1,1,239,136,0.5681253,"\int \frac{\sin ^5(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Sin[x]^5/(a + b*Cos[x] + c*Cos[x]^2),x]","\frac{3 c \cos (x) \left(c (4 a+7 c)-4 b^2\right)+\frac{6 \left(2 b^2 c (2 a+c)-2 b c (a+c) \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-2 c^2 (a+c)^2-b^4\right) \log \left(\sqrt{b^2-4 a c}-b-2 c \cos (x)\right)}{\sqrt{b^2-4 a c}}+\frac{6 \left(-2 b^2 c (2 a+c)-2 b c (a+c) \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+2 c^2 (a+c)^2+b^4\right) \log \left(\sqrt{b^2-4 a c}+b+2 c \cos (x)\right)}{\sqrt{b^2-4 a c}}+3 b c^2 \cos (2 x)+c^3 (-\cos (3 x))}{12 c^4}","\frac{b \left(b^2-2 c (a+c)\right) \log \left(a+b \cos (x)+c \cos ^2(x)\right)}{2 c^4}-\frac{\cos (x) \left(b^2-c (a+2 c)\right)}{c^3}+\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 \cos (x)}{\sqrt{b^2-4 a c}}\right)}{c^4 \sqrt{b^2-4 a c}}+\frac{b \cos ^2(x)}{2 c^2}-\frac{\cos ^3(x)}{3 c}",1,"(3*c*(-4*b^2 + c*(4*a + 7*c))*Cos[x] + 3*b*c^2*Cos[2*x] - c^3*Cos[3*x] + (6*(-b^4 - 2*c^2*(a + c)^2 + 2*b^2*c*(2*a + c) + b^3*Sqrt[b^2 - 4*a*c] - 2*b*c*(a + c)*Sqrt[b^2 - 4*a*c])*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*Cos[x]])/Sqrt[b^2 - 4*a*c] + (6*(b^4 + 2*c^2*(a + c)^2 - 2*b^2*c*(2*a + c) + b^3*Sqrt[b^2 - 4*a*c] - 2*b*c*(a + c)*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*Cos[x]])/Sqrt[b^2 - 4*a*c])/(12*c^4)","A",1
2,1,131,76,0.2706656,"\int \frac{\sin ^3(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Sin[x]^3/(a + b*Cos[x] + c*Cos[x]^2),x]","\frac{2 c \cos (x) \sqrt{b^2-4 a c}+\left(-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2\right) \log \left(\sqrt{b^2-4 a c}-b-2 c \cos (x)\right)-\left(b \sqrt{b^2-4 a c}-2 c (a+c)+b^2\right) \log \left(\sqrt{b^2-4 a c}+b+2 c \cos (x)\right)}{2 c^2 \sqrt{b^2-4 a c}}","-\frac{\left(b^2-2 c (a+c)\right) \tanh ^{-1}\left(\frac{b+2 c \cos (x)}{\sqrt{b^2-4 a c}}\right)}{c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left(a+b \cos (x)+c \cos ^2(x)\right)}{2 c^2}+\frac{\cos (x)}{c}",1,"(2*c*Sqrt[b^2 - 4*a*c]*Cos[x] + (b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c])*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*Cos[x]] - (b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*Cos[x]])/(2*c^2*Sqrt[b^2 - 4*a*c])","A",1
3,1,39,35,0.033711,"\int \frac{\sin (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Sin[x]/(a + b*Cos[x] + c*Cos[x]^2),x]","-\frac{2 \tan ^{-1}\left(\frac{b+2 c \cos (x)}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}","\frac{2 \tanh ^{-1}\left(\frac{b+2 c \cos (x)}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}",1,"(-2*ArcTan[(b + 2*c*Cos[x])/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c]","A",1
4,1,126,129,0.1985109,"\int \frac{\csc (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Csc[x]/(a + b*Cos[x] + c*Cos[x]^2),x]","-\frac{\sqrt{4 a c-b^2} \left(-b \log \left(a+b \cos (x)+c \cos ^2(x)\right)-((a-b+c) \log (1-\cos (x)))+(a+b+c) \log (\cos (x)+1)\right)+\left(4 c (a+c)-2 b^2\right) \tan ^{-1}\left(\frac{b+2 c \cos (x)}{\sqrt{4 a c-b^2}}\right)}{2 (a-b+c) (a+b+c) \sqrt{4 a c-b^2}}","-\frac{\left(-2 a c+b^2-2 c^2\right) \tanh ^{-1}\left(\frac{b+2 c \cos (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 \cos (x)+c \cos ^2(x)\right)}{2 (a-b+c) (a+b+c)}+\frac{\log (1-\cos (x))}{2 (a+b+c)}-\frac{\log (\cos (x)+1)}{2 (a-b+c)}",1,"-1/2*((-2*b^2 + 4*c*(a + c))*ArcTan[(b + 2*c*Cos[x])/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*(-((a - b + c)*Log[1 - Cos[x]]) + (a + b + c)*Log[1 + Cos[x]] - b*Log[a + b*Cos[x] + c*Cos[x]^2]))/((a - b + c)*(a + b + c)*Sqrt[-b^2 + 4*a*c])","A",1
