1,1,121,0,0.1723597,"\int \sin ^3(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Int[Sin[e + f*x]^3*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","\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","\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*x)/16 - (a^2*c*Cos[e + f*x]^3)/(3*f) + (a^2*c*Cos[e + f*x]^5)/(5*f) - (a^2*c*Cos[e + f*x]*Sin[e + f*x])/(16*f) - (a^2*c*Cos[e + f*x]*Sin[e + f*x]^3)/(24*f) + (a^2*c*Cos[e + f*x]*Sin[e + f*x]^5)/(6*f)","A",13,4,32,0.1250,1,"{2966, 2633, 2635, 8}"
2,1,96,0,0.142691,"\int \sin ^2(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Int[Sin[e + f*x]^2*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","\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","\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*x)/8 - (a^2*c*Cos[e + f*x]^3)/(3*f) + (a^2*c*Cos[e + f*x]^5)/(5*f) - (a^2*c*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a^2*c*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f)","A",11,4,32,0.1250,1,"{2966, 2635, 8, 2633}"
3,1,77,0,0.1024532,"\int \sin (e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Int[Sin[e + f*x]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","-\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","-\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*x)/8 - (a^2*c*Cos[e + f*x]^3)/(3*f) - (a^2*c*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a^2*c*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f)","A",10,5,30,0.1667,1,"{2966, 2638, 2635, 8, 2633}"
4,1,52,0,0.0601336,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Int[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","-\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","-\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,"(a^2*c*x)/2 - (a^2*c*Cos[e + f*x]^3)/(3*f) + (a^2*c*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",4,4,24,0.1667,1,"{2736, 2669, 2635, 8}"
5,1,63,0,0.0898987,"\int \csc (e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Int[Csc[e + f*x]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","\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","\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*x)/2 - (a^2*c*ArcTanh[Cos[e + f*x]])/f + (a^2*c*Cos[e + f*x])/f + (a^2*c*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",6,5,30,0.1667,1,"{2966, 3770, 2638, 2635, 8}"
6,1,53,0,0.1202185,"\int \csc ^2(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Int[Csc[e + f*x]^2*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),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","\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*ArcTanh[Cos[e + f*x]])/f + (a^2*c*Cos[e + f*x])/f - (a^2*c*Cot[e + f*x])/f","A",8,7,32,0.2188,1,"{2950, 2710, 2592, 321, 206, 3473, 8}"
7,1,64,0,0.1114569,"\int \csc ^3(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Int[Csc[e + f*x]^3*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","-\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","-\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,"-(a^2*c*x) + (a^2*c*ArcTanh[Cos[e + f*x]])/(2*f) - (a^2*c*Cot[e + f*x])/f - (a^2*c*Cot[e + f*x]*Csc[e + f*x])/(2*f)","A",7,5,32,0.1562,1,"{2966, 3770, 3767, 8, 3768}"
8,1,61,0,0.1620822,"\int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Int[Csc[e + f*x]^4*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","-\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}","-\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*ArcTanh[Cos[e + f*x]])/(2*f) - (a^2*c*Cot[e + f*x]^3)/(3*f) - (a^2*c*Cot[e + f*x]*Csc[e + f*x])/(2*f)","A",6,6,32,0.1875,1,"{2950, 2706, 2607, 30, 2611, 3770}"
9,1,86,0,0.1500487,"\int \csc ^5(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Int[Csc[e + f*x]^5*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","-\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}","-\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*ArcTanh[Cos[e + f*x]])/(8*f) - (a^2*c*Cot[e + f*x]^3)/(3*f) + (a^2*c*Cot[e + f*x]*Csc[e + f*x])/(8*f) - (a^2*c*Cot[e + f*x]*Csc[e + f*x]^3)/(4*f)","A",11,5,32,0.1562,1,"{2966, 3767, 8, 3768, 3770}"
10,1,105,0,0.1615476,"\int \csc ^6(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Int[Csc[e + f*x]^6*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","-\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}","-\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,"(a^2*c*ArcTanh[Cos[e + f*x]])/(8*f) - (a^2*c*Cot[e + f*x]^3)/(3*f) - (a^2*c*Cot[e + f*x]^5)/(5*f) + (a^2*c*Cot[e + f*x]*Csc[e + f*x])/(8*f) - (a^2*c*Cot[e + f*x]*Csc[e + f*x]^3)/(4*f)","A",11,4,32,0.1250,1,"{2966, 3768, 3770, 3767}"
11,1,130,0,0.1915304,"\int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Int[Csc[e + f*x]^7*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","-\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}","-\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,"(a^2*c*ArcTanh[Cos[e + f*x]])/(16*f) - (a^2*c*Cot[e + f*x]^3)/(3*f) - (a^2*c*Cot[e + f*x]^5)/(5*f) + (a^2*c*Cot[e + f*x]*Csc[e + f*x])/(16*f) + (a^2*c*Cot[e + f*x]*Csc[e + f*x]^3)/(24*f) - (a^2*c*Cot[e + f*x]*Csc[e + f*x]^5)/(6*f)","A",13,4,32,0.1250,1,"{2966, 3767, 3768, 3770}"
