1,1,77,121,0.1210729,"\int \sin ^3(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Integrate[Sin[e + f*x]^3*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","\frac{a^2 c (-15 \sin (2 (e+f x))-15 \sin (4 (e+f x))+5 \sin (6 (e+f x))-120 \cos (e+f x)-20 \cos (3 (e+f x))+12 \cos (5 (e+f x))+60 e+60 f x)}{960 f}","\frac{a^2 c \cos ^5(e+f x)}{5 f}-\frac{a^2 c \cos ^3(e+f x)}{3 f}+\frac{a^2 c \sin ^5(e+f x) \cos (e+f x)}{6 f}-\frac{a^2 c \sin ^3(e+f x) \cos (e+f x)}{24 f}-\frac{a^2 c \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} a^2 c x",1,"(a^2*c*(60*e + 60*f*x - 120*Cos[e + f*x] - 20*Cos[3*(e + f*x)] + 12*Cos[5*(e + f*x)] - 15*Sin[2*(e + f*x)] - 15*Sin[4*(e + f*x)] + 5*Sin[6*(e + f*x)]))/(960*f)","A",1
2,1,57,96,0.0902023,"\int \sin ^2(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Integrate[Sin[e + f*x]^2*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","\frac{a^2 c (-15 \sin (4 (e+f x))-60 \cos (e+f x)-10 \cos (3 (e+f x))+6 \cos (5 (e+f x))+60 e+60 f x)}{480 f}","\frac{a^2 c \cos ^5(e+f x)}{5 f}-\frac{a^2 c \cos ^3(e+f x)}{3 f}+\frac{a^2 c \sin ^3(e+f x) \cos (e+f x)}{4 f}-\frac{a^2 c \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a^2 c x",1,"(a^2*c*(60*e + 60*f*x - 60*Cos[e + f*x] - 10*Cos[3*(e + f*x)] + 6*Cos[5*(e + f*x)] - 15*Sin[4*(e + f*x)]))/(480*f)","A",1
3,1,47,77,0.1139244,"\int \sin (e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Integrate[Sin[e + f*x]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","\frac{a^2 c (-3 \sin (4 (e+f x))-24 \cos (e+f x)-8 \cos (3 (e+f x))+12 e+12 f x)}{96 f}","-\frac{a^2 c \cos ^3(e+f x)}{3 f}+\frac{a^2 c \sin ^3(e+f x) \cos (e+f x)}{4 f}-\frac{a^2 c \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a^2 c x",1,"(a^2*c*(12*e + 12*f*x - 24*Cos[e + f*x] - 8*Cos[3*(e + f*x)] - 3*Sin[4*(e + f*x)]))/(96*f)","A",1
4,1,43,52,0.3652401,"\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
5,1,61,63,0.0855803,"\int \csc (e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Integrate[Csc[e + f*x]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","\frac{a^2 c \left(\sin (2 (e+f x))+4 \cos (e+f x)+4 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-4 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)-2 e+2 f x\right)}{4 f}","\frac{a^2 c \cos (e+f x)}{f}+\frac{a^2 c \sin (e+f x) \cos (e+f x)}{2 f}-\frac{a^2 c \tanh ^{-1}(\cos (e+f x))}{f}+\frac{1}{2} a^2 c x",1,"(a^2*c*(-2*e + 2*f*x + 4*Cos[e + f*x] - 4*Log[Cos[(e + f*x)/2]] + 4*Log[Sin[(e + f*x)/2]] + Sin[2*(e + f*x)]))/(4*f)","A",1
6,1,97,53,0.0512128,"\int \csc ^2(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Integrate[Csc[e + f*x]^2*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","-\frac{a^2 c \sin (e) \sin (f x)}{f}+\frac{a^2 c \cos (e) \cos (f x)}{f}-\frac{a^2 c \cot (e+f x)}{f}+\frac{a^2 c \log \left(\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{f}-\frac{a^2 c \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{f}+a^2 (-c) x","\frac{a^2 c \cos (e+f x)}{f}-\frac{a^2 c \cot (e+f x)}{f}-\frac{a^2 c \tanh ^{-1}(\cos (e+f x))}{f}+a^2 (-c) x",1,"-(a^2*c*x) + (a^2*c*Cos[e]*Cos[f*x])/f - (a^2*c*Cot[e + f*x])/f - (a^2*c*Log[Cos[e/2 + (f*x)/2]])/f + (a^2*c*Log[Sin[e/2 + (f*x)/2]])/f - (a^2*c*Sin[e]*Sin[f*x])/f","A",1
7,1,95,64,0.7145229,"\int \csc ^3(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Integrate[Csc[e + f*x]^3*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","-\frac{a^2 c \left(-4 \tan \left(\frac{1}{2} (e+f x)\right)+4 \cot \left(\frac{1}{2} (e+f x)\right)+\csc ^2\left(\frac{1}{2} (e+f x)\right)-\sec ^2\left(\frac{1}{2} (e+f x)\right)+4 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-4 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)+8 e+8 f x\right)}{8 f}","-\frac{a^2 c \cot (e+f x)}{f}+\frac{a^2 c \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a^2 c \cot (e+f x) \csc (e+f x)}{2 f}+a^2 (-c) x",1,"-1/8*(a^2*c*(8*e + 8*f*x + 4*Cot[(e + f*x)/2] + Csc[(e + f*x)/2]^2 - 4*Log[Cos[(e + f*x)/2]] + 4*Log[Sin[(e + f*x)/2]] - Sec[(e + f*x)/2]^2 - 4*Tan[(e + f*x)/2]))/f","A",1
8,1,172,61,0.0717595,"\int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Integrate[Csc[e + f*x]^4*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","a^2 c \left(-\frac{\tan \left(\frac{1}{2} (e+f x)\right)}{6 f}+\frac{\cot \left(\frac{1}{2} (e+f x)\right)}{6 f}-\frac{\csc ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}-\frac{\log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}+\frac{\log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}-\frac{\cot \left(\frac{1}{2} (e+f x)\right) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{24 f}+\frac{\tan \left(\frac{1}{2} (e+f x)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{24 f}\right)","-\frac{a^2 c \cot ^3(e+f x)}{3 f}+\frac{a^2 c \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a^2 c \cot (e+f x) \csc (e+f x)}{2 f}",1,"a^2*c*(Cot[(e + f*x)/2]/(6*f) - Csc[(e + f*x)/2]^2/(8*f) - (Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2)/(24*f) + Log[Cos[(e + f*x)/2]]/(2*f) - Log[Sin[(e + f*x)/2]]/(2*f) + Sec[(e + f*x)/2]^2/(8*f) - Tan[(e + f*x)/2]/(6*f) + (Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(24*f))","B",1
9,1,179,86,0.061769,"\int \csc ^5(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Integrate[Csc[e + f*x]^5*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","\frac{a^2 c \cot (e+f x)}{3 f}-\frac{a^2 c \csc ^4\left(\frac{1}{2} (e+f x)\right)}{64 f}+\frac{a^2 c \csc ^2\left(\frac{1}{2} (e+f x)\right)}{32 f}+\frac{a^2 c \sec ^4\left(\frac{1}{2} (e+f x)\right)}{64 f}-\frac{a^2 c \sec ^2\left(\frac{1}{2} (e+f x)\right)}{32 f}-\frac{a^2 c \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{8 f}+\frac{a^2 c \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{8 f}-\frac{a^2 c \cot (e+f x) \csc ^2(e+f x)}{3 f}","-\frac{a^2 c \cot ^3(e+f x)}{3 f}+\frac{a^2 c \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{a^2 c \cot (e+f x) \csc ^3(e+f x)}{4 f}+\frac{a^2 c \cot (e+f x) \csc (e+f x)}{8 f}",1,"(a^2*c*Cot[e + f*x])/(3*f) + (a^2*c*Csc[(e + f*x)/2]^2)/(32*f) - (a^2*c*Csc[(e + f*x)/2]^4)/(64*f) - (a^2*c*Cot[e + f*x]*Csc[e + f*x]^2)/(3*f) + (a^2*c*Log[Cos[(e + f*x)/2]])/(8*f) - (a^2*c*Log[Sin[(e + f*x)/2]])/(8*f) - (a^2*c*Sec[(e + f*x)/2]^2)/(32*f) + (a^2*c*Sec[(e + f*x)/2]^4)/(64*f)","B",1
10,1,204,105,0.053157,"\int \csc ^6(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Integrate[Csc[e + f*x]^6*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","\frac{2 a^2 c \cot (e+f x)}{15 f}-\frac{a^2 c \csc ^4\left(\frac{1}{2} (e+f x)\right)}{64 f}+\frac{a^2 c \csc ^2\left(\frac{1}{2} (e+f x)\right)}{32 f}+\frac{a^2 c \sec ^4\left(\frac{1}{2} (e+f x)\right)}{64 f}-\frac{a^2 c \sec ^2\left(\frac{1}{2} (e+f x)\right)}{32 f}-\frac{a^2 c \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{8 f}+\frac{a^2 c \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{8 f}-\frac{a^2 c \cot (e+f x) \csc ^4(e+f x)}{5 f}+\frac{a^2 c \cot (e+f x) \csc ^2(e+f x)}{15 f}","-\frac{a^2 c \cot ^5(e+f x)}{5 f}-\frac{a^2 c \cot ^3(e+f x)}{3 f}+\frac{a^2 c \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{a^2 c \cot (e+f x) \csc ^3(e+f x)}{4 f}+\frac{a^2 c \cot (e+f x) \csc (e+f x)}{8 f}",1,"(2*a^2*c*Cot[e + f*x])/(15*f) + (a^2*c*Csc[(e + f*x)/2]^2)/(32*f) - (a^2*c*Csc[(e + f*x)/2]^4)/(64*f) + (a^2*c*Cot[e + f*x]*Csc[e + f*x]^2)/(15*f) - (a^2*c*Cot[e + f*x]*Csc[e + f*x]^4)/(5*f) + (a^2*c*Log[Cos[(e + f*x)/2]])/(8*f) - (a^2*c*Log[Sin[(e + f*x)/2]])/(8*f) - (a^2*c*Sec[(e + f*x)/2]^2)/(32*f) + (a^2*c*Sec[(e + f*x)/2]^4)/(64*f)","A",1
