1,1,117,49,0.321935,"\int (a+i a \cot (c+d x))^n \, dx","Integrate[(a + I*a*Cot[c + d*x])^n,x]","\frac{i (a+i a \cot (c+d x))^n \left(2 (n+1) \, _2F_1(1,n;n+1;i \cot (c+d x)+1)+(n+i n \cot (c+d x)) \left(\, _2F_1\left(1,n+1;n+2;\frac{1}{2} (i \cot (c+d x)+1)\right)-2 \, _2F_1(1,n+1;n+2;i \cot (c+d x)+1)\right)\right)}{4 d n (n+1)}","\frac{i (a+i a \cot (c+d x))^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (i \cot (c+d x)+1)\right)}{2 d n}",1,"((I/4)*(a + I*a*Cot[c + d*x])^n*(2*(1 + n)*Hypergeometric2F1[1, n, 1 + n, 1 + I*Cot[c + d*x]] + (n + I*n*Cot[c + d*x])*(Hypergeometric2F1[1, 1 + n, 2 + n, (1 + I*Cot[c + d*x])/2] - 2*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + I*Cot[c + d*x]])))/(d*n*(1 + n))","B",1
2,1,68,116,0.1868597,"\int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x)) \, dx","Integrate[(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x]),x]","-\frac{2 a e (e \cot (c+d x))^{3/2} \left(5 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)+3 \cot (c+d x) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\tan ^2(c+d x)\right)\right)}{15 d}","-\frac{\sqrt{2} a e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{d}+\frac{2 a e^2 \sqrt{e \cot (c+d x)}}{d}-\frac{2 a e (e \cot (c+d x))^{3/2}}{3 d}-\frac{2 a (e \cot (c+d x))^{5/2}}{5 d}",1,"(-2*a*e*(e*Cot[c + d*x])^(3/2)*(3*Cot[c + d*x]*Hypergeometric2F1[-5/4, 1, -1/4, -Tan[c + d*x]^2] + 5*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2]))/(15*d)","C",1
3,1,67,94,0.1121628,"\int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx","Integrate[(e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x]),x]","-\frac{2 a e \sqrt{e \cot (c+d x)} \left(3 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)+\cot (c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)\right)}{3 d}","-\frac{\sqrt{2} a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{d}-\frac{2 a e \sqrt{e \cot (c+d x)}}{d}-\frac{2 a (e \cot (c+d x))^{3/2}}{3 d}",1,"(-2*a*e*Sqrt[e*Cot[c + d*x]]*(Cot[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2] + 3*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2]))/(3*d)","C",1
4,1,154,71,0.2890118,"\int \sqrt{e \cot (c+d x)} (a+a \cot (c+d x)) \, dx","Integrate[Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x]),x]","-\frac{a \sqrt{e \cot (c+d x)} \left(8 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)+\sqrt{2} \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)\right)}{4 d}","\frac{\sqrt{2} a \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{d}-\frac{2 a \sqrt{e \cot (c+d x)}}{d}",1,"-1/4*(a*Sqrt[e*Cot[c + d*x]]*(8*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2] + Sqrt[2]*(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]]))/d","C",1
5,1,165,49,0.217147,"\int \frac{a+a \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx","Integrate[(a + a*Cot[c + d*x])/Sqrt[e*Cot[c + d*x]],x]","\frac{a \left(8 \tan ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)+3 \sqrt{2} \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)\right)}{12 d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}","\frac{\sqrt{2} a \tan ^{-1}\left(\frac{\sqrt{e} (1-\cot (c+d x))}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{d \sqrt{e}}",1,"(a*(3*Sqrt[2]*(-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]]) + 8*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2)))/(12*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])","C",1
6,1,191,75,0.2589381,"\int \frac{a+a \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx","Integrate[(a + a*Cot[c + d*x])/(e*Cot[c + d*x])^(3/2),x]","\frac{a \left(8 \tan ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)+6 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-6 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+24 \sqrt{\tan (c+d x)}+3 \sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-3 \sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{12 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}","\frac{2 a}{d e \sqrt{e \cot (c+d x)}}-\frac{\sqrt{2} a \tanh ^{-1}\left(\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{d e^{3/2}}",1,"(a*(6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 3*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 24*Sqrt[Tan[c + d*x]] + 8*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2)))/(12*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))","C",1
7,1,203,99,0.4190489,"\int \frac{a+a \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx","Integrate[(a + a*Cot[c + d*x])/(e*Cot[c + d*x])^(5/2),x]","\frac{a \left(-8 \tan ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)+6 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-6 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+8 \tan ^{\frac{3}{2}}(c+d x)+24 \sqrt{\tan (c+d x)}+3 \sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-3 \sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{12 d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}","-\frac{\sqrt{2} a \tan ^{-1}\left(\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{d e^{5/2}}+\frac{2 a}{d e^2 \sqrt{e \cot (c+d x)}}+\frac{2 a}{3 d e (e \cot (c+d x))^{3/2}}",1,"(a*(6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 3*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 24*Sqrt[Tan[c + d*x]] + 8*Tan[c + d*x]^(3/2) - 8*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2)))/(12*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))","C",1
8,1,187,269,1.1877879,"\int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^2 \, dx","Integrate[(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^2,x]","-\frac{a^2 (e \cot (c+d x))^{5/2} \left(20 \cot ^{\frac{7}{2}}(c+d x)+56 \cot ^{\frac{5}{2}}(c+d x)-280 \sqrt{\cot (c+d x)}-35 \sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+35 \sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-70 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)+70 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{70 d \cot ^{\frac{5}{2}}(c+d x)}","\frac{a^2 e^{5/2} \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{\sqrt{2} d}-\frac{a^2 e^{5/2} \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{\sqrt{2} d}+\frac{\sqrt{2} a^2 e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{d}-\frac{\sqrt{2} a^2 e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{d}+\frac{4 a^2 e^2 \sqrt{e \cot (c+d x)}}{d}-\frac{2 a^2 (e \cot (c+d x))^{7/2}}{7 d e}-\frac{4 a^2 (e \cot (c+d x))^{5/2}}{5 d}",1,"-1/70*(a^2*(e*Cot[c + d*x])^(5/2)*(-70*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + 70*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] - 280*Sqrt[Cot[c + d*x]] + 56*Cot[c + d*x]^(5/2) + 20*Cot[c + d*x]^(7/2) - 35*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] + 35*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(d*Cot[c + d*x]^(5/2))","A",1
9,1,52,246,0.3859531,"\int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2 \, dx","Integrate[(e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])^2,x]","-\frac{2 a^2 (e \cot (c+d x))^{3/2} \left(-10 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)+3 \cot (c+d x)+10\right)}{15 d}","\frac{a^2 e^{3/2} \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{\sqrt{2} d}-\frac{a^2 e^{3/2} \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{\sqrt{2} d}-\frac{\sqrt{2} a^2 e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{\sqrt{2} a^2 e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{d}-\frac{2 a^2 (e \cot (c+d x))^{5/2}}{5 d e}-\frac{4 a^2 (e \cot (c+d x))^{3/2}}{3 d}",1,"(-2*a^2*(e*Cot[c + d*x])^(3/2)*(10 + 3*Cot[c + d*x] - 10*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(15*d)","C",1
10,1,175,244,0.4317057,"\int \sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^2 \, dx","Integrate[Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^2,x]","-\frac{a^2 \sqrt{e \cot (c+d x)} \left(4 \cot ^{\frac{3}{2}}(c+d x)+24 \sqrt{\cot (c+d x)}+3 \sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-3 \sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+6 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-6 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{6 d \sqrt{\cot (c+d x)}}","-\frac{2 a^2 (e \cot (c+d x))^{3/2}}{3 d e}-\frac{4 a^2 \sqrt{e \cot (c+d x)}}{d}-\frac{a^2 \sqrt{e} \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{\sqrt{2} d}+\frac{a^2 \sqrt{e} \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{\sqrt{2} d}-\frac{\sqrt{2} a^2 \sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{\sqrt{2} a^2 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{d}",1,"-1/6*(a^2*Sqrt[e*Cot[c + d*x]]*(6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 24*Sqrt[Cot[c + d*x]] + 4*Cot[c + d*x]^(3/2) + 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 3*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(d*Sqrt[Cot[c + d*x]])","A",1
11,1,53,222,0.2574647,"\int \frac{(a+a \cot (c+d x))^2}{\sqrt{e \cot (c+d x)}} \, dx","Integrate[(a + a*Cot[c + d*x])^2/Sqrt[e*Cot[c + d*x]],x]","-\frac{2 a^2 \sqrt{e \cot (c+d x)} \left(2 \cot (c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)+3\right)}{3 d e}","-\frac{2 a^2 \sqrt{e \cot (c+d x)}}{d e}-\frac{a^2 \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{\sqrt{2} d \sqrt{e}}+\frac{a^2 \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{\sqrt{2} d \sqrt{e}}+\frac{\sqrt{2} a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e}}-\frac{\sqrt{2} a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{d \sqrt{e}}",1,"(-2*a^2*Sqrt[e*Cot[c + d*x]]*(3 + 2*Cot[c + d*x]*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(3*d*e)","C",1
12,1,236,222,1.8084652,"\int \frac{(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx","Integrate[(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(3/2),x]","\frac{a^2 (\cot (c+d x)+1)^2 \left(3 \sin (c+d x) \left(4 \cos (c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)+\sqrt{2} \sin (c+d x) \cot ^{\frac{3}{2}}(c+d x) \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)-4 \cos ^2(c+d x) \cot (c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)\right)}{6 d (e \cot (c+d x))^{3/2} (\sin (c+d x)+\cos (c+d x))^2}","\frac{a^2 \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{\sqrt{2} d e^{3/2}}-\frac{a^2 \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{\sqrt{2} d e^{3/2}}+\frac{\sqrt{2} a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2}}-\frac{\sqrt{2} a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{d e^{3/2}}+\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}}",1,"(a^2*(1 + Cot[c + d*x])^2*(-4*Cos[c + d*x]^2*Cot[c + d*x]*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] + 3*Sin[c + d*x]*(4*Cos[c + d*x]*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Sqrt[2]*Cot[c + d*x]^(3/2)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])*Sin[c + d*x])))/(6*d*(e*Cot[c + d*x])^(3/2)*(Cos[c + d*x] + Sin[c + d*x])^2)","C",1
13,1,233,247,1.261755,"\int \frac{(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx","Integrate[(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(5/2),x]","\frac{a^2 (\tan (c+d x)+1)^2 \left(48 \cos ^2(c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)+\sin (c+d x) \left(8 \cos (c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)+3 \sqrt{2} \sin (c+d x) \cot ^{\frac{5}{2}}(c+d x) \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)\right)}{12 d e^2 \sqrt{e \cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^2}","\frac{a^2 \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{\sqrt{2} d e^{5/2}}-\frac{a^2 \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{\sqrt{2} d e^{5/2}}-\frac{\sqrt{2} a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2}}+\frac{\sqrt{2} a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{d e^{5/2}}+\frac{4 a^2}{d e^2 \sqrt{e \cot (c+d x)}}+\frac{2 a^2}{3 d e (e \cot (c+d x))^{3/2}}",1,"(a^2*(48*Cos[c + d*x]^2*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Sin[c + d*x]*(8*Cos[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2] + 3*Sqrt[2]*Cot[c + d*x]^(5/2)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])*Sin[c + d*x]))*(1 + Tan[c + d*x])^2)/(12*d*e^2*Sqrt[e*Cot[c + d*x]]*(Cos[c + d*x] + Sin[c + d*x])^2)","C",1
14,1,141,249,0.4140812,"\int \frac{(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx","Integrate[(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(7/2),x]","\frac{2 a^2 \sin (c+d x) (\tan (c+d x)+1)^2 \left(10 \cos (c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)+15 \cos (c+d x) \cot (c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)+3 \sin (c+d x) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(c+d x)\right)\right)}{15 d e^3 \sqrt{e \cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^2}","-\frac{a^2 \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{\sqrt{2} d e^{7/2}}+\frac{a^2 \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{\sqrt{2} d e^{7/2}}-\frac{\sqrt{2} a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{d e^{7/2}}+\frac{\sqrt{2} a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{d e^{7/2}}+\frac{4 a^2}{3 d e^2 (e \cot (c+d x))^{3/2}}+\frac{2 a^2}{5 d e (e \cot (c+d x))^{5/2}}",1,"(2*a^2*Sin[c + d*x]*(10*Cos[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2] + 15*Cos[c + d*x]*Cot[c + d*x]*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + 3*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2]*Sin[c + d*x])*(1 + Tan[c + d*x])^2)/(15*d*e^3*Sqrt[e*Cot[c + d*x]]*(Cos[c + d*x] + Sin[c + d*x])^2)","C",1
15,1,729,186,6.1036853,"\int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \, dx","Integrate[(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3,x]","-\frac{4 \sin ^3(c+d x) \tan (c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)}{3 d (\sin (c+d x)+\cos (c+d x))^3}-\frac{2 \sin (c+d x) \cos ^2(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{9 d (\sin (c+d x)+\cos (c+d x))^3}-\frac{4 \sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{5 d (\sin (c+d x)+\cos (c+d x))^3}-\frac{6 \sin ^2(c+d x) \cos (c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{7 d (\sin (c+d x)+\cos (c+d x))^3}+\frac{\sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \cot ^{\frac{5}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3}-\frac{\sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \cot ^{\frac{5}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3}+\frac{\sqrt{2} \sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{d \cot ^{\frac{5}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3}-\frac{\sqrt{2} \sin ^3(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{d \cot ^{\frac{5}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3}+\frac{4 \sin ^3(c+d x) \tan ^2(c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{d (\sin (c+d x)+\cos (c+d x))^3}+\frac{4 \sin ^3(c+d x) \tan (c+d x) (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{5/2}}{3 d (\sin (c+d x)+\cos (c+d x))^3}","\frac{2 \sqrt{2} a^3 e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{d}+\frac{4 a^3 e^2 \sqrt{e \cot (c+d x)}}{d}-\frac{2 \left(a^3 \cot (c+d x)+a^3\right) (e \cot (c+d x))^{7/2}}{9 d e}-\frac{40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac{4 a^3 (e \cot (c+d x))^{5/2}}{5 d}+\frac{4 a^3 e (e \cot (c+d x))^{3/2}}{3 d}",1,"(-2*Cos[c + d*x]^2*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Sin[c + d*x])/(9*d*(Cos[c + d*x] + Sin[c + d*x])^3) - (6*Cos[c + d*x]*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Sin[c + d*x]^2)/(7*d*(Cos[c + d*x] + Sin[c + d*x])^3) - (4*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Sin[c + d*x]^3)/(5*d*(Cos[c + d*x] + Sin[c + d*x])^3) + (Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Sin[c + d*x]^3)/(d*Cot[c + d*x]^(5/2)*(Cos[c + d*x] + Sin[c + d*x])^3) - (Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Sin[c + d*x]^3)/(d*Cot[c + d*x]^(5/2)*(Cos[c + d*x] + Sin[c + d*x])^3) + ((e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^3)/(Sqrt[2]*d*Cot[c + d*x]^(5/2)*(Cos[c + d*x] + Sin[c + d*x])^3) - ((e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^3)/(Sqrt[2]*d*Cot[c + d*x]^(5/2)*(Cos[c + d*x] + Sin[c + d*x])^3) + (4*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Sin[c + d*x]^3*Tan[c + d*x])/(3*d*(Cos[c + d*x] + Sin[c + d*x])^3) - (4*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]*Sin[c + d*x]^3*Tan[c + d*x])/(3*d*(Cos[c + d*x] + Sin[c + d*x])^3) + (4*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3*Sin[c + d*x]^3*Tan[c + d*x]^2)/(d*(Cos[c + d*x] + Sin[c + d*x])^3)","C",1
