1,1,410,323,1.2873687,"\int \frac{\sin ^4(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[Sin[x]^4/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{-\frac{4 \left(2 i a^2 c^2-4 i a b^2 c-2 a b c \sqrt{4 a c-b^2}+b^3 \sqrt{4 a c-b^2}+i b^4\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b-i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}-\frac{4 \left(-2 i a^2 c^2+4 i a b^2 c-2 a b c \sqrt{4 a c-b^2}+b^3 \sqrt{4 a c-b^2}-i b^4\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b+i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}+2 c x (c-2 a)+4 b^2 x+4 b c \cos (x)-c^2 \sin (2 x)}{4 c^3}","-\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,"(4*b^2*x + 2*c*(-2*a + c)*x - (4*(I*b^4 - (4*I)*a*b^2*c + (2*I)*a^2*c^2 + b^3*Sqrt[-b^2 + 4*a*c] - 2*a*b*c*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b - I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]]) - (4*((-I)*b^4 + (4*I)*a*b^2*c - (2*I)*a^2*c^2 + b^3*Sqrt[-b^2 + 4*a*c] - 2*a*b*c*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b + I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]]) + 4*b*c*Cos[x] - c^2*Sin[2*x])/(4*c^3)","C",1
2,1,358,298,0.9627076,"\int \frac{\sin ^3(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[Sin[x]^3/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{\frac{\left(b^2 \sqrt{4 a c-b^2}-a c \sqrt{4 a c-b^2}-3 i a b c+i b^3\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b-i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}+\frac{\left(b^2 \sqrt{4 a c-b^2}-a c \sqrt{4 a c-b^2}+3 i a b c-i b^3\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b+i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}-b x-c \cos (x)}{c^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(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) + ((I*b^3 - (3*I)*a*b*c + b^2*Sqrt[-b^2 + 4*a*c] - a*c*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b - I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]]) + (((-I)*b^3 + (3*I)*a*b*c + b^2*Sqrt[-b^2 + 4*a*c] - a*c*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b + I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]]) - c*Cos[x])/c^2","C",1
3,1,310,253,0.6391245,"\int \frac{\sin ^2(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[Sin[x]^2/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{-\frac{\left(b \sqrt{4 a c-b^2}-2 i a c+i b^2\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b-i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}-\frac{\left(b \sqrt{4 a c-b^2}+2 i a c-i b^2\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b+i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}+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 - ((I*b^2 - (2*I)*a*c + b*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b - I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]]) - (((-I)*b^2 + (2*I)*a*c + b*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b + I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]]))/c","C",1
4,1,268,226,0.7154394,"\int \frac{\sin (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[Sin[x]/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{\frac{\left(\sqrt{4 a c-b^2}+i b\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b-i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}+\frac{\left(\sqrt{4 a c-b^2}-i b\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b+i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}}{\sqrt{2 a c-\frac{b^2}{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,"(((I*b + Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b - I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])])/Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]] + (((-I)*b + Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b + I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])])/Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])/Sqrt[-1/2*b^2 + 2*a*c]","C",1
5,1,233,221,0.5997839,"\int \frac{1}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[(a + b*Sin[x] + c*Sin[x]^2)^(-1),x]","-\frac{2 i c \left(\frac{\tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b-i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}-\frac{\tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b+i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{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*I)*c*(ArcTan[(2*c + (b - I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])]/Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]] - ArcTan[(2*c + (b + I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])]/Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]]))/Sqrt[-1/2*b^2 + 2*a*c]","C",1