12,1,165,0,0.3471586,"\int \sin ^2(c+d x) (a+a \sin (c+d x))^{3/2} (c-c \sin (c+d x)) \, dx","Int[Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2)*(c - c*Sin[c + d*x]),x]","\frac{2 a^2 c \sin ^3(c+d x) \cos (c+d x)}{63 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^2 c \cos (c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}+\frac{2 a c \sin ^3(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{9 d}-\frac{2 c \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{21 d}+\frac{4 a c \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{63 d}","-\frac{2 a^2 c \cos ^3(c+d x)}{21 d \sqrt{a \sin (c+d x)+a}}-\frac{8 a^3 c \cos ^3(c+d x)}{63 d (a \sin (c+d x)+a)^{3/2}}-\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,"(-2*a^2*c*Cos[c + d*x])/(9*d*Sqrt[a + a*Sin[c + d*x]]) + (2*a^2*c*Cos[c + d*x]*Sin[c + d*x]^3)/(63*d*Sqrt[a + a*Sin[c + d*x]]) + (4*a*c*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(63*d) + (2*a*c*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(9*d) - (2*c*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(21*d)","A",5,5,34,0.1471,1,"{2976, 2981, 2759, 2751, 2646}"
13,1,69,0,0.3061029,"\int \frac{\csc (e+f x) \sqrt{a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx","Int[(Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x]),x]","\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}","\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,"(-2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(c*f) + (2*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f)","A",5,5,34,0.1471,1,"{2934, 2773, 206, 2736, 2673}"
14,1,120,0,0.4428562,"\int \frac{\csc (e+f x)}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))} \, dx","Int[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)+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}","\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,"(-2*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(Sqrt[a]*c*f) + ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])]/(Sqrt[2]*Sqrt[a]*c*f) + (Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(a*c*f)","A",8,7,34,0.2059,1,"{2940, 2736, 2673, 2985, 2649, 206, 2773}"
15,1,103,0,0.4681322,"\int \frac{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx","Int[(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x]),x]","\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}","\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*Sqrt[a]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(c*f) + (2*Sec[e + f*x]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(c*f)","A",6,6,40,0.1500,1,"{2928, 2775, 205, 2930, 12, 30}"
16,1,43,0,0.2029307,"\int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c-c \sin (e+f x))} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c - c*Sin[e + f*x])),x]","\frac{2 \sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}{c f g}","\frac{2 \sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}{c f g}",1,"(2*Sec[e + f*x]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(c*f*g)","A",3,3,40,0.07500,1,"{2930, 12, 30}"
17,1,114,0,0.4899687,"\int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))} \, dx","Int[Sqrt[g*Sin[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)+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}","\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,"(Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[2]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[2]*Sqrt[a]*c*f) + (Sec[e + f*x]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(a*c*f)","A",6,6,40,0.1500,1,"{2936, 2782, 205, 2930, 12, 30}"
18,1,118,0,0.4959291,"\int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))} \, dx","Int[1/(Sqrt[g*Sin[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)+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}}","\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,"-(ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[2]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])]/(Sqrt[2]*Sqrt[a]*c*f*Sqrt[g])) + (Sec[e + f*x]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(a*c*f*g)","A",6,6,40,0.1500,1,"{2938, 2782, 205, 2930, 12, 30}"
19,1,46,0,0.1707715,"\int \csc (e+f x) \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)} \, dx","Int[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)+a} \sqrt{c-c \sin (e+f x)} \log (\sin (e+f x))}{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[Sin[e + f*x]]*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])/f","A",2,2,36,0.05556,1,"{2948, 3475}"
20,1,102,0,0.4604312,"\int \frac{\csc (e+f x) \sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(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)+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)}}","\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,"-((a*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (Log[Sin[e + f*x]]*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])/(c*f)","A",6,6,36,0.1667,1,"{2942, 2737, 2667, 31, 2948, 3475}"
21,1,100,0,0.4473351,"\int \frac{\csc (e+f x) \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[(Csc[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/Sqrt[a + a*Sin[e + f*x]],x]","\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)}}","\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,"-((c*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (Log[Sin[e + f*x]]*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])/(a*f)","A",6,6,36,0.1667,1,"{2942, 2737, 2667, 31, 2948, 3475}"