16,1,332,160,2.8102798,"\int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx","Integrate[(e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])^3,x]","\frac{a^3 \sin (c+d x) (\cot (c+d x)+1)^3 (e \cot (c+d x))^{3/2} \left(280 \sin ^2(c+d x) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-60 \cos ^2(c+d x) \cot ^{\frac{3}{2}}(c+d x)-280 \sin ^2(c+d x) \cot ^{\frac{3}{2}}(c+d x)-126 \sin (2 (c+d x)) \cot ^{\frac{3}{2}}(c+d x)+840 \sin ^2(c+d x) \sqrt{\cot (c+d x)}+105 \sqrt{2} \sin ^2(c+d x) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-105 \sqrt{2} \sin ^2(c+d x) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+210 \sqrt{2} \sin ^2(c+d x) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-210 \sqrt{2} \sin ^2(c+d x) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{210 d \cot ^{\frac{3}{2}}(c+d x) (\sin (c+d x)+\cos (c+d x))^3}","-\frac{2 \sqrt{2} a^3 e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{d}-\frac{32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac{4 a^3 (e \cot (c+d x))^{3/2}}{3 d}+\frac{4 a^3 e \sqrt{e \cot (c+d x)}}{d}-\frac{2 \left(a^3 \cot (c+d x)+a^3\right) (e \cot (c+d x))^{5/2}}{7 d e}",1,"(a^3*(e*Cot[c + d*x])^(3/2)*(1 + Cot[c + d*x])^3*Sin[c + d*x]*(-60*Cos[c + d*x]^2*Cot[c + d*x]^(3/2) + 210*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sin[c + d*x]^2 - 210*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sin[c + d*x]^2 + 840*Sqrt[Cot[c + d*x]]*Sin[c + d*x]^2 - 280*Cot[c + d*x]^(3/2)*Sin[c + d*x]^2 + 280*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]*Sin[c + d*x]^2 + 105*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 - 105*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 - 126*Cot[c + d*x]^(3/2)*Sin[2*(c + d*x)]))/(210*d*Cot[c + d*x]^(3/2)*(Cos[c + d*x] + Sin[c + d*x])^3)","C",1
17,1,315,138,1.5526004,"\int \sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx","Integrate[Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^3,x]","-\frac{a^3 \sin (c+d x) (\cot (c+d x)+1)^3 \sqrt{e \cot (c+d x)} \left(3 \left(4 \cos ^2(c+d x) \sqrt{\cot (c+d x)}+40 \sin ^2(c+d x) \sqrt{\cot (c+d x)}+10 \sin (2 (c+d x)) \sqrt{\cot (c+d x)}+5 \sqrt{2} \sin ^2(c+d x) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-5 \sqrt{2} \sin ^2(c+d x) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+10 \sqrt{2} \sin ^2(c+d x) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-10 \sqrt{2} \sin ^2(c+d x) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)-20 \sin (2 (c+d x)) \sqrt{\cot (c+d x)} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)\right)}{30 d \sqrt{\cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^3}","-\frac{8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac{4 a^3 \sqrt{e \cot (c+d x)}}{d}-\frac{2 \left(a^3 \cot (c+d x)+a^3\right) (e \cot (c+d x))^{3/2}}{5 d e}-\frac{2 \sqrt{2} a^3 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{d}",1,"-1/30*(a^3*Sqrt[e*Cot[c + d*x]]*(1 + Cot[c + d*x])^3*Sin[c + d*x]*(-20*Sqrt[Cot[c + d*x]]*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]*Sin[2*(c + d*x)] + 3*(4*Cos[c + d*x]^2*Sqrt[Cot[c + d*x]] + 10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sin[c + d*x]^2 - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sin[c + d*x]^2 + 40*Sqrt[Cot[c + d*x]]*Sin[c + d*x]^2 + 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 - 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 + 10*Sqrt[Cot[c + d*x]]*Sin[2*(c + d*x)])))/(d*Sqrt[Cot[c + d*x]]*(Cos[c + d*x] + Sin[c + d*x])^3)","C",1
18,1,292,117,5.1710146,"\int \frac{(a+a \cot (c+d x))^3}{\sqrt{e \cot (c+d x)}} \, dx","Integrate[(a + a*Cot[c + d*x])^3/Sqrt[e*Cot[c + d*x]],x]","-\frac{a^3 \sin (c+d x) (\cot (c+d x)+1)^3 \left(8 \cos ^2(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)+18 \sin (2 (c+d x))+4 \cos ^2(c+d x)+3 \sqrt{2} \sin ^2(c+d x) \sqrt{\cot (c+d x)} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-3 \sqrt{2} \sin ^2(c+d x) \sqrt{\cot (c+d x)} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+6 \sqrt{2} \sin ^2(c+d x) \sqrt{\cot (c+d x)} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-6 \sqrt{2} \sin ^2(c+d x) \sqrt{\cot (c+d x)} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{6 d \sqrt{e \cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^3}","-\frac{16 a^3 \sqrt{e \cot (c+d x)}}{3 d e}-\frac{2 \left(a^3 \cot (c+d x)+a^3\right) \sqrt{e \cot (c+d x)}}{3 d e}+\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left(\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{d \sqrt{e}}",1,"-1/6*(a^3*(1 + Cot[c + d*x])^3*Sin[c + d*x]*(4*Cos[c + d*x]^2 + 8*Cos[c + d*x]^2*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] + 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sin[c + d*x]^2 - 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sin[c + d*x]^2 + 3*Sqrt[2]*Sqrt[Cot[c + d*x]]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 - 3*Sqrt[2]*Sqrt[Cot[c + d*x]]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 + 18*Sin[2*(c + d*x)]))/(d*Sqrt[e*Cot[c + d*x]]*(Cos[c + d*x] + Sin[c + d*x])^3)","C",1
19,1,311,114,2.8938954,"\int \frac{(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx","Integrate[(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(3/2),x]","\frac{a^3 (\cot (c+d x)+1)^3 \left(\sin (c+d x) \left(2 \sin (2 (c+d x)) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)-4 \cos ^2(c+d x)+\sqrt{2} \sin ^2(c+d x) \cot ^{\frac{3}{2}}(c+d x) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \sin ^2(c+d x) \cot ^{\frac{3}{2}}(c+d x) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \sin ^2(c+d x) \cot ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \sin ^2(c+d x) \cot ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)-4 \cos ^3(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)\right)}{2 d (e \cot (c+d x))^{3/2} (\sin (c+d x)+\cos (c+d x))^3}","\frac{2 \sqrt{2} a^3 \tan ^{-1}\left(\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{d e^{3/2}}-\frac{4 a^3 \sqrt{e \cot (c+d x)}}{d e^2}+\frac{2 \left(a^3 \cot (c+d x)+a^3\right)}{d e \sqrt{e \cot (c+d x)}}",1,"(a^3*(1 + Cot[c + d*x])^3*(-4*Cos[c + d*x]^3*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] + Sin[c + d*x]*(-4*Cos[c + d*x]^2 + 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cot[c + d*x]^(3/2)*Sin[c + d*x]^2 - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cot[c + d*x]^(3/2)*Sin[c + d*x]^2 + Sqrt[2]*Cot[c + d*x]^(3/2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 - Sqrt[2]*Cot[c + d*x]^(3/2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 + 2*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2]*Sin[2*(c + d*x)])))/(2*d*(e*Cot[c + d*x])^(3/2)*(Cos[c + d*x] + Sin[c + d*x])^3)","C",1
20,1,417,117,6.1091794,"\int \frac{(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{5/2}} \, dx","Integrate[(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(5/2),x]","-\frac{2 \cos ^3(c+d x) \cot (c+d x) (a \cot (c+d x)+a)^3 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)}{3 d (e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x))^3}+\frac{6 \sin (c+d x) \cos ^2(c+d x) (a \cot (c+d x)+a)^3 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)}{d (e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x))^3}+\frac{2 \sin ^2(c+d x) \cos (c+d x) (a \cot (c+d x)+a)^3 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)}{3 d (e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x))^3}+\frac{3 \sin ^3(c+d x) \cot ^{\frac{5}{2}}(c+d x) (a \cot (c+d x)+a)^3 \left(\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{4 d (e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x))^3}","-\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left(\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{d e^{5/2}}+\frac{16 a^3}{3 d e^2 \sqrt{e \cot (c+d x)}}+\frac{2 \left(a^3 \cot (c+d x)+a^3\right)}{3 d e (e \cot (c+d x))^{3/2}}",1,"(-2*Cos[c + d*x]^3*Cot[c + d*x]*(a + a*Cot[c + d*x])^3*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/(3*d*(e*Cot[c + d*x])^(5/2)*(Cos[c + d*x] + Sin[c + d*x])^3) + (6*Cos[c + d*x]^2*(a + a*Cot[c + d*x])^3*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2]*Sin[c + d*x])/(d*(e*Cot[c + d*x])^(5/2)*(Cos[c + d*x] + Sin[c + d*x])^3) + (2*Cos[c + d*x]*(a + a*Cot[c + d*x])^3*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2]*Sin[c + d*x]^2)/(3*d*(e*Cot[c + d*x])^(5/2)*(Cos[c + d*x] + Sin[c + d*x])^3) + (3*Cot[c + d*x]^(5/2)*(a + a*Cot[c + d*x])^3*(2*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])*Sin[c + d*x]^3)/(4*d*(e*Cot[c + d*x])^(5/2)*(Cos[c + d*x] + Sin[c + d*x])^3)","C",1
21,1,269,141,3.3447547,"\int \frac{(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx","Integrate[(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(7/2),x]","\frac{a^3 (\tan (c+d x)+1)^3 \left(120 \cos ^3(c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)+\sin (c+d x) \left(40 \cos ^2(c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)+\sin (c+d x) \left(8 \cos (c+d x) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(c+d x)\right)+5 \sqrt{2} \sin (c+d x) \cot ^{\frac{7}{2}}(c+d x) \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)\right)\right)}{20 d e^3 \sqrt{e \cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^3}","-\frac{2 \sqrt{2} a^3 \tan ^{-1}\left(\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{d e^{7/2}}+\frac{4 a^3}{d e^3 \sqrt{e \cot (c+d x)}}+\frac{8 a^3}{5 d e^2 (e \cot (c+d x))^{3/2}}+\frac{2 \left(a^3 \cot (c+d x)+a^3\right)}{5 d e (e \cot (c+d x))^{5/2}}",1,"(a^3*(120*Cos[c + d*x]^3*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Sin[c + d*x]*(40*Cos[c + d*x]^2*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2] + Sin[c + d*x]*(8*Cos[c + d*x]*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2] + 5*Sqrt[2]*Cot[c + d*x]^(7/2)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])*Sin[c + d*x])))*(1 + Tan[c + d*x])^3)/(20*d*e^3*Sqrt[e*Cot[c + d*x]]*(Cos[c + d*x] + Sin[c + d*x])^3)","C",1
22,1,174,165,1.9980276,"\int \frac{(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx","Integrate[(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(9/2),x]","\frac{2 a^3 \cos (c+d x) (\cot (c+d x)+1)^3 \left(35 \cos ^2(c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)+35 \cos ^2(c+d x) \cot (c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)+5 \sin ^2(c+d x) \, _2F_1\left(-\frac{7}{4},1;-\frac{3}{4};-\cot ^2(c+d x)\right)+\frac{21}{2} \sin (2 (c+d x)) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(c+d x)\right)\right)}{35 d (e \cot (c+d x))^{9/2} (\sin (c+d x)+\cos (c+d x))^3}","\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left(\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{d e^{9/2}}-\frac{4 a^3}{d e^4 \sqrt{e \cot (c+d x)}}+\frac{4 a^3}{3 d e^3 (e \cot (c+d x))^{3/2}}+\frac{32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac{2 \left(a^3 \cot (c+d x)+a^3\right)}{7 d e (e \cot (c+d x))^{7/2}}",1,"(2*a^3*Cos[c + d*x]*(1 + Cot[c + d*x])^3*(35*Cos[c + d*x]^2*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2] + 35*Cos[c + d*x]^2*Cot[c + d*x]*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + 5*Hypergeometric2F1[-7/4, 1, -3/4, -Cot[c + d*x]^2]*Sin[c + d*x]^2 + (21*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2]*Sin[2*(c + d*x)])/2))/(35*d*(e*Cot[c + d*x])^(9/2)*(Cos[c + d*x] + Sin[c + d*x])^3)","C",1
23,1,110,111,0.8835239,"\int \frac{(e \cot (c+d x))^{5/2}}{a+a \cot (c+d x)} \, dx","Integrate[(e*Cot[c + d*x])^(5/2)/(a + a*Cot[c + d*x]),x]","-\frac{(e \cot (c+d x))^{5/2} \left(4 \sqrt{\cot (c+d x)}+\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)\right)}{2 a d \cot ^{\frac{5}{2}}(c+d x)}","\frac{e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{a d}-\frac{e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{\sqrt{2} a d}-\frac{2 e^2 \sqrt{e \cot (c+d x)}}{a d}",1,"-1/2*((Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[Sqrt[Cot[c + d*x]]] + 4*Sqrt[Cot[c + d*x]])*(e*Cot[c + d*x])^(5/2))/(a*d*Cot[c + d*x]^(5/2))","A",1
24,1,107,87,4.0518694,"\int \frac{(e \cot (c+d x))^{3/2}}{a+a \cot (c+d x)} \, dx","Integrate[(e*Cot[c + d*x])^(3/2)/(a + a*Cot[c + d*x]),x]","-\frac{(e \cot (c+d x))^{3/2} \left(\sqrt{2} \left(\log \left(-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)+4 \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)\right)}{4 a d \cot ^{\frac{3}{2}}(c+d x)}","\frac{e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{\sqrt{2} a d}-\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{a d}",1,"-1/4*((e*Cot[c + d*x])^(3/2)*(4*ArcTan[Sqrt[Cot[c + d*x]]] + Sqrt[2]*(Log[-1 + Sqrt[2]*Sqrt[Cot[c + d*x]] - Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(a*d*Cot[c + d*x]^(3/2))","A",1
25,1,98,87,0.2509144,"\int \frac{\sqrt{e \cot (c+d x)}}{a+a \cot (c+d x)} \, dx","Integrate[Sqrt[e*Cot[c + d*x]]/(a + a*Cot[c + d*x]),x]","\frac{\sqrt{e \cot (c+d x)} \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)\right)}{2 a d \sqrt{\cot (c+d x)}}","\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{a d}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{\sqrt{2} a d}",1,"((Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 2*ArcTan[Sqrt[Cot[c + d*x]]])*Sqrt[e*Cot[c + d*x]])/(2*a*d*Sqrt[Cot[c + d*x]])","A",1
26,1,107,83,0.5159511,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx","Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])),x]","-\frac{\sqrt{\cot (c+d x)} \left(\sqrt{2} \left(\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right)\right)+4 \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)\right)}{4 a d \sqrt{e \cot (c+d x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{a d \sqrt{e}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{e} (\cot (c+d x)+1)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{\sqrt{2} a d \sqrt{e}}",1,"-1/4*(Sqrt[Cot[c + d*x]]*(4*ArcTan[Sqrt[Cot[c + d*x]]] + Sqrt[2]*(-Log[-1 + Sqrt[2]*Sqrt[Cot[c + d*x]] - Cot[c + d*x]] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(a*d*Sqrt[e*Cot[c + d*x]])","A",1
27,1,176,111,2.0335407,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx","Integrate[1/((e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])),x]","\frac{2 \sin ^4(c+d x) \left(\cot ^4(c+d x)+2 \cot ^2(c+d x)-\sqrt{2} \cot ^{\frac{5}{2}}(c+d x) \csc ^2(2 (c+d x)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)+\sqrt{2} \cot ^{\frac{5}{2}}(c+d x) \csc ^2(2 (c+d x)) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \cot ^{\frac{5}{2}}(c+d x) \csc ^2(2 (c+d x)) \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)+1\right)}{a d e \sqrt{e \cot (c+d x)}}","\frac{\tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{a d e^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{\sqrt{2} a d e^{3/2}}+\frac{2}{a d e \sqrt{e \cot (c+d x)}}",1,"(2*(1 + 2*Cot[c + d*x]^2 + Cot[c + d*x]^4 - Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cot[c + d*x]^(5/2)*Csc[2*(c + d*x)]^2 + Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cot[c + d*x]^(5/2)*Csc[2*(c + d*x)]^2 + 2*ArcTan[Sqrt[Cot[c + d*x]]]*Cot[c + d*x]^(5/2)*Csc[2*(c + d*x)]^2)*Sin[c + d*x]^4)/(a*d*e*Sqrt[e*Cot[c + d*x]])","A",1
28,1,131,135,1.3332893,"\int \frac{1}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))} \, dx","Integrate[1/((e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])),x]","\frac{8 (\tan (c+d x)-3)-3 \sqrt{2} \sqrt{\cot (c+d x)} \left(\log \left(-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)-12 \sqrt{\cot (c+d x)} \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)}{12 a d e^2 \sqrt{e \cot (c+d x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{a d e^{5/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{\sqrt{2} a d e^{5/2}}-\frac{2}{a d e^2 \sqrt{e \cot (c+d x)}}+\frac{2}{3 a d e (e \cot (c+d x))^{3/2}}",1,"(-12*ArcTan[Sqrt[Cot[c + d*x]]]*Sqrt[Cot[c + d*x]] - 3*Sqrt[2]*Sqrt[Cot[c + d*x]]*(Log[-1 + Sqrt[2]*Sqrt[Cot[c + d*x]] - Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]) + 8*(-3 + Tan[c + d*x]))/(12*a*d*e^2*Sqrt[e*Cot[c + d*x]])","A",1
29,1,224,281,2.0191238,"\int \frac{(e \cot (c+d x))^{5/2}}{(a+a \cot (c+d x))^2} \, dx","Integrate[(e*Cot[c + d*x])^(5/2)/(a + a*Cot[c + d*x])^2,x]","\frac{(e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x)) \left(2 \cot ^{\frac{3}{2}}(c+d x) \sec (c+d x)-\frac{1}{2} (\cot (c+d x)+1) \csc (c+d x) \left(\sqrt{2} \log \left(-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+12 \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)\right)\right)}{4 a^2 d \cot ^{\frac{5}{2}}(c+d x) (\cot (c+d x)+1)^2}","-\frac{e^{5/2} \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{4 \sqrt{2} a^2 d}+\frac{e^{5/2} \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{4 \sqrt{2} a^2 d}-\frac{3 e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{2 a^2 d}-\frac{e^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{2 \sqrt{2} a^2 d}+\frac{e^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{2 \sqrt{2} a^2 d}+\frac{e^2 \sqrt{e \cot (c+d x)}}{2 d \left(a^2 \cot (c+d x)+a^2\right)}",1,"((e*Cot[c + d*x])^(5/2)*(-1/2*((1 + Cot[c + d*x])*Csc[c + d*x]*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 12*ArcTan[Sqrt[Cot[c + d*x]]] + Sqrt[2]*Log[-1 + Sqrt[2]*Sqrt[Cot[c + d*x]] - Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])) + 2*Cot[c + d*x]^(3/2)*Sec[c + d*x])*(Cos[c + d*x] + Sin[c + d*x]))/(4*a^2*d*Cot[c + d*x]^(5/2)*(1 + Cot[c + d*x])^2)","A",1
