1,1,12,12,0.0070119,"\int \tan (c+d x) \, dx","Integrate[Tan[c + d*x],x]","-\frac{\log (\cos (c+d x))}{d}","-\frac{\log (\cos (c+d x))}{d}",1,"-(Log[Cos[c + d*x]]/d)","A",1
2,1,23,14,0.0071859,"\int \tan ^2(c+d x) \, dx","Integrate[Tan[c + d*x]^2,x]","\frac{\tan (c+d x)}{d}-\frac{\tan ^{-1}(\tan (c+d x))}{d}","\frac{\tan (c+d x)}{d}-x",1,"-(ArcTan[Tan[c + d*x]]/d) + Tan[c + d*x]/d","A",1
3,1,25,27,0.0250806,"\int \tan ^3(c+d x) \, dx","Integrate[Tan[c + d*x]^3,x]","\frac{\tan ^2(c+d x)+2 \log (\cos (c+d x))}{2 d}","\frac{\tan ^2(c+d x)}{2 d}+\frac{\log (\cos (c+d x))}{d}",1,"(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2)/(2*d)","A",1
4,1,38,28,0.0109937,"\int \tan ^4(c+d x) \, dx","Integrate[Tan[c + d*x]^4,x]","\frac{\tan ^{-1}(\tan (c+d x))}{d}+\frac{\tan ^3(c+d x)}{3 d}-\frac{\tan (c+d x)}{d}","\frac{\tan ^3(c+d x)}{3 d}-\frac{\tan (c+d x)}{d}+x",1,"ArcTan[Tan[c + d*x]]/d - Tan[c + d*x]/d + Tan[c + d*x]^3/(3*d)","A",1
5,1,37,43,0.0482282,"\int \tan ^5(c+d x) \, dx","Integrate[Tan[c + d*x]^5,x]","-\frac{-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))}{4 d}","\frac{\tan ^4(c+d x)}{4 d}-\frac{\tan ^2(c+d x)}{2 d}-\frac{\log (\cos (c+d x))}{d}",1,"-1/4*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4)/d","A",1
6,1,53,44,0.0159826,"\int \tan ^6(c+d x) \, dx","Integrate[Tan[c + d*x]^6,x]","-\frac{\tan ^{-1}(\tan (c+d x))}{d}+\frac{\tan ^5(c+d x)}{5 d}-\frac{\tan ^3(c+d x)}{3 d}+\frac{\tan (c+d x)}{d}","\frac{\tan ^5(c+d x)}{5 d}-\frac{\tan ^3(c+d x)}{3 d}+\frac{\tan (c+d x)}{d}-x",1,"-(ArcTan[Tan[c + d*x]]/d) + Tan[c + d*x]/d - Tan[c + d*x]^3/(3*d) + Tan[c + d*x]^5/(5*d)","A",1
7,1,47,57,0.1057946,"\int \tan ^7(c+d x) \, dx","Integrate[Tan[c + d*x]^7,x]","\frac{2 \tan ^6(c+d x)-3 \tan ^4(c+d x)+6 \tan ^2(c+d x)+12 \log (\cos (c+d x))}{12 d}","\frac{\tan ^6(c+d x)}{6 d}-\frac{\tan ^4(c+d x)}{4 d}+\frac{\tan ^2(c+d x)}{2 d}+\frac{\log (\cos (c+d x))}{d}",1,"(12*Log[Cos[c + d*x]] + 6*Tan[c + d*x]^2 - 3*Tan[c + d*x]^4 + 2*Tan[c + d*x]^6)/(12*d)","A",1
8,1,68,58,0.0120349,"\int \tan ^8(c+d x) \, dx","Integrate[Tan[c + d*x]^8,x]","\frac{\tan ^{-1}(\tan (c+d x))}{d}+\frac{\tan ^7(c+d x)}{7 d}-\frac{\tan ^5(c+d x)}{5 d}+\frac{\tan ^3(c+d x)}{3 d}-\frac{\tan (c+d x)}{d}","\frac{\tan ^7(c+d x)}{7 d}-\frac{\tan ^5(c+d x)}{5 d}+\frac{\tan ^3(c+d x)}{3 d}-\frac{\tan (c+d x)}{d}+x",1,"ArcTan[Tan[c + d*x]]/d - Tan[c + d*x]/d + Tan[c + d*x]^3/(3*d) - Tan[c + d*x]^5/(5*d) + Tan[c + d*x]^7/(7*d)","A",1
9,1,175,232,0.3851358,"\int (b \tan (c+d x))^{7/2} \, dx","Integrate[(b*Tan[c + d*x])^(7/2),x]","\frac{b^3 \sqrt{b \tan (c+d x)} \left(-10 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+10 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+8 \tan ^{\frac{5}{2}}(c+d x)-40 \sqrt{\tan (c+d x)}-5 \sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+5 \sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{20 d \sqrt{\tan (c+d x)}}","-\frac{b^{7/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right)}{\sqrt{2} d}+\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right)}{\sqrt{2} d}-\frac{b^{7/2} \log \left(\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} d}+\frac{b^{7/2} \log \left(\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} d}-\frac{2 b^3 \sqrt{b \tan (c+d x)}}{d}+\frac{2 b (b \tan (c+d x))^{5/2}}{5 d}",1,"(b^3*Sqrt[b*Tan[c + d*x]]*(-10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 40*Sqrt[Tan[c + d*x]] + 8*Tan[c + d*x]^(5/2)))/(20*d*Sqrt[Tan[c + d*x]])","A",1
10,1,40,212,0.0723219,"\int (b \tan (c+d x))^{5/2} \, dx","Integrate[(b*Tan[c + d*x])^(5/2),x]","-\frac{2 b (b \tan (c+d x))^{3/2} \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)-1\right)}{3 d}","\frac{b^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right)}{\sqrt{2} d}-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right)}{\sqrt{2} d}-\frac{b^{5/2} \log \left(\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} d}+\frac{b^{5/2} \log \left(\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} d}+\frac{2 b (b \tan (c+d x))^{3/2}}{3 d}",1,"(-2*b*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2])*(b*Tan[c + d*x])^(3/2))/(3*d)","C",1
11,1,159,210,0.1697764,"\int (b \tan (c+d x))^{3/2} \, dx","Integrate[(b*Tan[c + d*x])^(3/2),x]","\frac{(b \tan (c+d x))^{3/2} \left(2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+8 \sqrt{\tan (c+d x)}+\sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 d \tan ^{\frac{3}{2}}(c+d x)}","\frac{b^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right)}{\sqrt{2} d}-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right)}{\sqrt{2} d}+\frac{b^{3/2} \log \left(\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} d}-\frac{b^{3/2} \log \left(\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} d}+\frac{2 b \sqrt{b \tan (c+d x)}}{d}",1,"((2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 8*Sqrt[Tan[c + d*x]])*(b*Tan[c + d*x])^(3/2))/(4*d*Tan[c + d*x]^(3/2))","A",1
12,1,40,192,0.043739,"\int \sqrt{b \tan (c+d x)} \, dx","Integrate[Sqrt[b*Tan[c + d*x]],x]","\frac{2 (b \tan (c+d x))^{3/2} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)}{3 b d}","-\frac{\sqrt{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right)}{\sqrt{2} d}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right)}{\sqrt{2} d}+\frac{\sqrt{b} \log \left(\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} d}-\frac{\sqrt{b} \log \left(\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} d}",1,"(2*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*(b*Tan[c + d*x])^(3/2))/(3*b*d)","C",1
13,1,131,192,0.1088346,"\int \frac{1}{\sqrt{b \tan (c+d x)}} \, dx","Integrate[1/Sqrt[b*Tan[c + d*x]],x]","\frac{\sqrt{\tan (c+d x)} \left(-2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{2 \sqrt{2} d \sqrt{b \tan (c+d x)}}","-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right)}{\sqrt{2} \sqrt{b} d}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right)}{\sqrt{2} \sqrt{b} d}-\frac{\log \left(\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} \sqrt{b} d}+\frac{\log \left(\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} \sqrt{b} d}",1,"((-2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/(2*Sqrt[2]*d*Sqrt[b*Tan[c + d*x]])","A",1
14,1,38,212,0.0673512,"\int \frac{1}{(b \tan (c+d x))^{3/2}} \, dx","Integrate[(b*Tan[c + d*x])^(-3/2),x]","-\frac{2 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)}{b d \sqrt{b \tan (c+d x)}}","\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right)}{\sqrt{2} b^{3/2} d}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right)}{\sqrt{2} b^{3/2} d}-\frac{\log \left(\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} b^{3/2} d}+\frac{\log \left(\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} b^{3/2} d}-\frac{2}{b d \sqrt{b \tan (c+d x)}}",1,"(-2*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2])/(b*d*Sqrt[b*Tan[c + d*x]])","C",1
15,1,40,214,0.0903563,"\int \frac{1}{(b \tan (c+d x))^{5/2}} \, dx","Integrate[(b*Tan[c + d*x])^(-5/2),x]","-\frac{2 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)}{3 b d (b \tan (c+d x))^{3/2}}","\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right)}{\sqrt{2} b^{5/2} d}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right)}{\sqrt{2} b^{5/2} d}+\frac{\log \left(\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} b^{5/2} d}-\frac{\log \left(\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} b^{5/2} d}-\frac{2}{3 b d (b \tan (c+d x))^{3/2}}",1,"(-2*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2])/(3*b*d*(b*Tan[c + d*x])^(3/2))","C",1
16,1,40,234,0.1080096,"\int \frac{1}{(b \tan (c+d x))^{7/2}} \, dx","Integrate[(b*Tan[c + d*x])^(-7/2),x]","-\frac{2 \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\tan ^2(c+d x)\right)}{5 b d (b \tan (c+d x))^{5/2}}","-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right)}{\sqrt{2} b^{7/2} d}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right)}{\sqrt{2} b^{7/2} d}+\frac{\log \left(\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} b^{7/2} d}-\frac{\log \left(\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} b^{7/2} d}+\frac{2}{b^3 d \sqrt{b \tan (c+d x)}}-\frac{2}{5 b d (b \tan (c+d x))^{5/2}}",1,"(-2*Hypergeometric2F1[-5/4, 1, -1/4, -Tan[c + d*x]^2])/(5*b*d*(b*Tan[c + d*x])^(5/2))","C",1
17,1,38,243,0.0267265,"\int (b \tan (c+d x))^{4/3} \, dx","Integrate[(b*Tan[c + d*x])^(4/3),x]","-\frac{3 b \sqrt[3]{b \tan (c+d x)} \left(\, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\tan ^2(c+d x)\right)-1\right)}{d}","-\frac{b^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{b \tan (c+d x)}}{\sqrt[3]{b}}\right)}{d}+\frac{b^{4/3} \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[3]{b \tan (c+d x)}}{\sqrt[3]{b}}\right)}{2 d}-\frac{b^{4/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{b \tan (c+d x)}}{\sqrt[3]{b}}+\sqrt{3}\right)}{2 d}+\frac{\sqrt{3} b^{4/3} \log \left(b^{2/3}-\sqrt{3} \sqrt[3]{b} \sqrt[3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right)}{4 d}-\frac{\sqrt{3} b^{4/3} \log \left(b^{2/3}+\sqrt{3} \sqrt[3]{b} \sqrt[3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right)}{4 d}+\frac{3 b \sqrt[3]{b \tan (c+d x)}}{d}",1,"(-3*b*(-1 + Hypergeometric2F1[1/6, 1, 7/6, -Tan[c + d*x]^2])*(b*Tan[c + d*x])^(1/3))/d","C",1
18,1,40,224,0.0500841,"\int (b \tan (c+d x))^{2/3} \, dx","Integrate[(b*Tan[c + d*x])^(2/3),x]","\frac{3 (b \tan (c+d x))^{5/3} \, _2F_1\left(\frac{5}{6},1;\frac{11}{6};-\tan ^2(c+d x)\right)}{5 b d}","\frac{b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{b \tan (c+d x)}}{\sqrt[3]{b}}\right)}{d}-\frac{b^{2/3} \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[3]{b \tan (c+d x)}}{\sqrt[3]{b}}\right)}{2 d}+\frac{b^{2/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{b \tan (c+d x)}}{\sqrt[3]{b}}+\sqrt{3}\right)}{2 d}+\frac{\sqrt{3} b^{2/3} \log \left(b^{2/3}-\sqrt{3} \sqrt[3]{b} \sqrt[3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right)}{4 d}-\frac{\sqrt{3} b^{2/3} \log \left(b^{2/3}+\sqrt{3} \sqrt[3]{b} \sqrt[3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right)}{4 d}",1,"(3*Hypergeometric2F1[5/6, 1, 11/6, -Tan[c + d*x]^2]*(b*Tan[c + d*x])^(5/3))/(5*b*d)","C",1
19,1,40,131,0.0429767,"\int \sqrt[3]{b \tan (c+d x)} \, dx","Integrate[(b*Tan[c + d*x])^(1/3),x]","\frac{3 (b \tan (c+d x))^{4/3} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\tan ^2(c+d x)\right)}{4 b d}","-\frac{\sqrt{3} \sqrt[3]{b} \tan ^{-1}\left(\frac{b^{2/3}-2 (b \tan (c+d x))^{2/3}}{\sqrt{3} b^{2/3}}\right)}{2 d}-\frac{\sqrt[3]{b} \log \left(b^{2/3}+(b \tan (c+d x))^{2/3}\right)}{2 d}+\frac{\sqrt[3]{b} \log \left(-b^{2/3} (b \tan (c+d x))^{2/3}+b^{4/3}+(b \tan (c+d x))^{4/3}\right)}{4 d}",1,"(3*Hypergeometric2F1[2/3, 1, 5/3, -Tan[c + d*x]^2]*(b*Tan[c + d*x])^(4/3))/(4*b*d)","C",1
20,1,100,131,0.143174,"\int \frac{1}{\sqrt[3]{b \tan (c+d x)}} \, dx","Integrate[(b*Tan[c + d*x])^(-1/3),x]","\frac{\sqrt[3]{\tan (c+d x)} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{2 \tan ^{\frac{2}{3}}(c+d x)-1}{\sqrt{3}}\right)+2 \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)-\log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)\right)}{4 d \sqrt[3]{b \tan (c+d x)}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{b^{2/3}-2 (b \tan (c+d x))^{2/3}}{\sqrt{3} b^{2/3}}\right)}{2 \sqrt[3]{b} d}+\frac{\log \left(b^{2/3}+(b \tan (c+d x))^{2/3}\right)}{2 \sqrt[3]{b} d}-\frac{\log \left(-b^{2/3} (b \tan (c+d x))^{2/3}+b^{4/3}+(b \tan (c+d x))^{4/3}\right)}{4 \sqrt[3]{b} d}",1,"((2*Sqrt[3]*ArcTan[(-1 + 2*Tan[c + d*x]^(2/3))/Sqrt[3]] + 2*Log[1 + Tan[c + d*x]^(2/3)] - Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])*Tan[c + d*x]^(1/3))/(4*d*(b*Tan[c + d*x])^(1/3))","A",1
21,1,38,224,0.0291725,"\int \frac{1}{(b \tan (c+d x))^{2/3}} \, dx","Integrate[(b*Tan[c + d*x])^(-2/3),x]","\frac{3 \sqrt[3]{b \tan (c+d x)} \, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\tan ^2(c+d x)\right)}{b d}","\frac{\tan ^{-1}\left(\frac{\sqrt[3]{b \tan (c+d x)}}{\sqrt[3]{b}}\right)}{b^{2/3} d}-\frac{\tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[3]{b \tan (c+d x)}}{\sqrt[3]{b}}\right)}{2 b^{2/3} d}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{b \tan (c+d x)}}{\sqrt[3]{b}}+\sqrt{3}\right)}{2 b^{2/3} d}-\frac{\sqrt{3} \log \left(b^{2/3}-\sqrt{3} \sqrt[3]{b} \sqrt[3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right)}{4 b^{2/3} d}+\frac{\sqrt{3} \log \left(b^{2/3}+\sqrt{3} \sqrt[3]{b} \sqrt[3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right)}{4 b^{2/3} d}",1,"(3*Hypergeometric2F1[1/6, 1, 7/6, -Tan[c + d*x]^2]*(b*Tan[c + d*x])^(1/3))/(b*d)","C",1
22,1,38,245,0.0619419,"\int \frac{1}{(b \tan (c+d x))^{4/3}} \, dx","Integrate[(b*Tan[c + d*x])^(-4/3),x]","-\frac{3 \, _2F_1\left(-\frac{1}{6},1;\frac{5}{6};-\tan ^2(c+d x)\right)}{b d \sqrt[3]{b \tan (c+d x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{b \tan (c+d x)}}{\sqrt[3]{b}}\right)}{b^{4/3} d}+\frac{\tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[3]{b \tan (c+d x)}}{\sqrt[3]{b}}\right)}{2 b^{4/3} d}-\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{b \tan (c+d x)}}{\sqrt[3]{b}}+\sqrt{3}\right)}{2 b^{4/3} d}-\frac{\sqrt{3} \log \left(b^{2/3}-\sqrt{3} \sqrt[3]{b} \sqrt[3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right)}{4 b^{4/3} d}+\frac{\sqrt{3} \log \left(b^{2/3}+\sqrt{3} \sqrt[3]{b} \sqrt[3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right)}{4 b^{4/3} d}-\frac{3}{b d \sqrt[3]{b \tan (c+d x)}}",1,"(-3*Hypergeometric2F1[-1/6, 1, 5/6, -Tan[c + d*x]^2])/(b*d*(b*Tan[c + d*x])^(1/3))","C",1
23,1,53,50,0.0423242,"\int (b \tan (c+d x))^n \, dx","Integrate[(b*Tan[c + d*x])^n,x]","\frac{\tan (c+d x) (b \tan (c+d x))^n \, _2F_1\left(1,\frac{n+1}{2};\frac{n+1}{2}+1;-\tan ^2(c+d x)\right)}{d (n+1)}","\frac{(b \tan (c+d x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(c+d x)\right)}{b d (n+1)}",1,"(Hypergeometric2F1[1, (1 + n)/2, 1 + (1 + n)/2, -Tan[c + d*x]^2]*Tan[c + d*x]*(b*Tan[c + d*x])^n)/(d*(1 + n))","A",1
24,1,56,98,0.3760225,"\int \left(b \tan ^2(c+d x)\right)^{5/2} \, dx","Integrate[(b*Tan[c + d*x]^2)^(5/2),x]","-\frac{\cot (c+d x) \left(b \tan ^2(c+d x)\right)^{5/2} \left(2 \cot ^2(c+d x)+4 \cot ^4(c+d x) \log (\cos (c+d x))-1\right)}{4 d}","-\frac{b^2 \tan (c+d x) \sqrt{b \tan ^2(c+d x)}}{2 d}+\frac{b^2 \tan ^3(c+d x) \sqrt{b \tan ^2(c+d x)}}{4 d}-\frac{b^2 \cot (c+d x) \sqrt{b \tan ^2(c+d x)} \log (\cos (c+d x))}{d}",1,"-1/4*(Cot[c + d*x]*(-1 + 2*Cot[c + d*x]^2 + 4*Cot[c + d*x]^4*Log[Cos[c + d*x]])*(b*Tan[c + d*x]^2)^(5/2))/d","A",1
25,1,47,61,0.1099695,"\int \left(b \tan ^2(c+d x)\right)^{3/2} \, dx","Integrate[(b*Tan[c + d*x]^2)^(3/2),x]","\frac{\cot ^3(c+d x) \left(b \tan ^2(c+d x)\right)^{3/2} \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{2 d}","\frac{b \tan (c+d x) \sqrt{b \tan ^2(c+d x)}}{2 d}+\frac{b \cot (c+d x) \sqrt{b \tan ^2(c+d x)} \log (\cos (c+d x))}{d}",1,"(Cot[c + d*x]^3*(b*Tan[c + d*x]^2)^(3/2)*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d)","A",1
26,1,32,32,0.0405362,"\int \sqrt{b \tan ^2(c+d x)} \, dx","Integrate[Sqrt[b*Tan[c + d*x]^2],x]","-\frac{\cot (c+d x) \sqrt{b \tan ^2(c+d x)} \log (\cos (c+d x))}{d}","-\frac{\cot (c+d x) \sqrt{b \tan ^2(c+d x)} \log (\cos (c+d x))}{d}",1,"-((Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[b*Tan[c + d*x]^2])/d)","A",1
27,1,39,31,0.0865999,"\int \frac{1}{\sqrt{b \tan ^2(c+d x)}} \, dx","Integrate[1/Sqrt[b*Tan[c + d*x]^2],x]","\frac{\tan (c+d x) (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d \sqrt{b \tan ^2(c+d x)}}","\frac{\tan (c+d x) \log (\sin (c+d x))}{d \sqrt{b \tan ^2(c+d x)}}",1,"((Log[Cos[c + d*x]] + Log[Tan[c + d*x]])*Tan[c + d*x])/(d*Sqrt[b*Tan[c + d*x]^2])","A",1
28,1,56,66,0.3863432,"\int \frac{1}{\left(b \tan ^2(c+d x)\right)^{3/2}} \, dx","Integrate[(b*Tan[c + d*x]^2)^(-3/2),x]","-\frac{\tan ^3(c+d x) \left(\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right)}{2 d \left(b \tan ^2(c+d x)\right)^{3/2}}","-\frac{\cot (c+d x)}{2 b d \sqrt{b \tan ^2(c+d x)}}-\frac{\tan (c+d x) \log (\sin (c+d x))}{b d \sqrt{b \tan ^2(c+d x)}}",1,"-1/2*((Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]])*Tan[c + d*x]^3)/(d*(b*Tan[c + d*x]^2)^(3/2))","A",1
29,1,68,97,0.2710175,"\int \frac{1}{\left(b \tan ^2(c+d x)\right)^{5/2}} \, dx","Integrate[(b*Tan[c + d*x]^2)^(-5/2),x]","\frac{\tan ^5(c+d x) \left(-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right)}{4 d \left(b \tan ^2(c+d x)\right)^{5/2}}","-\frac{\cot ^3(c+d x)}{4 b^2 d \sqrt{b \tan ^2(c+d x)}}+\frac{\cot (c+d x)}{2 b^2 d \sqrt{b \tan ^2(c+d x)}}+\frac{\tan (c+d x) \log (\sin (c+d x))}{b^2 d \sqrt{b \tan ^2(c+d x)}}",1,"((2*Cot[c + d*x]^2 - Cot[c + d*x]^4 + 4*Log[Cos[c + d*x]] + 4*Log[Tan[c + d*x]])*Tan[c + d*x]^5)/(4*d*(b*Tan[c + d*x]^2)^(5/2))","A",1
30,1,199,364,0.8401675,"\int \left(b \tan ^3(c+d x)\right)^{5/2} \, dx","Integrate[(b*Tan[c + d*x]^3)^(5/2),x]","\frac{b \left(b \tan ^3(c+d x)\right)^{3/2} \left(-1170 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+1170 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+360 \tan ^{\frac{13}{2}}(c+d x)-520 \tan ^{\frac{9}{2}}(c+d x)+936 \tan ^{\frac{5}{2}}(c+d x)-4680 \sqrt{\tan (c+d x)}-585 \sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+585 \sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{2340 d \tan ^{\frac{9}{2}}(c+d x)}","-\frac{2 b^2 \tan ^3(c+d x) \sqrt{b \tan ^3(c+d x)}}{9 d}+\frac{2 b^2 \tan (c+d x) \sqrt{b \tan ^3(c+d x)}}{5 d}+\frac{2 b^2 \tan ^5(c+d x) \sqrt{b \tan ^3(c+d x)}}{13 d}-\frac{b^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{b^2 \sqrt{b \tan ^3(c+d x)} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b^2 \sqrt{b \tan ^3(c+d x)} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}",1,"(b*(b*Tan[c + d*x]^3)^(3/2)*(-1170*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 1170*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - 585*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 585*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 4680*Sqrt[Tan[c + d*x]] + 936*Tan[c + d*x]^(5/2) - 520*Tan[c + d*x]^(9/2) + 360*Tan[c + d*x]^(13/2)))/(2340*d*Tan[c + d*x]^(9/2))","A",1
31,1,54,286,0.0616396,"\int \left(b \tan ^3(c+d x)\right)^{3/2} \, dx","Integrate[(b*Tan[c + d*x]^3)^(3/2),x]","\frac{2 b \sqrt{b \tan ^3(c+d x)} \left(7 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)+3 \tan ^2(c+d x)-7\right)}{21 d}","-\frac{2 b \sqrt{b \tan ^3(c+d x)}}{3 d}+\frac{2 b \tan ^2(c+d x) \sqrt{b \tan ^3(c+d x)}}{7 d}-\frac{b \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \sqrt{b \tan ^3(c+d x)} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{b \sqrt{b \tan ^3(c+d x)} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}",1,"(2*b*Sqrt[b*Tan[c + d*x]^3]*(-7 + 7*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2] + 3*Tan[c + d*x]^2))/(21*d)","C",1
32,1,161,255,0.2540571,"\int \sqrt{b \tan ^3(c+d x)} \, dx","Integrate[Sqrt[b*Tan[c + d*x]^3],x]","\frac{\sqrt{b \tan ^3(c+d x)} \left(2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+8 \sqrt{\tan (c+d x)}+\sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{4 d \tan ^{\frac{3}{2}}(c+d x)}","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{\sqrt{b \tan ^3(c+d x)} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\sqrt{b \tan ^3(c+d x)} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}",1,"((2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 8*Sqrt[Tan[c + d*x]])*Sqrt[b*Tan[c + d*x]^3])/(4*d*Tan[c + d*x]^(3/2))","A",1
33,1,43,255,0.0345526,"\int \frac{1}{\sqrt{b \tan ^3(c+d x)}} \, dx","Integrate[1/Sqrt[b*Tan[c + d*x]^3],x]","-\frac{2 \tan (c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)}{d \sqrt{b \tan ^3(c+d x)}}","-\frac{2 \tan (c+d x)}{d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{2} d \sqrt{b \tan ^3(c+d x)}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{2} d \sqrt{b \tan ^3(c+d x)}}-\frac{\tan ^{\frac{3}{2}}(c+d x) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{\frac{3}{2}}(c+d x) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \sqrt{b \tan ^3(c+d x)}}",1,"(-2*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2]*Tan[c + d*x])/(d*Sqrt[b*Tan[c + d*x]^3])","C",1
34,1,45,298,0.0712655,"\int \frac{1}{\left(b \tan ^3(c+d x)\right)^{3/2}} \, dx","Integrate[(b*Tan[c + d*x]^3)^(-3/2),x]","-\frac{2 \tan (c+d x) \, _2F_1\left(-\frac{7}{4},1;-\frac{3}{4};-\tan ^2(c+d x)\right)}{7 d \left(b \tan ^3(c+d x)\right)^{3/2}}","\frac{2}{3 b d \sqrt{b \tan ^3(c+d x)}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}-\frac{\tan ^{\frac{3}{2}}(c+d x) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{\frac{3}{2}}(c+d x) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}-\frac{2 \cot ^2(c+d x)}{7 b d \sqrt{b \tan ^3(c+d x)}}",1,"(-2*Hypergeometric2F1[-7/4, 1, -3/4, -Tan[c + d*x]^2]*Tan[c + d*x])/(7*d*(b*Tan[c + d*x]^3)^(3/2))","C",1
35,1,45,364,0.0573067,"\int \frac{1}{\left(b \tan ^3(c+d x)\right)^{5/2}} \, dx","Integrate[(b*Tan[c + d*x]^3)^(-5/2),x]","-\frac{2 \tan (c+d x) \, _2F_1\left(-\frac{13}{4},1;-\frac{9}{4};-\tan ^2(c+d x)\right)}{13 d \left(b \tan ^3(c+d x)\right)^{5/2}}","\frac{2 \tan (c+d x)}{b^2 d \sqrt{b \tan ^3(c+d x)}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right) \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{2} b^2 d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right) \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{2} b^2 d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{\frac{3}{2}}(c+d x) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} b^2 d \sqrt{b \tan ^3(c+d x)}}-\frac{\tan ^{\frac{3}{2}}(c+d x) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} b^2 d \sqrt{b \tan ^3(c+d x)}}-\frac{2 \cot ^5(c+d x)}{13 b^2 d \sqrt{b \tan ^3(c+d x)}}+\frac{2 \cot ^3(c+d x)}{9 b^2 d \sqrt{b \tan ^3(c+d x)}}-\frac{2 \cot (c+d x)}{5 b^2 d \sqrt{b \tan ^3(c+d x)}}",1,"(-2*Hypergeometric2F1[-13/4, 1, -9/4, -Tan[c + d*x]^2]*Tan[c + d*x])/(13*d*(b*Tan[c + d*x]^3)^(5/2))","C",1
36,1,86,182,0.7551727,"\int \left(b \tan ^4(c+d x)\right)^{5/2} \, dx","Integrate[(b*Tan[c + d*x]^4)^(5/2),x]","\frac{\cot (c+d x) \left(b \tan ^4(c+d x)\right)^{5/2} \left(315 \cot ^8(c+d x)-105 \cot ^6(c+d x)+63 \cot ^4(c+d x)-45 \cot ^2(c+d x)-315 \tan ^{-1}(\tan (c+d x)) \cot ^9(c+d x)+35\right)}{315 d}","-\frac{b^2 \tan (c+d x) \sqrt{b \tan ^4(c+d x)}}{3 d}+\frac{b^2 \tan ^7(c+d x) \sqrt{b \tan ^4(c+d x)}}{9 d}-\frac{b^2 \tan ^5(c+d x) \sqrt{b \tan ^4(c+d x)}}{7 d}+\frac{b^2 \tan ^3(c+d x) \sqrt{b \tan ^4(c+d x)}}{5 d}-b^2 x \cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)}+\frac{b^2 \cot (c+d x) \sqrt{b \tan ^4(c+d x)}}{d}",1,"(Cot[c + d*x]*(35 - 45*Cot[c + d*x]^2 + 63*Cot[c + d*x]^4 - 105*Cot[c + d*x]^6 + 315*Cot[c + d*x]^8 - 315*ArcTan[Tan[c + d*x]]*Cot[c + d*x]^9)*(b*Tan[c + d*x]^4)^(5/2))/(315*d)","A",1
37,1,66,110,0.7717972,"\int \left(b \tan ^4(c+d x)\right)^{3/2} \, dx","Integrate[(b*Tan[c + d*x]^4)^(3/2),x]","\frac{\cot (c+d x) \left(b \tan ^4(c+d x)\right)^{3/2} \left(15 \cot ^4(c+d x)-5 \cot ^2(c+d x)-15 \tan ^{-1}(\tan (c+d x)) \cot ^5(c+d x)+3\right)}{15 d}","-\frac{b \tan (c+d x) \sqrt{b \tan ^4(c+d x)}}{3 d}+\frac{b \tan ^3(c+d x) \sqrt{b \tan ^4(c+d x)}}{5 d}-b x \cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)}+\frac{b \cot (c+d x) \sqrt{b \tan ^4(c+d x)}}{d}",1,"(Cot[c + d*x]*(3 - 5*Cot[c + d*x]^2 + 15*Cot[c + d*x]^4 - 15*ArcTan[Tan[c + d*x]]*Cot[c + d*x]^5)*(b*Tan[c + d*x]^4)^(3/2))/(15*d)","A",1
38,1,41,50,0.0968387,"\int \sqrt{b \tan ^4(c+d x)} \, dx","Integrate[Sqrt[b*Tan[c + d*x]^4],x]","-\frac{\cot (c+d x) \sqrt{b \tan ^4(c+d x)} \left(\tan ^{-1}(\tan (c+d x)) \cot (c+d x)-1\right)}{d}","\frac{\cot (c+d x) \sqrt{b \tan ^4(c+d x)}}{d}-x \cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)}",1,"-((Cot[c + d*x]*(-1 + ArcTan[Tan[c + d*x]]*Cot[c + d*x])*Sqrt[b*Tan[c + d*x]^4])/d)","A",1
39,1,43,51,0.0517347,"\int \frac{1}{\sqrt{b \tan ^4(c+d x)}} \, dx","Integrate[1/Sqrt[b*Tan[c + d*x]^4],x]","-\frac{\tan (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d \sqrt{b \tan ^4(c+d x)}}","-\frac{\tan (c+d x)}{d \sqrt{b \tan ^4(c+d x)}}-\frac{x \tan ^2(c+d x)}{\sqrt{b \tan ^4(c+d x)}}",1,"-((Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2]*Tan[c + d*x])/(d*Sqrt[b*Tan[c + d*x]^4]))","C",1
40,1,45,119,0.0479313,"\int \frac{1}{\left(b \tan ^4(c+d x)\right)^{3/2}} \, dx","Integrate[(b*Tan[c + d*x]^4)^(-3/2),x]","-\frac{\tan (c+d x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(c+d x)\right)}{5 d \left(b \tan ^4(c+d x)\right)^{3/2}}","-\frac{\tan (c+d x)}{b d \sqrt{b \tan ^4(c+d x)}}-\frac{x \tan ^2(c+d x)}{b \sqrt{b \tan ^4(c+d x)}}-\frac{\cot ^3(c+d x)}{5 b d \sqrt{b \tan ^4(c+d x)}}+\frac{\cot (c+d x)}{3 b d \sqrt{b \tan ^4(c+d x)}}",1,"-1/5*(Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2]*Tan[c + d*x])/(d*(b*Tan[c + d*x]^4)^(3/2))","C",1
41,1,45,183,0.0359582,"\int \frac{1}{\left(b \tan ^4(c+d x)\right)^{5/2}} \, dx","Integrate[(b*Tan[c + d*x]^4)^(-5/2),x]","-\frac{\tan (c+d x) \, _2F_1\left(-\frac{9}{2},1;-\frac{7}{2};-\tan ^2(c+d x)\right)}{9 d \left(b \tan ^4(c+d x)\right)^{5/2}}","-\frac{\tan (c+d x)}{b^2 d \sqrt{b \tan ^4(c+d x)}}-\frac{x \tan ^2(c+d x)}{b^2 \sqrt{b \tan ^4(c+d x)}}-\frac{\cot ^7(c+d x)}{9 b^2 d \sqrt{b \tan ^4(c+d x)}}+\frac{\cot ^5(c+d x)}{7 b^2 d \sqrt{b \tan ^4(c+d x)}}-\frac{\cot ^3(c+d x)}{5 b^2 d \sqrt{b \tan ^4(c+d x)}}+\frac{\cot (c+d x)}{3 b^2 d \sqrt{b \tan ^4(c+d x)}}",1,"-1/9*(Hypergeometric2F1[-9/2, 1, -7/2, -Tan[c + d*x]^2]*Tan[c + d*x])/(d*(b*Tan[c + d*x]^4)^(5/2))","C",1
42,1,57,59,0.0512095,"\int \left(b \tan ^p(c+d x)\right)^n \, dx","Integrate[(b*Tan[c + d*x]^p)^n,x]","\frac{\tan (c+d x) \left(b \tan ^p(c+d x)\right)^n \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(c+d x)\right)}{d n p+d}","\frac{\tan (c+d x) \left(b \tan ^p(c+d x)\right)^n \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(c+d x)\right)}{d (n p+1)}",1,"(Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[c + d*x]^2]*Tan[c + d*x]*(b*Tan[c + d*x]^p)^n)/(d + d*n*p)","A",1
43,1,49,59,0.0450351,"\int \left(b \tan ^2(c+d x)\right)^n \, dx","Integrate[(b*Tan[c + d*x]^2)^n,x]","\frac{\tan (c+d x) \left(b \tan ^2(c+d x)\right)^n \, _2F_1\left(1,n+\frac{1}{2};n+\frac{3}{2};-\tan ^2(c+d x)\right)}{2 d n+d}","\frac{\tan (c+d x) \left(b \tan ^2(c+d x)\right)^n \, _2F_1\left(1,\frac{1}{2} (2 n+1);\frac{1}{2} (2 n+3);-\tan ^2(c+d x)\right)}{d (2 n+1)}",1,"(Hypergeometric2F1[1, 1/2 + n, 3/2 + n, -Tan[c + d*x]^2]*Tan[c + d*x]*(b*Tan[c + d*x]^2)^n)/(d + 2*d*n)","A",1
44,1,55,57,0.04157,"\int \left(b \tan ^3(c+d x)\right)^n \, dx","Integrate[(b*Tan[c + d*x]^3)^n,x]","\frac{\tan (c+d x) \left(b \tan ^3(c+d x)\right)^n \, _2F_1\left(1,\frac{1}{2} (3 n+1);\frac{3 (n+1)}{2};-\tan ^2(c+d x)\right)}{3 d n+d}","\frac{\tan (c+d x) \left(b \tan ^3(c+d x)\right)^n \, _2F_1\left(1,\frac{1}{2} (3 n+1);\frac{3 (n+1)}{2};-\tan ^2(c+d x)\right)}{d (3 n+1)}",1,"(Hypergeometric2F1[1, (1 + 3*n)/2, (3*(1 + n))/2, -Tan[c + d*x]^2]*Tan[c + d*x]*(b*Tan[c + d*x]^3)^n)/(d + 3*d*n)","A",1
45,1,53,59,0.0406047,"\int \left(b \tan ^4(c+d x)\right)^n \, dx","Integrate[(b*Tan[c + d*x]^4)^n,x]","\frac{\tan (c+d x) \left(b \tan ^4(c+d x)\right)^n \, _2F_1\left(1,2 n+\frac{1}{2};2 n+\frac{3}{2};-\tan ^2(c+d x)\right)}{4 d n+d}","\frac{\tan (c+d x) \left(b \tan ^4(c+d x)\right)^n \, _2F_1\left(1,\frac{1}{2} (4 n+1);\frac{1}{2} (4 n+3);-\tan ^2(c+d x)\right)}{d (4 n+1)}",1,"(Hypergeometric2F1[1, 1/2 + 2*n, 3/2 + 2*n, -Tan[c + d*x]^2]*Tan[c + d*x]*(b*Tan[c + d*x]^4)^n)/(d + 4*d*n)","A",1
46,1,62,71,0.1062578,"\int \left(b \tan ^p(c+d x)\right)^{5/2} \, dx","Integrate[(b*Tan[c + d*x]^p)^(5/2),x]","\frac{2 \tan (c+d x) \left(b \tan ^p(c+d x)\right)^{5/2} \, _2F_1\left(1,\frac{1}{4} (5 p+2);\frac{1}{4} (5 p+6);-\tan ^2(c+d x)\right)}{d (5 p+2)}","\frac{2 b^2 \tan ^{2 p+1}(c+d x) \sqrt{b \tan ^p(c+d x)} \, _2F_1\left(1,\frac{1}{4} (5 p+2);\frac{1}{4} (5 p+6);-\tan ^2(c+d x)\right)}{d (5 p+2)}",1,"(2*Hypergeometric2F1[1, (2 + 5*p)/4, (6 + 5*p)/4, -Tan[c + d*x]^2]*Tan[c + d*x]*(b*Tan[c + d*x]^p)^(5/2))/(d*(2 + 5*p))","A",1
47,1,60,65,0.0715451,"\int \left(b \tan ^p(c+d x)\right)^{3/2} \, dx","Integrate[(b*Tan[c + d*x]^p)^(3/2),x]","\frac{2 \tan (c+d x) \left(b \tan ^p(c+d x)\right)^{3/2} \, _2F_1\left(1,\frac{1}{4} (3 p+2);\frac{3 (p+2)}{4};-\tan ^2(c+d x)\right)}{d (3 p+2)}","\frac{2 b \tan ^{p+1}(c+d x) \sqrt{b \tan ^p(c+d x)} \, _2F_1\left(1,\frac{1}{4} (3 p+2);\frac{3 (p+2)}{4};-\tan ^2(c+d x)\right)}{d (3 p+2)}",1,"(2*Hypergeometric2F1[1, (2 + 3*p)/4, (3*(2 + p))/4, -Tan[c + d*x]^2]*Tan[c + d*x]*(b*Tan[c + d*x]^p)^(3/2))/(d*(2 + 3*p))","A",1
48,1,56,56,0.0397963,"\int \sqrt{b \tan ^p(c+d x)} \, dx","Integrate[Sqrt[b*Tan[c + d*x]^p],x]","\frac{2 \tan (c+d x) \sqrt{b \tan ^p(c+d x)} \, _2F_1\left(1,\frac{p+2}{4};\frac{p+6}{4};-\tan ^2(c+d x)\right)}{d (p+2)}","\frac{2 \tan (c+d x) \sqrt{b \tan ^p(c+d x)} \, _2F_1\left(1,\frac{p+2}{4};\frac{p+6}{4};-\tan ^2(c+d x)\right)}{d (p+2)}",1,"(2*Hypergeometric2F1[1, (2 + p)/4, (6 + p)/4, -Tan[c + d*x]^2]*Tan[c + d*x]*Sqrt[b*Tan[c + d*x]^p])/(d*(2 + p))","A",1
49,1,60,62,0.0518154,"\int \frac{1}{\sqrt{b \tan ^p(c+d x)}} \, dx","Integrate[1/Sqrt[b*Tan[c + d*x]^p],x]","-\frac{2 \tan (c+d x) \, _2F_1\left(1,\frac{2-p}{4};\frac{6-p}{4};-\tan ^2(c+d x)\right)}{d (p-2) \sqrt{b \tan ^p(c+d x)}}","\frac{2 \tan (c+d x) \, _2F_1\left(1,\frac{2-p}{4};\frac{6-p}{4};-\tan ^2(c+d x)\right)}{d (2-p) \sqrt{b \tan ^p(c+d x)}}",1,"(-2*Hypergeometric2F1[1, (2 - p)/4, (6 - p)/4, -Tan[c + d*x]^2]*Tan[c + d*x])/(d*(-2 + p)*Sqrt[b*Tan[c + d*x]^p])","A",1
50,1,60,71,0.0702379,"\int \frac{1}{\left(b \tan ^p(c+d x)\right)^{3/2}} \, dx","Integrate[(b*Tan[c + d*x]^p)^(-3/2),x]","-\frac{2 \tan (c+d x) \, _2F_1\left(1,\frac{1}{4} (2-3 p);-\frac{3}{4} (p-2);-\tan ^2(c+d x)\right)}{d (3 p-2) \left(b \tan ^p(c+d x)\right)^{3/2}}","\frac{2 \tan ^{1-p}(c+d x) \, _2F_1\left(1,\frac{1}{4} (2-3 p);\frac{3 (2-p)}{4};-\tan ^2(c+d x)\right)}{b d (2-3 p) \sqrt{b \tan ^p(c+d x)}}",1,"(-2*Hypergeometric2F1[1, (2 - 3*p)/4, (-3*(-2 + p))/4, -Tan[c + d*x]^2]*Tan[c + d*x])/(d*(-2 + 3*p)*(b*Tan[c + d*x]^p)^(3/2))","A",1
51,1,62,71,0.071591,"\int \frac{1}{\left(b \tan ^p(c+d x)\right)^{5/2}} \, dx","Integrate[(b*Tan[c + d*x]^p)^(-5/2),x]","-\frac{2 \tan (c+d x) \, _2F_1\left(1,\frac{1}{4} (2-5 p);\frac{1}{4} (6-5 p);-\tan ^2(c+d x)\right)}{d (5 p-2) \left(b \tan ^p(c+d x)\right)^{5/2}}","\frac{2 \tan ^{1-2 p}(c+d x) \, _2F_1\left(1,\frac{1}{4} (2-5 p);\frac{1}{4} (6-5 p);-\tan ^2(c+d x)\right)}{b^2 d (2-5 p) \sqrt{b \tan ^p(c+d x)}}",1,"(-2*Hypergeometric2F1[1, (2 - 5*p)/4, (6 - 5*p)/4, -Tan[c + d*x]^2]*Tan[c + d*x])/(d*(-2 + 5*p)*(b*Tan[c + d*x]^p)^(5/2))","A",1
52,1,32,32,0.0225982,"\int \left(b \tan ^p(c+d x)\right)^{\frac{1}{p}} \, dx","Integrate[(b*Tan[c + d*x]^p)^p^(-1),x]","-\frac{\cot (c+d x) \log (\cos (c+d x)) \left(b \tan ^p(c+d x)\right)^{\frac{1}{p}}}{d}","-\frac{\cot (c+d x) \log (\cos (c+d x)) \left(b \tan ^p(c+d x)\right)^{\frac{1}{p}}}{d}",1,"-((Cot[c + d*x]*Log[Cos[c + d*x]]*(b*Tan[c + d*x]^p)^p^(-1))/d)","A",1
53,1,59,61,0.0529526,"\int \left(a (b \tan (c+d x))^p\right)^n \, dx","Integrate[(a*(b*Tan[c + d*x])^p)^n,x]","\frac{\tan (c+d x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(c+d x)\right) \left(a (b \tan (c+d x))^p\right)^n}{d n p+d}","\frac{\tan (c+d x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(c+d x)\right) \left(a (b \tan (c+d x))^p\right)^n}{d (n p+1)}",1,"(Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[c + d*x]^2]*Tan[c + d*x]*(a*(b*Tan[c + d*x])^p)^n)/(d + d*n*p)","A",1
54,1,122,257,0.241136,"\int \sin ^4(a+b x) \sqrt{d \tan (a+b x)} \, dx","Integrate[Sin[a + b*x]^4*Sqrt[d*Tan[a + b*x]],x]","-\frac{\sqrt{d \tan (a+b x)} \left(18 \sin (2 (a+b x))-2 \sin (4 (a+b x))+21 \sqrt{\sin (2 (a+b x))} \csc (a+b x) \sin ^{-1}(\cos (a+b x)-\sin (a+b x))+21 \sqrt{\sin (2 (a+b x))} \csc (a+b x) \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{64 b}","-\frac{\cos ^4(a+b x) (d \tan (a+b x))^{7/2}}{4 b d^3}-\frac{21 \sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{32 \sqrt{2} b}+\frac{21 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{32 \sqrt{2} b}+\frac{21 \sqrt{d} \log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{64 \sqrt{2} b}-\frac{21 \sqrt{d} \log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{64 \sqrt{2} b}-\frac{7 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d}",1,"-1/64*((21*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Csc[a + b*x]*Sqrt[Sin[2*(a + b*x)]] + 21*Csc[a + b*x]*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]] + 18*Sin[2*(a + b*x)] - 2*Sin[4*(a + b*x)])*Sqrt[d*Tan[a + b*x]])/b","A",1
55,1,104,227,0.1806986,"\int \sin ^2(a+b x) \sqrt{d \tan (a+b x)} \, dx","Integrate[Sin[a + b*x]^2*Sqrt[d*Tan[a + b*x]],x]","-\frac{\sqrt{\sin (2 (a+b x))} \sqrt{d \tan (a+b x)} \left(2 \sqrt{\sin (2 (a+b x))}+3 \csc (a+b x) \sin ^{-1}(\cos (a+b x)-\sin (a+b x))+3 \csc (a+b x) \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{8 b}","-\frac{3 \sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{4 \sqrt{2} b}+\frac{3 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{4 \sqrt{2} b}+\frac{3 \sqrt{d} \log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b}-\frac{3 \sqrt{d} \log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b}-\frac{\cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{2 b d}",1,"-1/8*((3*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Csc[a + b*x] + 3*Csc[a + b*x]*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]] + 2*Sqrt[Sin[2*(a + b*x)]])*Sqrt[Sin[2*(a + b*x)]]*Sqrt[d*Tan[a + b*x]])/b","A",1
56,1,18,18,0.0756524,"\int \csc ^2(a+b x) \sqrt{d \tan (a+b x)} \, dx","Integrate[Csc[a + b*x]^2*Sqrt[d*Tan[a + b*x]],x]","-\frac{2 d}{b \sqrt{d \tan (a+b x)}}","-\frac{2 d}{b \sqrt{d \tan (a+b x)}}",1,"(-2*d)/(b*Sqrt[d*Tan[a + b*x]])","A",1
57,1,30,41,0.1287366,"\int \csc ^4(a+b x) \sqrt{d \tan (a+b x)} \, dx","Integrate[Csc[a + b*x]^4*Sqrt[d*Tan[a + b*x]],x]","-\frac{2 d \left(\csc ^2(a+b x)+4\right)}{5 b \sqrt{d \tan (a+b x)}}","-\frac{2 d^3}{5 b (d \tan (a+b x))^{5/2}}-\frac{2 d}{b \sqrt{d \tan (a+b x)}}",1,"(-2*d*(4 + Csc[a + b*x]^2))/(5*b*Sqrt[d*Tan[a + b*x]])","A",1
58,1,50,63,0.169679,"\int \csc ^6(a+b x) \sqrt{d \tan (a+b x)} \, dx","Integrate[Csc[a + b*x]^6*Sqrt[d*Tan[a + b*x]],x]","\frac{2 d (20 \cos (2 (a+b x))-4 \cos (4 (a+b x))-21) \csc ^4(a+b x)}{45 b \sqrt{d \tan (a+b x)}}","-\frac{2 d^5}{9 b (d \tan (a+b x))^{9/2}}-\frac{4 d^3}{5 b (d \tan (a+b x))^{5/2}}-\frac{2 d}{b \sqrt{d \tan (a+b x)}}",1,"(2*d*(-21 + 20*Cos[2*(a + b*x)] - 4*Cos[4*(a + b*x)])*Csc[a + b*x]^4)/(45*b*Sqrt[d*Tan[a + b*x]])","A",1
59,1,139,105,1.9418051,"\int \sin ^3(a+b x) \sqrt{d \tan (a+b x)} \, dx","Integrate[Sin[a + b*x]^3*Sqrt[d*Tan[a + b*x]],x]","-\frac{\cos (2 (a+b x)) \sec (a+b x) \sqrt{d \tan (a+b x)} \left((\cos (2 (a+b x))-6) \sqrt{\tan (a+b x)} \sqrt{\sec ^2(a+b x)}-5 \sqrt[4]{-1} \sec ^2(a+b x) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)\right)}{6 b \sqrt{\tan (a+b x)} \left(\tan ^2(a+b x)-1\right) \sqrt{\sec ^2(a+b x)}}","-\frac{d \sin ^3(a+b x)}{3 b \sqrt{d \tan (a+b x)}}-\frac{5 d \sin (a+b x)}{6 b \sqrt{d \tan (a+b x)}}+\frac{5 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{12 b}",1,"-1/6*(Cos[2*(a + b*x)]*Sec[a + b*x]*(-5*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Sec[a + b*x]^2 + (-6 + Cos[2*(a + b*x)])*Sqrt[Sec[a + b*x]^2]*Sqrt[Tan[a + b*x]])*Sqrt[d*Tan[a + b*x]])/(b*Sqrt[Sec[a + b*x]^2]*Sqrt[Tan[a + b*x]]*(-1 + Tan[a + b*x]^2))","C",1
60,1,57,75,1.0486784,"\int \sin (a+b x) \sqrt{d \tan (a+b x)} \, dx","Integrate[Sin[a + b*x]*Sqrt[d*Tan[a + b*x]],x]","\frac{\cos (a+b x) \sqrt{d \tan (a+b x)} \left(\sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(a+b x)\right)-1\right)}{b}","\frac{\sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{2 b}-\frac{d \sin (a+b x)}{b \sqrt{d \tan (a+b x)}}",1,"(Cos[a + b*x]*(-1 + Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2])*Sqrt[d*Tan[a + b*x]])/b","C",1
61,1,73,47,0.1543396,"\int \csc (a+b x) \sqrt{d \tan (a+b x)} \, dx","Integrate[Csc[a + b*x]*Sqrt[d*Tan[a + b*x]],x]","-\frac{2 \sqrt[4]{-1} \cos (a+b x) \sqrt{\sec ^2(a+b x)} \sqrt{d \tan (a+b x)} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)}{b \sqrt{\tan (a+b x)}}","\frac{\sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{b}",1,"(-2*(-1)^(1/4)*Cos[a + b*x]*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Sqrt[Sec[a + b*x]^2]*Sqrt[d*Tan[a + b*x]])/(b*Sqrt[Tan[a + b*x]])","C",1
62,1,115,77,0.5901378,"\int \csc ^3(a+b x) \sqrt{d \tan (a+b x)} \, dx","Integrate[Csc[a + b*x]^3*Sqrt[d*Tan[a + b*x]],x]","\frac{2 \cos (2 (a+b x)) \csc ^3(a+b x) (d \tan (a+b x))^{3/2} \left(\sqrt{\sec ^2(a+b x)}+2 \sqrt[4]{-1} \tan ^{\frac{3}{2}}(a+b x) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)\right)}{3 b d \left(\tan ^2(a+b x)-1\right) \sqrt{\sec ^2(a+b x)}}","\frac{2 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{3 b}-\frac{2 d \csc (a+b x)}{3 b \sqrt{d \tan (a+b x)}}",1,"(2*Cos[2*(a + b*x)]*Csc[a + b*x]^3*(d*Tan[a + b*x])^(3/2)*(Sqrt[Sec[a + b*x]^2] + 2*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Tan[a + b*x]^(3/2)))/(3*b*d*Sqrt[Sec[a + b*x]^2]*(-1 + Tan[a + b*x]^2))","C",1