6,1,306,244,1.2963106,"\int \frac{\csc (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[Csc[x]/(a + b*Sin[x] + c*Sin[x]^2),x]","-\frac{\frac{c \left(\sqrt{4 a c-b^2}-i b\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b-i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}+\frac{c \left(\sqrt{4 a c-b^2}+i b\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b+i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}-\log \left(\sin \left(\frac{x}{2}\right)\right)+\log \left(\cos \left(\frac{x}{2}\right)\right)}{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,"-(((c*((-I)*b + Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b - I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]]) + (c*(I*b + Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b + I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]]) + Log[Cos[x/2]] - Log[Sin[x/2]])/a)","C",1
7,1,388,271,1.2743778,"\int \frac{\csc ^2(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[Csc[x]^2/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{\csc ^2(x) (-2 a-2 b \sin (x)+c \cos (2 x)-c) \left(-\frac{2 c \left(b \sqrt{4 a c-b^2}+2 i a c-i b^2\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b-i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}+\frac{2 i c \left(i b \sqrt{4 a c-b^2}+2 a c-b^2\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b+i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}-a \tan \left(\frac{x}{2}\right)+a \cot \left(\frac{x}{2}\right)+2 b \log \left(\sin \left(\frac{x}{2}\right)\right)-2 b \log \left(\cos \left(\frac{x}{2}\right)\right)\right)}{4 a^2 \left(a \csc ^2(x)+b \csc (x)+c\right)}","\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,"(Csc[x]^2*(-2*a - c + c*Cos[2*x] - 2*b*Sin[x])*((-2*c*((-I)*b^2 + (2*I)*a*c + b*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b - I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]]) + ((2*I)*c*(-b^2 + 2*a*c + I*b*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b + I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]]) + a*Cot[x/2] - 2*b*Log[Cos[x/2]] + 2*b*Log[Sin[x/2]] - a*Tan[x/2]))/(4*a^2*(c + b*Csc[x] + a*Csc[x]^2))","C",1
8,1,481,331,1.6241996,"\int \frac{\csc ^3(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[Csc[x]^3/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{\csc ^2(x) (-2 a-2 b \sin (x)+c \cos (2 x)-c) \left(-4 \left(a^2-2 a c+2 b^2\right) \log \left(\sin \left(\frac{x}{2}\right)\right)+4 \left(a^2-2 a c+2 b^2\right) \log \left(\cos \left(\frac{x}{2}\right)\right)+a^2 \csc ^2\left(\frac{x}{2}\right)-a^2 \sec ^2\left(\frac{x}{2}\right)+\frac{8 c \left(b^2 \sqrt{4 a c-b^2}-a c \sqrt{4 a c-b^2}+3 i a b c-i b^3\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b-i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}+\frac{8 c \left(b^2 \sqrt{4 a c-b^2}-a c \sqrt{4 a c-b^2}-3 i a b c+i b^3\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b+i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}+4 a b \tan \left(\frac{x}{2}\right)-4 a b \cot \left(\frac{x}{2}\right)\right)}{16 a^3 \left(a \csc ^2(x)+b \csc (x)+c\right)}","-\frac{\left(b^2-a c\right) \tanh ^{-1}(\cos (x))}{a^3}-\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{b \cot (x)}{a^2}-\frac{\tanh ^{-1}(\cos (x))}{2 a}-\frac{\cot (x) \csc (x)}{2 a}",1,"(Csc[x]^2*(-2*a - c + c*Cos[2*x] - 2*b*Sin[x])*((8*c*((-I)*b^3 + (3*I)*a*b*c + b^2*Sqrt[-b^2 + 4*a*c] - a*c*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b - I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]]) + (8*c*(I*b^3 - (3*I)*a*b*c + b^2*Sqrt[-b^2 + 4*a*c] - a*c*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b + I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]]) - 4*a*b*Cot[x/2] + a^2*Csc[x/2]^2 + 4*(a^2 + 2*b^2 - 2*a*c)*Log[Cos[x/2]] - 4*(a^2 + 2*b^2 - 2*a*c)*Log[Sin[x/2]] - a^2*Sec[x/2]^2 + 4*a*b*Tan[x/2]))/(16*a^3*(c + b*Csc[x] + a*Csc[x]^2))","C",1
9,1,73,76,0.1164417,"\int \frac{\cos ^3(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[Cos[x]^3/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{\frac{2 \left(b^2-2 c (a+c)\right) \tanh ^{-1}\left(\frac{b+2 c \sin (x)}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}+b \log \left(a+b \sin (x)+c \sin ^2(x)\right)-2 c \sin (x)}{2 c^2}","\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,"((2*(b^2 - 2*c*(a + c))*ArcTanh[(b + 2*c*Sin[x])/Sqrt[b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c] + b*Log[a + b*Sin[x] + c*Sin[x]^2] - 2*c*Sin[x])/(2*c^2)","A",1
10,1,314,230,0.4790524,"\int \frac{\cos ^2(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[Cos[x]^2/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{\frac{\left(b \sqrt{4 a c-b^2}-2 i c (a+c)+i b^2\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b-i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}+\frac{\left(b \sqrt{4 a c-b^2}+2 i c (a+c)-i b^2\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b+i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}-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 + ((I*b^2 - (2*I)*c*(a + c) + b*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b - I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]]) + (((-I)*b^2 + (2*I)*c*(a + c) + b*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b + I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]]))/c","C",1