30,1,312,279,2.8292411,"\int \frac{(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^2} \, dx","Integrate[(e*Cot[c + d*x])^(3/2)/(a + a*Cot[c + d*x])^2,x]","-\frac{\sin ^2(c+d x) (e \cot (c+d x))^{3/2} \left(4 \cot ^{\frac{7}{2}}(c+d x)-4 \cot ^{\frac{5}{2}}(c+d x)+4 \cot ^{\frac{3}{2}}(c+d x)-4 \sqrt{\cot (c+d x)}+\sqrt{2} \cos (2 (c+d x)) \csc ^4(c+d x) \log \left(-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right)-\sqrt{2} \cos (2 (c+d x)) \csc ^4(c+d x) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \sqrt{2} \cos (2 (c+d x)) \csc ^4(c+d x) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)+2 \sqrt{2} \cos (2 (c+d x)) \csc ^4(c+d x) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-4 \cos (2 (c+d x)) \csc ^4(c+d x) \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)\right)}{8 a^2 d \cot ^{\frac{3}{2}}(c+d x) \left(\cot ^2(c+d x)-1\right)}","-\frac{e^{3/2} \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{4 \sqrt{2} a^2 d}+\frac{e^{3/2} \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{4 \sqrt{2} a^2 d}+\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{2 a^2 d}+\frac{e^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{2 \sqrt{2} a^2 d}-\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{2 \sqrt{2} a^2 d}-\frac{e \sqrt{e \cot (c+d x)}}{2 d \left(a^2 \cot (c+d x)+a^2\right)}",1,"-1/8*((e*Cot[c + d*x])^(3/2)*(-4*Sqrt[Cot[c + d*x]] + 4*Cot[c + d*x]^(3/2) - 4*Cot[c + d*x]^(5/2) + 4*Cot[c + d*x]^(7/2) - 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cos[2*(c + d*x)]*Csc[c + d*x]^4 + 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cos[2*(c + d*x)]*Csc[c + d*x]^4 - 4*ArcTan[Sqrt[Cot[c + d*x]]]*Cos[2*(c + d*x)]*Csc[c + d*x]^4 + Sqrt[2]*Cos[2*(c + d*x)]*Csc[c + d*x]^4*Log[-1 + Sqrt[2]*Sqrt[Cot[c + d*x]] - Cot[c + d*x]] - Sqrt[2]*Cos[2*(c + d*x)]*Csc[c + d*x]^4*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])*Sin[c + d*x]^2)/(a^2*d*Cot[c + d*x]^(3/2)*(-1 + Cot[c + d*x]^2))","A",1
31,1,207,278,1.2990274,"\int \frac{\sqrt{e \cot (c+d x)}}{(a+a \cot (c+d x))^2} \, dx","Integrate[Sqrt[e*Cot[c + d*x]]/(a + a*Cot[c + d*x])^2,x]","\frac{\csc (c+d x) \sqrt{e \cot (c+d x)} (\sin (c+d x)+\cos (c+d x)) \left(\frac{1}{2} (\tan (c+d x)+1) \sqrt{\cot (c+d x)} \left(\sqrt{2} \log \left(-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+4 \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)\right)+2\right)}{4 a^2 d (\cot (c+d x)+1)^2}","\frac{\sqrt{e \cot (c+d x)}}{2 d \left(a^2 \cot (c+d x)+a^2\right)}+\frac{\sqrt{e} \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{4 \sqrt{2} a^2 d}-\frac{\sqrt{e} \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{4 \sqrt{2} a^2 d}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{2 a^2 d}+\frac{\sqrt{e} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{2 \sqrt{2} a^2 d}-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{2 \sqrt{2} a^2 d}",1,"(Sqrt[e*Cot[c + d*x]]*Csc[c + d*x]*(Cos[c + d*x] + Sin[c + d*x])*(2 + (Sqrt[Cot[c + d*x]]*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 4*ArcTan[Sqrt[Cot[c + d*x]]] + Sqrt[2]*Log[-1 + Sqrt[2]*Sqrt[Cot[c + d*x]] - Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])*(1 + Tan[c + d*x]))/2))/(4*a^2*d*(1 + Cot[c + d*x])^2)","A",1
32,1,337,281,0.8386987,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^2} \, dx","Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^2),x]","-\frac{\sqrt{\cot (c+d x)} \left(4 \sin (c+d x) \sqrt{\cot (c+d x)}-\sqrt{2} \cos (c+d x) \log \left(-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right)+\sqrt{2} \cos (c+d x) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+12 \cos (c+d x) \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)-\sqrt{2} \sin (c+d x) \log \left(-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right)+\sqrt{2} \sin (c+d x) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+12 \sin (c+d x) \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)+2 \sqrt{2} (\sin (c+d x)+\cos (c+d x)) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} (\sin (c+d x)+\cos (c+d x)) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{8 a^2 d \sqrt{e \cot (c+d x)} (\sin (c+d x)+\cos (c+d x))}","-\frac{\sqrt{e \cot (c+d x)}}{2 d e \left(a^2 \cot (c+d x)+a^2\right)}+\frac{\log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{4 \sqrt{2} a^2 d \sqrt{e}}-\frac{\log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{4 \sqrt{2} a^2 d \sqrt{e}}-\frac{3 \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{2 a^2 d \sqrt{e}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{2 \sqrt{2} a^2 d \sqrt{e}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{2 \sqrt{2} a^2 d \sqrt{e}}",1,"-1/8*(Sqrt[Cot[c + d*x]]*(12*ArcTan[Sqrt[Cot[c + d*x]]]*Cos[c + d*x] - Sqrt[2]*Cos[c + d*x]*Log[-1 + Sqrt[2]*Sqrt[Cot[c + d*x]] - Cot[c + d*x]] + Sqrt[2]*Cos[c + d*x]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] + 12*ArcTan[Sqrt[Cot[c + d*x]]]*Sin[c + d*x] + 4*Sqrt[Cot[c + d*x]]*Sin[c + d*x] - Sqrt[2]*Log[-1 + Sqrt[2]*Sqrt[Cot[c + d*x]] - Cot[c + d*x]]*Sin[c + d*x] + Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x] + 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*(Cos[c + d*x] + Sin[c + d*x]) - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*(Cos[c + d*x] + Sin[c + d*x])))/(a^2*d*Sqrt[e*Cot[c + d*x]]*(Cos[c + d*x] + Sin[c + d*x]))","A",1
33,1,203,306,1.3270668,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2} \, dx","Integrate[1/((e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])^2),x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \left(\frac{\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right)}{\sqrt{2}}-\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)+\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+10 \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)+\frac{2 (4 \sin (c+d x)+5 \cos (c+d x))}{\sqrt{\cot (c+d x)} (\sin (c+d x)+\cos (c+d x))}\right)}{4 a^2 d (e \cot (c+d x))^{3/2}}","-\frac{\log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{4 \sqrt{2} a^2 d e^{3/2}}+\frac{\log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{4 \sqrt{2} a^2 d e^{3/2}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{2 a^2 d e^{3/2}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{2 \sqrt{2} a^2 d e^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{2 \sqrt{2} a^2 d e^{3/2}}+\frac{5}{2 a^2 d e \sqrt{e \cot (c+d x)}}-\frac{1}{2 d e \left(a^2 \cot (c+d x)+a^2\right) \sqrt{e \cot (c+d x)}}",1,"(Cot[c + d*x]^(3/2)*(-(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]) + Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 10*ArcTan[Sqrt[Cot[c + d*x]]] + (-Log[-1 + Sqrt[2]*Sqrt[Cot[c + d*x]] - Cot[c + d*x]] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/Sqrt[2] + (2*(5*Cos[c + d*x] + 4*Sin[c + d*x]))/(Sqrt[Cot[c + d*x]]*(Cos[c + d*x] + Sin[c + d*x]))))/(4*a^2*d*(e*Cot[c + d*x])^(3/2))","A",1
34,1,467,331,6.3413491,"\int \frac{1}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^2} \, dx","Integrate[1/((e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^2),x]","\frac{\cot ^3(c+d x) \csc ^2(c+d x) (\sin (c+d x)+\cos (c+d x))^2 \left(-4 \tan (c+d x)+\frac{2}{3} \sec ^2(c+d x)-\frac{\sin (c+d x)}{2 (\sin (c+d x)+\cos (c+d x))}-\frac{2}{3}\right)}{d (a \cot (c+d x)+a)^2 (e \cot (c+d x))^{5/2}}+\frac{\cot ^{\frac{5}{2}}(c+d x) \csc ^2(c+d x) (\sin (c+d x)+\cos (c+d x))^2 \left(-\frac{16 (\cot (c+d x)+1) \csc ^3(c+d x) \sec (c+d x) \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)}{(\tan (c+d x)+1) \left(\cot ^2(c+d x)+1\right)^2}+\frac{\sin (2 (c+d x)) (\cot (c+d x)+1) \csc ^2(c+d x) \sec ^2(c+d x) \left(2 \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)-\sqrt{2} \left(\tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)\right)\right)}{2 (\tan (c+d x)+1) \left(\cot ^2(c+d x)+1\right)}+\frac{\cos (2 (c+d x)) \csc ^3(c+d x) \sec (c+d x) \left(\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right)\right)}{\sqrt{2} (\tan (c+d x)+1) (\cot (c+d x)-1) \left(\cot ^2(c+d x)+1\right)}\right)}{4 d (a \cot (c+d x)+a)^2 (e \cot (c+d x))^{5/2}}","-\frac{\log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{4 \sqrt{2} a^2 d e^{5/2}}+\frac{\log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{4 \sqrt{2} a^2 d e^{5/2}}-\frac{7 \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{2 a^2 d e^{5/2}}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{2 \sqrt{2} a^2 d e^{5/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{2 \sqrt{2} a^2 d e^{5/2}}-\frac{9}{2 a^2 d e^2 \sqrt{e \cot (c+d x)}}-\frac{1}{2 d e \left(a^2 \cot (c+d x)+a^2\right) (e \cot (c+d x))^{3/2}}+\frac{7}{6 a^2 d e (e \cot (c+d x))^{3/2}}",1,"(Cot[c + d*x]^3*Csc[c + d*x]^2*(Cos[c + d*x] + Sin[c + d*x])^2*(-2/3 + (2*Sec[c + d*x]^2)/3 - Sin[c + d*x]/(2*(Cos[c + d*x] + Sin[c + d*x])) - 4*Tan[c + d*x]))/(d*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^2) + (Cot[c + d*x]^(5/2)*Csc[c + d*x]^2*(Cos[c + d*x] + Sin[c + d*x])^2*((-16*ArcTan[Sqrt[Cot[c + d*x]]]*(1 + Cot[c + d*x])*Csc[c + d*x]^3*Sec[c + d*x])/((1 + Cot[c + d*x]^2)^2*(1 + Tan[c + d*x])) + (Cos[2*(c + d*x)]*Csc[c + d*x]^3*(-Log[-1 + Sqrt[2]*Sqrt[Cot[c + d*x]] - Cot[c + d*x]] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])*Sec[c + d*x])/(Sqrt[2]*(-1 + Cot[c + d*x])*(1 + Cot[c + d*x]^2)*(1 + Tan[c + d*x])) + ((-(Sqrt[2]*(-ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])) + 2*ArcTan[Sqrt[Cot[c + d*x]]])*(1 + Cot[c + d*x])*Csc[c + d*x]^2*Sec[c + d*x]^2*Sin[2*(c + d*x)])/(2*(1 + Cot[c + d*x]^2)*(1 + Tan[c + d*x]))))/(4*d*(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^2)","A",1
35,1,192,164,2.1343201,"\int \frac{(e \cot (c+d x))^{5/2}}{(a+a \cot (c+d x))^3} \, dx","Integrate[(e*Cot[c + d*x])^(5/2)/(a + a*Cot[c + d*x])^3,x]","\frac{\csc (c+d x) (e \cot (c+d x))^{5/2} (\sin (c+d x)+\cos (c+d x))^3 \left(\frac{\sec ^4(c+d x) (-5 \sin (2 (c+d x))+3 \cos (2 (c+d x))-3)}{(\tan (c+d x)+1)^2}-\frac{2 \csc (c+d x) \sec (c+d x) \left(\sqrt{2} \left(\log \left(-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)+\tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)\right)}{\cot ^{\frac{3}{2}}(c+d x)}\right)}{16 a^3 d (\cot (c+d x)+1)^3}","-\frac{e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{8 a^3 d}+\frac{e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{2 \sqrt{2} a^3 d}-\frac{5 e^2 \sqrt{e \cot (c+d x)}}{8 a^3 d (\cot (c+d x)+1)}+\frac{e^2 \sqrt{e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}",1,"((e*Cot[c + d*x])^(5/2)*Csc[c + d*x]*(Cos[c + d*x] + Sin[c + d*x])^3*((-2*Csc[c + d*x]*(ArcTan[Sqrt[Cot[c + d*x]]] + Sqrt[2]*(Log[-1 + Sqrt[2]*Sqrt[Cot[c + d*x]] - Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))*Sec[c + d*x])/Cot[c + d*x]^(3/2) + (Sec[c + d*x]^4*(-3 + 3*Cos[2*(c + d*x)] - 5*Sin[2*(c + d*x)]))/(1 + Tan[c + d*x])^2))/(16*a^3*d*(1 + Cot[c + d*x])^3)","A",1
36,1,131,164,2.0230099,"\int \frac{(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^3} \, dx","Integrate[(e*Cot[c + d*x])^(3/2)/(a + a*Cot[c + d*x])^3,x]","\frac{e \sqrt{e \cot (c+d x)} \left(\frac{2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+5 \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)}{\sqrt{\cot (c+d x)}}+\frac{\tan (c+d x)-\sec ^2(c+d x)+1}{(\tan (c+d x)+1)^2}\right)}{8 a^3 d}","\frac{5 e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{8 a^3 d}+\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{2 \sqrt{2} a^3 d}+\frac{e \sqrt{e \cot (c+d x)}}{8 d \left(a^3 \cot (c+d x)+a^3\right)}-\frac{e \sqrt{e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}",1,"(e*Sqrt[e*Cot[c + d*x]]*((2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 5*ArcTan[Sqrt[Cot[c + d*x]]])/Sqrt[Cot[c + d*x]] + (1 - Sec[c + d*x]^2 + Tan[c + d*x])/(1 + Tan[c + d*x])^2))/(8*a^3*d)","A",1
37,1,181,161,0.8183758,"\int \frac{\sqrt{e \cot (c+d x)}}{(a+a \cot (c+d x))^3} \, dx","Integrate[Sqrt[e*Cot[c + d*x]]/(a + a*Cot[c + d*x])^3,x]","-\frac{\sqrt{e \cot (c+d x)} \left(\sqrt{\cot (c+d x)} (-3 \sin (2 (c+d x))+5 \cos (2 (c+d x))-5)+2 (\sin (2 (c+d x))+1) \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)-2 \sqrt{2} (\sin (c+d x)+\cos (c+d x))^2 \left(\log \left(-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{16 a^3 d \sqrt{\cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^2}","\frac{3 \sqrt{e \cot (c+d x)}}{8 d \left(a^3 \cot (c+d x)+a^3\right)}-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{8 a^3 d}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{2 \sqrt{2} a^3 d}+\frac{\sqrt{e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}",1,"-1/16*(Sqrt[e*Cot[c + d*x]]*(-2*Sqrt[2]*(Log[-1 + Sqrt[2]*Sqrt[Cot[c + d*x]] - Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])*(Cos[c + d*x] + Sin[c + d*x])^2 + Sqrt[Cot[c + d*x]]*(-5 + 5*Cos[2*(c + d*x)] - 3*Sin[2*(c + d*x)]) + 2*ArcTan[Sqrt[Cot[c + d*x]]]*(1 + Sin[2*(c + d*x)])))/(a^3*d*Sqrt[Cot[c + d*x]]*(Cos[c + d*x] + Sin[c + d*x])^2)","A",1
38,1,217,165,1.270861,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^3} \, dx","Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^3),x]","\frac{\sqrt{\cot (c+d x)} \left(-9 \sqrt{\cot (c+d x)}+9 \cos (2 (c+d x)) \sqrt{\cot (c+d x)}-7 \sin (2 (c+d x)) \sqrt{\cot (c+d x)}-22 \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)-22 \sin (2 (c+d x)) \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)-4 \sqrt{2} (\sin (c+d x)+\cos (c+d x))^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)+4 \sqrt{2} (\sin (c+d x)+\cos (c+d x))^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{16 a^3 d \sqrt{e \cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^2}","-\frac{7 \sqrt{e \cot (c+d x)}}{8 a^3 d e (\cot (c+d x)+1)}-\frac{11 \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{8 a^3 d \sqrt{e}}-\frac{\tan ^{-1}\left(\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{2 \sqrt{2} a^3 d \sqrt{e}}-\frac{\sqrt{e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}",1,"(Sqrt[Cot[c + d*x]]*(-22*ArcTan[Sqrt[Cot[c + d*x]]] - 9*Sqrt[Cot[c + d*x]] + 9*Cos[2*(c + d*x)]*Sqrt[Cot[c + d*x]] - 4*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*(Cos[c + d*x] + Sin[c + d*x])^2 + 4*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*(Cos[c + d*x] + Sin[c + d*x])^2 - 22*ArcTan[Sqrt[Cot[c + d*x]]]*Sin[2*(c + d*x)] - 7*Sqrt[Cot[c + d*x]]*Sin[2*(c + d*x)]))/(16*a^3*d*Sqrt[e*Cot[c + d*x]]*(Cos[c + d*x] + Sin[c + d*x])^2)","A",1
39,1,156,189,1.2106726,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3} \, dx","Integrate[1/((e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])^3),x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \left(-2 \sqrt{2} \left(\log \left(-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)+62 \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)+\frac{45 \sin (2 (c+d x))+11 \cos (2 (c+d x))+43}{\sqrt{\cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^2}\right)}{16 a^3 d (e \cot (c+d x))^{3/2}}","\frac{31 \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{8 a^3 d e^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{2 \sqrt{2} a^3 d e^{3/2}}+\frac{27}{8 a^3 d e \sqrt{e \cot (c+d x)}}-\frac{9}{8 a^3 d e (\cot (c+d x)+1) \sqrt{e \cot (c+d x)}}-\frac{1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt{e \cot (c+d x)}}",1,"(Cot[c + d*x]^(3/2)*(62*ArcTan[Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*(Log[-1 + Sqrt[2]*Sqrt[Cot[c + d*x]] - Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]) + (43 + 11*Cos[2*(c + d*x)] + 45*Sin[2*(c + d*x)])/(Sqrt[Cot[c + d*x]]*(Cos[c + d*x] + Sin[c + d*x])^2)))/(16*a^3*d*(e*Cot[c + d*x])^(3/2))","A",1
40,1,167,215,3.1897727,"\int \frac{1}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3} \, dx","Integrate[1/((e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3),x]","\frac{\cot ^{\frac{5}{2}}(c+d x) \left(4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-118 \tan ^{-1}\left(\sqrt{\cot (c+d x)}\right)-\frac{\sqrt{\cot (c+d x)} \sec ^2(c+d x) (678 \cos (2 (c+d x))+679 \cot (c+d x)+77 \cos (3 (c+d x)) \csc (c+d x)+614)}{6 (\cot (c+d x)+1)^2}\right)}{16 a^3 d (e \cot (c+d x))^{5/2}}","-\frac{59 \tan ^{-1}\left(\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{8 a^3 d e^{5/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right)}{2 \sqrt{2} a^3 d e^{5/2}}-\frac{63}{8 a^3 d e^2 \sqrt{e \cot (c+d x)}}-\frac{11}{8 a^3 d e (\cot (c+d x)+1) (e \cot (c+d x))^{3/2}}+\frac{55}{24 a^3 d e (e \cot (c+d x))^{3/2}}-\frac{1}{4 a d e (a \cot (c+d x)+a)^2 (e \cot (c+d x))^{3/2}}",1,"(Cot[c + d*x]^(5/2)*(4*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 4*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] - 118*ArcTan[Sqrt[Cot[c + d*x]]] - (Sqrt[Cot[c + d*x]]*(614 + 678*Cos[2*(c + d*x)] + 679*Cot[c + d*x] + 77*Cos[3*(c + d*x)]*Csc[c + d*x])*Sec[c + d*x]^2)/(6*(1 + Cot[c + d*x])^2)))/(16*a^3*d*(e*Cot[c + d*x])^(5/2))","A",1