63,1,124,105,1.4182561,"\int \csc ^5(a+b x) \sqrt{d \tan (a+b x)} \, dx","Integrate[Csc[a + b*x]^5*Sqrt[d*Tan[a + b*x]],x]","-\frac{2 d \cos (2 (a+b x)) \csc ^3(a+b x) \left((\cos (2 (a+b x))-2) \sec ^2(a+b x)^{3/2}-4 \sqrt[4]{-1} \tan ^{\frac{7}{2}}(a+b x) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)\right)}{7 b \left(\tan ^2(a+b x)-1\right) \sqrt{\sec ^2(a+b x)} \sqrt{d \tan (a+b x)}}","-\frac{2 d \csc ^3(a+b x)}{7 b \sqrt{d \tan (a+b x)}}-\frac{4 d \csc (a+b x)}{7 b \sqrt{d \tan (a+b x)}}+\frac{4 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{7 b}",1,"(-2*d*Cos[2*(a + b*x)]*Csc[a + b*x]^3*((-2 + Cos[2*(a + b*x)])*(Sec[a + b*x]^2)^(3/2) - 4*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Tan[a + b*x]^(7/2)))/(7*b*Sqrt[Sec[a + b*x]^2]*Sqrt[d*Tan[a + b*x]]*(-1 + Tan[a + b*x]^2))","C",1
64,1,123,277,0.769727,"\int \sin ^4(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Sin[a + b*x]^4*(d*Tan[a + b*x])^(3/2),x]","-\frac{d \csc (a+b x) \sqrt{d \tan (a+b x)} \left(-143 \sin (a+b x)-14 \sin (3 (a+b x))+\sin (5 (a+b x))-45 \sqrt{\sin (2 (a+b x))} \sin ^{-1}(\cos (a+b x)-\sin (a+b x))+45 \sqrt{\sin (2 (a+b x))} \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{64 b}","\frac{45 d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{32 \sqrt{2} b}-\frac{45 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{32 \sqrt{2} b}+\frac{45 d^{3/2} \log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{64 \sqrt{2} b}-\frac{45 d^{3/2} \log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{64 \sqrt{2} b}-\frac{\cos ^4(a+b x) (d \tan (a+b x))^{9/2}}{4 b d^3}+\frac{45 d \sqrt{d \tan (a+b x)}}{16 b}-\frac{9 \cos ^2(a+b x) (d \tan (a+b x))^{5/2}}{16 b d}",1,"-1/64*(d*Csc[a + b*x]*(-143*Sin[a + b*x] - 45*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Sqrt[Sin[2*(a + b*x)]] + 45*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]] - 14*Sin[3*(a + b*x)] + Sin[5*(a + b*x)])*Sqrt[d*Tan[a + b*x]])/b","A",1
65,1,113,247,0.5471821,"\int \sin ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Sin[a + b*x]^2*(d*Tan[a + b*x])^(3/2),x]","\frac{d \csc (a+b x) \sqrt{d \tan (a+b x)} \left(17 \sin (a+b x)+\sin (3 (a+b x))+5 \sqrt{\sin (2 (a+b x))} \sin ^{-1}(\cos (a+b x)-\sin (a+b x))-5 \sqrt{\sin (2 (a+b x))} \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{8 b}","\frac{5 d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{4 \sqrt{2} b}-\frac{5 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{4 \sqrt{2} b}+\frac{5 d^{3/2} \log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b}-\frac{5 d^{3/2} \log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b}+\frac{5 d \sqrt{d \tan (a+b x)}}{2 b}-\frac{\cos ^2(a+b x) (d \tan (a+b x))^{5/2}}{2 b d}",1,"(d*Csc[a + b*x]*(17*Sin[a + b*x] + 5*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Sqrt[Sin[2*(a + b*x)]] - 5*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]] + Sin[3*(a + b*x)])*Sqrt[d*Tan[a + b*x]])/(8*b)","A",1
66,1,18,18,0.0550244,"\int \csc ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Csc[a + b*x]^2*(d*Tan[a + b*x])^(3/2),x]","\frac{2 d \sqrt{d \tan (a+b x)}}{b}","\frac{2 d \sqrt{d \tan (a+b x)}}{b}",1,"(2*d*Sqrt[d*Tan[a + b*x]])/b","A",1
67,1,30,41,0.0822791,"\int \csc ^4(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Csc[a + b*x]^4*(d*Tan[a + b*x])^(3/2),x]","-\frac{2 d \left(\csc ^2(a+b x)-4\right) \sqrt{d \tan (a+b x)}}{3 b}","\frac{2 d \sqrt{d \tan (a+b x)}}{b}-\frac{2 d^3}{3 b (d \tan (a+b x))^{3/2}}",1,"(-2*d*(-4 + Csc[a + b*x]^2)*Sqrt[d*Tan[a + b*x]])/(3*b)","A",1
68,1,42,63,0.1451026,"\int \csc ^6(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Csc[a + b*x]^6*(d*Tan[a + b*x])^(3/2),x]","-\frac{2 d \left(3 \csc ^4(a+b x)+8 \csc ^2(a+b x)-32\right) \sqrt{d \tan (a+b x)}}{21 b}","-\frac{2 d^5}{7 b (d \tan (a+b x))^{7/2}}-\frac{4 d^3}{3 b (d \tan (a+b x))^{3/2}}+\frac{2 d \sqrt{d \tan (a+b x)}}{b}",1,"(-2*d*(-32 + 8*Csc[a + b*x]^2 + 3*Csc[a + b*x]^4)*Sqrt[d*Tan[a + b*x]])/(21*b)","A",1
69,1,90,110,0.604743,"\int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Sin[a + b*x]^3*(d*Tan[a + b*x])^(3/2),x]","\frac{(d \tan (a+b x))^{3/2} \left(2 \cos (a+b x) (\cos (2 (a+b x))+13) \sqrt{\sec ^2(a+b x)}-28 \sec (a+b x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)\right)}{12 b \sqrt{\sec ^2(a+b x)}}","\frac{7 d^3 \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}-\frac{7 d^2 \sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2 b \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}+\frac{2 d \sin ^3(a+b x) \sqrt{d \tan (a+b x)}}{b}",1,"((-28*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sec[a + b*x] + 2*Cos[a + b*x]*(13 + Cos[2*(a + b*x)])*Sqrt[Sec[a + b*x]^2])*(d*Tan[a + b*x])^(3/2))/(12*b*Sqrt[Sec[a + b*x]^2])","C",1
70,1,58,76,0.2914879,"\int \sin (a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Sin[a + b*x]*(d*Tan[a + b*x])^(3/2),x]","-\frac{2 \cos (a+b x) (d \tan (a+b x))^{3/2} \left(\sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)-1\right)}{b}","\frac{2 d \sin (a+b x) \sqrt{d \tan (a+b x)}}{b}-\frac{3 d^2 \sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{b \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}",1,"(-2*Cos[a + b*x]*(-1 + Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2])*(d*Tan[a + b*x])^(3/2))/b","C",1
71,1,61,76,0.2914777,"\int \csc (a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Csc[a + b*x]*(d*Tan[a + b*x])^(3/2),x]","-\frac{2 \cos (a+b x) (d \tan (a+b x))^{3/2} \left(2 \sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)-3\right)}{3 b}","\frac{2 d \sin (a+b x) \sqrt{d \tan (a+b x)}}{b}-\frac{2 d^2 \sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{b \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}",1,"(-2*Cos[a + b*x]*(-3 + 2*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2])*(d*Tan[a + b*x])^(3/2))/(3*b)","C",1
72,1,71,102,0.5866217,"\int \csc ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Csc[a + b*x]^3*(d*Tan[a + b*x])^(3/2),x]","-\frac{2 \cos (a+b x) (d \tan (a+b x))^{3/2} \left(4 \sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)+3 \csc ^2(a+b x)-6\right)}{3 b}","-\frac{4 d^2 \cos (a+b x)}{b \sqrt{d \tan (a+b x)}}-\frac{4 d^2 \sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{b \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}+\frac{2 d \csc (a+b x) \sqrt{d \tan (a+b x)}}{b}",1,"(-2*Cos[a + b*x]*(-6 + 3*Csc[a + b*x]^2 + 4*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2])*(d*Tan[a + b*x])^(3/2))/(3*b)","C",1
73,1,142,277,0.57975,"\int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx","Integrate[Sin[a + b*x]^4*(d*Tan[a + b*x])^(5/2),x]","\frac{d (d \tan (a+b x))^{3/2} \left(204 \cos ^2(a+b x)-6 \sin (4 (a+b x)) \cot (a+b x)+231 \sqrt{\sin (2 (a+b x))} \cot (a+b x) \csc (a+b x) \sin ^{-1}(\cos (a+b x)-\sin (a+b x))+231 \sqrt{\sin (2 (a+b x))} \cot (a+b x) \csc (a+b x) \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)+128\right)}{192 b}","\frac{77 d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{32 \sqrt{2} b}-\frac{77 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{32 \sqrt{2} b}-\frac{77 d^{5/2} \log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{64 \sqrt{2} b}+\frac{77 d^{5/2} \log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{64 \sqrt{2} b}-\frac{\cos ^4(a+b x) (d \tan (a+b x))^{11/2}}{4 b d^3}+\frac{77 d (d \tan (a+b x))^{3/2}}{48 b}-\frac{11 \cos ^2(a+b x) (d \tan (a+b x))^{7/2}}{16 b d}",1,"(d*(128 + 204*Cos[a + b*x]^2 + 231*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Cot[a + b*x]*Csc[a + b*x]*Sqrt[Sin[2*(a + b*x)]] + 231*Cot[a + b*x]*Csc[a + b*x]*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]] - 6*Cot[a + b*x]*Sin[4*(a + b*x)])*(d*Tan[a + b*x])^(3/2))/(192*b)","A",1
74,1,126,247,0.4056728,"\int \sin ^2(a+b x) (d \tan (a+b x))^{5/2} \, dx","Integrate[Sin[a + b*x]^2*(d*Tan[a + b*x])^(5/2),x]","\frac{d (d \tan (a+b x))^{3/2} \left(12 \cos ^2(a+b x)+21 \sqrt{\sin (2 (a+b x))} \cot (a+b x) \csc (a+b x) \sin ^{-1}(\cos (a+b x)-\sin (a+b x))+21 \sqrt{\sin (2 (a+b x))} \cot (a+b x) \csc (a+b x) \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)+16\right)}{24 b}","\frac{7 d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{4 \sqrt{2} b}-\frac{7 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{4 \sqrt{2} b}-\frac{7 d^{5/2} \log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b}+\frac{7 d^{5/2} \log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b}+\frac{7 d (d \tan (a+b x))^{3/2}}{6 b}-\frac{\cos ^2(a+b x) (d \tan (a+b x))^{7/2}}{2 b d}",1,"(d*(16 + 12*Cos[a + b*x]^2 + 21*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Cot[a + b*x]*Csc[a + b*x]*Sqrt[Sin[2*(a + b*x)]] + 21*Cot[a + b*x]*Csc[a + b*x]*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]])*(d*Tan[a + b*x])^(3/2))/(24*b)","A",1
75,1,20,20,0.0806665,"\int \csc ^2(a+b x) (d \tan (a+b x))^{5/2} \, dx","Integrate[Csc[a + b*x]^2*(d*Tan[a + b*x])^(5/2),x]","\frac{2 d (d \tan (a+b x))^{3/2}}{3 b}","\frac{2 d (d \tan (a+b x))^{3/2}}{3 b}",1,"(2*d*(d*Tan[a + b*x])^(3/2))/(3*b)","A",1
76,1,32,41,0.114369,"\int \csc ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx","Integrate[Csc[a + b*x]^4*(d*Tan[a + b*x])^(5/2),x]","-\frac{2 d \left(3 \cot ^2(a+b x)-1\right) (d \tan (a+b x))^{3/2}}{3 b}","\frac{2 d (d \tan (a+b x))^{3/2}}{3 b}-\frac{2 d^3}{b \sqrt{d \tan (a+b x)}}",1,"(-2*d*(-1 + 3*Cot[a + b*x]^2)*(d*Tan[a + b*x])^(3/2))/(3*b)","A",1
77,1,42,63,0.2316029,"\int \csc ^6(a+b x) (d \tan (a+b x))^{5/2} \, dx","Integrate[Csc[a + b*x]^6*(d*Tan[a + b*x])^(5/2),x]","-\frac{2 d (d \tan (a+b x))^{3/2} \left(3 \cot ^2(a+b x) \left(\csc ^2(a+b x)+9\right)-5\right)}{15 b}","-\frac{2 d^5}{5 b (d \tan (a+b x))^{5/2}}-\frac{4 d^3}{b \sqrt{d \tan (a+b x)}}+\frac{2 d (d \tan (a+b x))^{3/2}}{3 b}",1,"(-2*d*(-5 + 3*Cot[a + b*x]^2*(9 + Csc[a + b*x]^2))*(d*Tan[a + b*x])^(3/2))/(15*b)","A",1
78,1,153,137,3.3033155,"\int \sin ^3(a+b x) (d \tan (a+b x))^{5/2} \, dx","Integrate[Sin[a + b*x]^3*(d*Tan[a + b*x])^(5/2),x]","-\frac{\csc (a+b x) \sqrt{\sec ^2(a+b x)} (d \tan (a+b x))^{5/2} \left((77 \cos (2 (a+b x))+22 \cos (4 (a+b x))-\cos (6 (a+b x))+22) \sqrt{\tan (a+b x)} \sqrt{\sec ^2(a+b x)}+120 \sqrt[4]{-1} \cos (2 (a+b x)) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)\right)}{48 b \tan ^{\frac{3}{2}}(a+b x) \left(\tan ^2(a+b x)-1\right)}","\frac{d^3 \sin ^3(a+b x)}{b \sqrt{d \tan (a+b x)}}+\frac{5 d^3 \sin (a+b x)}{2 b \sqrt{d \tan (a+b x)}}-\frac{5 d^2 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{4 b}+\frac{2 d \sin ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b}",1,"-1/48*(Csc[a + b*x]*Sqrt[Sec[a + b*x]^2]*(120*(-1)^(1/4)*Cos[2*(a + b*x)]*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1] + (22 + 77*Cos[2*(a + b*x)] + 22*Cos[4*(a + b*x)] - Cos[6*(a + b*x)])*Sqrt[Sec[a + b*x]^2]*Sqrt[Tan[a + b*x]])*(d*Tan[a + b*x])^(5/2))/(b*Tan[a + b*x]^(3/2)*(-1 + Tan[a + b*x]^2))","C",1
79,1,133,108,2.2396784,"\int \sin (a+b x) (d \tan (a+b x))^{5/2} \, dx","Integrate[Sin[a + b*x]*(d*Tan[a + b*x])^(5/2),x]","-\frac{\cos (2 (a+b x)) \csc (a+b x) \sqrt{\sec ^2(a+b x)} (d \tan (a+b x))^{5/2} \left((3 \cos (2 (a+b x))+7) \sqrt{\tan (a+b x)} \sqrt{\sec ^2(a+b x)}+10 \sqrt[4]{-1} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)\right)}{6 b \tan ^{\frac{3}{2}}(a+b x) \left(\tan ^2(a+b x)-1\right)}","\frac{5 d^3 \sin (a+b x)}{3 b \sqrt{d \tan (a+b x)}}-\frac{5 d^2 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{6 b}+\frac{2 d \sin (a+b x) (d \tan (a+b x))^{3/2}}{3 b}",1,"-1/6*(Cos[2*(a + b*x)]*Csc[a + b*x]*Sqrt[Sec[a + b*x]^2]*(10*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1] + (7 + 3*Cos[2*(a + b*x)])*Sqrt[Sec[a + b*x]^2]*Sqrt[Tan[a + b*x]])*(d*Tan[a + b*x])^(5/2))/(b*Tan[a + b*x]^(3/2)*(-1 + Tan[a + b*x]^2))","C",1
80,1,71,80,0.3876213,"\int \csc (a+b x) (d \tan (a+b x))^{5/2} \, dx","Integrate[Csc[a + b*x]*(d*Tan[a + b*x])^(5/2),x]","\frac{2 d^2 \cos (a+b x) \sqrt{d \tan (a+b x)} \left(\sec ^2(a+b x)-\sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(a+b x)\right)\right)}{3 b}","\frac{2 d \csc (a+b x) (d \tan (a+b x))^{3/2}}{3 b}-\frac{d^2 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{3 b}",1,"(2*d^2*Cos[a + b*x]*(Sec[a + b*x]^2 - Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2])*Sqrt[d*Tan[a + b*x]])/(3*b)","C",1
81,1,71,80,0.3497428,"\int \csc ^3(a+b x) (d \tan (a+b x))^{5/2} \, dx","Integrate[Csc[a + b*x]^3*(d*Tan[a + b*x])^(5/2),x]","\frac{2 d^2 \cos (a+b x) \sqrt{d \tan (a+b x)} \left(2 \sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(a+b x)\right)+\sec ^2(a+b x)\right)}{3 b}","\frac{2 d^2 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{3 b}+\frac{2 d \csc (a+b x) (d \tan (a+b x))^{3/2}}{3 b}",1,"(2*d^2*Cos[a + b*x]*(Sec[a + b*x]^2 + 2*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2])*Sqrt[d*Tan[a + b*x]])/(3*b)","C",1
82,1,110,110,0.4917575,"\int \csc ^5(a+b x) (d \tan (a+b x))^{5/2} \, dx","Integrate[Csc[a + b*x]^5*(d*Tan[a + b*x])^(5/2),x]","-\frac{2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2} \left(\cos (2 (a+b x)) \sqrt{\sec ^2(a+b x)}+2 \sqrt[4]{-1} \sin (2 (a+b x)) \sqrt{\tan (a+b x)} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)\right)}{3 b \sqrt{\sec ^2(a+b x)}}","-\frac{4 d^3 \csc (a+b x)}{3 b \sqrt{d \tan (a+b x)}}+\frac{4 d^2 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{3 b}+\frac{2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b}",1,"(-2*d*Csc[a + b*x]^3*(Cos[2*(a + b*x)]*Sqrt[Sec[a + b*x]^2] + 2*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Sin[2*(a + b*x)]*Sqrt[Tan[a + b*x]])*(d*Tan[a + b*x])^(3/2))/(3*b*Sqrt[Sec[a + b*x]^2])","C",1
83,1,130,140,1.6494853,"\int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx","Integrate[Csc[a + b*x]^7*(d*Tan[a + b*x])^(5/2),x]","-\frac{d^2 \csc (a+b x) \sqrt{d \tan (a+b x)} \left((10 \cos (2 (a+b x))-5 \cos (4 (a+b x))+1) \csc ^3(a+b x) \sec (a+b x) \sqrt{\sec ^2(a+b x)}+80 \sqrt[4]{-1} \sqrt{\tan (a+b x)} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)\right)}{21 b \sqrt{\sec ^2(a+b x)}}","-\frac{20 d^3 \csc ^3(a+b x)}{21 b \sqrt{d \tan (a+b x)}}-\frac{40 d^3 \csc (a+b x)}{21 b \sqrt{d \tan (a+b x)}}+\frac{40 d^2 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{21 b}+\frac{2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}",1,"-1/21*(d^2*Csc[a + b*x]*((1 + 10*Cos[2*(a + b*x)] - 5*Cos[4*(a + b*x)])*Csc[a + b*x]^3*Sec[a + b*x]*Sqrt[Sec[a + b*x]^2] + 80*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Sqrt[Tan[a + b*x]])*Sqrt[d*Tan[a + b*x]])/(b*Sqrt[Sec[a + b*x]^2])","C",1
84,1,122,257,0.7122352,"\int \frac{\sin ^4(a+b x)}{\sqrt{d \tan (a+b x)}} \, dx","Integrate[Sin[a + b*x]^4/Sqrt[d*Tan[a + b*x]],x]","\frac{\sec (a+b x) \left(-7 \sin (a+b x)-6 \sin (3 (a+b x))+\sin (5 (a+b x))-5 \sqrt{\sin (2 (a+b x))} \sin ^{-1}(\cos (a+b x)-\sin (a+b x))+5 \sqrt{\sin (2 (a+b x))} \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{64 b \sqrt{d \tan (a+b x)}}","-\frac{\cos ^4(a+b x) (d \tan (a+b x))^{5/2}}{4 b d^3}-\frac{5 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{32 \sqrt{2} b \sqrt{d}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{32 \sqrt{2} b \sqrt{d}}-\frac{5 \log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{64 \sqrt{2} b \sqrt{d}}+\frac{5 \log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{64 \sqrt{2} b \sqrt{d}}-\frac{5 \cos ^2(a+b x) \sqrt{d \tan (a+b x)}}{16 b d}",1,"(Sec[a + b*x]*(-7*Sin[a + b*x] - 5*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Sqrt[Sin[2*(a + b*x)]] + 5*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]] - 6*Sin[3*(a + b*x)] + Sin[5*(a + b*x)]))/(64*b*Sqrt[d*Tan[a + b*x]])","A",1
85,1,109,227,0.6447106,"\int \frac{\sin ^2(a+b x)}{\sqrt{d \tan (a+b x)}} \, dx","Integrate[Sin[a + b*x]^2/Sqrt[d*Tan[a + b*x]],x]","-\frac{\sec (a+b x) \left(\sin (a+b x)+\sin (3 (a+b x))+\sqrt{\sin (2 (a+b x))} \sin ^{-1}(\cos (a+b x)-\sin (a+b x))-\sqrt{\sin (2 (a+b x))} \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{8 b \sqrt{d \tan (a+b x)}}","-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{4 \sqrt{2} b \sqrt{d}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{4 \sqrt{2} b \sqrt{d}}-\frac{\log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b \sqrt{d}}+\frac{\log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b \sqrt{d}}-\frac{\cos ^2(a+b x) \sqrt{d \tan (a+b x)}}{2 b d}",1,"-1/8*(Sec[a + b*x]*(Sin[a + b*x] + ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Sqrt[Sin[2*(a + b*x)]] - Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]] + Sin[3*(a + b*x)]))/(b*Sqrt[d*Tan[a + b*x]])","A",1
86,1,20,20,0.1014399,"\int \frac{\csc ^2(a+b x)}{\sqrt{d \tan (a+b x)}} \, dx","Integrate[Csc[a + b*x]^2/Sqrt[d*Tan[a + b*x]],x]","-\frac{2 d}{3 b (d \tan (a+b x))^{3/2}}","-\frac{2 d}{3 b (d \tan (a+b x))^{3/2}}",1,"(-2*d)/(3*b*(d*Tan[a + b*x])^(3/2))","A",1
87,1,40,43,0.1304076,"\int \frac{\csc ^4(a+b x)}{\sqrt{d \tan (a+b x)}} \, dx","Integrate[Csc[a + b*x]^4/Sqrt[d*Tan[a + b*x]],x]","\frac{2 d (2 \cos (2 (a+b x))-5) \csc ^2(a+b x)}{21 b (d \tan (a+b x))^{3/2}}","-\frac{2 d^3}{7 b (d \tan (a+b x))^{7/2}}-\frac{2 d}{3 b (d \tan (a+b x))^{3/2}}",1,"(2*d*(-5 + 2*Cos[2*(a + b*x)])*Csc[a + b*x]^2)/(21*b*(d*Tan[a + b*x])^(3/2))","A",1
88,1,50,65,0.1592617,"\int \frac{\csc ^6(a+b x)}{\sqrt{d \tan (a+b x)}} \, dx","Integrate[Csc[a + b*x]^6/Sqrt[d*Tan[a + b*x]],x]","\frac{2 d (28 \cos (2 (a+b x))-4 \cos (4 (a+b x))-45) \csc ^4(a+b x)}{231 b (d \tan (a+b x))^{3/2}}","-\frac{2 d^5}{11 b (d \tan (a+b x))^{11/2}}-\frac{4 d^3}{7 b (d \tan (a+b x))^{7/2}}-\frac{2 d}{3 b (d \tan (a+b x))^{3/2}}",1,"(2*d*(-45 + 28*Cos[2*(a + b*x)] - 4*Cos[4*(a + b*x)])*Csc[a + b*x]^4)/(231*b*(d*Tan[a + b*x])^(3/2))","A",1
89,1,86,107,0.8734518,"\int \frac{\sin ^5(a+b x)}{\sqrt{d \tan (a+b x)}} \, dx","Integrate[Sin[a + b*x]^5/Sqrt[d*Tan[a + b*x]],x]","\frac{\sin (a+b x) \left(28 \tan (a+b x) \sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)-20 \sin (2 (a+b x))+3 \sin (4 (a+b x))\right)}{120 b \sqrt{d \tan (a+b x)}}","-\frac{d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}-\frac{7 d \sin ^3(a+b x)}{30 b (d \tan (a+b x))^{3/2}}+\frac{7 \sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{20 b \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}",1,"(Sin[a + b*x]*(-20*Sin[2*(a + b*x)] + 3*Sin[4*(a + b*x)] + 28*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2]*Tan[a + b*x]))/(120*b*Sqrt[d*Tan[a + b*x]])","C",1
90,1,98,79,0.75974,"\int \frac{\sin ^3(a+b x)}{\sqrt{d \tan (a+b x)}} \, dx","Integrate[Sin[a + b*x]^3/Sqrt[d*Tan[a + b*x]],x]","\frac{\sqrt{d \tan (a+b x)} \left(4 \tan (a+b x) \sec (a+b x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)-(\sin (a+b x)+\sin (3 (a+b x))) \sqrt{\sec ^2(a+b x)}\right)}{12 b d \sqrt{\sec ^2(a+b x)}}","\frac{\sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2 b \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}-\frac{d \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}",1,"(Sqrt[d*Tan[a + b*x]]*(-(Sqrt[Sec[a + b*x]^2]*(Sin[a + b*x] + Sin[3*(a + b*x)])) + 4*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sec[a + b*x]*Tan[a + b*x]))/(12*b*d*Sqrt[Sec[a + b*x]^2])","C",1
91,1,60,47,0.1269512,"\int \frac{\sin (a+b x)}{\sqrt{d \tan (a+b x)}} \, dx","Integrate[Sin[a + b*x]/Sqrt[d*Tan[a + b*x]],x]","\frac{2 \sin (a+b x) \sqrt{\sec ^2(a+b x)} \sqrt{d \tan (a+b x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)}{3 b d}","\frac{\sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{b \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}",1,"(2*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2]*Sin[a + b*x]*Sqrt[d*Tan[a + b*x]])/(3*b*d)","C",1
92,1,69,72,0.3186705,"\int \frac{\csc (a+b x)}{\sqrt{d \tan (a+b x)}} \, dx","Integrate[Csc[a + b*x]/Sqrt[d*Tan[a + b*x]],x]","-\frac{2 \cos (a+b x) \left(2 \tan ^2(a+b x) \sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)+3\right)}{3 b \sqrt{d \tan (a+b x)}}","-\frac{2 \cos (a+b x)}{b \sqrt{d \tan (a+b x)}}-\frac{2 \sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{b \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}",1,"(-2*Cos[a + b*x]*(3 + 2*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2]*Tan[a + b*x]^2))/(3*b*Sqrt[d*Tan[a + b*x]])","C",1
93,1,104,102,0.6830954,"\int \frac{\csc ^3(a+b x)}{\sqrt{d \tan (a+b x)}} \, dx","Integrate[Csc[a + b*x]^3/Sqrt[d*Tan[a + b*x]],x]","\frac{6 (\cos (2 (a+b x))-2) \cot (a+b x) \csc (a+b x) \sqrt{\sec ^2(a+b x)}-8 \tan ^2(a+b x) \sec (a+b x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)}{15 b \sqrt{\sec ^2(a+b x)} \sqrt{d \tan (a+b x)}}","-\frac{4 \cos (a+b x)}{5 b \sqrt{d \tan (a+b x)}}-\frac{2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}-\frac{4 \sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{5 b \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}",1,"(6*(-2 + Cos[2*(a + b*x)])*Cot[a + b*x]*Csc[a + b*x]*Sqrt[Sec[a + b*x]^2] - 8*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sec[a + b*x]*Tan[a + b*x]^2)/(15*b*Sqrt[Sec[a + b*x]^2]*Sqrt[d*Tan[a + b*x]])","C",1
94,1,123,257,0.3498584,"\int \frac{\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Sin[a + b*x]^4/(d*Tan[a + b*x])^(3/2),x]","\frac{\csc (a+b x) \sqrt{d \tan (a+b x)} \left(\cos (a+b x)-2 \cos (3 (a+b x))+\cos (5 (a+b x))-3 \sqrt{\sin (2 (a+b x))} \sin ^{-1}(\cos (a+b x)-\sin (a+b x))-3 \sqrt{\sin (2 (a+b x))} \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{64 b d^2}","-\frac{3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{32 \sqrt{2} b d^{3/2}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{32 \sqrt{2} b d^{3/2}}+\frac{3 \log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{64 \sqrt{2} b d^{3/2}}-\frac{3 \log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{64 \sqrt{2} b d^{3/2}}-\frac{\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}+\frac{3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}",1,"(Csc[a + b*x]*(Cos[a + b*x] - 2*Cos[3*(a + b*x)] + Cos[5*(a + b*x)] - 3*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Sqrt[Sin[2*(a + b*x)]] - 3*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]])*Sqrt[d*Tan[a + b*x]])/(64*b*d^2)","A",1
95,1,105,227,0.2540737,"\int \frac{\sin ^2(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Sin[a + b*x]^2/(d*Tan[a + b*x])^(3/2),x]","-\frac{\sqrt{\sin (2 (a+b x))} \sqrt{d \tan (a+b x)} \left(-2 \sqrt{\sin (2 (a+b x))}+\csc (a+b x) \sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\csc (a+b x) \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{8 b d^2}","-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{4 \sqrt{2} b d^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{4 \sqrt{2} b d^{3/2}}+\frac{\log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b d^{3/2}}-\frac{\log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b d^{3/2}}+\frac{\cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{2 b d^3}",1,"-1/8*((ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Csc[a + b*x] + Csc[a + b*x]*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]] - 2*Sqrt[Sin[2*(a + b*x)]])*Sqrt[Sin[2*(a + b*x)]]*Sqrt[d*Tan[a + b*x]])/(b*d^2)","A",1
96,1,20,20,0.1228168,"\int \frac{\csc ^2(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Csc[a + b*x]^2/(d*Tan[a + b*x])^(3/2),x]","-\frac{2 d}{5 b (d \tan (a+b x))^{5/2}}","-\frac{2 d}{5 b (d \tan (a+b x))^{5/2}}",1,"(-2*d)/(5*b*(d*Tan[a + b*x])^(5/2))","A",1
97,1,42,43,0.0932736,"\int \frac{\csc ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Csc[a + b*x]^4/(d*Tan[a + b*x])^(3/2),x]","\frac{2 \left(-5 \csc ^4(a+b x)+\csc ^2(a+b x)+4\right)}{45 b d \sqrt{d \tan (a+b x)}}","-\frac{2 d^3}{9 b (d \tan (a+b x))^{9/2}}-\frac{2 d}{5 b (d \tan (a+b x))^{5/2}}",1,"(2*(4 + Csc[a + b*x]^2 - 5*Csc[a + b*x]^4))/(45*b*d*Sqrt[d*Tan[a + b*x]])","A",1
98,1,54,65,0.1300632,"\int \frac{\csc ^6(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Csc[a + b*x]^6/(d*Tan[a + b*x])^(3/2),x]","\frac{-90 \csc ^6(a+b x)+10 \csc ^4(a+b x)+16 \csc ^2(a+b x)+64}{585 b d \sqrt{d \tan (a+b x)}}","-\frac{2 d^5}{13 b (d \tan (a+b x))^{13/2}}-\frac{4 d^3}{9 b (d \tan (a+b x))^{9/2}}-\frac{2 d}{5 b (d \tan (a+b x))^{5/2}}",1,"(64 + 16*Csc[a + b*x]^2 + 10*Csc[a + b*x]^4 - 90*Csc[a + b*x]^6)/(585*b*d*Sqrt[d*Tan[a + b*x]])","A",1
99,1,102,112,0.3790611,"\int \frac{\sin ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Sin[a + b*x]^3/(d*Tan[a + b*x])^(3/2),x]","-\frac{\csc (a+b x) \sqrt{d \tan (a+b x)} \left(\sin (4 (a+b x)) \sqrt{\sec ^2(a+b x)}+4 \sqrt[4]{-1} \sqrt{\tan (a+b x)} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)\right)}{24 b d^2 \sqrt{\sec ^2(a+b x)}}","\frac{\sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{12 b d^2}+\frac{\sin ^3(a+b x)}{3 b d \sqrt{d \tan (a+b x)}}-\frac{\sin (a+b x)}{6 b d \sqrt{d \tan (a+b x)}}",1,"-1/24*(Csc[a + b*x]*(Sqrt[Sec[a + b*x]^2]*Sin[4*(a + b*x)] + 4*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Sqrt[Tan[a + b*x]])*Sqrt[d*Tan[a + b*x]])/(b*d^2*Sqrt[Sec[a + b*x]^2])","C",1
100,1,126,79,0.769016,"\int \frac{\sin (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Sin[a + b*x]/(d*Tan[a + b*x])^(3/2),x]","\frac{\cos (2 (a+b x)) \tan ^{\frac{3}{2}}(a+b x) \sec (a+b x) \left(-\sqrt{\tan (a+b x)} \sqrt{\sec ^2(a+b x)}+\sqrt[4]{-1} \sec ^2(a+b x) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)\right)}{b \left(\tan ^2(a+b x)-1\right) \sqrt{\sec ^2(a+b x)} (d \tan (a+b x))^{3/2}}","\frac{\sin (a+b x)}{b d \sqrt{d \tan (a+b x)}}+\frac{\sqrt{\sin (2 a+2 b x)} \sec (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2 b d \sqrt{d \tan (a+b x)}}",1,"(Cos[2*(a + b*x)]*Sec[a + b*x]*((-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Sec[a + b*x]^2 - Sqrt[Sec[a + b*x]^2]*Sqrt[Tan[a + b*x]])*Tan[a + b*x]^(3/2))/(b*Sqrt[Sec[a + b*x]^2]*(d*Tan[a + b*x])^(3/2)*(-1 + Tan[a + b*x]^2))","C",1
101,1,110,82,0.7330496,"\int \frac{\csc (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Csc[a + b*x]/(d*Tan[a + b*x])^(3/2),x]","\frac{2 \cos (2 (a+b x)) \sec (a+b x) \sqrt{\sec ^2(a+b x)} \left(\sqrt{\sec ^2(a+b x)}-\sqrt[4]{-1} \tan ^{\frac{3}{2}}(a+b x) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)\right)}{3 b \left(\tan ^2(a+b x)-1\right) (d \tan (a+b x))^{3/2}}","-\frac{\sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{3 b d^2}-\frac{2 \csc (a+b x)}{3 b d \sqrt{d \tan (a+b x)}}",1,"(2*Cos[2*(a + b*x)]*Sec[a + b*x]*Sqrt[Sec[a + b*x]^2]*(Sqrt[Sec[a + b*x]^2] - (-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Tan[a + b*x]^(3/2)))/(3*b*(d*Tan[a + b*x])^(3/2)*(-1 + Tan[a + b*x]^2))","C",1
102,1,136,112,1.7091111,"\int \frac{\csc ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Csc[a + b*x]^3/(d*Tan[a + b*x])^(3/2),x]","\frac{\csc ^3(a+b x) \left((10 \cos (2 (a+b x))+\cos (4 (a+b x))+1) \sec ^2(a+b x)^{3/2}-8 \sqrt[4]{-1} \cos (2 (a+b x)) \tan ^{\frac{7}{2}}(a+b x) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)\right)}{42 b d \left(\tan ^2(a+b x)-1\right) \sqrt{\sec ^2(a+b x)} \sqrt{d \tan (a+b x)}}","-\frac{2 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{21 b d^2}-\frac{2 \csc ^3(a+b x)}{7 b d \sqrt{d \tan (a+b x)}}+\frac{2 \csc (a+b x)}{21 b d \sqrt{d \tan (a+b x)}}",1,"(Csc[a + b*x]^3*((1 + 10*Cos[2*(a + b*x)] + Cos[4*(a + b*x)])*(Sec[a + b*x]^2)^(3/2) - 8*(-1)^(1/4)*Cos[2*(a + b*x)]*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Tan[a + b*x]^(7/2)))/(42*b*d*Sqrt[Sec[a + b*x]^2]*Sqrt[d*Tan[a + b*x]]*(-1 + Tan[a + b*x]^2))","C",1
103,1,123,257,0.7657158,"\int \frac{\sin ^4(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx","Integrate[Sin[a + b*x]^4/(d*Tan[a + b*x])^(5/2),x]","-\frac{\csc (a+b x) \sqrt{d \tan (a+b x)} \left(\sin (a+b x)+2 \sin (3 (a+b x))+\sin (5 (a+b x))+3 \sqrt{\sin (2 (a+b x))} \sin ^{-1}(\cos (a+b x)-\sin (a+b x))-3 \sqrt{\sin (2 (a+b x))} \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{64 b d^3}","-\frac{3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{32 \sqrt{2} b d^{5/2}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{32 \sqrt{2} b d^{5/2}}-\frac{3 \log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{64 \sqrt{2} b d^{5/2}}+\frac{3 \log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{64 \sqrt{2} b d^{5/2}}-\frac{\cos ^4(a+b x) \sqrt{d \tan (a+b x)}}{4 b d^3}+\frac{\cos ^2(a+b x) \sqrt{d \tan (a+b x)}}{16 b d^3}",1,"-1/64*(Csc[a + b*x]*(Sin[a + b*x] + 3*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Sqrt[Sin[2*(a + b*x)]] - 3*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]] + 2*Sin[3*(a + b*x)] + Sin[5*(a + b*x)])*Sqrt[d*Tan[a + b*x]])/(b*d^3)","A",1
104,1,113,227,0.5496493,"\int \frac{\sin ^2(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx","Integrate[Sin[a + b*x]^2/(d*Tan[a + b*x])^(5/2),x]","\frac{\csc (a+b x) \sqrt{d \tan (a+b x)} \left(\sin (a+b x)+\sin (3 (a+b x))-3 \sqrt{\sin (2 (a+b x))} \sin ^{-1}(\cos (a+b x)-\sin (a+b x))+3 \sqrt{\sin (2 (a+b x))} \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{8 b d^3}","-\frac{3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{4 \sqrt{2} b d^{5/2}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{4 \sqrt{2} b d^{5/2}}-\frac{3 \log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b d^{5/2}}+\frac{3 \log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b d^{5/2}}+\frac{\cos ^2(a+b x) \sqrt{d \tan (a+b x)}}{2 b d^3}",1,"(Csc[a + b*x]*(Sin[a + b*x] - 3*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Sqrt[Sin[2*(a + b*x)]] + 3*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]] + Sin[3*(a + b*x)])*Sqrt[d*Tan[a + b*x]])/(8*b*d^3)","A",1
105,1,20,20,0.1650589,"\int \frac{\csc ^2(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx","Integrate[Csc[a + b*x]^2/(d*Tan[a + b*x])^(5/2),x]","-\frac{2 d}{7 b (d \tan (a+b x))^{7/2}}","-\frac{2 d}{7 b (d \tan (a+b x))^{7/2}}",1,"(-2*d)/(7*b*(d*Tan[a + b*x])^(7/2))","A",1
106,1,50,43,0.183192,"\int \frac{\csc ^4(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx","Integrate[Csc[a + b*x]^4/(d*Tan[a + b*x])^(5/2),x]","\frac{2 (2 \cos (2 (a+b x))-9) \cot ^4(a+b x) \csc ^2(a+b x) \sqrt{d \tan (a+b x)}}{77 b d^3}","-\frac{2 d^3}{11 b (d \tan (a+b x))^{11/2}}-\frac{2 d}{7 b (d \tan (a+b x))^{7/2}}",1,"(2*(-9 + 2*Cos[2*(a + b*x)])*Cot[a + b*x]^4*Csc[a + b*x]^2*Sqrt[d*Tan[a + b*x]])/(77*b*d^3)","A",1
107,1,60,65,0.2445395,"\int \frac{\csc ^6(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx","Integrate[Csc[a + b*x]^6/(d*Tan[a + b*x])^(5/2),x]","\frac{2 (44 \cos (2 (a+b x))-4 \cos (4 (a+b x))-117) \cot ^4(a+b x) \csc ^4(a+b x) \sqrt{d \tan (a+b x)}}{1155 b d^3}","-\frac{2 d^5}{15 b (d \tan (a+b x))^{15/2}}-\frac{4 d^3}{11 b (d \tan (a+b x))^{11/2}}-\frac{2 d}{7 b (d \tan (a+b x))^{7/2}}",1,"(2*(-117 + 44*Cos[2*(a + b*x)] - 4*Cos[4*(a + b*x)])*Cot[a + b*x]^4*Csc[a + b*x]^4*Sqrt[d*Tan[a + b*x]])/(1155*b*d^3)","A",1
108,1,122,144,1.5694584,"\int \frac{\sin ^7(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx","Integrate[Sin[a + b*x]^7/(d*Tan[a + b*x])^(5/2),x]","\frac{\sqrt{d \tan (a+b x)} \left(112 \tan (a+b x) \sec (a+b x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)-(15 \sin (a+b x)+29 \sin (3 (a+b x))+9 \sin (5 (a+b x))-5 \sin (7 (a+b x))) \sqrt{\sec ^2(a+b x)}\right)}{2240 b d^3 \sqrt{\sec ^2(a+b x)}}","\frac{3 \sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{40 b d^2 \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}+\frac{\sin ^7(a+b x)}{7 b d (d \tan (a+b x))^{3/2}}-\frac{3 \sin ^5(a+b x)}{70 b d (d \tan (a+b x))^{3/2}}-\frac{\sin ^3(a+b x)}{20 b d (d \tan (a+b x))^{3/2}}",1,"(Sqrt[d*Tan[a + b*x]]*(-(Sqrt[Sec[a + b*x]^2]*(15*Sin[a + b*x] + 29*Sin[3*(a + b*x)] + 9*Sin[5*(a + b*x)] - 5*Sin[7*(a + b*x)])) + 112*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sec[a + b*x]*Tan[a + b*x]))/(2240*b*d^3*Sqrt[Sec[a + b*x]^2])","C",1
109,1,100,114,1.0917869,"\int \frac{\sin ^5(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx","Integrate[Sin[a + b*x]^5/(d*Tan[a + b*x])^(5/2),x]","\frac{\sqrt{d \tan (a+b x)} \left(8 \tan (a+b x) \sec (a+b x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)-(\sin (3 (a+b x))+\sin (5 (a+b x))) \sqrt{\sec ^2(a+b x)}\right)}{80 b d^3 \sqrt{\sec ^2(a+b x)}}","\frac{3 \sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{20 b d^2 \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}+\frac{\sin ^5(a+b x)}{5 b d (d \tan (a+b x))^{3/2}}-\frac{\sin ^3(a+b x)}{10 b d (d \tan (a+b x))^{3/2}}",1,"(Sqrt[d*Tan[a + b*x]]*(-(Sqrt[Sec[a + b*x]^2]*(Sin[3*(a + b*x)] + Sin[5*(a + b*x)])) + 8*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sec[a + b*x]*Tan[a + b*x]))/(80*b*d^3*Sqrt[Sec[a + b*x]^2])","C",1
110,1,97,84,0.6301238,"\int \frac{\sin ^3(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx","Integrate[Sin[a + b*x]^3/(d*Tan[a + b*x])^(5/2),x]","\frac{\sqrt{d \tan (a+b x)} \left(4 \tan (a+b x) \sec (a+b x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)+(\sin (a+b x)+\sin (3 (a+b x))) \sqrt{\sec ^2(a+b x)}\right)}{12 b d^3 \sqrt{\sec ^2(a+b x)}}","\frac{\sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2 b d^2 \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}+\frac{\sin ^3(a+b x)}{3 b d (d \tan (a+b x))^{3/2}}",1,"(Sqrt[d*Tan[a + b*x]]*(Sqrt[Sec[a + b*x]^2]*(Sin[a + b*x] + Sin[3*(a + b*x)]) + 4*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sec[a + b*x]*Tan[a + b*x]))/(12*b*d^3*Sqrt[Sec[a + b*x]^2])","C",1
111,1,69,78,0.4074502,"\int \frac{\sin (a+b x)}{(d \tan (a+b x))^{5/2}} \, dx","Integrate[Sin[a + b*x]/(d*Tan[a + b*x])^(5/2),x]","-\frac{2 \cos (a+b x) \left(\tan ^2(a+b x) \sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)+1\right)}{b d^2 \sqrt{d \tan (a+b x)}}","-\frac{3 \sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{b d^2 \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}-\frac{2 \sin (a+b x)}{b d (d \tan (a+b x))^{3/2}}",1,"(-2*Cos[a + b*x]*(1 + Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2]*Tan[a + b*x]^2))/(b*d^2*Sqrt[d*Tan[a + b*x]])","C",1
112,1,105,110,1.7863926,"\int \frac{\csc (a+b x)}{(d \tan (a+b x))^{5/2}} \, dx","Integrate[Csc[a + b*x]/(d*Tan[a + b*x])^(5/2),x]","\frac{2 \sin (a+b x) \sqrt{d \tan (a+b x)} \left(2 \sec ^2(a+b x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)-\left(\csc ^4(a+b x)-4 \csc ^2(a+b x)+3\right) \sqrt{\sec ^2(a+b x)}\right)}{5 b d^3 \sqrt{\sec ^2(a+b x)}}","\frac{6 \cos (a+b x)}{5 b d^2 \sqrt{d \tan (a+b x)}}+\frac{6 \sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{5 b d^2 \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}-\frac{2 \csc (a+b x)}{5 b d (d \tan (a+b x))^{3/2}}",1,"(2*(2*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sec[a + b*x]^2 - (3 - 4*Csc[a + b*x]^2 + Csc[a + b*x]^4)*Sqrt[Sec[a + b*x]^2])*Sin[a + b*x]*Sqrt[d*Tan[a + b*x]])/(5*b*d^3*Sqrt[Sec[a + b*x]^2])","C",1
113,1,116,140,0.8387106,"\int \frac{\csc ^3(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx","Integrate[Csc[a + b*x]^3/(d*Tan[a + b*x])^(5/2),x]","\frac{2 \sin (a+b x) \sqrt{d \tan (a+b x)} \left(4 \sec ^2(a+b x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)+\left(-5 \csc ^6(a+b x)+8 \csc ^4(a+b x)+3 \csc ^2(a+b x)-6\right) \sqrt{\sec ^2(a+b x)}\right)}{45 b d^3 \sqrt{\sec ^2(a+b x)}}","\frac{4 \cos (a+b x)}{15 b d^2 \sqrt{d \tan (a+b x)}}+\frac{4 \sin (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{15 b d^2 \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}-\frac{2 \csc ^3(a+b x)}{9 b d (d \tan (a+b x))^{3/2}}+\frac{2 \csc (a+b x)}{15 b d (d \tan (a+b x))^{3/2}}",1,"(2*(4*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sec[a + b*x]^2 + (-6 + 3*Csc[a + b*x]^2 + 8*Csc[a + b*x]^4 - 5*Csc[a + b*x]^6)*Sqrt[Sec[a + b*x]^2])*Sin[a + b*x]*Sqrt[d*Tan[a + b*x]])/(45*b*d^3*Sqrt[Sec[a + b*x]^2])","C",1
114,1,51,68,0.1952688,"\int (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)} \, dx","Integrate[(a*Sin[e + f*x])^(5/2)*Sqrt[b*Tan[e + f*x]],x]","-\frac{a^2 \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)} (\sin (2 (e+f x))+8 \cot (e+f x))}{5 f}","-\frac{8 a^2 b \sqrt{a \sin (e+f x)}}{5 f \sqrt{b \tan (e+f x)}}-\frac{2 b (a \sin (e+f x))^{5/2}}{5 f \sqrt{b \tan (e+f x)}}",1,"-1/5*(a^2*Sqrt[a*Sin[e + f*x]]*(8*Cot[e + f*x] + Sin[2*(e + f*x)])*Sqrt[b*Tan[e + f*x]])/f","A",1
115,1,80,88,0.3183533,"\int (a \sin (e+f x))^{3/2} \sqrt{b \tan (e+f x)} \, dx","Integrate[(a*Sin[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]],x]","-\frac{2 a b \sqrt{a \sin (e+f x)} \left(\sin (e+f x) \sqrt[4]{\cos ^2(e+f x)}-2 F\left(\left.\frac{1}{2} \sin ^{-1}(\sin (e+f x))\right|2\right)\right)}{3 f \sqrt[4]{\cos ^2(e+f x)} \sqrt{b \tan (e+f x)}}","\frac{4 a^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \tan (e+f x)}}{3 f \sqrt{a \sin (e+f x)}}-\frac{2 b (a \sin (e+f x))^{3/2}}{3 f \sqrt{b \tan (e+f x)}}",1,"(-2*a*b*Sqrt[a*Sin[e + f*x]]*(-2*EllipticF[ArcSin[Sin[e + f*x]]/2, 2] + (Cos[e + f*x]^2)^(1/4)*Sin[e + f*x]))/(3*f*(Cos[e + f*x]^2)^(1/4)*Sqrt[b*Tan[e + f*x]])","A",1
116,1,30,30,0.1283774,"\int \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)} \, dx","Integrate[Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]],x]","-\frac{2 b \sqrt{a \sin (e+f x)}}{f \sqrt{b \tan (e+f x)}}","-\frac{2 b \sqrt{a \sin (e+f x)}}{f \sqrt{b \tan (e+f x)}}",1,"(-2*b*Sqrt[a*Sin[e + f*x]])/(f*Sqrt[b*Tan[e + f*x]])","A",1