11,1,204,130,0.0552368,"\int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Integrate[Csc[e + f*x]^7*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","\frac{2 a^2 c \cot (e+f x)}{15 f}-\frac{a^2 c \csc ^6\left(\frac{1}{2} (e+f x)\right)}{384 f}+\frac{a^2 c \csc ^2\left(\frac{1}{2} (e+f x)\right)}{64 f}+\frac{a^2 c \sec ^6\left(\frac{1}{2} (e+f x)\right)}{384 f}-\frac{a^2 c \sec ^2\left(\frac{1}{2} (e+f x)\right)}{64 f}-\frac{a^2 c \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{16 f}+\frac{a^2 c \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{16 f}-\frac{a^2 c \cot (e+f x) \csc ^4(e+f x)}{5 f}+\frac{a^2 c \cot (e+f x) \csc ^2(e+f x)}{15 f}","-\frac{a^2 c \cot ^5(e+f x)}{5 f}-\frac{a^2 c \cot ^3(e+f x)}{3 f}+\frac{a^2 c \tanh ^{-1}(\cos (e+f x))}{16 f}-\frac{a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f}+\frac{a^2 c \cot (e+f x) \csc ^3(e+f x)}{24 f}+\frac{a^2 c \cot (e+f x) \csc (e+f x)}{16 f}",1,"(2*a^2*c*Cot[e + f*x])/(15*f) + (a^2*c*Csc[(e + f*x)/2]^2)/(64*f) - (a^2*c*Csc[(e + f*x)/2]^6)/(384*f) + (a^2*c*Cot[e + f*x]*Csc[e + f*x]^2)/(15*f) - (a^2*c*Cot[e + f*x]*Csc[e + f*x]^4)/(5*f) + (a^2*c*Log[Cos[(e + f*x)/2]])/(16*f) - (a^2*c*Log[Sin[(e + f*x)/2]])/(16*f) - (a^2*c*Sec[(e + f*x)/2]^2)/(64*f) + (a^2*c*Sec[(e + f*x)/2]^6)/(384*f)","A",1
12,1,101,128,0.8112085,"\int \sin ^2(c+d x) (a+a \sin (c+d x))^{3/2} (c-c \sin (c+d x)) \, dx","Integrate[Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2)*(c - c*Sin[c + d*x]),x]","\frac{a c \sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (-69 \sin (c+d x)+7 \sin (3 (c+d x))+30 \cos (2 (c+d x))-62)}{126 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{8 a^3 c \cos ^3(c+d x)}{63 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 a^2 c \cos ^3(c+d x)}{21 d \sqrt{a \sin (c+d x)+a}}-\frac{2 c \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{9 d}+\frac{4 a c \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{21 d}",1,"(a*c*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*Sqrt[a*(1 + Sin[c + d*x])]*(-62 + 30*Cos[2*(c + d*x)] - 69*Sin[c + d*x] + 7*Sin[3*(c + d*x)]))/(126*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
13,1,157,69,0.3521215,"\int \frac{\csc (e+f x) \sqrt{a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx","Integrate[(Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x]),x]","\frac{\sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right) \left(\log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)+\sin \left(\frac{1}{2} (e+f x)\right) \left(\log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)+2\right)}{c f}","\frac{2 \sec (e+f x) \sqrt{a \sin (e+f x)+a}}{c f}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{c f}",1,"(Sec[e + f*x]*(2 + Cos[(e + f*x)/2]*(-Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + (Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])])/(c*f)","B",1
14,1,234,120,0.4596978,"\int \frac{\csc (e+f x)}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))} \, dx","Integrate[Csc[e + f*x]/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])),x]","\frac{\cos (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right) \left(\log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)-\sin \left(\frac{1}{2} (e+f x)\right) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+\sin \left(\frac{1}{2} (e+f x)\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+(1+i) (-1)^{3/4} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \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)-1\right)}{c f (\sin (e+f x)-1) \sqrt{a (\sin (e+f x)+1)}}","\frac{\sec (e+f x) \sqrt{a \sin (e+f x)+a}}{a c f}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} c f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{2} \sqrt{a} c f}",1,"(Cos[e + f*x]*(-1 + Cos[(e + f*x)/2]*(Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[1 - 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])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[(e + f*x)/2] + Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[(e + f*x)/2]))/(c*f*(-1 + Sin[e + f*x])*Sqrt[a*(1 + Sin[e + f*x])])","C",1
15,1,194,103,0.9289389,"\int \frac{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x]),x]","\frac{2 e^{i (e+f x)} \sqrt{a (\sin (e+f x)+1)} \sqrt{g \sin (e+f x)} \left(2 \left(-1+e^{2 i (e+f x)}\right)-i \left(e^{i (e+f x)}-i\right) \sqrt{-1+e^{2 i (e+f x)}} \tan ^{-1}\left(\sqrt{-1+e^{2 i (e+f x)}}\right)-\left(e^{i (e+f x)}-i\right) \sqrt{-1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\frac{e^{i (e+f x)}}{\sqrt{-1+e^{2 i (e+f x)}}}\right)\right)}{c f \left(-1+e^{4 i (e+f x)}\right)}","\frac{2 \sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}{c f}+\frac{2 \sqrt{a} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right)}{c f}",1,"(2*E^(I*(e + f*x))*(2*(-1 + E^((2*I)*(e + f*x))) - I*(-I + E^(I*(e + f*x)))*Sqrt[-1 + E^((2*I)*(e + f*x))]*ArcTan[Sqrt[-1 + E^((2*I)*(e + f*x))]] - (-I + E^(I*(e + f*x)))*Sqrt[-1 + E^((2*I)*(e + f*x))]*ArcTanh[E^(I*(e + f*x))/Sqrt[-1 + E^((2*I)*(e + f*x))]])*Sqrt[g*Sin[e + f*x]]*Sqrt[a*(1 + Sin[e + f*x])])/(c*(-1 + E^((4*I)*(e + f*x)))*f)","C",1
16,1,40,43,0.2148922,"\int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c-c \sin (e+f x))} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c - c*Sin[e + f*x])),x]","\frac{2 \tan (e+f x) \sqrt{a (\sin (e+f x)+1)}}{c f \sqrt{g \sin (e+f x)}}","\frac{2 \sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}{c f g}",1,"(2*Sqrt[a*(1 + Sin[e + f*x])]*Tan[e + f*x])/(c*f*Sqrt[g*Sin[e + f*x]])","A",1
17,1,133,114,0.3183607,"\int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))} \, dx","Integrate[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])),x]","\frac{\sqrt{\sin (e+f x)} \csc (2 (e+f x)) \sqrt{a (\sin (e+f x)+1)} \sqrt{g \sin (e+f x)} \left(2 \sqrt{c} \sqrt{\sin (e+f x)}-\sqrt{2} \sqrt{c-c \sin (e+f x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{\sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}}\right)\right)}{a c^{3/2} f}","\frac{\sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}{a c f}+\frac{\sqrt{g} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right)}{\sqrt{2} \sqrt{a} c f}",1,"(Csc[2*(e + f*x)]*Sqrt[Sin[e + f*x]]*Sqrt[g*Sin[e + f*x]]*Sqrt[a*(1 + Sin[e + f*x])]*(2*Sqrt[c]*Sqrt[Sin[e + f*x]] - Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[Sin[e + f*x]])/Sqrt[c - c*Sin[e + f*x]]]*Sqrt[c - c*Sin[e + f*x]]))/(a*c^(3/2)*f)","A",1
18,1,132,118,0.2734599,"\int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))} \, dx","Integrate[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])),x]","\frac{\sin ^{\frac{3}{2}}(e+f x) \csc (2 (e+f x)) \sqrt{a (\sin (e+f x)+1)} \left(2 \sqrt{c} \sqrt{\sin (e+f x)}+\sqrt{2} \sqrt{c-c \sin (e+f x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{\sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}}\right)\right)}{a c^{3/2} f \sqrt{g \sin (e+f x)}}","\frac{\sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}{a c f g}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right)}{\sqrt{2} \sqrt{a} c f \sqrt{g}}",1,"(Csc[2*(e + f*x)]*Sin[e + f*x]^(3/2)*Sqrt[a*(1 + Sin[e + f*x])]*(2*Sqrt[c]*Sqrt[Sin[e + f*x]] + Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[Sin[e + f*x]])/Sqrt[c - c*Sin[e + f*x]]]*Sqrt[c - c*Sin[e + f*x]]))/(a*c^(3/2)*f*Sqrt[g*Sin[e + f*x]])","A",1
19,1,62,46,0.1079894,"\int \csc (e+f x) \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)} \, dx","Integrate[Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]],x]","\frac{\sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} \left(\log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{f}","\frac{\sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)} \log (\sin (e+f x))}{f}",1,"((Log[Cos[(e + f*x)/2]] + Log[Sin[(e + f*x)/2]])*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])/f","A",1
20,1,144,102,1.3343606,"\int \frac{\csc (e+f x) \sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(Csc[e + f*x]*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) \sqrt{a (\sin (e+f x)+1)} \left(i \left(\log \left(1-e^{2 i (e+f x)}\right)-\log \left(1+e^{2 i (e+f x)}\right)\right)+2 \tan ^{-1}\left(e^{i (e+f x)}\right)\right)}{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{\sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)} \log (\sin (e+f x))}{c f}-\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)))*(2*ArcTan[E^(I*(e + f*x))] + I*(Log[1 - E^((2*I)*(e + f*x))] - Log[1 + E^((2*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