41,1,69,223,0.1657326,"\int \cot ^2(x) \sqrt{1+\cot (x)} \, dx","Integrate[Cot[x]^2*Sqrt[1 + Cot[x]],x]","-\frac{2}{3} (\cot (x)+1)^{3/2}-i \sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1-i}}\right)+i \sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1+i}}\right)","-\frac{2}{3} (\cot (x)+1)^{3/2}+\frac{\log \left(\cot (x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\cot (x)+1}+\sqrt{2}+1\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)}}-\frac{\log \left(\cot (x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\cot (x)+1}+\sqrt{2}+1\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)}}-\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\cot (x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)+\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 \sqrt{\cot (x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)",1,"(-I)*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]] + I*Sqrt[1 + I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]] - (2*(1 + Cot[x])^(3/2))/3","C",1
42,1,61,135,0.09054,"\int \cot (x) \sqrt{1+\cot (x)} \, dx","Integrate[Cot[x]*Sqrt[1 + Cot[x]],x]","-2 \sqrt{\cot (x)+1}+\sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1-i}}\right)+\sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1+i}}\right)","-2 \sqrt{\cot (x)+1}+\sqrt{\frac{1}{2} \left(\sqrt{2}-1\right)} \tan ^{-1}\left(\frac{\left(2-\sqrt{2}\right) \cot (x)-3 \sqrt{2}+4}{2 \sqrt{5 \sqrt{2}-7} \sqrt{\cot (x)+1}}\right)+\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\left(2+\sqrt{2}\right) \cot (x)+3 \sqrt{2}+4}{2 \sqrt{7+5 \sqrt{2}} \sqrt{\cot (x)+1}}\right)",1,"Sqrt[1 - I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]] + Sqrt[1 + I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]] - 2*Sqrt[1 + Cot[x]]","C",1
43,1,96,139,0.3175296,"\int \cot ^2(x) (1+\cot (x))^{3/2} \, dx","Integrate[Cot[x]^2*(1 + Cot[x])^(3/2),x]","\frac{\sin (x) \left(-\frac{2}{5} \sin (x) (\cot (x)+1)^{5/2} \left(2 \cot (x)+\csc ^2(x)-5\right)-2 \sin (x) (\cot (x)+1)^2 \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1-i}}\right)}{\sqrt{1-i}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1+i}}\right)}{\sqrt{1+i}}\right)\right)}{(\sin (x)+\cos (x))^2}","-\frac{2}{5} (\cot (x)+1)^{5/2}+2 \sqrt{\cot (x)+1}-\sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\left(1-\sqrt{2}\right) \cot (x)-2 \sqrt{2}+3}{\sqrt{2 \left(5 \sqrt{2}-7\right)} \sqrt{\cot (x)+1}}\right)-\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \cot (x)+2 \sqrt{2}+3}{\sqrt{2 \left(7+5 \sqrt{2}\right)} \sqrt{\cot (x)+1}}\right)",1,"(Sin[x]*(-2*(ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]]/Sqrt[1 - I] + ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]]/Sqrt[1 + I])*(1 + Cot[x])^2*Sin[x] - (2*(1 + Cot[x])^(5/2)*(-5 + 2*Cot[x] + Csc[x]^2)*Sin[x])/5))/(Cos[x] + Sin[x])^2","C",1
44,1,98,221,0.2597546,"\int \cot (x) (1+\cot (x))^{3/2} \, dx","Integrate[Cot[x]*(1 + Cot[x])^(3/2),x]","\frac{\sin (x) \left(-\frac{2}{3} (\cot (x)+1)^{3/2} (\cot (x)+4) (\sin (x)+\cos (x))+(1+i) \sin (x) (\cot (x)+1)^2 \left(\sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1+i}}\right)-i \sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1-i}}\right)\right)\right)}{(\sin (x)+\cos (x))^2}","-\frac{2}{3} (\cot (x)+1)^{3/2}-2 \sqrt{\cot (x)+1}-\frac{\log \left(\cot (x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\cot (x)+1}+\sqrt{2}+1\right)}{2 \sqrt{1+\sqrt{2}}}+\frac{\log \left(\cot (x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\cot (x)+1}+\sqrt{2}+1\right)}{2 \sqrt{1+\sqrt{2}}}-\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\cot (x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)+\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{\cot (x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)",1,"(Sin[x]*((1 + I)*((-I)*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]] + Sqrt[1 + I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]])*(1 + Cot[x])^2*Sin[x] - (2*(1 + Cot[x])^(3/2)*(4 + Cot[x])*(Cos[x] + Sin[x]))/3))/(Cos[x] + Sin[x])^2","C",1
45,1,67,214,0.1677236,"\int \frac{\cot ^2(x)}{\sqrt{1+\cot (x)}} \, dx","Integrate[Cot[x]^2/Sqrt[1 + Cot[x]],x]","-2 \sqrt{\cot (x)+1}+\frac{1}{2} (1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1-i}}\right)+\frac{1}{2} (1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1+i}}\right)","-2 \sqrt{\cot (x)+1}-\frac{\log \left(\cot (x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\cot (x)+1}+\sqrt{2}+1\right)}{4 \sqrt{1+\sqrt{2}}}+\frac{\log \left(\cot (x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\cot (x)+1}+\sqrt{2}+1\right)}{4 \sqrt{1+\sqrt{2}}}-\frac{1}{2} \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\cot (x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)+\frac{1}{2} \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{\cot (x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)",1,"((1 - I)^(3/2)*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]])/2 + ((1 + I)^(3/2)*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]])/2 - 2*Sqrt[1 + Cot[x]]","C",1
46,1,51,121,0.0756803,"\int \frac{\cot (x)}{\sqrt{1+\cot (x)}} \, dx","Integrate[Cot[x]/Sqrt[1 + Cot[x]],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1-i}}\right)}{\sqrt{1-i}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1+i}}\right)}{\sqrt{1+i}}","\frac{1}{2} \sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\left(1-\sqrt{2}\right) \cot (x)-2 \sqrt{2}+3}{\sqrt{2 \left(5 \sqrt{2}-7\right)} \sqrt{\cot (x)+1}}\right)+\frac{1}{2} \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \cot (x)+2 \sqrt{2}+3}{\sqrt{2 \left(7+5 \sqrt{2}\right)} \sqrt{\cot (x)+1}}\right)",1,"ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]]/Sqrt[1 - I] + ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]]/Sqrt[1 + I]","C",1
47,1,65,139,0.1317862,"\int \frac{\cot ^2(x)}{(1+\cot (x))^{3/2}} \, dx","Integrate[Cot[x]^2/(1 + Cot[x])^(3/2),x]","\frac{1}{\sqrt{\cot (x)+1}}+\frac{1}{2} \sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1-i}}\right)+\frac{1}{2} \sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1+i}}\right)","\frac{1}{\sqrt{\cot (x)+1}}+\frac{1}{2} \sqrt{\frac{1}{2} \left(\sqrt{2}-1\right)} \tan ^{-1}\left(\frac{\left(2-\sqrt{2}\right) \cot (x)-3 \sqrt{2}+4}{2 \sqrt{5 \sqrt{2}-7} \sqrt{\cot (x)+1}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\left(2+\sqrt{2}\right) \cot (x)+3 \sqrt{2}+4}{2 \sqrt{7+5 \sqrt{2}} \sqrt{\cot (x)+1}}\right)",1,"(Sqrt[1 - I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]])/2 + (Sqrt[1 + I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]])/2 + 1/Sqrt[1 + Cot[x]]","C",1
48,1,71,226,0.1431619,"\int \frac{\cot (x)}{(1+\cot (x))^{3/2}} \, dx","Integrate[Cot[x]/(1 + Cot[x])^(3/2),x]","-\frac{1}{\sqrt{\cot (x)+1}}+\frac{1}{2} i \sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1-i}}\right)-\frac{1}{2} i \sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1+i}}\right)","-\frac{1}{\sqrt{\cot (x)+1}}-\frac{\log \left(\cot (x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\cot (x)+1}+\sqrt{2}+1\right)}{4 \sqrt{2 \left(1+\sqrt{2}\right)}}+\frac{\log \left(\cot (x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\cot (x)+1}+\sqrt{2}+1\right)}{4 \sqrt{2 \left(1+\sqrt{2}\right)}}+\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\cot (x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 \sqrt{\cot (x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)",1,"(I/2)*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]] - (I/2)*Sqrt[1 + I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]] - 1/Sqrt[1 + Cot[x]]","C",1
49,1,75,143,0.4056388,"\int \frac{\cot ^2(x)}{(1+\cot (x))^{5/2}} \, dx","Integrate[Cot[x]^2/(1 + Cot[x])^(5/2),x]","\frac{-3 \cot (x)-2}{3 (\cot (x)+1)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1-i}}\right)}{2 \sqrt{1-i}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1+i}}\right)}{2 \sqrt{1+i}}","-\frac{1}{\sqrt{\cot (x)+1}}+\frac{1}{3 (\cot (x)+1)^{3/2}}+\frac{1}{4} \sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\left(1-\sqrt{2}\right) \cot (x)-2 \sqrt{2}+3}{\sqrt{2 \left(5 \sqrt{2}-7\right)} \sqrt{\cot (x)+1}}\right)+\frac{1}{4} \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \cot (x)+2 \sqrt{2}+3}{\sqrt{2 \left(7+5 \sqrt{2}\right)} \sqrt{\cot (x)+1}}\right)",1,"ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]]/(2*Sqrt[1 - I]) + ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]]/(2*Sqrt[1 + I]) + (-2 - 3*Cot[x])/(3*(1 + Cot[x])^(3/2))","C",1
50,1,69,216,0.2416366,"\int \frac{\cot (x)}{(1+\cot (x))^{5/2}} \, dx","Integrate[Cot[x]/(1 + Cot[x])^(5/2),x]","-\frac{1}{3 (\cot (x)+1)^{3/2}}-\frac{1}{4} (1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1-i}}\right)-\frac{1}{4} (1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\cot (x)+1}}{\sqrt{1+i}}\right)","-\frac{1}{3 (\cot (x)+1)^{3/2}}+\frac{\log \left(\cot (x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\cot (x)+1}+\sqrt{2}+1\right)}{8 \sqrt{1+\sqrt{2}}}-\frac{\log \left(\cot (x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\cot (x)+1}+\sqrt{2}+1\right)}{8 \sqrt{1+\sqrt{2}}}+\frac{1}{4} \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\cot (x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)-\frac{1}{4} \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{\cot (x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)",1,"-1/4*((1 - I)^(3/2)*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]]) - ((1 + I)^(3/2)*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]])/4 - 1/(3*(1 + Cot[x])^(3/2))","C",1
51,1,68,247,0.1313564,"\int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x)) \, dx","Integrate[(e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x]),x]","-\frac{2 e \sqrt{e \cot (c+d x)} \left(3 a \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)+b \cot (c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)\right)}{3 d}","-\frac{e^{3/2} (a-b) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{e^{3/2} (a-b) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{e^{3/2} (a+b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}+\frac{e^{3/2} (a+b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}-\frac{2 a e \sqrt{e \cot (c+d x)}}{d}-\frac{2 b (e \cot (c+d x))^{3/2}}{3 d}",1,"(-2*e*Sqrt[e*Cot[c + d*x]]*(b*Cot[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2] + 3*a*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2]))/(3*d)","C",1
52,1,155,226,0.2815511,"\int \sqrt{e \cot (c+d x)} (a+b \cot (c+d x)) \, dx","Integrate[Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x]),x]","-\frac{\sqrt{e \cot (c+d x)} \left(\sqrt{2} a \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)+8 b \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)\right)}{4 d}","-\frac{\sqrt{e} (a+b) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{\sqrt{e} (a+b) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{\sqrt{e} (a-b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{\sqrt{e} (a-b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}-\frac{2 b \sqrt{e \cot (c+d x)}}{d}",1,"-1/4*(Sqrt[e*Cot[c + d*x]]*(8*b*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2] + Sqrt[2]*a*(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]]))/d","C",1
53,1,166,208,0.2155224,"\int \frac{a+b \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx","Integrate[(a + b*Cot[c + d*x])/Sqrt[e*Cot[c + d*x]],x]","\frac{8 a \tan ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)+3 \sqrt{2} b \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)}{12 d \sqrt{\tan (c+d x)} \sqrt{e \cot (c+d x)}}","\frac{(a-b) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}-\frac{(a-b) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{(a+b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \sqrt{e}}-\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \sqrt{e}}",1,"(3*Sqrt[2]*b*(-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]]) + 8*a*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2))/(12*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])","C",1
54,1,196,229,0.3587936,"\int \frac{a+b \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx","Integrate[(a + b*Cot[c + d*x])/(e*Cot[c + d*x])^(3/2),x]","\frac{3 a \left(2 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\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)+8 b \tan ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)}{12 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}","\frac{(a+b) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}-\frac{(a+b) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}-\frac{(a-b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{3/2}}+\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{3/2}}+\frac{2 a}{d e \sqrt{e \cot (c+d x)}}",1,"(3*a*(2*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 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]]) + 8*b*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2))/(12*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))","C",1
55,1,196,252,0.7576746,"\int \frac{a+b \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx","Integrate[(a + b*Cot[c + d*x])/(e*Cot[c + d*x])^(5/2),x]","\frac{3 b \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)-8 a \tan ^{\frac{3}{2}}(c+d x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)-1\right)}{12 d \tan ^{\frac{5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}","-\frac{(a-b) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}+\frac{(a-b) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{(a+b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{5/2}}+\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{5/2}}+\frac{2 a}{3 d e (e \cot (c+d x))^{3/2}}+\frac{2 b}{d e^2 \sqrt{e \cot (c+d x)}}",1,"(3*b*(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]]) - 8*a*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2])*Tan[c + d*x]^(3/2))/(12*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))","C",1
56,1,224,317,1.9820122,"\int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2 \, dx","Integrate[(e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x])^2,x]","-\frac{(e \cot (c+d x))^{3/2} \left(\frac{1}{4} \left(a^2-b^2\right) \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)-\frac{4}{3} a b \cot ^{\frac{3}{2}}(c+d x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-1\right)+\frac{2}{5} b^2 \cot ^{\frac{5}{2}}(c+d x)\right)}{d \cot ^{\frac{3}{2}}(c+d x)}","-\frac{e^{3/2} \left(a^2-2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{e^{3/2} \left(a^2-2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{e^{3/2} \left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}+\frac{e^{3/2} \left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}-\frac{2 e \left(a^2-b^2\right) \sqrt{e \cot (c+d x)}}{d}-\frac{4 a b (e \cot (c+d x))^{3/2}}{3 d}-\frac{2 b^2 (e \cot (c+d x))^{5/2}}{5 d e}",1,"-(((e*Cot[c + d*x])^(3/2)*((2*b^2*Cot[c + d*x]^(5/2))/5 - (4*a*b*Cot[c + d*x]^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/3 + ((a^2 - b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/4))/(d*Cot[c + d*x]^(3/2)))","C",1
57,1,220,288,0.5690006,"\int \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2 \, dx","Integrate[Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^2,x]","-\frac{\sqrt{e \cot (c+d x)} \left(4 \left(a^2-b^2\right) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)+b \left(24 a \sqrt{\cot (c+d x)}+3 \sqrt{2} a \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-3 \sqrt{2} a \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+6 \sqrt{2} a \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-6 \sqrt{2} a \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+4 b \cot ^{\frac{3}{2}}(c+d x)\right)\right)}{6 d \sqrt{\cot (c+d x)}}","-\frac{\sqrt{e} \left(a^2+2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{\sqrt{e} \left(a^2+2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{\sqrt{e} \left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{\sqrt{e} \left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}-\frac{4 a b \sqrt{e \cot (c+d x)}}{d}-\frac{2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}",1,"-1/6*(Sqrt[e*Cot[c + d*x]]*(4*(a^2 - b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] + b*(6*Sqrt[2]*a*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 6*Sqrt[2]*a*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 24*a*Sqrt[Cot[c + d*x]] + 4*b*Cot[c + d*x]^(3/2) + 3*Sqrt[2]*a*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 3*Sqrt[2]*a*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(d*Sqrt[Cot[c + d*x]])","C",1
58,1,192,267,0.8903304,"\int \frac{(a+b \cot (c+d x))^2}{\sqrt{e \cot (c+d x)}} \, dx","Integrate[(a + b*Cot[c + d*x])^2/Sqrt[e*Cot[c + d*x]],x]","-\frac{\sqrt{\cot (c+d x)} \left(-\frac{\left(a^2-b^2\right) \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{2 \sqrt{2}}+\frac{4}{3} a b \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)+2 b^2 \sqrt{\cot (c+d x)}\right)}{d \sqrt{e \cot (c+d x)}}","\frac{\left(a^2-2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \sqrt{e}}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \sqrt{e}}-\frac{2 b^2 \sqrt{e \cot (c+d x)}}{d e}",1,"-((Sqrt[Cot[c + d*x]]*(2*b^2*Sqrt[Cot[c + d*x]] + (4*a*b*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/3 - ((a^2 - b^2)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(2*Sqrt[2])))/(d*Sqrt[e*Cot[c + d*x]]))","C",1