22,1,46,0,0.1935032,"\int \frac{\csc (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx","Int[Csc[e + f*x]/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]),x]","\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)}}","\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,"(Cos[e + f*x]*Log[Tan[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",3,3,36,0.08333,1,"{2946, 2620, 29}"
23,1,105,0,0.288795,"\int \frac{\csc (e+f x) \sqrt{a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx","Int[(Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c + d*Sin[e + f*x]),x]","\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}","\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,"(-2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(c*f) + (2*Sqrt[a]*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(c*Sqrt[c + d]*f)","A",5,4,33,0.1212,1,"{2934, 2773, 206, 208}"
24,1,165,0,0.464996,"\int \frac{\csc (e+f x)}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Int[Csc[e + f*x]/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","-\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}","-\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*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(Sqrt[a]*c*f) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*f) - (2*d^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*c*(c - d)*Sqrt[c + d]*f)","A",8,6,33,0.1818,1,"{2940, 2773, 208, 2985, 2649, 206}"
25,1,149,0,0.5082144,"\int \frac{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx","Int[(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(c + d*Sin[e + f*x]),x]","\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}","\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,"(-2*Sqrt[a]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(d*f) + (2*Sqrt[a]*Sqrt[c]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(d*Sqrt[c + d]*f)","A",5,4,39,0.1026,1,"{2928, 2775, 205, 2930}"
26,1,83,0,0.2282177,"\int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","-\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}}","-\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,"(-2*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[c]*Sqrt[c + d]*f*Sqrt[g])","A",2,2,39,0.05128,1,"{2930, 205}"
27,1,166,0,0.5154233,"\int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Int[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","\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}}","\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,"(Sqrt[2]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[2]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*f) - (2*Sqrt[c]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*Sqrt[c + d]*f)","A",5,4,39,0.1026,1,"{2936, 2782, 205, 2930}"
28,1,168,0,0.536077,"\int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","\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)}","\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,"-((Sqrt[2]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[2]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*f*Sqrt[g])) + (2*d*ArcTan[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*Sqrt[c]*(c - d)*Sqrt[c + d]*f*Sqrt[g])","A",5,4,39,0.1026,1,"{2938, 2782, 205, 2930}"
29,1,238,0,0.5033747,"\int \frac{\csc (e+f x) \sqrt{a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx","Int[(Csc[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(c + c*Sin[e + f*x]),x]","\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)}}","\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,"(EllipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(c*f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) - ((a - b)*EllipticF[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(c*f*Sqrt[a + b*Sin[e + f*x]]) + (2*a*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(c*f*Sqrt[a + b*Sin[e + f*x]]) + (Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(f*(c + c*Sin[e + f*x]))","A",9,9,33,0.2727,1,"{2935, 2807, 2805, 2768, 2752, 2663, 2661, 2655, 2653}"
30,1,246,0,0.4838786,"\int \frac{\csc (e+f x)}{\sqrt{a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx","Int[Csc[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])),x]","\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)}}","\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,"(EllipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*c*f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) - (EllipticF[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(c*f*Sqrt[a + b*Sin[e + f*x]]) + (2*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(c*f*Sqrt[a + b*Sin[e + f*x]]) + (Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*f*(c + c*Sin[e + f*x]))","A",9,9,33,0.2727,1,"{2941, 2807, 2805, 2768, 2752, 2663, 2661, 2655, 2653}"
31,1,267,0,0.4995233,"\int \frac{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx","Int[(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(c + c*Sin[e + f*x]),x]","\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}}","\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,"(2*Sqrt[g]*EllipticPi[b/(a + b), ArcSin[(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])/(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])], -((a - b)/(a + b))]*Sec[e + f*x]*Sqrt[(a*(1 - Sin[e + f*x]))/(a + b*Sin[e + f*x])]*Sqrt[(a*(1 + Sin[e + f*x]))/(a + b*Sin[e + f*x])]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*c*f) + (g*EllipticE[ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))]*Sqrt[Sin[e + f*x]/(1 + Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]])/(c*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x]))])","A",3,3,39,0.07692,1,"{2928, 2811, 2932}"
