1,1,56,98,0.3593451,"\int \left(b \tan ^2(e+f x)\right)^{5/2} \, dx","Integrate[(b*Tan[e + f*x]^2)^(5/2),x]","-\frac{\cot (e+f x) \left(b \tan ^2(e+f x)\right)^{5/2} \left(2 \cot ^2(e+f x)+4 \cot ^4(e+f x) \log (\cos (e+f x))-1\right)}{4 f}","-\frac{b^2 \tan (e+f x) \sqrt{b \tan ^2(e+f x)}}{2 f}+\frac{b^2 \tan ^3(e+f x) \sqrt{b \tan ^2(e+f x)}}{4 f}-\frac{b^2 \cot (e+f x) \sqrt{b \tan ^2(e+f x)} \log (\cos (e+f x))}{f}",1,"-1/4*(Cot[e + f*x]*(-1 + 2*Cot[e + f*x]^2 + 4*Cot[e + f*x]^4*Log[Cos[e + f*x]])*(b*Tan[e + f*x]^2)^(5/2))/f","A",1
2,1,47,61,0.1089028,"\int \left(b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[(b*Tan[e + f*x]^2)^(3/2),x]","\frac{\cot ^3(e+f x) \left(b \tan ^2(e+f x)\right)^{3/2} \left(\tan ^2(e+f x)+2 \log (\cos (e+f x))\right)}{2 f}","\frac{b \tan (e+f x) \sqrt{b \tan ^2(e+f x)}}{2 f}+\frac{b \cot (e+f x) \sqrt{b \tan ^2(e+f x)} \log (\cos (e+f x))}{f}",1,"(Cot[e + f*x]^3*(b*Tan[e + f*x]^2)^(3/2)*(2*Log[Cos[e + f*x]] + Tan[e + f*x]^2))/(2*f)","A",1
3,1,32,32,0.039498,"\int \sqrt{b \tan ^2(e+f x)} \, dx","Integrate[Sqrt[b*Tan[e + f*x]^2],x]","-\frac{\cot (e+f x) \sqrt{b \tan ^2(e+f x)} \log (\cos (e+f x))}{f}","-\frac{\cot (e+f x) \sqrt{b \tan ^2(e+f x)} \log (\cos (e+f x))}{f}",1,"-((Cot[e + f*x]*Log[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]^2])/f)","A",1
4,1,39,31,0.0814851,"\int \frac{1}{\sqrt{b \tan ^2(e+f x)}} \, dx","Integrate[1/Sqrt[b*Tan[e + f*x]^2],x]","\frac{\tan (e+f x) (\log (\tan (e+f x))+\log (\cos (e+f x)))}{f \sqrt{b \tan ^2(e+f x)}}","\frac{\tan (e+f x) \log (\sin (e+f x))}{f \sqrt{b \tan ^2(e+f x)}}",1,"((Log[Cos[e + f*x]] + Log[Tan[e + f*x]])*Tan[e + f*x])/(f*Sqrt[b*Tan[e + f*x]^2])","A",1
5,1,56,66,0.3603742,"\int \frac{1}{\left(b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[(b*Tan[e + f*x]^2)^(-3/2),x]","-\frac{\tan ^3(e+f x) \left(\cot ^2(e+f x)+2 \log (\tan (e+f x))+2 \log (\cos (e+f x))\right)}{2 f \left(b \tan ^2(e+f x)\right)^{3/2}}","-\frac{\cot (e+f x)}{2 b f \sqrt{b \tan ^2(e+f x)}}-\frac{\tan (e+f x) \log (\sin (e+f x))}{b f \sqrt{b \tan ^2(e+f x)}}",1,"-1/2*((Cot[e + f*x]^2 + 2*Log[Cos[e + f*x]] + 2*Log[Tan[e + f*x]])*Tan[e + f*x]^3)/(f*(b*Tan[e + f*x]^2)^(3/2))","A",1
6,1,68,97,0.2587133,"\int \frac{1}{\left(b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[(b*Tan[e + f*x]^2)^(-5/2),x]","\frac{\tan ^5(e+f x) \left(-\cot ^4(e+f x)+2 \cot ^2(e+f x)+4 \log (\tan (e+f x))+4 \log (\cos (e+f x))\right)}{4 f \left(b \tan ^2(e+f x)\right)^{5/2}}","-\frac{\cot ^3(e+f x)}{4 b^2 f \sqrt{b \tan ^2(e+f x)}}+\frac{\cot (e+f x)}{2 b^2 f \sqrt{b \tan ^2(e+f x)}}+\frac{\tan (e+f x) \log (\sin (e+f x))}{b^2 f \sqrt{b \tan ^2(e+f x)}}",1,"((2*Cot[e + f*x]^2 - Cot[e + f*x]^4 + 4*Log[Cos[e + f*x]] + 4*Log[Tan[e + f*x]])*Tan[e + f*x]^5)/(4*f*(b*Tan[e + f*x]^2)^(5/2))","A",1
7,1,199,364,0.8308006,"\int \left(b \tan ^3(e+f x)\right)^{5/2} \, dx","Integrate[(b*Tan[e + f*x]^3)^(5/2),x]","\frac{b \left(b \tan ^3(e+f x)\right)^{3/2} \left(-1170 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)+1170 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+360 \tan ^{\frac{13}{2}}(e+f x)-520 \tan ^{\frac{9}{2}}(e+f x)+936 \tan ^{\frac{5}{2}}(e+f x)-4680 \sqrt{\tan (e+f x)}-585 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+585 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)}{2340 f \tan ^{\frac{9}{2}}(e+f x)}","-\frac{2 b^2 \tan ^3(e+f x) \sqrt{b \tan ^3(e+f x)}}{9 f}+\frac{2 b^2 \tan (e+f x) \sqrt{b \tan ^3(e+f x)}}{5 f}+\frac{2 b^2 \tan ^5(e+f x) \sqrt{b \tan ^3(e+f x)}}{13 f}-\frac{b^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{b^2 \sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b^2 \sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{2 b^2 \cot (e+f x) \sqrt{b \tan ^3(e+f x)}}{f}",1,"(b*(b*Tan[e + f*x]^3)^(3/2)*(-1170*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] + 1170*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - 585*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] + 585*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 4680*Sqrt[Tan[e + f*x]] + 936*Tan[e + f*x]^(5/2) - 520*Tan[e + f*x]^(9/2) + 360*Tan[e + f*x]^(13/2)))/(2340*f*Tan[e + f*x]^(9/2))","A",1
8,1,54,286,0.0784055,"\int \left(b \tan ^3(e+f x)\right)^{3/2} \, dx","Integrate[(b*Tan[e + f*x]^3)^(3/2),x]","\frac{2 b \sqrt{b \tan ^3(e+f x)} \left(7 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)+3 \tan ^2(e+f x)-7\right)}{21 f}","-\frac{2 b \sqrt{b \tan ^3(e+f x)}}{3 f}+\frac{2 b \tan ^2(e+f x) \sqrt{b \tan ^3(e+f x)}}{7 f}-\frac{b \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b \sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{b \sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}",1,"(2*b*Sqrt[b*Tan[e + f*x]^3]*(-7 + 7*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2] + 3*Tan[e + f*x]^2))/(21*f)","C",1
9,1,161,255,0.2587768,"\int \sqrt{b \tan ^3(e+f x)} \, dx","Integrate[Sqrt[b*Tan[e + f*x]^3],x]","\frac{\sqrt{b \tan ^3(e+f x)} \left(2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+8 \sqrt{\tan (e+f x)}+\sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-\sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)}{4 f \tan ^{\frac{3}{2}}(e+f x)}","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{\sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{\sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{2 \cot (e+f x) \sqrt{b \tan ^3(e+f x)}}{f}",1,"((2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] + 8*Sqrt[Tan[e + f*x]])*Sqrt[b*Tan[e + f*x]^3])/(4*f*Tan[e + f*x]^(3/2))","A",1
10,1,43,255,0.033103,"\int \frac{1}{\sqrt{b \tan ^3(e+f x)}} \, dx","Integrate[1/Sqrt[b*Tan[e + f*x]^3],x]","-\frac{2 \tan (e+f x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(e+f x)\right)}{f \sqrt{b \tan ^3(e+f x)}}","-\frac{2 \tan (e+f x)}{f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \sqrt{b \tan ^3(e+f x)}}",1,"(-2*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[e + f*x]^2]*Tan[e + f*x])/(f*Sqrt[b*Tan[e + f*x]^3])","C",1
11,1,45,298,0.074358,"\int \frac{1}{\left(b \tan ^3(e+f x)\right)^{3/2}} \, dx","Integrate[(b*Tan[e + f*x]^3)^(-3/2),x]","-\frac{2 \tan (e+f x) \, _2F_1\left(-\frac{7}{4},1;-\frac{3}{4};-\tan ^2(e+f x)\right)}{7 f \left(b \tan ^3(e+f x)\right)^{3/2}}","\frac{2}{3 b f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} b f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} b f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^2(e+f x)}{7 b f \sqrt{b \tan ^3(e+f x)}}",1,"(-2*Hypergeometric2F1[-7/4, 1, -3/4, -Tan[e + f*x]^2]*Tan[e + f*x])/(7*f*(b*Tan[e + f*x]^3)^(3/2))","C",1
12,1,45,364,0.0569291,"\int \frac{1}{\left(b \tan ^3(e+f x)\right)^{5/2}} \, dx","Integrate[(b*Tan[e + f*x]^3)^(-5/2),x]","-\frac{2 \tan (e+f x) \, _2F_1\left(-\frac{13}{4},1;-\frac{9}{4};-\tan ^2(e+f x)\right)}{13 f \left(b \tan ^3(e+f x)\right)^{5/2}}","\frac{2 \tan (e+f x)}{b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \cot ^3(e+f x)}{9 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot (e+f x)}{5 b^2 f \sqrt{b \tan ^3(e+f x)}}",1,"(-2*Hypergeometric2F1[-13/4, 1, -9/4, -Tan[e + f*x]^2]*Tan[e + f*x])/(13*f*(b*Tan[e + f*x]^3)^(5/2))","C",1
13,1,86,182,0.7364991,"\int \left(b \tan ^4(e+f x)\right)^{5/2} \, dx","Integrate[(b*Tan[e + f*x]^4)^(5/2),x]","\frac{\cot (e+f x) \left(b \tan ^4(e+f x)\right)^{5/2} \left(315 \cot ^8(e+f x)-105 \cot ^6(e+f x)+63 \cot ^4(e+f x)-45 \cot ^2(e+f x)-315 \tan ^{-1}(\tan (e+f x)) \cot ^9(e+f x)+35\right)}{315 f}","-\frac{b^2 \tan (e+f x) \sqrt{b \tan ^4(e+f x)}}{3 f}+\frac{b^2 \tan ^7(e+f x) \sqrt{b \tan ^4(e+f x)}}{9 f}-\frac{b^2 \tan ^5(e+f x) \sqrt{b \tan ^4(e+f x)}}{7 f}+\frac{b^2 \tan ^3(e+f x) \sqrt{b \tan ^4(e+f x)}}{5 f}-b^2 x \cot ^2(e+f x) \sqrt{b \tan ^4(e+f x)}+\frac{b^2 \cot (e+f x) \sqrt{b \tan ^4(e+f x)}}{f}",1,"(Cot[e + f*x]*(35 - 45*Cot[e + f*x]^2 + 63*Cot[e + f*x]^4 - 105*Cot[e + f*x]^6 + 315*Cot[e + f*x]^8 - 315*ArcTan[Tan[e + f*x]]*Cot[e + f*x]^9)*(b*Tan[e + f*x]^4)^(5/2))/(315*f)","A",1
14,1,66,110,0.745044,"\int \left(b \tan ^4(e+f x)\right)^{3/2} \, dx","Integrate[(b*Tan[e + f*x]^4)^(3/2),x]","\frac{\cot (e+f x) \left(b \tan ^4(e+f x)\right)^{3/2} \left(15 \cot ^4(e+f x)-5 \cot ^2(e+f x)-15 \tan ^{-1}(\tan (e+f x)) \cot ^5(e+f x)+3\right)}{15 f}","-\frac{b \tan (e+f x) \sqrt{b \tan ^4(e+f x)}}{3 f}+\frac{b \tan ^3(e+f x) \sqrt{b \tan ^4(e+f x)}}{5 f}-b x \cot ^2(e+f x) \sqrt{b \tan ^4(e+f x)}+\frac{b \cot (e+f x) \sqrt{b \tan ^4(e+f x)}}{f}",1,"(Cot[e + f*x]*(3 - 5*Cot[e + f*x]^2 + 15*Cot[e + f*x]^4 - 15*ArcTan[Tan[e + f*x]]*Cot[e + f*x]^5)*(b*Tan[e + f*x]^4)^(3/2))/(15*f)","A",1
15,1,41,50,0.0960785,"\int \sqrt{b \tan ^4(e+f x)} \, dx","Integrate[Sqrt[b*Tan[e + f*x]^4],x]","-\frac{\cot (e+f x) \sqrt{b \tan ^4(e+f x)} \left(\tan ^{-1}(\tan (e+f x)) \cot (e+f x)-1\right)}{f}","\frac{\cot (e+f x) \sqrt{b \tan ^4(e+f x)}}{f}-x \cot ^2(e+f x) \sqrt{b \tan ^4(e+f x)}",1,"-((Cot[e + f*x]*(-1 + ArcTan[Tan[e + f*x]]*Cot[e + f*x])*Sqrt[b*Tan[e + f*x]^4])/f)","A",1
16,1,43,51,0.0525927,"\int \frac{1}{\sqrt{b \tan ^4(e+f x)}} \, dx","Integrate[1/Sqrt[b*Tan[e + f*x]^4],x]","-\frac{\tan (e+f x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(e+f x)\right)}{f \sqrt{b \tan ^4(e+f x)}}","-\frac{\tan (e+f x)}{f \sqrt{b \tan ^4(e+f x)}}-\frac{x \tan ^2(e+f x)}{\sqrt{b \tan ^4(e+f x)}}",1,"-((Hypergeometric2F1[-1/2, 1, 1/2, -Tan[e + f*x]^2]*Tan[e + f*x])/(f*Sqrt[b*Tan[e + f*x]^4]))","C",1
17,1,45,119,0.0501077,"\int \frac{1}{\left(b \tan ^4(e+f x)\right)^{3/2}} \, dx","Integrate[(b*Tan[e + f*x]^4)^(-3/2),x]","-\frac{\tan (e+f x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(e+f x)\right)}{5 f \left(b \tan ^4(e+f x)\right)^{3/2}}","-\frac{\tan (e+f x)}{b f \sqrt{b \tan ^4(e+f x)}}-\frac{x \tan ^2(e+f x)}{b \sqrt{b \tan ^4(e+f x)}}-\frac{\cot ^3(e+f x)}{5 b f \sqrt{b \tan ^4(e+f x)}}+\frac{\cot (e+f x)}{3 b f \sqrt{b \tan ^4(e+f x)}}",1,"-1/5*(Hypergeometric2F1[-5/2, 1, -3/2, -Tan[e + f*x]^2]*Tan[e + f*x])/(f*(b*Tan[e + f*x]^4)^(3/2))","C",1
18,1,45,183,0.0349403,"\int \frac{1}{\left(b \tan ^4(e+f x)\right)^{5/2}} \, dx","Integrate[(b*Tan[e + f*x]^4)^(-5/2),x]","-\frac{\tan (e+f x) \, _2F_1\left(-\frac{9}{2},1;-\frac{7}{2};-\tan ^2(e+f x)\right)}{9 f \left(b \tan ^4(e+f x)\right)^{5/2}}","-\frac{\tan (e+f x)}{b^2 f \sqrt{b \tan ^4(e+f x)}}-\frac{x \tan ^2(e+f x)}{b^2 \sqrt{b \tan ^4(e+f x)}}-\frac{\cot ^7(e+f x)}{9 b^2 f \sqrt{b \tan ^4(e+f x)}}+\frac{\cot ^5(e+f x)}{7 b^2 f \sqrt{b \tan ^4(e+f x)}}-\frac{\cot ^3(e+f x)}{5 b^2 f \sqrt{b \tan ^4(e+f x)}}+\frac{\cot (e+f x)}{3 b^2 f \sqrt{b \tan ^4(e+f x)}}",1,"-1/9*(Hypergeometric2F1[-9/2, 1, -7/2, -Tan[e + f*x]^2]*Tan[e + f*x])/(f*(b*Tan[e + f*x]^4)^(5/2))","C",1
19,1,62,71,0.1115938,"\int \left(b \tan ^n(e+f x)\right)^{5/2} \, dx","Integrate[(b*Tan[e + f*x]^n)^(5/2),x]","\frac{2 \tan (e+f x) \left(b \tan ^n(e+f x)\right)^{5/2} \, _2F_1\left(1,\frac{1}{4} (5 n+2);\frac{1}{4} (5 n+6);-\tan ^2(e+f x)\right)}{f (5 n+2)}","\frac{2 b^2 \tan ^{2 n+1}(e+f x) \sqrt{b \tan ^n(e+f x)} \, _2F_1\left(1,\frac{1}{4} (5 n+2);\frac{1}{4} (5 n+6);-\tan ^2(e+f x)\right)}{f (5 n+2)}",1,"(2*Hypergeometric2F1[1, (2 + 5*n)/4, (6 + 5*n)/4, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^n)^(5/2))/(f*(2 + 5*n))","A",1
20,1,60,65,0.0711581,"\int \left(b \tan ^n(e+f x)\right)^{3/2} \, dx","Integrate[(b*Tan[e + f*x]^n)^(3/2),x]","\frac{2 \tan (e+f x) \left(b \tan ^n(e+f x)\right)^{3/2} \, _2F_1\left(1,\frac{1}{4} (3 n+2);\frac{3 (n+2)}{4};-\tan ^2(e+f x)\right)}{f (3 n+2)}","\frac{2 b \tan ^{n+1}(e+f x) \sqrt{b \tan ^n(e+f x)} \, _2F_1\left(1,\frac{1}{4} (3 n+2);\frac{3 (n+2)}{4};-\tan ^2(e+f x)\right)}{f (3 n+2)}",1,"(2*Hypergeometric2F1[1, (2 + 3*n)/4, (3*(2 + n))/4, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^n)^(3/2))/(f*(2 + 3*n))","A",1
21,1,56,56,0.0394774,"\int \sqrt{b \tan ^n(e+f x)} \, dx","Integrate[Sqrt[b*Tan[e + f*x]^n],x]","\frac{2 \tan (e+f x) \sqrt{b \tan ^n(e+f x)} \, _2F_1\left(1,\frac{n+2}{4};\frac{n+6}{4};-\tan ^2(e+f x)\right)}{f (n+2)}","\frac{2 \tan (e+f x) \sqrt{b \tan ^n(e+f x)} \, _2F_1\left(1,\frac{n+2}{4};\frac{n+6}{4};-\tan ^2(e+f x)\right)}{f (n+2)}",1,"(2*Hypergeometric2F1[1, (2 + n)/4, (6 + n)/4, -Tan[e + f*x]^2]*Tan[e + f*x]*Sqrt[b*Tan[e + f*x]^n])/(f*(2 + n))","A",1
22,1,60,62,0.050911,"\int \frac{1}{\sqrt{b \tan ^n(e+f x)}} \, dx","Integrate[1/Sqrt[b*Tan[e + f*x]^n],x]","-\frac{2 \tan (e+f x) \, _2F_1\left(1,\frac{2-n}{4};\frac{6-n}{4};-\tan ^2(e+f x)\right)}{f (n-2) \sqrt{b \tan ^n(e+f x)}}","\frac{2 \tan (e+f x) \, _2F_1\left(1,\frac{2-n}{4};\frac{6-n}{4};-\tan ^2(e+f x)\right)}{f (2-n) \sqrt{b \tan ^n(e+f x)}}",1,"(-2*Hypergeometric2F1[1, (2 - n)/4, (6 - n)/4, -Tan[e + f*x]^2]*Tan[e + f*x])/(f*(-2 + n)*Sqrt[b*Tan[e + f*x]^n])","A",1
23,1,60,71,0.0731581,"\int \frac{1}{\left(b \tan ^n(e+f x)\right)^{3/2}} \, dx","Integrate[(b*Tan[e + f*x]^n)^(-3/2),x]","-\frac{2 \tan (e+f x) \, _2F_1\left(1,\frac{1}{4} (2-3 n);-\frac{3}{4} (n-2);-\tan ^2(e+f x)\right)}{f (3 n-2) \left(b \tan ^n(e+f x)\right)^{3/2}}","\frac{2 \tan ^{1-n}(e+f x) \, _2F_1\left(1,\frac{1}{4} (2-3 n);\frac{3 (2-n)}{4};-\tan ^2(e+f x)\right)}{b f (2-3 n) \sqrt{b \tan ^n(e+f x)}}",1,"(-2*Hypergeometric2F1[1, (2 - 3*n)/4, (-3*(-2 + n))/4, -Tan[e + f*x]^2]*Tan[e + f*x])/(f*(-2 + 3*n)*(b*Tan[e + f*x]^n)^(3/2))","A",1
24,1,62,71,0.0716309,"\int \frac{1}{\left(b \tan ^n(e+f x)\right)^{5/2}} \, dx","Integrate[(b*Tan[e + f*x]^n)^(-5/2),x]","-\frac{2 \tan (e+f x) \, _2F_1\left(1,\frac{1}{4} (2-5 n);\frac{1}{4} (6-5 n);-\tan ^2(e+f x)\right)}{f (5 n-2) \left(b \tan ^n(e+f x)\right)^{5/2}}","\frac{2 \tan ^{1-2 n}(e+f x) \, _2F_1\left(1,\frac{1}{4} (2-5 n);\frac{1}{4} (6-5 n);-\tan ^2(e+f x)\right)}{b^2 f (2-5 n) \sqrt{b \tan ^n(e+f x)}}",1,"(-2*Hypergeometric2F1[1, (2 - 5*n)/4, (6 - 5*n)/4, -Tan[e + f*x]^2]*Tan[e + f*x])/(f*(-2 + 5*n)*(b*Tan[e + f*x]^n)^(5/2))","A",1
25,1,57,59,0.0418989,"\int \left(b \tan ^n(e+f x)\right)^p \, dx","Integrate[(b*Tan[e + f*x]^n)^p,x]","\frac{\tan (e+f x) \left(b \tan ^n(e+f x)\right)^p \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right)}{f n p+f}","\frac{\tan (e+f x) \left(b \tan ^n(e+f x)\right)^p \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right)}{f (n p+1)}",1,"(Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^n)^p)/(f + f*n*p)","A",1
26,1,49,59,0.0484013,"\int \left(b \tan ^2(e+f x)\right)^p \, dx","Integrate[(b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p \, _2F_1\left(1,p+\frac{1}{2};p+\frac{3}{2};-\tan ^2(e+f x)\right)}{2 f p+f}","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p \, _2F_1\left(1,\frac{1}{2} (2 p+1);\frac{1}{2} (2 p+3);-\tan ^2(e+f x)\right)}{f (2 p+1)}",1,"(Hypergeometric2F1[1, 1/2 + p, 3/2 + p, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f + 2*f*p)","A",1
27,1,55,57,0.0419262,"\int \left(b \tan ^3(e+f x)\right)^p \, dx","Integrate[(b*Tan[e + f*x]^3)^p,x]","\frac{\tan (e+f x) \left(b \tan ^3(e+f x)\right)^p \, _2F_1\left(1,\frac{1}{2} (3 p+1);\frac{3 (p+1)}{2};-\tan ^2(e+f x)\right)}{3 f p+f}","\frac{\tan (e+f x) \left(b \tan ^3(e+f x)\right)^p \, _2F_1\left(1,\frac{1}{2} (3 p+1);\frac{3 (p+1)}{2};-\tan ^2(e+f x)\right)}{f (3 p+1)}",1,"(Hypergeometric2F1[1, (1 + 3*p)/2, (3*(1 + p))/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^3)^p)/(f + 3*f*p)","A",1
28,1,53,59,0.03295,"\int \left(b \tan ^4(e+f x)\right)^p \, dx","Integrate[(b*Tan[e + f*x]^4)^p,x]","\frac{\tan (e+f x) \left(b \tan ^4(e+f x)\right)^p \, _2F_1\left(1,2 p+\frac{1}{2};2 p+\frac{3}{2};-\tan ^2(e+f x)\right)}{4 f p+f}","\frac{\tan (e+f x) \left(b \tan ^4(e+f x)\right)^p \, _2F_1\left(1,\frac{1}{2} (4 p+1);\frac{1}{2} (4 p+3);-\tan ^2(e+f x)\right)}{f (4 p+1)}",1,"(Hypergeometric2F1[1, 1/2 + 2*p, 3/2 + 2*p, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^4)^p)/(f + 4*f*p)","A",1
29,1,32,32,0.0227671,"\int \left(b \tan ^n(e+f x)\right)^{\frac{1}{n}} \, dx","Integrate[(b*Tan[e + f*x]^n)^n^(-1),x]","-\frac{\cot (e+f x) \log (\cos (e+f x)) \left(b \tan ^n(e+f x)\right)^{\frac{1}{n}}}{f}","-\frac{\cot (e+f x) \log (\cos (e+f x)) \left(b \tan ^n(e+f x)\right)^{\frac{1}{n}}}{f}",1,"-((Cot[e + f*x]*Log[Cos[e + f*x]]*(b*Tan[e + f*x]^n)^n^(-1))/f)","A",1
30,1,104,70,0.0677462,"\int \sin ^5(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Sin[e + f*x]^5*(a + b*Tan[e + f*x]^2),x]","-\frac{5 a \cos (e+f x)}{8 f}+\frac{5 a \cos (3 (e+f x))}{48 f}-\frac{a \cos (5 (e+f x))}{80 f}+\frac{19 b \cos (e+f x)}{8 f}-\frac{3 b \cos (3 (e+f x))}{16 f}+\frac{b \cos (5 (e+f x))}{80 f}+\frac{b \sec (e+f x)}{f}","-\frac{(a-b) \cos ^5(e+f x)}{5 f}+\frac{(2 a-3 b) \cos ^3(e+f x)}{3 f}-\frac{(a-3 b) \cos (e+f x)}{f}+\frac{b \sec (e+f x)}{f}",1,"(-5*a*Cos[e + f*x])/(8*f) + (19*b*Cos[e + f*x])/(8*f) + (5*a*Cos[3*(e + f*x)])/(48*f) - (3*b*Cos[3*(e + f*x)])/(16*f) - (a*Cos[5*(e + f*x)])/(80*f) + (b*Cos[5*(e + f*x)])/(80*f) + (b*Sec[e + f*x])/f","A",1
31,1,72,48,0.0499806,"\int \sin ^3(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2),x]","-\frac{3 a \cos (e+f x)}{4 f}+\frac{a \cos (3 (e+f x))}{12 f}+\frac{7 b \cos (e+f x)}{4 f}-\frac{b \cos (3 (e+f x))}{12 f}+\frac{b \sec (e+f x)}{f}","\frac{(a-b) \cos ^3(e+f x)}{3 f}-\frac{(a-2 b) \cos (e+f x)}{f}+\frac{b \sec (e+f x)}{f}",1,"(-3*a*Cos[e + f*x])/(4*f) + (7*b*Cos[e + f*x])/(4*f) + (a*Cos[3*(e + f*x)])/(12*f) - (b*Cos[3*(e + f*x)])/(12*f) + (b*Sec[e + f*x])/f","A",1
32,1,46,28,0.0517337,"\int \sin (e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Sin[e + f*x]*(a + b*Tan[e + f*x]^2),x]","\frac{a \sin (e) \sin (f x)}{f}-\frac{a \cos (e) \cos (f x)}{f}+\frac{b \cos (e+f x)}{f}+\frac{b \sec (e+f x)}{f}","\frac{b \sec (e+f x)}{f}-\frac{(a-b) \cos (e+f x)}{f}",1,"-((a*Cos[e]*Cos[f*x])/f) + (b*Cos[e + f*x])/f + (b*Sec[e + f*x])/f + (a*Sin[e]*Sin[f*x])/f","A",1
33,1,51,25,0.0284554,"\int \csc (e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Csc[e + f*x]*(a + b*Tan[e + f*x]^2),x]","\frac{a \log \left(\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{f}-\frac{a \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{f}+\frac{b \sec (e+f x)}{f}","\frac{b \sec (e+f x)}{f}-\frac{a \tanh ^{-1}(\cos (e+f x))}{f}",1,"-((a*Log[Cos[e/2 + (f*x)/2]])/f) + (a*Log[Sin[e/2 + (f*x)/2]])/f + (b*Sec[e + f*x])/f","B",1
34,1,123,51,0.0461488,"\int \csc ^3(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Csc[e + f*x]^3*(a + b*Tan[e + f*x]^2),x]","-\frac{a \csc ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{a \sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{a \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}-\frac{a \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}+\frac{b \sec (e+f x)}{f}+\frac{b \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f}-\frac{b \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f}","-\frac{(a+2 b) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a \cot (e+f x) \csc (e+f x)}{2 f}+\frac{b \sec (e+f x)}{f}",1,"-1/8*(a*Csc[(e + f*x)/2]^2)/f - (a*Log[Cos[(e + f*x)/2]])/(2*f) - (b*Log[Cos[(e + f*x)/2]])/f + (a*Log[Sin[(e + f*x)/2]])/(2*f) + (b*Log[Sin[(e + f*x)/2]])/f + (a*Sec[(e + f*x)/2]^2)/(8*f) + (b*Sec[e + f*x])/f","B",1
35,1,276,79,6.0571005,"\int \csc ^5(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Csc[e + f*x]^5*(a + b*Tan[e + f*x]^2),x]","-\frac{a \csc ^4\left(\frac{1}{2} (e+f x)\right)}{64 f}-\frac{3 a \csc ^2\left(\frac{1}{2} (e+f x)\right)}{32 f}+\frac{a \sec ^4\left(\frac{1}{2} (e+f x)\right)}{64 f}+\frac{3 a \sec ^2\left(\frac{1}{2} (e+f x)\right)}{32 f}+\frac{3 a \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{8 f}-\frac{3 a \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{8 f}-\frac{b \csc ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{b \sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{3 b \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}-\frac{3 b \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}+\frac{b \sin \left(\frac{1}{2} (e+f x)\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}-\frac{b \sin \left(\frac{1}{2} (e+f x)\right)}{f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{3 (a+4 b) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{(5 a+4 b) \cot (e+f x) \csc (e+f x)}{8 f}-\frac{a \cot ^3(e+f x) \csc (e+f x)}{4 f}+\frac{b \sec (e+f x)}{f}",1,"(-3*a*Csc[(e + f*x)/2]^2)/(32*f) - (b*Csc[(e + f*x)/2]^2)/(8*f) - (a*Csc[(e + f*x)/2]^4)/(64*f) - (3*a*Log[Cos[(e + f*x)/2]])/(8*f) - (3*b*Log[Cos[(e + f*x)/2]])/(2*f) + (3*a*Log[Sin[(e + f*x)/2]])/(8*f) + (3*b*Log[Sin[(e + f*x)/2]])/(2*f) + (3*a*Sec[(e + f*x)/2]^2)/(32*f) + (b*Sec[(e + f*x)/2]^2)/(8*f) + (a*Sec[(e + f*x)/2]^4)/(64*f) + (b*Sin[(e + f*x)/2])/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])) - (b*Sin[(e + f*x)/2])/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","B",1
36,1,89,102,0.360165,"\int \sin ^6(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Sin[e + f*x]^6*(a + b*Tan[e + f*x]^2),x]","\frac{(141 b-45 a) \sin (2 (e+f x))+3 (3 a-5 b) \sin (4 (e+f x))-a \sin (6 (e+f x))+60 a e+60 a f x+b \sin (6 (e+f x))+192 b \tan (e+f x)-420 b e-420 b f x}{192 f}","-\frac{(a-b) \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{(13 a-19 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}-\frac{(11 a-29 b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} x (a-7 b)+\frac{b \tan (e+f x)}{f}",1,"(60*a*e - 420*b*e + 60*a*f*x - 420*b*f*x + (-45*a + 141*b)*Sin[2*(e + f*x)] + 3*(3*a - 5*b)*Sin[4*(e + f*x)] - a*Sin[6*(e + f*x)] + b*Sin[6*(e + f*x)] + 192*b*Tan[e + f*x])/(192*f)","A",1
37,1,58,74,0.3717903,"\int \sin ^4(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Sin[e + f*x]^4*(a + b*Tan[e + f*x]^2),x]","\frac{12 (a-5 b) (e+f x)-8 (a-2 b) \sin (2 (e+f x))+(a-b) \sin (4 (e+f x))+32 b \tan (e+f x)}{32 f}","\frac{(a-b) \sin (e+f x) \cos ^3(e+f x)}{4 f}-\frac{(5 a-9 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} x (a-5 b)+\frac{b \tan (e+f x)}{f}",1,"(12*(a - 5*b)*(e + f*x) - 8*(a - 2*b)*Sin[2*(e + f*x)] + (a - b)*Sin[4*(e + f*x)] + 32*b*Tan[e + f*x])/(32*f)","A",1
38,1,43,46,0.2285807,"\int \sin ^2(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2),x]","\frac{2 (a-3 b) (e+f x)+(b-a) \sin (2 (e+f x))+4 b \tan (e+f x)}{4 f}","-\frac{(a-b) \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} x (a-3 b)+\frac{b \tan (e+f x)}{f}",1,"(2*(a - 3*b)*(e + f*x) + (-a + b)*Sin[2*(e + f*x)] + 4*b*Tan[e + f*x])/(4*f)","A",1
39,1,28,19,0.0127205,"\int \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[a + b*Tan[e + f*x]^2,x]","a x-\frac{b \tan ^{-1}(\tan (e+f x))}{f}+\frac{b \tan (e+f x)}{f}","a x+\frac{b \tan (e+f x)}{f}-b x",1,"a*x - (b*ArcTan[Tan[e + f*x]])/f + (b*Tan[e + f*x])/f","A",1
40,1,24,24,0.0215855,"\int \csc ^2(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2),x]","\frac{b \tan (e+f x)}{f}-\frac{a \cot (e+f x)}{f}","\frac{b \tan (e+f x)}{f}-\frac{a \cot (e+f x)}{f}",1,"-((a*Cot[e + f*x])/f) + (b*Tan[e + f*x])/f","A",1
41,1,60,42,0.0771415,"\int \csc ^4(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Csc[e + f*x]^4*(a + b*Tan[e + f*x]^2),x]","-\frac{2 a \cot (e+f x)}{3 f}-\frac{a \cot (e+f x) \csc ^2(e+f x)}{3 f}+\frac{b \tan (e+f x)}{f}-\frac{b \cot (e+f x)}{f}","-\frac{(a+b) \cot (e+f x)}{f}-\frac{a \cot ^3(e+f x)}{3 f}+\frac{b \tan (e+f x)}{f}",1,"(-2*a*Cot[e + f*x])/(3*f) - (b*Cot[e + f*x])/f - (a*Cot[e + f*x]*Csc[e + f*x]^2)/(3*f) + (b*Tan[e + f*x])/f","A",1
42,1,106,64,0.0509326,"\int \csc ^6(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Csc[e + f*x]^6*(a + b*Tan[e + f*x]^2),x]","-\frac{8 a \cot (e+f x)}{15 f}-\frac{a \cot (e+f x) \csc ^4(e+f x)}{5 f}-\frac{4 a \cot (e+f x) \csc ^2(e+f x)}{15 f}+\frac{b \tan (e+f x)}{f}-\frac{5 b \cot (e+f x)}{3 f}-\frac{b \cot (e+f x) \csc ^2(e+f x)}{3 f}","-\frac{(2 a+b) \cot ^3(e+f x)}{3 f}-\frac{(a+2 b) \cot (e+f x)}{f}-\frac{a \cot ^5(e+f x)}{5 f}+\frac{b \tan (e+f x)}{f}",1,"(-8*a*Cot[e + f*x])/(15*f) - (5*b*Cot[e + f*x])/(3*f) - (4*a*Cot[e + f*x]*Csc[e + f*x]^2)/(15*f) - (b*Cot[e + f*x]*Csc[e + f*x]^2)/(3*f) - (a*Cot[e + f*x]*Csc[e + f*x]^4)/(5*f) + (b*Tan[e + f*x])/f","A",1
43,1,97,107,0.7227467,"\int \sin ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Sin[e + f*x]^5*(a + b*Tan[e + f*x]^2)^2,x]","\frac{-30 \left(5 a^2-38 a b+41 b^2\right) \cos (e+f x)+5 (5 a-13 b) (a-b) \cos (3 (e+f x))-3 (a-b)^2 \cos (5 (e+f x))+480 b (a-2 b) \sec (e+f x)+80 b^2 \sec ^3(e+f x)}{240 f}","-\frac{\left(a^2-6 a b+6 b^2\right) \cos (e+f x)}{f}-\frac{(a-b)^2 \cos ^5(e+f x)}{5 f}+\frac{2 (a-2 b) (a-b) \cos ^3(e+f x)}{3 f}+\frac{2 b (a-2 b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"(-30*(5*a^2 - 38*a*b + 41*b^2)*Cos[e + f*x] + 5*(5*a - 13*b)*(a - b)*Cos[3*(e + f*x)] - 3*(a - b)^2*Cos[5*(e + f*x)] + 480*(a - 2*b)*b*Sec[e + f*x] + 80*b^2*Sec[e + f*x]^3)/(240*f)","A",1
44,1,72,80,0.5263475,"\int \sin ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2)^2,x]","\frac{\left(-9 a^2+42 a b-33 b^2\right) \cos (e+f x)+(a-b)^2 \cos (3 (e+f x))+4 b \sec (e+f x) \left(6 a+b \sec ^2(e+f x)-9 b\right)}{12 f}","\frac{(a-b)^2 \cos ^3(e+f x)}{3 f}-\frac{(a-3 b) (a-b) \cos (e+f x)}{f}+\frac{b (2 a-3 b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"((-9*a^2 + 42*a*b - 33*b^2)*Cos[e + f*x] + (a - b)^2*Cos[3*(e + f*x)] + 4*b*Sec[e + f*x]*(6*a - 9*b + b*Sec[e + f*x]^2))/(12*f)","A",1
45,1,48,54,0.3068492,"\int \sin (e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^2,x]","\frac{b \sec (e+f x) \left(6 a+b \sec ^2(e+f x)-6 b\right)-3 (a-b)^2 \cos (e+f x)}{3 f}","-\frac{(a-b)^2 \cos (e+f x)}{f}+\frac{2 b (a-b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"(-3*(a - b)^2*Cos[e + f*x] + b*Sec[e + f*x]*(6*a - 6*b + b*Sec[e + f*x]^2))/(3*f)","A",1
46,1,66,52,0.1715799,"\int \csc (e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Csc[e + f*x]*(a + b*Tan[e + f*x]^2)^2,x]","\frac{3 a^2 \left(\log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+3 b (2 a-b) \sec (e+f x)+b^2 \sec ^3(e+f x)}{3 f}","-\frac{a^2 \tanh ^{-1}(\cos (e+f x))}{f}+\frac{b (2 a-b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"(3*a^2*(-Log[Cos[(e + f*x)/2]] + Log[Sin[(e + f*x)/2]]) + 3*(2*a - b)*b*Sec[e + f*x] + b^2*Sec[e + f*x]^3)/(3*f)","A",1
47,1,376,82,6.1304448,"\int \csc ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Csc[e + f*x]^3*(a + b*Tan[e + f*x]^2)^2,x]","\frac{\left(a^2+4 a b\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}+\frac{\left(-a^2-4 a b\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 f}-\frac{a^2 \csc ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{a^2 \sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 f}+\frac{b^2 \left(-\sin \left(\frac{1}{2} (e+f x)\right)\right)-12 a b \sin \left(\frac{1}{2} (e+f x)\right)}{6 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{12 a b \sin \left(\frac{1}{2} (e+f x)\right)+b^2 \sin \left(\frac{1}{2} (e+f x)\right)}{6 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{b^2 \sin \left(\frac{1}{2} (e+f x)\right)}{6 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{b^2}{12 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{b^2}{12 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{b^2 \sin \left(\frac{1}{2} (e+f x)\right)}{6 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\frac{a^2 \csc ^2(e+f x) \sec (e+f x)}{2 f}+\frac{a (a+4 b) \sec (e+f x)}{2 f}-\frac{a (a+4 b) \tanh ^{-1}(\cos (e+f x))}{2 f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"-1/8*(a^2*Csc[(e + f*x)/2]^2)/f + ((-a^2 - 4*a*b)*Log[Cos[(e + f*x)/2]])/(2*f) + ((a^2 + 4*a*b)*Log[Sin[(e + f*x)/2]])/(2*f) + (a^2*Sec[(e + f*x)/2]^2)/(8*f) + b^2/(12*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (b^2*Sin[(e + f*x)/2])/(6*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) - (b^2*Sin[(e + f*x)/2])/(6*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3) + b^2/(12*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (-12*a*b*Sin[(e + f*x)/2] - b^2*Sin[(e + f*x)/2])/(6*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) + (12*a*b*Sin[(e + f*x)/2] + b^2*Sin[(e + f*x)/2])/(6*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","B",1
48,1,447,123,6.1896632,"\int \csc ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Csc[e + f*x]^5*(a + b*Tan[e + f*x]^2)^2,x]","\frac{\left(3 a^2+24 a b+8 b^2\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{8 f}+\frac{\left(-3 a^2-24 a b-8 b^2\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{8 f}+\frac{\left(-3 a^2-8 a b\right) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{32 f}+\frac{\left(3 a^2+8 a b\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{32 f}-\frac{a^2 \csc ^4\left(\frac{1}{2} (e+f x)\right)}{64 f}+\frac{a^2 \sec ^4\left(\frac{1}{2} (e+f x)\right)}{64 f}+\frac{-12 a b \sin \left(\frac{1}{2} (e+f x)\right)-7 b^2 \sin \left(\frac{1}{2} (e+f x)\right)}{6 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{12 a b \sin \left(\frac{1}{2} (e+f x)\right)+7 b^2 \sin \left(\frac{1}{2} (e+f x)\right)}{6 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{b^2 \sin \left(\frac{1}{2} (e+f x)\right)}{6 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{b^2}{12 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{b^2}{12 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{b^2 \sin \left(\frac{1}{2} (e+f x)\right)}{6 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","\frac{\left(a^2+8 a b+4 b^2\right) \sec (e+f x)}{4 f}-\frac{\left(3 a^2+24 a b+8 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}-\frac{a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"((-3*a^2 - 8*a*b)*Csc[(e + f*x)/2]^2)/(32*f) - (a^2*Csc[(e + f*x)/2]^4)/(64*f) + ((-3*a^2 - 24*a*b - 8*b^2)*Log[Cos[(e + f*x)/2]])/(8*f) + ((3*a^2 + 24*a*b + 8*b^2)*Log[Sin[(e + f*x)/2]])/(8*f) + ((3*a^2 + 8*a*b)*Sec[(e + f*x)/2]^2)/(32*f) + (a^2*Sec[(e + f*x)/2]^4)/(64*f) + b^2/(12*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (b^2*Sin[(e + f*x)/2])/(6*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) - (b^2*Sin[(e + f*x)/2])/(6*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3) + b^2/(12*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (-12*a*b*Sin[(e + f*x)/2] - 7*b^2*Sin[(e + f*x)/2])/(6*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) + (12*a*b*Sin[(e + f*x)/2] + 7*b^2*Sin[(e + f*x)/2])/(6*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","B",1
49,1,96,122,1.4591056,"\int \sin ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Sin[e + f*x]^4*(a + b*Tan[e + f*x]^2)^2,x]","\frac{12 \left(3 a^2-30 a b+35 b^2\right) (e+f x)-24 \left(a^2-4 a b+3 b^2\right) \sin (2 (e+f x))+3 (a-b)^2 \sin (4 (e+f x))+32 b \tan (e+f x) \left(6 a+b \sec ^2(e+f x)-10 b\right)}{96 f}","-\frac{\left(a^2-10 a b+13 b^2\right) \tan (e+f x)}{4 f}+\frac{1}{8} x \left(3 a^2-30 a b+35 b^2\right)+\frac{(a-b)^2 \sin ^4(e+f x) \tan (e+f x)}{4 f}-\frac{(a-9 b) (a-b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"(12*(3*a^2 - 30*a*b + 35*b^2)*(e + f*x) - 24*(a^2 - 4*a*b + 3*b^2)*Sin[2*(e + f*x)] + 3*(a - b)^2*Sin[4*(e + f*x)] + 32*b*(6*a - 10*b + b*Sec[e + f*x]^2)*Tan[e + f*x])/(96*f)","A",1
50,1,71,85,0.7327443,"\int \sin ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2)^2,x]","\frac{6 \left(a^2-6 a b+5 b^2\right) (e+f x)-3 (a-b)^2 \sin (2 (e+f x))+4 b \tan (e+f x) \left(6 a+b \sec ^2(e+f x)-7 b\right)}{12 f}","-\frac{(a-5 b) (a-b) \tan (e+f x)}{2 f}+\frac{(a-b)^2 \sin ^2(e+f x) \tan (e+f x)}{2 f}+\frac{1}{2} x (a-5 b) (a-b)+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"(6*(a^2 - 6*a*b + 5*b^2)*(e + f*x) - 3*(a - b)^2*Sin[2*(e + f*x)] + 4*b*(6*a - 7*b + b*Sec[e + f*x]^2)*Tan[e + f*x])/(12*f)","A",1
51,1,73,46,0.6112931,"\int \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[(a + b*Tan[e + f*x]^2)^2,x]","\frac{\tan (e+f x) \left(b \left(6 a-b \left(3-\tan ^2(e+f x)\right)\right)+\frac{3 (a-b)^2 \tanh ^{-1}\left(\sqrt{-\tan ^2(e+f x)}\right)}{\sqrt{-\tan ^2(e+f x)}}\right)}{3 f}","\frac{b (2 a-b) \tan (e+f x)}{f}+x (a-b)^2+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"(Tan[e + f*x]*((3*(a - b)^2*ArcTanh[Sqrt[-Tan[e + f*x]^2]])/Sqrt[-Tan[e + f*x]^2] + b*(6*a - b*(3 - Tan[e + f*x]^2))))/(3*f)","A",1
52,1,44,46,0.515811,"\int \csc ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2)^2,x]","\frac{b \tan (e+f x) \left(6 a+b \sec ^2(e+f x)-b\right)-3 a^2 \cot (e+f x)}{3 f}","-\frac{a^2 \cot (e+f x)}{f}+\frac{2 a b \tan (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"(-3*a^2*Cot[e + f*x] + b*(6*a - b + b*Sec[e + f*x]^2)*Tan[e + f*x])/(3*f)","A",1
53,1,59,70,0.470326,"\int \csc ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Csc[e + f*x]^4*(a + b*Tan[e + f*x]^2)^2,x]","\frac{b \tan (e+f x) \left(6 a+b \sec ^2(e+f x)+2 b\right)-a \cot (e+f x) \left(a \csc ^2(e+f x)+2 a+6 b\right)}{3 f}","-\frac{a^2 \cot ^3(e+f x)}{3 f}+\frac{b (2 a+b) \tan (e+f x)}{f}-\frac{a (a+2 b) \cot (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"(-(a*Cot[e + f*x]*(2*a + 6*b + a*Csc[e + f*x]^2)) + b*(6*a + 2*b + b*Sec[e + f*x]^2)*Tan[e + f*x])/(3*f)","A",1
54,1,88,93,0.8062395,"\int \csc ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Csc[e + f*x]^6*(a + b*Tan[e + f*x]^2)^2,x]","\frac{5 b \tan (e+f x) \left(6 a+b \sec ^2(e+f x)+5 b\right)-\cot (e+f x) \left(3 a^2 \csc ^4(e+f x)+8 a^2+2 a (2 a+5 b) \csc ^2(e+f x)+50 a b+15 b^2\right)}{15 f}","-\frac{\left(a^2+4 a b+b^2\right) \cot (e+f x)}{f}-\frac{a^2 \cot ^5(e+f x)}{5 f}+\frac{2 b (a+b) \tan (e+f x)}{f}-\frac{2 a (a+b) \cot ^3(e+f x)}{3 f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"(-(Cot[e + f*x]*(8*a^2 + 50*a*b + 15*b^2 + 2*a*(2*a + 5*b)*Csc[e + f*x]^2 + 3*a^2*Csc[e + f*x]^4)) + 5*b*(6*a + 5*b + b*Sec[e + f*x]^2)*Tan[e + f*x])/(15*f)","A",1
55,1,177,117,3.0773037,"\int \frac{\sin ^5(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2),x]","\frac{\sqrt{a-b} \cos (e+f x) \left(4 \left(7 a^2-9 a b+2 b^2\right) \cos (2 (e+f x))-89 a^2-3 (a-b)^2 \cos (4 (e+f x))-42 a b+11 b^2\right)+120 a^2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a-b}-\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)+120 a^2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a-b}+\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{120 f (a-b)^{7/2}}","-\frac{a^2 \cos (e+f x)}{f (a-b)^3}-\frac{a^2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{f (a-b)^{7/2}}-\frac{\cos ^5(e+f x)}{5 f (a-b)}+\frac{(2 a-b) \cos ^3(e+f x)}{3 f (a-b)^2}",1,"(120*a^2*Sqrt[b]*ArcTan[(Sqrt[a - b] - Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]] + 120*a^2*Sqrt[b]*ArcTan[(Sqrt[a - b] + Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]] + Sqrt[a - b]*Cos[e + f*x]*(-89*a^2 - 42*a*b + 11*b^2 + 4*(7*a^2 - 9*a*b + 2*b^2)*Cos[2*(e + f*x)] - 3*(a - b)^2*Cos[4*(e + f*x)]))/(120*(a - b)^(7/2)*f)","A",1
56,1,149,84,0.6592633,"\int \frac{\sin ^3(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2),x]","\frac{(a-b) \cos (e+f x) ((a-b) \cos (2 (e+f x))-5 a-b)+6 a \sqrt{b} \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b}-\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)+6 a \sqrt{b} \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b}+\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{6 f (a-b)^3}","\frac{\cos ^3(e+f x)}{3 f (a-b)}-\frac{a \cos (e+f x)}{f (a-b)^2}-\frac{a \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}",1,"(6*a*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[a - b] - Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]] + 6*a*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[a - b] + Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]] + (a - b)*Cos[e + f*x]*(-5*a - b + (a - b)*Cos[2*(e + f*x)]))/(6*(a - b)^3*f)","A",1
57,1,121,60,0.2636478,"\int \frac{\sin (e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]/(a + b*Tan[e + f*x]^2),x]","\frac{(b-a) \cos (e+f x)+\sqrt{b} \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b}-\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)+\sqrt{b} \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b}+\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{f (a-b)^2}","-\frac{\cos (e+f x)}{f (a-b)}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}",1,"(Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[a - b] - Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]] + Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[a - b] + Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]] + (-a + b)*Cos[e + f*x])/((a - b)^2*f)","B",1
58,1,144,60,0.2427699,"\int \frac{\csc (e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]/(a + b*Tan[e + f*x]^2),x]","\frac{\sqrt{b} \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b}-\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)+\sqrt{b} \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b}+\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)-(a-b) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)}{a f (a-b)}","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{a f \sqrt{a-b}}-\frac{\tanh ^{-1}(\cos (e+f x))}{a f}",1,"(Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[a - b] - Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]] + Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[a - b] + Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]] - (a - b)*(Log[Cos[(e + f*x)/2]] - Log[Sin[(e + f*x)/2]]))/(a*(a - b)*f)","B",1
59,1,195,89,0.6533614,"\int \frac{\csc ^3(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2),x]","\frac{8 \sqrt{b} \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b}-\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)+8 \sqrt{b} \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b}+\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)-a \csc ^2\left(\frac{1}{2} (e+f x)\right)+a \sec ^2\left(\frac{1}{2} (e+f x)\right)+4 a \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-4 a \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)-8 b \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)+8 b \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{8 a^2 f}","-\frac{(a-2 b) \tanh ^{-1}(\cos (e+f x))}{2 a^2 f}-\frac{\sqrt{b} \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{a^2 f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f}",1,"(8*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[a - b] - Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]] + 8*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[a - b] + Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]] - a*Csc[(e + f*x)/2]^2 - 4*a*Log[Cos[(e + f*x)/2]] + 8*b*Log[Cos[(e + f*x)/2]] + 4*a*Log[Sin[(e + f*x)/2]] - 8*b*Log[Sin[(e + f*x)/2]] + a*Sec[(e + f*x)/2]^2)/(8*a^2*f)","B",1
60,1,326,130,6.2612953,"\int \frac{\csc ^5(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2),x]","\frac{\sqrt{b} (a-b)^{3/2} \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (e+f x)\right) \left(\sqrt{a-b} \cos \left(\frac{1}{2} (e+f x)\right)-\sqrt{a} \sin \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{b}}\right)}{a^3 f}+\frac{\sqrt{b} (a-b)^{3/2} \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (e+f x)\right) \left(\sqrt{a-b} \cos \left(\frac{1}{2} (e+f x)\right)+\sqrt{a} \sin \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{b}}\right)}{a^3 f}+\frac{(4 b-3 a) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{32 a^2 f}+\frac{(3 a-4 b) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{32 a^2 f}+\frac{\left(3 a^2-12 a b+8 b^2\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{8 a^3 f}+\frac{\left(-3 a^2+12 a b-8 b^2\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{8 a^3 f}-\frac{\csc ^4\left(\frac{1}{2} (e+f x)\right)}{64 a f}+\frac{\sec ^4\left(\frac{1}{2} (e+f x)\right)}{64 a f}","-\frac{\sqrt{b} (a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{a^3 f}-\frac{(5 a-4 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f}-\frac{\left(3 a^2-12 a b+8 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 a^3 f}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f}",1,"((a - b)^(3/2)*Sqrt[b]*ArcTan[(Sec[(e + f*x)/2]*(Sqrt[a - b]*Cos[(e + f*x)/2] - Sqrt[a]*Sin[(e + f*x)/2]))/Sqrt[b]])/(a^3*f) + ((a - b)^(3/2)*Sqrt[b]*ArcTan[(Sec[(e + f*x)/2]*(Sqrt[a - b]*Cos[(e + f*x)/2] + Sqrt[a]*Sin[(e + f*x)/2]))/Sqrt[b]])/(a^3*f) + ((-3*a + 4*b)*Csc[(e + f*x)/2]^2)/(32*a^2*f) - Csc[(e + f*x)/2]^4/(64*a*f) + ((-3*a^2 + 12*a*b - 8*b^2)*Log[Cos[(e + f*x)/2]])/(8*a^3*f) + ((3*a^2 - 12*a*b + 8*b^2)*Log[Sin[(e + f*x)/2]])/(8*a^3*f) + ((3*a - 4*b)*Sec[(e + f*x)/2]^2)/(32*a^2*f) + Sec[(e + f*x)/2]^4/(64*a*f)","B",1
61,1,140,178,0.5712796,"\int \frac{\sin ^6(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^6/(a + b*Tan[e + f*x]^2),x]","-\frac{192 a^{5/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)-12 \left(5 a^3+15 a^2 b-5 a b^2+b^3\right) (e+f x)+(a-b)^3 \sin (6 (e+f x))-3 (3 a-b) (a-b)^2 \sin (4 (e+f x))+3 (5 a-b) (3 a+b) (a-b) \sin (2 (e+f x))}{192 f (a-b)^4}","-\frac{a^{5/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{f (a-b)^4}-\frac{\left(11 a^2-4 a b+b^2\right) \sin (e+f x) \cos (e+f x)}{16 f (a-b)^3}+\frac{x \left(5 a^3+15 a^2 b-5 a b^2+b^3\right)}{16 (a-b)^4}+\frac{\sin ^3(e+f x) \cos ^3(e+f x)}{6 f (a-b)}+\frac{(3 a-b) \sin (e+f x) \cos ^3(e+f x)}{8 f (a-b)^2}",1,"-1/192*(-12*(5*a^3 + 15*a^2*b - 5*a*b^2 + b^3)*(e + f*x) + 192*a^(5/2)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] + 3*(a - b)*(5*a - b)*(3*a + b)*Sin[2*(e + f*x)] - 3*(a - b)^2*(3*a - b)*Sin[4*(e + f*x)] + (a - b)^3*Sin[6*(e + f*x)])/((a - b)^4*f)","A",1
62,1,99,129,0.2726825,"\int \frac{\sin ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2),x]","\frac{-32 a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)+4 \left(3 a^2+6 a b-b^2\right) (e+f x)+(a-b)^2 \sin (4 (e+f x))-8 a (a-b) \sin (2 (e+f x))}{32 f (a-b)^3}","-\frac{a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{f (a-b)^3}+\frac{x \left(3 a^2+6 a b-b^2\right)}{8 (a-b)^3}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b)}-\frac{(5 a-b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2}",1,"(4*(3*a^2 + 6*a*b - b^2)*(e + f*x) - 32*a^(3/2)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] - 8*a*(a - b)*Sin[2*(e + f*x)] + (a - b)^2*Sin[4*(e + f*x)])/(32*(a - b)^3*f)","A",1
63,1,69,82,0.1579161,"\int \frac{\sin ^2(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2),x]","\frac{2 (a+b) (e+f x)+(b-a) \sin (2 (e+f x))-4 \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{4 f (a-b)^2}","-\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{f (a-b)^2}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b)}+\frac{x (a+b)}{2 (a-b)^2}",1,"(2*(a + b)*(e + f*x) - 4*Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] + (-a + b)*Sin[2*(e + f*x)])/(4*(a - b)^2*f)","A",1
64,1,49,50,0.0627189,"\int \frac{1}{a+b \tan ^2(e+f x)} \, dx","Integrate[(a + b*Tan[e + f*x]^2)^(-1),x]","\frac{\tan ^{-1}(\tan (e+f x))-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{a}}}{a f-b f}","\frac{x}{a-b}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{a} f (a-b)}",1,"(ArcTan[Tan[e + f*x]] - (Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/Sqrt[a])/(a*f - b*f)","A",1
65,1,48,48,0.1050162,"\int \frac{\csc ^2(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2),x]","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{\cot (e+f x)}{a f}","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{\cot (e+f x)}{a f}",1,"-((Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(3/2)*f)) - Cot[e + f*x]/(a*f)","A",1
66,1,73,76,0.314944,"\int \frac{\csc ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2),x]","\frac{3 \sqrt{b} (b-a) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)-\sqrt{a} \cot (e+f x) \left(a \csc ^2(e+f x)+2 a-3 b\right)}{3 a^{5/2} f}","-\frac{\sqrt{b} (a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{(a-b) \cot (e+f x)}{a^2 f}-\frac{\cot ^3(e+f x)}{3 a f}",1,"(3*Sqrt[b]*(-a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] - Sqrt[a]*Cot[e + f*x]*(2*a - 3*b + a*Csc[e + f*x]^2))/(3*a^(5/2)*f)","A",1
67,1,103,105,0.8034587,"\int \frac{\csc ^6(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2),x]","\frac{-\sqrt{a} \cot (e+f x) \left(3 a^2 \csc ^4(e+f x)+8 a^2+a (4 a-5 b) \csc ^2(e+f x)-25 a b+15 b^2\right)-15 \sqrt{b} (a-b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{15 a^{7/2} f}","-\frac{\sqrt{b} (a-b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{7/2} f}-\frac{(a-b)^2 \cot (e+f x)}{a^3 f}-\frac{(2 a-b) \cot ^3(e+f x)}{3 a^2 f}-\frac{\cot ^5(e+f x)}{5 a f}",1,"(-15*(a - b)^2*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] - Sqrt[a]*Cot[e + f*x]*(8*a^2 - 25*a*b + 15*b^2 + a*(4*a - 5*b)*Csc[e + f*x]^2 + 3*a^2*Csc[e + f*x]^4))/(15*a^(7/2)*f)","A",1
68,1,215,204,3.8271839,"\int \frac{\sin ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2)^2,x]","\frac{\frac{(a-b) (5 (5 a+3 b) \cos (3 (e+f x))+3 (b-a) \cos (5 (e+f x)))-30 \cos (e+f x) \left(a^2 \left(\frac{8 b}{(a-b) \cos (2 (e+f x))+a+b}+5\right)+18 a b+b^2\right)}{(a-b)^4}+\frac{120 a \sqrt{b} (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{a-b}-\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{9/2}}+\frac{120 a \sqrt{b} (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{a-b}+\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{9/2}}}{240 f}","-\frac{\left(5 a^2+10 a b-b^2\right) \cos (e+f x)}{5 f (a-b)^4}-\frac{b \left(5 a^2+2 b^2\right) \sec (e+f x)}{10 f (a-b)^4 \left(a+b \sec ^2(e+f x)-b\right)}+\frac{(10 a-3 b) \cos ^3(e+f x)}{15 f (a-b)^3}-\frac{\cos ^5(e+f x)}{5 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)}-\frac{a \sqrt{b} (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 f (a-b)^{9/2}}",1,"((120*a*Sqrt[b]*(3*a + 4*b)*ArcTan[(Sqrt[a - b] - Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(9/2) + (120*a*Sqrt[b]*(3*a + 4*b)*ArcTan[(Sqrt[a - b] + Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(9/2) + (-30*Cos[e + f*x]*(18*a*b + b^2 + a^2*(5 + (8*b)/(a + b + (a - b)*Cos[2*(e + f*x)]))) + (a - b)*(5*(5*a + 3*b)*Cos[3*(e + f*x)] + 3*(-a + b)*Cos[5*(e + f*x)]))/(a - b)^4)/(240*f)","A",1
69,1,182,133,3.3161444,"\int \frac{\sin ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2)^2,x]","\frac{-\frac{\cos (e+f x) \left(\frac{12 a b}{(a-b) \cos (2 (e+f x))+a+b}+9 a+15 b\right)+(b-a) \cos (3 (e+f x))}{(a-b)^3}+\frac{6 \sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{a-b}-\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{7/2}}+\frac{6 \sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{a-b}+\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{7/2}}}{12 f}","\frac{\cos ^3(e+f x)}{3 f (a-b)^2}-\frac{(a+b) \cos (e+f x)}{f (a-b)^3}-\frac{a b \sec (e+f x)}{2 f (a-b)^3 \left(a+b \sec ^2(e+f x)-b\right)}-\frac{\sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 f (a-b)^{7/2}}",1,"((6*Sqrt[b]*(3*a + 2*b)*ArcTan[(Sqrt[a - b] - Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(7/2) + (6*Sqrt[b]*(3*a + 2*b)*ArcTan[(Sqrt[a - b] + Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(7/2) - (Cos[e + f*x]*(9*a + 15*b + (12*a*b)/(a + b + (a - b)*Cos[2*(e + f*x)])) + (-a + b)*Cos[3*(e + f*x)])/(a - b)^3)/(12*f)","A",1
70,1,146,101,0.8294602,"\int \frac{\sin (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Sin[e + f*x]/(a + b*Tan[e + f*x]^2)^2,x]","\frac{\frac{2 \cos (e+f x) \left(-\frac{b}{(a-b) \cos (2 (e+f x))+a+b}-1\right)}{(a-b)^2}+\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a-b}-\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{5/2}}+\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a-b}+\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{5/2}}}{2 f}","-\frac{3 \cos (e+f x)}{2 f (a-b)^2}+\frac{\cos (e+f x)}{2 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)}-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 f (a-b)^{5/2}}",1,"((3*Sqrt[b]*ArcTan[(Sqrt[a - b] - Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(5/2) + (3*Sqrt[b]*ArcTan[(Sqrt[a - b] + Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(5/2) + (2*Cos[e + f*x]*(-1 - b/(a + b + (a - b)*Cos[2*(e + f*x)])))/(a - b)^2)/(2*f)","A",1
71,1,184,110,0.9018143,"\int \frac{\csc (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Csc[e + f*x]/(a + b*Tan[e + f*x]^2)^2,x]","\frac{-\frac{2 a b \cos (e+f x)}{(a-b) ((a-b) \cos (2 (e+f x))+a+b)}+\frac{\sqrt{b} (3 a-2 b) \tan ^{-1}\left(\frac{\sqrt{a-b}-\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{3/2}}+\frac{\sqrt{b} (3 a-2 b) \tan ^{-1}\left(\frac{\sqrt{a-b}+\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{3/2}}+2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 a^2 f}","-\frac{\sqrt{b} (3 a-2 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 a^2 f (a-b)^{3/2}}-\frac{\tanh ^{-1}(\cos (e+f x))}{a^2 f}-\frac{b \sec (e+f x)}{2 a f (a-b) \left(a+b \sec ^2(e+f x)-b\right)}",1,"(((3*a - 2*b)*Sqrt[b]*ArcTan[(Sqrt[a - b] - Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(3/2) + ((3*a - 2*b)*Sqrt[b]*ArcTan[(Sqrt[a - b] + Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(3/2) - (2*a*b*Cos[e + f*x])/((a - b)*(a + b + (a - b)*Cos[2*(e + f*x)])) - 2*Log[Cos[(e + f*x)/2]] + 2*Log[Sin[(e + f*x)/2]])/(2*a^2*f)","A",1
72,1,325,147,6.3072594,"\int \frac{\csc ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2)^2,x]","\frac{(a-4 b) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 a^3 f}+\frac{(4 b-a) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 a^3 f}-\frac{\sqrt{b} (3 a-4 b) \sqrt{a-b} \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (e+f x)\right) \left(\sqrt{a-b} \cos \left(\frac{1}{2} (e+f x)\right)-\sqrt{a} \sin \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{b}}\right)}{2 a^3 f (b-a)}-\frac{\sqrt{b} (3 a-4 b) \sqrt{a-b} \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (e+f x)\right) \left(\sqrt{a-b} \cos \left(\frac{1}{2} (e+f x)\right)+\sqrt{a} \sin \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{b}}\right)}{2 a^3 f (b-a)}-\frac{b \cos (e+f x)}{a^2 f (a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b)}-\frac{\csc ^2\left(\frac{1}{2} (e+f x)\right)}{8 a^2 f}+\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 a^2 f}","-\frac{(a-4 b) \tanh ^{-1}(\cos (e+f x))}{2 a^3 f}-\frac{\sqrt{b} (3 a-4 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 a^3 f \sqrt{a-b}}-\frac{b \sec (e+f x)}{a^2 f \left(a+b \sec ^2(e+f x)-b\right)}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left(a+b \sec ^2(e+f x)-b\right)}",1,"-1/2*((3*a - 4*b)*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sec[(e + f*x)/2]*(Sqrt[a - b]*Cos[(e + f*x)/2] - Sqrt[a]*Sin[(e + f*x)/2]))/Sqrt[b]])/(a^3*(-a + b)*f) - ((3*a - 4*b)*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sec[(e + f*x)/2]*(Sqrt[a - b]*Cos[(e + f*x)/2] + Sqrt[a]*Sin[(e + f*x)/2]))/Sqrt[b]])/(2*a^3*(-a + b)*f) - (b*Cos[e + f*x])/(a^2*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])) - Csc[(e + f*x)/2]^2/(8*a^2*f) + ((-a + 4*b)*Log[Cos[(e + f*x)/2]])/(2*a^3*f) + ((a - 4*b)*Log[Sin[(e + f*x)/2]])/(2*a^3*f) + Sec[(e + f*x)/2]^2/(8*a^2*f)","B",1
73,1,392,210,6.3610123,"\int \frac{\csc ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2)^2,x]","\frac{3 \sqrt{b} (a-2 b) \sqrt{a-b} \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (e+f x)\right) \left(\sqrt{a-b} \cos \left(\frac{1}{2} (e+f x)\right)-\sqrt{a} \sin \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{b}}\right)}{2 a^4 f}+\frac{3 \sqrt{b} (a-2 b) \sqrt{a-b} \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (e+f x)\right) \left(\sqrt{a-b} \cos \left(\frac{1}{2} (e+f x)\right)+\sqrt{a} \sin \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{b}}\right)}{2 a^4 f}+\frac{b^2 \cos (e+f x)-a b \cos (e+f x)}{a^3 f (a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b)}+\frac{(8 b-3 a) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{32 a^3 f}+\frac{(3 a-8 b) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{32 a^3 f}-\frac{\csc ^4\left(\frac{1}{2} (e+f x)\right)}{64 a^2 f}+\frac{\sec ^4\left(\frac{1}{2} (e+f x)\right)}{64 a^2 f}+\frac{3 \left(a^2-8 a b+8 b^2\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{8 a^4 f}-\frac{3 \left(a^2-8 a b+8 b^2\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{8 a^4 f}","-\frac{3 \sqrt{b} (a-2 b) \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 a^4 f}-\frac{3 b (3 a-4 b) \sec (e+f x)}{8 a^3 f \left(a+b \sec ^2(e+f x)-b\right)}-\frac{(5 a-6 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left(a+b \sec ^2(e+f x)-b\right)}-\frac{3 \left(a^2-8 a b+8 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 a^4 f}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f \left(a+b \sec ^2(e+f x)-b\right)}",1,"(3*(a - 2*b)*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sec[(e + f*x)/2]*(Sqrt[a - b]*Cos[(e + f*x)/2] - Sqrt[a]*Sin[(e + f*x)/2]))/Sqrt[b]])/(2*a^4*f) + (3*(a - 2*b)*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sec[(e + f*x)/2]*(Sqrt[a - b]*Cos[(e + f*x)/2] + Sqrt[a]*Sin[(e + f*x)/2]))/Sqrt[b]])/(2*a^4*f) + (-(a*b*Cos[e + f*x]) + b^2*Cos[e + f*x])/(a^3*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])) + ((-3*a + 8*b)*Csc[(e + f*x)/2]^2)/(32*a^3*f) - Csc[(e + f*x)/2]^4/(64*a^2*f) - (3*(a^2 - 8*a*b + 8*b^2)*Log[Cos[(e + f*x)/2]])/(8*a^4*f) + (3*(a^2 - 8*a*b + 8*b^2)*Log[Sin[(e + f*x)/2]])/(8*a^4*f) + ((3*a - 8*b)*Sec[(e + f*x)/2]^2)/(32*a^3*f) + Sec[(e + f*x)/2]^4/(64*a^2*f)","A",1
74,1,136,196,1.6529633,"\int \frac{\sin ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2,x]","\frac{12 \left(a^2+6 a b+b^2\right) (e+f x)+(a-b)^2 \sin (4 (e+f x))-8 (a+b) (a-b) \sin (2 (e+f x))-48 \sqrt{a} \sqrt{b} (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)-\frac{16 a b (a-b) \sin (2 (e+f x))}{(a-b) \cos (2 (e+f x))+a+b}}{32 f (a-b)^4}","\frac{3 x \left(a^2+6 a b+b^2\right)}{8 (a-b)^4}-\frac{3 \sqrt{a} \sqrt{b} (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 f (a-b)^4}-\frac{3 b (3 a+b) \tan (e+f x)}{8 f (a-b)^3 \left(a+b \tan ^2(e+f x)\right)}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{(5 a+b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}",1,"(12*(a^2 + 6*a*b + b^2)*(e + f*x) - 48*Sqrt[a]*Sqrt[b]*(a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] - 8*(a - b)*(a + b)*Sin[2*(e + f*x)] - (16*a*(a - b)*b*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)]) + (a - b)^2*Sin[4*(e + f*x)])/(32*(a - b)^4*f)","A",1
75,1,111,138,1.7916035,"\int \frac{\sin ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{-2 (a+3 b) (e+f x)+(a-b) \sin (2 (e+f x))+\frac{2 \sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{2 b (a-b) \sin (2 (e+f x))}{(a-b) \cos (2 (e+f x))+a+b}}{4 f (a-b)^3}","-\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 \sqrt{a} f (a-b)^3}-\frac{b \tan (e+f x)}{f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x (a+3 b)}{2 (a-b)^3}",1,"-1/4*(-2*(a + 3*b)*(e + f*x) + (2*Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/Sqrt[a] + (a - b)*Sin[2*(e + f*x)] + (2*(a - b)*b*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)]))/((a - b)^3*f)","A",1
76,1,88,97,1.1127525,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[(a + b*Tan[e + f*x]^2)^(-2),x]","\frac{\frac{\sqrt{b} (b-3 a) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{3/2}}+\frac{b (b-a) \tan (e+f x)}{a \left(a+b \tan ^2(e+f x)\right)}+2 \tan ^{-1}(\tan (e+f x))}{2 f (a-b)^2}","-\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{3/2} f (a-b)^2}-\frac{b \tan (e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x}{(a-b)^2}",1,"(2*ArcTan[Tan[e + f*x]] + (Sqrt[b]*(-3*a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/a^(3/2) + (b*(-a + b)*Tan[e + f*x])/(a*(a + b*Tan[e + f*x]^2)))/(2*(a - b)^2*f)","A",1
77,1,83,82,0.6743092,"\int \frac{\csc ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2)^2,x]","\frac{\sqrt{a} \left(-\frac{b \sin (2 (e+f x))}{(a-b) \cos (2 (e+f x))+a+b}-2 \cot (e+f x)\right)-3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{5/2} f}","-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{5/2} f}-\frac{3 \cot (e+f x)}{2 a^2 f}+\frac{\cot (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)\right)}",1,"(-3*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] + Sqrt[a]*(-2*Cot[e + f*x] - (b*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)])))/(2*a^(5/2)*f)","A",1
78,1,112,116,0.9909208,"\int \frac{\csc ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2,x]","\frac{3 \sqrt{b} (5 b-3 a) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)+\sqrt{a} \left(\frac{3 b (b-a) \sin (2 (e+f x))}{(a-b) \cos (2 (e+f x))+a+b}-2 \cot (e+f x) \left(a \csc ^2(e+f x)+2 a-6 b\right)\right)}{6 a^{7/2} f}","-\frac{\sqrt{b} (3 a-5 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{7/2} f}-\frac{b (a-b) \tan (e+f x)}{2 a^3 f \left(a+b \tan ^2(e+f x)\right)}-\frac{(a-2 b) \cot (e+f x)}{a^3 f}-\frac{\cot ^3(e+f x)}{3 a^2 f}",1,"(3*Sqrt[b]*(-3*a + 5*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] + Sqrt[a]*(-2*Cot[e + f*x]*(2*a - 6*b + a*Csc[e + f*x]^2) + (3*b*(-a + b)*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)])))/(6*a^(7/2)*f)","A",1
79,1,151,182,1.8953004,"\int \frac{\csc ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2)^2,x]","\frac{\sqrt{a} \left(-2 \cot (e+f x) \left(3 a^2 \csc ^4(e+f x)+8 a^2+2 a (2 a-5 b) \csc ^2(e+f x)-50 a b+45 b^2\right)-\frac{15 b (a-b)^2 \sin (2 (e+f x))}{(a-b) \cos (2 (e+f x))+a+b}\right)-15 \sqrt{b} \left(3 a^2-10 a b+7 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{30 a^{9/2} f}","-\frac{\sqrt{b} (3 a-7 b) (a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{9/2} f}-\frac{(10 a-7 b) \cot ^3(e+f x)}{15 a^3 f}-\frac{b \left(5 a^2-10 a b+7 b^2\right) \tan (e+f x)}{10 a^4 f \left(a+b \tan ^2(e+f x)\right)}-\frac{\left(5 a^2-20 a b+14 b^2\right) \cot (e+f x)}{5 a^4 f}-\frac{\cot ^5(e+f x)}{5 a f \left(a+b \tan ^2(e+f x)\right)}",1,"(-15*Sqrt[b]*(3*a^2 - 10*a*b + 7*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] + Sqrt[a]*(-2*Cot[e + f*x]*(8*a^2 - 50*a*b + 45*b^2 + 2*a*(2*a - 5*b)*Csc[e + f*x]^2 + 3*a^2*Csc[e + f*x]^4) - (15*(a - b)^2*b*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)])))/(30*a^(9/2)*f)","A",1
80,1,278,264,5.8114443,"\int \frac{\sin ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\frac{(a-b) (5 (5 a+7 b) \cos (3 (e+f x))+3 (b-a) \cos (5 (e+f x)))-30 \cos (e+f x) \left(a^2 \left(-\frac{8 b^2}{((a-b) \cos (2 (e+f x))+a+b)^2}+\frac{18 b}{(a-b) \cos (2 (e+f x))+a+b}+5\right)+16 a b \left(\frac{b}{(a-b) \cos (2 (e+f x))+a+b}+2\right)+11 b^2\right)}{(a-b)^5}+\frac{30 \sqrt{b} \left(15 a^2+40 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b}-\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{11/2}}+\frac{30 \sqrt{b} \left(15 a^2+40 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b}+\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{11/2}}}{240 f}","-\frac{\left(5 a^2+20 a b+2 b^2\right) \cos (e+f x)}{5 f (a-b)^5}-\frac{b \left(35 a^2+40 a b+24 b^2\right) \sec (e+f x)}{40 f (a-b)^5 \left(a+b \sec ^2(e+f x)-b\right)}-\frac{b \left(5 a^2+4 b^2\right) \sec (e+f x)}{20 f (a-b)^4 \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{\sqrt{b} \left(15 a^2+40 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 f (a-b)^{11/2}}+\frac{(10 a-b) \cos ^3(e+f x)}{15 f (a-b)^4}-\frac{\cos ^5(e+f x)}{5 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^2}",1,"((30*Sqrt[b]*(15*a^2 + 40*a*b + 8*b^2)*ArcTan[(Sqrt[a - b] - Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(11/2) + (30*Sqrt[b]*(15*a^2 + 40*a*b + 8*b^2)*ArcTan[(Sqrt[a - b] + Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(11/2) + (-30*Cos[e + f*x]*(11*b^2 + 16*a*b*(2 + b/(a + b + (a - b)*Cos[2*(e + f*x)])) + a^2*(5 - (8*b^2)/(a + b + (a - b)*Cos[2*(e + f*x)])^2 + (18*b)/(a + b + (a - b)*Cos[2*(e + f*x)]))) + (a - b)*(5*(5*a + 7*b)*Cos[3*(e + f*x)] + 3*(-a + b)*Cos[5*(e + f*x)]))/(a - b)^5)/(240*f)","A",1