117,1,60,50,0.1183585,"\int \frac{\sqrt{b \tan (e+f x)}}{\sqrt{a \sin (e+f x)}} \, dx","Integrate[Sqrt[b*Tan[e + f*x]]/Sqrt[a*Sin[e + f*x]],x]","\frac{2 \cos (e+f x) \sqrt{b \tan (e+f x)} F\left(\left.\frac{1}{2} \sin ^{-1}(\sin (e+f x))\right|2\right)}{f \sqrt[4]{\cos ^2(e+f x)} \sqrt{a \sin (e+f x)}}","\frac{2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \tan (e+f x)}}{f \sqrt{a \sin (e+f x)}}",1,"(2*Cos[e + f*x]*EllipticF[ArcSin[Sin[e + f*x]]/2, 2]*Sqrt[b*Tan[e + f*x]])/(f*(Cos[e + f*x]^2)^(1/4)*Sqrt[a*Sin[e + f*x]])","A",1
118,1,72,107,0.2617904,"\int \frac{\sqrt{b \tan (e+f x)}}{(a \sin (e+f x))^{3/2}} \, dx","Integrate[Sqrt[b*Tan[e + f*x]]/(a*Sin[e + f*x])^(3/2),x]","-\frac{b \sqrt{a \sin (e+f x)} \left(\tan ^{-1}\left(\sqrt[4]{\cos ^2(e+f x)}\right)+\tanh ^{-1}\left(\sqrt[4]{\cos ^2(e+f x)}\right)\right)}{a^2 f \sqrt[4]{\cos ^2(e+f x)} \sqrt{b \tan (e+f x)}}","-\frac{\sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)} \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)}{a f \sqrt{a \sin (e+f x)}}-\frac{\sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)} \tanh ^{-1}\left(\sqrt{\cos (e+f x)}\right)}{a f \sqrt{a \sin (e+f x)}}",1,"-((b*(ArcTan[(Cos[e + f*x]^2)^(1/4)] + ArcTanh[(Cos[e + f*x]^2)^(1/4)])*Sqrt[a*Sin[e + f*x]])/(a^2*f*(Cos[e + f*x]^2)^(1/4)*Sqrt[b*Tan[e + f*x]]))","A",1
119,1,79,86,0.2468781,"\int \frac{\sqrt{b \tan (e+f x)}}{(a \sin (e+f x))^{5/2}} \, dx","Integrate[Sqrt[b*Tan[e + f*x]]/(a*Sin[e + f*x])^(5/2),x]","\frac{b \left(\sin (e+f x) F\left(\left.\frac{1}{2} \sin ^{-1}(\sin (e+f x))\right|2\right)-\sqrt[4]{\cos ^2(e+f x)}\right)}{a^2 f \sqrt[4]{\cos ^2(e+f x)} \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}","\frac{\sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \tan (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)}}-\frac{b}{a^2 f \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}",1,"(b*(-(Cos[e + f*x]^2)^(1/4) + EllipticF[ArcSin[Sin[e + f*x]]/2, 2]*Sin[e + f*x]))/(a^2*f*(Cos[e + f*x]^2)^(1/4)*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]])","A",1
120,1,99,126,0.3336599,"\int (a \sin (e+f x))^{5/2} (b \tan (e+f x))^{3/2} \, dx","Integrate[(a*Sin[e + f*x])^(5/2)*(b*Tan[e + f*x])^(3/2),x]","\frac{a^2 b \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)} \left(\cos ^2(e+f x)^{3/4} (\cos (2 (e+f x))+11)-12 \cos ^2(e+f x) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)\right)}{5 f \cos ^2(e+f x)^{3/4}}","-\frac{24 a^2 b^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{a \sin (e+f x)}}{5 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}+\frac{12 a^2 b \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}{5 f}-\frac{2 b (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}{5 f}",1,"(a^2*b*((Cos[e + f*x]^2)^(3/4)*(11 + Cos[2*(e + f*x)]) - 12*Cos[e + f*x]^2*Hypergeometric2F1[1/4, 1/2, 3/2, Sin[e + f*x]^2])*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]])/(5*f*(Cos[e + f*x]^2)^(3/4))","C",1
121,1,45,68,0.167797,"\int (a \sin (e+f x))^{3/2} (b \tan (e+f x))^{3/2} \, dx","Integrate[(a*Sin[e + f*x])^(3/2)*(b*Tan[e + f*x])^(3/2),x]","\frac{a^2 b (\cos (2 (e+f x))+7) \sqrt{b \tan (e+f x)}}{3 f \sqrt{a \sin (e+f x)}}","\frac{8 a^2 b \sqrt{b \tan (e+f x)}}{3 f \sqrt{a \sin (e+f x)}}-\frac{2 b (a \sin (e+f x))^{3/2} \sqrt{b \tan (e+f x)}}{3 f}",1,"(a^2*b*(7 + Cos[2*(e + f*x)])*Sqrt[b*Tan[e + f*x]])/(3*f*Sqrt[a*Sin[e + f*x]])","A",1
122,1,83,84,0.2051787,"\int \sqrt{a \sin (e+f x)} (b \tan (e+f x))^{3/2} \, dx","Integrate[Sqrt[a*Sin[e + f*x]]*(b*Tan[e + f*x])^(3/2),x]","\frac{2 b \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)} \left(\cos ^2(e+f x)^{3/4}-\cos ^2(e+f x) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)\right)}{f \cos ^2(e+f x)^{3/4}}","\frac{2 b \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}{f}-\frac{4 b^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{a \sin (e+f x)}}{f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}",1,"(2*b*((Cos[e + f*x]^2)^(3/4) - Cos[e + f*x]^2*Hypergeometric2F1[1/4, 1/2, 3/2, Sin[e + f*x]^2])*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]])/(f*(Cos[e + f*x]^2)^(3/4))","C",1
123,1,30,30,0.0654608,"\int \frac{(b \tan (e+f x))^{3/2}}{\sqrt{a \sin (e+f x)}} \, dx","Integrate[(b*Tan[e + f*x])^(3/2)/Sqrt[a*Sin[e + f*x]],x]","\frac{2 b \sqrt{b \tan (e+f x)}}{f \sqrt{a \sin (e+f x)}}","\frac{2 b \sqrt{b \tan (e+f x)}}{f \sqrt{a \sin (e+f x)}}",1,"(2*b*Sqrt[b*Tan[e + f*x]])/(f*Sqrt[a*Sin[e + f*x]])","A",1
124,1,92,90,0.2675394,"\int \frac{(b \tan (e+f x))^{3/2}}{(a \sin (e+f x))^{3/2}} \, dx","Integrate[(b*Tan[e + f*x])^(3/2)/(a*Sin[e + f*x])^(3/2),x]","\frac{(b \tan (e+f x))^{3/2} \left(2 \cos (e+f x) \cos ^2(e+f x)^{3/4}-\cos ^3(e+f x) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)\right)}{a f \cos ^2(e+f x)^{3/4} \sqrt{a \sin (e+f x)}}","\frac{2 b \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}{a^2 f}-\frac{2 b^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{a \sin (e+f x)}}{a^2 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}",1,"((2*Cos[e + f*x]*(Cos[e + f*x]^2)^(3/4) - Cos[e + f*x]^3*Hypergeometric2F1[1/4, 1/2, 3/2, Sin[e + f*x]^2])*(b*Tan[e + f*x])^(3/2))/(a*f*(Cos[e + f*x]^2)^(3/4)*Sqrt[a*Sin[e + f*x]])","C",1
125,1,104,145,0.3643642,"\int \frac{(b \tan (e+f x))^{3/2}}{(a \sin (e+f x))^{5/2}} \, dx","Integrate[(b*Tan[e + f*x])^(3/2)/(a*Sin[e + f*x])^(5/2),x]","\frac{b \sqrt{b \tan (e+f x)} \left(2 \cos ^2(e+f x)^{3/4}+\cos ^2(e+f x) \tan ^{-1}\left(\sqrt[4]{\cos ^2(e+f x)}\right)-\cos ^2(e+f x) \tanh ^{-1}\left(\sqrt[4]{\cos ^2(e+f x)}\right)\right)}{a^2 f \cos ^2(e+f x)^{3/4} \sqrt{a \sin (e+f x)}}","\frac{b^2 \sqrt{a \sin (e+f x)} \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)}{a^3 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{b^2 \sqrt{a \sin (e+f x)} \tanh ^{-1}\left(\sqrt{\cos (e+f x)}\right)}{a^3 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}+\frac{2 b \sqrt{b \tan (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)}}",1,"(b*(ArcTan[(Cos[e + f*x]^2)^(1/4)]*Cos[e + f*x]^2 - ArcTanh[(Cos[e + f*x]^2)^(1/4)]*Cos[e + f*x]^2 + 2*(Cos[e + f*x]^2)^(3/4))*Sqrt[b*Tan[e + f*x]])/(a^2*f*(Cos[e + f*x]^2)^(3/4)*Sqrt[a*Sin[e + f*x]])","A",1
126,1,100,123,0.5246516,"\int \frac{(a \sin (e+f x))^{9/2}}{\sqrt{b \tan (e+f x)}} \, dx","Integrate[(a*Sin[e + f*x])^(9/2)/Sqrt[b*Tan[e + f*x]],x]","\frac{a^4 \sin (2 (e+f x)) \sqrt{a \sin (e+f x)} \left(12 \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)+\cos ^2(e+f x)^{3/4} (5 \cos (2 (e+f x))-17)\right)}{90 f \cos ^2(e+f x)^{3/4} \sqrt{b \tan (e+f x)}}","\frac{8 a^4 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{a \sin (e+f x)}}{15 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{4 a^2 b (a \sin (e+f x))^{5/2}}{15 f (b \tan (e+f x))^{3/2}}-\frac{2 b (a \sin (e+f x))^{9/2}}{9 f (b \tan (e+f x))^{3/2}}",1,"(a^4*((Cos[e + f*x]^2)^(3/4)*(-17 + 5*Cos[2*(e + f*x)]) + 12*Hypergeometric2F1[1/4, 1/2, 3/2, Sin[e + f*x]^2])*Sqrt[a*Sin[e + f*x]]*Sin[2*(e + f*x)])/(90*f*(Cos[e + f*x]^2)^(3/4)*Sqrt[b*Tan[e + f*x]])","C",1
127,1,52,68,0.1640239,"\int \frac{(a \sin (e+f x))^{7/2}}{\sqrt{b \tan (e+f x)}} \, dx","Integrate[(a*Sin[e + f*x])^(7/2)/Sqrt[b*Tan[e + f*x]],x]","\frac{a^3 \cos (e+f x) (3 \cos (2 (e+f x))-11) \sqrt{a \sin (e+f x)}}{21 f \sqrt{b \tan (e+f x)}}","-\frac{8 a^2 b (a \sin (e+f x))^{3/2}}{21 f (b \tan (e+f x))^{3/2}}-\frac{2 b (a \sin (e+f x))^{7/2}}{7 f (b \tan (e+f x))^{3/2}}",1,"(a^3*Cos[e + f*x]*(-11 + 3*Cos[2*(e + f*x)])*Sqrt[a*Sin[e + f*x]])/(21*f*Sqrt[b*Tan[e + f*x]])","A",1
128,1,87,88,0.2377289,"\int \frac{(a \sin (e+f x))^{5/2}}{\sqrt{b \tan (e+f x)}} \, dx","Integrate[(a*Sin[e + f*x])^(5/2)/Sqrt[b*Tan[e + f*x]],x]","-\frac{a^2 \sin (2 (e+f x)) \sqrt{a \sin (e+f x)} \left(\cos ^2(e+f x)^{3/4}-\, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)\right)}{5 f \cos ^2(e+f x)^{3/4} \sqrt{b \tan (e+f x)}}","\frac{4 a^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{a \sin (e+f x)}}{5 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{2 b (a \sin (e+f x))^{5/2}}{5 f (b \tan (e+f x))^{3/2}}",1,"-1/5*(a^2*((Cos[e + f*x]^2)^(3/4) - Hypergeometric2F1[1/4, 1/2, 3/2, Sin[e + f*x]^2])*Sqrt[a*Sin[e + f*x]]*Sin[2*(e + f*x)])/(f*(Cos[e + f*x]^2)^(3/4)*Sqrt[b*Tan[e + f*x]])","C",1
129,1,32,32,0.13751,"\int \frac{(a \sin (e+f x))^{3/2}}{\sqrt{b \tan (e+f x)}} \, dx","Integrate[(a*Sin[e + f*x])^(3/2)/Sqrt[b*Tan[e + f*x]],x]","-\frac{2 b (a \sin (e+f x))^{3/2}}{3 f (b \tan (e+f x))^{3/2}}","-\frac{2 b (a \sin (e+f x))^{3/2}}{3 f (b \tan (e+f x))^{3/2}}",1,"(-2*b*(a*Sin[e + f*x])^(3/2))/(3*f*(b*Tan[e + f*x])^(3/2))","A",1
130,1,69,50,0.1534284,"\int \frac{\sqrt{a \sin (e+f x)}}{\sqrt{b \tan (e+f x)}} \, dx","Integrate[Sqrt[a*Sin[e + f*x]]/Sqrt[b*Tan[e + f*x]],x]","\frac{\sin (2 (e+f x)) \sqrt{a \sin (e+f x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)}{2 f \cos ^2(e+f x)^{3/4} \sqrt{b \tan (e+f x)}}","\frac{2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{a \sin (e+f x)}}{f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}",1,"(Hypergeometric2F1[1/4, 1/2, 3/2, Sin[e + f*x]^2]*Sqrt[a*Sin[e + f*x]]*Sin[2*(e + f*x)])/(2*f*(Cos[e + f*x]^2)^(3/4)*Sqrt[b*Tan[e + f*x]])","C",1
131,1,80,106,0.1463099,"\int \frac{1}{\sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}} \, dx","Integrate[1/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]]),x]","\frac{\sin (2 (e+f x)) \left(\tan ^{-1}\left(\sqrt[4]{\cos ^2(e+f x)}\right)-\tanh ^{-1}\left(\sqrt[4]{\cos ^2(e+f x)}\right)\right)}{2 f \cos ^2(e+f x)^{3/4} \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}","\frac{\sqrt{a \sin (e+f x)} \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)}{a f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{\sqrt{a \sin (e+f x)} \tanh ^{-1}\left(\sqrt{\cos (e+f x)}\right)}{a f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}",1,"((ArcTan[(Cos[e + f*x]^2)^(1/4)] - ArcTanh[(Cos[e + f*x]^2)^(1/4)])*Sin[2*(e + f*x)])/(2*f*(Cos[e + f*x]^2)^(3/4)*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]])","A",1
132,1,89,87,0.3462221,"\int \frac{1}{(a \sin (e+f x))^{3/2} \sqrt{b \tan (e+f x)}} \, dx","Integrate[1/((a*Sin[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]]),x]","-\frac{b \sqrt{a \sin (e+f x)} \left(\sin ^2(e+f x) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)+2 \cos ^2(e+f x)^{3/4}\right)}{2 a^2 f \cos ^2(e+f x)^{3/4} (b \tan (e+f x))^{3/2}}","-\frac{b \sqrt{a \sin (e+f x)}}{a^2 f (b \tan (e+f x))^{3/2}}-\frac{E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{a \sin (e+f x)}}{a^2 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}",1,"-1/2*(b*Sqrt[a*Sin[e + f*x]]*(2*(Cos[e + f*x]^2)^(3/4) + Hypergeometric2F1[1/4, 1/2, 3/2, Sin[e + f*x]^2]*Sin[e + f*x]^2))/(a^2*f*(Cos[e + f*x]^2)^(3/4)*(b*Tan[e + f*x])^(3/2))","C",1
133,1,112,146,0.6283953,"\int \frac{1}{(a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}} \, dx","Integrate[1/((a*Sin[e + f*x])^(5/2)*Sqrt[b*Tan[e + f*x]]),x]","\frac{-4 \cos ^2(e+f x)^{3/4} \cot (e+f x)+\sin (2 (e+f x)) \tan ^{-1}\left(\sqrt[4]{\cos ^2(e+f x)}\right)-\sin (2 (e+f x)) \tanh ^{-1}\left(\sqrt[4]{\cos ^2(e+f x)}\right)}{8 a^2 f \cos ^2(e+f x)^{3/4} \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}","\frac{\sqrt{a \sin (e+f x)} \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)}{4 a^3 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{\sqrt{a \sin (e+f x)} \tanh ^{-1}\left(\sqrt{\cos (e+f x)}\right)}{4 a^3 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{b}{2 a^2 f \sqrt{a \sin (e+f x)} (b \tan (e+f x))^{3/2}}",1,"(-4*(Cos[e + f*x]^2)^(3/4)*Cot[e + f*x] + ArcTan[(Cos[e + f*x]^2)^(1/4)]*Sin[2*(e + f*x)] - ArcTanh[(Cos[e + f*x]^2)^(1/4)]*Sin[2*(e + f*x)])/(8*a^2*f*(Cos[e + f*x]^2)^(3/4)*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]])","A",1
134,1,67,146,0.4295362,"\int \frac{(a \sin (e+f x))^{13/2}}{(b \tan (e+f x))^{3/2}} \, dx","Integrate[(a*Sin[e + f*x])^(13/2)/(b*Tan[e + f*x])^(3/2),x]","\frac{a^6 \cos ^2(e+f x) (340 \cos (2 (e+f x))-45 \cos (4 (e+f x))-551) \sqrt{a \sin (e+f x)}}{2340 b f \sqrt{b \tan (e+f x)}}","-\frac{64 a^6 \sqrt{a \sin (e+f x)}}{585 b f \sqrt{b \tan (e+f x)}}-\frac{16 a^4 (a \sin (e+f x))^{5/2}}{585 b f \sqrt{b \tan (e+f x)}}-\frac{2 a^2 (a \sin (e+f x))^{9/2}}{117 b f \sqrt{b \tan (e+f x)}}+\frac{2 (a \sin (e+f x))^{13/2}}{13 b f \sqrt{b \tan (e+f x)}}",1,"(a^6*Cos[e + f*x]^2*(-551 + 340*Cos[2*(e + f*x)] - 45*Cos[4*(e + f*x)])*Sqrt[a*Sin[e + f*x]])/(2340*b*f*Sqrt[b*Tan[e + f*x]])","A",1
135,1,57,109,0.2205886,"\int \frac{(a \sin (e+f x))^{9/2}}{(b \tan (e+f x))^{3/2}} \, dx","Integrate[(a*Sin[e + f*x])^(9/2)/(b*Tan[e + f*x])^(3/2),x]","\frac{a^4 \cos ^2(e+f x) (5 \cos (2 (e+f x))-13) \sqrt{a \sin (e+f x)}}{45 b f \sqrt{b \tan (e+f x)}}","-\frac{8 a^4 \sqrt{a \sin (e+f x)}}{45 b f \sqrt{b \tan (e+f x)}}-\frac{2 a^2 (a \sin (e+f x))^{5/2}}{45 b f \sqrt{b \tan (e+f x)}}+\frac{2 (a \sin (e+f x))^{9/2}}{9 b f \sqrt{b \tan (e+f x)}}",1,"(a^4*Cos[e + f*x]^2*(-13 + 5*Cos[2*(e + f*x)])*Sqrt[a*Sin[e + f*x]])/(45*b*f*Sqrt[b*Tan[e + f*x]])","A",1
136,1,45,32,0.136016,"\int \frac{(a \sin (e+f x))^{5/2}}{(b \tan (e+f x))^{3/2}} \, dx","Integrate[(a*Sin[e + f*x])^(5/2)/(b*Tan[e + f*x])^(3/2),x]","-\frac{2 a^2 \cos ^2(e+f x) \sqrt{a \sin (e+f x)}}{5 b f \sqrt{b \tan (e+f x)}}","-\frac{2 b (a \sin (e+f x))^{5/2}}{5 f (b \tan (e+f x))^{5/2}}",1,"(-2*a^2*Cos[e + f*x]^2*Sqrt[a*Sin[e + f*x]])/(5*b*f*Sqrt[b*Tan[e + f*x]])","A",1
137,1,88,141,0.3389261,"\int \frac{\sqrt{a \sin (e+f x)}}{(b \tan (e+f x))^{3/2}} \, dx","Integrate[Sqrt[a*Sin[e + f*x]]/(b*Tan[e + f*x])^(3/2),x]","\frac{\sqrt{a \sin (e+f x)} \left(2 \sqrt[4]{\cos ^2(e+f x)}-\tan ^{-1}\left(\sqrt[4]{\cos ^2(e+f x)}\right)-\tanh ^{-1}\left(\sqrt[4]{\cos ^2(e+f x)}\right)\right)}{b f \sqrt[4]{\cos ^2(e+f x)} \sqrt{b \tan (e+f x)}}","-\frac{a \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)} \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)}{b^2 f \sqrt{a \sin (e+f x)}}-\frac{a \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)} \tanh ^{-1}\left(\sqrt{\cos (e+f x)}\right)}{b^2 f \sqrt{a \sin (e+f x)}}+\frac{2 \sqrt{a \sin (e+f x)}}{b f \sqrt{b \tan (e+f x)}}",1,"((-ArcTan[(Cos[e + f*x]^2)^(1/4)] - ArcTanh[(Cos[e + f*x]^2)^(1/4)] + 2*(Cos[e + f*x]^2)^(1/4))*Sqrt[a*Sin[e + f*x]])/(b*f*(Cos[e + f*x]^2)^(1/4)*Sqrt[b*Tan[e + f*x]])","A",1
138,1,103,151,0.3459263,"\int \frac{1}{(a \sin (e+f x))^{3/2} (b \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a*Sin[e + f*x])^(3/2)*(b*Tan[e + f*x])^(3/2)),x]","\frac{\sin ^2(e+f x) \left(\tan ^{-1}\left(\sqrt[4]{\cos ^2(e+f x)}\right)-2 \sqrt[4]{\cos ^2(e+f x)} \csc ^2(e+f x)+\tanh ^{-1}\left(\sqrt[4]{\cos ^2(e+f x)}\right)\right)}{4 b f \sqrt[4]{\cos ^2(e+f x)} (a \sin (e+f x))^{3/2} \sqrt{b \tan (e+f x)}}","\frac{\sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)} \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)}{4 a b^2 f \sqrt{a \sin (e+f x)}}+\frac{\sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)} \tanh ^{-1}\left(\sqrt{\cos (e+f x)}\right)}{4 a b^2 f \sqrt{a \sin (e+f x)}}-\frac{1}{2 b f (a \sin (e+f x))^{3/2} \sqrt{b \tan (e+f x)}}",1,"((ArcTan[(Cos[e + f*x]^2)^(1/4)] + ArcTanh[(Cos[e + f*x]^2)^(1/4)] - 2*(Cos[e + f*x]^2)^(1/4)*Csc[e + f*x]^2)*Sin[e + f*x]^2)/(4*b*f*(Cos[e + f*x]^2)^(1/4)*(a*Sin[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]])","A",1
139,1,118,167,0.7672004,"\int \frac{(a \sin (e+f x))^{11/2}}{(b \tan (e+f x))^{3/2}} \, dx","Integrate[(a*Sin[e + f*x])^(11/2)/(b*Tan[e + f*x])^(3/2),x]","\frac{a^5 \tan ^2(e+f x) \sqrt{a \sin (e+f x)} \left(\sqrt[4]{\cos ^2(e+f x)} (-22 \cos (e+f x)-17 \cos (3 (e+f x))+7 \cos (5 (e+f x)))+64 \cot (e+f x) F\left(\left.\frac{1}{2} \sin ^{-1}(\sin (e+f x))\right|2\right)\right)}{616 f \sqrt[4]{\cos ^2(e+f x)} (b \tan (e+f x))^{3/2}}","\frac{8 a^6 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \tan (e+f x)}}{77 b^2 f \sqrt{a \sin (e+f x)}}-\frac{4 a^4 (a \sin (e+f x))^{3/2}}{77 b f \sqrt{b \tan (e+f x)}}-\frac{2 a^2 (a \sin (e+f x))^{7/2}}{77 b f \sqrt{b \tan (e+f x)}}+\frac{2 (a \sin (e+f x))^{11/2}}{11 b f \sqrt{b \tan (e+f x)}}",1,"(a^5*((Cos[e + f*x]^2)^(1/4)*(-22*Cos[e + f*x] - 17*Cos[3*(e + f*x)] + 7*Cos[5*(e + f*x)]) + 64*Cot[e + f*x]*EllipticF[ArcSin[Sin[e + f*x]]/2, 2])*Sqrt[a*Sin[e + f*x]]*Tan[e + f*x]^2)/(616*f*(Cos[e + f*x]^2)^(1/4)*(b*Tan[e + f*x])^(3/2))","A",1
140,1,97,130,0.3687568,"\int \frac{(a \sin (e+f x))^{7/2}}{(b \tan (e+f x))^{3/2}} \, dx","Integrate[(a*Sin[e + f*x])^(7/2)/(b*Tan[e + f*x])^(3/2),x]","\frac{a^3 \sqrt{a \sin (e+f x)} \left((5 \sin (e+f x)-3 \sin (3 (e+f x))) \sqrt[4]{\cos ^2(e+f x)}+8 F\left(\left.\frac{1}{2} \sin ^{-1}(\sin (e+f x))\right|2\right)\right)}{42 b f \sqrt[4]{\cos ^2(e+f x)} \sqrt{b \tan (e+f x)}}","\frac{4 a^4 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \tan (e+f x)}}{21 b^2 f \sqrt{a \sin (e+f x)}}-\frac{2 a^2 (a \sin (e+f x))^{3/2}}{21 b f \sqrt{b \tan (e+f x)}}+\frac{2 (a \sin (e+f x))^{7/2}}{7 b f \sqrt{b \tan (e+f x)}}",1,"(a^3*Sqrt[a*Sin[e + f*x]]*(8*EllipticF[ArcSin[Sin[e + f*x]]/2, 2] + (Cos[e + f*x]^2)^(1/4)*(5*Sin[e + f*x] - 3*Sin[3*(e + f*x)])))/(42*b*f*(Cos[e + f*x]^2)^(1/4)*Sqrt[b*Tan[e + f*x]])","A",1
141,1,80,93,0.2267782,"\int \frac{(a \sin (e+f x))^{3/2}}{(b \tan (e+f x))^{3/2}} \, dx","Integrate[(a*Sin[e + f*x])^(3/2)/(b*Tan[e + f*x])^(3/2),x]","\frac{2 a \sqrt{a \sin (e+f x)} \left(\sin (e+f x) \sqrt[4]{\cos ^2(e+f x)}+F\left(\left.\frac{1}{2} \sin ^{-1}(\sin (e+f x))\right|2\right)\right)}{3 b f \sqrt[4]{\cos ^2(e+f x)} \sqrt{b \tan (e+f x)}}","\frac{2 a^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \tan (e+f x)}}{3 b^2 f \sqrt{a \sin (e+f x)}}+\frac{2 (a \sin (e+f x))^{3/2}}{3 b f \sqrt{b \tan (e+f x)}}",1,"(2*a*Sqrt[a*Sin[e + f*x]]*(EllipticF[ArcSin[Sin[e + f*x]]/2, 2] + (Cos[e + f*x]^2)^(1/4)*Sin[e + f*x]))/(3*b*f*(Cos[e + f*x]^2)^(1/4)*Sqrt[b*Tan[e + f*x]])","A",1
142,1,79,86,0.1905759,"\int \frac{1}{\sqrt{a \sin (e+f x)} (b \tan (e+f x))^{3/2}} \, dx","Integrate[1/(Sqrt[a*Sin[e + f*x]]*(b*Tan[e + f*x])^(3/2)),x]","\frac{\sin (e+f x) \left(-F\left(\left.\frac{1}{2} \sin ^{-1}(\sin (e+f x))\right|2\right)\right)-\sqrt[4]{\cos ^2(e+f x)}}{b f \sqrt[4]{\cos ^2(e+f x)} \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}","-\frac{\sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \tan (e+f x)}}{b^2 f \sqrt{a \sin (e+f x)}}-\frac{1}{b f \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}",1,"(-(Cos[e + f*x]^2)^(1/4) - EllipticF[ArcSin[Sin[e + f*x]]/2, 2]*Sin[e + f*x])/(b*f*(Cos[e + f*x]^2)^(1/4)*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]])","A",1
143,1,96,130,0.3691345,"\int \frac{1}{(a \sin (e+f x))^{5/2} (b \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a*Sin[e + f*x])^(5/2)*(b*Tan[e + f*x])^(3/2)),x]","\frac{\sqrt[4]{\cos ^2(e+f x)} \left(1-2 \csc ^2(e+f x)\right)-\sin (e+f x) F\left(\left.\frac{1}{2} \sin ^{-1}(\sin (e+f x))\right|2\right)}{6 a^2 b f \sqrt[4]{\cos ^2(e+f x)} \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}","-\frac{\sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \tan (e+f x)}}{6 a^2 b^2 f \sqrt{a \sin (e+f x)}}+\frac{1}{6 a^2 b f \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{1}{3 b f (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}",1,"((Cos[e + f*x]^2)^(1/4)*(1 - 2*Csc[e + f*x]^2) - EllipticF[ArcSin[Sin[e + f*x]]/2, 2]*Sin[e + f*x])/(6*a^2*b*f*(Cos[e + f*x]^2)^(1/4)*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]])","A",1
144,1,106,167,0.405044,"\int \frac{1}{(a \sin (e+f x))^{9/2} (b \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a*Sin[e + f*x])^(9/2)*(b*Tan[e + f*x])^(3/2)),x]","\frac{\sqrt[4]{\cos ^2(e+f x)} \left(-12 \csc ^4(e+f x)+2 \csc ^2(e+f x)+5\right)-5 \sin (e+f x) F\left(\left.\frac{1}{2} \sin ^{-1}(\sin (e+f x))\right|2\right)}{60 a^4 b f \sqrt[4]{\cos ^2(e+f x)} \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}","-\frac{\sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \tan (e+f x)}}{12 a^4 b^2 f \sqrt{a \sin (e+f x)}}+\frac{1}{12 a^4 b f \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}+\frac{1}{30 a^2 b f (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}-\frac{1}{5 b f (a \sin (e+f x))^{9/2} \sqrt{b \tan (e+f x)}}",1,"((Cos[e + f*x]^2)^(1/4)*(5 + 2*Csc[e + f*x]^2 - 12*Csc[e + f*x]^4) - 5*EllipticF[ArcSin[Sin[e + f*x]]/2, 2]*Sin[e + f*x])/(60*a^4*b*f*(Cos[e + f*x]^2)^(1/4)*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]])","A",1
145,1,69,64,0.3983595,"\int (b \sin (e+f x))^{4/3} \sqrt{d \tan (e+f x)} \, dx","Integrate[(b*Sin[e + f*x])^(4/3)*Sqrt[d*Tan[e + f*x]],x]","\frac{3 \sin (2 (e+f x)) (b \sin (e+f x))^{4/3} \sqrt{d \tan (e+f x)} \, _2F_1\left(\frac{3}{4},\frac{17}{12};\frac{29}{12};\sin ^2(e+f x)\right)}{17 f \sqrt[4]{\cos ^2(e+f x)}}","\frac{6 \cos ^2(e+f x)^{3/4} (b \sin (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, _2F_1\left(\frac{3}{4},\frac{17}{12};\frac{29}{12};\sin ^2(e+f x)\right)}{17 d f}",1,"(3*Hypergeometric2F1[3/4, 17/12, 29/12, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(4/3)*Sin[2*(e + f*x)]*Sqrt[d*Tan[e + f*x]])/(17*f*(Cos[e + f*x]^2)^(1/4))","A",1
146,1,69,64,0.3511795,"\int \sqrt[3]{b \sin (e+f x)} \sqrt{d \tan (e+f x)} \, dx","Integrate[(b*Sin[e + f*x])^(1/3)*Sqrt[d*Tan[e + f*x]],x]","\frac{3 \sin (2 (e+f x)) \sqrt[3]{b \sin (e+f x)} \sqrt{d \tan (e+f x)} \, _2F_1\left(\frac{3}{4},\frac{11}{12};\frac{23}{12};\sin ^2(e+f x)\right)}{11 f \sqrt[4]{\cos ^2(e+f x)}}","\frac{6 \cos ^2(e+f x)^{3/4} \sqrt[3]{b \sin (e+f x)} (d \tan (e+f x))^{3/2} \, _2F_1\left(\frac{3}{4},\frac{11}{12};\frac{23}{12};\sin ^2(e+f x)\right)}{11 d f}",1,"(3*Hypergeometric2F1[3/4, 11/12, 23/12, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(1/3)*Sin[2*(e + f*x)]*Sqrt[d*Tan[e + f*x]])/(11*f*(Cos[e + f*x]^2)^(1/4))","A",1
147,1,64,64,0.3303336,"\int \frac{\sqrt{d \tan (e+f x)}}{\sqrt[3]{b \sin (e+f x)}} \, dx","Integrate[Sqrt[d*Tan[e + f*x]]/(b*Sin[e + f*x])^(1/3),x]","\frac{6 \cos ^2(e+f x)^{3/4} (d \tan (e+f x))^{3/2} \, _2F_1\left(\frac{7}{12},\frac{3}{4};\frac{19}{12};\sin ^2(e+f x)\right)}{7 d f \sqrt[3]{b \sin (e+f x)}}","\frac{6 \cos ^2(e+f x)^{3/4} (d \tan (e+f x))^{3/2} \, _2F_1\left(\frac{7}{12},\frac{3}{4};\frac{19}{12};\sin ^2(e+f x)\right)}{7 d f \sqrt[3]{b \sin (e+f x)}}",1,"(6*(Cos[e + f*x]^2)^(3/4)*Hypergeometric2F1[7/12, 3/4, 19/12, Sin[e + f*x]^2]*(d*Tan[e + f*x])^(3/2))/(7*d*f*(b*Sin[e + f*x])^(1/3))","A",1
148,1,67,62,0.323667,"\int \frac{\sqrt{d \tan (e+f x)}}{(b \sin (e+f x))^{4/3}} \, dx","Integrate[Sqrt[d*Tan[e + f*x]]/(b*Sin[e + f*x])^(4/3),x]","\frac{3 \sin (2 (e+f x)) \sqrt{d \tan (e+f x)} \, _2F_1\left(\frac{1}{12},\frac{3}{4};\frac{13}{12};\sin ^2(e+f x)\right)}{f \sqrt[4]{\cos ^2(e+f x)} (b \sin (e+f x))^{4/3}}","\frac{6 \cos ^2(e+f x)^{3/4} (d \tan (e+f x))^{3/2} \, _2F_1\left(\frac{1}{12},\frac{3}{4};\frac{13}{12};\sin ^2(e+f x)\right)}{d f (b \sin (e+f x))^{4/3}}",1,"(3*Hypergeometric2F1[1/12, 3/4, 13/12, Sin[e + f*x]^2]*Sin[2*(e + f*x)]*Sqrt[d*Tan[e + f*x]])/(f*(Cos[e + f*x]^2)^(1/4)*(b*Sin[e + f*x])^(4/3))","A",1
149,1,63,64,0.541076,"\int (b \sin (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx","Integrate[(b*Sin[e + f*x])^(4/3)*(d*Tan[e + f*x])^(3/2),x]","-\frac{2 d (b \sin (e+f x))^{4/3} \sqrt{d \tan (e+f x)} \left(\sqrt[4]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{4},\frac{11}{12};\frac{23}{12};\sin ^2(e+f x)\right)-1\right)}{f}","\frac{6 \cos ^2(e+f x)^{5/4} (b \sin (e+f x))^{4/3} (d \tan (e+f x))^{5/2} \, _2F_1\left(\frac{5}{4},\frac{23}{12};\frac{35}{12};\sin ^2(e+f x)\right)}{23 d f}",1,"(-2*d*(-1 + (Cos[e + f*x]^2)^(1/4)*Hypergeometric2F1[1/4, 11/12, 23/12, Sin[e + f*x]^2])*(b*Sin[e + f*x])^(4/3)*Sqrt[d*Tan[e + f*x]])/f","A",1
150,1,63,64,0.4317754,"\int \sqrt[3]{b \sin (e+f x)} (d \tan (e+f x))^{3/2} \, dx","Integrate[(b*Sin[e + f*x])^(1/3)*(d*Tan[e + f*x])^(3/2),x]","-\frac{2 d \sqrt[3]{b \sin (e+f x)} \sqrt{d \tan (e+f x)} \left(\sqrt[4]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{4},\frac{5}{12};\frac{17}{12};\sin ^2(e+f x)\right)-1\right)}{f}","\frac{6 \cos ^2(e+f x)^{5/4} \sqrt[3]{b \sin (e+f x)} (d \tan (e+f x))^{5/2} \, _2F_1\left(\frac{5}{4},\frac{17}{12};\frac{29}{12};\sin ^2(e+f x)\right)}{17 d f}",1,"(-2*d*(-1 + (Cos[e + f*x]^2)^(1/4)*Hypergeometric2F1[1/4, 5/12, 17/12, Sin[e + f*x]^2])*(b*Sin[e + f*x])^(1/3)*Sqrt[d*Tan[e + f*x]])/f","A",1
151,1,63,64,0.4594526,"\int \frac{(d \tan (e+f x))^{3/2}}{\sqrt[3]{b \sin (e+f x)}} \, dx","Integrate[(d*Tan[e + f*x])^(3/2)/(b*Sin[e + f*x])^(1/3),x]","-\frac{2 d \sqrt{d \tan (e+f x)} \left(\sqrt[4]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{12},\frac{1}{4};\frac{13}{12};\sin ^2(e+f x)\right)-1\right)}{f \sqrt[3]{b \sin (e+f x)}}","\frac{6 \cos ^2(e+f x)^{5/4} (d \tan (e+f x))^{5/2} \, _2F_1\left(\frac{13}{12},\frac{5}{4};\frac{25}{12};\sin ^2(e+f x)\right)}{13 d f \sqrt[3]{b \sin (e+f x)}}",1,"(-2*d*(-1 + (Cos[e + f*x]^2)^(1/4)*Hypergeometric2F1[1/12, 1/4, 13/12, Sin[e + f*x]^2])*Sqrt[d*Tan[e + f*x]])/(f*(b*Sin[e + f*x])^(1/3))","A",1
152,1,69,64,0.4875425,"\int \frac{(d \tan (e+f x))^{3/2}}{(b \sin (e+f x))^{4/3}} \, dx","Integrate[(d*Tan[e + f*x])^(3/2)/(b*Sin[e + f*x])^(4/3),x]","-\frac{2 d (b \sin (e+f x))^{2/3} \sqrt{d \tan (e+f x)} \left(4 \sqrt[4]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{4},\frac{7}{12};\frac{19}{12};\sin ^2(e+f x)\right)-7\right)}{7 b^2 f}","\frac{6 \cos ^2(e+f x)^{5/4} (d \tan (e+f x))^{5/2} \, _2F_1\left(\frac{7}{12},\frac{5}{4};\frac{19}{12};\sin ^2(e+f x)\right)}{7 d f (b \sin (e+f x))^{4/3}}",1,"(-2*d*(-7 + 4*(Cos[e + f*x]^2)^(1/4)*Hypergeometric2F1[1/4, 7/12, 19/12, Sin[e + f*x]^2])*(b*Sin[e + f*x])^(2/3)*Sqrt[d*Tan[e + f*x]])/(7*b^2*f)","A",1
153,1,65,64,0.4280148,"\int \sqrt{b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, dx","Integrate[Sqrt[b*Sin[e + f*x]]*(d*Tan[e + f*x])^(4/3),x]","-\frac{3 d \sqrt{b \sin (e+f x)} \sqrt[3]{d \tan (e+f x)} \left(\sqrt[4]{\sec ^2(e+f x)} \, _2F_1\left(\frac{5}{12},\frac{5}{4};\frac{17}{12};-\tan ^2(e+f x)\right)-1\right)}{f}","\frac{6 \cos ^2(e+f x)^{7/6} \sqrt{b \sin (e+f x)} (d \tan (e+f x))^{7/3} \, _2F_1\left(\frac{7}{6},\frac{17}{12};\frac{29}{12};\sin ^2(e+f x)\right)}{17 d f}",1,"(-3*d*(-1 + Hypergeometric2F1[5/12, 5/4, 17/12, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(1/4))*Sqrt[b*Sin[e + f*x]]*(d*Tan[e + f*x])^(1/3))/f","A",1
154,1,66,64,0.349743,"\int \sqrt{b \sin (e+f x)} \sqrt[3]{d \tan (e+f x)} \, dx","Integrate[Sqrt[b*Sin[e + f*x]]*(d*Tan[e + f*x])^(1/3),x]","\frac{6 \sqrt[4]{\sec ^2(e+f x)} \sqrt{b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, _2F_1\left(\frac{11}{12},\frac{5}{4};\frac{23}{12};-\tan ^2(e+f x)\right)}{11 d f}","\frac{6 \cos ^2(e+f x)^{2/3} \sqrt{b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, _2F_1\left(\frac{2}{3},\frac{11}{12};\frac{23}{12};\sin ^2(e+f x)\right)}{11 d f}",1,"(6*Hypergeometric2F1[11/12, 5/4, 23/12, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(1/4)*Sqrt[b*Sin[e + f*x]]*(d*Tan[e + f*x])^(4/3))/(11*d*f)","A",1
155,1,66,64,0.3378777,"\int \frac{\sqrt{b \sin (e+f x)}}{\sqrt[3]{d \tan (e+f x)}} \, dx","Integrate[Sqrt[b*Sin[e + f*x]]/(d*Tan[e + f*x])^(1/3),x]","\frac{6 \sqrt[4]{\sec ^2(e+f x)} \sqrt{b \sin (e+f x)} (d \tan (e+f x))^{2/3} \, _2F_1\left(\frac{7}{12},\frac{5}{4};\frac{19}{12};-\tan ^2(e+f x)\right)}{7 d f}","\frac{6 \sqrt[3]{\cos ^2(e+f x)} \sqrt{b \sin (e+f x)} (d \tan (e+f x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{7}{12};\frac{19}{12};\sin ^2(e+f x)\right)}{7 d f}",1,"(6*Hypergeometric2F1[7/12, 5/4, 19/12, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(1/4)*Sqrt[b*Sin[e + f*x]]*(d*Tan[e + f*x])^(2/3))/(7*d*f)","A",1
156,1,64,62,0.412181,"\int \frac{\sqrt{b \sin (e+f x)}}{(d \tan (e+f x))^{4/3}} \, dx","Integrate[Sqrt[b*Sin[e + f*x]]/(d*Tan[e + f*x])^(4/3),x]","\frac{6 \sqrt[4]{\sec ^2(e+f x)} \sqrt{b \sin (e+f x)} \, _2F_1\left(\frac{1}{12},\frac{5}{4};\frac{13}{12};-\tan ^2(e+f x)\right)}{d f \sqrt[3]{d \tan (e+f x)}}","\frac{6 \sqrt{b \sin (e+f x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{12};\frac{13}{12};\sin ^2(e+f x)\right)}{d f \sqrt[6]{\cos ^2(e+f x)} \sqrt[3]{d \tan (e+f x)}}",1,"(6*Hypergeometric2F1[1/12, 5/4, 13/12, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(1/4)*Sqrt[b*Sin[e + f*x]])/(d*f*(d*Tan[e + f*x])^(1/3))","A",1
157,1,85,64,0.6154607,"\int (b \sin (e+f x))^{3/2} (d \tan (e+f x))^{4/3} \, dx","Integrate[(b*Sin[e + f*x])^(3/2)*(d*Tan[e + f*x])^(4/3),x]","\frac{3 d (b \sin (e+f x))^{3/2} \sqrt[3]{d \tan (e+f x)} \left(\sqrt[4]{\sec ^2(e+f x)}-\sec ^2(e+f x) \, _2F_1\left(\frac{11}{12},\frac{7}{4};\frac{23}{12};-\tan ^2(e+f x)\right)\right)}{f \sqrt[4]{\sec ^2(e+f x)}}","\frac{6 \cos ^2(e+f x)^{7/6} (b \sin (e+f x))^{3/2} (d \tan (e+f x))^{7/3} \, _2F_1\left(\frac{7}{6},\frac{23}{12};\frac{35}{12};\sin ^2(e+f x)\right)}{23 d f}",1,"(3*d*(-(Hypergeometric2F1[11/12, 7/4, 23/12, -Tan[e + f*x]^2]*Sec[e + f*x]^2) + (Sec[e + f*x]^2)^(1/4))*(b*Sin[e + f*x])^(3/2)*(d*Tan[e + f*x])^(1/3))/(f*(Sec[e + f*x]^2)^(1/4))","A",1
158,1,72,64,0.5021873,"\int (b \sin (e+f x))^{3/2} \sqrt[3]{d \tan (e+f x)} \, dx","Integrate[(b*Sin[e + f*x])^(3/2)*(d*Tan[e + f*x])^(1/3),x]","\frac{6 \cos (e+f x) \sec ^2(e+f x)^{7/4} (b \sin (e+f x))^{5/2} \sqrt[3]{d \tan (e+f x)} \, _2F_1\left(\frac{17}{12},\frac{7}{4};\frac{29}{12};-\tan ^2(e+f x)\right)}{17 b f}","\frac{6 \cos ^2(e+f x)^{2/3} (b \sin (e+f x))^{3/2} (d \tan (e+f x))^{4/3} \, _2F_1\left(\frac{2}{3},\frac{17}{12};\frac{29}{12};\sin ^2(e+f x)\right)}{17 d f}",1,"(6*Cos[e + f*x]*Hypergeometric2F1[17/12, 7/4, 29/12, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(7/4)*(b*Sin[e + f*x])^(5/2)*(d*Tan[e + f*x])^(1/3))/(17*b*f)","A",1
159,1,67,64,0.7529961,"\int \frac{(b \sin (e+f x))^{3/2}}{\sqrt[3]{d \tan (e+f x)}} \, dx","Integrate[(b*Sin[e + f*x])^(3/2)/(d*Tan[e + f*x])^(1/3),x]","\frac{2 d (b \sin (e+f x))^{3/2} \left(\sec ^2(e+f x)^{3/4} \, _2F_1\left(\frac{1}{12},\frac{3}{4};\frac{13}{12};-\tan ^2(e+f x)\right)-1\right)}{3 f (d \tan (e+f x))^{4/3}}","\frac{6 \sqrt[3]{\cos ^2(e+f x)} (b \sin (e+f x))^{3/2} (d \tan (e+f x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{13}{12};\frac{25}{12};\sin ^2(e+f x)\right)}{13 d f}",1,"(2*d*(-1 + Hypergeometric2F1[1/12, 3/4, 13/12, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(3/4))*(b*Sin[e + f*x])^(3/2))/(3*f*(d*Tan[e + f*x])^(4/3))","A",1
160,1,70,64,0.3863901,"\int \frac{(b \sin (e+f x))^{3/2}}{(d \tan (e+f x))^{4/3}} \, dx","Integrate[(b*Sin[e + f*x])^(3/2)/(d*Tan[e + f*x])^(4/3),x]","\frac{2 (b \sin (e+f x))^{3/2} \left(2 \sec ^2(e+f x)^{3/4} \, _2F_1\left(\frac{7}{12},\frac{3}{4};\frac{19}{12};-\tan ^2(e+f x)\right)+7\right)}{21 d f \sqrt[3]{d \tan (e+f x)}}","\frac{6 (b \sin (e+f x))^{3/2} \, _2F_1\left(-\frac{1}{6},\frac{7}{12};\frac{19}{12};\sin ^2(e+f x)\right)}{7 d f \sqrt[6]{\cos ^2(e+f x)} \sqrt[3]{d \tan (e+f x)}}",1,"(2*(7 + 2*Hypergeometric2F1[7/12, 3/4, 19/12, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(3/4))*(b*Sin[e + f*x])^(3/2))/(21*d*f*(d*Tan[e + f*x])^(1/3))","A",1
161,1,53,48,0.0616678,"\int (a \sin (e+f x))^m \tan ^3(e+f x) \, dx","Integrate[(a*Sin[e + f*x])^m*Tan[e + f*x]^3,x]","\frac{\sin ^4(e+f x) (a \sin (e+f x))^m \, _2F_1\left(2,\frac{m+4}{2};\frac{m+4}{2}+1;\sin ^2(e+f x)\right)}{f (m+4)}","\frac{(a \sin (e+f x))^{m+4} \, _2F_1\left(2,\frac{m+4}{2};\frac{m+6}{2};\sin ^2(e+f x)\right)}{a^4 f (m+4)}",1,"(Hypergeometric2F1[2, (4 + m)/2, 1 + (4 + m)/2, Sin[e + f*x]^2]*Sin[e + f*x]^4*(a*Sin[e + f*x])^m)/(f*(4 + m))","A",1
162,1,53,48,0.0354942,"\int (a \sin (e+f x))^m \tan (e+f x) \, dx","Integrate[(a*Sin[e + f*x])^m*Tan[e + f*x],x]","\frac{\sin ^2(e+f x) (a \sin (e+f x))^m \, _2F_1\left(1,\frac{m+2}{2};\frac{m+2}{2}+1;\sin ^2(e+f x)\right)}{f (m+2)}","\frac{(a \sin (e+f x))^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right)}{a^2 f (m+2)}",1,"(Hypergeometric2F1[1, (2 + m)/2, 1 + (2 + m)/2, Sin[e + f*x]^2]*Sin[e + f*x]^2*(a*Sin[e + f*x])^m)/(f*(2 + m))","A",1
163,1,17,17,0.0100421,"\int \cot (e+f x) (a \sin (e+f x))^m \, dx","Integrate[Cot[e + f*x]*(a*Sin[e + f*x])^m,x]","\frac{(a \sin (e+f x))^m}{f m}","\frac{(a \sin (e+f x))^m}{f m}",1,"(a*Sin[e + f*x])^m/(f*m)","A",1
164,1,37,46,0.0592668,"\int \cot ^3(e+f x) (a \sin (e+f x))^m \, dx","Integrate[Cot[e + f*x]^3*(a*Sin[e + f*x])^m,x]","\frac{\left(m \csc ^2(e+f x)-m+2\right) (a \sin (e+f x))^m}{f (m-2) m}","-\frac{a^2 (a \sin (e+f x))^{m-2}}{f (2-m)}-\frac{(a \sin (e+f x))^m}{f m}",1,"((2 - m + m*Csc[e + f*x]^2)*(a*Sin[e + f*x])^m)/(f*(-2 + m)*m)","A",1
165,1,62,72,0.3324987,"\int \cot ^5(e+f x) (a \sin (e+f x))^m \, dx","Integrate[Cot[e + f*x]^5*(a*Sin[e + f*x])^m,x]","\frac{\left((m-2) m \csc ^4(e+f x)-2 (m-4) m \csc ^2(e+f x)+m^2-6 m+8\right) (a \sin (e+f x))^m}{f (m-4) (m-2) m}","-\frac{a^4 (a \sin (e+f x))^{m-4}}{f (4-m)}+\frac{2 a^2 (a \sin (e+f x))^{m-2}}{f (2-m)}+\frac{(a \sin (e+f x))^m}{f m}",1,"((8 - 6*m + m^2 - 2*(-4 + m)*m*Csc[e + f*x]^2 + (-2 + m)*m*Csc[e + f*x]^4)*(a*Sin[e + f*x])^m)/(f*(-4 + m)*(-2 + m)*m)","A",1
166,1,71,68,0.1391781,"\int (a \sin (e+f x))^m \tan ^4(e+f x) \, dx","Integrate[(a*Sin[e + f*x])^m*Tan[e + f*x]^4,x]","\frac{\sin ^4(e+f x) \sqrt{\cos ^2(e+f x)} \tan (e+f x) (a \sin (e+f x))^m \, _2F_1\left(\frac{5}{2},\frac{m+5}{2};\frac{m+7}{2};\sin ^2(e+f x)\right)}{f (m+5)}","\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) (a \sin (e+f x))^{m+5} \, _2F_1\left(\frac{5}{2},\frac{m+5}{2};\frac{m+7}{2};\sin ^2(e+f x)\right)}{a^5 f (m+5)}",1,"(Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[5/2, (5 + m)/2, (7 + m)/2, Sin[e + f*x]^2]*Sin[e + f*x]^4*(a*Sin[e + f*x])^m*Tan[e + f*x])/(f*(5 + m))","A",1
167,1,71,68,0.0860121,"\int (a \sin (e+f x))^m \tan ^2(e+f x) \, dx","Integrate[(a*Sin[e + f*x])^m*Tan[e + f*x]^2,x]","\frac{\sin ^2(e+f x) \sqrt{\cos ^2(e+f x)} \tan (e+f x) (a \sin (e+f x))^m \, _2F_1\left(\frac{3}{2},\frac{m+3}{2};\frac{m+5}{2};\sin ^2(e+f x)\right)}{f (m+3)}","\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) (a \sin (e+f x))^{m+3} \, _2F_1\left(\frac{3}{2},\frac{m+3}{2};\frac{m+5}{2};\sin ^2(e+f x)\right)}{a^3 f (m+3)}",1,"(Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[3/2, (3 + m)/2, (5 + m)/2, Sin[e + f*x]^2]*Sin[e + f*x]^2*(a*Sin[e + f*x])^m*Tan[e + f*x])/(f*(3 + m))","A",1
168,1,66,69,0.08098,"\int \cot ^2(e+f x) (a \sin (e+f x))^m \, dx","Integrate[Cot[e + f*x]^2*(a*Sin[e + f*x])^m,x]","\frac{a \sqrt{\cos ^2(e+f x)} \sec (e+f x) (a \sin (e+f x))^{m-1} \, _2F_1\left(-\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sin ^2(e+f x)\right)}{f (m-1)}","-\frac{a \cos (e+f x) (a \sin (e+f x))^{m-1} \, _2F_1\left(-\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sin ^2(e+f x)\right)}{f (1-m) \sqrt{\cos ^2(e+f x)}}",1,"(a*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-1/2, (-1 + m)/2, (1 + m)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(a*Sin[e + f*x])^(-1 + m))/(f*(-1 + m))","A",1
169,1,71,71,0.0705727,"\int \cot ^4(e+f x) (a \sin (e+f x))^m \, dx","Integrate[Cot[e + f*x]^4*(a*Sin[e + f*x])^m,x]","\frac{\sqrt{\cos ^2(e+f x)} \csc ^3(e+f x) \sec (e+f x) (a \sin (e+f x))^m \, _2F_1\left(-\frac{3}{2},\frac{m-3}{2};\frac{m-1}{2};\sin ^2(e+f x)\right)}{f (m-3)}","-\frac{a^3 \cos (e+f x) (a \sin (e+f x))^{m-3} \, _2F_1\left(-\frac{3}{2},\frac{m-3}{2};\frac{m-1}{2};\sin ^2(e+f x)\right)}{f (3-m) \sqrt{\cos ^2(e+f x)}}",1,"(Sqrt[Cos[e + f*x]^2]*Csc[e + f*x]^3*Hypergeometric2F1[-3/2, (-3 + m)/2, (-1 + m)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(a*Sin[e + f*x])^m)/(f*(-3 + m))","A",1
170,1,87,79,8.1388809,"\int (a \sin (e+f x))^m (b \tan (e+f x))^{3/2} \, dx","Integrate[(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(3/2),x]","\frac{2 (b \tan (e+f x))^{5/2} \sec ^2(e+f x)^{m/2} (a \sin (e+f x))^m \, _2F_1\left(\frac{m+2}{2},\frac{1}{4} (2 m+5);\frac{1}{4} (2 m+9);-\tan ^2(e+f x)\right)}{b f (2 m+5)}","\frac{2 \cos ^2(e+f x)^{5/4} (b \tan (e+f x))^{5/2} (a \sin (e+f x))^m \, _2F_1\left(\frac{5}{4},\frac{1}{4} (2 m+5);\frac{1}{4} (2 m+9);\sin ^2(e+f x)\right)}{b f (2 m+5)}",1,"(2*Hypergeometric2F1[(2 + m)/2, (5 + 2*m)/4, (9 + 2*m)/4, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(m/2)*(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(5/2))/(b*f*(5 + 2*m))","A",1
171,1,87,79,3.242152,"\int (a \sin (e+f x))^m \sqrt{b \tan (e+f x)} \, dx","Integrate[(a*Sin[e + f*x])^m*Sqrt[b*Tan[e + f*x]],x]","\frac{2 (b \tan (e+f x))^{3/2} \sec ^2(e+f x)^{m/2} (a \sin (e+f x))^m \, _2F_1\left(\frac{m+2}{2},\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(e+f x)\right)}{b f (2 m+3)}","\frac{2 \cos ^2(e+f x)^{3/4} (b \tan (e+f x))^{3/2} (a \sin (e+f x))^m \, _2F_1\left(\frac{3}{4},\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);\sin ^2(e+f x)\right)}{b f (2 m+3)}",1,"(2*Hypergeometric2F1[(2 + m)/2, (3 + 2*m)/4, (7 + 2*m)/4, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(m/2)*(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(3/2))/(b*f*(3 + 2*m))","A",1
172,1,87,79,3.0251959,"\int \frac{(a \sin (e+f x))^m}{\sqrt{b \tan (e+f x)}} \, dx","Integrate[(a*Sin[e + f*x])^m/Sqrt[b*Tan[e + f*x]],x]","\frac{2 \sqrt{b \tan (e+f x)} \sec ^2(e+f x)^{m/2} (a \sin (e+f x))^m \, _2F_1\left(\frac{m+2}{2},\frac{1}{4} (2 m+1);\frac{1}{4} (2 m+5);-\tan ^2(e+f x)\right)}{b f (2 m+1)}","\frac{2 \sqrt[4]{\cos ^2(e+f x)} \sqrt{b \tan (e+f x)} (a \sin (e+f x))^m \, _2F_1\left(\frac{1}{4},\frac{1}{4} (2 m+1);\frac{1}{4} (2 m+5);\sin ^2(e+f x)\right)}{b f (2 m+1)}",1,"(2*Hypergeometric2F1[(2 + m)/2, (1 + 2*m)/4, (5 + 2*m)/4, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(m/2)*(a*Sin[e + f*x])^m*Sqrt[b*Tan[e + f*x]])/(b*f*(1 + 2*m))","A",1
