1,1,323,0,3.1292922,"\int \frac{\sin ^4(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[Sin[x]^4/(a + b*Sin[x] + c*Sin[x]^2),x]","-\frac{\sqrt{2} \left(-\frac{2 a^2 c^2-4 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-2 a b c+b^3\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c^3 \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{\sqrt{2} \left(\frac{2 a^2 c^2-4 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-2 a b c+b^3\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c^3 \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{x \left(b^2-a c\right)}{c^3}+\frac{b \cos (x)}{c^2}+\frac{x}{2 c}-\frac{\sin (x) \cos (x)}{2 c}","-\frac{\sqrt{2} \left(-\frac{2 a^2 c^2-4 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-2 a b c+b^3\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c^3 \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{\sqrt{2} \left(\frac{2 a^2 c^2-4 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-2 a b c+b^3\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c^3 \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{x \left(b^2-a c\right)}{c^3}+\frac{b \cos (x)}{c^2}+\frac{x}{2 c}-\frac{\sin (x) \cos (x)}{2 c}",1,"x/(2*c) + ((b^2 - a*c)*x)/c^3 - (Sqrt[2]*(b^3 - 2*a*b*c - (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(c^3*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(b^3 - 2*a*b*c + (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(c^3*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) + (b*Cos[x])/c^2 - (Cos[x]*Sin[x])/(2*c)","A",12,8,19,0.4211,1,"{3256, 2638, 2635, 8, 3292, 2660, 618, 204}"
2,1,298,0,3.740614,"\int \frac{\sin ^3(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[Sin[x]^3/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{\sqrt{2} b \left(-\frac{b^2}{\sqrt{b^2-4 a c}}+\frac{3 a c}{\sqrt{b^2-4 a c}}-\frac{a c}{b}+b\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c^2 \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{\sqrt{2} b \left(\frac{b^2}{\sqrt{b^2-4 a c}}-\frac{3 a c}{\sqrt{b^2-4 a c}}-\frac{a c}{b}+b\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c^2 \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{b x}{c^2}-\frac{\cos (x)}{c}","\frac{\sqrt{2} b \left(-\frac{b^2}{\sqrt{b^2-4 a c}}+\frac{3 a c}{\sqrt{b^2-4 a c}}-\frac{a c}{b}+b\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c^2 \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{\sqrt{2} b \left(\frac{b^2}{\sqrt{b^2-4 a c}}-\frac{3 a c}{\sqrt{b^2-4 a c}}-\frac{a c}{b}+b\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c^2 \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{b x}{c^2}-\frac{\cos (x)}{c}",1,"-((b*x)/c^2) + (Sqrt[2]*b*(b - (a*c)/b - b^2/Sqrt[b^2 - 4*a*c] + (3*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(c^2*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*b*(b - (a*c)/b + b^2/Sqrt[b^2 - 4*a*c] - (3*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(c^2*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) - Cos[x]/c","A",10,6,19,0.3158,1,"{3256, 2638, 3292, 2660, 618, 204}"
3,1,253,0,1.0383605,"\int \frac{\sin ^2(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[Sin[x]^2/(a + b*Sin[x] + c*Sin[x]^2),x]","-\frac{\sqrt{2} \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{\sqrt{2} \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{x}{c}","-\frac{\sqrt{2} \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{\sqrt{2} \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{x}{c}",1,"x/c - (Sqrt[2]*(b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(c*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(c*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])","A",9,5,19,0.2632,1,"{3256, 3292, 2660, 618, 204}"
