d^3 + 147*d^4)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - 16*(4*c^4 - 21*c^3*d + 61*c^2*d^2 + 21*c*d^3 - 65*d^4)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - 2*d*Cos[e + f*x]*(16*c^3 - 84*c^2*d - 556*c*d^2 - 126*d^3 + 18*d^2*(2*c + 7*d)*Cos[2*(e + f*x)] + d*(4*c^2 - 336*c*d - 565*d^2)*Sin[e + f*x] + 15*d^3*Sin[3*(e + f*x)])))/(d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",1
499,1,246,258,1.6187121,"\int \frac{(a+a \sin (e+f x))^3}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])^3/Sqrt[c + d*Sin[e + f*x]],x]","-\frac{a^3 (\sin (e+f x)+1)^3 \left(-d \cos (e+f x) \left(8 c^2+2 d (c-15 d) \sin (e+f x)-30 c d+3 d^2 \cos (2 (e+f x))-3 d^2\right)-4 \left(4 c^3-15 c^2 d+26 c d^2-15 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+4 \left(4 c^3-11 c^2 d+12 c d^2+27 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{15 d^3 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6 \sqrt{c+d \sin (e+f x)}}","-\frac{4 a^3 (c-d) \left(4 c^2-11 c d+15 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left(4 c^2-15 c d+27 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) \sqrt{c+d \sin (e+f x)}}{5 d f}",1,"-1/15*(a^3*(1 + Sin[e + f*x])^3*(4*(4*c^3 - 11*c^2*d + 12*c*d^2 + 27*d^3)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - 4*(4*c^3 - 15*c^2*d + 26*c*d^2 - 15*d^3)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - d*Cos[e + f*x]*(8*c^2 - 30*c*d - 3*d^2 + 3*d^2*Cos[2*(e + f*x)] + 2*(c - 15*d)*d*Sin[e + f*x])))/(d^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Sin[e + f*x]])","A",1
500,1,234,270,1.4477847,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(3/2),x]","-\frac{2 a^3 (\sin (e+f x)+1)^3 \left(d \cos (e+f x) \left(4 c^2+d (c+d) \sin (e+f x)-5 c d+3 d^2\right)+2 \left(4 c^3-5 c^2 d-4 c d^2+5 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-2 \left(4 c^3-c^2 d-8 c d^2-3 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{3 d^3 f (c+d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6 \sqrt{c+d \sin (e+f x)}}","-\frac{4 a^3 \left(4 c^2-5 c d-3 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{4 a^3 (4 c-5 d) (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 (2 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d^2 f (c+d)}+\frac{2 (c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{d f (c+d) \sqrt{c+d \sin (e+f x)}}",1,"(-2*a^3*(1 + Sin[e + f*x])^3*(-2*(4*c^3 - c^2*d - 8*c*d^2 - 3*d^3)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + 2*(4*c^3 - 5*c^2*d - 4*c*d^2 + 5*d^3)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d*Cos[e + f*x]*(4*c^2 - 5*c*d + 3*d^2 + d*(c + d)*Sin[e + f*x])))/(3*d^3*(c + d)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Sin[e + f*x]])","A",1
501,1,232,280,1.5480317,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(5/2),x]","-\frac{2 a^3 (\sin (e+f x)+1)^3 \left(d (d-c) \cos (e+f x) \left(4 c^2+d (5 c+9 d) \sin (e+f x)+9 c d+d^2\right)+2 (c+d) \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{3/2} \left(\left(4 c^2+5 c d-3 d^2\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)+d^2 (c+5 d) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)}{3 d^3 f (c+d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6 (c+d \sin (e+f x))^{3/2}}","\frac{4 a^3 \left(4 c^2+5 c d-3 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{4 a^3 (c-d) (4 c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f (c+d) \sqrt{c+d \sin (e+f x)}}+\frac{8 a^3 (c-d) (c+2 d) \cos (e+f x)}{3 d^2 f (c+d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{2 (c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}",1,"(-2*a^3*(1 + Sin[e + f*x])^3*(2*(c + d)*(d^2*(c + 5*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (4*c^2 + 5*c*d - 3*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*((c + d*Sin[e + f*x])/(c + d))^(3/2) + d*(-c + d)*Cos[e + f*x]*(4*c^2 + 9*c*d + d^2 + d*(5*c + 9*d)*Sin[e + f*x])))/(3*d^3*(c + d)^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c + d*Sin[e + f*x])^(3/2))","A",1
502,1,298,336,2.1213539,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(7/2),x]","-\frac{2 a^3 (\sin (e+f x)+1)^3 \left(d \cos (e+f x) \left(4 c^4+15 c^3 d+2 d^2 \left(4 c^2+15 c d+27 d^2\right) \sin ^2(e+f x)+55 c^2 d^2+d \left(9 c^3+45 c^2 d+115 c d^2+15 d^3\right) \sin (e+f x)+15 c d^3+3 d^4\right)-2 (c+d \sin (e+f x))^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(\left(4 c^2+15 c d+27 d^2\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)+d^2 (c-15 d) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)}{15 d^3 f (c+d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6 (c+d \sin (e+f x))^{5/2}}","-\frac{4 a^3 \left(4 c^2+15 c d+27 d^2\right) \cos (e+f x)}{15 d^2 f (c+d)^3 \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left(4 c^2+11 c d+15 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 \left(4 c^2+15 c d+27 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f (c+d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-d) (c+3 d) \cos (e+f x)}{15 d^2 f (c+d)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 (c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}",1,"(-2*a^3*(1 + Sin[e + f*x])^3*(-2*((c - 15*d)*d^2*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (4*c^2 + 15*c*d + 27*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*(c + d*Sin[e + f*x])^2*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d*Cos[e + f*x]*(4*c^4 + 15*c^3*d + 55*c^2*d^2 + 15*c*d^3 + 3*d^4 + d*(9*c^3 + 45*c^2*d + 115*c*d^2 + 15*d^3)*Sin[e + f*x] + 2*d^2*(4*c^2 + 15*c*d + 27*d^2)*Sin[e + f*x]^2)))/(15*d^3*(c + d)^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c + d*Sin[e + f*x])^(5/2))","A",1
503,1,351,419,3.7005579,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(9/2),x]","-\frac{2 a^3 (\sin (e+f x)+1)^3 \left(d \cos (e+f x) \left(2 (c-d) \left(4 c^2+21 c d+65 d^2\right) (c+d) (c+d \sin (e+f x))^2+2 \left(4 c^3+21 c^2 d+62 c d^2-147 d^3\right) (c+d \sin (e+f x))^3-9 (c-d)^2 (3 c+7 d) (c+d)^2 (c+d \sin (e+f x))+15 (c-d)^3 (c+d)^3\right)-2 (c+d \sin (e+f x))^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(d^2 \left(c^2-126 c d+65 d^2\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+\left(4 c^3+21 c^2 d+62 c d^2-147 d^3\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)\right)}{105 d^3 f (c-d) (c+d)^4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6 (c+d \sin (e+f x))^{7/2}}","-\frac{4 a^3 \left(4 c^2+21 c d+65 d^2\right) \cos (e+f x)}{105 d^2 f (c+d)^3 (c+d \sin (e+f x))^{3/2}}+\frac{4 a^3 \left(4 c^2+21 c d+65 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f (c+d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 \left(4 c^3+21 c^2 d+62 c d^2-147 d^3\right) \cos (e+f x)}{105 d^2 f (c-d) (c+d)^4 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 \left(4 c^3+21 c^2 d+62 c d^2-147 d^3\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f (c-d) (c+d)^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-d) (c+4 d) \cos (e+f x)}{35 d^2 f (c+d)^2 (c+d \sin (e+f x))^{5/2}}+\frac{2 (c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}}",1,"(-2*a^3*(1 + Sin[e + f*x])^3*(-2*(d^2*(c^2 - 126*c*d + 65*d^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (4*c^3 + 21*c^2*d + 62*c*d^2 - 147*d^3)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*(c + d*Sin[e + f*x])^3*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d*Cos[e + f*x]*(15*(c - d)^3*(c + d)^3 - 9*(c - d)^2*(c + d)^2*(3*c + 7*d)*(c + d*Sin[e + f*x]) + 2*(c - d)*(c + d)*(4*c^2 + 21*c*d + 65*d^2)*(c + d*Sin[e + f*x])^2 + 2*(4*c^3 + 21*c^2*d + 62*c*d^2 - 147*d^3)*(c + d*Sin[e + f*x])^3)))/(105*(c - d)*d^3*(c + d)^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c + d*Sin[e + f*x])^(7/2))","A",1
504,1,298,246,1.3848362,"\int \frac{(c+d \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx","Integrate[(c + d*Sin[e + f*x])^(5/2)/(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)^2 \left(-d \left(15 c^2-12 c d+5 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+\left(3 c^2-20 c d+9 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)-2 d^2 \cos (e+f x) (c+d \sin (e+f x))-3 (c-d)^2 (c+d \sin (e+f x))+\frac{6 (c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right) (c+d \sin (e+f x))}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}\right)}{3 a f (\sin (e+f x)+1) \sqrt{c+d \sin (e+f x)}}","\frac{(3 c-5 d) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a f \sqrt{c+d \sin (e+f x)}}-\frac{\left(3 c^2-20 c d+9 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}+\frac{d (3 c-5 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a f}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(-3*(c - d)^2*(c + d*Sin[e + f*x]) - 2*d^2*Cos[e + f*x]*(c + d*Sin[e + f*x]) + (6*(c - d)^2*Sin[(e + f*x)/2]*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - d*(15*c^2 - 12*c*d + 5*d^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (3*c^2 - 20*c*d + 9*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(3*a*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])","A",1
505,1,223,186,1.5288788,"\int \frac{(c+d \sin (e+f x))^{3/2}}{a+a \sin (e+f x)} \, dx","Integrate[(c + d*Sin[e + f*x])^(3/2)/(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 (c-d) \sin \left(\frac{1}{2} (e+f x)\right) (c+d \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((c-d) \left((c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+c+d \sin (e+f x)\right)-\left(c^2-2 c d-3 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)}{a f (\sin (e+f x)+1) \sqrt{c+d \sin (e+f x)}}","\frac{\left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{c+d \sin (e+f x)}}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}-\frac{(c-3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*(c - d)*Sin[(e + f*x)/2]*(c + d*Sin[e + f*x]) - (Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-((c^2 - 2*c*d - 3*d^2)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (c - d)*(c + d*Sin[e + f*x] + (c + d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))))/(a*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])","A",1
506,1,201,170,1.1230746,"\int \frac{\sqrt{c+d \sin (e+f x)}}{a+a \sin (e+f x)} \, dx","Integrate[Sqrt[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(2 \sin \left(\frac{1}{2} (e+f x)\right) (c+d \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((c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-(c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+c+d \sin (e+f x)\right)\right)}{a f (\sin (e+f x)+1) \sqrt{c+d \sin (e+f x)}}","-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}+\frac{(c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*Sin[(e + f*x)/2]*(c + d*Sin[e + f*x]) - (Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x] - (c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (c + d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])))/(a*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])","A",1
507,1,210,181,1.1464501,"\int \frac{1}{(a+a \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[1/((a + a*Sin[e + f*x])*Sqrt[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 \sin \left(\frac{1}{2} (e+f x)\right) (c+d \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((c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-(c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+c+d \sin (e+f x)\right)\right)}{a f (c-d) (\sin (e+f x)+1) \sqrt{c+d \sin (e+f x)}}","-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}+\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*Sin[(e + f*x)/2]*(c + d*Sin[e + f*x]) - (Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x] - (c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (c - d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])))/(a*(c - d)*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])","A",1
508,1,264,244,2.0613897,"\int \frac{1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])*(c + d*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)^2 \left(-\left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+\left(c^2+4 c d+3 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+(c+3 d) (c+d \sin (e+f x))-\frac{2 \left((c+d)^2 \cos \left(\frac{1}{2} (e+f x)\right)+d \sin \left(\frac{1}{2} (e+f x)\right) ((c+3 d) \cos (e+f x)+2 (c+d))\right)}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}\right)}{a f (c-d)^2 (c+d) (\sin (e+f x)+1) \sqrt{c+d \sin (e+f x)}}","-\frac{d (c+3 d) \cos (e+f x)}{a f (c-d)^2 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f (c-d) \sqrt{c+d \sin (e+f x)}}-\frac{(c+3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f (c-d)^2 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*((-2*((c + d)^2*Cos[(e + f*x)/2] + d*(2*(c + d) + (c + 3*d)*Cos[e + f*x])*Sin[(e + f*x)/2]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (c + 3*d)*(c + d*Sin[e + f*x]) + (c^2 + 4*c*d + 3*d^2)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - (c^2 - d^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(a*(c - d)^2*(c + d)*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])","A",1
509,1,367,333,4.2625028,"\int \frac{1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])*(c + d*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)^2 \left(2 (c+d \sin (e+f x)) \left(\frac{3 \sin \left(\frac{1}{2} (e+f x)\right)}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}-\frac{\frac{d^2 \cos (e+f x) \left(8 c^2+d (7 c+3 d) \sin (e+f x)+3 c d-d^2\right)}{(c+d \sin (e+f x))^2}+3 c^2+13 c d+6 d^2}{(c+d)^2}\right)+\frac{\left(3 c^2+20 c d+9 d^2\right) (c+d \sin (e+f x))+d \left(15 c^2+12 c d+5 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+\left(3 c^2+20 c d+9 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{(c+d)^2}\right)}{3 a f (c-d)^3 (\sin (e+f x)+1) \sqrt{c+d \sin (e+f x)}}","-\frac{d \left(3 c^2+20 c d+9 d^2\right) \cos (e+f x)}{3 a f (c-d)^3 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left(3 c^2+20 c d+9 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a f (c-d)^3 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{d (3 c+5 d) \cos (e+f x)}{3 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}+\frac{(3 c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a f (c-d)^2 (c+d) \sqrt{c+d \sin (e+f x)}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(((3*c^2 + 20*c*d + 9*d^2)*(c + d*Sin[e + f*x]) + d*(15*c^2 + 12*c*d + 5*d^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (3*c^2 + 20*c*d + 9*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(c + d)^2 + 2*(c + d*Sin[e + f*x])*((3*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - (3*c^2 + 13*c*d + 6*d^2 + (d^2*Cos[e + f*x]*(8*c^2 + 3*c*d - d^2 + d*(7*c + 3*d)*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2)/(c + d)^2)))/(3*a*(c - d)^3*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])","A",1
510,1,310,256,2.6057099,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c + d*Sin[e + f*x])^(5/2)/(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)^4 \left(-\left(c^2+5 c d-6 d^2\right) (c+d \sin (e+f x))+\left(c^2+5 c d-12 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)+d^2 (5 d-11 c) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+\frac{(c-d) (c+d \sin (e+f x)) \left((3 c+11 d) \sin \left(\frac{1}{2} (e+f x)\right)-(c+6 d) \cos \left(\frac{3}{2} (e+f x)\right)+7 d \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)^3}\right)}{3 a^2 f (\sin (e+f x)+1)^2 \sqrt{c+d \sin (e+f x)}}","\frac{(c+5 d) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{c+d \sin (e+f x)}}-\frac{\left(c^2+5 c d-12 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-d) (c+5 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-((c^2 + 5*c*d - 6*d^2)*(c + d*Sin[e + f*x])) + ((c - d)*(7*d*Cos[(e + f*x)/2] - (c + 6*d)*Cos[(3*(e + f*x))/2] + (3*c + 11*d)*Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + d^2*(-11*c + 5*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (c^2 + 5*c*d - 12*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(3*a^2*f*(1 + Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]])","A",1
511,1,283,237,2.7705057,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c + d*Sin[e + f*x])^(3/2)/(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)^4 \left(-2 d^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-(c+3 d) (c+d \sin (e+f x))+\frac{(c+d \sin (e+f x)) \left((3 c+5 d) \sin \left(\frac{1}{2} (e+f x)\right)-(c+3 d) \cos \left(\frac{3}{2} (e+f x)\right)+4 d \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)^3}+(c+3 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)}{3 a^2 f (\sin (e+f x)+1)^2 \sqrt{c+d \sin (e+f x)}}","-\frac{(c+3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (\sin (e+f x)+1)}+\frac{(c+d) (c+2 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{c+d \sin (e+f x)}}-\frac{(c+3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-((c + 3*d)*(c + d*Sin[e + f*x])) + ((4*d*Cos[(e + f*x)/2] - (c + 3*d)*Cos[(3*(e + f*x))/2] + (3*c + 5*d)*Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 2*d^2*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (c + 3*d)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(3*a^2*f*(1 + Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]])","A",1
512,1,256,233,2.5485755,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+a \sin (e+f x))^2} \, dx","Integrate[Sqrt[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)^4 \left(-\left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c (c+d \sin (e+f x))+\frac{(c+d \sin (e+f x)) \left((3 c-d) \sin \left(\frac{1}{2} (e+f x)\right)-c \cos \left(\frac{3}{2} (e+f x)\right)+d \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)^3}+c (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{3 a^2 f (c-d) (\sin (e+f x)+1)^2 \sqrt{c+d \sin (e+f x)}}","-\frac{c \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (c-d) (\sin (e+f x)+1)}+\frac{(c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{c+d \sin (e+f x)}}-\frac{c \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-(c*(c + d*Sin[e + f*x])) + ((d*Cos[(e + f*x)/2] - c*Cos[(3*(e + f*x))/2] + (3*c - d)*Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + c*(c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - (c^2 - d^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(3*a^2*(c - d)*f*(1 + Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]])","A",1
513,1,290,257,2.4718726,"\int \frac{1}{(a+a \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^2*Sqrt[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)^4 \left(-2 d^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-(c-3 d) (c+d \sin (e+f x))-\frac{(c+d \sin (e+f x)) \left((7 d-3 c) \sin \left(\frac{1}{2} (e+f x)\right)+(c-3 d) \cos \left(\frac{3}{2} (e+f x)\right)+2 d \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)^3}+(c-3 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1)^2 \sqrt{c+d \sin (e+f x)}}","-\frac{(c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (c-d)^2 (\sin (e+f x)+1)}+\frac{(c-2 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d) \sqrt{c+d \sin (e+f x)}}-\frac{(c-3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\cos (e+f x) \sqrt{c+d \sin (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])^4*(-((c - 3*d)*(c + d*Sin[e + f*x])) - ((2*d*Cos[(e + f*x)/2] + (c - 3*d)*Cos[(3*(e + f*x))/2] + (-3*c + 7*d)*Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 2*d^2*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (c - 3*d)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]])","A",1
514,1,405,326,4.7526163,"\int \frac{1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^2*(c + d*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)^4 \left(\frac{\left(c^2-5 c d-12 d^2\right) (c+d \sin (e+f x))+\left(c^2-5 c d-12 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)-\left(d^2 (11 c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{c+d}+(c+d \sin (e+f x)) \left(-\frac{2 \left(c^2-5 c d-9 d^2\right)}{c+d}+\frac{6 d^3 \cos (e+f x)}{(c+d) (c+d \sin (e+f x))}+\frac{2 (c-6 d) \sin \left(\frac{1}{2} (e+f x)\right)}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}+\frac{d-c}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{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)^3}\right)\right)}{3 a^2 f (c-d)^3 (\sin (e+f x)+1)^2 \sqrt{c+d \sin (e+f x)}}","-\frac{d \left(c^2-5 c d-12 d^2\right) \cos (e+f x)}{3 a^2 f (c-d)^3 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\left(c^2-5 c d-12 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^3 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-5 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) \sqrt{c+d \sin (e+f x)}}+\frac{(c-5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt{c+d \sin (e+f x)}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*((c + d*Sin[e + f*x])*((-2*(c^2 - 5*c*d - 9*d^2))/(c + d) + (2*(c - d)*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (-c + d)/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (2*(c - 6*d)*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (6*d^3*Cos[e + f*x])/((c + d)*(c + d*Sin[e + f*x]))) + ((c^2 - 5*c*d - 12*d^2)*(c + d*Sin[e + f*x]) - d^2*(11*c + 5*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (c^2 - 5*c*d - 12*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(c + d)))/(3*a^2*(c - d)^3*f*(1 + Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]])","A",1
515,1,674,405,6.6761456,"\int \frac{1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2)),x]","\frac{d \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \left(-\frac{2 \left(26 c^2 d+28 c d^2+10 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{\sqrt{c+d \sin (e+f x)}}+\frac{2 \left(c^3-7 c^2 d-37 c d^2-21 d^3\right) \cos ^2(e+f x) \sqrt{c+d \sin (e+f x)}}{d \left(1-\sin ^2(e+f x)\right)}-\frac{\left(-c^3+7 c^2 d+37 c d^2+21 d^3\right) \left(\frac{2 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{\sqrt{c+d \sin (e+f x)}}-\frac{2 c \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{\sqrt{c+d \sin (e+f x)}}\right)}{d}\right)}{6 f (c-d)^4 (c+d)^2 (a \sin (e+f x)+a)^2}+\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \sqrt{c+d \sin (e+f x)} \left(-\frac{2 \left(c^3-7 c^2 d-27 c d^2-15 d^3\right)}{3 (c-d)^4 (c+d)^2}+\frac{2 d^3 \cos (e+f x)}{3 (c-d)^3 (c+d) (c+d \sin (e+f x))^2}+\frac{4 \left(5 c d^3 \cos (e+f x)+3 d^4 \cos (e+f x)\right)}{3 (c-d)^4 (c+d)^2 (c+d \sin (e+f x))}+\frac{2 \left(c \sin \left(\frac{1}{2} (e+f x)\right)-9 d \sin \left(\frac{1}{2} (e+f x)\right)\right)}{3 (c-d)^4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}-\frac{1}{3 (c-d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{2 \sin \left(\frac{1}{2} (e+f x)\right)}{3 (c-d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}\right)}{f (a \sin (e+f x)+a)^2}","-\frac{d (c+3 d) \left(c^2-10 c d-7 d^2\right) \cos (e+f x)}{3 a^2 f (c-d)^4 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{d \left(c^2-7 c d-10 d^2\right) \cos (e+f x)}{3 a^2 f (c-d)^3 (c+d) (c+d \sin (e+f x))^{3/2}}+\frac{\left(c^2-7 c d-10 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^3 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{(c+3 d) \left(c^2-10 c d-7 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^4 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-7 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*Sqrt[c + d*Sin[e + f*x]]*((-2*(c^3 - 7*c^2*d - 27*c*d^2 - 15*d^3))/(3*(c - d)^4*(c + d)^2) + (2*Sin[(e + f*x)/2])/(3*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3) - 1/(3*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (2*(c*Sin[(e + f*x)/2] - 9*d*Sin[(e + f*x)/2]))/(3*(c - d)^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) + (2*d^3*Cos[e + f*x])/(3*(c - d)^3*(c + d)*(c + d*Sin[e + f*x])^2) + (4*(5*c*d^3*Cos[e + f*x] + 3*d^4*Cos[e + f*x]))/(3*(c - d)^4*(c + d)^2*(c + d*Sin[e + f*x]))))/(f*(a + a*Sin[e + f*x])^2) + (d*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*((-2*(26*c^2*d + 28*c*d^2 + 10*d^3)*EllipticF[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]] + (2*(c^3 - 7*c^2*d - 37*c*d^2 - 21*d^3)*Cos[e + f*x]^2*Sqrt[c + d*Sin[e + f*x]])/(d*(1 - Sin[e + f*x]^2)) - ((-c^3 + 7*c^2*d + 37*c*d^2 + 21*d^3)*((2*(c + d)*EllipticE[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]] - (2*c*EllipticF[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]]))/d))/(6*(c - d)^4*(c + d)^2*f*(a + a*Sin[e + f*x])^2)","A",1
516,1,385,322,5.8133407,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^(5/2)/(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)^6 \left(-\left(4 c^2+15 c d+27 d^2\right) (c+d \sin (e+f x))-\frac{(c+d \sin (e+f x)) \left(\left(20 c^2+74 c d+90 d^2\right) \cos \left(\frac{3}{2} (e+f x)\right)+2 \sin \left(\frac{1}{2} (e+f x)\right) \left(2 \left(2 c^2+7 c d-9 d^2\right) \cos (e+f x)+\left(4 c^2+15 c d+27 d^2\right) \cos (2 (e+f x))-3 \left(6 c^2+11 c d+29 d^2\right)\right)-2 d (35 c+57 d) \cos \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}+\left(4 c^2+15 c d+27 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)+d^2 (c-15 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{30 a^3 f (\sin (e+f x)+1)^3 \sqrt{c+d \sin (e+f x)}}","-\frac{\left(4 c^2+15 c d+27 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{(c+d) \left(4 c^2+11 c d+15 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f \sqrt{c+d \sin (e+f x)}}-\frac{\left(4 c^2+15 c d+27 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}-\frac{2 (c-d) (c+3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a f (a \sin (e+f x)+a)^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-((4*c^2 + 15*c*d + 27*d^2)*(c + d*Sin[e + f*x])) - ((-2*d*(35*c + 57*d)*Cos[(e + f*x)/2] + (20*c^2 + 74*c*d + 90*d^2)*Cos[(3*(e + f*x))/2] + 2*(-3*(6*c^2 + 11*c*d + 29*d^2) + 2*(2*c^2 + 7*c*d - 9*d^2)*Cos[e + f*x] + (4*c^2 + 15*c*d + 27*d^2)*Cos[2*(e + f*x)])*Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + (c - 15*d)*d^2*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (4*c^2 + 15*c*d + 27*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(30*a^3*f*(1 + Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]])","A",1
517,1,441,323,6.2056727,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^(3/2)/(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)^6 \left(\frac{2 (c+d \sin (e+f x)) \left(\frac{\left(4 c^2+5 c d-3 d^2\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}{c-d}+6 (c-d) \sin \left(\frac{1}{2} (e+f x)\right)-2 (c+2 d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+4 (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+3 (d-c) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}-\frac{\left(4 c^2+5 c d-3 d^2\right) (c+d \sin (e+f x))+\left(-4 c^2-5 c d+3 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)-\left(d^2 (c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{c-d}\right)}{30 a^3 f (\sin (e+f x)+1)^3 \sqrt{c+d \sin (e+f x)}}","-\frac{\left(4 c^2+5 c d-3 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 f (c-d) \left(a^3 \sin (e+f x)+a^3\right)}-\frac{\left(4 c^2+5 c d-3 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(c+d) (4 c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f \sqrt{c+d \sin (e+f x)}}-\frac{2 (c+2 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a f (a \sin (e+f x)+a)^2}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*((2*(6*(c - d)*Sin[(e + f*x)/2] + 3*(-c + d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 4*(c + 2*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 2*(c + 2*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + ((4*c^2 + 5*c*d - 3*d^2)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/(c - d))*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - ((4*c^2 + 5*c*d - 3*d^2)*(c + d*Sin[e + f*x]) - d^2*(c + 5*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (-4*c^2 - 5*c*d + 3*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(c - d)))/(30*a^3*f*(1 + Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]])","A",1
518,1,449,334,5.8867171,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx","Integrate[Sqrt[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)^6 \left(-\left(4 c^2-5 c d-3 d^2\right) (c+d \sin (e+f x))+\frac{2 (c+d \sin (e+f x)) \left(\left(4 c^2-5 c d-3 d^2\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+6 (c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right)-(c-d) (2 c-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 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)^2-3 (c-d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}+\left(4 c^2-5 c d-3 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)+d^2 (c-5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{30 a^3 f (c-d)^2 (\sin (e+f x)+1)^3 \sqrt{c+d \sin (e+f x)}}","-\frac{\left(4 c^2-5 c d-3 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 f (c-d)^2 \left(a^3 \sin (e+f x)+a^3\right)}-\frac{\left(4 c^2-5 c d-3 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(4 c-5 d) (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d) \sqrt{c+d \sin (e+f x)}}-\frac{(2 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a f (c-d) (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-((4*c^2 - 5*c*d - 3*d^2)*(c + d*Sin[e + f*x])) + (2*(6*(c - d)^2*Sin[(e + f*x)/2] - 3*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(c - d)*(2*c - d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - (c - d)*(2*c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (4*c^2 - 5*c*d - 3*d^2)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + (c - 5*d)*d^2*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (4*c^2 - 5*c*d - 3*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(30*a^3*(c - d)^2*f*(1 + Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]])","A",1
519,1,445,344,6.3190882,"\int \frac{1}{(a+a \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^3*Sqrt[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)^6 \left(-\left(4 c^2-15 c d+27 d^2\right) (c+d \sin (e+f x))+\frac{2 (c+d \sin (e+f x)) \left(\left(4 c^2-15 c d+27 d^2\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+6 (c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right)-2 (c-3 d) (c-d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+4 (c-3 d) (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)^2-3 (c-d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}+\left(4 c^2-15 c d+27 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)+d^2 (c+15 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{30 a^3 f (c-d)^3 (\sin (e+f x)+1)^3 \sqrt{c+d \sin (e+f x)}}","-\frac{\left(4 c^2-15 c d+27 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right)}+\frac{\left(4 c^2-11 c d+15 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left(4 c^2-15 c d+27 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x) \sqrt{c+d \sin (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])^6*(-((4*c^2 - 15*c*d + 27*d^2)*(c + d*Sin[e + f*x])) + (2*(6*(c - d)^2*Sin[(e + f*x)/2] - 3*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 4*(c - 3*d)*(c - d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 2*(c - 3*d)*(c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (4*c^2 - 15*c*d + 27*d^2)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + d^2*(c + 15*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (4*c^2 - 15*c*d + 27*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(30*a^3*(c - d)^3*f*(1 + Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]])","A",1
520,1,745,423,6.5439522,"\int \frac{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2)),x]","\frac{d \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6 \left(-\frac{2 \left(-c^2 d-126 c d^2-65 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{\sqrt{c+d \sin (e+f x)}}+\frac{2 \left(4 c^3-21 c^2 d+62 c d^2+147 d^3\right) \cos ^2(e+f x) \sqrt{c+d \sin (e+f x)}}{d \left(1-\sin ^2(e+f x)\right)}-\frac{\left(-4 c^3+21 c^2 d-62 c d^2-147 d^3\right) \left(\frac{2 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{\sqrt{c+d \sin (e+f x)}}-\frac{2 c \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{\sqrt{c+d \sin (e+f x)}}\right)}{d}\right)}{60 f (c-d)^4 (c+d) (a \sin (e+f x)+a)^3}+\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6 \sqrt{c+d \sin (e+f x)} \left(\frac{4 c^2 \sin \left(\frac{1}{2} (e+f x)\right)-25 c d \sin \left(\frac{1}{2} (e+f x)\right)+87 d^2 \sin \left(\frac{1}{2} (e+f x)\right)}{15 (c-d)^4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}-\frac{4 c^3-21 c^2 d+62 c d^2+117 d^3}{15 (c-d)^4 (c+d)}-\frac{2 d^4 \cos (e+f x)}{(c-d)^4 (c+d) (c+d \sin (e+f x))}+\frac{2 \left(2 c \sin \left(\frac{1}{2} (e+f x)\right)-11 d \sin \left(\frac{1}{2} (e+f x)\right)\right)}{15 (c-d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{11 d-2 c}{15 (c-d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{1}{5 (c-d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{2 \sin \left(\frac{1}{2} (e+f x)\right)}{5 (c-d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}\right)}{f (a \sin (e+f x)+a)^3}","-\frac{\left(4 c^2-21 c d+65 d^2\right) \cos (e+f x)}{30 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right) \sqrt{c+d \sin (e+f x)}}+\frac{\left(4 c^2-21 c d+65 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{d \left(4 c^3-21 c^2 d+62 c d^2+147 d^3\right) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\left(4 c^3-21 c^2 d+62 c d^2+147 d^3\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^4 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-4 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt{c+d \sin (e+f x)}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Sin[e + f*x]]*(-1/15*(4*c^3 - 21*c^2*d + 62*c*d^2 + 117*d^3)/((c - d)^4*(c + d)) + (2*Sin[(e + f*x)/2])/(5*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) - 1/(5*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (-2*c + 11*d)/(15*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (2*(2*c*Sin[(e + f*x)/2] - 11*d*Sin[(e + f*x)/2]))/(15*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3) + (4*c^2*Sin[(e + f*x)/2] - 25*c*d*Sin[(e + f*x)/2] + 87*d^2*Sin[(e + f*x)/2])/(15*(c - d)^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) - (2*d^4*Cos[e + f*x])/((c - d)^4*(c + d)*(c + d*Sin[e + f*x]))))/(f*(a + a*Sin[e + f*x])^3) + (d*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*((-2*(-(c^2*d) - 126*c*d^2 - 65*d^3)*EllipticF[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]] + (2*(4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*Cos[e + f*x]^2*Sqrt[c + d*Sin[e + f*x]])/(d*(1 - Sin[e + f*x]^2)) - ((-4*c^3 + 21*c^2*d - 62*c*d^2 - 147*d^3)*((2*(c + d)*EllipticE[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]] - (2*c*EllipticF[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]]))/d))/(60*(c - d)^4*(c + d)*f*(a + a*Sin[e + f*x])^3)","A",1
521,1,828,518,6.9214474,"\int \frac{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2)),x]","\frac{\sqrt{c+d \sin (e+f x)} \left(-\frac{2 \cos (e+f x) d^4}{3 (c-d)^4 (c+d) (c+d \sin (e+f x))^2}-\frac{4 c^4-27 d c^3+111 d^2 c^2+449 d^3 c+267 d^4}{15 (c-d)^5 (c+d)^2}+\frac{4 \left(c \sin \left(\frac{1}{2} (e+f x)\right)-8 d \sin \left(\frac{1}{2} (e+f x)\right)\right)}{15 (c-d)^4 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{4 \sin \left(\frac{1}{2} (e+f x)\right) c^2-35 d \sin \left(\frac{1}{2} (e+f x)\right) c+177 d^2 \sin \left(\frac{1}{2} (e+f x)\right)}{15 (c-d)^5 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}-\frac{2 \left(9 \cos (e+f x) d^5+13 c \cos (e+f x) d^4\right)}{3 (c-d)^5 (c+d)^2 (c+d \sin (e+f x))}-\frac{2 (c-8 d)}{15 (c-d)^4 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{1}{5 (c-d)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{2 \sin \left(\frac{1}{2} (e+f x)\right)}{5 (c-d)^3 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}{f (\sin (e+f x) a+a)^3}+\frac{d \left(\frac{2 \left(4 c^4-27 d c^3+111 d^2 c^2+579 d^3 c+357 d^4\right) \sqrt{c+d \sin (e+f x)} \cos ^2(e+f x)}{d \left(1-\sin ^2(e+f x)\right)}-\frac{\left(-4 c^4+27 d c^3-111 d^2 c^2-579 d^3 c-357 d^4\right) \left(\frac{2 (c+d) E\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{\sqrt{c+d \sin (e+f x)}}-\frac{2 c F\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{\sqrt{c+d \sin (e+f x)}}\right)}{d}-\frac{2 \left(-165 d^4-471 c d^3-387 c^2 d^2-c^3 d\right) F\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{\sqrt{c+d \sin (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}{60 (c-d)^5 (c+d)^2 f (\sin (e+f x) a+a)^3}","-\frac{\left(4 c^2-27 c d+119 d^2\right) \cos (e+f x)}{30 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^{3/2}}-\frac{d \left(4 c^3-27 c^2 d+114 c d^2+165 d^3\right) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^{3/2}}+\frac{\left(4 c^3-27 c^2 d+114 c d^2+165 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^4 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{d \left(4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left(4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^5 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Sin[e + f*x]]*(-1/15*(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 449*c*d^3 + 267*d^4)/((c - d)^5*(c + d)^2) + (2*Sin[(e + f*x)/2])/(5*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) - 1/(5*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - (2*(c - 8*d))/(15*(c - d)^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (4*(c*Sin[(e + f*x)/2] - 8*d*Sin[(e + f*x)/2]))/(15*(c - d)^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3) + (4*c^2*Sin[(e + f*x)/2] - 35*c*d*Sin[(e + f*x)/2] + 177*d^2*Sin[(e + f*x)/2])/(15*(c - d)^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) - (2*d^4*Cos[e + f*x])/(3*(c - d)^4*(c + d)*(c + d*Sin[e + f*x])^2) - (2*(13*c*d^4*Cos[e + f*x] + 9*d^5*Cos[e + f*x]))/(3*(c - d)^5*(c + d)^2*(c + d*Sin[e + f*x]))))/(f*(a + a*Sin[e + f*x])^3) + (d*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*((-2*(-(c^3*d) - 387*c^2*d^2 - 471*c*d^3 - 165*d^4)*EllipticF[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]] + (2*(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*Cos[e + f*x]^2*Sqrt[c + d*Sin[e + f*x]])/(d*(1 - Sin[e + f*x]^2)) - ((-4*c^4 + 27*c^3*d - 111*c^2*d^2 - 579*c*d^3 - 357*d^4)*((2*(c + d)*EllipticE[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]] - (2*c*EllipticF[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]]))/d))/(60*(c - d)^5*(c + d)^2*f*(a + a*Sin[e + f*x])^3)","A",1
522,1,146,161,0.519358,"\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \, dx","Integrate[Sqrt[a + a*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(140 c^3+d \left(140 c^2+112 c d+47 d^2\right) \sin (e+f x)+280 c^2 d-6 d^2 (7 c+2 d) \cos (2 (e+f x))+266 c d^2-5 d^3 \sin (3 (e+f x))+76 d^3\right)}{70 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) \cos (e+f x)}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{12 d^2 (c+d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 a f}-\frac{2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt{a \sin (e+f x)+a}}-\frac{8 d (5 c-d) (c+d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{35 f}",1,"-1/70*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(140*c^3 + 280*c^2*d + 266*c*d^2 + 76*d^3 - 6*d^2*(7*c + 2*d)*Cos[2*(e + f*x)] + d*(140*c^2 + 112*c*d + 47*d^2)*Sin[e + f*x] - 5*d^3*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
523,1,111,112,0.2907397,"\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx","Integrate[Sqrt[a + a*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(30 c^2+4 d (5 c+2 d) \sin (e+f x)+40 c d-3 d^2 \cos (2 (e+f x))+19 d^2\right)}{15 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) \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{4 d (5 c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{2 d^2 \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*c^2 + 40*c*d + 19*d^2 - 3*d^2*Cos[2*(e + f*x)] + 4*d*(5*c + 2*d)*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
524,1,82,62,0.1237598,"\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x)) \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*(c + d*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 c+d \sin (e+f x)+2 d)}{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 c+d) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 d \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*c + 2*d + d*Sin[e + f*x]))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
525,1,65,26,0.036153,"\int \sqrt{a+a \sin (e+f x)} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]],x]","\frac{2 \sqrt{a (\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)}{f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{2 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}",1,"(2*(-Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])])/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","B",1
526,1,657,61,5.4853252,"\int \frac{\sqrt{a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x]),x]","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right) \sqrt{a (\sin (e+f x)+1)} \left(\text{RootSum}\left[\text{$\#$1}^4 d e^{2 i e}+2 i \text{$\#$1}^2 c e^{i e}-d\&,\frac{\text{$\#$1}^3 \left(-\sqrt{d}\right) e^{i e} f x \sqrt{c+d}-2 i \text{$\#$1}^3 \sqrt{d} e^{i e} \sqrt{c+d} \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)+\frac{(1-i) \text{$\#$1}^2 c f x}{\sqrt{e^{-i e}}}+\frac{(2+2 i) \text{$\#$1}^2 c \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)}{\sqrt{e^{-i e}}}-i \text{$\#$1} \sqrt{d} f x \sqrt{c+d}+2 \text{$\#$1} \sqrt{d} \sqrt{c+d} \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)-(2-2 i) d \sqrt{e^{-i e}} \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)+(1+i) d \sqrt{e^{-i e}} f x}{d-i \text{$\#$1}^2 c e^{i e}}\&\right]-i \text{RootSum}\left[\text{$\#$1}^4 d e^{2 i e}+2 i \text{$\#$1}^2 c e^{i e}-d\&,\frac{-i \text{$\#$1}^3 \sqrt{d} e^{i e} f x \sqrt{c+d}+2 \text{$\#$1}^3 \sqrt{d} e^{i e} \sqrt{c+d} \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)-\frac{(1+i) \text{$\#$1}^2 c f x}{\sqrt{e^{-i e}}}+\frac{(2-2 i) \text{$\#$1}^2 c \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)}{\sqrt{e^{-i e}}}+\text{$\#$1} \sqrt{d} f x \sqrt{c+d}+2 i \text{$\#$1} \sqrt{d} \sqrt{c+d} \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)+(2+2 i) d \sqrt{e^{-i e}} \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)+(1-i) d \sqrt{e^{-i e}} f x}{d-i \text{$\#$1}^2 c e^{i e}}\&\right]\right)}{\sqrt{d} f \sqrt{c+d} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{d} f \sqrt{c+d}}",1,"((1/8 + I/8)*(RootSum[-d + (2*I)*c*E^(I*e)*#1^2 + d*E^((2*I)*e)*#1^4 & , ((1 + I)*d*Sqrt[E^((-I)*e)]*f*x - (2 - 2*I)*d*Sqrt[E^((-I)*e)]*Log[E^((I/2)*f*x) - #1] - I*Sqrt[d]*Sqrt[c + d]*f*x*#1 + 2*Sqrt[d]*Sqrt[c + d]*Log[E^((I/2)*f*x) - #1]*#1 + ((1 - I)*c*f*x*#1^2)/Sqrt[E^((-I)*e)] + ((2 + 2*I)*c*Log[E^((I/2)*f*x) - #1]*#1^2)/Sqrt[E^((-I)*e)] - Sqrt[d]*Sqrt[c + d]*E^(I*e)*f*x*#1^3 - (2*I)*Sqrt[d]*Sqrt[c + d]*E^(I*e)*Log[E^((I/2)*f*x) - #1]*#1^3)/(d - I*c*E^(I*e)*#1^2) & ] - I*RootSum[-d + (2*I)*c*E^(I*e)*#1^2 + d*E^((2*I)*e)*#1^4 & , ((1 - I)*d*Sqrt[E^((-I)*e)]*f*x + (2 + 2*I)*d*Sqrt[E^((-I)*e)]*Log[E^((I/2)*f*x) - #1] + Sqrt[d]*Sqrt[c + d]*f*x*#1 + (2*I)*Sqrt[d]*Sqrt[c + d]*Log[E^((I/2)*f*x) - #1]*#1 - ((1 + I)*c*f*x*#1^2)/Sqrt[E^((-I)*e)] + ((2 - 2*I)*c*Log[E^((I/2)*f*x) - #1]*#1^2)/Sqrt[E^((-I)*e)] - I*Sqrt[d]*Sqrt[c + d]*E^(I*e)*f*x*#1^3 + 2*Sqrt[d]*Sqrt[c + d]*E^(I*e)*Log[E^((I/2)*f*x) - #1]*#1^3)/(d - I*c*E^(I*e)*#1^2) & ])*(Cos[e/2] + I*Sin[e/2])*Sqrt[a*(1 + Sin[e + f*x])])/(Sqrt[d]*Sqrt[c + d]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","C",1
527,1,871,105,6.0177676,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^2} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x])^2,x]","\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{a (\sin (e+f x)+1)} \left(\frac{\left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right) \left((-1+i) x \cos (e)+(1+i) x \sin (e)+\frac{\text{RootSum}\left[d e^{2 i e} \text{$\#$1}^4+2 i c e^{i e} \text{$\#$1}^2-d\&,\frac{-\sqrt{d} \sqrt{c+d} e^{i e} f x \text{$\#$1}^3-2 i \sqrt{d} \sqrt{c+d} e^{i e} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^3+\frac{(1-i) c f x \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\frac{(2+2 i) c \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^2}{\sqrt{e^{-i e}}}-i \sqrt{d} \sqrt{c+d} f x \text{$\#$1}+2 \sqrt{d} \sqrt{c+d} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}+(1+i) d \sqrt{e^{-i e}} f x-(2-2 i) d \sqrt{e^{-i e}} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right)}{d-i c e^{i e} \text{$\#$1}^2}\&\right] (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}{4 f}\right)}{\sqrt{d} (c+d)^{3/2} (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}+\frac{\left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right) \left((1-i) x \cos (e)-(1+i) x \sin (e)+\frac{\text{RootSum}\left[d e^{2 i e} \text{$\#$1}^4+2 i c e^{i e} \text{$\#$1}^2-d\&,\frac{-i \sqrt{d} \sqrt{c+d} e^{i e} f x \text{$\#$1}^3+2 \sqrt{d} \sqrt{c+d} e^{i e} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^3-\frac{(1+i) c f x \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\frac{(2-2 i) c \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\sqrt{d} \sqrt{c+d} f x \text{$\#$1}+2 i \sqrt{d} \sqrt{c+d} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}+(1-i) d \sqrt{e^{-i e}} f x+(2+2 i) d \sqrt{e^{-i e}} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right)}{d-i c e^{i e} \text{$\#$1}^2}\&\right] \sqrt{\cos (e)-i \sin (e)} (-i \cos (e)+\sin (e)-1)}{4 f}\right)}{\sqrt{d} (c+d)^{3/2} (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}-\frac{(2-2 i) \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)}{\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)}","-\frac{a \cos (e+f x)}{f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{d} f (c+d)^{3/2}}",1,"((1/4 + I/4)*Sqrt[a*(1 + Sin[e + f*x])]*(((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[d]*(c + d)^(3/2)*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]]) + ((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[d]*(c + d)^(3/2)*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]]) - ((2 - 2*I)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c + d)*f*(c + d*Sin[e + f*x]))))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])","C",1
528,1,920,154,7.5457436,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^3} \, dx","Integrate[Sqrt[a + a*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 \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{$\#$1}^4+2 i c e^{i e} \text{$\#$1}^2-d\&,\frac{-\sqrt{d} \sqrt{c+d} e^{i e} f x \text{$\#$1}^3-2 i \sqrt{d} \sqrt{c+d} e^{i e} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^3+\frac{(1-i) c f x \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\frac{(2+2 i) c \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^2}{\sqrt{e^{-i e}}}-i \sqrt{d} \sqrt{c+d} f x \text{$\#$1}+2 \sqrt{d} \sqrt{c+d} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}+(1+i) d \sqrt{e^{-i e}} f x-(2-2 i) d \sqrt{e^{-i e}} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right)}{d-i c e^{i e} \text{$\#$1}^2}\&\right] (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}{4 f}\right)}{\sqrt{d} (c+d)^{5/2} (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}+\frac{3 \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right) \left((1-i) x \cos (e)-(1+i) x \sin (e)+\frac{\text{RootSum}\left[d e^{2 i e} \text{$\#$1}^4+2 i c e^{i e} \text{$\#$1}^2-d\&,\frac{-i \sqrt{d} \sqrt{c+d} e^{i e} f x \text{$\#$1}^3+2 \sqrt{d} \sqrt{c+d} e^{i e} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^3-\frac{(1+i) c f x \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\frac{(2-2 i) c \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\sqrt{d} \sqrt{c+d} f x \text{$\#$1}+2 i \sqrt{d} \sqrt{c+d} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right) \text{$\#$1}+(1-i) d \sqrt{e^{-i e}} f x+(2+2 i) d \sqrt{e^{-i e}} \log \left(e^{\frac{i f x}{2}}-\text{$\#$1}\right)}{d-i c e^{i e} \text{$\#$1}^2}\&\right] \sqrt{\cos (e)-i \sin (e)} (-i \cos (e)+\sin (e)-1)}{4 f}\right)}{\sqrt{d} (c+d)^{5/2} (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}-\frac{(6-6 i) \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) \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)}{\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)}","-\frac{3 a \cos (e+f x)}{4 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{a \cos (e+f x)}{2 f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}-\frac{3 \sqrt{a} \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{d} f (c+d)^{5/2}}",1,"((1/16 + I/16)*Sqrt[a*(1 + Sin[e + f*x])]*((3*(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[d]*(c + d)^(5/2)*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]]) + (3*(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[d]*(c + d)^(5/2)*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]]) - ((4 - 4*I)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c + d)*f*(c + d*Sin[e + f*x])^2) - ((6 - 6*I)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c + d)^2*f*(c + d*Sin[e + f*x]))))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])","C",1
529,1,203,231,1.6928969,"\int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(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(840 c^3 \sin (e+f x)+4200 c^3-4 d \left(189 c^2+351 c d+137 d^2\right) \cos (2 (e+f x))+4536 c^2 d \sin (e+f x)+9828 c^2 d+4554 c d^2 \sin (e+f x)-270 c d^2 \sin (3 (e+f x))+8892 c d^2+1598 d^3 \sin (e+f x)-170 d^3 \sin (3 (e+f x))+35 d^3 \cos (4 (e+f x))+2689 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^2 (c-17 d) (c+d) \left(15 c^2+10 c d+7 d^2\right) \cos (e+f x)}{315 d f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (c-17 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-17 d) (c+d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}+\frac{8 a (c-17 d) (5 c-d) (c+d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 f}",1,"-1/1260*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(4200*c^3 + 9828*c^2*d + 8892*c*d^2 + 2689*d^3 - 4*d*(189*c^2 + 351*c*d + 137*d^2)*Cos[2*(e + f*x)] + 35*d^3*Cos[4*(e + f*x)] + 840*c^3*Sin[e + f*x] + 4536*c^2*d*Sin[e + f*x] + 4554*c*d^2*Sin[e + f*x] + 1598*d^3*Sin[e + f*x] - 270*c*d^2*Sin[3*(e + f*x)] - 170*d^3*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
530,1,136,157,0.8760803,"\int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(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(\left(140 c^2+504 c d+253 d^2\right) \sin (e+f x)+700 c^2-6 d (14 c+13 d) \cos (2 (e+f x))+1092 c d-15 d^2 \sin (3 (e+f x))+494 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{8 a^2 \left(35 c^2+42 c d+19 d^2\right) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a \left(35 c^2+42 c d+19 d^2\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{4 d (7 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac{2 d^2 \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*c^2 + 1092*c*d + 494*d^2 - 6*d*(14*c + 13*d)*Cos[2*(e + f*x)] + (140*c^2 + 504*c*d + 253*d^2)*Sin[e + f*x] - 15*d^2*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
531,1,101,101,0.3948062,"\int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(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) (2 (5 c+9 d) \sin (e+f x)+50 c-3 d \cos (2 (e+f x))+39 d)}{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 c+3 d) \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a (5 c+3 d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{2 d \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*c + 39*d - 3*d*Cos[2*(e + f*x)] + 2*(5*c + 9*d)*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
532,1,89,59,0.1415335,"\int (a+a \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2),x]","-\frac{(a (\sin (e+f x)+1))^{3/2} \left(-9 \sin \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{3}{2} (e+f x)\right)+9 \cos \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{3}{2} (e+f x)\right)\right)}{3 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}",1,"-1/3*((a*(1 + Sin[e + f*x]))^(3/2)*(9*Cos[(e + f*x)/2] + Cos[(3*(e + f*x))/2] - 9*Sin[(e + f*x)/2] + Sin[(3*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
533,1,233,98,2.2928067,"\int \frac{(a+a \sin (e+f x))^{3/2}}{c+d \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x]),x]","\frac{(a (\sin (e+f x)+1))^{3/2} \left(2 \sqrt{d} \sqrt{c+d} \sin \left(\frac{1}{2} (e+f x)\right)-2 \sqrt{d} \sqrt{c+d} \cos \left(\frac{1}{2} (e+f x)\right)+(c-d) \left(\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)-\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)\right)\right)}{d^{3/2} f \sqrt{c+d} \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) \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^2 \cos (e+f x)}{d f \sqrt{a \sin (e+f x)+a}}",1,"((-2*Sqrt[d]*Sqrt[c + d]*Cos[(e + f*x)/2] + (c - d)*(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]))] - 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)]) + 2*Sqrt[d]*Sqrt[c + d]*Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2))/(d^(3/2)*Sqrt[c + d]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","B",1
534,1,268,119,2.3511611,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^2} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^2,x]","-\frac{(a (\sin (e+f x)+1))^{3/2} \left(2 \sqrt{d} (c-d) \sqrt{c+d} \sin \left(\frac{1}{2} (e+f x)\right)-2 \sqrt{d} (c-d) \sqrt{c+d} \cos \left(\frac{1}{2} (e+f x)\right)+(c+3 d) (c+d \sin (e+f x)) \left(\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)-\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)\right)\right)}{2 d^{3/2} f (c+d)^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 (c+d \sin (e+f x))}","\frac{a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{a^{3/2} (c+3 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,"-1/2*((a*(1 + Sin[e + f*x]))^(3/2)*(-2*(c - d)*Sqrt[d]*Sqrt[c + d]*Cos[(e + f*x)/2] + 2*(c - d)*Sqrt[d]*Sqrt[c + d]*Sin[(e + f*x)/2] + (c + 3*d)*(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]))] - 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*Sin[e + f*x])))/(d^(3/2)*(c + d)^(3/2)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c + d*Sin[e + f*x]))","B",1
535,1,313,179,4.0607426,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^3} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^3,x]","\frac{(a (\sin (e+f x)+1))^{3/2} \left(\frac{4 \sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right) \left(-c^2+d (c+7 d) \sin (e+f x)+7 c d+2 d^2\right)}{(c+d)^2 (c+d \sin (e+f x))^2}-\frac{4 \sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right) \left(-c^2+d (c+7 d) \sin (e+f x)+7 c d+2 d^2\right)}{(c+d)^2 (c+d \sin (e+f x))^2}-\frac{2 (c+7 d) \left(\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)-\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)\right)}{(c+d)^{5/2}}\right)}{16 d^{3/2} 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} (c+7 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^2 (c+7 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^2 (c-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,"((a*(1 + Sin[e + f*x]))^(3/2)*((-2*(c + 7*d)*(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]))] - 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]*(-c^2 + 7*c*d + 2*d^2 + d*(c + 7*d)*Sin[e + f*x]))/((c + d)^2*(c + d*Sin[e + f*x])^2) + (4*Sqrt[d]*Sin[(e + f*x)/2]*(-c^2 + 7*c*d + 2*d^2 + d*(c + 7*d)*Sin[e + f*x]))/((c + d)^2*(c + d*Sin[e + f*x])^2)))/(16*d^(3/2)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
536,1,246,328,6.2669227,"\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3,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(51744 c^3 \sin (e+f x)+164472 c^3+199980 c^2 d \sin (e+f x)-5940 c^2 d \sin (3 (e+f x))+411840 c^2 d-8 \left(693 c^3+5940 c^2 d+8382 c d^2+3250 d^3\right) \cos (2 (e+f x))+205656 c d^2 \sin (e+f x)-17160 c d^2 \sin (3 (e+f x))+70 d^2 (33 c+32 d) \cos (4 (e+f x))+373098 c d^2+69890 d^3 \sin (e+f x)-8675 d^3 \sin (3 (e+f x))+315 d^3 \sin (5 (e+f x))+114640 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^3 \left(3 c^2-38 c d+355 d^2\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^3 (c+d) \left(15 c^2+10 c d+7 d^2\right) \left(3 c^2-38 c d+355 d^2\right) \cos (e+f x)}{3465 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{8 a^2 (5 c-d) (c+d) \left(3 c^2-38 c d+355 d^2\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3465 d f}-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}-\frac{4 a (c+d) \left(3 c^2-38 c d+355 d^2\right) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 f}",1,"-1/27720*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(164472*c^3 + 411840*c^2*d + 373098*c*d^2 + 114640*d^3 - 8*(693*c^3 + 5940*c^2*d + 8382*c*d^2 + 3250*d^3)*Cos[2*(e + f*x)] + 70*d^2*(33*c + 32*d)*Cos[4*(e + f*x)] + 51744*c^3*Sin[e + f*x] + 199980*c^2*d*Sin[e + f*x] + 205656*c*d^2*Sin[e + f*x] + 69890*d^3*Sin[e + f*x] - 5940*c^2*d*Sin[3*(e + f*x)] - 17160*c*d^2*Sin[3*(e + f*x)] - 8675*d^3*Sin[3*(e + f*x)] + 315*d^3*Sin[5*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
537,1,180,202,3.2990078,"\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^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(-4 \left(63 c^2+360 c d+254 d^2\right) \cos (2 (e+f x))+2352 c^2 \sin (e+f x)+7476 c^2+6060 c d \sin (e+f x)-180 c d \sin (3 (e+f x))+12480 c d+3116 d^2 \sin (e+f x)-260 d^2 \sin (3 (e+f x))+35 d^2 \cos (4 (e+f x))+5653 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{64 a^3 \left(21 c^2+30 c d+13 d^2\right) \cos (e+f x)}{315 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 \left(21 c^2+30 c d+13 d^2\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 f}-\frac{2 a \left(21 c^2+30 c d+13 d^2\right) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}-\frac{4 d (9 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{63 f}-\frac{2 d^2 \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*c^2 + 12480*c*d + 5653*d^2 - 4*(63*c^2 + 360*c*d + 254*d^2)*Cos[2*(e + f*x)] + 35*d^2*Cos[4*(e + f*x)] + 2352*c^2*Sin[e + f*x] + 6060*c*d*Sin[e + f*x] + 3116*d^2*Sin[e + f*x] - 180*c*d*Sin[3*(e + f*x)] - 260*d^2*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
538,1,119,138,1.4994405,"\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(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) ((392 c+505 d) \sin (e+f x)-6 (7 c+20 d) \cos (2 (e+f x))+1246 c-15 d \sin (3 (e+f x))+1040 d)}{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 c+5 d) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 (7 c+5 d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{2 a (7 c+5 d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac{2 d \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*c + 1040*d - 6*(7*c + 20*d)*Cos[2*(e + f*x)] + (392*c + 505*d)*Sin[e + f*x] - 15*d*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
539,1,117,89,0.3033186,"\int (a+a \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2),x]","-\frac{(a (\sin (e+f x)+1))^{5/2} \left(-150 \sin \left(\frac{1}{2} (e+f x)\right)+25 \sin \left(\frac{3}{2} (e+f x)\right)+3 \sin \left(\frac{5}{2} (e+f x)\right)+150 \cos \left(\frac{1}{2} (e+f x)\right)+25 \cos \left(\frac{3}{2} (e+f x)\right)-3 \cos \left(\frac{5}{2} (e+f x)\right)\right)}{30 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\frac{64 a^3 \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}",1,"-1/30*((a*(1 + Sin[e + f*x]))^(5/2)*(150*Cos[(e + f*x)/2] + 25*Cos[(3*(e + f*x))/2] - 3*Cos[(5*(e + f*x))/2] - 150*Sin[(e + f*x)/2] + 25*Sin[(3*(e + f*x))/2] + 3*Sin[(5*(e + f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
540,1,330,142,3.5967191,"\int \frac{(a+a \sin (e+f x))^{5/2}}{c+d \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x]),x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(6 \sqrt{d} (5 d-2 c) \sin \left(\frac{1}{2} (e+f x)\right)+6 \sqrt{d} (2 c-5 d) \cos \left(\frac{1}{2} (e+f x)\right)+\frac{3 (c-d)^2 \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)^2 \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 d^{3/2} \sin \left(\frac{3}{2} (e+f x)\right)-2 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)^5}","-\frac{2 a^{5/2} (c-d)^2 \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^3 (3 c-7 d) \cos (e+f x)}{3 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 d f}",1,"((a*(1 + Sin[e + f*x]))^(5/2)*(6*(2*c - 5*d)*Sqrt[d]*Cos[(e + f*x)/2] - 2*d^(3/2)*Cos[(3*(e + f*x))/2] - (3*(c - 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]))]))/Sqrt[c + d] + (3*(c - 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)]))/Sqrt[c + d] + 6*Sqrt[d]*(-2*c + 5*d)*Sin[(e + f*x)/2] - 2*d^(3/2)*Sin[(3*(e + f*x))/2]))/(6*d^(5/2)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","B",1
541,1,350,166,4.0862147,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^2} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^2,x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(\frac{\left(-3 c^2-2 c d+5 d^2\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(3 c^2+2 c d-5 d^2\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} (c-d)^2 \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 \sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)-8 \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)^5}","\frac{a^{5/2} (c-d) (3 c+5 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 (c+d)^{3/2}}-\frac{a^3 (3 c+d) \cos (e+f x)}{d^2 f (c+d) \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-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]))^(5/2)*(-8*Sqrt[d]*Cos[(e + f*x)/2] + ((3*c^2 + 2*c*d - 5*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^2 - 2*c*d + 5*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*Sqrt[d]*Sin[(e + f*x)/2] - (4*(c - d)^2*Sqrt[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])^5)","B",1
542,1,379,194,4.9171887,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^3} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^3,x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(-\frac{4 \sqrt{d} \left(-5 c^2-6 c d+11 d^2\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(3 c^2+10 c d+19 d^2\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(3 c^2+10 c d+19 d^2\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} (c-d)^2 \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)^5}","-\frac{a^{5/2} \left(3 c^2+10 c d+19 d^2\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{3 a^3 (c-d) (c+3 d) \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^2 (c-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]))^(5/2)*(-(((3*c^2 + 10*c*d + 19*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)) + ((3*c^2 + 10*c*d + 19*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*(c - d)^2*Sqrt[d]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c + d)*(c + d*Sin[e + f*x])^2) - (4*Sqrt[d]*(-5*c^2 - 6*c*d + 11*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])^5)","A",0
543,1,155,178,0.5913741,"\int \frac{(c+d \sin (e+f x))^3}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(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(-2 d \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-90 c^2-2 d (15 c-d) \sin (e+f x)+30 c d+3 d^2 \cos (2 (e+f x))-29 d^2\right)+(-60-60 i) (-1)^{3/4} (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)\right)}{30 f \sqrt{a (\sin (e+f x)+1)}}","-\frac{4 d \left(21 c^2-12 c d+7 d^2\right) \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{2 d^2 (9 c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 a f}-\frac{2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{2} (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}",1,"-1/30*((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((-60 - 60*I)*(-1)^(3/4)*(c - d)^3*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] - 2*d*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(-90*c^2 + 30*c*d - 29*d^2 + 3*d^2*Cos[2*(e + f*x)] - 2*(15*c - d)*d*Sin[e + f*x])))/(f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
544,1,125,123,0.3881129,"\int \frac{(c+d \sin (e+f x))^2}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(c + d*Sin[e + f*x])^2/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(d \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (6 c+d \sin (e+f x)-d)-(3+3 i) (-1)^{3/4} (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)\right)}{3 f \sqrt{a (\sin (e+f x)+1)}}","-\frac{4 d (3 c-d) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{2} (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 d^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 a f}",1,"(-2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((-3 - 3*I)*(-1)^(3/4)*(c - d)^2*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] + d*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(6*c - d + d*Sin[e + f*x])))/(3*f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