5,1,392,205,2.3256046,"\int \frac{\csc ^3(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Csc[x]^3/(a + b*Cos[x] + c*Cos[x]^2),x]","\frac{1}{8} \left(-\frac{4 \left(-2 b^2 c (2 a+c)-2 b c (a+c) \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+2 c^2 (a+c)^2+b^4\right) \log \left(\sqrt{b^2-4 a c}-b-2 c \cos (x)\right)}{\sqrt{b^2-4 a c} \left(a^2+2 a c-b^2+c^2\right)^2}-\frac{4 \left(2 b^2 c (2 a+c)-2 b c (a+c) \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-2 c^2 (a+c)^2-b^4\right) \log \left(\sqrt{b^2-4 a c}+b+2 c \cos (x)\right)}{\sqrt{b^2-4 a c} \left(a^2+2 a c-b^2+c^2\right)^2}+\frac{16 i x \left(b^3-2 b c (a+c)\right)}{(a-b+c)^2 (a+b+c)^2}+\frac{4 i \tan ^{-1}(\tan (x)) (a-2 b+3 c)}{(a-b+c)^2}-\frac{4 i \tan ^{-1}(\tan (x)) (a+2 b+3 c)}{(a+b+c)^2}-\frac{\csc ^2\left(\frac{x}{2}\right)}{a+b+c}+\frac{\sec ^2\left(\frac{x}{2}\right)}{a-b+c}+\frac{2 (a+2 b+3 c) \log \left(\sin ^2\left(\frac{x}{2}\right)\right)}{(a+b+c)^2}-\frac{2 (a-2 b+3 c) \log \left(\cos ^2\left(\frac{x}{2}\right)\right)}{(a-b+c)^2}\right)","-\frac{b \left(b^2-2 c (a+c)\right) \log \left(a+b \cos (x)+c \cos ^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 \cos (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-\cos (x))}{4 (a+b+c)^2}-\frac{(a-2 b+3 c) \log (\cos (x)+1)}{4 (a-b+c)^2}+\frac{\csc ^2(x) (b-(a+c) \cos (x))}{2 (a-b+c) (a+b+c)}",1,"(((16*I)*(b^3 - 2*b*c*(a + c))*x)/((a - b + c)^2*(a + b + c)^2) + ((4*I)*(a - 2*b + 3*c)*ArcTan[Tan[x]])/(a - b + c)^2 - ((4*I)*(a + 2*b + 3*c)*ArcTan[Tan[x]])/(a + b + c)^2 - Csc[x/2]^2/(a + b + c) - (2*(a - 2*b + 3*c)*Log[Cos[x/2]^2])/(a - b + c)^2 - (4*(b^4 + 2*c^2*(a + c)^2 - 2*b^2*c*(2*a + c) + b^3*Sqrt[b^2 - 4*a*c] - 2*b*c*(a + c)*Sqrt[b^2 - 4*a*c])*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*Cos[x]])/(Sqrt[b^2 - 4*a*c]*(a^2 - b^2 + 2*a*c + c^2)^2) - (4*(-b^4 - 2*c^2*(a + c)^2 + 2*b^2*c*(2*a + c) + b^3*Sqrt[b^2 - 4*a*c] - 2*b*c*(a + c)*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*Cos[x]])/(Sqrt[b^2 - 4*a*c]*(a^2 - b^2 + 2*a*c + c^2)^2) + (2*(a + 2*b + 3*c)*Log[Sin[x/2]^2])/(a + b + c)^2 + Sec[x/2]^2/(a - b + c))/8","C",1
6,1,374,388,0.8919637,"\int \frac{\sin ^4(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Sin[x]^4/(a + b*Cos[x] + c*Cos[x]^2),x]","\frac{\frac{4 \sqrt{2} \left(-2 b^2 c (2 a+c)-2 b c (a+c) \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+2 c^2 (a+c)^2+b^4\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b-2 c\right)}{\sqrt{-2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{-b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}-\frac{4 \sqrt{2} \left(2 b^2 c (2 a+c)-2 b c (a+c) \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-2 c^2 (a+c)^2-b^4\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}-b+2 c\right)}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}-2 c x (2 a+3 c)+4 b^2 x-4 b c \sin (x)+c^2 \sin (2 x)}{4 c^3}","\frac{x \left(b^2-c (a+2 c)\right)}{c^3}+\frac{2 \left(b^2 \left(b^2-2 c (a+c)\right)-b \sqrt{b^2-4 a c} \left(b^2-2 c (a+c)\right)-2 c \left(a b^2-c (a+c)^2\right)\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{c^3 \sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{2 \left(-2 b^2 c (2 a+c)-2 b c (a+c) \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+2 c^2 (a+c)^2+b^4\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{c^3 \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{b \sin (x)}{c^2}+\frac{x}{2 c}+\frac{\sin (x) \cos (x)}{2 c}",1,"(4*b^2*x - 2*c*(2*a + 3*c)*x + (4*Sqrt[2]*(b^4 + 2*c^2*(a + c)^2 - 2*b^2*c*(2*a + c) + b^3*Sqrt[b^2 - 4*a*c] - 2*b*c*(a + c)*Sqrt[b^2 - 4*a*c])*ArcTanh[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) - 2*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]]) - (4*Sqrt[2]*(-b^4 - 2*c^2*(a + c)^2 + 2*b^2*c*(2*a + c) + b^3*Sqrt[b^2 - 4*a*c] - 2*b*c*(a + c)*Sqrt[b^2 - 4*a*c])*ArcTanh[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) + 2*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]]) - 4*b*c*Sin[x] + c^2*Sin[2*x])/(4*c^3)","A",1