81,1,230,180,5.95367,"\int \frac{\sin ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\frac{2 \left(3 \cos (e+f x) \left(a \left(\frac{4 b^2}{((a-b) \cos (2 (e+f x))+a+b)^2}-\frac{9 b}{(a-b) \cos (2 (e+f x))+a+b}-3\right)+b \left(-\frac{4 b}{(a-b) \cos (2 (e+f x))+a+b}-9\right)\right)+(a-b) \cos (3 (e+f x))\right)}{(a-b)^4}+\frac{15 \sqrt{b} (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{a-b}-\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{9/2}}+\frac{15 \sqrt{b} (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{a-b}+\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{9/2}}}{24 f}","\frac{\cos ^3(e+f x)}{3 f (a-b)^3}-\frac{(a+2 b) \cos (e+f x)}{f (a-b)^4}-\frac{b (7 a+4 b) \sec (e+f x)}{8 f (a-b)^4 \left(a+b \sec ^2(e+f x)-b\right)}-\frac{a b \sec (e+f x)}{4 f (a-b)^3 \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{5 \sqrt{b} (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 f (a-b)^{9/2}}",1,"((15*Sqrt[b]*(3*a + 4*b)*ArcTan[(Sqrt[a - b] - Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(9/2) + (15*Sqrt[b]*(3*a + 4*b)*ArcTan[(Sqrt[a - b] + Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(9/2) + (2*(3*Cos[e + f*x]*(a*(-3 + (4*b^2)/(a + b + (a - b)*Cos[2*(e + f*x)])^2 - (9*b)/(a + b + (a - b)*Cos[2*(e + f*x)])) + b*(-9 - (4*b)/(a + b + (a - b)*Cos[2*(e + f*x)]))) + (a - b)*Cos[3*(e + f*x)]))/(a - b)^4)/(24*f)","A",1
82,1,170,138,1.7668849,"\int \frac{\sin (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Sin[e + f*x]/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\frac{2 \cos (e+f x) \left(\frac{4 b^2}{((a-b) \cos (2 (e+f x))+a+b)^2}-\frac{9 b}{(a-b) \cos (2 (e+f x))+a+b}-4\right)}{(a-b)^3}+\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a-b}-\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{7/2}}+\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a-b}+\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{7/2}}}{8 f}","-\frac{15 \cos (e+f x)}{8 f (a-b)^3}+\frac{5 \cos (e+f x)}{8 f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)}+\frac{\cos (e+f x)}{4 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 f (a-b)^{7/2}}",1,"((15*Sqrt[b]*ArcTan[(Sqrt[a - b] - Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(7/2) + (15*Sqrt[b]*ArcTan[(Sqrt[a - b] + Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(7/2) + (2*Cos[e + f*x]*(-4 + (4*b^2)/(a + b + (a - b)*Cos[2*(e + f*x)])^2 - (9*b)/(a + b + (a - b)*Cos[2*(e + f*x)])))/(a - b)^3)/(8*f)","A",1
83,1,247,166,3.439244,"\int \frac{\csc (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Csc[e + f*x]/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\frac{8 a^2 b^2 \cos (e+f x)}{(a-b)^2 ((a-b) \cos (2 (e+f x))+a+b)^2}+\frac{\sqrt{b} \left(15 a^2-20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b}-\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{5/2}}+\frac{\sqrt{b} \left(15 a^2-20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b}+\sqrt{a} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b}}\right)}{(a-b)^{5/2}}-\frac{2 a b (9 a-4 b) \cos (e+f x)}{(a-b)^2 ((a-b) \cos (2 (e+f x))+a+b)}+8 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-8 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{8 a^3 f}","-\frac{\tanh ^{-1}(\cos (e+f x))}{a^3 f}-\frac{b (7 a-4 b) \sec (e+f x)}{8 a^2 f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)}-\frac{\sqrt{b} \left(15 a^2-20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 a^3 f (a-b)^{5/2}}-\frac{b \sec (e+f x)}{4 a f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^2}",1,"((Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[a - b] - Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(5/2) + (Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[a - b] + Sqrt[a]*Tan[(e + f*x)/2])/Sqrt[b]])/(a - b)^(5/2) + (8*a^2*b^2*Cos[e + f*x])/((a - b)^2*(a + b + (a - b)*Cos[2*(e + f*x)])^2) - (2*a*(9*a - 4*b)*b*Cos[e + f*x])/((a - b)^2*(a + b + (a - b)*Cos[2*(e + f*x)])) - 8*Log[Cos[(e + f*x)/2]] + 8*Log[Sin[(e + f*x)/2]])/(8*a^3*f)","A",1
84,1,414,205,6.5494585,"\int \frac{\csc ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2)^3,x]","\frac{(a-6 b) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{2 a^4 f}+\frac{(6 b-a) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 a^4 f}+\frac{8 b^2 \cos (e+f x)-9 a b \cos (e+f x)}{4 a^3 f (a-b) (a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b)}-\frac{\csc ^2\left(\frac{1}{2} (e+f x)\right)}{8 a^3 f}+\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right)}{8 a^3 f}+\frac{b^2 \cos (e+f x)}{a^2 f (a-b) (a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b)^2}+\frac{\sqrt{b} \sqrt{a-b} \left(15 a^2-40 a b+24 b^2\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (e+f x)\right) \left(\sqrt{a-b} \cos \left(\frac{1}{2} (e+f x)\right)-\sqrt{a} \sin \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{b}}\right)}{8 a^4 f (b-a)^2}+\frac{\sqrt{b} \sqrt{a-b} \left(15 a^2-40 a b+24 b^2\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (e+f x)\right) \left(\sqrt{a-b} \cos \left(\frac{1}{2} (e+f x)\right)+\sqrt{a} \sin \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{b}}\right)}{8 a^4 f (b-a)^2}","-\frac{(a-6 b) \tanh ^{-1}(\cos (e+f x))}{2 a^4 f}-\frac{b (11 a-12 b) \sec (e+f x)}{8 a^3 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)}-\frac{3 b \sec (e+f x)}{4 a^2 f \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{\sqrt{b} \left(15 a^2-40 a b+24 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 a^4 f (a-b)^{3/2}}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left(a+b \sec ^2(e+f x)-b\right)^2}",1,"(Sqrt[a - b]*Sqrt[b]*(15*a^2 - 40*a*b + 24*b^2)*ArcTan[(Sec[(e + f*x)/2]*(Sqrt[a - b]*Cos[(e + f*x)/2] - Sqrt[a]*Sin[(e + f*x)/2]))/Sqrt[b]])/(8*a^4*(-a + b)^2*f) + (Sqrt[a - b]*Sqrt[b]*(15*a^2 - 40*a*b + 24*b^2)*ArcTan[(Sec[(e + f*x)/2]*(Sqrt[a - b]*Cos[(e + f*x)/2] + Sqrt[a]*Sin[(e + f*x)/2]))/Sqrt[b]])/(8*a^4*(-a + b)^2*f) + (b^2*Cos[e + f*x])/(a^2*(a - b)*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])^2) + (-9*a*b*Cos[e + f*x] + 8*b^2*Cos[e + f*x])/(4*a^3*(a - b)*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])) - Csc[(e + f*x)/2]^2/(8*a^3*f) + ((-a + 6*b)*Log[Cos[(e + f*x)/2]])/(2*a^4*f) + ((a - 6*b)*Log[Sin[(e + f*x)/2]])/(2*a^4*f) + Sec[(e + f*x)/2]^2/(8*a^3*f)","B",0
85,1,468,259,6.5755795,"\int \frac{\csc ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{3 \left(3 a b \cos (e+f x)-4 b^2 \cos (e+f x)\right)}{4 a^4 f (a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b)}-\frac{3 (a-4 b) \csc ^2\left(\frac{1}{2} (e+f x)\right)}{32 a^4 f}+\frac{3 (a-4 b) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{32 a^4 f}+\frac{b^2 \cos (e+f x)}{a^3 f (a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b)^2}-\frac{\csc ^4\left(\frac{1}{2} (e+f x)\right)}{64 a^3 f}+\frac{\sec ^4\left(\frac{1}{2} (e+f x)\right)}{64 a^3 f}+\frac{3 \left(a^2-12 a b+16 b^2\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{8 a^5 f}-\frac{3 \left(a^2-12 a b+16 b^2\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)}{8 a^5 f}-\frac{3 \sqrt{b} \sqrt{a-b} \left(5 a^2-20 a b+16 b^2\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (e+f x)\right) \left(\sqrt{a-b} \cos \left(\frac{1}{2} (e+f x)\right)-\sqrt{a} \sin \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{b}}\right)}{8 a^5 f (b-a)}-\frac{3 \sqrt{b} \sqrt{a-b} \left(5 a^2-20 a b+16 b^2\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (e+f x)\right) \left(\sqrt{a-b} \cos \left(\frac{1}{2} (e+f x)\right)+\sqrt{a} \sin \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{b}}\right)}{8 a^5 f (b-a)}","-\frac{3 b (a-2 b) \sec (e+f x)}{2 a^4 f \left(a+b \sec ^2(e+f x)-b\right)}-\frac{b (7 a-12 b) \sec (e+f x)}{8 a^3 f \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{3 \left(a^2-12 a b+16 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 a^5 f}-\frac{3 \sqrt{b} \left(5 a^2-20 a b+16 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 a^5 f \sqrt{a-b}}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f \left(a+b \sec ^2(e+f x)-b\right)^2}",1,"(-3*Sqrt[a - b]*Sqrt[b]*(5*a^2 - 20*a*b + 16*b^2)*ArcTan[(Sec[(e + f*x)/2]*(Sqrt[a - b]*Cos[(e + f*x)/2] - Sqrt[a]*Sin[(e + f*x)/2]))/Sqrt[b]])/(8*a^5*(-a + b)*f) - (3*Sqrt[a - b]*Sqrt[b]*(5*a^2 - 20*a*b + 16*b^2)*ArcTan[(Sec[(e + f*x)/2]*(Sqrt[a - b]*Cos[(e + f*x)/2] + Sqrt[a]*Sin[(e + f*x)/2]))/Sqrt[b]])/(8*a^5*(-a + b)*f) + (b^2*Cos[e + f*x])/(a^3*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])^2) - (3*(3*a*b*Cos[e + f*x] - 4*b^2*Cos[e + f*x]))/(4*a^4*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])) - (3*(a - 4*b)*Csc[(e + f*x)/2]^2)/(32*a^4*f) - Csc[(e + f*x)/2]^4/(64*a^3*f) - (3*(a^2 - 12*a*b + 16*b^2)*Log[Cos[(e + f*x)/2]])/(8*a^5*f) + (3*(a^2 - 12*a*b + 16*b^2)*Log[Sin[(e + f*x)/2]])/(8*a^5*f) + (3*(a - 4*b)*Sec[(e + f*x)/2]^2)/(32*a^4*f) + Sec[(e + f*x)/2]^4/(64*a^3*f)","A",0
86,1,194,250,0.9312218,"\int \frac{\sin ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2)^3,x]","\frac{12 \left(a^2+10 a b+5 b^2\right) (e+f x)-\frac{12 \sqrt{b} \left(5 a^2+10 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{16 a b^2 (a-b) \sin (2 (e+f x))}{((a-b) \cos (2 (e+f x))+a+b)^2}+(a-b)^2 \sin (4 (e+f x))-8 (a+2 b) (a-b) \sin (2 (e+f x))-\frac{4 b (9 a+5 b) (a-b) \sin (2 (e+f x))}{(a-b) \cos (2 (e+f x))+a+b}}{32 f (a-b)^5}","-\frac{3 \sqrt{b} \left(5 a^2+10 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 \sqrt{a} f (a-b)^5}+\frac{3 x \left(a^2+10 a b+5 b^2\right)}{8 (a-b)^5}-\frac{3 b (a+b) \tan (e+f x)}{2 f (a-b)^4 \left(a+b \tan ^2(e+f x)\right)}-\frac{b (7 a+5 b) \tan (e+f x)}{8 f (a-b)^3 \left(a+b \tan ^2(e+f x)\right)^2}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{(5 a+3 b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)^2}",1,"(12*(a^2 + 10*a*b + 5*b^2)*(e + f*x) - (12*Sqrt[b]*(5*a^2 + 10*a*b + b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/Sqrt[a] - 8*(a - b)*(a + 2*b)*Sin[2*(e + f*x)] + (16*a*(a - b)*b^2*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)])^2 - (4*(a - b)*b*(9*a + 5*b)*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)]) + (a - b)^2*Sin[4*(e + f*x)])/(32*(a - b)^5*f)","A",1
87,1,164,193,2.5632279,"\int \frac{\sin ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\frac{\sqrt{b} \left(-15 a^2-10 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{3/2}}+\frac{4 b^2 (a-b) \sin (2 (e+f x))}{((a-b) \cos (2 (e+f x))+a+b)^2}+4 (a+5 b) (e+f x)-2 (a-b) \sin (2 (e+f x))-\frac{b (a-b) (9 a+b) \sin (2 (e+f x))}{a ((a-b) \cos (2 (e+f x))+a+b)}}{8 f (a-b)^4}","-\frac{\sqrt{b} \left(15 a^2+10 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{3/2} f (a-b)^4}-\frac{b (11 a+b) \tan (e+f x)}{8 a f (a-b)^3 \left(a+b \tan ^2(e+f x)\right)}-\frac{3 b \tan (e+f x)}{4 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)^2}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x (a+5 b)}{2 (a-b)^4}",1,"(4*(a + 5*b)*(e + f*x) + (Sqrt[b]*(-15*a^2 - 10*a*b + b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/a^(3/2) - 2*(a - b)*Sin[2*(e + f*x)] + (4*(a - b)*b^2*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)])^2 - ((a - b)*b*(9*a + b)*Sin[2*(e + f*x)])/(a*(a + b + (a - b)*Cos[2*(e + f*x)])))/(8*(a - b)^4*f)","A",1
88,1,138,150,1.9682502,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[(a + b*Tan[e + f*x]^2)^(-3),x]","-\frac{\frac{b (7 a-3 b) (a-b) \tan (e+f x)}{a^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{5/2}}+\frac{2 b (a-b)^2 \tan (e+f x)}{a \left(a+b \tan ^2(e+f x)\right)^2}-8 \tan ^{-1}(\tan (e+f x))}{8 f (a-b)^3}","-\frac{b (7 a-3 b) \tan (e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{5/2} f (a-b)^3}-\frac{b \tan (e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x}{(a-b)^3}",1,"-1/8*(-8*ArcTan[Tan[e + f*x]] + (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/a^(5/2) + (2*(a - b)^2*b*Tan[e + f*x])/(a*(a + b*Tan[e + f*x]^2)^2) + ((7*a - 3*b)*(a - b)*b*Tan[e + f*x])/(a^2*(a + b*Tan[e + f*x]^2)))/((a - b)^3*f)","A",1
89,1,144,112,0.9933953,"\int \frac{\csc ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\frac{4 a^{3/2} b^2 \sin (2 (e+f x))}{(a-b) ((a-b) \cos (2 (e+f x))+a+b)^2}-15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)-\frac{\sqrt{a} b (9 a-7 b) \sin (2 (e+f x))}{(a-b) ((a-b) \cos (2 (e+f x))+a+b)}-8 \sqrt{a} \cot (e+f x)}{8 a^{7/2} f}","-\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{7/2} f}-\frac{15 \cot (e+f x)}{8 a^3 f}+\frac{5 \cot (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)\right)}+\frac{\cot (e+f x)}{4 a f \left(a+b \tan ^2(e+f x)\right)^2}",1,"(-15*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] - 8*Sqrt[a]*Cot[e + f*x] + (4*a^(3/2)*b^2*Sin[2*(e + f*x)])/((a - b)*(a + b + (a - b)*Cos[2*(e + f*x)])^2) - (Sqrt[a]*(9*a - 7*b)*b*Sin[2*(e + f*x)])/((a - b)*(a + b + (a - b)*Cos[2*(e + f*x)])))/(8*a^(7/2)*f)","A",1
90,1,146,154,1.8279621,"\int \frac{\csc ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\sqrt{a} \left(-\frac{3 b \sin (2 (e+f x)) \left(\left(9 a^2-20 a b+11 b^2\right) \cos (2 (e+f x))+9 a^2-6 a b-11 b^2\right)}{((a-b) \cos (2 (e+f x))+a+b)^2}-8 \cot (e+f x) \left(a \csc ^2(e+f x)+2 a-9 b\right)\right)+15 \sqrt{b} (7 b-3 a) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{24 a^{9/2} f}","-\frac{5 \sqrt{b} (3 a-7 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{9/2} f}-\frac{b (7 a-11 b) \tan (e+f x)}{8 a^4 f \left(a+b \tan ^2(e+f x)\right)}-\frac{(a-3 b) \cot (e+f x)}{a^4 f}-\frac{b (a-b) \tan (e+f x)}{4 a^3 f \left(a+b \tan ^2(e+f x)\right)^2}-\frac{\cot ^3(e+f x)}{3 a^3 f}",1,"(15*Sqrt[b]*(-3*a + 7*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] + Sqrt[a]*(-8*Cot[e + f*x]*(2*a - 9*b + a*Csc[e + f*x]^2) - (3*b*(9*a^2 - 6*a*b - 11*b^2 + (9*a^2 - 20*a*b + 11*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)])^2))/(24*a^(9/2)*f)","A",1
91,1,346,231,1.6762622,"\int \frac{\csc ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2)^3,x]","\frac{-960 \sqrt{b} \left(15 a^2-70 a b+63 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)-\frac{2 \sqrt{a} \cot (e+f x) \csc ^4(e+f x) \left(-128 a^4 \cos (6 (e+f x))+64 a^4 \cos (8 (e+f x))+1600 a^4+1788 a^3 b \cos (6 (e+f x))-863 a^3 b \cos (8 (e+f x))-165 a^3 b-8800 a^2 b^2 \cos (6 (e+f x))+2479 a^2 b^2 \cos (8 (e+f x))+637 a^2 b^2+4 \left(416 a^4-447 a^3 b-1400 a^2 b^2+13125 a b^3-13230 b^4\right) \cos (2 (e+f x))-4 \left(32 a^4-257 a^3 b-2821 a^2 b^2+8925 a b^3-6615 b^4\right) \cos (4 (e+f x))+14700 a b^3 \cos (6 (e+f x))-2625 a b^3 \cos (8 (e+f x))-28875 a b^3-7560 b^4 \cos (6 (e+f x))+945 b^4 \cos (8 (e+f x))+33075 b^4\right)}{((a-b) \cos (2 (e+f x))+a+b)^2}}{7680 a^{11/2} f}","-\frac{(10 a-9 b) \cot ^3(e+f x)}{15 a^4 f}-\frac{\sqrt{b} \left(15 a^2-70 a b+63 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{11/2} f}-\frac{b \left(35 a^2-110 a b+99 b^2\right) \tan (e+f x)}{40 a^5 f \left(a+b \tan ^2(e+f x)\right)}-\frac{\left(5 a^2-30 a b+27 b^2\right) \cot (e+f x)}{5 a^5 f}-\frac{b \left(5 a^2-10 a b+9 b^2\right) \tan (e+f x)}{20 a^4 f \left(a+b \tan ^2(e+f x)\right)^2}-\frac{\cot ^5(e+f x)}{5 a f \left(a+b \tan ^2(e+f x)\right)^2}",1,"(-960*Sqrt[b]*(15*a^2 - 70*a*b + 63*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] - (2*Sqrt[a]*(1600*a^4 - 165*a^3*b + 637*a^2*b^2 - 28875*a*b^3 + 33075*b^4 + 4*(416*a^4 - 447*a^3*b - 1400*a^2*b^2 + 13125*a*b^3 - 13230*b^4)*Cos[2*(e + f*x)] - 4*(32*a^4 - 257*a^3*b - 2821*a^2*b^2 + 8925*a*b^3 - 6615*b^4)*Cos[4*(e + f*x)] - 128*a^4*Cos[6*(e + f*x)] + 1788*a^3*b*Cos[6*(e + f*x)] - 8800*a^2*b^2*Cos[6*(e + f*x)] + 14700*a*b^3*Cos[6*(e + f*x)] - 7560*b^4*Cos[6*(e + f*x)] + 64*a^4*Cos[8*(e + f*x)] - 863*a^3*b*Cos[8*(e + f*x)] + 2479*a^2*b^2*Cos[8*(e + f*x)] - 2625*a*b^3*Cos[8*(e + f*x)] + 945*b^4*Cos[8*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^4)/(a + b + (a - b)*Cos[2*(e + f*x)])^2)/(7680*a^(11/2)*f)","A",1
92,1,208,161,3.2888445,"\int \sin ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\cos (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\sqrt{(a-b) \cos (2 (e+f x))+a+b} \left(4 \left(7 a^2-15 a b+8 b^2\right) \cos (2 (e+f x))-89 a^2-3 (a-b)^2 \cos (4 (e+f x))+254 a b-149 b^2\right)+120 \sqrt{2} \sqrt{b} (a-b)^2 \tanh ^{-1}\left(\frac{\sqrt{(a-b) \cos (2 (e+f x))+a+b}}{\sqrt{2} \sqrt{b}}\right)\right)}{120 \sqrt{2} f (a-b)^2 \sqrt{(a-b) \cos (2 (e+f x))+a+b}}","-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{5 f (a-b)}+\frac{2 (5 a-4 b) \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{15 f (a-b)^2}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}",1,"(Cos[e + f*x]*(120*Sqrt[2]*(a - b)^2*Sqrt[b]*ArcTanh[Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]/(Sqrt[2]*Sqrt[b])] + Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*(-89*a^2 + 254*a*b - 149*b^2 + 4*(7*a^2 - 15*a*b + 8*b^2)*Cos[2*(e + f*x)] - 3*(a - b)^2*Cos[4*(e + f*x)]))*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(120*Sqrt[2]*(a - b)^2*f*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])","A",1
93,1,170,113,1.0648732,"\int \sin ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\cos (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\sqrt{(a-b) \cos (2 (e+f x))+a+b} ((a-b) \cos (2 (e+f x))-5 a+7 b)+6 \sqrt{2} \sqrt{b} (a-b) \tanh ^{-1}\left(\frac{\sqrt{(a-b) \cos (2 (e+f x))+a+b}}{\sqrt{2} \sqrt{b}}\right)\right)}{6 \sqrt{2} f (a-b) \sqrt{(a-b) \cos (2 (e+f x))+a+b}}","\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{3 f (a-b)}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}",1,"(Cos[e + f*x]*(6*Sqrt[2]*(a - b)*Sqrt[b]*ArcTanh[Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]/(Sqrt[2]*Sqrt[b])] + Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*(-5*a + 7*b + (a - b)*Cos[2*(e + f*x)]))*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(6*Sqrt[2]*(a - b)*f*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])","A",1
94,1,140,72,0.5609574,"\int \sin (e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\sin (2 (e+f x)) \csc (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\sqrt{2} \sqrt{(a-b) \cos (2 (e+f x))+a+b}-2 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{(a-b) \cos (2 (e+f x))+a+b}}{\sqrt{2} \sqrt{b}}\right)\right)}{4 f \sqrt{(a-b) \cos (2 (e+f x))+a+b}}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f}",1,"-1/4*((-2*Sqrt[b]*ArcTanh[Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]/(Sqrt[2]*Sqrt[b])] + Sqrt[2]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])*Csc[e + f*x]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*Sin[2*(e + f*x)])/(f*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])","A",1
95,1,295,84,6.6549857,"\int \csc (e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(2 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)}{\sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)-\sqrt{a} \left(\tanh ^{-1}\left(\frac{a-(a-2 b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+\tanh ^{-1}\left(\frac{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)+2 b}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right)\right)}{2 f \sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}",1,"((2*Sqrt[b]*ArcTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]] - Sqrt[a]*(ArcTanh[(a - (a - 2*b)*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]))*Cos[e + f*x]*Sec[(e + f*x)/2]^2*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(2*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4])","B",1
96,1,460,127,3.7393478,"\int \csc ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\cot (e+f x) \csc (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\sqrt{2} \sqrt{a} \sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}-16 \sqrt{a} \sqrt{b} \sin ^2\left(\frac{1}{2} (e+f x)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)}{\sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+4 (a+b) \sin ^2\left(\frac{1}{2} (e+f x)\right) \tanh ^{-1}\left(\frac{a-(a-2 b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+4 a \sin ^2\left(\frac{1}{2} (e+f x)\right) \tanh ^{-1}\left(\frac{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)+2 b}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+4 b \sin ^2\left(\frac{1}{2} (e+f x)\right) \tanh ^{-1}\left(\frac{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)+2 b}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right)}{4 \sqrt{a} f \sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}}","-\frac{(a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 \sqrt{a} f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f}",1,"-1/4*(Cot[e + f*x]*Csc[e + f*x]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*(Sqrt[2]*Sqrt[a]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4] - 16*Sqrt[a]*Sqrt[b]*ArcTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]]*Sin[(e + f*x)/2]^2 + 4*(a + b)*ArcTanh[(a - (a - 2*b)*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]*Sin[(e + f*x)/2]^2 + 4*a*ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]*Sin[(e + f*x)/2]^2 + 4*b*ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]*Sin[(e + f*x)/2]^2))/(Sqrt[a]*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4])","B",1
97,1,1059,187,6.7036087,"\int \csc ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{\frac{\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\frac{(-3 a \cos (e+f x)-b \cos (e+f x)) \csc ^2(e+f x)}{8 a}-\frac{1}{4} \cot (e+f x) \csc ^3(e+f x)\right)}{f}+\frac{\frac{\left(3 a^2-2 b a-b^2\right) \left(2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{b} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)}{\sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)+\sqrt{b} \left(\tanh ^{-1}\left(\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+2 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a}{\sqrt{a} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)+\tanh ^{-1}\left(\frac{2 b+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)}{\sqrt{a} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)\right)\right) (\cos (e+f x)+1) \sqrt{\frac{\cos (2 (e+f x))+1}{(\cos (e+f x)+1)^2}} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}{\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)^2}}}{4 \sqrt{a} \sqrt{b} \sqrt{a+b+(a-b) \cos (2 (e+f x))} \sqrt{\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}-\frac{\left(3 a^2+14 b a-b^2\right) \left(2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{b} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)}{\sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)-\sqrt{b} \left(\tanh ^{-1}\left(\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+2 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a}{\sqrt{a} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)+\tanh ^{-1}\left(\frac{2 b+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)}{\sqrt{a} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)\right)\right) (\cos (e+f x)+1) \sqrt{\frac{\cos (2 (e+f x))+1}{(\cos (e+f x)+1)^2}} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}{\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)^2}}}{4 \sqrt{a} \sqrt{b} \sqrt{a+b+(a-b) \cos (2 (e+f x))} \sqrt{\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}}{8 a f}","-\frac{\left(3 a^2+6 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 a^{3/2} f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}-\frac{\cot (e+f x) \csc ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{4 f}-\frac{(3 a+b) \cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{8 a f}",1,"(Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(((-3*a*Cos[e + f*x] - b*Cos[e + f*x])*Csc[e + f*x]^2)/(8*a) - (Cot[e + f*x]*Csc[e + f*x]^3)/4))/f + (-1/4*((3*a^2 + 14*a*b - b^2)*(2*Sqrt[a]*ArcTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]] - Sqrt[b]*(ArcTanh[(a - a*Tan[(e + f*x)/2]^2 + 2*b*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]))*(1 + Cos[e + f*x])*Sqrt[(1 + Cos[2*(e + f*x)])/(1 + Cos[e + f*x])^2]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2)/(1 + Tan[(e + f*x)/2]^2)^2])/(Sqrt[a]*Sqrt[b]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]) + ((3*a^2 - 2*a*b - b^2)*(2*Sqrt[a]*ArcTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]] + Sqrt[b]*(ArcTanh[(a - a*Tan[(e + f*x)/2]^2 + 2*b*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]))*(1 + Cos[e + f*x])*Sqrt[(1 + Cos[2*(e + f*x)])/(1 + Cos[e + f*x])^2]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2)/(1 + Tan[(e + f*x)/2]^2)^2])/(4*Sqrt[a]*Sqrt[b]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]))/(8*a*f)","B",0
98,1,330,189,3.8017816,"\int \sin ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sin (2 (e+f x)) \sec ^2(e+f x) \left(-\left((a-b) \left(6 \left(a^2-3 a b+2 b^2\right) \cos (2 (e+f x))+7 a^2-(a-b)^2 \cos (4 (e+f x))-11 b^2\right)\right)+2 \sqrt{2} a \left(3 a^2-7 a b+4 b^2\right) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)-2 \sqrt{2} a \left(3 a^2-12 a b+8 b^2\right) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{32 \sqrt{2} f (a-b)^2 \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","\frac{\left(3 a^2-12 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f (a-b)^{3/2}}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}-\frac{(3 a-4 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f (a-b)}",1,"((-((a - b)*(7*a^2 - 11*b^2 + 6*(a^2 - 3*a*b + 2*b^2)*Cos[2*(e + f*x)] - (a - b)^2*Cos[4*(e + f*x)])) + 2*Sqrt[2]*a*(3*a^2 - 7*a*b + 4*b^2)*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1] - 2*Sqrt[2]*a*(3*a^2 - 12*a*b + 8*b^2)*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])*Sec[e + f*x]^2*Sin[2*(e + f*x)])/(32*Sqrt[2]*(a - b)^2*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])","C",1
99,1,273,128,3.835786,"\int \sin ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\sin (2 (e+f x)) \sec ^2(e+f x) \left((a-b) ((a-b) \cos (2 (e+f x))+a+b)+\sqrt{2} a (b-a) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)+\sqrt{2} a (a-2 b) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{4 \sqrt{2} f (a-b) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","\frac{(a-2 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f \sqrt{a-b}}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}",1,"-1/4*(((a - b)*(a + b + (a - b)*Cos[2*(e + f*x)]) + Sqrt[2]*a*(-a + b)*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1] + Sqrt[2]*a*(a - 2*b)*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])*Sec[e + f*x]^2*Sin[2*(e + f*x)])/(Sqrt[2]*(a - b)*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])","C",1
100,1,203,85,0.8930227,"\int \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{-i \sqrt{a-b} \log \left(-\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \tan ^2(e+f x)}+a-i b \tan (e+f x)\right)}{(a-b)^{3/2} (\tan (e+f x)+i)}\right)+i \sqrt{a-b} \log \left(\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \tan ^2(e+f x)}+a+i b \tan (e+f x)\right)}{(a-b)^{3/2} (\tan (e+f x)-i)}\right)+2 \sqrt{b} \log \left(\sqrt{b} \sqrt{a+b \tan ^2(e+f x)}+b \tan (e+f x)\right)}{2 f}","\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}",1,"((-I)*Sqrt[a - b]*Log[((-4*I)*(a - I*b*Tan[e + f*x] + Sqrt[a - b]*Sqrt[a + b*Tan[e + f*x]^2]))/((a - b)^(3/2)*(I + Tan[e + f*x]))] + I*Sqrt[a - b]*Log[((4*I)*(a + I*b*Tan[e + f*x] + Sqrt[a - b]*Sqrt[a + b*Tan[e + f*x]^2]))/((a - b)^(3/2)*(-I + Tan[e + f*x]))] + 2*Sqrt[b]*Log[b*Tan[e + f*x] + Sqrt[b]*Sqrt[a + b*Tan[e + f*x]^2]])/(2*f)","C",1
101,1,156,66,2.1799329,"\int \csc ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\tan (e+f x) \left(\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)-\sqrt{2} b \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{\sqrt{2} f \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}",1,"-((((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2 - Sqrt[2]*b*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])*Tan[e + f*x])/(Sqrt[2]*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]))","C",1
102,1,204,100,4.4099923,"\int \csc ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\tan (e+f x) \left(\csc ^4(e+f x) \left(4 \left(a^2-3 a b-b^2\right) \cos (2 (e+f x))+\left(-2 a^2+a b+b^2\right) \cos (4 (e+f x))+6 a^2+11 a b+3 b^2\right)-12 \sqrt{2} a b \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{12 \sqrt{2} a f \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 a f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}",1,"-1/12*(((6*a^2 + 11*a*b + 3*b^2 + 4*(a^2 - 3*a*b - b^2)*Cos[2*(e + f*x)] + (-2*a^2 + a*b + b^2)*Cos[4*(e + f*x)])*Csc[e + f*x]^4 - 12*Sqrt[2]*a*b*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])*Tan[e + f*x])/(Sqrt[2]*a*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])","C",1
103,1,287,141,3.500728,"\int \csc ^6(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^6*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\tan (e+f x) \left(\csc ^6(e+f x) \left(8 a^3 \cos (6 (e+f x))+80 a^3+a^2 b \cos (6 (e+f x))+198 a^2 b+\left(40 a^3-241 a^2 b-149 a b^2+30 b^3\right) \cos (2 (e+f x))+\left(-32 a^3+42 a^2 b+62 a b^2-12 b^3\right) \cos (4 (e+f x))-11 a b^2 \cos (6 (e+f x))+98 a b^2+2 b^3 \cos (6 (e+f x))-20 b^3\right)-240 \sqrt{2} a^2 b \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{240 \sqrt{2} a^2 f \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","-\frac{2 (5 a-b) \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{15 a^2 f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{5 a f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}",1,"-1/240*(((80*a^3 + 198*a^2*b + 98*a*b^2 - 20*b^3 + (40*a^3 - 241*a^2*b - 149*a*b^2 + 30*b^3)*Cos[2*(e + f*x)] + (-32*a^3 + 42*a^2*b + 62*a*b^2 - 12*b^3)*Cos[4*(e + f*x)] + 8*a^3*Cos[6*(e + f*x)] + a^2*b*Cos[6*(e + f*x)] - 11*a*b^2*Cos[6*(e + f*x)] + 2*b^3*Cos[6*(e + f*x)])*Csc[e + f*x]^6 - 240*Sqrt[2]*a^2*b*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])*Tan[e + f*x])/(Sqrt[2]*a^2*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])","C",1
104,1,233,227,5.5696084,"\int \sin ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sin[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\cos (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(2 \sqrt{(a-b) \cos (2 (e+f x))+a+b} \left(4 \left(7 a^2-20 a b+13 b^2\right) \cos (2 (e+f x))-89 a^2-3 (a-b)^2 \cos (4 (e+f x))+60 b (a-b) \sec ^2(e+f x)+474 a b-409 b^2\right)+120 \sqrt{2} \sqrt{b} \left(3 a^2-10 a b+7 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{(a-b) \cos (2 (e+f x))+a+b}}{\sqrt{2} \sqrt{b}}\right)\right)}{240 \sqrt{2} f (a-b) \sqrt{(a-b) \cos (2 (e+f x))+a+b}}","\frac{b (3 a-7 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f (a-b)}-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{5/2}}{5 f (a-b)}+\frac{2 \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{5/2}}{3 f (a-b)}-\frac{(3 a-7 b) \cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{3 f (a-b)}+\frac{\sqrt{b} (3 a-7 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}",1,"(Cos[e + f*x]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*(120*Sqrt[2]*Sqrt[b]*(3*a^2 - 10*a*b + 7*b^2)*ArcTanh[Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]/(Sqrt[2]*Sqrt[b])] + 2*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*(-89*a^2 + 474*a*b - 409*b^2 + 4*(7*a^2 - 20*a*b + 13*b^2)*Cos[2*(e + f*x)] - 3*(a - b)^2*Cos[4*(e + f*x)] + 60*(a - b)*b*Sec[e + f*x]^2)))/(240*Sqrt[2]*(a - b)*f*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])","A",1
105,1,188,186,1.8942469,"\int \sin ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sec (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\sqrt{(a-b) \cos (2 (e+f x))+a+b} (-8 (a-3 b) \cos (2 (e+f x))+(a-b) \cos (4 (e+f x))-9 a+37 b)+12 \sqrt{2} \sqrt{b} (3 a-5 b) \cos ^2(e+f x) \tanh ^{-1}\left(\frac{\sqrt{(a-b) \cos (2 (e+f x))+a+b}}{\sqrt{2} \sqrt{b}}\right)\right)}{24 \sqrt{2} f \sqrt{(a-b) \cos (2 (e+f x))+a+b}}","\frac{b (3 a-5 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f (a-b)}+\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{5/2}}{3 f (a-b)}-\frac{(3 a-5 b) \cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{3 f (a-b)}+\frac{\sqrt{b} (3 a-5 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}",1,"((12*Sqrt[2]*(3*a - 5*b)*Sqrt[b]*ArcTanh[Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]/(Sqrt[2]*Sqrt[b])]*Cos[e + f*x]^2 + Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*(-9*a + 37*b - 8*(a - 3*b)*Cos[2*(e + f*x)] + (a - b)*Cos[4*(e + f*x)]))*Sec[e + f*x]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(24*Sqrt[2]*f*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])","A",1
106,1,170,113,1.3335525,"\int \sin (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sec (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(6 \sqrt{2} \sqrt{b} (a-b) \cos ^2(e+f x) \tanh ^{-1}\left(\frac{\sqrt{(a-b) \cos (2 (e+f x))+a+b}}{\sqrt{2} \sqrt{b}}\right)-2 ((a-b) \cos (2 (e+f x))+a-2 b) \sqrt{(a-b) \cos (2 (e+f x))+a+b}\right)}{4 \sqrt{2} f \sqrt{(a-b) \cos (2 (e+f x))+a+b}}","\frac{3 b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f}-\frac{\cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{f}+\frac{3 \sqrt{b} (a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}",1,"((6*Sqrt[2]*(a - b)*Sqrt[b]*ArcTanh[Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]/(Sqrt[2]*Sqrt[b])]*Cos[e + f*x]^2 - 2*(a - 2*b + (a - b)*Cos[2*(e + f*x)])*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])*Sec[e + f*x]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(4*Sqrt[2]*f*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])","A",1
107,1,418,127,5.1857846,"\int \csc (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(-4 a^{3/2} \cos ^2(e+f x) \tanh ^{-1}\left(\frac{a-(a-2 b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)-4 a^{3/2} \cos ^2(e+f x) \tanh ^{-1}\left(\frac{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)+2 b}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+\sqrt{2} b \cos (e+f x) \sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}+\sqrt{2} b \sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}-4 \sqrt{b} (b-3 a) \cos ^2(e+f x) \tanh ^{-1}\left(\frac{\sqrt{b} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)}{\sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right)}{8 f \sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}+\frac{b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f}+\frac{\sqrt{b} (3 a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}",1,"(Sec[(e + f*x)/2]^2*(-4*Sqrt[b]*(-3*a + b)*ArcTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]]*Cos[e + f*x]^2 - 4*a^(3/2)*ArcTanh[(a - (a - 2*b)*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]*Cos[e + f*x]^2 - 4*a^(3/2)*ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]*Cos[e + f*x]^2 + Sqrt[2]*b*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4] + Sqrt[2]*b*Cos[e + f*x]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4])*Sec[e + f*x]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(8*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4])","B",1
108,1,1022,167,6.7650755,"\int \csc ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sqrt{\frac{\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\frac{1}{2} b \sec (e+f x)-\frac{1}{2} a \cot (e+f x) \csc (e+f x)\right)}{f}+\frac{\frac{\left(a^2-b^2\right) \left(2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{b} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)}{\sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)+\sqrt{b} \left(\tanh ^{-1}\left(\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+2 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a}{\sqrt{a} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)+\tanh ^{-1}\left(\frac{2 b+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)}{\sqrt{a} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)\right)\right) (\cos (e+f x)+1) \sqrt{\frac{\cos (2 (e+f x))+1}{(\cos (e+f x)+1)^2}} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}{\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)^2}}}{4 \sqrt{a} \sqrt{b} \sqrt{a+b+(a-b) \cos (2 (e+f x))} \sqrt{\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}-\frac{\left(a^2+6 b a+b^2\right) \left(2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{b} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)}{\sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)-\sqrt{b} \left(\tanh ^{-1}\left(\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+2 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a}{\sqrt{a} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)+\tanh ^{-1}\left(\frac{2 b+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)}{\sqrt{a} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)\right)\right) (\cos (e+f x)+1) \sqrt{\frac{\cos (2 (e+f x))+1}{(\cos (e+f x)+1)^2}} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}{\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)^2}}}{4 \sqrt{a} \sqrt{b} \sqrt{a+b+(a-b) \cos (2 (e+f x))} \sqrt{\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}}{2 f}","\frac{b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f}-\frac{\sqrt{a} (a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}+\frac{\sqrt{b} (3 a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}-\frac{\cot (e+f x) \csc (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{2 f}",1,"(Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(-1/2*(a*Cot[e + f*x]*Csc[e + f*x]) + (b*Sec[e + f*x])/2))/f + (-1/4*((a^2 + 6*a*b + b^2)*(2*Sqrt[a]*ArcTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]] - Sqrt[b]*(ArcTanh[(a - a*Tan[(e + f*x)/2]^2 + 2*b*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]))*(1 + Cos[e + f*x])*Sqrt[(1 + Cos[2*(e + f*x)])/(1 + Cos[e + f*x])^2]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2)/(1 + Tan[(e + f*x)/2]^2)^2])/(Sqrt[a]*Sqrt[b]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]) + ((a^2 - b^2)*(2*Sqrt[a]*ArcTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]] + Sqrt[b]*(ArcTanh[(a - a*Tan[(e + f*x)/2]^2 + 2*b*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]))*(1 + Cos[e + f*x])*Sqrt[(1 + Cos[2*(e + f*x)])/(1 + Cos[e + f*x])^2]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2)/(1 + Tan[(e + f*x)/2]^2)^2])/(4*Sqrt[a]*Sqrt[b]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]))/(2*f)","B",1
109,1,409,223,5.2383931,"\int \csc ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\cos (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(-\frac{3 \sec ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\cos ^2(e+f x) \sec ^4\left(\frac{1}{2} (e+f x)\right)} \left(\left(a^2+6 a b+b^2\right) \left(\tanh ^{-1}\left(\frac{a-(a-2 b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+\tanh ^{-1}\left(\frac{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)+2 b}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right)-8 \sqrt{a} \sqrt{b} (a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)}{\sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right)}{\sqrt{a} \sqrt{\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}-2 \csc ^2(e+f x) \left(2 a \csc ^2(e+f x)+3 a+5 b\right)+8 b \sec ^2(e+f x)\right)}{16 \sqrt{2} f}","-\frac{3 \left(a^2+6 a b+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 \sqrt{a} f}+\frac{3 (a+3 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{8 f}-\frac{3 (a+b) \csc ^2(e+f x) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{8 f}+\frac{3 \sqrt{b} (a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}-\frac{\cot (e+f x) \csc ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{4 f}",1,"(Cos[e + f*x]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*(-2*Csc[e + f*x]^2*(3*a + 5*b + 2*a*Csc[e + f*x]^2) + 8*b*Sec[e + f*x]^2 - (3*(-8*Sqrt[a]*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]] + (a^2 + 6*a*b + b^2)*(ArcTanh[(a - (a - 2*b)*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]))*Sec[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]^2*Sec[(e + f*x)/2]^4])/(Sqrt[a]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])))/(16*Sqrt[2]*f)","A",1
110,1,278,222,5.0112619,"\int \sin ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sin[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\frac{3 a \sin (2 (e+f x)) \csc ^2(e+f x) \left(\left(a^2-5 a b+4 b^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)-\left(a^2-8 a b+8 b^2\right) \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{\sqrt{2} b (a-b) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}}}+\frac{1}{4} ((18 b-8 a) \sin (2 (e+f x))+(a-b) \sin (4 (e+f x))+16 b \tan (e+f x))\right)}{8 \sqrt{2} f}","\frac{3 \left(a^2-8 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f \sqrt{a-b}}-\frac{3 (a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{3 (a-2 b) \sin ^2(e+f x) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{3 \sqrt{b} (a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\sin ^3(e+f x) \cos (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{4 f}",1,"(Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*((3*a*Csc[e + f*x]^2*((a^2 - 5*a*b + 4*b^2)*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1] - (a^2 - 8*a*b + 8*b^2)*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])*Sin[2*(e + f*x)])/(Sqrt[2]*(a - b)*b*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]) + ((-8*a + 18*b)*Sin[2*(e + f*x)] + (a - b)*Sin[4*(e + f*x)] + 16*b*Tan[e + f*x])/4))/(8*Sqrt[2]*f)","C",1
111,1,324,165,5.5936669,"\int \sin ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\sin (2 (e+f x)) \tan (e+f x) \sec ^2(e+f x) \left(\csc (e+f x) \sec (e+f x) \left(3 a^2+4 (a-b)^2 \cos (2 (e+f x))+(a-b)^2 \cos (4 (e+f x))-6 a b-5 b^2\right)-4 \sqrt{2} a (a-2 b) \cot (e+f x) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)+4 \sqrt{2} a (a-4 b) \cot (e+f x) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{16 \sqrt{2} f \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{(a-4 b) \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}+\frac{\sqrt{b} (3 a-4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\sin (e+f x) \cos (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{2 f}",1,"-1/16*(Sec[e + f*x]^2*(-4*Sqrt[2]*a*(a - 2*b)*Cot[e + f*x]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1] + 4*Sqrt[2]*a*(a - 4*b)*Cot[e + f*x]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1] + (3*a^2 - 6*a*b - 5*b^2 + 4*(a - b)^2*Cos[2*(e + f*x)] + (a - b)^2*Cos[4*(e + f*x)])*Csc[e + f*x]*Sec[e + f*x])*Sin[2*(e + f*x)]*Tan[e + f*x])/(Sqrt[2]*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])","C",1
112,1,233,125,1.4746733,"\int \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}-i (a-b)^{3/2} \log \left(-\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \tan ^2(e+f x)}+a-i b \tan (e+f x)\right)}{(a-b)^{5/2} (\tan (e+f x)+i)}\right)+i (a-b)^{3/2} \log \left(\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \tan ^2(e+f x)}+a+i b \tan (e+f x)\right)}{(a-b)^{5/2} (\tan (e+f x)-i)}\right)+\sqrt{b} (3 a-2 b) \log \left(\sqrt{b} \sqrt{a+b \tan ^2(e+f x)}+b \tan (e+f x)\right)}{2 f}","\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}+\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}",1,"((-I)*(a - b)^(3/2)*Log[((-4*I)*(a - I*b*Tan[e + f*x] + Sqrt[a - b]*Sqrt[a + b*Tan[e + f*x]^2]))/((a - b)^(5/2)*(I + Tan[e + f*x]))] + I*(a - b)^(3/2)*Log[((4*I)*(a + I*b*Tan[e + f*x] + Sqrt[a - b]*Sqrt[a + b*Tan[e + f*x]^2]))/((a - b)^(5/2)*(-I + Tan[e + f*x]))] + (3*a - 2*b)*Sqrt[b]*Log[b*Tan[e + f*x] + Sqrt[b]*Sqrt[a + b*Tan[e + f*x]^2]] + b*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)","C",1
113,1,220,100,2.6832724,"\int \csc ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\csc (e+f x) \sec ^3(e+f x) \left(-4 \left(2 a^2+b^2\right) \cos (2 (e+f x))-2 a^2 \cos (4 (e+f x))-6 a^2+a b \cos (4 (e+f x))+3 \sqrt{2} a b \sin ^2(2 (e+f x)) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)-a b+b^2 \cos (4 (e+f x))+3 b^2\right)}{8 \sqrt{2} f \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","\frac{3 b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}+\frac{3 a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{f}",1,"(Csc[e + f*x]*Sec[e + f*x]^3*(-6*a^2 - a*b + 3*b^2 - 4*(2*a^2 + b^2)*Cos[2*(e + f*x)] - 2*a^2*Cos[4*(e + f*x)] + a*b*Cos[4*(e + f*x)] + b^2*Cos[4*(e + f*x)] + 3*Sqrt[2]*a*b*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[2*(e + f*x)]^2))/(8*Sqrt[2]*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])","C",1
114,1,177,162,1.9300138,"\int \csc ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(-4 (a+2 b) \cot (e+f x)+\frac{3 \sqrt{2} (3 a+2 b) \cot (e+f x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{\sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}}}-2 a \cot (e+f x) \csc ^2(e+f x)+3 b \tan (e+f x)\right)}{6 \sqrt{2} f}","\frac{b (3 a+2 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 a f}+\frac{\sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{5/2}}{3 a f}-\frac{(3 a+2 b) \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 a f}",1,"(Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*(-4*(a + 2*b)*Cot[e + f*x] - 2*a*Cot[e + f*x]*Csc[e + f*x]^2 + (3*Sqrt[2]*(3*a + 2*b)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])/Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b] + 3*b*Tan[e + f*x]))/(6*Sqrt[2]*f)","C",1
115,1,213,196,2.169375,"\int \csc ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^6*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(-\frac{2 \left(8 a^2+34 a b+3 b^2\right) \cot (e+f x)}{a}-4 (2 a+3 b) \cot (e+f x) \csc ^2(e+f x)+\frac{15 \sqrt{2} (3 a+4 b) \cot (e+f x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{\sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}}}-6 a \cot (e+f x) \csc ^4(e+f x)+15 b \tan (e+f x)\right)}{30 \sqrt{2} f}","\frac{b (3 a+4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 a f}+\frac{\sqrt{b} (3 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{5/2}}{5 a f}-\frac{2 \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{5/2}}{3 a f}-\frac{(3 a+4 b) \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 a f}",1,"(Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*((-2*(8*a^2 + 34*a*b + 3*b^2)*Cot[e + f*x])/a - 4*(2*a + 3*b)*Cot[e + f*x]*Csc[e + f*x]^2 - 6*a*Cot[e + f*x]*Csc[e + f*x]^4 + (15*Sqrt[2]*(3*a + 4*b)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])/Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b] + 15*b*Tan[e + f*x]))/(30*Sqrt[2]*f)","C",1
116,1,112,144,2.1369289,"\int \frac{\sin ^5(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]^5/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\cos (e+f x) \left(4 \left(7 a^2-10 a b+3 b^2\right) \cos (2 (e+f x))-89 a^2-3 (a-b)^2 \cos (4 (e+f x))+34 a b-9 b^2\right) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}{120 \sqrt{2} f (a-b)^3}","-\frac{\left(15 a^2-10 a b+3 b^2\right) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{15 f (a-b)^3}-\frac{\cos ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{5 f (a-b)}+\frac{2 (5 a-3 b) \cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{15 f (a-b)^2}",1,"(Cos[e + f*x]*(-89*a^2 + 34*a*b - 9*b^2 + 4*(7*a^2 - 10*a*b + 3*b^2)*Cos[2*(e + f*x)] - 3*(a - b)^2*Cos[4*(e + f*x)])*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(120*Sqrt[2]*(a - b)^3*f)","A",1
117,1,74,88,1.4934152,"\int \frac{\sin ^3(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]^3/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\cos (e+f x) ((a-b) \cos (2 (e+f x))-5 a+b) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}{6 \sqrt{2} f (a-b)^2}","\frac{\cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{3 f (a-b)}-\frac{(3 a-b) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{3 f (a-b)^2}",1,"(Cos[e + f*x]*(-5*a + b + (a - b)*Cos[2*(e + f*x)])*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(6*Sqrt[2]*(a - b)^2*f)","A",1
118,1,52,37,0.5993106,"\int \frac{\sin (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\cos (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}{\sqrt{2} f (b-a)}","-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f (a-b)}",1,"(Cos[e + f*x]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(Sqrt[2]*(-a + b)*f)","A",1
119,1,226,42,2.6624279,"\int \frac{\csc (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\tanh ^{-1}\left(\frac{a-(a-2 b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+\tanh ^{-1}\left(\frac{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)+2 b}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right)}{2 \sqrt{a} f \sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{\sqrt{a} f}",1,"-1/2*((ArcTanh[(a - (a - 2*b)*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])])*Cos[e + f*x]*Sec[(e + f*x)/2]^2*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(Sqrt[a]*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4])","B",1
120,1,303,91,3.1963896,"\int \frac{\csc ^3(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]^3/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\cot (e+f x) \csc (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\sqrt{2} \sqrt{a} \sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}+4 (a-b) \sin ^2\left(\frac{1}{2} (e+f x)\right) \tanh ^{-1}\left(\frac{a-(a-2 b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+4 (a-b) \sin ^2\left(\frac{1}{2} (e+f x)\right) \tanh ^{-1}\left(\frac{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)+2 b}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right)}{4 a^{3/2} f \sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}}","-\frac{(a-b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 a^{3/2} f}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 a f}",1,"-1/4*(Cot[e + f*x]*Csc[e + f*x]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*(Sqrt[2]*Sqrt[a]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4] + 4*(a - b)*ArcTanh[(a - (a - 2*b)*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]*Sin[(e + f*x)/2]^2 + 4*(a - b)*ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]*Sin[(e + f*x)/2]^2))/(a^(3/2)*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4])","B",1
121,1,278,143,4.7564738,"\int \frac{\csc ^5(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]^5/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(-\sqrt{2} \sqrt{a} \cot (e+f x) \csc (e+f x) \left(2 a \csc ^2(e+f x)+3 a-3 b\right)-\frac{3 (a-b)^2 \cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) \left(\tanh ^{-1}\left(\frac{a-(a-2 b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+\tanh ^{-1}\left(\frac{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)+2 b}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right)}{\sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}}\right)}{16 a^{5/2} f}","-\frac{3 (a-b)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 a^{5/2} f}-\frac{(5 a-3 b) \cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{8 a^2 f}-\frac{\cot ^3(e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{4 a f}",1,"((-(Sqrt[2]*Sqrt[a]*Cot[e + f*x]*Csc[e + f*x]*(3*a - 3*b + 2*a*Csc[e + f*x]^2)) - (3*(a - b)^2*(ArcTanh[(a - (a - 2*b)*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])])*Cos[e + f*x]*Sec[(e + f*x)/2]^2)/Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4])*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(16*a^(5/2)*f)","A",1
122,1,314,146,4.2443348,"\int \frac{\sin ^4(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]^4/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\sin (2 (e+f x)) \sec ^2(e+f x) \left(6 \sqrt{2} a^3 \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)+(a-b) \left(2 \left(3 a^2-5 a b+2 b^2\right) \cos (2 (e+f x))+7 a^2-(a-b)^2 \cos (4 (e+f x))+8 a b-3 b^2\right)+6 \sqrt{2} a^2 (b-a) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{32 \sqrt{2} f (a-b)^3 \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","\frac{3 a^2 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f (a-b)^{5/2}}+\frac{\sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f (a-b)}-\frac{(5 a-2 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f (a-b)^2}",1,"-1/32*(((a - b)*(7*a^2 + 8*a*b - 3*b^2 + 2*(3*a^2 - 5*a*b + 2*b^2)*Cos[2*(e + f*x)] - (a - b)^2*Cos[4*(e + f*x)]) + 6*Sqrt[2]*a^2*(-a + b)*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1] + 6*Sqrt[2]*a^3*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])*Sec[e + f*x]^2*Sin[2*(e + f*x)])/(Sqrt[2]*(a - b)^3*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])","C",1
123,1,270,93,3.2561059,"\int \frac{\sin ^2(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]^2/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\sin (2 (e+f x)) \sec ^2(e+f x) \left(\sqrt{2} a^2 \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)+(a-b) ((a-b) \cos (2 (e+f x))+a+b)+\sqrt{2} a (b-a) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{4 \sqrt{2} f (a-b)^2 \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","\frac{a \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f (a-b)^{3/2}}-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f (a-b)}",1,"-1/4*(((a - b)*(a + b + (a - b)*Cos[2*(e + f*x)]) + Sqrt[2]*a*(-a + b)*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1] + Sqrt[2]*a^2*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])*Sec[e + f*x]^2*Sin[2*(e + f*x)])/(Sqrt[2]*(a - b)^2*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])","C",1
124,1,46,46,0.081133,"\int \frac{1}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[1/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}",1,"ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f)","A",1
125,1,49,30,0.2538269,"\int \frac{\csc ^2(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]^2/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\cot (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}{\sqrt{2} a f}","-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a f}",1,"-((Cot[e + f*x]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(Sqrt[2]*a*f))","A",1
126,1,68,74,0.4241542,"\int \frac{\csc ^4(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]^4/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\cot (e+f x) \left(a \csc ^2(e+f x)+2 a-2 b\right) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}{3 \sqrt{2} a^2 f}","-\frac{(3 a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^2 f}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a f}",1,"-1/3*(Cot[e + f*x]*(2*a - 2*b + a*Csc[e + f*x]^2)*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(Sqrt[2]*a^2*f)","A",1
127,1,90,123,1.9051138,"\int \frac{\csc ^6(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]^6/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\cot (e+f x) \left(3 a^2 \csc ^4(e+f x)+4 a (a-b) \csc ^2(e+f x)+8 (a-b)^2\right) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}{15 \sqrt{2} a^3 f}","-\frac{2 (5 a-2 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac{\left(15 a^2-20 a b+8 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^3 f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a f}",1,"-1/15*(Cot[e + f*x]*(8*(a - b)^2 + 4*a*(a - b)*Csc[e + f*x]^2 + 3*a^2*Csc[e + f*x]^4)*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(Sqrt[2]*a^3*f)","A",1
128,1,186,199,1.9374697,"\int \frac{\sin ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\sec (e+f x) \left(3 a^3 \cos (6 (e+f x))+150 a^3-9 a^2 b \cos (6 (e+f x))+1078 a^2 b+\left(125 a^3+169 a^2 b-329 a b^2+35 b^3\right) \cos (2 (e+f x))+9 a b^2 \cos (6 (e+f x))+338 a b^2-2 (a-b)^2 (11 a+b) \cos (4 (e+f x))-3 b^3 \cos (6 (e+f x))-30 b^3\right)}{240 \sqrt{2} f (a-b)^4 \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","-\frac{2 b \left(15 a^2+10 a b-b^2\right) \sec (e+f x)}{15 f (a-b)^4 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\left(15 a^2+10 a b-b^2\right) \cos (e+f x)}{15 f (a-b)^3 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\cos ^5(e+f x)}{5 f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}+\frac{2 (5 a-2 b) \cos ^3(e+f x)}{15 f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}}",1,"-1/240*((150*a^3 + 1078*a^2*b + 338*a*b^2 - 30*b^3 + (125*a^3 + 169*a^2*b - 329*a*b^2 + 35*b^3)*Cos[2*(e + f*x)] - 2*(a - b)^2*(11*a + b)*Cos[4*(e + f*x)] + 3*a^3*Cos[6*(e + f*x)] - 9*a^2*b*Cos[6*(e + f*x)] + 9*a*b^2*Cos[6*(e + f*x)] - 3*b^3*Cos[6*(e + f*x)])*Sec[e + f*x])/(Sqrt[2]*(a - b)^4*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])","A",1
129,1,106,131,1.1873034,"\int \frac{\sin ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\sec (e+f x) \left(8 \left(a^2-b^2\right) \cos (2 (e+f x))+9 a^2-(a-b)^2 \cos (4 (e+f x))+46 a b+9 b^2\right)}{12 \sqrt{2} f (a-b)^3 \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","-\frac{2 b (3 a+b) \sec (e+f x)}{3 f (a-b)^3 \sqrt{a+b \sec ^2(e+f x)-b}}+\frac{\cos ^3(e+f x)}{3 f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{(3 a+b) \cos (e+f x)}{3 f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}}",1,"-1/12*((9*a^2 + 46*a*b + 9*b^2 + 8*(a^2 - b^2)*Cos[2*(e + f*x)] - (a - b)^2*Cos[4*(e + f*x)])*Sec[e + f*x])/(Sqrt[2]*(a - b)^3*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])","A",1
130,1,72,76,1.6148335,"\int \frac{\sin (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\sec (e+f x) ((a-b) \cos (2 (e+f x))+a+3 b)}{\sqrt{2} f (a-b)^2 \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","-\frac{2 b \sec (e+f x)}{f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\cos (e+f x)}{f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}",1,"-(((a + 3*b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x])/(Sqrt[2]*(a - b)^2*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]))","A",1
131,1,333,84,5.7663323,"\int \frac{\csc (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\cos (e+f x) \sec ^6\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\sqrt{2} \sqrt{a} b (\cos (e+f x)+1) \sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}+(a-b) ((a-b) \cos (2 (e+f x))+a+b) \tanh ^{-1}\left(\frac{a-(a-2 b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+(a-b) ((a-b) \cos (2 (e+f x))+a+b) \tanh ^{-1}\left(\frac{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)+2 b}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right)}{2 a^{3/2} f (a-b) \left(\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)\right)^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{a^{3/2} f}-\frac{b \sec (e+f x)}{a f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}",1,"-1/2*(Cos[e + f*x]*Sec[(e + f*x)/2]^6*((a - b)*ArcTanh[(a - (a - 2*b)*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]*(a + b + (a - b)*Cos[2*(e + f*x)]) + (a - b)*ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]*(a + b + (a - b)*Cos[2*(e + f*x)]) + Sqrt[2]*Sqrt[a]*b*(1 + Cos[e + f*x])*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4])*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(a^(3/2)*(a - b)*f*((a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4)^(3/2))","B",1