21,1,145,100,1.3366004,"\int \frac{\csc (e+f x) \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(Csc[e + f*x]*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) \sqrt{c-c \sin (e+f x)} \left(2 \tan ^{-1}\left(e^{i (e+f x)}\right)-i \left(\log \left(1-e^{2 i (e+f x)}\right)-\log \left(1+e^{2 i (e+f x)}\right)\right)\right)}{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{\sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)} \log (\sin (e+f x))}{a f}-\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)))*(2*ArcTan[E^(I*(e + f*x))] - I*(Log[1 - E^((2*I)*(e + f*x))] - Log[1 + E^((2*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
22,1,63,46,0.182308,"\int \frac{\csc (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[Csc[e + f*x]/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]),x]","-\frac{\sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (\log (\cos (e+f x))-\log (\sin (e+f x)))}{a c f}","\frac{\cos (e+f x) \log (\tan (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"-(((Log[Cos[e + f*x]] - Log[Sin[e + f*x]])*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])/(a*c*f))","A",1
23,1,746,105,5.6465361,"\int \frac{\csc (e+f x) \sqrt{a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx","Integrate[(Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c + d*Sin[e + f*x]),x]","-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{a (\sin (e+f x)+1)} \left(\sqrt{d} \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right) \text{RootSum}\left[\text{$\#$1}^4 d e^{2 i e}+2 i \text{$\#$1}^2 c e^{i e}-d\&,\frac{\text{$\#$1}^3 \left(-\sqrt{d}\right) e^{i e} f x \sqrt{c+d}-2 i \text{$\#$1}^3 \sqrt{d} e^{i e} \sqrt{c+d} \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)+\frac{(1-i) \text{$\#$1}^2 c f x}{\sqrt{e^{-i e}}}+\frac{(2+2 i) \text{$\#$1}^2 c \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)}{\sqrt{e^{-i e}}}-i \text{$\#$1} \sqrt{d} f x \sqrt{c+d}+2 \text{$\#$1} \sqrt{d} \sqrt{c+d} \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)-(2-2 i) d \sqrt{e^{-i e}} \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)+(1+i) d \sqrt{e^{-i e}} f x}{\text{$\#$1}^2 (-c) e^{i e}-i d}\&\right]+\sqrt{d} \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right) \text{RootSum}\left[\text{$\#$1}^4 d e^{2 i e}+2 i \text{$\#$1}^2 c e^{i e}-d\&,\frac{-i \text{$\#$1}^3 \sqrt{d} e^{i e} f x \sqrt{c+d}+2 \text{$\#$1}^3 \sqrt{d} e^{i e} \sqrt{c+d} \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)-\frac{(1+i) \text{$\#$1}^2 c f x}{\sqrt{e^{-i e}}}+\frac{(2-2 i) \text{$\#$1}^2 c \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)}{\sqrt{e^{-i e}}}+\text{$\#$1} \sqrt{d} f x \sqrt{c+d}+2 i \text{$\#$1} \sqrt{d} \sqrt{c+d} \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)+(2+2 i) d \sqrt{e^{-i e}} \log \left(-\text{$\#$1}+e^{\frac{i f x}{2}}\right)+(1-i) d \sqrt{e^{-i e}} f x}{d-i \text{$\#$1}^2 c e^{i e}}\&\right]+(4+4 i) \sqrt{c+d} \left(\log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)\right)}{c 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} \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)}{c f \sqrt{c+d}}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{c f}",1,"((-1/8 + I/8)*((4 + 4*I)*Sqrt[c + d]*(Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + Sqrt[d]*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)/((-I)*d - c*E^(I*e)*#1^2) & ]*(Cos[e/2] + I*Sin[e/2]) + Sqrt[d]*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])])/(c*Sqrt[c + d]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","C",1
24,1,331,165,2.2365467,"\int \frac{\csc (e+f x)}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Integrate[Csc[e + f*x]/(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(d^{3/2} \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)-d^{3/2} \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+2 i) (-1)^{3/4} c \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)+d \sqrt{c+d} \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+(c-d) \sqrt{c+d} \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-c \sqrt{c+d} \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)}{c f (c-d) \sqrt{c+d} \sqrt{a (\sin (e+f x)+1)}}","-\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)}{\sqrt{a} c 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)}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} c f}",1,"-((((2 + 2*I)*(-1)^(3/4)*c*Sqrt[c + d]*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] + (c - d)*Sqrt[c + d]*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - c*Sqrt[c + d]*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + d*Sqrt[c + d]*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + d^(3/2)*Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] + Sqrt[d]*Cos[(e + f*x)/2] - Sqrt[d]*Sin[(e + f*x)/2])] - d^(3/2)*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*(c - d)*Sqrt[c + d]*f*Sqrt[a*(1 + Sin[e + f*x])]))","C",1
25,1,661,149,56.0974268,"\int \frac{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(c + d*Sin[e + f*x]),x]","\frac{\left(\frac{1}{2}+\frac{i}{2}\right) e^{-\frac{5}{2} i (e+f x)} \left(-1+e^{2 i (e+f x)}\right)^{5/2} \sqrt{a (\sin (e+f x)+1)} \sqrt{g \sin (e+f x)} \left(\left(\frac{c-d}{\sqrt{d^2-c^2}}+i\right) \sqrt{-1+e^{2 i (e+f x)}}+\left(\frac{d-c}{\sqrt{d^2-c^2}}+i\right) \sqrt{-1+e^{2 i (e+f x)}}+\frac{\left(\frac{c-d}{\sqrt{d^2-c^2}}+i\right) \left(\sqrt{2} \sqrt{c} \sqrt{c-i \sqrt{d^2-c^2}} \tan ^{-1}\left(\frac{d+\left(\sqrt{d^2-c^2}+i c\right) e^{i (e+f x)}}{\sqrt{2} \sqrt{c} \sqrt{c-i \sqrt{d^2-c^2}} \sqrt{-1+e^{2 i (e+f x)}}}\right)-\left(\sqrt{d^2-c^2}+i c\right) \tanh ^{-1}\left(\frac{e^{i (e+f x)}}{\sqrt{-1+e^{2 i (e+f x)}}}\right)\right)}{d}+\frac{\left(\frac{d-c}{\sqrt{d^2-c^2}}+i\right) \left(\sqrt{2} \sqrt{c} \sqrt{c+i \sqrt{d^2-c^2}} \tan ^{-1}\left(\frac{d-\left(\sqrt{d^2-c^2}-i c\right) e^{i (e+f x)}}{\sqrt{2} \sqrt{c} \sqrt{c+i \sqrt{d^2-c^2}} \sqrt{-1+e^{2 i (e+f x)}}}\right)+\left(\sqrt{d^2-c^2}-i c\right) \tanh ^{-1}\left(\frac{e^{i (e+f x)}}{\sqrt{-1+e^{2 i (e+f x)}}}\right)\right)}{d}-2 i \left(\sqrt{-1+e^{2 i (e+f x)}}-\tan ^{-1}\left(\sqrt{-1+e^{2 i (e+f x)}}\right)\right)\right)}{\sqrt{2} d f \left(-i e^{-i (e+f x)} \left(-1+e^{2 i (e+f x)}\right)\right)^{5/2} \sqrt{\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{a} \sqrt{c} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \sqrt{g} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right)}{d f \sqrt{c+d}}-\frac{2 \sqrt{a} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right)}{d f}",1,"((1/2 + I/2)*(-1 + E^((2*I)*(e + f*x)))^(5/2)*((I + (c - d)/Sqrt[-c^2 + d^2])*Sqrt[-1 + E^((2*I)*(e + f*x))] + (I + (-c + d)/Sqrt[-c^2 + d^2])*Sqrt[-1 + E^((2*I)*(e + f*x))] - (2*I)*(Sqrt[-1 + E^((2*I)*(e + f*x))] - ArcTan[Sqrt[-1 + E^((2*I)*(e + f*x))]]) + ((I + (-c + d)/Sqrt[-c^2 + d^2])*(Sqrt[2]*Sqrt[c]*Sqrt[c + I*Sqrt[-c^2 + d^2]]*ArcTan[(d - ((-I)*c + Sqrt[-c^2 + d^2])*E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[c]*Sqrt[c + I*Sqrt[-c^2 + d^2]]*Sqrt[-1 + E^((2*I)*(e + f*x))])] + ((-I)*c + Sqrt[-c^2 + d^2])*ArcTanh[E^(I*(e + f*x))/Sqrt[-1 + E^((2*I)*(e + f*x))]]))/d + ((I + (c - d)/Sqrt[-c^2 + d^2])*(Sqrt[2]*Sqrt[c]*Sqrt[c - I*Sqrt[-c^2 + d^2]]*ArcTan[(d + (I*c + Sqrt[-c^2 + d^2])*E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[c]*Sqrt[c - I*Sqrt[-c^2 + d^2]]*Sqrt[-1 + E^((2*I)*(e + f*x))])] - (I*c + Sqrt[-c^2 + d^2])*ArcTanh[E^(I*(e + f*x))/Sqrt[-1 + E^((2*I)*(e + f*x))]]))/d)*Sqrt[g*Sin[e + f*x]]*Sqrt[a*(1 + Sin[e + f*x])])/(Sqrt[2]*d*E^(((5*I)/2)*(e + f*x))*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^(5/2)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[Sin[e + f*x]])","C",0