173,1,225,79,5.1633439,"\int \frac{(a \sin (e+f x))^m}{(b \tan (e+f x))^{3/2}} \, dx","Integrate[(a*Sin[e + f*x])^m/(b*Tan[e + f*x])^(3/2),x]","\frac{\sec ^4(e+f x) \sec ^2(e+f x)^{\frac{m-4}{2}} (a \sin (e+f x))^m \left(\, _2F_1\left(\frac{m}{2},\frac{1}{4} (2 m-1);\frac{1}{4} (2 m+3);-\tan ^2(e+f x)\right)+\frac{\cos (2 (e+f x)) \sec ^2(e+f x) \left((2 m-1) \tan ^2(e+f x) \, _2F_1\left(\frac{m+2}{2},\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(e+f x)\right)-(2 m+3) \, _2F_1\left(\frac{m+2}{2},\frac{1}{4} (2 m-1);\frac{1}{4} (2 m+3);-\tan ^2(e+f x)\right)\right)}{(2 m+3) \left(\tan ^2(e+f x)-1\right)}\right)}{b f (2 m-1) \sqrt{b \tan (e+f x)}}","-\frac{2 (a \sin (e+f x))^m \, _2F_1\left(-\frac{1}{4},\frac{1}{4} (2 m-1);\frac{1}{4} (2 m+3);\sin ^2(e+f x)\right)}{b f (1-2 m) \sqrt[4]{\cos ^2(e+f x)} \sqrt{b \tan (e+f x)}}",1,"(Sec[e + f*x]^4*(Sec[e + f*x]^2)^((-4 + m)/2)*(a*Sin[e + f*x])^m*(Hypergeometric2F1[m/2, (-1 + 2*m)/4, (3 + 2*m)/4, -Tan[e + f*x]^2] + (Cos[2*(e + f*x)]*Sec[e + f*x]^2*(-((3 + 2*m)*Hypergeometric2F1[(2 + m)/2, (-1 + 2*m)/4, (3 + 2*m)/4, -Tan[e + f*x]^2]) + (-1 + 2*m)*Hypergeometric2F1[(2 + m)/2, (3 + 2*m)/4, (7 + 2*m)/4, -Tan[e + f*x]^2]*Tan[e + f*x]^2))/((3 + 2*m)*(-1 + Tan[e + f*x]^2))))/(b*f*(-1 + 2*m)*Sqrt[b*Tan[e + f*x]])","B",1
174,1,260,83,1.9592157,"\int (a \sin (e+f x))^m (b \tan (e+f x))^n \, dx","Integrate[(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^n,x]","\frac{(m+n+3) \sin (e+f x) (a \sin (e+f x))^m (b \tan (e+f x))^n F_1\left(\frac{1}{2} (m+n+1);n,m+1;\frac{1}{2} (m+n+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{f (m+n+1) \left((m+n+3) F_1\left(\frac{1}{2} (m+n+1);n,m+1;\frac{1}{2} (m+n+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+3);n,m+2;\frac{1}{2} (m+n+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{1}{2} (m+n+3);n+1,m+1;\frac{1}{2} (m+n+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{\cos ^2(e+f x)^{\frac{n+1}{2}} (a \sin (e+f x))^m (b \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\sin ^2(e+f x)\right)}{b f (m+n+1)}",1,"((3 + m + n)*AppellF1[(1 + m + n)/2, n, 1 + m, (3 + m + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[e + f*x]*(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^n)/(f*(1 + m + n)*((3 + m + n)*AppellF1[(1 + m + n)/2, n, 1 + m, (3 + m + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*((1 + m)*AppellF1[(3 + m + n)/2, n, 2 + m, (5 + m + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[(3 + m + n)/2, 1 + n, 1 + m, (5 + m + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))","C",0
175,1,916,50,4.7149221,"\int \sin ^4(e+f x) (b \tan (e+f x))^n \, dx","Integrate[Sin[e + f*x]^4*(b*Tan[e + f*x])^n,x]","\frac{64 (n+3) \left(F_1\left(\frac{n+1}{2};n,3;\frac{n+3}{2};\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{n+1}{2};n,4;\frac{n+3}{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+1}{2};n,5;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \cos ^7\left(\frac{1}{2} (e+f x)\right) \sin ^5\left(\frac{1}{2} (e+f x)\right) (b \tan (e+f x))^n}{f (n+1) \left((n+3) F_1\left(\frac{n+1}{2};n,3;\frac{n+3}{2};\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)+(n+3) F_1\left(\frac{n+1}{2};n,5;\frac{n+3}{2};\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)+2 \left(-2 n F_1\left(\frac{n+1}{2};n,4;\frac{n+3}{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)-6 F_1\left(\frac{n+1}{2};n,4;\frac{n+3}{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)-5 F_1\left(\frac{n+3}{2};n,6;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+n F_1\left(\frac{n+3}{2};n+1,3;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 n F_1\left(\frac{n+3}{2};n+1,4;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+n F_1\left(\frac{n+3}{2};n+1,5;\frac{n+5}{2};\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{n+3}{2};n,4;\frac{n+5}{2};\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)-8 F_1\left(\frac{n+3}{2};n,5;\frac{n+5}{2};\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)+5 F_1\left(\frac{n+3}{2};n,6;\frac{n+5}{2};\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 F_1\left(\frac{n+3}{2};n+1,3;\frac{n+5}{2};\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)+2 n F_1\left(\frac{n+3}{2};n+1,4;\frac{n+5}{2};\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 F_1\left(\frac{n+3}{2};n+1,5;\frac{n+5}{2};\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)\right)}","\frac{(b \tan (e+f x))^{n+5} \, _2F_1\left(3,\frac{n+5}{2};\frac{n+7}{2};-\tan ^2(e+f x)\right)}{b^5 f (n+5)}",1,"(64*(3 + n)*(AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*AppellF1[(1 + n)/2, n, 4, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[(1 + n)/2, n, 5, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Cos[(e + f*x)/2]^7*Sin[(e + f*x)/2]^5*(b*Tan[e + f*x])^n)/(f*(1 + n)*((3 + n)*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + Cos[e + f*x]) + (3 + n)*AppellF1[(1 + n)/2, n, 5, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + Cos[e + f*x]) + 2*(-5*AppellF1[(3 + n)/2, n, 6, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[(3 + n)/2, 1 + n, 3, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*n*AppellF1[(3 + n)/2, 1 + n, 4, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[(3 + n)/2, 1 + n, 5, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 6*AppellF1[(1 + n)/2, n, 4, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - 2*n*AppellF1[(1 + n)/2, n, 4, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + 3*AppellF1[(3 + n)/2, n, 4, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(-1 + Cos[e + f*x]) - 8*AppellF1[(3 + n)/2, n, 5, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(-1 + Cos[e + f*x]) + 5*AppellF1[(3 + n)/2, n, 6, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] - n*AppellF1[(3 + n)/2, 1 + n, 3, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] + 2*n*AppellF1[(3 + n)/2, 1 + n, 4, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] - n*AppellF1[(3 + n)/2, 1 + n, 5, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x])))","C",0
176,1,450,50,2.109276,"\int \sin ^2(e+f x) (b \tan (e+f x))^n \, dx","Integrate[Sin[e + f*x]^2*(b*Tan[e + f*x])^n,x]","\frac{16 (n+3) \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{n+1}{2};n,2;\frac{n+3}{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+1}{2};n,3;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) (b \tan (e+f x))^n}{f (n+1) \left(-2 (n+3) \cos ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(\frac{n+1}{2};n,3;\frac{n+3}{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+3}{2};n,3;\frac{n+5}{2};\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{n+3}{2};n,4;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+n \left(F_1\left(\frac{n+3}{2};n+1,3;\frac{n+5}{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+3}{2};n+1,2;\frac{n+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)+(n+3) (\cos (e+f x)+1) F_1\left(\frac{n+1}{2};n,2;\frac{n+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{(b \tan (e+f x))^{n+3} \, _2F_1\left(2,\frac{n+3}{2};\frac{n+5}{2};-\tan ^2(e+f x)\right)}{b^3 f (n+3)}",1,"(16*(3 + n)*(AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - AppellF1[(1 + n)/2, n, 3, (3 + n)/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*Tan[e + f*x])^n)/(f*(1 + n)*(-2*(3 + n)*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + 2*(2*AppellF1[(3 + n)/2, n, 3, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 3*AppellF1[(3 + n)/2, n, 4, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*(-AppellF1[(3 + n)/2, 1 + n, 2, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[(3 + n)/2, 1 + n, 3, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]))*(-1 + Cos[e + f*x]) + (3 + n)*AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + Cos[e + f*x])))","C",0
177,1,22,25,0.0731461,"\int \csc ^2(e+f x) (b \tan (e+f x))^n \, dx","Integrate[Csc[e + f*x]^2*(b*Tan[e + f*x])^n,x]","\frac{b (b \tan (e+f x))^{n-1}}{f (n-1)}","-\frac{b (b \tan (e+f x))^{n-1}}{f (1-n)}",1,"(b*(b*Tan[e + f*x])^(-1 + n))/(f*(-1 + n))","A",1
178,1,46,53,0.1583955,"\int \csc ^4(e+f x) (b \tan (e+f x))^n \, dx","Integrate[Csc[e + f*x]^4*(b*Tan[e + f*x])^n,x]","\frac{b \csc ^2(e+f x) (\cos (2 (e+f x))+n-2) (b \tan (e+f x))^{n-1}}{f (n-3) (n-1)}","-\frac{b^3 (b \tan (e+f x))^{n-3}}{f (3-n)}-\frac{b (b \tan (e+f x))^{n-1}}{f (1-n)}",1,"(b*(-2 + n + Cos[2*(e + f*x)])*Csc[e + f*x]^2*(b*Tan[e + f*x])^(-1 + n))/(f*(-3 + n)*(-1 + n))","A",1
179,1,69,80,0.2612203,"\int \csc ^6(e+f x) (b \tan (e+f x))^n \, dx","Integrate[Csc[e + f*x]^6*(b*Tan[e + f*x])^n,x]","\frac{b \csc ^4(e+f x) \left(2 (n-3) \cos (2 (e+f x))+\cos (4 (e+f x))+n^2-6 n+8\right) (b \tan (e+f x))^{n-1}}{f (n-5) (n-3) (n-1)}","-\frac{b^5 (b \tan (e+f x))^{n-5}}{f (5-n)}-\frac{2 b^3 (b \tan (e+f x))^{n-3}}{f (3-n)}-\frac{b (b \tan (e+f x))^{n-1}}{f (1-n)}",1,"(b*(8 - 6*n + n^2 + 2*(-3 + n)*Cos[2*(e + f*x)] + Cos[4*(e + f*x)])*Csc[e + f*x]^4*(b*Tan[e + f*x])^(-1 + n))/(f*(-5 + n)*(-3 + n)*(-1 + n))","A",1
180,1,456,78,2.8297266,"\int \sin ^3(e+f x) (b \tan (e+f x))^n \, dx","Integrate[Sin[e + f*x]^3*(b*Tan[e + f*x])^n,x]","\frac{4 (n+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}{2}+1;n,3;\frac{n}{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}{2}+1;n,4;\frac{n}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) (b \tan (e+f x))^n}{f (n+2) \left(-2 (n+4) \cos ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(\frac{n}{2}+1;n,4;\frac{n}{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}{2}+2;n,4;\frac{n}{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}{2}+2;n,5;\frac{n}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+n \left(F_1\left(\frac{n}{2}+2;n+1,4;\frac{n}{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}{2}+2;n+1,3;\frac{n}{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)+(n+4) (\cos (e+f x)+1) F_1\left(\frac{n}{2}+1;n,3;\frac{n}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}","\frac{\sin ^3(e+f x) \cos ^2(e+f x)^{\frac{n+1}{2}} (b \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+4}{2};\frac{n+6}{2};\sin ^2(e+f x)\right)}{b f (n+4)}",1,"(4*(4 + n)*(AppellF1[1 + n/2, n, 3, 2 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - AppellF1[1 + n/2, n, 4, 2 + n/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*Tan[e + f*x])^n)/(f*(2 + n)*(-2*(4 + n)*AppellF1[1 + n/2, n, 4, 2 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + 2*(3*AppellF1[2 + n/2, n, 4, 3 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 4*AppellF1[2 + n/2, n, 5, 3 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*(-AppellF1[2 + n/2, 1 + n, 3, 3 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[2 + n/2, 1 + n, 4, 3 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]))*(-1 + Cos[e + f*x]) + (4 + n)*AppellF1[1 + n/2, n, 3, 2 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + Cos[e + f*x])))","C",0
181,1,252,76,1.0901789,"\int \sin (e+f x) (b \tan (e+f x))^n \, dx","Integrate[Sin[e + f*x]*(b*Tan[e + f*x])^n,x]","\frac{8 (n+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}{2}+1;n,2;\frac{n}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) (b \tan (e+f x))^n}{f (n+2) \left(2 (\cos (e+f x)-1) \left(2 F_1\left(\frac{n}{2}+2;n,3;\frac{n}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{n}{2}+2;n+1,2;\frac{n}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)+(n+4) (\cos (e+f x)+1) F_1\left(\frac{n}{2}+1;n,2;\frac{n}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}","\frac{\sin (e+f x) \cos ^2(e+f x)^{\frac{n+1}{2}} (b \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right)}{b f (n+2)}",1,"(8*(4 + n)*AppellF1[1 + n/2, n, 2, 2 + n/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*Tan[e + f*x])^n)/(f*(2 + n)*(2*(2*AppellF1[2 + n/2, n, 3, 3 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[2 + n/2, 1 + n, 2, 3 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*(-1 + Cos[e + f*x]) + (4 + n)*AppellF1[1 + n/2, n, 2, 2 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + Cos[e + f*x])))","C",0
182,1,64,78,0.2113107,"\int \csc (e+f x) (b \tan (e+f x))^n \, dx","Integrate[Csc[e + f*x]*(b*Tan[e + f*x])^n,x]","\frac{(b \tan (e+f x))^n \, _2F_1\left(\frac{n}{2},n;\frac{n}{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}{f n}","-\frac{\cos (e+f x) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n \, _2F_1\left(\frac{1-n}{2},\frac{2-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n)}",1,"(Hypergeometric2F1[n/2, n, 1 + n/2, Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*(b*Tan[e + f*x])^n)/(f*n)","A",0
183,1,1242,78,15.522534,"\int \csc ^3(e+f x) (b \tan (e+f x))^n \, dx","Integrate[Csc[e + f*x]^3*(b*Tan[e + f*x])^n,x]","\frac{\cot ^2\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(\frac{n}{2}-1,n;\frac{n}{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 (b \tan (e+f x))^n}{f (4 n-8)}+\frac{(n+4) F_1\left(\frac{n}{2}+1;n,1;\frac{n}{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) (b \tan (e+f x))^n}{4 f (n+2) \left(2 \left(F_1\left(\frac{n}{2}+2;n,2;\frac{n}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{n}{2}+2;n+1,1;\frac{n}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) (\cos (e+f x)-1)+(n+4) F_1\left(\frac{n}{2}+1;n,1;\frac{n}{2}+2;\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(\frac{n}{2}+1,n;\frac{n}{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 \tan ^2\left(\frac{1}{2} (e+f x)\right) (b \tan (e+f x))^n}{f (4 n+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 \left((n+2) \, _2F_1\left(\frac{n}{2},n;\frac{n}{2}+1;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{n}{2}+1;n,1;\frac{n}{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(e+f x) (b \tan (e+f x))^n}{8 f n (n+2) \left(\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+2) \, _2F_1\left(\frac{n}{2},n;\frac{n}{2}+1;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{n}{2}+1;n,1;\frac{n}{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(e+f x) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^{n-1}}{2 (n+2)}+\frac{\sec ^2(e+f x) \left((n+2) \, _2F_1\left(\frac{n}{2},n;\frac{n}{2}+1;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{n}{2}+1;n,1;\frac{n}{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-1}(e+f x) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^n}{2 (n+2)}+\frac{\left(-n F_1\left(\frac{n}{2}+1;n,1;\frac{n}{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 (n+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}-\, _2F_1\left(\frac{n}{2},n;\frac{n}{2}+1;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \sec \left(\frac{1}{2} (e+f x)\right)-n \tan ^2\left(\frac{1}{2} (e+f x)\right) \left(\frac{\left(\frac{n}{2}+1\right) n F_1\left(\frac{n}{2}+2;n+1,1;\frac{n}{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}{2}+2}-\frac{\left(\frac{n}{2}+1\right) F_1\left(\frac{n}{2}+2;n,2;\frac{n}{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}{2}+2}\right)\right) \tan ^n(e+f x) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^n}{2 n (n+2)}\right)}","-\frac{\cos (e+f x) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n \, _2F_1\left(\frac{1-n}{2},\frac{4-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n)}",1,"(Cot[(e + f*x)/2]^2*Hypergeometric2F1[-1 + n/2, n, n/2, Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*(b*Tan[e + f*x])^n)/(f*(-8 + 4*n)) + ((4 + n)*AppellF1[1 + n/2, n, 1, 2 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[e + f*x]^2*(b*Tan[e + f*x])^n)/(4*f*(2 + n)*(2*(AppellF1[2 + n/2, n, 2, 3 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[2 + n/2, 1 + n, 1, 3 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*(-1 + Cos[e + f*x]) + (4 + n)*AppellF1[1 + n/2, n, 1, 2 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + Cos[e + f*x]))) + (Hypergeometric2F1[1 + n/2, n, 2 + n/2, Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2*(b*Tan[e + f*x])^n)/(f*(8 + 4*n)) + (Cot[(e + f*x)/2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*((2 + n)*Hypergeometric2F1[n/2, n, 1 + n/2, Tan[(e + f*x)/2]^2] - n*AppellF1[1 + n/2, n, 1, 2 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)*Tan[e + f*x]^n*(b*Tan[e + f*x])^n)/(8*f*n*(2 + n)*(((Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Sec[e + f*x]^2*((2 + n)*Hypergeometric2F1[n/2, n, 1 + n/2, Tan[(e + f*x)/2]^2] - n*AppellF1[1 + n/2, n, 1, 2 + n/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))/(2*(2 + n)) + ((Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + n)*(-(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)*Hypergeometric2F1[n/2, n, 1 + n/2, Tan[(e + f*x)/2]^2] - n*AppellF1[1 + n/2, n, 1, 2 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)*Tan[e + f*x]^n)/(2*(2 + n)) + ((Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*(-(n*AppellF1[1 + n/2, n, 1, 2 + n/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*Tan[(e + f*x)/2]^2*(-(((1 + n/2)*AppellF1[2 + n/2, n, 2, 3 + n/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/2)) + ((1 + n/2)*n*AppellF1[2 + n/2, 1 + n, 1, 3 + n/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/2)) + (n*(2 + n)*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*(-Hypergeometric2F1[n/2, n, 1 + n/2, Tan[(e + f*x)/2]^2] + (1 - Tan[(e + f*x)/2]^2)^(-n)))/2)*Tan[e + f*x]^n)/(2*n*(2 + n))))","C",0
184,1,1516,78,17.7547428,"\int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx","Integrate[Csc[e + f*x]^5*(b*Tan[e + f*x])^n,x]","\frac{3 \cot ^2\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(\frac{n}{2}-1,n;\frac{n}{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 (b \tan (e+f x))^n}{16 f (n-2)}+\frac{\cot ^2\left(\frac{1}{2} (e+f x)\right) \left((n-2) \, _2F_1\left(\frac{n}{2}-2,n;\frac{n}{2}-1;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cot ^2\left(\frac{1}{2} (e+f x)\right)+(n-4) \, _2F_1\left(\frac{n}{2}-1,n;\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^n (b \tan (e+f x))^n}{16 f (n-4) (n-2)}+\frac{3 (n+4) F_1\left(\frac{n}{2}+1;n,1;\frac{n}{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) (b \tan (e+f x))^n}{16 f (n+2) \left(2 \left(F_1\left(\frac{n}{2}+2;n,2;\frac{n}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{n}{2}+2;n+1,1;\frac{n}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) (\cos (e+f x)-1)+(n+4) F_1\left(\frac{n}{2}+1;n,1;\frac{n}{2}+2;\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{3 \, _2F_1\left(\frac{n}{2}+1,n;\frac{n}{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 \tan ^2\left(\frac{1}{2} (e+f x)\right) (b \tan (e+f x))^n}{16 f (n+2)}+\frac{\left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^n \tan ^2\left(\frac{1}{2} (e+f x)\right) \left((n+2) \, _2F_1\left(\frac{n}{2}+2,n;\frac{n}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^2\left(\frac{1}{2} (e+f x)\right)+(n+4) \, _2F_1\left(\frac{n}{2}+1,n;\frac{n}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) (b \tan (e+f x))^n}{16 f (n+2) (n+4)}+\frac{9 \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 \left((n+2) \, _2F_1\left(\frac{n}{2},n;\frac{n}{2}+1;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{n}{2}+1;n,1;\frac{n}{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(e+f x) (b \tan (e+f x))^n}{128 f n (n+2) \left(\frac{3 \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+2) \, _2F_1\left(\frac{n}{2},n;\frac{n}{2}+1;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{n}{2}+1;n,1;\frac{n}{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(e+f x) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^{n-1}}{8 (n+2)}+\frac{3 \sec ^2(e+f x) \left((n+2) \, _2F_1\left(\frac{n}{2},n;\frac{n}{2}+1;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{n}{2}+1;n,1;\frac{n}{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-1}(e+f x) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^n}{8 (n+2)}+\frac{3 \left(-n F_1\left(\frac{n}{2}+1;n,1;\frac{n}{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 (n+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}-\, _2F_1\left(\frac{n}{2},n;\frac{n}{2}+1;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \sec \left(\frac{1}{2} (e+f x)\right)-n \tan ^2\left(\frac{1}{2} (e+f x)\right) \left(\frac{\left(\frac{n}{2}+1\right) n F_1\left(\frac{n}{2}+2;n+1,1;\frac{n}{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}{2}+2}-\frac{\left(\frac{n}{2}+1\right) F_1\left(\frac{n}{2}+2;n,2;\frac{n}{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}{2}+2}\right)\right) \tan ^n(e+f x) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^n}{8 n (n+2)}\right)}","-\frac{\cos (e+f x) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n \, _2F_1\left(\frac{1-n}{2},\frac{6-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n)}",1,"(3*Cot[(e + f*x)/2]^2*Hypergeometric2F1[-1 + n/2, n, n/2, Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*(b*Tan[e + f*x])^n)/(16*f*(-2 + n)) + (Cot[(e + f*x)/2]^2*((-2 + n)*Cot[(e + f*x)/2]^2*Hypergeometric2F1[-2 + n/2, n, -1 + n/2, Tan[(e + f*x)/2]^2] + (-4 + n)*Hypergeometric2F1[-1 + n/2, n, n/2, Tan[(e + f*x)/2]^2])*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*(b*Tan[e + f*x])^n)/(16*f*(-4 + n)*(-2 + n)) + (3*(4 + n)*AppellF1[1 + n/2, n, 1, 2 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[e + f*x]^2*(b*Tan[e + f*x])^n)/(16*f*(2 + n)*(2*(AppellF1[2 + n/2, n, 2, 3 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[2 + n/2, 1 + n, 1, 3 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*(-1 + Cos[e + f*x]) + (4 + n)*AppellF1[1 + n/2, n, 1, 2 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + Cos[e + f*x]))) + (3*Hypergeometric2F1[1 + n/2, n, 2 + n/2, Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2*(b*Tan[e + f*x])^n)/(16*f*(2 + n)) + ((Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2*((4 + n)*Hypergeometric2F1[1 + n/2, n, 2 + n/2, Tan[(e + f*x)/2]^2] + (2 + n)*Hypergeometric2F1[2 + n/2, n, 3 + n/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)*(b*Tan[e + f*x])^n)/(16*f*(2 + n)*(4 + n)) + (9*Cot[(e + f*x)/2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*((2 + n)*Hypergeometric2F1[n/2, n, 1 + n/2, Tan[(e + f*x)/2]^2] - n*AppellF1[1 + n/2, n, 1, 2 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)*Tan[e + f*x]^n*(b*Tan[e + f*x])^n)/(128*f*n*(2 + n)*((3*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Sec[e + f*x]^2*((2 + n)*Hypergeometric2F1[n/2, n, 1 + n/2, Tan[(e + f*x)/2]^2] - n*AppellF1[1 + n/2, n, 1, 2 + n/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))/(8*(2 + n)) + (3*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + n)*(-(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)*Hypergeometric2F1[n/2, n, 1 + n/2, Tan[(e + f*x)/2]^2] - n*AppellF1[1 + n/2, n, 1, 2 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)*Tan[e + f*x]^n)/(8*(2 + n)) + (3*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*(-(n*AppellF1[1 + n/2, n, 1, 2 + n/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*Tan[(e + f*x)/2]^2*(-(((1 + n/2)*AppellF1[2 + n/2, n, 2, 3 + n/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/2)) + ((1 + n/2)*n*AppellF1[2 + n/2, 1 + n, 1, 3 + n/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/2)) + (n*(2 + n)*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*(-Hypergeometric2F1[n/2, n, 1 + n/2, Tan[(e + f*x)/2]^2] + (1 - Tan[(e + f*x)/2]^2)^(-n)))/2)*Tan[e + f*x]^n)/(8*n*(2 + n))))","C",0
185,1,297,89,2.5644291,"\int (a \sin (e+f x))^{3/2} (b \tan (e+f x))^n \, dx","Integrate[(a*Sin[e + f*x])^(3/2)*(b*Tan[e + f*x])^n,x]","\frac{8 (2 n+9) \sin \left(\frac{1}{2} (e+f x)\right) \cos ^3\left(\frac{1}{2} (e+f x)\right) (a \sin (e+f x))^{3/2} F_1\left(\frac{n}{2}+\frac{5}{4};n,\frac{5}{2};\frac{n}{2}+\frac{9}{4};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) (b \tan (e+f x))^n}{f (2 n+5) \left(2 (2 n+9) \cos ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(\frac{n}{2}+\frac{5}{4};n,\frac{5}{2};\frac{n}{2}+\frac{9}{4};\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(5 F_1\left(\frac{n}{2}+\frac{9}{4};n,\frac{7}{2};\frac{n}{2}+\frac{13}{4};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 n F_1\left(\frac{n}{2}+\frac{9}{4};n+1,\frac{5}{2};\frac{n}{2}+\frac{13}{4};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}","\frac{2 (a \sin (e+f x))^{3/2} \cos ^2(e+f x)^{\frac{n+1}{2}} (b \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\sin ^2(e+f x)\right)}{b f (2 n+5)}",1,"(8*(9 + 2*n)*AppellF1[5/4 + n/2, n, 5/2, 9/4 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^3*Sin[(e + f*x)/2]*(a*Sin[e + f*x])^(3/2)*(b*Tan[e + f*x])^n)/(f*(5 + 2*n)*(2*(9 + 2*n)*AppellF1[5/4 + n/2, n, 5/2, 9/4 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + 2*(5*AppellF1[9/4 + n/2, n, 7/2, 13/4 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*n*AppellF1[9/4 + n/2, 1 + n, 5/2, 13/4 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*(-1 + Cos[e + f*x])))","C",0
186,1,91,89,1.6171787,"\int \sqrt{a \sin (e+f x)} (b \tan (e+f x))^n \, dx","Integrate[Sqrt[a*Sin[e + f*x]]*(b*Tan[e + f*x])^n,x]","\frac{\sin (2 (e+f x)) \sqrt{a \sin (e+f x)} \cos ^2(e+f x)^{\frac{n-1}{2}} (b \tan (e+f x))^n \, _2F_1\left(\frac{n+1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\sin ^2(e+f x)\right)}{f (2 n+3)}","\frac{2 \sqrt{a \sin (e+f x)} \cos ^2(e+f x)^{\frac{n+1}{2}} (b \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\sin ^2(e+f x)\right)}{b f (2 n+3)}",1,"((Cos[e + f*x]^2)^((-1 + n)/2)*Hypergeometric2F1[(1 + n)/2, (3 + 2*n)/4, (7 + 2*n)/4, Sin[e + f*x]^2]*Sqrt[a*Sin[e + f*x]]*Sin[2*(e + f*x)]*(b*Tan[e + f*x])^n)/(f*(3 + 2*n))","A",1
187,1,89,89,1.3080428,"\int \frac{(b \tan (e+f x))^n}{\sqrt{a \sin (e+f x)}} \, dx","Integrate[(b*Tan[e + f*x])^n/Sqrt[a*Sin[e + f*x]],x]","\frac{\sin (2 (e+f x)) \cos ^2(e+f x)^{\frac{n-1}{2}} (b \tan (e+f x))^n \, _2F_1\left(\frac{n+1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\sin ^2(e+f x)\right)}{(2 f n+f) \sqrt{a \sin (e+f x)}}","\frac{2 \cos ^2(e+f x)^{\frac{n+1}{2}} (b \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\sin ^2(e+f x)\right)}{b f (2 n+1) \sqrt{a \sin (e+f x)}}",1,"((Cos[e + f*x]^2)^((-1 + n)/2)*Hypergeometric2F1[(1 + n)/2, (1 + 2*n)/4, (5 + 2*n)/4, Sin[e + f*x]^2]*Sin[2*(e + f*x)]*(b*Tan[e + f*x])^n)/((f + 2*f*n)*Sqrt[a*Sin[e + f*x]])","A",1
188,1,90,89,1.9561178,"\int \frac{(b \tan (e+f x))^n}{(a \sin (e+f x))^{3/2}} \, dx","Integrate[(b*Tan[e + f*x])^n/(a*Sin[e + f*x])^(3/2),x]","\frac{2 b \sqrt{a \sin (e+f x)} \cos ^2(e+f x)^{\frac{n-1}{2}} (b \tan (e+f x))^{n-1} \, _2F_1\left(\frac{n+1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\sin ^2(e+f x)\right)}{a^2 f (2 n-1)}","-\frac{2 \cos ^2(e+f x)^{\frac{n+1}{2}} (b \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\sin ^2(e+f x)\right)}{b f (1-2 n) (a \sin (e+f x))^{3/2}}",1,"(2*b*(Cos[e + f*x]^2)^((-1 + n)/2)*Hypergeometric2F1[(1 + n)/2, (-1 + 2*n)/4, (3 + 2*n)/4, Sin[e + f*x]^2]*Sqrt[a*Sin[e + f*x]]*(b*Tan[e + f*x])^(-1 + n))/(a^2*f*(-1 + 2*n))","A",1
189,1,81,86,0.5834112,"\int (a \cos (e+f x))^m (b \tan (e+f x))^n \, dx","Integrate[(a*Cos[e + f*x])^m*(b*Tan[e + f*x])^n,x]","\frac{\tan (e+f x) \sec ^2(e+f x)^{m/2} (a \cos (e+f x))^m (b \tan (e+f x))^n \, _2F_1\left(\frac{m+2}{2},\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{f (n+1)}","\frac{(a \cos (e+f x))^m (b \tan (e+f x))^{n+1} \cos ^2(e+f x)^{\frac{1}{2} (-m+n+1)} \, _2F_1\left(\frac{n+1}{2},\frac{1}{2} (-m+n+1);\frac{n+3}{2};\sin ^2(e+f x)\right)}{b f (n+1)}",1,"((a*Cos[e + f*x])^m*Hypergeometric2F1[(2 + m)/2, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]*(b*Tan[e + f*x])^n)/(f*(1 + n))","A",1
190,1,66,63,0.0720606,"\int (a \tan (e+f x))^m (b \tan (e+f x))^n \, dx","Integrate[(a*Tan[e + f*x])^m*(b*Tan[e + f*x])^n,x]","\frac{\tan (e+f x) (a \tan (e+f x))^m (b \tan (e+f x))^n \, _2F_1\left(1,\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+1)+1;-\tan ^2(e+f x)\right)}{f (m+n+1)}","\frac{(a \tan (e+f x))^{m+1} (b \tan (e+f x))^n \, _2F_1\left(1,\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);-\tan ^2(e+f x)\right)}{a f (m+n+1)}",1,"(Hypergeometric2F1[1, (1 + m + n)/2, 1 + (1 + m + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(a*Tan[e + f*x])^m*(b*Tan[e + f*x])^n)/(f*(1 + m + n))","A",1
191,1,45,232,0.0784026,"\int \sqrt{d \cot (e+f x)} \tan ^4(e+f x) \, dx","Integrate[Sqrt[d*Cot[e + f*x]]*Tan[e + f*x]^4,x]","\frac{2 \tan ^3(e+f x) \sqrt{d \cot (e+f x)} \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(e+f x)\right)}{5 f}","\frac{2 d^3}{5 f (d \cot (e+f x))^{5/2}}-\frac{2 d}{f \sqrt{d \cot (e+f x)}}-\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}",1,"(2*Sqrt[d*Cot[e + f*x]]*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[e + f*x]^2]*Tan[e + f*x]^3)/(5*f)","C",1
192,1,45,214,0.0452568,"\int \sqrt{d \cot (e+f x)} \tan ^3(e+f x) \, dx","Integrate[Sqrt[d*Cot[e + f*x]]*Tan[e + f*x]^3,x]","\frac{2 \tan ^2(e+f x) \sqrt{d \cot (e+f x)} \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(e+f x)\right)}{3 f}","\frac{2 d^2}{3 f (d \cot (e+f x))^{3/2}}-\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}+\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}",1,"(2*Sqrt[d*Cot[e + f*x]]*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[e + f*x]^2]*Tan[e + f*x]^2)/(3*f)","C",1
193,1,36,210,0.0517176,"\int \sqrt{d \cot (e+f x)} \tan ^2(e+f x) \, dx","Integrate[Sqrt[d*Cot[e + f*x]]*Tan[e + f*x]^2,x]","\frac{2 d \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(e+f x)\right)}{f \sqrt{d \cot (e+f x)}}","\frac{2 d}{f \sqrt{d \cot (e+f x)}}+\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}+\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}",1,"(2*d*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[e + f*x]^2])/(f*Sqrt[d*Cot[e + f*x]])","C",1
194,1,132,192,0.1857235,"\int \sqrt{d \cot (e+f x)} \tan (e+f x) \, dx","Integrate[Sqrt[d*Cot[e + f*x]]*Tan[e + f*x],x]","\frac{d \sqrt{\cot (e+f x)} \left(\log \left(\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right)-\log \left(\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (e+f x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (e+f x)}+1\right)\right)}{2 \sqrt{2} f \sqrt{d \cot (e+f x)}}","\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}",1,"(d*Sqrt[Cot[e + f*x]]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[e + f*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[e + f*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]]))/(2*Sqrt[2]*f*Sqrt[d*Cot[e + f*x]])","A",1
195,1,40,192,0.0417703,"\int \sqrt{d \cot (e+f x)} \, dx","Integrate[Sqrt[d*Cot[e + f*x]],x]","-\frac{2 (d \cot (e+f x))^{3/2} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(e+f x)\right)}{3 d f}","-\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}",1,"(-2*(d*Cot[e + f*x])^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[e + f*x]^2])/(3*d*f)","C",1
196,1,162,209,0.2528754,"\int \cot (e+f x) \sqrt{d \cot (e+f x)} \, dx","Integrate[Cot[e + f*x]*Sqrt[d*Cot[e + f*x]],x]","-\frac{(d \cot (e+f x))^{3/2} \left(8 \sqrt{\cot (e+f x)}+\sqrt{2} \log \left(\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right)-\sqrt{2} \log \left(\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (e+f x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (e+f x)}+1\right)\right)}{4 d f \cot ^{\frac{3}{2}}(e+f x)}","-\frac{2 \sqrt{d \cot (e+f x)}}{f}-\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}+\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}",1,"-1/4*((d*Cot[e + f*x])^(3/2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[e + f*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[e + f*x]]] + 8*Sqrt[Cot[e + f*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]]))/(d*f*Cot[e + f*x]^(3/2))","A",1
197,1,42,214,0.0492515,"\int \cot ^2(e+f x) \sqrt{d \cot (e+f x)} \, dx","Integrate[Cot[e + f*x]^2*Sqrt[d*Cot[e + f*x]],x]","\frac{2 (d \cot (e+f x))^{3/2} \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(e+f x)\right)-1\right)}{3 d f}","-\frac{2 (d \cot (e+f x))^{3/2}}{3 d f}+\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}+\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}",1,"(2*(d*Cot[e + f*x])^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[e + f*x]^2]))/(3*d*f)","C",1
198,1,172,231,0.4517436,"\int \cot ^3(e+f x) \sqrt{d \cot (e+f x)} \, dx","Integrate[Cot[e + f*x]^3*Sqrt[d*Cot[e + f*x]],x]","\frac{\sqrt{d \cot (e+f x)} \left(-8 \cot ^{\frac{5}{2}}(e+f x)+40 \sqrt{\cot (e+f x)}+5 \sqrt{2} \log \left(\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right)-5 \sqrt{2} \log \left(\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right)+10 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (e+f x)}\right)-10 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (e+f x)}+1\right)\right)}{20 f \sqrt{\cot (e+f x)}}","-\frac{2 (d \cot (e+f x))^{5/2}}{5 d^2 f}+\frac{2 \sqrt{d \cot (e+f x)}}{f}+\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{\sqrt{d} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}",1,"(Sqrt[d*Cot[e + f*x]]*(10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[e + f*x]]] - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[e + f*x]]] + 40*Sqrt[Cot[e + f*x]] - 8*Cot[e + f*x]^(5/2) + 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]] - 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]]))/(20*f*Sqrt[Cot[e + f*x]])","A",1
199,1,45,234,0.0659086,"\int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx","Integrate[(d*Cot[e + f*x])^(3/2)*Tan[e + f*x]^5,x]","\frac{2 \tan ^4(e+f x) (d \cot (e+f x))^{3/2} \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(e+f x)\right)}{5 f}","-\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}-\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}+\frac{2 d^4}{5 f (d \cot (e+f x))^{5/2}}-\frac{2 d^2}{f \sqrt{d \cot (e+f x)}}",1,"(2*(d*Cot[e + f*x])^(3/2)*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[e + f*x]^2]*Tan[e + f*x]^4)/(5*f)","C",1
200,1,45,214,0.0465026,"\int (d \cot (e+f x))^{3/2} \tan ^4(e+f x) \, dx","Integrate[(d*Cot[e + f*x])^(3/2)*Tan[e + f*x]^4,x]","\frac{2 \tan ^3(e+f x) (d \cot (e+f x))^{3/2} \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(e+f x)\right)}{3 f}","-\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}+\frac{2 d^3}{3 f (d \cot (e+f x))^{3/2}}",1,"(2*(d*Cot[e + f*x])^(3/2)*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[e + f*x]^2]*Tan[e + f*x]^3)/(3*f)","C",1
201,1,38,212,0.0490758,"\int (d \cot (e+f x))^{3/2} \tan ^3(e+f x) \, dx","Integrate[(d*Cot[e + f*x])^(3/2)*Tan[e + f*x]^3,x]","\frac{2 d^2 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(e+f x)\right)}{f \sqrt{d \cot (e+f x)}}","\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}+\frac{2 d^2}{f \sqrt{d \cot (e+f x)}}",1,"(2*d^2*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[e + f*x]^2])/(f*Sqrt[d*Cot[e + f*x]])","C",1
202,1,134,192,0.0262028,"\int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx","Integrate[(d*Cot[e + f*x])^(3/2)*Tan[e + f*x]^2,x]","\frac{d^2 \sqrt{\cot (e+f x)} \left(\log \left(\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right)-\log \left(\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (e+f x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (e+f x)}+1\right)\right)}{2 \sqrt{2} f \sqrt{d \cot (e+f x)}}","\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}-\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}",1,"(d^2*Sqrt[Cot[e + f*x]]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[e + f*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[e + f*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]]))/(2*Sqrt[2]*f*Sqrt[d*Cot[e + f*x]])","A",1
203,1,37,192,0.0096315,"\int (d \cot (e+f x))^{3/2} \tan (e+f x) \, dx","Integrate[(d*Cot[e + f*x])^(3/2)*Tan[e + f*x],x]","-\frac{2 (d \cot (e+f x))^{3/2} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(e+f x)\right)}{3 f}","-\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}-\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}",1,"(-2*(d*Cot[e + f*x])^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[e + f*x]^2])/(3*f)","C",1
204,1,159,210,0.1907745,"\int (d \cot (e+f x))^{3/2} \, dx","Integrate[(d*Cot[e + f*x])^(3/2),x]","-\frac{(d \cot (e+f x))^{3/2} \left(8 \sqrt{\cot (e+f x)}+\sqrt{2} \log \left(\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right)-\sqrt{2} \log \left(\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (e+f x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (e+f x)}+1\right)\right)}{4 f \cot ^{\frac{3}{2}}(e+f x)}","-\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}-\frac{2 d \sqrt{d \cot (e+f x)}}{f}",1,"-1/4*((d*Cot[e + f*x])^(3/2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[e + f*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[e + f*x]]] + 8*Sqrt[Cot[e + f*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]]))/(f*Cot[e + f*x]^(3/2))","A",1
205,1,39,211,0.0592699,"\int \cot (e+f x) (d \cot (e+f x))^{3/2} \, dx","Integrate[Cot[e + f*x]*(d*Cot[e + f*x])^(3/2),x]","\frac{2 (d \cot (e+f x))^{3/2} \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(e+f x)\right)-1\right)}{3 f}","\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}-\frac{2 (d \cot (e+f x))^{3/2}}{3 f}",1,"(2*(d*Cot[e + f*x])^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[e + f*x]^2]))/(3*f)","C",1