132,1,308,127,4.6102174,"\int \frac{\csc ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\frac{(a-3 b) \cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\tanh ^{-1}\left(\frac{a-(a-2 b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+\tanh ^{-1}\left(\frac{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)+2 b}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right)}{2 a^{5/2} \sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}}+\frac{\csc ^2(e+f x) \sec (e+f x) ((a-3 b) \cos (2 (e+f x))+a+3 b)}{\sqrt{2} a^2 \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}}{2 f}","-\frac{(a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 a^{5/2} f}-\frac{3 b \sec (e+f x)}{2 a^2 f \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \sqrt{a+b \sec ^2(e+f x)-b}}",1,"-1/2*(((a + 3*b + (a - 3*b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2*Sec[e + f*x])/(Sqrt[2]*a^2*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]) + ((a - 3*b)*(ArcTanh[(a - (a - 2*b)*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])])*Cos[e + f*x]*Sec[(e + f*x)/2]^2*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(2*a^(5/2)*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4]))/f","B",1
133,1,350,187,5.0909066,"\int \frac{\csc ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\frac{\csc ^4(e+f x) \sec (e+f x) \left(\left(-8 a^2+52 a b-60 b^2\right) \cos (2 (e+f x))+(a-b) (3 (a-5 b) \cos (4 (e+f x))-11 a-45 b)\right)}{4 \sqrt{2} a^3 \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}-\frac{3 (a-5 b) (a-b) \cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\tanh ^{-1}\left(\frac{a-(a-2 b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+\tanh ^{-1}\left(\frac{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)+2 b}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right)}{2 a^{7/2} \sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}}}{8 f}","-\frac{3 (a-5 b) (a-b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 a^{7/2} f}-\frac{b (13 a-15 b) \sec (e+f x)}{8 a^3 f \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{5 (a-b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f \sqrt{a+b \sec ^2(e+f x)-b}}",1,"((((-8*a^2 + 52*a*b - 60*b^2)*Cos[2*(e + f*x)] + (a - b)*(-11*a - 45*b + 3*(a - 5*b)*Cos[4*(e + f*x)]))*Csc[e + f*x]^4*Sec[e + f*x])/(4*Sqrt[2]*a^3*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]) - (3*(a - 5*b)*(a - b)*(ArcTanh[(a - (a - 2*b)*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])])*Cos[e + f*x]*Sec[(e + f*x)/2]^2*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(2*a^(7/2)*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4]))/(8*f)","A",1
134,1,325,187,3.590653,"\int \frac{\sin ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sin (2 (e+f x)) \sec ^2(e+f x) \left(-(a-b) \left(\left(6 a^2-2 a b-4 b^2\right) \cos (2 (e+f x))+7 a^2-(a-b)^2 \cos (4 (e+f x))+48 a b+5 b^2\right)+6 \sqrt{2} a \left(a^2+3 a b-4 b^2\right) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)-6 \sqrt{2} a^2 (a+4 b) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{32 \sqrt{2} f (a-b)^4 \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","-\frac{b (13 a+2 b) \tan (e+f x)}{8 f (a-b)^3 \sqrt{a+b \tan ^2(e+f x)}}+\frac{3 a (a+4 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f (a-b)^{7/2}}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}-\frac{5 a \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}",1,"((-((a - b)*(7*a^2 + 48*a*b + 5*b^2 + (6*a^2 - 2*a*b - 4*b^2)*Cos[2*(e + f*x)] - (a - b)^2*Cos[4*(e + f*x)])) + 6*Sqrt[2]*a*(a^2 + 3*a*b - 4*b^2)*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1] - 6*Sqrt[2]*a^2*(a + 4*b)*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])*Sec[e + f*x]^2*Sin[2*(e + f*x)])/(32*Sqrt[2]*(a - b)^4*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])","C",1
135,1,282,134,2.9179088,"\int \frac{\sin ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\sin (2 (e+f x)) \sec ^2(e+f x) \left(-\sqrt{2} \left(a^2+a b-2 b^2\right) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)+(a-b) ((a-b) \cos (2 (e+f x))+a+5 b)+\sqrt{2} a (a+2 b) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{4 \sqrt{2} f (a-b)^3 \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","-\frac{3 b \tan (e+f x)}{2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{(a+2 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f (a-b)^{5/2}}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"-1/4*(((a - b)*(a + 5*b + (a - b)*Cos[2*(e + f*x)]) - Sqrt[2]*(a^2 + a*b - 2*b^2)*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1] + Sqrt[2]*a*(a + 2*b)*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])*Sec[e + f*x]^2*Sin[2*(e + f*x)])/(Sqrt[2]*(a - b)^3*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])","C",1
136,1,214,85,7.4591026,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[(a + b*Tan[e + f*x]^2)^(-3/2),x]","\frac{4 \sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)} \left(\frac{15 \left(3 a+2 b \tan ^2(e+f x)\right) \left(a \sqrt{\frac{(a-b) \sin ^2(2 (e+f x)) \left(a+b \tan ^2(e+f x)\right)}{a^2}}-2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)\right)}{\left(\frac{(a-b) \sin ^2(2 (e+f x)) \left(a+b \tan ^2(e+f x)\right)}{a^2}\right)^{3/2}}+a (a-b) \tan ^2(e+f x) \, _2F_1\left(2,2;\frac{7}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right)\right)}{15 a^4 f}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \tan (e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"(4*Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2]*(a*(a - b)*Hypergeometric2F1[2, 2, 7/2, ((a - b)*Sin[e + f*x]^2)/a]*Tan[e + f*x]^2 + (15*(3*a + 2*b*Tan[e + f*x]^2)*(-2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*(a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2) + a*Sqrt[((a - b)*Sin[2*(e + f*x)]^2*(a + b*Tan[e + f*x]^2))/a^2]))/(((a - b)*Sin[2*(e + f*x)]^2*(a + b*Tan[e + f*x]^2))/a^2)^(3/2)))/(15*a^4*f)","C",0
137,1,74,62,0.7497363,"\int \frac{\csc ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\csc (e+f x) \sec (e+f x) ((a-2 b) \cos (2 (e+f x))+a+2 b)}{\sqrt{2} a^2 f \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","-\frac{2 b \tan (e+f x)}{a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot (e+f x)}{a f \sqrt{a+b \tan ^2(e+f x)}}",1,"-(((a + 2*b + (a - 2*b)*Cos[2*(e + f*x)])*Csc[e + f*x]*Sec[e + f*x])/(Sqrt[2]*a^2*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]))","A",1
138,1,119,114,0.9287336,"\int \frac{\csc ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\csc ^3(e+f x) \sec (e+f x) \left(-2 \left(a^2-6 a b+8 b^2\right) \cos (2 (e+f x))+\left(a^2-5 a b+4 b^2\right) \cos (4 (e+f x))-3 a^2-7 a b+12 b^2\right)}{6 \sqrt{2} a^3 f \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","-\frac{2 b (3 a-4 b) \tan (e+f x)}{3 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{(3 a-4 b) \cot (e+f x)}{3 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^3(e+f x)}{3 a f \sqrt{a+b \tan ^2(e+f x)}}",1,"((-3*a^2 - 7*a*b + 12*b^2 - 2*(a^2 - 6*a*b + 8*b^2)*Cos[2*(e + f*x)] + (a^2 - 5*a*b + 4*b^2)*Cos[4*(e + f*x)])*Csc[e + f*x]^3*Sec[e + f*x])/(6*Sqrt[2]*a^3*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])","A",1
139,1,135,171,1.4649556,"\int \frac{\csc ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\cot (e+f x) \left(3 a^2 \csc ^4(e+f x)+8 a^2+a (4 a-9 b) \csc ^2(e+f x)-41 a b+33 b^2\right)+\frac{15 b (a-b)^2 \sin (2 (e+f x))}{(a-b) \cos (2 (e+f x))+a+b}\right)}{15 \sqrt{2} a^4 f}","-\frac{2 (5 a-3 b) \cot ^3(e+f x)}{15 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{2 b \left(15 a^2-40 a b+24 b^2\right) \tan (e+f x)}{15 a^4 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left(15 a^2-40 a b+24 b^2\right) \cot (e+f x)}{15 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^5(e+f x)}{5 a f \sqrt{a+b \tan ^2(e+f x)}}",1,"-1/15*(Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*(Cot[e + f*x]*(8*a^2 - 41*a*b + 33*b^2 + a*(4*a - 9*b)*Csc[e + f*x]^2 + 3*a^2*Csc[e + f*x]^4) + (15*(a - b)^2*b*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)])))/(Sqrt[2]*a^4*f)","A",1
140,1,294,248,2.4339243,"\int \frac{\sin ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{\cos (e+f x) \left(-16 a^4 \cos (6 (e+f x))+3 a^4 \cos (8 (e+f x))+425 a^4+32 a^3 b \cos (6 (e+f x))-12 a^3 b \cos (8 (e+f x))+4700 a^3 b+18 a^2 b^2 \cos (8 (e+f x))+12 (a-b)^2 \left(7 a^2+50 a b+7 b^2\right) \cos (4 (e+f x))+6134 a^2 b^2+48 \left(11 a^4+106 a^3 b-106 a b^3-11 b^4\right) \cos (2 (e+f x))-32 a b^3 \cos (6 (e+f x))-12 a b^3 \cos (8 (e+f x))+4700 a b^3+16 b^4 \cos (6 (e+f x))+3 b^4 \cos (8 (e+f x))+425 b^4\right) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}{480 \sqrt{2} f (a-b)^5 ((a-b) \cos (2 (e+f x))+a+b)^2}","-\frac{8 b \left(5 a^2+10 a b+b^2\right) \sec (e+f x)}{15 f (a-b)^5 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{4 b \left(5 a^2+10 a b+b^2\right) \sec (e+f x)}{15 f (a-b)^4 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{\left(5 a^2+10 a b+b^2\right) \cos (e+f x)}{5 f (a-b)^3 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{\cos ^5(e+f x)}{5 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}+\frac{2 (5 a-b) \cos ^3(e+f x)}{15 f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}",1,"-1/480*(Cos[e + f*x]*(425*a^4 + 4700*a^3*b + 6134*a^2*b^2 + 4700*a*b^3 + 425*b^4 + 48*(11*a^4 + 106*a^3*b - 106*a*b^3 - 11*b^4)*Cos[2*(e + f*x)] + 12*(a - b)^2*(7*a^2 + 50*a*b + 7*b^2)*Cos[4*(e + f*x)] - 16*a^4*Cos[6*(e + f*x)] + 32*a^3*b*Cos[6*(e + f*x)] - 32*a*b^3*Cos[6*(e + f*x)] + 16*b^4*Cos[6*(e + f*x)] + 3*a^4*Cos[8*(e + f*x)] - 12*a^3*b*Cos[8*(e + f*x)] + 18*a^2*b^2*Cos[8*(e + f*x)] - 12*a*b^3*Cos[8*(e + f*x)] + 3*b^4*Cos[8*(e + f*x)])*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(Sqrt[2]*(a - b)^5*f*(a + b + (a - b)*Cos[2*(e + f*x)])^2)","A",1
141,1,205,168,1.5525614,"\int \frac{\sin ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{\cos (e+f x) \left(a^3 (-\cos (6 (e+f x)))+26 a^3+3 a^2 b \cos (6 (e+f x))+186 a^2 b+3 \left(11 a^3+63 a^2 b-31 a b^2-43 b^3\right) \cos (2 (e+f x))-3 a b^2 \cos (6 (e+f x))+190 a b^2+6 (a-b)^2 (a+3 b) \cos (4 (e+f x))+b^3 \cos (6 (e+f x))+110 b^3\right) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}{24 \sqrt{2} f (a-b)^4 ((a-b) \cos (2 (e+f x))+a+b)^2}","-\frac{8 b (a+b) \sec (e+f x)}{3 f (a-b)^4 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{4 b (a+b) \sec (e+f x)}{3 f (a-b)^3 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}+\frac{\cos ^3(e+f x)}{3 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{(a+b) \cos (e+f x)}{f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}",1,"-1/24*(Cos[e + f*x]*(26*a^3 + 186*a^2*b + 190*a*b^2 + 110*b^3 + 3*(11*a^3 + 63*a^2*b - 31*a*b^2 - 43*b^3)*Cos[2*(e + f*x)] + 6*(a - b)^2*(a + 3*b)*Cos[4*(e + f*x)] - a^3*Cos[6*(e + f*x)] + 3*a^2*b*Cos[6*(e + f*x)] - 3*a*b^2*Cos[6*(e + f*x)] + b^3*Cos[6*(e + f*x)])*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(Sqrt[2]*(a - b)^4*f*(a + b + (a - b)*Cos[2*(e + f*x)])^2)","A",1
142,1,124,118,1.2666732,"\int \frac{\sin (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{\cos (e+f x) \left(12 \left(a^2+2 a b-3 b^2\right) \cos (2 (e+f x))+3 (a-b)^2 \cos (4 (e+f x))+(3 a+5 b)^2\right) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}{6 \sqrt{2} f (a-b)^3 ((a-b) \cos (2 (e+f x))+a+b)^2}","-\frac{8 b \sec (e+f x)}{3 f (a-b)^3 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{4 b \sec (e+f x)}{3 f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{\cos (e+f x)}{f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}",1,"-1/6*(Cos[e + f*x]*((3*a + 5*b)^2 + 12*(a^2 + 2*a*b - 3*b^2)*Cos[2*(e + f*x)] + 3*(a - b)^2*Cos[4*(e + f*x)])*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(Sqrt[2]*(a - b)^3*f*(a + b + (a - b)*Cos[2*(e + f*x)])^2)","A",1
143,1,305,136,5.4589181,"\int \frac{\csc (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csc[e + f*x]/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{\cos (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(-\frac{2 \sqrt{2} \sqrt{a} b \left(3 \left(2 a^2-3 a b+b^2\right) \cos (2 (e+f x))+6 a^2+a b-3 b^2\right)}{(a-b)^2 ((a-b) \cos (2 (e+f x))+a+b)^2}-\frac{3 \sec ^2\left(\frac{1}{2} (e+f x)\right) \left(\tanh ^{-1}\left(\frac{a-(a-2 b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+\tanh ^{-1}\left(\frac{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)+2 b}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right)}{\sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}}\right)}{6 a^{5/2} f}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{a^{5/2} f}-\frac{b (5 a-3 b) \sec (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{b \sec (e+f x)}{3 a f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}",1,"(Cos[e + f*x]*((-2*Sqrt[2]*Sqrt[a]*b*(6*a^2 + a*b - 3*b^2 + 3*(2*a^2 - 3*a*b + b^2)*Cos[2*(e + f*x)]))/((a - b)^2*(a + b + (a - b)*Cos[2*(e + f*x)])^2) - (3*(ArcTanh[(a - (a - 2*b)*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])])*Sec[(e + f*x)/2]^2)/Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4])*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(6*a^(5/2)*f)","B",1
144,1,385,177,4.6626254,"\int \frac{\csc ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{\frac{\sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \left(8 a b^2 \cos (e+f x)-24 b (a-b) \cos (e+f x) ((a-b) \cos (2 (e+f x))+a+b)-3 (a-b) \cot (e+f x) \csc (e+f x) ((a-b) \cos (2 (e+f x))+a+b)^2\right)}{3 a^3 (a-b) ((a-b) \cos (2 (e+f x))+a+b)^2}-\frac{(a-5 b) \cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\tanh ^{-1}\left(\frac{a-(a-2 b) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)+\tanh ^{-1}\left(\frac{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)+2 b}{\sqrt{a} \sqrt{a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2+4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)}}\right)\right)}{2 a^{7/2} \sqrt{\sec ^4\left(\frac{1}{2} (e+f x)\right) ((a-b) \cos (2 (e+f x))+a+b)}}}{2 f}","-\frac{(a-5 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 a^{7/2} f}-\frac{b (13 a-15 b) \sec (e+f x)}{6 a^3 f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{5 b \sec (e+f x)}{6 a^2 f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}",1,"((Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(8*a*b^2*Cos[e + f*x] - 24*(a - b)*b*Cos[e + f*x]*(a + b + (a - b)*Cos[2*(e + f*x)]) - 3*(a - b)*(a + b + (a - b)*Cos[2*(e + f*x)])^2*Cot[e + f*x]*Csc[e + f*x]))/(3*a^3*(a - b)*(a + b + (a - b)*Cos[2*(e + f*x)])^2) - ((a - 5*b)*(ArcTanh[(a - (a - 2*b)*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])])*Cos[e + f*x]*Sec[(e + f*x)/2]^2*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(2*a^(7/2)*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2]^4]))/(2*f)","B",1
145,1,1142,237,6.9775595,"\int \frac{\csc ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{\sqrt{\frac{\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(-\frac{\cot (e+f x) \csc ^3(e+f x)}{4 a^3}+\frac{(11 b \cos (e+f x)-3 a \cos (e+f x)) \csc ^2(e+f x)}{8 a^4}-\frac{2 \left(2 a b \cos (e+f x)-3 b^2 \cos (e+f x)\right)}{a^4 (\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x)))}+\frac{4 b^2 \cos (e+f x)}{3 a^3 (\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x)))^2}\right)}{f}+\frac{\left(3 a^2-30 b a+35 b^2\right) \left(\frac{\left(2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{b} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)}{\sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)+\sqrt{b} \left(\tanh ^{-1}\left(\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+2 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a}{\sqrt{a} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)+\tanh ^{-1}\left(\frac{2 b+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)}{\sqrt{a} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)\right)\right) (\cos (e+f x)+1) \sqrt{\frac{\cos (2 (e+f x))+1}{(\cos (e+f x)+1)^2}} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}{\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)^2}}}{4 \sqrt{a} \sqrt{b} \sqrt{a+b+(a-b) \cos (2 (e+f x))} \sqrt{\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}-\frac{\left(2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{b} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)}{\sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)-\sqrt{b} \left(\tanh ^{-1}\left(\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+2 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a}{\sqrt{a} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)+\tanh ^{-1}\left(\frac{2 b+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)}{\sqrt{a} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)\right)\right) (\cos (e+f x)+1) \sqrt{\frac{\cos (2 (e+f x))+1}{(\cos (e+f x)+1)^2}} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}{\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)^2}}}{4 \sqrt{a} \sqrt{b} \sqrt{a+b+(a-b) \cos (2 (e+f x))} \sqrt{\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2} \sqrt{4 b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)^2}}\right)}{8 a^4 f}","-\frac{5 b (11 a-21 b) \sec (e+f x)}{24 a^4 f \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{b (23 a-35 b) \sec (e+f x)}{24 a^3 f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{\left(3 a^2-30 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 a^{9/2} f}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}",1,"(Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((4*b^2*Cos[e + f*x])/(3*a^3*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])^2) - (2*(2*a*b*Cos[e + f*x] - 3*b^2*Cos[e + f*x]))/(a^4*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])) + ((-3*a*Cos[e + f*x] + 11*b*Cos[e + f*x])*Csc[e + f*x]^2)/(8*a^4) - (Cot[e + f*x]*Csc[e + f*x]^3)/(4*a^3)))/f + ((3*a^2 - 30*a*b + 35*b^2)*(-1/4*((2*Sqrt[a]*ArcTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]] - Sqrt[b]*(ArcTanh[(a - a*Tan[(e + f*x)/2]^2 + 2*b*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]))*(1 + Cos[e + f*x])*Sqrt[(1 + Cos[2*(e + f*x)])/(1 + Cos[e + f*x])^2]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2)/(1 + Tan[(e + f*x)/2]^2)^2])/(Sqrt[a]*Sqrt[b]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]) + ((2*Sqrt[a]*ArcTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]] + Sqrt[b]*(ArcTanh[(a - a*Tan[(e + f*x)/2]^2 + 2*b*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]))*(1 + Cos[e + f*x])*Sqrt[(1 + Cos[2*(e + f*x)])/(1 + Cos[e + f*x])^2]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2)/(1 + Tan[(e + f*x)/2]^2)^2])/(4*Sqrt[a]*Sqrt[b]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])))/(8*a^4*f)","B",0
146,1,378,246,5.6692549,"\int \frac{\sin ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{\sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(-3 \sqrt{2} a b \left(3 a^2+24 a b+8 b^2\right) \sin (2 (e+f x)) \sin ^2(e+f x) \left(\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}\right)^{3/2} \left(2 (a-b) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)-2 a \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)-a (a-b) \left(64 a b^2 \sin (2 (e+f x))-64 b (3 a+2 b) \sin (2 (e+f x)) ((a-b) \cos (2 (e+f x))+a+b)-6 (4 a+7 b) \sin (2 (e+f x)) ((a-b) \cos (2 (e+f x))+a+b)^2+3 (a-b) \sin (4 (e+f x)) ((a-b) \cos (2 (e+f x))+a+b)^2\right)\right)}{96 \sqrt{2} a f (a-b)^5 ((a-b) \cos (2 (e+f x))+a+b)^2}","\frac{\left(3 a^2+24 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f (a-b)^{9/2}}-\frac{5 b (11 a+10 b) \tan (e+f x)}{24 f (a-b)^4 \sqrt{a+b \tan ^2(e+f x)}}-\frac{b (23 a+12 b) \tan (e+f x)}{24 f (a-b)^3 \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{(5 a+2 b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"-1/96*(Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*(-3*Sqrt[2]*a*b*(3*a^2 + 24*a*b + 8*b^2)*(((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)^(3/2)*(2*(a - b)*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1] - 2*a*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])*Sin[e + f*x]^2*Sin[2*(e + f*x)] - a*(a - b)*(64*a*b^2*Sin[2*(e + f*x)] - 64*b*(3*a + 2*b)*(a + b + (a - b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)] - 6*(4*a + 7*b)*(a + b + (a - b)*Cos[2*(e + f*x)])^2*Sin[2*(e + f*x)] + 3*(a - b)*(a + b + (a - b)*Cos[2*(e + f*x)])^2*Sin[4*(e + f*x)])))/(Sqrt[2]*a*(a - b)^5*f*(a + b + (a - b)*Cos[2*(e + f*x)])^2)","C",1
147,1,309,181,4.620226,"\int \frac{\sin ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{\sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(-(a-b) \sin (2 (e+f x)) \left(8 a b^2-4 b (6 a+b) ((a-b) \cos (2 (e+f x))+a+b)-3 a ((a-b) \cos (2 (e+f x))+a+b)^2\right)-\frac{3 a b (a+4 b) \sin (2 (e+f x)) \sin ^2(e+f x) \left(\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}\right)^{3/2} \left(2 (a-b) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)-2 a \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{\sqrt{2}}\right)}{12 \sqrt{2} a f (a-b)^4 ((a-b) \cos (2 (e+f x))+a+b)^2}","-\frac{b (13 a+2 b) \tan (e+f x)}{6 a f (a-b)^3 \sqrt{a+b \tan ^2(e+f x)}}-\frac{5 b \tan (e+f x)}{6 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{(a+4 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f (a-b)^{7/2}}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"-1/12*(Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*(-((a - b)*(8*a*b^2 - 4*b*(6*a + b)*(a + b + (a - b)*Cos[2*(e + f*x)]) - 3*a*(a + b + (a - b)*Cos[2*(e + f*x)])^2)*Sin[2*(e + f*x)]) - (3*a*b*(a + 4*b)*(((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)^(3/2)*(2*(a - b)*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1] - 2*a*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])*Sin[e + f*x]^2*Sin[2*(e + f*x)])/Sqrt[2]))/(Sqrt[2]*a*(a - b)^4*f*(a + b + (a - b)*Cos[2*(e + f*x)])^2)","C",1
148,1,1331,134,9.4567406,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[(a + b*Tan[e + f*x]^2)^(-5/2),x]","\frac{\cos (e+f x) \sin (e+f x) \left(\frac{840 (a-b)^2 b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^4(e+f x) \sin ^4(e+f x)}{a^4}+\frac{2100 (a-b)^2 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^2(e+f x) \sin ^4(e+f x)}{a^3}+\frac{1575 (a-b)^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \sin ^4(e+f x)}{a^2}-\frac{1680 (a-b) b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^4(e+f x) \sin ^2(e+f x)}{a^3}-\frac{4200 (a-b) b \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^2(e+f x) \sin ^2(e+f x)}{a^2}-\frac{3150 (a-b) \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \sin ^2(e+f x)}{a}+\frac{840 b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^4(e+f x)}{a^2}+\frac{2100 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^2(e+f x)}{a}+1575 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right)+\frac{72 b^2 \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{7/2} \tan ^4(e+f x) \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}{a^2}+\frac{24 b^2 \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{7/2} \tan ^4(e+f x) \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}{a^2}+\frac{1120 b^2 \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{3/2} \tan ^4(e+f x) \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}{a^2}+96 \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{7/2} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}+24 \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{7/2} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}+\frac{168 b \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{7/2} \tan ^2(e+f x) \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}{a}+\frac{48 b \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{7/2} \tan ^2(e+f x) \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}{a}+\frac{2800 b \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{3/2} \tan ^2(e+f x) \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}{a}+2100 \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{3/2} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}-\frac{840 b^2 \tan ^4(e+f x) \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}{a^2}-\frac{2100 b \tan ^2(e+f x) \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}{a}-1575 \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}\right)}{315 a^2 f \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{5/2} \sqrt{b \tan ^2(e+f x)+a} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}} \left(\frac{b \tan ^2(e+f x)}{a}+1\right)}","-\frac{b (5 a-2 b) \tan (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{b \tan (e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}",1,"(Cos[e + f*x]*Sin[e + f*x]*(1575*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]] - (3150*(a - b)*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sin[e + f*x]^2)/a + (1575*(a - b)^2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sin[e + f*x]^4)/a^2 + (2100*b*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Tan[e + f*x]^2)/a - (4200*(a - b)*b*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sin[e + f*x]^2*Tan[e + f*x]^2)/a^2 + (2100*(a - b)^2*b*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sin[e + f*x]^4*Tan[e + f*x]^2)/a^3 + (840*b^2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Tan[e + f*x]^4)/a^2 - (1680*(a - b)*b^2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sin[e + f*x]^2*Tan[e + f*x]^4)/a^3 + (840*(a - b)^2*b^2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sin[e + f*x]^4*Tan[e + f*x]^4)/a^4 + 2100*(((a - b)*Sin[e + f*x]^2)/a)^(3/2)*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a] + 96*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Sin[e + f*x]^2)/a]*(((a - b)*Sin[e + f*x]^2)/a)^(7/2)*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a] + 24*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Sin[e + f*x]^2)/a]*(((a - b)*Sin[e + f*x]^2)/a)^(7/2)*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a] + (2800*b*(((a - b)*Sin[e + f*x]^2)/a)^(3/2)*Tan[e + f*x]^2*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a])/a + (168*b*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Sin[e + f*x]^2)/a]*(((a - b)*Sin[e + f*x]^2)/a)^(7/2)*Tan[e + f*x]^2*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a])/a + (48*b*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Sin[e + f*x]^2)/a]*(((a - b)*Sin[e + f*x]^2)/a)^(7/2)*Tan[e + f*x]^2*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a])/a + (1120*b^2*(((a - b)*Sin[e + f*x]^2)/a)^(3/2)*Tan[e + f*x]^4*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a])/a^2 + (72*b^2*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Sin[e + f*x]^2)/a]*(((a - b)*Sin[e + f*x]^2)/a)^(7/2)*Tan[e + f*x]^4*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a])/a^2 + (24*b^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Sin[e + f*x]^2)/a]*(((a - b)*Sin[e + f*x]^2)/a)^(7/2)*Tan[e + f*x]^4*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a])/a^2 - 1575*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2] - (2100*b*Tan[e + f*x]^2*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2])/a - (840*b^2*Tan[e + f*x]^4*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2])/a^2))/(315*a^2*f*(((a - b)*Sin[e + f*x]^2)/a)^(5/2)*Sqrt[a + b*Tan[e + f*x]^2]*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a]*(1 + (b*Tan[e + f*x]^2)/a))","C",0
149,1,133,97,1.0485254,"\int \frac{\csc ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{\cot (e+f x) \left(4 \left(3 a^2-8 b^2\right) \cos (2 (e+f x))+\left(3 a^2-12 a b+8 b^2\right) \cos (4 (e+f x))+3 \left(3 a^2+4 a b+8 b^2\right)\right) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}{6 \sqrt{2} a^3 f ((a-b) \cos (2 (e+f x))+a+b)^2}","-\frac{8 b \tan (e+f x)}{3 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{4 b \tan (e+f x)}{3 a^2 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\cot (e+f x)}{a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"-1/6*((3*(3*a^2 + 4*a*b + 8*b^2) + 4*(3*a^2 - 8*b^2)*Cos[2*(e + f*x)] + (3*a^2 - 12*a*b + 8*b^2)*Cos[4*(e + f*x)])*Cot[e + f*x]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])/(Sqrt[2]*a^3*f*(a + b + (a - b)*Cos[2*(e + f*x)])^2)","A",1
150,1,140,146,1.1715684,"\int \frac{\csc ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{\sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\frac{2 b \sin (2 (e+f x)) \left(\left(-3 a^2+7 a b-4 b^2\right) \cos (2 (e+f x))-3 a^2+2 a b+4 b^2\right)}{((a-b) \cos (2 (e+f x))+a+b)^2}-\cot (e+f x) \left(a \csc ^2(e+f x)+2 a-8 b\right)\right)}{3 \sqrt{2} a^4 f}","-\frac{8 b (a-2 b) \tan (e+f x)}{3 a^4 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{4 b (a-2 b) \tan (e+f x)}{3 a^3 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{(a-2 b) \cot (e+f x)}{a^2 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\cot ^3(e+f x)}{3 a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"(Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*(-(Cot[e + f*x]*(2*a - 8*b + a*Csc[e + f*x]^2)) + (2*b*(-3*a^2 + 2*a*b + 4*b^2 + (-3*a^2 + 7*a*b - 4*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)])^2))/(3*Sqrt[2]*a^4*f)","A",1
151,1,174,219,2.3354907,"\int \frac{\csc ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{\sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(\frac{5 b (b-a) \sin (2 (e+f x)) \left(\left(6 a^2-17 a b+11 b^2\right) \cos (2 (e+f x))+6 a^2-7 a b-11 b^2\right)}{((a-b) \cos (2 (e+f x))+a+b)^2}-\cot (e+f x) \left(3 a^2 \csc ^4(e+f x)+8 a^2+2 a (2 a-7 b) \csc ^2(e+f x)-66 a b+73 b^2\right)\right)}{15 \sqrt{2} a^5 f}","-\frac{2 (5 a-4 b) \cot ^3(e+f x)}{15 a^2 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{8 b \left(5 a^2-20 a b+16 b^2\right) \tan (e+f x)}{15 a^5 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{4 b \left(5 a^2-20 a b+16 b^2\right) \tan (e+f x)}{15 a^4 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\left(5 a^2-20 a b+16 b^2\right) \cot (e+f x)}{5 a^3 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\cot ^5(e+f x)}{5 a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"(Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*(-(Cot[e + f*x]*(8*a^2 - 66*a*b + 73*b^2 + 2*a*(2*a - 7*b)*Csc[e + f*x]^2 + 3*a^2*Csc[e + f*x]^4)) + (5*b*(-a + b)*(6*a^2 - 7*a*b - 11*b^2 + (6*a^2 - 17*a*b + 11*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)])^2))/(15*Sqrt[2]*a^5*f)","A",1
152,1,292,92,2.1272073,"\int (d \sin (e+f x))^m \left(b \tan ^2(e+f x)\right)^p \, dx","Integrate[(d*Sin[e + f*x])^m*(b*Tan[e + f*x]^2)^p,x]","\frac{(m+2 p+3) \sin (e+f x) \left(b \tan ^2(e+f x)\right)^p (d \sin (e+f x))^m F_1\left(\frac{m}{2}+p+\frac{1}{2};2 p,m+1;\frac{m}{2}+p+\frac{3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{f (m+2 p+1) \left((m+2 p+3) F_1\left(\frac{m}{2}+p+\frac{1}{2};2 p,m+1;\frac{m}{2}+p+\frac{3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 \tan ^2\left(\frac{1}{2} (e+f x)\right) \left((m+1) F_1\left(\frac{m}{2}+p+\frac{3}{2};2 p,m+2;\frac{m}{2}+p+\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 p F_1\left(\frac{m}{2}+p+\frac{3}{2};2 p+1,m+1;\frac{m}{2}+p+\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}","\frac{\tan (e+f x) \cos ^2(e+f x)^{p+\frac{1}{2}} \left(b \tan ^2(e+f x)\right)^p (d \sin (e+f x))^m \, _2F_1\left(\frac{1}{2} (2 p+1),\frac{1}{2} (m+2 p+1);\frac{1}{2} (m+2 p+3);\sin ^2(e+f x)\right)}{f (m+2 p+1)}",1,"((3 + m + 2*p)*AppellF1[1/2 + m/2 + p, 2*p, 1 + m, 3/2 + m/2 + p, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[e + f*x]*(d*Sin[e + f*x])^m*(b*Tan[e + f*x]^2)^p)/(f*(1 + m + 2*p)*((3 + m + 2*p)*AppellF1[1/2 + m/2 + p, 2*p, 1 + m, 3/2 + m/2 + p, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*((1 + m)*AppellF1[3/2 + m/2 + p, 2*p, 2 + m, 5/2 + m/2 + p, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*p*AppellF1[3/2 + m/2 + p, 1 + 2*p, 1 + m, 5/2 + m/2 + p, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))","C",0
153,1,275,121,2.427357,"\int (d \sin (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[(d*Sin[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p,x]","\frac{a (m+3) \sin (e+f x) \cos (e+f x) (d \sin (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p F_1\left(\frac{m+1}{2};\frac{m+2}{2},-p;\frac{m+3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f (m+1) \left(\tan ^2(e+f x) \left(2 b p F_1\left(\frac{m+3}{2};\frac{m+2}{2},1-p;\frac{m+5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)-a (m+2) F_1\left(\frac{m+3}{2};\frac{m+4}{2},-p;\frac{m+5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)\right)+a (m+3) F_1\left(\frac{m+1}{2};\frac{m+2}{2},-p;\frac{m+3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)\right)}","\frac{\tan (e+f x) \sec ^2(e+f x)^{m/2} (d \sin (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2};\frac{m+2}{2},-p;\frac{m+3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f (m+1)}",1,"(a*(3 + m)*AppellF1[(1 + m)/2, (2 + m)/2, -p, (3 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Cos[e + f*x]*Sin[e + f*x]*(d*Sin[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + m)*(a*(3 + m)*AppellF1[(1 + m)/2, (2 + m)/2, -p, (3 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + (2*b*p*AppellF1[(3 + m)/2, (2 + m)/2, 1 - p, (5 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - a*(2 + m)*AppellF1[(3 + m)/2, (4 + m)/2, -p, (5 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2))","B",0
154,1,283,208,7.841319,"\int \sin ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Sin[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{2^{p+3} \sin ^4(e+f x) \cos (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\left(15 a^2-20 a b (p+1)+4 b^2 \left(p^2+3 p+2\right)\right) \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a-b}\right)+\frac{1}{4} ((a-b) \cos (2 (e+f x))+a+b) (3 (a-b) \cos (2 (e+f x))-17 a+b (4 p+11)) \left(\frac{a+b \tan ^2(e+f x)}{a-b}\right)^p\right)}{15 f (a-b)^2 \left(-2^{p+2} \cos (2 (e+f x)) \left(\frac{a+b \tan ^2(e+f x)}{a-b}\right)^p+2^p \cos (4 (e+f x)) \left(\frac{a+b \tan ^2(e+f x)}{a-b}\right)^p+3 \left(\frac{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{a-b}\right)^p\right)}","-\frac{\left(15 a^2-20 a b (p+1)+4 b^2 \left(p^2+3 p+2\right)\right) \cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a-b}\right)}{15 f (a-b)^2}-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{p+1}}{5 f (a-b)}+\frac{(10 a-2 b p-7 b) \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{p+1}}{15 f (a-b)^2}",1,"-1/15*(2^(3 + p)*Cos[e + f*x]*Sin[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p*((15*a^2 - 20*a*b*(1 + p) + 4*b^2*(2 + 3*p + p^2))*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/(a - b))] + ((a + b + (a - b)*Cos[2*(e + f*x)])*(-17*a + b*(11 + 4*p) + 3*(a - b)*Cos[2*(e + f*x)])*((a + b*Tan[e + f*x]^2)/(a - b))^p)/4))/((a - b)^2*f*(3*(((a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2)/(a - b))^p - 2^(2 + p)*Cos[2*(e + f*x)]*((a + b*Tan[e + f*x]^2)/(a - b))^p + 2^p*Cos[4*(e + f*x)]*((a + b*Tan[e + f*x]^2)/(a - b))^p))","A",0
155,1,184,140,4.1123978,"\int \sin ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\sin (e+f x) \tan (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left((2 b (p+1)-3 a) \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a-b}\right)+\left(\frac{a+b \tan ^2(e+f x)}{a-b}\right)^p \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)\right)}{f \left(3 a \sec ^2(e+f x) \left(\frac{a+b \sec ^2(e+f x)-b}{a-b}\right)^p-3 (a-b) \left(\frac{a+b \tan ^2(e+f x)}{a-b}\right)^{p+1}\right)}","\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{p+1}}{3 f (a-b)}-\frac{(3 a-2 b (p+1)) \cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a-b}\right)}{3 f (a-b)}",1,"(Sin[e + f*x]*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p*((-3*a + 2*b*(1 + p))*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/(a - b))] + (a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2)*((a + b*Tan[e + f*x]^2)/(a - b))^p))/(f*(3*a*Sec[e + f*x]^2*((a - b + b*Sec[e + f*x]^2)/(a - b))^p - 3*(a - b)*((a + b*Tan[e + f*x]^2)/(a - b))^(1 + p)))","A",0
156,1,80,79,0.9109228,"\int \sin (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\cos (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{a+b \sec ^2(e+f x)-b}{a-b}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a-b}\right)}{f}","-\frac{\cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a-b}\right)}{f}",1,"-((Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/(a - b))]*(a + b*Tan[e + f*x]^2)^p)/(f*((a - b + b*Sec[e + f*x]^2)/(a - b))^p))","A",1
157,1,1215,88,15.1247876,"\int \csc (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\csc (e+f x) \left(b \tan ^2(e+f x)+a\right)^{2 p} \left(\frac{2 F_1\left(-p-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2}-p;-\cot ^2(e+f x),-\frac{a \cot ^2(e+f x)}{b}\right) \left(\frac{a \cot ^2(e+f x)}{b}+1\right)^{-p} \sqrt{\sec ^2(e+f x)}}{(2 p+1) \sqrt{\csc ^2(e+f x)}}-F_1\left(1;\frac{1}{2},-p;2;-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right) \tan ^2(e+f x) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}\right)}{2 f \left(b p \sec ^2(e+f x) \tan (e+f x) \left(\frac{2 F_1\left(-p-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2}-p;-\cot ^2(e+f x),-\frac{a \cot ^2(e+f x)}{b}\right) \left(\frac{a \cot ^2(e+f x)}{b}+1\right)^{-p} \sqrt{\sec ^2(e+f x)}}{(2 p+1) \sqrt{\csc ^2(e+f x)}}-F_1\left(1;\frac{1}{2},-p;2;-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right) \tan ^2(e+f x) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}\right) \left(b \tan ^2(e+f x)+a\right)^{p-1}+\frac{1}{2} \left(\frac{4 a p F_1\left(-p-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2}-p;-\cot ^2(e+f x),-\frac{a \cot ^2(e+f x)}{b}\right) \cot (e+f x) \sqrt{\csc ^2(e+f x)} \sqrt{\sec ^2(e+f x)} \left(\frac{a \cot ^2(e+f x)}{b}+1\right)^{-p-1}}{b (2 p+1)}+\frac{2 F_1\left(-p-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2}-p;-\cot ^2(e+f x),-\frac{a \cot ^2(e+f x)}{b}\right) \sqrt{\sec ^2(e+f x)} \tan (e+f x) \left(\frac{a \cot ^2(e+f x)}{b}+1\right)^{-p}}{(2 p+1) \sqrt{\csc ^2(e+f x)}}+\frac{2 \left(-\frac{2 a \left(-p-\frac{1}{2}\right) p F_1\left(\frac{1}{2}-p;-\frac{1}{2},1-p;\frac{3}{2}-p;-\cot ^2(e+f x),-\frac{a \cot ^2(e+f x)}{b}\right) \cot (e+f x) \csc ^2(e+f x)}{b \left(\frac{1}{2}-p\right)}-\frac{\left(-p-\frac{1}{2}\right) F_1\left(\frac{1}{2}-p;\frac{1}{2},-p;\frac{3}{2}-p;-\cot ^2(e+f x),-\frac{a \cot ^2(e+f x)}{b}\right) \cot (e+f x) \csc ^2(e+f x)}{\frac{1}{2}-p}\right) \sqrt{\sec ^2(e+f x)} \left(\frac{a \cot ^2(e+f x)}{b}+1\right)^{-p}}{(2 p+1) \sqrt{\csc ^2(e+f x)}}+\frac{2 F_1\left(-p-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2}-p;-\cot ^2(e+f x),-\frac{a \cot ^2(e+f x)}{b}\right) \cot (e+f x) \sqrt{\sec ^2(e+f x)} \left(\frac{a \cot ^2(e+f x)}{b}+1\right)^{-p}}{(2 p+1) \sqrt{\csc ^2(e+f x)}}+\frac{2 b p F_1\left(1;\frac{1}{2},-p;2;-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right) \sec ^2(e+f x) \tan ^3(e+f x) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p-1}}{a}-2 F_1\left(1;\frac{1}{2},-p;2;-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right) \sec ^2(e+f x) \tan (e+f x) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}-\tan ^2(e+f x) \left(\frac{b p F_1\left(2;\frac{1}{2},1-p;3;-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right) \sec ^2(e+f x) \tan (e+f x)}{a}-\frac{1}{2} F_1\left(2;\frac{3}{2},-p;3;-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right) \sec ^2(e+f x) \tan (e+f x)\right) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}\right) \left(b \tan ^2(e+f x)+a\right)^p\right)}","-\frac{\sec (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};\sec ^2(e+f x),-\frac{b \sec ^2(e+f x)}{a-b}\right)}{f}",1,"(Csc[e + f*x]*(a + b*Tan[e + f*x]^2)^(2*p)*((2*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -((a*Cot[e + f*x]^2)/b)]*Sqrt[Sec[e + f*x]^2])/((1 + 2*p)*(1 + (a*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2]) - (AppellF1[1, 1/2, -p, 2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^2)/(1 + (b*Tan[e + f*x]^2)/a)^p))/(2*f*(b*p*Sec[e + f*x]^2*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^(-1 + p)*((2*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -((a*Cot[e + f*x]^2)/b)]*Sqrt[Sec[e + f*x]^2])/((1 + 2*p)*(1 + (a*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2]) - (AppellF1[1, 1/2, -p, 2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^2)/(1 + (b*Tan[e + f*x]^2)/a)^p) + ((a + b*Tan[e + f*x]^2)^p*((2*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -((a*Cot[e + f*x]^2)/b)]*Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])/((1 + 2*p)*(1 + (a*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2]) + (4*a*p*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -((a*Cot[e + f*x]^2)/b)]*Cot[e + f*x]*(1 + (a*Cot[e + f*x]^2)/b)^(-1 - p)*Sqrt[Csc[e + f*x]^2]*Sqrt[Sec[e + f*x]^2])/(b*(1 + 2*p)) + (2*((-2*a*(-1/2 - p)*p*AppellF1[1/2 - p, -1/2, 1 - p, 3/2 - p, -Cot[e + f*x]^2, -((a*Cot[e + f*x]^2)/b)]*Cot[e + f*x]*Csc[e + f*x]^2)/(b*(1/2 - p)) - ((-1/2 - p)*AppellF1[1/2 - p, 1/2, -p, 3/2 - p, -Cot[e + f*x]^2, -((a*Cot[e + f*x]^2)/b)]*Cot[e + f*x]*Csc[e + f*x]^2)/(1/2 - p))*Sqrt[Sec[e + f*x]^2])/((1 + 2*p)*(1 + (a*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2]) + (2*AppellF1[-1/2 - p, -1/2, -p, 1/2 - p, -Cot[e + f*x]^2, -((a*Cot[e + f*x]^2)/b)]*Sqrt[Sec[e + f*x]^2]*Tan[e + f*x])/((1 + 2*p)*(1 + (a*Cot[e + f*x]^2)/b)^p*Sqrt[Csc[e + f*x]^2]) + (2*b*p*AppellF1[1, 1/2, -p, 2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x]^3*(1 + (b*Tan[e + f*x]^2)/a)^(-1 - p))/a - (2*AppellF1[1, 1/2, -p, 2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/(1 + (b*Tan[e + f*x]^2)/a)^p - (Tan[e + f*x]^2*((b*p*AppellF1[2, 1/2, 1 - p, 3, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/a - (AppellF1[2, 3/2, -p, 3, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/2))/(1 + (b*Tan[e + f*x]^2)/a)^p))/2))","B",0
158,1,252,92,20.7447264,"\int \csc ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p,x]","\frac{b (2 p-3) \cot (e+f x) \csc (e+f x) \left(a+b \tan ^2(e+f x)\right)^p F_1\left(\frac{1}{2}-p;-\frac{1}{2},-p;\frac{3}{2}-p;-\cot ^2(e+f x),-\frac{a \cot ^2(e+f x)}{b}\right)}{f (2 p-1) \left(b (2 p-3) F_1\left(\frac{1}{2}-p;-\frac{1}{2},-p;\frac{3}{2}-p;-\cot ^2(e+f x),-\frac{a \cot ^2(e+f x)}{b}\right)-\cot ^2(e+f x) \left(2 a p F_1\left(\frac{3}{2}-p;-\frac{1}{2},1-p;\frac{5}{2}-p;-\cot ^2(e+f x),-\frac{a \cot ^2(e+f x)}{b}\right)+b F_1\left(\frac{3}{2}-p;\frac{1}{2},-p;\frac{5}{2}-p;-\cot ^2(e+f x),-\frac{a \cot ^2(e+f x)}{b}\right)\right)\right)}","\frac{\sec ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} F_1\left(\frac{3}{2};2,-p;\frac{5}{2};\sec ^2(e+f x),-\frac{b \sec ^2(e+f x)}{a-b}\right)}{3 f}",1,"(b*(-3 + 2*p)*AppellF1[1/2 - p, -1/2, -p, 3/2 - p, -Cot[e + f*x]^2, -((a*Cot[e + f*x]^2)/b)]*Cot[e + f*x]*Csc[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(-1 + 2*p)*(b*(-3 + 2*p)*AppellF1[1/2 - p, -1/2, -p, 3/2 - p, -Cot[e + f*x]^2, -((a*Cot[e + f*x]^2)/b)] - (2*a*p*AppellF1[3/2 - p, -1/2, 1 - p, 5/2 - p, -Cot[e + f*x]^2, -((a*Cot[e + f*x]^2)/b)] + b*AppellF1[3/2 - p, 1/2, -p, 5/2 - p, -Cot[e + f*x]^2, -((a*Cot[e + f*x]^2)/b)])*Cot[e + f*x]^2))","B",0
159,1,3698,83,18.7794981,"\int \sin ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p,x]","\text{Result too large to show}","\frac{\tan ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{3}{2};2,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{3 f}",1,"(3*a*Cos[e + f*x]^3*Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^p*(AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]/(-3*a*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - 2*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - 2*a*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2) + (AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2)/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2))*(-1/4*(Cos[2*(e + f*x)]^3*(a + b*Tan[e + f*x]^2)^p) + (I/4)*Sin[2*(e + f*x)]*(a + b*Tan[e + f*x]^2)^p + (Sin[2*(e + f*x)]^2*(a + b*Tan[e + f*x]^2)^p)/2 - (I/4)*Sin[2*(e + f*x)]^3*(a + b*Tan[e + f*x]^2)^p + Cos[2*(e + f*x)]^2*((a + b*Tan[e + f*x]^2)^p/2 - (I/4)*Sin[2*(e + f*x)]*(a + b*Tan[e + f*x]^2)^p) + Cos[2*(e + f*x)]*(-1/4*(a + b*Tan[e + f*x]^2)^p - (Sin[2*(e + f*x)]^2*(a + b*Tan[e + f*x]^2)^p)/4)))/(f*(6*a*b*p*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(-1 + p)*(AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]/(-3*a*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - 2*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - 2*a*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2) + (AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2)/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) + 3*a*Cos[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p*(AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]/(-3*a*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - 2*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - 2*a*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2) + (AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2)/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) - 9*a*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p*(AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]/(-3*a*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - 2*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - 2*a*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2) + (AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2)/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) + 3*a*Cos[e + f*x]^3*Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^p*(((2*b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (4*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/3)/(-3*a*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - 2*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - 2*a*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2) + (2*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (Sec[e + f*x]^2*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(-4*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - 2*a*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Sec[e + f*x]^2*Tan[e + f*x] - 3*a*((2*b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (4*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/3) - 2*Tan[e + f*x]^2*(b*p*((-6*b*(1 - p)*AppellF1[5/2, 2, 2 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a) - (12*AppellF1[5/2, 3, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/5) - 2*a*((6*b*p*AppellF1[5/2, 3, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a) - (18*AppellF1[5/2, 4, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(-3*a*AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - 2*(b*p*AppellF1[3/2, 2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - 2*a*AppellF1[3/2, 3, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2)^2 - (AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*(4*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e + f*x] + 3*a*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a)) - a*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2)))","C",0
160,1,192,78,0.5857423,"\int \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[(a + b*Tan[e + f*x]^2)^p,x]","\frac{3 a \sin (2 (e+f x)) \left(a+b \tan ^2(e+f x)\right)^p F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}{4 f \tan ^2(e+f x) \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right)+6 a f F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}",1,"(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sin[2*(e + f*x)]*(a + b*Tan[e + f*x]^2)^p)/(6*a*f*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 4*f*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)","B",0
161,1,68,68,0.7208232,"\int \csc ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right)}{f}","-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right)}{f}",1,"-((Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p))","A",1
162,1,111,120,1.5764154,"\int \csc ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(-\left((3 a+b (2 p-1)) \tan ^2(e+f x) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right)\right)-a-b \tan ^2(e+f x)\right)}{3 a f}","-\frac{(3 a-b (1-2 p)) \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right)}{3 a f}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{3 a f}",1,"(Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p*(-a - b*Tan[e + f*x]^2 - ((3*a + b*(-1 + 2*p))*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^2)/(1 + (b*Tan[e + f*x]^2)/a)^p))/(3*a*f)","A",1
163,1,141,180,1.3399559,"\int \csc ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]^6*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} \left(15 \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right)+3 \cot ^4(e+f x) \, _2F_1\left(-\frac{5}{2},-p;-\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a}\right)+10 \cot ^2(e+f x) \, _2F_1\left(-\frac{3}{2},-p;-\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right)\right)}{15 f}","-\frac{\left(15 a^2-b (1-2 p) (10 a-b (3-2 p))\right) \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right)}{15 a^2 f}-\frac{(10 a-b (3-2 p)) \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{15 a^2 f}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{5 a f}",1,"-1/15*(Cot[e + f*x]*(3*Cot[e + f*x]^4*Hypergeometric2F1[-5/2, -p, -3/2, -((b*Tan[e + f*x]^2)/a)] + 10*Cot[e + f*x]^2*Hypergeometric2F1[-3/2, -p, -1/2, -((b*Tan[e + f*x]^2)/a)] + 15*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)])*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",1
164,1,295,98,2.027432,"\int (d \sin (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(d*Sin[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{(m+n p+3) \sin (e+f x) (d \sin (e+f x))^m F_1\left(\frac{1}{2} (m+n p+1);n p,m+1;\frac{1}{2} (m+n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (m+n p+1) \left((m+n p+3) F_1\left(\frac{1}{2} (m+n p+1);n p,m+1;\frac{1}{2} (m+n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 \tan ^2\left(\frac{1}{2} (e+f x)\right) \left((m+1) F_1\left(\frac{1}{2} (m+n p+3);n p,m+2;\frac{1}{2} (m+n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n p F_1\left(\frac{1}{2} (m+n p+3);n p+1,m+1;\frac{1}{2} (m+n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}","\frac{\tan (e+f x) (d \sin (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (m+n p+1);\frac{1}{2} (m+n p+3);\sin ^2(e+f x)\right)}{f (m+n p+1)}",1,"((3 + m + n*p)*AppellF1[(1 + m + n*p)/2, n*p, 1 + m, (3 + m + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[e + f*x]*(d*Sin[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + m + n*p)*((3 + m + n*p)*AppellF1[(1 + m + n*p)/2, n*p, 1 + m, (3 + m + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*((1 + m)*AppellF1[(3 + m + n*p)/2, n*p, 2 + m, (5 + m + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*p*AppellF1[(3 + m + n*p)/2, 1 + n*p, 1 + m, (5 + m + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))","C",0
165,1,517,63,2.4711615,"\int \sin ^2(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Sin[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{8 (2 n p+6) \sin ^3\left(\frac{1}{2} (e+f x)\right) \cos ^5\left(\frac{1}{2} (e+f x)\right) \left(F_1\left(\frac{1}{2} (n p+1);n p,2;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-F_1\left(\frac{1}{2} (n p+1);n p,3;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1) \left(2 (n p+3) \cos ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(\frac{1}{2} (n p+1);n p,2;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 (n p+3) \cos ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(\frac{1}{2} (n p+1);n p,3;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+2 (\cos (e+f x)-1) \left(2 F_1\left(\frac{1}{2} (n p+3);n p,3;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-3 F_1\left(\frac{1}{2} (n p+3);n p,4;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+n p \left(F_1\left(\frac{1}{2} (n p+3);n p+1,3;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-F_1\left(\frac{1}{2} (n p+3);n p+1,2;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)\right)}","\frac{\tan ^3(e+f x) \, _2F_1\left(2,\frac{1}{2} (n p+3);\frac{1}{2} (n p+5);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+3)}",1,"(8*(6 + 2*n*p)*(AppellF1[(1 + n*p)/2, n*p, 2, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - AppellF1[(1 + n*p)/2, n*p, 3, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Cos[(e + f*x)/2]^5*Sin[(e + f*x)/2]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p)*(2*(3 + n*p)*AppellF1[(1 + n*p)/2, n*p, 2, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - 2*(3 + n*p)*AppellF1[(1 + n*p)/2, n*p, 3, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + 2*(2*AppellF1[(3 + n*p)/2, n*p, 3, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 3*AppellF1[(3 + n*p)/2, n*p, 4, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*p*(-AppellF1[(3 + n*p)/2, 1 + n*p, 2, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[(3 + n*p)/2, 1 + n*p, 3, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]))*(-1 + Cos[e + f*x])))","C",0
166,1,59,61,0.0475704,"\int \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f n p+f}","\frac{\tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f + f*n*p)","A",1
167,1,31,33,0.0522374,"\int \csc ^2(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Csc[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\cot (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p-1)}","-\frac{\cot (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (1-n p)}",1,"(Cot[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(-1 + n*p))","A",1
168,1,59,69,0.1568256,"\int \csc ^4(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Csc[e + f*x]^4*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\cot (e+f x) \csc ^2(e+f x) (\cos (2 (e+f x))+n p-2) \left(b (c \tan (e+f x))^n\right)^p}{f (n p-3) (n p-1)}","-\frac{\cot ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (3-n p)}-\frac{\cot (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (1-n p)}",1,"((-2 + n*p + Cos[2*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p)/(f*(-3 + n*p)*(-1 + n*p))","A",1
169,1,89,104,0.2954241,"\int \csc ^6(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Csc[e + f*x]^6*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\cot (e+f x) \csc ^4(e+f x) \left(2 (n p-3) \cos (2 (e+f x))+\cos (4 (e+f x))+n^2 p^2-6 n p+8\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p-5) (n p-3) (n p-1)}","-\frac{\cot ^5(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (5-n p)}-\frac{2 \cot ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (3-n p)}-\frac{\cot (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (1-n p)}",1,"((8 - 6*n*p + n^2*p^2 + 2*(-3 + n*p)*Cos[2*(e + f*x)] + Cos[4*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^4*(b*(c*Tan[e + f*x])^n)^p)/(f*(-5 + n*p)*(-3 + n*p)*(-1 + n*p))","A",1
170,1,506,93,2.8422182,"\int \sin ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Sin[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{4 (n p+4) \sin \left(\frac{1}{2} (e+f x)\right) \sin ^3(e+f x) \cos ^3\left(\frac{1}{2} (e+f x)\right) \left(F_1\left(\frac{n p}{2}+1;n p,3;\frac{n p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-F_1\left(\frac{n p}{2}+1;n p,4;\frac{n p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+2) \left(2 (n p+4) \cos ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(\frac{n p}{2}+1;n p,3;\frac{n p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 (n p+4) \cos ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(\frac{n p}{2}+1;n p,4;\frac{n p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+2 (\cos (e+f x)-1) \left(3 F_1\left(\frac{n p}{2}+2;n p,4;\frac{n p}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-4 F_1\left(\frac{n p}{2}+2;n p,5;\frac{n p}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+n p \left(F_1\left(\frac{n p}{2}+2;n p+1,4;\frac{n p}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-F_1\left(\frac{n p}{2}+2;n p+1,3;\frac{n p}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)\right)}","\frac{\sin ^3(e+f x) \tan (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (n p+4);\frac{1}{2} (n p+6);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+4)}",1,"(4*(4 + n*p)*(AppellF1[1 + (n*p)/2, n*p, 3, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - AppellF1[1 + (n*p)/2, n*p, 4, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Cos[(e + f*x)/2]^3*Sin[(e + f*x)/2]*Sin[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(2 + n*p)*(2*(4 + n*p)*AppellF1[1 + (n*p)/2, n*p, 3, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - 2*(4 + n*p)*AppellF1[1 + (n*p)/2, n*p, 4, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + 2*(3*AppellF1[2 + (n*p)/2, n*p, 4, 3 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 4*AppellF1[2 + (n*p)/2, n*p, 5, 3 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*p*(-AppellF1[2 + (n*p)/2, 1 + n*p, 3, 3 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[2 + (n*p)/2, 1 + n*p, 4, 3 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]))*(-1 + Cos[e + f*x])))","C",0
171,1,284,91,1.3079235,"\int \sin (e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Sin[e + f*x]*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{8 (n p+4) \sin ^2\left(\frac{1}{2} (e+f x)\right) \cos ^4\left(\frac{1}{2} (e+f x)\right) F_1\left(\frac{n p}{2}+1;n p,2;\frac{n p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+2) \left(2 (n p+4) \cos ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(\frac{n p}{2}+1;n p,2;\frac{n p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+2 (\cos (e+f x)-1) \left(2 F_1\left(\frac{n p}{2}+2;n p,3;\frac{n p}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n p F_1\left(\frac{n p}{2}+2;n p+1,2;\frac{n p}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}","\frac{\sin (e+f x) \tan (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+2)}",1,"(8*(4 + n*p)*AppellF1[1 + (n*p)/2, n*p, 2, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^4*Sin[(e + f*x)/2]^2*(b*(c*Tan[e + f*x])^n)^p)/(f*(2 + n*p)*(2*(4 + n*p)*AppellF1[1 + (n*p)/2, n*p, 2, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + 2*(2*AppellF1[2 + (n*p)/2, n*p, 3, 3 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*p*AppellF1[2 + (n*p)/2, 1 + n*p, 2, 3 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*(-1 + Cos[e + f*x])))","C",0