26,1,436,83,54.7183709,"\int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","\frac{\left(\frac{1}{4}+\frac{i}{4}\right) g \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{3}{2} (e+f x)\right)-i \sin \left(\frac{3}{2} (e+f x)\right)\right) (i \sin (2 (e+f x))+\cos (2 (e+f x))-1)^{3/2} \left(\sqrt{c+i \sqrt{d^2-c^2}} \left(\sqrt{d^2-c^2}+i c-i d\right) \tan ^{-1}\left(\frac{d-\left(\sqrt{d^2-c^2}-i c\right) (\cos (e+f x)+i \sin (e+f x))}{\sqrt{2} \sqrt{c} \sqrt{c+i \sqrt{d^2-c^2}} \sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))-1}}\right)+\sqrt{c-i \sqrt{d^2-c^2}} \left(\sqrt{d^2-c^2}-i c+i d\right) \tan ^{-1}\left(\frac{d+\left(\sqrt{d^2-c^2}+i c\right) (\cos (e+f x)+i \sin (e+f x))}{\sqrt{2} \sqrt{c} \sqrt{c-i \sqrt{d^2-c^2}} \sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))-1}}\right)\right)}{\sqrt{2} \sqrt{c} d f \sqrt{d^2-c^2} (g \sin (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \sqrt{g} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right)}{\sqrt{c} f \sqrt{g} \sqrt{c+d}}",1,"((1/4 + I/4)*g*(Sqrt[c + I*Sqrt[-c^2 + d^2]]*(I*c - I*d + Sqrt[-c^2 + d^2])*ArcTan[(d - ((-I)*c + Sqrt[-c^2 + d^2])*(Cos[e + f*x] + I*Sin[e + f*x]))/(Sqrt[2]*Sqrt[c]*Sqrt[c + I*Sqrt[-c^2 + d^2]]*Sqrt[-1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]])] + Sqrt[c - I*Sqrt[-c^2 + d^2]]*((-I)*c + I*d + Sqrt[-c^2 + d^2])*ArcTan[(d + (I*c + Sqrt[-c^2 + d^2])*(Cos[e + f*x] + I*Sin[e + f*x]))/(Sqrt[2]*Sqrt[c]*Sqrt[c - I*Sqrt[-c^2 + d^2]]*Sqrt[-1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]])])*Sqrt[a*(1 + Sin[e + f*x])]*(Cos[(3*(e + f*x))/2] - I*Sin[(3*(e + f*x))/2])*(-1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])^(3/2))/(Sqrt[2]*Sqrt[c]*d*Sqrt[-c^2 + d^2]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(g*Sin[e + f*x])^(3/2))","C",0
27,1,61028,166,39.3957655,"\int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Integrate[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","\text{Result too large to show}","\frac{\sqrt{2} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right)}{\sqrt{a} f (c-d)}-\frac{2 \sqrt{c} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \sqrt{g} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right)}{\sqrt{a} f (c-d) \sqrt{c+d}}",1,"Result too large to show","C",0
28,1,99621,168,40.3665571,"\int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Integrate[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","\text{Result too large to show}","\frac{2 d \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \sqrt{g} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right)}{\sqrt{a} \sqrt{c} f \sqrt{g} (c-d) \sqrt{c+d}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right)}{\sqrt{a} f \sqrt{g} (c-d)}",1,"Result too large to show","C",0
29,1,611,238,6.6360278,"\int \frac{\csc (e+f x) \sqrt{a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx","Integrate[(Csc[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(c + c*Sin[e + f*x]),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) \sqrt{a+b \sin (e+f x)}}{f (c \sin (e+f x)+c)}+\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\frac{2 i b \cos (e+f x) \cos (2 (e+f x)) \sqrt{\frac{b-b \sin (e+f x)}{a+b}} \sqrt{-\frac{b \sin (e+f x)+b}{a-b}} \left(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)+b \left(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)-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)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(e+f x)} \left(-2 a^2+4 a (a+b \sin (e+f x))-2 (a+b \sin (e+f x))^2+b^2\right) \sqrt{-\frac{a^2-2 a (a+b \sin (e+f x))+(a+b \sin (e+f x))^2-b^2}{b^2}}}+\frac{2 \sin (2 (e+f x)) \cot (e+f x) \sqrt{a+b \sin (e+f x)}}{1-\sin ^2(e+f x)}-\frac{4 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)}{\sqrt{a+b \sin (e+f x)}}+\frac{2 (-4 a-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)}{\sqrt{a+b \sin (e+f x)}}\right)}{4 f (c \sin (e+f x)+c)}","\frac{\cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f (c \sin (e+f x)+c)}-\frac{(a-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)}{c 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)}{c f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}+\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)}{c f \sqrt{a+b \sin (e+f x)}}",1,"(-2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[a + b*Sin[e + f*x]])/(f*(c + c*Sin[e + f*x])) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*((-4*b*EllipticF[(-e + Pi/2 - f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/Sqrt[a + b*Sin[e + f*x]] + (2*(-4*a - b)*EllipticPi[2, (-e + Pi/2 - f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/Sqrt[a + b*Sin[e + f*x]] + ((2*I)*b*Cos[e + f*x]*Cos[2*(e + f*x)]*(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)]))*Sqrt[(b - b*Sin[e + f*x])/(a + b)]*Sqrt[-((b + b*Sin[e + f*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[e + f*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[e + f*x]) - 2*(a + b*Sin[e + f*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[e + f*x]) + (a + b*Sin[e + f*x])^2)/b^2)]) + (2*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sin[2*(e + f*x)])/(1 - Sin[e + f*x]^2)))/(4*f*(c + c*Sin[e + f*x]))","C",0
30,1,625,246,6.6636166,"\int \frac{\csc (e+f x)}{\sqrt{a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx","Integrate[Csc[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])),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) \sqrt{a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}-\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(-\frac{2 i b \cos (e+f x) \cos (2 (e+f x)) \sqrt{\frac{b-b \sin (e+f x)}{a+b}} \sqrt{-\frac{b \sin (e+f x)+b}{a-b}} \left(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)+b \left(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)-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)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(e+f x)} \left(-2 a^2+4 a (a+b \sin (e+f x))-2 (a+b \sin (e+f x))^2+b^2\right) \sqrt{-\frac{a^2-2 a (a+b \sin (e+f x))+(a+b \sin (e+f x))^2-b^2}{b^2}}}-\frac{2 \sin (2 (e+f x)) \cot (e+f x) \sqrt{a+b \sin (e+f x)}}{1-\sin ^2(e+f x)}+\frac{4 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)}{\sqrt{a+b \sin (e+f x)}}-\frac{2 (3 b-4 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)}{\sqrt{a+b \sin (e+f x)}}\right)}{4 f (a-b) (c \sin (e+f x)+c)}","\frac{\cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}-\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)}{c 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)}{c f (a-b) \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}+\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)}{c f \sqrt{a+b \sin (e+f x)}}",1,"(-2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[a + b*Sin[e + f*x]])/((a - b)*f*(c + c*Sin[e + f*x])) - ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*((4*b*EllipticF[(-e + Pi/2 - f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/Sqrt[a + b*Sin[e + f*x]] - (2*(-4*a + 3*b)*EllipticPi[2, (-e + Pi/2 - f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/Sqrt[a + b*Sin[e + f*x]] - ((2*I)*b*Cos[e + f*x]*Cos[2*(e + f*x)]*(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)]))*Sqrt[(b - b*Sin[e + f*x])/(a + b)]*Sqrt[-((b + b*Sin[e + f*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[e + f*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[e + f*x]) - 2*(a + b*Sin[e + f*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[e + f*x]) + (a + b*Sin[e + f*x])^2)/b^2)]) - (2*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sin[2*(e + f*x)])/(1 - Sin[e + f*x]^2)))/(4*(a - b)*f*(c + c*Sin[e + f*x]))","C",0
31,1,13199,267,34.4826117,"\int \frac{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(c + c*Sin[e + f*x]),x]","\text{Result too large to show}","\frac{g \sqrt{\frac{\sin (e+f x)}{\sin (e+f x)+1}} \sqrt{a+b \sin (e+f x)} E\left(\sin ^{-1}\left(\frac{\cos (e+f x)}{\sin (e+f x)+1}\right)|-\frac{a-b}{a+b}\right)}{c f \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}+\frac{2 \sqrt{g} \sec (e+f x) \sqrt{\frac{a (1-\sin (e+f x))}{a+b \sin (e+f x)}} \sqrt{\frac{a (\sin (e+f x)+1)}{a+b \sin (e+f x)}} (a+b \sin (e+f x)) \Pi \left(\frac{b}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{a+b \sin (e+f x)}}\right)|-\frac{a-b}{a+b}\right)}{c f \sqrt{a+b}}",1,"Result too large to show","C",0
