1,1,19,11,0.0149407,"\int \cot (a+b x) \, dx","Integrate[Cot[a + b*x],x]","\frac{\log (\tan (a+b x))+\log (\cos (a+b x))}{b}","\frac{\log (\sin (a+b x))}{b}",1,"(Log[Cos[a + b*x]] + Log[Tan[a + b*x]])/b","A",1
2,1,29,15,0.0159592,"\int \cot ^2(a+b x) \, dx","Integrate[Cot[a + b*x]^2,x]","-\frac{\cot (a+b x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(a+b x)\right)}{b}","-\frac{\cot (a+b x)}{b}-x",1,"-((Cot[a + b*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[a + b*x]^2])/b)","C",1
3,1,34,28,0.0922222,"\int \cot ^3(a+b x) \, dx","Integrate[Cot[a + b*x]^3,x]","-\frac{\cot ^2(a+b x)+2 \log (\tan (a+b x))+2 \log (\cos (a+b x))}{2 b}","-\frac{\cot ^2(a+b x)}{2 b}-\frac{\log (\sin (a+b x))}{b}",1,"-1/2*(Cot[a + b*x]^2 + 2*Log[Cos[a + b*x]] + 2*Log[Tan[a + b*x]])/b","A",1
4,1,33,27,0.0128665,"\int \cot ^4(a+b x) \, dx","Integrate[Cot[a + b*x]^4,x]","-\frac{\cot ^3(a+b x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(a+b x)\right)}{3 b}","-\frac{\cot ^3(a+b x)}{3 b}+\frac{\cot (a+b x)}{b}+x",1,"-1/3*(Cot[a + b*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[a + b*x]^2])/b","C",1
5,1,46,42,0.1089034,"\int \cot ^5(a+b x) \, dx","Integrate[Cot[a + b*x]^5,x]","\frac{-\cot ^4(a+b x)+2 \cot ^2(a+b x)+4 \log (\tan (a+b x))+4 \log (\cos (a+b x))}{4 b}","-\frac{\cot ^4(a+b x)}{4 b}+\frac{\cot ^2(a+b x)}{2 b}+\frac{\log (\sin (a+b x))}{b}",1,"(2*Cot[a + b*x]^2 - Cot[a + b*x]^4 + 4*Log[Cos[a + b*x]] + 4*Log[Tan[a + b*x]])/(4*b)","A",1
6,1,33,45,0.0299174,"\int \cot ^6(a+b x) \, dx","Integrate[Cot[a + b*x]^6,x]","-\frac{\cot ^5(a+b x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(a+b x)\right)}{5 b}","-\frac{\cot ^5(a+b x)}{5 b}+\frac{\cot ^3(a+b x)}{3 b}-\frac{\cot (a+b x)}{b}-x",1,"-1/5*(Cot[a + b*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[a + b*x]^2])/b","C",1
7,1,56,58,0.3050547,"\int \cot ^7(a+b x) \, dx","Integrate[Cot[a + b*x]^7,x]","-\frac{2 \cot ^6(a+b x)-3 \cot ^4(a+b x)+6 \cot ^2(a+b x)+12 \log (\tan (a+b x))+12 \log (\cos (a+b x))}{12 b}","-\frac{\cot ^6(a+b x)}{6 b}+\frac{\cot ^4(a+b x)}{4 b}-\frac{\cot ^2(a+b x)}{2 b}-\frac{\log (\sin (a+b x))}{b}",1,"-1/12*(6*Cot[a + b*x]^2 - 3*Cot[a + b*x]^4 + 2*Cot[a + b*x]^6 + 12*Log[Cos[a + b*x]] + 12*Log[Tan[a + b*x]])/b","A",1
8,1,33,57,0.0097176,"\int \cot ^8(a+b x) \, dx","Integrate[Cot[a + b*x]^8,x]","-\frac{\cot ^7(a+b x) \, _2F_1\left(-\frac{7}{2},1;-\frac{5}{2};-\tan ^2(a+b x)\right)}{7 b}","-\frac{\cot ^7(a+b x)}{7 b}+\frac{\cot ^5(a+b x)}{5 b}-\frac{\cot ^3(a+b x)}{3 b}+\frac{\cot (a+b x)}{b}+x",1,"-1/7*(Cot[a + b*x]^7*Hypergeometric2F1[-7/2, 1, -5/2, -Tan[a + b*x]^2])/b","C",1
9,1,175,232,0.520684,"\int (c \cot (a+b x))^{7/2} \, dx","Integrate[(c*Cot[a + b*x])^(7/2),x]","\frac{c^3 \sqrt{c \cot (a+b x)} \left(-8 \cot ^{\frac{5}{2}}(a+b x)+40 \sqrt{\cot (a+b x)}+5 \sqrt{2} \log \left(\cot (a+b x)-\sqrt{2} \sqrt{\cot (a+b x)}+1\right)-5 \sqrt{2} \log \left(\cot (a+b x)+\sqrt{2} \sqrt{\cot (a+b x)}+1\right)+10 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (a+b x)}\right)-10 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (a+b x)}+1\right)\right)}{20 b \sqrt{\cot (a+b x)}}","\frac{c^{7/2} \log \left(\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b}-\frac{c^{7/2} \log \left(\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b}+\frac{c^{7/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right)}{\sqrt{2} b}-\frac{c^{7/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right)}{\sqrt{2} b}+\frac{2 c^3 \sqrt{c \cot (a+b x)}}{b}-\frac{2 c (c \cot (a+b x))^{5/2}}{5 b}",1,"(c^3*Sqrt[c*Cot[a + b*x]]*(10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[a + b*x]]] - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[a + b*x]]] + 40*Sqrt[Cot[a + b*x]] - 8*Cot[a + b*x]^(5/2) + 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[a + b*x]] + Cot[a + b*x]] - 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[a + b*x]] + Cot[a + b*x]]))/(20*b*Sqrt[Cot[a + b*x]])","A",1
10,1,40,212,0.0770535,"\int (c \cot (a+b x))^{5/2} \, dx","Integrate[(c*Cot[a + b*x])^(5/2),x]","\frac{2 c (c \cot (a+b x))^{3/2} \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(a+b x)\right)-1\right)}{3 b}","\frac{c^{5/2} \log \left(\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b}-\frac{c^{5/2} \log \left(\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b}-\frac{c^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right)}{\sqrt{2} b}+\frac{c^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right)}{\sqrt{2} b}-\frac{2 c (c \cot (a+b x))^{3/2}}{3 b}",1,"(2*c*(c*Cot[a + b*x])^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[a + b*x]^2]))/(3*b)","C",1