172,1,77,81,0.2031917,"\int \csc (e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Csc[e + f*x]*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\, _2F_1\left(\frac{n p}{2},n p;\frac{n p}{2}+1;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^{n p} \left(b (c \tan (e+f x))^n\right)^p}{f n p}","\frac{\sec (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \, _2F_1\left(\frac{n p}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+2);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f n p}",1,"(Hypergeometric2F1[(n*p)/2, n*p, 1 + (n*p)/2, Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(n*p)*(b*(c*Tan[e + f*x])^n)^p)/(f*n*p)","A",0
173,1,1399,92,16.2801467,"\int \csc ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Csc[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\cot ^2\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(n p,\frac{n p}{2}-1;\frac{n p}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^{n p} \left(b (c \tan (e+f x))^n\right)^p}{f (4 n p-8)}+\frac{(n p+4) F_1\left(\frac{n p}{2}+1;n p,1;\frac{n p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \sin ^2(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{8 f (n p+2) \left((n p+4) F_1\left(\frac{n p}{2}+1;n p,1;\frac{n p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos ^2\left(\frac{1}{2} (e+f x)\right)+2 n p F_1\left(\frac{n p}{2}+2;n p+1,1;\frac{n p}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \sin ^2\left(\frac{1}{2} (e+f x)\right)+F_1\left(\frac{n p}{2}+2;n p,2;\frac{n p}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) (\cos (e+f x)-1)\right)}+\frac{\, _2F_1\left(n p,\frac{n p}{2}+1;\frac{n p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^{n p} \tan ^2\left(\frac{1}{2} (e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (4 n p+8)}+\frac{\cot \left(\frac{1}{2} (e+f x)\right) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^{n p} \left((n p+2) \, _2F_1\left(\frac{n p}{2},n p;\frac{n p}{2}+1;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n p F_1\left(\frac{n p}{2}+1;n p,1;\frac{n p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^{n p}(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{8 f n p (n p+2) \left(\frac{\left(-n p F_1\left(\frac{n p}{2}+1;n p,1;\frac{n p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan \left(\frac{1}{2} (e+f x)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)+\frac{1}{2} n p (n p+2) \csc \left(\frac{1}{2} (e+f x)\right) \left(\left(1-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)^{-n p}-\, _2F_1\left(\frac{n p}{2},n p;\frac{n p}{2}+1;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \sec \left(\frac{1}{2} (e+f x)\right)-n p \tan ^2\left(\frac{1}{2} (e+f x)\right) \left(\frac{n p \left(\frac{n p}{2}+1\right) F_1\left(\frac{n p}{2}+2;n p+1,1;\frac{n p}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right) \tan \left(\frac{1}{2} (e+f x)\right)}{\frac{n p}{2}+2}-\frac{\left(\frac{n p}{2}+1\right) F_1\left(\frac{n p}{2}+2;n p,2;\frac{n p}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right) \tan \left(\frac{1}{2} (e+f x)\right)}{\frac{n p}{2}+2}\right)\right) \tan ^{n p}(e+f x) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^{n p}}{2 n p (n p+2)}+\frac{\sec ^2(e+f x) \left((n p+2) \, _2F_1\left(\frac{n p}{2},n p;\frac{n p}{2}+1;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n p F_1\left(\frac{n p}{2}+1;n p,1;\frac{n p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^{n p-1}(e+f x) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^{n p}}{2 (n p+2)}+\frac{\left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) \tan \left(\frac{1}{2} (e+f x)\right)-\sec ^2\left(\frac{1}{2} (e+f x)\right) \sin (e+f x)\right) \left((n p+2) \, _2F_1\left(\frac{n p}{2},n p;\frac{n p}{2}+1;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n p F_1\left(\frac{n p}{2}+1;n p,1;\frac{n p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^{n p}(e+f x) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^{n p-1}}{2 (n p+2)}\right)}","-\frac{\csc ^2(e+f x) \sec (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \, _2F_1\left(\frac{1}{2} (n p-2),\frac{1}{2} (n p+1);\frac{n p}{2};\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (2-n p)}",1,"(Cot[(e + f*x)/2]^2*Hypergeometric2F1[n*p, -1 + (n*p)/2, (n*p)/2, Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(n*p)*(b*(c*Tan[e + f*x])^n)^p)/(f*(-8 + 4*n*p)) + ((4 + n*p)*AppellF1[1 + (n*p)/2, n*p, 1, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p)/(8*f*(2 + n*p)*((4 + n*p)*AppellF1[1 + (n*p)/2, n*p, 1, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + AppellF1[2 + (n*p)/2, n*p, 2, 3 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(-1 + Cos[e + f*x]) + 2*n*p*AppellF1[2 + (n*p)/2, 1 + n*p, 1, 3 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[(e + f*x)/2]^2)) + (Hypergeometric2F1[n*p, 1 + (n*p)/2, 2 + (n*p)/2, Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(n*p)*Tan[(e + f*x)/2]^2*(b*(c*Tan[e + f*x])^n)^p)/(f*(8 + 4*n*p)) + (Cot[(e + f*x)/2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(n*p)*((2 + n*p)*Hypergeometric2F1[(n*p)/2, n*p, 1 + (n*p)/2, Tan[(e + f*x)/2]^2] - n*p*AppellF1[1 + (n*p)/2, n*p, 1, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)*Tan[e + f*x]^(n*p)*(b*(c*Tan[e + f*x])^n)^p)/(8*f*n*p*(2 + n*p)*(((Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + n*p)*(-(Sec[(e + f*x)/2]^2*Sin[e + f*x]) + Cos[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])*((2 + n*p)*Hypergeometric2F1[(n*p)/2, n*p, 1 + (n*p)/2, Tan[(e + f*x)/2]^2] - n*p*AppellF1[1 + (n*p)/2, n*p, 1, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)*Tan[e + f*x]^(n*p))/(2*(2 + n*p)) + ((Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(n*p)*(-(n*p*AppellF1[1 + (n*p)/2, n*p, 1, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) - n*p*Tan[(e + f*x)/2]^2*(-(((1 + (n*p)/2)*AppellF1[2 + (n*p)/2, n*p, 2, 3 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2 + (n*p)/2)) + (n*p*(1 + (n*p)/2)*AppellF1[2 + (n*p)/2, 1 + n*p, 1, 3 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2 + (n*p)/2)) + (n*p*(2 + n*p)*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*(-Hypergeometric2F1[(n*p)/2, n*p, 1 + (n*p)/2, Tan[(e + f*x)/2]^2] + (1 - Tan[(e + f*x)/2]^2)^(-(n*p))))/2)*Tan[e + f*x]^(n*p))/(2*n*p*(2 + n*p)) + ((Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(n*p)*Sec[e + f*x]^2*((2 + n*p)*Hypergeometric2F1[(n*p)/2, n*p, 1 + (n*p)/2, Tan[(e + f*x)/2]^2] - n*p*AppellF1[1 + (n*p)/2, n*p, 1, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)*Tan[e + f*x]^(-1 + n*p))/(2*(2 + n*p))))","C",0
174,0,0,28,2.8008208,"\int (d \sin (e+f x))^m \left(a+b \tan ^n(e+f x)\right)^p \, dx","Integrate[(d*Sin[e + f*x])^m*(a + b*Tan[e + f*x]^n)^p,x]","\int (d \sin (e+f x))^m \left(a+b \tan ^n(e+f x)\right)^p \, dx","\text{Int}\left((d \sin (e+f x))^m \left(a+b \tan ^n(e+f x)\right)^p,x\right)",0,"Integrate[(d*Sin[e + f*x])^m*(a + b*Tan[e + f*x]^n)^p, x]","A",-1
175,1,81,99,0.5163303,"\int (d \cos (e+f x))^m \left(b \tan ^2(e+f x)\right)^p \, dx","Integrate[(d*Cos[e + f*x])^m*(b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \sec ^2(e+f x)^{m/2} \left(b \tan ^2(e+f x)\right)^p (d \cos (e+f x))^m \, _2F_1\left(\frac{m}{2}+1,p+\frac{1}{2};p+\frac{3}{2};-\tan ^2(e+f x)\right)}{f (2 p+1)}","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p (d \cos (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (-m+2 p+1)} \, _2F_1\left(\frac{1}{2} (2 p+1),\frac{1}{2} (-m+2 p+1);\frac{1}{2} (2 p+3);\sin ^2(e+f x)\right)}{f (2 p+1)}",1,"((d*Cos[e + f*x])^m*Hypergeometric2F1[1 + m/2, 1/2 + p, 3/2 + p, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f*(1 + 2*p))","A",1
176,1,2033,108,16.8809793,"\int (d \cos (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p,x]","\text{Result too large to show}","\frac{\tan (e+f x) \sec ^2(e+f x)^{m/2} (d \cos (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{m+2}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}",1,"(3*a*AppellF1[1/2, (2 + m)/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(d*Cos[e + f*x])^m*(Sec[e + f*x]^2)^(-1 - m/2)*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^(2*p))/(f*(3*a*AppellF1[1/2, (2 + m)/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, (2 + m)/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - a*(2 + m)*AppellF1[3/2, (4 + m)/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2)*((6*a*b*p*AppellF1[1/2, (2 + m)/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(-1 + p))/((Sec[e + f*x]^2)^(m/2)*(3*a*AppellF1[1/2, (2 + m)/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, (2 + m)/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - a*(2 + m)*AppellF1[3/2, (4 + m)/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2)) + (3*a*AppellF1[1/2, (2 + m)/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(a + b*Tan[e + f*x]^2)^p)/((Sec[e + f*x]^2)^(m/2)*(3*a*AppellF1[1/2, (2 + m)/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, (2 + m)/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - a*(2 + m)*AppellF1[3/2, (4 + m)/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2)) + (6*a*(-1 - m/2)*AppellF1[1/2, (2 + m)/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(Sec[e + f*x]^2)^(-1 - m/2)*Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p)/(3*a*AppellF1[1/2, (2 + m)/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, (2 + m)/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - a*(2 + m)*AppellF1[3/2, (4 + m)/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2) + (3*a*(Sec[e + f*x]^2)^(-1 - m/2)*Tan[e + f*x]*((2*b*p*AppellF1[3/2, (2 + m)/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - ((2 + m)*AppellF1[3/2, 1 + (2 + m)/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/3)*(a + b*Tan[e + f*x]^2)^p)/(3*a*AppellF1[1/2, (2 + m)/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, (2 + m)/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - a*(2 + m)*AppellF1[3/2, (4 + m)/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2) - (3*a*AppellF1[1/2, (2 + m)/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(Sec[e + f*x]^2)^(-1 - m/2)*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p*(2*(2*b*p*AppellF1[3/2, (2 + m)/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - a*(2 + m)*AppellF1[3/2, (4 + m)/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Sec[e + f*x]^2*Tan[e + f*x] + 3*a*((2*b*p*AppellF1[3/2, (2 + m)/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - ((2 + m)*AppellF1[3/2, 1 + (2 + m)/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/3) + Tan[e + f*x]^2*(2*b*p*((-6*b*(1 - p)*AppellF1[5/2, (2 + m)/2, 2 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a) - (3*(2 + m)*AppellF1[5/2, 1 + (2 + m)/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/5) - a*(2 + m)*((6*b*p*AppellF1[5/2, (4 + m)/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a) - (3*(4 + m)*AppellF1[5/2, 1 + (4 + m)/2, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*a*AppellF1[1/2, (2 + m)/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, (2 + m)/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - a*(2 + m)*AppellF1[3/2, (4 + m)/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2)^2))","B",0
177,1,91,101,0.5289984,"\int (d \cos (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(d*Cos[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \sec ^2(e+f x)^{m/2} (d \cos (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(\frac{m+2}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right)}{f (n p+1)}","\frac{\tan (e+f x) (d \cos (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (-m+n p+1)} \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (-m+n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right)}{f (n p+1)}",1,"((d*Cos[e + f*x])^m*Hypergeometric2F1[(2 + m)/2, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))","A",1
178,0,0,57,2.7770232,"\int (d \cos (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(d*Cos[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int (d \cos (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","(d \cos (e+f x))^m \left(\frac{\sec (e+f x)}{d}\right)^m \text{Int}\left(\left(\frac{\sec (e+f x)}{d}\right)^{-m} \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Integrate[(d*Cos[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",-1
179,1,46,65,0.1964427,"\int \left(a+a \tan ^2(c+d x)\right)^4 \, dx","Integrate[(a + a*Tan[c + d*x]^2)^4,x]","\frac{a^4 \left(\frac{1}{7} \tan ^7(c+d x)+\frac{3}{5} \tan ^5(c+d x)+\tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{a^4 \tan ^7(c+d x)}{7 d}+\frac{3 a^4 \tan ^5(c+d x)}{5 d}+\frac{a^4 \tan ^3(c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}",1,"(a^4*(Tan[c + d*x] + Tan[c + d*x]^3 + (3*Tan[c + d*x]^5)/5 + Tan[c + d*x]^7/7))/d","A",1
180,1,38,50,0.1025548,"\int \left(a+a \tan ^2(c+d x)\right)^3 \, dx","Integrate[(a + a*Tan[c + d*x]^2)^3,x]","\frac{a^3 \left(\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{a^3 \tan ^5(c+d x)}{5 d}+\frac{2 a^3 \tan ^3(c+d x)}{3 d}+\frac{a^3 \tan (c+d x)}{d}",1,"(a^3*(Tan[c + d*x] + (2*Tan[c + d*x]^3)/3 + Tan[c + d*x]^5/5))/d","A",1
181,1,26,32,0.0423871,"\int \left(a+a \tan ^2(c+d x)\right)^2 \, dx","Integrate[(a + a*Tan[c + d*x]^2)^2,x]","\frac{a^2 \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}",1,"(a^2*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
182,1,26,31,0.0257729,"\int \frac{1}{a+a \tan ^2(c+d x)} \, dx","Integrate[(a + a*Tan[c + d*x]^2)^(-1),x]","\frac{2 (c+d x)+\sin (2 (c+d x))}{4 a d}","\frac{\sin (c+d x) \cos (c+d x)}{2 a d}+\frac{x}{2 a}",1,"(2*(c + d*x) + Sin[2*(c + d*x)])/(4*a*d)","A",1
183,1,36,55,0.0428781,"\int \frac{1}{\left(a+a \tan ^2(c+d x)\right)^2} \, dx","Integrate[(a + a*Tan[c + d*x]^2)^(-2),x]","\frac{12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x))}{32 a^2 d}","\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{3 x}{8 a^2}",1,"(12*(c + d*x) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)])/(32*a^2*d)","A",1
184,1,46,79,0.0391989,"\int \frac{1}{\left(a+a \tan ^2(c+d x)\right)^3} \, dx","Integrate[(a + a*Tan[c + d*x]^2)^(-3),x]","\frac{45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))+60 c+60 d x}{192 a^3 d}","\frac{\sin (c+d x) \cos ^5(c+d x)}{6 a^3 d}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}+\frac{5 \sin (c+d x) \cos (c+d x)}{16 a^3 d}+\frac{5 x}{16 a^3}",1,"(60*c + 60*d*x + 45*Sin[2*(c + d*x)] + 9*Sin[4*(c + d*x)] + Sin[6*(c + d*x)])/(192*a^3*d)","A",1
185,1,63,74,0.2583061,"\int \tan ^5(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2),x]","\frac{3 (a-b) \tan ^4(e+f x)-6 (a-b) \tan ^2(e+f x)+12 (b-a) \log (\cos (e+f x))+2 b \tan ^6(e+f x)}{12 f}","\frac{(a-b) \tan ^4(e+f x)}{4 f}-\frac{(a-b) \tan ^2(e+f x)}{2 f}-\frac{(a-b) \log (\cos (e+f x))}{f}+\frac{b \tan ^6(e+f x)}{6 f}",1,"(12*(-a + b)*Log[Cos[e + f*x]] - 6*(a - b)*Tan[e + f*x]^2 + 3*(a - b)*Tan[e + f*x]^4 + 2*b*Tan[e + f*x]^6)/(12*f)","A",1
186,1,65,53,0.1682897,"\int \tan ^3(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2),x]","\frac{a \left(\tan ^2(e+f x)+2 \log (\cos (e+f x))\right)}{2 f}-\frac{b \left(-\tan ^4(e+f x)+2 \tan ^2(e+f x)+4 \log (\cos (e+f x))\right)}{4 f}","\frac{(a-b) \tan ^2(e+f x)}{2 f}+\frac{(a-b) \log (\cos (e+f x))}{f}+\frac{b \tan ^4(e+f x)}{4 f}",1,"(a*(2*Log[Cos[e + f*x]] + Tan[e + f*x]^2))/(2*f) - (b*(4*Log[Cos[e + f*x]] + 2*Tan[e + f*x]^2 - Tan[e + f*x]^4))/(4*f)","A",1
187,1,40,34,0.0720228,"\int \tan (e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Tan[e + f*x]*(a + b*Tan[e + f*x]^2),x]","\frac{b \left(\tan ^2(e+f x)+2 \log (\cos (e+f x))\right)}{2 f}-\frac{a \log (\cos (e+f x))}{f}","\frac{b \tan ^2(e+f x)}{2 f}-\frac{(a-b) \log (\cos (e+f x))}{f}",1,"-((a*Log[Cos[e + f*x]])/f) + (b*(2*Log[Cos[e + f*x]] + Tan[e + f*x]^2))/(2*f)","A",1
188,1,34,26,0.0411873,"\int \cot (e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Cot[e + f*x]*(a + b*Tan[e + f*x]^2),x]","\frac{a (\log (\tan (e+f x))+\log (\cos (e+f x)))}{f}-\frac{b \log (\cos (e+f x))}{f}","\frac{a \log (\sin (e+f x))}{f}-\frac{b \log (\cos (e+f x))}{f}",1,"-((b*Log[Cos[e + f*x]])/f) + (a*(Log[Cos[e + f*x]] + Log[Tan[e + f*x]]))/f","A",1
189,1,56,34,0.1593813,"\int \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2),x]","\frac{b (\log (\tan (e+f x))+\log (\cos (e+f x)))}{f}-\frac{a \left(\cot ^2(e+f x)+2 \log (\tan (e+f x))+2 \log (\cos (e+f x))\right)}{2 f}","-\frac{(a-b) \log (\sin (e+f x))}{f}-\frac{a \cot ^2(e+f x)}{2 f}",1,"(b*(Log[Cos[e + f*x]] + Log[Tan[e + f*x]]))/f - (a*(Cot[e + f*x]^2 + 2*Log[Cos[e + f*x]] + 2*Log[Tan[e + f*x]]))/(2*f)","A",1
190,1,56,53,0.2287309,"\int \cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2),x]","\frac{2 (a-b) \cot ^2(e+f x)+4 (a-b) (\log (\tan (e+f x))+\log (\cos (e+f x)))-a \cot ^4(e+f x)}{4 f}","\frac{(a-b) \cot ^2(e+f x)}{2 f}+\frac{(a-b) \log (\sin (e+f x))}{f}-\frac{a \cot ^4(e+f x)}{4 f}",1,"(2*(a - b)*Cot[e + f*x]^2 - a*Cot[e + f*x]^4 + 4*(a - b)*(Log[Cos[e + f*x]] + Log[Tan[e + f*x]]))/(4*f)","A",1
191,1,129,80,0.0501419,"\int \tan ^6(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2),x]","-\frac{a \tan ^{-1}(\tan (e+f x))}{f}+\frac{a \tan ^5(e+f x)}{5 f}-\frac{a \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f}+\frac{b \tan ^{-1}(\tan (e+f x))}{f}+\frac{b \tan ^7(e+f x)}{7 f}-\frac{b \tan ^5(e+f x)}{5 f}+\frac{b \tan ^3(e+f x)}{3 f}-\frac{b \tan (e+f x)}{f}","\frac{(a-b) \tan ^5(e+f x)}{5 f}-\frac{(a-b) \tan ^3(e+f x)}{3 f}+\frac{(a-b) \tan (e+f x)}{f}-x (a-b)+\frac{b \tan ^7(e+f x)}{7 f}",1,"-((a*ArcTan[Tan[e + f*x]])/f) + (b*ArcTan[Tan[e + f*x]])/f + (a*Tan[e + f*x])/f - (b*Tan[e + f*x])/f - (a*Tan[e + f*x]^3)/(3*f) + (b*Tan[e + f*x]^3)/(3*f) + (a*Tan[e + f*x]^5)/(5*f) - (b*Tan[e + f*x]^5)/(5*f) + (b*Tan[e + f*x]^7)/(7*f)","A",1
192,1,97,60,0.0440162,"\int \tan ^4(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Tan[e + f*x]^4*(a + b*Tan[e + f*x]^2),x]","\frac{a \tan ^{-1}(\tan (e+f x))}{f}+\frac{a \tan ^3(e+f x)}{3 f}-\frac{a \tan (e+f x)}{f}-\frac{b \tan ^{-1}(\tan (e+f x))}{f}+\frac{b \tan ^5(e+f x)}{5 f}-\frac{b \tan ^3(e+f x)}{3 f}+\frac{b \tan (e+f x)}{f}","\frac{(a-b) \tan ^3(e+f x)}{3 f}-\frac{(a-b) \tan (e+f x)}{f}+x (a-b)+\frac{b \tan ^5(e+f x)}{5 f}",1,"(a*ArcTan[Tan[e + f*x]])/f - (b*ArcTan[Tan[e + f*x]])/f - (a*Tan[e + f*x])/f + (b*Tan[e + f*x])/f + (a*Tan[e + f*x]^3)/(3*f) - (b*Tan[e + f*x]^3)/(3*f) + (b*Tan[e + f*x]^5)/(5*f)","A",1
193,1,65,40,0.0285174,"\int \tan ^2(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2),x]","-\frac{a \tan ^{-1}(\tan (e+f x))}{f}+\frac{a \tan (e+f x)}{f}+\frac{b \tan ^{-1}(\tan (e+f x))}{f}+\frac{b \tan ^3(e+f x)}{3 f}-\frac{b \tan (e+f x)}{f}","\frac{(a-b) \tan (e+f x)}{f}-x (a-b)+\frac{b \tan ^3(e+f x)}{3 f}",1,"-((a*ArcTan[Tan[e + f*x]])/f) + (b*ArcTan[Tan[e + f*x]])/f + (a*Tan[e + f*x])/f - (b*Tan[e + f*x])/f + (b*Tan[e + f*x]^3)/(3*f)","A",1
194,1,28,19,0.006645,"\int \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[a + b*Tan[e + f*x]^2,x]","a x-\frac{b \tan ^{-1}(\tan (e+f x))}{f}+\frac{b \tan (e+f x)}{f}","a x+\frac{b \tan (e+f x)}{f}-b x",1,"a*x - (b*ArcTan[Tan[e + f*x]])/f + (b*Tan[e + f*x])/f","A",1
195,1,34,21,0.0186506,"\int \cot ^2(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2),x]","b x-\frac{a \cot (e+f x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(e+f x)\right)}{f}","-(x (a-b))-\frac{a \cot (e+f x)}{f}",1,"b*x - (a*Cot[e + f*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[e + f*x]^2])/f","C",1
196,1,65,39,0.040559,"\int \cot ^4(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2),x]","-\frac{a \cot ^3(e+f x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(e+f x)\right)}{3 f}-\frac{b \cot (e+f x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(e+f x)\right)}{f}","\frac{(a-b) \cot (e+f x)}{f}+x (a-b)-\frac{a \cot ^3(e+f x)}{3 f}",1,"-1/3*(a*Cot[e + f*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[e + f*x]^2])/f - (b*Cot[e + f*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[e + f*x]^2])/f","C",1
197,1,69,61,0.0507205,"\int \cot ^6(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Integrate[Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2),x]","-\frac{a \cot ^5(e+f x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(e+f x)\right)}{5 f}-\frac{b \cot ^3(e+f x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(e+f x)\right)}{3 f}","\frac{(a-b) \cot ^3(e+f x)}{3 f}-\frac{(a-b) \cot (e+f x)}{f}-x (a-b)-\frac{a \cot ^5(e+f x)}{5 f}",1,"-1/5*(a*Cot[e + f*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[e + f*x]^2])/f - (b*Cot[e + f*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[e + f*x]^2])/(3*f)","C",1
198,1,89,105,0.3497984,"\int \tan ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2)^2,x]","\frac{4 b (2 a-b) \tan ^6(e+f x)+6 (a-b)^2 \tan ^4(e+f x)-12 (a-b)^2 \tan ^2(e+f x)-24 (a-b)^2 \log (\cos (e+f x))+3 b^2 \tan ^8(e+f x)}{24 f}","\frac{b (2 a-b) \tan ^6(e+f x)}{6 f}+\frac{(a-b)^2 \tan ^4(e+f x)}{4 f}-\frac{(a-b)^2 \tan ^2(e+f x)}{2 f}-\frac{(a-b)^2 \log (\cos (e+f x))}{f}+\frac{b^2 \tan ^8(e+f x)}{8 f}",1,"(-24*(a - b)^2*Log[Cos[e + f*x]] - 12*(a - b)^2*Tan[e + f*x]^2 + 6*(a - b)^2*Tan[e + f*x]^4 + 4*(2*a - b)*b*Tan[e + f*x]^6 + 3*b^2*Tan[e + f*x]^8)/(24*f)","A",1
199,1,72,82,0.2836535,"\int \tan ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^2,x]","\frac{3 b (2 a-b) \tan ^4(e+f x)+6 (a-b)^2 \tan ^2(e+f x)+12 (a-b)^2 \log (\cos (e+f x))+2 b^2 \tan ^6(e+f x)}{12 f}","\frac{b (2 a-b) \tan ^4(e+f x)}{4 f}+\frac{(a-b)^2 \tan ^2(e+f x)}{2 f}+\frac{(a-b)^2 \log (\cos (e+f x))}{f}+\frac{b^2 \tan ^6(e+f x)}{6 f}",1,"(12*(a - b)^2*Log[Cos[e + f*x]] + 6*(a - b)^2*Tan[e + f*x]^2 + 3*(2*a - b)*b*Tan[e + f*x]^4 + 2*b^2*Tan[e + f*x]^6)/(12*f)","A",1
200,1,54,62,0.2173975,"\int \tan (e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^2,x]","\frac{2 b (2 a-b) \tan ^2(e+f x)-4 (a-b)^2 \log (\cos (e+f x))+b^2 \tan ^4(e+f x)}{4 f}","\frac{b (a-b) \tan ^2(e+f x)}{2 f}+\frac{\left(a+b \tan ^2(e+f x)\right)^2}{4 f}-\frac{(a-b)^2 \log (\cos (e+f x))}{f}",1,"(-4*(a - b)^2*Log[Cos[e + f*x]] + 2*(2*a - b)*b*Tan[e + f*x]^2 + b^2*Tan[e + f*x]^4)/(4*f)","A",1
201,1,65,51,0.1272278,"\int \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^2,x]","\frac{a^2 (\log (\tan (e+f x))+\log (\cos (e+f x)))}{f}-\frac{2 a b \log (\cos (e+f x))}{f}+\frac{b^2 \left(\tan ^2(e+f x)+2 \log (\cos (e+f x))\right)}{2 f}","\frac{a^2 \log (\tan (e+f x))}{f}+\frac{(a-b)^2 \log (\cos (e+f x))}{f}+\frac{b^2 \tan ^2(e+f x)}{2 f}",1,"(-2*a*b*Log[Cos[e + f*x]])/f + (a^2*(Log[Cos[e + f*x]] + Log[Tan[e + f*x]]))/f + (b^2*(2*Log[Cos[e + f*x]] + Tan[e + f*x]^2))/(2*f)","A",1
202,1,51,56,0.2532001,"\int \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^2 \cot ^2(e+f x)+2 a (a-2 b) \log (\tan (e+f x))+2 (a-b)^2 \log (\cos (e+f x))}{2 f}","-\frac{a^2 \cot ^2(e+f x)}{2 f}-\frac{a (a-2 b) \log (\tan (e+f x))}{f}-\frac{(a-b)^2 \log (\cos (e+f x))}{f}",1,"-1/2*(a^2*Cot[e + f*x]^2 + 2*(a - b)^2*Log[Cos[e + f*x]] + 2*a*(a - 2*b)*Log[Tan[e + f*x]])/f","A",1
203,1,61,76,0.3021616,"\int \cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^2,x]","\frac{-a^2 \cot ^4(e+f x)+2 a (a-2 b) \cot ^2(e+f x)+4 (a-b)^2 (\log (\tan (e+f x))+\log (\cos (e+f x)))}{4 f}","-\frac{a^2 \cot ^4(e+f x)}{4 f}+\frac{a (a-2 b) \cot ^2(e+f x)}{2 f}+\frac{(a-b)^2 \log (\tan (e+f x))}{f}+\frac{(a-b)^2 \log (\cos (e+f x))}{f}",1,"(2*a*(a - 2*b)*Cot[e + f*x]^2 - a^2*Cot[e + f*x]^4 + 4*(a - b)^2*(Log[Cos[e + f*x]] + Log[Tan[e + f*x]]))/(4*f)","A",1
204,1,243,113,0.0838644,"\int \tan ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^2 \tan ^{-1}(\tan (e+f x))}{f}+\frac{a^2 \tan ^5(e+f x)}{5 f}-\frac{a^2 \tan ^3(e+f x)}{3 f}+\frac{a^2 \tan (e+f x)}{f}+\frac{2 a b \tan ^{-1}(\tan (e+f x))}{f}+\frac{2 a b \tan ^7(e+f x)}{7 f}-\frac{2 a b \tan ^5(e+f x)}{5 f}+\frac{2 a b \tan ^3(e+f x)}{3 f}-\frac{2 a b \tan (e+f x)}{f}-\frac{b^2 \tan ^{-1}(\tan (e+f x))}{f}+\frac{b^2 \tan ^9(e+f x)}{9 f}-\frac{b^2 \tan ^7(e+f x)}{7 f}+\frac{b^2 \tan ^5(e+f x)}{5 f}-\frac{b^2 \tan ^3(e+f x)}{3 f}+\frac{b^2 \tan (e+f x)}{f}","\frac{b (2 a-b) \tan ^7(e+f x)}{7 f}+\frac{(a-b)^2 \tan ^5(e+f x)}{5 f}-\frac{(a-b)^2 \tan ^3(e+f x)}{3 f}+\frac{(a-b)^2 \tan (e+f x)}{f}-x (a-b)^2+\frac{b^2 \tan ^9(e+f x)}{9 f}",1,"-((a^2*ArcTan[Tan[e + f*x]])/f) + (2*a*b*ArcTan[Tan[e + f*x]])/f - (b^2*ArcTan[Tan[e + f*x]])/f + (a^2*Tan[e + f*x])/f - (2*a*b*Tan[e + f*x])/f + (b^2*Tan[e + f*x])/f - (a^2*Tan[e + f*x]^3)/(3*f) + (2*a*b*Tan[e + f*x]^3)/(3*f) - (b^2*Tan[e + f*x]^3)/(3*f) + (a^2*Tan[e + f*x]^5)/(5*f) - (2*a*b*Tan[e + f*x]^5)/(5*f) + (b^2*Tan[e + f*x]^5)/(5*f) + (2*a*b*Tan[e + f*x]^7)/(7*f) - (b^2*Tan[e + f*x]^7)/(7*f) + (b^2*Tan[e + f*x]^9)/(9*f)","B",1
205,1,190,91,0.0780151,"\int \tan ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Tan[e + f*x]^4*(a + b*Tan[e + f*x]^2)^2,x]","\frac{a^2 \tan ^{-1}(\tan (e+f x))}{f}+\frac{a^2 \tan ^3(e+f x)}{3 f}-\frac{a^2 \tan (e+f x)}{f}-\frac{2 a b \tan ^{-1}(\tan (e+f x))}{f}+\frac{2 a b \tan ^5(e+f x)}{5 f}-\frac{2 a b \tan ^3(e+f x)}{3 f}+\frac{2 a b \tan (e+f x)}{f}+\frac{b^2 \tan ^{-1}(\tan (e+f x))}{f}+\frac{b^2 \tan ^7(e+f x)}{7 f}-\frac{b^2 \tan ^5(e+f x)}{5 f}+\frac{b^2 \tan ^3(e+f x)}{3 f}-\frac{b^2 \tan (e+f x)}{f}","\frac{b (2 a-b) \tan ^5(e+f x)}{5 f}+\frac{(a-b)^2 \tan ^3(e+f x)}{3 f}-\frac{(a-b)^2 \tan (e+f x)}{f}+x (a-b)^2+\frac{b^2 \tan ^7(e+f x)}{7 f}",1,"(a^2*ArcTan[Tan[e + f*x]])/f - (2*a*b*ArcTan[Tan[e + f*x]])/f + (b^2*ArcTan[Tan[e + f*x]])/f - (a^2*Tan[e + f*x])/f + (2*a*b*Tan[e + f*x])/f - (b^2*Tan[e + f*x])/f + (a^2*Tan[e + f*x]^3)/(3*f) - (2*a*b*Tan[e + f*x]^3)/(3*f) + (b^2*Tan[e + f*x]^3)/(3*f) + (2*a*b*Tan[e + f*x]^5)/(5*f) - (b^2*Tan[e + f*x]^5)/(5*f) + (b^2*Tan[e + f*x]^7)/(7*f)","B",1
206,1,137,69,0.0516466,"\int \tan ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^2 \tan ^{-1}(\tan (e+f x))}{f}+\frac{a^2 \tan (e+f x)}{f}+\frac{2 a b \tan ^{-1}(\tan (e+f x))}{f}+\frac{2 a b \tan ^3(e+f x)}{3 f}-\frac{2 a b \tan (e+f x)}{f}-\frac{b^2 \tan ^{-1}(\tan (e+f x))}{f}+\frac{b^2 \tan ^5(e+f x)}{5 f}-\frac{b^2 \tan ^3(e+f x)}{3 f}+\frac{b^2 \tan (e+f x)}{f}","\frac{b (2 a-b) \tan ^3(e+f x)}{3 f}+\frac{(a-b)^2 \tan (e+f x)}{f}-x (a-b)^2+\frac{b^2 \tan ^5(e+f x)}{5 f}",1,"-((a^2*ArcTan[Tan[e + f*x]])/f) + (2*a*b*ArcTan[Tan[e + f*x]])/f - (b^2*ArcTan[Tan[e + f*x]])/f + (a^2*Tan[e + f*x])/f - (2*a*b*Tan[e + f*x])/f + (b^2*Tan[e + f*x])/f + (2*a*b*Tan[e + f*x]^3)/(3*f) - (b^2*Tan[e + f*x]^3)/(3*f) + (b^2*Tan[e + f*x]^5)/(5*f)","A",1
207,1,73,46,0.5867859,"\int \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[(a + b*Tan[e + f*x]^2)^2,x]","\frac{\tan (e+f x) \left(b \left(6 a-b \left(3-\tan ^2(e+f x)\right)\right)+\frac{3 (a-b)^2 \tanh ^{-1}\left(\sqrt{-\tan ^2(e+f x)}\right)}{\sqrt{-\tan ^2(e+f x)}}\right)}{3 f}","\frac{b (2 a-b) \tan (e+f x)}{f}+x (a-b)^2+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"(Tan[e + f*x]*((3*(a - b)^2*ArcTanh[Sqrt[-Tan[e + f*x]^2]])/Sqrt[-Tan[e + f*x]^2] + b*(6*a - b*(3 - Tan[e + f*x]^2))))/(3*f)","A",1
208,1,66,38,0.1147859,"\int \cot ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^2 \cot (e+f x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(e+f x)\right)}{f}+2 a b x-\frac{b^2 \tan ^{-1}(\tan (e+f x))}{f}+\frac{b^2 \tan (e+f x)}{f}","-\frac{a^2 \cot (e+f x)}{f}-x (a-b)^2+\frac{b^2 \tan (e+f x)}{f}",1,"2*a*b*x - (b^2*ArcTan[Tan[e + f*x]])/f - (a^2*Cot[e + f*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[e + f*x]^2])/f + (b^2*Tan[e + f*x])/f","C",1
209,1,71,44,1.3079638,"\int \cot ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{\cot (e+f x) \left(a \left(a \cot ^2(e+f x)-3 a+6 b\right)+3 (a-b)^2 \sqrt{-\tan ^2(e+f x)} \tanh ^{-1}\left(\sqrt{-\tan ^2(e+f x)}\right)\right)}{3 f}","-\frac{a^2 \cot ^3(e+f x)}{3 f}+\frac{a (a-2 b) \cot (e+f x)}{f}+x (a-b)^2",1,"-1/3*(Cot[e + f*x]*(a*(-3*a + 6*b + a*Cot[e + f*x]^2) + 3*(a - b)^2*ArcTanh[Sqrt[-Tan[e + f*x]^2]]*Sqrt[-Tan[e + f*x]^2]))/f","A",1
210,1,104,68,0.1011323,"\int \cot ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Integrate[Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^2 \cot ^5(e+f x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(e+f x)\right)}{5 f}-\frac{2 a b \cot ^3(e+f x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(e+f x)\right)}{3 f}-\frac{b^2 \cot (e+f x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(e+f x)\right)}{f}","-\frac{a^2 \cot ^5(e+f x)}{5 f}+\frac{a (a-2 b) \cot ^3(e+f x)}{3 f}-\frac{(a-b)^2 \cot (e+f x)}{f}-x (a-b)^2",1,"-1/5*(a^2*Cot[e + f*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[e + f*x]^2])/f - (2*a*b*Cot[e + f*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[e + f*x]^2])/(3*f) - (b^2*Cot[e + f*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[e + f*x]^2])/f","C",1
211,1,64,71,0.1669726,"\int \frac{\tan ^5(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2),x]","\frac{-\frac{a^2 \log \left(a+b \tan ^2(e+f x)\right)}{b^2 (a-b)}-\frac{2 \log (\cos (e+f x))}{a-b}+\frac{\tan ^2(e+f x)}{b}}{2 f}","-\frac{a^2 \log \left(a+b \tan ^2(e+f x)\right)}{2 b^2 f (a-b)}-\frac{\log (\cos (e+f x))}{f (a-b)}+\frac{\tan ^2(e+f x)}{2 b f}",1,"((-2*Log[Cos[e + f*x]])/(a - b) - (a^2*Log[a + b*Tan[e + f*x]^2])/((a - b)*b^2) + Tan[e + f*x]^2/b)/(2*f)","A",1
212,1,41,50,0.0336062,"\int \frac{\tan ^3(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2),x]","\frac{a \log \left(a+b \tan ^2(e+f x)\right)+2 b \log (\cos (e+f x))}{2 a b f-2 b^2 f}","\frac{a \log \left(a+b \tan ^2(e+f x)\right)}{2 b f (a-b)}+\frac{\log (\cos (e+f x))}{f (a-b)}",1,"(2*b*Log[Cos[e + f*x]] + a*Log[a + b*Tan[e + f*x]^2])/(2*a*b*f - 2*b^2*f)","A",1
213,1,37,36,0.0423906,"\int \frac{\tan (e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]/(a + b*Tan[e + f*x]^2),x]","-\frac{\log \left(a+b \tan ^2(e+f x)\right)+2 \log (\cos (e+f x))}{2 f (a-b)}","-\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)}",1,"-1/2*(2*Log[Cos[e + f*x]] + Log[a + b*Tan[e + f*x]^2])/((a - b)*f)","A",1
214,1,57,64,0.0601471,"\int \frac{\cot (e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]/(a + b*Tan[e + f*x]^2),x]","\frac{b \log \left(a+b \tan ^2(e+f x)\right)+2 (a-b) \log (\tan (e+f x))+2 a \log (\cos (e+f x))}{2 a f (a-b)}","\frac{b \log \left(a+b \tan ^2(e+f x)\right)}{2 a f (a-b)}+\frac{\log (\cos (e+f x))}{f (a-b)}+\frac{\log (\tan (e+f x))}{a f}",1,"(2*a*Log[Cos[e + f*x]] + 2*(a - b)*Log[Tan[e + f*x]] + b*Log[a + b*Tan[e + f*x]^2])/(2*a*(a - b)*f)","A",1
215,1,63,89,0.255054,"\int \frac{\cot ^3(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2),x]","-\frac{\frac{b^2 \log \left(a \cot ^2(e+f x)+b\right)}{a^2 (a-b)}+\frac{2 \log (\sin (e+f x))}{a-b}+\frac{\cot ^2(e+f x)}{a}}{2 f}","-\frac{b^2 \log \left(a+b \tan ^2(e+f x)\right)}{2 a^2 f (a-b)}-\frac{(a+b) \log (\tan (e+f x))}{a^2 f}-\frac{\log (\cos (e+f x))}{f (a-b)}-\frac{\cot ^2(e+f x)}{2 a f}",1,"-1/2*(Cot[e + f*x]^2/a + (b^2*Log[b + a*Cot[e + f*x]^2])/(a^2*(a - b)) + (2*Log[Sin[e + f*x]])/(a - b))/f","A",1
216,1,83,115,0.3721816,"\int \frac{\cot ^5(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2),x]","-\frac{-\frac{b^3 \log \left(a \cot ^2(e+f x)+b\right)}{a^3 (a-b)}-\frac{(a+b) \cot ^2(e+f x)}{a^2}-\frac{2 \log (\sin (e+f x))}{a-b}+\frac{\cot ^4(e+f x)}{2 a}}{2 f}","\frac{b^3 \log \left(a+b \tan ^2(e+f x)\right)}{2 a^3 f (a-b)}+\frac{(a+b) \cot ^2(e+f x)}{2 a^2 f}+\frac{\left(a^2+a b+b^2\right) \log (\tan (e+f x))}{a^3 f}+\frac{\log (\cos (e+f x))}{f (a-b)}-\frac{\cot ^4(e+f x)}{4 a f}",1,"-1/2*(-(((a + b)*Cot[e + f*x]^2)/a^2) + Cot[e + f*x]^4/(2*a) - (b^3*Log[b + a*Cot[e + f*x]^2])/(a^3*(a - b)) - (2*Log[Sin[e + f*x]])/(a - b))/f","A",1
217,1,92,85,0.8392963,"\int \frac{\tan ^6(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2),x]","\frac{\sqrt{b} \left((a-b) \tan (e+f x) \left(3 a-b \sec ^2(e+f x)+4 b\right)+3 b^2 (e+f x)\right)-3 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{3 b^{5/2} f (b-a)}","\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{b^{5/2} f (a-b)}-\frac{(a+b) \tan (e+f x)}{b^2 f}-\frac{x}{a-b}+\frac{\tan ^3(e+f x)}{3 b f}",1,"(-3*a^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] + Sqrt[b]*(3*b^2*(e + f*x) + (a - b)*(3*a + 4*b - b*Sec[e + f*x]^2)*Tan[e + f*x]))/(3*b^(5/2)*(-a + b)*f)","A",1
218,1,70,63,0.2891104,"\int \frac{\tan ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2),x]","-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{b^{3/2} f (a-b)}+\frac{e+f x}{f (a-b)}+\frac{\tan (e+f x)}{b f}","-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{b^{3/2} f (a-b)}+\frac{x}{a-b}+\frac{\tan (e+f x)}{b f}",1,"(e + f*x)/((a - b)*f) - (a^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/((a - b)*b^(3/2)*f) + Tan[e + f*x]/(b*f)","A",1
219,1,49,50,0.0288514,"\int \frac{\tan ^2(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2),x]","\frac{\tan ^{-1}(\tan (e+f x))-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{b}}}{b f-a f}","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{b} f (a-b)}-\frac{x}{a-b}",1,"(ArcTan[Tan[e + f*x]] - (Sqrt[a]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/Sqrt[b])/(-(a*f) + b*f)","A",1
220,1,49,50,0.0496551,"\int \frac{1}{a+b \tan ^2(e+f x)} \, dx","Integrate[(a + b*Tan[e + f*x]^2)^(-1),x]","\frac{\tan ^{-1}(\tan (e+f x))-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{a}}}{a f-b f}","\frac{x}{a-b}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{a} f (a-b)}",1,"(ArcTan[Tan[e + f*x]] - (Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/Sqrt[a])/(a*f - b*f)","A",1
221,1,68,64,0.2594153,"\int \frac{\cot ^2(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2),x]","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)-\sqrt{a} ((a-b) \cot (e+f x)+a (e+f x))}{a^{3/2} f (a-b)}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{3/2} f (a-b)}-\frac{x}{a-b}-\frac{\cot (e+f x)}{a f}",1,"(b^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] - Sqrt[a]*(a*(e + f*x) + (a - b)*Cot[e + f*x]))/(a^(3/2)*(a - b)*f)","A",1
222,1,92,84,0.69619,"\int \frac{\cot ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2),x]","\frac{\sqrt{a} \left(3 a^2 (e+f x)-(a-b) \cot (e+f x) \left(a \csc ^2(e+f x)-4 a-3 b\right)\right)-3 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{3 a^{5/2} f (a-b)}","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{5/2} f (a-b)}+\frac{(a+b) \cot (e+f x)}{a^2 f}+\frac{x}{a-b}-\frac{\cot ^3(e+f x)}{3 a f}",1,"(-3*b^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] + Sqrt[a]*(3*a^2*(e + f*x) - (a - b)*Cot[e + f*x]*(-4*a - 3*b + a*Csc[e + f*x]^2)))/(3*a^(5/2)*(a - b)*f)","A",1
223,1,121,113,1.9243201,"\int \frac{\cot ^6(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2),x]","\frac{\sqrt{a} \left(-15 a^3 (e+f x)-(a-b) \cot (e+f x) \left(3 a^2 \csc ^4(e+f x)+23 a^2-a (11 a+5 b) \csc ^2(e+f x)+20 a b+15 b^2\right)\right)+15 b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{15 a^{7/2} f (a-b)}","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{7/2} f (a-b)}+\frac{(a+b) \cot ^3(e+f x)}{3 a^2 f}-\frac{\left(a^2+a b+b^2\right) \cot (e+f x)}{a^3 f}-\frac{x}{a-b}-\frac{\cot ^5(e+f x)}{5 a f}",1,"(15*b^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] + Sqrt[a]*(-15*a^3*(e + f*x) - (a - b)*Cot[e + f*x]*(23*a^2 + 20*a*b + 15*b^2 - a*(11*a + 5*b)*Csc[e + f*x]^2 + 3*a^2*Csc[e + f*x]^4)))/(15*a^(7/2)*(a - b)*f)","A",1
224,1,73,90,0.7203966,"\int \frac{\tan ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2)^2,x]","\frac{\frac{a^2 (a-b)}{b^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{a (a-2 b) \log \left(a+b \tan ^2(e+f x)\right)}{b^2}-2 \log (\cos (e+f x))}{2 f (a-b)^2}","\frac{a^2}{2 b^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{a (a-2 b) \log \left(a+b \tan ^2(e+f x)\right)}{2 b^2 f (a-b)^2}-\frac{\log (\cos (e+f x))}{f (a-b)^2}",1,"(-2*Log[Cos[e + f*x]] + (a*(a - 2*b)*Log[a + b*Tan[e + f*x]^2])/b^2 + (a^2*(a - b))/(b^2*(a + b*Tan[e + f*x]^2)))/(2*(a - b)^2*f)","A",1
225,1,61,69,0.5793013,"\int \frac{\tan ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2)^2,x]","\frac{\frac{a (b-a)}{b \left(a+b \tan ^2(e+f x)\right)}+\log \left(a+b \tan ^2(e+f x)\right)+2 \log (\cos (e+f x))}{2 f (a-b)^2}","\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^2}-\frac{a}{2 b f (a-b) \left(a+b \tan ^2(e+f x)\right)}",1,"(2*Log[Cos[e + f*x]] + Log[a + b*Tan[e + f*x]^2] + (a*(-a + b))/(b*(a + b*Tan[e + f*x]^2)))/(2*(a - b)^2*f)","A",1
226,1,57,65,0.6568929,"\int \frac{\tan (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Tan[e + f*x]/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{\frac{b-a}{a+b \tan ^2(e+f x)}+\log \left(a+b \tan ^2(e+f x)\right)+2 \log (\cos (e+f x))}{2 f (a-b)^2}","\frac{1}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^2}",1,"-1/2*(2*Log[Cos[e + f*x]] + Log[a + b*Tan[e + f*x]^2] + (-a + b)/(a + b*Tan[e + f*x]^2))/((a - b)^2*f)","A",1
227,1,90,103,2.1049144,"\int \frac{\cot (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Cot[e + f*x]/(a + b*Tan[e + f*x]^2)^2,x]","\frac{\frac{\frac{b \left(\frac{a (b-a)}{a+b \tan ^2(e+f x)}+(2 a-b) \log \left(a+b \tan ^2(e+f x)\right)\right)}{(a-b)^2}+2 \log (\tan (e+f x))}{a^2}+\frac{2 \log (\cos (e+f x))}{(a-b)^2}}{2 f}","\frac{b (2 a-b) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^2 f (a-b)^2}+\frac{\log (\tan (e+f x))}{a^2 f}-\frac{b}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{\log (\cos (e+f x))}{f (a-b)^2}",1,"((2*Log[Cos[e + f*x]])/(a - b)^2 + (2*Log[Tan[e + f*x]] + (b*((2*a - b)*Log[a + b*Tan[e + f*x]^2] + (a*(-a + b))/(a + b*Tan[e + f*x]^2)))/(a - b)^2)/a^2)/(2*f)","A",1
228,1,98,132,0.8826738,"\int \frac{\cot ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{\frac{b^3}{a^3 (a-b) \left(a \cot ^2(e+f x)+b\right)}+\frac{b^2 (3 a-2 b) \log \left(a \cot ^2(e+f x)+b\right)}{a^3 (a-b)^2}+\frac{\cot ^2(e+f x)}{a^2}+\frac{2 \log (\sin (e+f x))}{(a-b)^2}}{2 f}","-\frac{b^2 (3 a-2 b) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^3 f (a-b)^2}-\frac{(a+2 b) \log (\tan (e+f x))}{a^3 f}+\frac{b^2}{2 a^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{\cot ^2(e+f x)}{2 a^2 f}-\frac{\log (\cos (e+f x))}{f (a-b)^2}",1,"-1/2*(Cot[e + f*x]^2/a^2 + b^3/(a^3*(a - b)*(b + a*Cot[e + f*x]^2)) + ((3*a - 2*b)*b^2*Log[b + a*Cot[e + f*x]^2])/(a^3*(a - b)^2) + (2*Log[Sin[e + f*x]])/(a - b)^2)/f","A",1
229,1,121,161,1.0921662,"\int \frac{\cot ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{-\frac{b^4}{a^4 (a-b) \left(a \cot ^2(e+f x)+b\right)}-\frac{b^3 (4 a-3 b) \log \left(a \cot ^2(e+f x)+b\right)}{a^4 (a-b)^2}-\frac{(a+2 b) \cot ^2(e+f x)}{a^3}+\frac{\cot ^4(e+f x)}{2 a^2}-\frac{2 \log (\sin (e+f x))}{(a-b)^2}}{2 f}","\frac{b^3 (4 a-3 b) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^4 f (a-b)^2}-\frac{b^3}{2 a^3 f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{(a+2 b) \cot ^2(e+f x)}{2 a^3 f}-\frac{\cot ^4(e+f x)}{4 a^2 f}+\frac{\left(a^2+2 a b+3 b^2\right) \log (\tan (e+f x))}{a^4 f}+\frac{\log (\cos (e+f x))}{f (a-b)^2}",1,"-1/2*(-(((a + 2*b)*Cot[e + f*x]^2)/a^3) + Cot[e + f*x]^4/(2*a^2) - b^4/(a^4*(a - b)*(b + a*Cot[e + f*x]^2)) - ((4*a - 3*b)*b^3*Log[b + a*Cot[e + f*x]^2])/(a^4*(a - b)^2) - (2*Log[Sin[e + f*x]])/(a - b)^2)/f","A",1
230,1,118,130,1.319207,"\int \frac{\tan ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2)^2,x]","\frac{-\frac{a^{3/2} (3 a-5 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{b^{5/2} (a-b)^2}+\frac{a^2 \sin (2 (e+f x))}{b^2 (a-b) ((a-b) \cos (2 (e+f x))+a+b)}-\frac{2 (e+f x)}{(a-b)^2}+\frac{2 \tan (e+f x)}{b^2}}{2 f}","-\frac{a^{3/2} (3 a-5 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 b^{5/2} f (a-b)^2}+\frac{(3 a-2 b) \tan (e+f x)}{2 b^2 f (a-b)}-\frac{a \tan ^3(e+f x)}{2 b f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{x}{(a-b)^2}",1,"((-2*(e + f*x))/(a - b)^2 - (a^(3/2)*(3*a - 5*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/((a - b)^2*b^(5/2)) + (a^2*Sin[2*(e + f*x)])/((a - b)*b^2*(a + b + (a - b)*Cos[2*(e + f*x)])) + (2*Tan[e + f*x])/b^2)/(2*f)","A",1
231,1,94,95,0.8164603,"\int \frac{\tan ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2,x]","\frac{\frac{\sqrt{a} (a-3 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{b^{3/2}}-\frac{a (a-b) \sin (2 (e+f x))}{b ((a-b) \cos (2 (e+f x))+a+b)}+2 (e+f x)}{2 f (a-b)^2}","\frac{\sqrt{a} (a-3 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 b^{3/2} f (a-b)^2}-\frac{a \tan (e+f x)}{2 b f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x}{(a-b)^2}",1,"(2*(e + f*x) + (Sqrt[a]*(a - 3*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/b^(3/2) - (a*(a - b)*Sin[2*(e + f*x)])/(b*(a + b + (a - b)*Cos[2*(e + f*x)])))/(2*(a - b)^2*f)","A",1
232,1,87,90,0.5879587,"\int \frac{\tan ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2)^2,x]","\frac{\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}+\frac{(a-b) \sin (2 (e+f x))}{(a-b) \cos (2 (e+f x))+a+b}-2 (e+f x)}{2 f (a-b)^2}","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b} f (a-b)^2}+\frac{\tan (e+f x)}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{x}{(a-b)^2}",1,"(-2*(e + f*x) + ((a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) + ((a - b)*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)]))/(2*(a - b)^2*f)","A",1
233,1,88,97,1.0435093,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[(a + b*Tan[e + f*x]^2)^(-2),x]","\frac{\frac{\sqrt{b} (b-3 a) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{3/2}}+\frac{b (b-a) \tan (e+f x)}{a \left(a+b \tan ^2(e+f x)\right)}+2 \tan ^{-1}(\tan (e+f x))}{2 f (a-b)^2}","-\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{3/2} f (a-b)^2}-\frac{b \tan (e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x}{(a-b)^2}",1,"(2*ArcTan[Tan[e + f*x]] + (Sqrt[b]*(-3*a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/a^(3/2) + (b*(-a + b)*Tan[e + f*x])/(a*(a + b*Tan[e + f*x]^2)))/(2*(a - b)^2*f)","A",1
234,1,117,128,3.2157786,"\int \frac{\cot ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2)^2,x]","\frac{\frac{b^{3/2} (5 a-3 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{5/2} (a-b)^2}+\frac{\frac{b^2 (a-b) \sin (2 (e+f x))}{a^2 ((a-b) \cos (2 (e+f x))+a+b)}-2 (e+f x)}{(a-b)^2}-\frac{2 \cot (e+f x)}{a^2}}{2 f}","\frac{b^{3/2} (5 a-3 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{5/2} f (a-b)^2}-\frac{(2 a-3 b) \cot (e+f x)}{2 a^2 f (a-b)}-\frac{b \cot (e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{x}{(a-b)^2}",1,"(((5*a - 3*b)*b^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(5/2)*(a - b)^2) - (2*Cot[e + f*x])/a^2 + (-2*(e + f*x) + ((a - b)*b^2*Sin[2*(e + f*x)])/(a^2*(a + b + (a - b)*Cos[2*(e + f*x)])))/(a - b)^2)/(2*f)","A",1
235,1,137,169,3.6013149,"\int \frac{\cot ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2,x]","\frac{\frac{3 b^{5/2} (5 b-7 a) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{7/2} (a-b)^2}+\frac{3 \left(2 (e+f x)-\frac{b^3 (a-b) \sin (2 (e+f x))}{a^3 ((a-b) \cos (2 (e+f x))+a+b)}\right)}{(a-b)^2}-\frac{2 \cot (e+f x) \left(a \csc ^2(e+f x)-4 a-6 b\right)}{a^3}}{6 f}","-\frac{b^{5/2} (7 a-5 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{7/2} f (a-b)^2}-\frac{(2 a-5 b) \cot ^3(e+f x)}{6 a^2 f (a-b)}+\frac{\left(2 a^2+2 a b-5 b^2\right) \cot (e+f x)}{2 a^3 f (a-b)}-\frac{b \cot ^3(e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x}{(a-b)^2}",1,"((3*b^(5/2)*(-7*a + 5*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(7/2)*(a - b)^2) - (2*Cot[e + f*x]*(-4*a - 6*b + a*Csc[e + f*x]^2))/a^3 + (3*(2*(e + f*x) - ((a - b)*b^3*Sin[2*(e + f*x)])/(a^3*(a + b + (a - b)*Cos[2*(e + f*x)]))))/(a - b)^2)/(6*f)","A",1
236,1,231,218,6.281678,"\int \frac{\cot ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Integrate[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^2,x]","\frac{b^{7/2} (9 a-7 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{9/2} f (a-b)^2}+\frac{b^4 \sin (2 (e+f x))}{2 a^4 f (a-b) (a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b)}+\frac{\csc ^3(e+f x) (11 a \cos (e+f x)+10 b \cos (e+f x))}{15 a^3 f}-\frac{\cot (e+f x) \csc ^4(e+f x)}{5 a^2 f}+\frac{\csc (e+f x) \left(-23 a^2 \cos (e+f x)-40 a b \cos (e+f x)-45 b^2 \cos (e+f x)\right)}{15 a^4 f}-\frac{e+f x}{f (a-b)^2}","\frac{b^{7/2} (9 a-7 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{9/2} f (a-b)^2}-\frac{(2 a-7 b) \cot ^5(e+f x)}{10 a^2 f (a-b)}+\frac{\left(2 a^2+2 a b-7 b^2\right) \cot ^3(e+f x)}{6 a^3 f (a-b)}-\frac{\left(2 a^3+2 a^2 b+2 a b^2-7 b^3\right) \cot (e+f x)}{2 a^4 f (a-b)}-\frac{b \cot ^5(e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{x}{(a-b)^2}",1,"-((e + f*x)/((a - b)^2*f)) + ((9*a - 7*b)*b^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(9/2)*(a - b)^2*f) + ((-23*a^2*Cos[e + f*x] - 40*a*b*Cos[e + f*x] - 45*b^2*Cos[e + f*x])*Csc[e + f*x])/(15*a^4*f) + ((11*a*Cos[e + f*x] + 10*b*Cos[e + f*x])*Csc[e + f*x]^3)/(15*a^3*f) - (Cot[e + f*x]*Csc[e + f*x]^4)/(5*a^2*f) + (b^4*Sin[2*(e + f*x)])/(2*a^4*(a - b)*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)]))","A",1
237,1,97,108,1.1562765,"\int \frac{\tan ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\frac{a^2 (a-b)^2}{b^2 \left(a+b \tan ^2(e+f x)\right)^2}-\frac{2 a (a-2 b) (a-b)}{b^2 \left(a+b \tan ^2(e+f x)\right)}-2 \log \left(a+b \tan ^2(e+f x)\right)-4 \log (\cos (e+f x))}{4 f (a-b)^3}","\frac{a^2}{4 b^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{a (a-2 b)}{2 b^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^3}",1,"(-4*Log[Cos[e + f*x]] - 2*Log[a + b*Tan[e + f*x]^2] + (a^2*(a - b)^2)/(b^2*(a + b*Tan[e + f*x]^2)^2) - (2*a*(a - 2*b)*(a - b))/(b^2*(a + b*Tan[e + f*x]^2)))/(4*(a - b)^3*f)","A",1
238,1,87,97,0.8141433,"\int \frac{\tan ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2)^3,x]","\frac{-\frac{a (a-b)^2}{b \left(a+b \tan ^2(e+f x)\right)^2}-\frac{2 (a-b)}{a+b \tan ^2(e+f x)}+2 \log \left(a+b \tan ^2(e+f x)\right)+4 \log (\cos (e+f x))}{4 f (a-b)^3}","-\frac{a}{4 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{1}{2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^3}",1,"(4*Log[Cos[e + f*x]] + 2*Log[a + b*Tan[e + f*x]^2] - (a*(a - b)^2)/(b*(a + b*Tan[e + f*x]^2)^2) - (2*(a - b))/(a + b*Tan[e + f*x]^2))/(4*(a - b)^3*f)","A",1
239,1,82,93,0.7475927,"\int \frac{\tan (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Tan[e + f*x]/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\frac{(a-b)^2}{\left(a+b \tan ^2(e+f x)\right)^2}+\frac{2 (a-b)}{a+b \tan ^2(e+f x)}-2 \log \left(a+b \tan ^2(e+f x)\right)-4 \log (\cos (e+f x))}{4 f (a-b)^3}","\frac{1}{2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{1}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^3}",1,"(-4*Log[Cos[e + f*x]] - 2*Log[a + b*Tan[e + f*x]^2] + (a - b)^2/(a + b*Tan[e + f*x]^2)^2 + (2*(a - b))/(a + b*Tan[e + f*x]^2))/(4*(a - b)^3*f)","A",1
240,1,126,148,1.8249917,"\int \frac{\cot (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Cot[e + f*x]/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\frac{\frac{b \left(2 \left(3 a^2-3 a b+b^2\right) \log \left(a+b \tan ^2(e+f x)\right)-\frac{a (a-b) \left(2 b (2 a-b) \tan ^2(e+f x)+a (5 a-3 b)\right)}{\left(a+b \tan ^2(e+f x)\right)^2}\right)}{(a-b)^3}+4 \log (\tan (e+f x))}{a^3}+\frac{4 \log (\cos (e+f x))}{(a-b)^3}}{4 f}","\frac{\log (\tan (e+f x))}{a^3 f}-\frac{b (2 a-b)}{2 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{b \left(3 a^2-3 a b+b^2\right) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^3 f (a-b)^3}-\frac{b}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{\log (\cos (e+f x))}{f (a-b)^3}",1,"((4*Log[Cos[e + f*x]])/(a - b)^3 + (4*Log[Tan[e + f*x]] + (b*(2*(3*a^2 - 3*a*b + b^2)*Log[a + b*Tan[e + f*x]^2] - (a*(a - b)*(a*(5*a - 3*b) + 2*(2*a - b)*b*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]^2)^2))/(a - b)^3)/a^3)/(4*f)","A",1
241,1,144,181,2.0629759,"\int \frac{\cot ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{-\frac{b^4}{2 a^4 (a-b) \left(a \cot ^2(e+f x)+b\right)^2}+\frac{b^3 (4 a-3 b)}{a^4 (a-b)^2 \left(a \cot ^2(e+f x)+b\right)}+\frac{\cot ^2(e+f x)}{a^3}+\frac{b^2 \left(6 a^2-8 a b+3 b^2\right) \log \left(a \cot ^2(e+f x)+b\right)}{a^4 (a-b)^3}+\frac{2 \log (\sin (e+f x))}{(a-b)^3}}{2 f}","-\frac{(a+3 b) \log (\tan (e+f x))}{a^4 f}+\frac{b^2 (3 a-2 b)}{2 a^3 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{\cot ^2(e+f x)}{2 a^3 f}+\frac{b^2}{4 a^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{b^2 \left(6 a^2-8 a b+3 b^2\right) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^4 f (a-b)^3}-\frac{\log (\cos (e+f x))}{f (a-b)^3}",1,"-1/2*(Cot[e + f*x]^2/a^3 - b^4/(2*a^4*(a - b)*(b + a*Cot[e + f*x]^2)^2) + ((4*a - 3*b)*b^3)/(a^4*(a - b)^2*(b + a*Cot[e + f*x]^2)) + (b^2*(6*a^2 - 8*a*b + 3*b^2)*Log[b + a*Cot[e + f*x]^2])/(a^4*(a - b)^3) + (2*Log[Sin[e + f*x]])/(a - b)^3)/f","A",1
242,1,178,210,2.6536529,"\int \frac{\cot ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\frac{(a+3 b) \cot ^2(e+f x)}{a^4}-\frac{\cot ^4(e+f x)}{2 a^3}+\frac{4 \left(a^2+3 a b+6 b^2\right) \log (\tan (e+f x))+\frac{b^3 \left(2 \left(10 a^2-15 a b+6 b^2\right) \log \left(a+b \tan ^2(e+f x)\right)-\frac{a (a-b) \left(2 b (4 a-3 b) \tan ^2(e+f x)+a (9 a-7 b)\right)}{\left(a+b \tan ^2(e+f x)\right)^2}\right)}{(a-b)^3}}{2 a^5}+\frac{2 \log (\cos (e+f x))}{(a-b)^3}}{2 f}","-\frac{b^3 (4 a-3 b)}{2 a^4 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{(a+3 b) \cot ^2(e+f x)}{2 a^4 f}-\frac{b^3}{4 a^3 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{\cot ^4(e+f x)}{4 a^3 f}+\frac{\left(a^2+3 a b+6 b^2\right) \log (\tan (e+f x))}{a^5 f}+\frac{b^3 \left(10 a^2-15 a b+6 b^2\right) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^5 f (a-b)^3}+\frac{\log (\cos (e+f x))}{f (a-b)^3}",1,"(((a + 3*b)*Cot[e + f*x]^2)/a^4 - Cot[e + f*x]^4/(2*a^3) + (2*Log[Cos[e + f*x]])/(a - b)^3 + (4*(a^2 + 3*a*b + 6*b^2)*Log[Tan[e + f*x]] + (b^3*(2*(10*a^2 - 15*a*b + 6*b^2)*Log[a + b*Tan[e + f*x]^2] - (a*(a - b)*(a*(9*a - 7*b) + 2*(4*a - 3*b)*b*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]^2)^2))/(a - b)^3)/(2*a^5))/(2*f)","A",1
243,1,142,153,2.374227,"\int \frac{\tan ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2)^3,x]","\frac{-\frac{a (a-b) \sin (2 (e+f x)) \left(3 \left(a^2-4 a b+3 b^2\right) \cos (2 (e+f x))+3 a^2-2 a b-9 b^2\right)}{b^2 ((a-b) \cos (2 (e+f x))+a+b)^2}+\frac{\sqrt{a} \left(3 a^2-10 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{b^{5/2}}-8 (e+f x)}{8 f (a-b)^3}","\frac{\sqrt{a} \left(3 a^2-10 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 b^{5/2} f (a-b)^3}-\frac{a (3 a-7 b) \tan (e+f x)}{8 b^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{a \tan ^3(e+f x)}{4 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{x}{(a-b)^3}",1,"(-8*(e + f*x) + (Sqrt[a]*(3*a^2 - 10*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/b^(5/2) - (a*(a - b)*(3*a^2 - 2*a*b - 9*b^2 + 3*(a^2 - 4*a*b + 3*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(b^2*(a + b + (a - b)*Cos[2*(e + f*x)])^2))/(8*(a - b)^3*f)","A",1
244,1,136,145,2.0858933,"\int \frac{\tan ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^3,x]","\frac{-\frac{(a-b) \sin (2 (e+f x)) \left(\left(a^2+4 a b-5 b^2\right) \cos (2 (e+f x))+a^2+2 a b+5 b^2\right)}{b ((a-b) \cos (2 (e+f x))+a+b)^2}+\frac{\left(a^2-6 a b-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}+8 (e+f x)}{8 f (a-b)^3}","\frac{\left(a^2-6 a b-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 \sqrt{a} b^{3/2} f (a-b)^3}+\frac{(a-5 b) \tan (e+f x)}{8 b f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{a \tan (e+f x)}{4 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x}{(a-b)^3}",1,"(8*(e + f*x) + ((a^2 - 6*a*b - 3*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2)) - ((a - b)*(a^2 + 2*a*b + 5*b^2 + (a^2 + 4*a*b - 5*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(b*(a + b + (a - b)*Cos[2*(e + f*x)])^2))/(8*(a - b)^3*f)","A",1
245,1,139,144,2.1129177,"\int \frac{\tan ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\frac{(a-b) \sin (2 (e+f x)) \left(\left(5 a^2-4 a b-b^2\right) \cos (2 (e+f x))+5 a^2+2 a b+b^2\right)}{a ((a-b) \cos (2 (e+f x))+a+b)^2}+\frac{\left(3 a^2+6 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{3/2} \sqrt{b}}-8 (e+f x)}{8 f (a-b)^3}","\frac{\left(3 a^2+6 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{3/2} \sqrt{b} f (a-b)^3}+\frac{(3 a+b) \tan (e+f x)}{8 a f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{\tan (e+f x)}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{x}{(a-b)^3}",1,"(-8*(e + f*x) + ((3*a^2 + 6*a*b - b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(3/2)*Sqrt[b]) + ((a - b)*(5*a^2 + 2*a*b + b^2 + (5*a^2 - 4*a*b - b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(a*(a + b + (a - b)*Cos[2*(e + f*x)])^2))/(8*(a - b)^3*f)","A",1
246,1,138,150,1.9368295,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[(a + b*Tan[e + f*x]^2)^(-3),x]","-\frac{\frac{b (7 a-3 b) (a-b) \tan (e+f x)}{a^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{5/2}}+\frac{2 b (a-b)^2 \tan (e+f x)}{a \left(a+b \tan ^2(e+f x)\right)^2}-8 \tan ^{-1}(\tan (e+f x))}{8 f (a-b)^3}","-\frac{b (7 a-3 b) \tan (e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{5/2} f (a-b)^3}-\frac{b \tan (e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x}{(a-b)^3}",1,"-1/8*(-8*ArcTan[Tan[e + f*x]] + (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/a^(5/2) + (2*(a - b)^2*b*Tan[e + f*x])/(a*(a + b*Tan[e + f*x]^2)^2) + ((7*a - 3*b)*(a - b)*b*Tan[e + f*x])/(a^2*(a + b*Tan[e + f*x]^2)))/((a - b)^3*f)","A",1
247,1,174,189,2.3950682,"\int \frac{\cot ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\frac{b^2 (13 a-7 b) \sin (2 (e+f x))}{a^3 (a-b)^2 ((a-b) \cos (2 (e+f x))+a+b)}-\frac{8 \cot (e+f x)}{a^3}-\frac{4 b^3 \sin (2 (e+f x))}{a^2 (a-b)^2 ((a-b) \cos (2 (e+f x))+a+b)^2}+\frac{b^{3/2} \left(35 a^2-42 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{7/2} (a-b)^3}+\frac{8 (e+f x)}{(b-a)^3}}{8 f}","-\frac{b (9 a-5 b) \cot (e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{b^{3/2} \left(35 a^2-42 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{7/2} f (a-b)^3}-\frac{\left(8 a^2-27 a b+15 b^2\right) \cot (e+f x)}{8 a^3 f (a-b)^2}-\frac{b \cot (e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{x}{(a-b)^3}",1,"((8*(e + f*x))/(-a + b)^3 + (b^(3/2)*(35*a^2 - 42*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(7/2)*(a - b)^3) - (8*Cot[e + f*x])/a^3 - (4*b^3*Sin[2*(e + f*x)])/(a^2*(a - b)^2*(a + b + (a - b)*Cos[2*(e + f*x)])^2) + ((13*a - 7*b)*b^2*Sin[2*(e + f*x)])/(a^3*(a - b)^2*(a + b + (a - b)*Cos[2*(e + f*x)])))/(8*f)","A",1
248,1,184,240,4.9778661,"\int \frac{\cot ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^3,x]","\frac{-\frac{8 \cot (e+f x) \left(a \csc ^2(e+f x)-4 a-9 b\right)}{a^4}-\frac{3 b^{5/2} \left(63 a^2-90 a b+35 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{9/2} (a-b)^3}+\frac{3 \left(8 (e+f x)-\frac{b^3 (a-b) \sin (2 (e+f x)) \left(\left(17 a^2-28 a b+11 b^2\right) \cos (2 (e+f x))+17 a^2+2 a b-11 b^2\right)}{a^4 ((a-b) \cos (2 (e+f x))+a+b)^2}\right)}{(a-b)^3}}{24 f}","-\frac{b (11 a-7 b) \cot ^3(e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{b^{5/2} \left(63 a^2-90 a b+35 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{9/2} f (a-b)^3}-\frac{\left(8 a^2-55 a b+35 b^2\right) \cot ^3(e+f x)}{24 a^3 f (a-b)^2}+\frac{\left(8 a^3+8 a^2 b-55 a b^2+35 b^3\right) \cot (e+f x)}{8 a^4 f (a-b)^2}-\frac{b \cot ^3(e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x}{(a-b)^3}",1,"((-3*b^(5/2)*(63*a^2 - 90*a*b + 35*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(9/2)*(a - b)^3) - (8*Cot[e + f*x]*(-4*a - 9*b + a*Csc[e + f*x]^2))/a^4 + (3*(8*(e + f*x) - ((a - b)*b^3*(17*a^2 + 2*a*b - 11*b^2 + (17*a^2 - 28*a*b + 11*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(a^4*(a + b + (a - b)*Cos[2*(e + f*x)])^2)))/(a - b)^3)/(24*f)","A",1
249,1,949,297,6.3734939,"\int \frac{\cot ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Integrate[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\left(-3184 \cos (e+f x) a^7-1536 \cos (3 (e+f x)) a^7-704 \cos (5 (e+f x)) a^7-536 \cos (7 (e+f x)) a^7-184 \cos (9 (e+f x)) a^7-720 (e+f x) \sin (e+f x) a^7-480 (e+f x) \sin (3 (e+f x)) a^7+480 (e+f x) \sin (5 (e+f x)) a^7+120 (e+f x) \sin (7 (e+f x)) a^7-120 (e+f x) \sin (9 (e+f x)) a^7+7440 b \cos (e+f x) a^6+7648 b \cos (3 (e+f x)) a^6+2656 b \cos (5 (e+f x)) a^6+248 b \cos (7 (e+f x)) a^6+440 b \cos (9 (e+f x)) a^6-3360 b (e+f x) \sin (e+f x) a^6+1920 b (e+f x) \sin (5 (e+f x)) a^6-1200 b (e+f x) \sin (7 (e+f x)) a^6+240 b (e+f x) \sin (9 (e+f x)) a^6-12000 b^2 \cos (e+f x) a^5-2912 b^2 \cos (3 (e+f x)) a^5-4128 b^2 \cos (5 (e+f x)) a^5+768 b^2 \cos (7 (e+f x)) a^5-160 b^2 \cos (9 (e+f x)) a^5-15120 b^2 (e+f x) \sin (e+f x) a^5+10080 b^2 (e+f x) \sin (3 (e+f x)) a^5-4320 b^2 (e+f x) \sin (5 (e+f x)) a^5+1080 b^2 (e+f x) \sin (7 (e+f x)) a^5-120 b^2 (e+f x) \sin (9 (e+f x)) a^5+10240 b^3 \cos (e+f x) a^4-1152 b^3 \cos (3 (e+f x)) a^4-3712 b^3 \cos (5 (e+f x)) a^4+128 b^3 \cos (7 (e+f x)) a^4+640 b^3 \cos (9 (e+f x)) a^4+6450 b^4 \cos (e+f x) a^3-14872 b^4 \cos (3 (e+f x)) a^3+5504 b^4 \cos (5 (e+f x)) a^3+6553 b^4 \cos (7 (e+f x)) a^3-3635 b^4 \cos (9 (e+f x)) a^3+714 b^5 \cos (e+f x) a^2-12796 b^5 \cos (3 (e+f x)) a^2+27684 b^5 \cos (5 (e+f x)) a^2-21441 b^5 \cos (7 (e+f x)) a^2+5839 b^5 \cos (9 (e+f x)) a^2-22890 b^6 \cos (e+f x) a+52080 b^6 \cos (3 (e+f x)) a-46200 b^6 \cos (5 (e+f x)) a+20895 b^6 \cos (7 (e+f x)) a-3885 b^6 \cos (9 (e+f x)) a+13230 b^7 \cos (e+f x)-26460 b^7 \cos (3 (e+f x))+18900 b^7 \cos (5 (e+f x))-6615 b^7 \cos (7 (e+f x))+945 b^7 \cos (9 (e+f x))\right) \csc ^5(e+f x)}{7680 a^5 (a-b)^3 f (\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x)))^2}+\frac{b^{7/2} \left(99 a^2-154 b a+63 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{11/2} (a-b)^3 f}","-\frac{b (13 a-9 b) \cot ^5(e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{b^{7/2} \left(99 a^2-154 a b+63 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{11/2} f (a-b)^3}-\frac{\left(8 a^2-91 a b+63 b^2\right) \cot ^5(e+f x)}{40 a^3 f (a-b)^2}+\frac{\left(8 a^3+8 a^2 b-91 a b^2+63 b^3\right) \cot ^3(e+f x)}{24 a^4 f (a-b)^2}-\frac{\left(8 a^4+8 a^3 b+8 a^2 b^2-91 a b^3+63 b^4\right) \cot (e+f x)}{8 a^5 f (a-b)^2}-\frac{b \cot ^5(e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{x}{(a-b)^3}",1,"(b^(7/2)*(99*a^2 - 154*a*b + 63*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(11/2)*(a - b)^3*f) + (Csc[e + f*x]^5*(-3184*a^7*Cos[e + f*x] + 7440*a^6*b*Cos[e + f*x] - 12000*a^5*b^2*Cos[e + f*x] + 10240*a^4*b^3*Cos[e + f*x] + 6450*a^3*b^4*Cos[e + f*x] + 714*a^2*b^5*Cos[e + f*x] - 22890*a*b^6*Cos[e + f*x] + 13230*b^7*Cos[e + f*x] - 1536*a^7*Cos[3*(e + f*x)] + 7648*a^6*b*Cos[3*(e + f*x)] - 2912*a^5*b^2*Cos[3*(e + f*x)] - 1152*a^4*b^3*Cos[3*(e + f*x)] - 14872*a^3*b^4*Cos[3*(e + f*x)] - 12796*a^2*b^5*Cos[3*(e + f*x)] + 52080*a*b^6*Cos[3*(e + f*x)] - 26460*b^7*Cos[3*(e + f*x)] - 704*a^7*Cos[5*(e + f*x)] + 2656*a^6*b*Cos[5*(e + f*x)] - 4128*a^5*b^2*Cos[5*(e + f*x)] - 3712*a^4*b^3*Cos[5*(e + f*x)] + 5504*a^3*b^4*Cos[5*(e + f*x)] + 27684*a^2*b^5*Cos[5*(e + f*x)] - 46200*a*b^6*Cos[5*(e + f*x)] + 18900*b^7*Cos[5*(e + f*x)] - 536*a^7*Cos[7*(e + f*x)] + 248*a^6*b*Cos[7*(e + f*x)] + 768*a^5*b^2*Cos[7*(e + f*x)] + 128*a^4*b^3*Cos[7*(e + f*x)] + 6553*a^3*b^4*Cos[7*(e + f*x)] - 21441*a^2*b^5*Cos[7*(e + f*x)] + 20895*a*b^6*Cos[7*(e + f*x)] - 6615*b^7*Cos[7*(e + f*x)] - 184*a^7*Cos[9*(e + f*x)] + 440*a^6*b*Cos[9*(e + f*x)] - 160*a^5*b^2*Cos[9*(e + f*x)] + 640*a^4*b^3*Cos[9*(e + f*x)] - 3635*a^3*b^4*Cos[9*(e + f*x)] + 5839*a^2*b^5*Cos[9*(e + f*x)] - 3885*a*b^6*Cos[9*(e + f*x)] + 945*b^7*Cos[9*(e + f*x)] - 720*a^7*(e + f*x)*Sin[e + f*x] - 3360*a^6*b*(e + f*x)*Sin[e + f*x] - 15120*a^5*b^2*(e + f*x)*Sin[e + f*x] - 480*a^7*(e + f*x)*Sin[3*(e + f*x)] + 10080*a^5*b^2*(e + f*x)*Sin[3*(e + f*x)] + 480*a^7*(e + f*x)*Sin[5*(e + f*x)] + 1920*a^6*b*(e + f*x)*Sin[5*(e + f*x)] - 4320*a^5*b^2*(e + f*x)*Sin[5*(e + f*x)] + 120*a^7*(e + f*x)*Sin[7*(e + f*x)] - 1200*a^6*b*(e + f*x)*Sin[7*(e + f*x)] + 1080*a^5*b^2*(e + f*x)*Sin[7*(e + f*x)] - 120*a^7*(e + f*x)*Sin[9*(e + f*x)] + 240*a^6*b*(e + f*x)*Sin[9*(e + f*x)] - 120*a^5*b^2*(e + f*x)*Sin[9*(e + f*x)]))/(7680*a^5*(a - b)^3*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])^2)","B",1