32,1,4679,116,40.2471281,"\int \frac{\sqrt{a+b \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c+c \sin (e+f x))} \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + c*Sin[e + f*x])),x]","\text{Result too large to show}","-\frac{\sqrt{\frac{\sin (e+f x)}{\sin (e+f x)+1}} \sqrt{a+b \sin (e+f x)} E\left(\sin ^{-1}\left(\frac{\cos (e+f x)}{\sin (e+f x)+1}\right)|-\frac{a-b}{a+b}\right)}{c f \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}",1,"(-2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(f*Sqrt[g*Sin[e + f*x]]*(c + c*Sin[e + f*x])) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sqrt[Sin[e + f*x]]*((a*Sqrt[Sin[e + f*x]])/(2*Sqrt[a + b*Sin[e + f*x]]) - (b*Sqrt[Sin[e + f*x]])/(2*Sqrt[a + b*Sin[e + f*x]]) + (a*Cot[(e + f*x)/2]*Sqrt[Sin[e + f*x]])/(2*Sqrt[a + b*Sin[e + f*x]]) + (b*Cot[(e + f*x)/2]*Sqrt[Sin[e + f*x]])/(2*Sqrt[a + b*Sin[e + f*x]]) - (b*Cos[(3*(e + f*x))/2]*Csc[(e + f*x)/2]*Sqrt[Sin[e + f*x]])/(2*Sqrt[a + b*Sin[e + f*x]]) + (b*Csc[(e + f*x)/2]*Sqrt[Sin[e + f*x]]*Sin[(3*(e + f*x))/2])/(2*Sqrt[a + b*Sin[e + f*x]]))*Sqrt[(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + f*x)/2]^2)]*((Tan[(e + f*x)/2]*(1 + Tan[(e + f*x)/2]))/(1 + Tan[(e + f*x)/2]^2) + (Sqrt[-a^2 + b^2]*Sqrt[(a*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))/(a^2 - b^2)]*(EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2] + EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))))/(f*Sqrt[g*Sin[e + f*x]]*(c + c*Sin[e + f*x])*Sqrt[Tan[(e + f*x)/2]/(2 + 2*Tan[(e + f*x)/2]^2)]*(-1/2*(Sqrt[(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + f*x)/2]^2)]*((-2*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]^2)/(2 + 2*Tan[(e + f*x)/2]^2)^2 + Sec[(e + f*x)/2]^2/(2*(2 + 2*Tan[(e + f*x)/2]^2)))*((Tan[(e + f*x)/2]*(1 + Tan[(e + f*x)/2]))/(1 + Tan[(e + f*x)/2]^2) + (Sqrt[-a^2 + b^2]*Sqrt[(a*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))/(a^2 - b^2)]*(EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2] + EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))))/(Tan[(e + f*x)/2]/(2 + 2*Tan[(e + f*x)/2]^2))^(3/2) + (((b*Sec[(e + f*x)/2]^2 + a*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(1 + Tan[(e + f*x)/2]^2) - (Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))/(1 + Tan[(e + f*x)/2]^2)^2)*((Tan[(e + f*x)/2]*(1 + Tan[(e + f*x)/2]))/(1 + Tan[(e + f*x)/2]^2) + (Sqrt[-a^2 + b^2]*Sqrt[(a*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))/(a^2 - b^2)]*(EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2] + EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))))/(2*Sqrt[Tan[(e + f*x)/2]/(2 + 2*Tan[(e + f*x)/2]^2)]*Sqrt[(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + f*x)/2]^2)]) + (Sqrt[(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + f*x)/2]^2)]*(-((Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]^2*(1 + Tan[(e + f*x)/2]))/(1 + Tan[(e + f*x)/2]^2)^2) + (Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2*(1 + Tan[(e + f*x)/2]^2)) + (Sec[(e + f*x)/2]^2*(1 + Tan[(e + f*x)/2]))/(2*(1 + Tan[(e + f*x)/2]^2)) + (a*Sqrt[-a^2 + b^2]*(b*Sec[(e + f*x)/2]^2 + a*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])*(EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2] + EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(2*(a^2 - b^2)*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2)*Sqrt[(a*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))/(a^2 - b^2)]) - (Sqrt[-a^2 + b^2]*(b*Sec[(e + f*x)/2]^2 + a*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])*Sqrt[(a*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))/(a^2 - b^2)]*(EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2] + EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2)^2) - (a*Sqrt[-a^2 + b^2]*Sec[(e + f*x)/2]^2*Sqrt[(a*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))/(a^2 - b^2)]*(EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2] + EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(4*(-b + Sqrt[-a^2 + b^2])*((a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2]))^(3/2)*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2)) + (Sqrt[-a^2 + b^2]*Sqrt[(a*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))/(a^2 - b^2)]*((EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Sec[(e + f*x)/2]^2)/2 - (a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sec[(e + f*x)/2]^2*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])/(4*(b + Sqrt[-a^2 + b^2])*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]) + (a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sec[(e + f*x)/2]^2*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))])/(4*(-b + Sqrt[-a^2 + b^2])*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]) - (a*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]*Sqrt[1 - (-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])/(4*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/(2*Sqrt[-a^2 + b^2])]) + (a*Sec[(e + f*x)/2]^2*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))])/(4*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(2*Sqrt[-a^2 + b^2])]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])])))/(Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))))/Sqrt[Tan[(e + f*x)/2]/(2 + 2*Tan[(e + f*x)/2]^2)]))","B",0
33,1,5708,252,33.6169197,"\int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx","Integrate[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])),x]","\text{Result too large to show}","\frac{g \sqrt{\frac{\sin (e+f x)}{\sin (e+f x)+1}} \sqrt{a+b \sin (e+f x)} E\left(\sin ^{-1}\left(\frac{\cos (e+f x)}{\sin (e+f x)+1}\right)|-\frac{a-b}{a+b}\right)}{c f (a-b) \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}-\frac{2 \sqrt{g} \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{c f (a-b)}",1,"Result too large to show","B",0
34,1,1659,256,10.1070045,"\int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx","Integrate[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(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)^2 \sqrt{\sin (e+f x)} \left(\frac{4 a (a-b) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}} \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{(a+b) \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}+4 a^2 \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{b \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}\right)-2 b \left(\frac{\sqrt{a+b \sin (e+f x)} \cos (e+f x)}{b \sqrt{\sin (e+f x)}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{b \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}\right)}{b}+\frac{i \cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \csc (e+f x) E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{\sqrt{\sin (e+f x)}}\right)|-\frac{2 a}{-a-b}\right) \sqrt{a+b \sin (e+f x)}}{b \sqrt{\cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \csc (e+f x)} \sqrt{\frac{\csc (e+f x) (a+b \sin (e+f x))}{a+b}}}\right)+\frac{2 b \cot (e+f x) \left(\frac{\sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}{2 b}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\sin (e+f x)}}{\sqrt{a+b \sin (e+f x)}}\right)}{2 b^{3/2}}\right) \sin (2 (e+f x))}{1-\sin ^2(e+f x)}+\frac{2 a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\sin (e+f x)}}{\sqrt{a+b \sin (e+f x)}}\right) \cos ^2(e+f x)}{\sqrt{b} \left(1-\sin ^2(e+f x)\right)}\right)}{2 (a-b) f \sqrt{g \sin (e+f x)} (\sin (e+f x) c+c)}-\frac{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) \sin (e+f x) \sqrt{a+b \sin (e+f x)}}{(a-b) f \sqrt{g \sin (e+f x)} (\sin (e+f x) c+c)}","\frac{2 b \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{a c f \sqrt{g} (a-b)}-\frac{\sqrt{\frac{\sin (e+f x)}{\sin (e+f x)+1}} \sqrt{a+b \sin (e+f x)} E\left(\sin ^{-1}\left(\frac{\cos (e+f x)}{\sin (e+f x)+1}\right)|-\frac{a-b}{a+b}\right)}{c f (a-b) \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}",1,"(-2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*f*Sqrt[g*Sin[e + f*x]]*(c + c*Sin[e + f*x])) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sqrt[Sin[e + f*x]]*((4*a*(a - b)*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + (2*a*ArcTanh[(Sqrt[b]*Sqrt[Sin[e + f*x]])/Sqrt[a + b*Sin[e + f*x]]]*Cos[e + f*x]^2)/(Sqrt[b]*(1 - Sin[e + f*x]^2)) + 4*a^2*((Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])) - 2*b*((Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Sin[e + f*x]]) + (I*Cos[(-e + Pi/2 - f*x)/2]*Csc[e + f*x]*EllipticE[I*ArcSinh[Sin[(-e + Pi/2 - f*x)/2]/Sqrt[Sin[e + f*x]]], (-2*a)/(-a - b)]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Cos[(-e + Pi/2 - f*x)/2]^2*Csc[e + f*x]]*Sqrt[(Csc[e + f*x]*(a + b*Sin[e + f*x]))/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])))/b) + (2*b*Cot[e + f*x]*(-1/2*(a*ArcTanh[(Sqrt[b]*Sqrt[Sin[e + f*x]])/Sqrt[a + b*Sin[e + f*x]]])/b^(3/2) + (Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(2*b))*Sin[2*(e + f*x)])/(1 - Sin[e + f*x]^2)))/(2*(a - b)*f*Sqrt[g*Sin[e + f*x]]*(c + c*Sin[e + f*x]))","C",1