59,1,218,267,0.3298896,"\int \frac{(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx","Integrate[(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(3/2),x]","-\frac{\cot ^{\frac{3}{2}}(c+d x) \left(-\frac{2 \left(a^2-b^2\right) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)}{\sqrt{\cot (c+d x)}}+4 a b \left(\frac{1}{2} \left(\frac{\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2}}-\frac{\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2}}\right)+\frac{1}{2} \left(\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2}}\right)\right)-\frac{2 b^2}{\sqrt{\cot (c+d x)}}\right)}{d (e \cot (c+d x))^{3/2}}","\frac{\left(a^2+2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{3/2}}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{3/2}}+\frac{2 a^2}{d e \sqrt{e \cot (c+d x)}}",1,"-((Cot[c + d*x]^(3/2)*((-2*b^2)/Sqrt[Cot[c + d*x]] - (2*(a^2 - b^2)*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2])/Sqrt[Cot[c + d*x]] + 4*a*b*((-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2])/2 + (-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2]))/2)))/(d*(e*Cot[c + d*x])^(3/2)))","C",1
60,1,82,291,0.3007501,"\int \frac{(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx","Integrate[(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(5/2),x]","\frac{2 \left(\left(a^2-b^2\right) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)+b \left(6 a \cot (c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)+b\right)\right)}{3 d e (e \cot (c+d x))^{3/2}}","-\frac{\left(a^2-2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}+\frac{\left(a^2-2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{5/2}}+\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{5/2}}+\frac{2 a^2}{3 d e (e \cot (c+d x))^{3/2}}+\frac{4 a b}{d e^2 \sqrt{e \cot (c+d x)}}",1,"(2*((a^2 - b^2)*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2] + b*(b + 6*a*Cot[c + d*x]*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2])))/(3*d*e*(e*Cot[c + d*x])^(3/2))","C",1
61,1,85,322,0.3582767,"\int \frac{(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx","Integrate[(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(7/2),x]","\frac{2 \left(3 \left(a^2-b^2\right) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(c+d x)\right)+b \left(10 a \cot (c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)+3 b\right)\right)}{15 d e (e \cot (c+d x))^{5/2}}","-\frac{\left(a^2+2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{7/2}}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{7/2}}-\frac{2 \left(a^2-b^2\right)}{d e^3 \sqrt{e \cot (c+d x)}}+\frac{2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac{4 a b}{3 d e^2 (e \cot (c+d x))^{3/2}}",1,"(2*(3*(a^2 - b^2)*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2] + b*(3*b + 10*a*Cot[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2])))/(15*d*e*(e*Cot[c + d*x])^(5/2))","C",1
62,1,251,372,3.0613453,"\int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3 \, dx","Integrate[(e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x])^3,x]","-\frac{(e \cot (c+d x))^{3/2} \left(\frac{2}{3} b \left(b^2-3 a^2\right) \cot ^{\frac{3}{2}}(c+d x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-1\right)+\frac{1}{4} a \left(a^2-3 b^2\right) \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)+\frac{6}{5} a b^2 \cot ^{\frac{5}{2}}(c+d x)+\frac{2}{7} b^3 \cot ^{\frac{7}{2}}(c+d x)\right)}{d \cot ^{\frac{3}{2}}(c+d x)}","-\frac{e^{3/2} (a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{e^{3/2} (a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}-\frac{e^{3/2} (a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}+\frac{e^{3/2} (a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}-\frac{2 b \left(3 a^2-b^2\right) (e \cot (c+d x))^{3/2}}{3 d}-\frac{2 a e \left(a^2-3 b^2\right) \sqrt{e \cot (c+d x)}}{d}-\frac{2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}-\frac{32 a b^2 (e \cot (c+d x))^{5/2}}{35 d e}",1,"-(((e*Cot[c + d*x])^(3/2)*((6*a*b^2*Cot[c + d*x]^(5/2))/5 + (2*b^3*Cot[c + d*x]^(7/2))/7 + (2*b*(-3*a^2 + b^2)*Cot[c + d*x]^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/3 + (a*(a^2 - 3*b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/4))/(d*Cot[c + d*x]^(3/2)))","C",1
63,1,247,342,2.5934368,"\int \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^3 \, dx","Integrate[Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^3,x]","-\frac{\sqrt{e \cot (c+d x)} \left(\frac{2}{3} a \left(a^2-3 b^2\right) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-\frac{1}{4} b \left(b^2-3 a^2\right) \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)+2 a b^2 \cot ^{\frac{3}{2}}(c+d x)+\frac{2}{5} b^3 \cot ^{\frac{5}{2}}(c+d x)\right)}{d \sqrt{\cot (c+d x)}}","-\frac{2 b \left(3 a^2-b^2\right) \sqrt{e \cot (c+d x)}}{d}-\frac{\sqrt{e} (a-b) \left(a^2+4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{\sqrt{e} (a-b) \left(a^2+4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d}+\frac{\sqrt{e} (a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d}-\frac{\sqrt{e} (a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d}-\frac{8 a b^2 (e \cot (c+d x))^{3/2}}{5 d e}-\frac{2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}",1,"-((Sqrt[e*Cot[c + d*x]]*(2*a*b^2*Cot[c + d*x]^(3/2) + (2*b^3*Cot[c + d*x]^(5/2))/5 + (2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/3 - (b*(-3*a^2 + b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/4))/(d*Sqrt[Cot[c + d*x]]))","C",1
64,1,216,313,1.0341044,"\int \frac{(a+b \cot (c+d x))^3}{\sqrt{e \cot (c+d x)}} \, dx","Integrate[(a + b*Cot[c + d*x])^3/Sqrt[e*Cot[c + d*x]],x]","-\frac{2 \sqrt{\cot (c+d x)} \left(-b \left(b^2-3 a^2\right) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-\frac{3 a \left(a^2-3 b^2\right) \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{4 \sqrt{2}}+9 a b^2 \sqrt{\cot (c+d x)}+b^3 \cot ^{\frac{3}{2}}(c+d x)\right)}{3 d \sqrt{e \cot (c+d x)}}","\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \sqrt{e}}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \sqrt{e}}-\frac{2 b^2 \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}-\frac{16 a b^2 \sqrt{e \cot (c+d x)}}{3 d e}",1,"(-2*Sqrt[Cot[c + d*x]]*(9*a*b^2*Sqrt[Cot[c + d*x]] + b^3*Cot[c + d*x]^(3/2) - b*(-3*a^2 + b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] - (3*a*(a^2 - 3*b^2)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(4*Sqrt[2])))/(3*d*Sqrt[e*Cot[c + d*x]])","C",1
65,1,193,313,3.3901334,"\int \frac{(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx","Integrate[(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(3/2),x]","\frac{2 \left(a \left(a^2-3 b^2\right) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)-\frac{b \left(b^2-3 a^2\right) \sqrt{\cot (c+d x)} \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{4 \sqrt{2}}+3 a b^2+b^3 (-\cot (c+d x))\right)}{d e \sqrt{e \cot (c+d x)}}","\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2}}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{3/2}}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{3/2}}-\frac{2 b \left(a^2+b^2\right) \sqrt{e \cot (c+d x)}}{d e^2}+\frac{2 a^2 (a+b \cot (c+d x))}{d e \sqrt{e \cot (c+d x)}}",1,"(2*(3*a*b^2 - b^3*Cot[c + d*x] + a*(a^2 - 3*b^2)*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] - (b*(-3*a^2 + b^2)*Sqrt[Cot[c + d*x]]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(4*Sqrt[2])))/(d*e*Sqrt[e*Cot[c + d*x]])","C",1
66,1,104,313,0.3759963,"\int \frac{(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{5/2}} \, dx","Integrate[(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(5/2),x]","\frac{-6 b \left(b^2-3 a^2\right) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)+2 a \left(a^2-3 b^2\right) \tan (c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)+6 b^2 (a \tan (c+d x)+b)}{3 d e^2 \sqrt{e \cot (c+d x)}}","-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2}}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{5/2}}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{5/2}}+\frac{16 a^2 b}{3 d e^2 \sqrt{e \cot (c+d x)}}+\frac{2 a^2 (a+b \cot (c+d x))}{3 d e (e \cot (c+d x))^{3/2}}",1,"(-6*b*(-3*a^2 + b^2)*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + 2*a*(a^2 - 3*b^2)*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2]*Tan[c + d*x] + 6*b^2*(b + a*Tan[c + d*x]))/(3*d*e^2*Sqrt[e*Cot[c + d*x]])","C",1
67,1,108,343,0.6118811,"\int \frac{(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx","Integrate[(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(7/2),x]","\frac{2 \left(3 a \left(a^2-3 b^2\right) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(c+d x)\right)+b \left(5 \left(3 a^2-b^2\right) \cot (c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)+b (9 a+5 b \cot (c+d x))\right)\right)}{15 d e (e \cot (c+d x))^{5/2}}","-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{7/2}}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{7/2}}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{7/2}}-\frac{2 a \left(a^2-3 b^2\right)}{d e^3 \sqrt{e \cot (c+d x)}}+\frac{8 a^2 b}{5 d e^2 (e \cot (c+d x))^{3/2}}+\frac{2 a^2 (a+b \cot (c+d x))}{5 d e (e \cot (c+d x))^{5/2}}",1,"(2*(3*a*(a^2 - 3*b^2)*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2] + b*(b*(9*a + 5*b*Cot[c + d*x]) + 5*(3*a^2 - b^2)*Cot[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2])))/(15*d*e*(e*Cot[c + d*x])^(5/2))","C",1
68,1,116,377,0.677014,"\int \frac{(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx","Integrate[(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(9/2),x]","\frac{2 \tan ^4(c+d x) \sqrt{e \cot (c+d x)} \left(5 a \left(a^2-3 b^2\right) \, _2F_1\left(-\frac{7}{4},1;-\frac{3}{4};-\cot ^2(c+d x)\right)+b \left(7 \left(3 a^2-b^2\right) \cot (c+d x) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(c+d x)\right)+b (15 a+7 b \cot (c+d x))\right)\right)}{35 d e^5}","\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{9/2}}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{9/2}}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{9/2}}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{9/2}}-\frac{2 b \left(3 a^2-b^2\right)}{d e^4 \sqrt{e \cot (c+d x)}}-\frac{2 a \left(a^2-3 b^2\right)}{3 d e^3 (e \cot (c+d x))^{3/2}}+\frac{32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac{2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}",1,"(2*Sqrt[e*Cot[c + d*x]]*(5*a*(a^2 - 3*b^2)*Hypergeometric2F1[-7/4, 1, -3/4, -Cot[c + d*x]^2] + b*(b*(15*a + 7*b*Cot[c + d*x]) + 7*(3*a^2 - b^2)*Cot[c + d*x]*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2]))*Tan[c + d*x]^4)/(35*d*e^5)","C",1
69,1,286,325,0.874419,"\int \frac{(e \cot (c+d x))^{5/2}}{a+b \cot (c+d x)} \, dx","Integrate[(e*Cot[c + d*x])^(5/2)/(a + b*Cot[c + d*x]),x]","\frac{(e \cot (c+d x))^{5/2} \left(8 a b^{3/2} \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-3 \left(-8 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)+8 a^2 \sqrt{b} \sqrt{\cot (c+d x)}+8 b^{5/2} \sqrt{\cot (c+d x)}+\sqrt{2} b^{5/2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} b^{5/2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} b^{5/2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} b^{5/2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{12 b^{3/2} d \left(a^2+b^2\right) \cot ^{\frac{5}{2}}(c+d x)}","\frac{e^{5/2} (a-b) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{e^{5/2} (a-b) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{e^{5/2} (a+b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{e^{5/2} (a+b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{2 a^{5/2} e^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{b^{3/2} d \left(a^2+b^2\right)}-\frac{2 e^2 \sqrt{e \cot (c+d x)}}{b d}",1,"((e*Cot[c + d*x])^(5/2)*(8*a*b^(3/2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] - 3*(2*Sqrt[2]*b^(5/2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*b^(5/2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] - 8*a^(5/2)*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]] + 8*a^2*Sqrt[b]*Sqrt[Cot[c + d*x]] + 8*b^(5/2)*Sqrt[Cot[c + d*x]] + Sqrt[2]*b^(5/2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*b^(5/2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(12*b^(3/2)*(a^2 + b^2)*d*Cot[c + d*x]^(5/2))","C",1
70,1,249,302,0.567193,"\int \frac{(e \cot (c+d x))^{3/2}}{a+b \cot (c+d x)} \, dx","Integrate[(e*Cot[c + d*x])^(3/2)/(a + b*Cot[c + d*x]),x]","-\frac{(e \cot (c+d x))^{3/2} \left(3 a \left(8 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)+\sqrt{2} \sqrt{b} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \sqrt{b} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \sqrt{b} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \sqrt{b} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)+8 b^{3/2} \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)\right)}{12 \sqrt{b} d \left(a^2+b^2\right) \cot ^{\frac{3}{2}}(c+d x)}","-\frac{e^{3/2} (a+b) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{e^{3/2} (a+b) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{e^{3/2} (a-b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{e^{3/2} (a-b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{2 a^{3/2} e^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{\sqrt{b} d \left(a^2+b^2\right)}",1,"-1/12*((e*Cot[c + d*x])^(3/2)*(8*b^(3/2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] + 3*a*(2*Sqrt[2]*Sqrt[b]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*Sqrt[b]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]] + Sqrt[2]*Sqrt[b]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Sqrt[b]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(Sqrt[b]*(a^2 + b^2)*d*Cot[c + d*x]^(3/2))","C",1
71,1,226,302,0.2827985,"\int \frac{\sqrt{e \cot (c+d x)}}{a+b \cot (c+d x)} \, dx","Integrate[Sqrt[e*Cot[c + d*x]]/(a + b*Cot[c + d*x]),x]","\frac{\sqrt{e \cot (c+d x)} \left(24 \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)-8 a \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)+3 \sqrt{2} b \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-3 \sqrt{2} b \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+6 \sqrt{2} b \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-6 \sqrt{2} b \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{12 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)}}","-\frac{\sqrt{e} (a-b) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{\sqrt{e} (a-b) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{2 \sqrt{a} \sqrt{b} \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{d \left(a^2+b^2\right)}+\frac{\sqrt{e} (a+b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{\sqrt{e} (a+b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}",1,"(Sqrt[e*Cot[c + d*x]]*(6*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 6*Sqrt[2]*b*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 24*Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]] - 8*a*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] + 3*Sqrt[2]*b*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 3*Sqrt[2]*b*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(12*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]])","C",1
72,1,248,302,0.2544473,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))} \, dx","Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])),x]","-\frac{\sqrt{\cot (c+d x)} \left(-\frac{2 b \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)}{3 \left(a^2+b^2\right)}-\frac{a \left(2 \sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+4 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{8 \left(a^2+b^2\right)}+\frac{2 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \left(a^2+b^2\right)}\right)}{d \sqrt{e \cot (c+d x)}}","\frac{(a+b) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e} \left(a^2+b^2\right)}-\frac{(a+b) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e} \left(a^2+b^2\right)}+\frac{(a-b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \sqrt{e} \left(a^2+b^2\right)}-\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \sqrt{e} \left(a^2+b^2\right)}-\frac{2 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{\sqrt{a} d \sqrt{e} \left(a^2+b^2\right)}",1,"-((Sqrt[Cot[c + d*x]]*((2*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)) - (2*b*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/(3*(a^2 + b^2)) - (a*(4*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 2*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(8*(a^2 + b^2))))/(d*Sqrt[e*Cot[c + d*x]]))","C",1