11,1,35,35,0.0141647,"\int \frac{\cos (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[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",1
12,1,119,128,0.2351784,"\int \frac{\sec (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[Sec[x]/(a + b*Sin[x] + c*Sin[x]^2),x]","-\frac{\sqrt{b^2-4 a c} \left(b \log \left(a+b \sin (x)+c \sin ^2(x)\right)+(a-b+c) \log (1-\sin (x))-(a+b+c) \log (\sin (x)+1)\right)+\left(4 c (a+c)-2 b^2\right) \tanh ^{-1}\left(\frac{b+2 c \sin (x)}{\sqrt{b^2-4 a c}}\right)}{2 (a-b+c) (a+b+c) \sqrt{b^2-4 a 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,"-1/2*((-2*b^2 + 4*c*(a + c))*ArcTanh[(b + 2*c*Sin[x])/Sqrt[b^2 - 4*a*c]] + Sqrt[b^2 - 4*a*c]*((a - b + c)*Log[1 - Sin[x]] - (a + b + c)*Log[1 + Sin[x]] + b*Log[a + b*Sin[x] + c*Sin[x]^2]))/((a - b + c)*(a + b + c)*Sqrt[b^2 - 4*a*c])","A",1
13,1,407,324,1.0080665,"\int \frac{\sec ^2(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[Sec[x]^2/(a + b*Sin[x] + c*Sin[x]^2),x]","-\frac{c \left(b \sqrt{4 a c-b^2}+2 i c (a+c)-i b^2\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b-i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \left(a^2+2 a c-b^2+c^2\right) \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}-\frac{c \left(b \sqrt{4 a c-b^2}-2 i c (a+c)+i b^2\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b+i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{2 a c-\frac{b^2}{2}} \left(a^2+2 a c-b^2+c^2\right) \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}+\frac{\sin \left(\frac{x}{2}\right)}{(a+b+c) \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)}+\frac{\sin \left(\frac{x}{2}\right)}{(a-b+c) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}","-\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,"-((c*((-I)*b^2 + (2*I)*c*(a + c) + b*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b - I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*(a^2 - b^2 + 2*a*c + c^2)*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])) - (c*(I*b^2 - (2*I)*c*(a + c) + b*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b + I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*(a^2 - b^2 + 2*a*c + c^2)*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]]) + Sin[x/2]/((a + b + c)*(Cos[x/2] - Sin[x/2])) + Sin[x/2]/((a - b + c)*(Cos[x/2] + Sin[x/2]))","C",1
14,1,202,206,0.7659418,"\int \frac{\sec ^3(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[Sec[x]^3/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{1}{4} \left(\frac{2 b \left(b^2-2 c (a+c)\right) \log \left(a+b \sin (x)+c \sin ^2(x)\right)}{\left(a^2+2 a c-b^2+c^2\right)^2}-\frac{4 \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{1}{(\sin (x)-1) (a+b+c)}-\frac{1}{(\sin (x)+1) (a-b+c)}-\frac{(a+2 b+3 c) \log (1-\sin (x))}{(a+b+c)^2}+\frac{(a-2 b+3 c) \log (\sin (x)+1)}{(a-b+c)^2}\right)","\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,"((-4*(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]])/(a + b + c)^2 + ((a - 2*b + 3*c)*Log[1 + Sin[x]])/(a - b + c)^2 + (2*b*(b^2 - 2*c*(a + c))*Log[a + b*Sin[x] + c*Sin[x]^2])/(a^2 - b^2 + 2*a*c + c^2)^2 - 1/((a + b + c)*(-1 + Sin[x])) - 1/((a - b + c)*(1 + Sin[x])))/4","A",1
15,1,15,21,0.0086472,"\int \frac{\cos (x)}{-6+\sin (x)+\sin ^2(x)} \, dx","Integrate[Cos[x]/(-6 + Sin[x] + Sin[x]^2),x]","-\frac{2}{5} \tanh ^{-1}\left(\frac{1}{5} (2 \sin (x)+1)\right)","\frac{1}{5} \log (2-\sin (x))-\frac{1}{5} \log (\sin (x)+3)",1,"(-2*ArcTanh[(1 + 2*Sin[x])/5])/5","A",1
16,1,26,17,0.0561931,"\int \frac{\cos (x)}{2-3 \sin (x)+\sin ^2(x)} \, dx","Integrate[Cos[x]/(2 - 3*Sin[x] + Sin[x]^2),x]","\log (2-\sin (x))-2 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)","\log (2-\sin (x))-\log (1-\sin (x))",1,"-2*Log[Cos[x/2] - Sin[x/2]] + Log[2 - Sin[x]]","A",1
17,1,30,21,0.0490803,"\int \frac{\cos (x)}{-5+4 \sin (x)+\sin ^2(x)} \, dx","Integrate[Cos[x]/(-5 + 4*Sin[x] + Sin[x]^2),x]","\frac{1}{6} \left(2 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log (\sin (x)+5)\right)","\frac{1}{6} \log (1-\sin (x))-\frac{1}{6} \log (\sin (x)+5)",1,"(2*Log[Cos[x/2] - Sin[x/2]] - Log[5 + Sin[x]])/6","A",1
18,1,9,9,0.0097635,"\int \frac{\cos (x)}{10-6 \sin (x)+\sin ^2(x)} \, dx","Integrate[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",1
19,1,5,5,0.0099283,"\int \frac{\cos (x)}{2+2 \sin (x)+\sin ^2(x)} \, dx","Integrate[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",1