35,1,567,123,3.1691366,"\int \csc (e+f x) \sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \, dx","Integrate[Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]],x]","-\frac{\sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)+i \sin \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{c+d \sin (e+f x)} \left(\sqrt{c} \log \left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) e^{-\frac{i e}{2}} f \left(2 i \sqrt{c} \sqrt{2 c e^{i (e+f x)}-i d \left(-1+e^{2 i (e+f x)}\right)}-\sqrt{2} c \left(-1+e^{i (e+f x)}\right)-i \sqrt{2} d \left(1+e^{i (e+f x)}\right)\right)}{c^{3/2} \left(1+e^{i (e+f x)}\right)}\right)+\sqrt{c} \log \left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) e^{-\frac{i e}{2}} f \left(2 \sqrt{c} \sqrt{2 c e^{i (e+f x)}-i d \left(-1+e^{2 i (e+f x)}\right)}+\sqrt{2} c \left(1+e^{i (e+f x)}\right)-i \sqrt{2} d \left(-1+e^{i (e+f x)}\right)\right)}{c^{3/2} \left(-1+e^{i (e+f x)}\right)}\right)-i \sqrt{d} \left(\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)}{d^{3/2}}\right)-\log \left(\frac{(1+i) \sqrt{2} \left((1-i) \sqrt{d} \sqrt{(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}+c+d \sin (e+f x)-i d \cos (e+f x)\right)}{\sqrt{d}}\right)\right)\right)}{f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}}","-\frac{2 \sqrt{a} \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)}{f}-\frac{2 \sqrt{a} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{f}",1,"-(((Sqrt[c]*Log[((1/2 + I/2)*(-(Sqrt[2]*c*(-1 + E^(I*(e + f*x)))) - I*Sqrt[2]*d*(1 + E^(I*(e + f*x))) + (2*I)*Sqrt[c]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/(c^(3/2)*E^((I/2)*e)*(1 + E^(I*(e + f*x))))] + Sqrt[c]*Log[((1/2 + I/2)*((-I)*Sqrt[2]*d*(-1 + E^(I*(e + f*x))) + Sqrt[2]*c*(1 + E^(I*(e + f*x))) + 2*Sqrt[c]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/(c^(3/2)*E^((I/2)*e)*(-1 + E^(I*(e + f*x))))] - I*Sqrt[d]*(Log[(2*((-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)/(d^(3/2)*E^((I/2)*(e + 2*f*x)))] - Log[((1 + I)*Sqrt[2]*(c - I*d*Cos[e + f*x] + d*Sin[e + f*x] + (1 - I)*Sqrt[d]*Sqrt[(Cos[e + f*x] + I*Sin[e + f*x])*(c + d*Sin[e + f*x])]))/Sqrt[d]]))*(Cos[(e + f*x)/2] + I*Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c + d*Sin[e + f*x]])/(f*(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])]))","C",0
36,1,367,61,1.9692346,"\int \frac{\csc (e+f x) \sqrt{a+a \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/Sqrt[c + d*Sin[e + f*x]],x]","-\frac{\sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-i \sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\log \left(-\frac{(1+i) e^{\frac{i e}{2}} f \left(-2 i \sqrt{c} \sqrt{2 c e^{i (e+f x)}-i d \left(-1+e^{2 i (e+f x)}\right)}+\sqrt{2} c \left(-1+e^{i (e+f x)}\right)+i \sqrt{2} d \left(1+e^{i (e+f x)}\right)\right)}{\sqrt{c} \left(1+e^{i (e+f x)}\right)}\right)+\log \left(\frac{(1+i) e^{\frac{i e}{2}} f \left(2 \sqrt{c} \sqrt{2 c e^{i (e+f x)}-i d \left(-1+e^{2 i (e+f x)}\right)}+\sqrt{2} c \left(1+e^{i (e+f x)}\right)-i \sqrt{2} d \left(-1+e^{i (e+f x)}\right)\right)}{\sqrt{c} \left(-1+e^{i (e+f x)}\right)}\right)\right) \sqrt{(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}}{\sqrt{c} 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} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{c} f}",1,"-(((Log[((-1 - I)*E^((I/2)*e)*(Sqrt[2]*c*(-1 + E^(I*(e + f*x))) + I*Sqrt[2]*d*(1 + E^(I*(e + f*x))) - (2*I)*Sqrt[c]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/(Sqrt[c]*(1 + E^(I*(e + f*x))))] + Log[((1 + I)*E^((I/2)*e)*((-I)*Sqrt[2]*d*(-1 + E^(I*(e + f*x))) + Sqrt[2]*c*(1 + E^(I*(e + f*x))) + 2*Sqrt[c]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/(Sqrt[c]*(-1 + E^(I*(e + f*x))))])*(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[c]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]))","C",1
37,1,472502,140,35.0997958,"\int \frac{\csc (e+f x) \sqrt{c+d \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(Csc[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[a + a*Sin[e + f*x]],x]","\text{Result too large to show}","\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}-\frac{2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}",1,"Result too large to show","C",0
38,1,309729,140,34.4923894,"\int \frac{\csc (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[Csc[e + f*x]/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]),x]","\text{Result too large to show}","\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}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} \sqrt{c} f}",1,"Result too large to show","C",0
39,1,178,181,1.1195397,"\int \frac{\sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx","Integrate[Sin[e + f*x]^2/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])),x]","\frac{-\frac{2 a \left(a^2 c+a b d-2 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)^{3/2} (b c-a d)^2}-\frac{a^2 \cos (e+f x)}{(a-b) (a+b) (a d-b c) (a+b \sin (e+f x))}+\frac{2 c^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 a \left(a^2 c+a b d-2 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)^{3/2} (b c-a d)^2}+\frac{a^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))}+\frac{2 c^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*a*(a^2*c - 2*b^2*c + a*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)^2) + (2*c^2*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^2*Sqrt[c^2 - d^2]) - (a^2*Cos[e + f*x])/((a - b)*(a + b)*(-(b*c) + a*d)*(a + b*Sin[e + f*x])))/f","A",1
40,1,179,154,3.766457,"\int \frac{\csc (e+f x) \sqrt{c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx","Integrate[(Csc[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(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)}{d-c}} \left(\Pi \left(\frac{c+d}{c};i \sinh ^{-1}\left(\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right)|\frac{c+d}{c-d}\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)\right)}{a f \sqrt{-\frac{1}{c+d}}}","\frac{2 c \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(2;\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{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)}{a f (a+b) \sqrt{c+d \sin (e+f x)}}",1,"((2*I)*(EllipticPi[(c + d)/c, I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - 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)])/(a*Sqrt[-(c + d)^(-1)]*f)","C",1
41,1,203,146,4.0627841,"\int \frac{\csc (e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[Csc[e + f*x]/((a + b*Sin[e + f*x])*Sqrt[c + d*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 c) \Pi \left(\frac{c+d}{c};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 \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)}{a c f \sqrt{-\frac{1}{c+d}} (b c-a d)}","\frac{2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(2;\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{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)}{a f (a+b) \sqrt{c+d \sin (e+f x)}}",1,"((-2*I)*((-(b*c) + a*d)*EllipticPi[(c + d)/c, I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + b*c*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))])/(a*c*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*f)","C",1
42,1,23019,254,29.7961842,"\int \frac{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}{c+d \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(c + d*Sin[e + f*x]),x]","\text{Result too large to show}","\frac{2 \sqrt{g} \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{d f}-\frac{2 (b c-a d) \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{a \csc (e+f x)+b}{a+b}} \Pi \left(\frac{2 c}{c+d};\sin ^{-1}\left(\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right)|\frac{2 a}{a+b}\right)}{d f (c+d) \sqrt{a+b \sin (e+f x)}}",1,"Result too large to show","C",0