4,1,226,0,0.548391,"\int \frac{\sin (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[Sin[x]/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{\sqrt{2} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{\sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{\sqrt{2} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{\sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}","\frac{\sqrt{2} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{\sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{\sqrt{2} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{\sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}",1,"(Sqrt[2]*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]] + (Sqrt[2]*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]","A",8,4,17,0.2353,1,"{3256, 2660, 618, 204}"
5,1,221,0,0.3970461,"\int \frac{1}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[(a + b*Sin[x] + c*Sin[x]^2)^(-1),x]","\frac{2 \sqrt{2} c \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{2 \sqrt{2} c \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}","\frac{2 \sqrt{2} c \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{2 \sqrt{2} c \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}",1,"(2*Sqrt[2]*c*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(Sqrt[b^2 - 4*a*c]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) - (2*Sqrt[2]*c*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(Sqrt[b^2 - 4*a*c]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])","A",7,4,14,0.2857,1,"{3248, 2660, 618, 204}"
6,1,244,0,0.7621388,"\int \frac{\csc (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[Csc[x]/(a + b*Sin[x] + c*Sin[x]^2),x]","-\frac{\sqrt{2} c \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{a \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{\sqrt{2} c \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{a \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{\tanh ^{-1}(\cos (x))}{a}","-\frac{\sqrt{2} c \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{a \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{\sqrt{2} c \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{a \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{\tanh ^{-1}(\cos (x))}{a}",1,"-((Sqrt[2]*c*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(a*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])) - (Sqrt[2]*c*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(a*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) - ArcTanh[Cos[x]]/a","A",10,6,17,0.3529,1,"{3256, 3770, 3292, 2660, 618, 204}"