206,1,172,232,0.2695779,"\int \cot ^2(e+f x) (d \cot (e+f x))^{3/2} \, dx","Integrate[Cot[e + f*x]^2*(d*Cot[e + f*x])^(3/2),x]","\frac{(d \cot (e+f x))^{3/2} \left(-8 \cot ^{\frac{5}{2}}(e+f x)+40 \sqrt{\cot (e+f x)}+5 \sqrt{2} \log \left(\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right)-5 \sqrt{2} \log \left(\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right)+10 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (e+f x)}\right)-10 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (e+f x)}+1\right)\right)}{20 f \cot ^{\frac{3}{2}}(e+f x)}","\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{d^{3/2} \log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}-\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}-\frac{2 (d \cot (e+f x))^{5/2}}{5 d f}+\frac{2 d \sqrt{d \cot (e+f x)}}{f}",1,"((d*Cot[e + f*x])^(3/2)*(10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[e + f*x]]] - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[e + f*x]]] + 40*Sqrt[Cot[e + f*x]] - 8*Cot[e + f*x]^(5/2) + 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]] - 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]]))/(20*f*Cot[e + f*x]^(3/2))","A",1
207,1,40,231,0.1123398,"\int \frac{\tan ^3(e+f x)}{\sqrt{d \cot (e+f x)}} \, dx","Integrate[Tan[e + f*x]^3/Sqrt[d*Cot[e + f*x]],x]","\frac{2 d^2 \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(e+f x)\right)}{5 f (d \cot (e+f x))^{5/2}}","\frac{2 d^2}{5 f (d \cot (e+f x))^{5/2}}-\frac{2}{f \sqrt{d \cot (e+f x)}}-\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} \sqrt{d} f}+\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} \sqrt{d} f}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} \sqrt{d} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} \sqrt{d} f}",1,"(2*d^2*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[e + f*x]^2])/(5*f*(d*Cot[e + f*x])^(5/2))","C",1
208,1,38,212,0.0925212,"\int \frac{\tan ^2(e+f x)}{\sqrt{d \cot (e+f x)}} \, dx","Integrate[Tan[e + f*x]^2/Sqrt[d*Cot[e + f*x]],x]","\frac{2 d \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(e+f x)\right)}{3 f (d \cot (e+f x))^{3/2}}","\frac{2 d}{3 f (d \cot (e+f x))^{3/2}}-\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} \sqrt{d} f}+\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} \sqrt{d} f}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} \sqrt{d} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} \sqrt{d} f}",1,"(2*d*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[e + f*x]^2])/(3*f*(d*Cot[e + f*x])^(3/2))","C",1
209,1,35,209,0.0682043,"\int \frac{\tan (e+f x)}{\sqrt{d \cot (e+f x)}} \, dx","Integrate[Tan[e + f*x]/Sqrt[d*Cot[e + f*x]],x]","\frac{2 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(e+f x)\right)}{f \sqrt{d \cot (e+f x)}}","\frac{2}{f \sqrt{d \cot (e+f x)}}+\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} \sqrt{d} f}-\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} \sqrt{d} f}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} \sqrt{d} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} \sqrt{d} f}",1,"(2*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[e + f*x]^2])/(f*Sqrt[d*Cot[e + f*x]])","C",1
210,1,131,192,0.0149416,"\int \frac{1}{\sqrt{d \cot (e+f x)}} \, dx","Integrate[1/Sqrt[d*Cot[e + f*x]],x]","\frac{\sqrt{\cot (e+f x)} \left(\log \left(\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right)-\log \left(\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (e+f x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (e+f x)}+1\right)\right)}{2 \sqrt{2} f \sqrt{d \cot (e+f x)}}","\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} \sqrt{d} f}-\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} \sqrt{d} f}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} \sqrt{d} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} \sqrt{d} f}",1,"(Sqrt[Cot[e + f*x]]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[e + f*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[e + f*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]]))/(2*Sqrt[2]*f*Sqrt[d*Cot[e + f*x]])","A",1
211,1,40,192,0.04083,"\int \frac{\cot (e+f x)}{\sqrt{d \cot (e+f x)}} \, dx","Integrate[Cot[e + f*x]/Sqrt[d*Cot[e + f*x]],x]","-\frac{2 (d \cot (e+f x))^{3/2} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(e+f x)\right)}{3 d^2 f}","-\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} \sqrt{d} f}+\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} \sqrt{d} f}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} \sqrt{d} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} \sqrt{d} f}",1,"(-2*(d*Cot[e + f*x])^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[e + f*x]^2])/(3*d^2*f)","C",1
212,1,159,212,0.1688522,"\int \frac{\cot ^2(e+f x)}{\sqrt{d \cot (e+f x)}} \, dx","Integrate[Cot[e + f*x]^2/Sqrt[d*Cot[e + f*x]],x]","-\frac{\sqrt{\cot (e+f x)} \left(8 \sqrt{\cot (e+f x)}+\sqrt{2} \log \left(\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right)-\sqrt{2} \log \left(\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (e+f x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (e+f x)}+1\right)\right)}{4 f \sqrt{d \cot (e+f x)}}","-\frac{2 \sqrt{d \cot (e+f x)}}{d f}-\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} \sqrt{d} f}+\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} \sqrt{d} f}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} \sqrt{d} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} \sqrt{d} f}",1,"-1/4*(Sqrt[Cot[e + f*x]]*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[e + f*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[e + f*x]]] + 8*Sqrt[Cot[e + f*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]]))/(f*Sqrt[d*Cot[e + f*x]])","A",1
213,1,47,214,0.0640615,"\int \frac{\cot ^3(e+f x)}{\sqrt{d \cot (e+f x)}} \, dx","Integrate[Cot[e + f*x]^3/Sqrt[d*Cot[e + f*x]],x]","\frac{2 \cot ^2(e+f x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(e+f x)\right)-1\right)}{3 f \sqrt{d \cot (e+f x)}}","-\frac{2 (d \cot (e+f x))^{3/2}}{3 d^2 f}+\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} \sqrt{d} f}-\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} \sqrt{d} f}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} \sqrt{d} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} \sqrt{d} f}",1,"(2*Cot[e + f*x]^2*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[e + f*x]^2]))/(3*f*Sqrt[d*Cot[e + f*x]])","C",1
214,1,38,232,0.1680739,"\int \frac{\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx","Integrate[Tan[e + f*x]^2/(d*Cot[e + f*x])^(3/2),x]","\frac{2 d \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(e+f x)\right)}{5 f (d \cot (e+f x))^{5/2}}","-\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}+\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} d^{3/2} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} d^{3/2} f}+\frac{2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac{2}{d f \sqrt{d \cot (e+f x)}}",1,"(2*d*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[e + f*x]^2])/(5*f*(d*Cot[e + f*x])^(5/2))","C",1
215,1,37,211,0.0778029,"\int \frac{\tan (e+f x)}{(d \cot (e+f x))^{3/2}} \, dx","Integrate[Tan[e + f*x]/(d*Cot[e + f*x])^(3/2),x]","\frac{2 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(e+f x)\right)}{3 f (d \cot (e+f x))^{3/2}}","-\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}+\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} d^{3/2} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} d^{3/2} f}+\frac{2}{3 f (d \cot (e+f x))^{3/2}}",1,"(2*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[e + f*x]^2])/(3*f*(d*Cot[e + f*x])^(3/2))","C",1
216,1,38,212,0.0056536,"\int \frac{1}{(d \cot (e+f x))^{3/2}} \, dx","Integrate[(d*Cot[e + f*x])^(-3/2),x]","\frac{2 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(e+f x)\right)}{d f \sqrt{d \cot (e+f x)}}","\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}-\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} d^{3/2} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} d^{3/2} f}+\frac{2}{d f \sqrt{d \cot (e+f x)}}",1,"(2*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[e + f*x]^2])/(d*f*Sqrt[d*Cot[e + f*x]])","C",1
217,1,134,192,0.0306287,"\int \frac{\cot (e+f x)}{(d \cot (e+f x))^{3/2}} \, dx","Integrate[Cot[e + f*x]/(d*Cot[e + f*x])^(3/2),x]","\frac{\sqrt{\cot (e+f x)} \left(\log \left(\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right)-\log \left(\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (e+f x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (e+f x)}+1\right)\right)}{2 \sqrt{2} d f \sqrt{d \cot (e+f x)}}","\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}-\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} d^{3/2} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} d^{3/2} f}",1,"(Sqrt[Cot[e + f*x]]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[e + f*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[e + f*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]]))/(2*Sqrt[2]*d*f*Sqrt[d*Cot[e + f*x]])","A",1
218,1,40,192,0.0098665,"\int \frac{\cot ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx","Integrate[Cot[e + f*x]^2/(d*Cot[e + f*x])^(3/2),x]","-\frac{2 (d \cot (e+f x))^{3/2} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(e+f x)\right)}{3 d^3 f}","-\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}+\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} d^{3/2} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} d^{3/2} f}",1,"(-2*(d*Cot[e + f*x])^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[e + f*x]^2])/(3*d^3*f)","C",1
219,1,159,212,0.1561063,"\int \frac{\cot ^3(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx","Integrate[Cot[e + f*x]^3/(d*Cot[e + f*x])^(3/2),x]","-\frac{\cot ^{\frac{3}{2}}(e+f x) \left(8 \sqrt{\cot (e+f x)}+\sqrt{2} \log \left(\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right)-\sqrt{2} \log \left(\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (e+f x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (e+f x)}+1\right)\right)}{4 f (d \cot (e+f x))^{3/2}}","-\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}+\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} d^{3/2} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} d^{3/2} f}-\frac{2 \sqrt{d \cot (e+f x)}}{d^2 f}",1,"-1/4*(Cot[e + f*x]^(3/2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[e + f*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[e + f*x]]] + 8*Sqrt[Cot[e + f*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]]))/(f*(d*Cot[e + f*x])^(3/2))","A",1
220,1,47,214,0.0888531,"\int \frac{\cot ^4(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx","Integrate[Cot[e + f*x]^4/(d*Cot[e + f*x])^(3/2),x]","\frac{2 \cot ^3(e+f x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(e+f x)\right)-1\right)}{3 f (d \cot (e+f x))^{3/2}}","\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}-\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} d^{3/2} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} d^{3/2} f}-\frac{2 (d \cot (e+f x))^{3/2}}{3 d^3 f}",1,"(2*Cot[e + f*x]^3*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[e + f*x]^2]))/(3*f*(d*Cot[e + f*x])^(3/2))","C",1
221,1,172,234,0.4373338,"\int \frac{\cot ^5(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx","Integrate[Cot[e + f*x]^5/(d*Cot[e + f*x])^(3/2),x]","\frac{\cot ^{\frac{3}{2}}(e+f x) \left(-8 \cot ^{\frac{5}{2}}(e+f x)+40 \sqrt{\cot (e+f x)}+5 \sqrt{2} \log \left(\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right)-5 \sqrt{2} \log \left(\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right)+10 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (e+f x)}\right)-10 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (e+f x)}+1\right)\right)}{20 f (d \cot (e+f x))^{3/2}}","\frac{\log \left(\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}-\frac{\log \left(\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} d^{3/2} f}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} d^{3/2} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} d^{3/2} f}-\frac{2 (d \cot (e+f x))^{5/2}}{5 d^4 f}+\frac{2 \sqrt{d \cot (e+f x)}}{d^2 f}",1,"(Cot[e + f*x]^(3/2)*(10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[e + f*x]]] - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[e + f*x]]] + 40*Sqrt[Cot[e + f*x]] - 8*Cot[e + f*x]^(5/2) + 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]] - 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]]))/(20*f*(d*Cot[e + f*x])^(3/2))","A",1
222,1,62,62,0.0943162,"\int \cot ^m(e+f x) \tan ^n(e+f x) \, dx","Integrate[Cot[e + f*x]^m*Tan[e + f*x]^n,x]","\frac{\cot ^m(e+f x) \tan ^{n+1}(e+f x) \, _2F_1\left(1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right)}{f (-m+n+1)}","\frac{\cot ^m(e+f x) \tan ^{n+1}(e+f x) \, _2F_1\left(1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right)}{f (-m+n+1)}",1,"(Cot[e + f*x]^m*Hypergeometric2F1[1, (1 - m + n)/2, (3 - m + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^(1 + n))/(f*(1 - m + n))","A",1
223,1,64,67,0.0667881,"\int \cot ^m(e+f x) (b \tan (e+f x))^n \, dx","Integrate[Cot[e + f*x]^m*(b*Tan[e + f*x])^n,x]","\frac{\cot ^{m-1}(e+f x) (b \tan (e+f x))^n \, _2F_1\left(1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right)}{f (-m+n+1)}","\frac{\cot ^m(e+f x) (b \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right)}{b f (-m+n+1)}",1,"(Cot[e + f*x]^(-1 + m)*Hypergeometric2F1[1, (1 - m + n)/2, (3 - m + n)/2, -Tan[e + f*x]^2]*(b*Tan[e + f*x])^n)/(f*(1 - m + n))","A",1
224,1,64,64,0.0626794,"\int (a \cot (e+f x))^m \tan ^n(e+f x) \, dx","Integrate[(a*Cot[e + f*x])^m*Tan[e + f*x]^n,x]","\frac{\tan ^{n+1}(e+f x) (a \cot (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right)}{f (-m+n+1)}","\frac{\tan ^{n+1}(e+f x) (a \cot (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right)}{f (-m+n+1)}",1,"((a*Cot[e + f*x])^m*Hypergeometric2F1[1, (1 - m + n)/2, (3 - m + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^(1 + n))/(f*(1 - m + n))","A",1
225,1,67,69,0.0946344,"\int (a \cot (e+f x))^m (b \tan (e+f x))^n \, dx","Integrate[(a*Cot[e + f*x])^m*(b*Tan[e + f*x])^n,x]","\frac{a (a \cot (e+f x))^{m-1} (b \tan (e+f x))^n \, _2F_1\left(1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right)}{f (-m+n+1)}","\frac{(a \cot (e+f x))^m (b \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right)}{b f (-m+n+1)}",1,"(a*(a*Cot[e + f*x])^(-1 + m)*Hypergeometric2F1[1, (1 - m + n)/2, (3 - m + n)/2, -Tan[e + f*x]^2]*(b*Tan[e + f*x])^n)/(f*(1 - m + n))","A",1
226,1,52,67,0.1950799,"\int \sec ^6(e+f x) \sqrt{d \tan (e+f x)} \, dx","Integrate[Sec[e + f*x]^6*Sqrt[d*Tan[e + f*x]],x]","\frac{2 (28 \cos (2 (e+f x))+4 \cos (4 (e+f x))+45) \sec ^4(e+f x) (d \tan (e+f x))^{3/2}}{231 d f}","\frac{2 (d \tan (e+f x))^{11/2}}{11 d^5 f}+\frac{4 (d \tan (e+f x))^{7/2}}{7 d^3 f}+\frac{2 (d \tan (e+f x))^{3/2}}{3 d f}",1,"(2*(45 + 28*Cos[2*(e + f*x)] + 4*Cos[4*(e + f*x)])*Sec[e + f*x]^4*(d*Tan[e + f*x])^(3/2))/(231*d*f)","A",1
227,1,34,45,0.1466918,"\int \sec ^4(e+f x) \sqrt{d \tan (e+f x)} \, dx","Integrate[Sec[e + f*x]^4*Sqrt[d*Tan[e + f*x]],x]","\frac{2 \left(3 \sec ^2(e+f x)+4\right) (d \tan (e+f x))^{3/2}}{21 d f}","\frac{2 (d \tan (e+f x))^{7/2}}{7 d^3 f}+\frac{2 (d \tan (e+f x))^{3/2}}{3 d f}",1,"(2*(4 + 3*Sec[e + f*x]^2)*(d*Tan[e + f*x])^(3/2))/(21*d*f)","A",1
228,1,22,22,0.0368323,"\int \sec ^2(e+f x) \sqrt{d \tan (e+f x)} \, dx","Integrate[Sec[e + f*x]^2*Sqrt[d*Tan[e + f*x]],x]","\frac{2 (d \tan (e+f x))^{3/2}}{3 d f}","\frac{2 (d \tan (e+f x))^{3/2}}{3 d f}",1,"(2*(d*Tan[e + f*x])^(3/2))/(3*d*f)","A",1
229,1,40,192,0.0389442,"\int \sqrt{d \tan (e+f x)} \, dx","Integrate[Sqrt[d*Tan[e + f*x]],x]","\frac{2 (d \tan (e+f x))^{3/2} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)}{3 d f}","-\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}+\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}+\frac{\sqrt{d} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}-\frac{\sqrt{d} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}",1,"(2*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(3/2))/(3*d*f)","C",1
230,1,102,227,0.1932088,"\int \cos ^2(e+f x) \sqrt{d \tan (e+f x)} \, dx","Integrate[Cos[e + f*x]^2*Sqrt[d*Tan[e + f*x]],x]","-\frac{\sqrt{\sin (2 (e+f x))} \sqrt{d \tan (e+f x)} \left(-2 \sqrt{\sin (2 (e+f x))}+\csc (e+f x) \sin ^{-1}(\cos (e+f x)-\sin (e+f x))+\csc (e+f x) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)\right)}{8 f}","-\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{4 \sqrt{2} f}+\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{4 \sqrt{2} f}+\frac{\sqrt{d} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{8 \sqrt{2} f}-\frac{\sqrt{d} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{8 \sqrt{2} f}+\frac{\cos ^2(e+f x) (d \tan (e+f x))^{3/2}}{2 d f}",1,"-1/8*((ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Csc[e + f*x] + Csc[e + f*x]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]] - 2*Sqrt[Sin[2*(e + f*x)]])*Sqrt[Sin[2*(e + f*x)]]*Sqrt[d*Tan[e + f*x]])/f","A",1
231,1,102,107,0.4429894,"\int \sec ^3(e+f x) \sqrt{d \tan (e+f x)} \, dx","Integrate[Sec[e + f*x]^3*Sqrt[d*Tan[e + f*x]],x]","\frac{2 \sqrt{d \tan (e+f x)} \left(3 \sqrt{\sec ^2(e+f x)} (2 \sin (e+f x)+\tan (e+f x) \sec (e+f x))-4 \tan (e+f x) \sec (e+f x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(e+f x)\right)\right)}{15 f \sqrt{\sec ^2(e+f x)}}","\frac{4 \cos (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac{2 \sec (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}-\frac{4 \cos (e+f x) E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (e+f x)}}{5 f \sqrt{\sin (2 e+2 f x)}}",1,"(2*Sqrt[d*Tan[e + f*x]]*(-4*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[e + f*x]^2]*Sec[e + f*x]*Tan[e + f*x] + 3*Sqrt[Sec[e + f*x]^2]*(2*Sin[e + f*x] + Sec[e + f*x]*Tan[e + f*x])))/(15*f*Sqrt[Sec[e + f*x]^2])","C",1
232,1,61,75,0.2768102,"\int \sec (e+f x) \sqrt{d \tan (e+f x)} \, dx","Integrate[Sec[e + f*x]*Sqrt[d*Tan[e + f*x]],x]","-\frac{2 \sin (e+f x) \sqrt{d \tan (e+f x)} \left(2 \sqrt{\sec ^2(e+f x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(e+f x)\right)-3\right)}{3 f}","\frac{2 \cos (e+f x) (d \tan (e+f x))^{3/2}}{d f}-\frac{2 \cos (e+f x) E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (e+f x)}}{f \sqrt{\sin (2 e+2 f x)}}",1,"(-2*(-3 + 2*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[e + f*x]^2]*Sqrt[Sec[e + f*x]^2])*Sin[e + f*x]*Sqrt[d*Tan[e + f*x]])/(3*f)","C",1
233,1,57,47,0.0993803,"\int \cos (e+f x) \sqrt{d \tan (e+f x)} \, dx","Integrate[Cos[e + f*x]*Sqrt[d*Tan[e + f*x]],x]","\frac{2 \sin (e+f x) \sqrt{\sec ^2(e+f x)} \sqrt{d \tan (e+f x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(e+f x)\right)}{3 f}","\frac{\cos (e+f x) E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (e+f x)}}{f \sqrt{\sin (2 e+2 f x)}}",1,"(2*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[e + f*x]^2]*Sqrt[Sec[e + f*x]^2]*Sin[e + f*x]*Sqrt[d*Tan[e + f*x]])/(3*f)","C",1
234,1,94,81,0.4565195,"\int \cos ^3(e+f x) \sqrt{d \tan (e+f x)} \, dx","Integrate[Cos[e + f*x]^3*Sqrt[d*Tan[e + f*x]],x]","\frac{\sqrt{d \tan (e+f x)} \left(4 \tan (e+f x) \sec (e+f x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(e+f x)\right)+(\sin (e+f x)+\sin (3 (e+f x))) \sqrt{\sec ^2(e+f x)}\right)}{12 f \sqrt{\sec ^2(e+f x)}}","\frac{\cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{3 d f}+\frac{\cos (e+f x) E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (e+f x)}}{2 f \sqrt{\sin (2 e+2 f x)}}",1,"(Sqrt[d*Tan[e + f*x]]*(Sqrt[Sec[e + f*x]^2]*(Sin[e + f*x] + Sin[3*(e + f*x)]) + 4*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[e + f*x]^2]*Sec[e + f*x]*Tan[e + f*x]))/(12*f*Sqrt[Sec[e + f*x]^2])","C",1
235,1,86,111,0.7774805,"\int \cos ^5(e+f x) \sqrt{d \tan (e+f x)} \, dx","Integrate[Cos[e + f*x]^5*Sqrt[d*Tan[e + f*x]],x]","\frac{\cos (e+f x) \sqrt{d \tan (e+f x)} \left(28 \tan (e+f x) \sqrt{\sec ^2(e+f x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(e+f x)\right)+20 \sin (2 (e+f x))+3 \sin (4 (e+f x))\right)}{120 f}","\frac{\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac{7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac{7 \cos (e+f x) E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (e+f x)}}{20 f \sqrt{\sin (2 e+2 f x)}}",1,"(Cos[e + f*x]*Sqrt[d*Tan[e + f*x]]*(20*Sin[2*(e + f*x)] + 3*Sin[4*(e + f*x)] + 28*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[e + f*x]^2]*Sqrt[Sec[e + f*x]^2]*Tan[e + f*x]))/(120*f)","C",1
236,1,52,67,0.1443766,"\int \sec ^6(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Sec[a + b*x]^6*(d*Tan[a + b*x])^(3/2),x]","\frac{2 d \left(45 \sec ^6(a+b x)-5 \sec ^4(a+b x)-8 \sec ^2(a+b x)-32\right) \sqrt{d \tan (a+b x)}}{585 b}","\frac{2 (d \tan (a+b x))^{13/2}}{13 b d^5}+\frac{4 (d \tan (a+b x))^{9/2}}{9 b d^3}+\frac{2 (d \tan (a+b x))^{5/2}}{5 b d}",1,"(2*d*(-32 - 8*Sec[a + b*x]^2 - 5*Sec[a + b*x]^4 + 45*Sec[a + b*x]^6)*Sqrt[d*Tan[a + b*x]])/(585*b)","A",1
237,1,42,45,0.1363493,"\int \sec ^4(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Sec[a + b*x]^4*(d*Tan[a + b*x])^(3/2),x]","\frac{2 d \left(5 \sec ^4(a+b x)-\sec ^2(a+b x)-4\right) \sqrt{d \tan (a+b x)}}{45 b}","\frac{2 (d \tan (a+b x))^{9/2}}{9 b d^3}+\frac{2 (d \tan (a+b x))^{5/2}}{5 b d}",1,"(2*d*(-4 - Sec[a + b*x]^2 + 5*Sec[a + b*x]^4)*Sqrt[d*Tan[a + b*x]])/(45*b)","A",1
238,1,22,22,0.0583811,"\int \sec ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Sec[a + b*x]^2*(d*Tan[a + b*x])^(3/2),x]","\frac{2 (d \tan (a+b x))^{5/2}}{5 b d}","\frac{2 (d \tan (a+b x))^{5/2}}{5 b d}",1,"(2*(d*Tan[a + b*x])^(5/2))/(5*b*d)","A",1
239,1,159,210,0.2666537,"\int (d \tan (a+b x))^{3/2} \, dx","Integrate[(d*Tan[a + b*x])^(3/2),x]","\frac{(d \tan (a+b x))^{3/2} \left(2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (a+b x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (a+b x)}+1\right)+8 \sqrt{\tan (a+b x)}+\sqrt{2} \log \left(\tan (a+b x)-\sqrt{2} \sqrt{\tan (a+b x)}+1\right)-\sqrt{2} \log \left(\tan (a+b x)+\sqrt{2} \sqrt{\tan (a+b x)}+1\right)\right)}{4 b \tan ^{\frac{3}{2}}(a+b x)}","\frac{d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{\sqrt{2} b}-\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{\sqrt{2} b}+\frac{d^{3/2} \log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{2 \sqrt{2} b}-\frac{d^{3/2} \log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{2 \sqrt{2} b}+\frac{2 d \sqrt{d \tan (a+b x)}}{b}",1,"((2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]] + 8*Sqrt[Tan[a + b*x]])*(d*Tan[a + b*x])^(3/2))/(4*b*Tan[a + b*x]^(3/2))","A",1
240,1,110,225,0.2767405,"\int \cos ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Cos[a + b*x]^2*(d*Tan[a + b*x])^(3/2),x]","-\frac{d \csc (a+b x) \sqrt{d \tan (a+b x)} \left(\sin (a+b x)+\sin (3 (a+b x))+\sqrt{\sin (2 (a+b x))} \sin ^{-1}(\cos (a+b x)-\sin (a+b x))-\sqrt{\sin (2 (a+b x))} \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{8 b}","-\frac{d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{4 \sqrt{2} b}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{4 \sqrt{2} b}-\frac{d^{3/2} \log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b}+\frac{d^{3/2} \log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b}-\frac{d \cos ^2(a+b x) \sqrt{d \tan (a+b x)}}{2 b}",1,"-1/8*(d*Csc[a + b*x]*(Sin[a + b*x] + ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Sqrt[Sin[2*(a + b*x)]] - Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]] + Sin[3*(a + b*x)])*Sqrt[d*Tan[a + b*x]])/b","A",1
241,1,90,136,0.8242335,"\int \sec ^5(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Sec[a + b*x]^5*(d*Tan[a + b*x])^(3/2),x]","-\frac{d \sec ^5(a+b x) \sqrt{d \tan (a+b x)} \left(16 \cos ^6(a+b x) \sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(a+b x)\right)+6 \cos (2 (a+b x))+\cos (4 (a+b x))-23\right)}{154 b}","-\frac{4 d^2 \sqrt{\sin (2 a+2 b x)} \sec (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{77 b \sqrt{d \tan (a+b x)}}+\frac{2 d \sec ^5(a+b x) \sqrt{d \tan (a+b x)}}{11 b}-\frac{2 d \sec ^3(a+b x) \sqrt{d \tan (a+b x)}}{77 b}-\frac{4 d \sec (a+b x) \sqrt{d \tan (a+b x)}}{77 b}",1,"-1/154*(d*Sec[a + b*x]^5*(-23 + 6*Cos[2*(a + b*x)] + Cos[4*(a + b*x)] + 16*Cos[a + b*x]^6*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2])*Sqrt[d*Tan[a + b*x]])/b","C",1
242,1,80,108,0.4754828,"\int \sec ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Sec[a + b*x]^3*(d*Tan[a + b*x])^(3/2),x]","-\frac{d \sec ^3(a+b x) \sqrt{d \tan (a+b x)} \left(4 \cos ^4(a+b x) \sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(a+b x)\right)+\cos (2 (a+b x))-5\right)}{21 b}","-\frac{2 d^2 \sqrt{\sin (2 a+2 b x)} \sec (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{21 b \sqrt{d \tan (a+b x)}}+\frac{2 d \sec ^3(a+b x) \sqrt{d \tan (a+b x)}}{7 b}-\frac{2 d \sec (a+b x) \sqrt{d \tan (a+b x)}}{21 b}",1,"-1/21*(d*Sec[a + b*x]^3*(-5 + Cos[2*(a + b*x)] + 4*Cos[a + b*x]^4*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2])*Sqrt[d*Tan[a + b*x]])/b","C",1
243,1,69,80,0.3137501,"\int \sec (a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Sec[a + b*x]*(d*Tan[a + b*x])^(3/2),x]","\frac{2 d \cos (a+b x) \sqrt{d \tan (a+b x)} \left(\sec ^2(a+b x)-\sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(a+b x)\right)\right)}{3 b}","\frac{2 d \sec (a+b x) \sqrt{d \tan (a+b x)}}{3 b}-\frac{d^2 \sqrt{\sin (2 a+2 b x)} \sec (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{3 b \sqrt{d \tan (a+b x)}}",1,"(2*d*Cos[a + b*x]*(Sec[a + b*x]^2 - Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2])*Sqrt[d*Tan[a + b*x]])/(3*b)","C",1
244,1,58,78,0.1446299,"\int \cos (a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Cos[a + b*x]*(d*Tan[a + b*x])^(3/2),x]","\frac{d \cos (a+b x) \sqrt{d \tan (a+b x)} \left(\sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(a+b x)\right)-1\right)}{b}","\frac{d^2 \sqrt{\sin (2 a+2 b x)} \sec (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2 b \sqrt{d \tan (a+b x)}}-\frac{d \cos (a+b x) \sqrt{d \tan (a+b x)}}{b}",1,"(d*Cos[a + b*x]*(-1 + Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2])*Sqrt[d*Tan[a + b*x]])/b","C",1
245,1,96,108,1.1271089,"\int \cos ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Cos[a + b*x]^3*(d*Tan[a + b*x])^(3/2),x]","-\frac{\cos (a+b x) (d \tan (a+b x))^{3/2} \left(\cos (2 (a+b x)) \sqrt{\tan (a+b x)}+\sqrt[4]{-1} \sqrt{\sec ^2(a+b x)} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)\right)}{6 b \tan ^{\frac{3}{2}}(a+b x)}","\frac{d^2 \sqrt{\sin (2 a+2 b x)} \sec (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{12 b \sqrt{d \tan (a+b x)}}-\frac{d \cos ^3(a+b x) \sqrt{d \tan (a+b x)}}{3 b}+\frac{d \cos (a+b x) \sqrt{d \tan (a+b x)}}{6 b}",1,"-1/6*(Cos[a + b*x]*((-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Sqrt[Sec[a + b*x]^2] + Cos[2*(a + b*x)]*Sqrt[Tan[a + b*x]])*(d*Tan[a + b*x])^(3/2))/(b*Tan[a + b*x]^(3/2))","C",1
246,1,131,136,2.4694311,"\int \cos ^5(a+b x) (d \tan (a+b x))^{3/2} \, dx","Integrate[Cos[a + b*x]^5*(d*Tan[a + b*x])^(3/2),x]","\frac{\cos (2 (a+b x)) \csc (a+b x) (d \tan (a+b x))^{3/2} \left((10 \cos (2 (a+b x))+3 \cos (4 (a+b x))-3) \sqrt{\tan (a+b x)}+10 \sqrt[4]{-1} \sqrt{\sec ^2(a+b x)} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)\right)}{120 b \sqrt{\tan (a+b x)} \left(\tan ^2(a+b x)-1\right)}","\frac{d^2 \sqrt{\sin (2 a+2 b x)} \sec (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{24 b \sqrt{d \tan (a+b x)}}-\frac{d \cos ^5(a+b x) \sqrt{d \tan (a+b x)}}{5 b}+\frac{d \cos ^3(a+b x) \sqrt{d \tan (a+b x)}}{30 b}+\frac{d \cos (a+b x) \sqrt{d \tan (a+b x)}}{12 b}",1,"(Cos[2*(a + b*x)]*Csc[a + b*x]*(10*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Sqrt[Sec[a + b*x]^2] + (-3 + 10*Cos[2*(a + b*x)] + 3*Cos[4*(a + b*x)])*Sqrt[Tan[a + b*x]])*(d*Tan[a + b*x])^(3/2))/(120*b*Sqrt[Tan[a + b*x]]*(-1 + Tan[a + b*x]^2))","C",1
247,1,52,67,0.443565,"\int \sec ^6(e+f x) (d \tan (e+f x))^{5/2} \, dx","Integrate[Sec[e + f*x]^6*(d*Tan[e + f*x])^(5/2),x]","\frac{2 (44 \cos (2 (e+f x))+4 \cos (4 (e+f x))+117) \sec ^4(e+f x) (d \tan (e+f x))^{7/2}}{1155 d f}","\frac{2 (d \tan (e+f x))^{15/2}}{15 d^5 f}+\frac{4 (d \tan (e+f x))^{11/2}}{11 d^3 f}+\frac{2 (d \tan (e+f x))^{7/2}}{7 d f}",1,"(2*(117 + 44*Cos[2*(e + f*x)] + 4*Cos[4*(e + f*x)])*Sec[e + f*x]^4*(d*Tan[e + f*x])^(7/2))/(1155*d*f)","A",1
248,1,42,45,0.2733526,"\int \sec ^4(e+f x) (d \tan (e+f x))^{5/2} \, dx","Integrate[Sec[e + f*x]^4*(d*Tan[e + f*x])^(5/2),x]","\frac{2 (2 \cos (2 (e+f x))+9) \sec ^2(e+f x) (d \tan (e+f x))^{7/2}}{77 d f}","\frac{2 (d \tan (e+f x))^{11/2}}{11 d^3 f}+\frac{2 (d \tan (e+f x))^{7/2}}{7 d f}",1,"(2*(9 + 2*Cos[2*(e + f*x)])*Sec[e + f*x]^2*(d*Tan[e + f*x])^(7/2))/(77*d*f)","A",1
249,1,22,22,0.06353,"\int \sec ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx","Integrate[Sec[e + f*x]^2*(d*Tan[e + f*x])^(5/2),x]","\frac{2 (d \tan (e+f x))^{7/2}}{7 d f}","\frac{2 (d \tan (e+f x))^{7/2}}{7 d f}",1,"(2*(d*Tan[e + f*x])^(7/2))/(7*d*f)","A",1
250,1,40,212,0.0435601,"\int (d \tan (e+f x))^{5/2} \, dx","Integrate[(d*Tan[e + f*x])^(5/2),x]","-\frac{2 d (d \tan (e+f x))^{3/2} \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)-1\right)}{3 f}","\frac{d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} f}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} f}-\frac{d^{5/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{d^{5/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{2 \sqrt{2} f}+\frac{2 d (d \tan (e+f x))^{3/2}}{3 f}",1,"(-2*d*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2])*(d*Tan[e + f*x])^(3/2))/(3*f)","C",1
251,1,107,225,0.1939568,"\int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx","Integrate[Cos[e + f*x]^2*(d*Tan[e + f*x])^(5/2),x]","-\frac{d^2 \sqrt{\sin (2 (e+f x))} \sqrt{d \tan (e+f x)} \left(2 \sqrt{\sin (2 (e+f x))}+3 \csc (e+f x) \sin ^{-1}(\cos (e+f x)-\sin (e+f x))+3 \csc (e+f x) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)\right)}{8 f}","-\frac{3 d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{4 \sqrt{2} f}+\frac{3 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{4 \sqrt{2} f}+\frac{3 d^{5/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{8 \sqrt{2} f}-\frac{3 d^{5/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{8 \sqrt{2} f}-\frac{d \cos ^2(e+f x) (d \tan (e+f x))^{3/2}}{2 f}",1,"-1/8*(d^2*(3*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Csc[e + f*x] + 3*Csc[e + f*x]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]] + 2*Sqrt[Sin[2*(e + f*x)]])*Sqrt[Sin[2*(e + f*x)]]*Sqrt[d*Tan[e + f*x]])/f","A",1
252,1,125,253,0.1965759,"\int \cos ^4(e+f x) (d \tan (e+f x))^{5/2} \, dx","Integrate[Cos[e + f*x]^4*(d*Tan[e + f*x])^(5/2),x]","-\frac{d^2 \sqrt{d \tan (e+f x)} \left(-2 \sin (2 (e+f x))+2 \sin (4 (e+f x))+3 \sqrt{\sin (2 (e+f x))} \csc (e+f x) \sin ^{-1}(\cos (e+f x)-\sin (e+f x))+3 \sqrt{\sin (2 (e+f x))} \csc (e+f x) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)\right)}{64 f}","-\frac{3 d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{32 \sqrt{2} f}+\frac{3 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{32 \sqrt{2} f}+\frac{3 d^{5/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{64 \sqrt{2} f}-\frac{3 d^{5/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{64 \sqrt{2} f}-\frac{d \cos ^4(e+f x) (d \tan (e+f x))^{3/2}}{4 f}+\frac{3 d \cos ^2(e+f x) (d \tan (e+f x))^{3/2}}{16 f}",1,"-1/64*(d^2*(3*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Csc[e + f*x]*Sqrt[Sin[2*(e + f*x)]] + 3*Csc[e + f*x]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] - 2*Sin[2*(e + f*x)] + 2*Sin[4*(e + f*x)])*Sqrt[d*Tan[e + f*x]])/f","A",1
253,1,79,109,0.5442041,"\int \frac{\sec ^5(e+f x)}{\sqrt{d \tan (e+f x)}} \, dx","Integrate[Sec[e + f*x]^5/Sqrt[d*Tan[e + f*x]],x]","\frac{2 \sin (e+f x) \left(4 \sqrt{\sec ^2(e+f x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(e+f x)\right)+(\cos (2 (e+f x))+2) \sec ^4(e+f x)\right)}{7 f \sqrt{d \tan (e+f x)}}","\frac{2 \sec ^3(e+f x) \sqrt{d \tan (e+f x)}}{7 d f}+\frac{4 \sec (e+f x) \sqrt{d \tan (e+f x)}}{7 d f}+\frac{4 \sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{7 f \sqrt{d \tan (e+f x)}}",1,"(2*((2 + Cos[2*(e + f*x)])*Sec[e + f*x]^4 + 4*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[e + f*x]^2]*Sqrt[Sec[e + f*x]^2])*Sin[e + f*x])/(7*f*Sqrt[d*Tan[e + f*x]])","C",1
254,1,68,79,0.269159,"\int \frac{\sec ^3(e+f x)}{\sqrt{d \tan (e+f x)}} \, dx","Integrate[Sec[e + f*x]^3/Sqrt[d*Tan[e + f*x]],x]","\frac{2 \sin (e+f x) \left(2 \sqrt{\sec ^2(e+f x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2(e+f x)\right)+\sec ^2(e+f x)\right)}{3 f \sqrt{d \tan (e+f x)}}","\frac{2 \sec (e+f x) \sqrt{d \tan (e+f x)}}{3 d f}+\frac{2 \sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{3 f \sqrt{d \tan (e+f x)}}",1,"(2*(Sec[e + f*x]^2 + 2*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[e + f*x]^2]*Sqrt[Sec[e + f*x]^2])*Sin[e + f*x])/(3*f*Sqrt[d*Tan[e + f*x]])","C",1
255,1,77,47,0.1287566,"\int \frac{\sec (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx","Integrate[Sec[e + f*x]/Sqrt[d*Tan[e + f*x]],x]","-\frac{2 \sqrt[4]{-1} \sqrt{\tan (e+f x)} \sec ^3(e+f x) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (e+f x)}\right)\right|-1\right)}{f \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{d \tan (e+f x)}}","\frac{\sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{f \sqrt{d \tan (e+f x)}}",1,"(-2*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[e + f*x]]], -1]*Sec[e + f*x]^3*Sqrt[Tan[e + f*x]])/(f*Sqrt[d*Tan[e + f*x]]*(1 + Tan[e + f*x]^2)^(3/2))","C",1
256,1,126,76,0.5531107,"\int \frac{\cos (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx","Integrate[Cos[e + f*x]/Sqrt[d*Tan[e + f*x]],x]","\frac{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \sec (e+f x) \left(-\sqrt{\tan (e+f x)} \sqrt{\sec ^2(e+f x)}+\sqrt[4]{-1} \sec ^2(e+f x) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (e+f x)}\right)\right|-1\right)\right)}{f \left(\tan ^2(e+f x)-1\right) \sqrt{\sec ^2(e+f x)} \sqrt{d \tan (e+f x)}}","\frac{\cos (e+f x) \sqrt{d \tan (e+f x)}}{d f}+\frac{\sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{2 f \sqrt{d \tan (e+f x)}}",1,"(Cos[2*(e + f*x)]*Sec[e + f*x]*((-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[e + f*x]]], -1]*Sec[e + f*x]^2 - Sqrt[Sec[e + f*x]^2]*Sqrt[Tan[e + f*x]])*Sqrt[Tan[e + f*x]])/(f*Sqrt[Sec[e + f*x]^2]*Sqrt[d*Tan[e + f*x]]*(-1 + Tan[e + f*x]^2))","C",1
257,1,94,109,1.0084824,"\int \frac{\cos ^3(e+f x)}{\sqrt{d \tan (e+f x)}} \, dx","Integrate[Cos[e + f*x]^3/Sqrt[d*Tan[e + f*x]],x]","\frac{11 \sin (e+f x)+\sin (3 (e+f x))-10 \sqrt[4]{-1} \cos (e+f x) \sqrt{\tan (e+f x)} \sqrt{\sec ^2(e+f x)} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (e+f x)}\right)\right|-1\right)}{12 f \sqrt{d \tan (e+f x)}}","\frac{\cos ^3(e+f x) \sqrt{d \tan (e+f x)}}{3 d f}+\frac{5 \cos (e+f x) \sqrt{d \tan (e+f x)}}{6 d f}+\frac{5 \sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{12 f \sqrt{d \tan (e+f x)}}",1,"(11*Sin[e + f*x] + Sin[3*(e + f*x)] - 10*(-1)^(1/4)*Cos[e + f*x]*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[e + f*x]]], -1]*Sqrt[Sec[e + f*x]^2]*Sqrt[Tan[e + f*x]])/(12*f*Sqrt[d*Tan[e + f*x]])","C",1
258,1,45,65,0.2059812,"\int \frac{\sec ^6(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Sec[a + b*x]^6/(d*Tan[a + b*x])^(3/2),x]","\frac{\tan ^2(a+b x) \left(6 \sec ^2(a+b x)+22\right)-42}{21 b d \sqrt{d \tan (a+b x)}}","\frac{2 (d \tan (a+b x))^{7/2}}{7 b d^5}+\frac{4 (d \tan (a+b x))^{3/2}}{3 b d^3}-\frac{2}{b d \sqrt{d \tan (a+b x)}}",1,"(-42 + (22 + 6*Sec[a + b*x]^2)*Tan[a + b*x]^2)/(21*b*d*Sqrt[d*Tan[a + b*x]])","A",1
259,1,32,43,0.1018691,"\int \frac{\sec ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Sec[a + b*x]^4/(d*Tan[a + b*x])^(3/2),x]","\frac{2 \left(\tan ^2(a+b x)-3\right)}{3 b d \sqrt{d \tan (a+b x)}}","\frac{2 (d \tan (a+b x))^{3/2}}{3 b d^3}-\frac{2}{b d \sqrt{d \tan (a+b x)}}",1,"(2*(-3 + Tan[a + b*x]^2))/(3*b*d*Sqrt[d*Tan[a + b*x]])","A",1
260,1,20,20,0.0607448,"\int \frac{\sec ^2(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Sec[a + b*x]^2/(d*Tan[a + b*x])^(3/2),x]","-\frac{2}{b d \sqrt{d \tan (a+b x)}}","-\frac{2}{b d \sqrt{d \tan (a+b x)}}",1,"-2/(b*d*Sqrt[d*Tan[a + b*x]])","A",1
261,1,38,212,0.0388048,"\int \frac{1}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[(d*Tan[a + b*x])^(-3/2),x]","-\frac{2 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(a+b x)\right)}{b d \sqrt{d \tan (a+b x)}}","\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{\sqrt{2} b d^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{\sqrt{2} b d^{3/2}}-\frac{\log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{2 \sqrt{2} b d^{3/2}}+\frac{\log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{2 \sqrt{2} b d^{3/2}}-\frac{2}{b d \sqrt{d \tan (a+b x)}}",1,"(-2*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[a + b*x]^2])/(b*d*Sqrt[d*Tan[a + b*x]])","C",1