11,1,159,210,0.2214151,"\int (c \cot (a+b x))^{3/2} \, dx","Integrate[(c*Cot[a + b*x])^(3/2),x]","-\frac{(c \cot (a+b x))^{3/2} \left(8 \sqrt{\cot (a+b x)}+\sqrt{2} \log \left(\cot (a+b x)-\sqrt{2} \sqrt{\cot (a+b x)}+1\right)-\sqrt{2} \log \left(\cot (a+b x)+\sqrt{2} \sqrt{\cot (a+b x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (a+b x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (a+b x)}+1\right)\right)}{4 b \cot ^{\frac{3}{2}}(a+b x)}","-\frac{c^{3/2} \log \left(\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b}+\frac{c^{3/2} \log \left(\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b}-\frac{c^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right)}{\sqrt{2} b}+\frac{c^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right)}{\sqrt{2} b}-\frac{2 c \sqrt{c \cot (a+b x)}}{b}",1,"-1/4*((c*Cot[a + b*x])^(3/2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[a + b*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[a + b*x]]] + 8*Sqrt[Cot[a + b*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[a + b*x]] + Cot[a + b*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[a + b*x]] + Cot[a + b*x]]))/(b*Cot[a + b*x]^(3/2))","A",1
12,1,40,192,0.039676,"\int \sqrt{c \cot (a+b x)} \, dx","Integrate[Sqrt[c*Cot[a + b*x]],x]","-\frac{2 (c \cot (a+b x))^{3/2} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(a+b x)\right)}{3 b c}","-\frac{\sqrt{c} \log \left(\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b}+\frac{\sqrt{c} \log \left(\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b}+\frac{\sqrt{c} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right)}{\sqrt{2} b}-\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right)}{\sqrt{2} b}",1,"(-2*(c*Cot[a + b*x])^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[a + b*x]^2])/(3*b*c)","C",1
13,1,131,192,0.0960436,"\int \frac{1}{\sqrt{c \cot (a+b x)}} \, dx","Integrate[1/Sqrt[c*Cot[a + b*x]],x]","\frac{\sqrt{\cot (a+b x)} \left(\log \left(\cot (a+b x)-\sqrt{2} \sqrt{\cot (a+b x)}+1\right)-\log \left(\cot (a+b x)+\sqrt{2} \sqrt{\cot (a+b x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (a+b x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (a+b x)}+1\right)\right)}{2 \sqrt{2} b \sqrt{c \cot (a+b x)}}","\frac{\log \left(\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b \sqrt{c}}-\frac{\log \left(\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b \sqrt{c}}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right)}{\sqrt{2} b \sqrt{c}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right)}{\sqrt{2} b \sqrt{c}}",1,"(Sqrt[Cot[a + b*x]]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[a + b*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[a + b*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[a + b*x]] + Cot[a + b*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[a + b*x]] + Cot[a + b*x]]))/(2*Sqrt[2]*b*Sqrt[c*Cot[a + b*x]])","A",1
14,1,38,212,0.0677582,"\int \frac{1}{(c \cot (a+b x))^{3/2}} \, dx","Integrate[(c*Cot[a + b*x])^(-3/2),x]","\frac{2 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(a+b x)\right)}{b c \sqrt{c \cot (a+b x)}}","\frac{\log \left(\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b c^{3/2}}-\frac{\log \left(\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b c^{3/2}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right)}{\sqrt{2} b c^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right)}{\sqrt{2} b c^{3/2}}+\frac{2}{b c \sqrt{c \cot (a+b x)}}",1,"(2*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[a + b*x]^2])/(b*c*Sqrt[c*Cot[a + b*x]])","C",1
15,1,40,214,0.0810539,"\int \frac{1}{(c \cot (a+b x))^{5/2}} \, dx","Integrate[(c*Cot[a + b*x])^(-5/2),x]","\frac{2 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(a+b x)\right)}{3 b c (c \cot (a+b x))^{3/2}}","-\frac{\log \left(\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b c^{5/2}}+\frac{\log \left(\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b c^{5/2}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right)}{\sqrt{2} b c^{5/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right)}{\sqrt{2} b c^{5/2}}+\frac{2}{3 b c (c \cot (a+b x))^{3/2}}",1,"(2*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[a + b*x]^2])/(3*b*c*(c*Cot[a + b*x])^(3/2))","C",1
16,1,40,234,0.1144541,"\int \frac{1}{(c \cot (a+b x))^{7/2}} \, dx","Integrate[(c*Cot[a + b*x])^(-7/2),x]","\frac{2 \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(a+b x)\right)}{5 b c (c \cot (a+b x))^{5/2}}","-\frac{\log \left(\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b c^{7/2}}+\frac{\log \left(\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right)}{2 \sqrt{2} b c^{7/2}}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right)}{\sqrt{2} b c^{7/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right)}{\sqrt{2} b c^{7/2}}-\frac{2}{b c^3 \sqrt{c \cot (a+b x)}}+\frac{2}{5 b c (c \cot (a+b x))^{5/2}}",1,"(2*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[a + b*x]^2])/(5*b*c*(c*Cot[a + b*x])^(5/2))","C",1
17,1,38,242,0.0317172,"\int (c \cot (a+b x))^{4/3} \, dx","Integrate[(c*Cot[a + b*x])^(4/3),x]","\frac{3 c \sqrt[3]{c \cot (a+b x)} \left(\, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\cot ^2(a+b x)\right)-1\right)}{b}","-\frac{\sqrt{3} c^{4/3} \log \left(-\sqrt{3} \sqrt[3]{c} \sqrt[3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right)}{4 b}+\frac{\sqrt{3} c^{4/3} \log \left(\sqrt{3} \sqrt[3]{c} \sqrt[3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right)}{4 b}+\frac{c^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{c \cot (a+b x)}}{\sqrt[3]{c}}\right)}{b}-\frac{c^{4/3} \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[3]{c \cot (a+b x)}}{\sqrt[3]{c}}\right)}{2 b}+\frac{c^{4/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c \cot (a+b x)}}{\sqrt[3]{c}}+\sqrt{3}\right)}{2 b}-\frac{3 c \sqrt[3]{c \cot (a+b x)}}{b}",1,"(3*c*(c*Cot[a + b*x])^(1/3)*(-1 + Hypergeometric2F1[1/6, 1, 7/6, -Cot[a + b*x]^2]))/b","C",1