73,1,198,325,0.4462779,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx","Integrate[1/((e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x])),x]","\frac{8 b^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\frac{b \cot (c+d x)}{a}\right)+a \left(8 a \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)+\sqrt{2} b \sqrt{\cot (c+d x)} \left(-\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)+2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{4 a d e \left(a^2+b^2\right) \sqrt{e \cot (c+d x)}}","\frac{(a-b) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2} \left(a^2+b^2\right)}-\frac{(a-b) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2} \left(a^2+b^2\right)}-\frac{(a+b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{3/2} \left(a^2+b^2\right)}+\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{3/2} \left(a^2+b^2\right)}+\frac{2 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{a^{3/2} d e^{3/2} \left(a^2+b^2\right)}+\frac{2}{a d e \sqrt{e \cot (c+d x)}}",1,"(8*b^2*Hypergeometric2F1[-1/2, 1, 1/2, -((b*Cot[c + d*x])/a)] + a*(8*a*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Sqrt[2]*b*Sqrt[Cot[c + d*x]]*(-2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] - Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(4*a*(a^2 + b^2)*d*e*Sqrt[e*Cot[c + d*x]])","C",1
74,1,109,351,0.2828306,"\int \frac{1}{(e \cot (c+d x))^{5/2} (a+b \cot (c+d x))} \, dx","Integrate[1/((e*Cot[c + d*x])^(5/2)*(a + b*Cot[c + d*x])),x]","\frac{2 \left(b^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\frac{b \cot (c+d x)}{a}\right)+a \left(a \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)-3 b \cot (c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)\right)\right)}{3 a d e \left(a^2+b^2\right) (e \cot (c+d x))^{3/2}}","-\frac{(a+b) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2} \left(a^2+b^2\right)}+\frac{(a+b) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{5/2} \left(a^2+b^2\right)}-\frac{(a-b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{5/2} \left(a^2+b^2\right)}+\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{5/2} \left(a^2+b^2\right)}-\frac{2 b}{a^2 d e^2 \sqrt{e \cot (c+d x)}}-\frac{2 b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{a^{5/2} d e^{5/2} \left(a^2+b^2\right)}+\frac{2}{3 a d e (e \cot (c+d x))^{3/2}}",1,"(2*(b^2*Hypergeometric2F1[-3/2, 1, -1/2, -((b*Cot[c + d*x])/a)] + a*(a*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2] - 3*b*Cot[c + d*x]*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2])))/(3*a*(a^2 + b^2)*d*e*(e*Cot[c + d*x])^(3/2))","C",1
75,1,445,437,6.1583793,"\int \frac{(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^2} \, dx","Integrate[(e*Cot[c + d*x])^(7/2)/(a + b*Cot[c + d*x])^2,x]","-\frac{(e \cot (c+d x))^{7/2} \left(\frac{2 b^2 \cot ^{\frac{9}{2}}(c+d x) \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};-\frac{b \cot (c+d x)}{a}\right)}{9 a^2 \left(a^2+b^2\right)}+\frac{4 a b \left(-7 \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-3 \cot ^{\frac{7}{2}}(c+d x)+7 \cot ^{\frac{3}{2}}(c+d x)\right)}{21 \left(a^2+b^2\right)^2}+\frac{4 a b \cot ^{\frac{7}{2}}(c+d x)}{7 \left(a^2+b^2\right)^2}-\frac{(a-b) (a+b) \left(-8 \cot ^{\frac{5}{2}}(c+d x)+40 \sqrt{\cot (c+d x)}+\frac{5}{2} \left(2 \sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+4 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)\right)}{20 \left(a^2+b^2\right)^2}-\frac{4 a^2 \left(3 \cot ^{\frac{5}{2}}(c+d x)-5 a \left(\frac{\cot ^{\frac{3}{2}}(c+d x)}{b}-\frac{3 a \left(\frac{\sqrt{\cot (c+d x)}}{b}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)}{b^{3/2}}\right)}{b}\right)\right)}{15 \left(a^2+b^2\right)^2}\right)}{d \cot ^{\frac{7}{2}}(c+d x)}","\frac{e^{7/2} \left(a^2+2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{e^{7/2} \left(a^2+2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{e^{7/2} \left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{e^{7/2} \left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{e^3 \left(3 a^2+2 b^2\right) \sqrt{e \cot (c+d x)}}{b^2 d \left(a^2+b^2\right)}+\frac{a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left(a^2+b^2\right) (a+b \cot (c+d x))}+\frac{a^{5/2} e^{7/2} \left(3 a^2+7 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{b^{5/2} d \left(a^2+b^2\right)^2}",1,"-(((e*Cot[c + d*x])^(7/2)*((4*a*b*Cot[c + d*x]^(7/2))/(7*(a^2 + b^2)^2) - (4*a^2*(3*Cot[c + d*x]^(5/2) - 5*a*((-3*a*(-((Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/b^(3/2)) + Sqrt[Cot[c + d*x]]/b))/b + Cot[c + d*x]^(3/2)/b)))/(15*(a^2 + b^2)^2) + (4*a*b*(7*Cot[c + d*x]^(3/2) - 3*Cot[c + d*x]^(7/2) - 7*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(21*(a^2 + b^2)^2) + (2*b^2*Cot[c + d*x]^(9/2)*Hypergeometric2F1[2, 9/2, 11/2, -((b*Cot[c + d*x])/a)])/(9*a^2*(a^2 + b^2)) - ((a - b)*(a + b)*(40*Sqrt[Cot[c + d*x]] - 8*Cot[c + d*x]^(5/2) + (5*(4*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 2*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/2))/(20*(a^2 + b^2)^2)))/(d*Cot[c + d*x]^(7/2)))","C",1
76,1,390,393,2.7950936,"\int \frac{(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^2} \, dx","Integrate[(e*Cot[c + d*x])^(5/2)/(a + b*Cot[c + d*x])^2,x]","-\frac{(e \cot (c+d x))^{5/2} \left(12 b^{7/2} \left(a^2+b^2\right) \cot ^{\frac{7}{2}}(c+d x) \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{b \cot (c+d x)}{a}\right)-28 a^2 b^{3/2} \left(a^2-b^2\right) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-7 a^2 \left(24 a^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)-24 a^3 \sqrt{b} \sqrt{\cot (c+d x)}+4 a^2 b^{3/2} \cot ^{\frac{3}{2}}(c+d x)-24 a b^{5/2} \sqrt{\cot (c+d x)}-3 \sqrt{2} a b^{5/2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+3 \sqrt{2} a b^{5/2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-6 \sqrt{2} a b^{5/2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)+6 \sqrt{2} a b^{5/2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+4 b^{7/2} \cot ^{\frac{3}{2}}(c+d x)\right)\right)}{42 a^2 b^{3/2} d \left(a^2+b^2\right)^2 \cot ^{\frac{5}{2}}(c+d x)}","\frac{e^{5/2} \left(a^2-2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{e^{5/2} \left(a^2-2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{e^{5/2} \left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{e^{5/2} \left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{a^2 e^2 \sqrt{e \cot (c+d x)}}{b d \left(a^2+b^2\right) (a+b \cot (c+d x))}-\frac{a^{3/2} e^{5/2} \left(a^2+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{b^{3/2} d \left(a^2+b^2\right)^2}",1,"-1/42*((e*Cot[c + d*x])^(5/2)*(-28*a^2*b^(3/2)*(a^2 - b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] + 12*b^(7/2)*(a^2 + b^2)*Cot[c + d*x]^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, -((b*Cot[c + d*x])/a)] - 7*a^2*(-6*Sqrt[2]*a*b^(5/2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + 6*Sqrt[2]*a*b^(5/2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 24*a^(7/2)*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]] - 24*a^3*Sqrt[b]*Sqrt[Cot[c + d*x]] - 24*a*b^(5/2)*Sqrt[Cot[c + d*x]] + 4*a^2*b^(3/2)*Cot[c + d*x]^(3/2) + 4*b^(7/2)*Cot[c + d*x]^(3/2) - 3*Sqrt[2]*a*b^(5/2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] + 3*Sqrt[2]*a*b^(5/2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(a^2*b^(3/2)*(a^2 + b^2)^2*d*Cot[c + d*x]^(5/2))","C",1
77,1,322,387,3.3262479,"\int \frac{(e \cot (c+d x))^{3/2}}{(a+b \cot (c+d x))^2} \, dx","Integrate[(e*Cot[c + d*x])^(3/2)/(a + b*Cot[c + d*x])^2,x]","-\frac{(e \cot (c+d x))^{3/2} \left(\frac{24 b^2 \left(a^2+b^2\right) \cot ^{\frac{5}{2}}(c+d x) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\frac{b \cot (c+d x)}{a}\right)}{a^2}-240 a^2 \left(\sqrt{\cot (c+d x)}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b}}\right)+80 a b \cot ^{\frac{3}{2}}(c+d x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-1\right)+80 a b \cot ^{\frac{3}{2}}(c+d x)+15 (a-b) (a+b) \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{60 d \left(a^2+b^2\right)^2 \cot ^{\frac{3}{2}}(c+d x)}","-\frac{e^{3/2} \left(a^2+2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{e^{3/2} \left(a^2+2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\sqrt{a} e^{3/2} \left(a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{\sqrt{b} d \left(a^2+b^2\right)^2}-\frac{e^{3/2} \left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{e^{3/2} \left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{a e \sqrt{e \cot (c+d x)}}{d \left(a^2+b^2\right) (a+b \cot (c+d x))}",1,"-1/60*((e*Cot[c + d*x])^(3/2)*(-240*a^2*(-((Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/Sqrt[b]) + Sqrt[Cot[c + d*x]]) + 80*a*b*Cot[c + d*x]^(3/2) + 80*a*b*Cot[c + d*x]^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]) + (24*b^2*(a^2 + b^2)*Cot[c + d*x]^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, -((b*Cot[c + d*x])/a)])/a^2 + 15*(a - b)*(a + b)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/((a^2 + b^2)^2*d*Cot[c + d*x]^(3/2))","C",1
78,1,401,386,6.098594,"\int \frac{\sqrt{e \cot (c+d x)}}{(a+b \cot (c+d x))^2} \, dx","Integrate[Sqrt[e*Cot[c + d*x]]/(a + b*Cot[c + d*x])^2,x]","-\frac{\sqrt{e \cot (c+d x)} \left(\frac{2 (a-b) (a+b) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)}{3 \left(a^2+b^2\right)^2}+\frac{4 a b \sqrt{\cot (c+d x)}}{\left(a^2+b^2\right)^2}-\frac{\sqrt{b} \left(\sqrt{a} \sqrt{b} \sqrt{\cot (c+d x)}-a \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)-b \cot (c+d x) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)\right)}{\sqrt{a} \left(a^2+b^2\right) (a+b \cot (c+d x))}-\frac{a b \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{2 \left(a^2+b^2\right)^2}-\frac{4 a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)}{\left(a^2+b^2\right)^2}\right)}{d \sqrt{\cot (c+d x)}}","\frac{b \sqrt{e \cot (c+d x)}}{d \left(a^2+b^2\right) (a+b \cot (c+d x))}-\frac{\sqrt{e} \left(a^2-2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\sqrt{e} \left(a^2-2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\sqrt{b} \sqrt{e} \left(3 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{\sqrt{a} d \left(a^2+b^2\right)^2}+\frac{\sqrt{e} \left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\sqrt{e} \left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}",1,"-((Sqrt[e*Cot[c + d*x]]*((-4*a^(3/2)*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/(a^2 + b^2)^2 + (4*a*b*Sqrt[Cot[c + d*x]])/(a^2 + b^2)^2 - (Sqrt[b]*(-(a*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]]) + Sqrt[a]*Sqrt[b]*Sqrt[Cot[c + d*x]] - b*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]]*Cot[c + d*x]))/(Sqrt[a]*(a^2 + b^2)*(a + b*Cot[c + d*x])) + (2*(a - b)*(a + b)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/(3*(a^2 + b^2)^2) - (a*b*(2*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(2*(a^2 + b^2)^2)))/(d*Sqrt[Cot[c + d*x]]))","C",1
79,1,300,394,2.8646766,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2} \, dx","Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^2),x]","-\frac{\sqrt{\cot (c+d x)} \left(\frac{24 b^2 \left(a^2+b^2\right) \sqrt{\cot (c+d x)} \left(\frac{a}{a+b \cot (c+d x)}+\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{\cot (c+d x)}}\right)}{a^2}+96 \sqrt{a} b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)-32 a b \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-6 \sqrt{2} (a-b) (a+b) \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{24 d \left(a^2+b^2\right)^2 \sqrt{e \cot (c+d x)}}","-\frac{b^2 \sqrt{e \cot (c+d x)}}{a d e \left(a^2+b^2\right) (a+b \cot (c+d x))}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e} \left(a^2+b^2\right)^2}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e} \left(a^2+b^2\right)^2}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \sqrt{e} \left(a^2+b^2\right)^2}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \sqrt{e} \left(a^2+b^2\right)^2}-\frac{b^{3/2} \left(5 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{a^{3/2} d \sqrt{e} \left(a^2+b^2\right)^2}",1,"-1/24*(Sqrt[Cot[c + d*x]]*(96*Sqrt[a]*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]] + (24*b^2*(a^2 + b^2)*Sqrt[Cot[c + d*x]]*((Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/(Sqrt[b]*Sqrt[Cot[c + d*x]]) + a/(a + b*Cot[c + d*x])))/a^2 - 32*a*b*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] - 6*Sqrt[2]*(a - b)*(a + b)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/((a^2 + b^2)^2*d*Sqrt[e*Cot[c + d*x]])","C",1
80,1,244,437,0.6300721,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2} \, dx","Integrate[1/((e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x])^2),x]","\frac{8 a^2 b^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\frac{b \cot (c+d x)}{a}\right)+4 b^2 \left(a^2+b^2\right) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};-\frac{b \cot (c+d x)}{a}\right)+a^2 \left(4 \left(a^2-b^2\right) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)+\sqrt{2} a b \sqrt{\cot (c+d x)} \left(-\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)+2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{2 a^2 d e \left(a^2+b^2\right)^2 \sqrt{e \cot (c+d x)}}","\frac{\left(a^2-2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2} \left(a^2+b^2\right)^2}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2} \left(a^2+b^2\right)^2}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{3/2} \left(a^2+b^2\right)^2}+\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{3/2} \left(a^2+b^2\right)^2}-\frac{b^2}{a d e \left(a^2+b^2\right) \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}+\frac{2 a^2+3 b^2}{a^2 d e \left(a^2+b^2\right) \sqrt{e \cot (c+d x)}}+\frac{b^{5/2} \left(7 a^2+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{a^{5/2} d e^{3/2} \left(a^2+b^2\right)^2}",1,"(8*a^2*b^2*Hypergeometric2F1[-1/2, 1, 1/2, -((b*Cot[c + d*x])/a)] + 4*b^2*(a^2 + b^2)*Hypergeometric2F1[-1/2, 2, 1/2, -((b*Cot[c + d*x])/a)] + a^2*(4*(a^2 - b^2)*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Sqrt[2]*a*b*Sqrt[Cot[c + d*x]]*(-2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] - Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(2*a^2*(a^2 + b^2)^2*d*e*Sqrt[e*Cot[c + d*x]])","C",1
81,1,556,529,6.2605544,"\int \frac{(e \cot (c+d x))^{9/2}}{(a+b \cot (c+d x))^3} \, dx","Integrate[(e*Cot[c + d*x])^(9/2)/(a + b*Cot[c + d*x])^3,x]","-\frac{(e \cot (c+d x))^{9/2} \left(\frac{4 b^2 \cot ^{\frac{11}{2}}(c+d x) \, _2F_1\left(2,\frac{11}{2};\frac{13}{2};-\frac{b \cot (c+d x)}{a}\right)}{11 a \left(a^2+b^2\right)^2}-\frac{2 a \left(a^2-3 b^2\right) \left(-7 \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-3 \cot ^{\frac{7}{2}}(c+d x)+7 \cot ^{\frac{3}{2}}(c+d x)\right)}{21 \left(a^2+b^2\right)^3}+\frac{2 b \left(3 a^2-b^2\right) \cot ^{\frac{9}{2}}(c+d x)}{9 \left(a^2+b^2\right)^3}-\frac{b \left(3 a^2-b^2\right) \left(40 \cot ^{\frac{9}{2}}(c+d x)-72 \cot ^{\frac{5}{2}}(c+d x)+360 \sqrt{\cot (c+d x)}+45 \left(\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)\right)}{180 \left(a^2+b^2\right)^3}-\frac{2 a \left(3 a^2-b^2\right) \left(15 \cot ^{\frac{7}{2}}(c+d x)-7 a \left(\frac{3 \cot ^{\frac{5}{2}}(c+d x)}{b}-\frac{5 a \left(\frac{\cot ^{\frac{3}{2}}(c+d x)}{b}-\frac{3 a \left(\frac{\sqrt{\cot (c+d x)}}{b}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)}{b^{3/2}}\right)}{b}\right)}{b}\right)\right)}{105 \left(a^2+b^2\right)^3}+\frac{2 b^2 \cot ^{\frac{11}{2}}(c+d x) \, _2F_1\left(3,\frac{11}{2};\frac{13}{2};-\frac{b \cot (c+d x)}{a}\right)}{11 a^3 \left(a^2+b^2\right)}\right)}{d \cot ^{\frac{9}{2}}(c+d x)}","-\frac{e^{9/2} (a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{e^{9/2} (a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{e^{9/2} (a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{e^{9/2} (a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{a^2 e^3 \left(5 a^2+13 b^2\right) (e \cot (c+d x))^{3/2}}{4 b^2 d \left(a^2+b^2\right)^2 (a+b \cot (c+d x))}+\frac{a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left(a^2+b^2\right) (a+b \cot (c+d x))^2}-\frac{e^4 \left(15 a^4+31 a^2 b^2+8 b^4\right) \sqrt{e \cot (c+d x)}}{4 b^3 d \left(a^2+b^2\right)^2}+\frac{a^{5/2} e^{9/2} \left(15 a^4+46 a^2 b^2+63 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{4 b^{7/2} d \left(a^2+b^2\right)^3}",1,"-(((e*Cot[c + d*x])^(9/2)*((2*b*(3*a^2 - b^2)*Cot[c + d*x]^(9/2))/(9*(a^2 + b^2)^3) - (2*a*(3*a^2 - b^2)*(15*Cot[c + d*x]^(7/2) - 7*a*((3*Cot[c + d*x]^(5/2))/b - (5*a*((-3*a*(-((Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/b^(3/2)) + Sqrt[Cot[c + d*x]]/b))/b + Cot[c + d*x]^(3/2)/b))/b)))/(105*(a^2 + b^2)^3) - (2*a*(a^2 - 3*b^2)*(7*Cot[c + d*x]^(3/2) - 3*Cot[c + d*x]^(7/2) - 7*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(21*(a^2 + b^2)^3) + (4*b^2*Cot[c + d*x]^(11/2)*Hypergeometric2F1[2, 11/2, 13/2, -((b*Cot[c + d*x])/a)])/(11*a*(a^2 + b^2)^2) + (2*b^2*Cot[c + d*x]^(11/2)*Hypergeometric2F1[3, 11/2, 13/2, -((b*Cot[c + d*x])/a)])/(11*a^3*(a^2 + b^2)) - (b*(3*a^2 - b^2)*(360*Sqrt[Cot[c + d*x]] - 72*Cot[c + d*x]^(5/2) + 40*Cot[c + d*x]^(9/2) + 45*(2*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(180*(a^2 + b^2)^3)))/(d*Cot[c + d*x]^(9/2)))","C",0