250,1,137,115,1.9137773,"\int \left(a+b \tan ^2(c+d x)\right)^4 \, dx","Integrate[(a + b*Tan[c + d*x]^2)^4,x]","\frac{\tan (c+d x) \left(b \left(35 b \left(6 a^2-4 a b+b^2\right) \tan ^2(c+d x)+105 \left(4 a^3-6 a^2 b+4 a b^2-b^3\right)+21 b^2 (4 a-b) \tan ^4(c+d x)+15 b^3 \tan ^6(c+d x)\right)+\frac{105 (a-b)^4 \tanh ^{-1}\left(\sqrt{-\tan ^2(c+d x)}\right)}{\sqrt{-\tan ^2(c+d x)}}\right)}{105 d}","\frac{b^2 \left(6 a^2-4 a b+b^2\right) \tan ^3(c+d x)}{3 d}+\frac{b (2 a-b) \left(2 a^2-2 a b+b^2\right) \tan (c+d x)}{d}+\frac{b^3 (4 a-b) \tan ^5(c+d x)}{5 d}+x (a-b)^4+\frac{b^4 \tan ^7(c+d x)}{7 d}",1,"(Tan[c + d*x]*((105*(a - b)^4*ArcTanh[Sqrt[-Tan[c + d*x]^2]])/Sqrt[-Tan[c + d*x]^2] + b*(105*(4*a^3 - 6*a^2*b + 4*a*b^2 - b^3) + 35*b*(6*a^2 - 4*a*b + b^2)*Tan[c + d*x]^2 + 21*(4*a - b)*b^2*Tan[c + d*x]^4 + 15*b^3*Tan[c + d*x]^6)))/(105*d)","A",1
251,1,102,77,0.92751,"\int \left(a+b \tan ^2(c+d x)\right)^3 \, dx","Integrate[(a + b*Tan[c + d*x]^2)^3,x]","\frac{\tan (c+d x) \left(b \left(45 a^2-15 a b \left(3-\tan ^2(c+d x)\right)+b^2 \left(3 \tan ^4(c+d x)-5 \tan ^2(c+d x)+15\right)\right)+\frac{15 (a-b)^3 \tanh ^{-1}\left(\sqrt{-\tan ^2(c+d x)}\right)}{\sqrt{-\tan ^2(c+d x)}}\right)}{15 d}","\frac{b \left(3 a^2-3 a b+b^2\right) \tan (c+d x)}{d}+\frac{b^2 (3 a-b) \tan ^3(c+d x)}{3 d}+x (a-b)^3+\frac{b^3 \tan ^5(c+d x)}{5 d}",1,"(Tan[c + d*x]*((15*(a - b)^3*ArcTanh[Sqrt[-Tan[c + d*x]^2]])/Sqrt[-Tan[c + d*x]^2] + b*(45*a^2 - 15*a*b*(3 - Tan[c + d*x]^2) + b^2*(15 - 5*Tan[c + d*x]^2 + 3*Tan[c + d*x]^4))))/(15*d)","A",1
252,1,73,46,0.596684,"\int \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Integrate[(a + b*Tan[c + d*x]^2)^2,x]","\frac{\tan (c+d x) \left(b \left(6 a-b \left(3-\tan ^2(c+d x)\right)\right)+\frac{3 (a-b)^2 \tanh ^{-1}\left(\sqrt{-\tan ^2(c+d x)}\right)}{\sqrt{-\tan ^2(c+d x)}}\right)}{3 d}","\frac{b (2 a-b) \tan (c+d x)}{d}+x (a-b)^2+\frac{b^2 \tan ^3(c+d x)}{3 d}",1,"(Tan[c + d*x]*((3*(a - b)^2*ArcTanh[Sqrt[-Tan[c + d*x]^2]])/Sqrt[-Tan[c + d*x]^2] + b*(6*a - b*(3 - Tan[c + d*x]^2))))/(3*d)","A",1
253,1,28,19,0.0069398,"\int \left(a+b \tan ^2(c+d x)\right) \, dx","Integrate[a + b*Tan[c + d*x]^2,x]","a x-\frac{b \tan ^{-1}(\tan (c+d x))}{d}+\frac{b \tan (c+d x)}{d}","a x+\frac{b \tan (c+d x)}{d}-b x",1,"a*x - (b*ArcTan[Tan[c + d*x]])/d + (b*Tan[c + d*x])/d","A",1
254,1,49,50,0.076186,"\int \frac{1}{a+b \tan ^2(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x]^2)^(-1),x]","\frac{\tan ^{-1}(\tan (c+d x))-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a}}}{a d-b d}","\frac{x}{a-b}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)}",1,"(ArcTan[Tan[c + d*x]] - (Sqrt[b]*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/Sqrt[a])/(a*d - b*d)","A",1
255,1,88,97,1.0466587,"\int \frac{1}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Integrate[(a + b*Tan[c + d*x]^2)^(-2),x]","\frac{\frac{\sqrt{b} (b-3 a) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{3/2}}+\frac{b (b-a) \tan (c+d x)}{a \left(a+b \tan ^2(c+d x)\right)}+2 \tan ^{-1}(\tan (c+d x))}{2 d (a-b)^2}","-\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^2}-\frac{b \tan (c+d x)}{2 a d (a-b) \left(a+b \tan ^2(c+d x)\right)}+\frac{x}{(a-b)^2}",1,"(2*ArcTan[Tan[c + d*x]] + (Sqrt[b]*(-3*a + b)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/a^(3/2) + (b*(-a + b)*Tan[c + d*x])/(a*(a + b*Tan[c + d*x]^2)))/(2*(a - b)^2*d)","A",1
256,1,138,150,1.9288109,"\int \frac{1}{\left(a+b \tan ^2(c+d x)\right)^3} \, dx","Integrate[(a + b*Tan[c + d*x]^2)^(-3),x]","-\frac{\frac{b (7 a-3 b) (a-b) \tan (c+d x)}{a^2 \left(a+b \tan ^2(c+d x)\right)}+\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{5/2}}+\frac{2 b (a-b)^2 \tan (c+d x)}{a \left(a+b \tan ^2(c+d x)\right)^2}-8 \tan ^{-1}(\tan (c+d x))}{8 d (a-b)^3}","-\frac{b (7 a-3 b) \tan (c+d x)}{8 a^2 d (a-b)^2 \left(a+b \tan ^2(c+d x)\right)}-\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a-b)^3}-\frac{b \tan (c+d x)}{4 a d (a-b) \left(a+b \tan ^2(c+d x)\right)^2}+\frac{x}{(a-b)^3}",1,"-1/8*(-8*ArcTan[Tan[c + d*x]] + (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/a^(5/2) + (2*(a - b)^2*b*Tan[c + d*x])/(a*(a + b*Tan[c + d*x]^2)^2) + ((7*a - 3*b)*(a - b)*b*Tan[c + d*x])/(a^2*(a + b*Tan[c + d*x]^2)))/((a - b)^3*d)","A",1
257,1,32,54,0.0770512,"\int \tan ^4(x) \sqrt{a+a \tan ^2(x)} \, dx","Integrate[Tan[x]^4*Sqrt[a + a*Tan[x]^2],x]","\frac{1}{8} \sqrt{a \sec ^2(x)} \left(2 \tan ^3(x)-3 \tan (x)+3 \cos (x) \tanh ^{-1}(\sin (x))\right)","\frac{1}{4} \tan ^3(x) \sqrt{a \sec ^2(x)}-\frac{3}{8} \tan (x) \sqrt{a \sec ^2(x)}+\frac{3}{8} \cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x))",1,"(Sqrt[a*Sec[x]^2]*(3*ArcTanh[Sin[x]]*Cos[x] - 3*Tan[x] + 2*Tan[x]^3))/8","A",1
258,1,20,30,0.0259686,"\int \tan ^3(x) \sqrt{a+a \tan ^2(x)} \, dx","Integrate[Tan[x]^3*Sqrt[a + a*Tan[x]^2],x]","\frac{1}{3} \left(\sec ^2(x)-3\right) \sqrt{a \sec ^2(x)}","\frac{\left(a \sec ^2(x)\right)^{3/2}}{3 a}-\sqrt{a \sec ^2(x)}",1,"(Sqrt[a*Sec[x]^2]*(-3 + Sec[x]^2))/3","A",1
259,1,24,36,0.0487055,"\int \tan ^2(x) \sqrt{a+a \tan ^2(x)} \, dx","Integrate[Tan[x]^2*Sqrt[a + a*Tan[x]^2],x]","\frac{1}{2} \sqrt{a \sec ^2(x)} \left(\tan (x)-\cos (x) \tanh ^{-1}(\sin (x))\right)","\frac{1}{2} \tan (x) \sqrt{a \sec ^2(x)}-\frac{1}{2} \cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x))",1,"(Sqrt[a*Sec[x]^2]*(-(ArcTanh[Sin[x]]*Cos[x]) + Tan[x]))/2","A",1
260,1,10,10,0.0087379,"\int \tan (x) \sqrt{a+a \tan ^2(x)} \, dx","Integrate[Tan[x]*Sqrt[a + a*Tan[x]^2],x]","\sqrt{a \sec ^2(x)}","\sqrt{a \sec ^2(x)}",1,"Sqrt[a*Sec[x]^2]","A",1
261,1,30,24,0.0194812,"\int \cot (x) \sqrt{a+a \tan ^2(x)} \, dx","Integrate[Cot[x]*Sqrt[a + a*Tan[x]^2],x]","\cos (x) \sqrt{a \sec ^2(x)} \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)","-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)",1,"Cos[x]*(-Log[Cos[x/2]] + Log[Sin[x/2]])*Sqrt[a*Sec[x]^2]","A",1
262,1,14,14,0.0140997,"\int \cot ^2(x) \sqrt{a+a \tan ^2(x)} \, dx","Integrate[Cot[x]^2*Sqrt[a + a*Tan[x]^2],x]","-\cot (x) \sqrt{a \sec ^2(x)}","-\cot (x) \sqrt{a \sec ^2(x)}",1,"-(Cot[x]*Sqrt[a*Sec[x]^2])","A",1
263,1,38,45,0.0843622,"\int \cot ^3(x) \sqrt{a+a \tan ^2(x)} \, dx","Integrate[Cot[x]^3*Sqrt[a + a*Tan[x]^2],x]","-\frac{1}{2} \cos (x) \sqrt{a \sec ^2(x)} \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)+\cot (x) \csc (x)\right)","\frac{1}{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)-\frac{1}{2} \cot ^2(x) \sqrt{a \sec ^2(x)}",1,"-1/2*(Cos[x]*(Cot[x]*Csc[x] - Log[Cos[x/2]] + Log[Sin[x/2]])*Sqrt[a*Sec[x]^2])","A",1
264,1,22,34,0.0231052,"\int \cot ^4(x) \sqrt{a+a \tan ^2(x)} \, dx","Integrate[Cot[x]^4*Sqrt[a + a*Tan[x]^2],x]","-\frac{1}{3} \cot (x) \left(\csc ^2(x)-3\right) \sqrt{a \sec ^2(x)}","\cot (x) \sqrt{a \sec ^2(x)}-\frac{1}{3} \cot (x) \csc ^2(x) \sqrt{a \sec ^2(x)}",1,"-1/3*(Cot[x]*(-3 + Csc[x]^2)*Sqrt[a*Sec[x]^2])","A",1
265,1,31,36,0.0304478,"\int \sqrt{a+a \tan ^2(c+d x)} \, dx","Integrate[Sqrt[a + a*Tan[c + d*x]^2],x]","\frac{\cos (c+d x) \sqrt{a \sec ^2(c+d x)} \tanh ^{-1}(\sin (c+d x))}{d}","\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec ^2(c+d x)}}\right)}{d}",1,"(ArcTanh[Sin[c + d*x]]*Cos[c + d*x]*Sqrt[a*Sec[c + d*x]^2])/d","A",1
266,1,22,32,0.0489229,"\int \tan ^3(x) \left(a+a \tan ^2(x)\right)^{3/2} \, dx","Integrate[Tan[x]^3*(a + a*Tan[x]^2)^(3/2),x]","\frac{1}{15} \left(3 \sec ^2(x)-5\right) \left(a \sec ^2(x)\right)^{3/2}","\frac{\left(a \sec ^2(x)\right)^{5/2}}{5 a}-\frac{1}{3} \left(a \sec ^2(x)\right)^{3/2}",1,"((a*Sec[x]^2)^(3/2)*(-5 + 3*Sec[x]^2))/15","A",1
267,1,34,59,0.0671138,"\int \tan ^2(x) \left(a+a \tan ^2(x)\right)^{3/2} \, dx","Integrate[Tan[x]^2*(a + a*Tan[x]^2)^(3/2),x]","\frac{1}{8} \left(a \sec ^2(x)\right)^{3/2} \left(2 \tan (x)-\sin (x) \cos (x)+\cos ^3(x) \left(-\tanh ^{-1}(\sin (x))\right)\right)","\frac{1}{4} a \tan (x) \sec ^2(x) \sqrt{a \sec ^2(x)}-\frac{1}{8} a \tan (x) \sqrt{a \sec ^2(x)}-\frac{1}{8} a \cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x))",1,"((a*Sec[x]^2)^(3/2)*(-(ArcTanh[Sin[x]]*Cos[x]^3) - Cos[x]*Sin[x] + 2*Tan[x]))/8","A",1
268,1,14,14,0.0145911,"\int \tan (x) \left(a+a \tan ^2(x)\right)^{3/2} \, dx","Integrate[Tan[x]*(a + a*Tan[x]^2)^(3/2),x]","\frac{1}{3} \left(a \sec ^2(x)\right)^{3/2}","\frac{1}{3} \left(a \sec ^2(x)\right)^{3/2}",1,"(a*Sec[x]^2)^(3/2)/3","A",1
269,1,34,37,0.0368611,"\int \cot (x) \left(a+a \tan ^2(x)\right)^{3/2} \, dx","Integrate[Cot[x]*(a + a*Tan[x]^2)^(3/2),x]","a \sqrt{a \sec ^2(x)} \left(\cos (x) \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)+1\right)","a \sqrt{a \sec ^2(x)}-a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)",1,"a*(1 + Cos[x]*(-Log[Cos[x/2]] + Log[Sin[x/2]]))*Sqrt[a*Sec[x]^2]","A",1
270,1,27,33,0.0268953,"\int \cot ^2(x) \left(a+a \tan ^2(x)\right)^{3/2} \, dx","Integrate[Cot[x]^2*(a + a*Tan[x]^2)^(3/2),x]","-a \cot (x) \sqrt{a \sec ^2(x)} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\sin ^2(x)\right)","a \cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x))-a \cot (x) \sqrt{a \sec ^2(x)}",1,"-(a*Cot[x]*Hypergeometric2F1[-1/2, 1, 1/2, Sin[x]^2]*Sqrt[a*Sec[x]^2])","C",1
271,1,43,68,0.0696931,"\int \left(a+a \tan ^2(c+d x)\right)^{3/2} \, dx","Integrate[(a + a*Tan[c + d*x]^2)^(3/2),x]","\frac{a \sqrt{a \sec ^2(c+d x)} \left(\tan (c+d x)+\cos (c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{2 d}","\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec ^2(c+d x)}}\right)}{2 d}+\frac{a \tan (c+d x) \sqrt{a \sec ^2(c+d x)}}{2 d}",1,"(a*Sqrt[a*Sec[c + d*x]^2]*(ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + Tan[c + d*x]))/(2*d)","A",1
272,1,65,98,0.2207324,"\int \left(a+a \tan ^2(c+d x)\right)^{5/2} \, dx","Integrate[(a + a*Tan[c + d*x]^2)^(5/2),x]","\frac{a^2 \cos (c+d x) \sqrt{a \sec ^2(c+d x)} \left(3 \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left(2 \sec ^2(c+d x)+3\right)\right)}{8 d}","\frac{3 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec ^2(c+d x)}}\right)}{8 d}+\frac{3 a^2 \tan (c+d x) \sqrt{a \sec ^2(c+d x)}}{8 d}+\frac{a \tan (c+d x) \left(a \sec ^2(c+d x)\right)^{3/2}}{4 d}",1,"(a^2*Cos[c + d*x]*Sqrt[a*Sec[c + d*x]^2]*(3*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(3 + 2*Sec[c + d*x]^2)*Tan[c + d*x]))/(8*d)","A",1
273,1,17,25,0.0247644,"\int \frac{\tan ^3(x)}{\sqrt{a+a \tan ^2(x)}} \, dx","Integrate[Tan[x]^3/Sqrt[a + a*Tan[x]^2],x]","\frac{\sec ^2(x)+1}{\sqrt{a \sec ^2(x)}}","\frac{\sqrt{a \sec ^2(x)}}{a}+\frac{1}{\sqrt{a \sec ^2(x)}}",1,"(1 + Sec[x]^2)/Sqrt[a*Sec[x]^2]","A",1
274,1,49,31,0.0378127,"\int \frac{\tan ^2(x)}{\sqrt{a+a \tan ^2(x)}} \, dx","Integrate[Tan[x]^2/Sqrt[a + a*Tan[x]^2],x]","-\frac{\sec (x) \left(\sin (x)+\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)\right)}{\sqrt{a \sec ^2(x)}}","\frac{\sec (x) \tanh ^{-1}(\sin (x))}{\sqrt{a \sec ^2(x)}}-\frac{\tan (x)}{\sqrt{a \sec ^2(x)}}",1,"-((Sec[x]*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]] + Sin[x]))/Sqrt[a*Sec[x]^2])","A",1
275,1,12,12,0.0126496,"\int \frac{\tan (x)}{\sqrt{a+a \tan ^2(x)}} \, dx","Integrate[Tan[x]/Sqrt[a + a*Tan[x]^2],x]","-\frac{1}{\sqrt{a \sec ^2(x)}}","-\frac{1}{\sqrt{a \sec ^2(x)}}",1,"-(1/Sqrt[a*Sec[x]^2])","A",1
276,1,32,35,0.0335087,"\int \frac{\cot (x)}{\sqrt{a+a \tan ^2(x)}} \, dx","Integrate[Cot[x]/Sqrt[a + a*Tan[x]^2],x]","\frac{\sec (x) \left(\cos (x)+\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)}{\sqrt{a \sec ^2(x)}}","\frac{1}{\sqrt{a \sec ^2(x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"((Cos[x] - Log[Cos[x/2]] + Log[Sin[x/2]])*Sec[x])/Sqrt[a*Sec[x]^2]","A",1
277,1,22,31,0.0280583,"\int \frac{\cot ^2(x)}{\sqrt{a+a \tan ^2(x)}} \, dx","Integrate[Cot[x]^2/Sqrt[a + a*Tan[x]^2],x]","\frac{-\tan (x)-\csc (x) \sec (x)}{\sqrt{a \sec ^2(x)}}","-\frac{\csc (x) \sec (x)}{\sqrt{a \sec ^2(x)}}-\frac{\tan (x)}{\sqrt{a \sec ^2(x)}}",1,"(-(Csc[x]*Sec[x]) - Tan[x])/Sqrt[a*Sec[x]^2]","A",1
278,1,23,30,0.0251582,"\int \frac{\tan ^3(x)}{\left(a+a \tan ^2(x)\right)^{3/2}} \, dx","Integrate[Tan[x]^3/(a + a*Tan[x]^2)^(3/2),x]","\frac{\cos (2 x)-5}{6 a \sqrt{a \sec ^2(x)}}","\frac{1}{3 \left(a \sec ^2(x)\right)^{3/2}}-\frac{1}{a \sqrt{a \sec ^2(x)}}",1,"(-5 + Cos[2*x])/(6*a*Sqrt[a*Sec[x]^2])","A",1
279,1,18,23,0.017198,"\int \frac{\tan ^2(x)}{\left(a+a \tan ^2(x)\right)^{3/2}} \, dx","Integrate[Tan[x]^2/(a + a*Tan[x]^2)^(3/2),x]","\frac{\tan ^3(x)}{3 \left(a \sec ^2(x)\right)^{3/2}}","\frac{\sin ^2(x) \tan (x)}{3 a \sqrt{a \sec ^2(x)}}",1,"Tan[x]^3/(3*(a*Sec[x]^2)^(3/2))","A",1
280,1,14,14,0.012215,"\int \frac{\tan (x)}{\left(a+a \tan ^2(x)\right)^{3/2}} \, dx","Integrate[Tan[x]/(a + a*Tan[x]^2)^(3/2),x]","-\frac{1}{3 \left(a \sec ^2(x)\right)^{3/2}}","-\frac{1}{3 \left(a \sec ^2(x)\right)^{3/2}}",1,"-1/3*1/(a*Sec[x]^2)^(3/2)","A",1
281,1,47,53,0.0612158,"\int \frac{\cot (x)}{\left(a+a \tan ^2(x)\right)^{3/2}} \, dx","Integrate[Cot[x]/(a + a*Tan[x]^2)^(3/2),x]","\frac{\cos (3 x) \sec (x)+12 \sec (x) \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)+15}{12 a \sqrt{a \sec ^2(x)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{1}{a \sqrt{a \sec ^2(x)}}+\frac{1}{3 \left(a \sec ^2(x)\right)^{3/2}}",1,"(15 + Cos[3*x]*Sec[x] + 12*(-Log[Cos[x/2]] + Log[Sin[x/2]])*Sec[x])/(12*a*Sqrt[a*Sec[x]^2])","A",1
282,1,31,60,0.0575755,"\int \frac{\cot ^2(x)}{\left(a+a \tan ^2(x)\right)^{3/2}} \, dx","Integrate[Cot[x]^2/(a + a*Tan[x]^2)^(3/2),x]","\frac{\sec ^3(x) \left(\sin ^3(x)-6 \sin (x)-3 \csc (x)\right)}{3 \left(a \sec ^2(x)\right)^{3/2}}","-\frac{\csc (x) \sec (x)}{a \sqrt{a \sec ^2(x)}}-\frac{2 \tan (x)}{a \sqrt{a \sec ^2(x)}}+\frac{\sin ^2(x) \tan (x)}{3 a \sqrt{a \sec ^2(x)}}",1,"(Sec[x]^3*(-3*Csc[x] - 6*Sin[x] + Sin[x]^3))/(3*(a*Sec[x]^2)^(3/2))","A",1
283,1,24,24,0.0508618,"\int \frac{1}{\sqrt{a+a \tan ^2(c+d x)}} \, dx","Integrate[1/Sqrt[a + a*Tan[c + d*x]^2],x]","\frac{\tan (c+d x)}{d \sqrt{a \sec ^2(c+d x)}}","\frac{\tan (c+d x)}{d \sqrt{a \sec ^2(c+d x)}}",1,"Tan[c + d*x]/(d*Sqrt[a*Sec[c + d*x]^2])","A",1
284,1,40,58,0.0620605,"\int \frac{1}{\left(a+a \tan ^2(c+d x)\right)^{3/2}} \, dx","Integrate[(a + a*Tan[c + d*x]^2)^(-3/2),x]","-\frac{\left(\sin ^2(c+d x)-3\right) \tan (c+d x)}{3 a d \sqrt{a \sec ^2(c+d x)}}","\frac{2 \tan (c+d x)}{3 a d \sqrt{a \sec ^2(c+d x)}}+\frac{\tan (c+d x)}{3 d \left(a \sec ^2(c+d x)\right)^{3/2}}",1,"-1/3*((-3 + Sin[c + d*x]^2)*Tan[c + d*x])/(a*d*Sqrt[a*Sec[c + d*x]^2])","A",1
285,1,52,88,0.1025573,"\int \frac{1}{\left(a+a \tan ^2(c+d x)\right)^{5/2}} \, dx","Integrate[(a + a*Tan[c + d*x]^2)^(-5/2),x]","\frac{\left(3 \sin ^4(c+d x)-10 \sin ^2(c+d x)+15\right) \tan (c+d x)}{15 a^2 d \sqrt{a \sec ^2(c+d x)}}","\frac{8 \tan (c+d x)}{15 a^2 d \sqrt{a \sec ^2(c+d x)}}+\frac{4 \tan (c+d x)}{15 a d \left(a \sec ^2(c+d x)\right)^{3/2}}+\frac{\tan (c+d x)}{5 d \left(a \sec ^2(c+d x)\right)^{5/2}}",1,"((15 - 10*Sin[c + d*x]^2 + 3*Sin[c + d*x]^4)*Tan[c + d*x])/(15*a^2*d*Sqrt[a*Sec[c + d*x]^2])","A",1
286,1,62,118,0.1753865,"\int \frac{1}{\left(a+a \tan ^2(c+d x)\right)^{7/2}} \, dx","Integrate[(a + a*Tan[c + d*x]^2)^(-7/2),x]","\frac{\left(-5 \sin ^6(c+d x)+21 \sin ^4(c+d x)-35 \sin ^2(c+d x)+35\right) \tan (c+d x)}{35 a^3 d \sqrt{a \sec ^2(c+d x)}}","\frac{16 \tan (c+d x)}{35 a^3 d \sqrt{a \sec ^2(c+d x)}}+\frac{8 \tan (c+d x)}{35 a^2 d \left(a \sec ^2(c+d x)\right)^{3/2}}+\frac{6 \tan (c+d x)}{35 a d \left(a \sec ^2(c+d x)\right)^{5/2}}+\frac{\tan (c+d x)}{7 d \left(a \sec ^2(c+d x)\right)^{7/2}}",1,"((35 - 35*Sin[c + d*x]^2 + 21*Sin[c + d*x]^4 - 5*Sin[c + d*x]^6)*Tan[c + d*x])/(35*a^3*d*Sqrt[a*Sec[c + d*x]^2])","A",1
287,1,52,22,0.0674721,"\int \left(1+\tan ^2(x)\right)^{3/2} \, dx","Integrate[(1 + Tan[x]^2)^(3/2),x]","\frac{1}{2} \cos (x) \sqrt{\sec ^2(x)} \left(\tan (x) \sec (x)-\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)\right)","\frac{1}{2} \tan (x) \sqrt{\sec ^2(x)}+\frac{1}{2} \sinh ^{-1}(\tan (x))",1,"(Cos[x]*Sqrt[Sec[x]^2]*(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]] + Sec[x]*Tan[x]))/2","B",1
288,1,44,3,0.0095188,"\int \sqrt{1+\tan ^2(x)} \, dx","Integrate[Sqrt[1 + Tan[x]^2],x]","\cos (x) \sqrt{\sec ^2(x)} \left(\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)\right)","\sinh ^{-1}(\tan (x))",1,"Cos[x]*(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])*Sqrt[Sec[x]^2]","B",1
289,1,11,11,0.0069395,"\int \frac{1}{\sqrt{1+\tan ^2(x)}} \, dx","Integrate[1/Sqrt[1 + Tan[x]^2],x]","\frac{\tan (x)}{\sqrt{\sec ^2(x)}}","\frac{\tan (x)}{\sqrt{\sec ^2(x)}}",1,"Tan[x]/Sqrt[Sec[x]^2]","A",1
290,1,72,35,0.0673594,"\int \left(-1-\tan ^2(x)\right)^{3/2} \, dx","Integrate[(-1 - Tan[x]^2)^(3/2),x]","\frac{1}{4} \cos (x) \sqrt{-\sec ^2(x)} \left(\frac{1}{\sin (x)-1}+\frac{1}{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2}+2 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-2 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)","\frac{1}{2} \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}\right)-\frac{1}{2} \tan (x) \sqrt{-\sec ^2(x)}",1,"(Cos[x]*Sqrt[-Sec[x]^2]*(2*Log[Cos[x/2] - Sin[x/2]] - 2*Log[Cos[x/2] + Sin[x/2]] + (Cos[x/2] + Sin[x/2])^(-2) + (-1 + Sin[x])^(-1)))/4","B",1
291,1,46,16,0.0072851,"\int \sqrt{-1-\tan ^2(x)} \, dx","Integrate[Sqrt[-1 - Tan[x]^2],x]","\cos (x) \sqrt{-\sec ^2(x)} \left(\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)\right)","-\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}\right)",1,"Cos[x]*(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])*Sqrt[-Sec[x]^2]","B",1
292,1,13,13,0.0070507,"\int \frac{1}{\sqrt{-1-\tan ^2(x)}} \, dx","Integrate[1/Sqrt[-1 - Tan[x]^2],x]","\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}","\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}",1,"Tan[x]/Sqrt[-Sec[x]^2]","A",1
293,1,109,117,1.3834067,"\int \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\frac{\sqrt{a+b \tan ^2(e+f x)} \left(-2 a^2+b (a-5 b) \tan ^2(e+f x)-5 a b+3 b^2 \tan ^4(e+f x)+15 b^2\right)}{b^2}-15 \sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{15 f}","\frac{\left(a+b \tan ^2(e+f x)\right)^{5/2}}{5 b^2 f}-\frac{(a+b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 b^2 f}+\frac{\sqrt{a+b \tan ^2(e+f x)}}{f}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"(-15*Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + (Sqrt[a + b*Tan[e + f*x]^2]*(-2*a^2 - 5*a*b + 15*b^2 + (a - 5*b)*b*Tan[e + f*x]^2 + 3*b^2*Tan[e + f*x]^4))/b^2)/(15*f)","A",1
294,1,82,88,0.3491961,"\int \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{a+b \tan ^2(e+f x)} \left(a+b \tan ^2(e+f x)-3 b\right)+3 b \sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{3 b f}","\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 b f}-\frac{\sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"(3*Sqrt[a - b]*b*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + Sqrt[a + b*Tan[e + f*x]^2]*(a - 3*b + b*Tan[e + f*x]^2))/(3*b*f)","A",1
295,1,59,62,0.0502423,"\int \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{a+b \tan ^2(e+f x)}-\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}","\frac{\sqrt{a+b \tan ^2(e+f x)}}{f}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"(-(Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]) + Sqrt[a + b*Tan[e + f*x]^2])/f","A",1
296,1,72,74,0.062485,"\int \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{f}","\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{f}",1,"(-(Sqrt[a]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]]) + Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f","A",1
297,1,115,115,0.3687253,"\int \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)-\sqrt{a} \left(2 \sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)+\cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}\right)}{2 \sqrt{a} f}","\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 \sqrt{a} f}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{\cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}",1,"((2*a - b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]] - Sqrt[a]*(2*Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2]))/(2*Sqrt[a]*f)","A",1
298,1,138,163,1.3194155,"\int \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\left(-8 a^2+4 a b+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)+\sqrt{a} \left(8 a \sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)-\cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)} \left(2 a \cot ^2(e+f x)-4 a+b\right)\right)}{8 a^{3/2} f}","-\frac{\left(8 a^2-4 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{3/2} f}+\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{\cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{(4 a-b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 a f}",1,"((-8*a^2 + 4*a*b + b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]] + Sqrt[a]*(8*a*Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] - Cot[e + f*x]^2*(-4*a + b + 2*a*Cot[e + f*x]^2)*Sqrt[a + b*Tan[e + f*x]^2]))/(8*a^(3/2)*f)","A",1
299,1,823,222,6.3348527,"\int \tan ^6(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]^6*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{-\frac{b \left(a^3+2 b a^2-8 b^3\right) \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac{4 b \left(8 b^3-8 a b^2\right) \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}-\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}\right)}{\sqrt{a+b+(a-b) \cos (2 (e+f x))}}}{8 b^2 f}+\frac{\sqrt{\frac{\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\frac{1}{6} \tan (e+f x) \sec ^4(e+f x)+\frac{(a \sin (e+f x)-14 b \sin (e+f x)) \sec ^3(e+f x)}{24 b}+\frac{\left(-3 \sin (e+f x) a^2-8 b \sin (e+f x) a+44 b^2 \sin (e+f x)\right) \sec (e+f x)}{48 b^2}\right)}{f}","\frac{\left(a^3+2 a^2 b+8 a b^2-16 b^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{16 b^{5/2} f}-\frac{(a-2 b) (a+4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{16 b^2 f}-\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{\tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{6 f}+\frac{(a-6 b) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{24 b f}",1,"(-((b*(a^3 + 2*a^2*b - 8*b^3)*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*(a + b + (a - b)*Cos[2*(e + f*x)]))) - (4*b*(-8*a*b^2 + 8*b^3)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])/(8*b^2*f) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sec[e + f*x]^3*(a*Sin[e + f*x] - 14*b*Sin[e + f*x]))/(24*b) + (Sec[e + f*x]*(-3*a^2*Sin[e + f*x] - 8*a*b*Sin[e + f*x] + 44*b^2*Sin[e + f*x]))/(48*b^2) + (Sec[e + f*x]^4*Tan[e + f*x])/6))/f","C",1
300,1,767,169,6.2519986,"\int \tan ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{\frac{a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b}{\cos (2 (e+f x))+1}} \left(\frac{\sec (e+f x) (a \sin (e+f x)-6 b \sin (e+f x))}{8 b}+\frac{1}{4} \tan (e+f x) \sec ^2(e+f x)\right)}{f}-\frac{-\frac{b \left(a^2-4 b^2\right) \sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{a ((a-b) \cos (2 (e+f x))+a+b)}-\frac{4 b \left(4 b^2-4 a b\right) \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \left(\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}-\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}\right)}{\sqrt{(a-b) \cos (2 (e+f x))+a+b}}}{4 b f}","-\frac{\left(a^2+4 a b-8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 b^{3/2} f}+\frac{(a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 b f}+\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{\tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}",1,"-1/4*(-((b*(a^2 - 4*b^2)*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*(a + b + (a - b)*Cos[2*(e + f*x)]))) - (4*b*(-4*a*b + 4*b^2)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])/(b*f) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sec[e + f*x]*(a*Sin[e + f*x] - 6*b*Sin[e + f*x]))/(8*b) + (Sec[e + f*x]^2*Tan[e + f*x])/4))/f","C",1
301,1,708,123,6.1775733,"\int \tan ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Tan[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{b^2 \sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{a f ((a-b) \cos (2 (e+f x))+a+b)}+\frac{\tan (e+f x) \sqrt{\frac{a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b}{\cos (2 (e+f x))+1}}}{2 f}+\frac{4 b (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \left(\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}-\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}\right)}{f \sqrt{(a-b) \cos (2 (e+f x))+a+b}}","\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}-\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{(a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 \sqrt{b} f}",1,"(b^2*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*f*(a + b + (a - b)*Cos[2*(e + f*x)])) + (4*(a - b)*b*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/(f*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Tan[e + f*x])/(2*f)","C",1
302,1,203,85,0.8836427,"\int \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{-i \sqrt{a-b} \log \left(-\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \tan ^2(e+f x)}+a-i b \tan (e+f x)\right)}{(a-b)^{3/2} (\tan (e+f x)+i)}\right)+i \sqrt{a-b} \log \left(\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \tan ^2(e+f x)}+a+i b \tan (e+f x)\right)}{(a-b)^{3/2} (\tan (e+f x)-i)}\right)+2 \sqrt{b} \log \left(\sqrt{b} \sqrt{a+b \tan ^2(e+f x)}+b \tan (e+f x)\right)}{2 f}","\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}",1,"((-I)*Sqrt[a - b]*Log[((-4*I)*(a - I*b*Tan[e + f*x] + Sqrt[a - b]*Sqrt[a + b*Tan[e + f*x]^2]))/((a - b)^(3/2)*(I + Tan[e + f*x]))] + I*Sqrt[a - b]*Log[((4*I)*(a + I*b*Tan[e + f*x] + Sqrt[a - b]*Sqrt[a + b*Tan[e + f*x]^2]))/((a - b)^(3/2)*(-I + Tan[e + f*x]))] + 2*Sqrt[b]*Log[b*Tan[e + f*x] + Sqrt[b]*Sqrt[a + b*Tan[e + f*x]^2]])/(2*f)","C",1
303,1,64,75,0.2835306,"\int \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\frac{(a-b) \tan ^2(e+f x)}{b \tan ^2(e+f x)+a}\right)}{f}","-\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}",1,"-((Cot[e + f*x]*Hypergeometric2F1[-1/2, 1, 1/2, -(((a - b)*Tan[e + f*x]^2)/(a + b*Tan[e + f*x]^2))]*Sqrt[a + b*Tan[e + f*x]^2])/f)","C",1
304,1,241,117,6.9833597,"\int \cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\cos ^2(e+f x) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)} \left(\frac{b \tan ^2(e+f x)}{a}+1\right) \left(\frac{\sec ^2(e+f x) \left(a-2 b \tan ^2(e+f x)\right) \left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}} \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right)+\sqrt{\frac{b \sin ^2(e+f x)}{a}+\cos ^2(e+f x)}\right)}{\left(a+b \tan ^2(e+f x)\right) \sqrt{\frac{b \sin ^2(e+f x)}{a}+\cos ^2(e+f x)}}-\frac{4 (a-b) \sin ^2(e+f x) \left(a+b \tan ^2(e+f x)\right) \, _2F_1\left(2,2;\frac{3}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right)}{a^2}\right)}{3 f}","\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 f}+\frac{(3 a-b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a f}",1,"-1/3*(Cos[e + f*x]^2*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2]*(1 + (b*Tan[e + f*x]^2)/a)*((Sec[e + f*x]^2*(ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sqrt[((a - b)*Sin[e + f*x]^2)/a] + Sqrt[Cos[e + f*x]^2 + (b*Sin[e + f*x]^2)/a])*(a - 2*b*Tan[e + f*x]^2))/(Sqrt[Cos[e + f*x]^2 + (b*Sin[e + f*x]^2)/a]*(a + b*Tan[e + f*x]^2)) - (4*(a - b)*Hypergeometric2F1[2, 2, 3/2, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2))/f","C",0
305,1,339,167,14.3870724,"\int \cot ^6(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^6*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\cos ^4(e+f x) \cot ^5(e+f x) \left(\frac{b \tan ^2(e+f x)}{a}+1\right) \left(8 (a-b) \tan ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^3 \, _3F_2\left(2,2,2;1,\frac{3}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right)+8 \tan ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \left(-2 a^2+a b \left(3 \tan ^2(e+f x)+2\right)-3 b^2 \tan ^2(e+f x)\right) \, _2F_1\left(2,2;\frac{3}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right)+\frac{a^2 \sec ^4(e+f x) \left(3 a^2-4 a b \tan ^2(e+f x)+8 b^2 \tan ^4(e+f x)\right) \left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}} \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right)+\sqrt{\frac{\cos ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)}{a}}\right)}{\sqrt{\frac{\cos ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)}{a}}}\right)}{15 a^3 f \sqrt{a+b \tan ^2(e+f x)}}","-\frac{\left(15 a^2-5 a b-2 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 f}+\frac{(5 a-b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a f}",1,"-1/15*(Cos[e + f*x]^4*Cot[e + f*x]^5*(1 + (b*Tan[e + f*x]^2)/a)*(8*(a - b)*HypergeometricPFQ[{2, 2, 2}, {1, 3/2}, ((a - b)*Sin[e + f*x]^2)/a]*Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^3 + 8*Hypergeometric2F1[2, 2, 3/2, ((a - b)*Sin[e + f*x]^2)/a]*Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^2*(-2*a^2 - 3*b^2*Tan[e + f*x]^2 + a*b*(2 + 3*Tan[e + f*x]^2)) + (a^2*Sec[e + f*x]^4*(3*a^2 - 4*a*b*Tan[e + f*x]^2 + 8*b^2*Tan[e + f*x]^4)*(ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sqrt[((a - b)*Sin[e + f*x]^2)/a] + Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a]))/Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a]))/(a^3*f*Sqrt[a + b*Tan[e + f*x]^2])","C",0
306,1,139,145,1.3940017,"\int \tan ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\frac{2 \left(a+b \tan ^2(e+f x)\right)^{7/2}}{7 b^2}-\frac{2 (a+b) \left(a+b \tan ^2(e+f x)\right)^{5/2}}{5 b^2}+\frac{2}{3} \left(a+b \tan ^2(e+f x)\right)^{3/2}+2 (a-b) \left(\sqrt{a+b \tan ^2(e+f x)}-\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)\right)}{2 f}","\frac{\left(a+b \tan ^2(e+f x)\right)^{7/2}}{7 b^2 f}-\frac{(a+b) \left(a+b \tan ^2(e+f x)\right)^{5/2}}{5 b^2 f}+\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 f}+\frac{(a-b) \sqrt{a+b \tan ^2(e+f x)}}{f}-\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"((2*(a + b*Tan[e + f*x]^2)^(3/2))/3 - (2*(a + b)*(a + b*Tan[e + f*x]^2)^(5/2))/(5*b^2) + (2*(a + b*Tan[e + f*x]^2)^(7/2))/(7*b^2) + 2*(a - b)*(-(Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]) + Sqrt[a + b*Tan[e + f*x]^2]))/(2*f)","A",1
307,1,112,116,0.890203,"\int \tan ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sqrt{a+b \tan ^2(e+f x)} \left(3 a^2+b (6 a-5 b) \tan ^2(e+f x)-20 a b+3 b^2 \tan ^4(e+f x)+15 b^2\right)+15 b (a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{15 b f}","\frac{\left(a+b \tan ^2(e+f x)\right)^{5/2}}{5 b f}-\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 f}-\frac{(a-b) \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"(15*(a - b)^(3/2)*b*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + Sqrt[a + b*Tan[e + f*x]^2]*(3*a^2 - 20*a*b + 15*b^2 + (6*a - 5*b)*b*Tan[e + f*x]^2 + 3*b^2*Tan[e + f*x]^4))/(15*b*f)","A",1
308,1,80,90,0.4008345,"\int \tan (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sqrt{a+b \tan ^2(e+f x)} \left(4 a+b \tan ^2(e+f x)-3 b\right)-3 (a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{3 f}","\frac{(a-b) \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 f}-\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"(-3*(a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + Sqrt[a + b*Tan[e + f*x]^2]*(4*a - 3*b + b*Tan[e + f*x]^2))/(3*f)","A",1
309,1,90,95,0.2461595,"\int \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{a^{3/2} \left(-\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)\right)+b \sqrt{a+b \tan ^2(e+f x)}+(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{b \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"(-(a^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]]) + (a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + b*Sqrt[a + b*Tan[e + f*x]^2])/f","A",1
310,1,109,116,0.3354288,"\int \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sqrt{a} (2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)-2 (a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)-a \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}","\frac{\sqrt{a} (2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 f}-\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{a \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}",1,"(Sqrt[a]*(2*a - 3*b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]] - 2*(a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] - a*Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)","A",1
311,1,140,161,1.4036609,"\int \cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\left(-8 a^2+12 a b-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)+\sqrt{a} \left(8 (a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)+\cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)} \left(-2 a \cot ^2(e+f x)+4 a-5 b\right)\right)}{8 \sqrt{a} f}","-\frac{\left(8 a^2-12 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 \sqrt{a} f}+\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{a \cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{(4 a-5 b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}",1,"((-8*a^2 + 12*a*b - 3*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]] + Sqrt[a]*(8*(a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + Cot[e + f*x]^2*(4*a - 5*b - 2*a*Cot[e + f*x]^2)*Sqrt[a + b*Tan[e + f*x]^2]))/(8*Sqrt[a]*f)","A",1
312,1,908,294,6.5495382,"\int \tan ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{-\frac{b \left(3 a^4+8 b a^3-16 b^2 a^2-64 b^3 a+64 b^4\right) \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac{4 b \left(-64 b^4+128 a b^3-64 a^2 b^2\right) \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}-\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}\right)}{\sqrt{a+b+(a-b) \cos (2 (e+f x))}}}{64 b^2 f}+\frac{\sqrt{\frac{\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\frac{1}{8} b \tan (e+f x) \sec ^6(e+f x)+\frac{1}{48} (9 a \sin (e+f x)-26 b \sin (e+f x)) \sec ^5(e+f x)+\frac{\left(3 \sin (e+f x) a^2-128 b \sin (e+f x) a+184 b^2 \sin (e+f x)\right) \sec ^3(e+f x)}{192 b}+\frac{\left(-9 \sin (e+f x) a^3-30 b \sin (e+f x) a^2+424 b^2 \sin (e+f x) a-400 b^3 \sin (e+f x)\right) \sec (e+f x)}{384 b^2}\right)}{f}","\frac{\left(3 a^2-56 a b+48 b^2\right) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{192 b f}-\frac{\left(3 a^3+8 a^2 b-80 a b^2+64 b^3\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{128 b^2 f}+\frac{\left(3 a^4+8 a^3 b+48 a^2 b^2-192 a b^3+128 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{128 b^{5/2} f}-\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{b \tan ^7(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{(9 a-8 b) \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{48 f}",1,"(-((b*(3*a^4 + 8*a^3*b - 16*a^2*b^2 - 64*a*b^3 + 64*b^4)*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*(a + b + (a - b)*Cos[2*(e + f*x)]))) - (4*b*(-64*a^2*b^2 + 128*a*b^3 - 64*b^4)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])/(64*b^2*f) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sec[e + f*x]^5*(9*a*Sin[e + f*x] - 26*b*Sin[e + f*x]))/48 + (Sec[e + f*x]^3*(3*a^2*Sin[e + f*x] - 128*a*b*Sin[e + f*x] + 184*b^2*Sin[e + f*x]))/(192*b) + (Sec[e + f*x]*(-9*a^3*Sin[e + f*x] - 30*a^2*b*Sin[e + f*x] + 424*a*b^2*Sin[e + f*x] - 400*b^3*Sin[e + f*x]))/(384*b^2) + (b*Sec[e + f*x]^6*Tan[e + f*x])/8))/f","C",1
313,1,833,224,6.4395038,"\int \tan ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Tan[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sqrt{\frac{\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\frac{1}{6} b \tan (e+f x) \sec ^4(e+f x)+\frac{7}{24} (a \sin (e+f x)-2 b \sin (e+f x)) \sec ^3(e+f x)+\frac{\left(3 \sin (e+f x) a^2-44 b \sin (e+f x) a+44 b^2 \sin (e+f x)\right) \sec (e+f x)}{48 b}\right)}{f}-\frac{-\frac{b \left(a^3-2 b a^2-8 b^2 a+8 b^3\right) \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac{4 b \left(-8 b^3+16 a b^2-8 a^2 b\right) \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}-\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}\right)}{\sqrt{a+b+(a-b) \cos (2 (e+f x))}}}{8 b f}","\frac{\left(a^2-10 a b+8 b^2\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{16 b f}-\frac{\left(a^3+6 a^2 b-24 a b^2+16 b^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{16 b^{3/2} f}+\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{b \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{6 f}+\frac{(7 a-6 b) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{24 f}",1,"-1/8*(-((b*(a^3 - 2*a^2*b - 8*a*b^2 + 8*b^3)*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*(a + b + (a - b)*Cos[2*(e + f*x)]))) - (4*b*(-8*a^2*b + 16*a*b^2 - 8*b^3)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])/(b*f) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((7*Sec[e + f*x]^3*(a*Sin[e + f*x] - 2*b*Sin[e + f*x]))/24 + (Sec[e + f*x]*(3*a^2*Sin[e + f*x] - 44*a*b*Sin[e + f*x] + 44*b^2*Sin[e + f*x]))/(48*b) + (b*Sec[e + f*x]^4*Tan[e + f*x])/6))/f","C",1
314,1,771,172,6.30942,"\int \tan ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\frac{b \left(a^2+4 a b-4 b^2\right) \sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{a ((a-b) \cos (2 (e+f x))+a+b)}+\frac{4 b \left(4 a^2-8 a b+4 b^2\right) \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \left(\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}-\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}\right)}{\sqrt{(a-b) \cos (2 (e+f x))+a+b}}}{4 f}+\frac{\sqrt{\frac{a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b}{\cos (2 (e+f x))+1}} \left(\frac{1}{8} \sec (e+f x) (5 a \sin (e+f x)-6 b \sin (e+f x))+\frac{1}{4} b \tan (e+f x) \sec ^2(e+f x)\right)}{f}","\frac{\left(3 a^2-12 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 \sqrt{b} f}+\frac{(5 a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}-\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{b \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}",1,"((b*(a^2 + 4*a*b - 4*b^2)*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*(a + b + (a - b)*Cos[2*(e + f*x)])) + (4*b*(4*a^2 - 8*a*b + 4*b^2)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])/(4*f) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sec[e + f*x]*(5*a*Sin[e + f*x] - 6*b*Sin[e + f*x]))/8 + (b*Sec[e + f*x]^2*Tan[e + f*x])/4))/f","C",1
315,1,233,125,1.4272354,"\int \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}-i (a-b)^{3/2} \log \left(-\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \tan ^2(e+f x)}+a-i b \tan (e+f x)\right)}{(a-b)^{5/2} (\tan (e+f x)+i)}\right)+i (a-b)^{3/2} \log \left(\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \tan ^2(e+f x)}+a+i b \tan (e+f x)\right)}{(a-b)^{5/2} (\tan (e+f x)-i)}\right)+\sqrt{b} (3 a-2 b) \log \left(\sqrt{b} \sqrt{a+b \tan ^2(e+f x)}+b \tan (e+f x)\right)}{2 f}","\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}+\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}",1,"((-I)*(a - b)^(3/2)*Log[((-4*I)*(a - I*b*Tan[e + f*x] + Sqrt[a - b]*Sqrt[a + b*Tan[e + f*x]^2]))/((a - b)^(5/2)*(I + Tan[e + f*x]))] + I*(a - b)^(3/2)*Log[((4*I)*(a + I*b*Tan[e + f*x] + Sqrt[a - b]*Sqrt[a + b*Tan[e + f*x]^2]))/((a - b)^(5/2)*(-I + Tan[e + f*x]))] + (3*a - 2*b)*Sqrt[b]*Log[b*Tan[e + f*x] + Sqrt[b]*Sqrt[a + b*Tan[e + f*x]^2]] + b*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)","C",1
316,1,724,114,6.2410792,"\int \cot ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{b \left(a^2-2 a b-b^2\right) \sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{a f ((a-b) \cos (2 (e+f x))+a+b)}+\frac{4 b \left(a^2-2 a b+b^2\right) \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \left(\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}-\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}\right)}{f \sqrt{(a-b) \cos (2 (e+f x))+a+b}}-\frac{a \cot (e+f x) \sqrt{\frac{a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b}{\cos (2 (e+f x))+1}}}{f}","\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{a \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}",1,"-((a*Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Cot[e + f*x])/f) + (b*(a^2 - 2*a*b - b^2)*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*f*(a + b + (a - b)*Cos[2*(e + f*x)])) + (4*b*(a^2 - 2*a*b + b^2)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/(f*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])","C",1
317,1,78,115,0.3413646,"\int \cot ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)} \left(a \cot ^2(e+f x)+b\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\frac{(a-b) \tan ^2(e+f x)}{b \tan ^2(e+f x)+a}\right)}{3 f}","\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{a \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 f}+\frac{(3 a-4 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 f}",1,"-1/3*(Cot[e + f*x]*(b + a*Cot[e + f*x]^2)*Hypergeometric2F1[-3/2, 1, -1/2, -(((a - b)*Tan[e + f*x]^2)/(a + b*Tan[e + f*x]^2))]*Sqrt[a + b*Tan[e + f*x]^2])/f","C",1
318,1,140,165,9.4959758,"\int \cot ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)} \left(a \cot ^2(e+f x)+b\right)^2 \left(2 (a-b) ((a-b) \cos (2 (e+f x))+a+b) \, _2F_1\left(2,2;\frac{1}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right)+a \left(3 a \cot ^2(e+f x)-2 b\right) \, _2F_1\left(1,1;-\frac{1}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right)\right)}{15 a^3 f}","-\frac{\left(15 a^2-20 a b+3 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a f}-\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{a \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 f}+\frac{(5 a-6 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 f}",1,"-1/15*(Cos[e + f*x]*(b + a*Cot[e + f*x]^2)^2*(a*(-2*b + 3*a*Cot[e + f*x]^2)*Hypergeometric2F1[1, 1, -1/2, ((a - b)*Sin[e + f*x]^2)/a] + 2*(a - b)*(a + b + (a - b)*Cos[2*(e + f*x)])*Hypergeometric2F1[2, 2, 1/2, ((a - b)*Sin[e + f*x]^2)/a])*Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(a^3*f)","C",0
319,1,259,170,1.3425883,"\int \left(a+b \tan ^2(c+d x)\right)^{5/2} \, dx","Integrate[(a + b*Tan[c + d*x]^2)^(5/2),x]","\frac{\sqrt{b} \left(15 a^2-20 a b+8 b^2\right) \log \left(\sqrt{b} \sqrt{a+b \tan ^2(c+d x)}+b \tan (c+d x)\right)+b \tan (c+d x) \sqrt{a+b \tan ^2(c+d x)} \left(9 a+2 b \tan ^2(c+d x)-4 b\right)-4 i (a-b)^{5/2} \log \left(-\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \tan ^2(c+d x)}+a-i b \tan (c+d x)\right)}{(a-b)^{7/2} (\tan (c+d x)+i)}\right)+4 i (a-b)^{5/2} \log \left(\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \tan ^2(c+d x)}+a+i b \tan (c+d x)\right)}{(a-b)^{7/2} (\tan (c+d x)-i)}\right)}{8 d}","\frac{\sqrt{b} \left(15 a^2-20 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a+b \tan ^2(c+d x)}}\right)}{8 d}+\frac{b \tan (c+d x) \left(a+b \tan ^2(c+d x)\right)^{3/2}}{4 d}+\frac{b (7 a-4 b) \tan (c+d x) \sqrt{a+b \tan ^2(c+d x)}}{8 d}+\frac{(a-b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (c+d x)}{\sqrt{a+b \tan ^2(c+d x)}}\right)}{d}",1,"((-4*I)*(a - b)^(5/2)*Log[((-4*I)*(a - I*b*Tan[c + d*x] + Sqrt[a - b]*Sqrt[a + b*Tan[c + d*x]^2]))/((a - b)^(7/2)*(I + Tan[c + d*x]))] + (4*I)*(a - b)^(5/2)*Log[((4*I)*(a + I*b*Tan[c + d*x] + Sqrt[a - b]*Sqrt[a + b*Tan[c + d*x]^2]))/((a - b)^(7/2)*(-I + Tan[c + d*x]))] + Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*Log[b*Tan[c + d*x] + Sqrt[b]*Sqrt[a + b*Tan[c + d*x]^2]] + b*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]^2]*(9*a - 4*b + 2*b*Tan[c + d*x]^2))/(8*d)","C",1
320,1,87,95,2.4516363,"\int \frac{\tan ^5(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^5/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\frac{2 \left(2 a-b \tan ^2(e+f x)+3 b\right) \sqrt{a+b \tan ^2(e+f x)}}{3 b^2}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}}{2 f}","\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 b^2 f}-\frac{(a+b) \sqrt{a+b \tan ^2(e+f x)}}{b^2 f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}",1,"-1/2*((2*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/Sqrt[a - b] + (2*(2*a + 3*b - b*Tan[e + f*x]^2)*Sqrt[a + b*Tan[e + f*x]^2])/(3*b^2))/f","A",1
321,1,62,64,0.2839021,"\int \frac{\tan ^3(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^3/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\frac{\sqrt{a+b \tan ^2(e+f x)}}{b}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}}{f}","\frac{\sqrt{a+b \tan ^2(e+f x)}}{b f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}",1,"(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/Sqrt[a - b] + Sqrt[a + b*Tan[e + f*x]^2]/b)/f","A",1
322,1,41,41,0.0300403,"\int \frac{\tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}",1,"-(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f))","A",1
323,1,72,74,0.0839095,"\int \frac{\cot (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a}}}{f}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}",1,"(-(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]]/Sqrt[a]) + ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/Sqrt[a - b])/f","A",1
324,1,135,116,0.7567107,"\int \frac{\cot ^3(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^3/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\left(2 a^2-a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)+\sqrt{a} \left((b-a) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}-2 a \sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)\right)}{2 a^{3/2} f (a-b)}","\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{3/2} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}-\frac{\cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 a f}",1,"((2*a^2 - a*b - b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]] + Sqrt[a]*(-2*a*Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + (-a + b)*Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2]))/(2*a^(3/2)*(a - b)*f)","A",1
325,1,162,166,1.9607539,"\int \frac{\cot ^5(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^5/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{a} \left(8 a^2 \sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)+(b-a) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)} \left(2 a \cot ^2(e+f x)-4 a-3 b\right)\right)+\left(-8 a^3+4 a^2 b+a b^2+3 b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{5/2} f (a-b)}","\frac{(4 a+3 b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 a^2 f}-\frac{\left(8 a^2+4 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{5/2} f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}-\frac{\cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 a f}",1,"((-8*a^3 + 4*a^2*b + a*b^2 + 3*b^3)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]] + Sqrt[a]*(8*a^2*Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + (-a + b)*Cot[e + f*x]^2*(-4*a - 3*b + 2*a*Cot[e + f*x]^2)*Sqrt[a + b*Tan[e + f*x]^2]))/(8*a^(5/2)*(a - b)*f)","A",1
326,1,768,177,6.3346697,"\int \frac{\tan ^6(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^6/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\frac{16 b^3 \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \left(\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}-\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}\right)}{\sqrt{(a-b) \cos (2 (e+f x))+a+b}}-\frac{b \left(3 a^2+4 a b+4 b^2\right) \sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{a ((a-b) \cos (2 (e+f x))+a+b)}}{4 b^2 f}+\frac{\sqrt{\frac{a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b}{\cos (2 (e+f x))+1}} \left(\frac{\tan (e+f x) \sec ^2(e+f x)}{4 b}-\frac{3 \sec (e+f x) (a \sin (e+f x)+2 b \sin (e+f x))}{8 b^2}\right)}{f}","\frac{\left(3 a^2+4 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 b^{5/2} f}-\frac{(3 a+4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 b^2 f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}+\frac{\tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 b f}",1,"(-((b*(3*a^2 + 4*a*b + 4*b^2)*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*(a + b + (a - b)*Cos[2*(e + f*x)]))) + (16*b^3*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])/(4*b^2*f) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((-3*Sec[e + f*x]*(a*Sin[e + f*x] + 2*b*Sin[e + f*x]))/(8*b^2) + (Sec[e + f*x]^2*Tan[e + f*x])/(4*b)))/f","C",1
327,1,713,125,6.2566585,"\int \frac{\tan ^4(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^4/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\tan (e+f x) \sqrt{\frac{a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b}{\cos (2 (e+f x))+1}}}{2 b f}-\frac{\frac{4 b^2 \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \left(\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}-\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}\right)}{\sqrt{(a-b) \cos (2 (e+f x))+a+b}}-\frac{b (a+b) \sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{a ((a-b) \cos (2 (e+f x))+a+b)}}{b f}","-\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 b^{3/2} f}+\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 b f}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}",1,"-((-((b*(a + b)*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*(a + b + (a - b)*Cos[2*(e + f*x)]))) + (4*b^2*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])/(b*f)) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Tan[e + f*x])/(2*b*f)","C",1
328,1,149,86,0.7879167,"\int \frac{\tan ^2(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^2/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{a \sin (2 (e+f x)) \csc ^2(e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{2 b f (a-b) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{\sqrt{b} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}",1,"(a*Csc[e + f*x]^2*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*Sin[2*(e + f*x)])/(2*(a - b)*b*f*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b])","C",1
329,1,46,46,0.0697573,"\int \frac{1}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[1/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}",1,"ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f)","A",1
330,1,212,78,9.4975196,"\int \frac{\cot ^2(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^2/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\cos ^2(e+f x) \cot (e+f x) \left(\frac{b \tan ^2(e+f x)}{a}+1\right) \left(\frac{4 \sin ^2(e+f x) \left(a^2+a b \left(\tan ^2(e+f x)-1\right)-b^2 \tan ^2(e+f x)\right) \, _2F_1\left(2,2;\frac{5}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right)}{3 a^2}+\frac{\sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \left(a+2 b \tan ^2(e+f x)\right)}{a \sqrt{\frac{\sin ^2(e+f x) \cos ^2(e+f x) \left(a^2+a b \left(\tan ^2(e+f x)-1\right)-b^2 \tan ^2(e+f x)\right)}{a^2}}}\right)}{f \sqrt{a+b \tan ^2(e+f x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a f}",1,"-((Cos[e + f*x]^2*Cot[e + f*x]*(1 + (b*Tan[e + f*x]^2)/a)*((4*Hypergeometric2F1[2, 2, 5/2, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*(a^2 - b^2*Tan[e + f*x]^2 + a*b*(-1 + Tan[e + f*x]^2)))/(3*a^2) + (ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*(a + 2*b*Tan[e + f*x]^2))/(a*Sqrt[(Cos[e + f*x]^2*Sin[e + f*x]^2*(a^2 - b^2*Tan[e + f*x]^2 + a*b*(-1 + Tan[e + f*x]^2)))/a^2])))/(f*Sqrt[a + b*Tan[e + f*x]^2]))","C",0
331,1,263,120,11.0712158,"\int \frac{\cot ^4(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^4/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\cos ^2(e+f x) \cot ^3(e+f x) \left(\frac{b \tan ^2(e+f x)}{a}+1\right) \left(-8 (a-b) \sin ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, _3F_2\left(2,2,2;1,\frac{5}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right)+\frac{6 a \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \left(a^2-4 a b \tan ^2(e+f x)-8 b^2 \tan ^4(e+f x)\right)}{\sqrt{\frac{(a-b) \sin ^2(2 (e+f x)) \left(a+b \tan ^2(e+f x)\right)}{a^2}}}-12 b (b-a) \tan ^4(e+f x) ((b-a) \cos (2 (e+f x))-a-b) \, _2F_1\left(2,2;\frac{5}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right)\right)}{9 a^3 f \sqrt{a+b \tan ^2(e+f x)}}","\frac{(3 a+2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^2 f}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a f}",1,"-1/9*(Cos[e + f*x]^2*Cot[e + f*x]^3*(1 + (b*Tan[e + f*x]^2)/a)*(-12*b*(-a + b)*(-a - b + (-a + b)*Cos[2*(e + f*x)])*Hypergeometric2F1[2, 2, 5/2, ((a - b)*Sin[e + f*x]^2)/a]*Tan[e + f*x]^4 - 8*(a - b)*HypergeometricPFQ[{2, 2, 2}, {1, 5/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2)^2 + (6*a*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*(a^2 - 4*a*b*Tan[e + f*x]^2 - 8*b^2*Tan[e + f*x]^4))/Sqrt[((a - b)*Sin[2*(e + f*x)]^2*(a + b*Tan[e + f*x]^2))/a^2]))/(a^3*f*Sqrt[a + b*Tan[e + f*x]^2])","C",0
332,1,794,170,16.3108557,"\int \frac{\cot ^6(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^6/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{\frac{a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b}{\cos (2 (e+f x))+1}} \left(\frac{\csc ^3(e+f x) (11 a \cos (e+f x)+4 b \cos (e+f x))}{15 a^2}+\frac{\csc (e+f x) \left(-23 a^2 \cos (e+f x)-14 a b \cos (e+f x)-8 b^2 \cos (e+f x)\right)}{15 a^3}-\frac{\cot (e+f x) \csc ^4(e+f x)}{5 a}\right)}{f}+\frac{b \sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{a f ((a-b) \cos (2 (e+f x))+a+b)}+\frac{4 b \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \left(\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}-\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}\right)}{f \sqrt{(a-b) \cos (2 (e+f x))+a+b}}","\frac{(5 a+4 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac{\left(15 a^2+10 a b+8 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^3 f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a f}",1,"(Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(((-23*a^2*Cos[e + f*x] - 14*a*b*Cos[e + f*x] - 8*b^2*Cos[e + f*x])*Csc[e + f*x])/(15*a^3) + ((11*a*Cos[e + f*x] + 4*b*Cos[e + f*x])*Csc[e + f*x]^3)/(15*a^2) - (Cot[e + f*x]*Csc[e + f*x]^4)/(5*a)))/f + (b*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*f*(a + b + (a - b)*Cos[2*(e + f*x)])) + (4*b*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/(f*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])","C",1
333,1,84,98,0.3776461,"\int \frac{\tan ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{b^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tan ^2(e+f x)+a}{a-b}\right)+(a-b) \left(2 a+b \tan ^2(e+f x)+b\right)}{b^2 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}","\frac{a^2}{b^2 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}+\frac{\sqrt{a+b \tan ^2(e+f x)}}{b^2 f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}",1,"(b^2*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[e + f*x]^2)/(a - b)] + (a - b)*(2*a + b + b*Tan[e + f*x]^2))/((a - b)*b^2*f*Sqrt[a + b*Tan[e + f*x]^2])","C",1
334,1,75,73,0.358908,"\int \frac{\tan ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\frac{a (b-a)}{b \sqrt{a+b \tan ^2(e+f x)}}+\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^2}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}-\frac{a}{b f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"(Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + (a*(-a + b))/(b*Sqrt[a + b*Tan[e + f*x]^2]))/((a - b)^2*f)","A",1
335,1,56,69,0.072988,"\int \frac{\tan (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{f (b-a) \sqrt{a+b \tan ^2(e+f x)}}","\frac{1}{f (a-b) \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}",1,"-(Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[e + f*x]^2)/(a - b)]/((-a + b)*f*Sqrt[a + b*Tan[e + f*x]^2]))","C",1
336,1,91,106,0.1357074,"\int \frac{\cot (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{(a-b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tan ^2(e+f x)}{a}+1\right)-a \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{b}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}",1,"(-(a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[e + f*x]^2)/(a - b)]) + (a - b)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Tan[e + f*x]^2)/a])/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2])","C",1
337,1,115,157,0.4281316,"\int \frac{\cot ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{(a-b) \left((2 a+3 b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tan ^2(e+f x)}{a}+1\right)+a \cot ^2(e+f x)\right)-2 a^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 a^2 f (b-a) \sqrt{a+b \tan ^2(e+f x)}}","\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{5/2} f}-\frac{b (a-3 b)}{2 a^2 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}-\frac{\cot ^2(e+f x)}{2 a f \sqrt{a+b \tan ^2(e+f x)}}",1,"(-2*a^2*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[e + f*x]^2)/(a - b)] + (a - b)*(a*Cot[e + f*x]^2 + (2*a + 3*b)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Tan[e + f*x]^2)/a]))/(2*a^2*(-a + b)*f*Sqrt[a + b*Tan[e + f*x]^2])","C",1
338,1,142,215,1.2270035,"\int \frac{\cot ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{8 a^3 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tan ^2(e+f x)+a}{a-b}\right)+(a-b) \left(a \cot ^2(e+f x) \left(2 a \cot ^2(e+f x)-4 a-5 b\right)-\left(8 a^2+12 a b+15 b^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tan ^2(e+f x)}{a}+1\right)\right)}{8 a^3 f (b-a) \sqrt{a+b \tan ^2(e+f x)}}","\frac{(4 a+5 b) \cot ^2(e+f x)}{8 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left(8 a^2+12 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{7/2} f}+\frac{b \left(4 a^2+3 a b-15 b^2\right)}{8 a^3 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}-\frac{\cot ^4(e+f x)}{4 a f \sqrt{a+b \tan ^2(e+f x)}}",1,"(8*a^3*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[e + f*x]^2)/(a - b)] + (a - b)*(a*Cot[e + f*x]^2*(-4*a - 5*b + 2*a*Cot[e + f*x]^2) - (8*a^2 + 12*a*b + 15*b^2)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Tan[e + f*x]^2)/a]))/(8*a^3*(-a + b)*f*Sqrt[a + b*Tan[e + f*x]^2])","C",1