7,1,271,0,0.8950266,"\int \frac{\csc ^2(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[Csc[x]^2/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{\sqrt{2} b c \left(\frac{b^2-2 a c}{b \sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{a^2 \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{\sqrt{2} b c \left(1-\frac{b^2-2 a c}{b \sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{a^2 \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{b \tanh ^{-1}(\cos (x))}{a^2}-\frac{\cot (x)}{a}","\frac{\sqrt{2} b c \left(\frac{b^2-2 a c}{b \sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{a^2 \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{\sqrt{2} b c \left(1-\frac{b^2-2 a c}{b \sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{a^2 \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{b \tanh ^{-1}(\cos (x))}{a^2}-\frac{\cot (x)}{a}",1,"(Sqrt[2]*b*c*(1 + (b^2 - 2*a*c)/(b*Sqrt[b^2 - 4*a*c]))*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(a^2*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*b*c*(1 - (b^2 - 2*a*c)/(b*Sqrt[b^2 - 4*a*c]))*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(a^2*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) + (b*ArcTanh[Cos[x]])/a^2 - Cot[x]/a","A",12,8,19,0.4211,1,"{3256, 3770, 3767, 8, 3292, 2660, 618, 204}"
8,1,331,0,3.2101537,"\int \frac{\csc ^3(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[Csc[x]^3/(a + b*Sin[x] + c*Sin[x]^2),x]","-\frac{\sqrt{2} c \left(\sqrt{b^2-4 a c} \left(b^2-a c\right)-3 a b c+b^3\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{a^3 \sqrt{b^2-4 a c} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{\sqrt{2} c \left(-\sqrt{b^2-4 a c} \left(b^2-a c\right)-3 a b c+b^3\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{a^3 \sqrt{b^2-4 a c} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{\left(b^2-a c\right) \tanh ^{-1}(\cos (x))}{a^3}+\frac{b \cot (x)}{a^2}-\frac{\tanh ^{-1}(\cos (x))}{2 a}-\frac{\cot (x) \csc (x)}{2 a}","-\frac{\sqrt{2} c \left(\sqrt{b^2-4 a c} \left(b^2-a c\right)-3 a b c+b^3\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{a^3 \sqrt{b^2-4 a c} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{\sqrt{2} c \left(-\sqrt{b^2-4 a c} \left(b^2-a c\right)-3 a b c+b^3\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{a^3 \sqrt{b^2-4 a c} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{\left(b^2-a c\right) \tanh ^{-1}(\cos (x))}{a^3}+\frac{b \cot (x)}{a^2}-\frac{\tanh ^{-1}(\cos (x))}{2 a}-\frac{\cot (x) \csc (x)}{2 a}",1,"-((Sqrt[2]*c*(b^3 - 3*a*b*c + Sqrt[b^2 - 4*a*c]*(b^2 - a*c))*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(a^3*Sqrt[b^2 - 4*a*c]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])) + (Sqrt[2]*c*(b^3 - 3*a*b*c - Sqrt[b^2 - 4*a*c]*(b^2 - a*c))*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(a^3*Sqrt[b^2 - 4*a*c]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) - ArcTanh[Cos[x]]/(2*a) - ((b^2 - a*c)*ArcTanh[Cos[x]])/a^3 + (b*Cot[x])/a^2 - (Cot[x]*Csc[x])/(2*a)","A",14,9,19,0.4737,1,"{3256, 3770, 3767, 8, 3768, 3292, 2660, 618, 204}"
9,1,76,0,0.142085,"\int \frac{\cos ^3(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[Cos[x]^3/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{\left(b^2-2 c (a+c)\right) \tanh ^{-1}\left(\frac{b+2 c \sin (x)}{\sqrt{b^2-4 a c}}\right)}{c^2 \sqrt{b^2-4 a c}}+\frac{b \log \left(a+b \sin (x)+c \sin ^2(x)\right)}{2 c^2}-\frac{\sin (x)}{c}","\frac{\left(b^2-2 c (a+c)\right) \tanh ^{-1}\left(\frac{b+2 c \sin (x)}{\sqrt{b^2-4 a c}}\right)}{c^2 \sqrt{b^2-4 a c}}+\frac{b \log \left(a+b \sin (x)+c \sin ^2(x)\right)}{2 c^2}-\frac{\sin (x)}{c}",1,"((b^2 - 2*c*(a + c))*ArcTanh[(b + 2*c*Sin[x])/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]) + (b*Log[a + b*Sin[x] + c*Sin[x]^2])/(2*c^2) - Sin[x]/c","A",7,6,19,0.3158,1,"{3258, 1657, 634, 618, 206, 628}"