82,1,525,476,6.1888513,"\int \frac{(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^3} \, dx","Integrate[(e*Cot[c + d*x])^(7/2)/(a + b*Cot[c + d*x])^3,x]","-\frac{(e \cot (c+d x))^{7/2} \left(\frac{4 b^2 \cot ^{\frac{9}{2}}(c+d x) \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};-\frac{b \cot (c+d x)}{a}\right)}{9 a \left(a^2+b^2\right)^2}+\frac{2 b \left(3 a^2-b^2\right) \left(-7 \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-3 \cot ^{\frac{7}{2}}(c+d x)+7 \cot ^{\frac{3}{2}}(c+d x)\right)}{21 \left(a^2+b^2\right)^3}+\frac{2 b \left(3 a^2-b^2\right) \cot ^{\frac{7}{2}}(c+d x)}{7 \left(a^2+b^2\right)^3}-\frac{a \left(a^2-3 b^2\right) \left(-8 \cot ^{\frac{5}{2}}(c+d x)+40 \sqrt{\cot (c+d x)}+\frac{5}{2} \left(2 \sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+4 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)\right)}{20 \left(a^2+b^2\right)^3}-\frac{2 a \left(3 a^2-b^2\right) \left(3 \cot ^{\frac{5}{2}}(c+d x)-5 a \left(\frac{\cot ^{\frac{3}{2}}(c+d x)}{b}-\frac{3 a \left(\frac{\sqrt{\cot (c+d x)}}{b}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)}{b^{3/2}}\right)}{b}\right)\right)}{15 \left(a^2+b^2\right)^3}+\frac{2 b^2 \cot ^{\frac{9}{2}}(c+d x) \, _2F_1\left(3,\frac{9}{2};\frac{11}{2};-\frac{b \cot (c+d x)}{a}\right)}{9 a^3 \left(a^2+b^2\right)}\right)}{d \cot ^{\frac{7}{2}}(c+d x)}","\frac{e^{7/2} (a-b) \left(a^2+4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{e^{7/2} (a-b) \left(a^2+4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{e^{7/2} (a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{e^{7/2} (a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{a^2 e^3 \left(3 a^2+11 b^2\right) \sqrt{e \cot (c+d x)}}{4 b^2 d \left(a^2+b^2\right)^2 (a+b \cot (c+d x))}+\frac{a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left(a^2+b^2\right) (a+b \cot (c+d x))^2}-\frac{a^{3/2} e^{7/2} \left(3 a^4+6 a^2 b^2+35 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{4 b^{5/2} d \left(a^2+b^2\right)^3}",1,"-(((e*Cot[c + d*x])^(7/2)*((2*b*(3*a^2 - b^2)*Cot[c + d*x]^(7/2))/(7*(a^2 + b^2)^3) - (2*a*(3*a^2 - b^2)*(3*Cot[c + d*x]^(5/2) - 5*a*((-3*a*(-((Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/b^(3/2)) + Sqrt[Cot[c + d*x]]/b))/b + Cot[c + d*x]^(3/2)/b)))/(15*(a^2 + b^2)^3) + (2*b*(3*a^2 - b^2)*(7*Cot[c + d*x]^(3/2) - 3*Cot[c + d*x]^(7/2) - 7*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(21*(a^2 + b^2)^3) + (4*b^2*Cot[c + d*x]^(9/2)*Hypergeometric2F1[2, 9/2, 11/2, -((b*Cot[c + d*x])/a)])/(9*a*(a^2 + b^2)^2) + (2*b^2*Cot[c + d*x]^(9/2)*Hypergeometric2F1[3, 9/2, 11/2, -((b*Cot[c + d*x])/a)])/(9*a^3*(a^2 + b^2)) - (a*(a^2 - 3*b^2)*(40*Sqrt[Cot[c + d*x]] - 8*Cot[c + d*x]^(5/2) + (5*(4*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 2*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/2))/(20*(a^2 + b^2)^3)))/(d*Cot[c + d*x]^(7/2)))","C",1
83,1,488,470,6.1993399,"\int \frac{(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^3} \, dx","Integrate[(e*Cot[c + d*x])^(5/2)/(a + b*Cot[c + d*x])^3,x]","-\frac{(e \cot (c+d x))^{5/2} \left(\frac{4 b^2 \cot ^{\frac{7}{2}}(c+d x) \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{b \cot (c+d x)}{a}\right)}{7 a \left(a^2+b^2\right)^2}+\frac{2 a \left(a^2-3 b^2\right) \left(\cot ^{\frac{3}{2}}(c+d x)-\cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)\right)}{3 \left(a^2+b^2\right)^3}+\frac{2 b \left(3 a^2-b^2\right) \cot ^{\frac{5}{2}}(c+d x)}{5 \left(a^2+b^2\right)^3}+\frac{b \left(3 a^2-b^2\right) \left(-8 \cot ^{\frac{5}{2}}(c+d x)+40 \sqrt{\cot (c+d x)}+\frac{5}{2} \left(2 \sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+4 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)\right)}{20 \left(a^2+b^2\right)^3}-\frac{2 a \left(3 a^2-b^2\right) \left(\cot ^{\frac{3}{2}}(c+d x)-3 a \left(\frac{\sqrt{\cot (c+d x)}}{b}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)}{b^{3/2}}\right)\right)}{3 \left(a^2+b^2\right)^3}+\frac{2 b^2 \cot ^{\frac{7}{2}}(c+d x) \, _2F_1\left(3,\frac{7}{2};\frac{9}{2};-\frac{b \cot (c+d x)}{a}\right)}{7 a^3 \left(a^2+b^2\right)}\right)}{d \cot ^{\frac{5}{2}}(c+d x)}","\frac{e^{5/2} (a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{e^{5/2} (a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{e^{5/2} (a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{e^{5/2} (a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{a e^2 \left(a^2+9 b^2\right) \sqrt{e \cot (c+d x)}}{4 b d \left(a^2+b^2\right)^2 (a+b \cot (c+d x))}+\frac{a^2 e^2 \sqrt{e \cot (c+d x)}}{2 b d \left(a^2+b^2\right) (a+b \cot (c+d x))^2}-\frac{\sqrt{a} e^{5/2} \left(a^4+18 a^2 b^2-15 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{4 b^{3/2} d \left(a^2+b^2\right)^3}",1,"-(((e*Cot[c + d*x])^(5/2)*((2*b*(3*a^2 - b^2)*Cot[c + d*x]^(5/2))/(5*(a^2 + b^2)^3) - (2*a*(3*a^2 - b^2)*(-3*a*(-((Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/b^(3/2)) + Sqrt[Cot[c + d*x]]/b) + Cot[c + d*x]^(3/2)))/(3*(a^2 + b^2)^3) + (2*a*(a^2 - 3*b^2)*(Cot[c + d*x]^(3/2) - Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(3*(a^2 + b^2)^3) + (4*b^2*Cot[c + d*x]^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, -((b*Cot[c + d*x])/a)])/(7*a*(a^2 + b^2)^2) + (2*b^2*Cot[c + d*x]^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, -((b*Cot[c + d*x])/a)])/(7*a^3*(a^2 + b^2)) + (b*(3*a^2 - b^2)*(40*Sqrt[Cot[c + d*x]] - 8*Cot[c + d*x]^(5/2) + (5*(4*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 2*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/2))/(20*(a^2 + b^2)^3)))/(d*Cot[c + d*x]^(5/2)))","C",1
84,1,518,461,6.1562678,"\int \frac{(e \cot (c+d x))^{3/2}}{(a+b \cot (c+d x))^3} \, dx","Integrate[(e*Cot[c + d*x])^(3/2)/(a + b*Cot[c + d*x])^3,x]","-\frac{(e \cot (c+d x))^{3/2} \left(\frac{4 b^2 \cot ^{\frac{5}{2}}(c+d x) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\frac{b \cot (c+d x)}{a}\right)}{5 a \left(a^2+b^2\right)^2}-\frac{2 b \left(3 a^2-b^2\right) \left(\cot ^{\frac{3}{2}}(c+d x)-\cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)\right)}{3 \left(a^2+b^2\right)^3}+\frac{2 b \left(3 a^2-b^2\right) \cot ^{\frac{3}{2}}(c+d x)}{3 \left(a^2+b^2\right)^3}-\frac{\frac{2 b^2 \cot ^2(c+d x)}{(a+b \cot (c+d x))^2}+\frac{3 b \cot (c+d x)}{a+b \cot (c+d x)}-\frac{3 \sqrt{b} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}}{4 b \left(a^2+b^2\right) \sqrt{\cot (c+d x)}}-\frac{2 a \left(3 a^2-b^2\right) \left(\sqrt{\cot (c+d x)}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b}}\right)}{\left(a^2+b^2\right)^3}+\frac{a \left(a^2-3 b^2\right) \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{4 \left(a^2+b^2\right)^3}\right)}{d \cot ^{\frac{3}{2}}(c+d x)}","-\frac{e^{3/2} (a-b) \left(a^2+4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{e^{3/2} (a-b) \left(a^2+4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{e^{3/2} (a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{e^{3/2} (a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{e \left(3 a^2-5 b^2\right) \sqrt{e \cot (c+d x)}}{4 d \left(a^2+b^2\right)^2 (a+b \cot (c+d x))}-\frac{a e \sqrt{e \cot (c+d x)}}{2 d \left(a^2+b^2\right) (a+b \cot (c+d x))^2}-\frac{e^{3/2} \left(3 a^4-26 a^2 b^2+3 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{4 \sqrt{a} \sqrt{b} d \left(a^2+b^2\right)^3}",1,"-(((e*Cot[c + d*x])^(3/2)*((-2*a*(3*a^2 - b^2)*(-((Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/Sqrt[b]) + Sqrt[Cot[c + d*x]]))/(a^2 + b^2)^3 + (2*b*(3*a^2 - b^2)*Cot[c + d*x]^(3/2))/(3*(a^2 + b^2)^3) - ((-3*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]]*Sqrt[Cot[c + d*x]])/Sqrt[a] + (2*b^2*Cot[c + d*x]^2)/(a + b*Cot[c + d*x])^2 + (3*b*Cot[c + d*x])/(a + b*Cot[c + d*x]))/(4*b*(a^2 + b^2)*Sqrt[Cot[c + d*x]]) - (2*b*(3*a^2 - b^2)*(Cot[c + d*x]^(3/2) - Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(3*(a^2 + b^2)^3) + (4*b^2*Cot[c + d*x]^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, -((b*Cot[c + d*x])/a)])/(5*a*(a^2 + b^2)^2) + (a*(a^2 - 3*b^2)*(2*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(4*(a^2 + b^2)^3)))/(d*Cot[c + d*x]^(3/2)))","C",1
85,1,483,463,6.185719,"\int \frac{\sqrt{e \cot (c+d x)}}{(a+b \cot (c+d x))^3} \, dx","Integrate[Sqrt[e*Cot[c + d*x]]/(a + b*Cot[c + d*x])^3,x]","-\frac{\sqrt{e \cot (c+d x)} \left(\frac{2 a \left(a^2-3 b^2\right) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)}{3 \left(a^2+b^2\right)^3}+\frac{2 b \left(3 a^2-b^2\right) \sqrt{\cot (c+d x)}}{\left(a^2+b^2\right)^3}-\frac{2 \sqrt{a} \sqrt{b} \left(3 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)}{\left(a^2+b^2\right)^3}-\frac{2 \sqrt{a} \sqrt{b} \left(\sqrt{a} \sqrt{b} \sqrt{\cot (c+d x)}-a \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)-b \cot (c+d x) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)\right)}{\left(a^2+b^2\right)^2 (a+b \cot (c+d x))}-\frac{b \left(3 a^2-b^2\right) \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{4 \left(a^2+b^2\right)^3}+\frac{2 b^2 \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};-\frac{b \cot (c+d x)}{a}\right)}{3 a^3 \left(a^2+b^2\right)}\right)}{d \sqrt{\cot (c+d x)}}","\frac{b \left(7 a^2-b^2\right) \sqrt{e \cot (c+d x)}}{4 a d \left(a^2+b^2\right)^2 (a+b \cot (c+d x))}+\frac{b \sqrt{e \cot (c+d x)}}{2 d \left(a^2+b^2\right) (a+b \cot (c+d x))^2}-\frac{\sqrt{e} (a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\sqrt{e} (a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\sqrt{e} (a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\sqrt{e} (a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\sqrt{b} \sqrt{e} \left(15 a^4-18 a^2 b^2-b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{4 a^{3/2} d \left(a^2+b^2\right)^3}",1,"-((Sqrt[e*Cot[c + d*x]]*((-2*Sqrt[a]*Sqrt[b]*(3*a^2 - b^2)*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/(a^2 + b^2)^3 + (2*b*(3*a^2 - b^2)*Sqrt[Cot[c + d*x]])/(a^2 + b^2)^3 - (2*Sqrt[a]*Sqrt[b]*(-(a*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]]) + Sqrt[a]*Sqrt[b]*Sqrt[Cot[c + d*x]] - b*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]]*Cot[c + d*x]))/((a^2 + b^2)^2*(a + b*Cot[c + d*x])) + (2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/(3*(a^2 + b^2)^3) + (2*b^2*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, -((b*Cot[c + d*x])/a)])/(3*a^3*(a^2 + b^2)) - (b*(3*a^2 - b^2)*(2*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(4*(a^2 + b^2)^3)))/(d*Sqrt[Cot[c + d*x]]))","C",1
86,1,411,476,6.1343734,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^3} \, dx","Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^3),x]","-\frac{\sqrt{\cot (c+d x)} \left(-\frac{2 b \left(3 a^2-b^2\right) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)}{3 \left(a^2+b^2\right)^3}+\frac{2 b^2 \sqrt{\cot (c+d x)} \left(\frac{a}{a+b \cot (c+d x)}+\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{\cot (c+d x)}}\right)}{a \left(a^2+b^2\right)^2}-\frac{a \left(a^2-3 b^2\right) \left(2 \sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+4 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{8 \left(a^2+b^2\right)^3}+\frac{2 b^{3/2} \left(3 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \left(a^2+b^2\right)^3}+\frac{2 b^2 \sqrt{\cot (c+d x)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};-\frac{b \cot (c+d x)}{a}\right)}{a^3 \left(a^2+b^2\right)}\right)}{d \sqrt{e \cot (c+d x)}}","-\frac{b^2 \left(11 a^2+3 b^2\right) \sqrt{e \cot (c+d x)}}{4 a^2 d e \left(a^2+b^2\right)^2 (a+b \cot (c+d x))}-\frac{b^2 \sqrt{e \cot (c+d x)}}{2 a d e \left(a^2+b^2\right) (a+b \cot (c+d x))^2}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e} \left(a^2+b^2\right)^3}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e} \left(a^2+b^2\right)^3}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \sqrt{e} \left(a^2+b^2\right)^3}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \sqrt{e} \left(a^2+b^2\right)^3}-\frac{b^{3/2} \left(35 a^4+6 a^2 b^2+3 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{4 a^{5/2} d \sqrt{e} \left(a^2+b^2\right)^3}",1,"-((Sqrt[Cot[c + d*x]]*((2*b^(3/2)*(3*a^2 - b^2)*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)^3) + (2*b^2*Sqrt[Cot[c + d*x]]*((Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/(Sqrt[b]*Sqrt[Cot[c + d*x]]) + a/(a + b*Cot[c + d*x])))/(a*(a^2 + b^2)^2) + (2*b^2*Sqrt[Cot[c + d*x]]*Hypergeometric2F1[1/2, 3, 3/2, -((b*Cot[c + d*x])/a)])/(a^3*(a^2 + b^2)) - (2*b*(3*a^2 - b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/(3*(a^2 + b^2)^3) - (a*(a^2 - 3*b^2)*(4*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 2*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(8*(a^2 + b^2)^3)))/(d*Sqrt[e*Cot[c + d*x]]))","C",1
87,1,303,529,1.7918809,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3} \, dx","Integrate[1/((e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x])^3),x]","-\frac{-8 a^2 b^2 \left(3 a^2-b^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\frac{b \cot (c+d x)}{a}\right)-16 a^2 b^2 \left(a^2+b^2\right) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};-\frac{b \cot (c+d x)}{a}\right)-8 b^2 \left(a^2+b^2\right)^2 \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};-\frac{b \cot (c+d x)}{a}\right)-8 a^4 \left(a^2-3 b^2\right) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)+\sqrt{2} a^3 b \left(3 a^2-b^2\right) \sqrt{\cot (c+d x)} \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{4 a^3 d e \left(a^2+b^2\right)^3 \sqrt{e \cot (c+d x)}}","\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2} \left(a^2+b^2\right)^3}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d e^{3/2} \left(a^2+b^2\right)^3}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d e^{3/2} \left(a^2+b^2\right)^3}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d e^{3/2} \left(a^2+b^2\right)^3}-\frac{b^2 \left(13 a^2+5 b^2\right)}{4 a^2 d e \left(a^2+b^2\right)^2 \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}-\frac{b^2}{2 a d e \left(a^2+b^2\right) \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2}+\frac{b^{5/2} \left(63 a^4+46 a^2 b^2+15 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right)}{4 a^{7/2} d e^{3/2} \left(a^2+b^2\right)^3}+\frac{8 a^4+31 a^2 b^2+15 b^4}{4 a^3 d e \left(a^2+b^2\right)^2 \sqrt{e \cot (c+d x)}}",1,"-1/4*(-8*a^2*b^2*(3*a^2 - b^2)*Hypergeometric2F1[-1/2, 1, 1/2, -((b*Cot[c + d*x])/a)] - 16*a^2*b^2*(a^2 + b^2)*Hypergeometric2F1[-1/2, 2, 1/2, -((b*Cot[c + d*x])/a)] - 8*b^2*(a^2 + b^2)^2*Hypergeometric2F1[-1/2, 3, 1/2, -((b*Cot[c + d*x])/a)] - 8*a^4*(a^2 - 3*b^2)*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Sqrt[2]*a^3*b*(3*a^2 - b^2)*Sqrt[Cot[c + d*x]]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(a^3*(a^2 + b^2)^3*d*e*Sqrt[e*Cot[c + d*x]])","C",1
88,1,118,167,0.3019932,"\int (a+b \cot (c+d x))^n \, dx","Integrate[(a + b*Cot[c + d*x])^n,x]","\frac{(a+b \cot (c+d x))^{n+1} \left((a+i b) \, _2F_1\left(1,n+1;n+2;\frac{a+b \cot (c+d x)}{a-i b}\right)-(a-i b) \, _2F_1\left(1,n+1;n+2;\frac{a+b \cot (c+d x)}{a+i b}\right)\right)}{2 d (n+1) (a-i b) (b-i a)}","\frac{b (a+b \cot (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \cot (c+d x)}{a+\sqrt{-b^2}}\right)}{2 \sqrt{-b^2} d (n+1) \left(a+\sqrt{-b^2}\right)}-\frac{b (a+b \cot (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \cot (c+d x)}{a-\sqrt{-b^2}}\right)}{2 \sqrt{-b^2} d (n+1) \left(a-\sqrt{-b^2}\right)}",1,"((a + b*Cot[c + d*x])^(1 + n)*((a + I*b)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Cot[c + d*x])/(a - I*b)] - (a - I*b)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Cot[c + d*x])/(a + I*b)]))/(2*(a - I*b)*((-I)*a + b)*d*(1 + n))","C",1
89,0,0,193,3.2521782,"\int (a+b \cot (e+f x))^m (d \tan (e+f x))^n \, dx","Integrate[(a + b*Cot[e + f*x])^m*(d*Tan[e + f*x])^n,x]","\int (a+b \cot (e+f x))^m (d \tan (e+f x))^n \, dx","-\frac{\cot (e+f x) (d \tan (e+f x))^n (a+b \cot (e+f x))^m \left(\frac{b \cot (e+f x)}{a}+1\right)^{-m} F_1\left(1-n;-m,1;2-n;-\frac{b \cot (e+f x)}{a},-i \cot (e+f x)\right)}{2 f (1-n)}-\frac{\cot (e+f x) (d \tan (e+f x))^n (a+b \cot (e+f x))^m \left(\frac{b \cot (e+f x)}{a}+1\right)^{-m} F_1\left(1-n;-m,1;2-n;-\frac{b \cot (e+f x)}{a},i \cot (e+f x)\right)}{2 f (1-n)}",1,"Integrate[(a + b*Cot[e + f*x])^m*(d*Tan[e + f*x])^n, x]","F",-1
90,1,45,45,0.0813168,"\int \frac{1+i \cot (c+d x)}{\sqrt{a+b \cot (c+d x)}} \, dx","Integrate[(1 + I*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]],x]","\frac{2 i \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}","\frac{2 i \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}",1,"((2*I)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)","A",1
91,1,70,45,1.7208107,"\int \frac{1-i \cot (c+d x)}{\sqrt{a+b \cot (c+d x)}} \, dx","Integrate[(1 - I*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]],x]","-\frac{2 i \tanh ^{-1}\left(\frac{\sqrt{a+\frac{i b \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}","-\frac{2 i \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}",1,"((-2*I)*ArcTanh[Sqrt[a + (I*b*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)","A",1