339,1,787,182,6.4449494,"\int \frac{\tan ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sqrt{\frac{a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b}{\cos (2 (e+f x))+1}} \left(\frac{\tan (e+f x)}{2 b^2}-\frac{a^2 \sin (2 (e+f x))}{b^2 (a-b) (a (-\cos (2 (e+f x)))-a+b \cos (2 (e+f x))-b)}\right)}{f}-\frac{-\frac{b \left(3 a^2-a b-b^2\right) \sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{a ((a-b) \cos (2 (e+f x))+a+b)}-\frac{4 b^3 \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \left(\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}-\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}\right)}{\sqrt{(a-b) \cos (2 (e+f x))+a+b}}}{b^2 f (a-b)}","-\frac{(3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 b^{5/2} f}+\frac{(3 a-b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 b^2 f (a-b)}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{a \tan ^3(e+f x)}{b f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"-((-((b*(3*a^2 - a*b - b^2)*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*(a + b + (a - b)*Cos[2*(e + f*x)]))) - (4*b^3*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])/((a - b)*b^2*f)) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(-((a^2*Sin[2*(e + f*x)])/((a - b)*b^2*(-a - b - a*Cos[2*(e + f*x)] + b*Cos[2*(e + f*x)]))) + Tan[e + f*x]/(2*b^2)))/f","C",0
340,1,250,123,3.1954311,"\int \frac{\tan ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{a \sin (2 (e+f x)) \sec ^2(e+f x) \left(\frac{(a-b) \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{\sqrt{2}}-\frac{b \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)}{\sqrt{2}}-a+b\right)}{\sqrt{2} b f (a-b)^2 \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{b^{3/2} f}-\frac{a \tan (e+f x)}{b f (a-b) \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}",1,"(a*(-a + b + ((a - b)*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])/Sqrt[2] - (b*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])/Sqrt[2])*Sec[e + f*x]^2*Sin[2*(e + f*x)])/(Sqrt[2]*(a - b)^2*b*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])","C",1
341,1,154,81,3.2268118,"\int \frac{\tan ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\tan (e+f x) \left((a-b) \sqrt{\frac{b \tan ^2(e+f x)}{a}+1}+\sqrt{\frac{(b-a) \tan ^2(e+f x)}{a}} \left(a \cot ^2(e+f x)+b\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{(b-a) \tan ^2(e+f x)}{a}}}{\sqrt{\frac{b \tan ^2(e+f x)}{a}+1}}\right)\right)}{f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)} \sqrt{\frac{b \tan ^2(e+f x)}{a}+1}}","\frac{\tan (e+f x)}{f (a-b) \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}",1,"(Tan[e + f*x]*(ArcTanh[Sqrt[((-a + b)*Tan[e + f*x]^2)/a]/Sqrt[1 + (b*Tan[e + f*x]^2)/a]]*(b + a*Cot[e + f*x]^2)*Sqrt[((-a + b)*Tan[e + f*x]^2)/a] + (a - b)*Sqrt[1 + (b*Tan[e + f*x]^2)/a]))/((a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2]*Sqrt[1 + (b*Tan[e + f*x]^2)/a])","A",1
342,1,214,85,6.2839041,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[(a + b*Tan[e + f*x]^2)^(-3/2),x]","\frac{4 \sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)} \left(\frac{15 \left(3 a+2 b \tan ^2(e+f x)\right) \left(a \sqrt{\frac{(a-b) \sin ^2(2 (e+f x)) \left(a+b \tan ^2(e+f x)\right)}{a^2}}-2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)\right)}{\left(\frac{(a-b) \sin ^2(2 (e+f x)) \left(a+b \tan ^2(e+f x)\right)}{a^2}\right)^{3/2}}+a (a-b) \tan ^2(e+f x) \, _2F_1\left(2,2;\frac{7}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right)\right)}{15 a^4 f}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \tan (e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"(4*Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2]*(a*(a - b)*Hypergeometric2F1[2, 2, 7/2, ((a - b)*Sin[e + f*x]^2)/a]*Tan[e + f*x]^2 + (15*(3*a + 2*b*Tan[e + f*x]^2)*(-2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*(a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2) + a*Sqrt[((a - b)*Sin[2*(e + f*x)]^2*(a + b*Tan[e + f*x]^2))/a^2]))/(((a - b)*Sin[2*(e + f*x)]^2*(a + b*Tan[e + f*x]^2))/a^2)^(3/2)))/(15*a^4*f)","C",0
343,1,882,128,13.5994895,"\int \frac{\cot ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\cos ^2(e+f x) \cot (e+f x) \left(\frac{8 (a-b) b^2 \, _2F_1\left(2,2;\frac{7}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x) \tan ^4(e+f x)}{5 a^3}+\frac{8 (a-b) b^2 \, _3F_2\left(2,2,2;1,\frac{7}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x) \tan ^4(e+f x)}{15 a^3}-\frac{8 b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^4(e+f x)}{a^2 \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{3/2} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}+\frac{8 b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^4(e+f x)}{a^2 \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}+\frac{8 b^2 \sec ^2(e+f x) \tan ^2(e+f x)}{a (a-b)}+\frac{8 (a-b) b \, _2F_1\left(2,2;\frac{7}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x) \tan ^2(e+f x)}{3 a^2}+\frac{16 (a-b) b \, _3F_2\left(2,2,2;1,\frac{7}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x) \tan ^2(e+f x)}{15 a^2}-\frac{12 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^2(e+f x)}{a \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{3/2} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}+\frac{12 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^2(e+f x)}{a \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}+\frac{3 a \csc ^2(e+f x)}{a-b}+\frac{12 b \sec ^2(e+f x)}{a-b}+\frac{16 (a-b) \, _2F_1\left(2,2;\frac{7}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x)}{15 a}+\frac{8 (a-b) \, _3F_2\left(2,2,2;1,\frac{7}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x)}{15 a}-\frac{3 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right)}{\left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{3/2} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}+\frac{3 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right)}{\sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}\right)}{a f \sqrt{b \tan ^2(e+f x)+a}}","-\frac{(a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a^2 f (a-b)}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \cot (e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"-((Cos[e + f*x]^2*Cot[e + f*x]*((3*a*Csc[e + f*x]^2)/(a - b) + (12*b*Sec[e + f*x]^2)/(a - b) + (16*(a - b)*Hypergeometric2F1[2, 2, 7/2, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2)/(15*a) + (8*(a - b)*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2)/(15*a) + (8*b^2*Sec[e + f*x]^2*Tan[e + f*x]^2)/(a*(a - b)) + (8*(a - b)*b*Hypergeometric2F1[2, 2, 7/2, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^2)/(3*a^2) + (16*(a - b)*b*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^2)/(15*a^2) + (8*(a - b)*b^2*Hypergeometric2F1[2, 2, 7/2, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^4)/(5*a^3) + (8*(a - b)*b^2*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^4)/(15*a^3) - (3*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]])/((((a - b)*Sin[e + f*x]^2)/a)^(3/2)*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a]) - (12*b*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Tan[e + f*x]^2)/(a*(((a - b)*Sin[e + f*x]^2)/a)^(3/2)*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a]) - (8*b^2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Tan[e + f*x]^4)/(a^2*(((a - b)*Sin[e + f*x]^2)/a)^(3/2)*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a]) + (3*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]])/Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2] + (12*b*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Tan[e + f*x]^2)/(a*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2]) + (8*b^2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Tan[e + f*x]^4)/(a^2*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2])))/(a*f*Sqrt[a + b*Tan[e + f*x]^2]))","C",0
344,1,802,184,16.4458205,"\int \frac{\cot ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{-\frac{b \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac{4 b \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}-\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}\right)}{\sqrt{a+b+(a-b) \cos (2 (e+f x))}}}{(a-b) f}+\frac{\sqrt{\frac{\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(-\frac{\sin (2 (e+f x)) b^3}{a^3 (a-b) (\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x)))}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a^2}+\frac{(4 a \cos (e+f x)+5 b \cos (e+f x)) \csc (e+f x)}{3 a^3}\right)}{f}","\frac{(3 a-4 b) (a+2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 f (a-b)}-\frac{(a-4 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^2 f (a-b)}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \cot ^3(e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"(-((b*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*(a + b + (a - b)*Cos[2*(e + f*x)]))) - (4*b*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])/((a - b)*f) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(((4*a*Cos[e + f*x] + 5*b*Cos[e + f*x])*Csc[e + f*x])/(3*a^3) - (Cot[e + f*x]*Csc[e + f*x]^2)/(3*a^2) - (b^3*Sin[2*(e + f*x)])/(a^3*(a - b)*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)]))))/f","C",1
345,1,850,252,16.5567142,"\int \frac{\cot ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sqrt{\frac{\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\frac{\sin (2 (e+f x)) b^4}{a^4 (a-b) (\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x)))}-\frac{\cot (e+f x) \csc ^4(e+f x)}{5 a^2}+\frac{(11 a \cos (e+f x)+9 b \cos (e+f x)) \csc ^3(e+f x)}{15 a^3}+\frac{\left(-23 \cos (e+f x) a^2-34 b \cos (e+f x) a-33 b^2 \cos (e+f x)\right) \csc (e+f x)}{15 a^4}\right)}{f}-\frac{-\frac{b \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac{4 b \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}-\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}\right)}{\sqrt{a+b+(a-b) \cos (2 (e+f x))}}}{(a-b) f}","-\frac{(a-6 b) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a^2 f (a-b)}+\frac{\left(5 a^2+4 a b-24 b^2\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^3 f (a-b)}-\frac{\left(15 a^3+10 a^2 b+8 a b^2-48 b^3\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^4 f (a-b)}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \cot ^5(e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"-((-((b*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*(a + b + (a - b)*Cos[2*(e + f*x)]))) - (4*b*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])/((a - b)*f)) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(((-23*a^2*Cos[e + f*x] - 34*a*b*Cos[e + f*x] - 33*b^2*Cos[e + f*x])*Csc[e + f*x])/(15*a^4) + ((11*a*Cos[e + f*x] + 9*b*Cos[e + f*x])*Csc[e + f*x]^3)/(15*a^3) - (Cot[e + f*x]*Csc[e + f*x]^4)/(5*a^2) + (b^4*Sin[2*(e + f*x)])/(a^4*(a - b)*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)]))))/f","C",0
346,1,91,115,0.4561179,"\int \frac{\tan ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{b^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \tan ^2(e+f x)+a}{a-b}\right)-(a-b) \left(2 a+3 b \tan ^2(e+f x)-b\right)}{3 b^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}","\frac{a^2}{3 b^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{a (a-2 b)}{b^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}",1,"(b^2*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[e + f*x]^2)/(a - b)] - (a - b)*(2*a - b + 3*b*Tan[e + f*x]^2))/(3*(a - b)*b^2*f*(a + b*Tan[e + f*x]^2)^(3/2))","C",1
347,1,84,103,0.3032891,"\int \frac{\tan ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{a (b-a)-3 b \left(a+b \tan ^2(e+f x)\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{3 b f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)^{3/2}}","-\frac{a}{3 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{1}{f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}",1,"(a*(-a + b) - 3*b*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x]^2))/(3*(a - b)^2*b*f*(a + b*Tan[e + f*x]^2)^(3/2))","C",1
348,1,58,99,0.1373486,"\int \frac{\tan (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{3 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}","\frac{1}{f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{1}{3 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}",1,"Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[e + f*x]^2)/(a - b)]/(3*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2))","C",1
349,1,94,147,0.370653,"\int \frac{\cot (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{(a-b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \tan ^2(e+f x)}{a}+1\right)-a \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{b (2 a-b)}{a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{b}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}",1,"(-(a*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[e + f*x]^2)/(a - b)]) + (a - b)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Tan[e + f*x]^2)/a])/(3*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2))","C",1
350,1,138,206,0.6202151,"\int \frac{\cot ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{\cot ^2(e+f x) \left((a-b) \left((2 a+5 b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \tan ^2(e+f x)}{a}+1\right)+3 a \cot ^2(e+f x)\right)-2 a^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \tan ^2(e+f x)+a}{a-b}\right)\right)}{6 a^2 f (b-a) \sqrt{a+b \tan ^2(e+f x)} \left(a \cot ^2(e+f x)+b\right)}","\frac{(2 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{7/2} f}-\frac{b (3 a-5 b)}{6 a^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{b \left(a^2-8 a b+5 b^2\right)}{2 a^3 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}-\frac{\cot ^2(e+f x)}{2 a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"(Cot[e + f*x]^2*(-2*a^2*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[e + f*x]^2)/(a - b)] + (a - b)*(3*a*Cot[e + f*x]^2 + (2*a + 5*b)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Tan[e + f*x]^2)/a])))/(6*a^2*(-a + b)*f*(b + a*Cot[e + f*x]^2)*Sqrt[a + b*Tan[e + f*x]^2])","C",1
351,1,165,272,1.9913661,"\int \frac{\cot ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{\cot ^2(e+f x) \left(8 a^3 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \tan ^2(e+f x)+a}{a-b}\right)+(a-b) \left(3 a \cot ^2(e+f x) \left(2 a \cot ^2(e+f x)-4 a-7 b\right)-\left(8 a^2+20 a b+35 b^2\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \tan ^2(e+f x)}{a}+1\right)\right)\right)}{24 a^3 f (b-a) \sqrt{a+b \tan ^2(e+f x)} \left(a \cot ^2(e+f x)+b\right)}","\frac{(4 a+7 b) \cot ^2(e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\left(8 a^2+20 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{9/2} f}+\frac{b \left(12 a^2+15 a b-35 b^2\right)}{24 a^3 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{b \left(4 a^3+3 a^2 b-50 a b^2+35 b^3\right)}{8 a^4 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}-\frac{\cot ^4(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"(Cot[e + f*x]^2*(8*a^3*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[e + f*x]^2)/(a - b)] + (a - b)*(3*a*Cot[e + f*x]^2*(-4*a - 7*b + 2*a*Cot[e + f*x]^2) - (8*a^2 + 20*a*b + 35*b^2)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Tan[e + f*x]^2)/a])))/(24*a^3*(-a + b)*f*(b + a*Cot[e + f*x]^2)*Sqrt[a + b*Tan[e + f*x]^2])","C",1
352,1,295,171,4.5761754,"\int \frac{\tan ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{\sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left(a^2 (a-b) \sin (2 (e+f x)) ((3 a-7 b) ((a-b) \cos (2 (e+f x))+a+b)+2 a b)-\frac{3 a^2 b \sin ^2(e+f x) \sin (2 (e+f x)) \left(\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}\right)^{3/2} \left(\left(a^2-3 a b+2 b^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)+b^2 \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right)\right)}{\sqrt{2}}\right)}{3 \sqrt{2} a b^2 f (a-b)^3 ((a-b) \cos (2 (e+f x))+a+b)^2}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{b^{5/2} f}-\frac{a (a-2 b) \tan (e+f x)}{b^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}-\frac{a \tan ^3(e+f x)}{3 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"-1/3*(Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2]*(a^2*(a - b)*(2*a*b + (3*a - 7*b)*(a + b + (a - b)*Cos[2*(e + f*x)]))*Sin[2*(e + f*x)] - (3*a^2*b*(((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)^(3/2)*((a^2 - 3*a*b + 2*b^2)*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1] + b^2*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1])*Sin[e + f*x]^2*Sin[2*(e + f*x)])/Sqrt[2]))/(Sqrt[2]*a*(a - b)^3*b^2*f*(a + b + (a - b)*Cos[2*(e + f*x)])^2)","C",1
353,1,260,131,6.0539941,"\int \frac{\tan ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{\tan ^5(e+f x) \left(\frac{b \tan ^2(e+f x)}{a}+1\right) \left(-\frac{\left(\frac{b \tan ^2(e+f x)}{a}-\tan ^2(e+f x)\right)^2}{3 \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^2}-\frac{\frac{b \tan ^2(e+f x)}{a}-\tan ^2(e+f x)}{\frac{b \tan ^2(e+f x)}{a}+1}+\frac{\sqrt{\frac{b \tan ^2(e+f x)}{a}-\tan ^2(e+f x)} \tanh ^{-1}\left(\frac{\sqrt{\frac{b \tan ^2(e+f x)}{a}-\tan ^2(e+f x)}}{\sqrt{\frac{b \tan ^2(e+f x)}{a}+1}}\right)}{\sqrt{\frac{b \tan ^2(e+f x)}{a}+1}}\right)}{a^2 f \sqrt{a+b \tan ^2(e+f x)} \left(\frac{b \tan ^2(e+f x)}{a}-\tan ^2(e+f x)\right)^3}","\frac{(a-4 b) \tan (e+f x)}{3 b f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{a \tan (e+f x)}{3 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}",1,"(Tan[e + f*x]^5*(1 + (b*Tan[e + f*x]^2)/a)*((ArcTanh[Sqrt[-Tan[e + f*x]^2 + (b*Tan[e + f*x]^2)/a]/Sqrt[1 + (b*Tan[e + f*x]^2)/a]]*Sqrt[-Tan[e + f*x]^2 + (b*Tan[e + f*x]^2)/a])/Sqrt[1 + (b*Tan[e + f*x]^2)/a] - (-Tan[e + f*x]^2 + (b*Tan[e + f*x]^2)/a)/(1 + (b*Tan[e + f*x]^2)/a) - (-Tan[e + f*x]^2 + (b*Tan[e + f*x]^2)/a)^2/(3*(1 + (b*Tan[e + f*x]^2)/a)^2)))/(a^2*f*Sqrt[a + b*Tan[e + f*x]^2]*(-Tan[e + f*x]^2 + (b*Tan[e + f*x]^2)/a)^3)","A",1
354,1,365,128,7.7742218,"\int \frac{\tan ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{\cos ^4(e+f x) \cot (e+f x) \left(12 (a-b)^3 \tan ^6(e+f x) \left(a+b \tan ^2(e+f x)\right) \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sqrt{\frac{\sin ^2(e+f x) \cos ^2(e+f x) \left(a^2+a b \left(\tan ^2(e+f x)-1\right)-b^2 \tan ^2(e+f x)\right)}{a^2}}+35 a \sec ^2(e+f x) \left(5 a+2 b \tan ^2(e+f x)\right) \left(a \sec ^2(e+f x) \left(a \left(\tan ^2(e+f x)-3\right)-4 b \tan ^2(e+f x)\right) \sqrt{\frac{\sin ^2(e+f x) \cos ^2(e+f x) \left(a^2+a b \left(\tan ^2(e+f x)-1\right)-b^2 \tan ^2(e+f x)\right)}{a^2}}+3 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \left(a+b \tan ^2(e+f x)\right)^2\right)\right)}{315 a^4 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)} \left(\frac{b \tan ^2(e+f x)}{a}+1\right) \sqrt{\frac{(a-b) \sin ^2(e+f x) \cos ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)}{a^2}}}","\frac{(2 a+b) \tan (e+f x)}{3 a f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tan (e+f x)}{3 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}",1,"(Cos[e + f*x]^4*Cot[e + f*x]*(12*(a - b)^3*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Sin[e + f*x]^2)/a]*Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)*Sqrt[(Cos[e + f*x]^2*Sin[e + f*x]^2*(a^2 - b^2*Tan[e + f*x]^2 + a*b*(-1 + Tan[e + f*x]^2)))/a^2] + 35*a*Sec[e + f*x]^2*(5*a + 2*b*Tan[e + f*x]^2)*(3*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*(a + b*Tan[e + f*x]^2)^2 + a*Sec[e + f*x]^2*(-4*b*Tan[e + f*x]^2 + a*(-3 + Tan[e + f*x]^2))*Sqrt[(Cos[e + f*x]^2*Sin[e + f*x]^2*(a^2 - b^2*Tan[e + f*x]^2 + a*b*(-1 + Tan[e + f*x]^2)))/a^2])))/(315*a^4*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2]*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2]*(1 + (b*Tan[e + f*x]^2)/a))","C",0
355,1,1331,134,7.6005535,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[(a + b*Tan[e + f*x]^2)^(-5/2),x]","\frac{\cos (e+f x) \sin (e+f x) \left(\frac{840 (a-b)^2 b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^4(e+f x) \sin ^4(e+f x)}{a^4}+\frac{2100 (a-b)^2 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^2(e+f x) \sin ^4(e+f x)}{a^3}+\frac{1575 (a-b)^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \sin ^4(e+f x)}{a^2}-\frac{1680 (a-b) b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^4(e+f x) \sin ^2(e+f x)}{a^3}-\frac{4200 (a-b) b \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^2(e+f x) \sin ^2(e+f x)}{a^2}-\frac{3150 (a-b) \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \sin ^2(e+f x)}{a}+\frac{840 b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^4(e+f x)}{a^2}+\frac{2100 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^2(e+f x)}{a}+1575 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right)+\frac{72 b^2 \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{7/2} \tan ^4(e+f x) \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}{a^2}+\frac{24 b^2 \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{7/2} \tan ^4(e+f x) \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}{a^2}+\frac{1120 b^2 \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{3/2} \tan ^4(e+f x) \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}{a^2}+96 \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{7/2} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}+24 \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{7/2} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}+\frac{168 b \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{7/2} \tan ^2(e+f x) \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}{a}+\frac{48 b \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{7/2} \tan ^2(e+f x) \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}{a}+\frac{2800 b \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{3/2} \tan ^2(e+f x) \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}{a}+2100 \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{3/2} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}-\frac{840 b^2 \tan ^4(e+f x) \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}{a^2}-\frac{2100 b \tan ^2(e+f x) \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}{a}-1575 \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}\right)}{315 a^2 f \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{5/2} \sqrt{b \tan ^2(e+f x)+a} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}} \left(\frac{b \tan ^2(e+f x)}{a}+1\right)}","-\frac{b (5 a-2 b) \tan (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{b \tan (e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}",1,"(Cos[e + f*x]*Sin[e + f*x]*(1575*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]] - (3150*(a - b)*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sin[e + f*x]^2)/a + (1575*(a - b)^2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sin[e + f*x]^4)/a^2 + (2100*b*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Tan[e + f*x]^2)/a - (4200*(a - b)*b*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sin[e + f*x]^2*Tan[e + f*x]^2)/a^2 + (2100*(a - b)^2*b*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sin[e + f*x]^4*Tan[e + f*x]^2)/a^3 + (840*b^2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Tan[e + f*x]^4)/a^2 - (1680*(a - b)*b^2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sin[e + f*x]^2*Tan[e + f*x]^4)/a^3 + (840*(a - b)^2*b^2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sin[e + f*x]^4*Tan[e + f*x]^4)/a^4 + 2100*(((a - b)*Sin[e + f*x]^2)/a)^(3/2)*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a] + 96*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Sin[e + f*x]^2)/a]*(((a - b)*Sin[e + f*x]^2)/a)^(7/2)*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a] + 24*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Sin[e + f*x]^2)/a]*(((a - b)*Sin[e + f*x]^2)/a)^(7/2)*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a] + (2800*b*(((a - b)*Sin[e + f*x]^2)/a)^(3/2)*Tan[e + f*x]^2*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a])/a + (168*b*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Sin[e + f*x]^2)/a]*(((a - b)*Sin[e + f*x]^2)/a)^(7/2)*Tan[e + f*x]^2*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a])/a + (48*b*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Sin[e + f*x]^2)/a]*(((a - b)*Sin[e + f*x]^2)/a)^(7/2)*Tan[e + f*x]^2*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a])/a + (1120*b^2*(((a - b)*Sin[e + f*x]^2)/a)^(3/2)*Tan[e + f*x]^4*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a])/a^2 + (72*b^2*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Sin[e + f*x]^2)/a]*(((a - b)*Sin[e + f*x]^2)/a)^(7/2)*Tan[e + f*x]^4*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a])/a^2 + (24*b^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Sin[e + f*x]^2)/a]*(((a - b)*Sin[e + f*x]^2)/a)^(7/2)*Tan[e + f*x]^4*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a])/a^2 - 1575*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2] - (2100*b*Tan[e + f*x]^2*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2])/a - (840*b^2*Tan[e + f*x]^4*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2])/a^2))/(315*a^2*f*(((a - b)*Sin[e + f*x]^2)/a)^(5/2)*Sqrt[a + b*Tan[e + f*x]^2]*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a]*(1 + (b*Tan[e + f*x]^2)/a))","C",0
356,1,1890,186,15.9178626,"\int \frac{\cot ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{\cos ^2(e+f x) \cot (e+f x) \left(\frac{176 (a-b) b^3 \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x) \tan ^6(e+f x)}{105 a^4}+\frac{32 (a-b) b^3 \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x) \tan ^6(e+f x)}{35 a^4}+\frac{16 (a-b) b^3 \, _4F_3\left(2,2,2,2;1,1,\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x) \tan ^6(e+f x)}{105 a^4}+\frac{16 b^3 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^6(e+f x)}{a^3 \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{5/2} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}+\frac{16 b^3 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^6(e+f x)}{a^3 \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}+\frac{64 b^3 \sec ^2(e+f x) \tan ^4(e+f x)}{3 a^2 (a-b)}+\frac{152 (a-b) b^2 \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x) \tan ^4(e+f x)}{35 a^3}+\frac{88 (a-b) b^2 \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x) \tan ^4(e+f x)}{35 a^3}+\frac{16 (a-b) b^2 \, _4F_3\left(2,2,2,2;1,1,\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x) \tan ^4(e+f x)}{35 a^3}+\frac{40 b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^4(e+f x)}{a^2 \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{5/2} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}-\frac{32 b^3 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \sec ^2(e+f x) \tan ^4(e+f x)}{a^2 (a-b) \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}+\frac{40 b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^4(e+f x)}{a^2 \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}+\frac{160 b^2 \sec ^2(e+f x) \tan ^2(e+f x)}{3 a (a-b)}+\frac{124 (a-b) b \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x) \tan ^2(e+f x)}{35 a^2}+\frac{16 (a-b) b \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x) \tan ^2(e+f x)}{7 a^2}+\frac{16 (a-b) b \, _4F_3\left(2,2,2,2;1,1,\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x) \tan ^2(e+f x)}{35 a^2}+\frac{30 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^2(e+f x)}{a \left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{5/2} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}-\frac{80 b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \sec ^2(e+f x) \tan ^2(e+f x)}{a (a-b) \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}+\frac{30 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \tan ^2(e+f x)}{a \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}-\frac{5 a^2 \csc ^4(e+f x)}{(a-b)^2}-\frac{40 b^2 \sec ^4(e+f x)}{(a-b)^2}+\frac{20 a \csc ^2(e+f x)}{3 (a-b)}-\frac{30 a b \csc ^2(e+f x) \sec ^2(e+f x)}{(a-b)^2}+\frac{40 b \sec ^2(e+f x)}{a-b}+\frac{92 (a-b) \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x)}{105 a}+\frac{24 (a-b) \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x)}{35 a}+\frac{16 (a-b) \, _4F_3\left(2,2,2,2;1,1,\frac{9}{2};\frac{(a-b) \sin ^2(e+f x)}{a}\right) \sin ^2(e+f x)}{105 a}-\frac{16 b^3 \left(\tan ^3(e+f x)+\tan (e+f x)\right)^2}{a (a-b)^2}+\frac{5 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right)}{\left(\frac{(a-b) \sin ^2(e+f x)}{a}\right)^{5/2} \sqrt{\frac{\cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a}}}-\frac{10 a \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \csc ^2(e+f x)}{(a-b) \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}-\frac{60 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right) \sec ^2(e+f x)}{(a-b) \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}+\frac{5 \sin ^{-1}\left(\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right)}{\sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)}{a^2}}}\right)}{a^2 f \sqrt{b \tan ^2(e+f x)+a} \left(\frac{b \tan ^2(e+f x)}{a}+1\right)}","-\frac{(a-4 b) (3 a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 f (a-b)^2}-\frac{b (7 a-4 b) \cot (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}-\frac{b \cot (e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"-((Cos[e + f*x]^2*Cot[e + f*x]*((20*a*Csc[e + f*x]^2)/(3*(a - b)) - (5*a^2*Csc[e + f*x]^4)/(a - b)^2 + (40*b*Sec[e + f*x]^2)/(a - b) - (30*a*b*Csc[e + f*x]^2*Sec[e + f*x]^2)/(a - b)^2 - (40*b^2*Sec[e + f*x]^4)/(a - b)^2 + (92*(a - b)*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2)/(105*a) + (24*(a - b)*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2)/(35*a) + (16*(a - b)*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2)/(105*a) + (160*b^2*Sec[e + f*x]^2*Tan[e + f*x]^2)/(3*a*(a - b)) + (124*(a - b)*b*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^2)/(35*a^2) + (16*(a - b)*b*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^2)/(7*a^2) + (16*(a - b)*b*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^2)/(35*a^2) + (64*b^3*Sec[e + f*x]^2*Tan[e + f*x]^4)/(3*a^2*(a - b)) + (152*(a - b)*b^2*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^4)/(35*a^3) + (88*(a - b)*b^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^4)/(35*a^3) + (16*(a - b)*b^2*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^4)/(35*a^3) + (176*(a - b)*b^3*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^6)/(105*a^4) + (32*(a - b)*b^3*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^6)/(35*a^4) + (16*(a - b)*b^3*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^6)/(105*a^4) + (5*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]])/((((a - b)*Sin[e + f*x]^2)/a)^(5/2)*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a]) + (30*b*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Tan[e + f*x]^2)/(a*(((a - b)*Sin[e + f*x]^2)/a)^(5/2)*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a]) + (40*b^2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Tan[e + f*x]^4)/(a^2*(((a - b)*Sin[e + f*x]^2)/a)^(5/2)*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a]) + (16*b^3*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Tan[e + f*x]^6)/(a^3*(((a - b)*Sin[e + f*x]^2)/a)^(5/2)*Sqrt[(Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a]) + (5*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]])/Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2] - (10*a*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Csc[e + f*x]^2)/((a - b)*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2]) - (60*b*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sec[e + f*x]^2)/((a - b)*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2]) + (30*b*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Tan[e + f*x]^2)/(a*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2]) - (80*b^2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sec[e + f*x]^2*Tan[e + f*x]^2)/(a*(a - b)*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2]) + (40*b^2*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Tan[e + f*x]^4)/(a^2*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2]) - (32*b^3*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Sec[e + f*x]^2*Tan[e + f*x]^4)/(a^2*(a - b)*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2]) + (16*b^3*ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]]*Tan[e + f*x]^6)/(a^3*Sqrt[((a - b)*Cos[e + f*x]^2*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2]) - (16*b^3*(Tan[e + f*x] + Tan[e + f*x]^3)^2)/(a*(a - b)^2)))/(a^2*f*Sqrt[a + b*Tan[e + f*x]^2]*(1 + (b*Tan[e + f*x]^2)/a)))","C",0
357,1,871,249,16.5907368,"\int \frac{\cot ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{-\frac{b \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac{4 b \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}-\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}\right)}{\sqrt{a+b+(a-b) \cos (2 (e+f x))}}}{(a-b)^2 f}+\frac{\sqrt{\frac{\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\frac{2 \sin (2 (e+f x)) b^4}{3 a^3 (a-b)^2 (\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x)))^2}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a^3}+\frac{4 (a \cos (e+f x)+2 b \cos (e+f x)) \csc (e+f x)}{3 a^4}-\frac{4 \left(3 a b^3 \sin (2 (e+f x))-2 b^4 \sin (2 (e+f x))\right)}{3 a^4 (a-b)^2 (\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x)))}\right)}{f}","-\frac{b (3 a-2 b) \cot ^3(e+f x)}{a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{(a-2 b) \left(3 a^2+8 a b-8 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^4 f (a-b)^2}-\frac{\left(a^2-12 a b+8 b^2\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 f (a-b)^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}-\frac{b \cot ^3(e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"(-((b*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*(a + b + (a - b)*Cos[2*(e + f*x)]))) - (4*b*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])/((a - b)^2*f) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((4*(a*Cos[e + f*x] + 2*b*Cos[e + f*x])*Csc[e + f*x])/(3*a^4) - (Cot[e + f*x]*Csc[e + f*x]^2)/(3*a^3) + (2*b^4*Sin[2*(e + f*x)])/(3*a^3*(a - b)^2*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])^2) - (4*(3*a*b^3*Sin[2*(e + f*x)] - 2*b^4*Sin[2*(e + f*x)]))/(3*a^4*(a - b)^2*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)]))))/f","C",0
358,1,921,327,16.7717706,"\int \frac{\cot ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{\sqrt{\frac{\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(-\frac{2 \sin (2 (e+f x)) b^5}{3 a^4 (a-b)^2 (\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x)))^2}-\frac{\cot (e+f x) \csc ^4(e+f x)}{5 a^3}+\frac{(11 a \cos (e+f x)+14 b \cos (e+f x)) \csc ^3(e+f x)}{15 a^4}+\frac{\left(-23 \cos (e+f x) a^2-54 b \cos (e+f x) a-73 b^2 \cos (e+f x)\right) \csc (e+f x)}{15 a^5}+\frac{15 a b^4 \sin (2 (e+f x))-11 b^5 \sin (2 (e+f x))}{3 a^5 (a-b)^2 (\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x)))}\right)}{f}-\frac{-\frac{b \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac{4 b \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left(\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}-\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \Pi \left(-\frac{b}{a-b};\left.\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right)\right|1\right) \sin ^4(e+f x)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}\right)}{\sqrt{a+b+(a-b) \cos (2 (e+f x))}}}{(a-b)^2 f}","-\frac{b (11 a-8 b) \cot ^5(e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left(a^2-22 a b+16 b^2\right) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a^3 f (a-b)^2}+\frac{\left(5 a^3+4 a^2 b-88 a b^2+64 b^3\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^4 f (a-b)^2}-\frac{\left(15 a^4+10 a^3 b+8 a^2 b^2-176 a b^3+128 b^4\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^5 f (a-b)^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}-\frac{b \cot ^5(e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"-((-((b*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*(a + b + (a - b)*Cos[2*(e + f*x)]))) - (4*b*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) - (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])/((a - b)^2*f)) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(((-23*a^2*Cos[e + f*x] - 54*a*b*Cos[e + f*x] - 73*b^2*Cos[e + f*x])*Csc[e + f*x])/(15*a^5) + ((11*a*Cos[e + f*x] + 14*b*Cos[e + f*x])*Csc[e + f*x]^3)/(15*a^4) - (Cot[e + f*x]*Csc[e + f*x]^4)/(5*a^3) - (2*b^5*Sin[2*(e + f*x)])/(3*a^4*(a - b)^2*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])^2) + (15*a*b^4*Sin[2*(e + f*x)] - 11*b^5*Sin[2*(e + f*x)])/(3*a^5*(a - b)^2*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)]))))/f","C",0
359,1,74,72,0.0894681,"\int (d \tan (e+f x))^m \left(b \tan ^2(e+f x)\right)^p \, dx","Integrate[(d*Tan[e + f*x])^m*(b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (m+2 p+1);\frac{1}{2} (m+2 p+1)+1;-\tan ^2(e+f x)\right)}{f (m+2 p+1)}","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (m+2 p+1);\frac{1}{2} (m+2 p+3);-\tan ^2(e+f x)\right)}{f (m+2 p+1)}",1,"(Hypergeometric2F1[1, (1 + m + 2*p)/2, 1 + (1 + m + 2*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(d*Tan[e + f*x])^m*(b*Tan[e + f*x]^2)^p)/(f*(1 + m + 2*p))","A",1
360,1,101,100,0.3089205,"\int (d \tan (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[(d*Tan[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) (d \tan (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}{f (m+1)}","\frac{(d \tan (e+f x))^{m+1} \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2};1,-p;\frac{m+3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{d f (m+1)}",1,"(AppellF1[(1 + m)/2, -p, 1, (3 + m)/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Tan[e + f*x]*(d*Tan[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + m)*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",0
361,1,106,129,0.9080233,"\int \tan ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \left(b^2 (p+2) \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)+(a-b) \left(a-b (p+1) \tan ^2(e+f x)+b (p+2)\right)\right)}{2 b^2 f (p+1) (p+2) (b-a)}","-\frac{(a+b) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{2 b^2 f (p+1)}+\frac{\left(a+b \tan ^2(e+f x)\right)^{p+2}}{2 b^2 f (p+2)}-\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}",1,"((a + b*Tan[e + f*x]^2)^(1 + p)*(b^2*(2 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)] + (a - b)*(a + b*(2 + p) - b*(1 + p)*Tan[e + f*x]^2)))/(2*b^2*(-a + b)*f*(1 + p)*(2 + p))","A",1
362,1,73,95,0.1358948,"\int \tan ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \left(b \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)+a-b\right)}{2 b f (p+1) (b-a)}","\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}+\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1}}{2 b f (p+1)}",1,"-1/2*((a - b + b*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)])*(a + b*Tan[e + f*x]^2)^(1 + p))/(b*(-a + b)*f*(1 + p))","A",1
363,1,63,63,0.0999343,"\int \tan (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}","-\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}",1,"-1/2*(Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x]^2)^(1 + p))/((a - b)*f*(1 + p))","A",1
364,1,98,118,0.1737027,"\int \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \left(a \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)+(b-a) \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)}{a}+1\right)\right)}{2 a f (p+1) (a-b)}","\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}-\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)}{a}+1\right)}{2 a f (p+1)}",1,"((a*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)] + (-a + b)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Tan[e + f*x]^2)/a])*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*a*(a - b)*f*(1 + p))","A",1
365,1,142,158,0.7407592,"\int \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\tan ^2(e+f x) \left(a \cot ^2(e+f x)+b\right) \left(a+b \tan ^2(e+f x)\right)^p \left(a^2 \left(-\, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)\right)-(a-b) \left((b p-a) \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)}{a}+1\right)+a (p+1) \cot ^2(e+f x)\right)\right)}{2 a^2 f (p+1) (a-b)}","\frac{(a-b p) \left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)}{a}+1\right)}{2 a^2 f (p+1)}-\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}-\frac{\cot ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{2 a f}",1,"((b + a*Cot[e + f*x]^2)*(-(a^2*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]) - (a - b)*(a*(1 + p)*Cot[e + f*x]^2 + (-a + b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Tan[e + f*x]^2)/a]))*Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p)/(2*a^2*(a - b)*f*(1 + p))","A",1
366,1,172,217,2.7579603,"\int \cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\tan ^2(e+f x) \left(a \cot ^2(e+f x)+b\right) \left(a+b \tan ^2(e+f x)\right)^p \left((a-b) \left(\left(2 a^2-2 a b p+b^2 (p-1) p\right) \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)}{a}+1\right)+a (p+1) \cot ^2(e+f x) \left(a \cot ^2(e+f x)-2 a+b (p-1)\right)\right)-2 a^3 \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)\right)}{4 a^3 f (p+1) (a-b)}","\frac{(2 a-b p+b) \cot ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{4 a^2 f}-\frac{\left(2 a^2-2 a b p-b^2 (1-p) p\right) \left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)}{a}+1\right)}{4 a^3 f (p+1)}+\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}-\frac{\cot ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{4 a f}",1,"-1/4*((b + a*Cot[e + f*x]^2)*(-2*a^3*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)] + (a - b)*(a*(1 + p)*Cot[e + f*x]^2*(-2*a + b*(-1 + p) + a*Cot[e + f*x]^2) + (2*a^2 - 2*a*b*p + b^2*(-1 + p)*p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Tan[e + f*x]^2)/a]))*Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p)/(a^3*(a - b)*f*(1 + p))","A",1
367,0,0,83,3.5385044,"\int \tan ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^p,x]","\int \tan ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","\frac{\tan ^7(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{7}{2};1,-p;\frac{9}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{7 f}",1,"Integrate[Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^p, x]","F",-1
368,1,1896,83,16.5917252,"\int \tan ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Tan[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{2 \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a}\right) \tan (e+f x) \left(b \tan ^2(e+f x)+a\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}}{f}+\frac{\tan (e+f x) \left(b \tan ^2(e+f x)+a\right)^p \left(\left(b \tan ^2(e+f x)+a\right) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^p+(b (2 p+3)-a) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a}\right)\right) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}}{b f (2 p+3)}+\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \cos (e+f x) \sin (e+f x) \left(b \tan ^2(e+f x)+a\right)^{2 p}}{f \left(2 \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)+3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \left(\frac{6 a b p F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \tan ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)^{p-1}}{2 \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)+3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}-\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \cos (e+f x) \sin (e+f x) \left(4 \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan (e+f x) \sec ^2(e+f x)+3 a \left(\frac{2 b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{3 a}-\frac{2}{3} F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)\right)+2 \tan ^2(e+f x) \left(b p \left(-\frac{6}{5} F_1\left(\frac{5}{2};1-p,2;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)-\frac{6 b (1-p) F_1\left(\frac{5}{2};2-p,1;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)}{5 a}\right)-a \left(\frac{6 b p F_1\left(\frac{5}{2};1-p,2;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{5 a}-\frac{12}{5} F_1\left(\frac{5}{2};-p,3;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)\right)\right)\right) \left(b \tan ^2(e+f x)+a\right)^p}{\left(2 \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)+3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right){}^2}+\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)^p}{2 \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)+3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}-\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)^p}{2 \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)+3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}+\frac{3 a \cos (e+f x) \sin (e+f x) \left(\frac{2 b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{3 a}-\frac{2}{3} F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)\right) \left(b \tan ^2(e+f x)+a\right)^p}{2 \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)+3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}\right)}","\frac{\tan ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{5}{2};1,-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{5 f}",1,"(-2*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p) + (Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p*((-a + b*(3 + 2*p))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/a)] + (a + b*Tan[e + f*x]^2)*(1 + (b*Tan[e + f*x]^2)/a)^p))/(b*f*(3 + 2*p)*(1 + (b*Tan[e + f*x]^2)/a)^p) + (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]*Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^(2*p))/(f*(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)*((6*a*b*p*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(-1 + p))/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p)/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p)/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (3*a*Cos[e + f*x]*Sin[e + f*x]*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3)*(a + b*Tan[e + f*x]^2)^p)/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]*Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^p*(4*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e + f*x] + 3*a*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a)) - a*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2))","B",0
369,1,1992,83,15.1031594,"\int \tan ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\tan ^3(e+f x) \left(b \tan ^2(e+f x)+a\right)^{2 p} \left(\, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a}\right) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}+\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \cos ^2(e+f x)}{2 \left(a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)-3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}\right)}{f \left(2 b p \sec ^2(e+f x) \tan ^2(e+f x) \left(\, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a}\right) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}+\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \cos ^2(e+f x)}{2 \left(a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)-3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}\right) \left(b \tan ^2(e+f x)+a\right)^{p-1}+\sec ^2(e+f x) \left(\, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a}\right) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}+\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \cos ^2(e+f x)}{2 \left(a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)-3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}\right) \left(b \tan ^2(e+f x)+a\right)^p+\tan (e+f x) \left(-\frac{2 b p \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a}\right) \sec ^2(e+f x) \tan (e+f x) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p-1}}{a}+\csc (e+f x) \sec (e+f x) \left(\left(\frac{b \tan ^2(e+f x)}{a}+1\right)^p-\, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a}\right)\right) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}-\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \cos ^2(e+f x) \left(4 \left(a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan (e+f x) \sec ^2(e+f x)-3 a \left(\frac{2 b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{3 a}-\frac{2}{3} F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)\right)+2 \tan ^2(e+f x) \left(a \left(\frac{6 b p F_1\left(\frac{5}{2};1-p,2;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{5 a}-\frac{12}{5} F_1\left(\frac{5}{2};-p,3;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)\right)-b p \left(-\frac{6}{5} F_1\left(\frac{5}{2};1-p,2;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)-\frac{6 b (1-p) F_1\left(\frac{5}{2};2-p,1;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)}{5 a}\right)\right)\right)}{\left(2 \left(a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)-3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right){}^2}-\frac{6 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \cos (e+f x) \sin (e+f x)}{2 \left(a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)-3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}+\frac{3 a \cos ^2(e+f x) \left(\frac{2 b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{3 a}-\frac{2}{3} F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)\right)}{2 \left(a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)-3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}\right) \left(b \tan ^2(e+f x)+a\right)^p\right)}","\frac{\tan ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{3}{2};1,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{3 f}",1,"(Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(2*p)*(Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/a)]/(1 + (b*Tan[e + f*x]^2)/a)^p + (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]^2)/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)))/(f*(2*b*p*Sec[e + f*x]^2*Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(-1 + p)*(Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/a)]/(1 + (b*Tan[e + f*x]^2)/a)^p + (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]^2)/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) + Sec[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p*(Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/a)]/(1 + (b*Tan[e + f*x]^2)/a)^p + (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]^2)/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) + Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p*((-2*b*p*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x]*(1 + (b*Tan[e + f*x]^2)/a)^(-1 - p))/a - (6*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]*Sin[e + f*x])/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (3*a*Cos[e + f*x]^2*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (Csc[e + f*x]*Sec[e + f*x]*(-Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/a)] + (1 + (b*Tan[e + f*x]^2)/a)^p))/(1 + (b*Tan[e + f*x]^2)/a)^p - (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]^2*(4*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e + f*x] - 3*a*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(-(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a))) + a*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2)))","B",0
370,1,192,78,0.4869158,"\int \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[(a + b*Tan[e + f*x]^2)^p,x]","\frac{3 a \sin (2 (e+f x)) \left(a+b \tan ^2(e+f x)\right)^p F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}{4 f \tan ^2(e+f x) \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right)+6 a f F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}",1,"(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sin[2*(e + f*x)]*(a + b*Tan[e + f*x]^2)^p)/(6*a*f*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 4*f*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)","B",0
371,1,1989,79,14.9339976,"\int \cot ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\cot ^3(e+f x) \left(b \tan ^2(e+f x)+a\right)^{2 p} \left(\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sin ^2(e+f x)}{2 \left(a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)-3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}-\, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}\right)}{f \left(2 b p \sec ^2(e+f x) \left(\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sin ^2(e+f x)}{2 \left(a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)-3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}-\, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}\right) \left(b \tan ^2(e+f x)+a\right)^{p-1}-\csc ^2(e+f x) \left(\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sin ^2(e+f x)}{2 \left(a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)-3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}-\, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}\right) \left(b \tan ^2(e+f x)+a\right)^p+\cot (e+f x) \left(\frac{2 b p \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right) \sec ^2(e+f x) \tan (e+f x) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p-1}}{a}-\csc (e+f x) \sec (e+f x) \left(\, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right)-\left(\frac{b \tan ^2(e+f x)}{a}+1\right)^p\right) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}-\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sin ^2(e+f x) \left(4 \left(a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan (e+f x) \sec ^2(e+f x)-3 a \left(\frac{2 b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{3 a}-\frac{2}{3} F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)\right)+2 \tan ^2(e+f x) \left(a \left(\frac{6 b p F_1\left(\frac{5}{2};1-p,2;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{5 a}-\frac{12}{5} F_1\left(\frac{5}{2};-p,3;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)\right)-b p \left(-\frac{6}{5} F_1\left(\frac{5}{2};1-p,2;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)-\frac{6 b (1-p) F_1\left(\frac{5}{2};2-p,1;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)}{5 a}\right)\right)\right)}{\left(2 \left(a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)-3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right){}^2}+\frac{6 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \cos (e+f x) \sin (e+f x)}{2 \left(a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)-3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}+\frac{3 a \sin ^2(e+f x) \left(\frac{2 b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{3 a}-\frac{2}{3} F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)\right)}{2 \left(a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)-3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}\right) \left(b \tan ^2(e+f x)+a\right)^p\right)}","-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{1}{2};1,-p;\frac{1}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}",1,"(Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(2*p)*(-(Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]/(1 + (b*Tan[e + f*x]^2)/a)^p) + (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sin[e + f*x]^2)/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)))/(f*(2*b*p*Sec[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(-1 + p)*(-(Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]/(1 + (b*Tan[e + f*x]^2)/a)^p) + (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sin[e + f*x]^2)/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) - Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p*(-(Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]/(1 + (b*Tan[e + f*x]^2)/a)^p) + (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sin[e + f*x]^2)/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) + Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^p*((2*b*p*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x]*(1 + (b*Tan[e + f*x]^2)/a)^(-1 - p))/a + (6*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]*Sin[e + f*x])/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (3*a*Sin[e + f*x]^2*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (Csc[e + f*x]*Sec[e + f*x]*(Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)] - (1 + (b*Tan[e + f*x]^2)/a)^p))/(1 + (b*Tan[e + f*x]^2)/a)^p - (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sin[e + f*x]^2*(4*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e + f*x] - 3*a*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(-(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a))) + a*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2)))","B",0
372,1,1887,83,6.8668706,"\int \cot ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p,x]","\frac{2 \cot (e+f x) \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right) \left(b \tan ^2(e+f x)+a\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}}{f}+\frac{\cot ^3(e+f x) \left(b \tan ^2(e+f x)+a\right)^p \left(-\left((3 a+b (2 p-1)) \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right) \tan ^2(e+f x) \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p}\right)-b \tan ^2(e+f x)-a\right)}{3 a f}+\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \cos (e+f x) \sin (e+f x) \left(b \tan ^2(e+f x)+a\right)^{2 p}}{f \left(2 \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)+3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \left(\frac{6 a b p F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \tan ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)^{p-1}}{2 \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)+3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}-\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \cos (e+f x) \sin (e+f x) \left(4 \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan (e+f x) \sec ^2(e+f x)+3 a \left(\frac{2 b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{3 a}-\frac{2}{3} F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)\right)+2 \tan ^2(e+f x) \left(b p \left(-\frac{6}{5} F_1\left(\frac{5}{2};1-p,2;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)-\frac{6 b (1-p) F_1\left(\frac{5}{2};2-p,1;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)}{5 a}\right)-a \left(\frac{6 b p F_1\left(\frac{5}{2};1-p,2;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{5 a}-\frac{12}{5} F_1\left(\frac{5}{2};-p,3;\frac{7}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)\right)\right)\right) \left(b \tan ^2(e+f x)+a\right)^p}{\left(2 \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)+3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right){}^2}+\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \cos ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)^p}{2 \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)+3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}-\frac{3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sin ^2(e+f x) \left(b \tan ^2(e+f x)+a\right)^p}{2 \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)+3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}+\frac{3 a \cos (e+f x) \sin (e+f x) \left(\frac{2 b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)}{3 a}-\frac{2}{3} F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right) \sec ^2(e+f x) \tan (e+f x)\right) \left(b \tan ^2(e+f x)+a\right)^p}{2 \left(b p F_1\left(\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)-a F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right) \tan ^2(e+f x)+3 a F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}\right)}","-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{3}{2};1,-p;-\frac{1}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{3 f}",1,"(2*Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p) + (Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p*(-a - b*Tan[e + f*x]^2 - ((3*a + b*(-1 + 2*p))*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^2)/(1 + (b*Tan[e + f*x]^2)/a)^p))/(3*a*f) + (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]*Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^(2*p))/(f*(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)*((6*a*b*p*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(-1 + p))/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p)/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p)/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (3*a*Cos[e + f*x]*Sin[e + f*x]*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3)*(a + b*Tan[e + f*x]^2)^p)/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) - (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]*Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^p*(4*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e + f*x] + 3*a*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f*x]^2*(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a)) - a*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2))","B",0