10,1,230,0,0.5851442,"\int \frac{\cos ^2(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[Cos[x]^2/(a + b*Sin[x] + c*Sin[x]^2),x]","-\frac{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2} \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c \sqrt{b^2-4 a c}}+\frac{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2} \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c \sqrt{b^2-4 a c}}-\frac{x}{c}","-\frac{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2} \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c \sqrt{b^2-4 a c}}+\frac{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2} \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{c \sqrt{b^2-4 a c}}-\frac{x}{c}",1,"-(x/c) - (Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(c*Sqrt[b^2 - 4*a*c]) + (Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(c*Sqrt[b^2 - 4*a*c])","A",9,5,19,0.2632,1,"{3266, 3292, 2660, 618, 204}"
11,1,35,0,0.0449551,"\int \frac{\cos (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[Cos[x]/(a + b*Sin[x] + c*Sin[x]^2),x]","-\frac{2 \tanh ^{-1}\left(\frac{b+2 c \sin (x)}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}","-\frac{2 \tanh ^{-1}\left(\frac{b+2 c \sin (x)}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}",1,"(-2*ArcTanh[(b + 2*c*Sin[x])/Sqrt[b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c]","A",3,3,17,0.1765,1,"{3258, 618, 206}"
12,1,128,0,0.1748499,"\int \frac{\sec (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[Sec[x]/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{\left(-2 a c+b^2-2 c^2\right) \tanh ^{-1}\left(\frac{b+2 c \sin (x)}{\sqrt{b^2-4 a c}}\right)}{(a-b+c) (a+b+c) \sqrt{b^2-4 a c}}-\frac{b \log \left(a+b \sin (x)+c \sin ^2(x)\right)}{2 (a-b+c) (a+b+c)}-\frac{\log (1-\sin (x))}{2 (a+b+c)}+\frac{\log (\sin (x)+1)}{2 (a-b+c)}","\frac{\left(-2 a c+b^2-2 c^2\right) \tanh ^{-1}\left(\frac{b+2 c \sin (x)}{\sqrt{b^2-4 a c}}\right)}{(a-b+c) (a+b+c) \sqrt{b^2-4 a c}}-\frac{b \log \left(a+b \sin (x)+c \sin ^2(x)\right)}{2 (a-b+c) (a+b+c)}-\frac{\log (1-\sin (x))}{2 (a+b+c)}+\frac{\log (\sin (x)+1)}{2 (a-b+c)}",1,"((b^2 - 2*a*c - 2*c^2)*ArcTanh[(b + 2*c*Sin[x])/Sqrt[b^2 - 4*a*c]])/((a - b + c)*(a + b + c)*Sqrt[b^2 - 4*a*c]) - Log[1 - Sin[x]]/(2*(a + b + c)) + Log[1 + Sin[x]]/(2*(a - b + c)) - (b*Log[a + b*Sin[x] + c*Sin[x]^2])/(2*(a - b + c)*(a + b + c))","A",9,8,17,0.4706,1,"{3258, 981, 634, 618, 206, 628, 633, 31}"
13,1,324,0,2.267671,"\int \frac{\sec ^2(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[Sec[x]^2/(a + b*Sin[x] + c*Sin[x]^2),x]","-\frac{\sqrt{2} b c \left(\frac{b^2-2 c (a+c)}{b \sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{(a-b+c) (a+b+c) \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{\sqrt{2} b c \left(1-\frac{b^2-2 c (a+c)}{b \sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{(a-b+c) (a+b+c) \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{\cos (x)}{2 (1-\sin (x)) (a+b+c)}-\frac{\cos (x)}{2 (\sin (x)+1) (a-b+c)}","-\frac{\sqrt{2} b c \left(\frac{b^2-2 c (a+c)}{b \sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{(a-b+c) (a+b+c) \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}-\frac{\sqrt{2} b c \left(1-\frac{b^2-2 c (a+c)}{b \sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{(a-b+c) (a+b+c) \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{\cos (x)}{2 (1-\sin (x)) (a+b+c)}-\frac{\cos (x)}{2 (\sin (x)+1) (a-b+c)}",1,"-((Sqrt[2]*b*c*(1 + (b^2 - 2*c*(a + c))/(b*Sqrt[b^2 - 4*a*c]))*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/((a - b + c)*(a + b + c)*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])) - (Sqrt[2]*b*c*(1 - (b^2 - 2*c*(a + c))/(b*Sqrt[b^2 - 4*a*c]))*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/((a - b + c)*(a + b + c)*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) + Cos[x]/(2*(a + b + c)*(1 - Sin[x])) - Cos[x]/(2*(a - b + c)*(1 + Sin[x]))","A",11,6,19,0.3158,1,"{3266, 2648, 3292, 2660, 618, 204}"