32,1,116,0,0.2043397,"\int \frac{\sqrt{a+b \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c+c \sin (e+f x))} \, dx","Int[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + c*Sin[e + f*x])),x]","-\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)}}}","-\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,"-((EllipticE[ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))]*Sqrt[Sin[e + f*x]/(1 + Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]])/(c*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x]))]))","A",1,1,39,0.02564,1,"{2932}"
33,1,252,0,0.5112181,"\int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx","Int[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])),x]","\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)}","\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,"(g*EllipticE[ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))]*Sqrt[Sin[e + f*x]/(1 + Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*c*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x]))]) - (2*Sqrt[a + b]*Sqrt[g]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/((a - b)*c*f)","A",3,3,39,0.07692,1,"{2936, 2816, 2932}"
34,1,256,0,0.5199025,"\int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx","Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])),x]","\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)}}}","\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,"-((EllipticE[ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))]*Sqrt[Sin[e + f*x]/(1 + Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*c*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x]))])) + (2*b*Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(a*(a - b)*c*f*Sqrt[g])","A",3,3,39,0.07692,1,"{2938, 2816, 2932}"
35,1,123,0,0.4591745,"\int \csc (e+f x) \sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \, dx","Int[Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]],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}","-\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,"(-2*Sqrt[a]*Sqrt[d]*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/f - (2*Sqrt[a]*Sqrt[c]*ArcTanh[(Sqrt[a]*Sqrt[c]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/f","A",5,5,35,0.1429,1,"{2949, 2775, 205, 2943, 206}"
36,1,61,0,0.1880521,"\int \frac{\csc (e+f x) \sqrt{a+a \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[(Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/Sqrt[c + d*Sin[e + f*x]],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}","-\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,"(-2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[c]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[c]*f)","A",2,2,35,0.05714,1,"{2943, 206}"
37,1,140,0,0.5023799,"\int \frac{\csc (e+f x) \sqrt{c+d \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[(Csc[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[a + a*Sin[e + f*x]],x]","\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}","\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,"(-2*Sqrt[c]*ArcTanh[(Sqrt[a]*Sqrt[c]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f) + (Sqrt[2]*Sqrt[c - d]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f)","A",5,5,35,0.1429,1,"{2944, 2782, 208, 2943, 206}"
38,1,140,0,0.4707655,"\int \frac{\csc (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx","Int[Csc[e + f*x]/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]),x]","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f \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}","\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,"(-2*ArcTanh[(Sqrt[a]*Sqrt[c]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*Sqrt[c]*f) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*Sqrt[c - d]*f)","A",5,5,35,0.1429,1,"{2947, 2782, 208, 2943, 206}"
39,1,181,0,0.5087031,"\int \frac{\sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx","Int[Sin[e + f*x]^2/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])),x]","-\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}","-\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*f) + (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]*f) + (a^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x]))","A",8,5,33,0.1515,1,"{3056, 3001, 2660, 618, 204}"
40,1,154,0,0.486799,"\int \frac{\csc (e+f x) \sqrt{c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx","Int[(Csc[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(a + b*Sin[e + f*x]),x]","\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)}}","\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*c*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]]) - (2*(b*c - a*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])","A",5,3,33,0.09091,1,"{2935, 2807, 2805}"
41,1,146,0,0.4883433,"\int \frac{\csc (e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx","Int[Csc[e + f*x]/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]),x]","\frac{2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(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)}}","\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*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]]) - (2*b*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])","A",5,3,33,0.09091,1,"{2941, 2807, 2805}"