43,1,8202,250,29.2807895,"\int \frac{\sqrt{a+b \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","\text{Result too large to show}","\frac{2 (b c-a d) \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{a \csc (e+f x)+b}{a+b}} \Pi \left(\frac{2 c}{c+d};\sin ^{-1}\left(\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right)|\frac{2 a}{a+b}\right)}{c f g (c+d) \sqrt{a+b \sin (e+f x)}}-\frac{2 \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{c f \sqrt{g}}",1,"Result too large to show","B",0
44,1,3427,114,29.1756709,"\int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Integrate[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","\text{Result too large to show}","\frac{2 \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{a \csc (e+f x)+b}{a+b}} \Pi \left(\frac{2 c}{c+d};\sin ^{-1}\left(\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right)|\frac{2 a}{a+b}\right)}{f (c+d) \sqrt{a+b \sin (e+f x)}}",1,"(a*Sqrt[-a^2 + b^2]*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])])*Sqrt[Sin[e + f*x]]*Sqrt[g*Sin[e + f*x]]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)])/((b + Sqrt[-a^2 + b^2])^2*(b*c - a*d)*Sqrt[-c^2 + d^2]*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]*((a^2*Sqrt[-a^2 + b^2]*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])])*Sec[(e + f*x)/2]^2*Sqrt[Sin[e + f*x]]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)])/(4*(b + Sqrt[-a^2 + b^2])^3*(b*c - a*d)*Sqrt[-c^2 + d^2]*Sqrt[a + b*Sin[e + f*x]]*(-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])))^(3/2)) - (a*b*Sqrt[-a^2 + b^2]*Cos[e + f*x]*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])])*Sqrt[Sin[e + f*x]]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)])/(2*(b + Sqrt[-a^2 + b^2])^2*(b*c - a*d)*Sqrt[-c^2 + d^2]*(a + b*Sin[e + f*x])^(3/2)*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]) + (a*Sqrt[-a^2 + b^2]*Cos[e + f*x]*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])])*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)])/(2*(b + Sqrt[-a^2 + b^2])^2*(b*c - a*d)*Sqrt[-c^2 + d^2]*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]) + (a*Sqrt[-a^2 + b^2]*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])])*Sqrt[Sin[e + f*x]]*((a*b*Cos[e + f*x]*Sec[(e + f*x)/2]^2)/(a^2 - b^2) + (a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x])*Tan[(e + f*x)/2])/(a^2 - b^2)))/(2*(b + Sqrt[-a^2 + b^2])^2*(b*c - a*d)*Sqrt[-c^2 + d^2]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]) + (a*Sqrt[-a^2 + b^2]*Sqrt[Sin[e + f*x]]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*((a*(a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*Sec[(e + f*x)/2]^2)/(4*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(2*Sqrt[-a^2 + b^2])]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])]*(1 - (c*(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2]))/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]))) + (a*(-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*Sec[(e + f*x)/2]^2)/(4*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(2*Sqrt[-a^2 + b^2])]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])]*(1 - (c*(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2]))/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2]))))))/((b + Sqrt[-a^2 + b^2])^2*(b*c - a*d)*Sqrt[-c^2 + d^2]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))])))","B",0
45,1,4935,246,30.0563649,"\int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Integrate[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","\text{Result too large to show}","-\frac{2 d \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{a \csc (e+f x)+b}{a+b}} \Pi \left(\frac{2 c}{c+d};\sin ^{-1}\left(\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right)|\frac{2 a}{a+b}\right)}{c f g (c+d) \sqrt{a+b \sin (e+f x)}}-\frac{2 \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{a c f \sqrt{g}}",1,"(-4*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^4*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] - a*d*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]))*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])))^(3/2))/(a^2*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*f*Sin[e + f*x]^(3/2)*Sqrt[g*Sin[e + f*x]]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])*((3*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^2*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] - a*d*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]))*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))])/(a*(b + Sqrt[-a^2 + b^2])*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(3/2)*Sqrt[a + b*Sin[e + f*x]]) + (2*b*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^4*Cos[e + f*x]*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] - a*d*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]))*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])))^(3/2))/(a^2*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(3/2)*(a + b*Sin[e + f*x])^(3/2)) + (6*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^4*Cos[e + f*x]*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] - a*d*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]))*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])))^(3/2))/(a^2*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(5/2)*Sqrt[a + b*Sin[e + f*x]]) + (8*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^3*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] - a*d*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]))*Sin[(e + f*x)/2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])))^(3/2))/(a^2*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(3/2)*Sqrt[a + b*Sin[e + f*x]]) - (2*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^4*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] - a*d*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]))*(-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])))^(3/2)*((a*b*Cos[e + f*x]*Sec[(e + f*x)/2]^2)/(a^2 - b^2) + (a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x])*Tan[(e + f*x)/2])/(a^2 - b^2)))/(a^2*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(3/2)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]) - (4*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^4*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])))^(3/2)*(-1/2*(a*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*Sec[(e + f*x)/2]^2)/(Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(2*Sqrt[-a^2 + b^2])]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])]) - a*d*((a*(a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*Sec[(e + f*x)/2]^2)/(4*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(2*Sqrt[-a^2 + b^2])]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])]*(1 - (c*(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2]))/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]))) + (a*(-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*Sec[(e + f*x)/2]^2)/(4*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(2*Sqrt[-a^2 + b^2])]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])]*(1 - (c*(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2]))/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])))))))/(a^2*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(3/2)*Sqrt[a + b*Sin[e + f*x]])))","B",0
46,1,23019,254,30.5511404,"\int \frac{\sqrt{g \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(a + b*Sin[e + f*x]),x]","\text{Result too large to show}","\frac{2 (b c-a d) \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{c \csc (e+f x)+d}{c+d}} \Pi \left(\frac{2 a}{a+b};\sin ^{-1}\left(\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right)|\frac{2 c}{c+d}\right)}{b f (a+b) \sqrt{c+d \sin (e+f x)}}+\frac{2 \sqrt{g} \sqrt{c+d} \tan (e+f x) \sqrt{\frac{c (1-\csc (e+f x))}{c+d}} \sqrt{\frac{c (\csc (e+f x)+1)}{c-d}} \Pi \left(\frac{c+d}{d};\sin ^{-1}\left(\frac{\sqrt{g} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{g \sin (e+f x)}}\right)|-\frac{c+d}{c-d}\right)}{b f}",1,"Result too large to show","C",0