18,1,40,225,0.047434,"\int (c \cot (a+b x))^{2/3} \, dx","Integrate[(c*Cot[a + b*x])^(2/3),x]","-\frac{3 (c \cot (a+b x))^{5/3} \, _2F_1\left(\frac{5}{6},1;\frac{11}{6};-\cot ^2(a+b x)\right)}{5 b c}","-\frac{\sqrt{3} c^{2/3} \log \left(-\sqrt{3} \sqrt[3]{c} \sqrt[3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right)}{4 b}+\frac{\sqrt{3} c^{2/3} \log \left(\sqrt{3} \sqrt[3]{c} \sqrt[3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right)}{4 b}-\frac{c^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{c \cot (a+b x)}}{\sqrt[3]{c}}\right)}{b}+\frac{c^{2/3} \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[3]{c \cot (a+b x)}}{\sqrt[3]{c}}\right)}{2 b}-\frac{c^{2/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c \cot (a+b x)}}{\sqrt[3]{c}}+\sqrt{3}\right)}{2 b}",1,"(-3*(c*Cot[a + b*x])^(5/3)*Hypergeometric2F1[5/6, 1, 11/6, -Cot[a + b*x]^2])/(5*b*c)","C",1
19,1,40,131,0.0400707,"\int \sqrt[3]{c \cot (a+b x)} \, dx","Integrate[(c*Cot[a + b*x])^(1/3),x]","-\frac{3 (c \cot (a+b x))^{4/3} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\cot ^2(a+b x)\right)}{4 b c}","\frac{\sqrt[3]{c} \log \left((c \cot (a+b x))^{2/3}+c^{2/3}\right)}{2 b}-\frac{\sqrt[3]{c} \log \left(-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}+c^{4/3}\right)}{4 b}+\frac{\sqrt{3} \sqrt[3]{c} \tan ^{-1}\left(\frac{c^{2/3}-2 (c \cot (a+b x))^{2/3}}{\sqrt{3} c^{2/3}}\right)}{2 b}",1,"(-3*(c*Cot[a + b*x])^(4/3)*Hypergeometric2F1[2/3, 1, 5/3, -Cot[a + b*x]^2])/(4*b*c)","C",1
20,1,98,131,0.1553293,"\int \frac{1}{\sqrt[3]{c \cot (a+b x)}} \, dx","Integrate[(c*Cot[a + b*x])^(-1/3),x]","\frac{\sqrt[3]{\cot (a+b x)} \left(-2 \log \left(\cot ^{\frac{2}{3}}(a+b x)+1\right)+\log \left(\cot ^{\frac{4}{3}}(a+b x)-\cot ^{\frac{2}{3}}(a+b x)+1\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 \cot ^{\frac{2}{3}}(a+b x)-1}{\sqrt{3}}\right)\right)}{4 b \sqrt[3]{c \cot (a+b x)}}","-\frac{\log \left((c \cot (a+b x))^{2/3}+c^{2/3}\right)}{2 b \sqrt[3]{c}}+\frac{\log \left(-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}+c^{4/3}\right)}{4 b \sqrt[3]{c}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{c^{2/3}-2 (c \cot (a+b x))^{2/3}}{\sqrt{3} c^{2/3}}\right)}{2 b \sqrt[3]{c}}",1,"(Cot[a + b*x]^(1/3)*(-2*Sqrt[3]*ArcTan[(-1 + 2*Cot[a + b*x]^(2/3))/Sqrt[3]] - 2*Log[1 + Cot[a + b*x]^(2/3)] + Log[1 - Cot[a + b*x]^(2/3) + Cot[a + b*x]^(4/3)]))/(4*b*(c*Cot[a + b*x])^(1/3))","A",1
21,1,38,225,0.0279225,"\int \frac{1}{(c \cot (a+b x))^{2/3}} \, dx","Integrate[(c*Cot[a + b*x])^(-2/3),x]","-\frac{3 \sqrt[3]{c \cot (a+b x)} \, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\cot ^2(a+b x)\right)}{b c}","\frac{\sqrt{3} \log \left(-\sqrt{3} \sqrt[3]{c} \sqrt[3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right)}{4 b c^{2/3}}-\frac{\sqrt{3} \log \left(\sqrt{3} \sqrt[3]{c} \sqrt[3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right)}{4 b c^{2/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{c \cot (a+b x)}}{\sqrt[3]{c}}\right)}{b c^{2/3}}+\frac{\tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[3]{c \cot (a+b x)}}{\sqrt[3]{c}}\right)}{2 b c^{2/3}}-\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{c \cot (a+b x)}}{\sqrt[3]{c}}+\sqrt{3}\right)}{2 b c^{2/3}}",1,"(-3*(c*Cot[a + b*x])^(1/3)*Hypergeometric2F1[1/6, 1, 7/6, -Cot[a + b*x]^2])/(b*c)","C",1
22,1,38,244,0.0593453,"\int \frac{1}{(c \cot (a+b x))^{4/3}} \, dx","Integrate[(c*Cot[a + b*x])^(-4/3),x]","\frac{3 \, _2F_1\left(-\frac{1}{6},1;\frac{5}{6};-\cot ^2(a+b x)\right)}{b c \sqrt[3]{c \cot (a+b x)}}","\frac{\sqrt{3} \log \left(-\sqrt{3} \sqrt[3]{c} \sqrt[3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right)}{4 b c^{4/3}}-\frac{\sqrt{3} \log \left(\sqrt{3} \sqrt[3]{c} \sqrt[3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right)}{4 b c^{4/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{c \cot (a+b x)}}{\sqrt[3]{c}}\right)}{b c^{4/3}}-\frac{\tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[3]{c \cot (a+b x)}}{\sqrt[3]{c}}\right)}{2 b c^{4/3}}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{c \cot (a+b x)}}{\sqrt[3]{c}}+\sqrt{3}\right)}{2 b c^{4/3}}+\frac{3}{b c \sqrt[3]{c \cot (a+b x)}}",1,"(3*Hypergeometric2F1[-1/6, 1, 5/6, -Cot[a + b*x]^2])/(b*c*(c*Cot[a + b*x])^(1/3))","C",1
23,1,48,46,0.0451092,"\int \cot ^n(a+b x) \, dx","Integrate[Cot[a + b*x]^n,x]","-\frac{\cot ^{n+1}(a+b x) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+1}{2}+1;-\cot ^2(a+b x)\right)}{b (n+1)}","-\frac{\cot ^{n+1}(a+b x) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\cot ^2(a+b x)\right)}{b (n+1)}",1,"-((Cot[a + b*x]^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, 1 + (1 + n)/2, -Cot[a + b*x]^2])/(b*(1 + n)))","A",1