373,0,0,83,4.6213343,"\int \cot ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2)^p,x]","\int \cot ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{5}{2};1,-p;-\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{5 f}",1,"Integrate[Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2)^p, x]","F",-1
374,1,224,255,1.0372914,"\int \left(a+b \tan ^3(c+d x)\right)^4 \, dx","Integrate[(a + b*Tan[c + d*x]^3)^4,x]","\frac{-1386 b^2 \left(b^2-6 a^2\right) \tan ^5(c+d x)+2310 b^2 \left(b^2-6 a^2\right) \tan ^3(c+d x)+13860 a b \left(a^2-b^2\right) \tan ^2(c+d x)-6930 b^2 \left(b^2-6 a^2\right) \tan (c+d x)+3465 a b^3 \tan ^8(c+d x)-4620 a b^3 \tan ^6(c+d x)+6930 a b^3 \tan ^4(c+d x)-3465 i \left((a-i b)^4 \log (-\tan (c+d x)+i)-(a+i b)^4 \log (\tan (c+d x)+i)\right)+630 b^4 \tan ^{11}(c+d x)-770 b^4 \tan ^9(c+d x)+990 b^4 \tan ^7(c+d x)}{6930 d}","\frac{b^2 \left(6 a^2-b^2\right) \tan ^5(c+d x)}{5 d}-\frac{b^2 \left(6 a^2-b^2\right) \tan ^3(c+d x)}{3 d}+\frac{2 a b \left(a^2-b^2\right) \tan ^2(c+d x)}{d}+\frac{b^2 \left(6 a^2-b^2\right) \tan (c+d x)}{d}+\frac{4 a b \left(a^2-b^2\right) \log (\cos (c+d x))}{d}+x \left(a^4-6 a^2 b^2+b^4\right)+\frac{a b^3 \tan ^8(c+d x)}{2 d}-\frac{2 a b^3 \tan ^6(c+d x)}{3 d}+\frac{a b^3 \tan ^4(c+d x)}{d}+\frac{b^4 \tan ^{11}(c+d x)}{11 d}-\frac{b^4 \tan ^9(c+d x)}{9 d}+\frac{b^4 \tan ^7(c+d x)}{7 d}",1,"((-3465*I)*((a - I*b)^4*Log[I - Tan[c + d*x]] - (a + I*b)^4*Log[I + Tan[c + d*x]]) - 6930*b^2*(-6*a^2 + b^2)*Tan[c + d*x] + 13860*a*b*(a^2 - b^2)*Tan[c + d*x]^2 + 2310*b^2*(-6*a^2 + b^2)*Tan[c + d*x]^3 + 6930*a*b^3*Tan[c + d*x]^4 - 1386*b^2*(-6*a^2 + b^2)*Tan[c + d*x]^5 - 4620*a*b^3*Tan[c + d*x]^6 + 990*b^4*Tan[c + d*x]^7 + 3465*a*b^3*Tan[c + d*x]^8 - 770*b^4*Tan[c + d*x]^9 + 630*b^4*Tan[c + d*x]^11)/(6930*d)","C",1
375,1,160,168,0.5080881,"\int \left(a+b \tan ^3(c+d x)\right)^3 \, dx","Integrate[(a + b*Tan[c + d*x]^3)^3,x]","\frac{-60 b \left(b^2-3 a^2\right) \tan ^2(c+d x)+72 a b^2 \tan ^5(c+d x)-120 a b^2 \tan ^3(c+d x)+360 a b^2 \tan (c+d x)+60 \left(i (a+i b)^3 \log (\tan (c+d x)+i)-i (a-i b)^3 \log (-\tan (c+d x)+i)\right)+15 b^3 \tan ^8(c+d x)-20 b^3 \tan ^6(c+d x)+30 b^3 \tan ^4(c+d x)}{120 d}","\frac{b \left(3 a^2-b^2\right) \tan ^2(c+d x)}{2 d}+\frac{b \left(3 a^2-b^2\right) \log (\cos (c+d x))}{d}+a x \left(a^2-3 b^2\right)+\frac{3 a b^2 \tan ^5(c+d x)}{5 d}-\frac{a b^2 \tan ^3(c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^8(c+d x)}{8 d}-\frac{b^3 \tan ^6(c+d x)}{6 d}+\frac{b^3 \tan ^4(c+d x)}{4 d}",1,"(60*((-I)*(a - I*b)^3*Log[I - Tan[c + d*x]] + I*(a + I*b)^3*Log[I + Tan[c + d*x]]) + 360*a*b^2*Tan[c + d*x] - 60*b*(-3*a^2 + b^2)*Tan[c + d*x]^2 - 120*a*b^2*Tan[c + d*x]^3 + 30*b^3*Tan[c + d*x]^4 + 72*a*b^2*Tan[c + d*x]^5 - 20*b^3*Tan[c + d*x]^6 + 15*b^3*Tan[c + d*x]^8)/(120*d)","C",1
376,1,107,89,0.5186869,"\int \left(a+b \tan ^3(c+d x)\right)^2 \, dx","Integrate[(a + b*Tan[c + d*x]^3)^2,x]","\frac{30 a b \tan ^2(c+d x)-15 i \left((a-i b)^2 \log (-\tan (c+d x)+i)-(a+i b)^2 \log (\tan (c+d x)+i)\right)+6 b^2 \tan ^5(c+d x)-10 b^2 \tan ^3(c+d x)+30 b^2 \tan (c+d x)}{30 d}","x \left(a^2-b^2\right)+\frac{a b \tan ^2(c+d x)}{d}+\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \tan ^5(c+d x)}{5 d}-\frac{b^2 \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan (c+d x)}{d}",1,"((-15*I)*((a - I*b)^2*Log[I - Tan[c + d*x]] - (a + I*b)^2*Log[I + Tan[c + d*x]]) + 30*b^2*Tan[c + d*x] + 30*a*b*Tan[c + d*x]^2 - 10*b^2*Tan[c + d*x]^3 + 6*b^2*Tan[c + d*x]^5)/(30*d)","C",1
377,1,30,32,0.0889128,"\int \left(a+b \tan ^3(c+d x)\right) \, dx","Integrate[a + b*Tan[c + d*x]^3,x]","a x+\frac{b \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{2 d}","a x+\frac{b \tan ^2(c+d x)}{2 d}+\frac{b \log (\cos (c+d x))}{d}",1,"a*x + (b*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d)","A",1
378,1,278,256,0.6631905,"\int \frac{1}{a+b \tan ^3(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x]^3)^(-1),x]","\frac{-b^{5/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right)-3 a^{2/3} b \tan ^2(c+d x) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{b \tan ^3(c+d x)}{a}\right)-2 a^{2/3} b \log \left(a+b \tan ^3(c+d x)\right)+3 a^{2/3} b \log (-\tan (c+d x)+i)+3 a^{2/3} b \log (\tan (c+d x)+i)-3 i a^{5/3} \log (-\tan (c+d x)+i)+3 i a^{5/3} \log (\tan (c+d x)+i)-2 \sqrt{3} b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tan (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)+2 b^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tan (c+d x)\right)}{6 a^{2/3} d \left(a^2+b^2\right)}","-\frac{b \log \left(a \cos ^3(c+d x)+b \sin ^3(c+d x)\right)}{3 d \left(a^2+b^2\right)}+\frac{a x}{a^2+b^2}+\frac{\sqrt[3]{b} \left(a^{4/3}-b^{4/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tan (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} d \left(a^2+b^2\right)}-\frac{\sqrt[3]{b} \left(a^{4/3}+b^{4/3}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right)}{6 a^{2/3} d \left(a^2+b^2\right)}+\frac{\sqrt[3]{b} \left(a^{4/3}+b^{4/3}\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tan (c+d x)\right)}{3 a^{2/3} d \left(a^2+b^2\right)}",1,"(-2*Sqrt[3]*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))] - (3*I)*a^(5/3)*Log[I - Tan[c + d*x]] + 3*a^(2/3)*b*Log[I - Tan[c + d*x]] + (3*I)*a^(5/3)*Log[I + Tan[c + d*x]] + 3*a^(2/3)*b*Log[I + Tan[c + d*x]] + 2*b^(5/3)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]] - b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x] + b^(2/3)*Tan[c + d*x]^2] - 2*a^(2/3)*b*Log[a + b*Tan[c + d*x]^3] - 3*a^(2/3)*b*Hypergeometric2F1[2/3, 1, 5/3, -((b*Tan[c + d*x]^3)/a)]*Tan[c + d*x]^2)/(6*a^(2/3)*(a^2 + b^2)*d)","C",1
379,1,575,558,6.3115873,"\int \frac{1}{\left(a+b \tan ^3(c+d x)\right)^2} \, dx","Integrate[(a + b*Tan[c + d*x]^3)^(-2),x]","-\frac{b (a-b) (a+b) \tan ^2(c+d x) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{b \tan ^3(c+d x)}{a}\right)}{2 a d \left(a^2+b^2\right)^2}-\frac{b \tan ^2(c+d x) \, _2F_1\left(\frac{2}{3},2;\frac{5}{3};-\frac{b \tan ^3(c+d x)}{a}\right)}{2 a d \left(a^2+b^2\right)}+\frac{b^2 \tan (c+d x)}{3 a d \left(a^2+b^2\right) \left(a+b \tan ^3(c+d x)\right)}+\frac{b}{3 d \left(a^2+b^2\right) \left(a+b \tan ^3(c+d x)\right)}-\frac{2 a b \log \left(a+b \tan ^3(c+d x)\right)}{3 d \left(a^2+b^2\right)^2}+\frac{2 \sqrt[3]{a} b^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tan (c+d x)\right)}{3 d \left(a^2+b^2\right)^2}-\frac{\sqrt[3]{a} \left(b^{5/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right)+2 \sqrt{3} b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tan (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)\right)}{3 d \left(a^2+b^2\right)^2}+\frac{\frac{2 b^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tan (c+d x)\right)}{a^{2/3}}-\frac{b^{5/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right)+2 \sqrt{3} b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tan (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{a^{2/3}}}{9 a d \left(a^2+b^2\right)}-\frac{i \log (-\tan (c+d x)+i)}{2 d (a-i b)^2}+\frac{i \log (\tan (c+d x)+i)}{2 d (a+i b)^2}","\frac{b (\tan (c+d x) (b-a \tan (c+d x))+a)}{3 a d \left(a^2+b^2\right) \left(a+b \tan ^3(c+d x)\right)}-\frac{2 a b \log \left(a \cos ^3(c+d x)+b \sin ^3(c+d x)\right)}{3 d \left(a^2+b^2\right)^2}+\frac{x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^2}+\frac{\sqrt[3]{b} \left(a^{4/3}-2 b^{4/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tan (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} d \left(a^2+b^2\right)}+\frac{\sqrt[3]{b} \left(-2 a^{2/3} b^{4/3}+a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tan (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} d \left(a^2+b^2\right)^2}-\frac{\sqrt[3]{b} \left(a^{4/3}+2 b^{4/3}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right)}{18 a^{5/3} d \left(a^2+b^2\right)}-\frac{\sqrt[3]{b} \left(2 a^{2/3} b^{4/3}+a^2-b^2\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right)}{6 \sqrt[3]{a} d \left(a^2+b^2\right)^2}+\frac{\sqrt[3]{b} \left(a^{4/3}+2 b^{4/3}\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tan (c+d x)\right)}{9 a^{5/3} d \left(a^2+b^2\right)}+\frac{\sqrt[3]{b} \left(2 a^{2/3} b^{4/3}+a^2-b^2\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tan (c+d x)\right)}{3 \sqrt[3]{a} d \left(a^2+b^2\right)^2}",1,"((-1/2*I)*Log[I - Tan[c + d*x]])/((a - I*b)^2*d) + ((I/2)*Log[I + Tan[c + d*x]])/((a + I*b)^2*d) + (2*a^(1/3)*b^(5/3)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/(3*(a^2 + b^2)^2*d) - (a^(1/3)*(2*Sqrt[3]*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))] + b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x] + b^(2/3)*Tan[c + d*x]^2]))/(3*(a^2 + b^2)^2*d) + ((2*b^(5/3)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/a^(2/3) - (2*Sqrt[3]*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))] + b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x] + b^(2/3)*Tan[c + d*x]^2])/a^(2/3))/(9*a*(a^2 + b^2)*d) - (2*a*b*Log[a + b*Tan[c + d*x]^3])/(3*(a^2 + b^2)^2*d) - ((a - b)*b*(a + b)*Hypergeometric2F1[2/3, 1, 5/3, -((b*Tan[c + d*x]^3)/a)]*Tan[c + d*x]^2)/(2*a*(a^2 + b^2)^2*d) - (b*Hypergeometric2F1[2/3, 2, 5/3, -((b*Tan[c + d*x]^3)/a)]*Tan[c + d*x]^2)/(2*a*(a^2 + b^2)*d) + b/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x]^3)) + (b^2*Tan[c + d*x])/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]^3))","C",1
380,1,57,37,0.0263063,"\int \frac{1}{1+\tan ^3(x)} \, dx","Integrate[(1 + Tan[x]^3)^(-1),x]","-\frac{1}{3} \log \left(\tan ^2(x)-\tan (x)+1\right)+\left(\frac{1}{4}-\frac{i}{4}\right) \log (-\tan (x)+i)+\left(\frac{1}{4}+\frac{i}{4}\right) \log (\tan (x)+i)+\frac{1}{6} \log (\tan (x)+1)","\frac{x}{2}-\frac{1}{3} \log \left(\tan ^2(x)-\tan (x)+1\right)+\frac{1}{6} \log (\tan (x)+1)-\frac{1}{2} \log (\cos (x))",1,"(1/4 - I/4)*Log[I - Tan[x]] + (1/4 + I/4)*Log[I + Tan[x]] + Log[1 + Tan[x]]/6 - Log[1 - Tan[x] + Tan[x]^2]/3","C",1
381,1,196,216,5.3847823,"\int \left(a+b \tan ^4(c+d x)\right)^4 \, dx","Integrate[(a + b*Tan[c + d*x]^4)^4,x]","\frac{b \tan (c+d x) \left(6435 b \left(6 a^2+4 a b+b^2\right) \tan ^6(c+d x)-9009 b \left(6 a^2+4 a b+b^2\right) \tan ^4(c+d x)+15015 \left(4 a^3+6 a^2 b+4 a b^2+b^3\right) \tan ^2(c+d x)-45045 \left(4 a^3+6 a^2 b+4 a b^2+b^3\right)+4095 b^2 (4 a+b) \tan ^{10}(c+d x)-5005 b^2 (4 a+b) \tan ^8(c+d x)+3003 b^3 \tan ^{14}(c+d x)-3465 b^3 \tan ^{12}(c+d x)\right)}{45045 d}+\frac{(a+b)^4 \tan ^{-1}(\tan (c+d x))}{d}","\frac{b^2 \left(6 a^2+4 a b+b^2\right) \tan ^7(c+d x)}{7 d}-\frac{b^2 \left(6 a^2+4 a b+b^2\right) \tan ^5(c+d x)}{5 d}+\frac{b (2 a+b) \left(2 a^2+2 a b+b^2\right) \tan ^3(c+d x)}{3 d}-\frac{b (2 a+b) \left(2 a^2+2 a b+b^2\right) \tan (c+d x)}{d}+\frac{b^3 (4 a+b) \tan ^{11}(c+d x)}{11 d}-\frac{b^3 (4 a+b) \tan ^9(c+d x)}{9 d}+x (a+b)^4+\frac{b^4 \tan ^{15}(c+d x)}{15 d}-\frac{b^4 \tan ^{13}(c+d x)}{13 d}",1,"((a + b)^4*ArcTan[Tan[c + d*x]])/d + (b*Tan[c + d*x]*(-45045*(4*a^3 + 6*a^2*b + 4*a*b^2 + b^3) + 15015*(4*a^3 + 6*a^2*b + 4*a*b^2 + b^3)*Tan[c + d*x]^2 - 9009*b*(6*a^2 + 4*a*b + b^2)*Tan[c + d*x]^4 + 6435*b*(6*a^2 + 4*a*b + b^2)*Tan[c + d*x]^6 - 5005*b^2*(4*a + b)*Tan[c + d*x]^8 + 4095*b^2*(4*a + b)*Tan[c + d*x]^10 - 3465*b^3*Tan[c + d*x]^12 + 3003*b^3*Tan[c + d*x]^14))/(45045*d)","A",1
382,1,128,144,1.0655873,"\int \left(a+b \tan ^4(c+d x)\right)^3 \, dx","Integrate[(a + b*Tan[c + d*x]^4)^3,x]","\frac{b \tan (c+d x) \left(1155 \left(3 a^2+3 a b+b^2\right) \tan ^2(c+d x)-3465 \left(3 a^2+3 a b+b^2\right)+495 b (3 a+b) \tan ^6(c+d x)-693 b (3 a+b) \tan ^4(c+d x)+315 b^2 \tan ^{10}(c+d x)-385 b^2 \tan ^8(c+d x)\right)}{3465 d}+\frac{(a+b)^3 \tan ^{-1}(\tan (c+d x))}{d}","\frac{b \left(3 a^2+3 a b+b^2\right) \tan ^3(c+d x)}{3 d}-\frac{b \left(3 a^2+3 a b+b^2\right) \tan (c+d x)}{d}+\frac{b^2 (3 a+b) \tan ^7(c+d x)}{7 d}-\frac{b^2 (3 a+b) \tan ^5(c+d x)}{5 d}+x (a+b)^3+\frac{b^3 \tan ^{11}(c+d x)}{11 d}-\frac{b^3 \tan ^9(c+d x)}{9 d}",1,"((a + b)^3*ArcTan[Tan[c + d*x]])/d + (b*Tan[c + d*x]*(-3465*(3*a^2 + 3*a*b + b^2) + 1155*(3*a^2 + 3*a*b + b^2)*Tan[c + d*x]^2 - 693*b*(3*a + b)*Tan[c + d*x]^4 + 495*b*(3*a + b)*Tan[c + d*x]^6 - 385*b^2*Tan[c + d*x]^8 + 315*b^2*Tan[c + d*x]^10))/(3465*d)","A",1
383,1,75,82,0.5767616,"\int \left(a+b \tan ^4(c+d x)\right)^2 \, dx","Integrate[(a + b*Tan[c + d*x]^4)^2,x]","\frac{105 (a+b)^2 \tan ^{-1}(\tan (c+d x))+b \tan (c+d x) \left(35 (2 a+b) \tan ^2(c+d x)-105 (2 a+b)+15 b \tan ^6(c+d x)-21 b \tan ^4(c+d x)\right)}{105 d}","\frac{b (2 a+b) \tan ^3(c+d x)}{3 d}-\frac{b (2 a+b) \tan (c+d x)}{d}+x (a+b)^2+\frac{b^2 \tan ^7(c+d x)}{7 d}-\frac{b^2 \tan ^5(c+d x)}{5 d}",1,"(105*(a + b)^2*ArcTan[Tan[c + d*x]] + b*Tan[c + d*x]*(-105*(2*a + b) + 35*(2*a + b)*Tan[c + d*x]^2 - 21*b*Tan[c + d*x]^4 + 15*b*Tan[c + d*x]^6))/(105*d)","A",1
384,1,44,35,0.0268805,"\int \left(a+b \tan ^4(c+d x)\right) \, dx","Integrate[a + b*Tan[c + d*x]^4,x]","a x+\frac{b \tan ^{-1}(\tan (c+d x))}{d}+\frac{b \tan ^3(c+d x)}{3 d}-\frac{b \tan (c+d x)}{d}","a x+\frac{b \tan ^3(c+d x)}{3 d}-\frac{b \tan (c+d x)}{d}+b x",1,"a*x + (b*ArcTan[Tan[c + d*x]])/d - (b*Tan[c + d*x])/d + (b*Tan[c + d*x]^3)/(3*d)","A",1
385,1,228,302,0.5716232,"\int \frac{1}{a+b \tan ^4(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x]^4)^(-1),x]","\frac{8 a^{3/4} \tan ^{-1}(\tan (c+d x))+\sqrt{2} \sqrt[4]{b} \left(2 \left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)-2 \left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}+1\right)-\left(\sqrt{a}+\sqrt{b}\right) \left(\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)\right)\right)}{8 a^{3/4} d (a+b)}","\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} d (a+b)}-\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} d (a+b)}-\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b}\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{4 \sqrt{2} a^{3/4} d (a+b)}+\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b}\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{4 \sqrt{2} a^{3/4} d (a+b)}+\frac{x}{a+b}",1,"(8*a^(3/4)*ArcTan[Tan[c + d*x]] + Sqrt[2]*b^(1/4)*(2*(Sqrt[a] - Sqrt[b])*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)] - 2*(Sqrt[a] - Sqrt[b])*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)] - (Sqrt[a] + Sqrt[b])*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])))/(8*a^(3/4)*(a + b)*d)","A",1
386,1,598,648,6.2925119,"\int \frac{1}{\left(a+b \tan ^4(c+d x)\right)^2} \, dx","Integrate[(a + b*Tan[c + d*x]^4)^(-2),x]","-\frac{b \tan ^3(c+d x) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{b \tan ^4(c+d x)}{a}\right)}{3 a^2 d (a+b)}-\frac{3 \left(\frac{2 \left(\frac{\sqrt{2} b^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)}{\sqrt[4]{a}}-\frac{\sqrt{2} b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{a}}\right)}{\sqrt{a}}+\frac{\frac{\sqrt{2} b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{\sqrt[4]{a}}-\frac{\sqrt{2} b^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{\sqrt[4]{a}}}{\sqrt{a}}\right)}{32 a d (a+b)}+\frac{\tan ^{-1}(\tan (c+d x))}{d (a+b)^2}+\frac{\left(\sqrt{a}-\sqrt{b}\right) \left(\frac{\sqrt{2} \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)}{\sqrt[4]{a}}-\frac{\sqrt{2} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{a}}\right)}{4 \sqrt{a} d (a+b)^2}+\frac{b \tan (c+d x)}{4 a d (a+b) \left(a+b \tan ^4(c+d x)\right)}-\frac{\left(\sqrt{a}+\sqrt{b}\right) \left(\frac{\sqrt{2} \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{\sqrt[4]{a}}-\frac{\sqrt{2} \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{\sqrt[4]{a}}\right)}{8 \sqrt{a} d (a+b)^2}","\frac{\sqrt[4]{b} \left(\sqrt{a}-3 \sqrt{b}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} d (a+b)}+\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} d (a+b)^2}-\frac{\sqrt[4]{b} \left(\sqrt{a}-3 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} d (a+b)}-\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} d (a+b)^2}-\frac{\sqrt[4]{b} \left(\sqrt{a}+3 \sqrt{b}\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{16 \sqrt{2} a^{7/4} d (a+b)}-\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b}\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{4 \sqrt{2} a^{3/4} d (a+b)^2}+\frac{\sqrt[4]{b} \left(\sqrt{a}+3 \sqrt{b}\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{16 \sqrt{2} a^{7/4} d (a+b)}+\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b}\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{4 \sqrt{2} a^{3/4} d (a+b)^2}+\frac{b \tan (c+d x) \left(1-\tan ^2(c+d x)\right)}{4 a d (a+b) \left(a+b \tan ^4(c+d x)\right)}+\frac{x}{(a+b)^2}",1,"ArcTan[Tan[c + d*x]]/((a + b)^2*d) + ((Sqrt[a] - Sqrt[b])*((Sqrt[2]*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/a^(1/4) - (Sqrt[2]*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/a^(1/4)))/(4*Sqrt[a]*(a + b)^2*d) - ((Sqrt[a] + Sqrt[b])*((Sqrt[2]*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/a^(1/4) - (Sqrt[2]*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/a^(1/4)))/(8*Sqrt[a]*(a + b)^2*d) - (3*((2*((Sqrt[2]*b^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/a^(1/4) - (Sqrt[2]*b^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/a^(1/4)))/Sqrt[a] + ((Sqrt[2]*b^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/a^(1/4) - (Sqrt[2]*b^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/a^(1/4))/Sqrt[a]))/(32*a*(a + b)*d) - (b*Hypergeometric2F1[3/4, 2, 7/4, -((b*Tan[c + d*x]^4)/a)]*Tan[c + d*x]^3)/(3*a^2*(a + b)*d) + (b*Tan[c + d*x])/(4*a*(a + b)*d*(a + b*Tan[c + d*x]^4))","C",1
387,1,219,650,0.8117326,"\int \sqrt{a+b \tan ^4(c+d x)} \, dx","Integrate[Sqrt[a + b*Tan[c + d*x]^4],x]","\frac{\sqrt{\frac{b \tan ^4(c+d x)}{a}+1} \left(\sqrt{a} \sqrt{b} E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right)\right|-1\right)+\left(\sqrt{a}-i \sqrt{b}\right) \left(\left(\sqrt{b}-i \sqrt{a}\right) \Pi \left(-\frac{i \sqrt{a}}{\sqrt{b}};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right)\right|-1\right)-\sqrt{b} F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right)\right|-1\right)\right)\right)}{d \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b \tan ^4(c+d x)}}","\frac{\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a+b \tan ^4(c+d x)}}\right)}{2 d}+\frac{\sqrt{b} \tan (c+d x) \sqrt{a+b \tan ^4(c+d x)}}{d \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}-\frac{\sqrt[4]{b} (a+b) \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} d \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(c+d x)}}+\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} d \sqrt{a+b \tan ^4(c+d x)}}-\frac{\sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{d \sqrt{a+b \tan ^4(c+d x)}}+\frac{\left(\sqrt{a}+\sqrt{b}\right) (a+b) \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{b} d \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(c+d x)}}",1,"((Sqrt[a]*Sqrt[b]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[c + d*x]], -1] + (Sqrt[a] - I*Sqrt[b])*(-(Sqrt[b]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[c + d*x]], -1]) + ((-I)*Sqrt[a] + Sqrt[b])*EllipticPi[((-I)*Sqrt[a])/Sqrt[b], I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[c + d*x]], -1]))*Sqrt[1 + (b*Tan[c + d*x]^4)/a])/(Sqrt[(I*Sqrt[b])/Sqrt[a]]*d*Sqrt[a + b*Tan[c + d*x]^4])","C",1
388,1,106,348,0.4083236,"\int \frac{1}{\sqrt{a+b \tan ^4(c+d x)}} \, dx","Integrate[1/Sqrt[a + b*Tan[c + d*x]^4],x]","-\frac{i \sqrt{\frac{b \tan ^4(c+d x)}{a}+1} \Pi \left(-\frac{i \sqrt{a}}{\sqrt{b}};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right)\right|-1\right)}{d \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b \tan ^4(c+d x)}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a+b \tan ^4(c+d x)}}\right)}{2 d \sqrt{a+b}}-\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} d \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(c+d x)}}+\frac{\left(\sqrt{a}+\sqrt{b}\right) \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{b} d \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(c+d x)}}",1,"((-I)*EllipticPi[((-I)*Sqrt[a])/Sqrt[b], I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[c + d*x]], -1]*Sqrt[1 + (b*Tan[c + d*x]^4)/a])/(Sqrt[(I*Sqrt[b])/Sqrt[a]]*d*Sqrt[a + b*Tan[c + d*x]^4])","C",1
389,1,145,103,4.0943651,"\int \tan ^3(x) \sqrt{a+b \tan ^4(x)} \, dx","Integrate[Tan[x]^3*Sqrt[a + b*Tan[x]^4],x]","\frac{1}{4} \left(\frac{\frac{a^{3/2} \sqrt{\frac{b \tan ^4(x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a}}\right)}{\sqrt{b}}+\left(\tan ^2(x)-2\right) \left(a+b \tan ^4(x)\right)}{\sqrt{a+b \tan ^4(x)}}+2 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)+2 \sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)\right)","-\frac{1}{4} \left(2-\tan ^2(x)\right) \sqrt{a+b \tan ^4(x)}+\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)}{4 \sqrt{b}}+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)",1,"(2*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] + 2*Sqrt[a + b]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] + ((-2 + Tan[x]^2)*(a + b*Tan[x]^4) + (a^(3/2)*ArcSinh[(Sqrt[b]*Tan[x]^2)/Sqrt[a]]*Sqrt[1 + (b*Tan[x]^4)/a])/Sqrt[b])/Sqrt[a + b*Tan[x]^4])/4","A",1
390,1,86,90,0.0451281,"\int \tan (x) \sqrt{a+b \tan ^4(x)} \, dx","Integrate[Tan[x]*Sqrt[a + b*Tan[x]^4],x]","\frac{1}{2} \left(\sqrt{a+b \tan ^4(x)}-\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)-\sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)\right)","\frac{1}{2} \sqrt{a+b \tan ^4(x)}-\frac{1}{2} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)-\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)",1,"(-(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]]) - Sqrt[a + b]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] + Sqrt[a + b*Tan[x]^4])/2","A",1
391,1,98,102,0.0871094,"\int \cot (x) \sqrt{a+b \tan ^4(x)} \, dx","Integrate[Cot[x]*Sqrt[a + b*Tan[x]^4],x]","\frac{1}{2} \left(-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)+\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)+\sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)\right)","-\frac{1}{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)+\frac{1}{2} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] + Sqrt[a + b]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] - Sqrt[a]*ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]])/2","A",1
392,1,550,643,17.8763841,"\int \tan ^2(x) \sqrt{a+b \tan ^4(x)} \, dx","Integrate[Tan[x]^2*Sqrt[a + b*Tan[x]^4],x]","\left(\frac{\tan (x)}{3}-\frac{1}{2} \sin (2 x)\right) \sqrt{\frac{4 a \cos (2 x)+a \cos (4 x)+3 a-4 b \cos (2 x)+b \cos (4 x)+3 b}{4 \cos (2 x)+\cos (4 x)+3}}+\frac{3 b \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan ^5(x)+3 a \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (x)+\left(3 \sqrt{a} \sqrt{b}-2 i a-3 i b\right) \left(\tan ^2(x)+1\right) \sqrt{\frac{b \tan ^4(x)}{a}+1} F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (x)\right)\right|-1\right)-3 \sqrt{a} \sqrt{b} \left(\tan ^2(x)+1\right) \sqrt{\frac{b \tan ^4(x)}{a}+1} E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (x)\right)\right|-1\right)+3 i a \sqrt{\frac{b \tan ^4(x)}{a}+1} \Pi \left(-\frac{i \sqrt{a}}{\sqrt{b}};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (x)\right)\right|-1\right)+3 i b \sqrt{\frac{b \tan ^4(x)}{a}+1} \Pi \left(-\frac{i \sqrt{a}}{\sqrt{b}};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (x)\right)\right|-1\right)+3 i a \tan ^2(x) \sqrt{\frac{b \tan ^4(x)}{a}+1} \Pi \left(-\frac{i \sqrt{a}}{\sqrt{b}};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (x)\right)\right|-1\right)+3 i b \tan ^2(x) \sqrt{\frac{b \tan ^4(x)}{a}+1} \Pi \left(-\frac{i \sqrt{a}}{\sqrt{b}};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (x)\right)\right|-1\right)}{3 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left(\tan ^2(x)+1\right) \sqrt{a+b \tan ^4(x)}}","\frac{a^{3/4} \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{b} \sqrt{a+b \tan ^4(x)}}+\frac{1}{3} \tan (x) \sqrt{a+b \tan ^4(x)}-\frac{1}{2} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a+b \tan ^4(x)}}\right)-\frac{\sqrt{b} \tan (x) \sqrt{a+b \tan ^4(x)}}{\sqrt{a}+\sqrt{b} \tan ^2(x)}+\frac{\sqrt[4]{b} (a+b) \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(x)}}-\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt{a+b \tan ^4(x)}}+\frac{\sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{\sqrt{a+b \tan ^4(x)}}-\frac{\left(\sqrt{a}+\sqrt{b}\right) (a+b) \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(x)}}",1,"Sqrt[(3*a + 3*b + 4*a*Cos[2*x] - 4*b*Cos[2*x] + a*Cos[4*x] + b*Cos[4*x])/(3 + 4*Cos[2*x] + Cos[4*x])]*(-1/2*Sin[2*x] + Tan[x]/3) + (3*a*Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[x] + 3*Sqrt[(I*Sqrt[b])/Sqrt[a]]*b*Tan[x]^5 + (3*I)*a*EllipticPi[((-I)*Sqrt[a])/Sqrt[b], I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[x]], -1]*Sqrt[1 + (b*Tan[x]^4)/a] + (3*I)*b*EllipticPi[((-I)*Sqrt[a])/Sqrt[b], I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[x]], -1]*Sqrt[1 + (b*Tan[x]^4)/a] + (3*I)*a*EllipticPi[((-I)*Sqrt[a])/Sqrt[b], I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[x]], -1]*Tan[x]^2*Sqrt[1 + (b*Tan[x]^4)/a] + (3*I)*b*EllipticPi[((-I)*Sqrt[a])/Sqrt[b], I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[x]], -1]*Tan[x]^2*Sqrt[1 + (b*Tan[x]^4)/a] - 3*Sqrt[a]*Sqrt[b]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[x]], -1]*(1 + Tan[x]^2)*Sqrt[1 + (b*Tan[x]^4)/a] + ((-2*I)*a + 3*Sqrt[a]*Sqrt[b] - (3*I)*b)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[x]], -1]*(1 + Tan[x]^2)*Sqrt[1 + (b*Tan[x]^4)/a])/(3*Sqrt[(I*Sqrt[b])/Sqrt[a]]*(1 + Tan[x]^2)*Sqrt[a + b*Tan[x]^4])","C",1
393,1,324,148,6.0776036,"\int \tan ^3(x) \left(a+b \tan ^4(x)\right)^{3/2} \, dx","Integrate[Tan[x]^3*(a + b*Tan[x]^4)^(3/2),x]","\frac{1}{2} a \tan ^2(x) \sqrt{a+b \tan ^4(x)} \left(\frac{b \tan ^4(x)}{a}+1\right)^2 \left(\frac{1}{4} \left(\frac{1}{\frac{b \tan ^4(x)}{a}+1}+\frac{3}{2 \left(\frac{b \tan ^4(x)}{a}+1\right)^2}\right)+\frac{3 \sqrt{a} \cot ^2(x) \sinh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a}}\right)}{8 \sqrt{b} \left(\frac{b \tan ^4(x)}{a}+1\right)^{5/2}}\right)+\frac{1}{2} \left(-\frac{1}{3} \left(a+b \tan ^4(x)\right)^{3/2}-(a+b) \left(\sqrt{a+b \tan ^4(x)}-\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)-\sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)\right)+b \tan ^2(x) \sqrt{a+b \tan ^4(x)} \left(\frac{b \tan ^4(x)}{a}+1\right) \left(\frac{1}{2 \left(\frac{b \tan ^4(x)}{a}+1\right)}+\frac{\sqrt{a} \cot ^2(x) \sinh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a}}\right)}{2 \sqrt{b} \left(\frac{b \tan ^4(x)}{a}+1\right)^{3/2}}\right)\right)","\frac{\left(3 a^2+12 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)}{16 \sqrt{b}}-\frac{1}{24} \left(4-3 \tan ^2(x)\right) \left(a+b \tan ^4(x)\right)^{3/2}-\frac{1}{16} \left(8 (a+b)-(3 a+4 b) \tan ^2(x)\right) \sqrt{a+b \tan ^4(x)}+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)",1,"(-1/3*(a + b*Tan[x]^4)^(3/2) - (a + b)*(-(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]]) - Sqrt[a + b]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] + Sqrt[a + b*Tan[x]^4]) + b*Tan[x]^2*Sqrt[a + b*Tan[x]^4]*(1 + (b*Tan[x]^4)/a)*((Sqrt[a]*ArcSinh[(Sqrt[b]*Tan[x]^2)/Sqrt[a]]*Cot[x]^2)/(2*Sqrt[b]*(1 + (b*Tan[x]^4)/a)^(3/2)) + 1/(2*(1 + (b*Tan[x]^4)/a))))/2 + (a*Tan[x]^2*Sqrt[a + b*Tan[x]^4]*(1 + (b*Tan[x]^4)/a)^2*((3*Sqrt[a]*ArcSinh[(Sqrt[b]*Tan[x]^2)/Sqrt[a]]*Cot[x]^2)/(8*Sqrt[b]*(1 + (b*Tan[x]^4)/a)^(5/2)) + (3/(2*(1 + (b*Tan[x]^4)/a)^2) + (1 + (b*Tan[x]^4)/a)^(-1))/4))/2","B",1
394,1,166,126,4.8426163,"\int \tan (x) \left(a+b \tan ^4(x)\right)^{3/2} \, dx","Integrate[Tan[x]*(a + b*Tan[x]^4)^(3/2),x]","\frac{1}{12} \left(\sqrt{a+b \tan ^4(x)} \left(8 a+2 b \tan ^4(x)-3 b \tan ^2(x)+6 b\right)-6 (a+b)^{3/2} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)-6 \sqrt{b} (a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)-\frac{3 \sqrt{a} \sqrt{b} \sqrt{a+b \tan ^4(x)} \sinh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a}}\right)}{\sqrt{\frac{b \tan ^4(x)}{a}+1}}\right)","\frac{1}{6} \left(a+b \tan ^4(x)\right)^{3/2}+\frac{1}{4} \left(2 (a+b)-b \tan ^2(x)\right) \sqrt{a+b \tan ^4(x)}-\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)-\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)",1,"(-6*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] - 6*(a + b)^(3/2)*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] + Sqrt[a + b*Tan[x]^4]*(8*a + 6*b - 3*b*Tan[x]^2 + 2*b*Tan[x]^4) - (3*Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Tan[x]^2)/Sqrt[a]]*Sqrt[a + b*Tan[x]^4])/Sqrt[1 + (b*Tan[x]^4)/a])/12","A",1
395,1,190,155,3.23163,"\int \cot (x) \left(a+b \tan ^4(x)\right)^{3/2} \, dx","Integrate[Cot[x]*(a + b*Tan[x]^4)^(3/2),x]","\frac{1}{4} \left(-2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)-2 b \sqrt{a+b \tan ^4(x)}+b \tan ^2(x) \sqrt{a+b \tan ^4(x)}+2 \sqrt{b} (a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)+2 (a+b)^{3/2} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)+\frac{\sqrt{a} \sqrt{b} \sqrt{a+b \tan ^4(x)} \sinh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a}}\right)}{\sqrt{\frac{b \tan ^4(x)}{a}+1}}\right)","-\frac{1}{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)+\frac{1}{2} a \sqrt{a+b \tan ^4(x)}-\frac{1}{4} \left(2 (a+b)-b \tan ^2(x)\right) \sqrt{a+b \tan ^4(x)}+\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)",1,"(2*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] + 2*(a + b)^(3/2)*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] - 2*a^(3/2)*ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]] - 2*b*Sqrt[a + b*Tan[x]^4] + b*Tan[x]^2*Sqrt[a + b*Tan[x]^4] + (Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Tan[x]^2)/Sqrt[a]]*Sqrt[a + b*Tan[x]^4])/Sqrt[1 + (b*Tan[x]^4)/a])/4","A",1
396,1,74,74,0.0604247,"\int \frac{\tan ^3(x)}{\sqrt{a+b \tan ^4(x)}} \, dx","Integrate[Tan[x]^3/Sqrt[a + b*Tan[x]^4],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}",1,"ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]]/(2*Sqrt[b]) + ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*Sqrt[a + b])","A",1
397,1,41,41,0.0174797,"\int \frac{\tan (x)}{\sqrt{a+b \tan ^4(x)}} \, dx","Integrate[Tan[x]/Sqrt[a + b*Tan[x]^4],x]","-\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}","-\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}",1,"-1/2*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/Sqrt[a + b]","A",1
398,1,70,70,0.0644177,"\int \frac{\cot (x)}{\sqrt{a+b \tan ^4(x)}} \, dx","Integrate[Cot[x]/Sqrt[a + b*Tan[x]^4],x]","\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)}{2 \sqrt{a}}","\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)}{2 \sqrt{a}}",1,"ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*Sqrt[a + b]) - ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]]/(2*Sqrt[a])","A",1
399,1,122,291,2.3596985,"\int \frac{\tan ^2(x)}{\sqrt{a+b \tan ^4(x)}} \, dx","Integrate[Tan[x]^2/Sqrt[a + b*Tan[x]^4],x]","-\frac{i \sqrt{\frac{b \tan ^4(x)}{a}+1} \left(F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (x)\right)\right|-1\right)-\Pi \left(-\frac{i \sqrt{a}}{\sqrt{b}};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (x)\right)\right|-1\right)\right)}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b \tan ^4(x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(x)}}-\frac{\left(\sqrt{a}+\sqrt{b}\right) \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(x)}}",1,"((-I)*(EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[x]], -1] - EllipticPi[((-I)*Sqrt[a])/Sqrt[b], I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[x]], -1])*Sqrt[1 + (b*Tan[x]^4)/a])/(Sqrt[(I*Sqrt[b])/Sqrt[a]]*Sqrt[a + b*Tan[x]^4])","C",1
400,1,67,71,0.3342476,"\int \frac{\tan ^3(x)}{\left(a+b \tan ^4(x)\right)^{3/2}} \, dx","Integrate[Tan[x]^3/(a + b*Tan[x]^4)^(3/2),x]","\frac{1}{2} \left(\frac{\tan ^2(x)-1}{(a+b) \sqrt{a+b \tan ^4(x)}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{(a+b)^{3/2}}\right)","\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{3/2}}-\frac{1-\tan ^2(x)}{2 (a+b) \sqrt{a+b \tan ^4(x)}}",1,"(ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(a + b)^(3/2) + (-1 + Tan[x]^2)/((a + b)*Sqrt[a + b*Tan[x]^4]))/2","A",1
401,1,73,74,0.3075962,"\int \frac{\tan (x)}{\left(a+b \tan ^4(x)\right)^{3/2}} \, dx","Integrate[Tan[x]/(a + b*Tan[x]^4)^(3/2),x]","\frac{1}{2} \left(\frac{a+b \tan ^2(x)}{a (a+b) \sqrt{a+b \tan ^4(x)}}-\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{(a+b)^{3/2}}\right)","\frac{a+b \tan ^2(x)}{2 a (a+b) \sqrt{a+b \tan ^4(x)}}-\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{3/2}}",1,"(-(ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(a + b)^(3/2)) + (a + b*Tan[x]^2)/(a*(a + b)*Sqrt[a + b*Tan[x]^4]))/2","A",1
402,1,108,121,0.6195749,"\int \frac{\cot (x)}{\left(a+b \tan ^4(x)\right)^{3/2}} \, dx","Integrate[Cot[x]/(a + b*Tan[x]^4)^(3/2),x]","\frac{1}{2} \left(\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \tan ^4(x)}{a}+1\right)}{a \sqrt{a+b \tan ^4(x)}}-\frac{a+b \tan ^2(x)}{a (a+b) \sqrt{a+b \tan ^4(x)}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{(a+b)^{3/2}}\right)","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)}{2 a^{3/2}}+\frac{1}{2 a \sqrt{a+b \tan ^4(x)}}-\frac{a+b \tan ^2(x)}{2 a (a+b) \sqrt{a+b \tan ^4(x)}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{3/2}}",1,"(ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(a + b)^(3/2) + Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Tan[x]^4)/a]/(a*Sqrt[a + b*Tan[x]^4]) - (a + b*Tan[x]^2)/(a*(a + b)*Sqrt[a + b*Tan[x]^4]))/2","C",1
403,1,104,109,0.8676773,"\int \frac{\tan ^3(x)}{\left(a+b \tan ^4(x)\right)^{5/2}} \, dx","Integrate[Tan[x]^3/(a + b*Tan[x]^4)^(5/2),x]","\frac{1}{6} \left(\frac{3 a^2 \tan ^2(x)+b (2 a-b) \tan ^6(x)-3 a b \tan ^4(x)-a (4 a+b)}{a (a+b)^2 \left(a+b \tan ^4(x)\right)^{3/2}}+\frac{3 \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{(a+b)^{5/2}}\right)","-\frac{(b-2 a) \tan ^2(x)+3 a}{6 a (a+b)^2 \sqrt{a+b \tan ^4(x)}}-\frac{1-\tan ^2(x)}{6 (a+b) \left(a+b \tan ^4(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{5/2}}",1,"((3*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])])/(a + b)^(5/2) + (-(a*(4*a + b)) + 3*a^2*Tan[x]^2 - 3*a*b*Tan[x]^4 + (2*a - b)*b*Tan[x]^6)/(a*(a + b)^2*(a + b*Tan[x]^4)^(3/2)))/6","A",1
404,1,113,117,0.8333718,"\int \frac{\tan (x)}{\left(a+b \tan ^4(x)\right)^{5/2}} \, dx","Integrate[Tan[x]/(a + b*Tan[x]^4)^(5/2),x]","\frac{1}{6} \left(\frac{3 a^2 b \tan ^4(x)+a^2 (4 a+b)+b^2 (5 a+2 b) \tan ^6(x)+3 a b (2 a+b) \tan ^2(x)}{a^2 (a+b)^2 \left(a+b \tan ^4(x)\right)^{3/2}}-\frac{3 \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{(a+b)^{5/2}}\right)","\frac{3 a^2+b (5 a+2 b) \tan ^2(x)}{6 a^2 (a+b)^2 \sqrt{a+b \tan ^4(x)}}+\frac{a+b \tan ^2(x)}{6 a (a+b) \left(a+b \tan ^4(x)\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{5/2}}",1,"((-3*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])])/(a + b)^(5/2) + (a^2*(4*a + b) + 3*a*b*(2*a + b)*Tan[x]^2 + 3*a^2*b*Tan[x]^4 + b^2*(5*a + 2*b)*Tan[x]^6)/(a^2*(a + b)^2*(a + b*Tan[x]^4)^(3/2)))/6","A",1
405,1,149,183,1.5433249,"\int \frac{\cot (x)}{\left(a+b \tan ^4(x)\right)^{5/2}} \, dx","Integrate[Cot[x]/(a + b*Tan[x]^4)^(5/2),x]","\frac{1}{6} \left(-\frac{3 a^2 b \tan ^4(x)+a^2 (4 a+b)+b^2 (5 a+2 b) \tan ^6(x)+3 a b (2 a+b) \tan ^2(x)}{a^2 (a+b)^2 \left(a+b \tan ^4(x)\right)^{3/2}}+\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \tan ^4(x)}{a}+1\right)}{a \left(a+b \tan ^4(x)\right)^{3/2}}+\frac{3 \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{(a+b)^{5/2}}\right)","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)}{2 a^{5/2}}+\frac{1}{2 a^2 \sqrt{a+b \tan ^4(x)}}-\frac{3 a^2+b (5 a+2 b) \tan ^2(x)}{6 a^2 (a+b)^2 \sqrt{a+b \tan ^4(x)}}+\frac{1}{6 a \left(a+b \tan ^4(x)\right)^{3/2}}-\frac{a+b \tan ^2(x)}{6 a (a+b) \left(a+b \tan ^4(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{5/2}}",1,"((3*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])])/(a + b)^(5/2) + Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Tan[x]^4)/a]/(a*(a + b*Tan[x]^4)^(3/2)) - (a^2*(4*a + b) + 3*a*b*(2*a + b)*Tan[x]^2 + 3*a^2*b*Tan[x]^4 + b^2*(5*a + 2*b)*Tan[x]^6)/(a^2*(a + b)^2*(a + b*Tan[x]^4)^(3/2)))/6","C",1
406,1,151,212,1.4302707,"\int (d \tan (e+f x))^m \left(a+b \sqrt{c \tan (e+f x)}\right)^2 \, dx","Integrate[(d*Tan[e + f*x])^m*(a + b*Sqrt[c*Tan[e + f*x]])^2,x]","\frac{\tan (e+f x) (d \tan (e+f x))^m \left(\frac{a^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(e+f x)\right)}{m+1}+b \left(\frac{4 a \sqrt{c \tan (e+f x)} \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(e+f x)\right)}{2 m+3}+\frac{b c \tan (e+f x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(e+f x)\right)}{m+2}\right)\right)}{f}","\frac{\left(a^2-b^2 \sqrt{-c^2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1)}+\frac{\left(a^2+b^2 \sqrt{-c^2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1)}+\frac{4 a b (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(e+f x)\right)}{c f (2 m+3)}",1,"(Tan[e + f*x]*(d*Tan[e + f*x])^m*((a^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[e + f*x]^2])/(1 + m) + b*((b*c*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[e + f*x]^2]*Tan[e + f*x])/(2 + m) + (4*a*Hypergeometric2F1[1, (3 + 2*m)/4, (7 + 2*m)/4, -Tan[e + f*x]^2]*Sqrt[c*Tan[e + f*x]])/(3 + 2*m))))/f","A",1
407,1,304,121,0.6531535,"\int (d \tan (e+f x))^m \left(a+b \sqrt{c \tan (e+f x)}\right) \, dx","Integrate[(d*Tan[e + f*x])^m*(a + b*Sqrt[c*Tan[e + f*x]]),x]","\frac{\tan (e+f x) (d \tan (e+f x))^m \left(\left(a-b \sqrt[4]{-c^2}\right) \, _2F_1\left(1,2 (m+1);2 m+3;-\frac{\sqrt{c \tan (e+f x)}}{\sqrt[4]{-c^2}}\right)+\left(a+i b \sqrt[4]{-c^2}\right) \, _2F_1\left(1,2 (m+1);2 m+3;-\frac{i \sqrt{c \tan (e+f x)}}{\sqrt[4]{-c^2}}\right)+a \, _2F_1\left(1,2 (m+1);2 m+3;\frac{i \sqrt{c \tan (e+f x)}}{\sqrt[4]{-c^2}}\right)+a \, _2F_1\left(1,2 (m+1);2 m+3;\frac{\sqrt{c \tan (e+f x)}}{\sqrt[4]{-c^2}}\right)-i b \sqrt[4]{-c^2} \, _2F_1\left(1,2 (m+1);2 m+3;\frac{i \sqrt{c \tan (e+f x)}}{\sqrt[4]{-c^2}}\right)+b \sqrt[4]{-c^2} \, _2F_1\left(1,2 (m+1);2 m+3;\frac{\sqrt{c \tan (e+f x)}}{\sqrt[4]{-c^2}}\right)\right)}{4 f (m+1)}","\frac{a \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(e+f x)\right)}{f (m+1)}+\frac{2 b (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(e+f x)\right)}{c f (2 m+3)}",1,"(((a - b*(-c^2)^(1/4))*Hypergeometric2F1[1, 2*(1 + m), 3 + 2*m, -(Sqrt[c*Tan[e + f*x]]/(-c^2)^(1/4))] + (a + I*b*(-c^2)^(1/4))*Hypergeometric2F1[1, 2*(1 + m), 3 + 2*m, ((-I)*Sqrt[c*Tan[e + f*x]])/(-c^2)^(1/4)] + a*Hypergeometric2F1[1, 2*(1 + m), 3 + 2*m, (I*Sqrt[c*Tan[e + f*x]])/(-c^2)^(1/4)] - I*b*(-c^2)^(1/4)*Hypergeometric2F1[1, 2*(1 + m), 3 + 2*m, (I*Sqrt[c*Tan[e + f*x]])/(-c^2)^(1/4)] + a*Hypergeometric2F1[1, 2*(1 + m), 3 + 2*m, Sqrt[c*Tan[e + f*x]]/(-c^2)^(1/4)] + b*(-c^2)^(1/4)*Hypergeometric2F1[1, 2*(1 + m), 3 + 2*m, Sqrt[c*Tan[e + f*x]]/(-c^2)^(1/4)])*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(4*f*(1 + m))","C",1
408,1,385,460,6.3124216,"\int \frac{(d \tan (e+f x))^m}{a+b \sqrt{c \tan (e+f x)}} \, dx","Integrate[(d*Tan[e + f*x])^m/(a + b*Sqrt[c*Tan[e + f*x]]),x]","\frac{2 c (c \tan (e+f x))^{-m} (d \tan (e+f x))^m \left(\frac{b^4 (c \tan (e+f x))^{m+1} \, _2F_1\left(1,2 (m+1);2 m+3;-\frac{b \sqrt{c \tan (e+f x)}}{a}\right)}{2 a (m+1) \left(a^4+b^4 c^2\right)}-\frac{b^3 (c \tan (e+f x))^{\frac{1}{2} (2 m+5)} \, _2F_1\left(1,\frac{1}{4} (2 m+5);\frac{1}{4} (2 m+9);-\tan ^2(e+f x)\right)}{c^2 (2 m+5) \left(a^4+b^4 c^2\right)}+\frac{a b^2 (c \tan (e+f x))^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(e+f x)\right)}{2 c^2 (m+2) \left(a^4+b^4 c^2\right)}+\frac{a^3 (c \tan (e+f x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(e+f x)\right)}{2 c^2 (m+1) \left(a^4+b^4 c^2\right)}-\frac{a^2 b (c \tan (e+f x))^{\frac{1}{2} (2 m+3)} \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(e+f x)\right)}{c^2 (2 m+3) \left(a^4+b^4 c^2\right)}\right)}{f}","\frac{b^4 c^2 \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,2 (m+1);2 m+3;-\frac{b \sqrt{c \tan (e+f x)}}{a}\right)}{a f (m+1) \left(a^4+b^4 c^2\right)}-\frac{b \left(a^2-b^2 \sqrt{-c^2}\right) (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{c f (2 m+3) \left(a^4+b^4 c^2\right)}-\frac{b \left(a^2+b^2 \sqrt{-c^2}\right) (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{c f (2 m+3) \left(a^4+b^4 c^2\right)}+\frac{a \left(a^2-b^2 \sqrt{-c^2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1) \left(a^4+b^4 c^2\right)}+\frac{a \left(a^2+b^2 \sqrt{-c^2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1) \left(a^4+b^4 c^2\right)}",1,"(2*c*(d*Tan[e + f*x])^m*((a^3*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[e + f*x]^2]*(c*Tan[e + f*x])^(1 + m))/(2*c^2*(a^4 + b^4*c^2)*(1 + m)) + (b^4*Hypergeometric2F1[1, 2*(1 + m), 3 + 2*m, -((b*Sqrt[c*Tan[e + f*x]])/a)]*(c*Tan[e + f*x])^(1 + m))/(2*a*(a^4 + b^4*c^2)*(1 + m)) + (a*b^2*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[e + f*x]^2]*(c*Tan[e + f*x])^(2 + m))/(2*c^2*(a^4 + b^4*c^2)*(2 + m)) - (a^2*b*Hypergeometric2F1[1, (3 + 2*m)/4, (7 + 2*m)/4, -Tan[e + f*x]^2]*(c*Tan[e + f*x])^((3 + 2*m)/2))/(c^2*(a^4 + b^4*c^2)*(3 + 2*m)) - (b^3*Hypergeometric2F1[1, (5 + 2*m)/4, (9 + 2*m)/4, -Tan[e + f*x]^2]*(c*Tan[e + f*x])^((5 + 2*m)/2))/(c^2*(a^4 + b^4*c^2)*(5 + 2*m))))/(f*(c*Tan[e + f*x])^m)","A",1
409,1,381,617,6.4119206,"\int \frac{(d \tan (e+f x))^m}{\left(a+b \sqrt{c \tan (e+f x)}\right)^2} \, dx","Integrate[(d*Tan[e + f*x])^m/(a + b*Sqrt[c*Tan[e + f*x]])^2,x]","\frac{c (d \tan (e+f x))^m \left(\frac{4 a b \left(b^4 c^2-a^4\right) (c \tan (e+f x))^{3/2} \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(e+f x)\right)}{c^2 (2 m+3)}+\frac{b^2 \left(3 a^4-b^4 c^2\right) \tan ^2(e+f x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\tan ^2(e+f x)\right)}{m+2}-\frac{8 a^3 b^3 (c \tan (e+f x))^{5/2} \, _2F_1\left(1,\frac{1}{4} (2 m+5);\frac{1}{4} (2 m+9);-\tan ^2(e+f x)\right)}{c^2 (2 m+5)}+\frac{4 a^2 b^4 c \tan (e+f x) \, _2F_1\left(1,2 (m+1);2 m+3;-\frac{b \sqrt{c \tan (e+f x)}}{a}\right)}{m+1}+\frac{a^2 \left(a^4-3 b^4 c^2\right) \tan (e+f x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(e+f x)\right)}{c (m+1)}+\frac{b^4 c \left(a^4+b^4 c^2\right) \tan (e+f x) \, _2F_1\left(2,2 (m+1);2 m+3;-\frac{b \sqrt{c \tan (e+f x)}}{a}\right)}{a^2 (m+1)}\right)}{f \left(a^4+b^4 c^2\right)^2}","\frac{4 a^2 b^4 c^2 \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,2 (m+1);2 m+3;-\frac{b \sqrt{c \tan (e+f x)}}{a}\right)}{f (m+1) \left(a^4+b^4 c^2\right)^2}+\frac{b^4 c^2 \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(2,2 (m+1);2 m+3;-\frac{b \sqrt{c \tan (e+f x)}}{a}\right)}{a^2 f (m+1) \left(a^4+b^4 c^2\right)}-\frac{2 a b \left(a^4-2 a^2 b^2 \sqrt{-c^2}-b^4 c^2\right) (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{c f (2 m+3) \left(a^4+b^4 c^2\right)^2}-\frac{2 a b \left(a^4+2 a^2 b^2 \sqrt{-c^2}-b^4 c^2\right) (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{c f (2 m+3) \left(a^4+b^4 c^2\right)^2}+\frac{\left(a^6-3 a^4 b^2 \sqrt{-c^2}-3 a^2 b^4 c^2-b^6 \left(-c^2\right)^{3/2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1) \left(a^4+b^4 c^2\right)^2}+\frac{\left(a^6+3 a^4 b^2 \sqrt{-c^2}-3 a^2 b^4 c^2+b^6 \left(-c^2\right)^{3/2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1) \left(a^4+b^4 c^2\right)^2}",1,"(c*(d*Tan[e + f*x])^m*((a^2*(a^4 - 3*b^4*c^2)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[e + f*x]^2]*Tan[e + f*x])/(c*(1 + m)) + (4*a^2*b^4*c*Hypergeometric2F1[1, 2*(1 + m), 3 + 2*m, -((b*Sqrt[c*Tan[e + f*x]])/a)]*Tan[e + f*x])/(1 + m) + (b^4*c*(a^4 + b^4*c^2)*Hypergeometric2F1[2, 2*(1 + m), 3 + 2*m, -((b*Sqrt[c*Tan[e + f*x]])/a)]*Tan[e + f*x])/(a^2*(1 + m)) + (b^2*(3*a^4 - b^4*c^2)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^2)/(2 + m) + (4*a*b*(-a^4 + b^4*c^2)*Hypergeometric2F1[1, (3 + 2*m)/4, (7 + 2*m)/4, -Tan[e + f*x]^2]*(c*Tan[e + f*x])^(3/2))/(c^2*(3 + 2*m)) - (8*a^3*b^3*Hypergeometric2F1[1, (5 + 2*m)/4, (9 + 2*m)/4, -Tan[e + f*x]^2]*(c*Tan[e + f*x])^(5/2))/(c^2*(5 + 2*m))))/((a^4 + b^4*c^2)^2*f)","A",1
410,1,76,74,0.0846529,"\int (d \tan (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(d*Tan[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) (d \tan (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(1,\frac{1}{2} (m+n p+1);\frac{1}{2} (m+n p+1)+1;-\tan ^2(e+f x)\right)}{f (m+n p+1)}","\frac{\tan (e+f x) (d \tan (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(1,\frac{1}{2} (m+n p+1);\frac{1}{2} (m+n p+3);-\tan ^2(e+f x)\right)}{f (m+n p+1)}",1,"(Hypergeometric2F1[1, (1 + m + n*p)/2, 1 + (1 + m + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(d*Tan[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + m + n*p))","A",1
411,1,65,63,0.0851543,"\int \tan ^2(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Tan[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan ^3(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+3);\frac{1}{2} (n p+3)+1;-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+3)}","\frac{\tan ^3(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+3);\frac{1}{2} (n p+5);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+3)}",1,"(Hypergeometric2F1[1, (3 + n*p)/2, 1 + (3 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 + n*p))","A",1
412,1,59,61,0.0418113,"\int \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f n p+f}","\frac{\tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f + f*n*p)","A",1
413,1,61,63,0.0511909,"\int \cot ^2(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Cot[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\cot (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-1);\frac{1}{2} (n p+1);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p-1)}","-\frac{\cot (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-1);\frac{1}{2} (n p+1);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (1-n p)}",1,"(Cot[e + f*x]*Hypergeometric2F1[1, (-1 + n*p)/2, (1 + n*p)/2, -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(-1 + n*p))","A",1
414,1,65,65,0.082638,"\int \cot ^4(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Cot[e + f*x]^4*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\cot ^3(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-3);\frac{1}{2} (n p-3)+1;-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p-3)}","-\frac{\cot ^3(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-3);\frac{1}{2} (n p-1);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (3-n p)}",1,"(Cot[e + f*x]^3*Hypergeometric2F1[1, (-3 + n*p)/2, 1 + (-3 + n*p)/2, -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(-3 + n*p))","A",1
415,1,65,65,0.1020497,"\int \cot ^6(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Cot[e + f*x]^6*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\cot ^5(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-5);\frac{1}{2} (n p-5)+1;-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p-5)}","-\frac{\cot ^5(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-5);\frac{1}{2} (n p-3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (5-n p)}",1,"(Cot[e + f*x]^5*Hypergeometric2F1[1, (-5 + n*p)/2, 1 + (-5 + n*p)/2, -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(-5 + n*p))","A",1
416,1,61,63,0.0685222,"\int \tan ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Tan[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan ^4(e+f x) \, _2F_1\left(1,\frac{n p}{2}+2;\frac{n p}{2}+3;-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+4)}","\frac{\tan ^4(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+4);\frac{1}{2} (n p+6);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+4)}",1,"(Hypergeometric2F1[1, 2 + (n*p)/2, 3 + (n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^4*(b*(c*Tan[e + f*x])^n)^p)/(f*(4 + n*p))","A",1
417,1,61,63,0.0625652,"\int \tan (e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan ^2(e+f x) \, _2F_1\left(1,\frac{n p}{2}+1;\frac{n p}{2}+2;-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+2)}","\frac{\tan ^2(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+2)}",1,"(Hypergeometric2F1[1, 1 + (n*p)/2, 2 + (n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p)/(f*(2 + n*p))","A",1
418,1,50,50,0.0172151,"\int \cot (e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Cot[e + f*x]*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\, _2F_1\left(1,\frac{n p}{2};\frac{n p}{2}+1;-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f n p}","\frac{\, _2F_1\left(1,\frac{n p}{2};\frac{n p}{2}+1;-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f n p}",1,"(Hypergeometric2F1[1, (n*p)/2, 1 + (n*p)/2, -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*n*p)","A",1
419,1,59,62,0.0594901,"\int \cot ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Cot[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\cot ^2(e+f x) \, _2F_1\left(1,\frac{n p}{2}-1;\frac{n p}{2};-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p-2)}","-\frac{\cot ^2(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-2);\frac{n p}{2};-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (2-n p)}",1,"(Cot[e + f*x]^2*Hypergeometric2F1[1, -1 + (n*p)/2, (n*p)/2, -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(-2 + n*p))","A",1
420,0,0,30,3.7634168,"\int (d \tan (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(d*Tan[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int (d \tan (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left((d \tan (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Integrate[(d*Tan[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",-1
421,1,70,78,0.1583587,"\int (d \cot (e+f x))^m \left(b \tan ^2(e+f x)\right)^p \, dx","Integrate[(d*Cot[e + f*x])^m*(b*Tan[e + f*x]^2)^p,x]","-\frac{d \left(b \tan ^2(e+f x)\right)^p (d \cot (e+f x))^{m-1} \, _2F_1\left(1,-\frac{m}{2}+p+\frac{1}{2};-\frac{m}{2}+p+\frac{3}{2};-\tan ^2(e+f x)\right)}{f (m-2 p-1)}","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p (d \cot (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (-m+2 p+1);\frac{1}{2} (-m+2 p+3);-\tan ^2(e+f x)\right)}{f (-m+2 p+1)}",1,"-((d*(d*Cot[e + f*x])^(-1 + m)*Hypergeometric2F1[1, 1/2 - m/2 + p, 3/2 - m/2 + p, -Tan[e + f*x]^2]*(b*Tan[e + f*x]^2)^p)/(f*(-1 + m - 2*p)))","A",1
422,1,265,107,2.5791688,"\int (d \cot (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[(d*Cot[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{a (m-3) \cos ^2(e+f x) \cot (e+f x) (d \cot (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p F_1\left(\frac{1-m}{2};-p,1;\frac{3-m}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)}{f (m-1) \left(-2 b p F_1\left(\frac{3-m}{2};1-p,1;\frac{5-m}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)+2 a F_1\left(\frac{3-m}{2};-p,2;\frac{5-m}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)+a (m-3) \cot ^2(e+f x) F_1\left(\frac{1-m}{2};-p,1;\frac{3-m}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right)\right)}","\frac{\tan (e+f x) (d \cot (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1-m}{2};1,-p;\frac{3-m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f (1-m)}",1,"-((a*(-3 + m)*AppellF1[(1 - m)/2, -p, 1, (3 - m)/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]^2*Cot[e + f*x]*(d*Cot[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p)/(f*(-1 + m)*(-2*b*p*AppellF1[(3 - m)/2, 1 - p, 1, (5 - m)/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*a*AppellF1[(3 - m)/2, -p, 2, (5 - m)/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + a*(-3 + m)*AppellF1[(1 - m)/2, -p, 1, (3 - m)/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cot[e + f*x]^2)))","B",0
423,1,77,80,0.1235968,"\int (d \cot (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(d*Cot[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{d (d \cot (e+f x))^{m-1} \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(1,\frac{1}{2} (-m+n p+1);\frac{1}{2} (-m+n p+3);-\tan ^2(e+f x)\right)}{f (-m+n p+1)}","\frac{\tan (e+f x) (d \cot (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(1,\frac{1}{2} (-m+n p+1);\frac{1}{2} (-m+n p+3);-\tan ^2(e+f x)\right)}{f (-m+n p+1)}",1,"(d*(d*Cot[e + f*x])^(-1 + m)*Hypergeometric2F1[1, (1 - m + n*p)/2, (3 - m + n*p)/2, -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - m + n*p))","A",1
424,0,0,57,9.8643235,"\int (d \cot (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(d*Cot[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int (d \cot (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\left(\frac{\tan (e+f x)}{d}\right)^m (d \cot (e+f x))^m \text{Int}\left(\left(\frac{\tan (e+f x)}{d}\right)^{-m} \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Integrate[(d*Cot[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",-1
425,1,93,70,0.0753841,"\int \sec ^3(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Tan[c + d*x]^2),x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}-\frac{b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{b \tan (c+d x) \sec (c+d x)}{8 d}","\frac{(4 a-b) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(4 a-b) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) - (b*ArcTanh[Sin[c + d*x]])/(8*d) + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d) - (b*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",1
426,1,48,42,0.0280498,"\int \sec (c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Tan[c + d*x]^2),x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b \tan (c+d x) \sec (c+d x)}{2 d}","\frac{(2 a-b) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*ArcTanh[Sin[c + d*x]])/d - (b*ArcTanh[Sin[c + d*x]])/(2*d) + (b*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
427,1,47,28,0.0287076,"\int \cos (c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Tan[c + d*x]^2),x]","\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","\frac{(a-b) \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b*ArcTanh[Sin[c + d*x]])/d + (a*Cos[d*x]*Sin[c])/d + (a*Cos[c]*Sin[d*x])/d - (b*Sin[c + d*x])/d","A",1
428,1,44,32,0.0172594,"\int \cos ^3(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Tan[c + d*x]^2),x]","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin ^3(c+d x)}{3 d}","\frac{a \sin (c+d x)}{d}-\frac{(a-b) \sin ^3(c+d x)}{3 d}",1,"(a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d) + (b*Sin[c + d*x]^3)/(3*d)","A",1
429,1,52,54,0.190135,"\int \cos ^5(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Tan[c + d*x]^2),x]","\frac{\sin (c+d x) (4 (7 a-2 b) \cos (2 (c+d x))+3 (a-b) \cos (4 (c+d x))+89 a+11 b)}{120 d}","\frac{(a-b) \sin ^5(c+d x)}{5 d}-\frac{(2 a-b) \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}",1,"((89*a + 11*b + 4*(7*a - 2*b)*Cos[2*(c + d*x)] + 3*(a - b)*Cos[4*(c + d*x)])*Sin[c + d*x])/(120*d)","A",1
430,1,75,76,0.3099114,"\int \cos ^7(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^7*(a + b*Tan[c + d*x]^2),x]","\frac{\sin (c+d x) ((897 a-113 b) \cos (2 (c+d x))+6 (27 a-13 b) \cos (4 (c+d x))+15 a \cos (6 (c+d x))+2286 a-15 b \cos (6 (c+d x))+206 b)}{3360 d}","-\frac{(a-b) \sin ^7(c+d x)}{7 d}+\frac{(3 a-2 b) \sin ^5(c+d x)}{5 d}-\frac{(3 a-b) \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}",1,"((2286*a + 206*b + (897*a - 113*b)*Cos[2*(c + d*x)] + 6*(27*a - 13*b)*Cos[4*(c + d*x)] + 15*a*Cos[6*(c + d*x)] - 15*b*Cos[6*(c + d*x)])*Sin[c + d*x])/(3360*d)","A",1
431,1,75,68,0.2639694,"\int \sec ^6(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^6*(a + b*Tan[c + d*x]^2),x]","\frac{\tan (c+d x) \left(21 a \tan ^4(c+d x)+70 a \tan ^2(c+d x)+105 a+15 b \sec ^6(c+d x)-3 b \sec ^4(c+d x)-4 b \sec ^2(c+d x)-8 b\right)}{105 d}","\frac{(a+2 b) \tan ^5(c+d x)}{5 d}+\frac{(2 a+b) \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \tan ^7(c+d x)}{7 d}",1,"(Tan[c + d*x]*(105*a - 8*b - 4*b*Sec[c + d*x]^2 - 3*b*Sec[c + d*x]^4 + 15*b*Sec[c + d*x]^6 + 70*a*Tan[c + d*x]^2 + 21*a*Tan[c + d*x]^4))/(105*d)","A",1
432,1,53,46,0.1463467,"\int \sec ^4(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^4*(a + b*Tan[c + d*x]^2),x]","\frac{\tan (c+d x) \left(5 a \tan ^2(c+d x)+15 a+3 b \sec ^4(c+d x)-b \sec ^2(c+d x)-2 b\right)}{15 d}","\frac{(a+b) \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \tan ^5(c+d x)}{5 d}",1,"(Tan[c + d*x]*(15*a - 2*b - b*Sec[c + d*x]^2 + 3*b*Sec[c + d*x]^4 + 5*a*Tan[c + d*x]^2))/(15*d)","A",1
433,1,28,28,0.0123663,"\int \sec ^2(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Tan[c + d*x]^2),x]","\frac{a \tan (c+d x)}{d}+\frac{b \tan ^3(c+d x)}{3 d}","\frac{a \tan (c+d x)}{d}+\frac{b \tan ^3(c+d x)}{3 d}",1,"(a*Tan[c + d*x])/d + (b*Tan[c + d*x]^3)/(3*d)","A",1
434,1,32,33,0.0580107,"\int \cos ^2(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Tan[c + d*x]^2),x]","\frac{2 (a+b) (c+d x)+(a-b) \sin (2 (c+d x))}{4 d}","\frac{(a-b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a+b)",1,"(2*(a + b)*(c + d*x) + (a - b)*Sin[2*(c + d*x)])/(4*d)","A",1
435,1,46,61,0.131206,"\int \cos ^4(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Tan[c + d*x]^2),x]","\frac{(a-b) \sin (4 (c+d x))+12 a (c+d x)+8 a \sin (2 (c+d x))+4 b d x}{32 d}","\frac{(a-b) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{(3 a+b) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (3 a+b)",1,"(4*b*d*x + 12*a*(c + d*x) + 8*a*Sin[2*(c + d*x)] + (a - b)*Sin[4*(c + d*x)])/(32*d)","A",1
436,1,74,87,0.2119739,"\int \cos ^6(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + b*Tan[c + d*x]^2),x]","\frac{3 (15 a+b) \sin (2 (c+d x))+(9 a-3 b) \sin (4 (c+d x))+a \sin (6 (c+d x))+60 a c+60 a d x-b \sin (6 (c+d x))+12 b d x}{192 d}","\frac{(a-b) \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{(5 a+b) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{(5 a+b) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x (5 a+b)",1,"(60*a*c + 60*a*d*x + 12*b*d*x + 3*(15*a + b)*Sin[2*(c + d*x)] + (9*a - 3*b)*Sin[4*(c + d*x)] + a*Sin[6*(c + d*x)] - b*Sin[6*(c + d*x)])/(192*d)","A",1