47,1,3429,114,28.767378,"\int \frac{\sqrt{g \sin (e+f x)}}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[Sqrt[g*Sin[e + f*x]]/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]),x]","\text{Result too large to show}","\frac{2 \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{c \csc (e+f x)+d}{c+d}} \Pi \left(\frac{2 a}{a+b};\sin ^{-1}\left(\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right)|\frac{2 c}{c+d}\right)}{f (a+b) \sqrt{c+d \sin (e+f x)}}",1,"-((c*Sqrt[-c^2 + d^2]*((-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) - Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])] + (a*c + (-b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) + Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])])*Sqrt[Sin[e + f*x]]*Sqrt[g*Sin[e + f*x]]*Sqrt[(c*Sec[(e + f*x)/2]^2*(c + d*Sin[e + f*x]))/(c^2 - d^2)])/(Sqrt[-a^2 + b^2]*(b*c - a*d)*(d + Sqrt[-c^2 + d^2])^2*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])*Sqrt[-((c*Tan[(e + f*x)/2])/(d + Sqrt[-c^2 + d^2]))]*(-1/4*(c^2*Sqrt[-c^2 + d^2]*((-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) - Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])] + (a*c + (-b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) + Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])])*Sec[(e + f*x)/2]^2*Sqrt[Sin[e + f*x]]*Sqrt[(c*Sec[(e + f*x)/2]^2*(c + d*Sin[e + f*x]))/(c^2 - d^2)])/(Sqrt[-a^2 + b^2]*(b*c - a*d)*(d + Sqrt[-c^2 + d^2])^3*Sqrt[c + d*Sin[e + f*x]]*(-((c*Tan[(e + f*x)/2])/(d + Sqrt[-c^2 + d^2])))^(3/2)) + (c*d*Sqrt[-c^2 + d^2]*Cos[e + f*x]*((-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) - Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])] + (a*c + (-b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) + Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])])*Sqrt[Sin[e + f*x]]*Sqrt[(c*Sec[(e + f*x)/2]^2*(c + d*Sin[e + f*x]))/(c^2 - d^2)])/(2*Sqrt[-a^2 + b^2]*(b*c - a*d)*(d + Sqrt[-c^2 + d^2])^2*(c + d*Sin[e + f*x])^(3/2)*Sqrt[-((c*Tan[(e + f*x)/2])/(d + Sqrt[-c^2 + d^2]))]) - (c*Sqrt[-c^2 + d^2]*Cos[e + f*x]*((-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) - Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])] + (a*c + (-b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) + Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])])*Sqrt[(c*Sec[(e + f*x)/2]^2*(c + d*Sin[e + f*x]))/(c^2 - d^2)])/(2*Sqrt[-a^2 + b^2]*(b*c - a*d)*(d + Sqrt[-c^2 + d^2])^2*Sqrt[Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*Sqrt[-((c*Tan[(e + f*x)/2])/(d + Sqrt[-c^2 + d^2]))]) - (c*Sqrt[-c^2 + d^2]*((-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) - Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])] + (a*c + (-b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) + Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])])*Sqrt[Sin[e + f*x]]*((c*d*Cos[e + f*x]*Sec[(e + f*x)/2]^2)/(c^2 - d^2) + (c*Sec[(e + f*x)/2]^2*(c + d*Sin[e + f*x])*Tan[(e + f*x)/2])/(c^2 - d^2)))/(2*Sqrt[-a^2 + b^2]*(b*c - a*d)*(d + Sqrt[-c^2 + d^2])^2*Sqrt[c + d*Sin[e + f*x]]*Sqrt[(c*Sec[(e + f*x)/2]^2*(c + d*Sin[e + f*x]))/(c^2 - d^2)]*Sqrt[-((c*Tan[(e + f*x)/2])/(d + Sqrt[-c^2 + d^2]))]) - (c*Sqrt[-c^2 + d^2]*Sqrt[Sin[e + f*x]]*Sqrt[(c*Sec[(e + f*x)/2]^2*(c + d*Sin[e + f*x]))/(c^2 - d^2)]*((c*(-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*Sec[(e + f*x)/2]^2)/(4*Sqrt[2]*Sqrt[-c^2 + d^2]*Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]*Sqrt[1 - (d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/(2*Sqrt[-c^2 + d^2])]*Sqrt[1 - (d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/(d + Sqrt[-c^2 + d^2])]*(1 - (a*(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2]))/(-(b*c) - Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])))) + (c*(a*c + (-b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*Sec[(e + f*x)/2]^2)/(4*Sqrt[2]*Sqrt[-c^2 + d^2]*Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]*Sqrt[1 - (d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/(2*Sqrt[-c^2 + d^2])]*Sqrt[1 - (d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/(d + Sqrt[-c^2 + d^2])]*(1 - (a*(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2]))/(-(b*c) + Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2]))))))/(Sqrt[-a^2 + b^2]*(b*c - a*d)*(d + Sqrt[-c^2 + d^2])^2*Sqrt[c + d*Sin[e + f*x]]*Sqrt[-((c*Tan[(e + f*x)/2])/(d + Sqrt[-c^2 + d^2]))]))))","B",0
48,1,274,391,0.267982,"\int \csc (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \, dx","Integrate[Csc[e + f*x]*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{(b c-a d) (\sin (e+f x)-1)}{(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))}} \left(\Pi \left(\frac{a (c+d)}{(a+b) c};\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)-\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)\right)}{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)}{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{a (c+d)}{(a+b) c};\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 \sqrt{a+b}}",1,"(-2*Sqrt[c + d]*(EllipticPi[(a*(c + d))/((a + b)*c), 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))] - 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]*f)","A",1
49,1,197,198,0.1464462,"\int \frac{\csc (e+f x) \sqrt{a+b \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx","Integrate[(Csc[e + f*x]*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{a (c+d)}{(a+b) c};\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)}{c 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{a (c+d)}{(a+b) c};\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)}{c f \sqrt{a+b}}",1,"(-2*Sqrt[c + d]*EllipticPi[(a*(c + d))/((a + b)*c), 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]*c*f)","A",1
50,1,374,398,2.3349008,"\int \frac{\csc (e+f x)}{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[Csc[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]),x]","\frac{2 \sec (e+f x) \left(-\frac{b (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))}} \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)}{b c-a d}-\frac{(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))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{a (c+d)}{(a+b) c};\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)}{c}\right)}{a f \sqrt{a+b} \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))}} 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)}{a f \sqrt{c+d} (b c-a d)}-\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{a (c+d)}{(a+b) c};\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)}{a c f \sqrt{a+b}}",1,"(2*Sec[e + f*x]*(-(((c + d)*EllipticPi[(a*(c + d))/((a + b)*c), 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))]*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]))/c) - (b*(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))]*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*c - a*d)))/(a*Sqrt[a + b]*Sqrt[c + d]*f)","A",1
51,1,168,157,1.1194525,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x))^p (c-c \sin (e+f x))^n \, dx","Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])^p*(c - c*Sin[e + f*x])^n,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} (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n (A+B \sin (e+f x))^p \left(\frac{A+B \sin (e+f x)}{A+B}\right)^{-p} F_1\left(n+\frac{1}{2};\frac{1}{2}-m,-p;n+\frac{3}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 B \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{A+B}\right)}{2 f n+f}","\frac{2^{n+\frac{1}{2}} \sec (e+f x) (1-\sin (e+f x))^{\frac{1}{2}-n} (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^n (A+B \sin (e+f x))^p \left(\frac{A+B \sin (e+f x)}{A-B}\right)^{-p} F_1\left(m+\frac{1}{2};\frac{1}{2}-n,-p;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{B (\sin (e+f x)+1)}{A-B}\right)}{a f (2 m+1)}",1,"(-2*AppellF1[1/2 + n, 1/2 - m, -p, 3/2 + n, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*B*Sin[(2*e - Pi + 2*f*x)/4]^2)/(A + B)]*Cot[(2*e + Pi + 2*f*x)/4]*(a*(1 + Sin[e + f*x]))^m*(A + B*Sin[e + f*x])^p*(c - c*Sin[e + f*x])^n*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(1/2 - m))/((f + 2*f*n)*((A + B*Sin[e + f*x])/(A + B))^p)","A",0