24,1,54,51,0.0628816,"\int (b \cot (c+d x))^n \, dx","Integrate[(b*Cot[c + d*x])^n,x]","-\frac{\cot (c+d x) (b \cot (c+d x))^n \, _2F_1\left(1,\frac{n+1}{2};\frac{n+1}{2}+1;-\cot ^2(c+d x)\right)}{d (n+1)}","-\frac{(b \cot (c+d x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\cot ^2(c+d x)\right)}{b d (n+1)}",1,"-((Cot[c + d*x]*(b*Cot[c + d*x])^n*Hypergeometric2F1[1, (1 + n)/2, 1 + (1 + n)/2, -Cot[c + d*x]^2])/(d*(1 + n)))","A",1
25,1,27,36,0.0208015,"\int \left(a \cot ^2(x)\right)^{3/2} \, dx","Integrate[(a*Cot[x]^2)^(3/2),x]","-\frac{1}{2} a \tan (x) \sqrt{a \cot ^2(x)} \left(\csc ^2(x)+2 \log (\sin (x))\right)","-\frac{1}{2} a \cot (x) \sqrt{a \cot ^2(x)}-a \tan (x) \sqrt{a \cot ^2(x)} \log (\sin (x))",1,"-1/2*(a*Sqrt[a*Cot[x]^2]*(Csc[x]^2 + 2*Log[Sin[x]])*Tan[x])","A",1
26,1,16,16,0.007174,"\int \sqrt{a \cot ^2(x)} \, dx","Integrate[Sqrt[a*Cot[x]^2],x]","\tan (x) \sqrt{a \cot ^2(x)} \log (\sin (x))","\tan (x) \sqrt{a \cot ^2(x)} \log (\sin (x))",1,"Sqrt[a*Cot[x]^2]*Log[Sin[x]]*Tan[x]","A",1
27,1,17,17,0.0082341,"\int \frac{1}{\sqrt{a \cot ^2(x)}} \, dx","Integrate[1/Sqrt[a*Cot[x]^2],x]","-\frac{\cot (x) \log (\cos (x))}{\sqrt{a \cot ^2(x)}}","-\frac{\cot (x) \log (\cos (x))}{\sqrt{a \cot ^2(x)}}",1,"-((Cot[x]*Log[Cos[x]])/Sqrt[a*Cot[x]^2])","A",1
28,1,30,39,0.0306716,"\int \frac{1}{\left(a \cot ^2(x)\right)^{3/2}} \, dx","Integrate[(a*Cot[x]^2)^(-3/2),x]","\frac{\csc (x) \sec (x)+2 \cot (x) \log (\cos (x))}{2 a \sqrt{a \cot ^2(x)}}","\frac{\tan (x)}{2 a \sqrt{a \cot ^2(x)}}+\frac{\cot (x) \log (\cos (x))}{a \sqrt{a \cot ^2(x)}}",1,"(2*Cot[x]*Log[Cos[x]] + Csc[x]*Sec[x])/(2*a*Sqrt[a*Cot[x]^2])","A",1
29,1,39,200,0.0549828,"\int \left(a \cot ^3(x)\right)^{3/2} \, dx","Integrate[(a*Cot[x]^3)^(3/2),x]","-\frac{2}{21} a \sqrt{a \cot ^3(x)} \left(7 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(x)\right)+3 \cot ^2(x)-7\right)","\frac{2}{3} a \sqrt{a \cot ^3(x)}-\frac{2}{7} a \cot ^2(x) \sqrt{a \cot ^3(x)}-\frac{a \sqrt{a \cot ^3(x)} \log \left(\cot (x)-\sqrt{2} \sqrt{\cot (x)}+1\right)}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{a \sqrt{a \cot ^3(x)} \log \left(\cot (x)+\sqrt{2} \sqrt{\cot (x)}+1\right)}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{a \sqrt{a \cot ^3(x)} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (x)}\right)}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}-\frac{a \sqrt{a \cot ^3(x)} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (x)}+1\right)}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}",1,"(-2*a*Sqrt[a*Cot[x]^3]*(-7 + 3*Cot[x]^2 + 7*Hypergeometric2F1[3/4, 1, 7/4, -Cot[x]^2]))/21","C",1
30,1,122,176,0.1114476,"\int \sqrt{a \cot ^3(x)} \, dx","Integrate[Sqrt[a*Cot[x]^3],x]","-\frac{\sqrt{a \cot ^3(x)} \left(8 \sqrt{\cot (x)}+\sqrt{2} \log \left(\cot (x)-\sqrt{2} \sqrt{\cot (x)}+1\right)-\sqrt{2} \log \left(\cot (x)+\sqrt{2} \sqrt{\cot (x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (x)}+1\right)\right)}{4 \cot ^{\frac{3}{2}}(x)}","-2 \tan (x) \sqrt{a \cot ^3(x)}-\frac{\sqrt{a \cot ^3(x)} \log \left(\cot (x)-\sqrt{2} \sqrt{\cot (x)}+1\right)}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{\sqrt{a \cot ^3(x)} \log \left(\cot (x)+\sqrt{2} \sqrt{\cot (x)}+1\right)}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}-\frac{\sqrt{a \cot ^3(x)} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (x)}\right)}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{\sqrt{a \cot ^3(x)} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (x)}+1\right)}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}",1,"-1/4*(Sqrt[a*Cot[x]^3]*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[x]]] + 8*Sqrt[Cot[x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[x]] + Cot[x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[x]] + Cot[x]]))/Cot[x]^(3/2)","A",1
31,1,28,176,0.0120608,"\int \frac{1}{\sqrt{a \cot ^3(x)}} \, dx","Integrate[1/Sqrt[a*Cot[x]^3],x]","\frac{2 \cot (x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(x)\right)}{\sqrt{a \cot ^3(x)}}","\frac{2 \cot (x)}{\sqrt{a \cot ^3(x)}}+\frac{\cot ^{\frac{3}{2}}(x) \log \left(\cot (x)-\sqrt{2} \sqrt{\cot (x)}+1\right)}{2 \sqrt{2} \sqrt{a \cot ^3(x)}}-\frac{\cot ^{\frac{3}{2}}(x) \log \left(\cot (x)+\sqrt{2} \sqrt{\cot (x)}+1\right)}{2 \sqrt{2} \sqrt{a \cot ^3(x)}}-\frac{\cot ^{\frac{3}{2}}(x) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (x)}\right)}{\sqrt{2} \sqrt{a \cot ^3(x)}}+\frac{\cot ^{\frac{3}{2}}(x) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (x)}+1\right)}{\sqrt{2} \sqrt{a \cot ^3(x)}}",1,"(2*Cot[x]*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[x]^2])/Sqrt[a*Cot[x]^3]","C",1
32,1,30,212,0.015214,"\int \frac{1}{\left(a \cot ^3(x)\right)^{3/2}} \, dx","Integrate[(a*Cot[x]^3)^(-3/2),x]","\frac{2 \cot (x) \, _2F_1\left(-\frac{7}{4},1;-\frac{3}{4};-\cot ^2(x)\right)}{7 \left(a \cot ^3(x)\right)^{3/2}}","-\frac{2}{3 a \sqrt{a \cot ^3(x)}}+\frac{2 \tan ^2(x)}{7 a \sqrt{a \cot ^3(x)}}+\frac{\cot ^{\frac{3}{2}}(x) \log \left(\cot (x)-\sqrt{2} \sqrt{\cot (x)}+1\right)}{2 \sqrt{2} a \sqrt{a \cot ^3(x)}}-\frac{\cot ^{\frac{3}{2}}(x) \log \left(\cot (x)+\sqrt{2} \sqrt{\cot (x)}+1\right)}{2 \sqrt{2} a \sqrt{a \cot ^3(x)}}+\frac{\cot ^{\frac{3}{2}}(x) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (x)}\right)}{\sqrt{2} a \sqrt{a \cot ^3(x)}}-\frac{\cot ^{\frac{3}{2}}(x) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (x)}+1\right)}{\sqrt{2} a \sqrt{a \cot ^3(x)}}",1,"(2*Cot[x]*Hypergeometric2F1[-7/4, 1, -3/4, -Cot[x]^2])/(7*(a*Cot[x]^3)^(3/2))","C",1