437,1,875,128,8.3329796,"\int \sec ^3(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Integrate[Sec[c + d*x]^3*(a + b*Tan[c + d*x]^2)^2,x]","\frac{\sin (c+d x) \left(380 (a-b)^2 \, _4F_3\left(\frac{3}{2},2,2,2;1,1,\frac{9}{2};\sin ^2(c+d x)\right) \sqrt{\sin ^2(c+d x)} \sin ^{10}(c+d x)+128 (a-b)^2 \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{9}{2};\sin ^2(c+d x)\right) \sqrt{\sin ^2(c+d x)} \sin ^{10}(c+d x)+16 (a-b)^2 \, _6F_5\left(\frac{3}{2},2,2,2,2,2;1,1,1,1,\frac{9}{2};\sin ^2(c+d x)\right) \sqrt{\sin ^2(c+d x)} \sin ^{10}(c+d x)+525 (a-b)^2 \tanh ^{-1}\left(\sqrt{\sin ^2(c+d x)}\right) \sin ^8(c+d x)-968 a (a-b) \, _4F_3\left(\frac{3}{2},2,2,2;1,1,\frac{9}{2};\sin ^2(c+d x)\right) \sqrt{\sin ^2(c+d x)} \sin ^8(c+d x)-288 a (a-b) \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{9}{2};\sin ^2(c+d x)\right) \sqrt{\sin ^2(c+d x)} \sin ^8(c+d x)-32 a (a-b) \, _6F_5\left(\frac{3}{2},2,2,2,2,2;1,1,1,1,\frac{9}{2};\sin ^2(c+d x)\right) \sqrt{\sin ^2(c+d x)} \sin ^8(c+d x)-19845 (a-b)^2 \tanh ^{-1}\left(\sqrt{\sin ^2(c+d x)}\right) \sin ^6(c+d x)-1365 a (a-b) \tanh ^{-1}\left(\sqrt{\sin ^2(c+d x)}\right) \sin ^6(c+d x)+8855 (a-b)^2 \sqrt{\sin ^2(c+d x)} \sin ^6(c+d x)+620 a^2 \, _4F_3\left(\frac{3}{2},2,2,2;1,1,\frac{9}{2};\sin ^2(c+d x)\right) \sqrt{\sin ^2(c+d x)} \sin ^6(c+d x)+160 a^2 \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{9}{2};\sin ^2(c+d x)\right) \sqrt{\sin ^2(c+d x)} \sin ^6(c+d x)+16 a^2 \, _6F_5\left(\frac{3}{2},2,2,2,2,2;1,1,1,1,\frac{9}{2};\sin ^2(c+d x)\right) \sqrt{\sin ^2(c+d x)} \sin ^6(c+d x)+1680 a^2 \tanh ^{-1}\left(\sqrt{\sin ^2(c+d x)}\right) \sin ^4(c+d x)+32970 (a-b)^2 \tanh ^{-1}\left(\sqrt{\sin ^2(c+d x)}\right) \sin ^4(c+d x)+54180 a (a-b) \tanh ^{-1}\left(\sqrt{\sin ^2(c+d x)}\right) \sin ^4(c+d x)-32970 (a-b)^2 \sqrt{\sin ^2(c+d x)} \sin ^4(c+d x)-23555 a (a-b) \sqrt{\sin ^2(c+d x)} \sin ^4(c+d x)-36855 a^2 \tanh ^{-1}\left(\sqrt{\sin ^2(c+d x)}\right) \sin ^2(c+d x)-91875 a (a-b) \tanh ^{-1}\left(\sqrt{\sin ^2(c+d x)}\right) \sin ^2(c+d x)+14980 a^2 \sin ^2(c+d x)^{3/2}+91875 a (a-b) \sin ^2(c+d x)^{3/2}+65625 a^2 \tanh ^{-1}\left(\sqrt{\sin ^2(c+d x)}\right)-65625 a^2 \sqrt{\sin ^2(c+d x)}\right)}{2520 d \sin ^2(c+d x)^{5/2}}","\frac{\left(8 a^2-4 a b+b^2\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left(8 a^2-4 a b+b^2\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{b (8 a-3 b) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{b \tan (c+d x) \sec ^5(c+d x) \left(a-(a-b) \sin ^2(c+d x)\right)}{6 d}",1,"(Sin[c + d*x]*(65625*a^2*ArcTanh[Sqrt[Sin[c + d*x]^2]] - 36855*a^2*ArcTanh[Sqrt[Sin[c + d*x]^2]]*Sin[c + d*x]^2 - 91875*a*(a - b)*ArcTanh[Sqrt[Sin[c + d*x]^2]]*Sin[c + d*x]^2 + 1680*a^2*ArcTanh[Sqrt[Sin[c + d*x]^2]]*Sin[c + d*x]^4 + 54180*a*(a - b)*ArcTanh[Sqrt[Sin[c + d*x]^2]]*Sin[c + d*x]^4 + 32970*(a - b)^2*ArcTanh[Sqrt[Sin[c + d*x]^2]]*Sin[c + d*x]^4 - 1365*a*(a - b)*ArcTanh[Sqrt[Sin[c + d*x]^2]]*Sin[c + d*x]^6 - 19845*(a - b)^2*ArcTanh[Sqrt[Sin[c + d*x]^2]]*Sin[c + d*x]^6 + 525*(a - b)^2*ArcTanh[Sqrt[Sin[c + d*x]^2]]*Sin[c + d*x]^8 - 65625*a^2*Sqrt[Sin[c + d*x]^2] - 23555*a*(a - b)*Sin[c + d*x]^4*Sqrt[Sin[c + d*x]^2] - 32970*(a - b)^2*Sin[c + d*x]^4*Sqrt[Sin[c + d*x]^2] + 8855*(a - b)^2*Sin[c + d*x]^6*Sqrt[Sin[c + d*x]^2] + 620*a^2*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, Sin[c + d*x]^2]*Sin[c + d*x]^6*Sqrt[Sin[c + d*x]^2] + 160*a^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 9/2}, Sin[c + d*x]^2]*Sin[c + d*x]^6*Sqrt[Sin[c + d*x]^2] + 16*a^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 9/2}, Sin[c + d*x]^2]*Sin[c + d*x]^6*Sqrt[Sin[c + d*x]^2] - 968*a*(a - b)*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, Sin[c + d*x]^2]*Sin[c + d*x]^8*Sqrt[Sin[c + d*x]^2] - 288*a*(a - b)*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 9/2}, Sin[c + d*x]^2]*Sin[c + d*x]^8*Sqrt[Sin[c + d*x]^2] - 32*a*(a - b)*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 9/2}, Sin[c + d*x]^2]*Sin[c + d*x]^8*Sqrt[Sin[c + d*x]^2] + 380*(a - b)^2*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, Sin[c + d*x]^2]*Sin[c + d*x]^10*Sqrt[Sin[c + d*x]^2] + 128*(a - b)^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 9/2}, Sin[c + d*x]^2]*Sin[c + d*x]^10*Sqrt[Sin[c + d*x]^2] + 16*(a - b)^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 9/2}, Sin[c + d*x]^2]*Sin[c + d*x]^10*Sqrt[Sin[c + d*x]^2] + 14980*a^2*(Sin[c + d*x]^2)^(3/2) + 91875*a*(a - b)*(Sin[c + d*x]^2)^(3/2)))/(2520*d*(Sin[c + d*x]^2)^(5/2))","C",0
438,1,347,96,7.3341013,"\int \sec (c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Integrate[Sec[c + d*x]*(a + b*Tan[c + d*x]^2)^2,x]","\frac{\csc ^3(c+d x) \left(128 \sin ^6(c+d x) \left(\frac{1}{2} a^2 (5 \cos (2 (c+d x))+9) \cos ^2(c+d x)+b \sin ^2(c+d x) \left(5 a \cos (2 (c+d x))+7 a+5 b \sin ^2(c+d x)\right)\right) \, _4F_3\left(\frac{3}{2},2,2,2;1,1,\frac{9}{2};\sin ^2(c+d x)\right)+128 \sin ^6(c+d x) \left((b-a) \sin ^2(c+d x)+a\right)^2 \, _5F_4\left(\frac{3}{2},2,2,2,2;1,1,1,\frac{9}{2};\sin ^2(c+d x)\right)+35 \left(\left(-3161 a^2+5108 a b-1947 b^2\right) \sin ^4(c+d x)+\frac{3 \tanh ^{-1}\left(\sqrt{\sin ^2(c+d x)}\right) \left(\left(-400 a^2+778 a b-378 b^2\right) \sin ^6(c+d x)+\left(1674 a^2-2286 a b+649 b^2\right) \sin ^4(c+d x)+1125 a^2+9 (a-b)^2 \sin ^8(c+d x)-2 a (1172 a-875 b) \sin ^2(c+d x)\right)}{\sqrt{\sin ^2(c+d x)}}-3375 a^2+485 (a-b)^2 \sin ^6(c+d x)+3 a (1969 a-1750 b) \sin ^2(c+d x)\right)\right)}{6720 d}","\frac{\left(8 a^2-8 a b+3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 b (2 a-b) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x) \left(a-(a-b) \sin ^2(c+d x)\right)}{4 d}",1,"(Csc[c + d*x]^3*(128*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 9/2}, Sin[c + d*x]^2]*Sin[c + d*x]^6*(a + (-a + b)*Sin[c + d*x]^2)^2 + 128*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, Sin[c + d*x]^2]*Sin[c + d*x]^6*((a^2*Cos[c + d*x]^2*(9 + 5*Cos[2*(c + d*x)]))/2 + b*Sin[c + d*x]^2*(7*a + 5*a*Cos[2*(c + d*x)] + 5*b*Sin[c + d*x]^2)) + 35*(-3375*a^2 + 3*a*(1969*a - 1750*b)*Sin[c + d*x]^2 + (-3161*a^2 + 5108*a*b - 1947*b^2)*Sin[c + d*x]^4 + 485*(a - b)^2*Sin[c + d*x]^6 + (3*ArcTanh[Sqrt[Sin[c + d*x]^2]]*(1125*a^2 - 2*a*(1172*a - 875*b)*Sin[c + d*x]^2 + (1674*a^2 - 2286*a*b + 649*b^2)*Sin[c + d*x]^4 + (-400*a^2 + 778*a*b - 378*b^2)*Sin[c + d*x]^6 + 9*(a - b)^2*Sin[c + d*x]^8))/Sqrt[Sin[c + d*x]^2])))/(6720*d)","C",0
439,1,66,62,0.4599122,"\int \cos (c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Integrate[Cos[c + d*x]*(a + b*Tan[c + d*x]^2)^2,x]","\frac{\tan (c+d x) \sec (c+d x) \left(a^2+(a-b)^2 \cos (2 (c+d x))-2 a b+2 b^2\right)+b (4 a-3 b) \tanh ^{-1}(\sin (c+d x))}{2 d}","\frac{(a-b)^2 \sin (c+d x)}{d}+\frac{b (4 a-3 b) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^2 \tan (c+d x) \sec (c+d x)}{2 d}",1,"((4*a - 3*b)*b*ArcTanh[Sin[c + d*x]] + (a^2 - 2*a*b + 2*b^2 + (a - b)^2*Cos[2*(c + d*x)])*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
440,1,71,56,0.4302332,"\int \cos ^3(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Integrate[Cos[c + d*x]^3*(a + b*Tan[c + d*x]^2)^2,x]","\frac{\sin (c+d x) \left(\frac{3 b^2 \tanh ^{-1}\left(\sqrt{\sin ^2(c+d x)}\right)}{\sqrt{\sin ^2(c+d x)}}-(a-b) \left((a-b) \sin ^2(c+d x)-3 (a+b)\right)\right)}{3 d}","\frac{\left(a^2-b^2\right) \sin (c+d x)}{d}-\frac{(a-b)^2 \sin ^3(c+d x)}{3 d}+\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(Sin[c + d*x]*((3*b^2*ArcTanh[Sqrt[Sin[c + d*x]^2]])/Sqrt[Sin[c + d*x]^2] - (a - b)*(-3*(a + b) + (a - b)*Sin[c + d*x]^2)))/(3*d)","A",1
441,1,52,57,0.1636408,"\int \cos ^5(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Integrate[Cos[c + d*x]^5*(a + b*Tan[c + d*x]^2)^2,x]","\frac{15 a^2 \sin (c+d x)+3 (a-b)^2 \sin ^5(c+d x)-10 a (a-b) \sin ^3(c+d x)}{15 d}","\frac{a^2 \sin (c+d x)}{d}+\frac{(a-b)^2 \sin ^5(c+d x)}{5 d}-\frac{2 a (a-b) \sin ^3(c+d x)}{3 d}",1,"(15*a^2*Sin[c + d*x] - 10*a*(a - b)*Sin[c + d*x]^3 + 3*(a - b)^2*Sin[c + d*x]^5)/(15*d)","A",1
442,1,77,86,0.3774533,"\int \cos ^7(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Integrate[Cos[c + d*x]^7*(a + b*Tan[c + d*x]^2)^2,x]","\frac{21 \left(3 a^2-4 a b+b^2\right) \sin ^5(c+d x)+105 a^2 \sin (c+d x)-15 (a-b)^2 \sin ^7(c+d x)-35 a (3 a-2 b) \sin ^3(c+d x)}{105 d}","\frac{a^2 \sin (c+d x)}{d}-\frac{(a-b)^2 \sin ^7(c+d x)}{7 d}+\frac{(a-b) (3 a-b) \sin ^5(c+d x)}{5 d}-\frac{a (3 a-2 b) \sin ^3(c+d x)}{3 d}",1,"(105*a^2*Sin[c + d*x] - 35*a*(3*a - 2*b)*Sin[c + d*x]^3 + 21*(3*a^2 - 4*a*b + b^2)*Sin[c + d*x]^5 - 15*(a - b)^2*Sin[c + d*x]^7)/(105*d)","A",1
443,1,116,114,0.6177302,"\int \cos ^9(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Integrate[Cos[c + d*x]^9*(a + b*Tan[c + d*x]^2)^2,x]","\frac{630 \left(63 a^2+14 a b+3 b^2\right) \sin (c+d x)+420 \left(21 a^2-b^2\right) \sin (3 (c+d x))+252 \left(9 a^2-4 a b-b^2\right) \sin (5 (c+d x))+35 (a-b)^2 \sin (9 (c+d x))+45 (9 a-b) (a-b) \sin (7 (c+d x))}{80640 d}","\frac{\left(6 a^2-6 a b+b^2\right) \sin ^5(c+d x)}{5 d}+\frac{a^2 \sin (c+d x)}{d}+\frac{(a-b)^2 \sin ^9(c+d x)}{9 d}-\frac{2 (a-b) (2 a-b) \sin ^7(c+d x)}{7 d}-\frac{2 a (2 a-b) \sin ^3(c+d x)}{3 d}",1,"(630*(63*a^2 + 14*a*b + 3*b^2)*Sin[c + d*x] + 420*(21*a^2 - b^2)*Sin[3*(c + d*x)] + 252*(9*a^2 - 4*a*b - b^2)*Sin[5*(c + d*x)] + 45*(a - b)*(9*a - b)*Sin[7*(c + d*x)] + 35*(a - b)^2*Sin[9*(c + d*x)])/(80640*d)","A",1
444,1,106,96,0.3894349,"\int \sec ^6(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Integrate[Sec[c + d*x]^6*(a + b*Tan[c + d*x]^2)^2,x]","\frac{\tan (c+d x) \left(3 \left(21 a^2-6 a b+b^2\right) \sec ^4(c+d x)+4 \left(21 a^2-6 a b+b^2\right) \sec ^2(c+d x)+8 \left(21 a^2-6 a b+b^2\right)+10 b (9 a-5 b) \sec ^6(c+d x)+35 b^2 \sec ^8(c+d x)\right)}{315 d}","\frac{\left(a^2+4 a b+b^2\right) \tan ^5(c+d x)}{5 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 b (a+b) \tan ^7(c+d x)}{7 d}+\frac{2 a (a+b) \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^9(c+d x)}{9 d}",1,"((8*(21*a^2 - 6*a*b + b^2) + 4*(21*a^2 - 6*a*b + b^2)*Sec[c + d*x]^2 + 3*(21*a^2 - 6*a*b + b^2)*Sec[c + d*x]^4 + 10*(9*a - 5*b)*b*Sec[c + d*x]^6 + 35*b^2*Sec[c + d*x]^8)*Tan[c + d*x])/(315*d)","A",1
445,1,83,74,0.5563325,"\int \sec ^4(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Integrate[Sec[c + d*x]^4*(a + b*Tan[c + d*x]^2)^2,x]","\frac{\tan (c+d x) \left(\left(35 a^2-14 a b+3 b^2\right) \sec ^2(c+d x)+70 a^2+6 b (7 a-4 b) \sec ^4(c+d x)-28 a b+15 b^2 \sec ^6(c+d x)+6 b^2\right)}{105 d}","\frac{a^2 \tan (c+d x)}{d}+\frac{b (2 a+b) \tan ^5(c+d x)}{5 d}+\frac{a (a+2 b) \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^7(c+d x)}{7 d}",1,"((70*a^2 - 28*a*b + 6*b^2 + (35*a^2 - 14*a*b + 3*b^2)*Sec[c + d*x]^2 + 6*(7*a - 4*b)*b*Sec[c + d*x]^4 + 15*b^2*Sec[c + d*x]^6)*Tan[c + d*x])/(105*d)","A",1
446,1,49,49,0.1449612,"\int \sec ^2(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Integrate[Sec[c + d*x]^2*(a + b*Tan[c + d*x]^2)^2,x]","\frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^5(c+d x)}{5 d}","\frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^5(c+d x)}{5 d}",1,"(a^2*Tan[c + d*x])/d + (2*a*b*Tan[c + d*x]^3)/(3*d) + (b^2*Tan[c + d*x]^5)/(5*d)","A",1
447,1,55,55,0.4210867,"\int \cos ^2(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Integrate[Cos[c + d*x]^2*(a + b*Tan[c + d*x]^2)^2,x]","\frac{2 \left(a^2+2 a b-3 b^2\right) (c+d x)+(a-b)^2 \sin (2 (c+d x))+4 b^2 \tan (c+d x)}{4 d}","\frac{(a-b)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a+3 b) (a-b)+\frac{b^2 \tan (c+d x)}{d}",1,"(2*(a^2 + 2*a*b - 3*b^2)*(c + d*x) + (a - b)^2*Sin[2*(c + d*x)] + 4*b^2*Tan[c + d*x])/(4*d)","A",1
448,1,65,87,0.3308918,"\int \cos ^4(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Integrate[Cos[c + d*x]^4*(a + b*Tan[c + d*x]^2)^2,x]","\frac{4 \left(3 a^2+2 a b+3 b^2\right) (c+d x)+8 \left(a^2-b^2\right) \sin (2 (c+d x))+(a-b)^2 \sin (4 (c+d x))}{32 d}","\frac{3 \left(a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(3 a^2+2 a b+3 b^2\right)+\frac{(a-b) \sin (c+d x) \cos ^3(c+d x) \left(a+b \tan ^2(c+d x)\right)}{4 d}",1,"(4*(3*a^2 + 2*a*b + 3*b^2)*(c + d*x) + 8*(a^2 - b^2)*Sin[2*(c + d*x)] + (a - b)^2*Sin[4*(c + d*x)])/(32*d)","A",1
449,1,87,122,0.3741408,"\int \cos ^6(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Integrate[Cos[c + d*x]^6*(a + b*Tan[c + d*x]^2)^2,x]","\frac{12 (b+(1-2 i) a) (b+(1+2 i) a) (c+d x)+(a-b)^2 \sin (6 (c+d x))+3 (3 a+b) (a-b) \sin (4 (c+d x))+3 (5 a-b) (3 a+b) \sin (2 (c+d x))}{192 d}","\frac{\left(5 a^2+2 a b+b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(5 a^2+2 a b+b^2\right)+\frac{(a-b) (5 a+3 b) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{(a-b) \sin (c+d x) \cos ^5(c+d x) \left(a+b \tan ^2(c+d x)\right)}{6 d}",1,"(12*((1 - 2*I)*a + b)*((1 + 2*I)*a + b)*(c + d*x) + 3*(5*a - b)*(3*a + b)*Sin[2*(c + d*x)] + 3*(a - b)*(3*a + b)*Sin[4*(c + d*x)] + (a - b)^2*Sin[6*(c + d*x)])/(192*d)","C",1
450,1,207,90,1.3823162,"\int \frac{\sec ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Tan[c + d*x]^2),x]","\frac{-\frac{2 (a-b)^{3/2} \log \left(\sqrt{a}-\sqrt{a-b} \sin (c+d x)\right)}{\sqrt{a}}+\frac{2 (a-b)^{3/2} \log \left(\sqrt{a-b} \sin (c+d x)+\sqrt{a}\right)}{\sqrt{a}}+2 (2 a-3 b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 (3 b-2 a) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{b}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}}{4 b^2 d}","\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^2 d}-\frac{(2 a-3 b) \tanh ^{-1}(\sin (c+d x))}{2 b^2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d}",1,"(2*(2*a - 3*b)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(-2*a + 3*b)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - (2*(a - b)^(3/2)*Log[Sqrt[a] - Sqrt[a - b]*Sin[c + d*x]])/Sqrt[a] + (2*(a - b)^(3/2)*Log[Sqrt[a] + Sqrt[a - b]*Sin[c + d*x]])/Sqrt[a] + b/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - b/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(4*b^2*d)","B",1
451,1,53,59,0.10739,"\int \frac{\sec ^3(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Tan[c + d*x]^2),x]","\frac{\tanh ^{-1}(\sin (c+d x))-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a}}}{b d}","\frac{\tanh ^{-1}(\sin (c+d x))}{b d}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b d}",1,"(ArcTanh[Sin[c + d*x]] - (Sqrt[a - b]*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/Sqrt[a])/(b*d)","A",1
452,1,40,40,0.0475518,"\int \frac{\sec (c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[Sec[c + d*x]/(a + b*Tan[c + d*x]^2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d \sqrt{a-b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d \sqrt{a-b}}",1,"ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a - b]*d)","A",1
453,1,60,60,0.1017134,"\int \frac{\cos (c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[Cos[c + d*x]/(a + b*Tan[c + d*x]^2),x]","\frac{\sin (c+d x)}{d (a-b)}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^{3/2}}","\frac{\sin (c+d x)}{d (a-b)}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^{3/2}}",1,"-((b*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(3/2)*d)) + Sin[c + d*x]/((a - b)*d)","A",1
454,1,115,88,0.5101903,"\int \frac{\cos ^3(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Tan[c + d*x]^2),x]","\frac{\frac{6 b^2 \left(\log \left(\sqrt{a-b} \sin (c+d x)+\sqrt{a}\right)-\log \left(\sqrt{a}-\sqrt{a-b} \sin (c+d x)\right)\right)}{\sqrt{a} (a-b)^{5/2}}+\frac{3 (3 a-7 b) \sin (c+d x)}{(a-b)^2}+\frac{\sin (3 (c+d x))}{a-b}}{12 d}","\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^{5/2}}-\frac{\sin ^3(c+d x)}{3 d (a-b)}+\frac{(a-2 b) \sin (c+d x)}{d (a-b)^2}",1,"((6*b^2*(-Log[Sqrt[a] - Sqrt[a - b]*Sin[c + d*x]] + Log[Sqrt[a] + Sqrt[a - b]*Sin[c + d*x]]))/(Sqrt[a]*(a - b)^(5/2)) + (3*(3*a - 7*b)*Sin[c + d*x])/(a - b)^2 + Sin[3*(c + d*x)]/(a - b))/(12*d)","A",1
455,1,148,126,1.8214318,"\int \frac{\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[Cos[c + d*x]^5/(a + b*Tan[c + d*x]^2),x]","\frac{\frac{30 \left(5 a^2-16 a b+19 b^2\right) \sin (c+d x)}{(a-b)^3}+\frac{120 b^3 \left(\log \left(\sqrt{a}-\sqrt{a-b} \sin (c+d x)\right)-\log \left(\sqrt{a-b} \sin (c+d x)+\sqrt{a}\right)\right)}{\sqrt{a} (a-b)^{7/2}}+\frac{5 (5 a-9 b) \sin (3 (c+d x))}{(a-b)^2}+\frac{3 \sin (5 (c+d x))}{a-b}}{240 d}","\frac{\left(a^2-3 a b+3 b^2\right) \sin (c+d x)}{d (a-b)^3}-\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^{7/2}}+\frac{\sin ^5(c+d x)}{5 d (a-b)}-\frac{(2 a-3 b) \sin ^3(c+d x)}{3 d (a-b)^2}",1,"((120*b^3*(Log[Sqrt[a] - Sqrt[a - b]*Sin[c + d*x]] - Log[Sqrt[a] + Sqrt[a - b]*Sin[c + d*x]]))/(Sqrt[a]*(a - b)^(7/2)) + (30*(5*a^2 - 16*a*b + 19*b^2)*Sin[c + d*x])/(a - b)^3 + (5*(5*a - 9*b)*Sin[3*(c + d*x)])/(a - b)^2 + (3*Sin[5*(c + d*x)])/(a - b))/(240*d)","A",1
456,1,103,108,0.9029974,"\int \frac{\sec ^8(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[Sec[c + d*x]^8/(a + b*Tan[c + d*x]^2),x]","\frac{\sqrt{b} \tan (c+d x) \left(15 a^2-b (5 a-9 b) \sec ^2(c+d x)-40 a b+3 b^2 \sec ^4(c+d x)+33 b^2\right)-\frac{15 (a-b)^3 \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a}}}{15 b^{7/2} d}","\frac{\left(a^2-3 a b+3 b^2\right) \tan (c+d x)}{b^3 d}-\frac{(a-b)^3 \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2} d}-\frac{(a-3 b) \tan ^3(c+d x)}{3 b^2 d}+\frac{\tan ^5(c+d x)}{5 b d}",1,"((-15*(a - b)^3*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/Sqrt[a] + Sqrt[b]*(15*a^2 - 40*a*b + 33*b^2 - (5*a - 9*b)*b*Sec[c + d*x]^2 + 3*b^2*Sec[c + d*x]^4)*Tan[c + d*x])/(15*b^(7/2)*d)","A",1
457,1,74,77,0.3365486,"\int \frac{\sec ^6(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[Sec[c + d*x]^6/(a + b*Tan[c + d*x]^2),x]","\frac{\frac{3 (a-b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a}}+\sqrt{b} \tan (c+d x) \left(-3 a+b \sec ^2(c+d x)+5 b\right)}{3 b^{5/2} d}","\frac{(a-b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2} d}-\frac{(a-2 b) \tan (c+d x)}{b^2 d}+\frac{\tan ^3(c+d x)}{3 b d}",1,"((3*(a - b)^2*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/Sqrt[a] + Sqrt[b]*(-3*a + 5*b + b*Sec[c + d*x]^2)*Tan[c + d*x])/(3*b^(5/2)*d)","A",1
458,1,52,52,0.1451071,"\int \frac{\sec ^4(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Tan[c + d*x]^2),x]","\frac{\tan (c+d x)}{b d}-\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2} d}","\frac{\tan (c+d x)}{b d}-\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2} d}",1,"-(((a - b)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2)*d)) + Tan[c + d*x]/(b*d)","A",1
459,1,32,32,0.0559722,"\int \frac{\sec ^2(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Tan[c + d*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d}",1,"ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*d)","A",1
460,1,78,83,0.1882971,"\int \frac{\cos ^2(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Tan[c + d*x]^2),x]","\frac{4 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)+\sqrt{a} (2 (a-3 b) (c+d x)+(a-b) \sin (2 (c+d x)))}{4 \sqrt{a} d (a-b)^2}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^2}+\frac{\sin (c+d x) \cos (c+d x)}{2 d (a-b)}+\frac{x (a-3 b)}{2 (a-b)^2}",1,"(4*b^(3/2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]] + Sqrt[a]*(2*(a - 3*b)*(c + d*x) + (a - b)*Sin[2*(c + d*x)]))/(4*Sqrt[a]*(a - b)^2*d)","A",1
461,1,113,129,0.4232969,"\int \frac{\cos ^4(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Tan[c + d*x]^2),x]","\frac{\sqrt{a} \left(4 \left(3 a^2-10 a b+15 b^2\right) (c+d x)+8 \left(a^2-3 a b+2 b^2\right) \sin (2 (c+d x))+(a-b)^2 \sin (4 (c+d x))\right)-32 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{32 \sqrt{a} d (a-b)^3}","\frac{x \left(3 a^2-10 a b+15 b^2\right)}{8 (a-b)^3}-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^3}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 d (a-b)}+\frac{(3 a-7 b) \sin (c+d x) \cos (c+d x)}{8 d (a-b)^2}",1,"(-32*b^(5/2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]] + Sqrt[a]*(4*(3*a^2 - 10*a*b + 15*b^2)*(c + d*x) + 8*(a^2 - 3*a*b + 2*b^2)*Sin[2*(c + d*x)] + (a - b)^2*Sin[4*(c + d*x)]))/(32*Sqrt[a]*(a - b)^3*d)","A",1
462,1,254,167,4.0820309,"\int \frac{\sec ^7(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Integrate[Sec[c + d*x]^7/(a + b*Tan[c + d*x]^2)^2,x]","\frac{-\frac{(4 a+b) (a-b)^{3/2} \log \left(\sqrt{a}-\sqrt{a-b} \sin (c+d x)\right)}{a^{3/2}}+\frac{(4 a+b) (a-b)^{3/2} \log \left(\sqrt{a-b} \sin (c+d x)+\sqrt{a}\right)}{a^{3/2}}+\frac{4 b (a-b)^2 \sin (c+d x)}{a ((a-b) \cos (2 (c+d x))+a+b)}+2 (4 a-5 b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 (5 b-4 a) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{b}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}}{4 b^3 d}","\frac{(4 a+b) (a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^3 d}-\frac{(4 a-5 b) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}+\frac{(2 a-b) (a-b) \sin (c+d x)}{2 a b^2 d \left(a-(a-b) \sin ^2(c+d x)\right)}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d \left(a-(a-b) \sin ^2(c+d x)\right)}",1,"(2*(4*a - 5*b)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(-4*a + 5*b)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - ((a - b)^(3/2)*(4*a + b)*Log[Sqrt[a] - Sqrt[a - b]*Sin[c + d*x]])/a^(3/2) + ((a - b)^(3/2)*(4*a + b)*Log[Sqrt[a] + Sqrt[a - b]*Sin[c + d*x]])/a^(3/2) + b/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - b/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*(a - b)^2*b*Sin[c + d*x])/(a*(a + b + (a - b)*Cos[2*(c + d*x)])))/(4*b^3*d)","A",1
463,1,191,109,0.855555,"\int \frac{\sec ^5(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Tan[c + d*x]^2)^2,x]","\frac{\frac{\sqrt{a-b} (2 a+b) \log \left(\sqrt{a}-\sqrt{a-b} \sin (c+d x)\right)}{a^{3/2}}+\frac{\left(-2 a^2+a b+b^2\right) \log \left(\sqrt{a-b} \sin (c+d x)+\sqrt{a}\right)}{a^{3/2} \sqrt{a-b}}+\frac{4 b (b-a) \sin (c+d x)}{a ((a-b) \cos (2 (c+d x))+a+b)}-4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 b^2 d}","-\frac{\sqrt{a-b} (2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^2 d}-\frac{(a-b) \sin (c+d x)}{2 a b d \left(a-(a-b) \sin ^2(c+d x)\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^2 d}",1,"(-4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (Sqrt[a - b]*(2*a + b)*Log[Sqrt[a] - Sqrt[a - b]*Sin[c + d*x]])/a^(3/2) + ((-2*a^2 + a*b + b^2)*Log[Sqrt[a] + Sqrt[a - b]*Sin[c + d*x]])/(a^(3/2)*Sqrt[a - b]) + (4*b*(-a + b)*Sin[c + d*x])/(a*(a + b + (a - b)*Cos[2*(c + d*x)])))/(4*b^2*d)","A",1
464,1,75,79,0.2455076,"\int \frac{\sec ^3(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Tan[c + d*x]^2)^2,x]","\frac{\frac{\sqrt{a} \sin (c+d x)}{(b-a) \sin ^2(c+d x)+a}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a-b}}}{2 a^{3/2} d}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d \sqrt{a-b}}+\frac{\sin (c+d x)}{2 a d \left(a-(a-b) \sin ^2(c+d x)\right)}",1,"(ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]]/Sqrt[a - b] + (Sqrt[a]*Sin[c + d*x])/(a + (-a + b)*Sin[c + d*x]^2))/(2*a^(3/2)*d)","A",1
465,1,92,94,0.252827,"\int \frac{\sec (c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Integrate[Sec[c + d*x]/(a + b*Tan[c + d*x]^2)^2,x]","\frac{\frac{\sqrt{a} b \sin (c+d x)}{(a-b) \left((a-b) \sin ^2(c+d x)-a\right)}+\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{(a-b)^{3/2}}}{2 a^{3/2} d}","\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^{3/2}}-\frac{b \sin (c+d x)}{2 a d (a-b) \left(a-(a-b) \sin ^2(c+d x)\right)}",1,"(((2*a - b)*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(a - b)^(3/2) + (Sqrt[a]*b*Sin[c + d*x])/((a - b)*(-a + (a - b)*Sin[c + d*x]^2)))/(2*a^(3/2)*d)","A",1
466,1,119,114,0.555283,"\int \frac{\cos (c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Integrate[Cos[c + d*x]/(a + b*Tan[c + d*x]^2)^2,x]","\frac{-\frac{\sqrt{a} \sin (c+d x) \left(a^2+a (a-b) \cos (2 (c+d x))+a b+b^2\right)}{(a-b)^2 \left((a-b) \sin ^2(c+d x)-a\right)}-\frac{b (4 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{(a-b)^{5/2}}}{2 a^{3/2} d}","-\frac{b (4 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^{5/2}}+\frac{b^2 \sin (c+d x)}{2 a d (a-b)^2 \left(a-(a-b) \sin ^2(c+d x)\right)}+\frac{\sin (c+d x)}{d (a-b)^2}",1,"(-(((4*a - b)*b*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(a - b)^(5/2)) - (Sqrt[a]*(a^2 + a*b + b^2 + a*(a - b)*Cos[2*(c + d*x)])*Sin[c + d*x])/((a - b)^2*(-a + (a - b)*Sin[c + d*x]^2)))/(2*a^(3/2)*d)","A",1
467,1,147,143,1.6158884,"\int \frac{\cos ^3(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Tan[c + d*x]^2)^2,x]","\frac{\frac{3 b^2 (b-6 a) \left(\log \left(\sqrt{a}-\sqrt{a-b} \sin (c+d x)\right)-\log \left(\sqrt{a-b} \sin (c+d x)+\sqrt{a}\right)\right)}{a^{3/2} (a-b)^{7/2}}+\frac{3 \sin (c+d x) \left(-\frac{4 b^3}{a ((a-b) \cos (2 (c+d x))+a+b)}+3 a-11 b\right)}{(a-b)^3}+\frac{\sin (3 (c+d x))}{(a-b)^2}}{12 d}","\frac{b^2 (6 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^{7/2}}-\frac{b^3 \sin (c+d x)}{2 a d (a-b)^3 \left(a-(a-b) \sin ^2(c+d x)\right)}-\frac{\sin ^3(c+d x)}{3 d (a-b)^2}+\frac{(a-3 b) \sin (c+d x)}{d (a-b)^3}",1,"((3*b^2*(-6*a + b)*(Log[Sqrt[a] - Sqrt[a - b]*Sin[c + d*x]] - Log[Sqrt[a] + Sqrt[a - b]*Sin[c + d*x]]))/(a^(3/2)*(a - b)^(7/2)) + (3*(3*a - 11*b - (4*b^3)/(a*(a + b + (a - b)*Cos[2*(c + d*x)])))*Sin[c + d*x])/(a - b)^3 + Sin[3*(c + d*x)]/(a - b)^2)/(12*d)","A",1
468,1,135,127,0.7928955,"\int \frac{\sec ^8(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Integrate[Sec[c + d*x]^8/(a + b*Tan[c + d*x]^2)^2,x]","\frac{\frac{3 (a-b)^2 (5 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{3/2}}+4 \sqrt{b} (4 b-3 a) \tan (c+d x)+\frac{3 \sqrt{b} (b-a)^3 \sin (2 (c+d x))}{a ((a-b) \cos (2 (c+d x))+a+b)}+2 b^{3/2} \tan (c+d x) \sec ^2(c+d x)}{6 b^{7/2} d}","\frac{(5 a+b) (a-b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{7/2} d}-\frac{(a-b)^3 \tan (c+d x)}{2 a b^3 d \left(a+b \tan ^2(c+d x)\right)}-\frac{(2 a-3 b) \tan (c+d x)}{b^3 d}+\frac{\tan ^3(c+d x)}{3 b^2 d}",1,"((3*(a - b)^2*(5*a + b)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/a^(3/2) + (3*Sqrt[b]*(-a + b)^3*Sin[2*(c + d*x)])/(a*(a + b + (a - b)*Cos[2*(c + d*x)])) + 4*Sqrt[b]*(-3*a + 4*b)*Tan[c + d*x] + 2*b^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(6*b^(7/2)*d)","A",1
469,1,104,104,0.6595432,"\int \frac{\sec ^6(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Integrate[Sec[c + d*x]^6/(a + b*Tan[c + d*x]^2)^2,x]","\frac{-\frac{(3 a+b) (a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{3/2}}+\frac{\sqrt{b} (a-b)^2 \sin (2 (c+d x))}{a ((a-b) \cos (2 (c+d x))+a+b)}+2 \sqrt{b} \tan (c+d x)}{2 b^{5/2} d}","-\frac{\left(3 a^2-2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{5/2} d}+\frac{(a-b)^2 \tan (c+d x)}{2 a b^2 d \left(a+b \tan ^2(c+d x)\right)}+\frac{\tan (c+d x)}{b^2 d}",1,"(-(((a - b)*(3*a + b)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/a^(3/2)) + ((a - b)^2*Sqrt[b]*Sin[2*(c + d*x)])/(a*(a + b + (a - b)*Cos[2*(c + d*x)])) + 2*Sqrt[b]*Tan[c + d*x])/(2*b^(5/2)*d)","A",1
470,1,83,77,0.3101588,"\int \frac{\sec ^4(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Tan[c + d*x]^2)^2,x]","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)+\frac{\sqrt{a} \sqrt{b} (b-a) \sin (2 (c+d x))}{(a-b) \cos (2 (c+d x))+a+b}}{2 a^{3/2} b^{3/2} d}","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2} d}-\frac{(a-b) \tan (c+d x)}{2 a b d \left(a+b \tan ^2(c+d x)\right)}",1,"((a + b)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]] + (Sqrt[a]*Sqrt[b]*(-a + b)*Sin[2*(c + d*x)])/(a + b + (a - b)*Cos[2*(c + d*x)]))/(2*a^(3/2)*b^(3/2)*d)","A",1
471,1,63,66,0.2805284,"\int \frac{\sec ^2(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Tan[c + d*x]^2)^2,x]","\frac{\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{b}}+\frac{\sqrt{a} \tan (c+d x)}{a+b \tan ^2(c+d x)}}{2 a^{3/2} d}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b} d}+\frac{\tan (c+d x)}{2 a d \left(a+b \tan ^2(c+d x)\right)}",1,"(ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]]/Sqrt[b] + (Sqrt[a]*Tan[c + d*x])/(a + b*Tan[c + d*x]^2))/(2*a^(3/2)*d)","A",1
472,1,116,148,1.2542637,"\int \frac{\cos ^2(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Tan[c + d*x]^2)^2,x]","\frac{-\frac{2 b^{3/2} (b-5 a) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{3/2}}+\frac{2 b^2 (a-b) \sin (2 (c+d x))}{a ((a-b) \cos (2 (c+d x))+a+b)}+2 (a-5 b) (c+d x)+(a-b) \sin (2 (c+d x))}{4 d (a-b)^3}","\frac{b^{3/2} (5 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^3}+\frac{b (a+b) \tan (c+d x)}{2 a d (a-b)^2 \left(a+b \tan ^2(c+d x)\right)}+\frac{\sin (c+d x) \cos (c+d x)}{2 d (a-b) \left(a+b \tan ^2(c+d x)\right)}+\frac{x (a-5 b)}{2 (a-b)^3}",1,"(2*(a - 5*b)*(c + d*x) - (2*b^(3/2)*(-5*a + b)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/a^(3/2) + (a - b)*Sin[2*(c + d*x)] + (2*(a - b)*b^2*Sin[2*(c + d*x)])/(a*(a + b + (a - b)*Cos[2*(c + d*x)])))/(4*(a - b)^3*d)","A",1
473,1,148,212,2.1208103,"\int \frac{\cos ^4(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Tan[c + d*x]^2)^2,x]","\frac{\frac{16 b^{5/2} (b-7 a) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{3/2}}+4 \left(3 a^2-14 a b+35 b^2\right) (c+d x)-\frac{16 b^3 (a-b) \sin (2 (c+d x))}{a ((a-b) \cos (2 (c+d x))+a+b)}+8 (a-3 b) (a-b) \sin (2 (c+d x))+(a-b)^2 \sin (4 (c+d x))}{32 d (a-b)^4}","-\frac{b^{5/2} (7 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^4}+\frac{x \left(3 a^2-14 a b+35 b^2\right)}{8 (a-b)^4}+\frac{b (a-4 b) (3 a+b) \tan (c+d x)}{8 a d (a-b)^3 \left(a+b \tan ^2(c+d x)\right)}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 d (a-b) \left(a+b \tan ^2(c+d x)\right)}+\frac{3 (a-3 b) \sin (c+d x) \cos (c+d x)}{8 d (a-b)^2 \left(a+b \tan ^2(c+d x)\right)}",1,"(4*(3*a^2 - 14*a*b + 35*b^2)*(c + d*x) + (16*b^(5/2)*(-7*a + b)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/a^(3/2) + 8*(a - 3*b)*(a - b)*Sin[2*(c + d*x)] - (16*(a - b)*b^3*Sin[2*(c + d*x)])/(a*(a + b + (a - b)*Cos[2*(c + d*x)])) + (a - b)^2*Sin[4*(c + d*x)])/(32*(a - b)^4*d)","A",1
474,1,81,95,0.1860799,"\int (d \sec (e+f x))^m \left(b \tan ^2(e+f x)\right)^p \, dx","Integrate[(d*Sec[e + f*x])^m*(b*Tan[e + f*x]^2)^p,x]","\frac{\cot (e+f x) \left(-\tan ^2(e+f x)\right)^{\frac{1}{2}-p} \left(b \tan ^2(e+f x)\right)^p (d \sec (e+f x))^m \, _2F_1\left(\frac{m}{2},\frac{1}{2}-p;\frac{m+2}{2};\sec ^2(e+f x)\right)}{f m}","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p (d \sec (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (m+2 p+1)} \, _2F_1\left(\frac{1}{2} (2 p+1),\frac{1}{2} (m+2 p+1);\frac{1}{2} (2 p+3);\sin ^2(e+f x)\right)}{f (2 p+1)}",1,"(Cot[e + f*x]*Hypergeometric2F1[m/2, 1/2 - p, (2 + m)/2, Sec[e + f*x]^2]*(d*Sec[e + f*x])^m*(-Tan[e + f*x]^2)^(1/2 - p)*(b*Tan[e + f*x]^2)^p)/(f*m)","A",1
475,1,2033,108,16.2299912,"\int (d \sec (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p,x]","\text{Result too large to show}","\frac{\tan (e+f x) \sec ^2(e+f x)^{-m/2} (d \sec (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1-\frac{m}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}",1,"(3*a*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(d*Sec[e + f*x])^m*(Sec[e + f*x]^2)^(-1 + m/2)*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^(2*p))/(f*(3*a*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + a*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2)*((6*a*b*p*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(-1 + p))/(3*a*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + a*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2) + (3*a*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(Sec[e + f*x]^2)^(m/2)*(a + b*Tan[e + f*x]^2)^p)/(3*a*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + a*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2) + (6*a*(-1 + m/2)*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(Sec[e + f*x]^2)^(-1 + m/2)*Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p)/(3*a*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + a*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2) + (3*a*(Sec[e + f*x]^2)^(-1 + m/2)*Tan[e + f*x]*((2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (2*(1 - m/2)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/3)*(a + b*Tan[e + f*x]^2)^p)/(3*a*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + a*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2) - (3*a*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(Sec[e + f*x]^2)^(-1 + m/2)*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p*(2*(2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + a*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Sec[e + f*x]^2*Tan[e + f*x] + 3*a*((2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (2*(1 - m/2)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/3) + Tan[e + f*x]^2*(2*b*p*((-6*b*(1 - p)*AppellF1[5/2, 1 - m/2, 2 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a) - (6*(1 - m/2)*AppellF1[5/2, 2 - m/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/5) + a*(-2 + m)*((6*b*p*AppellF1[5/2, 2 - m/2, 1 - p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a) - (6*(2 - m/2)*AppellF1[5/2, 3 - m/2, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(3*a*AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, 1 - m/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + a*(-2 + m)*AppellF1[3/2, 2 - m/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)])*Tan[e + f*x]^2)^2))","B",0
476,1,89,97,0.1833631,"\int (d \sec (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(d*Sec[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\cot (e+f x) (d \sec (e+f x))^m \left(-\tan ^2(e+f x)\right)^{\frac{1}{2} (1-n p)} \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(\frac{m}{2},\frac{1}{2} (1-n p);\frac{m+2}{2};\sec ^2(e+f x)\right)}{f m}","\frac{\tan (e+f x) (d \sec (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (m+n p+1)} \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (m+n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right)}{f (n p+1)}",1,"(Cot[e + f*x]*Hypergeometric2F1[m/2, (1 - n*p)/2, (2 + m)/2, Sec[e + f*x]^2]*(d*Sec[e + f*x])^m*(-Tan[e + f*x]^2)^((1 - n*p)/2)*(b*(c*Tan[e + f*x])^n)^p)/(f*m)","A",1
477,1,122,99,2.1798198,"\int \sec ^6(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Sec[e + f*x]^6*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\cot (e+f x) \left(b (c \tan (e+f x))^n\right)^p \left(\tan ^2(e+f x) \sec ^4(e+f x) \left(2 (n p+3) \cos (2 (e+f x))+\cos (4 (e+f x))+n^2 p^2+6 n p+8\right)+8 \left(-\tan ^2(e+f x)\right)^{\frac{1}{2} (1-n p)}\right)}{f (n p+1) (n p+3) (n p+5)}","\frac{\tan ^5(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+5)}+\frac{2 \tan ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+3)}+\frac{\tan (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Cot[e + f*x]*(b*(c*Tan[e + f*x])^n)^p*((8 + 6*n*p + n^2*p^2 + 2*(3 + n*p)*Cos[2*(e + f*x)] + Cos[4*(e + f*x)])*Sec[e + f*x]^4*Tan[e + f*x]^2 + 8*(-Tan[e + f*x]^2)^((1 - n*p)/2)))/(f*(1 + n*p)*(3 + n*p)*(5 + n*p))","A",1
478,1,87,65,2.129246,"\int \sec ^4(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Sec[e + f*x]^4*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\cot (e+f x) \left(2 \left(-\tan ^2(e+f x)\right)^{\frac{1}{2} (1-n p)}+\tan ^2(e+f x) \left((n p+1) \sec ^2(e+f x)+2\right)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1) (n p+3)}","\frac{\tan ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+3)}+\frac{\tan (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Cot[e + f*x]*(b*(c*Tan[e + f*x])^n)^p*((2 + (1 + n*p)*Sec[e + f*x]^2)*Tan[e + f*x]^2 + 2*(-Tan[e + f*x]^2)^((1 - n*p)/2)))/(f*(1 + n*p)*(3 + n*p))","A",1
479,1,31,31,0.0241642,"\int \sec ^2(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Sec[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}","\frac{\tan (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))","A",1
480,1,59,61,0.0375972,"\int \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f n p+f}","\frac{\tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f + f*n*p)","A",1
481,1,1060,61,5.5052744,"\int \cos ^2(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Cos[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{2 \left(F_1\left(\frac{1}{2} (n p+1);n p,1;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-4 F_1\left(\frac{1}{2} (n p+1);n p,2;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+4 F_1\left(\frac{1}{2} (n p+1);n p,3;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \cos ^2(e+f x) \tan \left(\frac{1}{2} (e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f \left(\frac{2 (n p+1) \left(-F_1\left(\frac{1}{2} (n p+3);n p,2;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+8 F_1\left(\frac{1}{2} (n p+3);n p,3;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-12 F_1\left(\frac{1}{2} (n p+3);n p,4;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+n p F_1\left(\frac{1}{2} (n p+3);n p+1,1;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-4 n p F_1\left(\frac{1}{2} (n p+3);n p+1,2;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+4 n p F_1\left(\frac{1}{2} (n p+3);n p+1,3;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \tan ^2\left(\frac{1}{2} (e+f x)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)}{n p+3}+\left(F_1\left(\frac{1}{2} (n p+1);n p,1;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-4 F_1\left(\frac{1}{2} (n p+1);n p,2;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+4 F_1\left(\frac{1}{2} (n p+1);n p,3;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)+n p \left(F_1\left(\frac{1}{2} (n p+1);n p,1;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-4 F_1\left(\frac{1}{2} (n p+1);n p,2;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+4 F_1\left(\frac{1}{2} (n p+1);n p,3;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \sec (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)-2 n p \left(F_1\left(\frac{1}{2} (n p+1);n p,1;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-4 F_1\left(\frac{1}{2} (n p+1);n p,2;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+4 F_1\left(\frac{1}{2} (n p+1);n p,3;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \sec (e+f x) \tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}","\frac{\tan (e+f x) \, _2F_1\left(2,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(2*(AppellF1[(1 + n*p)/2, n*p, 1, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 4*AppellF1[(1 + n*p)/2, n*p, 2, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*AppellF1[(1 + n*p)/2, n*p, 3, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Cos[e + f*x]^2*Tan[(e + f*x)/2]*(b*(c*Tan[e + f*x])^n)^p)/(f*((AppellF1[(1 + n*p)/2, n*p, 1, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 4*AppellF1[(1 + n*p)/2, n*p, 2, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*AppellF1[(1 + n*p)/2, n*p, 3, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2 + n*p*(AppellF1[(1 + n*p)/2, n*p, 1, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 4*AppellF1[(1 + n*p)/2, n*p, 2, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*AppellF1[(1 + n*p)/2, n*p, 3, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Sec[e + f*x] + (2*(1 + n*p)*(-AppellF1[(3 + n*p)/2, n*p, 2, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 8*AppellF1[(3 + n*p)/2, n*p, 3, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 12*AppellF1[(3 + n*p)/2, n*p, 4, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*p*AppellF1[(3 + n*p)/2, 1 + n*p, 1, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 4*n*p*AppellF1[(3 + n*p)/2, 1 + n*p, 2, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*n*p*AppellF1[(3 + n*p)/2, 1 + n*p, 3, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]^2)/(3 + n*p) - 2*n*p*(AppellF1[(1 + n*p)/2, n*p, 1, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 4*AppellF1[(1 + n*p)/2, n*p, 2, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*AppellF1[(1 + n*p)/2, n*p, 3, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[e + f*x]*Tan[(e + f*x)/2]^2))","C",0
482,1,81,93,0.1125559,"\int \sec ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Sec[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\csc (e+f x) \sec ^2(e+f x) \left(-\tan ^2(e+f x)\right)^{\frac{1}{2} (1-n p)} \, _2F_1\left(\frac{3}{2},\frac{1}{2} (1-n p);\frac{5}{2};\sec ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{3 f}","\frac{\tan (e+f x) \sec ^3(e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+4)} \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (n p+4);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Csc[e + f*x]*Hypergeometric2F1[3/2, (1 - n*p)/2, 5/2, Sec[e + f*x]^2]*Sec[e + f*x]^2*(-Tan[e + f*x]^2)^((1 - n*p)/2)*(b*(c*Tan[e + f*x])^n)^p)/(3*f)","A",1
483,1,70,91,0.0821039,"\int \sec (e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Sec[e + f*x]*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\csc (e+f x) \left(-\tan ^2(e+f x)\right)^{\frac{1}{2} (1-n p)} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{3}{2};\sec ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f}","\frac{\tan (e+f x) \sec (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+2)} \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (n p+2);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Csc[e + f*x]*Hypergeometric2F1[1/2, (1 - n*p)/2, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^((1 - n*p)/2)*(b*(c*Tan[e + f*x])^n)^p)/f","A",1
484,1,482,79,3.6077489,"\int \cos (e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Cos[e + f*x]*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{(n p+3) \sin (2 (e+f x)) \left(F_1\left(\frac{1}{2} (n p+1);n p,1;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 F_1\left(\frac{1}{2} (n p+1);n p,2;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \left(b (c \tan (e+f x))^n\right)^p}{2 f (n p+1) \left((n p+3) F_1\left(\frac{1}{2} (n p+1);n p,1;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 \left(\tan ^2\left(\frac{1}{2} (e+f x)\right) \left(F_1\left(\frac{1}{2} (n p+3);n p,2;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-4 F_1\left(\frac{1}{2} (n p+3);n p,3;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n p F_1\left(\frac{1}{2} (n p+3);n p+1,1;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+2 n p F_1\left(\frac{1}{2} (n p+3);n p+1,2;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)+(n p+3) F_1\left(\frac{1}{2} (n p+1);n p,2;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}","\frac{\sin (e+f x) \cos ^2(e+f x)^{\frac{n p}{2}} \, _2F_1\left(\frac{n p}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"((3 + n*p)*(AppellF1[(1 + n*p)/2, n*p, 1, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*AppellF1[(1 + n*p)/2, n*p, 2, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sin[2*(e + f*x)]*(b*(c*Tan[e + f*x])^n)^p)/(2*f*(1 + n*p)*((3 + n*p)*AppellF1[(1 + n*p)/2, n*p, 1, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*((3 + n*p)*AppellF1[(1 + n*p)/2, n*p, 2, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (AppellF1[(3 + n*p)/2, n*p, 2, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 4*AppellF1[(3 + n*p)/2, n*p, 3, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*p*AppellF1[(3 + n*p)/2, 1 + n*p, 1, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*p*AppellF1[(3 + n*p)/2, 1 + n*p, 2, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","C",0
485,1,1552,82,6.3724454,"\int \cos ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Cos[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{(2 n p+6) \left(F_1\left(\frac{1}{2} (n p+1);n p,1;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-6 F_1\left(\frac{1}{2} (n p+1);n p,2;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+12 F_1\left(\frac{1}{2} (n p+1);n p,3;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-8 F_1\left(\frac{1}{2} (n p+1);n p,4;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \cos ^3\left(\frac{1}{2} (e+f x)\right) \cos ^3(e+f x) \sin \left(\frac{1}{2} (e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1) \left((n p+3) F_1\left(\frac{1}{2} (n p+1);n p,1;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos ^2\left(\frac{1}{2} (e+f x)\right)-6 n p F_1\left(\frac{1}{2} (n p+1);n p,2;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos ^2\left(\frac{1}{2} (e+f x)\right)-18 F_1\left(\frac{1}{2} (n p+1);n p,2;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos ^2\left(\frac{1}{2} (e+f x)\right)+12 n p F_1\left(\frac{1}{2} (n p+1);n p,3;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos ^2\left(\frac{1}{2} (e+f x)\right)+36 F_1\left(\frac{1}{2} (n p+1);n p,3;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos ^2\left(\frac{1}{2} (e+f x)\right)-8 (n p+3) F_1\left(\frac{1}{2} (n p+1);n p,4;\frac{1}{2} (n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos ^2\left(\frac{1}{2} (e+f x)\right)-F_1\left(\frac{1}{2} (n p+3);n p,2;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+12 F_1\left(\frac{1}{2} (n p+3);n p,3;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-36 F_1\left(\frac{1}{2} (n p+3);n p,4;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+32 F_1\left(\frac{1}{2} (n p+3);n p,5;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+n p F_1\left(\frac{1}{2} (n p+3);n p+1,1;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-6 n p F_1\left(\frac{1}{2} (n p+3);n p+1,2;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+12 n p F_1\left(\frac{1}{2} (n p+3);n p+1,3;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-8 n p F_1\left(\frac{1}{2} (n p+3);n p+1,4;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+F_1\left(\frac{1}{2} (n p+3);n p,2;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos (e+f x)-12 F_1\left(\frac{1}{2} (n p+3);n p,3;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos (e+f x)+36 F_1\left(\frac{1}{2} (n p+3);n p,4;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos (e+f x)-32 F_1\left(\frac{1}{2} (n p+3);n p,5;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos (e+f x)-n p F_1\left(\frac{1}{2} (n p+3);n p+1,1;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos (e+f x)+6 n p F_1\left(\frac{1}{2} (n p+3);n p+1,2;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos (e+f x)-12 n p F_1\left(\frac{1}{2} (n p+3);n p+1,3;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos (e+f x)+8 n p F_1\left(\frac{1}{2} (n p+3);n p+1,4;\frac{1}{2} (n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos (e+f x)\right)}","\frac{\sin (e+f x) \cos ^2(e+f x)^{\frac{n p}{2}} \, _2F_1\left(\frac{1}{2} (n p-2),\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"((6 + 2*n*p)*(AppellF1[(1 + n*p)/2, n*p, 1, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 6*AppellF1[(1 + n*p)/2, n*p, 2, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 12*AppellF1[(1 + n*p)/2, n*p, 3, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 8*AppellF1[(1 + n*p)/2, n*p, 4, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Cos[(e + f*x)/2]^3*Cos[e + f*x]^3*Sin[(e + f*x)/2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p)*(-AppellF1[(3 + n*p)/2, n*p, 2, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 12*AppellF1[(3 + n*p)/2, n*p, 3, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 36*AppellF1[(3 + n*p)/2, n*p, 4, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 32*AppellF1[(3 + n*p)/2, n*p, 5, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*p*AppellF1[(3 + n*p)/2, 1 + n*p, 1, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 6*n*p*AppellF1[(3 + n*p)/2, 1 + n*p, 2, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 12*n*p*AppellF1[(3 + n*p)/2, 1 + n*p, 3, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 8*n*p*AppellF1[(3 + n*p)/2, 1 + n*p, 4, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3 + n*p)*AppellF1[(1 + n*p)/2, n*p, 1, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - 18*AppellF1[(1 + n*p)/2, n*p, 2, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - 6*n*p*AppellF1[(1 + n*p)/2, n*p, 2, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + 36*AppellF1[(1 + n*p)/2, n*p, 3, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + 12*n*p*AppellF1[(1 + n*p)/2, n*p, 3, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - 8*(3 + n*p)*AppellF1[(1 + n*p)/2, n*p, 4, (3 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + AppellF1[(3 + n*p)/2, n*p, 2, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] - 12*AppellF1[(3 + n*p)/2, n*p, 3, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] + 36*AppellF1[(3 + n*p)/2, n*p, 4, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] - 32*AppellF1[(3 + n*p)/2, n*p, 5, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] - n*p*AppellF1[(3 + n*p)/2, 1 + n*p, 1, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] + 6*n*p*AppellF1[(3 + n*p)/2, 1 + n*p, 2, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] - 12*n*p*AppellF1[(3 + n*p)/2, 1 + n*p, 3, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] + 8*n*p*AppellF1[(3 + n*p)/2, 1 + n*p, 4, (5 + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]))","C",0
486,0,0,30,2.8446103,"\int (d \sec (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(d*Sec[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int (d \sec (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left((d \sec (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Integrate[(d*Sec[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",-1
487,0,0,28,6.0805787,"\int \sec ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Sec[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int \sec ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left(\sec ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Integrate[Sec[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",-1
488,0,0,26,1.872399,"\int \sec (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Sec[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int \sec (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left(\sec (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Integrate[Sec[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",-1
489,0,0,26,3.4822145,"\int \cos (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Cos[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int \cos (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left(\cos (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Integrate[Cos[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",-1
490,0,0,28,11.8732558,"\int \cos ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Cos[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int \cos ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left(\cos ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Integrate[Cos[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",-1
491,1,165,244,3.1324047,"\int \sec ^6(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Sec[e + f*x]^6*(a + b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \left(3 \tan ^4(e+f x) \, _2F_1\left(\frac{5}{n},-p;\frac{n+5}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)+10 \tan ^2(e+f x) \, _2F_1\left(\frac{3}{n},-p;\frac{n+3}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)+15 \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)\right)}{15 f}","\frac{\tan ^5(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{n},-p;\frac{n+5}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{5 f}+\frac{2 \tan ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{n},-p;\frac{n+3}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{3 f}+\frac{\tan (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{f}",1,"(Tan[e + f*x]*(15*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*(c*Tan[e + f*x])^n)/a)] + 10*Hypergeometric2F1[3/n, -p, (3 + n)/n, -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]^2 + 3*Hypergeometric2F1[5/n, -p, (5 + n)/n, -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]^4)*(a + b*(c*Tan[e + f*x])^n)^p)/(15*f*(1 + (b*(c*Tan[e + f*x])^n)/a)^p)","A",1
492,1,122,160,4.1152877,"\int \sec ^4(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Sec[e + f*x]^4*(a + b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \left(\tan ^2(e+f x) \, _2F_1\left(\frac{3}{n},-p;\frac{n+3}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)+3 \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)\right)}{3 f}","\frac{\tan ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{n},-p;\frac{n+3}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{3 f}+\frac{\tan (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{f}",1,"(Tan[e + f*x]*(3*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*(c*Tan[e + f*x])^n)/a)] + Hypergeometric2F1[3/n, -p, (3 + n)/n, -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]^2)*(a + b*(c*Tan[e + f*x])^n)^p)/(3*f*(1 + (b*(c*Tan[e + f*x])^n)/a)^p)","A",1
493,1,75,75,0.1268263,"\int \sec ^2(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Sec[e + f*x]^2*(a + b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{f}","\frac{\tan (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{f}",1,"(Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p)/(f*(1 + (b*(c*Tan[e + f*x])^n)/a)^p)","A",1
494,0,0,19,0.807184,"\int \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(a + b*(c*Tan[e + f*x])^n)^p,x]","\int \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left(\left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Integrate[(a + b*(c*Tan[e + f*x])^n)^p, x]","A",-1
495,0,0,28,8.6217305,"\int \cos ^2(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Integrate[Cos[e + f*x]^2*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int \cos ^2(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left(\cos ^2(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Integrate[Cos[e + f*x]^2*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",-1
496,1,299,98,2.0204015,"\int (d \csc (e+f x))^m \left(b \tan ^2(e+f x)\right)^p \, dx","Integrate[(d*Csc[e + f*x])^m*(b*Tan[e + f*x]^2)^p,x]","-\frac{d (m-2 p-3) \left(b \tan ^2(e+f x)\right)^p (d \csc (e+f x))^{m-1} F_1\left(-\frac{m}{2}+p+\frac{1}{2};2 p,1-m;-\frac{m}{2}+p+\frac{3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{f (m-2 p-1) \left(2 \tan ^2\left(\frac{1}{2} (e+f x)\right) \left(-\left((m-1) F_1\left(-\frac{m}{2}+p+\frac{3}{2};2 p,2-m;-\frac{m}{2}+p+\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)-2 p F_1\left(-\frac{m}{2}+p+\frac{3}{2};2 p+1,1-m;-\frac{m}{2}+p+\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)+(m-2 p-3) F_1\left(-\frac{m}{2}+p+\frac{1}{2};2 p,1-m;-\frac{m}{2}+p+\frac{3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}","\frac{\tan (e+f x) \cos ^2(e+f x)^{p+\frac{1}{2}} \left(b \tan ^2(e+f x)\right)^p (d \csc (e+f x))^m \, _2F_1\left(\frac{1}{2} (2 p+1),\frac{1}{2} (-m+2 p+1);\frac{1}{2} (-m+2 p+3);\sin ^2(e+f x)\right)}{f (-m+2 p+1)}",1,"-((d*(-3 + m - 2*p)*AppellF1[1/2 - m/2 + p, 2*p, 1 - m, 3/2 - m/2 + p, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(d*Csc[e + f*x])^(-1 + m)*(b*Tan[e + f*x]^2)^p)/(f*(-1 + m - 2*p)*((-3 + m - 2*p)*AppellF1[1/2 - m/2 + p, 2*p, 1 - m, 3/2 - m/2 + p, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*(-((-1 + m)*AppellF1[3/2 - m/2 + p, 2*p, 2 - m, 5/2 - m/2 + p, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]) - 2*p*AppellF1[3/2 - m/2 + p, 1 + 2*p, 1 - m, 5/2 - m/2 + p, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","C",0
497,1,292,127,3.6736483,"\int (d \csc (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \, dx","Integrate[(d*Csc[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{a (m-3) \cos ^2(e+f x) \cot (e+f x) (d \csc (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p F_1\left(\frac{1}{2}-\frac{m}{2};1-\frac{m}{2},-p;\frac{3}{2}-\frac{m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f (m-1) \left(-2 b p F_1\left(\frac{3}{2}-\frac{m}{2};1-\frac{m}{2},1-p;\frac{5}{2}-\frac{m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)-a (m-2) F_1\left(\frac{3}{2}-\frac{m}{2};2-\frac{m}{2},-p;\frac{5}{2}-\frac{m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)+a (m-3) \cot ^2(e+f x) F_1\left(\frac{1}{2}-\frac{m}{2};1-\frac{m}{2},-p;\frac{3}{2}-\frac{m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)\right)}","\frac{\tan (e+f x) \sec ^2(e+f x)^{-m/2} (d \csc (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1-m}{2};1-\frac{m}{2},-p;\frac{3-m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f (1-m)}",1,"-((a*(-3 + m)*AppellF1[1/2 - m/2, 1 - m/2, -p, 3/2 - m/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Cos[e + f*x]^2*Cot[e + f*x]*(d*Csc[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p)/(f*(-1 + m)*(-2*b*p*AppellF1[3/2 - m/2, 1 - m/2, 1 - p, 5/2 - m/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] - a*(-2 + m)*AppellF1[3/2 - m/2, 2 - m/2, -p, 5/2 - m/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)] + a*(-3 + m)*AppellF1[1/2 - m/2, 1 - m/2, -p, 3/2 - m/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Cot[e + f*x]^2)))","B",0
498,1,319,104,2.1871722,"\int (d \csc (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(d*Csc[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p,x]","-\frac{d (m-n p-3) (d \csc (e+f x))^{m-1} F_1\left(\frac{1}{2} (-m+n p+1);n p,1-m;\frac{1}{2} (-m+n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (m-n p-1) \left((m-n p-3) F_1\left(\frac{1}{2} (-m+n p+1);n p,1-m;\frac{1}{2} (-m+n p+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 \tan ^2\left(\frac{1}{2} (e+f x)\right) \left((m-1) F_1\left(\frac{1}{2} (-m+n p+3);n p,2-m;\frac{1}{2} (-m+n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+n p F_1\left(\frac{1}{2} (-m+n p+3);n p+1,1-m;\frac{1}{2} (-m+n p+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}","\frac{\tan (e+f x) (d \csc (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (-m+n p+1);\frac{1}{2} (-m+n p+3);\sin ^2(e+f x)\right)}{f (-m+n p+1)}",1,"-((d*(-3 + m - n*p)*AppellF1[(1 - m + n*p)/2, n*p, 1 - m, (3 - m + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(d*Csc[e + f*x])^(-1 + m)*(b*(c*Tan[e + f*x])^n)^p)/(f*(-1 + m - n*p)*((-3 + m - n*p)*AppellF1[(1 - m + n*p)/2, n*p, 1 - m, (3 - m + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*((-1 + m)*AppellF1[(3 - m + n*p)/2, n*p, 2 - m, (5 - m + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*p*AppellF1[(3 - m + n*p)/2, 1 + n*p, 1 - m, (5 - m + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","C",0
499,0,0,57,2.9079477,"\int (d \csc (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Integrate[(d*Csc[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int (d \csc (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\left(\frac{\sin (e+f x)}{d}\right)^m (d \csc (e+f x))^m \text{Int}\left(\left(\frac{\sin (e+f x)}{d}\right)^{-m} \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Integrate[(d*Csc[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",-1