7,1,238,260,0.6322909,"\int \frac{\sin ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Sin[x]^2/(a + b*Cos[x] + c*Cos[x]^2),x]","\frac{x \left(-\sqrt{b^2-4 a c}\right)-\frac{\left(b \sqrt{b^2-4 a c}-2 c (a+c)+b^2\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b-2 c\right)}{\sqrt{-2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{-\frac{1}{2} b \sqrt{b^2-4 a c}+c (a+c)-\frac{b^2}{2}}}+\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2} \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}-b+2 c\right)}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{c \sqrt{b^2-4 a c}}","\frac{2 \left(b-\frac{b^2-2 c (a+c)}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{c \sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}+\frac{2 \left(\frac{b^2-2 c (a+c)}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{c \sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{x}{c}",1,"(-(Sqrt[b^2 - 4*a*c]*x) - ((b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c])*ArcTanh[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) - 2*b*Sqrt[b^2 - 4*a*c]]])/Sqrt[-1/2*b^2 + c*(a + c) - (b*Sqrt[b^2 - 4*a*c])/2] + Sqrt[-2*b^2 + 4*c*(a + c) + 2*b*Sqrt[b^2 - 4*a*c]]*ArcTanh[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) + 2*b*Sqrt[b^2 - 4*a*c]]])/(c*Sqrt[b^2 - 4*a*c])","A",1
8,1,335,326,0.9732934,"\int \frac{\csc ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Csc[x]^2/(a + b*Cos[x] + c*Cos[x]^2),x]","\frac{\sqrt{2} c \left(b \sqrt{b^2-4 a c}+2 c (a+c)-b^2\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b-2 c\right)}{\sqrt{-2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b^2-4 a c} \left(a^2+2 a c-b^2+c^2\right) \sqrt{-b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}-\frac{\sqrt{2} c \left(b \sqrt{b^2-4 a c}-2 c (a+c)+b^2\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}-b+2 c\right)}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b^2-4 a c} \left(a^2+2 a c-b^2+c^2\right) \sqrt{b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}+\frac{\tan \left(\frac{x}{2}\right)}{2 (a-b+c)}-\frac{\cot \left(\frac{x}{2}\right)}{2 (a+b+c)}","-\frac{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) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{(a-b+c) (a+b+c) \sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{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) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{(a-b+c) (a+b+c) \sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{\sin (x)}{2 (1-\cos (x)) (a+b+c)}+\frac{\sin (x)}{2 (\cos (x)+1) (a-b+c)}",1,"(Sqrt[2]*c*(-b^2 + 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c])*ArcTanh[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) - 2*b*Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*(a^2 - b^2 + 2*a*c + c^2)*Sqrt[-b^2 + 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*c*(b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c])*ArcTanh[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) + 2*b*Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*(a^2 - b^2 + 2*a*c + c^2)*Sqrt[-b^2 + 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) - Cot[x/2]/(2*(a + b + c)) + Tan[x/2]/(2*(a - b + c))","A",1
9,1,19,21,0.0273611,"\int \frac{\sin (x)}{-2+\cos (x)+\cos ^2(x)} \, dx","Integrate[Sin[x]/(-2 + Cos[x] + Cos[x]^2),x]","\frac{1}{3} \left(\log (\cos (x)+2)-2 \log \left(\sin \left(\frac{x}{2}\right)\right)\right)","\frac{1}{3} \log (\cos (x)+2)-\frac{1}{3} \log (1-\cos (x))",1,"(Log[2 + Cos[x]] - 2*Log[Sin[x/2]])/3","A",1
10,1,29,23,0.0140463,"\int \frac{\sin (x)}{4-5 \cos (x)+\cos ^2(x)} \, dx","Integrate[Sin[x]/(4 - 5*Cos[x] + Cos[x]^2),x]","\frac{2}{3} \log \left(\sin \left(\frac{x}{2}\right)\right)-\frac{1}{3} \log \left(2 \sin ^2\left(\frac{x}{2}\right)+3\right)","\frac{1}{3} \log (1-\cos (x))-\frac{1}{3} \log (4-\cos (x))",1,"(2*Log[Sin[x/2]])/3 - Log[3 + 2*Sin[x/2]^2]/3","A",1
11,1,18,19,0.0264509,"\int \frac{\sin (x)}{3-2 \cos (x)+\cos ^2(x)} \, dx","Integrate[Sin[x]/(3 - 2*Cos[x] + Cos[x]^2),x]","-\frac{\tan ^{-1}\left(\frac{\cos (x)-1}{\sqrt{2}}\right)}{\sqrt{2}}","\frac{\tan ^{-1}\left(\frac{1-\cos (x)}{\sqrt{2}}\right)}{\sqrt{2}}",1,"-(ArcTan[(-1 + Cos[x])/Sqrt[2]]/Sqrt[2])","A",1