262,1,115,249,0.2921251,"\int \frac{\cos ^2(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Cos[a + b*x]^2/(d*Tan[a + b*x])^(3/2),x]","\frac{\csc (a+b x) \sqrt{d \tan (a+b x)} \left(-17 \cos (a+b x)+\cos (3 (a+b x))+5 \sqrt{\sin (2 (a+b x))} \sin ^{-1}(\cos (a+b x)-\sin (a+b x))+5 \sqrt{\sin (2 (a+b x))} \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{8 b d^2}","\frac{5 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right)}{4 \sqrt{2} b d^{3/2}}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right)}{4 \sqrt{2} b d^{3/2}}-\frac{5 \log \left(\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b d^{3/2}}+\frac{5 \log \left(\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right)}{8 \sqrt{2} b d^{3/2}}-\frac{5}{2 b d \sqrt{d \tan (a+b x)}}+\frac{\cos ^2(a+b x)}{2 b d \sqrt{d \tan (a+b x)}}",1,"(Csc[a + b*x]*(-17*Cos[a + b*x] + Cos[3*(a + b*x)] + 5*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Sqrt[Sin[2*(a + b*x)]] + 5*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]])*Sqrt[d*Tan[a + b*x]])/(8*b*d^2)","A",1
263,1,104,138,0.8977196,"\int \frac{\sec ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Sec[a + b*x]^5/(d*Tan[a + b*x])^(3/2),x]","\frac{2 \csc (a+b x) \sqrt{d \tan (a+b x)} \left(\sqrt{\sec ^2(a+b x)} \left(12 \sin ^2(a+b x)+\tan ^2(a+b x)-5\right)-8 \tan ^2(a+b x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)\right)}{5 b d^2 \sqrt{\sec ^2(a+b x)}}","\frac{24 \cos (a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3}+\frac{12 \sec (a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3}-\frac{24 \cos (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{5 b d^2 \sqrt{\sin (2 a+2 b x)}}-\frac{2 \sec ^3(a+b x)}{b d \sqrt{d \tan (a+b x)}}",1,"(2*Csc[a + b*x]*Sqrt[d*Tan[a + b*x]]*(-8*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Tan[a + b*x]^2 + Sqrt[Sec[a + b*x]^2]*(-5 + 12*Sin[a + b*x]^2 + Tan[a + b*x]^2)))/(5*b*d^2*Sqrt[Sec[a + b*x]^2])","C",1
264,1,93,104,0.4752696,"\int \frac{\sec ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Sec[a + b*x]^3/(d*Tan[a + b*x])^(3/2),x]","-\frac{2 \csc (a+b x) \sqrt{d \tan (a+b x)} \left(4 \tan ^2(a+b x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)+3 \cos (2 (a+b x)) \sqrt{\sec ^2(a+b x)}\right)}{3 b d^2 \sqrt{\sec ^2(a+b x)}}","\frac{4 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d^3}-\frac{4 \cos (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{b d^2 \sqrt{\sin (2 a+2 b x)}}-\frac{2 \sec (a+b x)}{b d \sqrt{d \tan (a+b x)}}",1,"(-2*Csc[a + b*x]*Sqrt[d*Tan[a + b*x]]*(3*Cos[2*(a + b*x)]*Sqrt[Sec[a + b*x]^2] + 4*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Tan[a + b*x]^2))/(3*b*d^2*Sqrt[Sec[a + b*x]^2])","C",1
265,1,69,78,0.3953519,"\int \frac{\sec (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Sec[a + b*x]/(d*Tan[a + b*x])^(3/2),x]","-\frac{2 \sin (a+b x) \left(2 \tan ^2(a+b x) \sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)+3\right)}{3 b (d \tan (a+b x))^{3/2}}","-\frac{2 \cos (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{b d^2 \sqrt{\sin (2 a+2 b x)}}-\frac{2 \cos (a+b x)}{b d \sqrt{d \tan (a+b x)}}",1,"(-2*Sin[a + b*x]*(3 + 2*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2]*Tan[a + b*x]^2))/(3*b*(d*Tan[a + b*x])^(3/2))","C",1
266,1,66,78,0.4124254,"\int \frac{\cos (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Cos[a + b*x]/(d*Tan[a + b*x])^(3/2),x]","-\frac{2 \sin (a+b x) \left(\tan ^2(a+b x) \sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)+1\right)}{b (d \tan (a+b x))^{3/2}}","-\frac{3 \cos (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{b d^2 \sqrt{\sin (2 a+2 b x)}}-\frac{2 \cos (a+b x)}{b d \sqrt{d \tan (a+b x)}}",1,"(-2*Sin[a + b*x]*(1 + Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2]*Tan[a + b*x]^2))/(b*(d*Tan[a + b*x])^(3/2))","C",1
267,1,77,112,0.5676914,"\int \frac{\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Cos[a + b*x]^3/(d*Tan[a + b*x])^(3/2),x]","\frac{\sin (a+b x) \left(-14 \tan ^2(a+b x) \sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)+\cos (2 (a+b x))-13\right)}{6 b (d \tan (a+b x))^{3/2}}","-\frac{7 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d^3}-\frac{7 \cos (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{2 b d^2 \sqrt{\sin (2 a+2 b x)}}-\frac{2 \cos ^3(a+b x)}{b d \sqrt{d \tan (a+b x)}}",1,"(Sin[a + b*x]*(-13 + Cos[2*(a + b*x)] - 14*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2]*Tan[a + b*x]^2))/(6*b*(d*Tan[a + b*x])^(3/2))","C",1
268,1,89,142,0.8519551,"\int \frac{\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx","Integrate[Cos[a + b*x]^5/(d*Tan[a + b*x])^(3/2),x]","\frac{\sin (a+b x) \left(-308 \tan ^2(a+b x) \sqrt{\sec ^2(a+b x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)+34 \cos (2 (a+b x))+3 \cos (4 (a+b x))-277\right)}{120 b (d \tan (a+b x))^{3/2}}","-\frac{11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3}-\frac{77 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{30 b d^3}-\frac{77 \cos (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{20 b d^2 \sqrt{\sin (2 a+2 b x)}}-\frac{2 \cos ^5(a+b x)}{b d \sqrt{d \tan (a+b x)}}",1,"(Sin[a + b*x]*(-277 + 34*Cos[2*(a + b*x)] + 3*Cos[4*(a + b*x)] - 308*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2]*Tan[a + b*x]^2))/(120*b*(d*Tan[a + b*x])^(3/2))","C",1
269,1,113,82,0.6872679,"\int \frac{\sec (a+b x)}{(d \tan (a+b x))^{5/2}} \, dx","Integrate[Sec[a + b*x]/(d*Tan[a + b*x])^(5/2),x]","\frac{2 \cos (2 (a+b x)) \csc (a+b x) \sqrt{\sec ^2(a+b x)} \left(\sqrt{\sec ^2(a+b x)}-\sqrt[4]{-1} \tan ^{\frac{3}{2}}(a+b x) F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} \sqrt{\tan (a+b x)}\right)\right|-1\right)\right)}{3 b d^2 \left(\tan ^2(a+b x)-1\right) \sqrt{d \tan (a+b x)}}","-\frac{\sqrt{\sin (2 a+2 b x)} \sec (a+b x) F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{3 b d^2 \sqrt{d \tan (a+b x)}}-\frac{2 \sec (a+b x)}{3 b d (d \tan (a+b x))^{3/2}}",1,"(2*Cos[2*(a + b*x)]*Csc[a + b*x]*Sqrt[Sec[a + b*x]^2]*(Sqrt[Sec[a + b*x]^2] - (-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Tan[a + b*x]^(3/2)))/(3*b*d^2*Sqrt[d*Tan[a + b*x]]*(-1 + Tan[a + b*x]^2))","C",1
270,1,103,110,1.5399337,"\int \frac{\sec ^3(a+b x)}{(d \tan (a+b x))^{7/2}} \, dx","Integrate[Sec[a + b*x]^3/(d*Tan[a + b*x])^(7/2),x]","-\frac{2 \sin (a+b x) \sqrt{d \tan (a+b x)} \left(4 \sec ^2(a+b x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right)+3 \left(\csc ^4(a+b x)+\csc ^2(a+b x)-2\right) \sqrt{\sec ^2(a+b x)}\right)}{15 b d^4 \sqrt{\sec ^2(a+b x)}}","-\frac{4 \cos (a+b x) E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{d \tan (a+b x)}}{5 b d^4 \sqrt{\sin (2 a+2 b x)}}-\frac{4 \cos (a+b x)}{5 b d^3 \sqrt{d \tan (a+b x)}}-\frac{2 \sec (a+b x)}{5 b d (d \tan (a+b x))^{5/2}}",1,"(-2*(4*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sec[a + b*x]^2 + 3*(-2 + Csc[a + b*x]^2 + Csc[a + b*x]^4)*Sqrt[Sec[a + b*x]^2])*Sin[a + b*x]*Sqrt[d*Tan[a + b*x]])/(15*b*d^4*Sqrt[Sec[a + b*x]^2])","C",1
271,1,77,53,0.2188921,"\int \sec ^{\frac{10}{3}}(e+f x) \sin ^2(e+f x) \, dx","Integrate[Sec[e + f*x]^(10/3)*Sin[e + f*x]^2,x]","\frac{3 \sqrt[3]{\sec (e+f x)} \left(2 \sin (e+f x) \sqrt[6]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)-3 \sin (e+f x)+\tan (e+f x) \sec (e+f x)\right)}{7 f}","\frac{3 \sin (e+f x) \sec ^{\frac{7}{3}}(e+f x) \, _2F_1\left(-\frac{7}{6},-\frac{1}{2};-\frac{1}{6};\cos ^2(e+f x)\right)}{7 f \sqrt{\sin ^2(e+f x)}}",1,"(3*Sec[e + f*x]^(1/3)*(-3*Sin[e + f*x] + 2*(Cos[e + f*x]^2)^(1/6)*Hypergeometric2F1[1/6, 1/2, 3/2, Sin[e + f*x]^2]*Sin[e + f*x] + Sec[e + f*x]*Tan[e + f*x]))/(7*f)","A",1
272,1,56,53,0.0863669,"\int \sec ^{\frac{8}{3}}(e+f x) \sin ^2(e+f x) \, dx","Integrate[Sec[e + f*x]^(8/3)*Sin[e + f*x]^2,x]","-\frac{3 \sin (e+f x) \sec ^{\frac{5}{3}}(e+f x) \left(\cos ^2(e+f x)^{5/6} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\sin ^2(e+f x)\right)-1\right)}{5 f}","\frac{3 \sin (e+f x) \sec ^{\frac{5}{3}}(e+f x) \, _2F_1\left(-\frac{5}{6},-\frac{1}{2};\frac{1}{6};\cos ^2(e+f x)\right)}{5 f \sqrt{\sin ^2(e+f x)}}",1,"(-3*(-1 + (Cos[e + f*x]^2)^(5/6)*Hypergeometric2F1[1/2, 5/6, 3/2, Sin[e + f*x]^2])*Sec[e + f*x]^(5/3)*Sin[e + f*x])/(5*f)","A",1
273,1,56,53,0.0886918,"\int \sec ^{\frac{7}{3}}(e+f x) \sin ^2(e+f x) \, dx","Integrate[Sec[e + f*x]^(7/3)*Sin[e + f*x]^2,x]","-\frac{3 \sin (e+f x) \sec ^{\frac{4}{3}}(e+f x) \left(\cos ^2(e+f x)^{2/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\sin ^2(e+f x)\right)-1\right)}{4 f}","\frac{3 \sin (e+f x) \sec ^{\frac{4}{3}}(e+f x) \, _2F_1\left(-\frac{2}{3},-\frac{1}{2};\frac{1}{3};\cos ^2(e+f x)\right)}{4 f \sqrt{\sin ^2(e+f x)}}",1,"(-3*(-1 + (Cos[e + f*x]^2)^(2/3)*Hypergeometric2F1[1/2, 2/3, 3/2, Sin[e + f*x]^2])*Sec[e + f*x]^(4/3)*Sin[e + f*x])/(4*f)","A",1
274,1,56,53,0.0842123,"\int \sec ^{\frac{5}{3}}(e+f x) \sin ^2(e+f x) \, dx","Integrate[Sec[e + f*x]^(5/3)*Sin[e + f*x]^2,x]","-\frac{3 \sin (e+f x) \sec ^{\frac{2}{3}}(e+f x) \left(\sqrt[3]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)-1\right)}{2 f}","\frac{3 \sin (e+f x) \sec ^{\frac{2}{3}}(e+f x) \, _2F_1\left(-\frac{1}{2},-\frac{1}{3};\frac{2}{3};\cos ^2(e+f x)\right)}{2 f \sqrt{\sin ^2(e+f x)}}",1,"(-3*(-1 + (Cos[e + f*x]^2)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, Sin[e + f*x]^2])*Sec[e + f*x]^(2/3)*Sin[e + f*x])/(2*f)","A",1
275,1,54,51,0.0745886,"\int \sec ^{\frac{4}{3}}(e+f x) \sin ^2(e+f x) \, dx","Integrate[Sec[e + f*x]^(4/3)*Sin[e + f*x]^2,x]","-\frac{3 \sin (e+f x) \sqrt[3]{\sec (e+f x)} \left(\sqrt[6]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)-1\right)}{f}","\frac{3 \sin (e+f x) \sqrt[3]{\sec (e+f x)} \, _2F_1\left(-\frac{1}{2},-\frac{1}{6};\frac{5}{6};\cos ^2(e+f x)\right)}{f \sqrt{\sin ^2(e+f x)}}",1,"(-3*(-1 + (Cos[e + f*x]^2)^(1/6)*Hypergeometric2F1[1/6, 1/2, 3/2, Sin[e + f*x]^2])*Sec[e + f*x]^(1/3)*Sin[e + f*x])/f","A",1
276,1,89,53,1.1090645,"\int \sec ^{\frac{16}{3}}(e+f x) \sin ^4(e+f x) \, dx","Integrate[Sec[e + f*x]^(16/3)*Sin[e + f*x]^4,x]","\frac{3 \sqrt[3]{\sec (e+f x)} \left(-18 \sin (e+f x) \sqrt[6]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)+27 \sin (e+f x)+\tan (e+f x) \sec (e+f x) \left(7 \sec ^2(e+f x)-16\right)\right)}{91 f}","\frac{3 \sin (e+f x) \sec ^{\frac{13}{3}}(e+f x) \, _2F_1\left(-\frac{13}{6},-\frac{3}{2};-\frac{7}{6};\cos ^2(e+f x)\right)}{13 f \sqrt{\sin ^2(e+f x)}}",1,"(3*Sec[e + f*x]^(1/3)*(27*Sin[e + f*x] - 18*(Cos[e + f*x]^2)^(1/6)*Hypergeometric2F1[1/6, 1/2, 3/2, Sin[e + f*x]^2]*Sin[e + f*x] + Sec[e + f*x]*(-16 + 7*Sec[e + f*x]^2)*Tan[e + f*x]))/(91*f)","A",1
277,1,78,53,0.8645732,"\int \sec ^{\frac{14}{3}}(e+f x) \sin ^4(e+f x) \, dx","Integrate[Sec[e + f*x]^(14/3)*Sin[e + f*x]^4,x]","\frac{3 \sin (e+f x) \left(\frac{9 \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\sin ^2(e+f x)\right)}{\sqrt[6]{\cos ^2(e+f x)}}-(7 \cos (2 (e+f x))+2) \sec ^4(e+f x)\right)}{55 f \sqrt[3]{\sec (e+f x)}}","\frac{3 \sin (e+f x) \sec ^{\frac{11}{3}}(e+f x) \, _2F_1\left(-\frac{11}{6},-\frac{3}{2};-\frac{5}{6};\cos ^2(e+f x)\right)}{11 f \sqrt{\sin ^2(e+f x)}}",1,"(3*((9*Hypergeometric2F1[1/2, 5/6, 3/2, Sin[e + f*x]^2])/(Cos[e + f*x]^2)^(1/6) - (2 + 7*Cos[2*(e + f*x)])*Sec[e + f*x]^4)*Sin[e + f*x])/(55*f*Sec[e + f*x]^(1/3))","A",1
278,1,77,53,0.7534068,"\int \sec ^{\frac{13}{3}}(e+f x) \sin ^4(e+f x) \, dx","Integrate[Sec[e + f*x]^(13/3)*Sin[e + f*x]^4,x]","\frac{3 \sin (e+f x) \left(\frac{9 \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\sin ^2(e+f x)\right)}{\sqrt[3]{\cos ^2(e+f x)}}+\left(4 \sec ^2(e+f x)-13\right) \sec ^2(e+f x)\right)}{40 f \sec ^{\frac{2}{3}}(e+f x)}","\frac{3 \sin (e+f x) \sec ^{\frac{10}{3}}(e+f x) \, _2F_1\left(-\frac{5}{3},-\frac{3}{2};-\frac{2}{3};\cos ^2(e+f x)\right)}{10 f \sqrt{\sin ^2(e+f x)}}",1,"(3*((9*Hypergeometric2F1[1/2, 2/3, 3/2, Sin[e + f*x]^2])/(Cos[e + f*x]^2)^(1/3) + Sec[e + f*x]^2*(-13 + 4*Sec[e + f*x]^2))*Sin[e + f*x])/(40*f*Sec[e + f*x]^(2/3))","A",1
279,1,78,53,0.1967963,"\int \sec ^{\frac{11}{3}}(e+f x) \sin ^4(e+f x) \, dx","Integrate[Sec[e + f*x]^(11/3)*Sin[e + f*x]^4,x]","\frac{3 \sec ^{\frac{2}{3}}(e+f x) \left(9 \sin (e+f x) \sqrt[3]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)-11 \sin (e+f x)+2 \tan (e+f x) \sec (e+f x)\right)}{16 f}","\frac{3 \sin (e+f x) \sec ^{\frac{8}{3}}(e+f x) \, _2F_1\left(-\frac{3}{2},-\frac{4}{3};-\frac{1}{3};\cos ^2(e+f x)\right)}{8 f \sqrt{\sin ^2(e+f x)}}",1,"(3*Sec[e + f*x]^(2/3)*(-11*Sin[e + f*x] + 9*(Cos[e + f*x]^2)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, Sin[e + f*x]^2]*Sin[e + f*x] + 2*Sec[e + f*x]*Tan[e + f*x]))/(16*f)","A",1
280,1,77,53,0.2537173,"\int \sec ^{\frac{10}{3}}(e+f x) \sin ^4(e+f x) \, dx","Integrate[Sec[e + f*x]^(10/3)*Sin[e + f*x]^4,x]","\frac{3 \sqrt[3]{\sec (e+f x)} \left(9 \sin (e+f x) \sqrt[6]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)-10 \sin (e+f x)+\tan (e+f x) \sec (e+f x)\right)}{7 f}","\frac{3 \sin (e+f x) \sec ^{\frac{7}{3}}(e+f x) \, _2F_1\left(-\frac{3}{2},-\frac{7}{6};-\frac{1}{6};\cos ^2(e+f x)\right)}{7 f \sqrt{\sin ^2(e+f x)}}",1,"(3*Sec[e + f*x]^(1/3)*(-10*Sin[e + f*x] + 9*(Cos[e + f*x]^2)^(1/6)*Hypergeometric2F1[1/6, 1/2, 3/2, Sin[e + f*x]^2]*Sin[e + f*x] + Sec[e + f*x]*Tan[e + f*x]))/(7*f)","A",1
281,1,80,57,0.2921342,"\int (d \sec (e+f x))^{4/3} \tan ^2(e+f x) \, dx","Integrate[(d*Sec[e + f*x])^(4/3)*Tan[e + f*x]^2,x]","\frac{3 d \sqrt[3]{d \sec (e+f x)} \left(2 \sin (e+f x) \sqrt[6]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)-3 \sin (e+f x)+\tan (e+f x) \sec (e+f x)\right)}{7 f}","\frac{\cos ^2(e+f x)^{13/6} \tan ^3(e+f x) (d \sec (e+f x))^{4/3} \, _2F_1\left(\frac{3}{2},\frac{13}{6};\frac{5}{2};\sin ^2(e+f x)\right)}{3 f}",1,"(3*d*(d*Sec[e + f*x])^(1/3)*(-3*Sin[e + f*x] + 2*(Cos[e + f*x]^2)^(1/6)*Hypergeometric2F1[1/6, 1/2, 3/2, Sin[e + f*x]^2]*Sin[e + f*x] + Sec[e + f*x]*Tan[e + f*x]))/(7*f)","A",1
282,1,80,57,0.3073206,"\int (d \sec (e+f x))^{2/3} \tan ^2(e+f x) \, dx","Integrate[(d*Sec[e + f*x])^(2/3)*Tan[e + f*x]^2,x]","\frac{3 (d \sec (e+f x))^{2/3} \left(2 \sqrt[6]{\cos ^2(e+f x)} \tan (e+f x)-\sin (2 (e+f x)) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\sin ^2(e+f x)\right)\right)}{10 f \sqrt[6]{\cos ^2(e+f x)}}","\frac{\cos ^2(e+f x)^{11/6} \tan ^3(e+f x) (d \sec (e+f x))^{2/3} \, _2F_1\left(\frac{3}{2},\frac{11}{6};\frac{5}{2};\sin ^2(e+f x)\right)}{3 f}",1,"(3*(d*Sec[e + f*x])^(2/3)*(-(Hypergeometric2F1[1/2, 5/6, 3/2, Sin[e + f*x]^2]*Sin[2*(e + f*x)]) + 2*(Cos[e + f*x]^2)^(1/6)*Tan[e + f*x]))/(10*f*(Cos[e + f*x]^2)^(1/6))","A",1
283,1,80,57,0.2813248,"\int \sqrt[3]{d \sec (e+f x)} \tan ^2(e+f x) \, dx","Integrate[(d*Sec[e + f*x])^(1/3)*Tan[e + f*x]^2,x]","\frac{3 \sqrt[3]{d \sec (e+f x)} \left(2 \sqrt[3]{\cos ^2(e+f x)} \tan (e+f x)-\sin (2 (e+f x)) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\sin ^2(e+f x)\right)\right)}{8 f \sqrt[3]{\cos ^2(e+f x)}}","\frac{\cos ^2(e+f x)^{5/3} \tan ^3(e+f x) \sqrt[3]{d \sec (e+f x)} \, _2F_1\left(\frac{3}{2},\frac{5}{3};\frac{5}{2};\sin ^2(e+f x)\right)}{3 f}",1,"(3*(d*Sec[e + f*x])^(1/3)*(-(Hypergeometric2F1[1/2, 2/3, 3/2, Sin[e + f*x]^2]*Sin[2*(e + f*x)]) + 2*(Cos[e + f*x]^2)^(1/3)*Tan[e + f*x]))/(8*f*(Cos[e + f*x]^2)^(1/3))","A",1
284,1,80,57,0.2148544,"\int \frac{\tan ^2(e+f x)}{\sqrt[3]{d \sec (e+f x)}} \, dx","Integrate[Tan[e + f*x]^2/(d*Sec[e + f*x])^(1/3),x]","\frac{3 \left(2 \cos ^2(e+f x)^{2/3} \tan (e+f x)-\sin (2 (e+f x)) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)\right)}{4 f \cos ^2(e+f x)^{2/3} \sqrt[3]{d \sec (e+f x)}}","\frac{\cos ^2(e+f x)^{4/3} \tan ^3(e+f x) \, _2F_1\left(\frac{4}{3},\frac{3}{2};\frac{5}{2};\sin ^2(e+f x)\right)}{3 f \sqrt[3]{d \sec (e+f x)}}",1,"(3*(-(Hypergeometric2F1[1/3, 1/2, 3/2, Sin[e + f*x]^2]*Sin[2*(e + f*x)]) + 2*(Cos[e + f*x]^2)^(2/3)*Tan[e + f*x]))/(4*f*(Cos[e + f*x]^2)^(2/3)*(d*Sec[e + f*x])^(1/3))","A",1
285,1,79,57,0.1954939,"\int \frac{\tan ^2(e+f x)}{(d \sec (e+f x))^{2/3}} \, dx","Integrate[Tan[e + f*x]^2/(d*Sec[e + f*x])^(2/3),x]","\frac{3 \cos ^2(e+f x)^{5/6} \tan (e+f x)-\frac{3}{2} \sin (2 (e+f x)) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)}{f \cos ^2(e+f x)^{5/6} (d \sec (e+f x))^{2/3}}","\frac{\cos ^2(e+f x)^{7/6} \tan ^3(e+f x) \, _2F_1\left(\frac{7}{6},\frac{3}{2};\frac{5}{2};\sin ^2(e+f x)\right)}{3 f (d \sec (e+f x))^{2/3}}",1,"((-3*Hypergeometric2F1[1/6, 1/2, 3/2, Sin[e + f*x]^2]*Sin[2*(e + f*x)])/2 + 3*(Cos[e + f*x]^2)^(5/6)*Tan[e + f*x])/(f*(Cos[e + f*x]^2)^(5/6)*(d*Sec[e + f*x])^(2/3))","A",1
286,1,92,57,1.1147555,"\int (d \sec (e+f x))^{4/3} \tan ^4(e+f x) \, dx","Integrate[(d*Sec[e + f*x])^(4/3)*Tan[e + f*x]^4,x]","\frac{3 d \sqrt[3]{d \sec (e+f x)} \left(-18 \sin (e+f x) \sqrt[6]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)+27 \sin (e+f x)+\tan (e+f x) \sec (e+f x) \left(7 \sec ^2(e+f x)-16\right)\right)}{91 f}","\frac{\cos ^2(e+f x)^{19/6} \tan ^5(e+f x) (d \sec (e+f x))^{4/3} \, _2F_1\left(\frac{5}{2},\frac{19}{6};\frac{7}{2};\sin ^2(e+f x)\right)}{5 f}",1,"(3*d*(d*Sec[e + f*x])^(1/3)*(27*Sin[e + f*x] - 18*(Cos[e + f*x]^2)^(1/6)*Hypergeometric2F1[1/6, 1/2, 3/2, Sin[e + f*x]^2]*Sin[e + f*x] + Sec[e + f*x]*(-16 + 7*Sec[e + f*x]^2)*Tan[e + f*x]))/(91*f)","A",1
287,1,69,57,0.1615573,"\int (d \sec (e+f x))^{2/3} \tan ^4(e+f x) \, dx","Integrate[(d*Sec[e + f*x])^(2/3)*Tan[e + f*x]^4,x]","\frac{3 \tan (e+f x) (d \sec (e+f x))^{2/3} \left(9 \cos ^2(e+f x)^{5/6} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\sin ^2(e+f x)\right)+5 \sec ^2(e+f x)-14\right)}{55 f}","\frac{\cos ^2(e+f x)^{17/6} \tan ^5(e+f x) (d \sec (e+f x))^{2/3} \, _2F_1\left(\frac{5}{2},\frac{17}{6};\frac{7}{2};\sin ^2(e+f x)\right)}{5 f}",1,"(3*(d*Sec[e + f*x])^(2/3)*(-14 + 9*(Cos[e + f*x]^2)^(5/6)*Hypergeometric2F1[1/2, 5/6, 3/2, Sin[e + f*x]^2] + 5*Sec[e + f*x]^2)*Tan[e + f*x])/(55*f)","A",1
288,1,69,57,0.1529531,"\int \sqrt[3]{d \sec (e+f x)} \tan ^4(e+f x) \, dx","Integrate[(d*Sec[e + f*x])^(1/3)*Tan[e + f*x]^4,x]","\frac{3 \tan (e+f x) \sqrt[3]{d \sec (e+f x)} \left(9 \cos ^2(e+f x)^{2/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\sin ^2(e+f x)\right)+4 \sec ^2(e+f x)-13\right)}{40 f}","\frac{\cos ^2(e+f x)^{8/3} \tan ^5(e+f x) \sqrt[3]{d \sec (e+f x)} \, _2F_1\left(\frac{5}{2},\frac{8}{3};\frac{7}{2};\sin ^2(e+f x)\right)}{5 f}",1,"(3*(d*Sec[e + f*x])^(1/3)*(-13 + 9*(Cos[e + f*x]^2)^(2/3)*Hypergeometric2F1[1/2, 2/3, 3/2, Sin[e + f*x]^2] + 4*Sec[e + f*x]^2)*Tan[e + f*x])/(40*f)","A",1
289,1,69,57,0.149159,"\int \frac{\tan ^4(e+f x)}{\sqrt[3]{d \sec (e+f x)}} \, dx","Integrate[Tan[e + f*x]^4/(d*Sec[e + f*x])^(1/3),x]","\frac{3 \tan (e+f x) \left(9 \sqrt[3]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)+2 \sec ^2(e+f x)-11\right)}{16 f \sqrt[3]{d \sec (e+f x)}}","\frac{\cos ^2(e+f x)^{7/3} \tan ^5(e+f x) \, _2F_1\left(\frac{7}{3},\frac{5}{2};\frac{7}{2};\sin ^2(e+f x)\right)}{5 f \sqrt[3]{d \sec (e+f x)}}",1,"(3*(-11 + 9*(Cos[e + f*x]^2)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, Sin[e + f*x]^2] + 2*Sec[e + f*x]^2)*Tan[e + f*x])/(16*f*(d*Sec[e + f*x])^(1/3))","A",1
290,1,67,57,0.1414533,"\int \frac{\tan ^4(e+f x)}{(d \sec (e+f x))^{2/3}} \, dx","Integrate[Tan[e + f*x]^4/(d*Sec[e + f*x])^(2/3),x]","\frac{3 \tan (e+f x) \left(9 \sqrt[6]{\cos ^2(e+f x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right)+\sec ^2(e+f x)-10\right)}{7 f (d \sec (e+f x))^{2/3}}","\frac{\cos ^2(e+f x)^{13/6} \tan ^5(e+f x) \, _2F_1\left(\frac{13}{6},\frac{5}{2};\frac{7}{2};\sin ^2(e+f x)\right)}{5 f (d \sec (e+f x))^{2/3}}",1,"(3*(-10 + 9*(Cos[e + f*x]^2)^(1/6)*Hypergeometric2F1[1/6, 1/2, 3/2, Sin[e + f*x]^2] + Sec[e + f*x]^2)*Tan[e + f*x])/(7*f*(d*Sec[e + f*x])^(2/3))","A",1
291,1,174,178,1.9222793,"\int (d \sec (e+f x))^{5/2} \sqrt{b \tan (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^(5/2)*Sqrt[b*Tan[e + f*x]],x]","\frac{b (d \sec (e+f x))^{5/2} \left(4 \sec ^{\frac{5}{2}}(e+f x)-4 \sqrt{\sec (e+f x)}+2 \sqrt[4]{\tan ^2(e+f x)} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)+\sqrt[4]{\tan ^2(e+f x)} \left(\log \left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}+1\right)-\log \left(1-\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)\right)\right)}{8 f \sec ^{\frac{5}{2}}(e+f x) \sqrt{b \tan (e+f x)}}","-\frac{\sqrt{b} d^3 \sqrt{b \tan (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{4 f \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{\sqrt{b} d^3 \sqrt{b \tan (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{4 f \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{d^2 (b \tan (e+f x))^{3/2} \sqrt{d \sec (e+f x)}}{2 b f}",1,"(b*(d*Sec[e + f*x])^(5/2)*(-4*Sqrt[Sec[e + f*x]] + 4*Sec[e + f*x]^(5/2) + 2*ArcTan[Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)]*(Tan[e + f*x]^2)^(1/4) + (-Log[1 - Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)] + Log[1 + Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)])*(Tan[e + f*x]^2)^(1/4)))/(8*f*Sec[e + f*x]^(5/2)*Sqrt[b*Tan[e + f*x]])","A",1
292,1,71,93,1.0030144,"\int (d \sec (e+f x))^{3/2} \sqrt{b \tan (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]],x]","\frac{d \sin (e+f x) \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)} \left(1-\frac{\, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right)}{\left(-\tan ^2(e+f x)\right)^{3/4}}\right)}{f}","\frac{d^2 (b \tan (e+f x))^{3/2}}{b f \sqrt{d \sec (e+f x)}}-\frac{d^2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{b \tan (e+f x)}}{f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}",1,"(d*Sqrt[d*Sec[e + f*x]]*Sin[e + f*x]*Sqrt[b*Tan[e + f*x]]*(1 - Hypergeometric2F1[-1/4, 1/4, 3/4, Sec[e + f*x]^2]/(-Tan[e + f*x]^2)^(3/4)))/f","C",1
293,1,136,132,0.734182,"\int \sqrt{d \sec (e+f x)} \sqrt{b \tan (e+f x)} \, dx","Integrate[Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]],x]","\frac{b \sqrt[4]{\tan ^2(e+f x)} \sqrt{d \sec (e+f x)} \left(2 \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)-\log \left(1-\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)+\log \left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}+1\right)\right)}{2 f \sqrt{\sec (e+f x)} \sqrt{b \tan (e+f x)}}","\frac{\sqrt{b} d \sqrt{b \tan (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{f \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{\sqrt{b} d \sqrt{b \tan (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{f \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)}}",1,"(b*(2*ArcTan[Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)] - Log[1 - Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)] + Log[1 + Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)])*Sqrt[d*Sec[e + f*x]]*(Tan[e + f*x]^2)^(1/4))/(2*f*Sqrt[Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])","A",1
294,1,62,55,0.5176674,"\int \frac{\sqrt{b \tan (e+f x)}}{\sqrt{d \sec (e+f x)}} \, dx","Integrate[Sqrt[b*Tan[e + f*x]]/Sqrt[d*Sec[e + f*x]],x]","-\frac{2 b \sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right)}{f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}","\frac{2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{b \tan (e+f x)}}{f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}",1,"(-2*b*Hypergeometric2F1[-1/4, 1/4, 3/4, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4))/(f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])","C",1
295,1,34,34,0.1230831,"\int \frac{\sqrt{b \tan (e+f x)}}{(d \sec (e+f x))^{3/2}} \, dx","Integrate[Sqrt[b*Tan[e + f*x]]/(d*Sec[e + f*x])^(3/2),x]","\frac{2 (b \tan (e+f x))^{3/2}}{3 b f (d \sec (e+f x))^{3/2}}","\frac{2 (b \tan (e+f x))^{3/2}}{3 b f (d \sec (e+f x))^{3/2}}",1,"(2*(b*Tan[e + f*x])^(3/2))/(3*b*f*(d*Sec[e + f*x])^(3/2))","A",1
296,1,79,95,0.6833959,"\int \frac{\sqrt{b \tan (e+f x)}}{(d \sec (e+f x))^{5/2}} \, dx","Integrate[Sqrt[b*Tan[e + f*x]]/(d*Sec[e + f*x])^(5/2),x]","-\frac{b \left(4 \sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right)+\cos (2 (e+f x))-1\right)}{5 d^2 f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}","\frac{4 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{b \tan (e+f x)}}{5 d^2 f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{2 (b \tan (e+f x))^{3/2}}{5 b f (d \sec (e+f x))^{5/2}}",1,"-1/5*(b*(-1 + Cos[2*(e + f*x)] + 4*Hypergeometric2F1[-1/4, 1/4, 3/4, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4)))/(d^2*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])","C",1
297,1,53,72,0.1679729,"\int \frac{\sqrt{b \tan (e+f x)}}{(d \sec (e+f x))^{7/2}} \, dx","Integrate[Sqrt[b*Tan[e + f*x]]/(d*Sec[e + f*x])^(7/2),x]","\frac{(19 \sin (e+f x)+3 \sin (3 (e+f x))) \sqrt{b \tan (e+f x)}}{42 d^3 f \sqrt{d \sec (e+f x)}}","\frac{8 (b \tan (e+f x))^{3/2}}{21 b d^2 f (d \sec (e+f x))^{3/2}}+\frac{2 (b \tan (e+f x))^{3/2}}{7 b f (d \sec (e+f x))^{7/2}}",1,"((19*Sin[e + f*x] + 3*Sin[3*(e + f*x)])*Sqrt[b*Tan[e + f*x]])/(42*d^3*f*Sqrt[d*Sec[e + f*x]])","A",1
298,1,92,132,0.9656297,"\int \frac{\sqrt{b \tan (e+f x)}}{(d \sec (e+f x))^{9/2}} \, dx","Integrate[Sqrt[b*Tan[e + f*x]]/(d*Sec[e + f*x])^(9/2),x]","\frac{b \sin ^2(e+f x) (5 \cos (2 (e+f x))+17)-24 b \sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right)}{45 d^4 f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}","\frac{8 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{b \tan (e+f x)}}{15 d^4 f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{4 (b \tan (e+f x))^{3/2}}{15 b d^2 f (d \sec (e+f x))^{5/2}}+\frac{2 (b \tan (e+f x))^{3/2}}{9 b f (d \sec (e+f x))^{9/2}}",1,"(b*(17 + 5*Cos[2*(e + f*x)])*Sin[e + f*x]^2 - 24*b*Hypergeometric2F1[-1/4, 1/4, 3/4, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4))/(45*d^4*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])","C",1
299,1,95,131,0.7803245,"\int (d \sec (e+f x))^{5/2} (b \tan (e+f x))^{3/2} \, dx","Integrate[(d*Sec[e + f*x])^(5/2)*(b*Tan[e + f*x])^(3/2),x]","\frac{b d^2 \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)} \left(\, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};\sec ^2(e+f x)\right)+\sqrt[4]{-\tan ^2(e+f x)} \left(2 \sec ^2(e+f x)-1\right)\right)}{6 f \sqrt[4]{-\tan ^2(e+f x)}}","-\frac{b^2 d^2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \sec (e+f x)}}{6 f \sqrt{b \tan (e+f x)}}-\frac{b d^2 \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}{6 f}+\frac{b \sqrt{b \tan (e+f x)} (d \sec (e+f x))^{5/2}}{3 f}",1,"(b*d^2*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]]*(Hypergeometric2F1[1/4, 3/4, 5/4, Sec[e + f*x]^2] + (-1 + 2*Sec[e + f*x]^2)*(-Tan[e + f*x]^2)^(1/4)))/(6*f*(-Tan[e + f*x]^2)^(1/4))","C",1
300,1,129,169,6.3630718,"\int (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2} \, dx","Integrate[(d*Sec[e + f*x])^(3/2)*(b*Tan[e + f*x])^(3/2),x]","\frac{b \sqrt{b \tan (e+f x)} (d \sec (e+f x))^{3/2} \left(2 \sqrt[4]{\tan ^2(e+f x)} \sec ^{\frac{3}{2}}(e+f x)+\tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)-\tanh ^{-1}\left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)\right)}{4 f \sqrt[4]{\tan ^2(e+f x)} \sec ^{\frac{3}{2}}(e+f x)}","-\frac{b^{3/2} d \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{4 f \sqrt{b \tan (e+f x)}}-\frac{b^{3/2} d \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{4 f \sqrt{b \tan (e+f x)}}+\frac{b \sqrt{b \tan (e+f x)} (d \sec (e+f x))^{3/2}}{2 f}",1,"(b*(d*Sec[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]]*(ArcTan[Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)] - ArcTanh[Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)] + 2*Sec[e + f*x]^(3/2)*(Tan[e + f*x]^2)^(1/4)))/(4*f*Sec[e + f*x]^(3/2)*(Tan[e + f*x]^2)^(1/4))","A",1
301,1,105,88,0.7597294,"\int \sqrt{d \sec (e+f x)} (b \tan (e+f x))^{3/2} \, dx","Integrate[Sqrt[d*Sec[e + f*x]]*(b*Tan[e + f*x])^(3/2),x]","\frac{b \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)} \left(\sec ^{\frac{3}{2}}(e+f x)-\frac{\sec ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{2}}\right)}{f \sec ^{\frac{3}{2}}(e+f x)}","\frac{b \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}{f}-\frac{b^2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \sec (e+f x)}}{f \sqrt{b \tan (e+f x)}}",1,"(b*Sqrt[d*Sec[e + f*x]]*(Sec[e + f*x]^(3/2) - (Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Sqrt[1 + Sec[e + f*x]])/Sqrt[2])*Sqrt[b*Tan[e + f*x]])/(f*Sec[e + f*x]^(3/2))","C",1
302,1,64,167,4.2889958,"\int \frac{(b \tan (e+f x))^{3/2}}{\sqrt{d \sec (e+f x)}} \, dx","Integrate[(b*Tan[e + f*x])^(3/2)/Sqrt[d*Sec[e + f*x]],x]","\frac{2 (b \tan (e+f x))^{5/2} \, _2F_1\left(-\frac{1}{4},-\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right)}{b f \left(-\tan ^2(e+f x)\right)^{5/4} \sqrt{d \sec (e+f x)}}","\frac{b^{3/2} d (b \tan (e+f x))^{3/2} \tan ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{f (b \sin (e+f x))^{3/2} (d \sec (e+f x))^{3/2}}+\frac{b^{3/2} d (b \tan (e+f x))^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{f (b \sin (e+f x))^{3/2} (d \sec (e+f x))^{3/2}}-\frac{2 d \csc (e+f x) (b \tan (e+f x))^{3/2}}{f (d \sec (e+f x))^{3/2}}",1,"(2*Hypergeometric2F1[-1/4, -1/4, 3/4, Sec[e + f*x]^2]*(b*Tan[e + f*x])^(5/2))/(b*f*Sqrt[d*Sec[e + f*x]]*(-Tan[e + f*x]^2)^(5/4))","C",1
303,1,98,96,0.5901786,"\int \frac{(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{3/2}} \, dx","Integrate[(b*Tan[e + f*x])^(3/2)/(d*Sec[e + f*x])^(3/2),x]","-\frac{2 b \sqrt{b \tan (e+f x)} \left(\sqrt{\sec (e+f x)+1}-\sqrt{2} \sec ^{\frac{3}{2}}(e+f x) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}{3 f \sqrt{\sec (e+f x)+1} (d \sec (e+f x))^{3/2}}","\frac{2 b^2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \sec (e+f x)}}{3 d^2 f \sqrt{b \tan (e+f x)}}-\frac{2 b \sqrt{b \tan (e+f x)}}{3 f (d \sec (e+f x))^{3/2}}",1,"(-2*b*(-(Sqrt[2]*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[(e + f*x)/2]^2]*Sec[e + f*x]^(3/2)) + Sqrt[1 + Sec[e + f*x]])*Sqrt[b*Tan[e + f*x]])/(3*f*(d*Sec[e + f*x])^(3/2)*Sqrt[1 + Sec[e + f*x]])","C",1
304,1,141,34,1.373543,"\int \frac{(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{5/2}} \, dx","Integrate[(b*Tan[e + f*x])^(3/2)/(d*Sec[e + f*x])^(5/2),x]","-\frac{b \sec ^{\frac{3}{2}}(e+f x) \sqrt{b \tan (e+f x)} \left(-\sqrt{\sec (e+f x)+1} \sec ^2\left(\frac{1}{2} (e+f x)\right)+\sqrt{\frac{1}{\cos (e+f x)+1}} \cos (3 (e+f x)) \sec ^{\frac{3}{2}}(e+f x)+\sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{\sec (e+f x)}\right)}{10 f \sqrt{\frac{1}{\cos (e+f x)+1}} (d \sec (e+f x))^{5/2}}","\frac{2 (b \tan (e+f x))^{5/2}}{5 b f (d \sec (e+f x))^{5/2}}",1,"-1/10*(b*Sec[e + f*x]^(3/2)*(Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[Sec[e + f*x]] + Sqrt[(1 + Cos[e + f*x])^(-1)]*Cos[3*(e + f*x)]*Sec[e + f*x]^(3/2) - Sec[(e + f*x)/2]^2*Sqrt[1 + Sec[e + f*x]])*Sqrt[b*Tan[e + f*x]])/(f*Sqrt[(1 + Cos[e + f*x])^(-1)]*(d*Sec[e + f*x])^(5/2))","B",1
305,1,105,131,1.3024084,"\int \frac{(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{7/2}} \, dx","Integrate[(b*Tan[e + f*x])^(3/2)/(d*Sec[e + f*x])^(7/2),x]","-\frac{b \sqrt{b \tan (e+f x)} \left(4 \sec ^2(e+f x) \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};\sec ^2(e+f x)\right)+(3 \cos (2 (e+f x))+1) \sqrt[4]{-\tan ^2(e+f x)}\right)}{21 d^2 f \sqrt[4]{-\tan ^2(e+f x)} (d \sec (e+f x))^{3/2}}","\frac{4 b^2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \sec (e+f x)}}{21 d^4 f \sqrt{b \tan (e+f x)}}+\frac{2 b \sqrt{b \tan (e+f x)}}{21 d^2 f (d \sec (e+f x))^{3/2}}-\frac{2 b \sqrt{b \tan (e+f x)}}{7 f (d \sec (e+f x))^{7/2}}",1,"-1/21*(b*Sqrt[b*Tan[e + f*x]]*(4*Hypergeometric2F1[1/4, 3/4, 5/4, Sec[e + f*x]^2]*Sec[e + f*x]^2 + (1 + 3*Cos[2*(e + f*x)])*(-Tan[e + f*x]^2)^(1/4)))/(d^2*f*(d*Sec[e + f*x])^(3/2)*(-Tan[e + f*x]^2)^(1/4))","C",1
306,1,158,103,3.3554028,"\int \frac{(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{9/2}} \, dx","Integrate[(b*Tan[e + f*x])^(3/2)/(d*Sec[e + f*x])^(9/2),x]","-\frac{b \sqrt{\sec (e+f x)} \sqrt{b \tan (e+f x)} \left(-21 \sqrt{\sec (e+f x)+1} \sec ^2\left(\frac{1}{2} (e+f x)\right)+\sqrt{\frac{1}{\cos (e+f x)+1}} (21 \cos (3 (e+f x))+5 \cos (5 (e+f x))) \sec ^{\frac{3}{2}}(e+f x)+16 \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{\sec (e+f x)}\right)}{360 d^3 f \sqrt{\frac{1}{\cos (e+f x)+1}} (d \sec (e+f x))^{3/2}}","\frac{8 b \sqrt{b \tan (e+f x)}}{45 d^4 f \sqrt{d \sec (e+f x)}}+\frac{2 b \sqrt{b \tan (e+f x)}}{45 d^2 f (d \sec (e+f x))^{5/2}}-\frac{2 b \sqrt{b \tan (e+f x)}}{9 f (d \sec (e+f x))^{9/2}}",1,"-1/360*(b*Sqrt[Sec[e + f*x]]*(16*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[Sec[e + f*x]] + Sqrt[(1 + Cos[e + f*x])^(-1)]*(21*Cos[3*(e + f*x)] + 5*Cos[5*(e + f*x)])*Sec[e + f*x]^(3/2) - 21*Sec[(e + f*x)/2]^2*Sqrt[1 + Sec[e + f*x]])*Sqrt[b*Tan[e + f*x]])/(d^3*f*Sqrt[(1 + Cos[e + f*x])^(-1)]*(d*Sec[e + f*x])^(3/2))","A",1
307,1,189,208,3.1508662,"\int (d \sec (e+f x))^{5/2} (b \tan (e+f x))^{5/2} \, dx","Integrate[(d*Sec[e + f*x])^(5/2)*(b*Tan[e + f*x])^(5/2),x]","\frac{b^3 (d \sec (e+f x))^{5/2} \left(16 \sec ^{\frac{9}{2}}(e+f x)-28 \sec ^{\frac{5}{2}}(e+f x)+12 \sqrt{\sec (e+f x)}-6 \sqrt[4]{\tan ^2(e+f x)} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)+3 \sqrt[4]{\tan ^2(e+f x)} \left(\log \left(1-\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)-\log \left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}+1\right)\right)\right)}{64 f \sec ^{\frac{5}{2}}(e+f x) \sqrt{b \tan (e+f x)}}","\frac{3 b^{5/2} d^3 \sqrt{b \tan (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{32 f \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{3 b^{5/2} d^3 \sqrt{b \tan (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{32 f \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{3 b d^2 (b \tan (e+f x))^{3/2} \sqrt{d \sec (e+f x)}}{16 f}+\frac{b (b \tan (e+f x))^{3/2} (d \sec (e+f x))^{5/2}}{4 f}",1,"(b^3*(d*Sec[e + f*x])^(5/2)*(12*Sqrt[Sec[e + f*x]] - 28*Sec[e + f*x]^(5/2) + 16*Sec[e + f*x]^(9/2) - 6*ArcTan[Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)]*(Tan[e + f*x]^2)^(1/4) + 3*(Log[1 - Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)] - Log[1 + Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)])*(Tan[e + f*x]^2)^(1/4)))/(64*f*Sec[e + f*x]^(5/2)*Sqrt[b*Tan[e + f*x]])","A",1
308,1,93,131,2.3522425,"\int (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{5/2} \, dx","Integrate[(d*Sec[e + f*x])^(3/2)*(b*Tan[e + f*x])^(5/2),x]","\frac{b^3 d^2 \left(-3 \sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right)+2 \sec ^4(e+f x)-5 \sec ^2(e+f x)+3\right)}{6 f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}","\frac{b^2 d^2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{b \tan (e+f x)}}{2 f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{b d^2 (b \tan (e+f x))^{3/2}}{2 f \sqrt{d \sec (e+f x)}}+\frac{b (b \tan (e+f x))^{3/2} (d \sec (e+f x))^{3/2}}{3 f}",1,"(b^3*d^2*(3 - 5*Sec[e + f*x]^2 + 2*Sec[e + f*x]^4 - 3*Hypergeometric2F1[-1/4, 1/4, 3/4, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4)))/(6*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])","C",1
309,1,182,169,1.751165,"\int \sqrt{d \sec (e+f x)} (b \tan (e+f x))^{5/2} \, dx","Integrate[Sqrt[d*Sec[e + f*x]]*(b*Tan[e + f*x])^(5/2),x]","\frac{\csc ^3(e+f x) (b \tan (e+f x))^{5/2} \sqrt{d \sec (e+f x)} \left(4 \sec ^{\frac{5}{2}}(e+f x)-4 \sqrt{\sec (e+f x)}-6 \sqrt[4]{\tan ^2(e+f x)} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)+3 \sqrt[4]{\tan ^2(e+f x)} \left(\log \left(1-\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)-\log \left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}+1\right)\right)\right)}{8 f \sec ^{\frac{7}{2}}(e+f x)}","\frac{3 b^{5/2} d \sqrt{b \tan (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{4 f \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{3 b^{5/2} d \sqrt{b \tan (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{4 f \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{b (b \tan (e+f x))^{3/2} \sqrt{d \sec (e+f x)}}{2 f}",1,"(Csc[e + f*x]^3*Sqrt[d*Sec[e + f*x]]*(b*Tan[e + f*x])^(5/2)*(-4*Sqrt[Sec[e + f*x]] + 4*Sec[e + f*x]^(5/2) - 6*ArcTan[Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)]*(Tan[e + f*x]^2)^(1/4) + 3*(Log[1 - Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)] - Log[1 + Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)])*(Tan[e + f*x]^2)^(1/4)))/(8*f*Sec[e + f*x]^(7/2))","A",1
310,1,74,88,0.7441714,"\int \frac{(b \tan (e+f x))^{5/2}}{\sqrt{d \sec (e+f x)}} \, dx","Integrate[(b*Tan[e + f*x])^(5/2)/Sqrt[d*Sec[e + f*x]],x]","\frac{b^3 \left(3 \sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right)+\tan ^2(e+f x)\right)}{f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}","\frac{b (b \tan (e+f x))^{3/2}}{f \sqrt{d \sec (e+f x)}}-\frac{3 b^2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{b \tan (e+f x)}}{f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}",1,"(b^3*(Tan[e + f*x]^2 + 3*Hypergeometric2F1[-1/4, 1/4, 3/4, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4)))/(f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])","C",1
311,1,181,168,1.1045319,"\int \frac{(b \tan (e+f x))^{5/2}}{(d \sec (e+f x))^{3/2}} \, dx","Integrate[(b*Tan[e + f*x])^(5/2)/(d*Sec[e + f*x])^(3/2),x]","\frac{\csc ^3(e+f x) (b \tan (e+f x))^{5/2} \sqrt{d \sec (e+f x)} \left(-4 \sin ^2(e+f x) \sqrt{\sec (e+f x)}+6 \sqrt[4]{\tan ^2(e+f x)} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)+3 \sqrt[4]{\tan ^2(e+f x)} \left(\log \left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}+1\right)-\log \left(1-\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)\right)\right)}{6 d^2 f \sec ^{\frac{7}{2}}(e+f x)}","-\frac{b^{5/2} \sqrt{b \tan (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{d f \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{b^{5/2} \sqrt{b \tan (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{d f \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{2 b (b \tan (e+f x))^{3/2}}{3 f (d \sec (e+f x))^{3/2}}",1,"(Csc[e + f*x]^3*Sqrt[d*Sec[e + f*x]]*(b*Tan[e + f*x])^(5/2)*(-4*Sqrt[Sec[e + f*x]]*Sin[e + f*x]^2 + 6*ArcTan[Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)]*(Tan[e + f*x]^2)^(1/4) + 3*(-Log[1 - Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)] + Log[1 + Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)])*(Tan[e + f*x]^2)^(1/4)))/(6*d^2*f*Sec[e + f*x]^(7/2))","A",1