545,1,106,79,0.2117921,"\int \frac{c+d \sin (e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(c + d*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(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} (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)}{f \sqrt{a (\sin (e+f x)+1)}}","-\frac{\sqrt{2} (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 d \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)*(c - d)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] + d*(-Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))/(f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
546,1,73,47,0.0462347,"\int \frac{1}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[1/Sqrt[a + a*Sin[e + f*x]],x]","\frac{(2+2 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \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)}{f \sqrt{a (\sin (e+f x)+1)}}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}",1,"((2 + 2*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
547,1,215,123,1.800665,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Integrate[1/(Sqrt[a + a*Sin[e + f*x]]*(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(\sqrt{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) (-1)^{3/4} \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)}{f (c-d) \sqrt{c+d} \sqrt{a (\sin (e+f x)+1)}}","\frac{2 \sqrt{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} f (c-d) \sqrt{c+d}}-\frac{\sqrt{2} \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)}",1,"(((2 + 2*I)*(-1)^(3/4)*Sqrt[c + d]*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] + Sqrt[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[c + d]*f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
548,1,324,175,3.5610772,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2} \, dx","Integrate[1/(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{4 d (c-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))}+\frac{\sqrt{d} (3 c+d) \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} (3 c+d) \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}}+(8+8 i) (-1)^{3/4} \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)}{4 f (c-d)^2 \sqrt{a (\sin (e+f x)+1)}}","\frac{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{\sqrt{2} \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}+\frac{\sqrt{d} (3 c+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} f (c-d)^2 (c+d)^{3/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((8 + 8*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] + (Sqrt[d]*(3*c + 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])]))/(c + d)^(3/2) - (Sqrt[d]*(3*c + 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])]))/(c + d)^(3/2) + (4*(c - d)*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
549,1,414,247,4.9787395,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3} \, dx","Integrate[1/(Sqrt[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{\sqrt{d} \left(15 c^2+10 c d+7 d^2\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)}{(c-d)^3 (c+d)^{5/2}}+\frac{\sqrt{d} \left(15 c^2+10 c d+7 d^2\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)}{(d-c)^3 (c+d)^{5/2}}+\frac{4 d (7 c+d) \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)^2 (c+d \sin (e+f x))}+\frac{8 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) (c+d \sin (e+f x))^2}+\frac{(32+32 i) (-1)^{3/4} \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)}{(c-d)^3}\right)}{16 f \sqrt{a (\sin (e+f x)+1)}}","\frac{d (7 c+d) \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{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{\sqrt{d} \left(15 c^2+10 c d+7 d^2\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} f (c-d)^3 (c+d)^{5/2}}-\frac{\sqrt{2} \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,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(((32 + 32*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])])/(c - d)^3 + (Sqrt[d]*(15*c^2 + 10*c*d + 7*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])]))/((c - d)^3*(c + d)^(5/2)) + (Sqrt[d]*(15*c^2 + 10*c*d + 7*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])]))/((-c + d)^3*(c + d)^(5/2)) + (8*d*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c - d)*(c + d)*(c + d*Sin[e + f*x])^2) + (4*d*(7*c + d)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c - d)^2*(c + d)^2*(c + d*Sin[e + f*x]))))/(16*f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
550,1,328,192,0.5500121,"\int \frac{(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(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(-18 d^2 (2 c-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+18 d^2 (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+6 (c-d)^3 \sin \left(\frac{1}{2} (e+f x)\right)-3 (c-d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+(3+3 i) (-1)^{3/4} (c+11 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 \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 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)^2-2 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)^2\right)}{6 f (a (\sin (e+f x)+1))^{3/2}}","-\frac{(c+11 d) (c-d)^2 \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 c-7 d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{6 a^2 f}+\frac{d \left(3 c^2-24 c d+13 d^2\right) \cos (e+f x)}{3 a f \sqrt{a \sin (e+f x)+a}}-\frac{(c-d) \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*(c - d)^3*Sin[(e + f*x)/2] - 3*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (3 + 3*I)*(-1)^(3/4)*(c - d)^2*(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 - 18*(2*c - d)*d^2*Cos[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 2*d^3*Cos[(3*(e + f*x))/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 18*(2*c - d)*d^2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 2*d^3*(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
551,1,239,138,0.3391618,"\int \frac{(c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(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((1+i) (-1)^{3/4} \left(c^2+6 c d-7 d^2\right) \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 (c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right)-(c-d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-4 d^2 \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 d^2 \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{(c-d) (c+7 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{(c-d) \cos (e+f x) (c+d \sin (e+f x))}{2 f (a \sin (e+f x)+a)^{3/2}}+\frac{d (c-5 d) \cos (e+f x)}{2 a f \sqrt{a \sin (e+f x)+a}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*(c - d)^2*Sin[(e + f*x)/2] - (c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (1 + I)*(-1)^(3/4)*(c^2 + 6*c*d - 7*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])^2 - 4*d^2*Cos[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 4*d^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))","C",1
552,1,150,87,0.198654,"\int \frac{c+d \sin (e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(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 (c-d) \sin \left(\frac{1}{2} (e+f x)\right)+(d-c) \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} (c+3 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 (a (\sin (e+f x)+1))^{3/2}}","-\frac{(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}-\frac{(c-d) \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*(c - d)*Sin[(e + f*x)/2] + (-c + d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (1 + I)*(-1)^(3/4)*(c + 3*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))/(2*f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
553,1,108,77,0.1751932,"\int \frac{1}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(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(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+(1+i) (-1)^{3/4} (\sin (e+f x)+1) \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{\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{\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])*(-Cos[(e + f*x)/2] + Sin[(e + f*x)/2] + (1 + I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(1 + Sin[e + f*x])))/(2*f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
554,1,385,164,1.8077639,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))} \, dx","Integrate[1/((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(-\frac{d^{3/2} \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{d^{3/2} \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}}+2 (c-d) \sin \left(\frac{1}{2} (e+f x)\right)-(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} (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)\right)}{2 f (c-d)^2 (a (\sin (e+f x)+1))^{3/2}}","-\frac{2 d^{3/2} \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{(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 (c-d)^2}-\frac{\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*(c - d)*Sin[(e + f*x)/2] - (c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (1 + I)*(-1)^(3/4)*(c - 5*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 - (d^(3/2)*(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)/Sqrt[c + d] + (d^(3/2)*(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)/Sqrt[c + d]))/(2*(c - d)^2*f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
555,1,491,243,4.5590288,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2} \, dx","Integrate[1/((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{d^{3/2} (5 c+3 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)}{(d-c)^3 (c+d)^{3/2}}+\frac{d^{3/2} (5 c+3 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)}{(c-d)^3 (c+d)^{3/2}}-\frac{4 d^2 \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)^2 (c+d) (c+d \sin (e+f x))}+\frac{4 \sin \left(\frac{1}{2} (e+f x)\right)}{(c-d)^2}-\frac{2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{(c-d)^2}+\frac{(2+2 i) (-1)^{3/4} (c-9 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)}{(c-d)^3}\right)}{4 f (a (\sin (e+f x)+1))^{3/2}}","-\frac{d^{3/2} (5 c+3 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)^3 (c+d)^{3/2}}-\frac{(c-9 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 (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{\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*Sin[(e + f*x)/2])/(c - d)^2 - (2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(c - d)^2 + ((2 + 2*I)*(-1)^(3/4)*(c - 9*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)/(c - d)^3 + (d^(3/2)*(5*c + 3*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)/((-c + d)^3*(c + d)^(3/2)) + (d^(3/2)*(5*c + 3*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)/((c - d)^3*(c + d)^(3/2)) - (4*d^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/((c - d)^2*(c + d)*(c + d*Sin[e + f*x]))))/(4*f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
556,1,570,318,6.5242613,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx","Integrate[1/((a + a*Sin[e + f*x])^(3/2)*(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^{3/2} \left(35 c^2+42 c d+19 d^2\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)^{5/2}}+\frac{d^{3/2} \left(35 c^2+42 c d+19 d^2\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)^{5/2}}-\frac{4 d^2 (c-d) (11 c+5 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)^2 (c+d \sin (e+f x))}-\frac{8 d^2 (c-d)^2 \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}+16 (c-d) \sin \left(\frac{1}{2} (e+f x)\right)-8 (c-d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+(8+8 i) (-1)^{3/4} (c-13 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)}{16 f (c-d)^4 (a (\sin (e+f x)+1))^{3/2}}","-\frac{d^{3/2} \left(35 c^2+42 c d+19 d^2\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{(c-13 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 (2 c+d) (c+7 d) \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 (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{\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(16*(c - d)*Sin[(e + f*x)/2] - 8*(c - d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (8 + 8*I)*(-1)^(3/4)*(c - 13*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 - (d^(3/2)*(35*c^2 + 42*c*d + 19*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)^(5/2) + (d^(3/2)*(35*c^2 + 42*c*d + 19*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)^(5/2) - (8*(c - d)^2*d^2*(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])^2) - (4*(c - d)*d^2*(11*c + 5*d)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/((c + d)^2*(c + d*Sin[e + f*x]))))/(16*(c - d)^4*f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
557,1,400,194,0.7493187,"\int \frac{(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(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(11 c^3 \sin \left(\frac{1}{2} (e+f x)\right)-3 c^3 \sin \left(\frac{3}{2} (e+f x)\right)-11 c^3 \cos \left(\frac{1}{2} (e+f x)\right)-3 c^3 \cos \left(\frac{3}{2} (e+f x)\right)-9 c^2 d \sin \left(\frac{1}{2} (e+f x)\right)-15 c^2 d \sin \left(\frac{3}{2} (e+f x)\right)+9 c^2 d \cos \left(\frac{1}{2} (e+f x)\right)-15 c^2 d \cos \left(\frac{3}{2} (e+f x)\right)+(6+6 i) (-1)^{3/4} \left(c^3+5 c^2 d+19 c d^2-25 d^3\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)-15 c d^2 \sin \left(\frac{1}{2} (e+f x)\right)+39 c d^2 \sin \left(\frac{3}{2} (e+f x)\right)+15 c d^2 \cos \left(\frac{1}{2} (e+f x)\right)+39 c d^2 \cos \left(\frac{3}{2} (e+f x)\right)+45 d^3 \sin \left(\frac{1}{2} (e+f x)\right)-69 d^3 \sin \left(\frac{3}{2} (e+f x)\right)-16 d^3 \sin \left(\frac{5}{2} (e+f x)\right)-45 d^3 \cos \left(\frac{1}{2} (e+f x)\right)-69 d^3 \cos \left(\frac{3}{2} (e+f x)\right)+16 d^3 \cos \left(\frac{5}{2} (e+f x)\right)\right)}{32 f (a (\sin (e+f x)+1))^{5/2}}","-\frac{3 \left(c^2+6 c d+25 d^2\right) (c-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{d^2 (c-9 d) \cos (e+f x)}{4 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{(3 c+13 d) (c-d)^2 \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-11*c^3*Cos[(e + f*x)/2] + 9*c^2*d*Cos[(e + f*x)/2] + 15*c*d^2*Cos[(e + f*x)/2] - 45*d^3*Cos[(e + f*x)/2] - 3*c^3*Cos[(3*(e + f*x))/2] - 15*c^2*d*Cos[(3*(e + f*x))/2] + 39*c*d^2*Cos[(3*(e + f*x))/2] - 69*d^3*Cos[(3*(e + f*x))/2] + 16*d^3*Cos[(5*(e + f*x))/2] + 11*c^3*Sin[(e + f*x)/2] - 9*c^2*d*Sin[(e + f*x)/2] - 15*c*d^2*Sin[(e + f*x)/2] + 45*d^3*Sin[(e + f*x)/2] + (6 + 6*I)*(-1)^(3/4)*(c^3 + 5*c^2*d + 19*c*d^2 - 25*d^3)*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*c^3*Sin[(3*(e + f*x))/2] - 15*c^2*d*Sin[(3*(e + f*x))/2] + 39*c*d^2*Sin[(3*(e + f*x))/2] - 69*d^3*Sin[(3*(e + f*x))/2] - 16*d^3*Sin[(5*(e + f*x))/2]))/(32*f*(a*(1 + Sin[e + f*x]))^(5/2))","C",1
558,1,252,147,0.5428193,"\int \frac{(c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(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 \left(3 c^2+10 c d-13 d^2\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)^2+(1+i) (-1)^{3/4} \left(3 c^2+10 c d+19 d^2\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)+8 (c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right)-(c-d) (3 c+13 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)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{16 f (a (\sin (e+f x)+1))^{5/2}}","-\frac{\left(3 c^2+10 c d+19 d^2\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{(c-d) \cos (e+f x) (c+d \sin (e+f x))}{4 f (a \sin (e+f x)+a)^{5/2}}-\frac{3 (c-d) (c+3 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])*(8*(c - d)^2*Sin[(e + f*x)/2] - 4*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(3*c^2 + 10*c*d - 13*d^2)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - (c - d)*(3*c + 13*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (1 + I)*(-1)^(3/4)*(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))/(16*f*(a*(1 + Sin[e + f*x]))^(5/2))","C",1
559,1,227,126,0.353839,"\int \frac{c+d \sin (e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(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 (c-d) \sin \left(\frac{1}{2} (e+f x)\right)-(3 c+5 d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+2 (3 c+5 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 (d-c) \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 c+5 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 c+5 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 c+5 d) \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac{(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*(c - d)*Sin[(e + f*x)/2] + 4*(-c + d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(3*c + 5*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - (3*c + 5*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (1 + I)*(-1)^(3/4)*(3*c + 5*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
560,1,196,107,0.1542969,"\int \frac{1}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(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 \sin \left(\frac{1}{2} (e+f x)\right)-3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+6 \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 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+(3+3 i) (-1)^{3/4} \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 \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 \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac{\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*Sin[(e + f*x)/2] - 4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 6*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (3 + 3*I)*(-1)^(3/4)*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
561,1,501,218,3.2891673,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))} \, dx","Integrate[1/((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(\frac{(1+i) (-1)^{3/4} \left(3 c^2-14 c d+43 d^2\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)}{(c-d)^3}+\frac{8 d^{5/2} \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)}{(c-d)^3 \sqrt{c+d}}+\frac{8 d^{5/2} \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)}{(d-c)^3 \sqrt{c+d}}+\frac{8 \sin \left(\frac{1}{2} (e+f x)\right)}{c-d}+\frac{(11 d-3 c) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}{(c-d)^2}+\frac{2 (3 c-11 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}{(c-d)^2}-\frac{4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{c-d}\right)}{16 f (a (\sin (e+f x)+1))^{5/2}}","-\frac{\left(3 c^2-14 c d+43 d^2\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^{5/2} \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 c-11 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2}}-\frac{\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*Sin[(e + f*x)/2])/(c - d) - (4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(c - d) + (2*(3*c - 11*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/(c - d)^2 + ((-3*c + 11*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/(c - d)^2 + ((1 + I)*(-1)^(3/4)*(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)/(c - d)^3 + (8*d^(5/2)*(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])^4)/((c - d)^3*Sqrt[c + d]) + (8*d^(5/2)*(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])^4)/((-c + d)^3*Sqrt[c + d])))/(16*f*(a*(1 + Sin[e + f*x]))^(5/2))","C",1
562,1,570,313,5.8378383,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx","Integrate[1/((a + a*Sin[e + f*x])^(5/2)*(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((1+i) (-1)^{3/4} \left(3 c^2-22 c d+115 d^2\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{4 d^{5/2} (7 c+5 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)}{(c+d)^{3/2}}-\frac{4 d^{5/2} (7 c+5 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)}{(c+d)^{3/2}}+\frac{16 d^3 (c-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)^4}{(c+d) (c+d \sin (e+f x))}+8 (c-d)^2 \sin \left(\frac{1}{2} (e+f x)\right)+(c-d) (19 d-3 c) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+2 (3 c-19 d) (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)^2-4 (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)^4 (a (\sin (e+f x)+1))^{5/2}}","-\frac{\left(3 c^2-22 c d+115 d^2\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^{5/2} (7 c+5 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)^4 (c+d)^{3/2}}-\frac{d (c-7 d) (3 c+5 d) \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 (c-5 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{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(8*(c - d)^2*Sin[(e + f*x)/2] - 4*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(3*c - 19*d)*(c - d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (c - d)*(-3*c + 19*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (1 + I)*(-1)^(3/4)*(3*c^2 - 22*c*d + 115*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 + (4*d^(5/2)*(7*c + 5*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)/(c + d)^(3/2) - (4*d^(5/2)*(7*c + 5*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)/(c + d)^(3/2) + (16*(c - d)*d^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/((c + d)*(c + d*Sin[e + f*x]))))/(16*(c - d)^4*f*(a*(1 + Sin[e + f*x]))^(5/2))","C",1
563,1,958,400,9.3596283,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx","Integrate[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3),x]","\frac{3 \left(3 \cos \left(\frac{1}{2} (e+f x)\right) d^4-3 \sin \left(\frac{1}{2} (e+f x)\right) d^4+5 c \cos \left(\frac{1}{2} (e+f x)\right) d^3-5 c \sin \left(\frac{1}{2} (e+f x)\right) d^3\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)^2 f (a (\sin (e+f x)+1))^{5/2} (c+d \sin (e+f x))}+\frac{\left(d^3 \cos \left(\frac{1}{2} (e+f x)\right)-d^3 \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)^5}{2 (c-d)^3 (c+d) f (a (\sin (e+f x)+1))^{5/2} (c+d \sin (e+f x))^2}+\frac{(3+3 i) \left(c^2-10 d c+73 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^5-80 \sqrt[4]{-1} d c^4+160 \sqrt[4]{-1} d^2 c^3-160 \sqrt[4]{-1} d^3 c^2+80 \sqrt[4]{-1} d^4 c-16 \sqrt[4]{-1} d^5\right) f (a (\sin (e+f x)+1))^{5/2}}+\frac{3 d^{5/2} \left(21 c^2+30 d c+13 d^2\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}{16 (c-d)^5 (c+d)^{5/2} f (a (\sin (e+f x)+1))^{5/2}}+\frac{3 d^{5/2} \left(21 c^2+30 d c+13 d^2\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}{16 (d-c)^5 (c+d)^{5/2} f (a (\sin (e+f x)+1))^{5/2}}-\frac{3 (c-9 d) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}{16 (c-d)^4 f (a (\sin (e+f x)+1))^{5/2}}+\frac{3 \left(c \sin \left(\frac{1}{2} (e+f x)\right)-9 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)^3}{8 (c-d)^4 f (a (\sin (e+f x)+1))^{5/2}}-\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}{4 (c-d)^3 f (a (\sin (e+f x)+1))^{5/2}}+\frac{\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 (c-d)^3 f (a (\sin (e+f x)+1))^{5/2}}","-\frac{3 \left(c^2-10 c d+73 d^2\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{3 d^{5/2} \left(21 c^2+30 c d+13 d^2\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{3 d (c+3 d) \left(c^2-10 c d-7 d^2\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{d \left(3 c^2-20 c d-31 d^2\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{(3 c-19 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{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}",1,"(Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(2*(c - d)^3*f*(a*(1 + Sin[e + f*x]))^(5/2)) - (Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2/(4*(c - d)^3*f*(a*(1 + Sin[e + f*x]))^(5/2)) - (3*(c - 9*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/(16*(c - d)^4*f*(a*(1 + Sin[e + f*x]))^(5/2)) + ((3 + 3*I)*(c^2 - 10*c*d + 73*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)) + (3*d^(5/2)*(21*c^2 + 30*c*d + 13*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)/(16*(c - d)^5*(c + d)^(5/2)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + (3*d^(5/2)*(21*c^2 + 30*c*d + 13*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)/(16*(-c + d)^5*(c + d)^(5/2)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + (3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c*Sin[(e + f*x)/2] - 9*d*Sin[(e + f*x)/2]))/(8*(c - d)^4*f*(a*(1 + Sin[e + f*x]))^(5/2)) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(d^3*Cos[(e + f*x)/2] - d^3*Sin[(e + f*x)/2]))/(2*(c - d)^3*(c + d)*f*(a*(1 + Sin[e + f*x]))^(5/2)*(c + d*Sin[e + f*x])^2) + (3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(5*c*d^3*Cos[(e + f*x)/2] + 3*d^4*Cos[(e + f*x)/2] - 5*c*d^3*Sin[(e + f*x)/2] - 3*d^4*Sin[(e + f*x)/2]))/(4*(c - d)^4*(c + d)^2*f*(a*(1 + Sin[e + f*x]))^(5/2)*(c + d*Sin[e + f*x]))","C",0
564,1,391,203,3.7250466,"\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2),x]","-\frac{\sqrt{a (\sin (e+f x)+1)} \left(2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sin (e+f x)) \left(33 c^2+2 d (13 c+5 d) \sin (e+f x)+40 c d-4 d^2 \cos (2 (e+f x))+19 d^2\right)+\frac{15 (c+d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+i \cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\log \left(\frac{e^{-i e} \left(2 \sqrt{d} \sqrt{2 c e^{i (e+f x)}-i d \left(-1+e^{2 i (e+f x)}\right)}+2 \sqrt[4]{-1} c-2 (-1)^{3/4} d e^{i (e+f x)}\right)}{\sqrt{d}}\right)-\log \left(\frac{2 f e^{\frac{1}{2} i (e-2 f x)} \left(i \sqrt{d} \sqrt{2 c e^{i (e+f x)}-i d \left(-1+e^{2 i (e+f x)}\right)}+\sqrt[4]{-1} c e^{i (e+f x)}+(-1)^{3/4} d\right)}{\sqrt{d}}\right)\right) \sqrt{(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}}{\sqrt{d}}\right)}{48 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}","-\frac{5 a (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{8 f \sqrt{a \sin (e+f x)+a}}-\frac{5 a (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{5 \sqrt{a} (c+d)^3 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{8 \sqrt{d} f}",1,"-1/48*(Sqrt[a*(1 + Sin[e + f*x])]*((15*(c + d)^3*(Log[(2*(-1)^(1/4)*c - 2*(-1)^(3/4)*d*E^(I*(e + f*x)) + 2*Sqrt[d]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])/(Sqrt[d]*E^(I*e))] - Log[(2*E^((I/2)*(e - 2*f*x))*((-1)^(3/4)*d + (-1)^(1/4)*c*E^(I*(e + f*x)) + I*Sqrt[d]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/Sqrt[d]])*(I*Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[(Cos[e + f*x] + I*Sin[e + f*x])*(c + d*Sin[e + f*x])])/Sqrt[d] + 2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c + d*Sin[e + f*x])*(33*c^2 + 40*c*d + 19*d^2 - 4*d^2*Cos[2*(e + f*x)] + 2*d*(13*c + 5*d)*Sin[e + f*x])))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])","C",1
565,1,365,156,2.0462788,"\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2),x]","\frac{\sqrt{a (\sin (e+f x)+1)} \left(-2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sin (e+f x)) (5 c+2 d \sin (e+f x)+3 d)-\frac{3 i (c+d)^2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-i \sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\log \left(\frac{e^{-i e} \left(2 \sqrt{d} \sqrt{2 c e^{i (e+f x)}-i d \left(-1+e^{2 i (e+f x)}\right)}+2 \sqrt[4]{-1} c-2 (-1)^{3/4} d e^{i (e+f x)}\right)}{\sqrt{d}}\right)-\log \left(\frac{2 f e^{\frac{1}{2} i (e-2 f x)} \left(i \sqrt{d} \sqrt{2 c e^{i (e+f x)}-i d \left(-1+e^{2 i (e+f x)}\right)}+\sqrt[4]{-1} c e^{i (e+f x)}+(-1)^{3/4} d\right)}{\sqrt{d}}\right)\right) \sqrt{(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}}{\sqrt{d}}\right)}{8 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}","-\frac{3 a (c+d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}-\frac{3 \sqrt{a} (c+d)^2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{4 \sqrt{d} f}",1,"(Sqrt[a*(1 + Sin[e + f*x])]*(((-3*I)*(c + d)^2*(Log[(2*(-1)^(1/4)*c - 2*(-1)^(3/4)*d*E^(I*(e + f*x)) + 2*Sqrt[d]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])/(Sqrt[d]*E^(I*e))] - Log[(2*E^((I/2)*(e - 2*f*x))*((-1)^(3/4)*d + (-1)^(1/4)*c*E^(I*(e + f*x)) + I*Sqrt[d]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/Sqrt[d]])*(Cos[(e + f*x)/2] - I*Sin[(e + f*x)/2])*Sqrt[(Cos[e + f*x] + I*Sin[e + f*x])*(c + d*Sin[e + f*x])])/Sqrt[d] - 2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c + d*Sin[e + f*x])*(5*c + 3*d + 2*d*Sin[e + f*x])))/(8*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])","C",1
566,1,350,105,1.3130402,"\int \sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]],x]","\frac{\sqrt{a (\sin (e+f x)+1)} \left(-\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sin (e+f x))}{f}-\frac{i (c+d) \left(\cos \left(\frac{1}{2} (e+f x)\right)-i \sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\log \left(\frac{e^{-i e} \left(2 \sqrt{d} \sqrt{2 c e^{i (e+f x)}-i d \left(-1+e^{2 i (e+f x)}\right)}+2 \sqrt[4]{-1} c-2 (-1)^{3/4} d e^{i (e+f x)}\right)}{\sqrt{d}}\right)-\log \left(\frac{2 f e^{\frac{1}{2} i (e-2 f x)} \left(i \sqrt{d} \sqrt{2 c e^{i (e+f x)}-i d \left(-1+e^{2 i (e+f x)}\right)}+\sqrt[4]{-1} c e^{i (e+f x)}+(-1)^{3/4} d\right)}{\sqrt{d}}\right)\right) \sqrt{(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}}{\sqrt{d} f}\right)}{2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}","-\frac{a \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{a} (c+d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{d} f}",1,"(Sqrt[a*(1 + Sin[e + f*x])]*((-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/f - (I*(c + d)*(Log[(2*(-1)^(1/4)*c - 2*(-1)^(3/4)*d*E^(I*(e + f*x)) + 2*Sqrt[d]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])/(Sqrt[d]*E^(I*e))] - Log[(2*E^((I/2)*(e - 2*f*x))*((-1)^(3/4)*d + (-1)^(1/4)*c*E^(I*(e + f*x)) + I*Sqrt[d]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/Sqrt[d]])*(Cos[(e + f*x)/2] - I*Sin[(e + f*x)/2])*Sqrt[(Cos[e + f*x] + I*Sin[e + f*x])*(c + d*Sin[e + f*x])])/(Sqrt[d]*f)))/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])","C",1
567,1,305,61,1.1547983,"\int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]],x]","-\frac{i \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-i \sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\log \left(\frac{e^{-i e} \left(2 \sqrt{d} \sqrt{2 c e^{i (e+f x)}-i d \left(-1+e^{2 i (e+f x)}\right)}+2 \sqrt[4]{-1} c-2 (-1)^{3/4} d e^{i (e+f x)}\right)}{\sqrt{d}}\right)-\log \left(-\frac{(1+i) f e^{\frac{1}{2} i (e-2 f x)} \left(-\sqrt{d} \sqrt{2 c e^{i (e+f x)}-i d \left(-1+e^{2 i (e+f x)}\right)}+(-1)^{3/4} c e^{i (e+f x)}-\sqrt[4]{-1} d\right)}{\sqrt{d}}\right)\right) \sqrt{(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}}{\sqrt{d} f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}","-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{d} f}",1,"((-I)*(Log[(2*(-1)^(1/4)*c - 2*(-1)^(3/4)*d*E^(I*(e + f*x)) + 2*Sqrt[d]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])/(Sqrt[d]*E^(I*e))] - Log[((-1 - I)*E^((I/2)*(e - 2*f*x))*(-((-1)^(1/4)*d) + (-1)^(3/4)*c*E^(I*(e + f*x)) - Sqrt[d]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/Sqrt[d]])*(Cos[(e + f*x)/2] - I*Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[(Cos[e + f*x] + I*Sin[e + f*x])*(c + d*Sin[e + f*x])])/(Sqrt[d]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])","C",1
568,1,84,45,0.1932559,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x])^(3/2),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)}{f (c+d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}","-\frac{2 a \cos (e+f x)}{f (c+d) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}",1,"(-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])])/((c + d)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])","A",1
569,1,100,95,0.2747868,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x])^(5/2),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 c+2 d \sin (e+f x)+d)}{3 f (c+d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sin (e+f x))^{3/2}}","-\frac{4 a \cos (e+f x)}{3 f (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a \cos (e+f x)}{3 f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}",1,"(-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(3*c + d + 2*d*Sin[e + f*x]))/(3*(c + d)^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x])^(3/2))","A",1
570,1,128,142,0.3942199,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x])^(7/2),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) \left(15 c^2+4 d (5 c+d) \sin (e+f x)+10 c d+8 d^2 \sin ^2(e+f x)+3 d^2\right)}{15 f (c+d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sin (e+f x))^{5/2}}","-\frac{16 a \cos (e+f x)}{15 f (c+d)^3 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{8 a \cos (e+f x)}{15 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a \cos (e+f x)}{5 f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}",1,"(-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(15*c^2 + 10*c*d + 3*d^2 + 4*d*(5*c + d)*Sin[e + f*x] + 8*d^2*Sin[e + f*x]^2))/(15*(c + d)^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x])^(5/2))","A",1
571,1,318,285,1.3668099,"\int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(5/2),x]","\frac{(a (\sin (e+f x)+1))^{3/2} \left(-\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)} \left(15 c^3+2 d \left(59 c^2+190 c d+93 d^2\right) \sin (e+f x)+455 c^2 d-4 d^2 (17 c+15 d) \cos (2 (e+f x))+653 c d^2-12 d^3 \sin (3 (e+f x))+285 d^3\right)}{3 d}-\frac{5 (c-15 d) (c+d)^3 \left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)-\log \left(\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)\right)}{d^{3/2}}\right)}{128 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","\frac{5 a^{3/2} (c-15 d) (c+d)^3 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{64 d^{3/2} f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt{a \sin (e+f x)+a}}+\frac{5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt{a \sin (e+f x)+a}}+\frac{5 a^2 (c-15 d) (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{64 d f \sqrt{a \sin (e+f x)+a}}",1,"((a*(1 + Sin[e + f*x]))^(3/2)*((-5*(c - 15*d)*(c + d)^3*(2*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] - Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]]))/d^(3/2) - (2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]*(15*c^3 + 455*c^2*d + 653*c*d^2 + 285*d^3 - 4*d^2*(17*c + 15*d)*Cos[2*(e + f*x)] + 2*d*(59*c^2 + 190*c*d + 93*d^2)*Sin[e + f*x] - 12*d^3*Sin[3*(e + f*x)]))/(3*d)))/(128*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
572,1,281,228,0.8835801,"\int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2),x]","\frac{(a (\sin (e+f x)+1))^{3/2} \left(\frac{(c-11 d) (c+d)^2 \left(-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)+\log \left(\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)-\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)\right)}{d^{3/2}}-\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)} \left(3 c^2+2 d (7 c+11 d) \sin (e+f x)+52 c d-4 d^2 \cos (2 (e+f x))+37 d^2\right)}{3 d}\right)}{16 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} (c-11 d) (c+d)^2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{8 d^{3/2} f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-11 d) (c+d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{8 d f \sqrt{a \sin (e+f x)+a}}",1,"((a*(1 + Sin[e + f*x]))^(3/2)*(((c - 11*d)*(c + d)^2*(-2*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] - ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]]))/d^(3/2) - (2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]*(3*c^2 + 52*c*d + 37*d^2 - 4*d^2*Cos[2*(e + f*x)] + 2*d*(7*c + 11*d)*Sin[e + f*x]))/(3*d)))/(16*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
573,1,247,171,0.5931022,"\int (a+a \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]],x]","\frac{(a (\sin (e+f x)+1))^{3/2} \left(\frac{(c-7 d) (c+d) \left(-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)+\log \left(\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)-\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)\right)}{d^{3/2}}-\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)} (c+2 d \sin (e+f x)+7 d)}{d}\right)}{8 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} (c-7 d) (c+d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{4 d^{3/2} f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 d f \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-7 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 d f \sqrt{a \sin (e+f x)+a}}",1,"((a*(1 + Sin[e + f*x]))^(3/2)*(((c - 7*d)*(c + d)*(-2*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] - ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]]))/d^(3/2) - (2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]*(c + 7*d + 2*d*Sin[e + f*x]))/d))/(8*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
574,1,301,111,0.5780076,"\int \frac{(a+a \sin (e+f x))^{3/2}}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/Sqrt[c + d*Sin[e + f*x]],x]","\frac{(a (\sin (e+f x)+1))^{3/2} \left(2 \sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right) \sqrt{c+d \sin (e+f x)}-2 (c-3 d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)-2 \sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right) \sqrt{c+d \sin (e+f x)}+c \log \left(\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)-3 d \log \left(\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)-(c-3 d) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)\right)}{2 d^{3/2} 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} (c-3 d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{d^{3/2} f}-\frac{a^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{d f \sqrt{a \sin (e+f x)+a}}",1,"((a*(1 + Sin[e + f*x]))^(3/2)*(-2*(c - 3*d)*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] - (c - 3*d)*ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + c*Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]] - 3*d*Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]] - 2*Sqrt[d]*Cos[(e + f*x)/2]*Sqrt[c + d*Sin[e + f*x]] + 2*Sqrt[d]*Sin[(e + f*x)/2]*Sqrt[c + d*Sin[e + f*x]]))/(2*d^(3/2)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","B",1
575,1,377,117,7.4052101,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(3/2),x]","\frac{(a (\sin (e+f x)+1))^{3/2} \left(-2 c \sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)+2 c \sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)+2 (c+d) \sqrt{c+d \sin (e+f x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)-d \sqrt{c+d \sin (e+f x)} \log \left(\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)-c \sqrt{c+d \sin (e+f x)} \log \left(\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+(c+d) \sqrt{c+d \sin (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)+2 d^{3/2} \sin \left(\frac{1}{2} (e+f x)\right)-2 d^{3/2} \cos \left(\frac{1}{2} (e+f x)\right)\right)}{d^{3/2} f (c+d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \sqrt{c+d \sin (e+f x)}}","\frac{2 a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{d^{3/2} f}",1,"((a*(1 + Sin[e + f*x]))^(3/2)*(2*c*Sqrt[d]*Cos[(e + f*x)/2] - 2*d^(3/2)*Cos[(e + f*x)/2] - 2*c*Sqrt[d]*Sin[(e + f*x)/2] + 2*d^(3/2)*Sin[(e + f*x)/2] + 2*(c + d)*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]]*Sqrt[c + d*Sin[e + f*x]] + (c + d)*ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]]*Sqrt[c + d*Sin[e + f*x]] - c*Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]]*Sqrt[c + d*Sin[e + f*x]] - d*Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]]*Sqrt[c + d*Sin[e + f*x]]))/(d^(3/2)*(c + d)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c + d*Sin[e + f*x]])","B",1
576,1,104,115,0.5944771,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(5/2),x]","-\frac{2 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) ((c+5 d) \sin (e+f x)+5 c+d)}{3 f (c+d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sin (e+f x))^{3/2}}","\frac{2 a^2 (c-d) \cos (e+f x)}{3 d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a^2 (c+5 d) \cos (e+f x)}{3 d f (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}",1,"(-2*a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(5*c + d + (c + 5*d)*Sin[e + f*x]))/(3*(c + d)^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x])^(3/2))","A",1
577,1,140,172,0.9039914,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{7/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(7/2),x]","-\frac{2 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(\left(5 c^2+46 c d+9 d^2\right) \sin (e+f x)+25 c^2-d (c+9 d) \cos (2 (e+f x))+13 c d+12 d^2\right)}{15 f (c+d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sin (e+f x))^{5/2}}","-\frac{4 a^2 (c+9 d) \cos (e+f x)}{15 d f (c+d)^3 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a^2 (c+9 d) \cos (e+f x)}{15 d f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}+\frac{2 a^2 (c-d) \cos (e+f x)}{5 d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}",1,"(-2*a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(25*c^2 + 13*c*d + 12*d^2 - d*(c + 9*d)*Cos[2*(e + f*x)] + (5*c^2 + 46*c*d + 9*d^2)*Sin[e + f*x]))/(15*(c + d)^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x])^(5/2))","A",1
578,1,193,229,1.5282486,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(9/2),x]","-\frac{2 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(175 c^3-2 d \left(7 c^2+92 c d+13 d^2\right) \cos (2 (e+f x))+147 c^2 d+\left(35 c^3+469 c^2 d+191 c d^2+117 d^3\right) \sin (e+f x)-2 c d^2 \sin (3 (e+f x))+253 c d^2-26 d^3 \sin (3 (e+f x))+41 d^3\right)}{105 f (c+d)^4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sin (e+f x))^{7/2}}","-\frac{16 a^2 (c+13 d) \cos (e+f x)}{105 d f (c+d)^4 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{8 a^2 (c+13 d) \cos (e+f x)}{105 d f (c+d)^3 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a^2 (c+13 d) \cos (e+f x)}{35 d f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac{2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}",1,"(-2*a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(175*c^3 + 147*c^2*d + 253*c*d^2 + 41*d^3 - 2*d*(7*c^2 + 92*c*d + 13*d^2)*Cos[2*(e + f*x)] + (35*c^3 + 469*c^2*d + 191*c*d^2 + 117*d^3)*Sin[e + f*x] - 2*c*d^2*Sin[3*(e + f*x)] - 26*d^3*Sin[3*(e + f*x)]))/(105*(c + d)^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x])^(7/2))","A",1
579,1,395,377,2.804279,"\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(5/2),x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(\frac{\left(3 c^2-34 c d+283 d^2\right) (c+d)^3 \left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)-\log \left(\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)\right)}{d^{5/2}}+\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)} \left(45 c^4-30 c^3 d \sin (e+f x)-390 c^3 d-3322 c^2 d^2 \sin (e+f x)+4 d^2 \left(93 c^2+488 c d+331 d^2\right) \cos (2 (e+f x))-8396 c^2 d^2-7774 c d^3 \sin (e+f x)+252 c d^3 \sin (3 (e+f x))-12762 c d^3-3874 d^4 \sin (e+f x)+348 d^4 \sin (3 (e+f x))-48 d^4 \cos (4 (e+f x))-5521 d^4\right)}{15 d^2}\right)}{256 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)^3 \left(3 c^2-34 c d+283 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{128 d^{5/2} f}-\frac{a^3 \left(3 c^2-34 c d+283 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 (c+d) \left(3 c^2-34 c d+283 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 (c+d)^2 \left(3 c^2-34 c d+283 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{128 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}",1,"((a*(1 + Sin[e + f*x]))^(5/2)*(((c + d)^3*(3*c^2 - 34*c*d + 283*d^2)*(2*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] - Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]]))/d^(5/2) + (2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]*(45*c^4 - 390*c^3*d - 8396*c^2*d^2 - 12762*c*d^3 - 5521*d^4 + 4*d^2*(93*c^2 + 488*c*d + 331*d^2)*Cos[2*(e + f*x)] - 48*d^4*Cos[4*(e + f*x)] - 30*c^3*d*Sin[e + f*x] - 3322*c^2*d^2*Sin[e + f*x] - 7774*c*d^3*Sin[e + f*x] - 3874*d^4*Sin[e + f*x] + 252*c*d^3*Sin[3*(e + f*x)] + 348*d^4*Sin[3*(e + f*x)]))/(15*d^2)))/(256*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",0
580,1,327,312,1.7297989,"\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2),x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(\frac{\left(3 c^2-26 c d+163 d^2\right) (c+d)^2 \left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)-\log \left(\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)\right)}{d^{5/2}}+\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)} \left(9 c^3-2 d \left(3 c^2+158 c d+181 d^2\right) \sin (e+f x)-63 c^2 d+4 d^2 (9 c+23 d) \cos (2 (e+f x))-773 c d^2+12 d^3 \sin (3 (e+f x))-581 d^3\right)}{3 d^2}\right)}{128 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)^2 \left(3 c^2-26 c d+163 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{64 d^{5/2} f}-\frac{a^3 \left(3 c^2-26 c d+163 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 (c+d) \left(3 c^2-26 c d+163 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{64 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{a^3 (3 c-17 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}{4 d f}",1,"((a*(1 + Sin[e + f*x]))^(5/2)*(((c + d)^2*(3*c^2 - 26*c*d + 163*d^2)*(2*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] - Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]]))/d^(5/2) + (2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]*(9*c^3 - 63*c^2*d - 773*c*d^2 - 581*d^3 + 4*d^2*(9*c + 23*d)*Cos[2*(e + f*x)] - 2*d*(3*c^2 + 158*c*d + 181*d^2)*Sin[e + f*x] + 12*d^3*Sin[3*(e + f*x)]))/(3*d^2)))/(128*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",0
581,1,285,241,0.9141873,"\int (a+a \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]],x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)} \left(3 c^2-2 d (c+17 d) \sin (e+f x)-16 c d+4 d^2 \cos (2 (e+f x))-79 d^2\right)}{3 d^2}+\frac{(c+d) \left(c^2-6 c d+25 d^2\right) \left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)-\log \left(\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)\right)}{d^{5/2}}\right)}{16 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(c^2-6 c d+25 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{8 d^{5/2} f}-\frac{a^3 \left(c^2-6 c d+25 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{8 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{a^3 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}{3 d f}",1,"((a*(1 + Sin[e + f*x]))^(5/2)*(((c + d)*(c^2 - 6*c*d + 25*d^2)*(2*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] - Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]]))/d^(5/2) + (2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]*(3*c^2 - 16*c*d - 79*d^2 + 4*d^2*Cos[2*(e + f*x)] - 2*d*(c + 17*d)*Sin[e + f*x]))/(3*d^2)))/(16*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
582,1,256,178,0.7963582,"\int \frac{(a+a \sin (e+f x))^{5/2}}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/Sqrt[c + d*Sin[e + f*x]],x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(\frac{\left(3 c^2-10 c d+19 d^2\right) \left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)-\log \left(\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)\right)}{d^{5/2}}+\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (3 c-2 d \sin (e+f x)-11 d) \sqrt{c+d \sin (e+f x)}}{d^2}\right)}{8 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(3 c^2-10 c d+19 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{4 d^{5/2} f}+\frac{3 a^3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}{2 d f}",1,"((a*(1 + Sin[e + f*x]))^(5/2)*(((3*c^2 - 10*c*d + 19*d^2)*(2*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] - Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]]))/d^(5/2) + (2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(3*c - 11*d - 2*d*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/d^2))/(8*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
583,1,263,180,1.0121774,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(3/2),x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(\frac{(5 d-3 c) \left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)-\log \left(\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)\right)}{d^{5/2}}-\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(3 c^2+d (c+d) \sin (e+f x)-3 c d+2 d^2\right)}{d^2 (c+d) \sqrt{c+d \sin (e+f x)}}\right)}{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} (3 c-5 d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{d^{5/2} f}-\frac{a^3 (3 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{d^2 f (c+d) \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{d f (c+d) \sqrt{c+d \sin (e+f x)}}",1,"((a*(1 + Sin[e + f*x]))^(5/2)*(((-3*c + 5*d)*(2*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] - Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]]))/d^(5/2) - (2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(3*c^2 - 3*c*d + 2*d^2 + d*(c + d)*Sin[e + f*x]))/(d^2*(c + d)*Sqrt[c + d*Sin[e + f*x]])))/(2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
584,1,261,183,8.1556943,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(5/2),x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(\frac{2 (c-d) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(3 c^2+4 d (c+2 d) \sin (e+f x)+8 c d+d^2\right)}{3 d^2 (c+d)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)-\log \left(\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sqrt{c+d \sin (e+f x)}}\right)}{d^{5/2}}\right)}{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} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{d^{5/2} f}+\frac{2 a^3 (c-d) (3 c+7 d) \cos (e+f x)}{3 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}",1,"((a*(1 + Sin[e + f*x]))^(5/2)*((2*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] - Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]])/d^(5/2) + (2*(c - d)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(3*c^2 + 8*c*d + d^2 + 4*d*(c + 2*d)*Sin[e + f*x]))/(3*d^2*(c + d)^2*(c + d*Sin[e + f*x])^(3/2))))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
585,1,152,189,2.1144603,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{7/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(7/2),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(4 \left(7 c^2+46 c d+7 d^2\right) \sin (e+f x)-\left(3 c^2+14 c d+43 d^2\right) \cos (2 (e+f x))+89 c^2+42 c d+49 d^2\right)}{15 f (c+d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sin (e+f x))^{5/2}}","-\frac{2 a^3 \left(3 c^2+14 c d+43 d^2\right) \cos (e+f x)}{15 d^2 f (c+d)^3 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}+\frac{2 a^3 (c-d) (3 c+11 d) \cos (e+f x)}{15 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}",1,"-1/15*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(89*c^2 + 42*c*d + 49*d^2 - (3*c^2 + 14*c*d + 43*d^2)*Cos[2*(e + f*x)] + 4*(7*c^2 + 46*c*d + 7*d^2)*Sin[e + f*x]))/((c + d)^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x])^(5/2))","A",1
586,1,216,254,4.3088832,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{9/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/(c + d*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) \left(-623 c^3+3 c^2 d \sin (3 (e+f x))-495 c^2 d-\left(196 c^3+1865 c^2 d+694 c d^2+465 d^3\right) \sin (e+f x)+\left(21 c^3+157 c^2 d+827 c d^2+115 d^3\right) \cos (2 (e+f x))+22 c d^2 \sin (3 (e+f x))-977 c d^2+115 d^3 \sin (3 (e+f x))-145 d^3\right)}{105 f (c+d)^4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sin (e+f x))^{7/2}}","-\frac{4 a^3 \left(3 c^2+22 c d+115 d^2\right) \cos (e+f x)}{105 d^2 f (c+d)^4 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a^3 \left(3 c^2+22 c d+115 d^2\right) \cos (e+f x)}{105 d^2 f (c+d)^3 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}+\frac{6 a^3 (c-d) (c+5 d) \cos (e+f x)}{35 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}}",1,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(-623*c^3 - 495*c^2*d - 977*c*d^2 - 145*d^3 + (21*c^3 + 157*c^2*d + 827*c*d^2 + 115*d^3)*Cos[2*(e + f*x)] - (196*c^3 + 1865*c^2*d + 694*c*d^2 + 465*d^3)*Sin[e + f*x] + 3*c^2*d*Sin[3*(e + f*x)] + 22*c*d^2*Sin[3*(e + f*x)] + 115*d^3*Sin[3*(e + f*x)]))/(105*(c + d)^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x])^(7/2))","A",1
587,1,616,317,6.5529483,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(11/2),x]","\frac{(a (\sin (e+f x)+1))^{5/2} \sqrt{c+d \sin (e+f x)} \left(-\frac{16 \left(-c^2 \sin \left(\frac{1}{2} (e+f x)\right)+c^2 \cos \left(\frac{1}{2} (e+f x)\right)-10 c d \sin \left(\frac{1}{2} (e+f x)\right)+10 c d \cos \left(\frac{1}{2} (e+f x)\right)-73 d^2 \sin \left(\frac{1}{2} (e+f x)\right)+73 d^2 \cos \left(\frac{1}{2} (e+f x)\right)\right)}{315 d^2 (c+d)^5 (c+d \sin (e+f x))}-\frac{8 \left(-c^2 \sin \left(\frac{1}{2} (e+f x)\right)+c^2 \cos \left(\frac{1}{2} (e+f x)\right)-10 c d \sin \left(\frac{1}{2} (e+f x)\right)+10 c d \cos \left(\frac{1}{2} (e+f x)\right)-73 d^2 \sin \left(\frac{1}{2} (e+f x)\right)+73 d^2 \cos \left(\frac{1}{2} (e+f x)\right)\right)}{315 d^2 (c+d)^4 (c+d \sin (e+f x))^2}-\frac{2 \left(-c^2 \sin \left(\frac{1}{2} (e+f x)\right)+c^2 \cos \left(\frac{1}{2} (e+f x)\right)-10 c d \sin \left(\frac{1}{2} (e+f x)\right)+10 c d \cos \left(\frac{1}{2} (e+f x)\right)-73 d^2 \sin \left(\frac{1}{2} (e+f x)\right)+73 d^2 \cos \left(\frac{1}{2} (e+f x)\right)\right)}{105 d^2 (c+d)^3 (c+d \sin (e+f x))^3}-\frac{4 \left(5 c^2 \sin \left(\frac{1}{2} (e+f x)\right)-5 c^2 \cos \left(\frac{1}{2} (e+f x)\right)+8 c d \sin \left(\frac{1}{2} (e+f x)\right)-8 c d \cos \left(\frac{1}{2} (e+f x)\right)-13 d^2 \sin \left(\frac{1}{2} (e+f x)\right)+13 d^2 \cos \left(\frac{1}{2} (e+f x)\right)\right)}{63 d^2 (c+d)^2 (c+d \sin (e+f x))^4}-\frac{2 \left(-c^2 \sin \left(\frac{1}{2} (e+f x)\right)+c^2 \cos \left(\frac{1}{2} (e+f x)\right)+2 c d \sin \left(\frac{1}{2} (e+f x)\right)-2 c d \cos \left(\frac{1}{2} (e+f x)\right)-d^2 \sin \left(\frac{1}{2} (e+f x)\right)+d^2 \cos \left(\frac{1}{2} (e+f x)\right)\right)}{9 d^2 (c+d) (c+d \sin (e+f x))^5}\right)}{f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\frac{16 a^3 \left(c^2+10 c d+73 d^2\right) \cos (e+f x)}{315 d^2 f (c+d)^5 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{8 a^3 \left(c^2+10 c d+73 d^2\right) \cos (e+f x)}{315 d^2 f (c+d)^4 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a^3 \left(c^2+10 c d+73 d^2\right) \cos (e+f x)}{105 d^2 f (c+d)^3 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac{2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}}",1,"((a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c + d*Sin[e + f*x]]*((-2*(c^2*Cos[(e + f*x)/2] - 2*c*d*Cos[(e + f*x)/2] + d^2*Cos[(e + f*x)/2] - c^2*Sin[(e + f*x)/2] + 2*c*d*Sin[(e + f*x)/2] - d^2*Sin[(e + f*x)/2]))/(9*d^2*(c + d)*(c + d*Sin[e + f*x])^5) - (4*(-5*c^2*Cos[(e + f*x)/2] - 8*c*d*Cos[(e + f*x)/2] + 13*d^2*Cos[(e + f*x)/2] + 5*c^2*Sin[(e + f*x)/2] + 8*c*d*Sin[(e + f*x)/2] - 13*d^2*Sin[(e + f*x)/2]))/(63*d^2*(c + d)^2*(c + d*Sin[e + f*x])^4) - (2*(c^2*Cos[(e + f*x)/2] + 10*c*d*Cos[(e + f*x)/2] + 73*d^2*Cos[(e + f*x)/2] - c^2*Sin[(e + f*x)/2] - 10*c*d*Sin[(e + f*x)/2] - 73*d^2*Sin[(e + f*x)/2]))/(105*d^2*(c + d)^3*(c + d*Sin[e + f*x])^3) - (8*(c^2*Cos[(e + f*x)/2] + 10*c*d*Cos[(e + f*x)/2] + 73*d^2*Cos[(e + f*x)/2] - c^2*Sin[(e + f*x)/2] - 10*c*d*Sin[(e + f*x)/2] - 73*d^2*Sin[(e + f*x)/2]))/(315*d^2*(c + d)^4*(c + d*Sin[e + f*x])^2) - (16*(c^2*Cos[(e + f*x)/2] + 10*c*d*Cos[(e + f*x)/2] + 73*d^2*Cos[(e + f*x)/2] - c^2*Sin[(e + f*x)/2] - 10*c*d*Sin[(e + f*x)/2] - 73*d^2*Sin[(e + f*x)/2]))/(315*d^2*(c + d)^5*(c + d*Sin[e + f*x]))))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
588,1,1893,249,17.6087401,"\int \frac{(c+d \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(c + d*Sin[e + f*x])^(5/2)/Sqrt[a + a*Sin[e + f*x]],x]","\frac{\left(\sqrt{2} \log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right) (c-d)^{5/2}-\sqrt{2} \log \left(c-d+(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right) (c-d)^{5/2}-\frac{1}{8} i \sqrt{d} \left(15 c^2-10 d c+7 d^2\right) \log \left(\frac{2 \left(c-i \left(d+(1+i) \sqrt{2} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)} \sqrt{d}\right)+(d-i c) \tan \left(\frac{1}{2} (e+f x)\right)\right)}{d^{3/2} \left(15 c^2-10 d c+7 d^2\right) \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)}\right)+\frac{1}{8} i \sqrt{d} \left(15 c^2-10 d c+7 d^2\right) \log \left(\frac{2 \left(c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{d^{3/2} \left(15 c^2-10 d c+7 d^2\right) \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\frac{c^3}{\left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}+\frac{15 d \sin (e+f x) c^2}{8 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}-\frac{9 d c^2}{8 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}-\frac{5 d^2 \sin (e+f x) c}{4 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}+\frac{7 d^2 c}{4 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}+\frac{7 d^3 \sin (e+f x)}{8 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}-\frac{d^3}{8 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}\right)}{f \sqrt{a (\sin (e+f x)+1)} \left(\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right) (c-d)^{5/2}}{\sqrt{2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)}-\frac{\sqrt{2} \left(\frac{1}{2} (d-c) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}+\frac{\sqrt{c-d} d \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{c+d \sin (e+f x)}}\right) (c-d)^{5/2}}{c-d+(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}}-\frac{i d^2 \left(15 c^2-10 d c+7 d^2\right)^2 \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right) \left(\frac{2 \left(\frac{1}{2} (d-i c) \sec ^2\left(\frac{1}{2} (e+f x)\right)-i \left(\frac{(1+i) \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}} d^{3/2}}{\sqrt{2} \sqrt{c+d \sin (e+f x)}}+\frac{(1+i) \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)} \sqrt{d}}{\sqrt{2}}\right)\right)}{d^{3/2} \left(15 c^2-10 d c+7 d^2\right) \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)}-\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right) \left(c-i \left(d+(1+i) \sqrt{2} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)} \sqrt{d}\right)+(d-i c) \tan \left(\frac{1}{2} (e+f x)\right)\right)}{d^{3/2} \left(15 c^2-10 d c+7 d^2\right) \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)^2}\right)}{16 \left(c-i \left(d+(1+i) \sqrt{2} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)} \sqrt{d}\right)+(d-i c) \tan \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{i d^2 \left(15 c^2-10 d c+7 d^2\right)^2 \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right) \left(\frac{2 \left(\frac{1}{2} (i c+d) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\frac{(1+i) \sqrt{d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}}{\sqrt{2}}+\frac{(1+i) d^{3/2} \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{2} \sqrt{c+d \sin (e+f x)}}\right)}{d^{3/2} \left(15 c^2-10 d c+7 d^2\right) \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)}-\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right) \left(c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{d^{3/2} \left(15 c^2-10 d c+7 d^2\right) \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)^2}\right)}{16 \left(c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}\right)}+\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)} \left(-\frac{1}{4} \cos \left(\frac{3}{2} (e+f x)\right) d^2-\frac{1}{4} \sin \left(\frac{3}{2} (e+f x)\right) d^2+\frac{1}{4} (2 d-9 c) \cos \left(\frac{1}{2} (e+f x)\right) d-\frac{1}{4} (2 d-9 c) \sin \left(\frac{1}{2} (e+f x)\right) d\right)}{f \sqrt{a (\sin (e+f x)+1)}}","-\frac{\sqrt{d} \left(15 c^2-10 c d+7 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{4 \sqrt{a} f}-\frac{d \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}-\frac{d (7 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{2} (c-d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]*((d*(-9*c + 2*d)*Cos[(e + f*x)/2])/4 - (d^2*Cos[(3*(e + f*x))/2])/4 - (d*(-9*c + 2*d)*Sin[(e + f*x)/2])/4 - (d^2*Sin[(3*(e + f*x))/2])/4))/(f*Sqrt[a*(1 + Sin[e + f*x])]) + ((Sqrt[2]*(c - d)^(5/2)*Log[1 + Tan[(e + f*x)/2]] - Sqrt[2]*(c - d)^(5/2)*Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]] - (I/8)*Sqrt[d]*(15*c^2 - 10*c*d + 7*d^2)*Log[(2*(c - I*(d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]]) + ((-I)*c + d)*Tan[(e + f*x)/2]))/(d^(3/2)*(15*c^2 - 10*c*d + 7*d^2)*(I + Tan[(e + f*x)/2]))] + (I/8)*Sqrt[d]*(15*c^2 - 10*c*d + 7*d^2)*Log[(2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))/(d^(3/2)*(15*c^2 - 10*c*d + 7*d^2)*(-I + Tan[(e + f*x)/2]))])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c^3/((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) - (9*c^2*d)/(8*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (7*c*d^2)/(4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) - d^3/(8*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (15*c^2*d*Sin[e + f*x])/(8*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) - (5*c*d^2*Sin[e + f*x])/(4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (7*d^3*Sin[e + f*x])/(8*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])))/(f*Sqrt[a*(1 + Sin[e + f*x])]*(((c - d)^(5/2)*Sec[(e + f*x)/2]^2)/(Sqrt[2]*(1 + Tan[(e + f*x)/2])) - (Sqrt[2]*(c - d)^(5/2)*(((-c + d)*Sec[(e + f*x)/2]^2)/2 + (Sqrt[c - d]*d*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/Sqrt[c + d*Sin[e + f*x]] + Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]))/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]) - ((I/16)*d^2*(15*c^2 - 10*c*d + 7*d^2)^2*(I + Tan[(e + f*x)/2])*((2*((((-I)*c + d)*Sec[(e + f*x)/2]^2)/2 - I*(((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2])))/(d^(3/2)*(15*c^2 - 10*c*d + 7*d^2)*(I + Tan[(e + f*x)/2])) - (Sec[(e + f*x)/2]^2*(c - I*(d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]]) + ((-I)*c + d)*Tan[(e + f*x)/2]))/(d^(3/2)*(15*c^2 - 10*c*d + 7*d^2)*(I + Tan[(e + f*x)/2])^2)))/(c - I*(d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]]) + ((-I)*c + d)*Tan[(e + f*x)/2]) + ((I/16)*d^2*(15*c^2 - 10*c*d + 7*d^2)^2*(-I + Tan[(e + f*x)/2])*((2*(((I*c + d)*Sec[(e + f*x)/2]^2)/2 + ((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2]))/(d^(3/2)*(15*c^2 - 10*c*d + 7*d^2)*(-I + Tan[(e + f*x)/2])) - (Sec[(e + f*x)/2]^2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))/(d^(3/2)*(15*c^2 - 10*c*d + 7*d^2)*(-I + Tan[(e + f*x)/2])^2)))/(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2])))","C",0
589,1,1639,188,17.2124162,"\int \frac{(c+d \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(c + d*Sin[e + f*x])^(3/2)/Sqrt[a + a*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) \sqrt{c+d \sin (e+f x)} \left(d \sin \left(\frac{1}{2} (e+f x)\right)-d \cos \left(\frac{1}{2} (e+f x)\right)\right)}{f \sqrt{a (\sin (e+f x)+1)}}+\frac{\left(\sqrt{2} \log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right) (c-d)^{3/2}-\sqrt{2} \log \left(c-d+(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right) (c-d)^{3/2}+\frac{1}{2} i \sqrt{d} (d-3 c) \left(\log \left(\frac{2 i \left(i c+d+(c+i d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{d^{3/2} (d-3 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)}\right)-\log \left(-\frac{2 \left(c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{d^{3/2} (d-3 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)}\right)\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\frac{c^2}{\left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}+\frac{3 d \sin (e+f x) c}{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}-\frac{d c}{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}-\frac{d^2 \sin (e+f x)}{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}+\frac{d^2}{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}\right)}{f \sqrt{a (\sin (e+f x)+1)} \left(\frac{(c-d)^{3/2} \sec ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)}+\frac{1}{2} i \sqrt{d} (d-3 c) \left(\frac{d^{3/2} (d-3 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right) \left(\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right) \left(c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{d^{3/2} (d-3 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)^2}-\frac{2 \left(\frac{1}{2} (i c+d) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\frac{(1+i) \sqrt{d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}}{\sqrt{2}}+\frac{(1+i) d^{3/2} \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{2} \sqrt{c+d \sin (e+f x)}}\right)}{d^{3/2} (d-3 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)}\right)}{2 \left(c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}-\frac{i d^{3/2} (d-3 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right) \left(\frac{2 i \left(\frac{1}{2} (c+i d) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\frac{(1+i) \sqrt{d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}}{\sqrt{2}}+\frac{(1+i) d^{3/2} \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{2} \sqrt{c+d \sin (e+f x)}}\right)}{d^{3/2} (d-3 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)}-\frac{i \sec ^2\left(\frac{1}{2} (e+f x)\right) \left(i c+d+(c+i d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{d^{3/2} (d-3 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)^2}\right)}{2 \left(i c+d+(c+i d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}\right)-\frac{\sqrt{2} (c-d)^{3/2} \left(\frac{1}{2} (d-c) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}+\frac{\sqrt{c-d} d \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{c+d \sin (e+f x)}}\right)}{c-d+(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}}\right)}","-\frac{d \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{d} (3 c-d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}-\frac{\sqrt{2} (c-d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-(d*Cos[(e + f*x)/2]) + d*Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a*(1 + Sin[e + f*x])]) + ((Sqrt[2]*(c - d)^(3/2)*Log[1 + Tan[(e + f*x)/2]] - Sqrt[2]*(c - d)^(3/2)*Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]] + (I/2)*Sqrt[d]*(-3*c + d)*(Log[((2*I)*(I*c + d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (c + I*d)*Tan[(e + f*x)/2]))/(d^(3/2)*(-3*c + d)*(I + Tan[(e + f*x)/2]))] - Log[(-2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))/(d^(3/2)*(-3*c + d)*(-I + Tan[(e + f*x)/2]))]))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c^2/((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) - (c*d)/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + d^2/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (3*c*d*Sin[e + f*x])/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) - (d^2*Sin[e + f*x])/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])))/(f*Sqrt[a*(1 + Sin[e + f*x])]*(((c - d)^(3/2)*Sec[(e + f*x)/2]^2)/(Sqrt[2]*(1 + Tan[(e + f*x)/2])) - (Sqrt[2]*(c - d)^(3/2)*(((-c + d)*Sec[(e + f*x)/2]^2)/2 + (Sqrt[c - d]*d*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/Sqrt[c + d*Sin[e + f*x]] + Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]))/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]) + (I/2)*Sqrt[d]*(-3*c + d)*(((-1/2*I)*d^(3/2)*(-3*c + d)*(I + Tan[(e + f*x)/2])*(((2*I)*(((c + I*d)*Sec[(e + f*x)/2]^2)/2 + ((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2]))/(d^(3/2)*(-3*c + d)*(I + Tan[(e + f*x)/2])) - (I*Sec[(e + f*x)/2]^2*(I*c + d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (c + I*d)*Tan[(e + f*x)/2]))/(d^(3/2)*(-3*c + d)*(I + Tan[(e + f*x)/2])^2)))/(I*c + d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (c + I*d)*Tan[(e + f*x)/2]) + (d^(3/2)*(-3*c + d)*(-I + Tan[(e + f*x)/2])*((-2*(((I*c + d)*Sec[(e + f*x)/2]^2)/2 + ((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2]))/(d^(3/2)*(-3*c + d)*(-I + Tan[(e + f*x)/2])) + (Sec[(e + f*x)/2]^2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))/(d^(3/2)*(-3*c + d)*(-I + Tan[(e + f*x)/2])^2)))/(2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2])))))","C",0