33,1,39,70,0.1421037,"\int \left(a \cot ^4(x)\right)^{3/2} \, dx","Integrate[(a*Cot[x]^4)^(3/2),x]","-\frac{1}{15} \tan ^6(x) \left(a \cot ^4(x)\right)^{3/2} \left(15 x+\cot (x) \left(3 \csc ^4(x)-11 \csc ^2(x)+23\right)\right)","\frac{1}{3} a \cot (x) \sqrt{a \cot ^4(x)}-\frac{1}{5} a \cot ^3(x) \sqrt{a \cot ^4(x)}-a x \tan ^2(x) \sqrt{a \cot ^4(x)}-a \tan (x) \sqrt{a \cot ^4(x)}",1,"-1/15*((a*Cot[x]^4)^(3/2)*(15*x + Cot[x]*(23 - 11*Csc[x]^2 + 3*Csc[x]^4))*Tan[x]^6)","A",1
34,1,20,32,0.0161153,"\int \sqrt{a \cot ^4(x)} \, dx","Integrate[Sqrt[a*Cot[x]^4],x]","\tan ^2(x) (x+\cot (x)) \left(-\sqrt{a \cot ^4(x)}\right)","-x \tan ^2(x) \sqrt{a \cot ^4(x)}-\tan (x) \sqrt{a \cot ^4(x)}",1,"-(Sqrt[a*Cot[x]^4]*(x + Cot[x])*Tan[x]^2)","A",1
35,1,21,31,0.0234563,"\int \frac{1}{\sqrt{a \cot ^4(x)}} \, dx","Integrate[1/Sqrt[a*Cot[x]^4],x]","\frac{\cot (x)-x \cot ^2(x)}{\sqrt{a \cot ^4(x)}}","\frac{\cot (x)}{\sqrt{a \cot ^4(x)}}-\frac{x \cot ^2(x)}{\sqrt{a \cot ^4(x)}}",1,"(Cot[x] - x*Cot[x]^2)/Sqrt[a*Cot[x]^4]","A",1
36,1,42,77,0.110715,"\int \frac{1}{\left(a \cot ^4(x)\right)^{3/2}} \, dx","Integrate[(a*Cot[x]^4)^(-3/2),x]","\frac{-15 x \cot ^2(x)+23 \cot (x)+\csc (x) \sec (x) \left(3 \sec ^2(x)-11\right)}{15 a \sqrt{a \cot ^4(x)}}","\frac{\cot (x)}{a \sqrt{a \cot ^4(x)}}-\frac{x \cot ^2(x)}{a \sqrt{a \cot ^4(x)}}+\frac{\tan ^3(x)}{5 a \sqrt{a \cot ^4(x)}}-\frac{\tan (x)}{3 a \sqrt{a \cot ^4(x)}}",1,"(23*Cot[x] - 15*x*Cot[x]^2 + Csc[x]*Sec[x]*(-11 + 3*Sec[x]^2))/(15*a*Sqrt[a*Cot[x]^4])","A",1
37,1,58,60,0.0552856,"\int \left(b \cot ^p(c+d x)\right)^n \, dx","Integrate[(b*Cot[c + d*x]^p)^n,x]","-\frac{\cot (c+d x) \left(b \cot ^p(c+d x)\right)^n \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\cot ^2(c+d x)\right)}{d n p+d}","-\frac{\cot (c+d x) \left(b \cot ^p(c+d x)\right)^n \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\cot ^2(c+d x)\right)}{d (n p+1)}",1,"-((Cot[c + d*x]*(b*Cot[c + d*x]^p)^n*Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Cot[c + d*x]^2])/(d + d*n*p))","A",1
38,1,60,62,0.0483789,"\int \left(a (b \cot (c+d x))^p\right)^n \, dx","Integrate[(a*(b*Cot[c + d*x])^p)^n,x]","-\frac{\cot (c+d x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\cot ^2(c+d x)\right) \left(a (b \cot (c+d x))^p\right)^n}{d n p+d}","-\frac{\cot (c+d x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\cot ^2(c+d x)\right) \left(a (b \cot (c+d x))^p\right)^n}{d (n p+1)}",1,"-((Cot[c + d*x]*(a*(b*Cot[c + d*x])^p)^n*Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Cot[c + d*x]^2])/(d + d*n*p))","A",1
39,1,289,87,1.8023871,"\int (b \cot (e+f x))^n (a \sin (e+f x))^m \, dx","Integrate[(b*Cot[e + f*x])^n*(a*Sin[e + f*x])^m,x]","\frac{(m-n+3) \sin (e+f x) (a \sin (e+f x))^m (b \cot (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(n F_1\left(\frac{1}{2} (m-n+3);1-n,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)+(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)\right)\right)}","-\frac{(a \sin (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^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};\cos ^2(e+f x)\right)}{b f (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]*(b*Cot[e + f*x])^n*Sin[e + f*x]*(a*Sin[e + f*x])^m)/(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*(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] + (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])*Tan[(e + f*x)/2]^2))","C",0
40,1,83,84,0.5464844,"\int (a \cos (e+f x))^m (b \cot (e+f x))^n \, dx","Integrate[(a*Cos[e + f*x])^m*(b*Cot[e + f*x])^n,x]","-\frac{b \sec ^2(e+f x)^{m/2} (a \cos (e+f x))^m (b \cot (e+f x))^{n-1} \, _2F_1\left(\frac{m+2}{2},\frac{1-n}{2};\frac{3-n}{2};-\tan ^2(e+f x)\right)}{f (n-1)}","-\frac{\sin ^2(e+f x)^{\frac{n+1}{2}} (a \cos (e+f x))^m (b \cot (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(e+f x)\right)}{b f (m+n+1)}",1,"-((b*(a*Cos[e + f*x])^m*(b*Cot[e + f*x])^(-1 + n)*Hypergeometric2F1[(2 + m)/2, (1 - n)/2, (3 - n)/2, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(m/2))/(f*(-1 + n)))","A",1
41,1,67,64,0.0919911,"\int (a \cot (e+f x))^m (b \cot (e+f x))^n \, dx","Integrate[(a*Cot[e + f*x])^m*(b*Cot[e + f*x])^n,x]","-\frac{\cot (e+f x) (a \cot (e+f x))^m (b \cot (e+f x))^n \, _2F_1\left(1,\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+1)+1;-\cot ^2(e+f x)\right)}{f (m+n+1)}","-\frac{(a \cot (e+f x))^{m+1} (b \cot (e+f x))^n \, _2F_1\left(1,\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);-\cot ^2(e+f x)\right)}{a f (m+n+1)}",1,"-((Cot[e + f*x]*(a*Cot[e + f*x])^m*(b*Cot[e + f*x])^n*Hypergeometric2F1[1, (1 + m + n)/2, 1 + (1 + m + n)/2, -Cot[e + f*x]^2])/(f*(1 + m + n)))","A",1
