1,1,49,0,0.0279005,"\int (a+i a \cot (c+d x))^n \, dx","Int[(a + I*a*Cot[c + d*x])^n,x]","\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}","\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/2)*(a + I*a*Cot[c + d*x])^n*Hypergeometric2F1[1, n, 1 + n, (1 + I*Cot[c + d*x])/2])/(d*n)","A",2,2,15,0.1333,1,"{3481, 68}"
2,1,116,0,0.1573819,"\int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x)) \, dx","Int[(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x]),x]","\frac{2 a e^2 \sqrt{e \cot (c+d x)}}{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 (e \cot (c+d x))^{3/2}}{3 d}-\frac{2 a (e \cot (c+d x))^{5/2}}{5 d}","\frac{2 a e^2 \sqrt{e \cot (c+d x)}}{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 (e \cot (c+d x))^{3/2}}{3 d}-\frac{2 a (e \cot (c+d x))^{5/2}}{5 d}",1,"-((Sqrt[2]*a*e^(5/2)*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d) + (2*a*e^2*Sqrt[e*Cot[c + d*x]])/d - (2*a*e*(e*Cot[c + d*x])^(3/2))/(3*d) - (2*a*(e*Cot[c + d*x])^(5/2))/(5*d)","A",5,3,23,0.1304,1,"{3528, 3532, 208}"
3,1,94,0,0.1162955,"\int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx","Int[(e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x]),x]","-\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}","-\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,"-((Sqrt[2]*a*e^(3/2)*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d) - (2*a*e*Sqrt[e*Cot[c + d*x]])/d - (2*a*(e*Cot[c + d*x])^(3/2))/(3*d)","A",4,3,23,0.1304,1,"{3528, 3532, 205}"
4,1,71,0,0.0778965,"\int \sqrt{e \cot (c+d x)} (a+a \cot (c+d x)) \, dx","Int[Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x]),x]","\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}","\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,"(Sqrt[2]*a*Sqrt[e]*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d - (2*a*Sqrt[e*Cot[c + d*x]])/d","A",3,3,23,0.1304,1,"{3528, 3532, 208}"
5,1,49,0,0.0442104,"\int \frac{a+a \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx","Int[(a + a*Cot[c + d*x])/Sqrt[e*Cot[c + d*x]],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}}","\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,"(Sqrt[2]*a*ArcTan[(Sqrt[e]*(1 - Cot[c + d*x]))/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*Sqrt[e])","A",2,2,23,0.08696,1,"{3532, 205}"
6,1,75,0,0.0858923,"\int \frac{a+a \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx","Int[(a + a*Cot[c + d*x])/(e*Cot[c + d*x])^(3/2),x]","\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}}","\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,"-((Sqrt[2]*a*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*e^(3/2))) + (2*a)/(d*e*Sqrt[e*Cot[c + d*x]])","A",3,3,23,0.1304,1,"{3529, 3532, 208}"
7,1,99,0,0.1346291,"\int \frac{a+a \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx","Int[(a + a*Cot[c + d*x])/(e*Cot[c + d*x])^(5/2),x]","\frac{2 a}{d e^2 \sqrt{e \cot (c+d x)}}-\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}{3 d e (e \cot (c+d x))^{3/2}}","\frac{2 a}{d e^2 \sqrt{e \cot (c+d x)}}-\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}{3 d e (e \cot (c+d x))^{3/2}}",1,"-((Sqrt[2]*a*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*e^(5/2))) + (2*a)/(3*d*e*(e*Cot[c + d*x])^(3/2)) + (2*a)/(d*e^2*Sqrt[e*Cot[c + d*x]])","A",4,3,23,0.1304,1,"{3529, 3532, 205}"
8,1,269,0,0.2880154,"\int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^2 \, dx","Int[(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^2,x]","\frac{4 a^2 e^2 \sqrt{e \cot (c+d x)}}{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{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{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}","\frac{4 a^2 e^2 \sqrt{e \cot (c+d x)}}{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{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{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,"(Sqrt[2]*a^2*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/d - (Sqrt[2]*a^2*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/d + (4*a^2*e^2*Sqrt[e*Cot[c + d*x]])/d - (4*a^2*(e*Cot[c + d*x])^(5/2))/(5*d) - (2*a^2*(e*Cot[c + d*x])^(7/2))/(7*d*e) + (a^2*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d) - (a^2*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d)","A",16,12,25,0.4800,1,"{3543, 12, 16, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204}"