92,1,67,59,0.1297149,"\int \frac{A+B \cot (c+d x)}{a+b \cot (c+d x)} \, dx","Integrate[(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x]),x]","-\frac{2 (a A+b B) \tan ^{-1}(\cot (c+d x))+(A b-a B) \left(2 \log (a+b \cot (c+d x))-\log \left(\csc ^2(c+d x)\right)\right)}{2 d \left(a^2+b^2\right)}","\frac{x (a A+b B)}{a^2+b^2}-\frac{(A b-a B) \log (a \sin (c+d x)+b \cos (c+d x))}{d \left(a^2+b^2\right)}",1,"-1/2*(2*(a*A + b*B)*ArcTan[Cot[c + d*x]] + (A*b - a*B)*(2*Log[a + b*Cot[c + d*x]] - Log[Csc[c + d*x]^2]))/((a^2 + b^2)*d)","A",1
93,1,144,111,1.92867,"\int \frac{A+B \cot (c+d x)}{(a+b \cot (c+d x))^2} \, dx","Integrate[(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^2,x]","\frac{\frac{2 b (a B-A b)}{a \left(a^2+b^2\right) (a \tan (c+d x)+b)}+\frac{2 \left(a^2 B-2 a A b-b^2 B\right) \log (a \tan (c+d x)+b)}{\left(a^2+b^2\right)^2}-\frac{(B+i A) \log (-\tan (c+d x)+i)}{(a-i b)^2}+\frac{i (A+i B) \log (\tan (c+d x)+i)}{(a+i b)^2}}{2 d}","\frac{A b-a B}{d \left(a^2+b^2\right) (a+b \cot (c+d x))}-\frac{\left(a^2 (-B)+2 a A b+b^2 B\right) \log (a \sin (c+d x)+b \cos (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{x \left(a^2 A+2 a b B-A b^2\right)}{\left(a^2+b^2\right)^2}",1,"(-(((I*A + B)*Log[I - Tan[c + d*x]])/(a - I*b)^2) + (I*(A + I*B)*Log[I + Tan[c + d*x]])/(a + I*b)^2 + (2*(-2*a*A*b + a^2*B - b^2*B)*Log[b + a*Tan[c + d*x]])/(a^2 + b^2)^2 + (2*b*(-(A*b) + a*B))/(a*(a^2 + b^2)*(b + a*Tan[c + d*x])))/(2*d)","C",1
94,1,202,175,5.0033547,"\int \frac{A+B \cot (c+d x)}{(a+b \cot (c+d x))^3} \, dx","Integrate[(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^3,x]","\frac{\frac{2 \left(a^3 B-3 a^2 A b-3 a b^2 B+A b^3\right) \log (a \tan (c+d x)+b)-\frac{b \left(a^2+b^2\right) \left(\left(-4 a^4 B+6 a^3 A b+2 a A b^3\right) \tan (c+d x)+b \left(-3 a^3 B+5 a^2 A b+a b^2 B+A b^3\right)\right)}{a^2 (a \tan (c+d x)+b)^2}}{\left(a^2+b^2\right)^3}-\frac{i (A-i B) \log (-\tan (c+d x)+i)}{(a-i b)^3}+\frac{i (A+i B) \log (\tan (c+d x)+i)}{(a+i b)^3}}{2 d}","\frac{A b-a B}{2 d \left(a^2+b^2\right) (a+b \cot (c+d x))^2}+\frac{a^2 (-B)+2 a A b+b^2 B}{d \left(a^2+b^2\right)^2 (a+b \cot (c+d x))}-\frac{\left(a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right) \log (a \sin (c+d x)+b \cos (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{x \left(a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right)}{\left(a^2+b^2\right)^3}",1,"(((-I)*(A - I*B)*Log[I - Tan[c + d*x]])/(a - I*b)^3 + (I*(A + I*B)*Log[I + Tan[c + d*x]])/(a + I*b)^3 + (2*(-3*a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*Log[b + a*Tan[c + d*x]] - (b*(a^2 + b^2)*(b*(5*a^2*A*b + A*b^3 - 3*a^3*B + a*b^2*B) + (6*a^3*A*b + 2*a*A*b^3 - 4*a^4*B)*Tan[c + d*x]))/(a^2*(b + a*Tan[c + d*x])^2))/(a^2 + b^2)^3)/(2*d)","C",1
95,1,379,188,1.7950707,"\int (a+b \cot (c+d x))^{5/2} (A+B \cot (c+d x)) \, dx","Integrate[(a + b*Cot[c + d*x])^(5/2)*(A + B*Cot[c + d*x]),x]","-\frac{2 \left(\left(a^2 B+2 a A b-b^2 B\right) \sqrt{a+b \cot (c+d x)}+\frac{\sqrt{a-\sqrt{-b^2}} \left(a^3 \left(A b-\sqrt{-b^2} B\right)-3 a^2 b \left(A \sqrt{-b^2}+b B\right)+3 a b^2 \left(\sqrt{-b^2} B-A b\right)+b^3 \left(A \sqrt{-b^2}+b B\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{2 \left(a \sqrt{-b^2}+b^2\right)}+\frac{\left(-\left(a^3 \left(A b+\sqrt{-b^2} B\right)\right)+3 a^2 b \left(b B-A \sqrt{-b^2}\right)+3 a b^2 \left(A b+\sqrt{-b^2} B\right)+b^3 \left(A \sqrt{-b^2}-b B\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{2 \sqrt{-b^2} \sqrt{a+\sqrt{-b^2}}}+\frac{1}{3} (a B+A b) (a+b \cot (c+d x))^{3/2}+\frac{1}{5} B (a+b \cot (c+d x))^{5/2}\right)}{d}","-\frac{2 \left(a^2 B+2 a A b-b^2 B\right) \sqrt{a+b \cot (c+d x)}}{d}-\frac{2 (a B+A b) (a+b \cot (c+d x))^{3/2}}{3 d}+\frac{(a-i b)^{5/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{(a+i b)^{5/2} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{2 B (a+b \cot (c+d x))^{5/2}}{5 d}",1,"(-2*((Sqrt[a - Sqrt[-b^2]]*(-3*a^2*b*(A*Sqrt[-b^2] + b*B) + b^3*(A*Sqrt[-b^2] + b*B) + a^3*(A*b - Sqrt[-b^2]*B) + 3*a*b^2*(-(A*b) + Sqrt[-b^2]*B))*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(2*(b^2 + a*Sqrt[-b^2])) + ((b^3*(A*Sqrt[-b^2] - b*B) + 3*a^2*b*(-(A*Sqrt[-b^2]) + b*B) - a^3*(A*b + Sqrt[-b^2]*B) + 3*a*b^2*(A*b + Sqrt[-b^2]*B))*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(2*Sqrt[-b^2]*Sqrt[a + Sqrt[-b^2]]) + (2*a*A*b + a^2*B - b^2*B)*Sqrt[a + b*Cot[c + d*x]] + ((A*b + a*B)*(a + b*Cot[c + d*x])^(3/2))/3 + (B*(a + b*Cot[c + d*x])^(5/2))/5))/d","B",1
96,1,294,150,0.973719,"\int (a+b \cot (c+d x))^{3/2} (A+B \cot (c+d x)) \, dx","Integrate[(a + b*Cot[c + d*x])^(3/2)*(A + B*Cot[c + d*x]),x]","-\frac{\frac{3 \sqrt{a-\sqrt{-b^2}} \left(a^2 \left(A b-\sqrt{-b^2} B\right)-2 a b \left(A \sqrt{-b^2}+b B\right)+b^2 \left(\sqrt{-b^2} B-A b\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{a \sqrt{-b^2}+b^2}+\frac{3 \left(-\left(a^2 \left(A b+\sqrt{-b^2} B\right)\right)+2 a b \left(b B-A \sqrt{-b^2}\right)+b^2 \left(A b+\sqrt{-b^2} B\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{-b^2} \sqrt{a+\sqrt{-b^2}}}+6 (a B+A b) \sqrt{a+b \cot (c+d x)}+2 B (a+b \cot (c+d x))^{3/2}}{3 d}","-\frac{2 (a B+A b) \sqrt{a+b \cot (c+d x)}}{d}+\frac{(a-i b)^{3/2} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{(a+i b)^{3/2} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{2 B (a+b \cot (c+d x))^{3/2}}{3 d}",1,"-1/3*((3*Sqrt[a - Sqrt[-b^2]]*(-2*a*b*(A*Sqrt[-b^2] + b*B) + a^2*(A*b - Sqrt[-b^2]*B) + b^2*(-(A*b) + Sqrt[-b^2]*B))*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(b^2 + a*Sqrt[-b^2]) + (3*(2*a*b*(-(A*Sqrt[-b^2]) + b*B) - a^2*(A*b + Sqrt[-b^2]*B) + b^2*(A*b + Sqrt[-b^2]*B))*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a + Sqrt[-b^2]]) + 6*(A*b + a*B)*Sqrt[a + b*Cot[c + d*x]] + 2*B*(a + b*Cot[c + d*x])^(3/2))/d","A",1
97,1,212,122,0.5669627,"\int \sqrt{a+b \cot (c+d x)} (A+B \cot (c+d x)) \, dx","Integrate[Sqrt[a + b*Cot[c + d*x]]*(A + B*Cot[c + d*x]),x]","-\frac{\frac{\left(a A b-a \sqrt{-b^2} B-A \sqrt{-b^2} b+b^2 (-B)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{-b^2} \sqrt{a-\sqrt{-b^2}}}-\frac{\left(a A b+a \sqrt{-b^2} B+A \sqrt{-b^2} b+b^2 (-B)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{-b^2} \sqrt{a+\sqrt{-b^2}}}+2 B \sqrt{a+b \cot (c+d x)}}{d}","\frac{\sqrt{a-i b} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{\sqrt{a+i b} (-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{2 B \sqrt{a+b \cot (c+d x)}}{d}",1,"-((((a*A*b - A*b*Sqrt[-b^2] - b^2*B - a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) - ((a*A*b + A*b*Sqrt[-b^2] - b^2*B + a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a + Sqrt[-b^2]]) + 2*B*Sqrt[a + b*Cot[c + d*x]])/d)","A",1
98,1,253,151,3.9587563,"\int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{5/2} \, dx","Integrate[(-a + b*Cot[c + d*x])*(a + b*Cot[c + d*x])^(5/2),x]","\frac{\sin (c+d x) (b \cot (c+d x)-a) (a+b \cot (c+d x))^{5/2} \left(\frac{2 b \left(-4 a^2+2 a b \cot (c+d x)+b^2 \csc ^2(c+d x)-6 b^2\right)}{(a+b \cot (c+d x))^2}+\frac{5 i \left(a^2+b^2\right) \left((a-i b)^2 \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)-\sqrt{a-i b} (a+i b)^2 \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)\right)}{\sqrt{a-i b} \sqrt{a+i b} (a+b \cot (c+d x))^{5/2}}\right)}{5 d (a \sin (c+d x)-b \cos (c+d x))}","\frac{2 b \left(a^2+b^2\right) \sqrt{a+b \cot (c+d x)}}{d}-\frac{2 b (a+b \cot (c+d x))^{5/2}}{5 d}-\frac{(-b+i a) (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{(a+i b)^{5/2} (b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"((-a + b*Cot[c + d*x])*(a + b*Cot[c + d*x])^(5/2)*(((5*I)*(a^2 + b^2)*((a - I*b)^2*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]] - Sqrt[a - I*b]*(a + I*b)^2*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]]))/(Sqrt[a - I*b]*Sqrt[a + I*b]*(a + b*Cot[c + d*x])^(5/2)) + (2*b*(-4*a^2 - 6*b^2 + 2*a*b*Cot[c + d*x] + b^2*Csc[c + d*x]^2))/(a + b*Cot[c + d*x])^2)*Sin[c + d*x])/(5*d*(-(b*Cos[c + d*x]) + a*Sin[c + d*x]))","A",1
99,1,178,408,1.9377022,"\int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx","Integrate[(-a + b*Cot[c + d*x])*(a + b*Cot[c + d*x])^(3/2),x]","\frac{\sin ^2(c+d x) (b \cot (c+d x)-a) (a+b \cot (c+d x)) \left(3 i \sqrt{a-i b} \left(a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)-3 i \sqrt{a+i b} \left(a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)+2 b (a+b \cot (c+d x))^{3/2}\right)}{3 a^2 d \sin ^2(c+d x)-3 b^2 d \cos ^2(c+d x)}","\frac{b \left(a^2+b^2\right) \log \left(-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \cot (c+d x)}+\sqrt{a^2+b^2}+a+b \cot (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \left(a^2+b^2\right) \log \left(\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \cot (c+d x)}+\sqrt{a^2+b^2}+a+b \cot (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \left(a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{b \left(a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{2 b (a+b \cot (c+d x))^{3/2}}{3 d}",1,"((-a + b*Cot[c + d*x])*(a + b*Cot[c + d*x])*((3*I)*Sqrt[a - I*b]*(a^2 + b^2)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]] - (3*I)*Sqrt[a + I*b]*(a^2 + b^2)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]] + 2*b*(a + b*Cot[c + d*x])^(3/2))*Sin[c + d*x]^2)/(-3*b^2*d*Cos[c + d*x]^2 + 3*a^2*d*Sin[c + d*x]^2)","C",1
100,1,158,422,1.055063,"\int (-a+b \cot (c+d x)) \sqrt{a+b \cot (c+d x)} \, dx","Integrate[(-a + b*Cot[c + d*x])*Sqrt[a + b*Cot[c + d*x]],x]","\frac{\sin (c+d x) (b \cot (c+d x)-a) \left(\frac{i \left(a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}-\frac{i \left(a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}+2 b \sqrt{a+b \cot (c+d x)}\right)}{d (a \sin (c+d x)-b \cos (c+d x))}","-\frac{b \sqrt{a^2+b^2} \log \left(-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \cot (c+d x)}+\sqrt{a^2+b^2}+a+b \cot (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \sqrt{a^2+b^2} \log \left(\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \cot (c+d x)}+\sqrt{a^2+b^2}+a+b \cot (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{b \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{2 b \sqrt{a+b \cot (c+d x)}}{d}",1,"((-a + b*Cot[c + d*x])*((I*(a^2 + b^2)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] - (I*(a^2 + b^2)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b] + 2*b*Sqrt[a + b*Cot[c + d*x]])*Sin[c + d*x])/(d*(-(b*Cos[c + d*x]) + a*Sin[c + d*x]))","C",1
101,1,154,102,0.6110305,"\int \frac{A+B \cot (c+d x)}{\sqrt{a+b \cot (c+d x)}} \, dx","Integrate[(A + B*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]],x]","\frac{\sin (c+d x) (A+B \cot (c+d x)) \left(\sqrt{a+i b} (B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)+\sqrt{a-i b} (B-i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)\right)}{d \sqrt{a-i b} \sqrt{a+i b} (A \sin (c+d x)+B \cos (c+d x))}","\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}-\frac{(-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}",1,"((Sqrt[a + I*b]*(I*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]] + Sqrt[a - I*b]*((-I)*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])*(A + B*Cot[c + d*x])*Sin[c + d*x])/(Sqrt[a - I*b]*Sqrt[a + I*b]*d*(B*Cos[c + d*x] + A*Sin[c + d*x]))","A",1
102,1,226,138,1.6742007,"\int \frac{A+B \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^(3/2),x]","-\frac{\frac{\left(a A b-a \sqrt{-b^2} B+A \sqrt{-b^2} b+b^2 B\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{-b^2} \sqrt{a-\sqrt{-b^2}}}-\frac{\left(a A b+a \sqrt{-b^2} B-A \sqrt{-b^2} b+b^2 B\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{-b^2} \sqrt{a+\sqrt{-b^2}}}+\frac{2 (a B-A b)}{\sqrt{a+b \cot (c+d x)}}}{d \left(a^2+b^2\right)}","\frac{2 (A b-a B)}{d \left(a^2+b^2\right) \sqrt{a+b \cot (c+d x)}}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}-\frac{(-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"-((((a*A*b + A*b*Sqrt[-b^2] + b^2*B - a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) - ((a*A*b - A*b*Sqrt[-b^2] + b^2*B + a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a + Sqrt[-b^2]]) + (2*(-(A*b) + a*B))/Sqrt[a + b*Cot[c + d*x]])/((a^2 + b^2)*d))","A",1
103,1,319,185,3.5296049,"\int \frac{A+B \cot (c+d x)}{(a+b \cot (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^(5/2),x]","-\frac{\frac{2 \left(a^2+b^2\right) (a B-A b)}{(a+b \cot (c+d x))^{3/2}}+\frac{6 \left(a^2 B-2 a A b-b^2 B\right)}{\sqrt{a+b \cot (c+d x)}}+\frac{3 \left(a^2 \left(A b-\sqrt{-b^2} B\right)+2 a b \left(A \sqrt{-b^2}+b B\right)+b^2 \left(\sqrt{-b^2} B-A b\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{-b^2} \sqrt{a-\sqrt{-b^2}}}+\frac{3 \left(-\left(a^2 \left(A b+\sqrt{-b^2} B\right)\right)+2 a b \left(A \sqrt{-b^2}-b B\right)+b^2 \left(A b+\sqrt{-b^2} B\right)\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{-b^2} \sqrt{a+\sqrt{-b^2}}}}{3 d \left(a^2+b^2\right)^2}","\frac{2 (A b-a B)}{3 d \left(a^2+b^2\right) (a+b \cot (c+d x))^{3/2}}+\frac{2 \left(a^2 (-B)+2 a A b+b^2 B\right)}{d \left(a^2+b^2\right)^2 \sqrt{a+b \cot (c+d x)}}+\frac{(B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}-\frac{(-B+i A) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"-1/3*((3*(2*a*b*(A*Sqrt[-b^2] + b*B) + a^2*(A*b - Sqrt[-b^2]*B) + b^2*(-(A*b) + Sqrt[-b^2]*B))*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) + (3*(2*a*b*(A*Sqrt[-b^2] - b*B) - a^2*(A*b + Sqrt[-b^2]*B) + b^2*(A*b + Sqrt[-b^2]*B))*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a + Sqrt[-b^2]]) + (2*(a^2 + b^2)*(-(A*b) + a*B))/(a + b*Cot[c + d*x])^(3/2) + (6*(-2*a*A*b + a^2*B - b^2*B))/Sqrt[a + b*Cot[c + d*x]])/((a^2 + b^2)^2*d)","A",1
104,1,146,102,0.331053,"\int \frac{-a+b \cot (c+d x)}{\sqrt{a+b \cot (c+d x)}} \, dx","Integrate[(-a + b*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]],x]","\frac{b \left(\left(a+\sqrt{-b^2}\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)-\left(a-\sqrt{-b^2}\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)\right)}{\sqrt{-b^2} d \sqrt{a-\sqrt{-b^2}} \sqrt{a+\sqrt{-b^2}}}","\frac{(b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}-\frac{(-b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}",1,"(b*((a + Sqrt[-b^2])^(3/2)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - Sqrt[-b^2]]] - (a - Sqrt[-b^2])^(3/2)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + Sqrt[-b^2]]]))/(Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]*Sqrt[a + Sqrt[-b^2]]*d)","A",1
105,1,216,132,1.4732493,"\int \frac{-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx","Integrate[(-a + b*Cot[c + d*x])/(a + b*Cot[c + d*x])^(3/2),x]","\frac{\sin (c+d x) (a-b \cot (c+d x)) \left(\sqrt{a-i b} \left(i (a-i b)^2 \sqrt{a+b \cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)-4 a b \sqrt{a+i b}\right)-i (a+i b)^{5/2} \sqrt{a+b \cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)\right)}{d (a-i b)^{3/2} (a+i b)^{3/2} \sqrt{a+b \cot (c+d x)} (a \sin (c+d x)-b \cos (c+d x))}","-\frac{4 a b}{d \left(a^2+b^2\right) \sqrt{a+b \cot (c+d x)}}-\frac{(-b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}+\frac{(b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"((a - b*Cot[c + d*x])*((-I)*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]]*Sqrt[a + b*Cot[c + d*x]] + Sqrt[a - I*b]*(-4*a*Sqrt[a + I*b]*b + I*(a - I*b)^2*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]]*Sqrt[a + b*Cot[c + d*x]]))*Sin[c + d*x])/((a - I*b)^(3/2)*(a + I*b)^(3/2)*d*Sqrt[a + b*Cot[c + d*x]]*(-(b*Cos[c + d*x]) + a*Sin[c + d*x]))","A",1
106,1,232,174,5.9380856,"\int \frac{-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{5/2}} \, dx","Integrate[(-a + b*Cot[c + d*x])/(a + b*Cot[c + d*x])^(5/2),x]","\frac{\sin (c+d x) (b \cot (c+d x)-a) \left(-\frac{2 b (a+b \cot (c+d x)) \left(-11 a^3+\left(3 b^3-9 a^2 b\right) \cot (c+d x)+a b^2\right)}{\left(a^2+b^2\right)^2}+\frac{3 i (a+b \cot (c+d x))^{5/2} \left((a+i b)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)-(a-i b)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)\right)}{(a-i b)^{5/2} (a+i b)^{5/2}}\right)}{3 d (a+b \cot (c+d x))^{5/2} (a \sin (c+d x)-b \cos (c+d x))}","-\frac{2 b \left(3 a^2-b^2\right)}{d \left(a^2+b^2\right)^2 \sqrt{a+b \cot (c+d x)}}-\frac{4 a b}{3 d \left(a^2+b^2\right) (a+b \cot (c+d x))^{3/2}}-\frac{(-b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}+\frac{(b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"((-a + b*Cot[c + d*x])*(((3*I)*((a + I*b)^(7/2)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]] - (a - I*b)^(7/2)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])*(a + b*Cot[c + d*x])^(5/2))/((a - I*b)^(5/2)*(a + I*b)^(5/2)) - (2*b*(a + b*Cot[c + d*x])*(-11*a^3 + a*b^2 + (-9*a^2*b + 3*b^3)*Cot[c + d*x]))/(a^2 + b^2)^2)*Sin[c + d*x])/(3*d*(a + b*Cot[c + d*x])^(5/2)*(-(b*Cos[c + d*x]) + a*Sin[c + d*x]))","A",1