312,1,81,96,0.7669774,"\int \frac{(b \tan (e+f x))^{5/2}}{(d \sec (e+f x))^{5/2}} \, dx","Integrate[(b*Tan[e + f*x])^(5/2)/(d*Sec[e + f*x])^(5/2),x]","\frac{b^3 \left(-6 \sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right)+\cos (2 (e+f x))-1\right)}{5 d^2 f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}","\frac{6 b^2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{b \tan (e+f x)}}{5 d^2 f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{2 b (b \tan (e+f x))^{3/2}}{5 f (d \sec (e+f x))^{5/2}}",1,"(b^3*(-1 + Cos[2*(e + f*x)] - 6*Hypergeometric2F1[-1/4, 1/4, 3/4, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4)))/(5*d^2*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])","C",1
313,1,45,34,0.1638613,"\int \frac{(b \tan (e+f x))^{5/2}}{(d \sec (e+f x))^{7/2}} \, dx","Integrate[(b*Tan[e + f*x])^(5/2)/(d*Sec[e + f*x])^(7/2),x]","\frac{2 b^2 \sin ^3(e+f x) \sqrt{b \tan (e+f x)}}{7 d^3 f \sqrt{d \sec (e+f x)}}","\frac{2 (b \tan (e+f x))^{7/2}}{7 b f (d \sec (e+f x))^{7/2}}",1,"(2*b^2*Sin[e + f*x]^3*Sqrt[b*Tan[e + f*x]])/(7*d^3*f*Sqrt[d*Sec[e + f*x]])","A",1
314,1,99,131,0.9854525,"\int \frac{(b \tan (e+f x))^{5/2}}{(d \sec (e+f x))^{9/2}} \, dx","Integrate[(b*Tan[e + f*x])^(5/2)/(d*Sec[e + f*x])^(9/2),x]","-\frac{b^2 \sin (2 (e+f x)) \sqrt{b \tan (e+f x)} \left(12 \sqrt[4]{-\tan ^2(e+f x)} \csc ^2(e+f x) \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right)+5 \cos (2 (e+f x))-1\right)}{90 d^4 f \sqrt{d \sec (e+f x)}}","\frac{4 b^2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{b \tan (e+f x)}}{15 d^4 f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{2 b (b \tan (e+f x))^{3/2}}{15 d^2 f (d \sec (e+f x))^{5/2}}-\frac{2 b (b \tan (e+f x))^{3/2}}{9 f (d \sec (e+f x))^{9/2}}",1,"-1/90*(b^2*Sin[2*(e + f*x)]*Sqrt[b*Tan[e + f*x]]*(-1 + 5*Cos[2*(e + f*x)] + 12*Csc[e + f*x]^2*Hypergeometric2F1[-1/4, 1/4, 3/4, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4)))/(d^4*f*Sqrt[d*Sec[e + f*x]])","C",1
315,1,136,178,6.5078833,"\int \frac{(d \sec (e+f x))^{7/2}}{\sqrt{b \tan (e+f x)}} \, dx","Integrate[(d*Sec[e + f*x])^(7/2)/Sqrt[b*Tan[e + f*x]],x]","\frac{d^3 \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)} \left(2 \sqrt[4]{\tan ^2(e+f x)} \sec ^{\frac{3}{2}}(e+f x)-3 \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)+3 \tanh ^{-1}\left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)\right)}{4 b f \sqrt[4]{\tan ^2(e+f x)} \sqrt{\sec (e+f x)}}","\frac{3 d^3 \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{4 \sqrt{b} f \sqrt{b \tan (e+f x)}}+\frac{3 d^3 \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{4 \sqrt{b} f \sqrt{b \tan (e+f x)}}+\frac{d^2 \sqrt{b \tan (e+f x)} (d \sec (e+f x))^{3/2}}{2 b f}",1,"(d^3*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]]*(-3*ArcTan[Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)] + 3*ArcTanh[Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)] + 2*Sec[e + f*x]^(3/2)*(Tan[e + f*x]^2)^(1/4)))/(4*b*f*Sqrt[Sec[e + f*x]]*(Tan[e + f*x]^2)^(1/4))","A",1
316,1,83,92,2.3645445,"\int \frac{(d \sec (e+f x))^{5/2}}{\sqrt{b \tan (e+f x)}} \, dx","Integrate[(d*Sec[e + f*x])^(5/2)/Sqrt[b*Tan[e + f*x]],x]","\frac{d^2 \sqrt{d \sec (e+f x)} \left(\sin (e+f x) \cos (e+f x) \sec ^2(e+f x)^{3/4} \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};-\tan ^2(e+f x)\right)+\tan (e+f x)\right)}{f \sqrt{b \tan (e+f x)}}","\frac{d^2 \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}{b f}+\frac{d^2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \sec (e+f x)}}{f \sqrt{b \tan (e+f x)}}",1,"(d^2*Sqrt[d*Sec[e + f*x]]*(Cos[e + f*x]*Hypergeometric2F1[1/4, 3/4, 5/4, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(3/4)*Sin[e + f*x] + Tan[e + f*x]))/(f*Sqrt[b*Tan[e + f*x]])","C",1
317,1,105,131,4.4203952,"\int \frac{(d \sec (e+f x))^{3/2}}{\sqrt{b \tan (e+f x)}} \, dx","Integrate[(d*Sec[e + f*x])^(3/2)/Sqrt[b*Tan[e + f*x]],x]","-\frac{\sqrt{b \tan (e+f x)} (d \sec (e+f x))^{3/2} \left(\tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)-\tanh ^{-1}\left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)\right)}{b f \sqrt[4]{\tan ^2(e+f x)} \sec ^{\frac{3}{2}}(e+f x)}","\frac{d \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{\sqrt{b} f \sqrt{b \tan (e+f x)}}+\frac{d \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{\sqrt{b} f \sqrt{b \tan (e+f x)}}",1,"-(((ArcTan[Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)] - ArcTanh[Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)])*(d*Sec[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]])/(b*f*Sec[e + f*x]^(3/2)*(Tan[e + f*x]^2)^(1/4)))","A",1
318,1,89,55,0.464309,"\int \frac{\sqrt{d \sec (e+f x)}}{\sqrt{b \tan (e+f x)}} \, dx","Integrate[Sqrt[d*Sec[e + f*x]]/Sqrt[b*Tan[e + f*x]],x]","\frac{2 \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{b f \sqrt{\sec (e+f x)} \sqrt{\cos ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x)}}","\frac{2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \sec (e+f x)}}{f \sqrt{b \tan (e+f x)}}",1,"(2*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[(e + f*x)/2]^2]*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])/(b*f*Sqrt[Sec[e + f*x]]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]])","C",1
319,1,32,32,0.3993989,"\int \frac{1}{\sqrt{d \sec (e+f x)} \sqrt{b \tan (e+f x)}} \, dx","Integrate[1/(Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]]),x]","\frac{2 \sqrt{b \tan (e+f x)}}{b f \sqrt{d \sec (e+f x)}}","\frac{2 \sqrt{b \tan (e+f x)}}{b f \sqrt{d \sec (e+f x)}}",1,"(2*Sqrt[b*Tan[e + f*x]])/(b*f*Sqrt[d*Sec[e + f*x]])","A",1
320,1,91,95,1.0133643,"\int \frac{1}{(d \sec (e+f x))^{3/2} \sqrt{b \tan (e+f x)}} \, dx","Integrate[1/((d*Sec[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]]),x]","\frac{2 \sqrt{b \tan (e+f x)} \left(\sqrt[4]{-\tan ^2(e+f x)}-2 \sec ^2(e+f x) \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};\sec ^2(e+f x)\right)\right)}{3 b f \sqrt[4]{-\tan ^2(e+f x)} (d \sec (e+f x))^{3/2}}","\frac{4 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \sec (e+f x)}}{3 d^2 f \sqrt{b \tan (e+f x)}}+\frac{2 \sqrt{b \tan (e+f x)}}{3 b f (d \sec (e+f x))^{3/2}}",1,"(2*Sqrt[b*Tan[e + f*x]]*(-2*Hypergeometric2F1[1/4, 3/4, 5/4, Sec[e + f*x]^2]*Sec[e + f*x]^2 + (-Tan[e + f*x]^2)^(1/4)))/(3*b*f*(d*Sec[e + f*x])^(3/2)*(-Tan[e + f*x]^2)^(1/4))","C",1
321,1,112,72,1.1759394,"\int \frac{1}{(d \sec (e+f x))^{5/2} \sqrt{b \tan (e+f x)}} \, dx","Integrate[1/((d*Sec[e + f*x])^(5/2)*Sqrt[b*Tan[e + f*x]]),x]","\frac{\sqrt{\frac{1}{\cos (e+f x)+1}} \cos (2 (e+f x)) \tan (e+f x)+9 \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)} \sqrt{\sec (e+f x)+1}}{5 d^2 f \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}","\frac{8 \sqrt{b \tan (e+f x)}}{5 b d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \sqrt{b \tan (e+f x)}}{5 b f (d \sec (e+f x))^{5/2}}",1,"(9*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2] + Sqrt[(1 + Cos[e + f*x])^(-1)]*Cos[2*(e + f*x)]*Tan[e + f*x])/(5*d^2*f*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])","A",1
322,1,211,171,1.0948704,"\int \frac{(d \sec (e+f x))^{5/2}}{(b \tan (e+f x))^{3/2}} \, dx","Integrate[(d*Sec[e + f*x])^(5/2)/(b*Tan[e + f*x])^(3/2),x]","-\frac{d^3 \sin (e+f x) \left(-4 \csc ^2(e+f x)+16 \csc ^2(2 (e+f x))-2 \sqrt[4]{\tan ^2(e+f x)} \sec ^{\frac{3}{2}}(e+f x) \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)+\sqrt[4]{\tan ^2(e+f x)} \sec ^{\frac{3}{2}}(e+f x) \log \left(1-\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)-\sqrt[4]{\tan ^2(e+f x)} \sec ^{\frac{3}{2}}(e+f x) \log \left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}+1\right)\right)}{2 f (b \tan (e+f x))^{3/2} \sqrt{d \sec (e+f x)}}","-\frac{d^3 \sqrt{b \tan (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{b^{3/2} f \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{d^3 \sqrt{b \tan (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{b^{3/2} f \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{2 d^2 \sqrt{d \sec (e+f x)}}{b f \sqrt{b \tan (e+f x)}}",1,"-1/2*(d^3*Sin[e + f*x]*(-4*Csc[e + f*x]^2 + 16*Csc[2*(e + f*x)]^2 - 2*ArcTan[Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)]*Sec[e + f*x]^(3/2)*(Tan[e + f*x]^2)^(1/4) + Log[1 - Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)]*Sec[e + f*x]^(3/2)*(Tan[e + f*x]^2)^(1/4) - Log[1 + Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)]*Sec[e + f*x]^(3/2)*(Tan[e + f*x]^2)^(1/4)))/(f*Sqrt[d*Sec[e + f*x]]*(b*Tan[e + f*x])^(3/2))","A",1
323,1,70,97,0.6327408,"\int \frac{(d \sec (e+f x))^{3/2}}{(b \tan (e+f x))^{3/2}} \, dx","Integrate[(d*Sec[e + f*x])^(3/2)/(b*Tan[e + f*x])^(3/2),x]","\frac{2 d^2 \left(\sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right)-1\right)}{b f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}","-\frac{2 d^2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{b \tan (e+f x)}}{b^2 f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{2 d^2}{b f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}",1,"(2*d^2*(-1 + Hypergeometric2F1[-1/4, 1/4, 3/4, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4)))/(b*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])","C",1
324,1,32,32,0.1169814,"\int \frac{\sqrt{d \sec (e+f x)}}{(b \tan (e+f x))^{3/2}} \, dx","Integrate[Sqrt[d*Sec[e + f*x]]/(b*Tan[e + f*x])^(3/2),x]","-\frac{2 \sqrt{d \sec (e+f x)}}{b f \sqrt{b \tan (e+f x)}}","-\frac{2 \sqrt{d \sec (e+f x)}}{b f \sqrt{b \tan (e+f x)}}",1,"(-2*Sqrt[d*Sec[e + f*x]])/(b*f*Sqrt[b*Tan[e + f*x]])","A",1
325,1,67,91,0.5545563,"\int \frac{1}{\sqrt{d \sec (e+f x)} (b \tan (e+f x))^{3/2}} \, dx","Integrate[1/(Sqrt[d*Sec[e + f*x]]*(b*Tan[e + f*x])^(3/2)),x]","\frac{4 \sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right)-2}{b f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}","-\frac{4 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{b \tan (e+f x)}}{b^2 f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{2}{b f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}",1,"(-2 + 4*Hypergeometric2F1[-1/4, 1/4, 3/4, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4))/(b*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])","C",1
326,1,52,72,0.1779191,"\int \frac{1}{(d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}} \, dx","Integrate[1/((d*Sec[e + f*x])^(3/2)*(b*Tan[e + f*x])^(3/2)),x]","\frac{(\cos (2 (e+f x))-7) \sec ^2(e+f x)}{3 b f \sqrt{b \tan (e+f x)} (d \sec (e+f x))^{3/2}}","\frac{2}{3 b f \sqrt{b \tan (e+f x)} (d \sec (e+f x))^{3/2}}-\frac{8 \sqrt{d \sec (e+f x)}}{3 b d^2 f \sqrt{b \tan (e+f x)}}",1,"((-7 + Cos[2*(e + f*x)])*Sec[e + f*x]^2)/(3*b*f*(d*Sec[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]])","A",1
327,1,81,130,0.6576422,"\int \frac{1}{(d \sec (e+f x))^{5/2} (b \tan (e+f x))^{3/2}} \, dx","Integrate[1/((d*Sec[e + f*x])^(5/2)*(b*Tan[e + f*x])^(3/2)),x]","\frac{24 \sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right)+\cos (2 (e+f x))-11}{5 b d^2 f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}","-\frac{12 (b \tan (e+f x))^{3/2}}{5 b^3 f (d \sec (e+f x))^{5/2}}-\frac{24 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{b \tan (e+f x)}}{5 b^2 d^2 f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{2}{b f \sqrt{b \tan (e+f x)} (d \sec (e+f x))^{5/2}}",1,"(-11 + Cos[2*(e + f*x)] + 24*Hypergeometric2F1[-1/4, 1/4, 3/4, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4))/(5*b*d^2*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])","C",1
328,1,144,172,1.2146576,"\int \frac{(d \sec (e+f x))^{7/2}}{(b \tan (e+f x))^{5/2}} \, dx","Integrate[(d*Sec[e + f*x])^(7/2)/(b*Tan[e + f*x])^(5/2),x]","-\frac{d^4 \sqrt{b \tan (e+f x)} \left(2 \sqrt[4]{\tan ^2(e+f x)} \csc ^2(e+f x)+3 \sqrt{\sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)-3 \sqrt{\sec (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{\sec (e+f x)}}{\sqrt[4]{\tan ^2(e+f x)}}\right)\right)}{3 b^3 f \sqrt[4]{\tan ^2(e+f x)} \sqrt{d \sec (e+f x)}}","\frac{d^3 \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{b^{5/2} f \sqrt{b \tan (e+f x)}}+\frac{d^3 \sqrt{b \sin (e+f x)} \sqrt{d \sec (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{b \sin (e+f x)}}{\sqrt{b}}\right)}{b^{5/2} f \sqrt{b \tan (e+f x)}}-\frac{2 d^2 (d \sec (e+f x))^{3/2}}{3 b f (b \tan (e+f x))^{3/2}}",1,"-1/3*(d^4*Sqrt[b*Tan[e + f*x]]*(3*ArcTan[Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)]*Sqrt[Sec[e + f*x]] - 3*ArcTanh[Sqrt[Sec[e + f*x]]/(Tan[e + f*x]^2)^(1/4)]*Sqrt[Sec[e + f*x]] + 2*Csc[e + f*x]^2*(Tan[e + f*x]^2)^(1/4)))/(b^3*f*Sqrt[d*Sec[e + f*x]]*(Tan[e + f*x]^2)^(1/4))","A",1
329,1,116,101,0.4399511,"\int \frac{(d \sec (e+f x))^{5/2}}{(b \tan (e+f x))^{5/2}} \, dx","Integrate[(d*Sec[e + f*x])^(5/2)/(b*Tan[e + f*x])^(5/2),x]","\frac{2 d^3 \sqrt{b \tan (e+f x)} \left(\sqrt{2} \sqrt{\sec (e+f x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-\cot (e+f x) \csc (e+f x) \sqrt{\sec (e+f x)+1}\right)}{3 b^3 f \sqrt{\sec (e+f x)+1} \sqrt{d \sec (e+f x)}}","\frac{2 d^2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \sec (e+f x)}}{3 b^2 f \sqrt{b \tan (e+f x)}}-\frac{2 d^2 \sqrt{d \sec (e+f x)}}{3 b f (b \tan (e+f x))^{3/2}}",1,"(2*d^3*(Sqrt[2]*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[(e + f*x)/2]^2]*Sqrt[Sec[e + f*x]] - Cot[e + f*x]*Csc[e + f*x]*Sqrt[1 + Sec[e + f*x]])*Sqrt[b*Tan[e + f*x]])/(3*b^3*f*Sqrt[d*Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]])","C",1
330,1,34,34,0.1463011,"\int \frac{(d \sec (e+f x))^{3/2}}{(b \tan (e+f x))^{5/2}} \, dx","Integrate[(d*Sec[e + f*x])^(3/2)/(b*Tan[e + f*x])^(5/2),x]","-\frac{2 (d \sec (e+f x))^{3/2}}{3 b f (b \tan (e+f x))^{3/2}}","-\frac{2 (d \sec (e+f x))^{3/2}}{3 b f (b \tan (e+f x))^{3/2}}",1,"(-2*(d*Sec[e + f*x])^(3/2))/(3*b*f*(b*Tan[e + f*x])^(3/2))","A",1
331,1,70,95,0.7233563,"\int \frac{\sqrt{d \sec (e+f x)}}{(b \tan (e+f x))^{5/2}} \, dx","Integrate[Sqrt[d*Sec[e + f*x]]/(b*Tan[e + f*x])^(5/2),x]","-\frac{2 \sqrt{d \sec (e+f x)} \left(2 \left(-\tan ^2(e+f x)\right)^{3/4} \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};\sec ^2(e+f x)\right)+1\right)}{3 b f (b \tan (e+f x))^{3/2}}","-\frac{4 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \sec (e+f x)}}{3 b^2 f \sqrt{b \tan (e+f x)}}-\frac{2 \sqrt{d \sec (e+f x)}}{3 b f (b \tan (e+f x))^{3/2}}",1,"(-2*Sqrt[d*Sec[e + f*x]]*(1 + 2*Hypergeometric2F1[1/4, 3/4, 5/4, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(3/4)))/(3*b*f*(b*Tan[e + f*x])^(3/2))","C",1
332,1,110,69,0.8464565,"\int \frac{1}{\sqrt{d \sec (e+f x)} (b \tan (e+f x))^{5/2}} \, dx","Integrate[1/(Sqrt[d*Sec[e + f*x]]*(b*Tan[e + f*x])^(5/2)),x]","-\frac{2 \left(3 \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)+1} \sqrt{\sec (e+f x)}+\sqrt{\frac{1}{\cos (e+f x)+1}} \csc (e+f x) \sec (e+f x)\right)}{3 b^2 f \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}","-\frac{8 \sqrt{b \tan (e+f x)}}{3 b^3 f \sqrt{d \sec (e+f x)}}-\frac{2}{3 b f (b \tan (e+f x))^{3/2} \sqrt{d \sec (e+f x)}}",1,"(-2*(Sqrt[(1 + Cos[e + f*x])^(-1)]*Csc[e + f*x]*Sec[e + f*x] + 3*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]))/(3*b^2*f*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])","A",1
333,1,112,132,2.3876059,"\int \frac{1}{(d \sec (e+f x))^{3/2} (b \tan (e+f x))^{5/2}} \, dx","Integrate[1/((d*Sec[e + f*x])^(3/2)*(b*Tan[e + f*x])^(5/2)),x]","\frac{\left(-\tan ^2(e+f x)\right)^{3/4} \csc ^2(e+f x) \sqrt{b \tan (e+f x)} \left(\sqrt[4]{-\tan ^2(e+f x)} \left(\cos (2 (e+f x))+2 \csc ^2(e+f x)-1\right)-8 \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};\sec ^2(e+f x)\right)\right)}{3 b^3 f (d \sec (e+f x))^{3/2}}","-\frac{4 \sqrt{b \tan (e+f x)}}{3 b^3 f (d \sec (e+f x))^{3/2}}-\frac{8 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \sec (e+f x)}}{3 b^2 d^2 f \sqrt{b \tan (e+f x)}}-\frac{2}{3 b f (b \tan (e+f x))^{3/2} (d \sec (e+f x))^{3/2}}",1,"(Csc[e + f*x]^2*Sqrt[b*Tan[e + f*x]]*(-Tan[e + f*x]^2)^(3/4)*(-8*Hypergeometric2F1[1/4, 3/4, 5/4, Sec[e + f*x]^2] + (-1 + Cos[2*(e + f*x)] + 2*Csc[e + f*x]^2)*(-Tan[e + f*x]^2)^(1/4)))/(3*b^3*f*(d*Sec[e + f*x])^(3/2))","C",1
334,1,159,106,3.1224362,"\int \frac{1}{(d \sec (e+f x))^{5/2} (b \tan (e+f x))^{5/2}} \, dx","Integrate[1/((d*Sec[e + f*x])^(5/2)*(b*Tan[e + f*x])^(5/2)),x]","\frac{-6 \sqrt{\frac{1}{\cos (e+f x)+1}} (2 \cos (2 (e+f x))-1) \tan (e+f x)-228 \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)+1} \sqrt{\sec (e+f x)}+\sqrt{\frac{1}{\cos (e+f x)+1}} (3 \cos (2 (e+f x))-43) \csc (e+f x) \sec (e+f x)}{60 b^2 d^2 f \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}","-\frac{64 \sqrt{b \tan (e+f x)}}{15 b^3 d^2 f \sqrt{d \sec (e+f x)}}-\frac{16 \sqrt{b \tan (e+f x)}}{15 b^3 f (d \sec (e+f x))^{5/2}}-\frac{2}{3 b f (b \tan (e+f x))^{3/2} (d \sec (e+f x))^{5/2}}",1,"(Sqrt[(1 + Cos[e + f*x])^(-1)]*(-43 + 3*Cos[2*(e + f*x)])*Csc[e + f*x]*Sec[e + f*x] - 228*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2] - 6*Sqrt[(1 + Cos[e + f*x])^(-1)]*(-1 + 2*Cos[2*(e + f*x)])*Tan[e + f*x])/(60*b^2*d^2*f*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])","A",1
335,1,64,64,0.0996182,"\int (b \sec (e+f x))^{4/3} \sqrt{d \tan (e+f x)} \, dx","Integrate[(b*Sec[e + f*x])^(4/3)*Sqrt[d*Tan[e + f*x]],x]","\frac{3 d \sqrt[4]{-\tan ^2(e+f x)} (b \sec (e+f x))^{4/3} \, _2F_1\left(\frac{1}{4},\frac{2}{3};\frac{5}{3};\sec ^2(e+f x)\right)}{4 f \sqrt{d \tan (e+f x)}}","\frac{2 \cos ^2(e+f x)^{17/12} (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, _2F_1\left(\frac{3}{4},\frac{17}{12};\frac{7}{4};\sin ^2(e+f x)\right)}{3 d f}",1,"(3*d*Hypergeometric2F1[1/4, 2/3, 5/3, Sec[e + f*x]^2]*(b*Sec[e + f*x])^(4/3)*(-Tan[e + f*x]^2)^(1/4))/(4*f*Sqrt[d*Tan[e + f*x]])","A",1
336,1,62,64,0.0812658,"\int \sqrt[3]{b \sec (e+f x)} \sqrt{d \tan (e+f x)} \, dx","Integrate[(b*Sec[e + f*x])^(1/3)*Sqrt[d*Tan[e + f*x]],x]","\frac{3 d \sqrt[4]{-\tan ^2(e+f x)} \sqrt[3]{b \sec (e+f x)} \, _2F_1\left(\frac{1}{6},\frac{1}{4};\frac{7}{6};\sec ^2(e+f x)\right)}{f \sqrt{d \tan (e+f x)}}","\frac{2 \cos ^2(e+f x)^{11/12} \sqrt[3]{b \sec (e+f x)} (d \tan (e+f x))^{3/2} \, _2F_1\left(\frac{3}{4},\frac{11}{12};\frac{7}{4};\sin ^2(e+f x)\right)}{3 d f}",1,"(3*d*Hypergeometric2F1[1/6, 1/4, 7/6, Sec[e + f*x]^2]*(b*Sec[e + f*x])^(1/3)*(-Tan[e + f*x]^2)^(1/4))/(f*Sqrt[d*Tan[e + f*x]])","A",1
337,1,62,64,0.0964197,"\int \frac{\sqrt{d \tan (e+f x)}}{\sqrt[3]{b \sec (e+f x)}} \, dx","Integrate[Sqrt[d*Tan[e + f*x]]/(b*Sec[e + f*x])^(1/3),x]","-\frac{3 d \sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{4};\frac{5}{6};\sec ^2(e+f x)\right)}{f \sqrt[3]{b \sec (e+f x)} \sqrt{d \tan (e+f x)}}","\frac{2 \cos ^2(e+f x)^{7/12} (d \tan (e+f x))^{3/2} \, _2F_1\left(\frac{7}{12},\frac{3}{4};\frac{7}{4};\sin ^2(e+f x)\right)}{3 d f \sqrt[3]{b \sec (e+f x)}}",1,"(-3*d*Hypergeometric2F1[-1/6, 1/4, 5/6, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4))/(f*(b*Sec[e + f*x])^(1/3)*Sqrt[d*Tan[e + f*x]])","A",1
338,1,64,64,0.1204898,"\int \frac{\sqrt{d \tan (e+f x)}}{(b \sec (e+f x))^{4/3}} \, dx","Integrate[Sqrt[d*Tan[e + f*x]]/(b*Sec[e + f*x])^(4/3),x]","-\frac{3 d \sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{2}{3},\frac{1}{4};\frac{1}{3};\sec ^2(e+f x)\right)}{4 f (b \sec (e+f x))^{4/3} \sqrt{d \tan (e+f x)}}","\frac{2 \sqrt[12]{\cos ^2(e+f x)} (d \tan (e+f x))^{3/2} \, _2F_1\left(\frac{1}{12},\frac{3}{4};\frac{7}{4};\sin ^2(e+f x)\right)}{3 d f (b \sec (e+f x))^{4/3}}",1,"(-3*d*Hypergeometric2F1[-2/3, 1/4, 1/3, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4))/(4*f*(b*Sec[e + f*x])^(4/3)*Sqrt[d*Tan[e + f*x]])","A",1
339,1,64,64,0.1540649,"\int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx","Integrate[(b*Sec[e + f*x])^(4/3)*(d*Tan[e + f*x])^(3/2),x]","\frac{3 d (b \sec (e+f x))^{4/3} \sqrt{d \tan (e+f x)} \, _2F_1\left(-\frac{1}{4},\frac{2}{3};\frac{5}{3};\sec ^2(e+f x)\right)}{4 f \sqrt[4]{-\tan ^2(e+f x)}}","\frac{2 \cos ^2(e+f x)^{23/12} (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{5/2} \, _2F_1\left(\frac{5}{4},\frac{23}{12};\frac{9}{4};\sin ^2(e+f x)\right)}{5 d f}",1,"(3*d*Hypergeometric2F1[-1/4, 2/3, 5/3, Sec[e + f*x]^2]*(b*Sec[e + f*x])^(4/3)*Sqrt[d*Tan[e + f*x]])/(4*f*(-Tan[e + f*x]^2)^(1/4))","A",1
340,1,62,64,0.1376115,"\int \sqrt[3]{b \sec (e+f x)} (d \tan (e+f x))^{3/2} \, dx","Integrate[(b*Sec[e + f*x])^(1/3)*(d*Tan[e + f*x])^(3/2),x]","\frac{3 d \sqrt[3]{b \sec (e+f x)} \sqrt{d \tan (e+f x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{6};\frac{7}{6};\sec ^2(e+f x)\right)}{f \sqrt[4]{-\tan ^2(e+f x)}}","\frac{2 \cos ^2(e+f x)^{17/12} \sqrt[3]{b \sec (e+f x)} (d \tan (e+f x))^{5/2} \, _2F_1\left(\frac{5}{4},\frac{17}{12};\frac{9}{4};\sin ^2(e+f x)\right)}{5 d f}",1,"(3*d*Hypergeometric2F1[-1/4, 1/6, 7/6, Sec[e + f*x]^2]*(b*Sec[e + f*x])^(1/3)*Sqrt[d*Tan[e + f*x]])/(f*(-Tan[e + f*x]^2)^(1/4))","A",1
341,1,69,64,0.0750254,"\int \frac{(d \tan (e+f x))^{3/2}}{\sqrt[3]{b \sec (e+f x)}} \, dx","Integrate[(d*Tan[e + f*x])^(3/2)/(b*Sec[e + f*x])^(1/3),x]","\frac{3 \left(-\tan ^2(e+f x)\right)^{3/4} \cot ^3(e+f x) (d \tan (e+f x))^{3/2} \, _2F_1\left(-\frac{1}{4},-\frac{1}{6};\frac{5}{6};\sec ^2(e+f x)\right)}{f \sqrt[3]{b \sec (e+f x)}}","\frac{2 \cos ^2(e+f x)^{13/12} (d \tan (e+f x))^{5/2} \, _2F_1\left(\frac{13}{12},\frac{5}{4};\frac{9}{4};\sin ^2(e+f x)\right)}{5 d f \sqrt[3]{b \sec (e+f x)}}",1,"(3*Cot[e + f*x]^3*Hypergeometric2F1[-1/4, -1/6, 5/6, Sec[e + f*x]^2]*(d*Tan[e + f*x])^(3/2)*(-Tan[e + f*x]^2)^(3/4))/(f*(b*Sec[e + f*x])^(1/3))","A",1
342,1,71,64,0.0800904,"\int \frac{(d \tan (e+f x))^{3/2}}{(b \sec (e+f x))^{4/3}} \, dx","Integrate[(d*Tan[e + f*x])^(3/2)/(b*Sec[e + f*x])^(4/3),x]","\frac{3 \left(-\tan ^2(e+f x)\right)^{3/4} \cot ^3(e+f x) (d \tan (e+f x))^{3/2} \, _2F_1\left(-\frac{2}{3},-\frac{1}{4};\frac{1}{3};\sec ^2(e+f x)\right)}{4 f (b \sec (e+f x))^{4/3}}","\frac{2 \cos ^2(e+f x)^{7/12} (d \tan (e+f x))^{5/2} \, _2F_1\left(\frac{7}{12},\frac{5}{4};\frac{9}{4};\sin ^2(e+f x)\right)}{5 d f (b \sec (e+f x))^{4/3}}",1,"(3*Cot[e + f*x]^3*Hypergeometric2F1[-2/3, -1/4, 1/3, Sec[e + f*x]^2]*(d*Tan[e + f*x])^(3/2)*(-Tan[e + f*x]^2)^(3/4))/(4*f*(b*Sec[e + f*x])^(4/3))","A",1
343,1,62,64,0.1436073,"\int \sqrt{b \sec (e+f x)} (d \tan (e+f x))^{4/3} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]*(d*Tan[e + f*x])^(4/3),x]","\frac{2 d \sqrt{b \sec (e+f x)} \sqrt[3]{d \tan (e+f x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{4};\frac{5}{4};\sec ^2(e+f x)\right)}{f \sqrt[6]{-\tan ^2(e+f x)}}","\frac{3 \cos ^2(e+f x)^{17/12} \sqrt{b \sec (e+f x)} (d \tan (e+f x))^{7/3} \, _2F_1\left(\frac{7}{6},\frac{17}{12};\frac{13}{6};\sin ^2(e+f x)\right)}{7 d f}",1,"(2*d*Hypergeometric2F1[-1/6, 1/4, 5/4, Sec[e + f*x]^2]*Sqrt[b*Sec[e + f*x]]*(d*Tan[e + f*x])^(1/3))/(f*(-Tan[e + f*x]^2)^(1/6))","A",1
344,1,62,64,0.0863936,"\int \sqrt{b \sec (e+f x)} \sqrt[3]{d \tan (e+f x)} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]*(d*Tan[e + f*x])^(1/3),x]","\frac{2 d \sqrt[3]{-\tan ^2(e+f x)} \sqrt{b \sec (e+f x)} \, _2F_1\left(\frac{1}{4},\frac{1}{3};\frac{5}{4};\sec ^2(e+f x)\right)}{f (d \tan (e+f x))^{2/3}}","\frac{3 \cos ^2(e+f x)^{11/12} \sqrt{b \sec (e+f x)} (d \tan (e+f x))^{4/3} \, _2F_1\left(\frac{2}{3},\frac{11}{12};\frac{5}{3};\sin ^2(e+f x)\right)}{4 d f}",1,"(2*d*Hypergeometric2F1[1/4, 1/3, 5/4, Sec[e + f*x]^2]*Sqrt[b*Sec[e + f*x]]*(-Tan[e + f*x]^2)^(1/3))/(f*(d*Tan[e + f*x])^(2/3))","A",1
345,1,62,64,0.1180966,"\int \frac{\sqrt{b \sec (e+f x)}}{\sqrt[3]{d \tan (e+f x)}} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]/(d*Tan[e + f*x])^(1/3),x]","\frac{2 d \left(-\tan ^2(e+f x)\right)^{2/3} \sqrt{b \sec (e+f x)} \, _2F_1\left(\frac{1}{4},\frac{2}{3};\frac{5}{4};\sec ^2(e+f x)\right)}{f (d \tan (e+f x))^{4/3}}","\frac{3 \cos ^2(e+f x)^{7/12} \sqrt{b \sec (e+f x)} (d \tan (e+f x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{7}{12};\frac{4}{3};\sin ^2(e+f x)\right)}{2 d f}",1,"(2*d*Hypergeometric2F1[1/4, 2/3, 5/4, Sec[e + f*x]^2]*Sqrt[b*Sec[e + f*x]]*(-Tan[e + f*x]^2)^(2/3))/(f*(d*Tan[e + f*x])^(4/3))","A",1
346,1,62,62,0.2333183,"\int \frac{\sqrt{b \sec (e+f x)}}{(d \tan (e+f x))^{4/3}} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]/(d*Tan[e + f*x])^(4/3),x]","\frac{2 d \left(-\tan ^2(e+f x)\right)^{7/6} \sqrt{b \sec (e+f x)} \, _2F_1\left(\frac{1}{4},\frac{7}{6};\frac{5}{4};\sec ^2(e+f x)\right)}{f (d \tan (e+f x))^{7/3}}","-\frac{3 \sqrt[12]{\cos ^2(e+f x)} \sqrt{b \sec (e+f x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{12};\frac{5}{6};\sin ^2(e+f x)\right)}{d f \sqrt[3]{d \tan (e+f x)}}",1,"(2*d*Hypergeometric2F1[1/4, 7/6, 5/4, Sec[e + f*x]^2]*Sqrt[b*Sec[e + f*x]]*(-Tan[e + f*x]^2)^(7/6))/(f*(d*Tan[e + f*x])^(7/3))","A",1
347,1,64,64,0.1453414,"\int (b \sec (e+f x))^{3/2} (d \tan (e+f x))^{4/3} \, dx","Integrate[(b*Sec[e + f*x])^(3/2)*(d*Tan[e + f*x])^(4/3),x]","\frac{2 d (b \sec (e+f x))^{3/2} \sqrt[3]{d \tan (e+f x)} \, _2F_1\left(-\frac{1}{6},\frac{3}{4};\frac{7}{4};\sec ^2(e+f x)\right)}{3 f \sqrt[6]{-\tan ^2(e+f x)}}","\frac{3 \cos ^2(e+f x)^{23/12} (b \sec (e+f x))^{3/2} (d \tan (e+f x))^{7/3} \, _2F_1\left(\frac{7}{6},\frac{23}{12};\frac{13}{6};\sin ^2(e+f x)\right)}{7 d f}",1,"(2*d*Hypergeometric2F1[-1/6, 3/4, 7/4, Sec[e + f*x]^2]*(b*Sec[e + f*x])^(3/2)*(d*Tan[e + f*x])^(1/3))/(3*f*(-Tan[e + f*x]^2)^(1/6))","A",1
348,1,64,64,0.0877288,"\int (b \sec (e+f x))^{3/2} \sqrt[3]{d \tan (e+f x)} \, dx","Integrate[(b*Sec[e + f*x])^(3/2)*(d*Tan[e + f*x])^(1/3),x]","\frac{2 d \sqrt[3]{-\tan ^2(e+f x)} (b \sec (e+f x))^{3/2} \, _2F_1\left(\frac{1}{3},\frac{3}{4};\frac{7}{4};\sec ^2(e+f x)\right)}{3 f (d \tan (e+f x))^{2/3}}","\frac{3 \cos ^2(e+f x)^{17/12} (b \sec (e+f x))^{3/2} (d \tan (e+f x))^{4/3} \, _2F_1\left(\frac{2}{3},\frac{17}{12};\frac{5}{3};\sin ^2(e+f x)\right)}{4 d f}",1,"(2*d*Hypergeometric2F1[1/3, 3/4, 7/4, Sec[e + f*x]^2]*(b*Sec[e + f*x])^(3/2)*(-Tan[e + f*x]^2)^(1/3))/(3*f*(d*Tan[e + f*x])^(2/3))","A",1
349,1,64,64,0.1069849,"\int \frac{(b \sec (e+f x))^{3/2}}{\sqrt[3]{d \tan (e+f x)}} \, dx","Integrate[(b*Sec[e + f*x])^(3/2)/(d*Tan[e + f*x])^(1/3),x]","\frac{2 d \left(-\tan ^2(e+f x)\right)^{2/3} (b \sec (e+f x))^{3/2} \, _2F_1\left(\frac{2}{3},\frac{3}{4};\frac{7}{4};\sec ^2(e+f x)\right)}{3 f (d \tan (e+f x))^{4/3}}","\frac{3 \cos ^2(e+f x)^{13/12} (b \sec (e+f x))^{3/2} (d \tan (e+f x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{13}{12};\frac{4}{3};\sin ^2(e+f x)\right)}{2 d f}",1,"(2*d*Hypergeometric2F1[2/3, 3/4, 7/4, Sec[e + f*x]^2]*(b*Sec[e + f*x])^(3/2)*(-Tan[e + f*x]^2)^(2/3))/(3*f*(d*Tan[e + f*x])^(4/3))","A",1
350,1,64,62,0.2105617,"\int \frac{(b \sec (e+f x))^{3/2}}{(d \tan (e+f x))^{4/3}} \, dx","Integrate[(b*Sec[e + f*x])^(3/2)/(d*Tan[e + f*x])^(4/3),x]","\frac{2 d \left(-\tan ^2(e+f x)\right)^{7/6} (b \sec (e+f x))^{3/2} \, _2F_1\left(\frac{3}{4},\frac{7}{6};\frac{7}{4};\sec ^2(e+f x)\right)}{3 f (d \tan (e+f x))^{7/3}}","-\frac{3 \cos ^2(e+f x)^{7/12} (b \sec (e+f x))^{3/2} \, _2F_1\left(-\frac{1}{6},\frac{7}{12};\frac{5}{6};\sin ^2(e+f x)\right)}{d f \sqrt[3]{d \tan (e+f x)}}",1,"(2*d*Hypergeometric2F1[3/4, 7/6, 7/4, Sec[e + f*x]^2]*(b*Sec[e + f*x])^(3/2)*(-Tan[e + f*x]^2)^(7/6))/(3*f*(d*Tan[e + f*x])^(7/3))","A",1
351,1,47,67,0.3765799,"\int (b \sec (e+f x))^m \tan ^5(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^m*Tan[e + f*x]^5,x]","\frac{\left(\frac{\sec ^4(e+f x)}{m+4}-\frac{2 \sec ^2(e+f x)}{m+2}+\frac{1}{m}\right) (b \sec (e+f x))^m}{f}","\frac{(b \sec (e+f x))^{m+4}}{b^4 f (m+4)}-\frac{2 (b \sec (e+f x))^{m+2}}{b^2 f (m+2)}+\frac{(b \sec (e+f x))^m}{f m}",1,"((b*Sec[e + f*x])^m*(m^(-1) - (2*Sec[e + f*x]^2)/(2 + m) + Sec[e + f*x]^4/(4 + m)))/f","A",1
352,1,34,43,0.1159922,"\int (b \sec (e+f x))^m \tan ^3(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^m*Tan[e + f*x]^3,x]","\frac{\left(\frac{\sec ^2(e+f x)}{m+2}-\frac{1}{m}\right) (b \sec (e+f x))^m}{f}","\frac{(b \sec (e+f x))^{m+2}}{b^2 f (m+2)}-\frac{(b \sec (e+f x))^m}{f m}",1,"((b*Sec[e + f*x])^m*(-m^(-1) + Sec[e + f*x]^2/(2 + m)))/f","A",1
353,1,17,17,0.0233908,"\int (b \sec (e+f x))^m \tan (e+f x) \, dx","Integrate[(b*Sec[e + f*x])^m*Tan[e + f*x],x]","\frac{(b \sec (e+f x))^m}{f m}","\frac{(b \sec (e+f x))^m}{f m}",1,"(b*Sec[e + f*x])^m/(f*m)","A",1
354,1,124,40,0.8225439,"\int \cot (e+f x) (b \sec (e+f x))^m \, dx","Integrate[Cot[e + f*x]*(b*Sec[e + f*x])^m,x]","\frac{b \sec ^2\left(\frac{1}{2} (e+f x)\right) (b \sec (e+f x))^{m-1} \left((\cos (e+f x)+1) \, _2F_1(1,1-m;2-m;\cos (e+f x))-2^m \sec ^2\left(\frac{1}{2} (e+f x)\right)^{-m} \, _2F_1\left(1-m,1-m;2-m;\frac{1}{2} \cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}{4 f (m-1)}","-\frac{(b \sec (e+f x))^m \, _2F_1\left(1,\frac{m}{2};\frac{m+2}{2};\sec ^2(e+f x)\right)}{f m}",1,"(b*Sec[(e + f*x)/2]^2*((1 + Cos[e + f*x])*Hypergeometric2F1[1, 1 - m, 2 - m, Cos[e + f*x]] - (2^m*Hypergeometric2F1[1 - m, 1 - m, 2 - m, (Cos[e + f*x]*Sec[(e + f*x)/2]^2)/2])/(Sec[(e + f*x)/2]^2)^m)*(b*Sec[e + f*x])^(-1 + m))/(4*f*(-1 + m))","B",1
355,1,815,39,12.2341938,"\int \cot ^3(e+f x) (b \sec (e+f x))^m \, dx","Integrate[Cot[e + f*x]^3*(b*Sec[e + f*x])^m,x]","\frac{\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right) \left(2^m \, _2F_1\left(1-m,1-m;2-m;\frac{1}{2} \cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)^{-m}-(\cos (e+f x)+1) \, _2F_1(1,1-m;2-m;\cos (e+f x))\right) (b \sec (e+f x))^m}{4 f (m-1)}-\frac{2 \cot \left(\frac{1}{2} (e+f x)\right) \cot (e+f x) \csc ^2(e+f x) \left(F_1\left(1;m,-m;2;\cot ^2\left(\frac{1}{2} (e+f x)\right),-\cot ^2\left(\frac{1}{2} (e+f x)\right)\right) \cot ^4\left(\frac{1}{2} (e+f x)\right) \left(-\cos (e+f x) \csc ^2\left(\frac{1}{2} (e+f x)\right)\right)^m \sec ^2\left(\frac{1}{2} (e+f x)\right)^m+F_1\left(1;m,-m;2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \csc ^2\left(\frac{1}{2} (e+f x)\right)^m \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^m\right) (b \sec (e+f x))^m}{f \left(-m F_1\left(2;m,1-m;3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\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)^{m+1} \sec (e+f x) \csc ^2\left(\frac{1}{2} (e+f x)\right)^m-m F_1\left(2;m+1,-m;3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\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)^{m+1} \sec (e+f x) \csc ^2\left(\frac{1}{2} (e+f x)\right)^m-2 F_1\left(1;m,-m;2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\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)^m \csc ^2\left(\frac{1}{2} (e+f x)\right)^{m+1}+m F_1\left(2;m,1-m;3;\cot ^2\left(\frac{1}{2} (e+f x)\right),-\cot ^2\left(\frac{1}{2} (e+f x)\right)\right) \cot ^8\left(\frac{1}{2} (e+f x)\right) \left(-\cos (e+f x) \csc ^2\left(\frac{1}{2} (e+f x)\right)\right)^m \sec ^2\left(\frac{1}{2} (e+f x)\right)^{m+1}+m F_1\left(2;m+1,-m;3;\cot ^2\left(\frac{1}{2} (e+f x)\right),-\cot ^2\left(\frac{1}{2} (e+f x)\right)\right) \cot ^8\left(\frac{1}{2} (e+f x)\right) \left(-\cos (e+f x) \csc ^2\left(\frac{1}{2} (e+f x)\right)\right)^m \sec ^2\left(\frac{1}{2} (e+f x)\right)^{m+1}+2 F_1\left(1;m,-m;2;\cot ^2\left(\frac{1}{2} (e+f x)\right),-\cot ^2\left(\frac{1}{2} (e+f x)\right)\right) \cot ^6\left(\frac{1}{2} (e+f x)\right) \left(-\cos (e+f x) \csc ^2\left(\frac{1}{2} (e+f x)\right)\right)^m \sec ^2\left(\frac{1}{2} (e+f x)\right)^{m+1}\right)}","\frac{(b \sec (e+f x))^m \, _2F_1\left(2,\frac{m}{2};\frac{m+2}{2};\sec ^2(e+f x)\right)}{f m}",1,"(Cos[e + f*x]*Sec[(e + f*x)/2]^2*(-((1 + Cos[e + f*x])*Hypergeometric2F1[1, 1 - m, 2 - m, Cos[e + f*x]]) + (2^m*Hypergeometric2F1[1 - m, 1 - m, 2 - m, (Cos[e + f*x]*Sec[(e + f*x)/2]^2)/2])/(Sec[(e + f*x)/2]^2)^m)*(b*Sec[e + f*x])^m)/(4*f*(-1 + m)) - (2*Cot[(e + f*x)/2]*Cot[e + f*x]*Csc[e + f*x]^2*(AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^4*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^m + AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m)*(b*Sec[e + f*x])^m)/(f*(2*AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^6*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) + m*AppellF1[2, m, 1 - m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^8*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) + m*AppellF1[2, 1 + m, -m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^8*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) - 2*AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^(1 + m)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m - m*AppellF1[2, m, 1 - m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(1 + m)*Sec[e + f*x] - m*AppellF1[2, 1 + m, -m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(1 + m)*Sec[e + f*x]))","C",0
356,1,2138,40,21.6282213,"\int \cot ^5(e+f x) (b \sec (e+f x))^m \, dx","Integrate[Cot[e + f*x]^5*(b*Sec[e + f*x])^m,x]","\text{Result too large to show}","-\frac{(b \sec (e+f x))^m \, _2F_1\left(3,\frac{m}{2};\frac{m+2}{2};\sec ^2(e+f x)\right)}{f m}",1,"(Cos[e + f*x]*Sec[(e + f*x)/2]^2*((1 + Cos[e + f*x])*Hypergeometric2F1[1, 1 - m, 2 - m, Cos[e + f*x]] - (2^m*Hypergeometric2F1[1 - m, 1 - m, 2 - m, (Cos[e + f*x]*Sec[(e + f*x)/2]^2)/2])/(Sec[(e + f*x)/2]^2)^m)*(b*Sec[e + f*x])^m)/(4*f*(-1 + m)) + (3*Cot[(e + f*x)/2]*Cot[e + f*x]*Csc[e + f*x]^4*(4*AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^6*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^m + AppellF1[2, m, -m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^8*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^m + (AppellF1[2, m, -m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^2)*(Csc[(e + f*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m)*(b*Sec[e + f*x])^m)/(2*f*(-6*AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^8*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) - 3*m*AppellF1[2, m, 1 - m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^10*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) - 3*AppellF1[2, m, -m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^10*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) - 3*m*AppellF1[2, 1 + m, -m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^10*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) - m*AppellF1[3, m, 1 - m, 4, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^12*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) - m*AppellF1[3, 1 + m, -m, 4, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^12*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) + 3*m*AppellF1[2, m, 1 - m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^(1 + m)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m + 3*AppellF1[2, m, -m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^(1 + m)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m + 3*m*AppellF1[2, 1 + m, -m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^(1 + m)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m + 6*AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^2*(Csc[(e + f*x)/2]^2)^(1 + m)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m + m*AppellF1[3, m, 1 - m, 4, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^m*Sec[(e + f*x)/2]^2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m + m*AppellF1[3, 1 + m, -m, 4, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^m*Sec[(e + f*x)/2]^2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m)) + (4*Cot[(e + f*x)/2]*Cot[e + f*x]*Csc[e + f*x]^2*(AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^4*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^m + AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m)*(b*Sec[e + f*x])^m)/(f*(2*AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^6*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) + m*AppellF1[2, m, 1 - m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^8*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) + m*AppellF1[2, 1 + m, -m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^8*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) - 2*AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^(1 + m)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m - m*AppellF1[2, m, 1 - m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(1 + m)*Sec[e + f*x] - m*AppellF1[2, 1 + m, -m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(1 + m)*Sec[e + f*x]))","C",0
357,1,110,63,0.2554854,"\int (b \sec (e+f x))^m \tan ^4(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^m*Tan[e + f*x]^4,x]","\frac{\sin (2 (e+f x)) \cos ^2(e+f x)^{\frac{m-1}{2}} (b \sec (e+f x))^m \left(\, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{3}{2};\sin ^2(e+f x)\right)-2 \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{3}{2};\sin ^2(e+f x)\right)+\, _2F_1\left(\frac{1}{2},\frac{m+5}{2};\frac{3}{2};\sin ^2(e+f x)\right)\right)}{2 f}","\frac{\tan ^5(e+f x) \cos ^2(e+f x)^{\frac{m+5}{2}} (b \sec (e+f x))^m \, _2F_1\left(\frac{5}{2},\frac{m+5}{2};\frac{7}{2};\sin ^2(e+f x)\right)}{5 f}",1,"((Cos[e + f*x]^2)^((-1 + m)/2)*(Hypergeometric2F1[1/2, (1 + m)/2, 3/2, Sin[e + f*x]^2] - 2*Hypergeometric2F1[1/2, (3 + m)/2, 3/2, Sin[e + f*x]^2] + Hypergeometric2F1[1/2, (5 + m)/2, 3/2, Sin[e + f*x]^2])*(b*Sec[e + f*x])^m*Sin[2*(e + f*x)])/(2*f)","A",1