9,1,246,0,0.2345112,"\int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2 \, dx","Int[(e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])^2,x]","\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}","\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,"-((Sqrt[2]*a^2*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/d) + (Sqrt[2]*a^2*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/d - (4*a^2*(e*Cot[c + d*x])^(3/2))/(3*d) - (2*a^2*(e*Cot[c + d*x])^(5/2))/(5*d*e) + (a^2*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d) - (a^2*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d)","A",15,12,25,0.4800,1,"{3543, 12, 16, 3473, 3476, 329, 297, 1162, 617, 204, 1165, 628}"
10,1,244,0,0.2265575,"\int \sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^2 \, dx","Int[Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^2,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}","-\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,"-((Sqrt[2]*a^2*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/d) + (Sqrt[2]*a^2*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/d - (4*a^2*Sqrt[e*Cot[c + d*x]])/d - (2*a^2*(e*Cot[c + d*x])^(3/2))/(3*d*e) - (a^2*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d) + (a^2*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d)","A",15,12,25,0.4800,1,"{3543, 12, 16, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204}"
11,1,222,0,0.1996336,"\int \frac{(a+a \cot (c+d x))^2}{\sqrt{e \cot (c+d x)}} \, dx","Int[(a + a*Cot[c + d*x])^2/Sqrt[e*Cot[c + d*x]],x]","-\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}}","-\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,"(Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*Sqrt[e]) - (Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*Sqrt[e]) - (2*a^2*Sqrt[e*Cot[c + d*x]])/(d*e) - (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*Sqrt[e]) + (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*Sqrt[e])","A",14,11,25,0.4400,1,"{3543, 12, 16, 3476, 329, 297, 1162, 617, 204, 1165, 628}"
12,1,222,0,0.2084591,"\int \frac{(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx","Int[(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(3/2),x]","\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)}}","\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,"(Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*e^(3/2)) - (Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*e^(3/2)) + (2*a^2)/(d*e*Sqrt[e*Cot[c + d*x]]) + (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*e^(3/2)) - (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*e^(3/2))","A",13,10,25,0.4000,1,"{3542, 12, 3476, 329, 211, 1165, 628, 1162, 617, 204}"
13,1,247,0,0.2374261,"\int \frac{(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx","Int[(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(5/2),x]","\frac{4 a^2}{d e^2 \sqrt{e \cot (c+d x)}}+\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{2 a^2}{3 d e (e \cot (c+d x))^{3/2}}","\frac{4 a^2}{d e^2 \sqrt{e \cot (c+d x)}}+\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{2 a^2}{3 d e (e \cot (c+d x))^{3/2}}",1,"-((Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*e^(5/2))) + (Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*e^(5/2)) + (2*a^2)/(3*d*e*(e*Cot[c + d*x])^(3/2)) + (4*a^2)/(d*e^2*Sqrt[e*Cot[c + d*x]]) + (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*e^(5/2)) - (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*e^(5/2))","A",14,11,25,0.4400,1,"{3542, 12, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628}"