590,1,1251,141,15.1473048,"\int \frac{\sqrt{c+d \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + a*Sin[e + f*x]],x]","\frac{\left(\sqrt{2} \sqrt{c-d} \log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\sqrt{2} \sqrt{c-d} \log \left(c-d+(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)-i \sqrt{d} \left(\log \left(\frac{2 \left(c-i d+(d-i c) \tan \left(\frac{1}{2} (e+f x)\right)+(1-i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{d^{3/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)}\right)-\log \left(\frac{2 \left(c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{d^{3/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)}\right)\right)\right) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a (\sin (e+f x)+1)} \left(\frac{\sqrt{c-d} \sec ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)}-i \sqrt{d} \left(\frac{d^{3/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right) \left(\frac{2 \left(\frac{1}{2} (d-i c) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\frac{(1-i) \sqrt{d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}}{\sqrt{2}}+\frac{(1-i) d^{3/2} \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{2} \sqrt{c+d \sin (e+f x)}}\right)}{d^{3/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)}-\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right) \left(c-i d+(d-i c) \tan \left(\frac{1}{2} (e+f x)\right)+(1-i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{d^{3/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)^2}\right)}{2 \left(c-i d+(d-i c) \tan \left(\frac{1}{2} (e+f x)\right)+(1-i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}-\frac{d^{3/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right) \left(\frac{2 \left(\frac{1}{2} (i c+d) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\frac{(1+i) \sqrt{d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}}{\sqrt{2}}+\frac{(1+i) d^{3/2} \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{2} \sqrt{c+d \sin (e+f x)}}\right)}{d^{3/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)}-\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right) \left(c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{d^{3/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)^2}\right)}{2 \left(c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}\right)-\frac{\sqrt{2} \sqrt{c-d} \left(\frac{1}{2} (d-c) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}+\frac{\sqrt{c-d} d \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{c+d \sin (e+f x)}}\right)}{c-d+(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}}\right)}","-\frac{2 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}-\frac{\sqrt{2} \sqrt{c-d} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}",1,"((Sqrt[2]*Sqrt[c - d]*Log[1 + Tan[(e + f*x)/2]] - Sqrt[2]*Sqrt[c - d]*Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]] - I*Sqrt[d]*(Log[(2*(c - I*d + (1 - I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + ((-I)*c + d)*Tan[(e + f*x)/2]))/(d^(3/2)*(I + Tan[(e + f*x)/2]))] - Log[(2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))/(d^(3/2)*(-I + Tan[(e + f*x)/2]))]))*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a*(1 + Sin[e + f*x])]*((Sqrt[c - d]*Sec[(e + f*x)/2]^2)/(Sqrt[2]*(1 + Tan[(e + f*x)/2])) - (Sqrt[2]*Sqrt[c - d]*(((-c + d)*Sec[(e + f*x)/2]^2)/2 + (Sqrt[c - d]*d*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/Sqrt[c + d*Sin[e + f*x]] + Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]))/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]) - I*Sqrt[d]*((d^(3/2)*(I + Tan[(e + f*x)/2])*((2*((((-I)*c + d)*Sec[(e + f*x)/2]^2)/2 + ((1 - I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 - I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2]))/(d^(3/2)*(I + Tan[(e + f*x)/2])) - (Sec[(e + f*x)/2]^2*(c - I*d + (1 - I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + ((-I)*c + d)*Tan[(e + f*x)/2]))/(d^(3/2)*(I + Tan[(e + f*x)/2])^2)))/(2*(c - I*d + (1 - I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + ((-I)*c + d)*Tan[(e + f*x)/2])) - (d^(3/2)*(-I + Tan[(e + f*x)/2])*((2*(((I*c + d)*Sec[(e + f*x)/2]^2)/2 + ((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2]))/(d^(3/2)*(-I + Tan[(e + f*x)/2])) - (Sec[(e + f*x)/2]^2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))/(d^(3/2)*(-I + Tan[(e + f*x)/2])^2)))/(2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2])))))","C",0
591,1,283,79,4.1019191,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[1/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]),x]","\frac{\log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left((d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d\right)}{f \sqrt{a (\sin (e+f x)+1)} \sqrt{c+d \sin (e+f x)} \left(\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{2 \tan \left(\frac{1}{2} (e+f x)\right)+2}-\frac{\frac{\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} (c \sin (e+f x)+d \cos (e+f x)+d)}{\sqrt{c+d \sin (e+f x)}}-\frac{1}{2} (c-d) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d}\right)}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f \sqrt{c-d}}",1,"(Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]])/(f*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c + d*Sin[e + f*x]]*(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2])))","B",0
592,1,306,131,6.4820482,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx","Integrate[1/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)),x]","\frac{\frac{2 d \cos (e+f x)}{c+d}+\frac{\log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left((d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d\right)}{\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{2 \tan \left(\frac{1}{2} (e+f x)\right)+2}-\frac{\frac{\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} (c \sin (e+f x)+d \cos (e+f x)+d)}{\sqrt{c+d \sin (e+f x)}}-\frac{1}{2} (c-d) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d}}}{f (c-d) \sqrt{a (\sin (e+f x)+1)} \sqrt{c+d \sin (e+f x)}}","\frac{2 d \cos (e+f x)}{f \left(c^2-d^2\right) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f (c-d)^{3/2}}",1,"((2*d*Cos[e + f*x])/(c + d) + (Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]])/(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2])))/((c - d)*f*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c + d*Sin[e + f*x]])","B",0
593,1,387,191,6.4508503,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}} \, dx","Integrate[1/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)),x]","\frac{\frac{2 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) \left(6 c^2+d (5 c+d) \sin (e+f x)+c d-d^2\right)}{(c+d)^2 (c+d \sin (e+f x))}+\frac{3 \left(\log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left((d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d\right)\right)}{\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{2 \tan \left(\frac{1}{2} (e+f x)\right)+2}-\frac{\frac{\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} (c \sin (e+f x)+d \cos (e+f x)+d)}{\sqrt{c+d \sin (e+f x)}}-\frac{1}{2} (c-d) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d}}}{3 f (c-d)^2 \sqrt{a (\sin (e+f x)+1)} \sqrt{c+d \sin (e+f x)}}","\frac{2 d (5 c+d) \cos (e+f x)}{3 f \left(c^2-d^2\right)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}+\frac{2 d \cos (e+f x)}{3 f \left(c^2-d^2\right) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f (c-d)^{5/2}}",1,"((2*d*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(6*c^2 + c*d - d^2 + d*(5*c + d)*Sin[e + f*x]))/((c + d)^2*(c + d*Sin[e + f*x])) + (3*(Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]]))/(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2])))/(3*(c - d)^2*f*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c + d*Sin[e + f*x]])","B",0
594,1,1844,251,17.0202929,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\left(-\frac{(c-d)^2}{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}-d^2 \cos \left(\frac{1}{2} (e+f x)\right)+d^2 \sin \left(\frac{1}{2} (e+f x)\right)+\frac{\sin \left(\frac{1}{2} (e+f x)\right) c^2-2 d \sin \left(\frac{1}{2} (e+f x)\right) c+d^2 \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}\right) \sqrt{c+d \sin (e+f x)} \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{f (a (\sin (e+f x)+1))^{3/2}}+\frac{\left(\frac{(c+9 d) \log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right) (c-d)^{3/2}}{\sqrt{2}}-\frac{(c+9 d) \log \left(c-d+(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right) (c-d)^{3/2}}{\sqrt{2}}+i (5 c-3 d) d^{3/2} \log \left(-\frac{i \left(-i c+d+(c-i d) \tan \left(\frac{1}{2} (e+f x)\right)+(1-i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{d^{5/2} (3 d-5 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)}\right)+i d^{3/2} (3 d-5 c) \log \left(\frac{i \left(i c+d+(c+i d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{d^{5/2} (3 d-5 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)}\right)\right) \left(\frac{c^3}{4 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}+\frac{7 d c^2}{4 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}+\frac{5 d^2 \sin (e+f x) c}{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}-\frac{7 d^2 c}{4 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}-\frac{3 d^3 \sin (e+f x)}{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}+\frac{3 d^3}{4 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{f (a (\sin (e+f x)+1))^{3/2} \left(-\frac{(5 c-3 d) (3 d-5 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right) \left(\frac{i \sec ^2\left(\frac{1}{2} (e+f x)\right) \left(-i c+d+(c-i d) \tan \left(\frac{1}{2} (e+f x)\right)+(1-i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{2 d^{5/2} (3 d-5 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)^2}-\frac{i \left(\frac{1}{2} (c-i d) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\frac{(1-i) \sqrt{d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}}{\sqrt{2}}+\frac{(1-i) d^{3/2} \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{2} \sqrt{c+d \sin (e+f x)}}\right)}{d^{5/2} (3 d-5 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)}\right) d^4}{-i c+d+(c-i d) \tan \left(\frac{1}{2} (e+f x)\right)+(1-i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}}+\frac{(3 d-5 c)^2 \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right) \left(\frac{i \left(\frac{1}{2} (c+i d) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\frac{(1+i) \sqrt{d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}}{\sqrt{2}}+\frac{(1+i) d^{3/2} \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{2} \sqrt{c+d \sin (e+f x)}}\right)}{d^{5/2} (3 d-5 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)}-\frac{i \sec ^2\left(\frac{1}{2} (e+f x)\right) \left(i c+d+(c+i d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{2 d^{5/2} (3 d-5 c) \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)^2}\right) d^4}{i c+d+(c+i d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}}+\frac{(c-d)^{3/2} (c+9 d) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{2 \sqrt{2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)}-\frac{(c-d)^{3/2} (c+9 d) \left(\frac{1}{2} (d-c) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}+\frac{\sqrt{c-d} d \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{2} \left(c-d+(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}\right)}","-\frac{d^{3/2} (5 c-3 d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{a^{3/2} f}-\frac{(c+9 d) (c-d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f (a \sin (e+f x)+a)^{3/2}}+\frac{d (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 a f \sqrt{a \sin (e+f x)+a}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(-(d^2*Cos[(e + f*x)/2]) + d^2*Sin[(e + f*x)/2] - (c - d)^2/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) + (c^2*Sin[(e + f*x)/2] - 2*c*d*Sin[(e + f*x)/2] + d^2*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)*Sqrt[c + d*Sin[e + f*x]])/(f*(a*(1 + Sin[e + f*x]))^(3/2)) + ((((c - d)^(3/2)*(c + 9*d)*Log[1 + Tan[(e + f*x)/2]])/Sqrt[2] + I*(5*c - 3*d)*d^(3/2)*Log[((-I)*((-I)*c + d + (1 - I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (c - I*d)*Tan[(e + f*x)/2]))/(d^(5/2)*(-5*c + 3*d)*(-I + Tan[(e + f*x)/2]))] + I*d^(3/2)*(-5*c + 3*d)*Log[(I*(I*c + d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (c + I*d)*Tan[(e + f*x)/2]))/(d^(5/2)*(-5*c + 3*d)*(I + Tan[(e + f*x)/2]))] - ((c - d)^(3/2)*(c + 9*d)*Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]])/Sqrt[2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c^3/(4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (7*c^2*d)/(4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) - (7*c*d^2)/(4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (3*d^3)/(4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (5*c*d^2*Sin[e + f*x])/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) - (3*d^3*Sin[e + f*x])/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])))/(f*(a*(1 + Sin[e + f*x]))^(3/2)*(((c - d)^(3/2)*(c + 9*d)*Sec[(e + f*x)/2]^2)/(2*Sqrt[2]*(1 + Tan[(e + f*x)/2])) - ((c - d)^(3/2)*(c + 9*d)*(((-c + d)*Sec[(e + f*x)/2]^2)/2 + (Sqrt[c - d]*d*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/Sqrt[c + d*Sin[e + f*x]] + Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]))/(Sqrt[2]*(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2])) - ((5*c - 3*d)*d^4*(-5*c + 3*d)*(-I + Tan[(e + f*x)/2])*(((-I)*(((c - I*d)*Sec[(e + f*x)/2]^2)/2 + ((1 - I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 - I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2]))/(d^(5/2)*(-5*c + 3*d)*(-I + Tan[(e + f*x)/2])) + ((I/2)*Sec[(e + f*x)/2]^2*((-I)*c + d + (1 - I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (c - I*d)*Tan[(e + f*x)/2]))/(d^(5/2)*(-5*c + 3*d)*(-I + Tan[(e + f*x)/2])^2)))/((-I)*c + d + (1 - I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (c - I*d)*Tan[(e + f*x)/2]) + (d^4*(-5*c + 3*d)^2*(I + Tan[(e + f*x)/2])*((I*(((c + I*d)*Sec[(e + f*x)/2]^2)/2 + ((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2]))/(d^(5/2)*(-5*c + 3*d)*(I + Tan[(e + f*x)/2])) - ((I/2)*Sec[(e + f*x)/2]^2*(I*c + d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (c + I*d)*Tan[(e + f*x)/2]))/(d^(5/2)*(-5*c + 3*d)*(I + Tan[(e + f*x)/2])^2)))/(I*c + d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (c + I*d)*Tan[(e + f*x)/2])))","C",0
595,1,1625,194,17.1169381,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(c + d*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\left(\frac{d-c}{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{c \sin \left(\frac{1}{2} (e+f x)\right)-d \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}\right) \sqrt{c+d \sin (e+f x)} \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{f (a (\sin (e+f x)+1))^{3/2}}+\frac{\left(-4 i \sqrt{c-d} \left(\log \left(\frac{c-i \left(d+(1+i) \sqrt{2} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)} \sqrt{d}\right)+(d-i c) \tan \left(\frac{1}{2} (e+f x)\right)}{2 d^{5/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)}\right)-\log \left(\frac{c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}}{2 d^{5/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)}\right)\right) d^{3/2}+\sqrt{2} \left(c^2+4 d c-5 d^2\right) \log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\sqrt{2} \left(c^2+4 d c-5 d^2\right) \log \left(c-d+(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)\right) \left(\frac{c^2}{4 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}+\frac{d c}{\left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}+\frac{d^2 \sin (e+f x)}{\left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}-\frac{d^2}{4 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{f (a (\sin (e+f x)+1))^{3/2} \left(\frac{\left(c^2+4 d c-5 d^2\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)}-4 i \sqrt{c-d} d^{3/2} \left(\frac{2 d^{5/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right) \left(\frac{\frac{1}{2} (d-i c) \sec ^2\left(\frac{1}{2} (e+f x)\right)-i \left(\frac{(1+i) \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}} d^{3/2}}{\sqrt{2} \sqrt{c+d \sin (e+f x)}}+\frac{(1+i) \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)} \sqrt{d}}{\sqrt{2}}\right)}{2 d^{5/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)}-\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right) \left(c-i \left(d+(1+i) \sqrt{2} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)} \sqrt{d}\right)+(d-i c) \tan \left(\frac{1}{2} (e+f x)\right)\right)}{4 d^{5/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)^2}\right)}{c-i \left(d+(1+i) \sqrt{2} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)} \sqrt{d}\right)+(d-i c) \tan \left(\frac{1}{2} (e+f x)\right)}-\frac{2 d^{5/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right) \left(\frac{\frac{1}{2} (i c+d) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\frac{(1+i) \sqrt{d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}}{\sqrt{2}}+\frac{(1+i) d^{3/2} \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{2} \sqrt{c+d \sin (e+f x)}}}{2 d^{5/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)}-\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right) \left(c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{4 d^{5/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)^2}\right)}{c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}}\right)-\frac{\sqrt{2} \left(c^2+4 d c-5 d^2\right) \left(\frac{1}{2} (d-c) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}+\frac{\sqrt{c-d} d \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{c+d \sin (e+f x)}}\right)}{c-d+(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}}\right)}","-\frac{2 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{a^{3/2} f}-\frac{\sqrt{c-d} (c+5 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*((-c + d)/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) + (c*Sin[(e + f*x)/2] - d*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)*Sqrt[c + d*Sin[e + f*x]])/(f*(a*(1 + Sin[e + f*x]))^(3/2)) + ((Sqrt[2]*(c^2 + 4*c*d - 5*d^2)*Log[1 + Tan[(e + f*x)/2]] - Sqrt[2]*(c^2 + 4*c*d - 5*d^2)*Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]] - (4*I)*Sqrt[c - d]*d^(3/2)*(Log[(c - I*(d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]]) + ((-I)*c + d)*Tan[(e + f*x)/2])/(2*d^(5/2)*(I + Tan[(e + f*x)/2]))] - Log[(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2])/(2*d^(5/2)*(-I + Tan[(e + f*x)/2]))]))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c^2/(4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (c*d)/((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) - d^2/(4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (d^2*Sin[e + f*x])/((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])))/(f*(a*(1 + Sin[e + f*x]))^(3/2)*(((c^2 + 4*c*d - 5*d^2)*Sec[(e + f*x)/2]^2)/(Sqrt[2]*(1 + Tan[(e + f*x)/2])) - (Sqrt[2]*(c^2 + 4*c*d - 5*d^2)*(((-c + d)*Sec[(e + f*x)/2]^2)/2 + (Sqrt[c - d]*d*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/Sqrt[c + d*Sin[e + f*x]] + Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]))/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]) - (4*I)*Sqrt[c - d]*d^(3/2)*((2*d^(5/2)*(I + Tan[(e + f*x)/2])*(((((-I)*c + d)*Sec[(e + f*x)/2]^2)/2 - I*(((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2]))/(2*d^(5/2)*(I + Tan[(e + f*x)/2])) - (Sec[(e + f*x)/2]^2*(c - I*(d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]]) + ((-I)*c + d)*Tan[(e + f*x)/2]))/(4*d^(5/2)*(I + Tan[(e + f*x)/2])^2)))/(c - I*(d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]]) + ((-I)*c + d)*Tan[(e + f*x)/2]) - (2*d^(5/2)*(-I + Tan[(e + f*x)/2])*((((I*c + d)*Sec[(e + f*x)/2]^2)/2 + ((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2])/(2*d^(5/2)*(-I + Tan[(e + f*x)/2])) - (Sec[(e + f*x)/2]^2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))/(4*d^(5/2)*(-I + Tan[(e + f*x)/2])^2)))/(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))))","C",0
596,1,372,126,5.3705766,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[Sqrt[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)^2 \left(\frac{(c+d) \left(\log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left((d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d\right)\right)}{\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{2 \tan \left(\frac{1}{2} (e+f x)\right)+2}-\frac{\frac{\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} (c \sin (e+f x)+d \cos (e+f x)+d)}{\sqrt{c+d \sin (e+f x)}}-\frac{1}{2} (c-d) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d}}-\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sin (e+f x))}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}\right)}{4 f (a (\sin (e+f x)+1))^{3/2} \sqrt{c+d \sin (e+f x)}}","-\frac{(c+d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f \sqrt{c-d}}-\frac{\cos (e+f x) \sqrt{c+d \sin (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*((-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + ((c + d)*(Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]]))/(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]))))/(4*f*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c + d*Sin[e + f*x]])","B",0
597,1,381,135,5.6709208,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^(3/2)*Sqrt[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)^2 \left(\frac{(c-3 d) \left(\log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left((d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d\right)\right)}{\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{2 \tan \left(\frac{1}{2} (e+f x)\right)+2}-\frac{\frac{\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} (c \sin (e+f x)+d \cos (e+f x)+d)}{\sqrt{c+d \sin (e+f x)}}-\frac{1}{2} (c-d) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d}}-\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sin (e+f x))}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}\right)}{4 f (c-d) (a (\sin (e+f x)+1))^{3/2} \sqrt{c+d \sin (e+f x)}}","-\frac{(c-3 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^{3/2}}-\frac{\cos (e+f x) \sqrt{c+d \sin (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*((-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + ((c - 3*d)*(Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]]))/(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]))))/(4*(c - d)*f*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c + d*Sin[e + f*x]])","B",0
598,1,401,197,6.236893,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^(3/2)*(c + d*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)^2 \left(\frac{(c-7 d) \left(\log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left((d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d\right)\right)}{\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{2 \tan \left(\frac{1}{2} (e+f x)\right)+2}-\frac{\frac{\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} (c \sin (e+f x)+d \cos (e+f x)+d)}{\sqrt{c+d \sin (e+f x)}}-\frac{1}{2} (c-d) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d}}-\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(c^2+d (c+5 d) \sin (e+f x)+c d+4 d^2\right)}{(c+d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}\right)}{4 f (c-d)^2 (a (\sin (e+f x)+1))^{3/2} \sqrt{c+d \sin (e+f x)}}","-\frac{(c-7 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^{5/2}}-\frac{d (c+5 d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} \sqrt{c+d \sin (e+f x)}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*((-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c^2 + c*d + 4*d^2 + d*(c + 5*d)*Sin[e + f*x]))/((c + d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) + ((c - 7*d)*(Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]]))/(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]))))/(4*(c - d)^2*f*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c + d*Sin[e + f*x]])","B",0
599,1,478,271,9.4532831,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^(3/2)*(c + d*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)^2 \left(\frac{3 (c-11 d) \left(\log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left((d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d\right)\right)}{\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{2 \tan \left(\frac{1}{2} (e+f x)\right)+2}-\frac{\frac{\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} (c \sin (e+f x)+d \cos (e+f x)+d)}{\sqrt{c+d \sin (e+f x)}}-\frac{1}{2} (c-d) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d}}-\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(6 c^4+12 c^3 d-d^2 \left(3 c^2+38 c d+19 d^2\right) \cos (2 (e+f x))+81 c^2 d^2+12 d \left(c^3+8 c^2 d+9 c d^2+2 d^3\right) \sin (e+f x)+70 c d^3+11 d^4\right)}{(c+d)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sin (e+f x))}\right)}{12 f (c-d)^3 (a (\sin (e+f x)+1))^{3/2} \sqrt{c+d \sin (e+f x)}}","-\frac{(c-11 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^{7/2}}-\frac{d \left(3 c^2+38 c d+19 d^2\right) \cos (e+f x)}{6 a f (c-d)^3 (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{d (3 c+7 d) \cos (e+f x)}{6 a f (c-d)^2 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(-(((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(6*c^4 + 12*c^3*d + 81*c^2*d^2 + 70*c*d^3 + 11*d^4 - d^2*(3*c^2 + 38*c*d + 19*d^2)*Cos[2*(e + f*x)] + 12*d*(c^3 + 8*c^2*d + 9*c*d^2 + 2*d^3)*Sin[e + f*x]))/((c + d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))) + (3*(c - 11*d)*(Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]]))/(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]))))/(12*(c - d)^3*f*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c + d*Sin[e + f*x]])","A",0
600,1,1845,260,17.4353617,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^(5/2),x]","\frac{\left(-\frac{(c-d)^2}{4 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}-\frac{3 (c+5 d) (c-d)}{16 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{3 \left(\sin \left(\frac{1}{2} (e+f x)\right) c^2+4 d \sin \left(\frac{1}{2} (e+f x)\right) c-5 d^2 \sin \left(\frac{1}{2} (e+f x)\right)\right)}{8 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{\sin \left(\frac{1}{2} (e+f x)\right) c^2-2 d \sin \left(\frac{1}{2} (e+f x)\right) c+d^2 \sin \left(\frac{1}{2} (e+f x)\right)}{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}\right) \sqrt{c+d \sin (e+f x)} \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}{f (a (\sin (e+f x)+1))^{5/2}}+\frac{\left(-32 i \sqrt{c-d} \left(\log \left(\frac{c-i \left(d+(1+i) \sqrt{2} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)} \sqrt{d}\right)+(d-i c) \tan \left(\frac{1}{2} (e+f x)\right)}{16 d^{7/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)}\right)-\log \left(\frac{c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}}{16 d^{7/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)}\right)\right) d^{5/2}+\sqrt{2} \left(3 c^3+11 d c^2+29 d^2 c-43 d^3\right) \log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\sqrt{2} \left(3 c^3+11 d c^2+29 d^2 c-43 d^3\right) \log \left(c-d+(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)\right) \left(\frac{3 c^3}{32 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}+\frac{11 d c^2}{32 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}+\frac{29 d^2 c}{32 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}+\frac{d^3 \sin (e+f x)}{\left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}-\frac{11 d^3}{32 \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)}}\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}{f (a (\sin (e+f x)+1))^{5/2} \left(-32 i \sqrt{c-d} \left(\frac{16 d^{7/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right) \left(\frac{\frac{1}{2} (d-i c) \sec ^2\left(\frac{1}{2} (e+f x)\right)-i \left(\frac{(1+i) \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}} d^{3/2}}{\sqrt{2} \sqrt{c+d \sin (e+f x)}}+\frac{(1+i) \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)} \sqrt{d}}{\sqrt{2}}\right)}{16 d^{7/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)}-\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right) \left(c-i \left(d+(1+i) \sqrt{2} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)} \sqrt{d}\right)+(d-i c) \tan \left(\frac{1}{2} (e+f x)\right)\right)}{32 d^{7/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+i\right)^2}\right)}{c-i \left(d+(1+i) \sqrt{2} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)} \sqrt{d}\right)+(d-i c) \tan \left(\frac{1}{2} (e+f x)\right)}-\frac{16 d^{7/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right) \left(\frac{\frac{1}{2} (i c+d) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\frac{(1+i) \sqrt{d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}}{\sqrt{2}}+\frac{(1+i) d^{3/2} \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{2} \sqrt{c+d \sin (e+f x)}}}{16 d^{7/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)}-\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right) \left(c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}\right)}{32 d^{7/2} \left(\tan \left(\frac{1}{2} (e+f x)\right)-i\right)^2}\right)}{c+i d+(i c+d) \tan \left(\frac{1}{2} (e+f x)\right)+(1+i) \sqrt{2} \sqrt{d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}}\right) d^{5/2}+\frac{\left(3 c^3+11 d c^2+29 d^2 c-43 d^3\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{2} \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)}-\frac{\sqrt{2} \left(3 c^3+11 d c^2+29 d^2 c-43 d^3\right) \left(\frac{1}{2} (d-c) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sin (e+f x) \sqrt{c+d \sin (e+f x)}+\frac{\sqrt{c-d} d \cos (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}}}{\sqrt{c+d \sin (e+f x)}}\right)}{c-d+(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}}\right)}","-\frac{\sqrt{c-d} \left(3 c^2+14 c d+43 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f}-\frac{2 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{a^{5/2} f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}-\frac{(c-d) (3 c+11 d) \cos (e+f x) \sqrt{c+d \sin (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])^5*(-1/4*(c - d)^2/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - (3*(c - d)*(c + 5*d))/(16*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) + (3*(c^2*Sin[(e + f*x)/2] + 4*c*d*Sin[(e + f*x)/2] - 5*d^2*Sin[(e + f*x)/2]))/(8*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (c^2*Sin[(e + f*x)/2] - 2*c*d*Sin[(e + f*x)/2] + d^2*Sin[(e + f*x)/2])/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4))*Sqrt[c + d*Sin[e + f*x]])/(f*(a*(1 + Sin[e + f*x]))^(5/2)) + ((Sqrt[2]*(3*c^3 + 11*c^2*d + 29*c*d^2 - 43*d^3)*Log[1 + Tan[(e + f*x)/2]] - Sqrt[2]*(3*c^3 + 11*c^2*d + 29*c*d^2 - 43*d^3)*Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]] - (32*I)*Sqrt[c - d]*d^(5/2)*(Log[(c - I*(d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]]) + ((-I)*c + d)*Tan[(e + f*x)/2])/(16*d^(7/2)*(I + Tan[(e + f*x)/2]))] - Log[(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2])/(16*d^(7/2)*(-I + Tan[(e + f*x)/2]))]))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*((3*c^3)/(32*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (11*c^2*d)/(32*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (29*c*d^2)/(32*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) - (11*d^3)/(32*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (d^3*Sin[e + f*x])/((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])))/(f*(a*(1 + Sin[e + f*x]))^(5/2)*(((3*c^3 + 11*c^2*d + 29*c*d^2 - 43*d^3)*Sec[(e + f*x)/2]^2)/(Sqrt[2]*(1 + Tan[(e + f*x)/2])) - (Sqrt[2]*(3*c^3 + 11*c^2*d + 29*c*d^2 - 43*d^3)*(((-c + d)*Sec[(e + f*x)/2]^2)/2 + (Sqrt[c - d]*d*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/Sqrt[c + d*Sin[e + f*x]] + Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]))/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]) - (32*I)*Sqrt[c - d]*d^(5/2)*((16*d^(7/2)*(I + Tan[(e + f*x)/2])*(((((-I)*c + d)*Sec[(e + f*x)/2]^2)/2 - I*(((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2]))/(16*d^(7/2)*(I + Tan[(e + f*x)/2])) - (Sec[(e + f*x)/2]^2*(c - I*(d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]]) + ((-I)*c + d)*Tan[(e + f*x)/2]))/(32*d^(7/2)*(I + Tan[(e + f*x)/2])^2)))/(c - I*(d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]]) + ((-I)*c + d)*Tan[(e + f*x)/2]) - (16*d^(7/2)*(-I + Tan[(e + f*x)/2])*((((I*c + d)*Sec[(e + f*x)/2]^2)/2 + ((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2])/(16*d^(7/2)*(-I + Tan[(e + f*x)/2])) - (Sec[(e + f*x)/2]^2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))/(32*d^(7/2)*(-I + Tan[(e + f*x)/2])^2)))/(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))))","C",0
601,1,396,184,7.2194711,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(c + d*Sin[e + f*x])^(3/2)/(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)^4 \left(\frac{3 (c+d)^2 \left(\log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left((d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d\right)\right)}{\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{2 \tan \left(\frac{1}{2} (e+f x)\right)+2}-\frac{\frac{\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} (c \sin (e+f x)+d \cos (e+f x)+d)}{\sqrt{c+d \sin (e+f x)}}-\frac{1}{2} (c-d) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d}}-\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (c+d \sin (e+f x)) ((3 c+7 d) \sin (e+f x)+7 c+3 d)}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}\right)}{32 f (a (\sin (e+f x)+1))^{5/2} \sqrt{c+d \sin (e+f x)}}","-\frac{3 (c+d)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f \sqrt{c-d}}-\frac{(3 c+7 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*((-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c + d*Sin[e + f*x])*(7*c + 3*d + (3*c + 7*d)*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (3*(c + d)^2*(Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]]))/(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]))))/(32*f*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c + d*Sin[e + f*x]])","B",0
602,1,412,191,7.5992442,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[Sqrt[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)^4 \left(\frac{\left(3 c^2-2 c d-5 d^2\right) \left(\log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left((d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d\right)\right)}{\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{2 \tan \left(\frac{1}{2} (e+f x)\right)+2}-\frac{\frac{\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} (c \sin (e+f x)+d \cos (e+f x)+d)}{\sqrt{c+d \sin (e+f x)}}-\frac{1}{2} (c-d) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d}}-\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) ((3 c-d) \sin (e+f x)+7 c-5 d) (c+d \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}\right)}{32 f (c-d) (a (\sin (e+f x)+1))^{5/2} \sqrt{c+d \sin (e+f x)}}","-\frac{(3 c-5 d) (c+d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^{3/2}}-\frac{(3 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{16 a f (c-d) (a \sin (e+f x)+a)^{3/2}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*((-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(7*c - 5*d + (3*c - d)*Sin[e + f*x])*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + ((3*c^2 - 2*c*d - 5*d^2)*(Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]]))/(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]))))/(32*(c - d)*f*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c + d*Sin[e + f*x]])","B",0
603,1,411,201,7.1331512,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^(5/2)*Sqrt[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)^4 \left(\frac{\left(3 c^2-10 c d+19 d^2\right) \left(\log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left((d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d\right)\right)}{\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{2 \tan \left(\frac{1}{2} (e+f x)\right)+2}-\frac{\frac{\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} (c \sin (e+f x)+d \cos (e+f x)+d)}{\sqrt{c+d \sin (e+f x)}}-\frac{1}{2} (c-d) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d}}-\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (3 (c-3 d) \sin (e+f x)+7 c-13 d) (c+d \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}\right)}{32 f (c-d)^2 (a (\sin (e+f x)+1))^{5/2} \sqrt{c+d \sin (e+f x)}}","-\frac{\left(3 c^2-10 c d+19 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^{5/2}}-\frac{3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2}}-\frac{\cos (e+f x) \sqrt{c+d \sin (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])^4*((-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(7*c - 13*d + 3*(c - 3*d)*Sin[e + f*x])*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + ((3*c^2 - 10*c*d + 19*d^2)*(Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]]))/(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]))))/(32*(c - d)^2*f*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c + d*Sin[e + f*x]])","B",0
604,1,462,270,8.9644973,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*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)^4 \left(\frac{3 \left(c^2-6 c d+25 d^2\right) \left(\log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left((d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d\right)\right)}{\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{2 \tan \left(\frac{1}{2} (e+f x)\right)+2}-\frac{\frac{\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} (c \sin (e+f x)+d \cos (e+f x)+d)}{\sqrt{c+d \sin (e+f x)}}-\frac{1}{2} (c-d) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d}}+\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-14 c^3+d \left(3 c^2-14 c d-49 d^2\right) \cos (2 (e+f x))+25 c^2 d+\left(-6 c^3+14 c^2 d+62 c d^2+170 d^3\right) \sin (e+f x)+56 c d^2+113 d^3\right)}{(c+d) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}\right)}{32 f (c-d)^3 (a (\sin (e+f x)+1))^{5/2} \sqrt{c+d \sin (e+f x)}}","-\frac{3 \left(c^2-6 c d+25 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^{7/2}}-\frac{d (c-7 d) (3 c+7 d) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{(3 c-13 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} \sqrt{c+d \sin (e+f x)}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(-14*c^3 + 25*c^2*d + 56*c*d^2 + 113*d^3 + d*(3*c^2 - 14*c*d - 49*d^2)*Cos[2*(e + f*x)] + (-6*c^3 + 14*c^2*d + 62*c*d^2 + 170*d^3)*Sin[e + f*x]))/((c + d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3) + (3*(c^2 - 6*c*d + 25*d^2)*(Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]]))/(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]))))/(32*(c - d)^3*f*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c + d*Sin[e + f*x]])","A",0
605,1,717,355,10.4275139,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx","Integrate[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(5/2)),x]","\frac{\left(3 c^2-26 c d+163 d^2\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \left(\log \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left((d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d\right)\right)}{32 f (c-d)^4 (a (\sin (e+f x)+1))^{5/2} \sqrt{c+d \sin (e+f x)} \left(\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{2 \tan \left(\frac{1}{2} (e+f x)\right)+2}-\frac{\frac{\sqrt{c-d} \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} (c \sin (e+f x)+d \cos (e+f x)+d)}{\sqrt{c+d \sin (e+f x)}}-\frac{1}{2} (c-d) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)+2 \sqrt{c-d} \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{c+d \sin (e+f x)}+c-d}\right)}+\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5 \sqrt{c+d \sin (e+f x)} \left(\frac{2 \left(d^3 \cos \left(\frac{1}{2} (e+f x)\right)-d^3 \sin \left(\frac{1}{2} (e+f x)\right)\right)}{3 (c-d)^3 (c+d) (c+d \sin (e+f x))^2}+\frac{2 \left(-11 c d^3 \sin \left(\frac{1}{2} (e+f x)\right)+11 c d^3 \cos \left(\frac{1}{2} (e+f x)\right)-7 d^4 \sin \left(\frac{1}{2} (e+f x)\right)+7 d^4 \cos \left(\frac{1}{2} (e+f x)\right)\right)}{3 (c-d)^4 (c+d)^2 (c+d \sin (e+f x))}+\frac{25 d-3 c}{16 (c-d)^4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{3 c \sin \left(\frac{1}{2} (e+f x)\right)-25 d \sin \left(\frac{1}{2} (e+f x)\right)}{8 (c-d)^4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{1}{4 (c-d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{\sin \left(\frac{1}{2} (e+f x)\right)}{2 (c-d)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4}\right)}{f (a (\sin (e+f x)+1))^{5/2}}","-\frac{\left(3 c^2-26 c d+163 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^{9/2}}-\frac{d \left(9 c^2-54 c d-95 d^2\right) \cos (e+f x)}{48 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{d \left(9 c^3-57 c^2 d-493 c d^2-299 d^3\right) \cos (e+f x)}{48 a^2 f (c-d)^4 (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{(3 c-17 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))^{3/2}}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}",1,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sqrt[c + d*Sin[e + f*x]]*(Sin[(e + f*x)/2]/(2*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - 1/(4*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3) + (-3*c + 25*d)/(16*(c - d)^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) + (3*c*Sin[(e + f*x)/2] - 25*d*Sin[(e + f*x)/2])/(8*(c - d)^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (2*(d^3*Cos[(e + f*x)/2] - d^3*Sin[(e + f*x)/2]))/(3*(c - d)^3*(c + d)*(c + d*Sin[e + f*x])^2) + (2*(11*c*d^3*Cos[(e + f*x)/2] + 7*d^4*Cos[(e + f*x)/2] - 11*c*d^3*Sin[(e + f*x)/2] - 7*d^4*Sin[(e + f*x)/2]))/(3*(c - d)^4*(c + d)^2*(c + d*Sin[e + f*x]))))/(f*(a*(1 + Sin[e + f*x]))^(5/2)) + ((3*c^2 - 26*c*d + 163*d^2)*(Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/(32*(c - d)^4*f*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c + d*Sin[e + f*x]]*(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2])))","B",0
606,1,373,129,1.4148481,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Integrate[(a + a*Sin[e + f*x])^m*(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 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)}{f \left(\sin ^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)\right)}","\frac{\sqrt{2} \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)}}",1,"(6*(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)]*(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*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(-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)])*Sin[(2*e - Pi + 2*f*x)/4]^2))","B",0
607,1,3599,320,57.573425,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^3,x]","\text{Result too large to show}","-\frac{d \left(2 c^2 \left(m^2+6 m+8\right)-c d \left(-2 m^2-3 m+5\right)+d^2 (m+4)\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1) (m+2) (m+3)}-\frac{2^{m+\frac{1}{2}} \left(c^3 \left(m^3+6 m^2+11 m+6\right)+3 c^2 d m \left(m^2+5 m+6\right)+3 c d^2 \left(m^3+4 m^2+4 m+3\right)+d^3 m \left(m^2+3 m+5\right)\right) \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) (m+2) (m+3)}-\frac{d^2 (c (m+5)+d m) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (m+2) (m+3)}-\frac{d \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^2}{f (m+3)}",1,"(-22*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 - m)*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(-1/2 + m)*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^3*(945*(c + d)^3*Gamma[1/2 - m]*Hypergeometric2F1[1/2, 1/2 - m, 9/2, Sin[(-e + Pi/2 - f*x)/2]^2] - 1890*d*(c + d)^2*Gamma[1/2 - m]*Hypergeometric2F1[1/2, 1/2 - m, 9/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2 + 142*(c + d)^3*Gamma[3/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 11/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2 + 60*(c + d)^3*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 3/2 - m}, {1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2 + 8*(c + d)^3*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 2, 3/2 - m}, {1, 1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2 + 2268*d^2*(c + d)*Gamma[1/2 - m]*Hypergeometric2F1[1/2, 1/2 - m, 9/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^4 - 564*d*(c + d)^2*Gamma[3/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 11/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^4 - 312*d*(c + d)^2*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 3/2 - m}, {1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^4 - 48*d*(c + d)^2*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 2, 3/2 - m}, {1, 1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^4 - 1080*d^3*Gamma[1/2 - m]*Hypergeometric2F1[1/2, 1/2 - m, 9/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^6 + 744*d^2*(c + d)*Gamma[3/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 11/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^6 + 528*d^2*(c + d)*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 3/2 - m}, {1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^6 + 96*d^2*(c + d)*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 2, 3/2 - m}, {1, 1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^6 - 368*d^3*Gamma[3/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 11/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^8 - 288*d^3*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 3/2 - m}, {1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^8 - 64*d^3*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 2, 3/2 - m}, {1, 1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^8)*(a + a*Sin[e + f*x])^m*Tan[(-e + Pi/2 - f*x)/2])/(3*f*(3465*(c + d)^3*Gamma[1/2 - m]*Hypergeometric2F1[1/2, 1/2 - m, 9/2, Sin[(-e + Pi/2 - f*x)/2]^2] - 20790*d*(c + d)^2*Gamma[1/2 - m]*Hypergeometric2F1[1/2, 1/2 - m, 9/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2 - 385*(c + d)^3*(-1 + 2*m)*Gamma[1/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 11/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2 + 1562*(c + d)^3*Gamma[3/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 11/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2 + 660*(c + d)^3*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 3/2 - m}, {1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2 + 88*(c + d)^3*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 2, 3/2 - m}, {1, 1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2 + 41580*d^2*(c + d)*Gamma[1/2 - m]*Hypergeometric2F1[1/2, 1/2 - m, 9/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^4 + 770*d*(c + d)^2*(-1 + 2*m)*Gamma[1/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 11/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^4 - 10340*d*(c + d)^2*Gamma[3/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 11/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^4 - 142*(c + d)^3*(-3 + 2*m)*Gamma[3/2 - m]*Hypergeometric2F1[5/2, 5/2 - m, 13/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^4 - 5720*d*(c + d)^2*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 3/2 - m}, {1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^4 - 120*(c + d)^3*(-3 + 2*m)*Gamma[3/2 - m]*HypergeometricPFQ[{5/2, 3, 5/2 - m}, {2, 13/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^4 - 880*d*(c + d)^2*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 2, 3/2 - m}, {1, 1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^4 - 32*(c + d)^3*(-3 + 2*m)*Gamma[3/2 - m]*HypergeometricPFQ[{5/2, 3, 3, 5/2 - m}, {2, 2, 13/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^4 - 27720*d^3*Gamma[1/2 - m]*Hypergeometric2F1[1/2, 1/2 - m, 9/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^6 - 924*d^2*(c + d)*(-1 + 2*m)*Gamma[1/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 11/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^6 + 19096*d^2*(c + d)*Gamma[3/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 11/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^6 + 564*d*(c + d)^2*(-3 + 2*m)*Gamma[3/2 - m]*Hypergeometric2F1[5/2, 5/2 - m, 13/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^6 + 13552*d^2*(c + d)*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 3/2 - m}, {1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^6 + 624*d*(c + d)^2*(-3 + 2*m)*Gamma[3/2 - m]*HypergeometricPFQ[{5/2, 3, 5/2 - m}, {2, 13/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^6 + 2464*d^2*(c + d)*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 2, 3/2 - m}, {1, 1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^6 + 192*d*(c + d)^2*(-3 + 2*m)*Gamma[3/2 - m]*HypergeometricPFQ[{5/2, 3, 3, 5/2 - m}, {2, 2, 13/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^6 + 440*d^3*(-1 + 2*m)*Gamma[1/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 11/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^8 - 12144*d^3*Gamma[3/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 11/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^8 - 744*d^2*(c + d)*(-3 + 2*m)*Gamma[3/2 - m]*Hypergeometric2F1[5/2, 5/2 - m, 13/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^8 - 9504*d^3*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 3/2 - m}, {1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^8 - 1056*d^2*(c + d)*(-3 + 2*m)*Gamma[3/2 - m]*HypergeometricPFQ[{5/2, 3, 5/2 - m}, {2, 13/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^8 - 2112*d^3*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 2, 3/2 - m}, {1, 1, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^8 - 384*d^2*(c + d)*(-3 + 2*m)*Gamma[3/2 - m]*HypergeometricPFQ[{5/2, 3, 3, 5/2 - m}, {2, 2, 13/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^8 + 368*d^3*(-3 + 2*m)*Gamma[3/2 - m]*Hypergeometric2F1[5/2, 5/2 - m, 13/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^10 + 576*d^3*(-3 + 2*m)*Gamma[3/2 - m]*HypergeometricPFQ[{5/2, 3, 5/2 - m}, {2, 13/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^10 + 256*d^3*(-3 + 2*m)*Gamma[3/2 - m]*HypergeometricPFQ[{5/2, 3, 3, 5/2 - m}, {2, 2, 13/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^10))","B",0
608,1,1774,193,65.6480522,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2,x]","-\frac{2 \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)^{\frac{1}{2}-m} \left(1-\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{m-\frac{1}{2}} \left(-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d\right)^2 \left(4 \Gamma \left(\frac{3}{2}-m\right) \, _3F_2\left(\frac{3}{2},2,\frac{3}{2}-m;1,\frac{9}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d\right)^2 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+16 \Gamma \left(\frac{3}{2}-m\right) \, _2F_1\left(\frac{3}{2},\frac{3}{2}-m;\frac{9}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(c^2+d \left(2-3 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) c+d^2 \left(2 \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)-3 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+1\right)\right) \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+7 \Gamma \left(\frac{1}{2}-m\right) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{7}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(15 c^2+10 d \left(3-2 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) c+d^2 \left(12 \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)-20 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+15\right)\right)\right) (\sin (e+f x) a+a)^m \tan \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{f \left(4 \Gamma \left(\frac{3}{2}-m\right) \, _3F_2\left(\frac{3}{2},2,\frac{3}{2}-m;1,\frac{9}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d\right)^2 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+16 \Gamma \left(\frac{3}{2}-m\right) \, _2F_1\left(\frac{3}{2},\frac{3}{2}-m;\frac{9}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(c^2+d \left(2-3 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) c+d^2 \left(2 \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)-3 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+1\right)\right) \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+\frac{2}{3} \left(-4 (2 m-3) \Gamma \left(\frac{3}{2}-m\right) \, _3F_2\left(\frac{5}{2},3,\frac{5}{2}-m;2,\frac{11}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d\right)^2 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)-48 d \Gamma \left(\frac{3}{2}-m\right) \, _3F_2\left(\frac{3}{2},2,\frac{3}{2}-m;1,\frac{9}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d\right) \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)-8 (2 m-3) \Gamma \left(\frac{3}{2}-m\right) \, _2F_1\left(\frac{5}{2},\frac{5}{2}-m;\frac{11}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(c^2+d \left(2-3 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) c+d^2 \left(2 \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)-3 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+1\right)\right) \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+48 d \Gamma \left(\frac{3}{2}-m\right) \, _2F_1\left(\frac{3}{2},\frac{3}{2}-m;\frac{9}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(d \sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(4 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)-3\right)-3 c \sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+12 \Gamma \left(\frac{3}{2}-m\right) \, _3F_2\left(\frac{3}{2},2,\frac{3}{2}-m;1,\frac{9}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d\right)^2+84 d \Gamma \left(\frac{1}{2}-m\right) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{7}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(d \left(6 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)-5\right)-5 c\right)+48 \Gamma \left(\frac{3}{2}-m\right) \, _2F_1\left(\frac{3}{2},\frac{3}{2}-m;\frac{9}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(c^2+d \left(2-3 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) c+d^2 \left(2 \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)-3 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+1\right)\right)+3 \left(\frac{1}{2}-m\right) \Gamma \left(\frac{1}{2}-m\right) \, _2F_1\left(\frac{3}{2},\frac{3}{2}-m;\frac{9}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(15 c^2+10 d \left(3-2 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) c+d^2 \left(12 \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)-20 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+15\right)\right)\right) \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+7 \Gamma \left(\frac{1}{2}-m\right) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{7}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(15 c^2+10 d \left(3-2 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) c+d^2 \left(12 \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)-20 \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+15\right)\right)\right)}","-\frac{2^{m+\frac{1}{2}} \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) \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) (m+2)}+\frac{d (d-2 c (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1) (m+2)}-\frac{d^2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (m+2)}",1,"(-2*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 - m)*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(-1/2 + m)*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^2*(4*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 3/2 - m}, {1, 9/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^2 + 16*Gamma[3/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 9/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2*(c^2 + c*d*(2 - 3*Sin[(-e + Pi/2 - f*x)/2]^2) + d^2*(1 - 3*Sin[(-e + Pi/2 - f*x)/2]^2 + 2*Sin[(-e + Pi/2 - f*x)/2]^4)) + 7*Gamma[1/2 - m]*Hypergeometric2F1[1/2, 1/2 - m, 7/2, Sin[(-e + Pi/2 - f*x)/2]^2]*(15*c^2 + 10*c*d*(3 - 2*Sin[(-e + Pi/2 - f*x)/2]^2) + d^2*(15 - 20*Sin[(-e + Pi/2 - f*x)/2]^2 + 12*Sin[(-e + Pi/2 - f*x)/2]^4)))*(a + a*Sin[e + f*x])^m*Tan[(-e + Pi/2 - f*x)/2])/(f*(4*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 3/2 - m}, {1, 9/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^2 + 16*Gamma[3/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 9/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2*(c^2 + c*d*(2 - 3*Sin[(-e + Pi/2 - f*x)/2]^2) + d^2*(1 - 3*Sin[(-e + Pi/2 - f*x)/2]^2 + 2*Sin[(-e + Pi/2 - f*x)/2]^4)) + 7*Gamma[1/2 - m]*Hypergeometric2F1[1/2, 1/2 - m, 7/2, Sin[(-e + Pi/2 - f*x)/2]^2]*(15*c^2 + 10*c*d*(3 - 2*Sin[(-e + Pi/2 - f*x)/2]^2) + d^2*(15 - 20*Sin[(-e + Pi/2 - f*x)/2]^2 + 12*Sin[(-e + Pi/2 - f*x)/2]^4)) + (2*Sin[(-e + Pi/2 - f*x)/2]^2*(-48*d*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 3/2 - m}, {1, 9/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2) + 12*Gamma[3/2 - m]*HypergeometricPFQ[{3/2, 2, 3/2 - m}, {1, 9/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^2 - 4*(-3 + 2*m)*Gamma[3/2 - m]*HypergeometricPFQ[{5/2, 3, 5/2 - m}, {2, 11/2}, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^2 + 48*d*Gamma[3/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 9/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]*(-3*c*Sin[(-e + Pi/2 - f*x)/2] + d*Sin[(-e + Pi/2 - f*x)/2]*(-3 + 4*Sin[(-e + Pi/2 - f*x)/2]^2)) + 84*d*Gamma[1/2 - m]*Hypergeometric2F1[1/2, 1/2 - m, 7/2, Sin[(-e + Pi/2 - f*x)/2]^2]*(-5*c + d*(-5 + 6*Sin[(-e + Pi/2 - f*x)/2]^2)) + 48*Gamma[3/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 9/2, Sin[(-e + Pi/2 - f*x)/2]^2]*(c^2 + c*d*(2 - 3*Sin[(-e + Pi/2 - f*x)/2]^2) + d^2*(1 - 3*Sin[(-e + Pi/2 - f*x)/2]^2 + 2*Sin[(-e + Pi/2 - f*x)/2]^4)) - 8*(-3 + 2*m)*Gamma[3/2 - m]*Hypergeometric2F1[5/2, 5/2 - m, 11/2, Sin[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]^2*(c^2 + c*d*(2 - 3*Sin[(-e + Pi/2 - f*x)/2]^2) + d^2*(1 - 3*Sin[(-e + Pi/2 - f*x)/2]^2 + 2*Sin[(-e + Pi/2 - f*x)/2]^4)) + 3*(1/2 - m)*Gamma[1/2 - m]*Hypergeometric2F1[3/2, 3/2 - m, 9/2, Sin[(-e + Pi/2 - f*x)/2]^2]*(15*c^2 + 10*c*d*(3 - 2*Sin[(-e + Pi/2 - f*x)/2]^2) + d^2*(15 - 20*Sin[(-e + Pi/2 - f*x)/2]^2 + 12*Sin[(-e + Pi/2 - f*x)/2]^4))))/3))","B",0
609,1,275,117,1.8617672,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x)) \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c + d*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} c \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} d 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}} (c m+c+d 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{d \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)*d*(-(((-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]*c*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
610,1,90,74,0.141463,"\int (a+a \sin (e+f x))^m \, dx","Integrate[(a + a*Sin[e + f*x])^m,x]","\frac{\sqrt{2} \cos (e+f x) (a (\sin (e+f x)+1))^m \, _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 f m+f) \sqrt{1-\sin (e+f x)}}","-\frac{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)}{f}",1,"(Sqrt[2]*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]*(a*(1 + Sin[e + f*x]))^m)/((f + 2*f*m)*Sqrt[1 - Sin[e + f*x]])","A",1
611,1,363,100,1.1319913,"\int \frac{(a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^m/(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 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)}{f (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} \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)}{f (2 m+1) (c-d) \sqrt{1-\sin (e+f x)}}",1,"(-6*(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)]*(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*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(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))","B",0
612,1,363,100,1.2654264,"\int \frac{(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx","Integrate[(a + a*Sin[e + f*x])^m/(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 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)}{f (c+d \sin (e+f x))^2 \left(\sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \left(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)-(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)\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)}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},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)^2 \sqrt{1-\sin (e+f x)}}",1,"(-6*(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)]*(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*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(c + d*Sin[e + f*x])^2*(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)])*Sin[(2*e - Pi + 2*f*x)/4]^2))","B",0
613,1,363,100,1.4150216,"\int \frac{(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx","Integrate[(a + a*Sin[e + f*x])^m/(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 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)}{f (c+d \sin (e+f x))^3 \left(\sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \left(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)-(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)\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)\right)}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},3;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)^3 \sqrt{1-\sin (e+f x)}}",1,"(-6*(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)]*(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*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(c + d*Sin[e + f*x])^3*(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)])*Sin[(2*e - Pi + 2*f*x)/4]^2))","B",0
614,1,365,138,1.8961251,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2),x]","-\frac{3 \sqrt{2} (c+d) \sqrt{\sin (e+f x)+1} \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right) (a (\sin (e+f x)+1))^m (c+d \sin (e+f x))^{5/2} F_1\left(\frac{1}{2};\frac{1}{2}-m,-\frac{5}{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)}{f \sqrt{\cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)} \left(\sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \left(10 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{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)\right)-3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-\frac{5}{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)}","\frac{\sqrt{2} (c-d)^2 \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{5}{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}}}",1,"(-3*Sqrt[2]*(c + d)*AppellF1[1/2, 1/2 - m, -5/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)]*Sqrt[1 + Sin[e + f*x]]*(a*(1 + Sin[e + f*x]))^m*(c + d*Sin[e + f*x])^(5/2)*Tan[(2*e - Pi + 2*f*x)/4])/(f*Sqrt[Cos[(2*e - Pi + 2*f*x)/4]^2]*(-3*(c + d)*AppellF1[1/2, 1/2 - m, -5/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (10*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, -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)])*Sin[(2*e - Pi + 2*f*x)/4]^2))","B",0