358,1,6612,63,25.1579478,"\int (b \sec (e+f x))^m \tan ^2(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^m*Tan[e + f*x]^2,x]","\text{Result too large to show}","\frac{\tan ^3(e+f x) \cos ^2(e+f x)^{\frac{m+3}{2}} (b \sec (e+f x))^m \, _2F_1\left(\frac{3}{2},\frac{m+3}{2};\frac{5}{2};\sin ^2(e+f x)\right)}{3 f}",1,"Result too large to show","C",0
359,1,4872,59,21.3867689,"\int \cot ^2(e+f x) (b \sec (e+f x))^m \, dx","Integrate[Cot[e + f*x]^2*(b*Sec[e + f*x])^m,x]","\text{Result too large to show}","-\frac{\cot (e+f x) \cos ^2(e+f x)^{\frac{m-1}{2}} (b \sec (e+f x))^m \, _2F_1\left(-\frac{1}{2},\frac{m-1}{2};\frac{1}{2};\sin ^2(e+f x)\right)}{f}",1,"(Cot[(e + f*x)/2]*Cot[e + f*x]^2*(b*Sec[e + f*x])^m*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*(-(AppellF1[-1/2, m, -m, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m) + 3*(Sec[(e + f*x)/2]^2)^m*Tan[(e + f*x)/2]^2*((-4*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2)/(3*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + AppellF1[1/2, m, -m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]/(3*AppellF1[1/2, m, -m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*m*(AppellF1[3/2, m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + m, -m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))))/(2*f*(-1/4*(Csc[(e + f*x)/2]^2*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*(-(AppellF1[-1/2, m, -m, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m) + 3*(Sec[(e + f*x)/2]^2)^m*Tan[(e + f*x)/2]^2*((-4*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2)/(3*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + AppellF1[1/2, m, -m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]/(3*AppellF1[1/2, m, -m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*m*(AppellF1[3/2, m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + m, -m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))) + (Cot[(e + f*x)/2]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*(-((Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m*(-(m*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) - m*AppellF1[1/2, 1 + m, -m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])) - m*AppellF1[-1/2, m, -m, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + m)*(-(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]) + 3*(Sec[(e + f*x)/2]^2)^(1 + m)*Tan[(e + f*x)/2]*((-4*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2)/(3*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + AppellF1[1/2, m, -m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]/(3*AppellF1[1/2, m, -m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*m*(AppellF1[3/2, m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + m, -m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + 3*m*(Sec[(e + f*x)/2]^2)^m*Tan[(e + f*x)/2]^3*((-4*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2)/(3*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + AppellF1[1/2, m, -m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]/(3*AppellF1[1/2, m, -m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*m*(AppellF1[3/2, m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + m, -m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + 3*(Sec[(e + f*x)/2]^2)^m*Tan[(e + f*x)/2]^2*((4*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]*Sin[(e + f*x)/2])/(3*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (4*Cos[(e + f*x)/2]^2*(-1/3*((1 - m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + ((m*AppellF1[3/2, m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (m*AppellF1[3/2, 1 + m, -m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3)/(3*AppellF1[1/2, m, -m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*m*(AppellF1[3/2, m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + m, -m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (AppellF1[1/2, m, -m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(2*m*(AppellF1[3/2, m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + m, -m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*((m*AppellF1[3/2, m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (m*AppellF1[3/2, 1 + m, -m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*m*Tan[(e + f*x)/2]^2*((-3*(1 - m)*AppellF1[5/2, m, 2 - m, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (6*m*AppellF1[5/2, 1 + m, 1 - m, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + m)*AppellF1[5/2, 2 + m, -m, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5)))/(3*AppellF1[1/2, m, -m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*m*(AppellF1[3/2, m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + m, -m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (4*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2*(2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*Tan[(e + f*x)/2]^2*((-1 + m)*((-3*(2 - m)*AppellF1[5/2, m, 3 - m, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*m*AppellF1[5/2, 1 + m, 2 - m, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + m*((-3*(1 - m)*AppellF1[5/2, 1 + m, 2 - m, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + m)*AppellF1[5/2, 2 + m, 1 - m, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2)))/2 + (m*Cot[(e + f*x)/2]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + m)*(-(AppellF1[-1/2, m, -m, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m) + 3*(Sec[(e + f*x)/2]^2)^m*Tan[(e + f*x)/2]^2*((-4*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2)/(3*AppellF1[1/2, m, 1 - m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + m)*AppellF1[3/2, m, 2 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + AppellF1[1/2, m, -m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]/(3*AppellF1[1/2, m, -m, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*m*(AppellF1[3/2, m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + m, -m, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/2))","C",0
360,1,6532,63,26.0065095,"\int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx","Integrate[Cot[e + f*x]^4*(b*Sec[e + f*x])^m,x]","\text{Result too large to show}","-\frac{\cot ^3(e+f x) \cos ^2(e+f x)^{\frac{m-3}{2}} (b \sec (e+f x))^m \, _2F_1\left(-\frac{3}{2},\frac{m-3}{2};-\frac{1}{2};\sin ^2(e+f x)\right)}{3 f}",1,"Result too large to show","C",0
361,0,0,63,0.596626,"\int \cot ^6(e+f x) (b \sec (e+f x))^m \, dx","Integrate[Cot[e + f*x]^6*(b*Sec[e + f*x])^m,x]","\int \cot ^6(e+f x) (b \sec (e+f x))^m \, dx","-\frac{\cot ^5(e+f x) \cos ^2(e+f x)^{\frac{m-5}{2}} (b \sec (e+f x))^m \, _2F_1\left(-\frac{5}{2},\frac{m-5}{2};-\frac{3}{2};\sin ^2(e+f x)\right)}{5 f}",1,"Integrate[Cot[e + f*x]^6*(b*Sec[e + f*x])^m, x]","F",-1
362,1,80,82,0.1338798,"\int (a \sec (e+f x))^m (b \tan (e+f x))^n \, dx","Integrate[(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^n,x]","\frac{b \left(-\tan ^2(e+f x)\right)^{\frac{1-n}{2}} (a \sec (e+f x))^m (b \tan (e+f x))^{n-1} \, _2F_1\left(\frac{m}{2},\frac{1-n}{2};\frac{m+2}{2};\sec ^2(e+f x)\right)}{f m}","\frac{(a \sec (e+f x))^m (b \tan (e+f x))^{n+1} \cos ^2(e+f x)^{\frac{1}{2} (m+n+1)} \, _2F_1\left(\frac{n+1}{2},\frac{1}{2} (m+n+1);\frac{n+3}{2};\sin ^2(e+f x)\right)}{b f (n+1)}",1,"(b*Hypergeometric2F1[m/2, (1 - n)/2, (2 + m)/2, Sec[e + f*x]^2]*(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(-1 + n)*(-Tan[e + f*x]^2)^((1 - n)/2))/(f*m)","A",1
363,1,101,74,2.1007629,"\int \sec ^6(a+b x) (d \tan (a+b x))^n \, dx","Integrate[Sec[a + b*x]^6*(d*Tan[a + b*x])^n,x]","\frac{d (d \tan (a+b x))^{n-1} \left(\tan ^2(a+b x) \sec ^4(a+b x) \left(2 (n+3) \cos (2 (a+b x))+\cos (4 (a+b x))+n^2+6 n+8\right)+8 \left(-\tan ^2(a+b x)\right)^{\frac{1-n}{2}}\right)}{b (n+1) (n+3) (n+5)}","\frac{(d \tan (a+b x))^{n+5}}{b d^5 (n+5)}+\frac{2 (d \tan (a+b x))^{n+3}}{b d^3 (n+3)}+\frac{(d \tan (a+b x))^{n+1}}{b d (n+1)}",1,"(d*(d*Tan[a + b*x])^(-1 + n)*((8 + 6*n + n^2 + 2*(3 + n)*Cos[2*(a + b*x)] + Cos[4*(a + b*x)])*Sec[a + b*x]^4*Tan[a + b*x]^2 + 8*(-Tan[a + b*x]^2)^((1 - n)/2)))/(b*(1 + n)*(3 + n)*(5 + n))","A",1
364,1,78,49,1.1466707,"\int \sec ^4(a+b x) (d \tan (a+b x))^n \, dx","Integrate[Sec[a + b*x]^4*(d*Tan[a + b*x])^n,x]","\frac{d (d \tan (a+b x))^{n-1} \left(2 \left(-\tan ^2(a+b x)\right)^{\frac{1-n}{2}}+\tan ^2(a+b x) \sec ^2(a+b x) (\cos (2 (a+b x))+n+2)\right)}{b (n+1) (n+3)}","\frac{(d \tan (a+b x))^{n+3}}{b d^3 (n+3)}+\frac{(d \tan (a+b x))^{n+1}}{b d (n+1)}",1,"(d*(d*Tan[a + b*x])^(-1 + n)*((2 + n + Cos[2*(a + b*x)])*Sec[a + b*x]^2*Tan[a + b*x]^2 + 2*(-Tan[a + b*x]^2)^((1 - n)/2)))/(b*(1 + n)*(3 + n))","A",1
365,1,25,24,0.0191373,"\int \sec ^2(a+b x) (d \tan (a+b x))^n \, dx","Integrate[Sec[a + b*x]^2*(d*Tan[a + b*x])^n,x]","\frac{\tan (a+b x) (d \tan (a+b x))^n}{b (n+1)}","\frac{(d \tan (a+b x))^{n+1}}{b d (n+1)}",1,"(Tan[a + b*x]*(d*Tan[a + b*x])^n)/(b*(1 + n))","A",1
366,1,53,50,0.0407861,"\int (d \tan (a+b x))^n \, dx","Integrate[(d*Tan[a + b*x])^n,x]","\frac{\tan (a+b x) (d \tan (a+b x))^n \, _2F_1\left(1,\frac{n+1}{2};\frac{n+1}{2}+1;-\tan ^2(a+b x)\right)}{b (n+1)}","\frac{(d \tan (a+b x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(a+b x)\right)}{b d (n+1)}",1,"(Hypergeometric2F1[1, (1 + n)/2, 1 + (1 + n)/2, -Tan[a + b*x]^2]*Tan[a + b*x]*(d*Tan[a + b*x])^n)/(b*(1 + n))","A",1
367,1,939,50,4.0957098,"\int \cos ^2(a+b x) (d \tan (a+b x))^n \, dx","Integrate[Cos[a + b*x]^2*(d*Tan[a + b*x])^n,x]","\frac{2 \left(F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-4 F_1\left(\frac{n+1}{2};n,2;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+4 F_1\left(\frac{n+1}{2};n,3;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right) \cos ^2(a+b x) \tan \left(\frac{1}{2} (a+b x)\right) (d \tan (a+b x))^n}{b \left(\frac{2 (n+1) \left(-F_1\left(\frac{n+3}{2};n,2;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+8 F_1\left(\frac{n+3}{2};n,3;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-12 F_1\left(\frac{n+3}{2};n,4;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+n F_1\left(\frac{n+3}{2};n+1,1;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-4 n F_1\left(\frac{n+3}{2};n+1,2;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+4 n F_1\left(\frac{n+3}{2};n+1,3;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right) \tan ^2\left(\frac{1}{2} (a+b x)\right) \sec ^2\left(\frac{1}{2} (a+b x)\right)}{n+3}+\left(F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-4 F_1\left(\frac{n+1}{2};n,2;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+4 F_1\left(\frac{n+1}{2};n,3;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right) \sec ^2\left(\frac{1}{2} (a+b x)\right)+n \left(F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-4 F_1\left(\frac{n+1}{2};n,2;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+4 F_1\left(\frac{n+1}{2};n,3;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right) \sec (a+b x) \sec ^2\left(\frac{1}{2} (a+b x)\right)-2 n \left(F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-4 F_1\left(\frac{n+1}{2};n,2;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+4 F_1\left(\frac{n+1}{2};n,3;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right) \sec (a+b x) \tan ^2\left(\frac{1}{2} (a+b x)\right)\right)}","\frac{(d \tan (a+b x))^{n+1} \, _2F_1\left(2,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(a+b x)\right)}{b d (n+1)}",1,"(2*(AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 4*AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 4*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Cos[a + b*x]^2*Tan[(a + b*x)/2]*(d*Tan[a + b*x])^n)/(b*((AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 4*AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 4*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Sec[(a + b*x)/2]^2 + n*(AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 4*AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 4*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Sec[(a + b*x)/2]^2*Sec[a + b*x] + (2*(1 + n)*(-AppellF1[(3 + n)/2, n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 8*AppellF1[(3 + n)/2, n, 3, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 12*AppellF1[(3 + n)/2, n, 4, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + n*AppellF1[(3 + n)/2, 1 + n, 1, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 4*n*AppellF1[(3 + n)/2, 1 + n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 4*n*AppellF1[(3 + n)/2, 1 + n, 3, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2]^2)/(3 + n) - 2*n*(AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 4*AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 4*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Sec[a + b*x]*Tan[(a + b*x)/2]^2))","C",0
368,1,1712,50,12.6292905,"\int \cos ^4(a+b x) (d \tan (a+b x))^n \, dx","Integrate[Cos[a + b*x]^4*(d*Tan[a + b*x])^n,x]","-\frac{8 (n+3) \left(F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-8 \left(F_1\left(\frac{n+1}{2};n,2;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-3 F_1\left(\frac{n+1}{2};n,3;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+4 F_1\left(\frac{n+1}{2};n,4;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-2 F_1\left(\frac{n+1}{2};n,5;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right)\right) \cos ^3\left(\frac{1}{2} (a+b x)\right) \cos ^5(a+b x) \sin ^2\left(\frac{1}{2} (a+b x)\right) (d \tan (a+b x))^n}{b (n+1) \left((n+3) F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) (\cos (a+b x)+1)+2 \left(-8 n F_1\left(\frac{n+1}{2};n,2;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos ^2\left(\frac{1}{2} (a+b x)\right)-24 F_1\left(\frac{n+1}{2};n,2;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos ^2\left(\frac{1}{2} (a+b x)\right)+24 n F_1\left(\frac{n+1}{2};n,3;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos ^2\left(\frac{1}{2} (a+b x)\right)+72 F_1\left(\frac{n+1}{2};n,3;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos ^2\left(\frac{1}{2} (a+b x)\right)-32 n F_1\left(\frac{n+1}{2};n,4;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos ^2\left(\frac{1}{2} (a+b x)\right)-96 F_1\left(\frac{n+1}{2};n,4;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos ^2\left(\frac{1}{2} (a+b x)\right)+16 F_1\left(\frac{n+3}{2};n,3;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-72 F_1\left(\frac{n+3}{2};n,4;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+128 F_1\left(\frac{n+3}{2};n,5;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-80 F_1\left(\frac{n+3}{2};n,6;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+n F_1\left(\frac{n+3}{2};n+1,1;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-8 n F_1\left(\frac{n+3}{2};n+1,2;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+24 n F_1\left(\frac{n+3}{2};n+1,3;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-32 n F_1\left(\frac{n+3}{2};n+1,4;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+16 n F_1\left(\frac{n+3}{2};n+1,5;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+F_1\left(\frac{n+3}{2};n,2;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) (\cos (a+b x)-1)-16 F_1\left(\frac{n+3}{2};n,3;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)+72 F_1\left(\frac{n+3}{2};n,4;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)-128 F_1\left(\frac{n+3}{2};n,5;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)+80 F_1\left(\frac{n+3}{2};n,6;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)-n F_1\left(\frac{n+3}{2};n+1,1;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)+8 n F_1\left(\frac{n+3}{2};n+1,2;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)-24 n F_1\left(\frac{n+3}{2};n+1,3;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)+32 n F_1\left(\frac{n+3}{2};n+1,4;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)-16 n F_1\left(\frac{n+3}{2};n+1,5;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)+8 (n+3) F_1\left(\frac{n+1}{2};n,5;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) (\cos (a+b x)+1)\right)\right) \left(\sin \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{3}{2} (a+b x)\right)\right)}","\frac{(d \tan (a+b x))^{n+1} \, _2F_1\left(3,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(a+b x)\right)}{b d (n+1)}",1,"(-8*(3 + n)*(AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 8*(AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 3*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 4*AppellF1[(1 + n)/2, n, 4, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 2*AppellF1[(1 + n)/2, n, 5, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]))*Cos[(a + b*x)/2]^3*Cos[a + b*x]^5*Sin[(a + b*x)/2]^2*(d*Tan[a + b*x])^n)/(b*(1 + n)*((3 + n)*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(1 + Cos[a + b*x]) + 2*(16*AppellF1[(3 + n)/2, n, 3, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 72*AppellF1[(3 + n)/2, n, 4, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 128*AppellF1[(3 + n)/2, n, 5, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 80*AppellF1[(3 + n)/2, n, 6, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + n*AppellF1[(3 + n)/2, 1 + n, 1, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 8*n*AppellF1[(3 + n)/2, 1 + n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 24*n*AppellF1[(3 + n)/2, 1 + n, 3, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 32*n*AppellF1[(3 + n)/2, 1 + n, 4, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 16*n*AppellF1[(3 + n)/2, 1 + n, 5, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 24*AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 - 8*n*AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 + 72*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 + 24*n*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 - 96*AppellF1[(1 + n)/2, n, 4, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 - 32*n*AppellF1[(1 + n)/2, n, 4, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 + AppellF1[(3 + n)/2, n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(-1 + Cos[a + b*x]) - 16*AppellF1[(3 + n)/2, n, 3, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] + 72*AppellF1[(3 + n)/2, n, 4, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] - 128*AppellF1[(3 + n)/2, n, 5, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] + 80*AppellF1[(3 + n)/2, n, 6, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] - n*AppellF1[(3 + n)/2, 1 + n, 1, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] + 8*n*AppellF1[(3 + n)/2, 1 + n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] - 24*n*AppellF1[(3 + n)/2, 1 + n, 3, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] + 32*n*AppellF1[(3 + n)/2, 1 + n, 4, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] - 16*n*AppellF1[(3 + n)/2, 1 + n, 5, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] + 8*(3 + n)*AppellF1[(1 + n)/2, n, 5, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(1 + Cos[a + b*x])))*(Sin[(a + b*x)/2] - Sin[(3*(a + b*x))/2]))","C",0
369,1,72,78,0.1302695,"\int \sec ^5(a+b x) (d \tan (a+b x))^n \, dx","Integrate[Sec[a + b*x]^5*(d*Tan[a + b*x])^n,x]","\frac{d \sec ^5(a+b x) \left(-\tan ^2(a+b x)\right)^{\frac{1-n}{2}} (d \tan (a+b x))^{n-1} \, _2F_1\left(\frac{5}{2},\frac{1-n}{2};\frac{7}{2};\sec ^2(a+b x)\right)}{5 b}","\frac{\sec ^5(a+b x) \cos ^2(a+b x)^{\frac{n+6}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+6}{2};\frac{n+3}{2};\sin ^2(a+b x)\right)}{b d (n+1)}",1,"(d*Hypergeometric2F1[5/2, (1 - n)/2, 7/2, Sec[a + b*x]^2]*Sec[a + b*x]^5*(d*Tan[a + b*x])^(-1 + n)*(-Tan[a + b*x]^2)^((1 - n)/2))/(5*b)","A",1
370,1,72,78,0.1005435,"\int \sec ^3(a+b x) (d \tan (a+b x))^n \, dx","Integrate[Sec[a + b*x]^3*(d*Tan[a + b*x])^n,x]","\frac{d \sec ^3(a+b x) \left(-\tan ^2(a+b x)\right)^{\frac{1-n}{2}} (d \tan (a+b x))^{n-1} \, _2F_1\left(\frac{3}{2},\frac{1-n}{2};\frac{5}{2};\sec ^2(a+b x)\right)}{3 b}","\frac{\sec ^3(a+b x) \cos ^2(a+b x)^{\frac{n+4}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+4}{2};\frac{n+3}{2};\sin ^2(a+b x)\right)}{b d (n+1)}",1,"(d*Hypergeometric2F1[3/2, (1 - n)/2, 5/2, Sec[a + b*x]^2]*Sec[a + b*x]^3*(d*Tan[a + b*x])^(-1 + n)*(-Tan[a + b*x]^2)^((1 - n)/2))/(3*b)","A",1
371,1,64,76,0.0714393,"\int \sec (a+b x) (d \tan (a+b x))^n \, dx","Integrate[Sec[a + b*x]*(d*Tan[a + b*x])^n,x]","\frac{\csc (a+b x) \left(-\tan ^2(a+b x)\right)^{\frac{1-n}{2}} (d \tan (a+b x))^n \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};\sec ^2(a+b x)\right)}{b}","\frac{\sec (a+b x) \cos ^2(a+b x)^{\frac{n+2}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\sin ^2(a+b x)\right)}{b d (n+1)}",1,"(Csc[a + b*x]*Hypergeometric2F1[1/2, (1 - n)/2, 3/2, Sec[a + b*x]^2]*(d*Tan[a + b*x])^n*(-Tan[a + b*x]^2)^((1 - n)/2))/b","A",1
372,1,452,72,2.3884707,"\int \cos (a+b x) (d \tan (a+b x))^n \, dx","Integrate[Cos[a + b*x]*(d*Tan[a + b*x])^n,x]","-\frac{2 \sin \left(\frac{1}{2} (a+b x)\right) \cos \left(\frac{1}{2} (a+b x)\right) \cos (a+b x) \left(F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-2 F_1\left(\frac{n+1}{2};n,2;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right) (d \tan (a+b x))^n}{b (n+1) \left(\frac{\sec ^2\left(\frac{1}{2} (a+b x)\right) \left((n+3) (\cos (a+b x)+1) F_1\left(\frac{n+1}{2};n,2;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-(\cos (a+b x)-1) \left(F_1\left(\frac{n+3}{2};n,2;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-4 F_1\left(\frac{n+3}{2};n,3;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-n F_1\left(\frac{n+3}{2};n+1,1;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+2 n F_1\left(\frac{n+3}{2};n+1,2;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right)\right)}{n+3}-F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right)}","\frac{\cos (a+b x) \cos ^2(a+b x)^{n/2} (d \tan (a+b x))^{n+1} \, _2F_1\left(\frac{n}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(a+b x)\right)}{b d (n+1)}",1,"(-2*(AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 2*AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Cos[(a + b*x)/2]*Cos[a + b*x]*Sin[(a + b*x)/2]*(d*Tan[a + b*x])^n)/(b*(1 + n)*(-AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + ((-((AppellF1[(3 + n)/2, n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 4*AppellF1[(3 + n)/2, n, 3, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - n*AppellF1[(3 + n)/2, 1 + n, 1, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 2*n*AppellF1[(3 + n)/2, 1 + n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*(-1 + Cos[a + b*x])) + (3 + n)*AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(1 + Cos[a + b*x]))*Sec[(a + b*x)/2]^2)/(3 + n)))","C",0
373,1,1313,78,6.2316677,"\int \cos ^3(a+b x) (d \tan (a+b x))^n \, dx","Integrate[Cos[a + b*x]^3*(d*Tan[a + b*x])^n,x]","\frac{4 (n+3) \left(F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-6 F_1\left(\frac{n+1}{2};n,2;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+12 F_1\left(\frac{n+1}{2};n,3;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-8 F_1\left(\frac{n+1}{2};n,4;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right) \cos ^3\left(\frac{1}{2} (a+b x)\right) \cos ^3(a+b x) \sin \left(\frac{1}{2} (a+b x)\right) (d \tan (a+b x))^n}{b (n+1) \left((n+3) F_1\left(\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) (\cos (a+b x)+1)-2 \left(6 n F_1\left(\frac{n+1}{2};n,2;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos ^2\left(\frac{1}{2} (a+b x)\right)+18 F_1\left(\frac{n+1}{2};n,2;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos ^2\left(\frac{1}{2} (a+b x)\right)+8 (n+3) F_1\left(\frac{n+1}{2};n,4;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos ^2\left(\frac{1}{2} (a+b x)\right)+F_1\left(\frac{n+3}{2};n,2;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-12 F_1\left(\frac{n+3}{2};n,3;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+36 F_1\left(\frac{n+3}{2};n,4;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-32 F_1\left(\frac{n+3}{2};n,5;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-n F_1\left(\frac{n+3}{2};n+1,1;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+6 n F_1\left(\frac{n+3}{2};n+1,2;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-12 n F_1\left(\frac{n+3}{2};n+1,3;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+8 n F_1\left(\frac{n+3}{2};n+1,4;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-F_1\left(\frac{n+3}{2};n,2;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)+12 F_1\left(\frac{n+3}{2};n,3;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)-36 F_1\left(\frac{n+3}{2};n,4;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)+32 F_1\left(\frac{n+3}{2};n,5;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)+n F_1\left(\frac{n+3}{2};n+1,1;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)-6 n F_1\left(\frac{n+3}{2};n+1,2;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)+12 n F_1\left(\frac{n+3}{2};n+1,3;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)-8 n F_1\left(\frac{n+3}{2};n+1,4;\frac{n+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos (a+b x)-6 (n+3) F_1\left(\frac{n+1}{2};n,3;\frac{n+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) (\cos (a+b x)+1)\right)\right)}","\frac{\cos ^3(a+b x) \cos ^2(a+b x)^{\frac{n-2}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left(\frac{n-2}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(a+b x)\right)}{b d (n+1)}",1,"(4*(3 + n)*(AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 6*AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 12*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 8*AppellF1[(1 + n)/2, n, 4, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Cos[(a + b*x)/2]^3*Cos[a + b*x]^3*Sin[(a + b*x)/2]*(d*Tan[a + b*x])^n)/(b*(1 + n)*((3 + n)*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(1 + Cos[a + b*x]) - 2*(AppellF1[(3 + n)/2, n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 12*AppellF1[(3 + n)/2, n, 3, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 36*AppellF1[(3 + n)/2, n, 4, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 32*AppellF1[(3 + n)/2, n, 5, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - n*AppellF1[(3 + n)/2, 1 + n, 1, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 6*n*AppellF1[(3 + n)/2, 1 + n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 12*n*AppellF1[(3 + n)/2, 1 + n, 3, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 8*n*AppellF1[(3 + n)/2, 1 + n, 4, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 18*AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 + 6*n*AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 + 8*(3 + n)*AppellF1[(1 + n)/2, n, 4, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 - AppellF1[(3 + n)/2, n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] + 12*AppellF1[(3 + n)/2, n, 3, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] - 36*AppellF1[(3 + n)/2, n, 4, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] + 32*AppellF1[(3 + n)/2, n, 5, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] + n*AppellF1[(3 + n)/2, 1 + n, 1, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] - 6*n*AppellF1[(3 + n)/2, 1 + n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] + 12*n*AppellF1[(3 + n)/2, 1 + n, 3, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] - 8*n*AppellF1[(3 + n)/2, 1 + n, 4, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] - 6*(3 + n)*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(1 + Cos[a + b*x]))))","C",0
374,1,52,40,0.0983912,"\int (b \csc (e+f x))^m \tan ^3(e+f x) \, dx","Integrate[(b*Csc[e + f*x])^m*Tan[e + f*x]^3,x]","-\frac{\sin ^4(e+f x) (b \csc (e+f x))^m \, _2F_1\left(2,2-\frac{m}{2};3-\frac{m}{2};\sin ^2(e+f x)\right)}{f (m-4)}","-\frac{(b \csc (e+f x))^m \, _2F_1\left(2,\frac{m}{2};\frac{m+2}{2};\csc ^2(e+f x)\right)}{f m}",1,"-(((b*Csc[e + f*x])^m*Hypergeometric2F1[2, 2 - m/2, 3 - m/2, Sin[e + f*x]^2]*Sin[e + f*x]^4)/(f*(-4 + m)))","A",1
375,1,52,39,0.0489803,"\int (b \csc (e+f x))^m \tan (e+f x) \, dx","Integrate[(b*Csc[e + f*x])^m*Tan[e + f*x],x]","-\frac{\sin ^2(e+f x) (b \csc (e+f x))^m \, _2F_1\left(1,1-\frac{m}{2};2-\frac{m}{2};\sin ^2(e+f x)\right)}{f (m-2)}","\frac{(b \csc (e+f x))^m \, _2F_1\left(1,\frac{m}{2};\frac{m+2}{2};\csc ^2(e+f x)\right)}{f m}",1,"-(((b*Csc[e + f*x])^m*Hypergeometric2F1[1, 1 - m/2, 2 - m/2, Sin[e + f*x]^2]*Sin[e + f*x]^2)/(f*(-2 + m)))","A",1
376,1,18,18,0.0173909,"\int \cot (e+f x) (b \csc (e+f x))^m \, dx","Integrate[Cot[e + f*x]*(b*Csc[e + f*x])^m,x]","-\frac{(b \csc (e+f x))^m}{f m}","-\frac{(b \csc (e+f x))^m}{f m}",1,"-((b*Csc[e + f*x])^m/(f*m))","A",1
377,1,36,43,0.0843414,"\int \cot ^3(e+f x) (b \csc (e+f x))^m \, dx","Integrate[Cot[e + f*x]^3*(b*Csc[e + f*x])^m,x]","\frac{\left(-m \csc ^2(e+f x)+m+2\right) (b \csc (e+f x))^m}{f m (m+2)}","\frac{(b \csc (e+f x))^m}{f m}-\frac{(b \csc (e+f x))^{m+2}}{b^2 f (m+2)}",1,"((b*Csc[e + f*x])^m*(2 + m - m*Csc[e + f*x]^2))/(f*m*(2 + m))","A",1
378,1,63,69,0.302546,"\int \cot ^5(e+f x) (b \csc (e+f x))^m \, dx","Integrate[Cot[e + f*x]^5*(b*Csc[e + f*x])^m,x]","-\frac{\left(m (m+2) \csc ^4(e+f x)-2 m (m+4) \csc ^2(e+f x)+m^2+6 m+8\right) (b \csc (e+f x))^m}{f m (m+2) (m+4)}","-\frac{(b \csc (e+f x))^{m+4}}{b^4 f (m+4)}+\frac{2 (b \csc (e+f x))^{m+2}}{b^2 f (m+2)}-\frac{(b \csc (e+f x))^m}{f m}",1,"-(((b*Csc[e + f*x])^m*(8 + 6*m + m^2 - 2*m*(4 + m)*Csc[e + f*x]^2 + m*(2 + m)*Csc[e + f*x]^4))/(f*m*(2 + m)*(4 + m)))","A",1
379,1,212,63,1.3816221,"\int (b \csc (e+f x))^m \tan ^4(e+f x) \, dx","Integrate[(b*Csc[e + f*x])^m*Tan[e + f*x]^4,x]","-\frac{\tan (e+f x) \sec ^2(e+f x)^{-m/2} (b \csc (e+f x))^m \left(\sqrt{\sin ^2(e+f x)} \, _2F_1\left(\frac{1-m}{2},-\frac{m}{2}-1;\frac{3-m}{2};-\tan ^2(e+f x)\right)-2 \sqrt{\sin ^2(e+f x)} \, _2F_1\left(\frac{1-m}{2},-\frac{m}{2};\frac{3-m}{2};-\tan ^2(e+f x)\right)+(m-1) \cos ^2(e+f x) \sin ^2(e+f x)^{m/2} \sec ^2(e+f x)^{m/2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{3}{2};\cos ^2(e+f x)\right)\right)}{f (m-1) \sqrt{\sin ^2(e+f x)}}","\frac{\tan ^3(e+f x) \sin ^2(e+f x)^{\frac{m-3}{2}} (b \csc (e+f x))^m \, _2F_1\left(-\frac{3}{2},\frac{m-3}{2};-\frac{1}{2};\cos ^2(e+f x)\right)}{3 f}",1,"-(((b*Csc[e + f*x])^m*(Hypergeometric2F1[(1 - m)/2, -1 - m/2, (3 - m)/2, -Tan[e + f*x]^2]*Sqrt[Sin[e + f*x]^2] - 2*Hypergeometric2F1[(1 - m)/2, -1/2*m, (3 - m)/2, -Tan[e + f*x]^2]*Sqrt[Sin[e + f*x]^2] + (-1 + m)*Cos[e + f*x]^2*Hypergeometric2F1[1/2, (1 + m)/2, 3/2, Cos[e + f*x]^2]*(Sec[e + f*x]^2)^(m/2)*(Sin[e + f*x]^2)^(m/2))*Tan[e + f*x])/(f*(-1 + m)*(Sec[e + f*x]^2)^(m/2)*Sqrt[Sin[e + f*x]^2]))","B",1
380,1,79,58,0.6319258,"\int (b \csc (e+f x))^m \tan ^2(e+f x) \, dx","Integrate[(b*Csc[e + f*x])^m*Tan[e + f*x]^2,x]","\frac{\tan ^3(e+f x) \sec ^2(e+f x)^{-m/2} (b \csc (e+f x))^m \, _2F_1\left(1-\frac{m}{2},\frac{3}{2}-\frac{m}{2};\frac{5}{2}-\frac{m}{2};-\tan ^2(e+f x)\right)}{f (3-m)}","\frac{\tan (e+f x) \sin ^2(e+f x)^{\frac{m-1}{2}} (b \csc (e+f x))^m \, _2F_1\left(-\frac{1}{2},\frac{m-1}{2};\frac{1}{2};\cos ^2(e+f x)\right)}{f}",1,"((b*Csc[e + f*x])^m*Hypergeometric2F1[1 - m/2, 3/2 - m/2, 5/2 - m/2, -Tan[e + f*x]^2]*Tan[e + f*x]^3)/(f*(3 - m)*(Sec[e + f*x]^2)^(m/2))","A",1
381,1,186,63,1.1886533,"\int \cot ^2(e+f x) (b \csc (e+f x))^m \, dx","Integrate[Cot[e + f*x]^2*(b*Csc[e + f*x])^m,x]","-\frac{\tan \left(\frac{1}{2} (e+f x)\right) \sec ^2\left(\frac{1}{2} (e+f x)\right)^{-m} (b \csc (e+f x))^m \left(-4 (m+1) \, _2F_1\left(1-m,\frac{1}{2}-\frac{m}{2};\frac{3}{2}-\frac{m}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+(m+1) \, _2F_1\left(\frac{1}{2}-\frac{m}{2},-m;\frac{3}{2}-\frac{m}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+(m-1) \cot ^2\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(-\frac{m}{2}-\frac{1}{2},-m;\frac{1}{2}-\frac{m}{2};-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}{2 f \left(m^2-1\right)}","-\frac{\cot ^3(e+f x) \sin ^2(e+f x)^{\frac{m+3}{2}} (b \csc (e+f x))^m \, _2F_1\left(\frac{3}{2},\frac{m+3}{2};\frac{5}{2};\cos ^2(e+f x)\right)}{3 f}",1,"-1/2*((b*Csc[e + f*x])^m*(-4*(1 + m)*Hypergeometric2F1[1 - m, 1/2 - m/2, 3/2 - m/2, -Tan[(e + f*x)/2]^2] + (-1 + m)*Cot[(e + f*x)/2]^2*Hypergeometric2F1[-1/2 - m/2, -m, 1/2 - m/2, -Tan[(e + f*x)/2]^2] + (1 + m)*Hypergeometric2F1[1/2 - m/2, -m, 3/2 - m/2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2])/(f*(-1 + m^2)*(Sec[(e + f*x)/2]^2)^m)","B",1
382,1,106,63,0.2622257,"\int \cot ^4(e+f x) (b \csc (e+f x))^m \, dx","Integrate[Cot[e + f*x]^4*(b*Csc[e + f*x])^m,x]","-\frac{\cot (e+f x) \sin ^2(e+f x)^{\frac{m+1}{2}} (b \csc (e+f x))^m \left(\, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{3}{2};\cos ^2(e+f x)\right)-2 \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{3}{2};\cos ^2(e+f x)\right)+\, _2F_1\left(\frac{1}{2},\frac{m+5}{2};\frac{3}{2};\cos ^2(e+f x)\right)\right)}{f}","-\frac{\cot ^5(e+f x) \sin ^2(e+f x)^{\frac{m+5}{2}} (b \csc (e+f x))^m \, _2F_1\left(\frac{5}{2},\frac{m+5}{2};\frac{7}{2};\cos ^2(e+f x)\right)}{5 f}",1,"-((Cot[e + f*x]*(b*Csc[e + f*x])^m*(Hypergeometric2F1[1/2, (1 + m)/2, 3/2, Cos[e + f*x]^2] - 2*Hypergeometric2F1[1/2, (3 + m)/2, 3/2, Cos[e + f*x]^2] + Hypergeometric2F1[1/2, (5 + m)/2, 3/2, Cos[e + f*x]^2])*(Sin[e + f*x]^2)^((1 + m)/2))/f)","A",1
383,1,87,79,5.5844445,"\int (b \csc (e+f x))^m (d \tan (e+f x))^{3/2} \, dx","Integrate[(b*Csc[e + f*x])^m*(d*Tan[e + f*x])^(3/2),x]","-\frac{2 (d \tan (e+f x))^{5/2} \sec ^2(e+f x)^{-m/2} (b \csc (e+f x))^m \, _2F_1\left(\frac{1}{4} (5-2 m),1-\frac{m}{2};\frac{1}{4} (9-2 m);-\tan ^2(e+f x)\right)}{d f (2 m-5)}","\frac{2 \cos ^2(e+f x)^{5/4} (d \tan (e+f x))^{5/2} (b \csc (e+f x))^m \, _2F_1\left(\frac{5}{4},\frac{1}{4} (5-2 m);\frac{1}{4} (9-2 m);\sin ^2(e+f x)\right)}{d f (5-2 m)}",1,"(-2*(b*Csc[e + f*x])^m*Hypergeometric2F1[(5 - 2*m)/4, 1 - m/2, (9 - 2*m)/4, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(5/2))/(d*f*(-5 + 2*m)*(Sec[e + f*x]^2)^(m/2))","A",1
384,1,87,79,3.2015352,"\int (b \csc (e+f x))^m \sqrt{d \tan (e+f x)} \, dx","Integrate[(b*Csc[e + f*x])^m*Sqrt[d*Tan[e + f*x]],x]","-\frac{2 (d \tan (e+f x))^{3/2} \sec ^2(e+f x)^{-m/2} (b \csc (e+f x))^m \, _2F_1\left(\frac{1}{4} (3-2 m),1-\frac{m}{2};\frac{1}{4} (7-2 m);-\tan ^2(e+f x)\right)}{d f (2 m-3)}","\frac{2 \cos ^2(e+f x)^{3/4} (d \tan (e+f x))^{3/2} (b \csc (e+f x))^m \, _2F_1\left(\frac{3}{4},\frac{1}{4} (3-2 m);\frac{1}{4} (7-2 m);\sin ^2(e+f x)\right)}{d f (3-2 m)}",1,"(-2*(b*Csc[e + f*x])^m*Hypergeometric2F1[(3 - 2*m)/4, 1 - m/2, (7 - 2*m)/4, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(3/2))/(d*f*(-3 + 2*m)*(Sec[e + f*x]^2)^(m/2))","A",1
385,1,87,79,1.2641334,"\int \frac{(b \csc (e+f x))^m}{\sqrt{d \tan (e+f x)}} \, dx","Integrate[(b*Csc[e + f*x])^m/Sqrt[d*Tan[e + f*x]],x]","-\frac{2 \sqrt{d \tan (e+f x)} \sec ^2(e+f x)^{-m/2} (b \csc (e+f x))^m \, _2F_1\left(\frac{1}{4} (1-2 m),1-\frac{m}{2};\frac{1}{4} (5-2 m);-\tan ^2(e+f x)\right)}{d f (2 m-1)}","\frac{2 \sqrt[4]{\cos ^2(e+f x)} \sqrt{d \tan (e+f x)} (b \csc (e+f x))^m \, _2F_1\left(\frac{1}{4},\frac{1}{4} (1-2 m);\frac{1}{4} (5-2 m);\sin ^2(e+f x)\right)}{d f (1-2 m)}",1,"(-2*(b*Csc[e + f*x])^m*Hypergeometric2F1[(1 - 2*m)/4, 1 - m/2, (5 - 2*m)/4, -Tan[e + f*x]^2]*Sqrt[d*Tan[e + f*x]])/(d*f*(-1 + 2*m)*(Sec[e + f*x]^2)^(m/2))","A",1
386,1,87,79,3.5837467,"\int \frac{(b \csc (e+f x))^m}{(d \tan (e+f x))^{3/2}} \, dx","Integrate[(b*Csc[e + f*x])^m/(d*Tan[e + f*x])^(3/2),x]","-\frac{2 \sec ^2(e+f x)^{-m/2} (b \csc (e+f x))^m \, _2F_1\left(\frac{1}{4} (-2 m-1),1-\frac{m}{2};\frac{1}{4} (3-2 m);-\tan ^2(e+f x)\right)}{d f (2 m+1) \sqrt{d \tan (e+f x)}}","-\frac{2 (b \csc (e+f x))^m \, _2F_1\left(-\frac{1}{4},\frac{1}{4} (-2 m-1);\frac{1}{4} (3-2 m);\sin ^2(e+f x)\right)}{d f (2 m+1) \sqrt[4]{\cos ^2(e+f x)} \sqrt{d \tan (e+f x)}}",1,"(-2*(b*Csc[e + f*x])^m*Hypergeometric2F1[(-1 - 2*m)/4, 1 - m/2, (3 - 2*m)/4, -Tan[e + f*x]^2])/(d*f*(1 + 2*m)*(Sec[e + f*x]^2)^(m/2)*Sqrt[d*Tan[e + f*x]])","A",1
387,1,287,89,2.0419196,"\int (a \csc (e+f x))^m (b \tan (e+f x))^n \, dx","Integrate[(a*Csc[e + f*x])^m*(b*Tan[e + f*x])^n,x]","-\frac{a (m-n-3) (a \csc (e+f x))^{m-1} (b \tan (e+f x))^n F_1\left(\frac{1}{2} (-m+n+1);n,1-m;\frac{1}{2} (-m+n+3);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{f (m-n-1) \left((m-n-3) F_1\left(\frac{1}{2} (-m+n+1);n,1-m;\frac{1}{2} (-m+n+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+3);n,2-m;\frac{1}{2} (-m+n+5);\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+n F_1\left(\frac{1}{2} (-m+n+3);n+1,1-m;\frac{1}{2} (-m+n+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{\cos ^2(e+f x)^{\frac{n+1}{2}} (a \csc (e+f x))^m (b \tan (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);\sin ^2(e+f x)\right)}{b f (-m+n+1)}",1,"-((a*(-3 + m - n)*AppellF1[(1 - m + n)/2, n, 1 - m, (3 - m + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(a*Csc[e + f*x])^(-1 + m)*(b*Tan[e + f*x])^n)/(f*(-1 + m - n)*((-3 + m - n)*AppellF1[(1 - m + n)/2, n, 1 - m, (3 - m + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*((-1 + m)*AppellF1[(3 - m + n)/2, n, 2 - m, (5 - m + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[(3 - m + n)/2, 1 + n, 1 - m, (5 - m + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","C",0