14,1,206,0,0.4998361,"\int \frac{\sec ^3(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[Sec[x]^3/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{b \left(b^2-2 c (a+c)\right) \log \left(a+b \sin (x)+c \sin ^2(x)\right)}{2 \left(a^2+2 a c-b^2+c^2\right)^2}-\frac{\left(-2 b^2 c (2 a+c)+2 c^2 (a+c)^2+b^4\right) \tanh ^{-1}\left(\frac{b+2 c \sin (x)}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(a^2+2 a c-b^2+c^2\right)^2}-\frac{(a+2 b+3 c) \log (1-\sin (x))}{4 (a+b+c)^2}+\frac{(a-2 b+3 c) \log (\sin (x)+1)}{4 (a-b+c)^2}-\frac{\sec ^2(x) (b-(a+c) \sin (x))}{2 (a-b+c) (a+b+c)}","\frac{b \left(b^2-2 c (a+c)\right) \log \left(a+b \sin (x)+c \sin ^2(x)\right)}{2 \left(a^2+2 a c-b^2+c^2\right)^2}-\frac{\left(-2 b^2 c (2 a+c)+2 c^2 (a+c)^2+b^4\right) \tanh ^{-1}\left(\frac{b+2 c \sin (x)}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(a^2+2 a c-b^2+c^2\right)^2}-\frac{(a+2 b+3 c) \log (1-\sin (x))}{4 (a+b+c)^2}+\frac{(a-2 b+3 c) \log (\sin (x)+1)}{4 (a-b+c)^2}-\frac{\sec ^2(x) (b-(a+c) \sin (x))}{2 (a-b+c) (a+b+c)}",1,"-(((b^4 + 2*c^2*(a + c)^2 - 2*b^2*c*(2*a + c))*ArcTanh[(b + 2*c*Sin[x])/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(a^2 - b^2 + 2*a*c + c^2)^2)) - ((a + 2*b + 3*c)*Log[1 - Sin[x]])/(4*(a + b + c)^2) + ((a - 2*b + 3*c)*Log[1 + Sin[x]])/(4*(a - b + c)^2) + (b*(b^2 - 2*c*(a + c))*Log[a + b*Sin[x] + c*Sin[x]^2])/(2*(a^2 - b^2 + 2*a*c + c^2)^2) - (Sec[x]^2*(b - (a + c)*Sin[x]))/(2*(a - b + c)*(a + b + c))","A",10,9,19,0.4737,1,"{3258, 976, 1074, 634, 618, 206, 628, 633, 31}"
15,1,21,0,0.0261708,"\int \frac{\cos (x)}{-6+\sin (x)+\sin ^2(x)} \, dx","Int[Cos[x]/(-6 + Sin[x] + Sin[x]^2),x]","\frac{1}{5} \log (2-\sin (x))-\frac{1}{5} \log (\sin (x)+3)","\frac{1}{5} \log (2-\sin (x))-\frac{1}{5} \log (\sin (x)+3)",1,"Log[2 - Sin[x]]/5 - Log[3 + Sin[x]]/5","A",4,3,13,0.2308,1,"{3258, 616, 31}"
16,1,17,0,0.0270485,"\int \frac{\cos (x)}{2-3 \sin (x)+\sin ^2(x)} \, dx","Int[Cos[x]/(2 - 3*Sin[x] + Sin[x]^2),x]","\log (2-\sin (x))-\log (1-\sin (x))","\log (2-\sin (x))-\log (1-\sin (x))",1,"-Log[1 - Sin[x]] + Log[2 - Sin[x]]","A",4,3,15,0.2000,1,"{3258, 616, 31}"
17,1,21,0,0.0276304,"\int \frac{\cos (x)}{-5+4 \sin (x)+\sin ^2(x)} \, dx","Int[Cos[x]/(-5 + 4*Sin[x] + Sin[x]^2),x]","\frac{1}{6} \log (1-\sin (x))-\frac{1}{6} \log (\sin (x)+5)","\frac{1}{6} \log (1-\sin (x))-\frac{1}{6} \log (\sin (x)+5)",1,"Log[1 - Sin[x]]/6 - Log[5 + Sin[x]]/6","A",4,3,15,0.2000,1,"{3258, 616, 31}"
18,1,9,0,0.0285221,"\int \frac{\cos (x)}{10-6 \sin (x)+\sin ^2(x)} \, dx","Int[Cos[x]/(10 - 6*Sin[x] + Sin[x]^2),x]","-\tan ^{-1}(3-\sin (x))","-\tan ^{-1}(3-\sin (x))",1,"-ArcTan[3 - Sin[x]]","A",3,3,15,0.2000,1,"{3258, 618, 204}"
19,1,5,0,0.026073,"\int \frac{\cos (x)}{2+2 \sin (x)+\sin ^2(x)} \, dx","Int[Cos[x]/(2 + 2*Sin[x] + Sin[x]^2),x]","\tan ^{-1}(\sin (x)+1)","\tan ^{-1}(\sin (x)+1)",1,"ArcTan[1 + Sin[x]]","A",3,3,15,0.2000,1,"{3258, 617, 204}"