42,1,83,90,0.4737344,"\int (b \cot (e+f x))^n (a \sec (e+f x))^m \, dx","Integrate[(b*Cot[e + f*x])^n*(a*Sec[e + f*x])^m,x]","-\frac{b \sec ^2(e+f x)^{-m/2} (a \sec (e+f x))^m (b \cot (e+f x))^{n-1} \, _2F_1\left(1-\frac{m}{2},\frac{1-n}{2};\frac{3-n}{2};-\tan ^2(e+f x)\right)}{f (n-1)}","-\frac{\sin ^2(e+f x)^{\frac{n+1}{2}} (a \sec (e+f x))^m (b \cot (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);\cos ^2(e+f x)\right)}{b f (-m+n+1)}",1,"-((b*(b*Cot[e + f*x])^(-1 + n)*Hypergeometric2F1[1 - m/2, (1 - n)/2, (3 - n)/2, -Tan[e + f*x]^2]*(a*Sec[e + f*x])^m)/(f*(-1 + n)*(Sec[e + f*x]^2)^(m/2)))","A",1
43,1,73,76,0.2518665,"\int (d \cot (e+f x))^n \csc ^6(e+f x) \, dx","Integrate[(d*Cot[e + f*x])^n*Csc[e + f*x]^6,x]","-\frac{\cot (e+f x) \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) (d \cot (e+f x))^n}{f (n+1) (n+3) (n+5)}","-\frac{(d \cot (e+f x))^{n+5}}{d^5 f (n+5)}-\frac{2 (d \cot (e+f x))^{n+3}}{d^3 f (n+3)}-\frac{(d \cot (e+f x))^{n+1}}{d f (n+1)}",1,"-(((8 + 6*n + n^2 - 2*(3 + n)*Cos[2*(e + f*x)] + Cos[4*(e + f*x)])*Cot[e + f*x]*(d*Cot[e + f*x])^n*Csc[e + f*x]^4)/(f*(1 + n)*(3 + n)*(5 + n)))","A",1
44,1,45,51,0.1312153,"\int (d \cot (e+f x))^n \csc ^4(e+f x) \, dx","Integrate[(d*Cot[e + f*x])^n*Csc[e + f*x]^4,x]","-\frac{\cot (e+f x) \left((n+1) \csc ^2(e+f x)+2\right) (d \cot (e+f x))^n}{f (n+1) (n+3)}","-\frac{(d \cot (e+f x))^{n+3}}{d^3 f (n+3)}-\frac{(d \cot (e+f x))^{n+1}}{d f (n+1)}",1,"-((Cot[e + f*x]*(d*Cot[e + f*x])^n*(2 + (1 + n)*Csc[e + f*x]^2))/(f*(1 + n)*(3 + n)))","A",1
45,1,26,25,0.0186858,"\int (d \cot (e+f x))^n \csc ^2(e+f x) \, dx","Integrate[(d*Cot[e + f*x])^n*Csc[e + f*x]^2,x]","-\frac{\cot (e+f x) (d \cot (e+f x))^n}{f (n+1)}","-\frac{(d \cot (e+f x))^{n+1}}{d f (n+1)}",1,"-((Cot[e + f*x]*(d*Cot[e + f*x])^n)/(f*(1 + n)))","A",1
46,1,509,51,3.0841824,"\int (d \cot (e+f x))^n \sin ^2(e+f x) \, dx","Integrate[(d*Cot[e + f*x])^n*Sin[e + f*x]^2,x]","-\frac{4 (n-3) \sin \left(\frac{1}{2} (e+f x)\right) \sin ^2(e+f x) \cos ^3\left(\frac{1}{2} (e+f x)\right) \left(F_1\left(\frac{1}{2}-\frac{n}{2};-n,2;\frac{3}{2}-\frac{n}{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{1}{2}-\frac{n}{2};-n,3;\frac{3}{2}-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) (d \cot (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{1}{2}-\frac{n}{2};-n,2;\frac{3}{2}-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 (n-3) \cos ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(\frac{1}{2}-\frac{n}{2};-n,3;\frac{3}{2}-\frac{n}{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(n F_1\left(\frac{3}{2}-\frac{n}{2};1-n,2;\frac{5}{2}-\frac{n}{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{3}{2}-\frac{n}{2};1-n,3;\frac{5}{2}-\frac{n}{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{3}{2}-\frac{n}{2};-n,3;\frac{5}{2}-\frac{n}{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{3}{2}-\frac{n}{2};-n,4;\frac{5}{2}-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}","-\frac{(d \cot (e+f x))^{n+1} \, _2F_1\left(2,\frac{n+1}{2};\frac{n+3}{2};-\cot ^2(e+f x)\right)}{d f (n+1)}",1,"(-4*(-3 + n)*(AppellF1[1/2 - n/2, -n, 2, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - AppellF1[1/2 - n/2, -n, 3, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Cos[(e + f*x)/2]^3*(d*Cot[e + f*x])^n*Sin[(e + f*x)/2]*Sin[e + f*x]^2)/(f*(-1 + n)*(2*(-3 + n)*AppellF1[1/2 - n/2, -n, 2, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - 2*(-3 + n)*AppellF1[1/2 - n/2, -n, 3, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - 2*(n*AppellF1[3/2 - n/2, 1 - n, 2, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[3/2 - n/2, 1 - n, 3, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*AppellF1[3/2 - n/2, -n, 3, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 3*AppellF1[3/2 - n/2, -n, 4, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*(-1 + Cos[e + f*x])))","C",0
47,1,1099,51,7.6579326,"\int (d \cot (e+f x))^n \sin ^4(e+f x) \, dx","Integrate[(d*Cot[e + f*x])^n*Sin[e + f*x]^4,x]","\frac{2 (n-3) \left(F_1\left(\frac{1}{2}-\frac{n}{2};-n,3;\frac{3}{2}-\frac{n}{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{1}{2}-\frac{n}{2};-n,4;\frac{3}{2}-\frac{n}{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{1}{2}-\frac{n}{2};-n,5;\frac{3}{2}-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \cos ^3\left(\frac{1}{2} (e+f x)\right) (d \cot (e+f x))^n \sin \left(\frac{1}{2} (e+f x)\right) \sin ^4(e+f x)}{f (n-1) \left(-n F_1\left(\frac{1}{2}-\frac{n}{2};-n,3;\frac{3}{2}-\frac{n}{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)+3 F_1\left(\frac{1}{2}-\frac{n}{2};-n,3;\frac{3}{2}-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \cos ^2\left(\frac{1}{2} (e+f x)\right)+2 n F_1\left(\frac{1}{2}-\frac{n}{2};-n,4;\frac{3}{2}-\frac{n}{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{1}{2}-\frac{n}{2};-n,4;\frac{3}{2}-\frac{n}{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)-n F_1\left(\frac{1}{2}-\frac{n}{2};-n,5;\frac{3}{2}-\frac{n}{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)+3 F_1\left(\frac{1}{2}-\frac{n}{2};-n,5;\frac{3}{2}-\frac{n}{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)+4 n F_1\left(\frac{3}{2}-\frac{n}{2};1-n,4;\frac{5}{2}-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \sin ^2\left(\frac{1}{2} (e+f x)\right)-3 F_1\left(\frac{3}{2}-\frac{n}{2};-n,4;\frac{5}{2}-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+8 F_1\left(\frac{3}{2}-\frac{n}{2};-n,5;\frac{5}{2}-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-5 F_1\left(\frac{3}{2}-\frac{n}{2};-n,6;\frac{5}{2}-\frac{n}{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{3}{2}-\frac{n}{2};1-n,3;\frac{5}{2}-\frac{n}{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 F_1\left(\frac{3}{2}-\frac{n}{2};1-n,5;\frac{5}{2}-\frac{n}{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)+3 F_1\left(\frac{3}{2}-\frac{n}{2};-n,4;\frac{5}{2}-\frac{n}{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)-8 F_1\left(\frac{3}{2}-\frac{n}{2};-n,5;\frac{5}{2}-\frac{n}{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)+5 F_1\left(\frac{3}{2}-\frac{n}{2};-n,6;\frac{5}{2}-\frac{n}{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)}","-\frac{(d \cot (e+f x))^{n+1} \, _2F_1\left(3,\frac{n+1}{2};\frac{n+3}{2};-\cot ^2(e+f x)\right)}{d f (n+1)}",1,"(2*(-3 + n)*(AppellF1[1/2 - n/2, -n, 3, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*AppellF1[1/2 - n/2, -n, 4, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[1/2 - n/2, -n, 5, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Cos[(e + f*x)/2]^3*(d*Cot[e + f*x])^n*Sin[(e + f*x)/2]*Sin[e + f*x]^4)/(f*(-1 + n)*(-3*AppellF1[3/2 - n/2, -n, 4, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 8*AppellF1[3/2 - n/2, -n, 5, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 5*AppellF1[3/2 - n/2, -n, 6, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 3*AppellF1[1/2 - n/2, -n, 3, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - n*AppellF1[1/2 - n/2, -n, 3, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - 6*AppellF1[1/2 - n/2, -n, 4, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + 2*n*AppellF1[1/2 - n/2, -n, 4, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + 3*AppellF1[1/2 - n/2, -n, 5, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - n*AppellF1[1/2 - n/2, -n, 5, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + n*AppellF1[3/2 - n/2, 1 - n, 3, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(-1 + Cos[e + f*x]) + n*AppellF1[3/2 - n/2, 1 - n, 5, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(-1 + Cos[e + f*x]) + 3*AppellF1[3/2 - n/2, -n, 4, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] - 8*AppellF1[3/2 - n/2, -n, 5, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] + 5*AppellF1[3/2 - n/2, -n, 6, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] + 4*n*AppellF1[3/2 - n/2, 1 - n, 4, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[(e + f*x)/2]^2))","C",0
48,1,784,79,14.0772529,"\int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx","Integrate[(d*Cot[e + f*x])^n*Csc[e + f*x]^3,x]","\frac{8 (n-4) \sin ^4\left(\frac{1}{2} (e+f x)\right) \cos ^6\left(\frac{1}{2} (e+f x)\right) \csc ^2(e+f x) (d \cot (e+f x))^n \left(n F_1\left(1-\frac{n}{2};-n,1;2-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-(n-2) \cot ^2\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(-n,-\frac{n}{2};1-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}{f (n-2) n \left(-8 n \sin ^4\left(\frac{1}{2} (e+f x)\right) F_1\left(2-\frac{n}{2};1-n,1;3-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-8 \sin ^4\left(\frac{1}{2} (e+f x)\right) F_1\left(2-\frac{n}{2};-n,2;3-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+(n-4) \left(4 \cos ^4\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-\sin ^2(e+f x) F_1\left(1-\frac{n}{2};-n,1;2-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}+\frac{(n-4) \sin ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(1-\frac{n}{2};-n,1;2-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) (d \cot (e+f x))^n}{f (4-2 n) \left(2 \tan ^2\left(\frac{1}{2} (e+f x)\right) \left(n F_1\left(2-\frac{n}{2};1-n,1;3-\frac{n}{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(2-\frac{n}{2};-n,2;3-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)+(n-4) F_1\left(1-\frac{n}{2};-n,1;2-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}-\frac{\cot ^2\left(\frac{1}{2} (e+f x)\right) (d \cot (e+f x))^n \, _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}}{f (4 n+8)}+\frac{\tan ^2\left(\frac{1}{2} (e+f x)\right) (d \cot (e+f x))^n \, _2F_1\left(1-\frac{n}{2},-n;2-\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}}{f (8-4 n)}","-\frac{\csc ^3(e+f x) \sin ^2(e+f x)^{\frac{n+4}{2}} (d \cot (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+4}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{d f (n+1)}",1,"-((Cot[(e + f*x)/2]^2*(d*Cot[e + f*x])^n*Hypergeometric2F1[-1 - n/2, -n, -1/2*n, Tan[(e + f*x)/2]^2])/(f*(8 + 4*n)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n)) + (8*(-4 + n)*Cos[(e + f*x)/2]^6*(d*Cot[e + f*x])^n*Csc[e + f*x]^2*(n*AppellF1[1 - n/2, -n, 1, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - (-2 + n)*Cot[(e + f*x)/2]^2*Hypergeometric2F1[-n, -1/2*n, 1 - n/2, Tan[(e + f*x)/2]^2])*Sin[(e + f*x)/2]^4)/(f*(-2 + n)*n*(-8*n*AppellF1[2 - n/2, 1 - n, 1, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[(e + f*x)/2]^4 - 8*AppellF1[2 - n/2, -n, 2, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[(e + f*x)/2]^4 + (-4 + n)*(4*Cos[(e + f*x)/2]^4*(Cos[e + f*x]*Sec[(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]*Sin[e + f*x]^2))) + ((d*Cot[e + f*x])^n*Hypergeometric2F1[1 - n/2, -n, 2 - n/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)/(f*(8 - 4*n)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n) + ((-4 + n)*AppellF1[1 - n/2, -n, 1, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(d*Cot[e + f*x])^n*Sin[(e + f*x)/2]^2)/(f*(4 - 2*n)*((-4 + n)*AppellF1[1 - n/2, -n, 1, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^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] + AppellF1[2 - n/2, -n, 2, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))","C",0
49,1,69,77,0.1544491,"\int (d \cot (e+f x))^n \csc (e+f x) \, dx","Integrate[(d*Cot[e + f*x])^n*Csc[e + f*x],x]","-\frac{(d \cot (e+f x))^n \, _2F_1\left(-n,-\frac{n}{2};1-\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}}{f n}","-\frac{\csc (e+f x) \sin ^2(e+f x)^{\frac{n+2}{2}} (d \cot (e+f x))^{n+1} \, _2F_1\left(\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{d f (n+1)}",1,"-(((d*Cot[e + f*x])^n*Hypergeometric2F1[-n, -1/2*n, 1 - n/2, Tan[(e + f*x)/2]^2])/(f*n*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n))","A",1
50,1,264,73,1.0996144,"\int (d \cot (e+f x))^n \sin (e+f x) \, dx","Integrate[(d*Cot[e + f*x])^n*Sin[e + f*x],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(1-\frac{n}{2};-n,2;2-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) (d \cot (e+f x))^n}{f (n-2) \left(2 (n-4) \cos ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(1-\frac{n}{2};-n,2;2-\frac{n}{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(n F_1\left(2-\frac{n}{2};1-n,2;3-\frac{n}{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(2-\frac{n}{2};-n,3;3-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}","-\frac{\sin (e+f x) \sin ^2(e+f x)^{n/2} (d \cot (e+f x))^{n+1} \, _2F_1\left(\frac{n}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{d f (n+1)}",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*(d*Cot[e + f*x])^n*Sin[(e + f*x)/2]^2)/(f*(-2 + n)*(2*(-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]^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] + 2*AppellF1[2 - n/2, -n, 3, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*(-1 + Cos[e + f*x])))","C",0
51,1,477,79,2.4272298,"\int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx","Integrate[(d*Cot[e + f*x])^n*Sin[e + f*x]^3,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(1-\frac{n}{2};-n,3;2-\frac{n}{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(1-\frac{n}{2};-n,4;2-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) (d \cot (e+f x))^n}{f (n-2) \left(2 (n-4) \cos ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(1-\frac{n}{2};-n,3;2-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 (n-4) \cos ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(1-\frac{n}{2};-n,4;2-\frac{n}{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(n F_1\left(2-\frac{n}{2};1-n,3;3-\frac{n}{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(2-\frac{n}{2};1-n,4;3-\frac{n}{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(2-\frac{n}{2};-n,4;3-\frac{n}{2};\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(2-\frac{n}{2};-n,5;3-\frac{n}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}","-\frac{\sin ^3(e+f x) \sin ^2(e+f x)^{\frac{n-2}{2}} (d \cot (e+f x))^{n+1} \, _2F_1\left(\frac{n-2}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{d f (n+1)}",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*(d*Cot[e + f*x])^n*Sin[(e + f*x)/2]*Sin[e + f*x]^3)/(f*(-2 + n)*(2*(-4 + n)*AppellF1[1 - n/2, -n, 3, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - 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*(n*AppellF1[2 - n/2, 1 - n, 3, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[2 - n/2, 1 - n, 4, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/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])*(-1 + Cos[e + f*x])))","C",0
52,1,306,83,1.8896624,"\int (b \cot (e+f x))^n (a \csc (e+f x))^m \, dx","Integrate[(b*Cot[e + f*x])^n*(a*Csc[e + f*x])^m,x]","-\frac{a (m+n-3) (a \csc (e+f x))^{m-1} (b \cot (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(2 \tan ^2\left(\frac{1}{2} (e+f x)\right) \left(n F_1\left(\frac{1}{2} (-m-n+3);1-n,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)-(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)\right)+(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)\right)}","-\frac{(a \csc (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^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};\cos ^2(e+f x)\right)}{b f (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]*(b*Cot[e + f*x])^n*(a*Csc[e + f*x])^(-1 + m))/(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*(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] - (-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])*Tan[(e + f*x)/2]^2)))","C",0