615,1,365,136,1.2887913,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2),x]","-\frac{3 \sqrt{2} (c+d) \sqrt{\sin (e+f x)+1} \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right) (a (\sin (e+f x)+1))^m (c+d \sin (e+f x))^{3/2} 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)}{f \sqrt{\cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)} \left(\sin ^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)}","\frac{\sqrt{2} (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}}}",1,"(-3*Sqrt[2]*(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)]*Sqrt[1 + Sin[e + f*x]]*(a*(1 + Sin[e + f*x]))^m*(c + d*Sin[e + f*x])^(3/2)*Tan[(2*e - Pi + 2*f*x)/4])/(f*Sqrt[Cos[(2*e - Pi + 2*f*x)/4]^2]*(-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)])*Sin[(2*e - Pi + 2*f*x)/4]^2))","B",0
616,1,365,131,1.1033228,"\int (a+a \sin (e+f x))^m \sqrt{c+d \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]],x]","-\frac{3 \sqrt{2} (c+d) \sqrt{\sin (e+f x)+1} \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right) (a (\sin (e+f x)+1))^m \sqrt{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)}{f \sqrt{\cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)} \left(\sin ^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)}","\frac{\sqrt{2} \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}}}",1,"(-3*Sqrt[2]*(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)]*Sqrt[1 + Sin[e + f*x]]*(a*(1 + Sin[e + f*x]))^m*Sqrt[c + d*Sin[e + f*x]]*Tan[(2*e - Pi + 2*f*x)/4])/(f*Sqrt[Cos[(2*e - Pi + 2*f*x)/4]^2]*(-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)])*Sin[(2*e - Pi + 2*f*x)/4]^2))","B",0
617,1,373,131,1.1660652,"\int \frac{(a+a \sin (e+f x))^m}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])^m/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 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)}{f \sqrt{c+d \sin (e+f x)} \left(\sin ^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)}","\frac{\sqrt{2} \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)}}",1,"(-6*(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)]*(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*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*Sqrt[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)])*Sin[(2*e - Pi + 2*f*x)/4]^2))","B",0
618,1,373,138,1.3115542,"\int \frac{(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^m/(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 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)}{f (c+d \sin (e+f x))^{3/2} \left(\sin ^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)\right)}","\frac{\sqrt{2} \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)}}",1,"(-6*(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)]*(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*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(c + d*Sin[e + f*x])^(3/2)*(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)])*Sin[(2*e - Pi + 2*f*x)/4]^2))","B",0
619,1,373,138,1.5324908,"\int \frac{(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx","Integrate[(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(5/2),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 F_1\left(\frac{1}{2};\frac{1}{2}-m,\frac{5}{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)}{f (c+d \sin (e+f x))^{5/2} \left(\sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \left(10 d F_1\left(\frac{3}{2};\frac{1}{2}-m,\frac{7}{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{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)\right)+3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,\frac{5}{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)}","\frac{\sqrt{2} \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{5}{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)^2 \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}",1,"(-6*(c + d)*AppellF1[1/2, 1/2 - m, 5/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)]*(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*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(c + d*Sin[e + f*x])^(5/2)*(3*(c + d)*AppellF1[1/2, 1/2 - m, 5/2, 3/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)] + (10*d*AppellF1[3/2, 1/2 - m, 7/2, 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, 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)])*Sin[(2*e - Pi + 2*f*x)/4]^2))","B",0
620,1,238,62,1.4038772,"\int (1+\sin (e+f x))^m (3+5 \sin (e+f x))^{-1-m} \, dx","Integrate[(1 + Sin[e + f*x])^m*(3 + 5*Sin[e + f*x])^(-1 - m),x]","\frac{4^m (\cosh (m \log (4))-\sinh (m \log (4))) (\sin (e+f x)+1)^m (5 \sin (e+f x)+3)^{-m} (\sin (e+f x)+i \cos (e+f x)+1) \left(-\frac{2 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+\cos \left(\frac{1}{4} (2 e+2 f x+\pi )\right)}{\sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+2 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}\right)^m \, _2F_1\left(m+1,2 m+1;2 (m+1);\frac{4 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{2 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+\sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}\right)}{f (2 m+1) ((2+i) \sin (e+f x)+(-1+2 i) \cos (e+f x)+(2-i))}","-\frac{4^{-m-1} \cos (e+f x) \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{1-\sin (e+f x)}{4 (\sin (e+f x)+1)}\right)}{f (\sin (e+f x)+1)}",1,"(4^m*Hypergeometric2F1[1 + m, 1 + 2*m, 2*(1 + m), (4*Cos[(2*e - Pi + 2*f*x)/4])/(2*Cos[(2*e - Pi + 2*f*x)/4] + Sin[(2*e - Pi + 2*f*x)/4])]*(1 + Sin[e + f*x])^m*(1 + I*Cos[e + f*x] + Sin[e + f*x])*(-((2*Cos[(2*e - Pi + 2*f*x)/4] + Cos[(2*e + Pi + 2*f*x)/4])/(2*Cos[(2*e - Pi + 2*f*x)/4] + Sin[(2*e - Pi + 2*f*x)/4])))^m*(Cosh[m*Log[4]] - Sinh[m*Log[4]]))/(f*(1 + 2*m)*((2 - I) - (1 - 2*I)*Cos[e + f*x] + (2 + I)*Sin[e + f*x])*(3 + 5*Sin[e + f*x])^m)","C",0
621,1,88,64,0.4817913,"\int (1+\sin (e+f x))^m (3+4 \sin (e+f x))^{-1-m} \, dx","Integrate[(1 + Sin[e + f*x])^m*(3 + 4*Sin[e + f*x])^(-1 - m),x]","-\frac{2\ 7^{-m-1} \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (\sin (e+f x)+1)^m \sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^{-m} \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{1}{7} \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f}","-\frac{\left(\frac{7}{2}\right)^{-m-1} \cos (e+f x) \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{1-\sin (e+f x)}{7 (\sin (e+f x)+1)}\right)}{f (\sin (e+f x)+1)}",1,"(-2*7^(-1 - m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1 + m, 3/2, Tan[(2*e - Pi + 2*f*x)/4]^2/7]*(1 + Sin[e + f*x])^m)/(f*(Sin[(2*e + Pi + 2*f*x)/4]^2)^m)","A",0
622,1,45,28,0.0370989,"\int (1+\sin (e+f x))^m (3+3 \sin (e+f x))^{-1-m} \, dx","Integrate[(1 + Sin[e + f*x])^m*(3 + 3*Sin[e + f*x])^(-1 - m),x]","\frac{2\ 3^{-m-1} \sin \left(\frac{1}{2} (e+f x)\right)}{f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{3^{-m-1} \cos (e+f x)}{f (\sin (e+f x)+1)}",1,"(2*3^(-1 - m)*Sin[(e + f*x)/2])/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
623,1,131,122,0.5394001,"\int (1+\sin (e+f x))^m (3+2 \sin (e+f x))^{-1-m} \, dx","Integrate[(1 + Sin[e + f*x])^m*(3 + 2*Sin[e + f*x])^(-1 - m),x]","\frac{2\ 5^{-m-1} \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right) (\sin (e+f x)+1)^m (2 \sin (e+f x)+3)^{-m} \left((2 \sin (e+f x)+3) \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{1}{5} \cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f}","-\frac{2^{m+\frac{1}{2}} 5^{-m-\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left(\frac{\sin (e+f x)+1}{2 \sin (e+f x)+3}\right)^{\frac{1}{2}-m} (2 \sin (e+f x)+3)^{-m} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1-\sin (e+f x)}{2 (2 \sin (e+f x)+3)}\right)}{f}",1,"(2*5^(-1 - m)*Hypergeometric2F1[1/2, 1 + m, 3/2, -1/5*(Cos[(2*e + Pi + 2*f*x)/4]^2*Sec[(2*e - Pi + 2*f*x)/4]^2)]*(1 + Sin[e + f*x])^m*(Sec[(2*e - Pi + 2*f*x)/4]^2*(3 + 2*Sin[e + f*x]))^m*Tan[(2*e - Pi + 2*f*x)/4])/(f*(3 + 2*Sin[e + f*x])^m)","A",1
624,1,167,106,0.5706111,"\int (1+\sin (e+f x))^m (3+\sin (e+f x))^{-1-m} \, dx","Integrate[(1 + Sin[e + f*x])^m*(3 + Sin[e + f*x])^(-1 - m),x]","\frac{2^{-2 m-1} \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right) (\sin (e+f x)+1)^m \cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)^{-m} \left(\frac{\cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sin (e+f x)+3}\right)^m \left((\sin (e+f x)+3) \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{1}{2} \cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f}","-\frac{2^{-m-\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left(\frac{\sin (e+f x)+1}{\sin (e+f x)+3}\right)^{\frac{1}{2}-m} (\sin (e+f x)+3)^{-m} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1-\sin (e+f x)}{\sin (e+f x)+3}\right)}{f}",1,"(2^(-1 - 2*m)*Hypergeometric2F1[1/2, 1 + m, 3/2, -1/2*(Cos[(2*e + Pi + 2*f*x)/4]^2*Sec[(2*e - Pi + 2*f*x)/4]^2)]*(1 + Sin[e + f*x])^m*(Cos[(2*e - Pi + 2*f*x)/4]^2/(3 + Sin[e + f*x]))^m*(Sec[(2*e - Pi + 2*f*x)/4]^2*(3 + Sin[e + f*x]))^m*Tan[(2*e - Pi + 2*f*x)/4])/(f*(Cos[(2*e - Pi + 2*f*x)/4]^2)^m)","A",0
625,1,95,65,0.1339419,"\int 3^{-1-m} (1+\sin (e+f x))^m \, dx","Integrate[3^(-1 - m)*(1 + Sin[e + f*x])^m,x]","\frac{\sqrt{2} 3^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^m \, _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 f m+f) \sqrt{1-\sin (e+f x)}}","-\frac{2^{m+\frac{1}{2}} 3^{-m-1} \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f \sqrt{\sin (e+f x)+1}}",1,"(Sqrt[2]*3^(-1 - m)*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]*(1 + Sin[e + f*x])^m)/((f + 2*f*m)*Sqrt[1 - Sin[e + f*x]])","A",1
626,1,182,94,1.0424742,"\int (3-\sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx","Integrate[(3 - Sin[e + f*x])^(-1 - m)*(1 + Sin[e + f*x])^m,x]","-\frac{2^{\frac{1}{2}-m} \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (3-\sin (e+f x))^{-m} (\sin (e+f x)+1)^m \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}} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};-\frac{4 \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sin (e+f x)-3}\right) \left(-\frac{\cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sin (e+f x)-3}\right)^{\frac{1}{2}-m}}{f}","-\frac{\cos (e+f x) (3-\sin (e+f x))^{-m-1} \left(\frac{3-\sin (e+f x)}{\sin (e+f x)+1}\right)^{m+1} (\sin (e+f x)+1)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{2 (1-\sin (e+f x))}{\sin (e+f x)+1}\right)}{f}",1,"-((2^(1/2 - m)*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (-4*Sin[(2*e - Pi + 2*f*x)/4]^2)/(-3 + Sin[e + f*x])]*(-(Cos[(2*e - Pi + 2*f*x)/4]^2/(-3 + Sin[e + f*x])))^(1/2 - m)*(1 + Sin[e + f*x])^m*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(3 - Sin[e + f*x])^m))","A",1
627,1,177,114,0.8739116,"\int (3-2 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx","Integrate[(3 - 2*Sin[e + f*x])^(-1 - m)*(1 + Sin[e + f*x])^m,x]","-\frac{2 \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (3-2 \sin (e+f x))^{-m} (\sin (e+f x)+1)^m \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}} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{5 \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{3-2 \sin (e+f x)}\right) \left(-\frac{\cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{2 \sin (e+f x)-3}\right)^{\frac{1}{2}-m}}{f}","\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-2 \sin (e+f x))^{-m} (\sin (e+f x)+1)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (3-2 \sin (e+f x))}{\sin (e+f x)+1}\right)}{\sqrt{5} f m (1-\sin (e+f x))}",1,"(-2*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (5*Sin[(2*e - Pi + 2*f*x)/4]^2)/(3 - 2*Sin[e + f*x])]*(1 + Sin[e + f*x])^m*(-(Cos[(2*e - Pi + 2*f*x)/4]^2/(-3 + 2*Sin[e + f*x])))^(1/2 - m)*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(3 - 2*Sin[e + f*x])^m)","A",0
628,1,97,43,0.5210821,"\int (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx","Integrate[(3 - 3*Sin[e + f*x])^(-1 - m)*(1 + Sin[e + f*x])^m,x]","\frac{\sin \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (6-6 \sin (e+f x))^{-m} (\sin (e+f x)+1)^m \cos ^{-2 m-1}\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)^{2 m}}{3 (2 f m+f)}","\frac{\cos (e+f x) (3-3 \sin (e+f x))^{-m-1} (\sin (e+f x)+1)^m}{f (2 m+1)}",1,"(Cos[(2*e + Pi + 2*f*x)/4]^(-1 - 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*m)*(1 + Sin[e + f*x])^m*Sin[(2*e + Pi + 2*f*x)/4])/(3*(f + 2*f*m)*(6 - 6*Sin[e + f*x])^m)","B",1
629,1,176,83,0.8728809,"\int (3-4 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx","Integrate[(3 - 4*Sin[e + f*x])^(-1 - m)*(1 + Sin[e + f*x])^m,x]","\frac{2 \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (3-4 \sin (e+f x))^{-m} (\sin (e+f x)+1)^m \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}} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{7 \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{3-4 \sin (e+f x)}\right) \left(\frac{\cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{4 \sin (e+f x)-3}\right)^{\frac{1}{2}-m}}{f}","\frac{2^{m+1} \cos (e+f x) (3-4 \sin (e+f x))^{-m} (4 \sin (e+f x)-3)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{7 (1-\sin (e+f x))}{\sin (e+f x)+1}\right)}{f (\sin (e+f x)+1)}",1,"(2*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (7*Sin[(2*e - Pi + 2*f*x)/4]^2)/(3 - 4*Sin[e + f*x])]*(1 + Sin[e + f*x])^m*(Cos[(2*e - Pi + 2*f*x)/4]^2/(-3 + 4*Sin[e + f*x]))^(1/2 - m)*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(3 - 4*Sin[e + f*x])^m)","B",0
630,1,246,78,1.7648209,"\int (3-5 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx","Integrate[(3 - 5*Sin[e + f*x])^(-1 - m)*(1 + Sin[e + f*x])^m,x]","-\frac{2^{2 m-1} (\cosh (m \log (4))-\sinh (m \log (4))) (3-5 \sin (e+f x))^{-m} (\sin (e+f x)+1)^m (\sin (e+f x)+i \cos (e+f x)+1) \left(\frac{2 \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)-\cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{2 \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+\cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}\right)^m \, _2F_1\left(m+1,2 m+1;2 (m+1);\frac{2 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+2 \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}\right)}{f (2 m+1) ((1+2 i) \sin (e+f x)+(-2+i) \cos (e+f x)+(1-2 i))}","\frac{\cos (e+f x) (3-5 \sin (e+f x))^{-m} (5 \sin (e+f x)-3)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{4 (1-\sin (e+f x))}{\sin (e+f x)+1}\right)}{f (\sin (e+f x)+1)}",1,"-((2^(-1 + 2*m)*Hypergeometric2F1[1 + m, 1 + 2*m, 2*(1 + m), (2*Cos[(2*e - Pi + 2*f*x)/4])/(Cos[(2*e - Pi + 2*f*x)/4] + 2*Sin[(2*e - Pi + 2*f*x)/4])]*(1 + Sin[e + f*x])^m*(1 + I*Cos[e + f*x] + Sin[e + f*x])*((-Cos[(2*e - Pi + 2*f*x)/4] + 2*Sin[(2*e - Pi + 2*f*x)/4])/(Cos[(2*e - Pi + 2*f*x)/4] + 2*Sin[(2*e - Pi + 2*f*x)/4]))^m*(Cosh[m*Log[4]] - Sinh[m*Log[4]]))/(f*(1 + 2*m)*(3 - 5*Sin[e + f*x])^m*((1 - 2*I) - (2 - I)*Cos[e + f*x] + (1 + 2*I)*Sin[e + f*x])))","C",0
631,1,240,81,0.5544264,"\int (3+5 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(3 + 5*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{4^m (\cosh (m \log (4))-\sinh (m \log (4))) (5 \sin (e+f x)+3)^{-m} (\sin (e+f x)+i \cos (e+f x)+1) (a (\sin (e+f x)+1))^m \left(-\frac{2 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+\cos \left(\frac{1}{4} (2 e+2 f x+\pi )\right)}{\sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+2 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}\right)^m \, _2F_1\left(m+1,2 m+1;2 (m+1);\frac{4 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{2 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+\sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}\right)}{f (2 m+1) ((2+i) \sin (e+f x)+(-1+2 i) \cos (e+f x)+(2-i))}","-\frac{4^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{a-a \sin (e+f x)}{4 (\sin (e+f x) a+a)}\right)}{f}",1,"(4^m*Hypergeometric2F1[1 + m, 1 + 2*m, 2*(1 + m), (4*Cos[(2*e - Pi + 2*f*x)/4])/(2*Cos[(2*e - Pi + 2*f*x)/4] + Sin[(2*e - Pi + 2*f*x)/4])]*(a*(1 + Sin[e + f*x]))^m*(1 + I*Cos[e + f*x] + Sin[e + f*x])*(-((2*Cos[(2*e - Pi + 2*f*x)/4] + Cos[(2*e + Pi + 2*f*x)/4])/(2*Cos[(2*e - Pi + 2*f*x)/4] + Sin[(2*e - Pi + 2*f*x)/4])))^m*(Cosh[m*Log[4]] - Sinh[m*Log[4]]))/(f*(1 + 2*m)*((2 - I) - (1 - 2*I)*Cos[e + f*x] + (2 + I)*Sin[e + f*x])*(3 + 5*Sin[e + f*x])^m)","C",0
632,1,90,83,0.472164,"\int (3+4 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(3 + 4*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{2\ 7^{-m-1} \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)^{-m} (a (\sin (e+f x)+1))^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{1}{7} \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f}","-\frac{\left(\frac{7}{2}\right)^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{a-a \sin (e+f x)}{7 (\sin (e+f x) a+a)}\right)}{f}",1,"(-2*7^(-1 - m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1 + m, 3/2, Tan[(2*e - Pi + 2*f*x)/4]^2/7]*(a*(1 + Sin[e + f*x]))^m)/(f*(Sin[(2*e + Pi + 2*f*x)/4]^2)^m)","A",0
633,1,104,39,5.2398811,"\int (3+3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(3 + 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{2^{-m} 3^{-m-1} \cos \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (\sin (e+f x)+1)^{-m-1} \sin ^{-2 m-1}\left(\frac{1}{4} (2 e+2 f x+\pi )\right) (a (\sin (e+f x)+1))^m \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^{2 (m+1)}}{f}","-\frac{\cos (e+f x) (3 \sin (e+f x)+3)^{-m-1} (a \sin (e+f x)+a)^m}{f}",1,"-((3^(-1 - m)*Cos[(2*e + Pi + 2*f*x)/4]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^(2*(1 + m))*(1 + Sin[e + f*x])^(-1 - m)*(a*(1 + Sin[e + f*x]))^m*Sin[(2*e + Pi + 2*f*x)/4]^(-1 - 2*m))/(2^m*f))","B",1
634,1,179,83,0.6217882,"\int (3+2 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(3 + 2*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{2\ 5^{-m-1} \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (2 \sin (e+f x)+3)^{-m} \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((2 \sin (e+f x)+3) \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{1}{5} \cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f}","-\frac{\left(\frac{5}{2}\right)^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{a-a \sin (e+f x)}{5 (\sin (e+f x) a+a)}\right)}{f}",1,"(-2*5^(-1 - m)*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1 + m, 3/2, -1/5*(Cos[(2*e + Pi + 2*f*x)/4]^2*Sec[(2*e - Pi + 2*f*x)/4]^2)]*(a*(1 + Sin[e + f*x]))^m*(Sec[(2*e - Pi + 2*f*x)/4]^2*(3 + 2*Sin[e + f*x]))^m*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(3 + 2*Sin[e + f*x])^m)","B",1
635,1,166,81,0.6764642,"\int (3+\sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(3 + Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{2^{-2 m-1} \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)^{-m} (a (\sin (e+f x)+1))^m \left(\frac{\cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sin (e+f x)+3}\right)^m \left((\sin (e+f x)+3) \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{1}{2} \cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f}","-\frac{2^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{a-a \sin (e+f x)}{2 (\sin (e+f x) a+a)}\right)}{f}",1,"-((2^(-1 - 2*m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1 + m, 3/2, -1/2*(Cos[(2*e + Pi + 2*f*x)/4]^2*Sec[(2*e - Pi + 2*f*x)/4]^2)]*(a*(1 + Sin[e + f*x]))^m*(Cos[(2*e - Pi + 2*f*x)/4]^2/(3 + Sin[e + f*x]))^m*(Sec[(2*e - Pi + 2*f*x)/4]^2*(3 + Sin[e + f*x]))^m)/(f*(Sin[(2*e + Pi + 2*f*x)/4]^2)^m))","B",0
636,1,97,81,0.1582165,"\int 3^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[3^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{\sqrt{2} 3^{-m-1} \cos (e+f x) (a (\sin (e+f x)+1))^m \, _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 f m+f) \sqrt{1-\sin (e+f x)}}","-\frac{2^{m+\frac{1}{2}} 3^{-m-1} \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}",1,"(Sqrt[2]*3^(-1 - m)*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]*(a*(1 + Sin[e + f*x]))^m)/((f + 2*f*m)*Sqrt[1 - Sin[e + f*x]])","A",1
637,1,184,72,0.8104472,"\int (3-\sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(3 - Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{2^{\frac{1}{2}-m} \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (3-\sin (e+f x))^{-m} \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 \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};-\frac{4 \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sin (e+f x)-3}\right) \left(-\frac{\cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sin (e+f x)-3}\right)^{\frac{1}{2}-m}}{f}","-\frac{\cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{2 (a-a \sin (e+f x))}{\sin (e+f x) a+a}\right)}{f}",1,"-((2^(1/2 - m)*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (-4*Sin[(2*e - Pi + 2*f*x)/4]^2)/(-3 + Sin[e + f*x])]*(-(Cos[(2*e - Pi + 2*f*x)/4]^2/(-3 + Sin[e + f*x])))^(1/2 - m)*(a*(1 + Sin[e + f*x]))^m*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(3 - Sin[e + f*x])^m))","B",1
638,1,179,77,0.6307175,"\int (3-2 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(3 - 2*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{2 \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (3-2 \sin (e+f x))^{-m} \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 \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{5 \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{3-2 \sin (e+f x)}\right) \left(-\frac{\cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{2 \sin (e+f x)-3}\right)^{\frac{1}{2}-m}}{f}","-\frac{2^{m+1} \cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{5 (a-a \sin (e+f x))}{\sin (e+f x) a+a}\right)}{f}",1,"(-2*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (5*Sin[(2*e - Pi + 2*f*x)/4]^2)/(3 - 2*Sin[e + f*x])]*(a*(1 + Sin[e + f*x]))^m*(-(Cos[(2*e - Pi + 2*f*x)/4]^2/(-3 + 2*Sin[e + f*x])))^(1/2 - m)*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(3 - 2*Sin[e + f*x])^m)","B",0
639,1,99,45,0.6047228,"\int (3-3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(3 - 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{\sin \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (6-6 \sin (e+f x))^{-m} \cos ^{-2 m-1}\left(\frac{1}{4} (2 e+2 f x+\pi )\right) (a (\sin (e+f x)+1))^m \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{2 m}}{3 (2 f m+f)}","\frac{\cos (e+f x) (3-3 \sin (e+f x))^{-m-1} (a \sin (e+f x)+a)^m}{f (2 m+1)}",1,"(Cos[(2*e + Pi + 2*f*x)/4]^(-1 - 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*m)*(a*(1 + Sin[e + f*x]))^m*Sin[(2*e + Pi + 2*f*x)/4])/(3*(f + 2*f*m)*(6 - 6*Sin[e + f*x])^m)","B",1
640,1,178,115,0.6455503,"\int (3-4 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(3 - 4*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{2 \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (3-4 \sin (e+f x))^{-m} \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 \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{7 \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{3-4 \sin (e+f x)}\right) \left(\frac{\cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{4 \sin (e+f x)-3}\right)^{\frac{1}{2}-m}}{f}","\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-4 \sin (e+f x))^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;-\frac{2 (3-4 \sin (e+f x))}{\sin (e+f x)+1}\right)}{\sqrt{7} f m (1-\sin (e+f x))}",1,"(2*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (7*Sin[(2*e - Pi + 2*f*x)/4]^2)/(3 - 4*Sin[e + f*x])]*(a*(1 + Sin[e + f*x]))^m*(Cos[(2*e - Pi + 2*f*x)/4]^2/(-3 + 4*Sin[e + f*x]))^(1/2 - m)*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(3 - 4*Sin[e + f*x])^m)","A",0
641,1,248,113,0.5725864,"\int (3-5 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(3 - 5*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{2^{2 m-1} (\cosh (m \log (4))-\sinh (m \log (4))) (3-5 \sin (e+f x))^{-m} (\sin (e+f x)+i \cos (e+f x)+1) (a (\sin (e+f x)+1))^m \left(\frac{2 \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)-\cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{2 \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+\cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}\right)^m \, _2F_1\left(m+1,2 m+1;2 (m+1);\frac{2 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+2 \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}\right)}{f (2 m+1) ((1+2 i) \sin (e+f x)+(-2+i) \cos (e+f x)+(1-2 i))}","\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-5 \sin (e+f x))^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;-\frac{3-5 \sin (e+f x)}{\sin (e+f x)+1}\right)}{4 f m (1-\sin (e+f x))}",1,"-((2^(-1 + 2*m)*Hypergeometric2F1[1 + m, 1 + 2*m, 2*(1 + m), (2*Cos[(2*e - Pi + 2*f*x)/4])/(Cos[(2*e - Pi + 2*f*x)/4] + 2*Sin[(2*e - Pi + 2*f*x)/4])]*(a*(1 + Sin[e + f*x]))^m*(1 + I*Cos[e + f*x] + Sin[e + f*x])*((-Cos[(2*e - Pi + 2*f*x)/4] + 2*Sin[(2*e - Pi + 2*f*x)/4])/(Cos[(2*e - Pi + 2*f*x)/4] + 2*Sin[(2*e - Pi + 2*f*x)/4]))^m*(Cosh[m*Log[4]] - Sinh[m*Log[4]]))/(f*(1 + 2*m)*(3 - 5*Sin[e + f*x])^m*((1 - 2*I) - (2 - I)*Cos[e + f*x] + (1 + 2*I)*Sin[e + f*x])))","C",0
642,1,247,72,1.5331734,"\int (-3+5 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(-3 + 5*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{2^{2 m-1} (\cosh (m \log (4))-\sinh (m \log (4))) (5 \sin (e+f x)-3)^{-m} (\sin (e+f x)+i \cos (e+f x)+1) (a (\sin (e+f x)+1))^m \left(\frac{2 \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)-\cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{2 \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+\cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}\right)^m \, _2F_1\left(m+1,2 m+1;2 (m+1);\frac{2 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+2 \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}\right)}{f (2 m+1) ((1+2 i) \sin (e+f x)+(-2+i) \cos (e+f x)+(1-2 i))}","-\frac{\cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{4 (a-a \sin (e+f x))}{\sin (e+f x) a+a}\right)}{f}",1,"(2^(-1 + 2*m)*Hypergeometric2F1[1 + m, 1 + 2*m, 2*(1 + m), (2*Cos[(2*e - Pi + 2*f*x)/4])/(Cos[(2*e - Pi + 2*f*x)/4] + 2*Sin[(2*e - Pi + 2*f*x)/4])]*(a*(1 + Sin[e + f*x]))^m*(1 + I*Cos[e + f*x] + Sin[e + f*x])*((-Cos[(2*e - Pi + 2*f*x)/4] + 2*Sin[(2*e - Pi + 2*f*x)/4])/(Cos[(2*e - Pi + 2*f*x)/4] + 2*Sin[(2*e - Pi + 2*f*x)/4]))^m*(Cosh[m*Log[4]] - Sinh[m*Log[4]]))/(f*(1 + 2*m)*((1 - 2*I) - (2 - I)*Cos[e + f*x] + (1 + 2*I)*Sin[e + f*x])*(-3 + 5*Sin[e + f*x])^m)","C",0
643,1,154,77,0.6175602,"\int (-3+4 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(-3 + 4*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{2 \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (4 \sin (e+f x)-3)^{-m} \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 \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};7 \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right) \left((4 \sin (e+f x)-3) \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)^m}{f}","-\frac{2^{m+1} \cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{7 (a-a \sin (e+f x))}{\sin (e+f x) a+a}\right)}{f}",1,"(-2*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1 + m, 3/2, 7*Tan[(2*e - Pi + 2*f*x)/4]^2]*(a*(1 + Sin[e + f*x]))^m*(Sec[(2*e - Pi + 2*f*x)/4]^2*(-3 + 4*Sin[e + f*x]))^m*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(-3 + 4*Sin[e + f*x])^m)","A",0
644,1,110,45,0.6637689,"\int (-3+3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(-3 + 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{2^{-m} 3^{-m-1} \sin \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (\sin (e+f x)-1)^{-m-1} \cos ^{-2 m-1}\left(\frac{1}{4} (2 e+2 f x+\pi )\right) (a (\sin (e+f x)+1))^m \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{2 (m+1)}}{2 f m+f}","\frac{\cos (e+f x) (3 \sin (e+f x)-3)^{-m-1} (a \sin (e+f x)+a)^m}{f (2 m+1)}",1,"(3^(-1 - m)*Cos[(2*e + Pi + 2*f*x)/4]^(-1 - 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(1 + m))*(-1 + Sin[e + f*x])^(-1 - m)*(a*(1 + Sin[e + f*x]))^m*Sin[(2*e + Pi + 2*f*x)/4])/(2^m*(f + 2*f*m))","B",1
645,1,155,117,0.6287443,"\int (-3+2 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(-3 + 2*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{2 \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (2 \sin (e+f x)-3)^{-m} \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 \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-5 \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right) \left((2 \sin (e+f x)-3) \left(-\sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)\right)^m}{f}","-\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (2 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (3-2 \sin (e+f x))}{\sin (e+f x)+1}\right)}{\sqrt{5} f m (1-\sin (e+f x))}",1,"(2*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1 + m, 3/2, -5*Tan[(2*e - Pi + 2*f*x)/4]^2]*(a*(1 + Sin[e + f*x]))^m*(-(Sec[(2*e - Pi + 2*f*x)/4]^2*(-3 + 2*Sin[e + f*x])))^m*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(-3 + 2*Sin[e + f*x])^m)","A",0
646,1,155,116,0.7051116,"\int (-3+\sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(-3 + Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{2^{-m} \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (\sin (e+f x)-3)^{-m} \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 \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-2 \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right) \left((\sin (e+f x)-3) \left(-\sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)\right)^m}{f}","-\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (\sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{3-\sin (e+f x)}{\sin (e+f x)+1}\right)}{2 \sqrt{2} f m (1-\sin (e+f x))}",1,"((Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1 + m, 3/2, -2*Tan[(2*e - Pi + 2*f*x)/4]^2]*(-(Sec[(2*e - Pi + 2*f*x)/4]^2*(-3 + Sin[e + f*x])))^m*(a*(1 + Sin[e + f*x]))^m*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(2^m*f*(-3 + Sin[e + f*x])^m)","A",0
647,1,97,81,0.1680567,"\int (-3)^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(-3)^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{\sqrt{2} (-3)^{-m-1} \cos (e+f x) (a (\sin (e+f x)+1))^m \, _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 f m+f) \sqrt{1-\sin (e+f x)}}","-\frac{(-3)^{-m-1} 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)}{f}",1,"((-3)^(-1 - m)*Sqrt[2]*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]*(a*(1 + Sin[e + f*x]))^m)/((f + 2*f*m)*Sqrt[1 - Sin[e + f*x]])","A",1
648,1,131,119,0.9120502,"\int (-3-\sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(-3 - Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{4^{-m} \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (-\sin (e+f x)-3)^{-m} (\sin (e+f x)+3)^{m-\frac{1}{2}} \sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^{\frac{1}{2}-m} (a (\sin (e+f x)+1))^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{2 \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{\sin (e+f x)+3}\right)}{f}","-\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (-\sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{\sin (e+f x)+3}{2 (\sin (e+f x)+1)}\right)}{2 \sqrt{2} f m (1-\sin (e+f x))}",1,"(Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (2*Sin[(2*e - Pi + 2*f*x)/4]^2)/(3 + Sin[e + f*x])]*(a*(1 + Sin[e + f*x]))^m*(3 + Sin[e + f*x])^(-1/2 + m)*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(4^m*f*(-3 - Sin[e + f*x])^m)","A",0
649,1,186,119,1.152001,"\int (-3-2 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(-3 - 2*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{2\ 5^{-m-\frac{1}{2}} \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (-2 \sin (e+f x)-3)^{-m} \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 \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{\sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{2 \sin (e+f x)+3}\right) \left(\frac{\cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{2 \sin (e+f x)+3}\right)^{\frac{1}{2}-m}}{f}","-\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (-2 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (2 \sin (e+f x)+3)}{5 (\sin (e+f x)+1)}\right)}{\sqrt{5} f m (1-\sin (e+f x))}",1,"(2*5^(-1/2 - m)*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, Sin[(2*e - Pi + 2*f*x)/4]^2/(3 + 2*Sin[e + f*x])]*(a*(1 + Sin[e + f*x]))^m*(Cos[(2*e - Pi + 2*f*x)/4]^2/(3 + 2*Sin[e + f*x]))^(1/2 - m)*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(-3 - 2*Sin[e + f*x])^m)","A",1
650,1,106,39,0.5213206,"\int (-3-3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(-3 - 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{2^{-m} 3^{-m-1} \cos \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (-\sin (e+f x)-1)^{-m-1} \sin ^{-2 m-1}\left(\frac{1}{4} (2 e+2 f x+\pi )\right) (a (\sin (e+f x)+1))^m \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^{2 (m+1)}}{f}","-\frac{\cos (e+f x) (-3 \sin (e+f x)-3)^{-m-1} (a \sin (e+f x)+a)^m}{f}",1,"-((3^(-1 - m)*Cos[(2*e + Pi + 2*f*x)/4]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^(2*(1 + m))*(-1 - Sin[e + f*x])^(-1 - m)*(a*(1 + Sin[e + f*x]))^m*Sin[(2*e + Pi + 2*f*x)/4]^(-1 - 2*m))/(2^m*f))","B",1
651,1,187,117,1.1617529,"\int (-3-4 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(-3 - 4*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{2\ 7^{-m-\frac{1}{2}} \cot \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (-4 \sin (e+f x)-3)^{-m} \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 \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};-\frac{\sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{4 \sin (e+f x)+3}\right) \left(\frac{\cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{4 \sin (e+f x)+3}\right)^{\frac{1}{2}-m}}{f}","\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (-4 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (4 \sin (e+f x)+3)}{7 (\sin (e+f x)+1)}\right)}{\sqrt{7} f m (1-\sin (e+f x))}",1,"(2*7^(-1/2 - m)*(Cos[(2*e - Pi + 2*f*x)/4]^2)^(-1/2 + m)*Cot[(2*e + Pi + 2*f*x)/4]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, -(Sin[(2*e - Pi + 2*f*x)/4]^2/(3 + 4*Sin[e + f*x]))]*(a*(1 + Sin[e + f*x]))^m*(Cos[(2*e - Pi + 2*f*x)/4]^2/(3 + 4*Sin[e + f*x]))^(1/2 - m)*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/(f*(-3 - 4*Sin[e + f*x])^m)","A",1
652,1,241,115,1.5594623,"\int (-3-5 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(-3 - 5*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{4^m (\cosh (m \log (4))-\sinh (m \log (4))) (-5 \sin (e+f x)-3)^{-m} (\sin (e+f x)+i \cos (e+f x)+1) (a (\sin (e+f x)+1))^m \left(-\frac{2 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+\cos \left(\frac{1}{4} (2 e+2 f x+\pi )\right)}{\sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+2 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}\right)^m \, _2F_1\left(m+1,2 m+1;2 (m+1);\frac{4 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{2 \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+\sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}\right)}{f (2 m+1) ((2+i) \sin (e+f x)+(-1+2 i) \cos (e+f x)+(2-i))}","\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (-5 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{5 \sin (e+f x)+3}{4 (\sin (e+f x)+1)}\right)}{4 f m (1-\sin (e+f x))}",1,"-((4^m*Hypergeometric2F1[1 + m, 1 + 2*m, 2*(1 + m), (4*Cos[(2*e - Pi + 2*f*x)/4])/(2*Cos[(2*e - Pi + 2*f*x)/4] + Sin[(2*e - Pi + 2*f*x)/4])]*(a*(1 + Sin[e + f*x]))^m*(1 + I*Cos[e + f*x] + Sin[e + f*x])*(-((2*Cos[(2*e - Pi + 2*f*x)/4] + Cos[(2*e + Pi + 2*f*x)/4])/(2*Cos[(2*e - Pi + 2*f*x)/4] + Sin[(2*e - Pi + 2*f*x)/4])))^m*(Cosh[m*Log[4]] - Sinh[m*Log[4]]))/(f*(1 + 2*m)*(-3 - 5*Sin[e + f*x])^m*((2 - I) - (1 - 2*I)*Cos[e + f*x] + (2 + I)*Sin[e + f*x])))","C",0
653,1,194,116,1.4900171,"\int (d \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Integrate[(d*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{(1-i) 2^m (\cosh (m \log (2))-\sinh (m \log (2))) (\cos (e+f x)-i (\sin (e+f x)+1)) (a (\sin (e+f x)+1))^m (d \sin (e+f x))^{-m} ((1-i) (-i \sin (e+f x)+\cos (e+f x)+1))^m ((1+i) (i \sin (e+f x)-\cos (e+f x)+1))^{-m} \, _2F_1\left(m+1,2 m+1;2 (m+1);\sqrt{2} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right) \csc \left(\frac{1}{2} (e+f x)\right)\right)}{d f (2 m+1) (-i \sin (e+f x)+\cos (e+f x)-1)}","-\frac{\cos (e+f x) \left(\frac{\sin (e+f x)+1}{1-\sin (e+f x)}\right)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m (d \sin (e+f x))^{-m} \, _2F_1\left(\frac{1}{2}-m,-m;1-m;-\frac{2 \sin (e+f x)}{1-\sin (e+f x)}\right)}{d f m (\sin (e+f x)+1)}",1,"((1 - I)*2^m*Hypergeometric2F1[1 + m, 1 + 2*m, 2*(1 + m), Sqrt[2]*Cos[(2*e - Pi + 2*f*x)/4]*Csc[(e + f*x)/2]]*((1 - I)*(1 + Cos[e + f*x] - I*Sin[e + f*x]))^m*(a*(1 + Sin[e + f*x]))^m*(Cos[e + f*x] - I*(1 + Sin[e + f*x]))*(Cosh[m*Log[2]] - Sinh[m*Log[2]]))/(d*f*(1 + 2*m)*(-1 + Cos[e + f*x] - I*Sin[e + f*x])*((1 + I)*(1 - Cos[e + f*x] + I*Sin[e + f*x]))^m*(d*Sin[e + f*x])^m)","C",0
654,1,187,129,1.3987921,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx","Integrate[(a + a*Sin[e + f*x])^m*(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-1} \, _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)^{-m-\frac{1}{2}}}{f}","-\frac{a 2^{m+\frac{1}{2}} \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]*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])]*(a*(1 + Sin[e + f*x]))^m*(((c + d)*Cos[(2*e - Pi + 2*f*x)/4]^2)/(c + d*Sin[e + f*x]))^(-1/2 - m)*(c + d*Sin[e + f*x])^(-1 - m)*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/f","A",0
655,0,0,107,36.0558756,"\int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx","Integrate[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n,x]","\int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx","-\frac{8 \sqrt{2} a^3 \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])^3*(c + d*Sin[e + f*x])^n, x]","F",-1
656,0,0,107,16.6307501,"\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","Integrate[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]","\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","-\frac{4 \sqrt{2} 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} 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])^2*(c + d*Sin[e + f*x])^n, x]","F",-1
657,0,0,105,5.6632399,"\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Integrate[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","-\frac{2 \sqrt{2} a \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}}",1,"Integrate[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n, x]","F",-1
658,1,120,104,0.2795863,"\int (c+d \sin (e+f x))^n \, dx","Integrate[(c + d*Sin[e + f*x])^n,x]","\frac{\sec (e+f x) \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} \sqrt{\frac{d (\sin (e+f x)+1)}{d-c}} (c+d \sin (e+f x))^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right)}{d f (n+1)}","-\frac{\sqrt{2} \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}}",1,"(AppellF1[1 + n, 1/2, 1/2, 2 + n, (c + d*Sin[e + f*x])/(c - d), (c + d*Sin[e + f*x])/(c + d)]*Sec[e + f*x]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[(d*(1 + Sin[e + f*x]))/(-c + d)]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n))","A",0
659,0,0,107,2.6959625,"\int \frac{(c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","Integrate[(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x]),x]","\int \frac{(c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","-\frac{\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}}",1,"Integrate[(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x]), x]","F",-1
660,0,0,109,7.0119829,"\int \frac{(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^2,x]","\int \frac{(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","-\frac{\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}}",1,"Integrate[(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^2, x]","F",-1
661,0,0,109,13.1318968,"\int \frac{(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^3,x]","\int \frac{(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx","-\frac{\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{7}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{4 \sqrt{2} a^3 f \sqrt{\sin (e+f x)+1}}",1,"Integrate[(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^3, x]","F",-1
662,1,190,257,32.7774633,"\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^n \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^n,x]","\frac{a^2 (\sin (e+f x)-1) \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} (c+d \sin (e+f x))^n \left(\left(3 c^2-2 c d (4 n+7)+d^2 \left(16 n^2+56 n+43\right)\right) \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};-\frac{d (\sin (e+f x)-1)}{c+d}\right)-(3 c-d (4 n+11)) (c+d \sin (e+f x))+d (2 n+3) (\sin (e+f x)+1) (c+d \sin (e+f x))\right)}{d^2 f \left(n+\frac{5}{2}\right) (2 n+3)}","-\frac{2 a^3 \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^3 (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^2 \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,"(a^2*Sec[e + f*x]*(-1 + Sin[e + f*x])*Sqrt[a*(1 + Sin[e + f*x])]*(c + d*Sin[e + f*x])^n*(-((3*c - d*(11 + 4*n))*(c + d*Sin[e + f*x])) + d*(3 + 2*n)*(1 + Sin[e + f*x])*(c + d*Sin[e + f*x]) + ((3*c^2 - 2*c*d*(7 + 4*n) + d^2*(43 + 56*n + 16*n^2))*Hypergeometric2F1[1/2, -n, 3/2, -((d*(-1 + Sin[e + f*x]))/(c + d))])/((c + d*Sin[e + f*x])/(c + d))^n))/(d^2*f*(5/2 + n)*(3 + 2*n))","A",1
663,1,133,160,7.6379385,"\int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^n \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^n,x]","-\frac{2 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((d (4 n+5)-c) \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};-\frac{d (\sin (e+f x)-1)}{c+d}\right)+(c+d) \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{n+1}\right)}{d f (2 n+3) \sqrt{a (\sin (e+f x)+1)}}","\frac{2 a^2 (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 \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,"(-2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^n*((-c + d*(5 + 4*n))*Hypergeometric2F1[1/2, -n, 3/2, -((d*(-1 + Sin[e + f*x]))/(c + d))] + (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
664,0,0,85,4.6291161,"\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^n \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n,x]","\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^n \, dx","-\frac{2 a \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,"Integrate[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x]","F",-1
665,1,236,99,2.8174376,"\int \frac{(c+d \sin (e+f x))^n}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(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 \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}-\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{\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};-n,1;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d},\frac{1}{2} (1-\sin (e+f x))\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*(-((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*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]))","B",0
666,1,319,104,4.8103733,"\int \frac{(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(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^2 \sqrt{2-2 \sin (e+f x)} (\sin (e+f x)+1)^2 \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 a (\sin (e+f x)+1) \sqrt{1-\frac{2}{\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)}{8 a^3 f \sqrt{a (\sin (e+f x)+1)}}","-\frac{\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};-n,2;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d},\frac{1}{2} (1-\sin (e+f x))\right)}{2 a f \sqrt{a \sin (e+f x)+a}}",1,"(Sec[e + f*x]*(c + d*Sin[e + f*x])^n*((a^2*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])^2)/((c + d*Sin[e + f*x])/(c - d))^n - (4*a*(1 + Sin[e + f*x])*Sqrt[1 - 2/(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
667,1,414,104,9.5431211,"\int \frac{(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^(5/2),x]","\frac{\sec (e+f x) (c+d \sin (e+f x))^n \left(a^3 \sqrt{2-2 \sin (e+f x)} (\sin (e+f x)+1)^3 \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 a^2 (\sin (e+f x)+1) \sqrt{1-\frac{2}{\sin (e+f x)+1}} \left(\frac{c-d}{d \sin (e+f x)+d}+1\right)^{-n} \left(a \left(4 n^2-8 n+3\right) (\sin (e+f x)+1)^2 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 (2 n+1) \left(2 a (2 n-1) F_1\left(\frac{3}{2}-n;-\frac{1}{2},-n;\frac{5}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{d-c}{\sin (e+f x) d+d}\right)+a (2 n-3) (\sin (e+f x)+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)\right)}{(2 n-3) (2 n-1) (2 n+1)}\right)}{16 a^4 f (a (\sin (e+f x)+1))^{3/2}}","-\frac{\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};-n,3;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d},\frac{1}{2} (1-\sin (e+f x))\right)}{4 a^2 f \sqrt{a \sin (e+f x)+a}}",1,"(Sec[e + f*x]*(c + d*Sin[e + f*x])^n*((a^3*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])^3)/((c + d*Sin[e + f*x])/(c - d))^n - (4*a^2*(1 + Sin[e + f*x])*Sqrt[1 - 2/(1 + Sin[e + f*x])]*(a*(3 - 8*n + 4*n^2)*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])^2 + 2*(1 + 2*n)*(2*a*(-1 + 2*n)*AppellF1[3/2 - n, -1/2, -n, 5/2 - n, 2/(1 + Sin[e + f*x]), (-c + d)/(d + d*Sin[e + f*x])] + a*(-3 + 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])]*(1 + Sin[e + f*x]))))/((-3 + 2*n)*(-1 + 2*n)*(1 + 2*n)*(1 + (c - d)/(d + d*Sin[e + f*x]))^n)))/(16*a^4*f*(a*(1 + Sin[e + f*x]))^(3/2))","B",0
668,1,1736,107,6.4306066,"\int (a+a \sin (e+f x)) \sqrt[3]{c+d \sin (e+f x)} \, dx","Integrate[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(1/3),x]","a \left(\frac{c \sec (e) \left(-\frac{F_1\left(-\frac{2}{3};-\frac{1}{2},-\frac{1}{2};\frac{1}{3};-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(1-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}\right)},-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}-1\right)}\right) \cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1} \sqrt{\frac{\cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} d+\sqrt{\cot ^2(e)+1} d}{d \sqrt{\cot ^2(e)+1}-c \csc (e)}} \sqrt{\frac{d \sqrt{\cot ^2(e)+1}-d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1}}{\sqrt{\cot ^2(e)+1} d+c \csc (e)}} \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)^{2/3}}-\frac{\frac{3 d \sin (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac{\cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1}}}{\left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)^{2/3}}\right) (\sin (e+f x)+1)}{4 f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{d \sec (e) \left(-\frac{F_1\left(-\frac{2}{3};-\frac{1}{2},-\frac{1}{2};\frac{1}{3};-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(1-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}\right)},-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}-1\right)}\right) \cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1} \sqrt{\frac{\cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} d+\sqrt{\cot ^2(e)+1} d}{d \sqrt{\cot ^2(e)+1}-c \csc (e)}} \sqrt{\frac{d \sqrt{\cot ^2(e)+1}-d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1}}{\sqrt{\cot ^2(e)+1} d+c \csc (e)}} \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)^{2/3}}-\frac{\frac{3 d \sin (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac{\cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1}}}{\left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)^{2/3}}\right) (\sin (e+f x)+1)}{f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{\sqrt[3]{c+d \sin (e+f x)} \left(-\frac{3 \cos (e) \cos (f x)}{4 f}+\frac{3 \sin (e) \sin (f x)}{4 f}+\frac{3 (c+4 d) \tan (e)}{4 d f}\right) (\sin (e+f x)+1)}{\left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{3 c F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(1-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}\right)},-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}-1\right)}\right) \sec (e) \sec \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\frac{d \sqrt{\tan ^2(e)+1}-d \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} d+c \sec (e)}} \sqrt{\frac{\sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1} d+\sqrt{\tan ^2(e)+1} d}{d \sqrt{\tan ^2(e)+1}-c \sec (e)}} \sqrt[3]{c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}} (\sin (e+f x)+1)}{d f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2 \sqrt{\tan ^2(e)+1}}+\frac{3 F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(1-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}\right)},-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}-1\right)}\right) \sec (e) \sec \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\frac{d \sqrt{\tan ^2(e)+1}-d \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} d+c \sec (e)}} \sqrt{\frac{\sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1} d+\sqrt{\tan ^2(e)+1} d}{d \sqrt{\tan ^2(e)+1}-c \sec (e)}} \sqrt[3]{c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}} (\sin (e+f x)+1)}{4 f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2 \sqrt{\tan ^2(e)+1}}\right)","-\frac{2 \sqrt{2} a \cos (e+f x) \sqrt[3]{c+d \sin (e+f x)} F_1\left(\frac{1}{2};-\frac{1}{2},-\frac{1}{3};\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} \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}}}",1,"a*((c*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-2/3, -1/2, -1/2, 1/3, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e])^(2/3))) - ((3*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e])^(2/3)))/(4*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (d*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-2/3, -1/2, -1/2, 1/3, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e])^(2/3))) - ((3*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e])^(2/3)))/(f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + ((1 + Sin[e + f*x])*(c + d*Sin[e + f*x])^(1/3)*((-3*Cos[e]*Cos[f*x])/(4*f) + (3*Sin[e]*Sin[f*x])/(4*f) + (3*(c + 4*d)*Tan[e])/(4*d*f)))/(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2 + (3*AppellF1[1/3, 1/2, 1/2, 4/3, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])^(1/3))/(4*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]) + (3*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])^(1/3))/(d*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]))","B",0
669,1,886,107,6.272318,"\int \frac{a+a \sin (e+f x)}{\sqrt[3]{c+d \sin (e+f x)}} \, dx","Integrate[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(1/3),x]","a \left(\frac{\sec (e) \left(-\frac{F_1\left(-\frac{1}{3};-\frac{1}{2},-\frac{1}{2};\frac{2}{3};-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(1-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}\right)},-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}-1\right)}\right) \cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1} \sqrt{\frac{\cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} d+\sqrt{\cot ^2(e)+1} d}{d \sqrt{\cot ^2(e)+1}-c \csc (e)}} \sqrt{\frac{d \sqrt{\cot ^2(e)+1}-d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1}}{\sqrt{\cot ^2(e)+1} d+c \csc (e)}} \sqrt[3]{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}-\frac{\frac{3 d \sin (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{2 \left(d^2 \cos ^2(e)+d^2 \sin ^2(e)\right)}-\frac{\cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1}}}{\sqrt[3]{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}\right) (\sin (e+f x)+1)}{f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{3 (c+d \sin (e+f x))^{2/3} \tan (e) (\sin (e+f x)+1)}{2 d f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{3 F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(1-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}\right)},-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}-1\right)}\right) \sec (e) \sec \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\frac{d \sqrt{\tan ^2(e)+1}-d \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} d+c \sec (e)}} \sqrt{\frac{\sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1} d+\sqrt{\tan ^2(e)+1} d}{d \sqrt{\tan ^2(e)+1}-c \sec (e)}} \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)^{2/3} (\sin (e+f x)+1)}{2 d f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2 \sqrt{\tan ^2(e)+1}}\right)","-\frac{2 \sqrt{2} a \cos (e+f x) \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}} F_1\left(\frac{1}{2};-\frac{1}{2},\frac{1}{3};\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} \sqrt[3]{c+d \sin (e+f x)}}",1,"a*((Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/3, -1/2, -1/2, 2/3, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e])^(1/3))) - ((3*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(2*(d^2*Cos[e]^2 + d^2*Sin[e]^2)) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e])^(1/3)))/(f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (3*(1 + Sin[e + f*x])*(c + d*Sin[e + f*x])^(2/3)*Tan[e])/(2*d*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (3*AppellF1[2/3, 1/2, 1/2, 5/3, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])^(2/3))/(2*d*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]))","B",0
670,1,942,112,6.4180718,"\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^{4/3}} \, dx","Integrate[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(4/3),x]","a \left(\frac{(c+d \sin (e+f x))^{2/3} \left(\frac{3 \csc (e) (c \cos (e)+d \sin (f x))}{d (c+d) f (c+d \sin (e+f x))}-\frac{3 \csc (e) \sec (e)}{d (c+d) f}\right) (\sin (e+f x)+1)}{\left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}-\frac{2 \sec (e) \left(-\frac{F_1\left(-\frac{1}{3};-\frac{1}{2},-\frac{1}{2};\frac{2}{3};-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(1-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}\right)},-\frac{\csc (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{d \sqrt{\cot ^2(e)+1} \left(-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}-1\right)}\right) \cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1} \sqrt{\frac{\cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} d+\sqrt{\cot ^2(e)+1} d}{d \sqrt{\cot ^2(e)+1}-c \csc (e)}} \sqrt{\frac{d \sqrt{\cot ^2(e)+1}-d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1}}{\sqrt{\cot ^2(e)+1} d+c \csc (e)}} \sqrt[3]{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}-\frac{\frac{3 d \sin (e) \left(c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)\right)}{2 \left(d^2 \cos ^2(e)+d^2 \sin ^2(e)\right)}-\frac{\cot (e) \sin \left(f x-\tan ^{-1}(\cot (e))\right)}{\sqrt{\cot ^2(e)+1}}}{\sqrt[3]{c+d \cos \left(f x-\tan ^{-1}(\cot (e))\right) \sqrt{\cot ^2(e)+1} \sin (e)}}\right) (\sin (e+f x)+1)}{(c+d) f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2}+\frac{3 F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(1-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}\right)},-\frac{\sec (e) \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)}{d \sqrt{\tan ^2(e)+1} \left(-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}-1\right)}\right) \sec (e) \sec \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\frac{d \sqrt{\tan ^2(e)+1}-d \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} d+c \sec (e)}} \sqrt{\frac{\sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1} d+\sqrt{\tan ^2(e)+1} d}{d \sqrt{\tan ^2(e)+1}-c \sec (e)}} \left(c+d \cos (e) \sin \left(f x+\tan ^{-1}(\tan (e))\right) \sqrt{\tan ^2(e)+1}\right)^{2/3} (\sin (e+f x)+1)}{2 d (c+d) f \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)+\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)^2 \sqrt{\tan ^2(e)+1}}\right)","-\frac{2 \sqrt{2} a \cos (e+f x) \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}} F_1\left(\frac{1}{2};-\frac{1}{2},\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f (c+d) \sqrt{\sin (e+f x)+1} \sqrt[3]{c+d \sin (e+f x)}}",1,"a*(((1 + Sin[e + f*x])*(c + d*Sin[e + f*x])^(2/3)*((-3*Csc[e]*Sec[e])/(d*(c + d)*f) + (3*Csc[e]*(c*Cos[e] + d*Sin[f*x]))/(d*(c + d)*f*(c + d*Sin[e + f*x]))))/(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2 - (2*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/3, -1/2, -1/2, 2/3, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e])^(1/3))) - ((3*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(2*(d^2*Cos[e]^2 + d^2*Sin[e]^2)) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e])^(1/3)))/((c + d)*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (3*AppellF1[2/3, 1/2, 1/2, 5/3, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])^(2/3))/(2*d*(c + d)*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]))","B",0
671,1,143,171,0.6786862,"\int (a+b \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx","Integrate[(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^3,x]","\frac{-24 \left(3 a d \left(4 c^2+d^2\right)+b \left(4 c^3+9 c d^2\right)\right) \cos (e+f x)+3 \left(-8 d \left(3 a c d+b \left(3 c^2+d^2\right)\right) \sin (2 (e+f x))+4 (e+f x) \left(8 a c^3+12 a c d^2+12 b c^2 d+3 b d^3\right)+b d^3 \sin (4 (e+f x))\right)+8 d^2 (a d+3 b c) \cos (3 (e+f x))}{96 f}","-\frac{d \left(20 a c d+6 b c^2+9 b d^2\right) \sin (e+f x) \cos (e+f x)}{24 f}-\frac{\left(4 a d \left(4 c^2+d^2\right)+3 b \left(c^3+4 c d^2\right)\right) \cos (e+f x)}{6 f}+\frac{1}{8} x \left(8 a c^3+12 a c d^2+12 b c^2 d+3 b d^3\right)-\frac{(4 a d+3 b c) \cos (e+f x) (c+d \sin (e+f x))^2}{12 f}-\frac{b \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}",1,"(-24*(3*a*d*(4*c^2 + d^2) + b*(4*c^3 + 9*c*d^2))*Cos[e + f*x] + 8*d^2*(3*b*c + a*d)*Cos[3*(e + f*x)] + 3*(4*(8*a*c^3 + 12*b*c^2*d + 12*a*c*d^2 + 3*b*d^3)*(e + f*x) - 8*d*(3*a*c*d + b*(3*c^2 + d^2))*Sin[2*(e + f*x)] + b*d^3*Sin[4*(e + f*x)]))/(96*f)","A",1
672,1,90,106,0.3067304,"\int (a+b \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx","Integrate[(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]","\frac{6 (e+f x) \left(a \left(2 c^2+d^2\right)+2 b c d\right)-3 \left(8 a c d+4 b c^2+3 b d^2\right) \cos (e+f x)-3 d (a d+2 b c) \sin (2 (e+f x))+b d^2 \cos (3 (e+f x))}{12 f}","-\frac{2 \left(3 a c d+b \left(c^2+d^2\right)\right) \cos (e+f x)}{3 f}+\frac{1}{2} x \left(a \left(2 c^2+d^2\right)+2 b c d\right)-\frac{d (3 a d+2 b c) \sin (e+f x) \cos (e+f x)}{6 f}-\frac{b \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}",1,"(6*(2*b*c*d + a*(2*c^2 + d^2))*(e + f*x) - 3*(4*b*c^2 + 8*a*c*d + 3*b*d^2)*Cos[e + f*x] + b*d^2*Cos[3*(e + f*x)] - 3*d*(2*b*c + a*d)*Sin[2*(e + f*x)])/(12*f)","A",1
673,1,52,53,0.0874313,"\int (a+b \sin (e+f x)) (c+d \sin (e+f x)) \, dx","Integrate[(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]","\frac{-4 (a d+b c) \cos (e+f x)+4 a c f x-b d \sin (2 (e+f x))+2 b d e+2 b d f x}{4 f}","-\frac{(a d+b c) \cos (e+f x)}{f}+\frac{1}{2} x (2 a c+b d)-\frac{b d \sin (e+f x) \cos (e+f x)}{2 f}",1,"(2*b*d*e + 4*a*c*f*x + 2*b*d*f*x - 4*(b*c + a*d)*Cos[e + f*x] - b*d*Sin[2*(e + f*x)])/(4*f)","A",1
674,1,27,16,0.0063629,"\int (a+b \sin (e+f x)) \, dx","Integrate[a + b*Sin[e + f*x],x]","a x+\frac{b \sin (e) \sin (f x)}{f}-\frac{b \cos (e) \cos (f x)}{f}","a x-\frac{b \cos (e+f x)}{f}",1,"a*x - (b*Cos[e]*Cos[f*x])/f + (b*Sin[e]*Sin[f*x])/f","A",1
675,1,67,65,0.1341893,"\int \frac{a+b \sin (e+f x)}{c+d \sin (e+f x)} \, dx","Integrate[(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x]),x]","\frac{\frac{(2 a d-2 b c) \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}}+b (e+f x)}{d f}","\frac{b x}{d}-\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)}{d f \sqrt{c^2-d^2}}",1,"(b*(e + f*x) + ((-2*b*c + 2*a*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2])/(d*f)","A",1
676,1,96,98,0.3005773,"\int \frac{a+b \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx","Integrate[(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^2,x]","\frac{\frac{2 (a c-b d) \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}}+\frac{(a d-b c) \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))}}{f}","\frac{2 (a c-b d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{3/2}}-\frac{(b c-a d) \cos (e+f x)}{f \left(c^2-d^2\right) (c+d \sin (e+f x))}",1,"((2*(a*c - b*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(3/2) + ((-(b*c) + a*d)*Cos[e + f*x])/((c - d)*(c + d)*(c + d*Sin[e + f*x])))/f","A",1
677,1,157,164,0.6170468,"\int \frac{a+b \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx","Integrate[(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^3,x]","\frac{\frac{2 \left(a \left(2 c^2+d^2\right)-3 b c d\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)^{5/2}}-\frac{\left(b \left(c^2+2 d^2\right)-3 a c d\right) \cos (e+f x)}{(c-d)^2 (c+d)^2 (c+d \sin (e+f x))}+\frac{(a d-b c) \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))^2}}{2 f}","-\frac{\left(3 b c d-a \left(2 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)}{f \left(c^2-d^2\right)^{5/2}}+\frac{\left(3 a c d-b \left(c^2+2 d^2\right)\right) \cos (e+f x)}{2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))}-\frac{(b c-a d) \cos (e+f x)}{2 f \left(c^2-d^2\right) (c+d \sin (e+f x))^2}",1,"((2*(-3*b*c*d + a*(2*c^2 + d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(5/2) + ((-(b*c) + a*d)*Cos[e + f*x])/((c - d)*(c + d)*(c + d*Sin[e + f*x])^2) - ((-3*a*c*d + b*(c^2 + 2*d^2))*Cos[e + f*x])/((c - d)^2*(c + d)^2*(c + d*Sin[e + f*x])))/(2*f)","A",1
678,1,249,314,1.3483601,"\int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx","Integrate[(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3,x]","\frac{10 d \left(4 a^2 d^2+24 a b c d+b^2 \left(12 c^2+5 d^2\right)\right) \cos (3 (e+f x))-60 \left(6 a^2 \left(4 c^2 d+d^3\right)+4 a b c \left(4 c^2+9 d^2\right)+b^2 d \left(18 c^2+5 d^2\right)\right) \cos (e+f x)+15 \left(4 (e+f x) \left(4 a^2 \left(2 c^3+3 c d^2\right)+6 a b d \left(4 c^2+d^2\right)+b^2 c \left(4 c^2+9 d^2\right)\right)-8 \left(3 a^2 c d^2+2 a b d \left(3 c^2+d^2\right)+b^2 \left(c^3+3 c d^2\right)\right) \sin (2 (e+f x))+b d^2 (2 a d+3 b c) \sin (4 (e+f x))\right)-6 b^2 d^3 \cos (5 (e+f x))}{480 f}","-\frac{\left(20 a^2 d^2 \left(4 c^2+d^2\right)+30 a b c d \left(c^2+4 d^2\right)-\left(b^2 \left(3 c^4-52 c^2 d^2-16 d^4\right)\right)\right) \cos (e+f x)}{30 d f}-\frac{\left(100 a^2 c d^2+30 a b d \left(2 c^2+3 d^2\right)-\left(b^2 \left(6 c^3-71 c d^2\right)\right)\right) \sin (e+f x) \cos (e+f x)}{120 f}+\frac{1}{8} x \left(4 a^2 \left(2 c^3+3 c d^2\right)+6 a b d \left(4 c^2+d^2\right)+b^2 c \left(4 c^2+9 d^2\right)\right)-\frac{\left(4 d^2 \left(5 a^2+4 b^2\right)-3 b c (b c-10 a d)\right) \cos (e+f x) (c+d \sin (e+f x))^2}{60 d f}+\frac{b (b c-10 a d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac{b^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}",1,"(-60*(b^2*d*(18*c^2 + 5*d^2) + 4*a*b*c*(4*c^2 + 9*d^2) + 6*a^2*(4*c^2*d + d^3))*Cos[e + f*x] + 10*d*(24*a*b*c*d + 4*a^2*d^2 + b^2*(12*c^2 + 5*d^2))*Cos[3*(e + f*x)] - 6*b^2*d^3*Cos[5*(e + f*x)] + 15*(4*(6*a*b*d*(4*c^2 + d^2) + b^2*c*(4*c^2 + 9*d^2) + 4*a^2*(2*c^3 + 3*c*d^2))*(e + f*x) - 8*(3*a^2*c*d^2 + 2*a*b*d*(3*c^2 + d^2) + b^2*(c^3 + 3*c*d^2))*Sin[2*(e + f*x)] + b*d^2*(3*b*c + 2*a*d)*Sin[4*(e + f*x)]))/(480*f)","A",1
679,1,160,217,0.7816458,"\int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2 \, dx","Integrate[(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2,x]","\frac{3 \left(4 (e+f x) \left(4 a^2 \left(2 c^2+d^2\right)+16 a b c d+b^2 \left(4 c^2+3 d^2\right)\right)-8 \left(a^2 d^2+4 a b c d+b^2 \left(c^2+d^2\right)\right) \sin (2 (e+f x))+b^2 d^2 \sin (4 (e+f x))\right)-48 \left(4 a^2 c d+a b \left(4 c^2+3 d^2\right)+3 b^2 c d\right) \cos (e+f x)+16 b d (a d+b c) \cos (3 (e+f x))}{96 f}","\frac{1}{8} x \left(4 a^2 \left(2 c^2+d^2\right)+16 a b c d+b^2 \left(4 c^2+3 d^2\right)\right)-\frac{\left(a^3 \left(-d^2\right)+8 a^2 b c d+4 a b^2 \left(3 c^2+2 d^2\right)+8 b^3 c d\right) \cos (e+f x)}{6 b f}-\frac{\left(2 a d (8 b c-a d)+3 b^2 \left(4 c^2+3 d^2\right)\right) \sin (e+f x) \cos (e+f x)}{24 f}-\frac{d (8 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^2}{12 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^3}{4 b f}",1,"(-48*(4*a^2*c*d + 3*b^2*c*d + a*b*(4*c^2 + 3*d^2))*Cos[e + f*x] + 16*b*d*(b*c + a*d)*Cos[3*(e + f*x)] + 3*(4*(16*a*b*c*d + 4*a^2*(2*c^2 + d^2) + b^2*(4*c^2 + 3*d^2))*(e + f*x) - 8*(4*a*b*c*d + a^2*d^2 + b^2*(c^2 + d^2))*Sin[2*(e + f*x)] + b^2*d^2*Sin[4*(e + f*x)]))/(96*f)","A",1