42,1,254,0,0.5202633,"\int \frac{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}{c+d \sin (e+f x)} \, dx","Int[(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(c + d*Sin[e + f*x]),x]","\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)}}","\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,"(2*Sqrt[a + b]*Sqrt[g]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticPi[(a + b)/b, ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(d*f) - (2*(b*c - a*d)*Sqrt[-Cot[e + f*x]^2]*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*a)/(a + b)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(d*(c + d)*f*Sqrt[a + b*Sin[e + f*x]])","A",3,3,39,0.07692,1,"{2929, 2809, 2937}"
43,1,250,0,0.5256476,"\int \frac{\sqrt{a+b \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Int[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","\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}}","\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,"(-2*Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(c*f*Sqrt[g]) + (2*(b*c - a*d)*Sqrt[-Cot[e + f*x]^2]*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*a)/(a + b)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(c*(c + d)*f*g*Sqrt[a + b*Sin[e + f*x]])","A",3,3,39,0.07692,1,"{2933, 2816, 2937}"
44,1,114,0,0.2067892,"\int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Int[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","\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)}}","\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,"(2*Sqrt[-Cot[e + f*x]^2]*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*a)/(a + b)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/((c + d)*f*Sqrt[a + b*Sin[e + f*x]])","A",1,1,39,0.02564,1,"{2937}"
45,1,246,0,0.5235023,"\int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","-\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}}","-\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,"(-2*Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(a*c*f*Sqrt[g]) - (2*d*Sqrt[-Cot[e + f*x]^2]*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*a)/(a + b)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(c*(c + d)*f*g*Sqrt[a + b*Sin[e + f*x]])","A",3,3,39,0.07692,1,"{2939, 2816, 2937}"
46,1,254,0,0.5046761,"\int \frac{\sqrt{g \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx","Int[(Sqrt[g*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(a + b*Sin[e + f*x]),x]","\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}","\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,"(2*Sqrt[c + d]*Sqrt[g]*Sqrt[(c*(1 - Csc[e + f*x]))/(c + d)]*Sqrt[(c*(1 + Csc[e + f*x]))/(c - d)]*EllipticPi[(c + d)/d, ArcSin[(Sqrt[g]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[g*Sin[e + f*x]])], -((c + d)/(c - d))]*Tan[e + f*x])/(b*f) + (2*(b*c - a*d)*Sqrt[-Cot[e + f*x]^2]*Sqrt[(d + c*Csc[e + f*x])/(c + d)]*EllipticPi[(2*a)/(a + b), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*c)/(c + d)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(b*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])","A",3,3,39,0.07692,1,"{2929, 2809, 2937}"
47,1,114,0,0.205679,"\int \frac{\sqrt{g \sin (e+f x)}}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx","Int[Sqrt[g*Sin[e + f*x]]/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]),x]","\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)}}","\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,"(2*Sqrt[-Cot[e + f*x]^2]*Sqrt[(d + c*Csc[e + f*x])/(c + d)]*EllipticPi[(2*a)/(a + b), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*c)/(c + d)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/((a + b)*f*Sqrt[c + d*Sin[e + f*x]])","A",1,1,39,0.02564,1,"{2937}"
48,1,391,0,0.5351881,"\int \csc (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \, dx","Int[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) (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}}","\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))]*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) + (2*Sqrt[c + d]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*f)","A",3,3,35,0.08571,1,"{2949, 2811, 2945}"
49,1,198,0,0.1956147,"\int \frac{\csc (e+f x) \sqrt{a+b \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[(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) (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",1,1,35,0.02857,1,"{2945}"
50,1,398,0,0.5548504,"\int \frac{\csc (e+f x)}{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx","Int[Csc[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]),x]","-\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}}","-\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*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]))/(a*Sqrt[a + b]*c*f) - (2*b*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(a*Sqrt[c + d]*(b*c - a*d)*f)","A",3,3,35,0.08571,1,"{2947, 2818, 2945}"
51,1,157,0,0.2781957,"\int (a+a \sin (e+f x))^m (A+B \sin (e+f x))^p (c-c \sin (e+f x))^n \, dx","Int[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])^p*(c - c*Sin[e + f*x])^n,x]","\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)}","\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^(1/2 + n)*AppellF1[1/2 + m, 1/2 - n, -p, 3/2 + m, (1 + Sin[e + f*x])/2, -((B*(1 + Sin[e + f*x]))/(A - B))]*Sec[e + f*x]*(1 - Sin[e + f*x])^(1/2 - n)*(a + a*Sin[e + f*x])^(1 + m)*(A + B*Sin[e + f*x])^p*(c - c*Sin[e + f*x])^n)/(a*f*(1 + 2*m)*((A + B*Sin[e + f*x])/(A - B))^p)","A",4,4,38,0.1053,1,"{3008, 140, 139, 138}"