12,1,34,36,0.0765274,"\int \frac{\sin (x)}{\left(13-4 \cos (x)+\cos ^2(x)\right)^2} \, dx","Integrate[Sin[x]/(13 - 4*Cos[x] + Cos[x]^2)^2,x]","-\frac{\cos (x)-2}{18 \left(\cos ^2(x)-4 \cos (x)+13\right)}-\frac{1}{54} \tan ^{-1}\left(\frac{1}{3} (\cos (x)-2)\right)","\frac{2-\cos (x)}{18 \left(\cos ^2(x)-4 \cos (x)+13\right)}-\frac{1}{54} \tan ^{-1}\left(\frac{1}{3} (\cos (x)-2)\right)",1,"-1/54*ArcTan[(-2 + Cos[x])/3] - (-2 + Cos[x])/(18*(13 - 4*Cos[x] + Cos[x]^2))","A",1
13,1,356,326,1.1324022,"\int \frac{\cos ^4(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Cos[x]^4/(a + b*Cos[x] + c*Cos[x]^2),x]","\frac{\frac{4 \sqrt{2} \left(2 a^2 c^2-4 a b^2 c-2 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+b^4\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b-2 c\right)}{\sqrt{-2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{-b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}-\frac{4 \sqrt{2} \left(-2 a^2 c^2+4 a b^2 c-2 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-b^4\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}-b+2 c\right)}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}+2 c x (c-2 a)+4 b^2 x-4 b c \sin (x)+c^2 \sin (2 x)}{4 c^3}","-\frac{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) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{c^3 \sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{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) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{c^3 \sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}+\frac{x \left(b^2-a c\right)}{c^3}-\frac{b \sin (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*Sqrt[2]*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 2*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTanh[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) - 2*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]]) - (4*Sqrt[2]*(-b^4 + 4*a*b^2*c - 2*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 2*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTanh[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) + 2*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]]) - 4*b*c*Sin[x] + c^2*Sin[2*x])/(4*c^3)","A",1
14,1,309,299,0.890172,"\int \frac{\cos ^3(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Cos[x]^3/(a + b*Cos[x] + c*Cos[x]^2),x]","\frac{-\frac{\sqrt{2} \left(b^2 \sqrt{b^2-4 a c}-a c \sqrt{b^2-4 a c}-3 a b c+b^3\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b-2 c\right)}{\sqrt{-2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{-b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}+\frac{\sqrt{2} \left(b^2 \sqrt{b^2-4 a c}-a c \sqrt{b^2-4 a c}+3 a b c-b^3\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}-b+2 c\right)}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}-b x+c \sin (x)}{c^2}","\frac{2 \left(\frac{3 a b c}{\sqrt{b^2-4 a c}}-\frac{b^3}{\sqrt{b^2-4 a c}}-a c+b^2\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{c^2 \sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}+\frac{2 \left(-\frac{3 a b c}{\sqrt{b^2-4 a c}}+\frac{b^3}{\sqrt{b^2-4 a c}}-a c+b^2\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{c^2 \sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{b x}{c^2}+\frac{\sin (x)}{c}",1,"(-(b*x) - (Sqrt[2]*(b^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*ArcTanh[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) - 2*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]]) + (Sqrt[2]*(-b^3 + 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*ArcTanh[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) + 2*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]]) + c*Sin[x])/c^2","A",1