680,1,90,107,0.2969456,"\int (a+b \sin (e+f x))^2 (c+d \sin (e+f x)) \, dx","Integrate[(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x]),x]","\frac{6 (e+f x) \left(2 a^2 c+2 a b d+b^2 c\right)-3 \left(4 a^2 d+8 a b c+3 b^2 d\right) \cos (e+f x)-3 b (2 a d+b c) \sin (2 (e+f x))+b^2 d \cos (3 (e+f x))}{12 f}","-\frac{2 \left(a^2 d+3 a b c+b^2 d\right) \cos (e+f x)}{3 f}+\frac{1}{2} x \left(2 a^2 c+2 a b d+b^2 c\right)-\frac{b (2 a d+3 b c) \sin (e+f x) \cos (e+f x)}{6 f}-\frac{d \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}",1,"(6*(2*a^2*c + b^2*c + 2*a*b*d)*(e + f*x) - 3*(8*a*b*c + 4*a^2*d + 3*b^2*d)*Cos[e + f*x] + b^2*d*Cos[3*(e + f*x)] - 3*b*(b*c + 2*a*d)*Sin[2*(e + f*x)])/(12*f)","A",1
681,1,46,50,0.0970834,"\int (a+b \sin (e+f x))^2 \, dx","Integrate[(a + b*Sin[e + f*x])^2,x]","-\frac{-2 \left(2 a^2+b^2\right) (e+f x)+8 a b \cos (e+f x)+b^2 \sin (2 (e+f x))}{4 f}","\frac{1}{2} x \left(2 a^2+b^2\right)-\frac{2 a b \cos (e+f x)}{f}-\frac{b^2 \sin (e+f x) \cos (e+f x)}{2 f}",1,"-1/4*(-2*(2*a^2 + b^2)*(e + f*x) + 8*a*b*Cos[e + f*x] + b^2*Sin[2*(e + f*x)])/f","A",1
682,1,89,93,0.2124514,"\int \frac{(a+b \sin (e+f x))^2}{c+d \sin (e+f x)} \, dx","Integrate[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x]),x]","-\frac{-\frac{2 (b c-a d)^2 \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}}+b (e+f x) (b c-2 a d)+b^2 d \cos (e+f x)}{d^2 f}","\frac{2 (b c-a d)^2 \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{b x (b c-2 a d)}{d^2}-\frac{b^2 \cos (e+f x)}{d f}",1,"-((b*(b*c - 2*a*d)*(e + f*x) - (2*(b*c - a*d)^2*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] + b^2*d*Cos[e + f*x])/(d^2*f))","A",1
683,1,134,129,0.5396756,"\int \frac{(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^2} \, dx","Integrate[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^2,x]","\frac{-\frac{2 \left(-a^2 c d^2+2 a b d^3+b^2 \left(c^3-2 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}}+\frac{d (b c-a d)^2 \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))}+b^2 (e+f x)}{d^2 f}","-\frac{2 (b c-a d) \left(a c d+b \left(c^2-2 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{(b c-a d)^2 \cos (e+f x)}{d f \left(c^2-d^2\right) (c+d \sin (e+f x))}+\frac{b^2 x}{d^2}",1,"(b^2*(e + f*x) - (2*(-(a^2*c*d^2) + 2*a*b*d^3 + b^2*(c^3 - 2*c*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(3/2) + (d*(b*c - a*d)^2*Cos[e + f*x])/((c - d)*(c + d)*(c + d*Sin[e + f*x])))/(d^2*f)","A",1
684,1,202,196,0.9403156,"\int \frac{(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^3} \, dx","Integrate[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^3,x]","\frac{\frac{2 \left(a^2 \left(2 c^2+d^2\right)-6 a b c d+b^2 \left(c^2+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)}{\left(c^2-d^2\right)^{5/2}}-\frac{\left(-3 a^2 c d^2+2 a b d \left(c^2+2 d^2\right)+b^2 \left(c^3-4 c d^2\right)\right) \cos (e+f x)}{d (c-d)^2 (c+d)^2 (c+d \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x)}{d (c-d) (c+d) (c+d \sin (e+f x))^2}}{2 f}","-\frac{\left(-\left(a^2 \left(2 c^2+d^2\right)\right)+6 a b c d-b^2 \left(c^2+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)}{f \left(c^2-d^2\right)^{5/2}}+\frac{(b c-a d)^2 \cos (e+f x)}{2 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^2}-\frac{\left(3 a c d+b \left(c^2-4 d^2\right)\right) (b c-a d) \cos (e+f x)}{2 d f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))}",1,"((2*(-6*a*b*c*d + a^2*(2*c^2 + d^2) + b^2*(c^2 + 2*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(5/2) + ((b*c - a*d)^2*Cos[e + f*x])/((c - d)*d*(c + d)*(c + d*Sin[e + f*x])^2) - ((-3*a^2*c*d^2 + 2*a*b*d*(c^2 + 2*d^2) + b^2*(c^3 - 4*c*d^2))*Cos[e + f*x])/((c - d)^2*d*(c + d)^2*(c + d*Sin[e + f*x])))/(2*f)","A",1
685,1,346,305,1.4432942,"\int \frac{(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx","Integrate[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^4,x]","\frac{\frac{12 \left(a^2 \left(2 c^3+3 c d^2\right)-2 a b d \left(4 c^2+d^2\right)+b^2 c \left(c^2+4 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)^{7/2}}+\frac{\cos (e+f x) \left(d \left(-a^2 d^2 \left(11 c^2+4 d^2\right)+a b \left(4 c^3 d+26 c d^3\right)+b^2 \left(c^4-10 c^2 d^2-6 d^4\right)\right) \cos (2 (e+f x))-6 \left(-a^2 c d^2 \left(9 c^2+d^2\right)-2 a b d \left(-2 c^4-9 c^2 d^2+d^4\right)+b^2 \left(c^5-9 c^3 d^2-2 c d^4\right)\right) \sin (e+f x)+36 a^2 c^4 d+a^2 c^2 d^3+8 a^2 d^5-24 a b c^5-44 a b c^3 d^2-22 a b c d^4+25 b^2 c^4 d+14 b^2 c^2 d^3+6 b^2 d^5\right)}{\left(c^2-d^2\right)^3 (c+d \sin (e+f x))^3}}{12 f}","-\frac{\left(-\left(a^2 \left(2 c^3+3 c d^2\right)\right)+2 a b d \left(4 c^2+d^2\right)-b^2 c \left(c^2+4 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)}{f \left(c^2-d^2\right)^{7/2}}+\frac{\left(a^2 d^2 \left(11 c^2+4 d^2\right)-a b \left(4 c^3 d+26 c d^3\right)-\left(b^2 \left(c^4-10 c^2 d^2-6 d^4\right)\right)\right) \cos (e+f x)}{6 d f \left(c^2-d^2\right)^3 (c+d \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x)}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^3}-\frac{\left(5 a c d+b \left(c^2-6 d^2\right)\right) (b c-a d) \cos (e+f x)}{6 d f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^2}",1,"((12*(-2*a*b*d*(4*c^2 + d^2) + b^2*c*(c^2 + 4*d^2) + a^2*(2*c^3 + 3*c*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(7/2) + (Cos[e + f*x]*(-24*a*b*c^5 + 36*a^2*c^4*d + 25*b^2*c^4*d - 44*a*b*c^3*d^2 + a^2*c^2*d^3 + 14*b^2*c^2*d^3 - 22*a*b*c*d^4 + 8*a^2*d^5 + 6*b^2*d^5 + d*(-(a^2*d^2*(11*c^2 + 4*d^2)) + a*b*(4*c^3*d + 26*c*d^3) + b^2*(c^4 - 10*c^2*d^2 - 6*d^4))*Cos[2*(e + f*x)] - 6*(-(a^2*c*d^2*(9*c^2 + d^2)) - 2*a*b*d*(-2*c^4 - 9*c^2*d^2 + d^4) + b^2*(c^5 - 9*c^3*d^2 - 2*c*d^4))*Sin[e + f*x]))/((c^2 - d^2)^3*(c + d*Sin[e + f*x])^3))/(12*f)","A",1
686,1,552,400,1.1760762,"\int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx","Integrate[(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3,x]","\frac{960 a^3 c^3 e+960 a^3 c^3 f x-720 a^3 c d^2 \sin (2 (e+f x))+1440 a^3 c d^2 e+1440 a^3 c d^2 f x-2160 a^2 b c^2 d \sin (2 (e+f x))+4320 a^2 b c^2 d e+4320 a^2 b c^2 d f x-720 a^2 b d^3 \sin (2 (e+f x))+90 a^2 b d^3 \sin (4 (e+f x))+1080 a^2 b d^3 e+1080 a^2 b d^3 f x-360 \left(2 a^3 \left(4 c^2 d+d^3\right)+2 a^2 b c \left(4 c^2+9 d^2\right)+a b^2 d \left(18 c^2+5 d^2\right)+b^3 c \left(2 c^2+5 d^2\right)\right) \cos (e+f x)+20 \left(4 a^3 d^3+36 a^2 b c d^2+3 a b^2 d \left(12 c^2+5 d^2\right)+b^3 \left(4 c^3+15 c d^2\right)\right) \cos (3 (e+f x))-720 a b^2 c^3 \sin (2 (e+f x))+1440 a b^2 c^3 e+1440 a b^2 c^3 f x-2160 a b^2 c d^2 \sin (2 (e+f x))+270 a b^2 c d^2 \sin (4 (e+f x))+3240 a b^2 c d^2 e+3240 a b^2 c d^2 f x-36 a b^2 d^3 \cos (5 (e+f x))-720 b^3 c^2 d \sin (2 (e+f x))+90 b^3 c^2 d \sin (4 (e+f x))+1080 b^3 c^2 d e+1080 b^3 c^2 d f x-36 b^3 c d^2 \cos (5 (e+f x))-225 b^3 d^3 \sin (2 (e+f x))+45 b^3 d^3 \sin (4 (e+f x))-5 b^3 d^3 \sin (6 (e+f x))+300 b^3 d^3 e+300 b^3 d^3 f x}{960 f}","\frac{(a d+b c) \left(a^2 d^2+8 a b c d+b^2 \left(c^2+6 d^2\right)\right) \cos ^3(e+f x)}{3 f}-\frac{3 b d \left(a^2 d^2+3 a b c d+b^2 c^2\right) \sin ^3(e+f x) \cos (e+f x)}{4 f}-\frac{\left(24 a^3 c d^2+18 a^2 b d \left(4 c^2+d^2\right)+6 a b^2 c \left(4 c^2+9 d^2\right)+b^3 d \left(18 c^2+5 d^2\right)\right) \sin (e+f x) \cos (e+f x)}{16 f}-\frac{\left(a^3 \left(3 c^2 d+d^3\right)+3 a^2 b c \left(c^2+3 d^2\right)+3 a b^2 d \left(3 c^2+d^2\right)+b^3 c \left(c^2+3 d^2\right)\right) \cos (e+f x)}{f}+\frac{1}{16} x \left(8 a^3 \left(2 c^3+3 c d^2\right)+18 a^2 b d \left(4 c^2+d^2\right)+6 a b^2 c \left(4 c^2+9 d^2\right)+b^3 d \left(18 c^2+5 d^2\right)\right)-\frac{3 b^2 d^2 (a d+b c) \cos ^5(e+f x)}{5 f}-\frac{b^3 d^3 \sin ^5(e+f x) \cos (e+f x)}{6 f}-\frac{5 b^3 d^3 \sin ^3(e+f x) \cos (e+f x)}{24 f}",1,"(960*a^3*c^3*e + 1440*a*b^2*c^3*e + 4320*a^2*b*c^2*d*e + 1080*b^3*c^2*d*e + 1440*a^3*c*d^2*e + 3240*a*b^2*c*d^2*e + 1080*a^2*b*d^3*e + 300*b^3*d^3*e + 960*a^3*c^3*f*x + 1440*a*b^2*c^3*f*x + 4320*a^2*b*c^2*d*f*x + 1080*b^3*c^2*d*f*x + 1440*a^3*c*d^2*f*x + 3240*a*b^2*c*d^2*f*x + 1080*a^2*b*d^3*f*x + 300*b^3*d^3*f*x - 360*(b^3*c*(2*c^2 + 5*d^2) + a*b^2*d*(18*c^2 + 5*d^2) + 2*a^2*b*c*(4*c^2 + 9*d^2) + 2*a^3*(4*c^2*d + d^3))*Cos[e + f*x] + 20*(36*a^2*b*c*d^2 + 4*a^3*d^3 + 3*a*b^2*d*(12*c^2 + 5*d^2) + b^3*(4*c^3 + 15*c*d^2))*Cos[3*(e + f*x)] - 36*b^3*c*d^2*Cos[5*(e + f*x)] - 36*a*b^2*d^3*Cos[5*(e + f*x)] - 720*a*b^2*c^3*Sin[2*(e + f*x)] - 2160*a^2*b*c^2*d*Sin[2*(e + f*x)] - 720*b^3*c^2*d*Sin[2*(e + f*x)] - 720*a^3*c*d^2*Sin[2*(e + f*x)] - 2160*a*b^2*c*d^2*Sin[2*(e + f*x)] - 720*a^2*b*d^3*Sin[2*(e + f*x)] - 225*b^3*d^3*Sin[2*(e + f*x)] + 90*b^3*c^2*d*Sin[4*(e + f*x)] + 270*a*b^2*c*d^2*Sin[4*(e + f*x)] + 90*a^2*b*d^3*Sin[4*(e + f*x)] + 45*b^3*d^3*Sin[4*(e + f*x)] - 5*b^3*d^3*Sin[6*(e + f*x)])/(960*f)","A",1
687,1,246,315,1.5850092,"\int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^2 \, dx","Integrate[(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2,x]","\frac{10 b \left(12 a^2 d^2+24 a b c d+b^2 \left(4 c^2+5 d^2\right)\right) \cos (3 (e+f x))+15 \left(4 (e+f x) \left(4 a^3 \left(2 c^2+d^2\right)+24 a^2 b c d+3 a b^2 \left(4 c^2+3 d^2\right)+6 b^3 c d\right)-8 \left(a^3 d^2+6 a^2 b c d+3 a b^2 \left(c^2+d^2\right)+2 b^3 c d\right) \sin (2 (e+f x))+b^2 d (3 a d+2 b c) \sin (4 (e+f x))\right)-60 \left(16 a^3 c d+6 a^2 b \left(4 c^2+3 d^2\right)+36 a b^2 c d+b^3 \left(6 c^2+5 d^2\right)\right) \cos (e+f x)-6 b^3 d^2 \cos (5 (e+f x))}{480 f}","-\frac{\left(-6 a^3 d^2+60 a^2 b c d+a b^2 \left(100 c^2+71 d^2\right)+90 b^3 c d\right) \sin (e+f x) \cos (e+f x)}{120 f}+\frac{1}{8} x \left(4 a^3 \left(2 c^2+d^2\right)+24 a^2 b c d+3 a b^2 \left(4 c^2+3 d^2\right)+6 b^3 c d\right)-\frac{\left(-3 a^4 d^2+30 a^3 b c d+4 a^2 b^2 \left(20 c^2+13 d^2\right)+120 a b^3 c d+4 b^4 \left(5 c^2+4 d^2\right)\right) \cos (e+f x)}{30 b f}-\frac{\left(3 a d (10 b c-a d)+4 b^2 \left(5 c^2+4 d^2\right)\right) \cos (e+f x) (a+b \sin (e+f x))^2}{60 b f}-\frac{d (10 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^3}{20 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^4}{5 b f}",1,"(-60*(16*a^3*c*d + 36*a*b^2*c*d + 6*a^2*b*(4*c^2 + 3*d^2) + b^3*(6*c^2 + 5*d^2))*Cos[e + f*x] + 10*b*(24*a*b*c*d + 12*a^2*d^2 + b^2*(4*c^2 + 5*d^2))*Cos[3*(e + f*x)] - 6*b^3*d^2*Cos[5*(e + f*x)] + 15*(4*(24*a^2*b*c*d + 6*b^3*c*d + 4*a^3*(2*c^2 + d^2) + 3*a*b^2*(4*c^2 + 3*d^2))*(e + f*x) - 8*(6*a^2*b*c*d + 2*b^3*c*d + a^3*d^2 + 3*a*b^2*(c^2 + d^2))*Sin[2*(e + f*x)] + b^2*d*(2*b*c + 3*a*d)*Sin[4*(e + f*x)]))/(480*f)","A",1
688,1,142,171,0.6571242,"\int (a+b \sin (e+f x))^3 (c+d \sin (e+f x)) \, dx","Integrate[(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x]),x]","\frac{3 \left(-8 b \left(3 a^2 d+3 a b c+b^2 d\right) \sin (2 (e+f x))+4 (e+f x) \left(8 a^3 c+12 a^2 b d+12 a b^2 c+3 b^3 d\right)+b^3 d \sin (4 (e+f x))\right)-24 \left(4 a^3 d+12 a^2 b c+9 a b^2 d+3 b^3 c\right) \cos (e+f x)+8 b^2 (3 a d+b c) \cos (3 (e+f x))}{96 f}","-\frac{b \left(6 a^2 d+20 a b c+9 b^2 d\right) \sin (e+f x) \cos (e+f x)}{24 f}-\frac{\left(3 a^3 d+16 a^2 b c+12 a b^2 d+4 b^3 c\right) \cos (e+f x)}{6 f}+\frac{1}{8} x \left(8 a^3 c+12 a^2 b d+12 a b^2 c+3 b^3 d\right)-\frac{(3 a d+4 b c) \cos (e+f x) (a+b \sin (e+f x))^2}{12 f}-\frac{d \cos (e+f x) (a+b \sin (e+f x))^3}{4 f}",1,"(-24*(12*a^2*b*c + 3*b^3*c + 4*a^3*d + 9*a*b^2*d)*Cos[e + f*x] + 8*b^2*(b*c + 3*a*d)*Cos[3*(e + f*x)] + 3*(4*(8*a^3*c + 12*a*b^2*c + 12*a^2*b*d + 3*b^3*d)*(e + f*x) - 8*b*(3*a*b*c + 3*a^2*d + b^2*d)*Sin[2*(e + f*x)] + b^3*d*Sin[4*(e + f*x)]))/(96*f)","A",1
689,1,71,90,0.1723871,"\int (a+b \sin (e+f x))^3 \, dx","Integrate[(a + b*Sin[e + f*x])^3,x]","\frac{6 a \left(2 a^2+3 b^2\right) (e+f x)-9 b \left(4 a^2+b^2\right) \cos (e+f x)-9 a b^2 \sin (2 (e+f x))+b^3 \cos (3 (e+f x))}{12 f}","-\frac{2 b \left(4 a^2+b^2\right) \cos (e+f x)}{3 f}+\frac{1}{2} a x \left(2 a^2+3 b^2\right)-\frac{5 a b^2 \sin (e+f x) \cos (e+f x)}{6 f}-\frac{b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}",1,"(6*a*(2*a^2 + 3*b^2)*(e + f*x) - 9*b*(4*a^2 + b^2)*Cos[e + f*x] + b^3*Cos[3*(e + f*x)] - 9*a*b^2*Sin[2*(e + f*x)])/(12*f)","A",1
690,1,137,156,0.372225,"\int \frac{(a+b \sin (e+f x))^3}{c+d \sin (e+f x)} \, dx","Integrate[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x]),x]","\frac{2 b (e+f x) \left(6 a^2 d^2-6 a b c d+b^2 \left(2 c^2+d^2\right)\right)+4 b^2 d (b c-3 a d) \cos (e+f x)-\frac{8 (b c-a d)^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}}+b^3 \left(-d^2\right) \sin (2 (e+f x))}{4 d^3 f}","-\frac{b x \left(-6 a^2 d^2+6 a b c d-\left(b^2 \left(2 c^2+d^2\right)\right)\right)}{2 d^3}+\frac{b^2 (2 b c-5 a d) \cos (e+f x)}{2 d^2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f}-\frac{2 (b c-a d)^3 \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}}",1,"(2*b*(-6*a*b*c*d + 6*a^2*d^2 + b^2*(2*c^2 + d^2))*(e + f*x) - (8*(b*c - a*d)^3*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] + 4*b^2*d*(b*c - 3*a*d)*Cos[e + f*x] - b^3*d^2*Sin[2*(e + f*x)])/(4*d^3*f)","A",1
691,1,152,208,1.1000613,"\int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^2} \, dx","Integrate[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^2,x]","\frac{-b^2 (e+f x) (2 b c-3 a d)+\frac{2 (b c-a d)^2 \left(a c d+2 b c^2-3 b d^2\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}}+\frac{d (a d-b c)^3 \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))}+b^3 (-d) \cos (e+f x)}{d^3 f}","\frac{b \left(-a^2 d^2+2 a b c d-\left(b^2 \left(2 c^2-d^2\right)\right)\right) \cos (e+f x)}{d^2 f \left(c^2-d^2\right)}-\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left(c^2-d^2\right) (c+d \sin (e+f x))}+\frac{2 (b c-a d)^2 \left(a c d+2 b c^2-3 b d^2\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}}",1,"(-(b^2*(2*b*c - 3*a*d)*(e + f*x)) + (2*(b*c - a*d)^2*(2*b*c^2 + a*c*d - 3*b*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(3/2) - b^3*d*Cos[e + f*x] + (d*(-(b*c) + a*d)^3*Cos[e + f*x])/((c - d)*(c + d)*(c + d*Sin[e + f*x])))/(d^3*f)","A",1
692,1,521,255,2.3510358,"\int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx","Integrate[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^3,x]","\frac{\frac{3 a^3 c d^5 \sin (2 (e+f x))-3 a^2 b c^2 d^4 \sin (2 (e+f x))-6 a^2 b d^6 \sin (2 (e+f x))-3 a b^2 c^3 d^3 \sin (2 (e+f x))+12 a b^2 c d^5 \sin (2 (e+f x))-2 d (b c-a d)^2 \left(-4 a c^2 d+a d^3-2 b c^3+5 b c d^2\right) \cos (e+f x)+4 b^3 c^6 e+4 b^3 c^6 f x+8 b^3 c^5 d e \sin (e+f x)+8 b^3 c^5 d f x \sin (e+f x)+3 b^3 c^4 d^2 \sin (2 (e+f x))-6 b^3 c^4 d^2 e-6 b^3 c^4 d^2 f x-16 b^3 c^3 d^3 e \sin (e+f x)-16 b^3 c^3 d^3 f x \sin (e+f x)-6 b^3 c^2 d^4 \sin (2 (e+f x))-2 b^3 \left(d^3-c^2 d\right)^2 (e+f x) \cos (2 (e+f x))+8 b^3 c d^5 e \sin (e+f x)+8 b^3 c d^5 f x \sin (e+f x)+2 b^3 d^6 e+2 b^3 d^6 f x}{\left(c^2-d^2\right)^2 (c+d \sin (e+f x))^2}-\frac{4 \left(-a^3 d^3 \left(2 c^2+d^2\right)+9 a^2 b c d^4-3 a b^2 d^3 \left(c^2+2 d^2\right)+b^3 \left(2 c^5-5 c^3 d^2+6 c d^4\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)^{5/2}}}{4 d^3 f}","-\frac{\left(-a^3 d^3 \left(2 c^2+d^2\right)+9 a^2 b c d^4-3 a b^2 d^3 \left(c^2+2 d^2\right)+b^3 \left(2 c^5-5 c^3 d^2+6 c d^4\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)^{5/2}}+\frac{(b c-a d)^2 \left(3 a c d+2 b c^2-5 b d^2\right) \cos (e+f x)}{2 d^2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^2}+\frac{b^3 x}{d^3}",1,"((-4*(9*a^2*b*c*d^4 - a^3*d^3*(2*c^2 + d^2) - 3*a*b^2*d^3*(c^2 + 2*d^2) + b^3*(2*c^5 - 5*c^3*d^2 + 6*c*d^4))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(5/2) + (4*b^3*c^6*e - 6*b^3*c^4*d^2*e + 2*b^3*d^6*e + 4*b^3*c^6*f*x - 6*b^3*c^4*d^2*f*x + 2*b^3*d^6*f*x - 2*d*(b*c - a*d)^2*(-2*b*c^3 - 4*a*c^2*d + 5*b*c*d^2 + a*d^3)*Cos[e + f*x] - 2*b^3*(-(c^2*d) + d^3)^2*(e + f*x)*Cos[2*(e + f*x)] + 8*b^3*c^5*d*e*Sin[e + f*x] - 16*b^3*c^3*d^3*e*Sin[e + f*x] + 8*b^3*c*d^5*e*Sin[e + f*x] + 8*b^3*c^5*d*f*x*Sin[e + f*x] - 16*b^3*c^3*d^3*f*x*Sin[e + f*x] + 8*b^3*c*d^5*f*x*Sin[e + f*x] + 3*b^3*c^4*d^2*Sin[2*(e + f*x)] - 3*a*b^2*c^3*d^3*Sin[2*(e + f*x)] - 3*a^2*b*c^2*d^4*Sin[2*(e + f*x)] - 6*b^3*c^2*d^4*Sin[2*(e + f*x)] + 3*a^3*c*d^5*Sin[2*(e + f*x)] + 12*a*b^2*c*d^5*Sin[2*(e + f*x)] - 6*a^2*b*d^6*Sin[2*(e + f*x)])/((c^2 - d^2)^2*(c + d*Sin[e + f*x])^2))/(4*d^3*f)","B",1
693,1,345,325,5.2671823,"\int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx","Integrate[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^4,x]","\frac{\frac{6 \left(a^3 \left(2 c^3+3 c d^2\right)-3 a^2 b d \left(4 c^2+d^2\right)+3 a b^2 c \left(c^2+4 d^2\right)-b^3 d \left(3 c^2+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)}{\left(c^2-d^2\right)^{7/2}}+\frac{\left(-a^3 d^3 \left(11 c^2+4 d^2\right)+3 a^2 b c d^2 \left(2 c^2+13 d^2\right)+3 a b^2 d \left(c^4-10 c^2 d^2-6 d^4\right)+b^3 \left(2 c^5-5 c^3 d^2+18 c d^4\right)\right) \cos (e+f x)}{d^2 \left(d^2-c^2\right)^3 (c+d \sin (e+f x))}+\frac{2 (b c-a d)^3 \cos (e+f x)}{d^2 \left(d^2-c^2\right) (c+d \sin (e+f x))^3}+\frac{\left(5 a c d+4 b c^2-9 b d^2\right) (b c-a d)^2 \cos (e+f x)}{d^2 \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^2}}{6 f}","-\frac{(a c-b d) \left(-\left(a^2 \left(2 c^2+3 d^2\right)\right)+10 a b c d-b^2 \left(3 c^2+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)}{f \left(c^2-d^2\right)^{7/2}}-\frac{\left(a^2 d^2 \left(11 c^2+4 d^2\right)+5 a b c d \left(c^2-7 d^2\right)+b^2 \left(2 c^4-5 c^2 d^2+18 d^4\right)\right) (b c-a d) \cos (e+f x)}{6 d^2 f \left(c^2-d^2\right)^3 (c+d \sin (e+f x))}+\frac{\left(5 a c d+2 b c^2-7 b d^2\right) (b c-a d)^2 \cos (e+f x)}{6 d^2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^2}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^3}",1,"((6*(-3*a^2*b*d*(4*c^2 + d^2) - b^3*d*(3*c^2 + 2*d^2) + 3*a*b^2*c*(c^2 + 4*d^2) + a^3*(2*c^3 + 3*c*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(7/2) + (2*(b*c - a*d)^3*Cos[e + f*x])/(d^2*(-c^2 + d^2)*(c + d*Sin[e + f*x])^3) + ((b*c - a*d)^2*(4*b*c^2 + 5*a*c*d - 9*b*d^2)*Cos[e + f*x])/(d^2*(c^2 - d^2)^2*(c + d*Sin[e + f*x])^2) + ((-(a^3*d^3*(11*c^2 + 4*d^2)) + 3*a^2*b*c*d^2*(2*c^2 + 13*d^2) + 3*a*b^2*d*(c^4 - 10*c^2*d^2 - 6*d^4) + b^3*(2*c^5 - 5*c^3*d^2 + 18*c*d^4))*Cos[e + f*x])/(d^2*(-c^2 + d^2)^3*(c + d*Sin[e + f*x])))/(6*f)","A",1
694,1,52,54,0.0749594,"\int \frac{\frac{b B}{a}+B \sin (x)}{a+b \sin (x)} \, dx","Integrate[((b*B)/a + B*Sin[x])/(a + b*Sin[x]),x]","\frac{B \left(a x-2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)\right)}{a b}","\frac{B x}{b}-\frac{2 B \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a b}",1,"(B*(a*x - 2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]]))/(a*b)","A",1
695,1,6,6,0.0004689,"\int \frac{\frac{a B}{b}+B \sin (x)}{a+b \sin (x)} \, dx","Integrate[((a*B)/b + B*Sin[x])/(a + b*Sin[x]),x]","\frac{B x}{b}","\frac{B x}{b}",1,"(B*x)/b","A",1
696,1,12,12,0.0312894,"\int \frac{a+b \sin (x)}{(b+a \sin (x))^2} \, dx","Integrate[(a + b*Sin[x])/(b + a*Sin[x])^2,x]","-\frac{\cos (x)}{a \sin (x)+b}","-\frac{\cos (x)}{a \sin (x)+b}",1,"-(Cos[x]/(b + a*Sin[x]))","A",1
697,1,28,34,0.0303448,"\int \frac{2-\sin (x)}{2+\sin (x)} \, dx","Integrate[(2 - Sin[x])/(2 + Sin[x]),x]","\frac{8 \tan ^{-1}\left(\frac{2 \tan \left(\frac{x}{2}\right)+1}{\sqrt{3}}\right)}{\sqrt{3}}-x","\frac{4 x}{\sqrt{3}}-x+\frac{8 \tan ^{-1}\left(\frac{\cos (x)}{\sin (x)+\sqrt{3}+2}\right)}{\sqrt{3}}",1,"-x + (8*ArcTan[(1 + 2*Tan[x/2])/Sqrt[3]])/Sqrt[3]","A",1
698,1,203,235,0.5820866,"\int \frac{(c+d \sin (e+f x))^4}{a+b \sin (e+f x)} \, dx","Integrate[(c + d*Sin[e + f*x])^4/(a + b*Sin[e + f*x]),x]","\frac{-3 b d^2 \left(4 a^2 d^2-16 a b c d+3 b^2 \left(8 c^2+d^2\right)\right) \cos (e+f x)+\frac{24 (b c-a d)^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-6 d (e+f x) \left(2 a^3 d^3-8 a^2 b c d^2+a b^2 d \left(12 c^2+d^2\right)-4 b^3 c \left(2 c^2+d^2\right)\right)-3 b^2 d^3 (4 b c-a d) \sin (2 (e+f x))+b^3 d^4 \cos (3 (e+f x))}{12 b^4 f}","\frac{2 (b c-a d)^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 f \sqrt{a^2-b^2}}+\frac{d^2 \left(-3 a^2 d^2+12 a b c d-\left(b^2 \left(17 c^2+2 d^2\right)\right)\right) \cos (e+f x)}{3 b^3 f}+\frac{d x \left(-2 a^3 d^3+8 a^2 b c d^2-a b^2 d \left(12 c^2+d^2\right)+4 b^3 c \left(2 c^2+d^2\right)\right)}{2 b^4}-\frac{d^3 (8 b c-3 a d) \sin (e+f x) \cos (e+f x)}{6 b^2 f}-\frac{d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}",1,"(-6*d*(-8*a^2*b*c*d^2 + 2*a^3*d^3 - 4*b^3*c*(2*c^2 + d^2) + a*b^2*d*(12*c^2 + d^2))*(e + f*x) + (24*(b*c - a*d)^4*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 3*b*d^2*(-16*a*b*c*d + 4*a^2*d^2 + 3*b^2*(8*c^2 + d^2))*Cos[e + f*x] + b^3*d^4*Cos[3*(e + f*x)] - 3*b^2*d^3*(4*b*c - a*d)*Sin[2*(e + f*x)])/(12*b^4*f)","A",1
699,1,138,156,0.3499295,"\int \frac{(c+d \sin (e+f x))^3}{a+b \sin (e+f x)} \, dx","Integrate[(c + d*Sin[e + f*x])^3/(a + b*Sin[e + f*x]),x]","\frac{2 d (e+f x) \left(2 a^2 d^2-6 a b c d+b^2 \left(6 c^2+d^2\right)\right)+\frac{8 (b c-a d)^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-4 b d^2 (3 b c-a d) \cos (e+f x)-b^2 d^3 \sin (2 (e+f x))}{4 b^3 f}","-\frac{d x \left(-2 a^2 d^2+6 a b c d-\left(b^2 \left(6 c^2+d^2\right)\right)\right)}{2 b^3}+\frac{2 (b c-a d)^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 f \sqrt{a^2-b^2}}-\frac{d^2 (5 b c-2 a d) \cos (e+f x)}{2 b^2 f}-\frac{d^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f}",1,"(2*d*(-6*a*b*c*d + 2*a^2*d^2 + b^2*(6*c^2 + d^2))*(e + f*x) + (8*(b*c - a*d)^3*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 4*b*d^2*(3*b*c - a*d)*Cos[e + f*x] - b^2*d^3*Sin[2*(e + f*x)])/(4*b^3*f)","A",1
700,1,90,93,0.1583884,"\int \frac{(c+d \sin (e+f x))^2}{a+b \sin (e+f x)} \, dx","Integrate[(c + d*Sin[e + f*x])^2/(a + b*Sin[e + f*x]),x]","\frac{\frac{2 (b c-a d)^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+d (e+f x) (2 b c-a d)-b d^2 \cos (e+f x)}{b^2 f}","\frac{2 (b c-a d)^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 f \sqrt{a^2-b^2}}+\frac{d x (2 b c-a d)}{b^2}-\frac{d^2 \cos (e+f x)}{b f}",1,"(d*(2*b*c - a*d)*(e + f*x) + (2*(b*c - a*d)^2*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - b*d^2*Cos[e + f*x])/(b^2*f)","A",1
701,1,67,65,0.1019545,"\int \frac{c+d \sin (e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[(c + d*Sin[e + f*x])/(a + b*Sin[e + f*x]),x]","\frac{\frac{2 (b c-a d) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+d (e+f x)}{b f}","\frac{2 (b c-a d) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b f \sqrt{a^2-b^2}}+\frac{d x}{b}",1,"(d*(e + f*x) + (2*(b*c - a*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2])/(b*f)","A",1
702,1,47,47,0.0317616,"\int \frac{1}{a+b \sin (e+f x)} \, dx","Integrate[(a + b*Sin[e + f*x])^(-1),x]","\frac{2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2}}","\frac{2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2}}",1,"(2*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*f)","A",1
703,1,104,117,0.1832302,"\int \frac{1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx","Integrate[1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])),x]","\frac{\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{2 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}}}{b c f-a d f}","\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2} (b c-a d)}-\frac{2 d \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \sqrt{c^2-d^2} (b c-a d)}",1,"((2*b*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (2*d*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2])/(b*c*f - a*d*f)","A",1
704,1,165,185,0.9271753,"\int \frac{1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^2} \, dx","Integrate[1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2),x]","\frac{\frac{2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{2 d \left(a c d+b \left(d^2-2 c^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}}+\frac{d^2 (a d-b c) \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))}}{f (b c-a d)^2}","\frac{2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2} (b c-a d)^2}+\frac{2 d \left(a c d-b \left(2 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)}{f \left(c^2-d^2\right)^{3/2} (b c-a d)^2}-\frac{d^2 \cos (e+f x)}{f \left(c^2-d^2\right) (b c-a d) (c+d \sin (e+f x))}",1,"((2*b^2*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (2*d*(a*c*d + b*(-2*c^2 + d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(3/2) + (d^2*(-(b*c) + a*d)*Cos[e + f*x])/((c - d)*(c + d)*(c + d*Sin[e + f*x])))/((b*c - a*d)^2*f)","A",1
705,1,263,284,2.2167942,"\int \frac{1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx","Integrate[1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^3),x]","\frac{-\frac{2 d \left(a^2 d^2 \left(2 c^2+d^2\right)-6 a b c^3 d+b^2 \left(6 c^4-5 c^2 d^2+2 d^4\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)^{5/2}}+\frac{4 b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{d^2 (b c-a d) \left(3 a c d-5 b c^2+2 b d^2\right) \cos (e+f x)}{(c-d)^2 (c+d)^2 (c+d \sin (e+f x))}-\frac{d^2 (b c-a d)^2 \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))^2}}{2 f (b c-a d)^3}","\frac{d \left(-a^2 d^2 \left(2 c^2+d^2\right)+6 a b c^3 d-b^2 \left(6 c^4-5 c^2 d^2+2 d^4\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{5/2} (b c-a d)^3}+\frac{2 b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2} (b c-a d)^3}-\frac{d^2 \left(-3 a c d+5 b c^2-2 b d^2\right) \cos (e+f x)}{2 f \left(c^2-d^2\right)^2 (b c-a d)^2 (c+d \sin (e+f x))}-\frac{d^2 \cos (e+f x)}{2 f \left(c^2-d^2\right) (b c-a d) (c+d \sin (e+f x))^2}",1,"((4*b^3*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (2*d*(-6*a*b*c^3*d + a^2*d^2*(2*c^2 + d^2) + b^2*(6*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(5/2) - (d^2*(b*c - a*d)^2*Cos[e + f*x])/((c - d)*(c + d)*(c + d*Sin[e + f*x])^2) + (d^2*(b*c - a*d)*(-5*b*c^2 + 3*a*c*d + 2*b*d^2)*Cos[e + f*x])/((c - d)^2*(c + d)^2*(c + d*Sin[e + f*x])))/(2*(b*c - a*d)^3*f)","A",1
706,1,199,306,2.0155524,"\int \frac{(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^2} \, dx","Integrate[(c + d*Sin[e + f*x])^4/(a + b*Sin[e + f*x])^2,x]","-\frac{-2 d^2 (e+f x) \left(6 a^2 d^2-16 a b c d+b^2 \left(12 c^2+d^2\right)\right)+\frac{8 (a d-b c)^3 \left(3 a^2 d+a b c-4 b^2 d\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}}+8 b d^3 (2 b c-a d) \cos (e+f x)-\frac{4 b (b c-a d)^4 \cos (e+f x)}{(a-b) (a+b) (a+b \sin (e+f x))}+b^2 d^4 \sin (2 (e+f x))}{4 b^4 f}","\frac{d^2 \left(-3 a^2 d^2+4 a b c d-\left(b^2 \left(2 c^2-d^2\right)\right)\right) \sin (e+f x) \cos (e+f x)}{2 b^2 f \left(a^2-b^2\right)}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}-\frac{d^2 x \left(-6 a^2 d^2+16 a b c d-\left(b^2 \left(12 c^2+d^2\right)\right)\right)}{2 b^4}+\frac{2 \left(3 a^2 d+a b c-4 b^2 d\right) (b c-a d)^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 f \left(a^2-b^2\right)^{3/2}}+\frac{d (2 b c-a d) \left(-3 a^2 d^2+2 a b c d-\left(b^2 \left(c^2-2 d^2\right)\right)\right) \cos (e+f x)}{b^3 f \left(a^2-b^2\right)}",1,"-1/4*(-2*d^2*(-16*a*b*c*d + 6*a^2*d^2 + b^2*(12*c^2 + d^2))*(e + f*x) + (8*(-(b*c) + a*d)^3*(a*b*c + 3*a^2*d - 4*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 8*b*d^3*(2*b*c - a*d)*Cos[e + f*x] - (4*b*(b*c - a*d)^4*Cos[e + f*x])/((a - b)*(a + b)*(a + b*Sin[e + f*x])) + b^2*d^4*Sin[2*(e + f*x)])/(b^4*f)","A",1
707,1,151,205,1.1030384,"\int \frac{(c+d \sin (e+f x))^3}{(a+b \sin (e+f x))^2} \, dx","Integrate[(c + d*Sin[e + f*x])^3/(a + b*Sin[e + f*x])^2,x]","\frac{\frac{2 (b c-a d)^2 \left(2 a^2 d+a b c-3 b^2 d\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}}+d^2 (e+f x) (3 b c-2 a d)+\frac{b (b c-a d)^3 \cos (e+f x)}{(a-b) (a+b) (a+b \sin (e+f x))}-b d^3 \cos (e+f x)}{b^3 f}","\frac{d \left(-2 a^2 d^2+2 a b c d-\left(b^2 \left(c^2-d^2\right)\right)\right) \cos (e+f x)}{b^2 f \left(a^2-b^2\right)}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{2 (b c-a d)^2 \left(2 a^2 d+a b c-3 b^2 d\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{d^2 x (3 b c-2 a d)}{b^3}",1,"(d^2*(3*b*c - 2*a*d)*(e + f*x) + (2*(b*c - a*d)^2*(a*b*c + 2*a^2*d - 3*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - b*d^3*Cos[e + f*x] + (b*(b*c - a*d)^3*Cos[e + f*x])/((a - b)*(a + b)*(a + b*Sin[e + f*x])))/(b^3*f)","A",1
708,1,133,129,0.5606263,"\int \frac{(c+d \sin (e+f x))^2}{(a+b \sin (e+f x))^2} \, dx","Integrate[(c + d*Sin[e + f*x])^2/(a + b*Sin[e + f*x])^2,x]","\frac{-\frac{2 \left(a^3 d^2-a b^2 \left(c^2+2 d^2\right)+2 b^3 c d\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}}+\frac{b (b c-a d)^2 \cos (e+f x)}{(a-b) (a+b) (a+b \sin (e+f x))}+d^2 (e+f x)}{b^2 f}","\frac{2 (b c-a d) \left(a^2 d+a b c-2 b^2 d\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 f \left(a^2-b^2\right)^{3/2}}+\frac{(b c-a d)^2 \cos (e+f x)}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{d^2 x}{b^2}",1,"(d^2*(e + f*x) - (2*(2*b^3*c*d + a^3*d^2 - a*b^2*(c^2 + 2*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (b*(b*c - a*d)^2*Cos[e + f*x])/((a - b)*(a + b)*(a + b*Sin[e + f*x])))/(b^2*f)","A",1
709,1,96,97,0.2975521,"\int \frac{c+d \sin (e+f x)}{(a+b \sin (e+f x))^2} \, dx","Integrate[(c + d*Sin[e + f*x])/(a + b*Sin[e + f*x])^2,x]","\frac{\frac{2 (a c-b d) \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}}+\frac{(b c-a d) \cos (e+f x)}{(a-b) (a+b) (a+b \sin (e+f x))}}{f}","\frac{2 (a c-b d) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2}}+\frac{(b c-a d) \cos (e+f x)}{f \left(a^2-b^2\right) (a+b \sin (e+f x))}",1,"((2*(a*c - b*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + ((b*c - a*d)*Cos[e + f*x])/((a - b)*(a + b)*(a + b*Sin[e + f*x])))/f","A",1
710,1,82,83,0.1841232,"\int \frac{1}{(a+b \sin (e+f x))^2} \, dx","Integrate[(a + b*Sin[e + f*x])^(-2),x]","\frac{\frac{2 a \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}}+\frac{b \cos (e+f x)}{(a-b) (a+b) (a+b \sin (e+f x))}}{f}","\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2}}+\frac{b \cos (e+f x)}{f \left(a^2-b^2\right) (a+b \sin (e+f x))}",1,"((2*a*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (b*Cos[e + f*x])/((a - b)*(a + b)*(a + b*Sin[e + f*x])))/f","A",1
711,1,178,181,0.8670633,"\int \frac{1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx","Integrate[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])),x]","\frac{\frac{2 b \left(-2 a^2 d+a b c+b^2 d\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} (b c-a d)^2}-\frac{b^2 \cos (e+f x)}{(a-b) (a+b) (a d-b c) (a+b \sin (e+f x))}+\frac{2 d^2 \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} (b c-a d)^2}}{f}","\frac{2 b \left(-2 a^2 d+a b c+b^2 d\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2} (b c-a d)^2}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))}+\frac{2 d^2 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \sqrt{c^2-d^2} (b c-a d)^2}",1,"((2*b*(a*b*c - 2*a^2*d + b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c - a*d)^2) + (2*d^2*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^2*Sqrt[c^2 - d^2]) - (b^2*Cos[e + f*x])/((a - b)*(a + b)*(-(b*c) + a*d)*(a + b*Sin[e + f*x])))/f","A",1
712,1,227,290,2.9092572,"\int \frac{1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2} \, dx","Integrate[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2),x]","\frac{\frac{2 b^2 \left(-3 a^2 d+a b c+2 b^2 d\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}}+\frac{b^3 (b c-a d) \cos (e+f x)}{(a-b) (a+b) (a+b \sin (e+f x))}-\frac{2 d^2 \left(a c d-3 b c^2+2 b d^2\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}}+\frac{d^3 (b c-a d) \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))}}{f (b c-a d)^3}","\frac{d \left(a^2 d^2+b^2 \left(c^2-2 d^2\right)\right) \cos (e+f x)}{f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 (c+d \sin (e+f x))}+\frac{2 b^2 \left(-3 a^2 d+a b c+2 b^2 d\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2} (b c-a d)^3}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))}+\frac{2 d^2 \left(-a c d+3 b c^2-2 b d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{3/2} (b c-a d)^3}",1,"((2*b^2*(a*b*c - 3*a^2*d + 2*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - (2*d^2*(-3*b*c^2 + a*c*d + 2*b*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(3/2) + (b^3*(b*c - a*d)*Cos[e + f*x])/((a - b)*(a + b)*(a + b*Sin[e + f*x])) + (d^3*(b*c - a*d)*Cos[e + f*x])/((c - d)*(c + d)*(c + d*Sin[e + f*x])))/((b*c - a*d)^3*f)","A",1
713,1,346,458,6.5857408,"\int \frac{1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx","Integrate[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3),x]","\frac{\frac{2 d^2 \left(a^2 d^2 \left(2 c^2+d^2\right)+2 a b c d \left(d^2-4 c^2\right)+3 b^2 \left(4 c^4-5 c^2 d^2+2 d^4\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)^{5/2} (b c-a d)^4}+\frac{4 b^3 \left(-4 a^2 d+a b c+3 b^2 d\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} (b c-a d)^4}-\frac{2 b^4 \cos (e+f x)}{(a-b) (a+b) (a d-b c)^3 (a+b \sin (e+f x))}+\frac{d^3 \left(-3 a c d+7 b c^2-4 b d^2\right) \cos (e+f x)}{(c-d)^2 (c+d)^2 (b c-a d)^3 (c+d \sin (e+f x))}+\frac{d^3 \cos (e+f x)}{(c-d) (c+d) (b c-a d)^2 (c+d \sin (e+f x))^2}}{2 f}","\frac{d \left(a^2 d^2+b^2 \left(2 c^2-3 d^2\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 (c+d \sin (e+f x))^2}-\frac{d^2 \left(-a^2 d^2 \left(2 c^2+d^2\right)+2 a b c d \left(4 c^2-d^2\right)-3 b^2 \left(4 c^4-5 c^2 d^2+2 d^4\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{5/2} (b c-a d)^4}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac{2 b^3 \left(-4 a^2 d+a b c+3 b^2 d\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2} (b c-a d)^4}-\frac{\left(3 a^3 c d^4-a^2 b d^3 \left(7 c^2-4 d^2\right)-3 a b^2 c d^4-b^3 \left(2 c^4 d-11 c^2 d^3+6 d^5\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right) \left(c^2-d^2\right)^2 (b c-a d)^3 (c+d \sin (e+f x))}",1,"((4*b^3*(a*b*c - 4*a^2*d + 3*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c - a*d)^4) + (2*d^2*(2*a*b*c*d*(-4*c^2 + d^2) + a^2*d^2*(2*c^2 + d^2) + 3*b^2*(4*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^4*(c^2 - d^2)^(5/2)) - (2*b^4*Cos[e + f*x])/((a - b)*(a + b)*(-(b*c) + a*d)^3*(a + b*Sin[e + f*x])) + (d^3*Cos[e + f*x])/((c - d)*(c + d)*(b*c - a*d)^2*(c + d*Sin[e + f*x])^2) + (d^3*(7*b*c^2 - 3*a*c*d - 4*b*d^2)*Cos[e + f*x])/((c - d)^2*(c + d)^2*(b*c - a*d)^3*(c + d*Sin[e + f*x])))/(2*f)","A",1
714,1,341,534,3.8230914,"\int \frac{(c+d \sin (e+f x))^5}{(a+b \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^5/(a + b*Sin[e + f*x])^3,x]","\frac{2 d^3 (e+f x) \left(12 a^2 d^2-30 a b c d+b^2 \left(20 c^2+d^2\right)\right)+\frac{2 b (b c-a d)^4 \left(7 a^2 d+3 a b c-10 b^2 d\right) \cos (e+f x)}{\left(a^2-b^2\right)^2 (a+b \sin (e+f x))}-\frac{2 b (b c-a d)^5 \cos (e+f x)}{\left(b^2-a^2\right) (a+b \sin (e+f x))^2}+\frac{4 (b c-a d)^3 \left(12 a^4 d^2+6 a^3 b c d+a^2 b^2 \left(2 c^2-29 d^2\right)-12 a b^3 c d+b^4 \left(c^2+20 d^2\right)\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)^{5/2}}+2 b d^4 (3 a d-5 b c) (\cos (e+f x)-i \sin (e+f x))+2 b d^4 (3 a d-5 b c) (\cos (e+f x)+i \sin (e+f x))-b^2 d^5 \sin (2 (e+f x))}{4 b^5 f}","\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{(b c-a d)^2 \left(4 a^2 d+3 a b c-7 b^2 d\right) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}-\frac{d^3 x \left(-12 a^2 d^2+30 a b c d-\left(b^2 \left(20 c^2+d^2\right)\right)\right)}{2 b^5}+\frac{d^2 \left(-6 a^4 d^3+7 a^3 b c d^2+a^2 b^2 d \left(c^2+10 d^2\right)-a b^3 c \left(3 c^2+16 d^2\right)+b^4 d \left(8 c^2-d^2\right)\right) \sin (e+f x) \cos (e+f x)}{2 b^3 f \left(a^2-b^2\right)^2}+\frac{(b c-a d)^3 \left(12 a^4 d^2+6 a^3 b c d+a^2 b^2 \left(2 c^2-29 d^2\right)-12 a b^3 c d+b^4 \left(c^2+20 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 f \left(a^2-b^2\right)^{5/2}}-\frac{d \left(-12 a^5 d^4+30 a^4 b c d^3-a^3 b^2 d^2 \left(16 c^2-21 d^2\right)-a^2 b^3 c d \left(4 c^2+55 d^2\right)+a b^4 \left(6 c^4+43 c^2 d^2-6 d^4\right)-b^5 c d \left(17 c^2-10 d^2\right)\right) \cos (e+f x)}{2 b^4 f \left(a^2-b^2\right)^2}",1,"(2*d^3*(-30*a*b*c*d + 12*a^2*d^2 + b^2*(20*c^2 + d^2))*(e + f*x) + (4*(b*c - a*d)^3*(6*a^3*b*c*d - 12*a*b^3*c*d + 12*a^4*d^2 + a^2*b^2*(2*c^2 - 29*d^2) + b^4*(c^2 + 20*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 2*b*d^4*(-5*b*c + 3*a*d)*(Cos[e + f*x] - I*Sin[e + f*x]) + 2*b*d^4*(-5*b*c + 3*a*d)*(Cos[e + f*x] + I*Sin[e + f*x]) - (2*b*(b*c - a*d)^5*Cos[e + f*x])/((-a^2 + b^2)*(a + b*Sin[e + f*x])^2) + (2*b*(b*c - a*d)^4*(3*a*b*c + 7*a^2*d - 10*b^2*d)*Cos[e + f*x])/((a^2 - b^2)^2*(a + b*Sin[e + f*x])) - b^2*d^5*Sin[2*(e + f*x)])/(4*b^5*f)","C",1
715,1,894,318,4.1614075,"\int \frac{(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^4/(a + b*Sin[e + f*x])^3,x]","\frac{\frac{4 \left(6 d^2 a^4+4 b c d a^3+b^2 \left(2 c^2-15 d^2\right) a^2-10 b^3 c d a+b^4 \left(c^2+12 d^2\right)\right) \tan ^{-1}\left(\frac{b+a \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a^2-b^2}}\right) (b c-a d)^2}{\left(a^2-b^2\right)^{5/2}}+\frac{-12 d^4 e a^7-12 d^4 f x a^7+16 b c d^3 e a^6+16 b c d^3 f x a^6-24 b d^4 e \sin (e+f x) a^6-24 b d^4 f x \sin (e+f x) a^6+18 b^2 d^4 e a^5+18 b^2 d^4 f x a^5+32 b^2 c d^3 e \sin (e+f x) a^5+32 b^2 c d^3 f x \sin (e+f x) a^5-9 b^2 d^4 \sin (2 (e+f x)) a^5-24 b^3 c d^3 e a^4-24 b^3 c d^3 f x a^4+b^3 d^4 \cos (3 (e+f x)) a^4+48 b^3 d^4 e \sin (e+f x) a^4+48 b^3 d^4 f x \sin (e+f x) a^4+12 b^3 c d^3 \sin (2 (e+f x)) a^4-64 b^4 c d^3 e \sin (e+f x) a^3-64 b^4 c d^3 f x \sin (e+f x) a^3+16 b^4 d^4 \sin (2 (e+f x)) a^3-6 b^4 c^2 d^2 \sin (2 (e+f x)) a^3-2 b^5 d^4 \cos (3 (e+f x)) a^2-24 b^5 d^4 e \sin (e+f x) a^2-24 b^5 d^4 f x \sin (e+f x) a^2-24 b^5 c d^3 \sin (2 (e+f x)) a^2-4 b^5 c^3 d \sin (2 (e+f x)) a^2-6 b^6 d^4 e a-6 b^6 d^4 f x a+32 b^6 c d^3 e \sin (e+f x) a+32 b^6 c d^3 f x \sin (e+f x) a+3 b^6 c^4 \sin (2 (e+f x)) a-4 b^6 d^4 \sin (2 (e+f x)) a+24 b^6 c^2 d^2 \sin (2 (e+f x)) a+8 b^7 c d^3 e+8 b^7 c d^3 f x-b \left(12 d^4 a^6-16 b c d^3 a^5-21 b^2 d^4 a^4+8 b^3 c d \left(2 c^2+5 d^2\right) a^3+2 b^4 \left(-4 c^4-18 d^2 c^2+d^4\right) a^2+8 b^5 c^3 d a+b^6 \left(2 c^4+d^4\right)\right) \cos (e+f x)+2 b^2 \left(a^2-b^2\right)^2 d^3 (3 a d-4 b c) (e+f x) \cos (2 (e+f x))+b^7 d^4 \cos (3 (e+f x))-8 b^7 c^3 d \sin (2 (e+f x))}{\left(a^2-b^2\right)^2 (a+b \sin (e+f x))^2}}{4 b^4 f}","\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{d^2 \left(-3 a^2 d^2+2 a b c d-\left(b^2 \left(c^2-2 d^2\right)\right)\right) \cos (e+f x)}{2 b^3 f \left(a^2-b^2\right)}+\frac{3 (b c-a d)^3 \left(a^2 d+a b c-2 b^2 d\right) \cos (e+f x)}{2 b^3 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \left(6 a^4 d^2+4 a^3 b c d+a^2 b^2 \left(2 c^2-15 d^2\right)-10 a b^3 c d+b^4 \left(c^2+12 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 f \left(a^2-b^2\right)^{5/2}}+\frac{d^3 x (4 b c-3 a d)}{b^4}",1,"((4*(b*c - a*d)^2*(4*a^3*b*c*d - 10*a*b^3*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 15*d^2) + b^4*(c^2 + 12*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (16*a^6*b*c*d^3*e - 24*a^4*b^3*c*d^3*e + 8*b^7*c*d^3*e - 12*a^7*d^4*e + 18*a^5*b^2*d^4*e - 6*a*b^6*d^4*e + 16*a^6*b*c*d^3*f*x - 24*a^4*b^3*c*d^3*f*x + 8*b^7*c*d^3*f*x - 12*a^7*d^4*f*x + 18*a^5*b^2*d^4*f*x - 6*a*b^6*d^4*f*x - b*(8*a*b^5*c^3*d - 16*a^5*b*c*d^3 + 12*a^6*d^4 - 21*a^4*b^2*d^4 + 8*a^3*b^3*c*d*(2*c^2 + 5*d^2) + b^6*(2*c^4 + d^4) + 2*a^2*b^4*(-4*c^4 - 18*c^2*d^2 + d^4))*Cos[e + f*x] + 2*b^2*(a^2 - b^2)^2*d^3*(-4*b*c + 3*a*d)*(e + f*x)*Cos[2*(e + f*x)] + a^4*b^3*d^4*Cos[3*(e + f*x)] - 2*a^2*b^5*d^4*Cos[3*(e + f*x)] + b^7*d^4*Cos[3*(e + f*x)] + 32*a^5*b^2*c*d^3*e*Sin[e + f*x] - 64*a^3*b^4*c*d^3*e*Sin[e + f*x] + 32*a*b^6*c*d^3*e*Sin[e + f*x] - 24*a^6*b*d^4*e*Sin[e + f*x] + 48*a^4*b^3*d^4*e*Sin[e + f*x] - 24*a^2*b^5*d^4*e*Sin[e + f*x] + 32*a^5*b^2*c*d^3*f*x*Sin[e + f*x] - 64*a^3*b^4*c*d^3*f*x*Sin[e + f*x] + 32*a*b^6*c*d^3*f*x*Sin[e + f*x] - 24*a^6*b*d^4*f*x*Sin[e + f*x] + 48*a^4*b^3*d^4*f*x*Sin[e + f*x] - 24*a^2*b^5*d^4*f*x*Sin[e + f*x] + 3*a*b^6*c^4*Sin[2*(e + f*x)] - 4*a^2*b^5*c^3*d*Sin[2*(e + f*x)] - 8*b^7*c^3*d*Sin[2*(e + f*x)] - 6*a^3*b^4*c^2*d^2*Sin[2*(e + f*x)] + 24*a*b^6*c^2*d^2*Sin[2*(e + f*x)] + 12*a^4*b^3*c*d^3*Sin[2*(e + f*x)] - 24*a^2*b^5*c*d^3*Sin[2*(e + f*x)] - 9*a^5*b^2*d^4*Sin[2*(e + f*x)] + 16*a^3*b^4*d^4*Sin[2*(e + f*x)] - 4*a*b^6*d^4*Sin[2*(e + f*x)])/((a^2 - b^2)^2*(a + b*Sin[e + f*x])^2))/(4*b^4*f)","B",1
716,1,524,248,2.3551596,"\int \frac{(c+d \sin (e+f x))^3}{(a+b \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^3/(a + b*Sin[e + f*x])^3,x]","\frac{\frac{4 a^6 d^3 e+4 a^6 d^3 f x+8 a^5 b d^3 e \sin (e+f x)+8 a^5 b d^3 f x \sin (e+f x)+3 a^4 b^2 d^3 \sin (2 (e+f x))-6 a^4 b^2 d^3 e-6 a^4 b^2 d^3 f x-3 a^3 b^3 c d^2 \sin (2 (e+f x))-16 a^3 b^3 d^3 e \sin (e+f x)-16 a^3 b^3 d^3 f x \sin (e+f x)-3 a^2 b^4 c^2 d \sin (2 (e+f x))-6 a^2 b^4 d^3 \sin (2 (e+f x))-2 d^3 \left(b^3-a^2 b\right)^2 (e+f x) \cos (2 (e+f x))-2 b (b c-a d)^2 \left(-2 a^3 d-4 a^2 b c+5 a b^2 d+b^3 c\right) \cos (e+f x)+3 a b^5 c^3 \sin (2 (e+f x))+12 a b^5 c d^2 \sin (2 (e+f x))+8 a b^5 d^3 e \sin (e+f x)+8 a b^5 d^3 f x \sin (e+f x)-6 b^6 c^2 d \sin (2 (e+f x))+2 b^6 d^3 e+2 b^6 d^3 f x}{\left(a^2-b^2\right)^2 (a+b \sin (e+f x))^2}-\frac{4 \left(2 a^5 d^3-5 a^3 b^2 d^3-a^2 b^3 c \left(2 c^2+3 d^2\right)+3 a b^4 d \left(3 c^2+2 d^2\right)-b^5 c \left(c^2+6 d^2\right)\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)^{5/2}}}{4 b^3 f}","\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{(b c-a d)^2 \left(2 a^2 d+3 a b c-5 b^2 d\right) \cos (e+f x)}{2 b^2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{(b c-a d) \left(2 a^4 d^2+2 a^3 b c d+a^2 b^2 \left(2 c^2-5 d^2\right)-8 a b^3 c d+b^4 \left(c^2+6 d^2\right)\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)^{5/2}}+\frac{d^3 x}{b^3}",1,"((-4*(2*a^5*d^3 - 5*a^3*b^2*d^3 + 3*a*b^4*d*(3*c^2 + 2*d^2) - a^2*b^3*c*(2*c^2 + 3*d^2) - b^5*c*(c^2 + 6*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (4*a^6*d^3*e - 6*a^4*b^2*d^3*e + 2*b^6*d^3*e + 4*a^6*d^3*f*x - 6*a^4*b^2*d^3*f*x + 2*b^6*d^3*f*x - 2*b*(b*c - a*d)^2*(-4*a^2*b*c + b^3*c - 2*a^3*d + 5*a*b^2*d)*Cos[e + f*x] - 2*(-(a^2*b) + b^3)^2*d^3*(e + f*x)*Cos[2*(e + f*x)] + 8*a^5*b*d^3*e*Sin[e + f*x] - 16*a^3*b^3*d^3*e*Sin[e + f*x] + 8*a*b^5*d^3*e*Sin[e + f*x] + 8*a^5*b*d^3*f*x*Sin[e + f*x] - 16*a^3*b^3*d^3*f*x*Sin[e + f*x] + 8*a*b^5*d^3*f*x*Sin[e + f*x] + 3*a*b^5*c^3*Sin[2*(e + f*x)] - 3*a^2*b^4*c^2*d*Sin[2*(e + f*x)] - 6*b^6*c^2*d*Sin[2*(e + f*x)] - 3*a^3*b^3*c*d^2*Sin[2*(e + f*x)] + 12*a*b^5*c*d^2*Sin[2*(e + f*x)] + 3*a^4*b^2*d^3*Sin[2*(e + f*x)] - 6*a^2*b^4*d^3*Sin[2*(e + f*x)])/((a^2 - b^2)^2*(a + b*Sin[e + f*x])^2))/(4*b^3*f)","B",1
717,1,204,196,0.9464551,"\int \frac{(c+d \sin (e+f x))^2}{(a+b \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^2/(a + b*Sin[e + f*x])^3,x]","\frac{\frac{2 \left(a^2 \left(2 c^2+d^2\right)-6 a b c d+b^2 \left(c^2+2 d^2\right)\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)^{5/2}}-\frac{\left(a^3 d^2+2 a^2 b c d-a b^2 \left(3 c^2+4 d^2\right)+4 b^3 c d\right) \cos (e+f x)}{b (a-b)^2 (a+b)^2 (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x)}{b (a-b) (a+b) (a+b \sin (e+f x))^2}}{2 f}","-\frac{\left(-\left(a^2 \left(2 c^2+d^2\right)\right)+6 a b c d-b^2 \left(c^2+2 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2}}+\frac{(b c-a d)^2 \cos (e+f x)}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{\left(a^2 d+3 a b c-4 b^2 d\right) (b c-a d) \cos (e+f x)}{2 b f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}",1,"((2*(-6*a*b*c*d + a^2*(2*c^2 + d^2) + b^2*(c^2 + 2*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + ((b*c - a*d)^2*Cos[e + f*x])/((a - b)*b*(a + b)*(a + b*Sin[e + f*x])^2) - ((2*a^2*b*c*d + 4*b^3*c*d + a^3*d^2 - a*b^2*(3*c^2 + 4*d^2))*Cos[e + f*x])/((a - b)^2*b*(a + b)^2*(a + b*Sin[e + f*x])))/(2*f)","A",1
718,1,157,162,0.6256628,"\int \frac{c+d \sin (e+f x)}{(a+b \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])/(a + b*Sin[e + f*x])^3,x]","\frac{\frac{2 \left(2 a^2 c-3 a b d+b^2 c\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)^{5/2}}-\frac{\left(a^2 d-3 a b c+2 b^2 d\right) \cos (e+f x)}{(a-b)^2 (a+b)^2 (a+b \sin (e+f x))}+\frac{(b c-a d) \cos (e+f x)}{(a-b) (a+b) (a+b \sin (e+f x))^2}}{2 f}","\frac{\left(2 a^2 c-3 a b d+b^2 c\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2}}+\frac{\left(a^2 (-d)+3 a b c-2 b^2 d\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{(b c-a d) \cos (e+f x)}{2 f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}",1,"((2*(2*a^2*c + b^2*c - 3*a*b*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + ((b*c - a*d)*Cos[e + f*x])/((a - b)*(a + b)*(a + b*Sin[e + f*x])^2) - ((-3*a*b*c + a^2*d + 2*b^2*d)*Cos[e + f*x])/((a - b)^2*(a + b)^2*(a + b*Sin[e + f*x])))/(2*f)","A",1
719,1,114,131,0.3965173,"\int \frac{1}{(a+b \sin (e+f x))^3} \, dx","Integrate[(a + b*Sin[e + f*x])^(-3),x]","\frac{\frac{2 \left(2 a^2+b^2\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)^{5/2}}+\frac{b \cos (e+f x) \left(4 a^2+3 a b \sin (e+f x)-b^2\right)}{(a-b)^2 (a+b)^2 (a+b \sin (e+f x))^2}}{2 f}","\frac{\left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2}}+\frac{3 a b \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{b \cos (e+f x)}{2 f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}",1,"((2*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (b*Cos[e + f*x]*(4*a^2 - b^2 + 3*a*b*Sin[e + f*x]))/((a - b)^2*(a + b)^2*(a + b*Sin[e + f*x])^2))/(2*f)","A",1
720,1,275,285,2.3526175,"\int \frac{1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx","Integrate[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])),x]","\frac{\frac{b^2 \left(-5 a^2 d+3 a b c+2 b^2 d\right) \cos (e+f x)}{(a-b)^2 (a+b)^2 (b c-a d)^2 (a+b \sin (e+f x))}-\frac{2 b \left(6 a^4 d^2-6 a^3 b c d+a^2 b^2 \left(2 c^2-5 d^2\right)+b^4 \left(c^2+2 d^2\right)\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)^{5/2} (a d-b c)^3}-\frac{b^2 \cos (e+f x)}{(a-b) (a+b) (a d-b c) (a+b \sin (e+f x))^2}+\frac{4 d^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} (a d-b c)^3}}{2 f}","\frac{b^2 \left(-5 a^2 d+3 a b c+2 b^2 d\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x))}+\frac{b^2 \cos (e+f x)}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2}-\frac{b \left(-6 a^4 d^2+6 a^3 b c d-a^2 b^2 \left(2 c^2-5 d^2\right)-b^4 \left(c^2+2 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2} (b c-a d)^3}-\frac{2 d^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \sqrt{c^2-d^2} (b c-a d)^3}",1,"((-2*b*(-6*a^3*b*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 5*d^2) + b^4*(c^2 + 2*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(-(b*c) + a*d)^3) + (4*d^3*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((-(b*c) + a*d)^3*Sqrt[c^2 - d^2]) - (b^2*Cos[e + f*x])/((a - b)*(a + b)*(-(b*c) + a*d)*(a + b*Sin[e + f*x])^2) + (b^2*(3*a*b*c - 5*a^2*d + 2*b^2*d)*Cos[e + f*x])/((a - b)^2*(a + b)^2*(b*c - a*d)^2*(a + b*Sin[e + f*x])))/(2*f)","A",1
721,1,346,454,5.4724794,"\int \frac{1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^2} \, dx","Integrate[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2),x]","\frac{\frac{b^3 \left(7 a^2 d-3 a b c-4 b^2 d\right) \cos (e+f x)}{(a-b)^2 (a+b)^2 (a d-b c)^3 (a+b \sin (e+f x))}+\frac{2 b^2 \left(12 a^4 d^2-8 a^3 b c d+a^2 b^2 \left(2 c^2-15 d^2\right)+2 a b^3 c d+b^4 \left(c^2+6 d^2\right)\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)^{5/2} (b c-a d)^4}+\frac{b^3 \cos (e+f x)}{(a-b) (a+b) (b c-a d)^2 (a+b \sin (e+f x))^2}+\frac{4 d^3 \left(a c d-4 b c^2+3 b d^2\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 c-a d)^4}-\frac{2 d^4 \cos (e+f x)}{(c-d) (c+d) (b c-a d)^3 (c+d \sin (e+f x))}}{2 f}","\frac{3 b^2 \left(-2 a^2 d+a b c+b^2 d\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x)) (c+d \sin (e+f x))}+\frac{b^2 \cos (e+f x)}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac{d \left(2 a^4 d^3+a^2 b^2 d \left(7 c^2-11 d^2\right)-3 a b^3 c \left(c^2-d^2\right)-2 b^4 d \left(2 c^2-3 d^2\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right) (b c-a d)^3 (c+d \sin (e+f x))}-\frac{b^2 \left(-12 a^4 d^2+8 a^3 b c d-a^2 b^2 \left(2 c^2-15 d^2\right)-2 a b^3 c d-b^4 \left(c^2+6 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2} (b c-a d)^4}-\frac{2 d^3 \left(-a c d+4 b c^2-3 b d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{3/2} (b c-a d)^4}",1,"((2*b^2*(-8*a^3*b*c*d + 2*a*b^3*c*d + 12*a^4*d^2 + a^2*b^2*(2*c^2 - 15*d^2) + b^4*(c^2 + 6*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(b*c - a*d)^4) + (4*d^3*(-4*b*c^2 + a*c*d + 3*b*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^4*(c^2 - d^2)^(3/2)) + (b^3*Cos[e + f*x])/((a - b)*(a + b)*(b*c - a*d)^2*(a + b*Sin[e + f*x])^2) + (b^3*(-3*a*b*c + 7*a^2*d - 4*b^2*d)*Cos[e + f*x])/((a - b)^2*(a + b)^2*(-(b*c) + a*d)^3*(a + b*Sin[e + f*x])) - (2*d^4*Cos[e + f*x])/((c - d)*(c + d)*(b*c - a*d)^3*(c + d*Sin[e + f*x])))/(2*f)","A",1
722,1,1815,669,8.5275052,"\int \frac{1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx","Integrate[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3),x]","-\frac{\left(20 d^2 a^4-10 b c d a^3+2 b^2 c^2 a^2-29 b^2 d^2 a^2+4 b^3 c d a+b^4 c^2+12 b^4 d^2\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (e+f x)\right) \left(b \cos \left(\frac{1}{2} (e+f x)\right)+a \sin \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{a^2-b^2}}\right) b^3}{\left(a^2-b^2\right)^{5/2} (a d-b c)^5 f}-\frac{d^3 \left(20 b^2 c^4-10 a b d c^3+2 a^2 d^2 c^2-29 b^2 d^2 c^2+4 a b d^3 c+a^2 d^4+12 b^2 d^4\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)}{(b c-a d)^5 \left(c^2-d^2\right)^{5/2} f}+\frac{-8 d^7 \cos (e+f x) a^7+32 c^2 d^5 \cos (e+f x) a^7+12 c d^6 \sin (2 (e+f x)) a^7+68 b c d^6 \cos (e+f x) a^6-80 b c^3 d^4 \cos (e+f x) a^6-12 b c d^6 \cos (3 (e+f x)) a^6+16 b d^7 \sin (2 (e+f x)) a^6-4 b c^2 d^5 \sin (2 (e+f x)) a^6+38 b^2 d^7 \cos (e+f x) a^5-92 b^2 c^2 d^5 \cos (e+f x) a^5-22 b^2 d^7 \cos (3 (e+f x)) a^5+28 b^2 c^2 d^5 \cos (3 (e+f x)) a^5+38 b^2 c d^6 \sin (2 (e+f x)) a^5-80 b^2 c^3 d^4 \sin (2 (e+f x)) a^5-3 b^2 c d^6 \sin (4 (e+f x)) a^5-122 b^3 c d^6 \cos (e+f x) a^4+140 b^3 c^3 d^4 \cos (e+f x) a^4+10 b^3 c d^6 \cos (3 (e+f x)) a^4+20 b^3 c^3 d^4 \cos (3 (e+f x)) a^4-20 b^3 d^7 \sin (2 (e+f x)) a^4-10 b^3 c^2 d^5 \sin (2 (e+f x)) a^4-6 b^3 d^7 \sin (4 (e+f x)) a^4+9 b^3 c^2 d^5 \sin (4 (e+f x)) a^4-72 b^4 d^7 \cos (e+f x) a^3+48 b^4 c^2 d^5 \cos (e+f x) a^3+140 b^4 c^4 d^3 \cos (e+f x) a^3-80 b^4 c^6 d \cos (e+f x) a^3+64 b^4 d^7 \cos (3 (e+f x)) a^3-96 b^4 c^2 d^5 \cos (3 (e+f x)) a^3+20 b^4 c^4 d^3 \cos (3 (e+f x)) a^3-192 b^4 c d^6 \sin (2 (e+f x)) a^3+320 b^4 c^3 d^4 \sin (2 (e+f x)) a^3-80 b^4 c^5 d^2 \sin (2 (e+f x)) a^3+6 b^4 c d^6 \sin (4 (e+f x)) a^3+32 b^5 c^7 \cos (e+f x) a^2+12 b^5 c d^6 \cos (e+f x) a^2+48 b^5 c^3 d^4 \cos (e+f x) a^2-92 b^5 c^5 d^2 \cos (e+f x) a^2+44 b^5 c d^6 \cos (3 (e+f x)) a^2-96 b^5 c^3 d^4 \cos (3 (e+f x)) a^2+28 b^5 c^5 d^2 \cos (3 (e+f x)) a^2-26 b^5 d^7 \sin (2 (e+f x)) a^2+64 b^5 c^2 d^5 \sin (2 (e+f x)) a^2-10 b^5 c^4 d^3 \sin (2 (e+f x)) a^2-4 b^5 c^6 d \sin (2 (e+f x)) a^2+21 b^5 d^7 \sin (4 (e+f x)) a^2-36 b^5 c^2 d^5 \sin (4 (e+f x)) a^2+9 b^5 c^4 d^3 \sin (4 (e+f x)) a^2+36 b^6 d^7 \cos (e+f x) a+12 b^6 c^2 d^5 \cos (e+f x) a-122 b^6 c^4 d^3 \cos (e+f x) a+68 b^6 c^6 d \cos (e+f x) a-36 b^6 d^7 \cos (3 (e+f x)) a+44 b^6 c^2 d^5 \cos (3 (e+f x)) a+10 b^6 c^4 d^3 \cos (3 (e+f x)) a-12 b^6 c^6 d \cos (3 (e+f x)) a+12 b^6 c^7 \sin (2 (e+f x)) a+124 b^6 c d^6 \sin (2 (e+f x)) a-192 b^6 c^3 d^4 \sin (2 (e+f x)) a+38 b^6 c^5 d^2 \sin (2 (e+f x)) a-6 b^6 c d^6 \sin (4 (e+f x)) a+6 b^6 c^3 d^4 \sin (4 (e+f x)) a-3 b^6 c^5 d^2 \sin (4 (e+f x)) a-8 b^7 c^7 \cos (e+f x)+36 b^7 c d^6 \cos (e+f x)-72 b^7 c^3 d^4 \cos (e+f x)+38 b^7 c^5 d^2 \cos (e+f x)-36 b^7 c d^6 \cos (3 (e+f x))+64 b^7 c^3 d^4 \cos (3 (e+f x))-22 b^7 c^5 d^2 \cos (3 (e+f x))+24 b^7 d^7 \sin (2 (e+f x))-26 b^7 c^2 d^5 \sin (2 (e+f x))-20 b^7 c^4 d^3 \sin (2 (e+f x))+16 b^7 c^6 d \sin (2 (e+f x))-12 b^7 d^7 \sin (4 (e+f x))+21 b^7 c^2 d^5 \sin (4 (e+f x))-6 b^7 c^4 d^3 \sin (4 (e+f x))}{16 \left(a^2-b^2\right)^2 (a d-b c)^4 \left(c^2-d^2\right)^2 f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}","-\frac{d^3 \left(a^2 d^2 \left(2 c^2+d^2\right)-a b \left(10 c^3 d-4 c d^3\right)+b^2 \left(20 c^4-29 c^2 d^2+12 d^4\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{5/2} (b c-a d)^5}+\frac{b^2 \left(-7 a^2 d+3 a b c+4 b^2 d\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac{b^2 \cos (e+f x)}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac{d \left(a^4 d^3+2 a^2 b^2 d \left(4 c^2-5 d^2\right)-3 a b^3 c \left(c^2-d^2\right)-b^4 d \left(5 c^2-6 d^2\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right) (b c-a d)^3 (c+d \sin (e+f x))^2}-\frac{b^3 \left(-20 a^4 d^2+10 a^3 b c d-a^2 b^2 \left(2 c^2-29 d^2\right)-4 a b^3 c d-b^4 \left(c^2+12 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2} (b c-a d)^5}+\frac{3 d \left(a^5 c d^4-a^4 b \left(3 c^2 d^3-2 d^5\right)-2 a^3 b^2 c d^4-a^2 b^3 d \left(3 c^4-12 c^2 d^2+7 d^4\right)+a b^4 c \left(c^4-2 c^2 d^2+2 d^4\right)+b^5 d \left(2 c^4-7 c^2 d^2+4 d^4\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right)^2 (b c-a d)^4 (c+d \sin (e+f x))}",1,"-((b^3*(2*a^2*b^2*c^2 + b^4*c^2 - 10*a^3*b*c*d + 4*a*b^3*c*d + 20*a^4*d^2 - 29*a^2*b^2*d^2 + 12*b^4*d^2)*ArcTan[(Sec[(e + f*x)/2]*(b*Cos[(e + f*x)/2] + a*Sin[(e + f*x)/2]))/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(-(b*c) + a*d)^5*f)) - (d^3*(20*b^2*c^4 - 10*a*b*c^3*d + 2*a^2*c^2*d^2 - 29*b^2*c^2*d^2 + 4*a*b*c*d^3 + a^2*d^4 + 12*b^2*d^4)*ArcTan[(Sec[(e + f*x)/2]*(d*Cos[(e + f*x)/2] + c*Sin[(e + f*x)/2]))/Sqrt[c^2 - d^2]])/((b*c - a*d)^5*(c^2 - d^2)^(5/2)*f) + (32*a^2*b^5*c^7*Cos[e + f*x] - 8*b^7*c^7*Cos[e + f*x] - 80*a^3*b^4*c^6*d*Cos[e + f*x] + 68*a*b^6*c^6*d*Cos[e + f*x] - 92*a^2*b^5*c^5*d^2*Cos[e + f*x] + 38*b^7*c^5*d^2*Cos[e + f*x] + 140*a^3*b^4*c^4*d^3*Cos[e + f*x] - 122*a*b^6*c^4*d^3*Cos[e + f*x] - 80*a^6*b*c^3*d^4*Cos[e + f*x] + 140*a^4*b^3*c^3*d^4*Cos[e + f*x] + 48*a^2*b^5*c^3*d^4*Cos[e + f*x] - 72*b^7*c^3*d^4*Cos[e + f*x] + 32*a^7*c^2*d^5*Cos[e + f*x] - 92*a^5*b^2*c^2*d^5*Cos[e + f*x] + 48*a^3*b^4*c^2*d^5*Cos[e + f*x] + 12*a*b^6*c^2*d^5*Cos[e + f*x] + 68*a^6*b*c*d^6*Cos[e + f*x] - 122*a^4*b^3*c*d^6*Cos[e + f*x] + 12*a^2*b^5*c*d^6*Cos[e + f*x] + 36*b^7*c*d^6*Cos[e + f*x] - 8*a^7*d^7*Cos[e + f*x] + 38*a^5*b^2*d^7*Cos[e + f*x] - 72*a^3*b^4*d^7*Cos[e + f*x] + 36*a*b^6*d^7*Cos[e + f*x] - 12*a*b^6*c^6*d*Cos[3*(e + f*x)] + 28*a^2*b^5*c^5*d^2*Cos[3*(e + f*x)] - 22*b^7*c^5*d^2*Cos[3*(e + f*x)] + 20*a^3*b^4*c^4*d^3*Cos[3*(e + f*x)] + 10*a*b^6*c^4*d^3*Cos[3*(e + f*x)] + 20*a^4*b^3*c^3*d^4*Cos[3*(e + f*x)] - 96*a^2*b^5*c^3*d^4*Cos[3*(e + f*x)] + 64*b^7*c^3*d^4*Cos[3*(e + f*x)] + 28*a^5*b^2*c^2*d^5*Cos[3*(e + f*x)] - 96*a^3*b^4*c^2*d^5*Cos[3*(e + f*x)] + 44*a*b^6*c^2*d^5*Cos[3*(e + f*x)] - 12*a^6*b*c*d^6*Cos[3*(e + f*x)] + 10*a^4*b^3*c*d^6*Cos[3*(e + f*x)] + 44*a^2*b^5*c*d^6*Cos[3*(e + f*x)] - 36*b^7*c*d^6*Cos[3*(e + f*x)] - 22*a^5*b^2*d^7*Cos[3*(e + f*x)] + 64*a^3*b^4*d^7*Cos[3*(e + f*x)] - 36*a*b^6*d^7*Cos[3*(e + f*x)] + 12*a*b^6*c^7*Sin[2*(e + f*x)] - 4*a^2*b^5*c^6*d*Sin[2*(e + f*x)] + 16*b^7*c^6*d*Sin[2*(e + f*x)] - 80*a^3*b^4*c^5*d^2*Sin[2*(e + f*x)] + 38*a*b^6*c^5*d^2*Sin[2*(e + f*x)] - 10*a^2*b^5*c^4*d^3*Sin[2*(e + f*x)] - 20*b^7*c^4*d^3*Sin[2*(e + f*x)] - 80*a^5*b^2*c^3*d^4*Sin[2*(e + f*x)] + 320*a^3*b^4*c^3*d^4*Sin[2*(e + f*x)] - 192*a*b^6*c^3*d^4*Sin[2*(e + f*x)] - 4*a^6*b*c^2*d^5*Sin[2*(e + f*x)] - 10*a^4*b^3*c^2*d^5*Sin[2*(e + f*x)] + 64*a^2*b^5*c^2*d^5*Sin[2*(e + f*x)] - 26*b^7*c^2*d^5*Sin[2*(e + f*x)] + 12*a^7*c*d^6*Sin[2*(e + f*x)] + 38*a^5*b^2*c*d^6*Sin[2*(e + f*x)] - 192*a^3*b^4*c*d^6*Sin[2*(e + f*x)] + 124*a*b^6*c*d^6*Sin[2*(e + f*x)] + 16*a^6*b*d^7*Sin[2*(e + f*x)] - 20*a^4*b^3*d^7*Sin[2*(e + f*x)] - 26*a^2*b^5*d^7*Sin[2*(e + f*x)] + 24*b^7*d^7*Sin[2*(e + f*x)] - 3*a*b^6*c^5*d^2*Sin[4*(e + f*x)] + 9*a^2*b^5*c^4*d^3*Sin[4*(e + f*x)] - 6*b^7*c^4*d^3*Sin[4*(e + f*x)] + 6*a*b^6*c^3*d^4*Sin[4*(e + f*x)] + 9*a^4*b^3*c^2*d^5*Sin[4*(e + f*x)] - 36*a^2*b^5*c^2*d^5*Sin[4*(e + f*x)] + 21*b^7*c^2*d^5*Sin[4*(e + f*x)] - 3*a^5*b^2*c*d^6*Sin[4*(e + f*x)] + 6*a^3*b^4*c*d^6*Sin[4*(e + f*x)] - 6*a*b^6*c*d^6*Sin[4*(e + f*x)] - 6*a^4*b^3*d^7*Sin[4*(e + f*x)] + 21*a^2*b^5*d^7*Sin[4*(e + f*x)] - 12*b^7*d^7*Sin[4*(e + f*x)])/(16*(a^2 - b^2)^2*(-(b*c) + a*d)^4*(c^2 - d^2)^2*f*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2)","B",1
723,1,275,298,1.1050147,"\int (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx","Integrate[(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2),x]","\frac{-d \cos (e+f x) (c+d \sin (e+f x)) \left(6 d (7 a d+15 b c) \sin (e+f x)+154 a c d+90 b c^2-15 b d^2 \cos (2 (e+f x))+65 b d^2\right)-2 d \left(7 a \left(15 c^3+17 c d^2\right)+5 b d \left(27 c^2+5 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-2 \left(7 a d \left(23 c^2+9 d^2\right)+5 b \left(3 c^3+29 c d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{105 d f \sqrt{c+d \sin (e+f x)}}","-\frac{2 \left(56 a c d+15 b c^2+25 b d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 f}-\frac{2 \left(c^2-d^2\right) \left(56 a c d+15 b c^2+25 b d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(161 a c^2 d+63 a d^3+15 b c^3+145 b c d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 f}-\frac{2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}",1,"(-2*d*(5*b*d*(27*c^2 + 5*d^2) + 7*a*(15*c^3 + 17*c*d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - 2*(7*a*d*(23*c^2 + 9*d^2) + 5*b*(3*c^3 + 29*c*d^2))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - d*Cos[e + f*x]*(c + d*Sin[e + f*x])*(90*b*c^2 + 154*a*c*d + 65*b*d^2 - 15*b*d^2*Cos[2*(e + f*x)] + 6*d*(15*b*c + 7*a*d)*Sin[e + f*x]))/(105*d*f*Sqrt[c + d*Sin[e + f*x]])","A",1
724,1,218,235,0.7608445,"\int (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2} \, dx","Integrate[(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2),x]","\frac{-2 d \left(5 a \left(3 c^2+d^2\right)+12 b c d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-2 \left(20 a c d+3 b \left(c^2+3 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)-2 d \cos (e+f x) (c+d \sin (e+f x)) (5 a d+6 b c+3 b d \sin (e+f x))}{15 d f \sqrt{c+d \sin (e+f x)}}","-\frac{2 \left(c^2-d^2\right) (5 a d+3 b c) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(20 a c d+3 b \left(c^2+3 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (5 a d+3 b c) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 f}-\frac{2 b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}",1,"(-2*d*(12*b*c*d + 5*a*(3*c^2 + d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - 2*(20*a*c*d + 3*b*(c^2 + 3*d^2))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - 2*d*Cos[e + f*x]*(c + d*Sin[e + f*x])*(6*b*c + 5*a*d + 3*b*d*Sin[e + f*x]))/(15*d*f*Sqrt[c + d*Sin[e + f*x]])","A",1
725,1,152,181,0.6329794,"\int (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)} \, dx","Integrate[(a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]],x]","-\frac{2 \left((c+d) (3 a d+b c) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-b \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+b d \cos (e+f x) (c+d \sin (e+f x))\right)}{3 d f \sqrt{c+d \sin (e+f x)}}","\frac{2 (3 a d+b c) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 b \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \sqrt{c+d \sin (e+f x)}}-\frac{2 b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f}",1,"(-2*(b*d*Cos[e + f*x]*(c + d*Sin[e + f*x]) + (c + d)*(b*c + 3*a*d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - b*(c^2 - d^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(3*d*f*Sqrt[c + d*Sin[e + f*x]])","A",1
726,1,101,140,2.5573939,"\int \frac{a+b \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(a + b*Sin[e + f*x])/Sqrt[c + d*Sin[e + f*x]],x]","-\frac{2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left((a d-b c) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+b (c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{d f \sqrt{c+d \sin (e+f x)}}","\frac{2 b \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{c+d \sin (e+f x)}}",1,"(-2*(b*(c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (-(b*c) + a*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]])","A",1
727,1,159,195,0.5582087,"\int \frac{a+b \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^(3/2),x]","\frac{2 \left(d (a d-b c) \cos (e+f x)+(c+d) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-b \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{d f (c-d) (c+d) \sqrt{c+d \sin (e+f x)}}","-\frac{2 (b c-a d) \cos (e+f x)}{f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}-\frac{2 (b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 b \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{c+d \sin (e+f x)}}",1,"(2*(d*(-(b*c) + a*d)*Cos[e + f*x] + (c + d)*(b*c - a*d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - b*(c^2 - d^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/((c - d)*d*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])","A",1
728,1,199,285,1.5529515,"\int \frac{a+b \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx","Integrate[(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^(5/2),x]","\frac{2 \left(\frac{\left(\frac{c+d \sin (e+f x)}{c+d}\right)^{3/2} \left(\left(b \left(c^2+3 d^2\right)-4 a c d\right) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-(c-d) (b c-a d) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{d (c-d)^2}-\frac{\cos (e+f x) \left(d \left(b \left(c^2+3 d^2\right)-4 a c d\right) \sin (e+f x)+a d \left(d^2-5 c^2\right)+2 b c \left(c^2+d^2\right)\right)}{\left(c^2-d^2\right)^2}\right)}{3 f (c+d \sin (e+f x))^{3/2}}","\frac{2 \left(4 a c d-b \left(c^2+3 d^2\right)\right) \cos (e+f x)}{3 f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 (b c-a d) \cos (e+f x)}{3 f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}+\frac{2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(4 a c d-b \left(c^2+3 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \left(c^2-d^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(2*((((-4*a*c*d + b*(c^2 + 3*d^2))*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - (c - d)*(b*c - a*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*((c + d*Sin[e + f*x])/(c + d))^(3/2))/((c - d)^2*d) - (Cos[e + f*x]*(a*d*(-5*c^2 + d^2) + 2*b*c*(c^2 + d^2) + d*(-4*a*c*d + b*(c^2 + 3*d^2))*Sin[e + f*x]))/(c^2 - d^2)^2))/(3*f*(c + d*Sin[e + f*x])^(3/2))","A",1
729,1,297,369,3.0626225,"\int \frac{a+b \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx","Integrate[(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^(7/2),x]","\frac{2 \left(\frac{\left(\frac{c+d \sin (e+f x)}{c+d}\right)^{5/2} \left(\left(-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-(c-d) \left(-8 a c d+3 b c^2+5 b d^2\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{d (c-d)^3}+\frac{\cos (e+f x) \left(d^2 \left(23 a c^2 d+9 a d^3-3 b c^3-29 b c d^2\right) \sin ^2(e+f x)+d \left(54 a c^3 d+10 a c d^3-9 b c^4-60 b c^2 d^2+5 b d^4\right) \sin (e+f x)+a d \left(34 c^4-5 c^2 d^2+3 d^4\right)+b \left(-9 c^5-25 c^3 d^2+2 c d^4\right)\right)}{\left(c^2-d^2\right)^3}\right)}{15 f (c+d \sin (e+f x))^{5/2}}","-\frac{2 \left(-8 a c d+3 b c^2+5 b d^2\right) \cos (e+f x)}{15 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^{3/2}}-\frac{2 (b c-a d) \cos (e+f x)}{5 f \left(c^2-d^2\right) (c+d \sin (e+f x))^{5/2}}+\frac{2 \left(-8 a c d+3 b c^2+5 b d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left(-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right) \cos (e+f x)}{15 f \left(c^2-d^2\right)^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left(-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \left(c^2-d^2\right)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(2*((((3*b*c^3 - 23*a*c^2*d + 29*b*c*d^2 - 9*a*d^3)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - (c - d)*(3*b*c^2 - 8*a*c*d + 5*b*d^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*((c + d*Sin[e + f*x])/(c + d))^(5/2))/((c - d)^3*d) + (Cos[e + f*x]*(a*d*(34*c^4 - 5*c^2*d^2 + 3*d^4) + b*(-9*c^5 - 25*c^3*d^2 + 2*c*d^4) + d*(-9*b*c^4 + 54*a*c^3*d - 60*b*c^2*d^2 + 10*a*c*d^3 + 5*b*d^4)*Sin[e + f*x] + d^2*(-3*b*c^3 + 23*a*c^2*d - 29*b*c*d^2 + 9*a*d^3)*Sin[e + f*x]^2))/(c^2 - d^2)^3))/(15*f*(c + d*Sin[e + f*x])^(5/2))","A",1
730,1,382,451,1.8019391,"\int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx","Integrate[(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2),x]","\frac{8 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(\left(-21 a^2 d^2 \left(23 c^2+9 d^2\right)-30 a b d \left(3 c^3+29 c d^2\right)+b^2 \left(10 c^4-279 c^2 d^2-147 d^4\right)\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)-d^2 \left(21 a^2 \left(15 c^3+17 c d^2\right)+30 a b d \left(27 c^2+5 d^2\right)+b^2 c \left(155 c^2+261 d^2\right)\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)-d (c+d \sin (e+f x)) \left(2 d \left(126 a^2 d^2+540 a b c d+b^2 \left(150 c^2+133 d^2\right)\right) \sin (2 (e+f x))+2 \left(924 a^2 c d^2+30 a b d \left(36 c^2+23 d^2\right)+b^2 \left(20 c^3+747 c d^2\right)\right) \cos (e+f x)-10 b d^2 (18 a d+19 b c) \cos (3 (e+f x))-35 b^2 d^3 \sin (4 (e+f x))\right)}{1260 d^2 f \sqrt{c+d \sin (e+f x)}}","-\frac{4 \left(84 a^2 c d^2+15 a b d \left(3 c^2+5 d^2\right)-\left(b^2 \left(5 c^3-57 c d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d f}+\frac{4 \left(c^2-d^2\right) \left(-84 a^2 c d^2-45 a b c^2 d-75 a b d^3+5 b^2 c^3-57 b^2 c d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(21 a^2 d^2 \left(23 c^2+9 d^2\right)+30 a b d \left(3 c^3+29 c d^2\right)-\left(b^2 \left(10 c^4-279 c^2 d^2-147 d^4\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left(7 d^2 \left(9 a^2+7 b^2\right)-10 b c (b c-9 a d)\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac{4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}",1,"(8*(-(d^2*(30*a*b*d*(27*c^2 + 5*d^2) + b^2*c*(155*c^2 + 261*d^2) + 21*a^2*(15*c^3 + 17*c*d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]) + (-21*a^2*d^2*(23*c^2 + 9*d^2) - 30*a*b*d*(3*c^3 + 29*c*d^2) + b^2*(10*c^4 - 279*c^2*d^2 - 147*d^4))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - d*(c + d*Sin[e + f*x])*(2*(924*a^2*c*d^2 + 30*a*b*d*(36*c^2 + 23*d^2) + b^2*(20*c^3 + 747*c*d^2))*Cos[e + f*x] - 10*b*d^2*(19*b*c + 18*a*d)*Cos[3*(e + f*x)] + 2*d*(540*a*b*c*d + 126*a^2*d^2 + b^2*(150*c^2 + 133*d^2))*Sin[2*(e + f*x)] - 35*b^2*d^3*Sin[4*(e + f*x)]))/(1260*d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",1
731,1,292,347,1.20207,"\int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx","Integrate[(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2),x]","\frac{4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(-\left(d^2 \left(35 a^2 \left(3 c^2+d^2\right)+168 a b c d+b^2 \left(51 c^2+25 d^2\right)\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)-2 \left(70 a^2 c d^2+21 a b d \left(c^2+3 d^2\right)+b^2 \left(41 c d^2-3 c^3\right)\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)-d (c+d \sin (e+f x)) \left(\left(140 a^2 d^2+336 a b c d+b^2 \left(12 c^2+115 d^2\right)\right) \cos (e+f x)+3 b d (4 (7 a d+4 b c) \sin (2 (e+f x))-5 b d \cos (3 (e+f x)))\right)}{210 d^2 f \sqrt{c+d \sin (e+f x)}}","-\frac{2 \left(c^2-d^2\right) \left(35 a^2 d^2+42 a b c d-\left(b^2 \left(6 c^2-25 d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 \left(70 a^2 c d^2+21 a b d \left(c^2+3 d^2\right)-\left(b^2 \left(3 c^3-41 c d^2\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left(5 d^2 \left(7 a^2+5 b^2\right)-6 b c (b c-7 a d)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d f}+\frac{4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}",1,"(4*(-(d^2*(168*a*b*c*d + 35*a^2*(3*c^2 + d^2) + b^2*(51*c^2 + 25*d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]) - 2*(70*a^2*c*d^2 + 21*a*b*d*(c^2 + 3*d^2) + b^2*(-3*c^3 + 41*c*d^2))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - d*(c + d*Sin[e + f*x])*((336*a*b*c*d + 140*a^2*d^2 + b^2*(12*c^2 + 115*d^2))*Cos[e + f*x] + 3*b*d*(-5*b*d*Cos[3*(e + f*x)] + 4*(4*b*c + 7*a*d)*Sin[2*(e + f*x)])))/(210*d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",1
732,1,214,254,0.9167563,"\int (a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)} \, dx","Integrate[(a + b*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]],x]","\frac{2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(\left(-15 a^2 d^2-10 a b c d+b^2 \left(2 c^2-9 d^2\right)\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)-d^2 \left(15 a^2 c+10 a b d+7 b^2 c\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)-2 b d \cos (e+f x) (c+d \sin (e+f x)) (10 a d+b c+3 b d \sin (e+f x))}{15 d^2 f \sqrt{c+d \sin (e+f x)}}","\frac{2 \left(3 d^2 \left(5 a^2+3 b^2\right)-2 b c (b c-5 a d)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{4 b \left(c^2-d^2\right) (b c-5 a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 b (b c-5 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}",1,"(2*(-(d^2*(15*a^2*c + 7*b^2*c + 10*a*b*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]) + (-10*a*b*c*d - 15*a^2*d^2 + b^2*(2*c^2 - 9*d^2))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - 2*b*d*Cos[e + f*x]*(c + d*Sin[e + f*x])*(b*c + 10*a*d + 3*b*d*Sin[e + f*x]))/(15*d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",1
733,1,173,203,0.9028605,"\int \frac{(a+b \sin (e+f x))^2}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(a + b*Sin[e + f*x])^2/Sqrt[c + d*Sin[e + f*x]],x]","-\frac{2 \left(\left(3 a^2 d^2-6 a b c d+b^2 \left(2 c^2+d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-2 b (c+d) (b c-3 a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+b^2 d \cos (e+f x) (c+d \sin (e+f x))\right)}{3 d^2 f \sqrt{c+d \sin (e+f x)}}","\frac{2 \left(d^2 \left(3 a^2+b^2\right)+2 b c (b c-3 a d)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 b (b c-3 a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d f}",1,"(-2*(b^2*d*Cos[e + f*x]*(c + d*Sin[e + f*x]) - 2*b*(c + d)*(b*c - 3*a*d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (-6*a*b*c*d + 3*a^2*d^2 + b^2*(2*c^2 + d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(3*d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",1
734,1,172,228,0.8663894,"\int \frac{(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(3/2),x]","\frac{2 \left(\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(\left(-a^2 d^2+2 a b c d+b^2 \left(d^2-2 c^2\right)\right) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+2 b (c-d) (b c-a d) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{d (c-d)}+\frac{(b c-a d)^2 \cos (e+f x)}{c^2-d^2}\right)}{d f \sqrt{c+d \sin (e+f x)}}","\frac{2 \left(d^2 \left(a^2-b^2\right)-2 a b c d+2 b^2 c^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d^2 f \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d)^2 \cos (e+f x)}{d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}-\frac{4 b (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d^2 f \sqrt{c+d \sin (e+f x)}}",1,"(2*(((b*c - a*d)^2*Cos[e + f*x])/(c^2 - d^2) + (((2*a*b*c*d - a^2*d^2 + b^2*(-2*c^2 + d^2))*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + 2*b*(c - d)*(b*c - a*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((c - d)*d)))/(d*f*Sqrt[c + d*Sin[e + f*x]])","A",1
735,1,302,329,2.4821176,"\int \frac{(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{5/2}} \, dx","Integrate[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(5/2),x]","\frac{2 \left(\frac{(-c-d \sin (e+f x)) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(d^2 \left(a^2 \left(3 c^2+d^2\right)-8 a b c d+b^2 \left(c^2+3 d^2\right)\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-2 \left(-2 a^2 c d^2+a b d \left(c^2+3 d^2\right)+b^2 \left(c^3-3 c d^2\right)\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)}{(c-d)^2 (c+d)^2}-\frac{d (a d-b c) \cos (e+f x) \left(-2 d \left(2 a c d+b \left(c^2-3 d^2\right)\right) \sin (e+f x)-5 a c^2 d+a d^3-b c^3+5 b c d^2\right)}{\left(c^2-d^2\right)^2}\right)}{3 d^2 f (c+d \sin (e+f x))^{3/2}}","\frac{2 \left(-a^2 d^2+2 a b c d+b^2 \left(2 c^2-3 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{2 (b c-a d)^2 \cos (e+f x)}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}-\frac{4 \left(2 a c d+b \left(c^2-3 d^2\right)\right) (b c-a d) \cos (e+f x)}{3 d f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}-\frac{4 \left(2 a c d+b \left(c^2-3 d^2\right)\right) (b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \left(c^2-d^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(2*(((d^2*(-8*a*b*c*d + a^2*(3*c^2 + d^2) + b^2*(c^2 + 3*d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - 2*(-2*a^2*c*d^2 + a*b*d*(c^2 + 3*d^2) + b^2*(c^3 - 3*c*d^2))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*(-c - d*Sin[e + f*x])*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((c - d)^2*(c + d)^2) - (d*(-(b*c) + a*d)*Cos[e + f*x]*(-(b*c^3) - 5*a*c^2*d + 5*b*c*d^2 + a*d^3 - 2*d*(2*a*c*d + b*(c^2 - 3*d^2))*Sin[e + f*x]))/(c^2 - d^2)^2))/(3*d^2*f*(c + d*Sin[e + f*x])^(3/2))","A",1
736,1,424,460,4.9883014,"\int \frac{(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx","Integrate[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(7/2),x]","\frac{2 \left(\frac{d \cos (e+f x) \left(-2 \left(c^2-d^2\right) \left(-4 a^2 c d^2+a b d \left(3 c^2+5 d^2\right)+b^2 \left(c^3-5 c d^2\right)\right) (c+d \sin (e+f x))-\left(-a^2 d^2 \left(23 c^2+9 d^2\right)+a b \left(6 c^3 d+58 c d^3\right)+b^2 \left(2 c^4-19 c^2 d^2-15 d^4\right)\right) (c+d \sin (e+f x))^2+3 \left(c^2-d^2\right)^2 (b c-a d)^2\right)}{\left(c^2-d^2\right)^3}-\frac{\left(\frac{c+d \sin (e+f x)}{c+d}\right)^{5/2} \left(d^2 \left(a^2 \left(15 c^3+17 c d^2\right)-2 a b d \left(27 c^2+5 d^2\right)+b^2 c \left(7 c^2+25 d^2\right)\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-\left(-a^2 d^2 \left(23 c^2+9 d^2\right)+a b \left(6 c^3 d+58 c d^3\right)+b^2 \left(2 c^4-19 c^2 d^2-15 d^4\right)\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)}{(c-d)^3 (c+d)}\right)}{15 d^2 f (c+d \sin (e+f x))^{5/2}}","\frac{2 \left(a^2 d^2 \left(23 c^2+9 d^2\right)-a b \left(6 c^3 d+58 c d^3\right)-\left(b^2 \left(2 c^4-19 c^2 d^2-15 d^4\right)\right)\right) \cos (e+f x)}{15 d f \left(c^2-d^2\right)^3 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(a^2 d^2 \left(23 c^2+9 d^2\right)-a b \left(6 c^3 d+58 c d^3\right)-\left(b^2 \left(2 c^4-19 c^2 d^2-15 d^4\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \left(c^2-d^2\right)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d)^2 \cos (e+f x)}{5 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{5/2}}-\frac{4 \left(4 a c d+b \left(c^2-5 d^2\right)\right) (b c-a d) \cos (e+f x)}{15 d f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^{3/2}}+\frac{4 \left(4 a c d+b \left(c^2-5 d^2\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}",1,"(2*(-(((d^2*(-2*a*b*d*(27*c^2 + 5*d^2) + b^2*c*(7*c^2 + 25*d^2) + a^2*(15*c^3 + 17*c*d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - (-(a^2*d^2*(23*c^2 + 9*d^2)) + a*b*(6*c^3*d + 58*c*d^3) + b^2*(2*c^4 - 19*c^2*d^2 - 15*d^4))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*((c + d*Sin[e + f*x])/(c + d))^(5/2))/((c - d)^3*(c + d))) + (d*Cos[e + f*x]*(3*(b*c - a*d)^2*(c^2 - d^2)^2 - 2*(c^2 - d^2)*(-4*a^2*c*d^2 + a*b*d*(3*c^2 + 5*d^2) + b^2*(c^3 - 5*c*d^2))*(c + d*Sin[e + f*x]) - (-(a^2*d^2*(23*c^2 + 9*d^2)) + a*b*(6*c^3*d + 58*c*d^3) + b^2*(2*c^4 - 19*c^2*d^2 - 15*d^4))*(c + d*Sin[e + f*x])^2))/(c^2 - d^2)^3))/(15*d^2*f*(c + d*Sin[e + f*x])^(5/2))","A",1
737,1,545,642,2.6072092,"\int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx","Integrate[(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2),x]","\frac{d (c+d \sin (e+f x)) \left(5 b d^2 \left(1188 a^2 d^2+2508 a b c d+b^2 \left(452 c^2+513 d^2\right)\right) \cos (3 (e+f x))-4 d \left(1386 a^3 d^3+8910 a^2 b c d^2+33 a b^2 d \left(150 c^2+133 d^2\right)+5 b^3 \left(6 c^3+619 c d^2\right)\right) \sin (2 (e+f x))+2 \left(-20328 a^3 c d^3-990 a^2 b d^2 \left(36 c^2+23 d^2\right)-66 a b^2 d \left(20 c^3+747 c d^2\right)+5 b^3 \left(32 c^4-1866 c^2 d^2-1305 d^4\right)\right) \cos (e+f x)+70 b^2 d^3 (33 a d+23 b c) \sin (4 (e+f x))-315 b^3 d^4 \cos (5 (e+f x))\right)-16 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(d^2 \left(231 a^3 c d \left(15 c^2+17 d^2\right)+495 a^2 b d^2 \left(27 c^2+5 d^2\right)+33 a b^2 d \left(155 c^3+261 c d^2\right)+5 b^3 \left(2 c^4+663 c^2 d^2+135 d^4\right)\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+\left(231 a^3 d^3 \left(23 c^2+9 d^2\right)+495 a^2 b c d^2 \left(3 c^2+29 d^2\right)+33 a b^2 d \left(-10 c^4+279 c^2 d^2+147 d^4\right)+5 b^3 \left(8 c^5+51 c^3 d^2+741 c d^4\right)\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)}{27720 d^3 f \sqrt{c+d \sin (e+f x)}}","\frac{2 b \left(-297 a^2 d^2+66 a b c d-\left(b^2 \left(8 c^2+81 d^2\right)\right)\right) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac{2 \left(693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left(10 c^2-49 d^2\right)+5 b^3 \left(8 c^3+67 c d^2\right)\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}-\frac{2 \left(1848 a^3 c d^3+495 a^2 b d^2 \left(3 c^2+5 d^2\right)-66 a b^2 d \left(5 c^3-57 c d^2\right)+5 b^3 \left(8 c^4+57 c^2 d^2+135 d^4\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3465 d^2 f}-\frac{2 \left(c^2-d^2\right) \left(1848 a^3 c d^3+495 a^2 b d^2 \left(3 c^2+5 d^2\right)-66 a b^2 d \left(5 c^3-57 c d^2\right)+5 b^3 \left(8 c^4+57 c^2 d^2+135 d^4\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3465 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(231 a^3 d^3 \left(23 c^2+9 d^2\right)+495 a^2 b c d^2 \left(3 c^2+29 d^2\right)-33 a b^2 d \left(10 c^4-279 c^2 d^2-147 d^4\right)+5 b^3 \left(8 c^5+51 c^3 d^2+741 c d^4\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3465 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}",1,"(-16*(d^2*(495*a^2*b*d^2*(27*c^2 + 5*d^2) + 231*a^3*c*d*(15*c^2 + 17*d^2) + 33*a*b^2*d*(155*c^3 + 261*c*d^2) + 5*b^3*(2*c^4 + 663*c^2*d^2 + 135*d^4))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (231*a^3*d^3*(23*c^2 + 9*d^2) + 495*a^2*b*c*d^2*(3*c^2 + 29*d^2) + 33*a*b^2*d*(-10*c^4 + 279*c^2*d^2 + 147*d^4) + 5*b^3*(8*c^5 + 51*c^3*d^2 + 741*c*d^4))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d*(c + d*Sin[e + f*x])*(2*(-20328*a^3*c*d^3 - 990*a^2*b*d^2*(36*c^2 + 23*d^2) - 66*a*b^2*d*(20*c^3 + 747*c*d^2) + 5*b^3*(32*c^4 - 1866*c^2*d^2 - 1305*d^4))*Cos[e + f*x] + 5*b*d^2*(2508*a*b*c*d + 1188*a^2*d^2 + b^2*(452*c^2 + 513*d^2))*Cos[3*(e + f*x)] - 315*b^3*d^4*Cos[5*(e + f*x)] - 4*d*(8910*a^2*b*c*d^2 + 1386*a^3*d^3 + 33*a*b^2*d*(150*c^2 + 133*d^2) + 5*b^3*(6*c^3 + 619*c*d^2))*Sin[2*(e + f*x)] + 70*b^2*d^3*(23*b*c + 33*a*d)*Sin[4*(e + f*x)]))/(27720*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",1
738,1,410,496,2.4343037,"\int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx","Integrate[(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2),x]","\frac{d (c+d \sin (e+f x)) \left(b d \left(10 b d (27 a d+10 b c) \cos (3 (e+f x))-2 \sin (2 (e+f x)) \left(378 a^2 d^2+432 a b c d+b^2 \left(6 c^2+133 d^2\right)-35 b^2 d^2 \cos (2 (e+f x))\right)\right)-2 \left(420 a^3 d^3+1512 a^2 b c d^2+9 a b^2 d \left(12 c^2+115 d^2\right)+b^3 \left(402 c d^2-16 c^3\right)\right) \cos (e+f x)\right)-8 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(d^2 \left(105 a^3 d \left(3 c^2+d^2\right)+756 a^2 b c d^2+9 a b^2 d \left(51 c^2+25 d^2\right)+2 b^3 \left(c^3+93 c d^2\right)\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+\left(420 a^3 c d^3+189 a^2 b d^2 \left(c^2+3 d^2\right)+a b^2 \left(738 c d^3-54 c^3 d\right)+b^3 \left(8 c^4+33 c^2 d^2+147 d^4\right)\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)}{1260 d^3 f \sqrt{c+d \sin (e+f x)}}","\frac{2 b \left(-189 a^2 d^2+54 a b c d-\left(b^2 \left(8 c^2+49 d^2\right)\right)\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}-\frac{2 \left(105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left(6 c^2-25 d^2\right)+b^3 \left(8 c^3+39 c d^2\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}-\frac{2 \left(c^2-d^2\right) \left(105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left(6 c^2-25 d^2\right)+b^3 \left(8 c^3+39 c d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(420 a^3 c d^3+189 a^2 b d^2 \left(c^2+3 d^2\right)-a b^2 \left(54 c^3 d-738 c d^3\right)+b^3 \left(8 c^4+33 c^2 d^2+147 d^4\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}",1,"(-8*(d^2*(756*a^2*b*c*d^2 + 105*a^3*d*(3*c^2 + d^2) + 9*a*b^2*d*(51*c^2 + 25*d^2) + 2*b^3*(c^3 + 93*c*d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (420*a^3*c*d^3 + 189*a^2*b*d^2*(c^2 + 3*d^2) + a*b^2*(-54*c^3*d + 738*c*d^3) + b^3*(8*c^4 + 33*c^2*d^2 + 147*d^4))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d*(c + d*Sin[e + f*x])*(-2*(1512*a^2*b*c*d^2 + 420*a^3*d^3 + 9*a*b^2*d*(12*c^2 + 115*d^2) + b^3*(-16*c^3 + 402*c*d^2))*Cos[e + f*x] + b*d*(10*b*d*(10*b*c + 27*a*d)*Cos[3*(e + f*x)] - 2*(432*a*b*c*d + 378*a^2*d^2 + b^2*(6*c^2 + 133*d^2) - 35*b^2*d^2*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])))/(1260*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",1
739,1,306,375,1.4488811,"\int (a+b \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)} \, dx","Integrate[(a + b*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]],x]","\frac{b d (c+d \sin (e+f x)) \left(\left(-420 a^2 d^2-84 a b c d+b^2 \left(16 c^2-115 d^2\right)\right) \cos (e+f x)+3 b d (5 b d \cos (3 (e+f x))-2 (21 a d+b c) \sin (2 (e+f x)))\right)-4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(d^2 \left(105 a^3 c d+105 a^2 b d^2+147 a b^2 c d+b^3 \left(2 c^2+25 d^2\right)\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+\left(105 a^3 d^3+105 a^2 b c d^2+21 a b^2 d \left(9 d^2-2 c^2\right)+b^3 \left(8 c^3+19 c d^2\right)\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)}{210 d^3 f \sqrt{c+d \sin (e+f x)}}","\frac{2 b \left(-105 a^2 d^2+42 a b c d-\left(b^2 \left(8 c^2+25 d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d^2 f}+\frac{2 b \left(c^2-d^2\right) \left(-105 a^2 d^2+42 a b c d-\left(b^2 \left(8 c^2+25 d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(105 a^3 d^3+105 a^2 b c d^2-21 a b^2 d \left(2 c^2-9 d^2\right)+b^3 \left(8 c^3+19 c d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}",1,"(-4*(d^2*(105*a^3*c*d + 147*a*b^2*c*d + 105*a^2*b*d^2 + b^3*(2*c^2 + 25*d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (105*a^2*b*c*d^2 + 105*a^3*d^3 + 21*a*b^2*d*(-2*c^2 + 9*d^2) + b^3*(8*c^3 + 19*c*d^2))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + b*d*(c + d*Sin[e + f*x])*((-84*a*b*c*d - 420*a^2*d^2 + b^2*(16*c^2 - 115*d^2))*Cos[e + f*x] + 3*b*d*(5*b*d*Cos[3*(e + f*x)] - 2*(b*c + 21*a*d)*Sin[2*(e + f*x)])))/(210*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",1
740,1,219,302,1.1984077,"\int \frac{(a+b \sin (e+f x))^3}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(a + b*Sin[e + f*x])^3/Sqrt[c + d*Sin[e + f*x]],x]","\frac{-2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(d^2 \left(15 a^3 d+15 a b^2 d+2 b^3 c\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+b \left(45 a^2 d^2-30 a b c d+b^2 \left(8 c^2+9 d^2\right)\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)-2 b^2 d \cos (e+f x) (c+d \sin (e+f x)) (15 a d-4 b c+3 b d \sin (e+f x))}{15 d^3 f \sqrt{c+d \sin (e+f x)}}","-\frac{2 b \left(-45 a^2 d^2+30 a b c d-\left(b^2 \left(8 c^2+9 d^2\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left(-15 a^3 d^3+45 a^2 b c d^2-15 a b^2 d \left(2 c^2+d^2\right)+b^3 \left(8 c^3+7 c d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{8 b^2 (b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}{5 d f}",1,"(-2*(d^2*(2*b^3*c + 15*a^3*d + 15*a*b^2*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + b*(-30*a*b*c*d + 45*a^2*d^2 + b^2*(8*c^2 + 9*d^2))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - 2*b^2*d*Cos[e + f*x]*(c + d*Sin[e + f*x])*(-4*b*c + 15*a*d + 3*b*d*Sin[e + f*x]))/(15*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",1
741,1,311,361,2.0457686,"\int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(3/2),x]","\frac{2 \left(\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(d^2 \left(-3 a^3 c d+9 a^2 b d^2-9 a b^2 c d+b^3 \left(2 c^2+d^2\right)\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+\left(-3 a^3 d^3+9 a^2 b c d^2+9 a b^2 d \left(d^2-2 c^2\right)+b^3 \left(8 c^3-5 c d^2\right)\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)}{(c-d) (c+d)}-\frac{d \cos (e+f x) \left(3 a^3 d^3-9 a^2 b c d^2+9 a b^2 c^2 d+b^3 \left(c d^2-4 c^3\right)+b^3 d \left(d^2-c^2\right) \sin (e+f x)\right)}{d^2-c^2}\right)}{3 d^3 f \sqrt{c+d \sin (e+f x)}}","\frac{2 b \left(-3 a^2 d^2+6 a b c d-\left(b^2 \left(4 c^2-d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d^2 f \left(c^2-d^2\right)}-\frac{2 b \left(-9 a^2 d^2+18 a b c d-\left(b^2 \left(8 c^2+d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \sqrt{c+d \sin (e+f x)}}-\frac{2 \left(-3 a^3 d^3+9 a^2 b c d^2-9 a b^2 d \left(2 c^2-d^2\right)+b^3 \left(8 c^3-5 c d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}",1,"(2*(((d^2*(-3*a^3*c*d - 9*a*b^2*c*d + 9*a^2*b*d^2 + b^3*(2*c^2 + d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (9*a^2*b*c*d^2 - 3*a^3*d^3 + 9*a*b^2*d*(-2*c^2 + d^2) + b^3*(8*c^3 - 5*c*d^2))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((c - d)*(c + d)) - (d*Cos[e + f*x]*(9*a*b^2*c^2*d - 9*a^2*b*c*d^2 + 3*a^3*d^3 + b^3*(-4*c^3 + c*d^2) + b^3*d*(-c^2 + d^2)*Sin[e + f*x]))/(-c^2 + d^2)))/(3*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",1
742,1,357,391,3.7268104,"\int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx","Integrate[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(5/2),x]","\frac{2 \left(\frac{(-c-d \sin (e+f x)) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(\left(4 a^3 c d^3-3 a^2 b d^2 \left(c^2+3 d^2\right)-6 a b^2 c d \left(c^2-3 d^2\right)+b^3 \left(8 c^4-15 c^2 d^2+3 d^4\right)\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)+d^2 \left(a^3 d \left(3 c^2+d^2\right)-12 a^2 b c d^2+3 a b^2 d \left(c^2+3 d^2\right)+2 b^3 \left(c^3-3 c d^2\right)\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{(c-d)^2 (c+d)^2}-\frac{d (b c-a d)^2 \cos (e+f x) \left(d \left(-4 a c d-5 b c^2+9 b d^2\right) \sin (e+f x)-5 a c^2 d+a d^3-4 b c^3+8 b c d^2\right)}{\left(c^2-d^2\right)^2}\right)}{3 d^3 f (c+d \sin (e+f x))^{3/2}}","-\frac{2 \left(-a^2 d^2+2 a b c d+b^2 \left(8 c^2-9 d^2\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(4 a^3 c d^3-3 a^2 b d^2 \left(c^2+3 d^2\right)-6 a b^2 c d \left(c^2-3 d^2\right)+b^3 \left(8 c^4-15 c^2 d^2+3 d^4\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \left(c^2-d^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 \left(a c d+b \left(c^2-2 d^2\right)\right) (b c-a d)^2 \cos (e+f x)}{3 d^2 f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}",1,"(2*(((d^2*(-12*a^2*b*c*d^2 + a^3*d*(3*c^2 + d^2) + 3*a*b^2*d*(c^2 + 3*d^2) + 2*b^3*(c^3 - 3*c*d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (4*a^3*c*d^3 - 6*a*b^2*c*d*(c^2 - 3*d^2) - 3*a^2*b*d^2*(c^2 + 3*d^2) + b^3*(8*c^4 - 15*c^2*d^2 + 3*d^4))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*(-c - d*Sin[e + f*x])*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((c - d)^2*(c + d)^2) - (d*(b*c - a*d)^2*Cos[e + f*x]*(-4*b*c^3 - 5*a*c^2*d + 8*b*c*d^2 + a*d^3 + d*(-5*b*c^2 - 4*a*c*d + 9*b*d^2)*Sin[e + f*x]))/(c^2 - d^2)^2))/(3*d^3*f*(c + d*Sin[e + f*x])^(3/2))","A",1
743,1,584,532,5.2973428,"\int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx","Integrate[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(7/2),x]","\frac{2 \left(\frac{d (b c-a d) \cos (e+f x) \left(-d^2 \left(a^2 d^2 \left(23 c^2+9 d^2\right)+2 a b d \left(7 c^3-39 c d^2\right)+b^2 \left(8 c^4-21 c^2 d^2+45 d^4\right)\right) \cos (2 (e+f x))+2 d \left(2 a^2 c d^2 \left(27 c^2+5 d^2\right)+a b d \left(27 c^4-170 c^2 d^2+15 d^4\right)+b^2 \left(9 c^5-20 c^3 d^2+75 c d^4\right)\right) \sin (e+f x)+68 a^2 c^4 d^2+13 a^2 c^2 d^4+15 a^2 d^6+14 a b c^5 d-146 a b c^3 d^3-60 a b c d^5+8 b^2 c^6-2 b^2 c^4 d^2+45 b^2 c^2 d^4+45 b^2 d^6\right)}{2 \left(d^2-c^2\right)^3}+\frac{\left(\frac{c+d \sin (e+f x)}{c+d}\right)^{5/2} \left(d^2 \left(a^3 (-c) d \left(15 c^2+17 d^2\right)+3 a^2 b d^2 \left(27 c^2+5 d^2\right)-3 a b^2 d \left(7 c^3+25 c d^2\right)+b^3 \left(2 c^4+15 c^2 d^2+15 d^4\right)\right) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+\left(-a^3 d^3 \left(23 c^2+9 d^2\right)+3 a^2 b c d^2 \left(3 c^2+29 d^2\right)-3 a b^2 d \left(-2 c^4+19 c^2 d^2+15 d^4\right)+b^3 \left(8 c^5-21 c^3 d^2+45 c d^4\right)\right) \left((c+d) E\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)-c F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)\right)}{(c-d)^3 (c+d)}\right)}{15 d^3 f (c+d \sin (e+f x))^{5/2}}","-\frac{2 \left(a^2 d^2 \left(23 c^2+9 d^2\right)+2 a b d \left(7 c^3-39 c d^2\right)+b^2 \left(8 c^4-21 c^2 d^2+45 d^4\right)\right) (b c-a d) \cos (e+f x)}{15 d^2 f \left(c^2-d^2\right)^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left(a^2 d^2 \left(23 c^2+9 d^2\right)+2 a b d \left(7 c^3-39 c d^2\right)+b^2 \left(8 c^4-21 c^2 d^2+45 d^4\right)\right) (b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \left(c^2-d^2\right)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left(8 a^3 c d^3-3 a^2 b d^2 \left(3 c^2+5 d^2\right)-6 a b^2 c d \left(c^2-5 d^2\right)-\left(b^3 \left(8 c^4-15 c^2 d^2+15 d^4\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}+\frac{8 \left(2 a c d+b \left(c^2-3 d^2\right)\right) (b c-a d)^2 \cos (e+f x)}{15 d^2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{5/2}}",1,"(2*(((d^2*(3*a^2*b*d^2*(27*c^2 + 5*d^2) - a^3*c*d*(15*c^2 + 17*d^2) - 3*a*b^2*d*(7*c^3 + 25*c*d^2) + b^3*(2*c^4 + 15*c^2*d^2 + 15*d^4))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (-(a^3*d^3*(23*c^2 + 9*d^2)) + 3*a^2*b*c*d^2*(3*c^2 + 29*d^2) - 3*a*b^2*d*(-2*c^4 + 19*c^2*d^2 + 15*d^4) + b^3*(8*c^5 - 21*c^3*d^2 + 45*c*d^4))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*((c + d*Sin[e + f*x])/(c + d))^(5/2))/((c - d)^3*(c + d)) + (d*(b*c - a*d)*Cos[e + f*x]*(8*b^2*c^6 + 14*a*b*c^5*d + 68*a^2*c^4*d^2 - 2*b^2*c^4*d^2 - 146*a*b*c^3*d^3 + 13*a^2*c^2*d^4 + 45*b^2*c^2*d^4 - 60*a*b*c*d^5 + 15*a^2*d^6 + 45*b^2*d^6 - d^2*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 + 45*d^4))*Cos[2*(e + f*x)] + 2*d*(2*a^2*c*d^2*(27*c^2 + 5*d^2) + a*b*d*(27*c^4 - 170*c^2*d^2 + 15*d^4) + b^2*(9*c^5 - 20*c^3*d^2 + 75*c*d^4))*Sin[e + f*x]))/(2*(-c^2 + d^2)^3)))/(15*d^3*f*(c + d*Sin[e + f*x])^(5/2))","A",1
744,1,1127,716,6.9802615,"\int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx","Integrate[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(9/2),x]","\frac{\sqrt{c+d \sin (e+f x)} \left(-\frac{2 \left(-b^3 \cos (e+f x) c^3+3 a b^2 d \cos (e+f x) c^2-3 a^2 b d^2 \cos (e+f x) c+a^3 d^3 \cos (e+f x)\right)}{7 d^2 \left(d^2-c^2\right) (c+d \sin (e+f x))^4}-\frac{2 \left(8 b^3 \cos (e+f x) c^6+18 a b^2 d \cos (e+f x) c^5-23 b^3 d^2 \cos (e+f x) c^4+45 a^2 b d^2 \cos (e+f x) c^4-176 a^3 d^3 \cos (e+f x) c^3-372 a b^2 d^3 \cos (e+f x) c^3+294 b^3 d^4 \cos (e+f x) c^2+918 a^2 b d^4 \cos (e+f x) c^2-208 a^3 d^5 \cos (e+f x) c-798 a b^2 d^5 \cos (e+f x) c+105 b^3 d^6 \cos (e+f x)+189 a^2 b d^6 \cos (e+f x)\right)}{105 d^2 \left(d^2-c^2\right)^4 (c+d \sin (e+f x))}-\frac{2 \left(-8 b^3 \cos (e+f x) c^5-18 a b^2 d \cos (e+f x) c^4+17 b^3 d^2 \cos (e+f x) c^3-45 a^2 b d^2 \cos (e+f x) c^3+71 a^3 d^3 \cos (e+f x) c^2+201 a b^2 d^3 \cos (e+f x) c^2-105 b^3 d^4 \cos (e+f x) c-243 a^2 b d^4 \cos (e+f x) c+25 a^3 d^5 \cos (e+f x)+105 a b^2 d^5 \cos (e+f x)\right)}{105 d^2 \left(d^2-c^2\right)^3 (c+d \sin (e+f x))^2}-\frac{6 \left(-3 b^3 \cos (e+f x) c^4+2 a b^2 d \cos (e+f x) c^3+7 b^3 d^2 \cos (e+f x) c^2+5 a^2 b d^2 \cos (e+f x) c^2-4 a^3 d^3 \cos (e+f x) c-14 a b^2 d^3 \cos (e+f x) c+7 a^2 b d^4 \cos (e+f x)\right)}{35 d^2 \left(d^2-c^2\right)^2 (c+d \sin (e+f x))^3}\right)}{f}-\frac{-\frac{2 \left(-25 a^3 d^6-105 a b^2 d^6+210 b^3 c d^5+432 a^2 b c d^5-254 a^3 c^2 d^4-894 a b^2 c^2 d^4+172 b^3 c^3 d^3+720 a^2 b c^3 d^3-105 a^3 c^4 d^2-153 a b^2 c^4 d^2+2 b^3 c^5 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{\sqrt{c+d \sin (e+f x)}}-\frac{\left(8 b^3 c^6+18 a b^2 d c^5-23 b^3 d^2 c^4+45 a^2 b d^2 c^4-176 a^3 d^3 c^3-372 a b^2 d^3 c^3+294 b^3 d^4 c^2+918 a^2 b d^4 c^2-208 a^3 d^5 c-798 a b^2 d^5 c+105 b^3 d^6+189 a^2 b d^6\right) \left(\frac{2 (c+d) E\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{\sqrt{c+d \sin (e+f x)}}-\frac{2 c F\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{\sqrt{c+d \sin (e+f x)}}\right)}{d}}{105 (c-d)^4 d^2 (c+d)^4 f}","-\frac{2 \left(a^2 d^2 \left(71 c^2+25 d^2\right)+a b \left(26 c^3 d-218 c d^3\right)+b^2 \left(8 c^4-17 c^2 d^2+105 d^4\right)\right) (b c-a d) \cos (e+f x)}{105 d^2 f \left(c^2-d^2\right)^3 (c+d \sin (e+f x))^{3/2}}+\frac{2 \left(a^2 d^2 \left(71 c^2+25 d^2\right)+a b \left(26 c^3 d-218 c d^3\right)+b^2 \left(8 c^4-17 c^2 d^2+105 d^4\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \left(c^2-d^2\right)^3 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(16 a^3 c d^3 \left(11 c^2+13 d^2\right)-9 a^2 b d^2 \left(5 c^4+102 c^2 d^2+21 d^4\right)-6 a b^2 c d \left(3 c^4-62 c^2 d^2-133 d^4\right)-\left(b^3 \left(8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right)\right)\right) \cos (e+f x)}{105 d^2 f \left(c^2-d^2\right)^4 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(16 a^3 c d^3 \left(11 c^2+13 d^2\right)-9 a^2 b d^2 \left(5 c^4+102 c^2 d^2+21 d^4\right)-6 a b^2 c d \left(3 c^4-62 c^2 d^2-133 d^4\right)-\left(b^3 \left(8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \left(c^2-d^2\right)^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 \left(3 a c d+b \left(c^2-4 d^2\right)\right) (b c-a d)^2 \cos (e+f x)}{35 d^2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^{5/2}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{7/2}}",1,"(Sqrt[c + d*Sin[e + f*x]]*((-2*(-(b^3*c^3*Cos[e + f*x]) + 3*a*b^2*c^2*d*Cos[e + f*x] - 3*a^2*b*c*d^2*Cos[e + f*x] + a^3*d^3*Cos[e + f*x]))/(7*d^2*(-c^2 + d^2)*(c + d*Sin[e + f*x])^4) - (6*(-3*b^3*c^4*Cos[e + f*x] + 2*a*b^2*c^3*d*Cos[e + f*x] + 5*a^2*b*c^2*d^2*Cos[e + f*x] + 7*b^3*c^2*d^2*Cos[e + f*x] - 4*a^3*c*d^3*Cos[e + f*x] - 14*a*b^2*c*d^3*Cos[e + f*x] + 7*a^2*b*d^4*Cos[e + f*x]))/(35*d^2*(-c^2 + d^2)^2*(c + d*Sin[e + f*x])^3) - (2*(-8*b^3*c^5*Cos[e + f*x] - 18*a*b^2*c^4*d*Cos[e + f*x] - 45*a^2*b*c^3*d^2*Cos[e + f*x] + 17*b^3*c^3*d^2*Cos[e + f*x] + 71*a^3*c^2*d^3*Cos[e + f*x] + 201*a*b^2*c^2*d^3*Cos[e + f*x] - 243*a^2*b*c*d^4*Cos[e + f*x] - 105*b^3*c*d^4*Cos[e + f*x] + 25*a^3*d^5*Cos[e + f*x] + 105*a*b^2*d^5*Cos[e + f*x]))/(105*d^2*(-c^2 + d^2)^3*(c + d*Sin[e + f*x])^2) - (2*(8*b^3*c^6*Cos[e + f*x] + 18*a*b^2*c^5*d*Cos[e + f*x] + 45*a^2*b*c^4*d^2*Cos[e + f*x] - 23*b^3*c^4*d^2*Cos[e + f*x] - 176*a^3*c^3*d^3*Cos[e + f*x] - 372*a*b^2*c^3*d^3*Cos[e + f*x] + 918*a^2*b*c^2*d^4*Cos[e + f*x] + 294*b^3*c^2*d^4*Cos[e + f*x] - 208*a^3*c*d^5*Cos[e + f*x] - 798*a*b^2*c*d^5*Cos[e + f*x] + 189*a^2*b*d^6*Cos[e + f*x] + 105*b^3*d^6*Cos[e + f*x]))/(105*d^2*(-c^2 + d^2)^4*(c + d*Sin[e + f*x]))))/f - ((-2*(2*b^3*c^5*d - 105*a^3*c^4*d^2 - 153*a*b^2*c^4*d^2 + 720*a^2*b*c^3*d^3 + 172*b^3*c^3*d^3 - 254*a^3*c^2*d^4 - 894*a*b^2*c^2*d^4 + 432*a^2*b*c*d^5 + 210*b^3*c*d^5 - 25*a^3*d^6 - 105*a*b^2*d^6)*EllipticF[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]] - ((8*b^3*c^6 + 18*a*b^2*c^5*d + 45*a^2*b*c^4*d^2 - 23*b^3*c^4*d^2 - 176*a^3*c^3*d^3 - 372*a*b^2*c^3*d^3 + 918*a^2*b*c^2*d^4 + 294*b^3*c^2*d^4 - 208*a^3*c*d^5 - 798*a*b^2*c*d^5 + 189*a^2*b*d^6 + 105*b^3*d^6)*((2*(c + d)*EllipticE[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]] - (2*c*EllipticF[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]]))/d)/(105*(c - d)^4*d^2*(c + d)^4*f)","A",1
745,1,606,296,5.7722998,"\int \frac{(c+d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx","Integrate[(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x]),x]","\frac{\frac{2 i (3 a d-7 b c) \sec (e+f x) \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} \sqrt{\frac{d (\sin (e+f x)+1)}{d-c}} \left(d \left(d \left(b^2-2 a^2\right) \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)-2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)}{b^2 \sqrt{-\frac{1}{c+d}} (b c-a d)}-\frac{2 \left(-a d^3+6 b c^3+7 b c d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}+\frac{4 i \left(b \left(9 c^2+d^2\right)-2 a c d\right) \sec (e+f x) \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} \sqrt{\frac{d (\sin (e+f x)+1)}{d-c}} \left((a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)}{b \sqrt{-\frac{1}{c+d}} (b c-a d)}-4 d^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{6 b f}","-\frac{2 d \left(-3 a^2 d^2+6 a b c d-\left(b^2 \left(2 c^2+d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 b^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^3 f (a+b) \sqrt{c+d \sin (e+f x)}}+\frac{2 d (7 b c-3 a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 b^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 d^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b f}",1,"(((4*I)*(-2*a*c*d + b*(9*c^2 + d^2))*((-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] - a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sec[e + f*x]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[(d*(1 + Sin[e + f*x]))/(-c + d)])/(b*Sqrt[-(c + d)^(-1)]*(b*c - a*d)) + ((2*I)*(-7*b*c + 3*a*d)*(-2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (-2*a^2 + b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sec[e + f*x]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[(d*(1 + Sin[e + f*x]))/(-c + d)])/(b^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)) - 4*d^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]] - (2*(6*b*c^3 + 7*b*c*d^2 - a*d^3)*EllipticPi[(2*b)/(a + b), (-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]))/(6*b*f)","C",1
746,1,242,229,3.9632954,"\int \frac{(c+d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx","Integrate[(c + d*Sin[e + f*x])^(3/2)/(a + b*Sin[e + f*x]),x]","\frac{2 i \sec (e+f x) \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} \sqrt{-\frac{d (\sin (e+f x)+1)}{c-d}} \left((a d+b (d-2 c)) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+(b c-a d) \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+b (c-d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)}{b^2 f \sqrt{-\frac{1}{c+d}}}","\frac{2 d (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f \sqrt{c+d \sin (e+f x)}}+\frac{2 (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f (a+b) \sqrt{c+d \sin (e+f x)}}+\frac{2 d \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"((2*I)*(b*(c - d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (a*d + b*(-2*c + d))*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (b*c - a*d)*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sec[e + f*x]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[-((d*(1 + Sin[e + f*x]))/(c - d))])/(b^2*Sqrt[-(c + d)^(-1)]*f)","C",1
747,1,114,153,2.847807,"\int \frac{\sqrt{c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx","Integrate[Sqrt[c + d*Sin[e + f*x]]/(a + b*Sin[e + f*x]),x]","-\frac{2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left(d (a+b) F\left(\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)+(b c-a d) \Pi \left(\frac{2 b}{a+b};\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)\right)}{b f (a+b) \sqrt{c+d \sin (e+f x)}}","\frac{2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f (a+b) \sqrt{c+d \sin (e+f x)}}+\frac{2 d \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f \sqrt{c+d \sin (e+f x)}}",1,"(-2*((a + b)*d*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (b*c - a*d)*EllipticPi[(2*b)/(a + b), (-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])","A",1
748,1,74,75,0.1128489,"\int \frac{1}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[1/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]),x]","-\frac{2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)}{f (a+b) \sqrt{c+d \sin (e+f x)}}","\frac{2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a+b) \sqrt{c+d \sin (e+f x)}}",1,"(-2*EllipticPi[(2*b)/(a + b), (-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*f*Sqrt[c + d*Sin[e + f*x]])","A",1
749,1,617,220,6.8880336,"\int \frac{1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx","Integrate[1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)),x]","-\frac{\frac{4 d^2 \cos (e+f x)}{\left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{-\frac{2 i \sec (e+f x) \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} \sqrt{\frac{d (\sin (e+f x)+1)}{d-c}} \left(d \left(d \left(b^2-2 a^2\right) \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)-2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)}{b \sqrt{-\frac{1}{c+d}} (b c-a d)}+\frac{2 \left(-2 a c d+2 b c^2-3 b d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}+\frac{4 i (a d+b c) \sec (e+f x) \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} \sqrt{\frac{d (\sin (e+f x)+1)}{d-c}} \left((a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)}{b \sqrt{-\frac{1}{c+d}} (b c-a d)}}{(c-d) (c+d)}}{2 f (b c-a d)}","-\frac{2 d^2 \cos (e+f x)}{f \left(c^2-d^2\right) (b c-a d) \sqrt{c+d \sin (e+f x)}}-\frac{2 d \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(c^2-d^2\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 b \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a+b) (b c-a d) \sqrt{c+d \sin (e+f x)}}",1,"-1/2*((4*d^2*Cos[e + f*x])/((c^2 - d^2)*Sqrt[c + d*Sin[e + f*x]]) + (((4*I)*(b*c + a*d)*((-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] - a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sec[e + f*x]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[(d*(1 + Sin[e + f*x]))/(-c + d)])/(b*Sqrt[-(c + d)^(-1)]*(b*c - a*d)) - ((2*I)*(-2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (-2*a^2 + b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sec[e + f*x]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[(d*(1 + Sin[e + f*x]))/(-c + d)])/(b*Sqrt[-(c + d)^(-1)]*(b*c - a*d)) + (2*(2*b*c^2 - 2*a*c*d - 3*b*d^2)*EllipticPi[(2*b)/(a + b), (-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]))/((c - d)*(c + d)))/((b*c - a*d)*f)","C",1
750,1,1079,399,7.168812,"\int \frac{1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx","Integrate[1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2)),x]","\frac{\sqrt{c+d \sin (e+f x)} \left(\frac{2 \left(3 b \cos (e+f x) d^4+4 a c \cos (e+f x) d^3-7 b c^2 \cos (e+f x) d^2\right)}{3 (b c-a d)^2 \left(c^2-d^2\right)^2 (c+d \sin (e+f x))}-\frac{2 d^2 \cos (e+f x)}{3 (b c-a d) \left(c^2-d^2\right) (c+d \sin (e+f x))^2}\right)}{f}+\frac{-\frac{2 \left(6 b^2 c^4-12 a b d c^3+6 a^2 d^2 c^2-19 b^2 d^2 c^2+8 a b d^3 c+2 a^2 d^4+9 b^2 d^4\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left(8 a b d^4+8 a^2 c d^3+4 b^2 c d^3-8 a b c^2 d^2-12 b^2 c^3 d\right) \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left(-3 b^2 d^4-4 a b c d^3+7 b^2 c^2 d^2\right) \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{6 (c-d)^2 (c+d)^2 (b c-a d)^2 f}","\frac{2 b^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a+b) (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 d^2 \left(-4 a c d+7 b c^2-3 b d^2\right) \cos (e+f x)}{3 f \left(c^2-d^2\right)^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 d^2 \cos (e+f x)}{3 f \left(c^2-d^2\right) (b c-a d) (c+d \sin (e+f x))^{3/2}}+\frac{2 d \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 f \left(c^2-d^2\right) (b c-a d) \sqrt{c+d \sin (e+f x)}}-\frac{2 d \left(-4 a c d+7 b c^2-3 b d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 f \left(c^2-d^2\right)^2 (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(Sqrt[c + d*Sin[e + f*x]]*((-2*d^2*Cos[e + f*x])/(3*(b*c - a*d)*(c^2 - d^2)*(c + d*Sin[e + f*x])^2) + (2*(-7*b*c^2*d^2*Cos[e + f*x] + 4*a*c*d^3*Cos[e + f*x] + 3*b*d^4*Cos[e + f*x]))/(3*(b*c - a*d)^2*(c^2 - d^2)^2*(c + d*Sin[e + f*x]))))/f + ((-2*(6*b^2*c^4 - 12*a*b*c^3*d + 6*a^2*c^2*d^2 - 19*b^2*c^2*d^2 + 8*a*b*c*d^3 + 2*a^2*d^4 + 9*b^2*d^4)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(-12*b^2*c^3*d - 8*a*b*c^2*d^2 + 8*a^2*c*d^3 + 4*b^2*c*d^3 + 8*a*b*d^4)*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(7*b^2*c^2*d^2 - 4*a*b*c*d^3 - 3*b^2*d^4)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(6*(c - d)^2*(c + d)^2*(b*c - a*d)^2*f)","C",0
751,1,1109,534,8.11745,"\int \frac{(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^2} \, dx","Integrate[(c + d*Sin[e + f*x])^(7/2)/(a + b*Sin[e + f*x])^2,x]","\frac{\sqrt{c+d \sin (e+f x)} \left(\frac{-b^3 \cos (e+f x) c^3+3 a b^2 d \cos (e+f x) c^2-3 a^2 b d^2 \cos (e+f x) c+a^3 d^3 \cos (e+f x)}{b^2 \left(b^2-a^2\right) (a+b \sin (e+f x))}-\frac{2 d^3 \cos (e+f x)}{3 b^2}\right)}{f}-\frac{-\frac{2 \left(-12 a b^2 c^4+39 b^3 d c^3-45 a b^2 d^2 c^2+20 b^3 d^3 c+a^2 b d^3 c+5 a^3 d^4-8 a b^2 d^4\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left(4 b^3 d^4+8 a^2 b d^4+20 a^3 c d^3-56 a b^2 c d^3+72 b^3 c^2 d^2-36 a^2 b c^2 d^2-12 a b^2 c^3 d\right) \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left(-15 a^3 d^4+12 a b^2 d^4-20 b^3 c d^3+29 a^2 b c d^3-9 a b^2 c^2 d^2+3 b^3 c^3 d\right) \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{12 (a-b) b^2 (a+b) f}","\frac{d \left(-5 a^2 d^2+6 a b c d-\left(b^2 \left(3 c^2-2 d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b^2 f \left(a^2-b^2\right)}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{\left(5 a^2 d+2 a b c-7 b^2 d\right) (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^4 f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left(-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left(9 c^2 d-12 d^3\right)+b^3 \left(3 c^3-20 c d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 b^3 f \left(a^2-b^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left(-15 a^4 d^4+24 a^3 b c d^3+2 a^2 b^2 d^2 \left(c^2+8 d^2\right)-12 a b^3 c d \left(c^2+3 d^2\right)+b^4 \left(3 c^4+16 c^2 d^2+2 d^4\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 b^4 f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}",1,"(Sqrt[c + d*Sin[e + f*x]]*((-2*d^3*Cos[e + f*x])/(3*b^2) + (-(b^3*c^3*Cos[e + f*x]) + 3*a*b^2*c^2*d*Cos[e + f*x] - 3*a^2*b*c*d^2*Cos[e + f*x] + a^3*d^3*Cos[e + f*x])/(b^2*(-a^2 + b^2)*(a + b*Sin[e + f*x]))))/f - ((-2*(-12*a*b^2*c^4 + 39*b^3*c^3*d - 45*a*b^2*c^2*d^2 + a^2*b*c*d^3 + 20*b^3*c*d^3 + 5*a^3*d^4 - 8*a*b^2*d^4)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(-12*a*b^2*c^3*d - 36*a^2*b*c^2*d^2 + 72*b^3*c^2*d^2 + 20*a^3*c*d^3 - 56*a*b^2*c*d^3 + 8*a^2*b*d^4 + 4*b^3*d^4)*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(3*b^3*c^3*d - 9*a*b^2*c^2*d^2 + 29*a^2*b*c*d^3 - 20*b^3*c*d^3 - 15*a^3*d^4 + 12*a*b^2*d^4)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(12*(a - b)*b^2*(a + b)*f)","C",0
752,1,986,390,8.1842672,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^2} \, dx","Integrate[(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x])^2,x]","\frac{\sqrt{c+d \sin (e+f x)} \left(-b^2 \cos (e+f x) c^2+2 a b d \cos (e+f x) c-a^2 d^2 \cos (e+f x)\right)}{b \left(b^2-a^2\right) f (a+b \sin (e+f x))}+\frac{-\frac{2 \left(4 a b c^3-9 b^2 d c^2+6 a b d^2 c+a^2 d^3-2 b^2 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left(4 a b d^3+4 a^2 c d^2-12 b^2 c d^2+4 a b c^2 d\right) \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left(-3 a^2 d^3+2 b^2 d^3+2 a b c d^2-b^2 c^2 d\right) \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{4 (a-b) b (a+b) f}","-\frac{\left(-3 a^2 d^2+2 a b c d-\left(b^2 \left(c^2-2 d^2\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f \left(a^2-b^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{\left(3 a^2 d^2+2 a b c d-\left(b^2 \left(c^2+4 d^2\right)\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^3 f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{\left(3 a^2 d+2 a b c-5 b^2 d\right) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^3 f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}}",1,"((-(b^2*c^2*Cos[e + f*x]) + 2*a*b*c*d*Cos[e + f*x] - a^2*d^2*Cos[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/(b*(-a^2 + b^2)*f*(a + b*Sin[e + f*x])) + ((-2*(4*a*b*c^3 - 9*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3 - 2*b^2*d^3)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(4*a*b*c^2*d + 4*a^2*c*d^2 - 12*b^2*c*d^2 + 4*a*b*d^3)*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(-(b^2*c^2*d) + 2*a*b*c*d^2 - 3*a^2*d^3 + 2*b^2*d^3)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(4*(a - b)*b*(a + b)*f)","C",0
753,1,891,351,7.1579351,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^2} \, dx","Integrate[(c + d*Sin[e + f*x])^(3/2)/(a + b*Sin[e + f*x])^2,x]","\frac{\sqrt{c+d \sin (e+f x)} (b c \cos (e+f x)-a d \cos (e+f x))}{\left(a^2-b^2\right) f (a+b \sin (e+f x))}+\frac{-\frac{2 \left(4 a c^2-5 b d c+a d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left(4 a c d-4 b d^2\right) \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left(a d^2-b c d\right) \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{4 (a-b) (a+b) f}","\frac{\left(a^2 d^2+2 a b c d-\left(b^2 \left(c^2+2 d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{(b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f \left(a^2-b^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(b c-a d) \left(a^2 d+2 a b c-3 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}}",1,"((b*c*Cos[e + f*x] - a*d*Cos[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*f*(a + b*Sin[e + f*x])) + ((-2*(4*a*c^2 - 5*b*c*d + a*d^2)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(4*a*c*d - 4*b*d^2)*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(-(b*c*d) + a*d^2)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(4*(a - b)*(a + b)*f)","C",0
754,1,846,307,6.9466658,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+b \sin (e+f x))^2} \, dx","Integrate[Sqrt[c + d*Sin[e + f*x]]/(a + b*Sin[e + f*x])^2,x]","\frac{-\frac{2 (4 a c-b d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{8 i a \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}+\frac{2 i \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{4 (a-b) (a+b) f}-\frac{b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{\left(b^2-a^2\right) f (a+b \sin (e+f x))}","\frac{b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \left(a^2-b^2\right) (a+b \sin (e+f x))}-\frac{(b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left(a^2 (-d)+2 a b c-b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}}",1,"-((b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((-a^2 + b^2)*f*(a + b*Sin[e + f*x]))) + ((-2*(4*a*c - b*d)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((8*I)*a*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) + ((2*I)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(4*(a - b)*(a + b)*f)","C",0
755,1,871,325,7.4749157,"\int \frac{1}{(a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[1/((a + b*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]]),x]","\frac{-\frac{2 \left(4 d a^2-4 b c a-3 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}+\frac{8 i a \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{\sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{4 (a-b) (a+b) (a d-b c) f}-\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{\left(a^2-b^2\right) (a d-b c) f (a+b \sin (e+f x))}","\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))}-\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{b \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left(-3 a^2 d+2 a b c+b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a-b) (a+b)^2 (b c-a d) \sqrt{c+d \sin (e+f x)}}",1,"-((b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*(-(b*c) + a*d)*f*(a + b*Sin[e + f*x]))) + ((-2*(-4*a*b*c + 4*a^2*d - 3*b^2*d)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) + ((8*I)*a*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(4*(a - b)*(a + b)*(-(b*c) + a*d)*f)","C",0
756,1,1057,449,8.0663255,"\int \frac{1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx","Integrate[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2)),x]","\frac{\sqrt{c+d \sin (e+f x)} \left(\frac{\cos (e+f x) b^3}{\left(a^2-b^2\right) (a d-b c)^2 (a+b \sin (e+f x))}+\frac{2 d^3 \cos (e+f x)}{(b c-a d)^2 \left(c^2-d^2\right) (c+d \sin (e+f x))}\right)}{f}+\frac{-\frac{2 \left(4 c d^2 a^3+10 b d^3 a^2-8 b c^2 d a^2+4 b^2 c^3 a-8 b^2 c d^2 a-9 b^3 d^3+7 b^3 c^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left(-4 c d^2 b^3-8 a d^3 b^2+4 a c^2 d b^2+4 a^2 c d^2 b+4 a^3 d^3\right) \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left(3 d^3 b^3-c^2 d b^3-2 a^2 d^3 b\right) \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{4 (a-b) (a+b) (c-d) (c+d) (a d-b c)^2 f}","\frac{d \left(2 a^2 d^2+b^2 \left(c^2-3 d^2\right)\right) \cos (e+f x)}{f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left(2 a^2 d^2+b^2 \left(c^2-3 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}-\frac{b \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) (b c-a d) \sqrt{c+d \sin (e+f x)}}+\frac{b \left(-5 a^2 d+2 a b c+3 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a-b) (a+b)^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}",1,"(Sqrt[c + d*Sin[e + f*x]]*((b^3*Cos[e + f*x])/((a^2 - b^2)*(-(b*c) + a*d)^2*(a + b*Sin[e + f*x])) + (2*d^3*Cos[e + f*x])/((b*c - a*d)^2*(c^2 - d^2)*(c + d*Sin[e + f*x]))))/f + ((-2*(4*a*b^2*c^3 - 8*a^2*b*c^2*d + 7*b^3*c^2*d + 4*a^3*c*d^2 - 8*a*b^2*c*d^2 + 10*a^2*b*d^3 - 9*b^3*d^3)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(4*a*b^2*c^2*d + 4*a^2*b*c*d^2 - 4*b^3*c*d^2 + 4*a^3*d^3 - 8*a*b^2*d^3)*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(-(b^3*c^2*d) - 2*a^2*b*d^3 + 3*b^3*d^3)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(4*(a - b)*(a + b)*(c - d)*(c + d)*(-(b*c) + a*d)^2*f)","C",0
757,1,1319,661,9.0152849,"\int \frac{1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx","Integrate[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2)),x]","\frac{\sqrt{c+d \sin (e+f x)} \left(-\frac{\cos (e+f x) b^4}{\left(a^2-b^2\right) (a d-b c)^3 (a+b \sin (e+f x))}-\frac{4 \left(3 b \cos (e+f x) d^5+2 a c \cos (e+f x) d^4-5 b c^2 \cos (e+f x) d^3\right)}{3 (b c-a d)^3 \left(c^2-d^2\right)^2 (c+d \sin (e+f x))}+\frac{2 d^3 \cos (e+f x)}{3 (b c-a d)^2 \left(c^2-d^2\right) (c+d \sin (e+f x))^2}\right)}{f}+\frac{-\frac{2 \left(-12 a b^3 c^5-33 b^4 d c^4+36 a^2 b^2 d c^4+60 a b^3 d^2 c^3-36 a^3 b d^2 c^3+12 a^4 d^3 c^2+86 b^4 d^3 c^2-104 a^2 b^2 d^3 c^2-40 a b^3 d^4 c+28 a^3 b d^4 c+4 a^4 d^5-45 b^4 d^5+44 a^2 b^2 d^5\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left(-40 a b^3 d^5+28 a^3 b d^5+16 a^4 c d^4-20 b^4 c d^4+4 a^2 b^2 c d^4+52 a b^3 c^2 d^3-28 a^3 b c^2 d^3+36 b^4 c^3 d^2-36 a^2 b^2 c^3 d^2-12 a b^3 c^4 d\right) \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left(15 b^4 d^5-12 a^2 b^2 d^5+8 a b^3 c d^4-8 a^3 b c d^4-26 b^4 c^2 d^3+20 a^2 b^2 c^2 d^3+3 b^4 c^4 d\right) \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{12 (a-b) (a+b) (c-d)^2 (c+d)^2 (a d-b c)^3 f}","\frac{d \left(2 a^2 d^2+b^2 \left(3 c^2-5 d^2\right)\right) \cos (e+f x)}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left(2 a^2 d^2+b^2 \left(3 c^2-5 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}+\frac{b^2 \left(-7 a^2 d+2 a b c+5 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a-b) (a+b)^2 (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{\left(8 a^3 c d^3-4 a^2 b d^2 \left(5 c^2-3 d^2\right)-8 a b^2 c d^3-\left(b^3 \left(3 c^4-26 c^2 d^2+15 d^4\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right)^2 (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left(8 a^3 c d^4-4 a^2 b d^3 \left(5 c^2-3 d^2\right)-8 a b^2 c d^4-b^3 \left(3 c^4 d-26 c^2 d^3+15 d^5\right)\right) \cos (e+f x)}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right)^2 (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}",1,"(Sqrt[c + d*Sin[e + f*x]]*(-((b^4*Cos[e + f*x])/((a^2 - b^2)*(-(b*c) + a*d)^3*(a + b*Sin[e + f*x]))) + (2*d^3*Cos[e + f*x])/(3*(b*c - a*d)^2*(c^2 - d^2)*(c + d*Sin[e + f*x])^2) - (4*(-5*b*c^2*d^3*Cos[e + f*x] + 2*a*c*d^4*Cos[e + f*x] + 3*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 + ((-2*(-12*a*b^3*c^5 + 36*a^2*b^2*c^4*d - 33*b^4*c^4*d - 36*a^3*b*c^3*d^2 + 60*a*b^3*c^3*d^2 + 12*a^4*c^2*d^3 - 104*a^2*b^2*c^2*d^3 + 86*b^4*c^2*d^3 + 28*a^3*b*c*d^4 - 40*a*b^3*c*d^4 + 4*a^4*d^5 + 44*a^2*b^2*d^5 - 45*b^4*d^5)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(-12*a*b^3*c^4*d - 36*a^2*b^2*c^3*d^2 + 36*b^4*c^3*d^2 - 28*a^3*b*c^2*d^3 + 52*a*b^3*c^2*d^3 + 16*a^4*c*d^4 + 4*a^2*b^2*c*d^4 - 20*b^4*c*d^4 + 28*a^3*b*d^5 - 40*a*b^3*d^5)*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(3*b^4*c^4*d + 20*a^2*b^2*c^2*d^3 - 26*b^4*c^2*d^3 - 8*a^3*b*c*d^4 + 8*a*b^3*c*d^4 - 12*a^2*b^2*d^5 + 15*b^4*d^5)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(12*(a - b)*(a + b)*(c - d)^2*(c + d)^2*(-(b*c) + a*d)^3*f)","C",0
758,1,1526,816,8.7833331,"\int \frac{(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^(9/2)/(a + b*Sin[e + f*x])^3,x]","\frac{\sqrt{c+d \sin (e+f x)} \left(-\frac{2 \cos (e+f x) d^4}{3 b^3}+\frac{-11 d^4 \cos (e+f x) a^5+27 b c d^3 \cos (e+f x) a^4+17 b^2 d^4 \cos (e+f x) a^3-15 b^2 c^2 d^2 \cos (e+f x) a^3-51 b^3 c d^3 \cos (e+f x) a^2-7 b^3 c^3 d \cos (e+f x) a^2+6 b^4 c^4 \cos (e+f x) a+51 b^4 c^2 d^2 \cos (e+f x) a-17 b^5 c^3 d \cos (e+f x)}{4 b^3 \left(b^2-a^2\right)^2 (a+b \sin (e+f x))}+\frac{-b^4 \cos (e+f x) c^4+4 a b^3 d \cos (e+f x) c^3-6 a^2 b^2 d^2 \cos (e+f x) c^2+4 a^3 b d^3 \cos (e+f x) c-a^4 d^4 \cos (e+f x)}{2 b^3 \left(b^2-a^2\right) (a+b \sin (e+f x))^2}\right)}{f}-\frac{-\frac{2 \left(-24 c^5 b^5-104 c d^4 b^5-327 c^3 d^2 b^5+56 a d^5 b^4+501 a c^2 d^3 b^4+306 a c^4 d b^4-48 a^2 c^5 b^3-53 a^2 c d^4 b^3-177 a^2 c^3 d^2 b^3-73 a^3 d^5 b^2-105 a^3 c^2 d^3 b^2+13 a^4 c d^4 b+35 a^5 d^5\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left(140 c d^4 a^5+56 b d^5 a^4-228 b c^2 d^3 a^4-364 b^2 c d^4 a^3+36 b^2 c^3 d^2 a^3-112 b^3 d^5 a^2+276 b^3 c^2 d^3 a^2-60 b^3 c^4 d a^2+512 b^4 c d^4 a+252 b^4 c^3 d^2 a-16 b^5 d^5-480 b^5 c^2 d^3-12 b^5 c^4 d\right) \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left(104 c d^4 b^5-51 c^3 d^2 b^5-72 a d^5 b^4+153 a c^2 d^3 b^4+18 a c^4 d b^4-361 a^2 c d^4 b^3-21 a^2 c^3 d^2 b^3+195 a^3 d^5 b^2-45 a^3 c^2 d^3 b^2+185 a^4 c d^4 b-105 a^5 d^5\right) \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{48 (a-b)^2 b^3 (a+b)^2 f}","\frac{\left(35 d^2 a^4+20 b c d a^3+2 b^2 \left(4 c^2-43 d^2\right) a^2-44 b^3 c d a+b^4 \left(4 c^2+63 d^2\right)\right) \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} (b c-a d)^3}{4 (a-b)^2 b^5 (a+b)^3 f \sqrt{c+d \sin (e+f x)}}+\frac{\cos (e+f x) (c+d \sin (e+f x))^{5/2} (b c-a d)^2}{2 b \left(a^2-b^2\right) f (a+b \sin (e+f x))^2}+\frac{\left(7 d a^2+6 b c a-13 b^2 d\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2} (b c-a d)^2}{4 b^2 \left(a^2-b^2\right)^2 f (a+b \sin (e+f x))}+\frac{d \left(-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left(9 c^2+61 d^2\right) a^2-18 b^3 c \left(c^2+5 d^2\right) a+b^4 d \left(45 c^2-8 d^2\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{12 b^3 \left(a^2-b^2\right)^2 f}-\frac{\left(-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left(26 c^2+223 d^2\right) a^4-12 b^3 c d^2 \left(4 c^2+29 d^2\right) a^3-b^4 d \left(33 c^4+70 d^2 c^2+128 d^4\right) a^2+6 b^5 c \left(3 c^4+38 d^2 c^2+48 d^4\right) a-b^6 d \left(57 c^4+136 d^2 c^2+8 d^4\right)\right) F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{12 b^5 \left(a^2-b^2\right)^2 f \sqrt{c+d \sin (e+f x)}}+\frac{\left(-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left(3 c^2-13 d^2\right) a^3-b^3 c d \left(21 c^2+361 d^2\right) a^2+9 b^4 \left(2 c^4+17 d^2 c^2-8 d^4\right) a-b^5 c d \left(51 c^2-104 d^2\right)\right) E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right) \sqrt{c+d \sin (e+f x)}}{12 b^4 \left(a^2-b^2\right)^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(Sqrt[c + d*Sin[e + f*x]]*((-2*d^4*Cos[e + f*x])/(3*b^3) + (-(b^4*c^4*Cos[e + f*x]) + 4*a*b^3*c^3*d*Cos[e + f*x] - 6*a^2*b^2*c^2*d^2*Cos[e + f*x] + 4*a^3*b*c*d^3*Cos[e + f*x] - a^4*d^4*Cos[e + f*x])/(2*b^3*(-a^2 + b^2)*(a + b*Sin[e + f*x])^2) + (6*a*b^4*c^4*Cos[e + f*x] - 7*a^2*b^3*c^3*d*Cos[e + f*x] - 17*b^5*c^3*d*Cos[e + f*x] - 15*a^3*b^2*c^2*d^2*Cos[e + f*x] + 51*a*b^4*c^2*d^2*Cos[e + f*x] + 27*a^4*b*c*d^3*Cos[e + f*x] - 51*a^2*b^3*c*d^3*Cos[e + f*x] - 11*a^5*d^4*Cos[e + f*x] + 17*a^3*b^2*d^4*Cos[e + f*x])/(4*b^3*(-a^2 + b^2)^2*(a + b*Sin[e + f*x]))))/f - ((-2*(-48*a^2*b^3*c^5 - 24*b^5*c^5 + 306*a*b^4*c^4*d - 177*a^2*b^3*c^3*d^2 - 327*b^5*c^3*d^2 - 105*a^3*b^2*c^2*d^3 + 501*a*b^4*c^2*d^3 + 13*a^4*b*c*d^4 - 53*a^2*b^3*c*d^4 - 104*b^5*c*d^4 + 35*a^5*d^5 - 73*a^3*b^2*d^5 + 56*a*b^4*d^5)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(-60*a^2*b^3*c^4*d - 12*b^5*c^4*d + 36*a^3*b^2*c^3*d^2 + 252*a*b^4*c^3*d^2 - 228*a^4*b*c^2*d^3 + 276*a^2*b^3*c^2*d^3 - 480*b^5*c^2*d^3 + 140*a^5*c*d^4 - 364*a^3*b^2*c*d^4 + 512*a*b^4*c*d^4 + 56*a^4*b*d^5 - 112*a^2*b^3*d^5 - 16*b^5*d^5)*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(18*a*b^4*c^4*d - 21*a^2*b^3*c^3*d^2 - 51*b^5*c^3*d^2 - 45*a^3*b^2*c^2*d^3 + 153*a*b^4*c^2*d^3 + 185*a^4*b*c*d^4 - 361*a^2*b^3*c*d^4 + 104*b^5*c*d^4 - 105*a^5*d^5 + 195*a^3*b^2*d^5 - 72*a*b^4*d^5)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(48*(a - b)^2*b^3*(a + b)^2*f)","C",0
759,1,1323,605,8.351271,"\int \frac{(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^(7/2)/(a + b*Sin[e + f*x])^3,x]","\frac{\sqrt{c+d \sin (e+f x)} \left(\frac{-b^3 \cos (e+f x) c^3+3 a b^2 d \cos (e+f x) c^2-3 a^2 b d^2 \cos (e+f x) c+a^3 d^3 \cos (e+f x)}{2 b^2 \left(b^2-a^2\right) (a+b \sin (e+f x))^2}+\frac{7 d^3 \cos (e+f x) a^4-8 b c d^2 \cos (e+f x) a^3-13 b^2 d^3 \cos (e+f x) a^2-5 b^2 c^2 d \cos (e+f x) a^2+6 b^3 c^3 \cos (e+f x) a+26 b^3 c d^2 \cos (e+f x) a-13 b^4 c^2 d \cos (e+f x)}{4 b^2 \left(b^2-a^2\right)^2 (a+b \sin (e+f x))}\right)}{f}+\frac{-\frac{2 \left(8 c^4 b^4+8 d^4 b^4+57 c^2 d^2 b^4-50 a c d^3 b^3-78 a c^3 d b^3+16 a^2 c^4 b^2-7 a^2 d^4 b^2+33 a^2 c^2 d^2 b^2+8 a^3 c d^3 b+5 a^4 d^4\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left(20 c d^3 a^4+8 b d^4 a^3-8 b c^2 d^2 a^3-12 b^2 c d^3 a^2+20 b^2 c^3 d a^2-32 b^3 d^4 a-64 b^3 c^2 d^2 a+64 b^4 c d^3+4 b^4 c^3 d\right) \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left(-8 d^4 b^4+13 c^2 d^2 b^4-26 a c d^3 b^3-6 a c^3 d b^3+29 a^2 d^4 b^2+5 a^2 c^2 d^2 b^2+8 a^3 c d^3 b-15 a^4 d^4\right) \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{16 (a-b)^2 b^2 (a+b)^2 f}","\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{\left(5 a^2 d+6 a b c-11 b^2 d\right) (b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 b^2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{\left(15 a^4 d^2+12 a^3 b c d+2 a^2 b^2 \left(4 c^2-19 d^2\right)-36 a b^3 c d+b^4 \left(4 c^2+35 d^2\right)\right) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^4 f (a-b)^2 (a+b)^3 \sqrt{c+d \sin (e+f x)}}+\frac{3 \left(5 a^4 d^3+4 a^3 b c d^2+a^2 b^2 d \left(c^2-11 d^2\right)-2 a b^3 c \left(c^2+5 d^2\right)+b^4 d \left(5 c^2+8 d^2\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^4 f \left(a^2-b^2\right)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left(-15 a^4 d^3+8 a^3 b c d^2+a^2 b^2 d \left(5 c^2+29 d^2\right)-2 a b^3 c \left(3 c^2+13 d^2\right)+b^4 d \left(13 c^2-8 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^3 f \left(a^2-b^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(Sqrt[c + d*Sin[e + f*x]]*((-(b^3*c^3*Cos[e + f*x]) + 3*a*b^2*c^2*d*Cos[e + f*x] - 3*a^2*b*c*d^2*Cos[e + f*x] + a^3*d^3*Cos[e + f*x])/(2*b^2*(-a^2 + b^2)*(a + b*Sin[e + f*x])^2) + (6*a*b^3*c^3*Cos[e + f*x] - 5*a^2*b^2*c^2*d*Cos[e + f*x] - 13*b^4*c^2*d*Cos[e + f*x] - 8*a^3*b*c*d^2*Cos[e + f*x] + 26*a*b^3*c*d^2*Cos[e + f*x] + 7*a^4*d^3*Cos[e + f*x] - 13*a^2*b^2*d^3*Cos[e + f*x])/(4*b^2*(-a^2 + b^2)^2*(a + b*Sin[e + f*x]))))/f + ((-2*(16*a^2*b^2*c^4 + 8*b^4*c^4 - 78*a*b^3*c^3*d + 33*a^2*b^2*c^2*d^2 + 57*b^4*c^2*d^2 + 8*a^3*b*c*d^3 - 50*a*b^3*c*d^3 + 5*a^4*d^4 - 7*a^2*b^2*d^4 + 8*b^4*d^4)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(20*a^2*b^2*c^3*d + 4*b^4*c^3*d - 8*a^3*b*c^2*d^2 - 64*a*b^3*c^2*d^2 + 20*a^4*c*d^3 - 12*a^2*b^2*c*d^3 + 64*b^4*c*d^3 + 8*a^3*b*d^4 - 32*a*b^3*d^4)*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(-6*a*b^3*c^3*d + 5*a^2*b^2*c^2*d^2 + 13*b^4*c^2*d^2 + 8*a^3*b*c*d^3 - 26*a*b^3*c*d^3 - 15*a^4*d^4 + 29*a^2*b^2*d^4 - 8*b^4*d^4)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(16*(a - b)^2*b^2*(a + b)^2*f)","C",0
760,1,1149,549,8.0693757,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x])^3,x]","\frac{\sqrt{c+d \sin (e+f x)} \left(\frac{-b^2 \cos (e+f x) c^2+2 a b d \cos (e+f x) c-a^2 d^2 \cos (e+f x)}{2 b \left(b^2-a^2\right) (a+b \sin (e+f x))^2}-\frac{3 \left(d^2 \cos (e+f x) a^3+b c d \cos (e+f x) a^2-2 b^2 c^2 \cos (e+f x) a-3 b^2 d^2 \cos (e+f x) a+3 b^3 c d \cos (e+f x)\right)}{4 b \left(b^2-a^2\right)^2 (a+b \sin (e+f x))}\right)}{f}-\frac{-\frac{2 \left(-8 c^3 b^3-21 c d^2 b^3+5 a d^3 b^2+54 a c^2 d b^2-16 a^2 c^3 b-15 a^2 c d^2 b+a^3 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left(4 c d^2 a^3-8 b d^3 a^2-20 b c^2 d a^2+44 b^2 c d^2 a-16 b^3 d^3-4 b^3 c^2 d\right) \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left(-9 c d^2 b^3+9 a d^3 b^2+6 a c^2 d b^2-3 a^2 c d^2 b-3 a^3 d^3\right) \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{16 (a-b)^2 b (a+b)^2 f}","\frac{(b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{3 \left(a^2 d+2 a b c-3 b^2 d\right) (b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 b f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{3 \left(a^2 d+2 a b c-3 b^2 d\right) (b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^2 f \left(a^2-b^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left(3 a^4 d^2+4 a^3 b c d+2 a^2 b^2 \left(4 c^2-3 d^2\right)-28 a b^3 c d+b^4 \left(4 c^2+15 d^2\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^3 f (a-b)^2 (a+b)^3 \sqrt{c+d \sin (e+f x)}}+\frac{\left(3 a^4 d^3+4 a^3 b c d^2+a^2 b^2 d \left(7 c^2-5 d^2\right)-2 a b^3 c \left(3 c^2+11 d^2\right)+b^4 d \left(11 c^2+8 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^3 f \left(a^2-b^2\right)^2 \sqrt{c+d \sin (e+f x)}}",1,"(Sqrt[c + d*Sin[e + f*x]]*((-(b^2*c^2*Cos[e + f*x]) + 2*a*b*c*d*Cos[e + f*x] - a^2*d^2*Cos[e + f*x])/(2*b*(-a^2 + b^2)*(a + b*Sin[e + f*x])^2) - (3*(-2*a*b^2*c^2*Cos[e + f*x] + a^2*b*c*d*Cos[e + f*x] + 3*b^3*c*d*Cos[e + f*x] + a^3*d^2*Cos[e + f*x] - 3*a*b^2*d^2*Cos[e + f*x]))/(4*b*(-a^2 + b^2)^2*(a + b*Sin[e + f*x]))))/f - ((-2*(-16*a^2*b*c^3 - 8*b^3*c^3 + 54*a*b^2*c^2*d - 15*a^2*b*c*d^2 - 21*b^3*c*d^2 + a^3*d^3 + 5*a*b^2*d^3)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(-20*a^2*b*c^2*d - 4*b^3*c^2*d + 4*a^3*c*d^2 + 44*a*b^2*c*d^2 - 8*a^2*b*d^3 - 16*b^3*d^3)*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(6*a*b^2*c^2*d - 3*a^2*b*c*d^2 - 9*b^3*c*d^2 - 3*a^3*d^3 + 9*a*b^2*d^3)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(16*(a - b)^2*b*(a + b)^2*f)","C",0
761,1,1001,472,7.4308671,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^3} \, dx","Integrate[(c + d*Sin[e + f*x])^(3/2)/(a + b*Sin[e + f*x])^3,x]","\frac{\sqrt{c+d \sin (e+f x)} \left(\frac{b c \cos (e+f x)-a d \cos (e+f x)}{2 \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{-d \cos (e+f x) a^2+6 b c \cos (e+f x) a-5 b^2 d \cos (e+f x)}{4 \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}\right)}{f}+\frac{-\frac{2 \left(16 a^2 c^2+8 b^2 c^2-30 a b d c+5 a^2 d^2+b^2 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left(20 c d a^2-24 b d^2 a+4 b^2 c d\right) \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left(a^2 d^2+5 b^2 d^2-6 a b c d\right) \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{16 (a-b)^2 (a+b)^2 f}","\frac{\left(a^2 (-d)+6 a b c-5 b^2 d\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}-\frac{(b c-a d) \left(a^2 d+6 a b c-7 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^2 f \left(a^2-b^2\right)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left(a^2 (-d)+6 a b c-5 b^2 d\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b f \left(a^2-b^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left(a^4 d^2+4 a^3 b c d-2 a^2 b^2 \left(4 c^2+5 d^2\right)+20 a b^3 c d-b^4 \left(4 c^2+3 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^2 f (a-b)^2 (a+b)^3 \sqrt{c+d \sin (e+f x)}}",1,"(Sqrt[c + d*Sin[e + f*x]]*((b*c*Cos[e + f*x] - a*d*Cos[e + f*x])/(2*(a^2 - b^2)*(a + b*Sin[e + f*x])^2) + (6*a*b*c*Cos[e + f*x] - a^2*d*Cos[e + f*x] - 5*b^2*d*Cos[e + f*x])/(4*(a^2 - b^2)^2*(a + b*Sin[e + f*x]))))/f + ((-2*(16*a^2*c^2 + 8*b^2*c^2 - 30*a*b*c*d + 5*a^2*d^2 + b^2*d^2)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(20*a^2*c*d + 4*b^2*c*d - 24*a*b*d^2)*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(-6*a*b*c*d + a^2*d^2 + 5*b^2*d^2)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(16*(a - b)^2*(a + b)^2*f)","C",0
762,1,1038,487,7.7403116,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+b \sin (e+f x))^3} \, dx","Integrate[Sqrt[c + d*Sin[e + f*x]]/(a + b*Sin[e + f*x])^3,x]","\frac{\sqrt{c+d \sin (e+f x)} \left(\frac{b \cos (e+f x)}{2 \left(a^2-b^2\right) (a+b \sin (e+f x))^2}-\frac{-d \cos (e+f x) b^3+6 a c \cos (e+f x) b^2-5 a^2 d \cos (e+f x) b}{4 \left(a^2-b^2\right)^2 (a d-b c) (a+b \sin (e+f x))}\right)}{f}+\frac{-\frac{2 \left(16 c d a^3-16 b c^2 a^2-9 b d^2 a^2+14 b^2 c d a-8 b^3 c^2+3 b^3 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left(16 d^2 a^3-20 b c d a^2+8 b^2 d^2 a-4 b^3 c d\right) \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left(-d^2 b^3+6 a c d b^2-5 a^2 d^2 b\right) \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{16 (a-b)^2 (a+b)^2 (a d-b c) f}","\frac{b \left(-5 a^2 d+6 a b c-b^2 d\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \left(a^2-b^2\right)^2 (b c-a d) (a+b \sin (e+f x))}+\frac{b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}-\frac{3 \left(a^2 (-d)+2 a b c-b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b f \left(a^2-b^2\right)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left(-5 a^2 d+6 a b c-b^2 d\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left(-3 a^4 d^2+12 a^3 b c d-2 a^2 b^2 \left(4 c^2+5 d^2\right)+12 a b^3 c d-b^4 \left(4 c^2-d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b f (a-b)^2 (a+b)^3 (b c-a d) \sqrt{c+d \sin (e+f x)}}",1,"(Sqrt[c + d*Sin[e + f*x]]*((b*Cos[e + f*x])/(2*(a^2 - b^2)*(a + b*Sin[e + f*x])^2) - (6*a*b^2*c*Cos[e + f*x] - 5*a^2*b*d*Cos[e + f*x] - b^3*d*Cos[e + f*x])/(4*(a^2 - b^2)^2*(-(b*c) + a*d)*(a + b*Sin[e + f*x]))))/f + ((-2*(-16*a^2*b*c^2 - 8*b^3*c^2 + 16*a^3*c*d + 14*a*b^2*c*d - 9*a^2*b*d^2 + 3*b^3*d^2)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(-20*a^2*b*c*d - 4*b^3*c*d + 16*a^3*d^2 + 8*a*b^2*d^2)*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(6*a*b^2*c*d - 5*a^2*b*d^2 - b^3*d^2)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(16*(a - b)^2*(a + b)^2*(-(b*c) + a*d)*f)","C",0