15,1,264,255,0.5893803,"\int \frac{\cos ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Cos[x]^2/(a + b*Cos[x] + c*Cos[x]^2),x]","\frac{\frac{\sqrt{2} \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b-2 c\right)}{\sqrt{-2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{-b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}-\frac{\sqrt{2} \left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}-b+2 c\right)}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}+x}{c}","-\frac{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) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{c \sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{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) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{c \sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}+\frac{x}{c}",1,"(x + (Sqrt[2]*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTanh[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) - 2*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]]) - (Sqrt[2]*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTanh[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) + 2*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]]))/c","A",1
16,1,227,230,0.5704809,"\int \frac{\cos (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Cos[x]/(a + b*Cos[x] + c*Cos[x]^2),x]","\frac{\sqrt{2} \left(\frac{\left(\sqrt{b^2-4 a c}-b\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}-b+2 c\right)}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}-\frac{\left(\sqrt{b^2-4 a c}+b\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b-2 c\right)}{\sqrt{-2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{-b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}\right)}{\sqrt{b^2-4 a c}}","\frac{2 \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}+\frac{2 \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}",1,"(Sqrt[2]*(-(((b + Sqrt[b^2 - 4*a*c])*ArcTanh[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) - 2*b*Sqrt[b^2 - 4*a*c]]])/Sqrt[-b^2 + 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) + ((-b + Sqrt[b^2 - 4*a*c])*ArcTanh[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) + 2*b*Sqrt[b^2 - 4*a*c]]])/Sqrt[-b^2 + 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]))/Sqrt[b^2 - 4*a*c]","A",1
17,1,198,223,0.4116466,"\int \frac{1}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[(a + b*Cos[x] + c*Cos[x]^2)^(-1),x]","\frac{2 \sqrt{2} c \left(\frac{\tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b-2 c\right)}{\sqrt{-2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{-b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}+\frac{\tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}-b+2 c\right)}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}\right)}{\sqrt{b^2-4 a c}}","\frac{4 c \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{4 c \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}",1,"(2*Sqrt[2]*c*(ArcTanh[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) - 2*b*Sqrt[b^2 - 4*a*c]]]/Sqrt[-b^2 + 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]] + ArcTanh[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) + 2*b*Sqrt[b^2 - 4*a*c]]]/Sqrt[-b^2 + 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]))/Sqrt[b^2 - 4*a*c]","A",1
18,1,281,245,0.6660798,"\int \frac{\sec (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Sec[x]/(a + b*Cos[x] + c*Cos[x]^2),x]","\frac{\frac{\sqrt{2} c \left(\sqrt{b^2-4 a c}-b\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b-2 c\right)}{\sqrt{-2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\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}+b\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}-b+2 c\right)}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{a}","-\frac{2 c \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{a \sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{2 c \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{a \sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}+\frac{\tanh ^{-1}(\sin (x))}{a}",1,"((Sqrt[2]*c*(-b + Sqrt[b^2 - 4*a*c])*ArcTanh[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) - 2*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]]) - (Sqrt[2]*c*(b + Sqrt[b^2 - 4*a*c])*ArcTanh[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) + 2*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]]) - Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])/a","A",1