763,1,1069,503,7.8158856,"\int \frac{1}{(a+b \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[1/((a + b*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]]),x]","\frac{\sqrt{c+d \sin (e+f x)} \left(\frac{3 \left(d \cos (e+f x) b^4+2 a c \cos (e+f x) b^3-3 a^2 d \cos (e+f x) b^2\right)}{4 \left(a^2-b^2\right)^2 (a d-b c)^2 (a+b \sin (e+f x))}-\frac{b^2 \cos (e+f x)}{2 \left(a^2-b^2\right) (a d-b c) (a+b \sin (e+f x))^2}\right)}{f}+\frac{-\frac{2 \left(16 d^2 a^4-32 b c d a^3+16 b^2 c^2 a^2-19 b^2 d^2 a^2+2 b^3 c d a+8 b^4 c^2+9 b^4 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left(4 c d b^4+8 a d^2 b^3+20 a^2 c d b^2-32 a^3 d^2 b\right) \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left(-3 d^2 b^4-6 a c d b^3+9 a^2 d^2 b^2\right) \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{16 (a-b)^2 (a+b)^2 (a d-b c)^2 f}","\frac{3 b^2 \left(-3 a^2 d+2 a b c+b^2 d\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x))}+\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2}-\frac{\left(-7 a^2 d+6 a b c+b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 (b c-a d) \sqrt{c+d \sin (e+f x)}}+\frac{3 b \left(-3 a^2 d+2 a b c+b^2 d\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left(-15 a^4 d^2+20 a^3 b c d-2 a^2 b^2 \left(4 c^2-3 d^2\right)+4 a b^3 c d-b^4 \left(4 c^2+3 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f (a-b)^2 (a+b)^3 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}",1,"(Sqrt[c + d*Sin[e + f*x]]*(-1/2*(b^2*Cos[e + f*x])/((a^2 - b^2)*(-(b*c) + a*d)*(a + b*Sin[e + f*x])^2) + (3*(2*a*b^3*c*Cos[e + f*x] - 3*a^2*b^2*d*Cos[e + f*x] + b^4*d*Cos[e + f*x]))/(4*(a^2 - b^2)^2*(-(b*c) + a*d)^2*(a + b*Sin[e + f*x]))))/f + ((-2*(16*a^2*b^2*c^2 + 8*b^4*c^2 - 32*a^3*b*c*d + 2*a*b^3*c*d + 16*a^4*d^2 - 19*a^2*b^2*d^2 + 9*b^4*d^2)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(20*a^2*b^2*c*d + 4*b^4*c*d - 32*a^3*b*d^2 + 8*a*b^3*d^2)*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(-6*a*b^3*c*d + 9*a^2*b^2*d^2 - 3*b^4*d^2)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(16*(a - b)^2*(a + b)^2*(-(b*c) + a*d)^2*f)","C",0
764,1,1318,682,9.4103162,"\int \frac{1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx","Integrate[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2)),x]","\frac{\sqrt{c+d \sin (e+f x)} \left(-\frac{2 \cos (e+f x) d^4}{(b c-a d)^3 \left(c^2-d^2\right) (c+d \sin (e+f x))}-\frac{7 d \cos (e+f x) b^5+6 a c \cos (e+f x) b^4-13 a^2 d \cos (e+f x) b^3}{4 \left(a^2-b^2\right)^2 (a d-b c)^3 (a+b \sin (e+f x))}+\frac{b^3 \cos (e+f x)}{2 \left(a^2-b^2\right) (a d-b c)^2 (a+b \sin (e+f x))^2}\right)}{f}+\frac{-\frac{2 \left(16 c d^3 a^5+56 b d^4 a^4-48 b c^2 d^2 a^4-80 b^2 c d^3 a^3+48 b^2 c^3 d a^3-16 b^3 c^4 a^2-95 b^3 d^4 a^2+95 b^3 c^2 d^2 a^2+34 b^4 c d^3 a-18 b^4 c^3 d a-8 b^5 c^4+45 b^5 d^4-29 b^5 c^2 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left(16 d^4 a^5+16 b c d^3 a^4-80 b^2 d^4 a^3+48 b^2 c^2 d^2 a^3-12 b^3 c d^3 a^2-20 b^3 c^3 d a^2+40 b^4 d^4 a-24 b^4 c^2 d^2 a+20 b^5 c d^3-4 b^5 c^3 d\right) \cos (e+f x) \left((b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+a d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left(-15 d^4 b^5+7 c^2 d^2 b^5-6 a c d^3 b^4+6 a c^3 d b^4+29 a^2 d^4 b^3-13 a^2 c^2 d^2 b^3-8 a^4 d^4 b\right) \cos (e+f x) \cos (2 (e+f x)) \left(2 b (c-d) (b c-a d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)+d \left(\left(2 a^2-b^2\right) d \Pi \left(\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)-2 (a+b) (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\right)\right)\right) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left(-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{16 (a-b)^2 (a+b)^2 (c-d) (c+d) (a d-b c)^3 f}","\frac{b^2 \left(-11 a^2 d+6 a b c+5 b^2 d\right) \cos (e+f x)}{4 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}-\frac{b \left(-11 a^2 d+6 a b c+5 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{d \left(8 a^4 d^3+a^2 b^2 d \left(13 c^2-29 d^2\right)-6 a b^3 c \left(c^2-d^2\right)-b^4 d \left(7 c^2-15 d^2\right)\right) \cos (e+f x)}{4 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right) (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{\left(8 a^4 d^3+a^2 b^2 d \left(13 c^2-29 d^2\right)-6 a b^3 c \left(c^2-d^2\right)-b^4 d \left(7 c^2-15 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right) (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{b \left(-35 a^4 d^2+28 a^3 b c d-2 a^2 b^2 \left(4 c^2-19 d^2\right)-4 a b^3 c d-b^4 \left(4 c^2+15 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f (a-b)^2 (a+b)^3 (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}",1,"(Sqrt[c + d*Sin[e + f*x]]*((b^3*Cos[e + f*x])/(2*(a^2 - b^2)*(-(b*c) + a*d)^2*(a + b*Sin[e + f*x])^2) - (6*a*b^4*c*Cos[e + f*x] - 13*a^2*b^3*d*Cos[e + f*x] + 7*b^5*d*Cos[e + f*x])/(4*(a^2 - b^2)^2*(-(b*c) + a*d)^3*(a + b*Sin[e + f*x])) - (2*d^4*Cos[e + f*x])/((b*c - a*d)^3*(c^2 - d^2)*(c + d*Sin[e + f*x]))))/f + ((-2*(-16*a^2*b^3*c^4 - 8*b^5*c^4 + 48*a^3*b^2*c^3*d - 18*a*b^4*c^3*d - 48*a^4*b*c^2*d^2 + 95*a^2*b^3*c^2*d^2 - 29*b^5*c^2*d^2 + 16*a^5*c*d^3 - 80*a^3*b^2*c*d^3 + 34*a*b^4*c*d^3 + 56*a^4*b*d^4 - 95*a^2*b^3*d^4 + 45*b^5*d^4)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(-20*a^2*b^3*c^3*d - 4*b^5*c^3*d + 48*a^3*b^2*c^2*d^2 - 24*a*b^4*c^2*d^2 + 16*a^4*b*c*d^3 - 12*a^2*b^3*c*d^3 + 20*b^5*c*d^3 + 16*a^5*d^4 - 80*a^3*b^2*d^4 + 40*a*b^4*d^4)*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(6*a*b^4*c^3*d - 13*a^2*b^3*c^2*d^2 + 7*b^5*c^2*d^2 - 6*a*b^4*c*d^3 - 8*a^4*b*d^4 + 29*a^2*b^3*d^4 - 15*b^5*d^4)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b^2*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(16*(a - b)^2*(a + b)^2*(c - d)*(c + d)*(-(b*c) + a*d)^3*f)","C",0
765,1,1978,888,7.1380267,"\int \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2),x]","\frac{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \left(-\frac{1}{6} \sin (2 (e+f x)) d^2-\frac{(13 b c+a d) \cos (e+f x) d}{12 b}\right)}{f}-\frac{-\frac{4 (a d-b c) \left(-48 a b c^3-59 b^2 d c^2-58 a b d^2 c+a^2 d^3-16 b^2 d^3\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(-48 b^2 c^3-92 a b d c^2+4 a^2 d^2 c-76 b^2 d^2 c-28 a b d^3\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(-3 a^2 d^3+16 b^2 d^3+14 a b c d^2+33 b^2 c^2 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)}{48 b f}","-\frac{\cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)} d^2}{3 b f}-\frac{(13 b c-3 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} d}{12 b f}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(\left(33 c^2+16 d^2\right) b^2+14 a c d b-3 a^2 d^2\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))}{24 b^2 (b c-a d) f}+\frac{(a+b)^{3/2} \left(\left(33 c^2+26 d c+16 d^2\right) b^2-6 a d (2 c+d) b+3 a^2 d^2\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))}{24 b^3 \sqrt{c+d} f}-\frac{\left(\left(33 c^2+16 d^2\right) b^2+14 a c d b-3 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 b f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{c+d} \left(-5 \left(c^3+4 d^2 c\right) b^3-a d \left(15 c^2+4 d^2\right) b^2+5 a^2 c d^2 b-a^3 d^3\right) \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))}{8 b^3 \sqrt{a+b} f d}",1,"-1/48*((-4*(-(b*c) + a*d)*(-48*a*b*c^3 - 59*b^2*c^2*d - 58*a*b*c*d^2 + a^2*d^3 - 16*b^2*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)*(-48*b^2*c^3 - 92*a*b*c^2*d + 4*a^2*c*d^2 - 76*b^2*c*d^2 - 28*a*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*(33*b^2*c^2*d + 14*a*b*c*d^2 - 3*a^2*d^3 + 16*b^2*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)))/(b*f) + (Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*(-1/12*(d*(13*b*c + a*d)*Cos[e + f*x])/b - (d^2*Sin[2*(e + f*x)])/6))/f","B",0
766,1,1879,784,9.5405326,"\int \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2),x]","\frac{-\frac{4 (a d-b c) \left(8 a c^2+7 b d c+3 a d^2\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(8 b c^2+12 a d c+4 b d^2\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 d^2-5 b c 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)}{8 f}-\frac{d \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{2 f}","\frac{\sqrt{c+d} \left(-a^2 d^2+6 a b c d+b^2 \left(3 c^2+4 d^2\right)\right) \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))}} \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)}{4 b^2 d f \sqrt{a+b}}+\frac{(a+b)^{3/2} (-a d+5 b c+2 b d) \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)}{4 b^2 f \sqrt{c+d}}-\frac{b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a+b \sin (e+f x)}}+\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \sqrt{a+b \sin (e+f x)}}-\frac{(a d+5 b c) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} (a d+5 b c) \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)}{4 b f (b c-a d)}",1,"-1/2*(d*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/f + ((-4*(-(b*c) + a*d)*(8*a*c^2 + 7*b*c*d + 3*a*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)*(8*b*c^2 + 12*a*c*d + 4*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*(-5*b*c*d - a*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)))/(8*f)","B",0
767,1,228392,628,31.7074815,"\int \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]],x]","\text{Result too large to show}","-\frac{b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a+b \sin (e+f x)}}+\frac{(a+b)^{3/2} \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 \sqrt{c+d}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \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 (b c-a d)}+\frac{\sqrt{c+d} (a d+b c) \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))}} \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)}{b d f \sqrt{a+b}}",1,"Result too large to show","C",0
768,1,197,198,0.2159281,"\int \frac{\sqrt{a+b \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]],x]","\frac{2 \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{\frac{(a d-b c) (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))}} \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)}{d f \sqrt{a+b}}","\frac{2 \sqrt{c+d} \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))}} \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)}{d f \sqrt{a+b}}",1,"(2*Sqrt[c + d]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[((-(b*c) + a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x]))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*d*f)","A",1
769,1,263,409,7.4241009,"\int \frac{\sqrt{a+b \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x])^(3/2),x]","\frac{2 \left(-((b c-a d) \cos (e+f x))-\frac{\sqrt{2} \sqrt{\frac{a-b}{a+b}} (a+b) (c+d) \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right) \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))}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{a-b}{a+b}} \cos \left(\frac{1}{4} (2 e+2 f x+\pi )\right)}{\sqrt{\frac{a+b \sin (e+f x)}{a+b}}}\right)|\frac{2 (a d-b c)}{(a-b) (c+d)}\right)}{\sqrt{\frac{(a+b) (\sin (e+f x)+1)}{a+b \sin (e+f x)}}}\right)}{f (c-d) (c+d) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}","\frac{2 (a-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))}} 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 (c-d) \sqrt{c+d} (b c-a d)}-\frac{2 (a-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))}} 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 (c-d) \sqrt{c+d} (b c-a d)}",1,"(2*(-((b*c - a*d)*Cos[e + f*x]) - (Sqrt[2]*Sqrt[(a - b)/(a + b)]*(a + b)*(c + d)*Cos[(2*e - Pi + 2*f*x)/4]*EllipticE[ArcSin[(Sqrt[(a - b)/(a + b)]*Cos[(2*e + Pi + 2*f*x)/4])/Sqrt[(a + b*Sin[e + f*x])/(a + b)]], (2*(-(b*c) + a*d))/((a - b)*(c + d))]*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]))])/Sqrt[((a + b)*(1 + Sin[e + f*x]))/(a + b*Sin[e + f*x])]))/((c - d)*(c + d)*f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",1
770,1,2067,489,6.415802,"\int \frac{\sqrt{a+b \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x])^(5/2),x]","\text{Result too large to show}","\frac{2 d \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}+\frac{2 (a-b) \sqrt{a+b} \left(4 a c d-b \left(3 c^2+d^2\right)\right) \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))}} 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)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)^2}+\frac{2 (a-b) \sqrt{a+b} (3 c+d) \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)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)}",1,"(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((2*d*Cos[e + f*x])/(3*(c^2 - d^2)*(c + d*Sin[e + f*x])^2) + (2*(3*b*c^2*d*Cos[e + f*x] - 4*a*c*d^2*Cos[e + f*x] + b*d^3*Cos[e + f*x]))/(3*(b*c - a*d)*(c^2 - d^2)^2*(c + d*Sin[e + f*x]))))/f + ((-4*(-(b*c) + a*d)*(-3*a*b*c^3 + 3*a^2*c^2*d + b^2*c^2*d - a*b*c*d^2 + a^2*d^3 - b^2*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)*(-3*b^2*c^3 + a*b*c^2*d + 4*a^2*c*d^2 - b^2*c*d^2 - a*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*(3*b^2*c^2*d - 4*a*b*c*d^2 + b^2*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)))/(3*(c - d)^2*(c + d)^2*(-(b*c) + a*d)*f)","B",0
771,1,2091,1080,7.5275834,"\int (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx","Integrate[(a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(5/2),x]","\text{Result too large to show}","-\frac{d^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}{4 b f}-\frac{d (17 b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}}{24 b f}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(\left(15 c^3+284 d^2 c\right) b^3+a d \left(337 c^2+156 d^2\right) b^2+57 a^2 c d^2 b-9 a^3 d^3\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))}{192 b^2 d (b c-a d) f}-\frac{\sqrt{c+d} \left(\left(5 c^4-120 d^2 c^2-48 d^4\right) b^4-60 a c d \left(c^2+4 d^2\right) b^3-6 a^2 d^2 \left(15 c^2+4 d^2\right) b^2+20 a^3 c d^3 b-3 a^4 d^4\right) \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))}{64 b^3 \sqrt{a+b} d^2 f}-\frac{\left(\left(59 c^2+36 d^2\right) b^2+54 a c d b-9 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}{96 b f}+\frac{(a+b)^{3/2} \left(\left(15 c^3+118 d c^2+284 d^2 c+72 d^3\right) b^3+3 a d \left(73 c^2+36 d c+28 d^2\right) b^2-3 a^2 d^2 (17 c+6 d) b+9 a^3 d^3\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))}{192 b^3 d \sqrt{c+d} f}-\frac{\left(\left(15 c^3+284 d^2 c\right) b^3+a d \left(337 c^2+156 d^2\right) b^2+57 a^2 c d^2 b-9 a^3 d^3\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 b d f \sqrt{a+b \sin (e+f x)}}",1,"-1/384*((-4*(-(b*c) + a*d)*(-384*a^2*b*c^3 - 133*b^3*c^3 - 971*a*b^2*c^2*d - 451*a^2*b*c*d^2 - 356*b^3*c*d^2 + 3*a^3*d^3 - 228*a*b^2*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)*(-532*a*b^2*c^3 - 664*a^2*b*c^2*d - 644*b^3*c^2*d + 12*a^3*c*d^2 - 1160*a*b^2*c*d^2 - 228*a^2*b*d^3 - 144*b^3*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*(15*b^3*c^3 + 337*a*b^2*c^2*d + 57*a^2*b*c*d^2 + 284*b^3*c*d^2 - 9*a^3*d^3 + 156*a*b^2*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)))/(b*f) + (Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*(-1/96*((59*b^2*c^2 + 122*a*b*c*d + 3*a^2*d^2 + 42*b^2*d^2)*Cos[e + f*x])/b + (b*d^2*Cos[3*(e + f*x)])/16 - (d*(17*b*c + 9*a*d)*Sin[2*(e + f*x)])/48))/f","A",0
772,1,1952,870,6.3089225,"\int (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2} \, dx","Integrate[(a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2),x]","\frac{-\frac{4 (a d-b c) \left(48 a^2 c^2+17 b^2 c^2+82 a b d c+17 a^2 d^2+16 b^2 d^2\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(68 c d a^2+68 b c^2 a+52 b d^2 a+52 b^2 c d\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(-3 b^2 c^2-38 a b d c-3 a^2 d^2-16 b^2 d^2\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)}{48 f}+\frac{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \left(-\frac{7}{12} (b c+a d) \cos (e+f x)-\frac{1}{6} b d \sin (2 (e+f x))\right)}{f}","-\frac{\left(-\left(\left(3 c^2+14 d c+16 d^2\right) b^2\right)-6 a d (4 c+d) b+3 a^2 d^2\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)^{3/2}}{24 b^2 d \sqrt{c+d} f}+\frac{(c-d) \sqrt{c+d} \left(\left(3 c^2+16 d^2\right) b^2+38 a c d b+3 a^2 d^2\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)) \sqrt{a+b}}{24 b d (b c-a d) f}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}-\frac{(3 b c+7 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 f}-\frac{\left(\left(3 c^2+16 d^2\right) b^2+38 a c d b+3 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 d f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{c+d} (b c+a d) \left(-\left(\left(c^2-12 d^2\right) b^2\right)+10 a c d b-a^2 d^2\right) \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))}{8 b^2 d^2 f \sqrt{a+b}}",1,"((-4*(-(b*c) + a*d)*(48*a^2*c^2 + 17*b^2*c^2 + 82*a*b*c*d + 17*a^2*d^2 + 16*b^2*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)*(68*a*b*c^2 + 68*a^2*c*d + 52*b^2*c*d + 52*a*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*(-3*b^2*c^2 - 38*a*b*c*d - 3*a^2*d^2 - 16*b^2*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)))/(48*f) + (Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-7*(b*c + a*d)*Cos[e + f*x])/12 - (b*d*Sin[2*(e + f*x)])/6))/f","B",0
773,1,1879,740,9.4808891,"\int (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)} \, dx","Integrate[(a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]],x]","\frac{-\frac{4 (a d-b c) \left(8 c a^2+7 b d a+3 b^2 c\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(8 d a^2+12 b c a+4 b^2 d\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(-c b^2-5 a d b\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)}{8 f}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{2 f}","\frac{\sqrt{c+d} \left(3 a^2 d^2+6 a b c d-\left(b^2 \left(c^2-4 d^2\right)\right)\right) \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))}} \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)}{4 b d^2 f \sqrt{a+b}}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{2 f}-\frac{b (5 a d+b c) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 d f \sqrt{a+b \sin (e+f x)}}+\frac{(a+b)^{3/2} (3 a d+b (c+2 d)) \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)}{4 b d f \sqrt{c+d}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} (5 a d+b c) \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)}{4 d f (b c-a d)}",1,"-1/2*(b*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/f + ((-4*(-(b*c) + a*d)*(8*a^2*c + 3*b^2*c + 7*a*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]]) - 4*(-(b*c) + a*d)*(12*a*b*c + 8*a^2*d + 4*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]]) - (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*(-(b^2*c) - 5*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)))/(8*f)","B",0
774,1,222963,644,32.6933367,"\int \frac{(a+b \sin (e+f x))^{3/2}}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(a + b*Sin[e + f*x])^(3/2)/Sqrt[c + d*Sin[e + f*x]],x]","\text{Result too large to show}","\frac{\sqrt{a+b} (b (c-d)-2 a d) \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)}{d^2 f \sqrt{c+d}}-\frac{\sqrt{a+b} (b c-3 a d) \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)}{d^2 f \sqrt{c+d}}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f \sqrt{c+d \sin (e+f x)}}-\frac{b (a-b) \sqrt{a+b} \sqrt{c+d} \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))}} 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)}{d f (b c-a d)}",1,"Result too large to show","C",0
775,1,1896,600,9.5618066,"\int \frac{(a+b \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[(a + b*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(3/2),x]","\frac{-\frac{4 (a d-b c) \left(a^2 c-a b 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(a^2 d-b^2 d\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(b^2 c-a 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)}{(c-d) (c+d) f}-\frac{2 (b c \cos (e+f x)-a d \cos (e+f x)) \sqrt{a+b \sin (e+f x)}}{\left(c^2-d^2\right) f \sqrt{c+d \sin (e+f x)}}","-\frac{2 \sqrt{a+b} (a d+b (c-2 d)) \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)}{d^2 f (c-d) \sqrt{c+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)}{d^2 f \sqrt{c+d}}+\frac{2 (a-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))}} 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)}{d f (c-d) \sqrt{c+d}}",1,"(-2*(b*c*Cos[e + f*x] - a*d*Cos[e + f*x])*Sqrt[a + b*Sin[e + f*x]])/((c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((-4*(-(b*c) + a*d)*(a^2*c - a*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]]) - 4*(-(b*c) + a*d)*(a^2*d - 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]]) - (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*(b^2*c - 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)))/((c - d)*(c + d)*f)","B",0
776,1,2012,497,6.3372415,"\int \frac{(a+b \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{5/2}} \, dx","Integrate[(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 (b c-a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}+\frac{2 (a-b) \sqrt{a+b} (a (3 c+d)-b (c+3 d)) \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)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)}-\frac{8 (a-b) \sqrt{a+b} (a c-b d) \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))}} 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)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)}",1,"(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*(b*c*Cos[e + f*x] - a*d*Cos[e + f*x]))/(3*(c^2 - d^2)*(c + d*Sin[e + f*x])^2) - (8*(-(a*c*d*Cos[e + f*x]) + b*d^2*Cos[e + f*x]))/(3*(c^2 - d^2)^2*(c + d*Sin[e + f*x]))))/f + ((-4*(-(b*c) + a*d)*(3*a^2*c^2 + b^2*c^2 - 4*a*b*c*d + a^2*d^2 - b^2*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)*(4*a*b*c^2 + 4*a^2*c*d - 4*b^2*c*d - 4*a*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*(-4*a*b*c*d + 4*b^2*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)))/(3*(c - d)^2*(c + d)^2*f)","B",0
777,1,2276,1295,8.4711783,"\int (a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx","Integrate[(a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(5/2),x]","\text{Result too large to show}","-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac{3 b (b c-7 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac{\left(-\left(\left(15 c^2-64 d^2\right) b^2\right)+110 a c d b+93 a^2 d^2\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac{(a+b)^{3/2} \left(-\left(\left(45 c^4-30 d c^3-1692 d^2 c^2-1544 d^3 c-1024 d^4\right) b^4\right)+2 a d \left(165 c^3+917 d c^2+2392 d^2 c+516 d^3\right) b^3+30 a^2 d^2 \left(64 c^2+23 d c+22 d^2\right) b^2-30 a^3 d^3 (11 c+3 d) b+45 a^4 d^4\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))}{1920 b^3 d^2 \sqrt{c+d} f}-\frac{\left(-\left(\left(45 c^3-516 c d^2\right) b^3\right)+a d \left(345 c^2+772 d^2\right) b^2+917 a^2 c d^2 b+15 a^3 d^3\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{960 b d f}-\frac{\left(-\left(\left(45 c^4-1692 d^2 c^2-1024 d^4\right) b^4\right)+8 a d \left(45 c^3+791 d^2 c\right) b^3+2 a^2 d^2 \left(1877 c^2+846 d^2\right) b^2+360 a^3 c d^3 b-45 a^4 d^4\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{1920 b d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(-\left(\left(45 c^4-1692 d^2 c^2-1024 d^4\right) b^4\right)+8 a d \left(45 c^3+791 d^2 c\right) b^3+2 a^2 d^2 \left(1877 c^2+846 d^2\right) b^2+360 a^3 c d^3 b-45 a^4 d^4\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))}{1920 b^2 d^2 (b c-a d) f}-\frac{\sqrt{c+d} (b c+a d) \left(-\left(\left(3 c^4+40 d^2 c^2+240 d^4\right) b^4\right)+28 a c d \left(c^2-20 d^2\right) b^3-2 a^2 d^2 \left(89 c^2+20 d^2\right) b^2+28 a^3 c d^3 b-3 a^4 d^4\right) \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))}{128 b^3 \sqrt{a+b} d^3 f}",1,"((-4*(-(b*c) + a*d)*(-15*b^4*c^4 + 3840*a^3*b*c^3*d + 4456*a*b^3*c^3*d + 14702*a^2*b^2*c^2*d^2 + 3236*b^4*c^2*d^2 + 4456*a^3*b*c*d^3 + 10440*a*b^3*c*d^3 - 15*a^4*d^4 + 3236*a^2*b^2*d^4 + 1024*b^4*d^4)*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)*(-60*a*b^3*c^4 + 6364*a^2*b^2*c^3*d + 2292*b^4*c^3*d + 6364*a^3*b*c^2*d^2 + 17020*a*b^3*c^2*d^2 - 60*a^4*c*d^3 + 17020*a^2*b^2*c*d^3 + 4624*b^4*c*d^3 + 2292*a^3*b*d^4 + 4624*a*b^3*d^4)*((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*(45*b^4*c^4 - 360*a*b^3*c^3*d - 3754*a^2*b^2*c^2*d^2 - 1692*b^4*c^2*d^2 - 360*a^3*b*c*d^3 - 6328*a*b^3*c*d^3 + 45*a^4*d^4 - 1692*a^2*b^2*d^4 - 1024*b^4*d^4)*((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)))/(3840*b*d*f) + (Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*(-1/960*((15*b^3*c^3 + 1289*a*b^2*c^2*d + 1289*a^2*b*c*d^2 + 898*b^3*c*d^2 + 15*a^3*d^3 + 898*a*b^2*d^3)*Cos[e + f*x])/(b*d) + (21*b*d*(b*c + a*d)*Cos[3*(e + f*x)])/160 - ((93*b^2*c^2 + 362*a*b*c*d + 93*a^2*d^2 + 88*b^2*d^2)*Sin[2*(e + f*x)])/480 + (b^2*d^2*Sin[4*(e + f*x)])/40))/f","A",0
778,1,2091,1071,7.6026384,"\int (a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2} \, dx","Integrate[(a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac{b (3 b c-17 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac{(a+b)^{3/2} \left(\left(9 c^3-6 d c^2-156 d^2 c-72 d^3\right) b^3-a d \left(51 c^2+172 d c+212 d^2\right) b^2-15 a^2 d^2 (11 c+2 d) b+15 a^3 d^3\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))}{192 b^2 d^2 \sqrt{c+d} f}-\frac{\left(-9 \left(c^2-4 d^2\right) b^2+54 a c d b+59 a^2 d^2\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{96 d f}-\frac{\left(-\left(\left(9 c^3-156 c d^2\right) b^3\right)+a d \left(57 c^2+284 d^2\right) b^2+337 a^2 c d^2 b+15 a^3 d^3\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(-\left(\left(9 c^3-156 c d^2\right) b^3\right)+a d \left(57 c^2+284 d^2\right) b^2+337 a^2 c d^2 b+15 a^3 d^3\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))}{192 b d^2 (b c-a d) f}+\frac{\sqrt{c+d} \left(3 \left(c^2+4 d^2\right)^2 b^4-20 a c d \left(c^2-12 d^2\right) b^3+30 a^2 d^2 \left(3 c^2+4 d^2\right) b^2+60 a^3 c d^3 b-5 a^4 d^4\right) \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))}{64 b^2 \sqrt{a+b} d^3 f}",1,"((-4*(-(b*c) + a*d)*(-3*b^3*c^3 + 384*a^3*c^2*d + 451*a*b^2*c^2*d + 971*a^2*b*c*d^2 + 228*b^3*c*d^2 + 133*a^3*d^3 + 356*a*b^2*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)*(-12*a*b^2*c^3 + 664*a^2*b*c^2*d + 228*b^3*c^2*d + 532*a^3*c*d^2 + 1160*a*b^2*c*d^2 + 644*a^2*b*d^3 + 144*b^3*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*(9*b^3*c^3 - 57*a*b^2*c^2*d - 337*a^2*b*c*d^2 - 156*b^3*c*d^2 - 15*a^3*d^3 - 284*a*b^2*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)))/(384*d*f) + (Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*(-1/96*((3*b^2*c^2 + 122*a*b*c*d + 59*a^2*d^2 + 42*b^2*d^2)*Cos[e + f*x])/d + (b^2*d*Cos[3*(e + f*x)])/16 - (b*(9*b*c + 17*a*d)*Sin[2*(e + f*x)])/48))/f","A",0
779,1,1979,894,7.1048902,"\int (a+b \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)} \, dx","Integrate[(a + b*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]],x]","\frac{-\frac{4 (a d-b c) \left(48 c d a^3+59 b d^2 a^2+58 b^2 c d a-b^3 c^2+16 b^3 d^2\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(48 d^2 a^3+92 b c d a^2-4 b^2 c^2 a+76 b^2 d^2 a+28 b^3 c d\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(3 c^2 b^3-16 d^2 b^3-14 a c d b^2-33 a^2 d^2 b\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)}{48 d f}+\frac{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \left(-\frac{1}{6} \sin (2 (e+f x)) b^2-\frac{(b c+13 a d) \cos (e+f x) b}{12 d}\right)}{f}","-\frac{\cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} b^2}{3 d f}+\frac{(3 b c-13 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} b}{12 d f}-\frac{\left(-\left(\left(3 c^2-16 d^2\right) b^2\right)+14 a c d b+33 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)} b}{24 d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(-\left(\left(3 c^2-16 d^2\right) b^2\right)+14 a c d b+33 a^2 d^2\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))}{24 d^2 (b c-a d) f}+\frac{\sqrt{c+d} \left(\left(c^3+4 d^2 c\right) b^3-5 a d \left(c^2-4 d^2\right) b^2+15 a^2 c d^2 b+5 a^3 d^3\right) \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))}{8 \sqrt{a+b} d^3 f b}+\frac{(a+b)^{3/2} \left(-\left(\left(3 c^2-2 d c-16 d^2\right) b^2\right)+6 a d (2 c+3 d) b+15 a^2 d^2\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))}{24 d^2 \sqrt{c+d} f b}",1,"((-4*(-(b*c) + a*d)*(-(b^3*c^2) + 48*a^3*c*d + 58*a*b^2*c*d + 59*a^2*b*d^2 + 16*b^3*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)*(-4*a*b^2*c^2 + 92*a^2*b*c*d + 28*b^3*c*d + 48*a^3*d^2 + 76*a*b^2*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*(3*b^3*c^2 - 14*a*b^2*c*d - 33*a^2*b*d^2 - 16*b^3*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)))/(48*d*f) + (Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*(-1/12*(b*(b*c + 13*a*d)*Cos[e + f*x])/d - (b^2*Sin[2*(e + f*x)])/6))/f","B",0
780,1,1894,745,10.2463684,"\int \frac{(a+b \sin (e+f x))^{5/2}}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(a + b*Sin[e + f*x])^(5/2)/Sqrt[c + d*Sin[e + f*x]],x]","\frac{-\frac{4 (a d-b c) \left(8 d a^3+11 b^2 d a-b^3 c\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(4 d b^3-4 a c b^2+24 a^2 d b\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(3 b^3 c-9 a b^2 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)}{8 d f}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{2 d f}","-\frac{\sqrt{c+d} \left(-15 a^2 d^2+10 a b c d-\left(b^2 \left(3 c^2+4 d^2\right)\right)\right) \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))}} \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)}{4 d^3 f \sqrt{a+b}}+\frac{3 b^2 (b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 d^2 f \sqrt{a+b \sin (e+f x)}}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{2 d f}-\frac{(a+b)^{3/2} (-7 a d+3 b c-2 b d) \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)}{4 d^2 f \sqrt{c+d}}-\frac{3 b \sqrt{a+b} (c-d) \sqrt{c+d} (b c-3 a d) \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)}{4 d^2 f (b c-a d)}",1,"-1/2*(b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(d*f) + ((-4*(-(b*c) + a*d)*(-(b^3*c) + 8*a^3*d + 11*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)*(-4*a*b^2*c + 24*a^2*b*d + 4*b^3*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*(3*b^3*c - 9*a*b^2*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)))/(8*d*f)","B",0
781,1,2006,780,6.8138794,"\int \frac{(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[(a + b*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(3/2),x]","\text{Result too large to show}","\frac{b \left(-2 a^2 d^2+4 a b c d-\left(b^2 \left(3 c^2-d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{d^2 f \left(c^2-d^2\right) \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{a+b} \left(-2 a^2 d^2+4 a b c d-\left(b^2 \left(3 c^2-d^2\right)\right)\right) \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)}{d^2 f \sqrt{c+d} (b c-a d)}+\frac{2 (b c-a d)^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}-\frac{b \sqrt{c+d} (3 b c-5 a d) \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))}} \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)}{d^3 f \sqrt{a+b}}-\frac{(a+b)^{3/2} (2 a d-b (3 c+d)) \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)}{d^2 f (c+d)^{3/2}}",1,"(-2*(b^2*c^2*Cos[e + f*x] - 2*a*b*c*d*Cos[e + f*x] + a^2*d^2*Cos[e + f*x])*Sqrt[a + b*Sin[e + f*x]])/(d*(-c^2 + d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - ((-4*(-(b*c) + a*d)*(-(b^3*c^2) - 2*a^3*c*d - 2*a*b^2*c*d + 4*a^2*b*d^2 + b^3*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)*(-4*a*b^2*c^2 + 2*a^2*b*c*d - 2*b^3*c*d - 2*a^3*d^2 + 6*a*b^2*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*(3*b^3*c^2 - 4*a*b^2*c*d + 2*a^2*b*d^2 - b^3*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*(c - d)*d*(c + d)*f)","B",0
782,1,2169,737,6.9444724,"\int \frac{(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx","Integrate[(a + b*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 \sqrt{a+b} \left(a^2 d^2 (3 c+d)+a b d \left(3 c^2-4 c d-7 d^2\right)+b^2 \left(3 c^3-6 c^2 d-2 c d^2+9 d^3\right)\right) \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)}{3 d^3 f (c-d)^2 (c+d)^{3/2}}+\frac{2 b^2 \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)}{d^3 f \sqrt{c+d}}+\frac{2 (b c-a d)^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}+\frac{2 (a-b) \sqrt{a+b} \left(4 a c d+3 b c^2-7 b d^2\right) \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))}} 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)}{3 d^2 f (c-d)^2 (c+d)^{3/2}}",1,"(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*(b^2*c^2*Cos[e + f*x] - 2*a*b*c*d*Cos[e + f*x] + a^2*d^2*Cos[e + f*x]))/(3*d*(-c^2 + d^2)*(c + d*Sin[e + f*x])^2) - (2*(3*b^2*c^3*Cos[e + f*x] + a*b*c^2*d*Cos[e + f*x] - 4*a^2*c*d^2*Cos[e + f*x] - 7*b^2*c*d^2*Cos[e + f*x] + 7*a*b*d^3*Cos[e + f*x]))/(3*d*(-c^2 + d^2)^2*(c + d*Sin[e + f*x]))))/f + ((-4*(-(b*c) + a*d)*(-(b^3*c^3) + 3*a^3*c^2*d + 2*a*b^2*c^2*d - 8*a^2*b*c*d^2 + b^3*c*d^2 + a^3*d^3 + 2*a*b^2*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)*(-4*a*b^2*c^3 + 3*a^2*b*c^2*d + b^3*c^2*d + 4*a^3*c*d^2 - 7*a^2*b*d^3 + 3*b^3*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*(3*b^3*c^3 + a*b^2*c^2*d - 4*a^2*b*c*d^2 - 7*b^3*c*d^2 + 7*a*b^2*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)))/(3*(c - d)^2*d*(c + d)^2*f)","B",0
783,1,1894,772,10.3931213,"\int \frac{(c+d \sin (e+f x))^{5/2}}{\sqrt{a+b \sin (e+f x)}} \, dx","Integrate[(c + d*Sin[e + f*x])^(5/2)/Sqrt[a + b*Sin[e + f*x]],x]","\frac{-\frac{4 (a d-b c) \left(8 b c^3+11 b d^2 c-a d^3\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(4 b d^3-4 a c d^2+24 b c^2 d\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(3 a d^3-9 b c d^2\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)}{8 b f}-\frac{d^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{2 b f}","\frac{\sqrt{a+b} \left(3 a^2 d^2-a b d (7 c+3 d)+b^2 \left(8 c^2+9 c d+2 d^2\right)\right) \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)}{4 b^3 f \sqrt{c+d}}-\frac{\sqrt{c+d} \left(-3 a^2 d^2+10 a b c d-\left(b^2 \left(15 c^2+4 d^2\right)\right)\right) \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))}} \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)}{4 b^3 f \sqrt{a+b}}+\frac{3 d \sqrt{a+b} (c-d) \sqrt{c+d} (3 b c-a d) \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)}{4 b^2 f (b c-a d)}-\frac{d^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{2 b f}-\frac{3 d (3 b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 b f \sqrt{a+b \sin (e+f x)}}",1,"-1/2*(d^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(b*f) + ((-4*(-(b*c) + a*d)*(8*b*c^3 + 11*b*c*d^2 - a*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)*(24*b*c^2*d - 4*a*c*d^2 + 4*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*(-9*b*c*d^2 + 3*a*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)))/(8*b*f)","B",0
784,1,222963,644,32.4557215,"\int \frac{(c+d \sin (e+f x))^{3/2}}{\sqrt{a+b \sin (e+f x)}} \, dx","Integrate[(c + d*Sin[e + f*x])^(3/2)/Sqrt[a + b*Sin[e + f*x]],x]","\text{Result too large to show}","-\frac{\sqrt{a+b} (a d-b (2 c+d)) \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^2 f \sqrt{c+d}}+\frac{\sqrt{c+d} (3 b c-a d) \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))}} \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)}{b^2 f \sqrt{a+b}}-\frac{d \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a+b \sin (e+f x)}}+\frac{d \sqrt{a+b} (c-d) \sqrt{c+d} \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 (b c-a d)}",1,"Result too large to show","C",0
785,1,197,198,0.275296,"\int \frac{\sqrt{c+d \sin (e+f x)}}{\sqrt{a+b \sin (e+f x)}} \, dx","Integrate[Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]],x]","\frac{2 \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{(a d-b c) (\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 f \sqrt{c+d}}","\frac{2 \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 f \sqrt{c+d}}",1,"(2*Sqrt[a + b]*EllipticPi[((a + b)*d)/(b*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[((-(b*c) + a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x]))]*(c + d*Sin[e + f*x]))/(b*Sqrt[c + d]*f)","A",1
786,1,191,192,0.2355318,"\int \frac{1}{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]),x]","\frac{2 \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{(a d-b c) (\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{c+d} (b c-a d)}","\frac{2 \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))}} 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{c+d} (b c-a d)}",1,"(2*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[((-(b*c) + a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x]))]*(c + d*Sin[e + f*x]))/(Sqrt[c + d]*(b*c - a*d)*f)","A",1
787,1,90261,405,32.6585629,"\int \frac{1}{\sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx","Integrate[1/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)),x]","\text{Result too large to show}","\frac{2 \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))}} 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 (c-d) \sqrt{c+d} (b c-a d)}+\frac{2 d (a-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))}} 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 (c-d) \sqrt{c+d} (b c-a d)^2}",1,"Result too large to show","B",0
788,1,2102,521,6.7823342,"\int \frac{1}{\sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}} \, dx","Integrate[1/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)),x]","\text{Result too large to show}","-\frac{2 d^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f \left(c^2-d^2\right) (b c-a d) (c+d \sin (e+f x))^{3/2}}-\frac{2 \sqrt{a+b} \left(a d (3 c+d)-b \left(3 c^2+3 c d-2 d^2\right)\right) \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)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)^2}-\frac{4 d (a-b) \sqrt{a+b} \left(2 a c d-b \left(3 c^2-d^2\right)\right) \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))}} 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)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)^3}",1,"(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*d^2*Cos[e + f*x])/(3*(b*c - a*d)*(c^2 - d^2)*(c + d*Sin[e + f*x])^2) + (4*(-3*b*c^2*d^2*Cos[e + f*x] + 2*a*c*d^3*Cos[e + f*x] + b*d^4*Cos[e + f*x]))/(3*(b*c - a*d)^2*(c^2 - d^2)^2*(c + d*Sin[e + f*x]))))/f + ((-4*(-(b*c) + a*d)*(3*b^2*c^4 - 6*a*b*c^3*d + 3*a^2*c^2*d^2 - 5*b^2*c^2*d^2 + 2*a*b*c*d^3 + a^2*d^4 + 2*b^2*d^4)*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)*(-6*b^2*c^3*d - 2*a*b*c^2*d^2 + 4*a^2*c*d^3 + 2*b^2*c*d^3 + 2*a*b*d^4)*((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*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 - 2*b^2*d^4)*((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*(c - d)^2*(c + d)^2*(b*c - a*d)^2*f)","B",0
789,1,2005,822,6.8434016,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^{3/2}} \, dx","Integrate[(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x])^(3/2),x]","\text{Result too large to show}","\frac{2 \cos (e+f x) \sqrt{c+d \sin (e+f x)} (b c-a d)^2}{b \left(a^2-b^2\right) f \sqrt{a+b \sin (e+f x)}}+\frac{d \sqrt{c+d} (5 b c-3 a 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(-\left(\left(2 c^2-6 d c-d^2\right) b^2\right)-2 a d (c+3 d) b+3 a^2 d^2\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}+\frac{\left(-\left(\left(2 c^2-d^2\right) b^2\right)+4 a c d b-3 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (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(2 b^2 c^2-4 a b d c+3 a^2 d^2-b^2 d^2\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} f (b c-a d)}",1,"(-2*(b^2*c^2*Cos[e + f*x] - 2*a*b*c*d*Cos[e + f*x] + a^2*d^2*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*b*c^3 - 4*b^2*c^2*d + 2*a*b*c*d^2 + a^2*d^3 - b^2*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)*(2*b^2*c^3 - 2*a*b*c^2*d + 4*a^2*c*d^2 - 6*b^2*c*d^2 + 2*a*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*(-2*b^2*c^2*d + 4*a*b*c*d^2 - 3*a^2*d^3 + b^2*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)))/(2*(a - b)*b*(a + b)*f)","B",0
790,1,1896,600,9.451497,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx","Integrate[(c + d*Sin[e + f*x])^(3/2)/(a + b*Sin[e + f*x])^(3/2),x]","\frac{-\frac{4 (a d-b c) \left(a c^2-b c 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 c^2-b d^2\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 d^2-b c 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) f}-\frac{2 (a d \cos (e+f x)-b c \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} (a d+b (c-2 d)) \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^2 f (a-b) \sqrt{c+d}}+\frac{2 d \sqrt{c+d} \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))}} \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)}{b^2 f \sqrt{a+b}}+\frac{2 (c-d) \sqrt{c+d} \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}}",1,"(-2*(-(b*c*Cos[e + f*x]) + a*d*Cos[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*f*Sqrt[a + b*Sin[e + f*x]]) + ((-4*(-(b*c) + a*d)*(a*c^2 - b*c*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)*(b*c^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]]) - (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*(-(b*c*d) + a*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)))/((a - b)*(a + b)*f)","B",0
791,1,226,409,4.1642391,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+b \sin (e+f x))^{3/2}} \, dx","Integrate[Sqrt[c + d*Sin[e + f*x]]/(a + b*Sin[e + f*x])^(3/2),x]","-\frac{2 \sqrt{2} \cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right) \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \sqrt{c+d \sin (e+f x)} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{a-b}{a+b}} \cos \left(\frac{1}{4} (2 e+2 f x+\pi )\right)}{\sqrt{\frac{a+b \sin (e+f x)}{a+b}}}\right)|\frac{2 (a d-b c)}{(a-b) (c+d)}\right)}{f \sqrt{\frac{a-b}{a+b}} \sqrt{\frac{(a+b) (\sin (e+f x)+1)}{a+b \sin (e+f x)}} (a+b \sin (e+f x))^{3/2} \sqrt{\frac{(a+b) (c+d \sin (e+f x))}{(c+d) (a+b \sin (e+f x))}}}","\frac{2 \sqrt{a+b} (c-d) \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} \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)}",1,"(-2*Sqrt[2]*Cos[(2*e - Pi + 2*f*x)/4]*EllipticE[ArcSin[(Sqrt[(a - b)/(a + b)]*Cos[(2*e + Pi + 2*f*x)/4])/Sqrt[(a + b*Sin[e + f*x])/(a + b)]], (2*(-(b*c) + a*d))/((a - b)*(c + d))]*Sqrt[(a + b*Sin[e + f*x])/(a + b)]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[(a - b)/(a + b)]*f*Sqrt[((a + b)*(1 + Sin[e + f*x]))/(a + b*Sin[e + f*x])]*(a + b*Sin[e + f*x])^(3/2)*Sqrt[((a + b)*(c + d*Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x]))])","A",1
792,1,90261,405,32.7366487,"\int \frac{1}{(a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[1/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]),x]","\text{Result too large to show}","\frac{2 \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))}} 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 b (c-d) \sqrt{c+d} \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,"Result too large to show","B",0
793,1,2082,495,6.9181325,"\int \frac{1}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}} \, dx","Integrate[1/((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 \left(a^2 d^2+b^2 \left(c^2-2 d^2\right)\right) \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))}} 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^2 \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 (b (c-2 d)-a d) \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 \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*b^3*Cos[e + f*x])/((a^2 - b^2)*(-(b*c) + a*d)^2*(a + b*Sin[e + f*x])) + (2*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*b^2*c^3 - 2*a^2*b*c^2*d + 2*b^3*c^2*d + a^3*c*d^2 - 2*a*b^2*c*d^2 + 2*a^2*b*d^3 - 2*b^3*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)*(b^3*c^3 + a*b^2*c^2*d + a^2*b*c*d^2 - 2*b^3*c*d^2 + a^3*d^3 - 2*a*b^2*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*(-(b^3*c^2*d) - a^2*b*d^3 + 2*b^3*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
794,1,2350,681,7.6219757,"\int \frac{1}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}} \, dx","Integrate[1/((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^2 d^2+b^2 \left(3 c^2-4 d^2\right)\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 \left(a^2 d^2 (3 c+d)-6 a b d \left(c^2-d^2\right)+b^2 \left(3 c^3-9 c^2 d-6 c d^2+8 d^3\right)\right) \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)}{3 f \sqrt{a+b} (c-d)^2 (c+d)^{3/2} (b c-a d)^3}+\frac{2 b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac{2 \left(4 a^3 c d^3-a^2 b d^2 \left(9 c^2-5 d^2\right)-4 a b^2 c d^3-\left(b^3 \left(3 c^4-15 c^2 d^2+8 d^4\right)\right)\right) \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))}} 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)}{3 f \sqrt{a+b} (c-d)^2 (c+d)^{3/2} (b c-a d)^4}",1,"(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*b^4*Cos[e + f*x])/((a^2 - b^2)*(-(b*c) + a*d)^3*(a + b*Sin[e + f*x])) + (2*d^3*Cos[e + f*x])/(3*(b*c - a*d)^2*(c^2 - d^2)*(c + d*Sin[e + f*x])^2) - (2*(-9*b*c^2*d^3*Cos[e + f*x] + 4*a*c*d^4*Cos[e + f*x] + 5*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*b^3*c^5 + 9*a^2*b^2*c^4*d - 9*b^4*c^4*d - 9*a^3*b*c^3*d^2 + 15*a*b^3*c^3*d^2 + 3*a^4*c^2*d^3 - 20*a^2*b^2*c^2*d^3 + 17*b^4*c^2*d^3 + 5*a^3*b*c*d^4 - 8*a*b^3*c*d^4 + a^4*d^5 + 7*a^2*b^2*d^5 - 8*b^4*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*b^4*c^5 - 3*a*b^3*c^4*d - 9*a^2*b^2*c^3*d^2 + 15*b^4*c^3*d^2 - 5*a^3*b*c^2*d^3 + 11*a*b^3*c^2*d^3 + 4*a^4*c*d^4 + a^2*b^2*c*d^4 - 8*b^4*c*d^4 + 5*a^3*b*d^5 - 8*a*b^3*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*b^4*c^4*d + 9*a^2*b^2*c^2*d^3 - 15*b^4*c^2*d^3 - 4*a^3*b*c*d^4 + 4*a*b^3*c*d^4 - 5*a^2*b^2*d^5 + 8*b^4*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
795,1,2172,736,7.0904153,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^{5/2}} \, dx","Integrate[(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x])^(5/2),x]","\text{Result too large to show}","\frac{2 (b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^{3/2}}+\frac{2 (c-d) \sqrt{c+d} \left(3 a^2 d+4 a b c-7 b^2 d\right) \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)}{3 b^2 f (a-b)^2 (a+b)^{3/2}}+\frac{2 \left(3 a^3 d^2+3 a^2 b d (c-2 d)+a b^2 \left(3 c^2-4 c d-2 d^2\right)+b^3 \left(c^2-7 c d+9 d^2\right)\right) \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)}{3 b^3 f (a-b)^2 \sqrt{a+b} \sqrt{c+d}}+\frac{2 d^2 \sqrt{c+d} \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))}} \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)}{b^3 f \sqrt{a+b}}",1,"(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*(b^2*c^2*Cos[e + f*x] - 2*a*b*c*d*Cos[e + f*x] + a^2*d^2*Cos[e + f*x]))/(3*b*(-a^2 + b^2)*(a + b*Sin[e + f*x])^2) - (2*(-4*a*b^2*c^2*Cos[e + f*x] + a^2*b*c*d*Cos[e + f*x] + 7*b^3*c*d*Cos[e + f*x] + 3*a^3*d^2*Cos[e + f*x] - 7*a*b^2*d^2*Cos[e + f*x]))/(3*b*(-a^2 + b^2)^2*(a + b*Sin[e + f*x]))))/f - ((-4*(-(b*c) + a*d)*(-3*a^2*b*c^3 - b^3*c^3 + 8*a*b^2*c^2*d - 2*a^2*b*c*d^2 - 2*b^3*c*d^2 + a^3*d^3 - a*b^2*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)*(-4*a*b^2*c^3 - 3*a^2*b*c^2*d + 7*b^3*c^2*d + 4*a^3*c*d^2 - a^2*b*d^3 - 3*b^3*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*(4*a*b^2*c^2*d - a^2*b*c*d^2 - 7*b^3*c*d^2 - 3*a^3*d^3 + 7*a*b^2*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)))/(3*(a - b)^2*b*(a + b)^2*f)","B",0
796,1,2012,497,6.3489883,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{5/2}} \, dx","Integrate[(c + d*Sin[e + f*x])^(3/2)/(a + b*Sin[e + f*x])^(5/2),x]","\text{Result too large to show}","\frac{2 (b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f \left(a^2-b^2\right) (a+b \sin (e+f x))^{3/2}}+\frac{2 (c-d) (3 a c-a d+b c-3 b d) \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)}{3 f (a-b)^2 \sqrt{a+b} \sqrt{c+d} (b c-a d)}+\frac{8 (c-d) \sqrt{c+d} (a c-b d) \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)}{3 f (a-b)^2 (a+b)^{3/2} (b c-a d)}",1,"(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*(-(b*c*Cos[e + f*x]) + a*d*Cos[e + f*x]))/(3*(a^2 - b^2)*(a + b*Sin[e + f*x])^2) - (8*(-(a*b*c*Cos[e + f*x]) + b^2*d*Cos[e + f*x]))/(3*(a^2 - b^2)^2*(a + b*Sin[e + f*x]))))/f + ((-4*(-(b*c) + a*d)*(3*a^2*c^2 + b^2*c^2 - 4*a*b*c*d + a^2*d^2 - b^2*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)*(4*a*b*c^2 + 4*a^2*c*d - 4*b^2*c*d - 4*a*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*(-4*a*b*c*d + 4*b^2*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)))/(3*(a - b)^2*(a + b)^2*f)","B",0
797,1,2067,489,6.4278408,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+b \sin (e+f x))^{5/2}} \, dx","Integrate[Sqrt[c + d*Sin[e + f*x]]/(a + b*Sin[e + f*x])^(5/2),x]","\text{Result too large to show}","\frac{2 b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f \left(a^2-b^2\right) (a+b \sin (e+f x))^{3/2}}+\frac{2 (c-d) \sqrt{c+d} \left(-3 a^2 d+4 a b c-b^2 d\right) \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)}{3 f (a-b)^2 (a+b)^{3/2} (b c-a d)^2}+\frac{2 (3 a+b) (c-d) \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)}{3 f (a-b)^2 \sqrt{a+b} \sqrt{c+d} (b c-a d)}",1,"(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((2*b*Cos[e + f*x])/(3*(a^2 - b^2)*(a + b*Sin[e + f*x])^2) + (2*(-4*a*b^2*c*Cos[e + f*x] + 3*a^2*b*d*Cos[e + f*x] + b^3*d*Cos[e + f*x]))/(3*(a^2 - b^2)^2*(-(b*c) + a*d)*(a + b*Sin[e + f*x]))))/f + ((-4*(-(b*c) + a*d)*(-3*a^2*b*c^2 - b^3*c^2 + 3*a^3*c*d + a*b^2*c*d - a^2*b*d^2 + b^3*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)*(-4*a*b^2*c^2 - a^2*b*c*d + b^3*c*d + 3*a^3*d^2 + a*b^2*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*(4*a*b^2*c*d - 3*a^2*b*d^2 - b^3*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)))/(3*(a - b)^2*(a + b)^2*(-(b*c) + a*d)*f)","B",0
798,1,2102,516,6.4526292,"\int \frac{1}{(a+b \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[1/((a + b*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]]),x]","\text{Result too large to show}","\frac{2 b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^{3/2}}+\frac{2 \left(-3 a^2 d+3 a b (c-d)+b^2 (c+2 d)\right) \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)}{3 f (a-b)^2 \sqrt{a+b} \sqrt{c+d} (b c-a d)^2}+\frac{4 b (c-d) \sqrt{c+d} \left(-3 a^2 d+2 a b c+b^2 d\right) \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)}{3 f (a-b)^2 (a+b)^{3/2} (b c-a d)^3}",1,"(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*b^2*Cos[e + f*x])/(3*(a^2 - b^2)*(-(b*c) + a*d)*(a + b*Sin[e + f*x])^2) + (4*(2*a*b^3*c*Cos[e + f*x] - 3*a^2*b^2*d*Cos[e + f*x] + b^4*d*Cos[e + f*x]))/(3*(a^2 - b^2)^2*(-(b*c) + a*d)^2*(a + b*Sin[e + f*x]))))/f + ((-4*(-(b*c) + a*d)*(3*a^2*b^2*c^2 + b^4*c^2 - 6*a^3*b*c*d + 2*a*b^3*c*d + 3*a^4*d^2 - 5*a^2*b^2*d^2 + 2*b^4*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)*(4*a*b^3*c^2 - 2*a^2*b^2*c*d + 2*b^4*c*d - 6*a^3*b*d^2 + 2*a*b^3*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*(-4*a*b^3*c*d + 6*a^2*b^2*d^2 - 2*b^4*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)))/(3*(a - b)^2*(a + b)^2*(-(b*c) + a*d)^2*f)","B",0
799,1,2352,688,7.8155527,"\int \frac{1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx","Integrate[1/((a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2)),x]","\text{Result too large to show}","\frac{8 b^2 \left(-2 a^2 d+a b c+b^2 d\right) \cos (e+f x)}{3 f \left(a^2-b^2\right)^2 (b c-a d)^2 \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 b^2 \cos (e+f x)}{3 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(3 a^4 d^3+3 a^2 b^2 d \left(3 c^2-5 d^2\right)-4 a b^3 c \left(c^2-d^2\right)-b^4 d \left(5 c^2-8 d^2\right)\right) \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))}} 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)}{3 f \sqrt{a+b} \left(a^2-b^2\right) (c-d) \sqrt{c+d} (b c-a d)^4}-\frac{2 \left(-3 a^3 d^2+3 a^2 b d (2 c-3 d)-3 a b^2 \left(c^2-2 d^2\right)+b^3 \left(c^2-6 c d+8 d^2\right)\right) \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)}{3 f \sqrt{a+b} \left(a^2-b^2\right) (c-d) \sqrt{c+d} (b c-a d)^3}",1,"(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((2*b^3*Cos[e + f*x])/(3*(a^2 - b^2)*(-(b*c) + a*d)^2*(a + b*Sin[e + f*x])^2) - (2*(4*a*b^4*c*Cos[e + f*x] - 9*a^2*b^3*d*Cos[e + f*x] + 5*b^5*d*Cos[e + f*x]))/(3*(a^2 - b^2)^2*(-(b*c) + a*d)^3*(a + b*Sin[e + f*x])) - (2*d^4*Cos[e + f*x])/((b*c - a*d)^3*(c^2 - d^2)*(c + d*Sin[e + f*x]))))/f + ((-4*(-(b*c) + a*d)*(-3*a^2*b^3*c^4 - b^5*c^4 + 9*a^3*b^2*c^3*d - 5*a*b^4*c^3*d - 9*a^4*b*c^2*d^2 + 20*a^2*b^3*c^2*d^2 - 7*b^5*c^2*d^2 + 3*a^5*c*d^3 - 15*a^3*b^2*c*d^3 + 8*a*b^4*c*d^3 + 9*a^4*b*d^4 - 17*a^2*b^3*d^4 + 8*b^5*d^4)*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)*(-4*a*b^4*c^4 + 5*a^2*b^3*c^3*d - 5*b^5*c^3*d + 9*a^3*b^2*c^2*d^2 - a*b^4*c^2*d^2 + 3*a^4*b*c*d^3 - 11*a^2*b^3*c*d^3 + 8*b^5*c*d^3 + 3*a^5*d^4 - 15*a^3*b^2*d^4 + 8*a*b^4*d^4)*((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*(4*a*b^4*c^3*d - 9*a^2*b^3*c^2*d^2 + 5*b^5*c^2*d^2 - 4*a*b^4*c*d^3 - 3*a^4*b*d^4 + 15*a^2*b^3*d^4 - 8*b^5*d^4)*((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)^2*(a + b)^2*(c - d)*(c + d)*(-(b*c) + a*d)^3*f)","B",0
800,1,2669,941,8.9854287,"\int \frac{1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx","Integrate[1/((a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(5/2)),x]","\text{Result too large to show}","\frac{4 \left(-5 d a^2+2 b c a+3 b^2 d\right) \cos (e+f x) b^2}{3 \left(a^2-b^2\right)^2 (b c-a d)^2 f \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac{2 \cos (e+f x) b^2}{3 \left(a^2-b^2\right) (b c-a d) f (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}-\frac{8 \left(c d^4 a^5-b \left(3 c^2 d^3-2 d^5\right) a^4-2 b^2 c d^4 a^3-b^3 d \left(3 c^4-12 d^2 c^2+7 d^4\right) a^2+b^4 c \left(c^4-2 d^2 c^2+2 d^4\right) a+b^5 d \left(2 c^4-7 d^2 c^2+4 d^4\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} \left(a^2-b^2\right) (c-d)^2 (c+d)^{3/2} (b c-a d)^5 f}-\frac{2 \left(d^3 (3 c+d) a^4-9 b d^2 \left(c^2-d^2\right) a^3+b^2 d \left(9 c^3-18 d c^2-15 d^2 c+16 d^3\right) a^2-3 b^3 \left(c^4-5 d^2 c^2+4 d^4\right) a+b^4 \left(c^4-9 d c^3+16 d^2 c^2+12 d^3 c-16 d^4\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} \left(a^2-b^2\right) (c-d)^2 (c+d)^{3/2} (b c-a d)^4 f}-\frac{2 d \left(d^3 a^4+b^2 d \left(11 c^2-13 d^2\right) a^2-4 b^3 c \left(c^2-d^2\right) a-b^4 d \left(7 c^2-8 d^2\right)\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 \left(a^2-b^2\right)^2 (b c-a d)^3 \left(c^2-d^2\right) f (c+d \sin (e+f x))^{3/2}}",1,"(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*b^4*Cos[e + f*x])/(3*(a^2 - b^2)*(-(b*c) + a*d)^3*(a + b*Sin[e + f*x])^2) + (8*(a*b^5*c*Cos[e + f*x] - 3*a^2*b^4*d*Cos[e + f*x] + 2*b^6*d*Cos[e + f*x]))/(3*(a^2 - b^2)^2*(-(b*c) + a*d)^4*(a + b*Sin[e + f*x])) - (2*d^4*Cos[e + f*x])/(3*(b*c - a*d)^3*(c^2 - d^2)*(c + d*Sin[e + f*x])^2) + (8*(-3*b*c^2*d^4*Cos[e + f*x] + a*c*d^5*Cos[e + f*x] + 2*b*d^6*Cos[e + f*x]))/(3*(b*c - a*d)^4*(c^2 - d^2)^2*(c + d*Sin[e + f*x]))))/f + ((-4*(-(b*c) + a*d)*(3*a^2*b^4*c^6 + b^6*c^6 - 12*a^3*b^3*c^5*d + 8*a*b^5*c^5*d + 18*a^4*b^2*c^4*d^2 - 41*a^2*b^4*c^4*d^2 + 15*b^6*c^4*d^2 - 12*a^5*b*c^3*d^3 + 48*a^3*b^3*c^3*d^3 - 28*a*b^5*c^3*d^3 + 3*a^6*c^2*d^4 - 41*a^4*b^2*c^2*d^4 + 74*a^2*b^4*c^2*d^4 - 32*b^6*c^2*d^4 + 8*a^5*b*c*d^5 - 28*a^3*b^3*c*d^5 + 16*a*b^5*c*d^5 + a^6*d^6 + 15*a^4*b^2*d^6 - 32*a^2*b^4*d^6 + 16*b^6*d^6)*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)*(4*a*b^5*c^6 - 8*a^2*b^4*c^5*d + 8*b^6*c^5*d - 12*a^3*b^3*c^4*d^2 - 12*a^4*b^2*c^3*d^3 + 40*a^2*b^4*c^3*d^3 - 28*b^6*c^3*d^3 - 8*a^5*b*c^2*d^4 + 40*a^3*b^3*c^2*d^4 - 20*a*b^5*c^2*d^4 + 4*a^6*c*d^5 - 20*a^2*b^4*c*d^5 + 16*b^6*c*d^5 + 8*a^5*b*d^6 - 28*a^3*b^3*d^6 + 16*a*b^5*d^6)*((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*(-4*a*b^5*c^5*d + 12*a^2*b^4*c^4*d^2 - 8*b^6*c^4*d^2 + 8*a*b^5*c^3*d^3 + 12*a^4*b^2*c^2*d^4 - 48*a^2*b^4*c^2*d^4 + 28*b^6*c^2*d^4 - 4*a^5*b*c*d^5 + 8*a^3*b^3*c*d^5 - 8*a*b^5*c*d^5 - 8*a^4*b^2*d^6 + 28*a^2*b^4*d^6 - 16*b^6*d^6)*((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)^2*(a + b)^2*(c - d)^2*(c + d)^2*(-(b*c) + a*d)^4*f)","B",0
801,0,0,28,3.2527756,"\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Integrate[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]","\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","\text{Int}\left((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*(c + d*Sin[e + f*x])^n, x]","A",-1
802,0,0,311,18.7268811,"\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx","Integrate[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2,x]","\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx","-\frac{\sqrt{2} \cos (e+f x) \left(a d (a d-2 b c (m+2))+b^2 \left(c^2 (m+2)+d^2 (m+1)\right)\right) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\sin (e+f x)+1}}+\frac{\sqrt{2} d (a+b) \cos (e+f x) (a d-2 b c (m+2)) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\sin (e+f x)+1}}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{m+1}}{b f (m+2)}",1,"Integrate[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2, x]","F",-1
803,1,200,229,0.5710403,"\int (a+b \sin (e+f x))^m (c+d \sin (e+f x)) \, dx","Integrate[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]),x]","\frac{\sec (e+f x) \sqrt{-\frac{b (\sin (e+f x)-1)}{a+b}} \sqrt{\frac{b (\sin (e+f x)+1)}{b-a}} (a+b \sin (e+f x))^{m+1} \left((m+2) (b c-a d) F_1\left(m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{a+b \sin (e+f x)}{a-b},\frac{a+b \sin (e+f x)}{a+b}\right)+d (m+1) (a+b \sin (e+f x)) F_1\left(m+2;\frac{1}{2},\frac{1}{2};m+3;\frac{a+b \sin (e+f x)}{a-b},\frac{a+b \sin (e+f x)}{a+b}\right)\right)}{b^2 f (m+1) (m+2)}","-\frac{\sqrt{2} (b c-a d) \cos (e+f x) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{b f \sqrt{\sin (e+f x)+1}}-\frac{\sqrt{2} d (a+b) \cos (e+f x) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{b f \sqrt{\sin (e+f x)+1}}",1,"(Sec[e + f*x]*Sqrt[-((b*(-1 + Sin[e + f*x]))/(a + b))]*Sqrt[(b*(1 + Sin[e + f*x]))/(-a + b)]*(a + b*Sin[e + f*x])^(1 + m)*((b*c - a*d)*(2 + m)*AppellF1[1 + m, 1/2, 1/2, 2 + m, (a + b*Sin[e + f*x])/(a - b), (a + b*Sin[e + f*x])/(a + b)] + d*(1 + m)*AppellF1[2 + m, 1/2, 1/2, 3 + m, (a + b*Sin[e + f*x])/(a - b), (a + b*Sin[e + f*x])/(a + b)]*(a + b*Sin[e + f*x])))/(b^2*f*(1 + m)*(2 + m))","A",0
804,1,120,104,0.2552967,"\int (a+b \sin (e+f x))^m \, dx","Integrate[(a + b*Sin[e + f*x])^m,x]","\frac{\sec (e+f x) \sqrt{-\frac{b (\sin (e+f x)-1)}{a+b}} \sqrt{\frac{b (\sin (e+f x)+1)}{b-a}} (a+b \sin (e+f x))^{m+1} F_1\left(m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{a+b \sin (e+f x)}{a-b},\frac{a+b \sin (e+f x)}{a+b}\right)}{b f (m+1)}","-\frac{\sqrt{2} \cos (e+f x) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{f \sqrt{\sin (e+f x)+1}}",1,"(AppellF1[1 + m, 1/2, 1/2, 2 + m, (a + b*Sin[e + f*x])/(a - b), (a + b*Sin[e + f*x])/(a + b)]*Sec[e + f*x]*Sqrt[-((b*(-1 + Sin[e + f*x]))/(a + b))]*Sqrt[(b*(1 + Sin[e + f*x]))/(-a + b)]*(a + b*Sin[e + f*x])^(1 + m))/(b*f*(1 + m))","A",0
805,0,0,28,2.6483026,"\int \frac{(a+b \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx","Integrate[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]),x]","\int \frac{(a+b \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx","\text{Int}\left(\frac{(a+b \sin (e+f x))^m}{c+d \sin (e+f x)},x\right)",0,"Integrate[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x]","A",-1
806,0,0,28,4.8482475,"\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx","Integrate[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^2,x]","\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx","\text{Int}\left(\frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2},x\right)",0,"Integrate[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^2, x]","A",-1
807,0,0,28,15.869946,"\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx","Integrate[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^3,x]","\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx","\text{Int}\left(\frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^3},x\right)",0,"Integrate[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^3, x]","A",-1
808,0,0,30,34.7106918,"\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx","Integrate[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2),x]","\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx","\text{Int}\left((c+d \sin (e+f x))^{5/2} (a+b \sin (e+f x))^m,x\right)",0,"Integrate[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2), x]","A",-1
809,0,0,30,13.6156386,"\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \, dx","Integrate[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2),x]","\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \, dx","\text{Int}\left((c+d \sin (e+f x))^{3/2} (a+b \sin (e+f x))^m,x\right)",0,"Integrate[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2), x]","A",-1
810,0,0,30,0.4423796,"\int (a+b \sin (e+f x))^m \sqrt{c+d \sin (e+f x)} \, dx","Integrate[(a + b*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]],x]","\int (a+b \sin (e+f x))^m \sqrt{c+d \sin (e+f x)} \, dx","\text{Int}\left(\sqrt{c+d \sin (e+f x)} (a+b \sin (e+f x))^m,x\right)",0,"Integrate[(a + b*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]], x]","A",-1
811,0,0,30,3.3344881,"\int \frac{(a+b \sin (e+f x))^m}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(a + b*Sin[e + f*x])^m/Sqrt[c + d*Sin[e + f*x]],x]","\int \frac{(a+b \sin (e+f x))^m}{\sqrt{c+d \sin (e+f x)}} \, dx","\text{Int}\left(\frac{(a+b \sin (e+f x))^m}{\sqrt{c+d \sin (e+f x)}},x\right)",0,"Integrate[(a + b*Sin[e + f*x])^m/Sqrt[c + d*Sin[e + f*x]], x]","A",-1
812,0,0,30,4.5975466,"\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx","Integrate[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2),x]","\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx","\text{Int}\left(\frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}},x\right)",0,"Integrate[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2), x]","A",-1
813,0,0,30,9.6912959,"\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx","Integrate[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(5/2),x]","\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx","\text{Int}\left(\frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}},x\right)",0,"Integrate[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(5/2), x]","A",-1
814,1,493,272,11.572712,"\int (d \csc (e+f x))^n (a+a \sin (e+f x))^3 \, dx","Integrate[(d*Csc[e + f*x])^n*(a + a*Sin[e + f*x])^3,x]","\frac{2^{1-n} \tan \left(\frac{1}{2} (e+f x)\right) (a \sin (e+f x)+a)^3 \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)^{-n} \csc ^{-n}(e+f x) (d \csc (e+f x))^n \left(-\frac{15 \tan ^2\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(\frac{3-n}{2},4-n;\frac{5-n}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n-3}-\frac{6 \tan \left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(4-n,1-\frac{n}{2};2-\frac{n}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n-2}+\frac{\, _2F_1\left(4-n,\frac{1}{2}-\frac{n}{2};\frac{3}{2}-\frac{n}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{1-n}+\frac{\tan ^6\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(4-n,\frac{7}{2}-\frac{n}{2};\frac{9}{2}-\frac{n}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{7-n}-\frac{6 \tan ^5\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(4-n,3-\frac{n}{2};4-\frac{n}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n-6}-\frac{15 \tan ^4\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(4-n,\frac{5-n}{2};\frac{7-n}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n-5}-\frac{20 \tan ^3\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(4-n,2-\frac{n}{2};3-\frac{n}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n-4}\right) \left(\tan \left(\frac{1}{2} (e+f x)\right)+\cot \left(\frac{1}{2} (e+f x)\right)\right)^n}{f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^6}","\frac{a^3 d^4 (11-4 n) \cos (e+f x) (d \csc (e+f x))^{n-4} \, _2F_1\left(\frac{1}{2},\frac{4-n}{2};\frac{6-n}{2};\sin ^2(e+f x)\right)}{f (2-n) (4-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 d^3 (5-4 n) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right)}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (1-n) (2-n)}+\frac{d^3 \cot (e+f x) \left(a^3 \csc (e+f x)+a^3\right) (d \csc (e+f x))^{n-3}}{f (1-n)}",1,"(2^(1 - n)*(d*Csc[e + f*x])^n*(a + a*Sin[e + f*x])^3*Tan[(e + f*x)/2]*(Cot[(e + f*x)/2] + Tan[(e + f*x)/2])^n*(Hypergeometric2F1[4 - n, 1/2 - n/2, 3/2 - n/2, -Tan[(e + f*x)/2]^2]/(1 - n) - (6*Hypergeometric2F1[4 - n, 1 - n/2, 2 - n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(-2 + n) - (15*Hypergeometric2F1[(3 - n)/2, 4 - n, (5 - n)/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)/(-3 + n) - (20*Hypergeometric2F1[4 - n, 2 - n/2, 3 - n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^3)/(-4 + n) - (15*Hypergeometric2F1[4 - n, (5 - n)/2, (7 - n)/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^4)/(-5 + n) - (6*Hypergeometric2F1[4 - n, 3 - n/2, 4 - n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^5)/(-6 + n) + (Hypergeometric2F1[4 - n, 7/2 - n/2, 9/2 - n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^6)/(7 - n)))/(f*Csc[e + f*x]^n*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(1 + Tan[(e + f*x)/2]^2)^n)","A",0