19,1,348,275,1.1760083,"\int \frac{\sec ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Sec[x]^2/(a + b*Cos[x] + c*Cos[x]^2),x]","\frac{-\frac{\sqrt{2} c \left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b-2 c\right)}{\sqrt{-2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\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(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}-b+2 c\right)}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}+\frac{a \sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)}+\frac{a \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}+b \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-b \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{a^2}","\frac{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) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{a^2 \sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}+\frac{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) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{a^2 \sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{b \tanh ^{-1}(\sin (x))}{a^2}+\frac{\tan (x)}{a}",1,"(-((Sqrt[2]*c*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTanh[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) - 2*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]])) + (Sqrt[2]*c*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTanh[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) + 2*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]]) + b*Log[Cos[x/2] - Sin[x/2]] - b*Log[Cos[x/2] + Sin[x/2]] + (a*Sin[x/2])/(Cos[x/2] - Sin[x/2]) + (a*Sin[x/2])/(Cos[x/2] + Sin[x/2]))/a^2","A",1
20,1,446,334,3.0735434,"\int \frac{\sec ^3(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[Sec[x]^3/(a + b*Cos[x] + c*Cos[x]^2),x]","-\frac{2 \left(a^2-2 a c+2 b^2\right) \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-2 \left(a^2-2 a c+2 b^2\right) \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)+\frac{a^2}{\sin (x)-1}+\frac{a^2}{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2}+\frac{4 \sqrt{2} c \left(-b^2 \sqrt{b^2-4 a c}+a c \sqrt{b^2-4 a c}-3 a b c+b^3\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b-2 c\right)}{\sqrt{-2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{-b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}+\frac{4 \sqrt{2} c \left(b^2 \sqrt{b^2-4 a c}-a c \sqrt{b^2-4 a c}-3 a b c+b^3\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}-b+2 c\right)}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}+\frac{4 a b \sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)}+\frac{4 a b \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}}{4 a^3}","\frac{\left(b^2-a c\right) \tanh ^{-1}(\sin (x))}{a^3}-\frac{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) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{a^3 \sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}+\frac{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) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{a^3 \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{b \tan (x)}{a^2}+\frac{\tanh ^{-1}(\sin (x))}{2 a}+\frac{\tan (x) \sec (x)}{2 a}",1,"-1/4*((4*Sqrt[2]*c*(b^3 - 3*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + a*c*Sqrt[b^2 - 4*a*c])*ArcTanh[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) - 2*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]]) + (4*Sqrt[2]*c*(b^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*ArcTanh[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) + 2*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*(a^2 + 2*b^2 - 2*a*c)*Log[Cos[x/2] - Sin[x/2]] - 2*(a^2 + 2*b^2 - 2*a*c)*Log[Cos[x/2] + Sin[x/2]] + (4*a*b*Sin[x/2])/(Cos[x/2] - Sin[x/2]) + a^2/(Cos[x/2] + Sin[x/2])^2 + (4*a*b*Sin[x/2])/(Cos[x/2] + Sin[x/2]) + a^2/(-1 + Sin[x]))/a^3","A",1