815,1,342,203,6.34339,"\int (d \csc (e+f x))^n (a+a \sin (e+f x))^2 \, dx","Integrate[(d*Csc[e + f*x])^n*(a + a*Sin[e + f*x])^2,x]","\frac{2 \tan \left(\frac{1}{2} (e+f x)\right) (a \sin (e+f x)+a)^2 \sec ^2\left(\frac{1}{2} (e+f x)\right)^{-n} (d \csc (e+f x))^n \left(\frac{\, _2F_1\left(3-n,\frac{1}{2}-\frac{n}{2};\frac{3}{2}-\frac{n}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{1-n}+\tan \left(\frac{1}{2} (e+f x)\right) \left(\tan \left(\frac{1}{2} (e+f x)\right) \left(\frac{\tan ^2\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(3-n,\frac{5}{2}-\frac{n}{2};\frac{7}{2}-\frac{n}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{5-n}-\frac{4 \tan \left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(3-n,2-\frac{n}{2};3-\frac{n}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n-4}-\frac{6 \, _2F_1\left(\frac{3-n}{2},3-n;\frac{5-n}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{n-3}\right)-\frac{4 \, _2F_1\left(3-n,1-\frac{n}{2};2-\frac{n}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\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)^4}","\frac{a^2 d^3 (3-2 n) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right)}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{2 a^2 d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right)}{f (2-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}",1,"(2*(d*Csc[e + f*x])^n*(a + a*Sin[e + f*x])^2*Tan[(e + f*x)/2]*(Hypergeometric2F1[3 - n, 1/2 - n/2, 3/2 - n/2, -Tan[(e + f*x)/2]^2]/(1 - n) + Tan[(e + f*x)/2]*((-4*Hypergeometric2F1[3 - n, 1 - n/2, 2 - n/2, -Tan[(e + f*x)/2]^2])/(-2 + n) + Tan[(e + f*x)/2]*((-6*Hypergeometric2F1[(3 - n)/2, 3 - n, (5 - n)/2, -Tan[(e + f*x)/2]^2])/(-3 + n) - (4*Hypergeometric2F1[3 - n, 2 - n/2, 3 - n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(-4 + n) + (Hypergeometric2F1[3 - n, 5/2 - n/2, 7/2 - n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)/(5 - n)))))/(f*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)","A",1
816,1,280,149,1.6570622,"\int (d \csc (e+f x))^n (a+a \sin (e+f x)) \, dx","Integrate[(d*Csc[e + f*x])^n*(a + a*Sin[e + f*x]),x]","\frac{a 2^{n-1} \left(-1+e^{2 i (e+f x)}\right) e^{-i (e+f n x)} \left(\frac{i e^{i (e+f x)}}{-1+e^{2 i (e+f x)}}\right)^n (\csc (e+f x)+1) \left(e^{i e} (n-1) \left(n e^{i (e+f (n+1) x)} \, _2F_1\left(1,\frac{3-n}{2};\frac{n+3}{2};e^{2 i (e+f x)}\right)+2 i (n+1) e^{i f n x} \, _2F_1\left(1,1-\frac{n}{2};\frac{n+2}{2};e^{2 i (e+f x)}\right)\right)-n (n+1) e^{i f (n-1) x} \, _2F_1\left(1,\frac{1-n}{2};\frac{n+1}{2};e^{2 i (e+f x)}\right)\right) \csc ^{-n-1}(e+f x) (d \csc (e+f x))^n}{f (n-1) n (n+1) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","\frac{a d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right)}{f (2-n) \sqrt{\cos ^2(e+f x)}}+\frac{a d \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right)}{f (1-n) \sqrt{\cos ^2(e+f x)}}",1,"(2^(-1 + n)*a*((I*E^(I*(e + f*x)))/(-1 + E^((2*I)*(e + f*x))))^n*(-1 + E^((2*I)*(e + f*x)))*Csc[e + f*x]^(-1 - n)*(d*Csc[e + f*x])^n*(1 + Csc[e + f*x])*(-(E^(I*f*(-1 + n)*x)*n*(1 + n)*Hypergeometric2F1[1, (1 - n)/2, (1 + n)/2, E^((2*I)*(e + f*x))]) + E^(I*e)*(-1 + n)*(E^(I*(e + f*(1 + n)*x))*n*Hypergeometric2F1[1, (3 - n)/2, (3 + n)/2, E^((2*I)*(e + f*x))] + (2*I)*E^(I*f*n*x)*(1 + n)*Hypergeometric2F1[1, 1 - n/2, (2 + n)/2, E^((2*I)*(e + f*x))])))/(E^(I*(e + f*n*x))*f*(-1 + n)*n*(1 + n)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)","C",0
817,0,0,171,2.8004776,"\int \frac{(d \csc (e+f x))^n}{a+a \sin (e+f x)} \, dx","Integrate[(d*Csc[e + f*x])^n/(a + a*Sin[e + f*x]),x]","\int \frac{(d \csc (e+f x))^n}{a+a \sin (e+f x)} \, dx","\frac{d n \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right)}{a f (1-n) \sqrt{\cos ^2(e+f x)}}+\frac{\cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\sin ^2(e+f x)\right)}{a f \sqrt{\cos ^2(e+f x)}}-\frac{\cot (e+f x) (d \csc (e+f x))^n}{f (a \csc (e+f x)+a)}",1,"Integrate[(d*Csc[e + f*x])^n/(a + a*Sin[e + f*x]), x]","F",-1
818,0,0,231,4.7939447,"\int \frac{(d \csc (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","Integrate[(d*Csc[e + f*x])^n/(a + a*Sin[e + f*x])^2,x]","\int \frac{(d \csc (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","\frac{2 n \cos (e+f x) (d \csc (e+f x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-n-2);-\frac{n}{2};\sin ^2(e+f x)\right)}{3 a^2 d^2 f \sqrt{\cos ^2(e+f x)}}-\frac{2 n \cot (e+f x) (d \csc (e+f x))^{n+2}}{3 a^2 d^2 f (\csc (e+f x)+1)}-\frac{(2 n+1) \cos (e+f x) (d \csc (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-n-1);\frac{1-n}{2};\sin ^2(e+f x)\right)}{3 a^2 d f \sqrt{\cos ^2(e+f x)}}+\frac{\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 d^2 f (a \csc (e+f x)+a)^2}",1,"Integrate[(d*Csc[e + f*x])^n/(a + a*Sin[e + f*x])^2, x]","F",-1
819,1,2967,113,15.157881,"\int \left(c (d \sin (e+f x))^p\right)^n (a+a \sin (e+f x))^m \, dx","Integrate[(c*(d*Sin[e + f*x])^p)^n*(a + a*Sin[e + f*x])^m,x]","\text{Result too large to show}","-\frac{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 \sin ^{-n p}(e+f x) F_1\left(\frac{1}{2};-n p,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) \left(c (d \sin (e+f x))^p\right)^n}{f}",1,"(-3*AppellF1[1/2, -(n*p), 1 + m + n*p, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^(n*p)*(c*(d*Sin[e + f*x])^p)^n*(a + a*Sin[e + f*x])^m)/(f*(Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -(n*p), 1 + m + n*p, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*((1 + m + n*p)*AppellF1[3/2, -(n*p), 2 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + n*p*AppellF1[3/2, 1 - n*p, 1 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)*((-3*n*p*AppellF1[1/2, -(n*p), 1 + m + n*p, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]^2*Sin[e + f*x]^(-1 + n*p))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -(n*p), 1 + m + n*p, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*((1 + m + n*p)*AppellF1[3/2, -(n*p), 2 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + n*p*AppellF1[3/2, 1 - n*p, 1 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) + (3*AppellF1[1/2, -(n*p), 1 + m + n*p, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sin[e + f*x]^(1 + n*p))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -(n*p), 1 + m + n*p, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*((1 + m + n*p)*AppellF1[3/2, -(n*p), 2 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + n*p*AppellF1[3/2, 1 - n*p, 1 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) - (3*m*AppellF1[1/2, -(n*p), 1 + m + n*p, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^(n*p)*Tan[(-e + Pi/2 - f*x)/2])/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -(n*p), 1 + m + n*p, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*((1 + m + n*p)*AppellF1[3/2, -(n*p), 2 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + n*p*AppellF1[3/2, 1 - n*p, 1 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) + (3*Cos[e + f*x]*Sin[e + f*x]^(n*p)*(-1/3*((1 + m + n*p)*AppellF1[3/2, -(n*p), 2 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2]) - (n*p*AppellF1[3/2, 1 - n*p, 1 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/3))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -(n*p), 1 + m + n*p, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*((1 + m + n*p)*AppellF1[3/2, -(n*p), 2 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + n*p*AppellF1[3/2, 1 - n*p, 1 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) - (3*AppellF1[1/2, -(n*p), 1 + m + n*p, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^(n*p)*(-2*((1 + m + n*p)*AppellF1[3/2, -(n*p), 2 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + n*p*AppellF1[3/2, 1 - n*p, 1 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2] + 3*(-1/3*((1 + m + n*p)*AppellF1[3/2, -(n*p), 2 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2]) - (n*p*AppellF1[3/2, 1 - n*p, 1 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/3) - 2*Tan[(-e + Pi/2 - f*x)/2]^2*((1 + m + n*p)*((-3*(2 + m + n*p)*AppellF1[5/2, -(n*p), 3 + m + n*p, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5 - (3*n*p*AppellF1[5/2, 1 - n*p, 2 + m + n*p, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5) + n*p*((-3*(1 + m + n*p)*AppellF1[5/2, 1 - n*p, 2 + m + n*p, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5 + (3*(1 - n*p)*AppellF1[5/2, 2 - n*p, 1 + m + n*p, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5))))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -(n*p), 1 + m + n*p, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*((1 + m + n*p)*AppellF1[3/2, -(n*p), 2 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + n*p*AppellF1[3/2, 1 - n*p, 1 + m + n*p, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)^2)))","B",0
820,1,297,299,1.3659872,"\int \left(c (d \sin (e+f x))^p\right)^n (a+a \sin (e+f x))^3 \, dx","Integrate[(c*(d*Sin[e + f*x])^p)^n*(a + a*Sin[e + f*x])^3,x]","-\frac{a^3 \sin (e+f x) \cos (e+f x) \sqrt{\cos ^2(e+f x)} \left(\frac{1}{2} (n p+1) \sin (e+f x) \left(6 \left(n^2 p^2+7 n p+12\right) \, _2F_1\left(\frac{1}{2},\frac{n p}{2}+1;\frac{n p}{2}+2;\sin ^2(e+f x)\right)+2 (n p+2) \sin (e+f x) \left(3 (n p+4) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+3);\frac{1}{2} (n p+5);\sin ^2(e+f x)\right)+(n p+3) \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n p}{2}+2;\frac{n p}{2}+3;\sin ^2(e+f x)\right)\right)\right)+\left(n^3 p^3+9 n^2 p^2+26 n p+24\right) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) (n p+2) (n p+3) (n p+4) (\sin (e+f x)-1) (\sin (e+f x)+1)}","\frac{a^3 (4 n p+11) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) (n p+3) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 (4 n p+5) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{a^3 (2 n p+7) \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) (n p+3)}-\frac{\sin (e+f x) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+3)}",1,"-((a^3*Cos[e + f*x]*Sqrt[Cos[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n*((24 + 26*n*p + 9*n^2*p^2 + n^3*p^3)*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2] + ((1 + n*p)*Sin[e + f*x]*(6*(12 + 7*n*p + n^2*p^2)*Hypergeometric2F1[1/2, 1 + (n*p)/2, 2 + (n*p)/2, Sin[e + f*x]^2] + 2*(2 + n*p)*Sin[e + f*x]*(3*(4 + n*p)*Hypergeometric2F1[1/2, (3 + n*p)/2, (5 + n*p)/2, Sin[e + f*x]^2] + (3 + n*p)*Hypergeometric2F1[1/2, 2 + (n*p)/2, 3 + (n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x])))/2))/(f*(1 + n*p)*(2 + n*p)*(3 + n*p)*(4 + n*p)*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])))","A",1
821,1,222,222,0.6640263,"\int \left(c (d \sin (e+f x))^p\right)^n (a+a \sin (e+f x))^2 \, dx","Integrate[(c*(d*Sin[e + f*x])^p)^n*(a + a*Sin[e + f*x])^2,x]","-\frac{a^2 \sin (e+f x) \cos (e+f x) \sqrt{\cos ^2(e+f x)} \left(\left(n^2 p^2+5 n p+6\right) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right)+(n p+1) \sin (e+f x) \left(2 (n p+3) \, _2F_1\left(\frac{1}{2},\frac{n p}{2}+1;\frac{n p}{2}+2;\sin ^2(e+f x)\right)+(n p+2) \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+3);\frac{1}{2} (n p+5);\sin ^2(e+f x)\right)\right)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) (n p+2) (n p+3) (\sin (e+f x)-1) (\sin (e+f x)+1)}","\frac{2 a^2 \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 (2 n p+3) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{a^2 \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2)}",1,"-((a^2*Cos[e + f*x]*Sqrt[Cos[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n*((6 + 5*n*p + n^2*p^2)*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2] + (1 + n*p)*Sin[e + f*x]*(2*(3 + n*p)*Hypergeometric2F1[1/2, 1 + (n*p)/2, 2 + (n*p)/2, Sin[e + f*x]^2] + (2 + n*p)*Hypergeometric2F1[1/2, (3 + n*p)/2, (5 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x])))/(f*(1 + n*p)*(2 + n*p)*(3 + n*p)*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])))","A",1
822,1,270,163,1.5146722,"\int \left(c (d \sin (e+f x))^p\right)^n (a+a \sin (e+f x)) \, dx","Integrate[(c*(d*Sin[e + f*x])^p)^n*(a + a*Sin[e + f*x]),x]","\frac{a 2^{-n p-1} (\sin (e+f x)+1) \left(-i e^{-i (e+f x)} \left(-1+e^{2 i (e+f x)}\right)\right)^{n p+1} \left(2 \left(n^2 p^2-1\right) e^{i (e+f x)} \, _2F_1\left(1,\frac{n p}{2}+1;1-\frac{n p}{2};e^{2 i (e+f x)}\right)+i n p \left((n p-1) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (1-n p);e^{2 i (e+f x)}\right)-(n p+1) e^{2 i (e+f x)} \, _2F_1\left(1,\frac{1}{2} (n p+3);\frac{1}{2} (3-n p);e^{2 i (e+f x)}\right)\right)\right) \sin ^{-n p}(e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f n p (n p-1) (n p+1) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","\frac{a \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}+\frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) \sqrt{\cos ^2(e+f x)}}",1,"(2^(-1 - n*p)*a*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^(1 + n*p)*(2*E^(I*(e + f*x))*(-1 + n^2*p^2)*Hypergeometric2F1[1, 1 + (n*p)/2, 1 - (n*p)/2, E^((2*I)*(e + f*x))] + I*n*p*((-1 + n*p)*Hypergeometric2F1[1, (1 + n*p)/2, (1 - n*p)/2, E^((2*I)*(e + f*x))] - E^((2*I)*(e + f*x))*(1 + n*p)*Hypergeometric2F1[1, (3 + n*p)/2, (3 - n*p)/2, E^((2*I)*(e + f*x))]))*(c*(d*Sin[e + f*x])^p)^n*(1 + Sin[e + f*x]))/(f*n*p*(-1 + n*p)*(1 + n*p)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sin[e + f*x]^(n*p))","C",0
823,1,157,189,0.2716073,"\int \frac{\left(c (d \sin (e+f x))^p\right)^n}{a+a \sin (e+f x)} \, dx","Integrate[(c*(d*Sin[e + f*x])^p)^n/(a + a*Sin[e + f*x]),x]","\frac{\sin (e+f x) \cos (e+f x) \sqrt{\cos ^2(e+f x)} \left((n p+1) \sin (e+f x) \, _2F_1\left(\frac{3}{2},\frac{n p}{2}+1;\frac{n p}{2}+2;\sin ^2(e+f x)\right)-(n p+2) \, _2F_1\left(\frac{3}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right)\right) \left(c (d \sin (e+f x))^p\right)^n}{a f (n p+1) (n p+2) (\sin (e+f x)-1) (\sin (e+f x)+1)}","\frac{\cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n p}{2};\frac{1}{2} (n p+2);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{a f \sqrt{\cos ^2(e+f x)}}-\frac{n p \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{a f (n p+1) \sqrt{\cos ^2(e+f x)}}-\frac{\cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (a \sin (e+f x)+a)}",1,"(Cos[e + f*x]*Sqrt[Cos[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n*(-((2 + n*p)*Hypergeometric2F1[3/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]) + (1 + n*p)*Hypergeometric2F1[3/2, 1 + (n*p)/2, 2 + (n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]))/(a*f*(1 + n*p)*(2 + n*p)*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x]))","A",1
824,1,195,288,2.9035573,"\int \frac{\left(c (d \sin (e+f x))^p\right)^n}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c*(d*Sin[e + f*x])^p)^n/(a + a*Sin[e + f*x])^2,x]","\frac{\sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n \left(-\frac{2 \left(n^2 p^2-1\right) \sqrt{\cos ^2(e+f x)} \tan (e+f x) \sec (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n p}{2}+1;\frac{n p}{2}+2;\sin ^2(e+f x)\right)}{n p+2}+\frac{n p (2 n p-1) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right)}{(n p+1) \sqrt{\cos ^2(e+f x)}}+\frac{(2-2 n p) \sin (e+f x)-2 n p+3}{(\sin (e+f x)+1)^2}\right)}{3 a^2 f}","\frac{2 \left(1-n^2 p^2\right) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{3 a^2 f (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{n p (1-2 n p) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{3 a^2 f (n p+1) \sqrt{\cos ^2(e+f x)}}+\frac{2 (1-n p) \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{3 a^2 f (\sin (e+f x)+1)}+\frac{\sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{3 f (a \sin (e+f x)+a)^2}",1,"(Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n*((n*p*(-1 + 2*n*p)*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2])/((1 + n*p)*Sqrt[Cos[e + f*x]^2]) + (3 - 2*n*p + (2 - 2*n*p)*Sin[e + f*x])/(1 + Sin[e + f*x])^2 - (2*(-1 + n^2*p^2)*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/2, 1 + (n*p)/2, 2 + (n*p)/2, Sin[e + f*x]^2]*Sec[e + f*x]*Tan[e + f*x])/(2 + n*p)))/(3*a^2*f)","A",1
825,1,167,298,0.5650299,"\int (d \csc (e+f x))^n (a+b \sin (e+f x))^3 \, dx","Integrate[(d*Csc[e + f*x])^n*(a + b*Sin[e + f*x])^3,x]","-\frac{d \cos (e+f x) \sin ^2(e+f x)^{\frac{n-1}{2}} (d \csc (e+f x))^{n-1} \left(a^3 \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{3}{2};\cos ^2(e+f x)\right)+b \sqrt{\sin ^2(e+f x)} \csc (e+f x) \left(3 a^2 \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{3}{2};\cos ^2(e+f x)\right)+b^2 \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{3}{2};\cos ^2(e+f x)\right)\right)+3 a b^2 \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{3}{2};\cos ^2(e+f x)\right)\right)}{f}","\frac{b d^4 \left(3 a^2 (3-n)+b^2 (2-n)\right) \cos (e+f x) (d \csc (e+f x))^{n-4} \, _2F_1\left(\frac{1}{2},\frac{4-n}{2};\frac{6-n}{2};\sin ^2(e+f x)\right)}{f (2-n) (4-n) \sqrt{\cos ^2(e+f x)}}+\frac{a d^3 \left(a^2 (2-n)+3 b^2 (1-n)\right) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right)}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 b d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (1-n) (2-n)}+\frac{a^2 d^3 \cot (e+f x) (a \csc (e+f x)+b) (d \csc (e+f x))^{n-3}}{f (1-n)}",1,"-((d*Cos[e + f*x]*(d*Csc[e + f*x])^(-1 + n)*(Sin[e + f*x]^2)^((-1 + n)/2)*(3*a*b^2*Hypergeometric2F1[1/2, (-1 + n)/2, 3/2, Cos[e + f*x]^2] + a^3*Hypergeometric2F1[1/2, (1 + n)/2, 3/2, Cos[e + f*x]^2] + b*Csc[e + f*x]*(b^2*Hypergeometric2F1[1/2, (-2 + n)/2, 3/2, Cos[e + f*x]^2] + 3*a^2*Hypergeometric2F1[1/2, n/2, 3/2, Cos[e + f*x]^2])*Sqrt[Sin[e + f*x]^2]))/f)","A",1
826,1,135,213,0.3850177,"\int (d \csc (e+f x))^n (a+b \sin (e+f x))^2 \, dx","Integrate[(d*Csc[e + f*x])^n*(a + b*Sin[e + f*x])^2,x]","-\frac{d \cos (e+f x) \sin ^2(e+f x)^{\frac{n-1}{2}} (d \csc (e+f x))^{n-1} \left(a \left(a \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{3}{2};\cos ^2(e+f x)\right)+2 b \sqrt{\sin ^2(e+f x)} \csc (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{3}{2};\cos ^2(e+f x)\right)\right)+b^2 \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{3}{2};\cos ^2(e+f x)\right)\right)}{f}","\frac{d^3 \left(a^2 (2-n)+b^2 (1-n)\right) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right)}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}+\frac{2 a b d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right)}{f (2-n) \sqrt{\cos ^2(e+f x)}}",1,"-((d*Cos[e + f*x]*(d*Csc[e + f*x])^(-1 + n)*(Sin[e + f*x]^2)^((-1 + n)/2)*(b^2*Hypergeometric2F1[1/2, (-1 + n)/2, 3/2, Cos[e + f*x]^2] + a*(a*Hypergeometric2F1[1/2, (1 + n)/2, 3/2, Cos[e + f*x]^2] + 2*b*Csc[e + f*x]*Hypergeometric2F1[1/2, n/2, 3/2, Cos[e + f*x]^2]*Sqrt[Sin[e + f*x]^2])))/f)","A",1
827,1,105,149,0.2346387,"\int (d \csc (e+f x))^n (a+b \sin (e+f x)) \, dx","Integrate[(d*Csc[e + f*x])^n*(a + b*Sin[e + f*x]),x]","-\frac{d \cos (e+f x) \sin ^2(e+f x)^{\frac{n-1}{2}} (d \csc (e+f x))^{n-1} \left(a \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{3}{2};\cos ^2(e+f x)\right)+b \sqrt{\sin ^2(e+f x)} \csc (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{3}{2};\cos ^2(e+f x)\right)\right)}{f}","\frac{a d \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right)}{f (1-n) \sqrt{\cos ^2(e+f x)}}+\frac{b d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right)}{f (2-n) \sqrt{\cos ^2(e+f x)}}",1,"-((d*Cos[e + f*x]*(d*Csc[e + f*x])^(-1 + n)*(Sin[e + f*x]^2)^((-1 + n)/2)*(a*Hypergeometric2F1[1/2, (1 + n)/2, 3/2, Cos[e + f*x]^2] + b*Csc[e + f*x]*Hypergeometric2F1[1/2, n/2, 3/2, Cos[e + f*x]^2]*Sqrt[Sin[e + f*x]^2]))/f)","A",1
828,1,1665,204,16.8485612,"\int \frac{(d \csc (e+f x))^n}{a+b \sin (e+f x)} \, dx","Integrate[(d*Csc[e + f*x])^n/(a + b*Sin[e + f*x]),x]","-\frac{(d \csc (e+f x))^n \sec ^2(e+f x)^{-n/2} \left(\cot (e+f x) \sqrt{\sec ^2(e+f x)}\right)^n \tan (e+f x) \left(a b (n-2) F_1\left(\frac{1-n}{2};-\frac{n}{2},1;\frac{3-n}{2};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)+(n-1) \left(\left(a^2-b^2\right) F_1\left(1-\frac{n}{2};\frac{1}{2} (-n-1),1;2-\frac{n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-a^2 \, _2F_1\left(\frac{1}{2}-\frac{n}{2},1-\frac{n}{2};2-\frac{n}{2};-\tan ^2(e+f x)\right)\right) \tan (e+f x)\right)}{a^2 b f (n-2) (n-1) (a+b \sin (e+f x)) \left(-\frac{\left(\cot (e+f x) \sqrt{\sec ^2(e+f x)}\right)^n \left(a b (n-2) F_1\left(\frac{1-n}{2};-\frac{n}{2},1;\frac{3-n}{2};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)+(n-1) \left(\left(a^2-b^2\right) F_1\left(1-\frac{n}{2};\frac{1}{2} (-n-1),1;2-\frac{n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-a^2 \, _2F_1\left(\frac{1}{2}-\frac{n}{2},1-\frac{n}{2};2-\frac{n}{2};-\tan ^2(e+f x)\right)\right) \tan (e+f x)\right) \sec ^2(e+f x)^{1-\frac{n}{2}}}{a^2 b (n-2) (n-1)}+\frac{n \left(\cot (e+f x) \sqrt{\sec ^2(e+f x)}\right)^n \tan ^2(e+f x) \left(a b (n-2) F_1\left(\frac{1-n}{2};-\frac{n}{2},1;\frac{3-n}{2};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)+(n-1) \left(\left(a^2-b^2\right) F_1\left(1-\frac{n}{2};\frac{1}{2} (-n-1),1;2-\frac{n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-a^2 \, _2F_1\left(\frac{1}{2}-\frac{n}{2},1-\frac{n}{2};2-\frac{n}{2};-\tan ^2(e+f x)\right)\right) \tan (e+f x)\right) \sec ^2(e+f x)^{-n/2}}{a^2 b (n-2) (n-1)}-\frac{n \left(\cot (e+f x) \sqrt{\sec ^2(e+f x)}\right)^{n-1} \left(\sqrt{\sec ^2(e+f x)}-\csc ^2(e+f x) \sqrt{\sec ^2(e+f x)}\right) \tan (e+f x) \left(a b (n-2) F_1\left(\frac{1-n}{2};-\frac{n}{2},1;\frac{3-n}{2};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)+(n-1) \left(\left(a^2-b^2\right) F_1\left(1-\frac{n}{2};\frac{1}{2} (-n-1),1;2-\frac{n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-a^2 \, _2F_1\left(\frac{1}{2}-\frac{n}{2},1-\frac{n}{2};2-\frac{n}{2};-\tan ^2(e+f x)\right)\right) \tan (e+f x)\right) \sec ^2(e+f x)^{-n/2}}{a^2 b (n-2) (n-1)}-\frac{\left(\cot (e+f x) \sqrt{\sec ^2(e+f x)}\right)^n \tan (e+f x) \left((n-1) \left(\left(a^2-b^2\right) F_1\left(1-\frac{n}{2};\frac{1}{2} (-n-1),1;2-\frac{n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-a^2 \, _2F_1\left(\frac{1}{2}-\frac{n}{2},1-\frac{n}{2};2-\frac{n}{2};-\tan ^2(e+f x)\right)\right) \sec ^2(e+f x)+a b (n-2) \left(\frac{(1-n) n F_1\left(\frac{1-n}{2}+1;1-\frac{n}{2},1;\frac{3-n}{2}+1;-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan (e+f x) \sec ^2(e+f x)}{3-n}+\frac{2 \left(b^2-a^2\right) (1-n) F_1\left(\frac{1-n}{2}+1;-\frac{n}{2},2;\frac{3-n}{2}+1;-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan (e+f x) \sec ^2(e+f x)}{a^2 (3-n)}\right)+(n-1) \tan (e+f x) \left(\left(a^2-b^2\right) \left(\frac{2 \left(\frac{b^2}{a^2}-1\right) \left(1-\frac{n}{2}\right) F_1\left(2-\frac{n}{2};\frac{1}{2} (-n-1),2;3-\frac{n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{2-\frac{n}{2}}-\frac{(-n-1) \left(1-\frac{n}{2}\right) F_1\left(2-\frac{n}{2};\frac{1}{2} (-n-1)+1,1;3-\frac{n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{2-\frac{n}{2}}\right)-2 a^2 \left(1-\frac{n}{2}\right) \csc (e+f x) \sec (e+f x) \left(\left(\tan ^2(e+f x)+1\right)^{\frac{n}{2}-\frac{1}{2}}-\, _2F_1\left(\frac{1}{2}-\frac{n}{2},1-\frac{n}{2};2-\frac{n}{2};-\tan ^2(e+f x)\right)\right)\right)\right) \sec ^2(e+f x)^{-n/2}}{a^2 b (n-2) (n-1)}\right)}","\frac{b \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{n/2} (d \csc (e+f x))^{n+1} F_1\left(\frac{1}{2};\frac{n}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d f \left(a^2-b^2\right)}-\frac{a \cos (e+f x) \sin ^2(e+f x)^{\frac{n+1}{2}} (d \csc (e+f x))^{n+1} F_1\left(\frac{1}{2};\frac{n+1}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d f \left(a^2-b^2\right)}",1,"-(((d*Csc[e + f*x])^n*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*Tan[e + f*x]*(a*b*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 1, (3 - n)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] + (-1 + n)*((a^2 - b^2)*AppellF1[1 - n/2, (-1 - n)/2, 1, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - a^2*Hypergeometric2F1[1/2 - n/2, 1 - n/2, 2 - n/2, -Tan[e + f*x]^2])*Tan[e + f*x]))/(a^2*b*f*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2)*(a + b*Sin[e + f*x])*(-(((Sec[e + f*x]^2)^(1 - n/2)*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*(a*b*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 1, (3 - n)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] + (-1 + n)*((a^2 - b^2)*AppellF1[1 - n/2, (-1 - n)/2, 1, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - a^2*Hypergeometric2F1[1/2 - n/2, 1 - n/2, 2 - n/2, -Tan[e + f*x]^2])*Tan[e + f*x]))/(a^2*b*(-2 + n)*(-1 + n))) - (n*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^(-1 + n)*(Sqrt[Sec[e + f*x]^2] - Csc[e + f*x]^2*Sqrt[Sec[e + f*x]^2])*Tan[e + f*x]*(a*b*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 1, (3 - n)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] + (-1 + n)*((a^2 - b^2)*AppellF1[1 - n/2, (-1 - n)/2, 1, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - a^2*Hypergeometric2F1[1/2 - n/2, 1 - n/2, 2 - n/2, -Tan[e + f*x]^2])*Tan[e + f*x]))/(a^2*b*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2)) + (n*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*Tan[e + f*x]^2*(a*b*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 1, (3 - n)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] + (-1 + n)*((a^2 - b^2)*AppellF1[1 - n/2, (-1 - n)/2, 1, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - a^2*Hypergeometric2F1[1/2 - n/2, 1 - n/2, 2 - n/2, -Tan[e + f*x]^2])*Tan[e + f*x]))/(a^2*b*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2)) - ((Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*Tan[e + f*x]*((-1 + n)*((a^2 - b^2)*AppellF1[1 - n/2, (-1 - n)/2, 1, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - a^2*Hypergeometric2F1[1/2 - n/2, 1 - n/2, 2 - n/2, -Tan[e + f*x]^2])*Sec[e + f*x]^2 + a*b*(-2 + n)*(((1 - n)*n*AppellF1[1 + (1 - n)/2, 1 - n/2, 1, 1 + (3 - n)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 - n) + (2*(-a^2 + b^2)*(1 - n)*AppellF1[1 + (1 - n)/2, -1/2*n, 2, 1 + (3 - n)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Sec[e + f*x]^2*Tan[e + f*x])/(a^2*(3 - n))) + (-1 + n)*Tan[e + f*x]*((a^2 - b^2)*(-(((-1 - n)*(1 - n/2)*AppellF1[2 - n/2, 1 + (-1 - n)/2, 1, 3 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 - n/2)) + (2*(-1 + b^2/a^2)*(1 - n/2)*AppellF1[2 - n/2, (-1 - n)/2, 2, 3 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 - n/2)) - 2*a^2*(1 - n/2)*Csc[e + f*x]*Sec[e + f*x]*(-Hypergeometric2F1[1/2 - n/2, 1 - n/2, 2 - n/2, -Tan[e + f*x]^2] + (1 + Tan[e + f*x]^2)^(-1/2 + n/2)))))/(a^2*b*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2)))))","B",0
829,1,1872,321,19.0605935,"\int \frac{(d \csc (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx","Integrate[(d*Csc[e + f*x])^n/(a + b*Sin[e + f*x])^2,x]","\frac{(d \csc (e+f x))^n \sec ^2(e+f x)^{-n/2} \left(\cot (e+f x) \sqrt{\sec ^2(e+f x)}\right)^n \tan (e+f x) \left(2 b \left(a b (n-2) F_1\left(\frac{1-n}{2};-\frac{n}{2},2;\frac{3-n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)+\left(a^2-b^2\right) (n-1) F_1\left(1-\frac{n}{2};\frac{1}{2} (-n-1),2;2-\frac{n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)\right)-a \left(a^2+b^2\right) (n-2) F_1\left(\frac{1-n}{2};-\frac{n}{2},1;\frac{3-n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right)}{a^3 \left(a^2-b^2\right) f (n-2) (n-1) (a+b \sin (e+f x))^2 \left(\frac{\left(\cot (e+f x) \sqrt{\sec ^2(e+f x)}\right)^n \left(2 b \left(a b (n-2) F_1\left(\frac{1-n}{2};-\frac{n}{2},2;\frac{3-n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)+\left(a^2-b^2\right) (n-1) F_1\left(1-\frac{n}{2};\frac{1}{2} (-n-1),2;2-\frac{n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)\right)-a \left(a^2+b^2\right) (n-2) F_1\left(\frac{1-n}{2};-\frac{n}{2},1;\frac{3-n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right) \sec ^2(e+f x)^{1-\frac{n}{2}}}{a^3 \left(a^2-b^2\right) (n-2) (n-1)}-\frac{n \left(\cot (e+f x) \sqrt{\sec ^2(e+f x)}\right)^n \tan ^2(e+f x) \left(2 b \left(a b (n-2) F_1\left(\frac{1-n}{2};-\frac{n}{2},2;\frac{3-n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)+\left(a^2-b^2\right) (n-1) F_1\left(1-\frac{n}{2};\frac{1}{2} (-n-1),2;2-\frac{n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)\right)-a \left(a^2+b^2\right) (n-2) F_1\left(\frac{1-n}{2};-\frac{n}{2},1;\frac{3-n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right) \sec ^2(e+f x)^{-n/2}}{a^3 \left(a^2-b^2\right) (n-2) (n-1)}+\frac{n \left(\cot (e+f x) \sqrt{\sec ^2(e+f x)}\right)^{n-1} \left(\sqrt{\sec ^2(e+f x)}-\csc ^2(e+f x) \sqrt{\sec ^2(e+f x)}\right) \tan (e+f x) \left(2 b \left(a b (n-2) F_1\left(\frac{1-n}{2};-\frac{n}{2},2;\frac{3-n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)+\left(a^2-b^2\right) (n-1) F_1\left(1-\frac{n}{2};\frac{1}{2} (-n-1),2;2-\frac{n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)\right)-a \left(a^2+b^2\right) (n-2) F_1\left(\frac{1-n}{2};-\frac{n}{2},1;\frac{3-n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right) \sec ^2(e+f x)^{-n/2}}{a^3 \left(a^2-b^2\right) (n-2) (n-1)}+\frac{\left(\cot (e+f x) \sqrt{\sec ^2(e+f x)}\right)^n \tan (e+f x) \left(2 b \left(\left(a^2-b^2\right) (n-1) F_1\left(1-\frac{n}{2};\frac{1}{2} (-n-1),2;2-\frac{n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x)+a b (n-2) \left(\frac{(1-n) n F_1\left(\frac{1-n}{2}+1;1-\frac{n}{2},2;\frac{3-n}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)}{3-n}+\frac{4 \left(\frac{b^2}{a^2}-1\right) (1-n) F_1\left(\frac{1-n}{2}+1;-\frac{n}{2},3;\frac{3-n}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)}{3-n}\right)+\left(a^2-b^2\right) (n-1) \tan (e+f x) \left(\frac{4 \left(\frac{b^2}{a^2}-1\right) \left(1-\frac{n}{2}\right) F_1\left(2-\frac{n}{2};\frac{1}{2} (-n-1),3;3-\frac{n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{2-\frac{n}{2}}-\frac{(-n-1) \left(1-\frac{n}{2}\right) F_1\left(2-\frac{n}{2};\frac{1}{2} (-n-1)+1,2;3-\frac{n}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{2-\frac{n}{2}}\right)\right)-a \left(a^2+b^2\right) (n-2) \left(\frac{(1-n) n F_1\left(\frac{1-n}{2}+1;1-\frac{n}{2},1;\frac{3-n}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)}{3-n}+\frac{2 \left(\frac{b^2}{a^2}-1\right) (1-n) F_1\left(\frac{1-n}{2}+1;-\frac{n}{2},2;\frac{3-n}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)}{3-n}\right)\right) \sec ^2(e+f x)^{-n/2}}{a^3 \left(a^2-b^2\right) (n-2) (n-1)}\right)}","-\frac{a^2 \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{n+1}{2}} (d \csc (e+f x))^{n+2} F_1\left(\frac{1}{2};\frac{n+1}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^2 f \left(a^2-b^2\right)^2}+\frac{2 a b \cos (e+f x) \sin ^2(e+f x)^{\frac{n+2}{2}} (d \csc (e+f x))^{n+2} F_1\left(\frac{1}{2};\frac{n}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^2 f \left(a^2-b^2\right)^2}-\frac{b^2 \sin ^3(e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{n-1}{2}} (d \csc (e+f x))^{n+2} F_1\left(\frac{1}{2};\frac{n-1}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^2 f \left(a^2-b^2\right)^2}",1,"((d*Csc[e + f*x])^n*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*Tan[e + f*x]*(-(a*(a^2 + b^2)*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 1, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + 2*b*(a*b*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 2, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (a^2 - b^2)*(-1 + n)*AppellF1[1 - n/2, (-1 - n)/2, 2, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x])))/(a^3*(a^2 - b^2)*f*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2)*(a + b*Sin[e + f*x])^2*(((Sec[e + f*x]^2)^(1 - n/2)*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*(-(a*(a^2 + b^2)*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 1, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + 2*b*(a*b*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 2, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (a^2 - b^2)*(-1 + n)*AppellF1[1 - n/2, (-1 - n)/2, 2, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x])))/(a^3*(a^2 - b^2)*(-2 + n)*(-1 + n)) + (n*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^(-1 + n)*(Sqrt[Sec[e + f*x]^2] - Csc[e + f*x]^2*Sqrt[Sec[e + f*x]^2])*Tan[e + f*x]*(-(a*(a^2 + b^2)*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 1, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + 2*b*(a*b*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 2, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (a^2 - b^2)*(-1 + n)*AppellF1[1 - n/2, (-1 - n)/2, 2, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x])))/(a^3*(a^2 - b^2)*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2)) - (n*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*Tan[e + f*x]^2*(-(a*(a^2 + b^2)*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 1, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + 2*b*(a*b*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 2, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (a^2 - b^2)*(-1 + n)*AppellF1[1 - n/2, (-1 - n)/2, 2, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x])))/(a^3*(a^2 - b^2)*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2)) + ((Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*Tan[e + f*x]*(-(a*(a^2 + b^2)*(-2 + n)*(((1 - n)*n*AppellF1[1 + (1 - n)/2, 1 - n/2, 1, 1 + (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 - n) + (2*(-1 + b^2/a^2)*(1 - n)*AppellF1[1 + (1 - n)/2, -1/2*n, 2, 1 + (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 - n))) + 2*b*((a^2 - b^2)*(-1 + n)*AppellF1[1 - n/2, (-1 - n)/2, 2, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2 + a*b*(-2 + n)*(((1 - n)*n*AppellF1[1 + (1 - n)/2, 1 - n/2, 2, 1 + (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 - n) + (4*(-1 + b^2/a^2)*(1 - n)*AppellF1[1 + (1 - n)/2, -1/2*n, 3, 1 + (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 - n)) + (a^2 - b^2)*(-1 + n)*Tan[e + f*x]*(-(((-1 - n)*(1 - n/2)*AppellF1[2 - n/2, 1 + (-1 - n)/2, 2, 3 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 - n/2)) + (4*(-1 + b^2/a^2)*(1 - n/2)*AppellF1[2 - n/2, (-1 - n)/2, 3, 3 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 - n/2)))))/(a^3*(a^2 - b^2)*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2))))","B",0
830,1,2406,432,20.2195029,"\int \frac{(d \csc (e+f x))^n}{(a+b \sin (e+f x))^3} \, dx","Integrate[(d*Csc[e + f*x])^n/(a + b*Sin[e + f*x])^3,x]","\text{Result too large to show}","-\frac{3 a b^2 \sin ^4(e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{n-1}{2}} (d \csc (e+f x))^{n+3} F_1\left(\frac{1}{2};\frac{n-1}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^3 f \left(a^2-b^2\right)^3}+\frac{3 a^2 b \sin ^3(e+f x) \cos (e+f x) \sin ^2(e+f x)^{n/2} (d \csc (e+f x))^{n+3} F_1\left(\frac{1}{2};\frac{n}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^3 f \left(a^2-b^2\right)^3}+\frac{b^3 \sin ^3(e+f x) \cos (e+f x) \sin ^2(e+f x)^{n/2} (d \csc (e+f x))^{n+3} F_1\left(\frac{1}{2};\frac{n-2}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^3 f \left(a^2-b^2\right)^3}-\frac{a^3 \cos (e+f x) \sin ^2(e+f x)^{\frac{n+3}{2}} (d \csc (e+f x))^{n+3} F_1\left(\frac{1}{2};\frac{n+1}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^3 f \left(a^2-b^2\right)^3}",1,"((d*Csc[e + f*x])^n*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*Tan[e + f*x]*(-(a*(a^2 + 3*b^2)*(-2 + n)*AppellF1[(1 - n)/2, -1 - n/2, 2, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + b*(4*a*b*(-2 + n)*AppellF1[(1 - n)/2, -1 - n/2, 3, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (-1 + n)*((3*a^2 + b^2)*AppellF1[1 - n/2, (-1 - n)/2, 2, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 4*b^2*AppellF1[1 - n/2, (-1 - n)/2, 3, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])*Tan[e + f*x])))/(a^4*(a^2 - b^2)*f*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2)*(a + b*Sin[e + f*x])^3*(((Sec[e + f*x]^2)^(1 - n/2)*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*(-(a*(a^2 + 3*b^2)*(-2 + n)*AppellF1[(1 - n)/2, -1 - n/2, 2, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + b*(4*a*b*(-2 + n)*AppellF1[(1 - n)/2, -1 - n/2, 3, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (-1 + n)*((3*a^2 + b^2)*AppellF1[1 - n/2, (-1 - n)/2, 2, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 4*b^2*AppellF1[1 - n/2, (-1 - n)/2, 3, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])*Tan[e + f*x])))/(a^4*(a^2 - b^2)*(-2 + n)*(-1 + n)) + (n*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^(-1 + n)*(Sqrt[Sec[e + f*x]^2] - Csc[e + f*x]^2*Sqrt[Sec[e + f*x]^2])*Tan[e + f*x]*(-(a*(a^2 + 3*b^2)*(-2 + n)*AppellF1[(1 - n)/2, -1 - n/2, 2, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + b*(4*a*b*(-2 + n)*AppellF1[(1 - n)/2, -1 - n/2, 3, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (-1 + n)*((3*a^2 + b^2)*AppellF1[1 - n/2, (-1 - n)/2, 2, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 4*b^2*AppellF1[1 - n/2, (-1 - n)/2, 3, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])*Tan[e + f*x])))/(a^4*(a^2 - b^2)*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2)) - (n*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*Tan[e + f*x]^2*(-(a*(a^2 + 3*b^2)*(-2 + n)*AppellF1[(1 - n)/2, -1 - n/2, 2, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + b*(4*a*b*(-2 + n)*AppellF1[(1 - n)/2, -1 - n/2, 3, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (-1 + n)*((3*a^2 + b^2)*AppellF1[1 - n/2, (-1 - n)/2, 2, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 4*b^2*AppellF1[1 - n/2, (-1 - n)/2, 3, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])*Tan[e + f*x])))/(a^4*(a^2 - b^2)*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2)) + ((Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*Tan[e + f*x]*(-(a*(a^2 + 3*b^2)*(-2 + n)*((4*(-1 + b^2/a^2)*(1 - n)*AppellF1[1 + (1 - n)/2, -1 - n/2, 3, 1 + (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 - n) - (2*(1 - n)*(-1 - n/2)*AppellF1[1 + (1 - n)/2, -1/2*n, 2, 1 + (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 - n))) + b*((-1 + n)*((3*a^2 + b^2)*AppellF1[1 - n/2, (-1 - n)/2, 2, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 4*b^2*AppellF1[1 - n/2, (-1 - n)/2, 3, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])*Sec[e + f*x]^2 + 4*a*b*(-2 + n)*((6*(-1 + b^2/a^2)*(1 - n)*AppellF1[1 + (1 - n)/2, -1 - n/2, 4, 1 + (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 - n) - (2*(1 - n)*(-1 - n/2)*AppellF1[1 + (1 - n)/2, -1/2*n, 3, 1 + (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 - n)) + (-1 + n)*Tan[e + f*x]*((3*a^2 + b^2)*(-(((-1 - n)*(1 - n/2)*AppellF1[2 - n/2, 1 + (-1 - n)/2, 2, 3 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 - n/2)) + (4*(-1 + b^2/a^2)*(1 - n/2)*AppellF1[2 - n/2, (-1 - n)/2, 3, 3 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 - n/2)) - 4*b^2*(-(((-1 - n)*(1 - n/2)*AppellF1[2 - n/2, 1 + (-1 - n)/2, 3, 3 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 - n/2)) + (6*(-1 + b^2/a^2)*(1 - n/2)*AppellF1[2 - n/2, (-1 - n)/2, 4, 3 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 - n/2))))))/(a^4*(a^2 - b^2)*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2))))","B",0
831,0,0,56,2.5707763,"\int \left(c (d \sin (e+f x))^p\right)^n (a+b \sin (e+f x))^m \, dx","Integrate[(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x])^m,x]","\int \left(c (d \sin (e+f x))^p\right)^n (a+b \sin (e+f x))^m \, dx","(d \sin (e+f x))^{-n p} \left(c (d \sin (e+f x))^p\right)^n \text{Int}\left((a+b \sin (e+f x))^m (d \sin (e+f x))^{n p},x\right)",0,"Integrate[(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x])^m, x]","A",-1
832,1,230,323,1.0391451,"\int \left(c (d \sin (e+f x))^p\right)^n (a+b \sin (e+f x))^3 \, dx","Integrate[(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x])^3,x]","\frac{\sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n \left(\frac{b \left(3 a^2 (n p+3)+b^2 (n p+2)\right) \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n p}{2}+1;\frac{n p}{2}+2;\sin ^2(e+f x)\right)}{(n p+2) \sqrt{\cos ^2(e+f x)}}+\frac{a (n p+3) \left(a^2 (n p+2)+3 b^2 (n p+1)\right) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right)}{(n p+1) (n p+2) \sqrt{\cos ^2(e+f x)}}-b^2 (a+b \sin (e+f x))-\frac{a b^2 (2 n p+7)}{n p+2}\right)}{f (n p+3)}","\frac{b \left(3 a^2 (n p+3)+b^2 (n p+2)\right) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) (n p+3) \sqrt{\cos ^2(e+f x)}}+\frac{a \left(a^2 (n p+2)+3 b^2 (n p+1)\right) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{a b^2 (2 n p+7) \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) (n p+3)}-\frac{b^2 \sin (e+f x) \cos (e+f x) (a+b \sin (e+f x)) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+3)}",1,"(Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n*(-((a*b^2*(7 + 2*n*p))/(2 + n*p)) + (a*(3 + n*p)*(3*b^2*(1 + n*p) + a^2*(2 + n*p))*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2])/((1 + n*p)*(2 + n*p)*Sqrt[Cos[e + f*x]^2]) + (b*(b^2*(2 + n*p) + 3*a^2*(3 + n*p))*Hypergeometric2F1[1/2, 1 + (n*p)/2, 2 + (n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x])/((2 + n*p)*Sqrt[Cos[e + f*x]^2]) - b^2*(a + b*Sin[e + f*x])))/(f*(3 + n*p))","A",1
833,1,152,231,0.3497213,"\int \left(c (d \sin (e+f x))^p\right)^n (a+b \sin (e+f x))^2 \, dx","Integrate[(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x])^2,x]","-\frac{\cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-n p-1)} \left(c (d \sin (e+f x))^p\right)^n \left(a \left(a \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{3}{2};\cos ^2(e+f x)\right)+2 b \sqrt{\sin ^2(e+f x)} \, _2F_1\left(\frac{1}{2},-\frac{n p}{2};\frac{3}{2};\cos ^2(e+f x)\right)\right)+b^2 \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-n p-1);\frac{3}{2};\cos ^2(e+f x)\right)\right)}{f}","\frac{\left(a^2 (n p+2)+b^2 (n p+1)\right) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) (n p+2) \sqrt{\cos ^2(e+f x)}}+\frac{2 a b \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{b^2 \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2)}",1,"-((Cos[e + f*x]*(Sin[e + f*x]^2)^((-1 - n*p)/2)*(c*(d*Sin[e + f*x])^p)^n*(b^2*Hypergeometric2F1[1/2, (-1 - n*p)/2, 3/2, Cos[e + f*x]^2]*Sin[e + f*x] + a*(a*Hypergeometric2F1[1/2, (1 - n*p)/2, 3/2, Cos[e + f*x]^2]*Sin[e + f*x] + 2*b*Hypergeometric2F1[1/2, -1/2*(n*p), 3/2, Cos[e + f*x]^2]*Sqrt[Sin[e + f*x]^2])))/f)","A",1
834,1,129,163,0.2407892,"\int \left(c (d \sin (e+f x))^p\right)^n (a+b \sin (e+f x)) \, dx","Integrate[(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x]),x]","\frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) \left(c (d \sin (e+f x))^p\right)^n \left(a (n p+2) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right)+b (n p+1) \sin (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n p}{2}+1;\frac{n p}{2}+2;\sin ^2(e+f x)\right)\right)}{f (n p+1) (n p+2)}","\frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) \sqrt{\cos ^2(e+f x)}}+\frac{b \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}",1,"(Sqrt[Cos[e + f*x]^2]*(c*(d*Sin[e + f*x])^p)^n*(a*(2 + n*p)*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2] + b*(1 + n*p)*Hypergeometric2F1[1/2, 1 + (n*p)/2, 2 + (n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x])*Tan[e + f*x])/(f*(1 + n*p)*(2 + n*p))","A",1
835,1,1808,204,18.0029484,"\int \frac{\left(c (d \sin (e+f x))^p\right)^n}{a+b \sin (e+f x)} \, dx","Integrate[(c*(d*Sin[e + f*x])^p)^n/(a + b*Sin[e + f*x]),x]","\frac{\sec ^2(e+f x)^{\frac{n p}{2}} \left(c (d \sin (e+f x))^p\right)^n \tan (e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^{n p} \left(\left(a^2-b^2\right) (n p+1) F_1\left(\frac{n p}{2}+1;\frac{1}{2} (n p-1),1;\frac{n p}{2}+2;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)+a \left(b (n p+2) F_1\left(\frac{1}{2} (n p+1);\frac{n p}{2},1;\frac{1}{2} (n p+3);-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)-a (n p+1) \, _2F_1\left(\frac{n p}{2}+1,\frac{1}{2} (n p+1);\frac{n p}{2}+2;-\tan ^2(e+f x)\right) \tan (e+f x)\right)\right)}{a^2 b f (n p+1) (n p+2) (a+b \sin (e+f x)) \left(\frac{n p \tan ^2(e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^{n p} \left(\left(a^2-b^2\right) (n p+1) F_1\left(\frac{n p}{2}+1;\frac{1}{2} (n p-1),1;\frac{n p}{2}+2;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)+a \left(b (n p+2) F_1\left(\frac{1}{2} (n p+1);\frac{n p}{2},1;\frac{1}{2} (n p+3);-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)-a (n p+1) \, _2F_1\left(\frac{n p}{2}+1,\frac{1}{2} (n p+1);\frac{n p}{2}+2;-\tan ^2(e+f x)\right) \tan (e+f x)\right)\right) \sec ^2(e+f x)^{\frac{n p}{2}}}{a^2 b (n p+1) (n p+2)}+\frac{n p \tan (e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^{n p-1} \left(\sqrt{\sec ^2(e+f x)}-\frac{\tan ^2(e+f x)}{\sqrt{\sec ^2(e+f x)}}\right) \left(\left(a^2-b^2\right) (n p+1) F_1\left(\frac{n p}{2}+1;\frac{1}{2} (n p-1),1;\frac{n p}{2}+2;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)+a \left(b (n p+2) F_1\left(\frac{1}{2} (n p+1);\frac{n p}{2},1;\frac{1}{2} (n p+3);-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)-a (n p+1) \, _2F_1\left(\frac{n p}{2}+1,\frac{1}{2} (n p+1);\frac{n p}{2}+2;-\tan ^2(e+f x)\right) \tan (e+f x)\right)\right) \sec ^2(e+f x)^{\frac{n p}{2}}}{a^2 b (n p+1) (n p+2)}+\frac{\tan (e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^{n p} \left(\left(a^2-b^2\right) (n p+1) F_1\left(\frac{n p}{2}+1;\frac{1}{2} (n p-1),1;\frac{n p}{2}+2;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x)+\left(a^2-b^2\right) (n p+1) \tan (e+f x) \left(\frac{2 \left(\frac{b^2}{a^2}-1\right) \left(\frac{n p}{2}+1\right) F_1\left(\frac{n p}{2}+2;\frac{1}{2} (n p-1),2;\frac{n p}{2}+3;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{\frac{n p}{2}+2}-\frac{\left(\frac{n p}{2}+1\right) (n p-1) F_1\left(\frac{n p}{2}+2;\frac{1}{2} (n p-1)+1,1;\frac{n p}{2}+3;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{\frac{n p}{2}+2}\right)+a \left(-a (n p+1) \, _2F_1\left(\frac{n p}{2}+1,\frac{1}{2} (n p+1);\frac{n p}{2}+2;-\tan ^2(e+f x)\right) \sec ^2(e+f x)-2 a \left(\frac{n p}{2}+1\right) (n p+1) \left(\left(\tan ^2(e+f x)+1\right)^{\frac{1}{2} (-n p-1)}-\, _2F_1\left(\frac{n p}{2}+1,\frac{1}{2} (n p+1);\frac{n p}{2}+2;-\tan ^2(e+f x)\right)\right) \sec ^2(e+f x)+b (n p+2) \left(\frac{2 \left(b^2-a^2\right) (n p+1) F_1\left(\frac{1}{2} (n p+1)+1;\frac{n p}{2},2;\frac{1}{2} (n p+3)+1;-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \sec ^2(e+f x) \tan (e+f x)}{a^2 (n p+3)}-\frac{n p (n p+1) F_1\left(\frac{1}{2} (n p+1)+1;\frac{n p}{2}+1,1;\frac{1}{2} (n p+3)+1;-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \sec ^2(e+f x) \tan (e+f x)}{n p+3}\right)\right)\right) \sec ^2(e+f x)^{\frac{n p}{2}}}{a^2 b (n p+1) (n p+2)}+\frac{\left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^{n p} \left(\left(a^2-b^2\right) (n p+1) F_1\left(\frac{n p}{2}+1;\frac{1}{2} (n p-1),1;\frac{n p}{2}+2;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)+a \left(b (n p+2) F_1\left(\frac{1}{2} (n p+1);\frac{n p}{2},1;\frac{1}{2} (n p+3);-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)-a (n p+1) \, _2F_1\left(\frac{n p}{2}+1,\frac{1}{2} (n p+1);\frac{n p}{2}+2;-\tan ^2(e+f x)\right) \tan (e+f x)\right)\right) \sec ^2(e+f x)^{\frac{n p}{2}+1}}{a^2 b (n p+1) (n p+2)}\right)}","\frac{b \cos (e+f x) \sin ^2(e+f x)^{-\frac{n p}{2}} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};-\frac{n p}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{a \cot (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (1-n p)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (1-n p),1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}",1,"((Sec[e + f*x]^2)^((n*p)/2)*(c*(d*Sin[e + f*x])^p)^n*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(n*p)*((a^2 - b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 1, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x] + a*(b*(2 + n*p)*AppellF1[(1 + n*p)/2, (n*p)/2, 1, (3 + n*p)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] - a*(1 + n*p)*Hypergeometric2F1[1 + (n*p)/2, (1 + n*p)/2, 2 + (n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x])))/(a^2*b*f*(1 + n*p)*(2 + n*p)*(a + b*Sin[e + f*x])*(((Sec[e + f*x]^2)^(1 + (n*p)/2)*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(n*p)*((a^2 - b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 1, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x] + a*(b*(2 + n*p)*AppellF1[(1 + n*p)/2, (n*p)/2, 1, (3 + n*p)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] - a*(1 + n*p)*Hypergeometric2F1[1 + (n*p)/2, (1 + n*p)/2, 2 + (n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x])))/(a^2*b*(1 + n*p)*(2 + n*p)) + (n*p*(Sec[e + f*x]^2)^((n*p)/2)*Tan[e + f*x]^2*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(n*p)*((a^2 - b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 1, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x] + a*(b*(2 + n*p)*AppellF1[(1 + n*p)/2, (n*p)/2, 1, (3 + n*p)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] - a*(1 + n*p)*Hypergeometric2F1[1 + (n*p)/2, (1 + n*p)/2, 2 + (n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x])))/(a^2*b*(1 + n*p)*(2 + n*p)) + (n*p*(Sec[e + f*x]^2)^((n*p)/2)*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(-1 + n*p)*(Sqrt[Sec[e + f*x]^2] - Tan[e + f*x]^2/Sqrt[Sec[e + f*x]^2])*((a^2 - b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 1, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x] + a*(b*(2 + n*p)*AppellF1[(1 + n*p)/2, (n*p)/2, 1, (3 + n*p)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] - a*(1 + n*p)*Hypergeometric2F1[1 + (n*p)/2, (1 + n*p)/2, 2 + (n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x])))/(a^2*b*(1 + n*p)*(2 + n*p)) + ((Sec[e + f*x]^2)^((n*p)/2)*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(n*p)*((a^2 - b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 1, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2 + (a^2 - b^2)*(1 + n*p)*Tan[e + f*x]*((2*(-1 + b^2/a^2)*(1 + (n*p)/2)*AppellF1[2 + (n*p)/2, (-1 + n*p)/2, 2, 3 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 + (n*p)/2) - ((1 + (n*p)/2)*(-1 + n*p)*AppellF1[2 + (n*p)/2, 1 + (-1 + n*p)/2, 1, 3 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 + (n*p)/2)) + a*(-(a*(1 + n*p)*Hypergeometric2F1[1 + (n*p)/2, (1 + n*p)/2, 2 + (n*p)/2, -Tan[e + f*x]^2]*Sec[e + f*x]^2) + b*(2 + n*p)*((2*(-a^2 + b^2)*(1 + n*p)*AppellF1[1 + (1 + n*p)/2, (n*p)/2, 2, 1 + (3 + n*p)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Sec[e + f*x]^2*Tan[e + f*x])/(a^2*(3 + n*p)) - (n*p*(1 + n*p)*AppellF1[1 + (1 + n*p)/2, 1 + (n*p)/2, 1, 1 + (3 + n*p)/2, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + n*p)) - 2*a*(1 + (n*p)/2)*(1 + n*p)*Sec[e + f*x]^2*(-Hypergeometric2F1[1 + (n*p)/2, (1 + n*p)/2, 2 + (n*p)/2, -Tan[e + f*x]^2] + (1 + Tan[e + f*x]^2)^((-1 - n*p)/2)))))/(a^2*b*(1 + n*p)*(2 + n*p))))","B",0
836,1,1970,322,19.0168051,"\int \frac{\left(c (d \sin (e+f x))^p\right)^n}{(a+b \sin (e+f x))^2} \, dx","Integrate[(c*(d*Sin[e + f*x])^p)^n/(a + b*Sin[e + f*x])^2,x]","-\frac{\sec ^2(e+f x)^{\frac{n p}{2}} \left(c (d \sin (e+f x))^p\right)^n \tan (e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^{n p} \left(2 b \left(a^2-b^2\right) (n p+1) F_1\left(\frac{n p}{2}+1;\frac{1}{2} (n p-1),2;\frac{n p}{2}+2;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)-a (n p+2) \left(\left(a^2+b^2\right) F_1\left(\frac{1}{2} (n p+1);\frac{n p}{2},1;\frac{1}{2} (n p+3);-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-2 b^2 F_1\left(\frac{1}{2} (n p+1);\frac{n p}{2},2;\frac{1}{2} (n p+3);-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right)\right)}{a^3 \left(a^2-b^2\right) f (n p+1) (n p+2) (a+b \sin (e+f x))^2 \left(-\frac{n p \tan ^2(e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^{n p} \left(2 b \left(a^2-b^2\right) (n p+1) F_1\left(\frac{n p}{2}+1;\frac{1}{2} (n p-1),2;\frac{n p}{2}+2;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)-a (n p+2) \left(\left(a^2+b^2\right) F_1\left(\frac{1}{2} (n p+1);\frac{n p}{2},1;\frac{1}{2} (n p+3);-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-2 b^2 F_1\left(\frac{1}{2} (n p+1);\frac{n p}{2},2;\frac{1}{2} (n p+3);-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right)\right) \sec ^2(e+f x)^{\frac{n p}{2}}}{a^3 \left(a^2-b^2\right) (n p+1) (n p+2)}-\frac{n p \tan (e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^{n p-1} \left(2 b \left(a^2-b^2\right) (n p+1) F_1\left(\frac{n p}{2}+1;\frac{1}{2} (n p-1),2;\frac{n p}{2}+2;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)-a (n p+2) \left(\left(a^2+b^2\right) F_1\left(\frac{1}{2} (n p+1);\frac{n p}{2},1;\frac{1}{2} (n p+3);-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-2 b^2 F_1\left(\frac{1}{2} (n p+1);\frac{n p}{2},2;\frac{1}{2} (n p+3);-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right)\right) \left(\sqrt{\sec ^2(e+f x)}-\frac{\tan ^2(e+f x)}{\sqrt{\sec ^2(e+f x)}}\right) \sec ^2(e+f x)^{\frac{n p}{2}}}{a^3 \left(a^2-b^2\right) (n p+1) (n p+2)}-\frac{\tan (e+f x) \left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^{n p} \left(2 b \left(a^2-b^2\right) (n p+1) F_1\left(\frac{n p}{2}+1;\frac{1}{2} (n p-1),2;\frac{n p}{2}+2;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x)+2 b \left(a^2-b^2\right) (n p+1) \tan (e+f x) \left(\frac{4 \left(\frac{b^2}{a^2}-1\right) \left(\frac{n p}{2}+1\right) F_1\left(\frac{n p}{2}+2;\frac{1}{2} (n p-1),3;\frac{n p}{2}+3;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{\frac{n p}{2}+2}-\frac{\left(\frac{n p}{2}+1\right) (n p-1) F_1\left(\frac{n p}{2}+2;\frac{1}{2} (n p-1)+1,2;\frac{n p}{2}+3;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{\frac{n p}{2}+2}\right)-a (n p+2) \left(\left(a^2+b^2\right) \left(\frac{2 \left(\frac{b^2}{a^2}-1\right) (n p+1) F_1\left(\frac{1}{2} (n p+1)+1;\frac{n p}{2},2;\frac{1}{2} (n p+3)+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{n p+3}-\frac{n p (n p+1) F_1\left(\frac{1}{2} (n p+1)+1;\frac{n p}{2}+1,1;\frac{1}{2} (n p+3)+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{n p+3}\right)-2 b^2 \left(\frac{4 \left(\frac{b^2}{a^2}-1\right) (n p+1) F_1\left(\frac{1}{2} (n p+1)+1;\frac{n p}{2},3;\frac{1}{2} (n p+3)+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{n p+3}-\frac{n p (n p+1) F_1\left(\frac{1}{2} (n p+1)+1;\frac{n p}{2}+1,2;\frac{1}{2} (n p+3)+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{n p+3}\right)\right)\right) \sec ^2(e+f x)^{\frac{n p}{2}}}{a^3 \left(a^2-b^2\right) (n p+1) (n p+2)}-\frac{\left(\frac{\tan (e+f x)}{\sqrt{\sec ^2(e+f x)}}\right)^{n p} \left(2 b \left(a^2-b^2\right) (n p+1) F_1\left(\frac{n p}{2}+1;\frac{1}{2} (n p-1),2;\frac{n p}{2}+2;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)-a (n p+2) \left(\left(a^2+b^2\right) F_1\left(\frac{1}{2} (n p+1);\frac{n p}{2},1;\frac{1}{2} (n p+3);-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-2 b^2 F_1\left(\frac{1}{2} (n p+1);\frac{n p}{2},2;\frac{1}{2} (n p+3);-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right)\right) \sec ^2(e+f x)^{\frac{n p}{2}+1}}{a^3 \left(a^2-b^2\right) (n p+1) (n p+2)}\right)}","\frac{2 a b \cos (e+f x) \sin ^2(e+f x)^{-\frac{n p}{2}} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};-\frac{n p}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}-\frac{b^2 \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-n p-1)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (-n p-1),2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}-\frac{a^2 \cot (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (1-n p)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (1-n p),2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}",1,"-(((Sec[e + f*x]^2)^((n*p)/2)*(c*(d*Sin[e + f*x])^p)^n*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(n*p)*(-(a*(2 + n*p)*((a^2 + b^2)*AppellF1[(1 + n*p)/2, (n*p)/2, 1, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 2*b^2*AppellF1[(1 + n*p)/2, (n*p)/2, 2, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])) + 2*b*(a^2 - b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 2, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]))/(a^3*(a^2 - b^2)*f*(1 + n*p)*(2 + n*p)*(a + b*Sin[e + f*x])^2*(-(((Sec[e + f*x]^2)^(1 + (n*p)/2)*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(n*p)*(-(a*(2 + n*p)*((a^2 + b^2)*AppellF1[(1 + n*p)/2, (n*p)/2, 1, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 2*b^2*AppellF1[(1 + n*p)/2, (n*p)/2, 2, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])) + 2*b*(a^2 - b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 2, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]))/(a^3*(a^2 - b^2)*(1 + n*p)*(2 + n*p))) - (n*p*(Sec[e + f*x]^2)^((n*p)/2)*Tan[e + f*x]^2*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(n*p)*(-(a*(2 + n*p)*((a^2 + b^2)*AppellF1[(1 + n*p)/2, (n*p)/2, 1, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 2*b^2*AppellF1[(1 + n*p)/2, (n*p)/2, 2, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])) + 2*b*(a^2 - b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 2, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]))/(a^3*(a^2 - b^2)*(1 + n*p)*(2 + n*p)) - (n*p*(Sec[e + f*x]^2)^((n*p)/2)*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(-1 + n*p)*(-(a*(2 + n*p)*((a^2 + b^2)*AppellF1[(1 + n*p)/2, (n*p)/2, 1, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 2*b^2*AppellF1[(1 + n*p)/2, (n*p)/2, 2, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])) + 2*b*(a^2 - b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 2, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x])*(Sqrt[Sec[e + f*x]^2] - Tan[e + f*x]^2/Sqrt[Sec[e + f*x]^2]))/(a^3*(a^2 - b^2)*(1 + n*p)*(2 + n*p)) - ((Sec[e + f*x]^2)^((n*p)/2)*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(n*p)*(2*b*(a^2 - b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 2, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2 + 2*b*(a^2 - b^2)*(1 + n*p)*Tan[e + f*x]*((4*(-1 + b^2/a^2)*(1 + (n*p)/2)*AppellF1[2 + (n*p)/2, (-1 + n*p)/2, 3, 3 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 + (n*p)/2) - ((1 + (n*p)/2)*(-1 + n*p)*AppellF1[2 + (n*p)/2, 1 + (-1 + n*p)/2, 2, 3 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 + (n*p)/2)) - a*(2 + n*p)*((a^2 + b^2)*((2*(-1 + b^2/a^2)*(1 + n*p)*AppellF1[1 + (1 + n*p)/2, (n*p)/2, 2, 1 + (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + n*p) - (n*p*(1 + n*p)*AppellF1[1 + (1 + n*p)/2, 1 + (n*p)/2, 1, 1 + (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + n*p)) - 2*b^2*((4*(-1 + b^2/a^2)*(1 + n*p)*AppellF1[1 + (1 + n*p)/2, (n*p)/2, 3, 1 + (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + n*p) - (n*p*(1 + n*p)*AppellF1[1 + (1 + n*p)/2, 1 + (n*p)/2, 2, 1 + (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + n*p)))))/(a^3*(a^2 - b^2)*(1 + n*p)*(2 + n*p)))))","B",0
837,1,2570,428,20.5066339,"\int \frac{\left(c (d \sin (e+f x))^p\right)^n}{(a+b \sin (e+f x))^3} \, dx","Integrate[(c*(d*Sin[e + f*x])^p)^n/(a + b*Sin[e + f*x])^3,x]","\text{Result too large to show}","\frac{3 a^2 b \cos (e+f x) \sin ^2(e+f x)^{-\frac{n p}{2}} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};-\frac{n p}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}-\frac{3 a b^2 \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-n p-1)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (-n p-1),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}+\frac{b^3 \cos (e+f x) \sin ^2(e+f x)^{-\frac{n p}{2}} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (-n p-2),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}-\frac{a^3 \cot (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (1-n p)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (1-n p),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}",1,"-(((Sec[e + f*x]^2)^((n*p)/2)*(c*(d*Sin[e + f*x])^p)^n*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(n*p)*(-(a*(2 + n*p)*((a^2 + 3*b^2)*AppellF1[(1 + n*p)/2, -1 + (n*p)/2, 2, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 4*b^2*AppellF1[(1 + n*p)/2, -1 + (n*p)/2, 3, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])) + b*(3*a^2 + b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 2, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x] - 4*b^3*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 3, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]))/(a^4*(a^2 - b^2)*f*(1 + n*p)*(2 + n*p)*(a + b*Sin[e + f*x])^3*(-(((Sec[e + f*x]^2)^(1 + (n*p)/2)*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(n*p)*(-(a*(2 + n*p)*((a^2 + 3*b^2)*AppellF1[(1 + n*p)/2, -1 + (n*p)/2, 2, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 4*b^2*AppellF1[(1 + n*p)/2, -1 + (n*p)/2, 3, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])) + b*(3*a^2 + b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 2, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x] - 4*b^3*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 3, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]))/(a^4*(a^2 - b^2)*(1 + n*p)*(2 + n*p))) - (n*p*(Sec[e + f*x]^2)^((n*p)/2)*Tan[e + f*x]^2*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(n*p)*(-(a*(2 + n*p)*((a^2 + 3*b^2)*AppellF1[(1 + n*p)/2, -1 + (n*p)/2, 2, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 4*b^2*AppellF1[(1 + n*p)/2, -1 + (n*p)/2, 3, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])) + b*(3*a^2 + b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 2, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x] - 4*b^3*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 3, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]))/(a^4*(a^2 - b^2)*(1 + n*p)*(2 + n*p)) - (n*p*(Sec[e + f*x]^2)^((n*p)/2)*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(-1 + n*p)*(-(a*(2 + n*p)*((a^2 + 3*b^2)*AppellF1[(1 + n*p)/2, -1 + (n*p)/2, 2, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 4*b^2*AppellF1[(1 + n*p)/2, -1 + (n*p)/2, 3, (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])) + b*(3*a^2 + b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 2, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x] - 4*b^3*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 3, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x])*(Sqrt[Sec[e + f*x]^2] - Tan[e + f*x]^2/Sqrt[Sec[e + f*x]^2]))/(a^4*(a^2 - b^2)*(1 + n*p)*(2 + n*p)) - ((Sec[e + f*x]^2)^((n*p)/2)*Tan[e + f*x]*(Tan[e + f*x]/Sqrt[Sec[e + f*x]^2])^(n*p)*(b*(3*a^2 + b^2)*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 2, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2 - 4*b^3*(1 + n*p)*AppellF1[1 + (n*p)/2, (-1 + n*p)/2, 3, 2 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2 + b*(3*a^2 + b^2)*(1 + n*p)*Tan[e + f*x]*((4*(-1 + b^2/a^2)*(1 + (n*p)/2)*AppellF1[2 + (n*p)/2, (-1 + n*p)/2, 3, 3 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 + (n*p)/2) - ((1 + (n*p)/2)*(-1 + n*p)*AppellF1[2 + (n*p)/2, 1 + (-1 + n*p)/2, 2, 3 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 + (n*p)/2)) - 4*b^3*(1 + n*p)*Tan[e + f*x]*((6*(-1 + b^2/a^2)*(1 + (n*p)/2)*AppellF1[2 + (n*p)/2, (-1 + n*p)/2, 4, 3 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 + (n*p)/2) - ((1 + (n*p)/2)*(-1 + n*p)*AppellF1[2 + (n*p)/2, 1 + (-1 + n*p)/2, 3, 3 + (n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 + (n*p)/2)) - a*(2 + n*p)*((a^2 + 3*b^2)*((-2*(-1 + (n*p)/2)*(1 + n*p)*AppellF1[1 + (1 + n*p)/2, (n*p)/2, 2, 1 + (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + n*p) + (4*(-1 + b^2/a^2)*(1 + n*p)*AppellF1[1 + (1 + n*p)/2, -1 + (n*p)/2, 3, 1 + (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + n*p)) - 4*b^2*((-2*(-1 + (n*p)/2)*(1 + n*p)*AppellF1[1 + (1 + n*p)/2, (n*p)/2, 3, 1 + (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + n*p) + (6*(-1 + b^2/a^2)*(1 + n*p)*AppellF1[1 + (1 + n*p)/2, -1 + (n*p)/2, 4, 1 + (3 + n*p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 + n*p)))))/(a^4*(a^2 - b^2)*(1 + n*p)*(2 + n*p)))))","B",0