14,1,249,0,0.2372866,"\int \frac{(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx","Int[(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(7/2),x]","\frac{4 a^2}{3 d e^2 (e \cot (c+d x))^{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^{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{2 a^2}{5 d e (e \cot (c+d x))^{5/2}}","\frac{4 a^2}{3 d e^2 (e \cot (c+d x))^{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^{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{2 a^2}{5 d e (e \cot (c+d x))^{5/2}}",1,"-((Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*e^(7/2))) + (Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*e^(7/2)) + (2*a^2)/(5*d*e*(e*Cot[c + d*x])^(5/2)) + (4*a^2)/(3*d*e^2*(e*Cot[c + d*x])^(3/2)) - (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*e^(7/2)) + (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*e^(7/2))","A",14,11,25,0.4400,1,"{3542, 12, 3474, 3476, 329, 211, 1165, 628, 1162, 617, 204}"
15,1,186,0,0.3004418,"\int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \, dx","Int[(e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3,x]","\frac{4 a^3 e^2 \sqrt{e \cot (c+d x)}}{d}+\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{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}","\frac{4 a^3 e^2 \sqrt{e \cot (c+d x)}}{d}+\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{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*Sqrt[2]*a^3*e^(5/2)*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d + (4*a^3*e^2*Sqrt[e*Cot[c + d*x]])/d + (4*a^3*e*(e*Cot[c + d*x])^(3/2))/(3*d) - (4*a^3*(e*Cot[c + d*x])^(5/2))/(5*d) - (40*a^3*(e*Cot[c + d*x])^(7/2))/(63*d*e) - (2*(e*Cot[c + d*x])^(7/2)*(a^3 + a^3*Cot[c + d*x]))/(9*d*e)","A",7,5,25,0.2000,1,"{3566, 3630, 3528, 3532, 205}"
16,1,160,0,0.258923,"\int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx","Int[(e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])^3,x]","-\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}","-\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,"(-2*Sqrt[2]*a^3*e^(3/2)*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d + (4*a^3*e*Sqrt[e*Cot[c + d*x]])/d - (4*a^3*(e*Cot[c + d*x])^(3/2))/(3*d) - (32*a^3*(e*Cot[c + d*x])^(5/2))/(35*d*e) - (2*(e*Cot[c + d*x])^(5/2)*(a^3 + a^3*Cot[c + d*x]))/(7*d*e)","A",6,5,25,0.2000,1,"{3566, 3630, 3528, 3532, 208}"
17,1,138,0,0.2030405,"\int \sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx","Int[Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^3,x]","-\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}","-\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,"(-2*Sqrt[2]*a^3*Sqrt[e]*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d - (4*a^3*Sqrt[e*Cot[c + d*x]])/d - (8*a^3*(e*Cot[c + d*x])^(3/2))/(5*d*e) - (2*(e*Cot[c + d*x])^(3/2)*(a^3 + a^3*Cot[c + d*x]))/(5*d*e)","A",5,5,25,0.2000,1,"{3566, 3630, 3528, 3532, 205}"
18,1,117,0,0.1690998,"\int \frac{(a+a \cot (c+d x))^3}{\sqrt{e \cot (c+d x)}} \, dx","Int[(a + a*Cot[c + d*x])^3/Sqrt[e*Cot[c + d*x]],x]","-\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}}","-\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,"(2*Sqrt[2]*a^3*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*Sqrt[e]) - (16*a^3*Sqrt[e*Cot[c + d*x]])/(3*d*e) - (2*Sqrt[e*Cot[c + d*x]]*(a^3 + a^3*Cot[c + d*x]))/(3*d*e)","A",4,4,25,0.1600,1,"{3566, 3630, 3532, 208}"
19,1,114,0,0.1753592,"\int \frac{(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx","Int[(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(3/2),x]","-\frac{4 a^3 \sqrt{e \cot (c+d x)}}{d e^2}+\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{2 \left(a^3 \cot (c+d x)+a^3\right)}{d e \sqrt{e \cot (c+d x)}}","-\frac{4 a^3 \sqrt{e \cot (c+d x)}}{d e^2}+\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{2 \left(a^3 \cot (c+d x)+a^3\right)}{d e \sqrt{e \cot (c+d x)}}",1,"(2*Sqrt[2]*a^3*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*e^(3/2)) - (4*a^3*Sqrt[e*Cot[c + d*x]])/(d*e^2) + (2*(a^3 + a^3*Cot[c + d*x]))/(d*e*Sqrt[e*Cot[c + d*x]])","A",4,4,25,0.1600,1,"{3565, 3630, 3532, 205}"
20,1,117,0,0.1880378,"\int \frac{(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{5/2}} \, dx","Int[(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(5/2),x]","\frac{16 a^3}{3 d e^2 \sqrt{e \cot (c+d x)}}-\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{2 \left(a^3 \cot (c+d x)+a^3\right)}{3 d e (e \cot (c+d x))^{3/2}}","\frac{16 a^3}{3 d e^2 \sqrt{e \cot (c+d x)}}-\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{2 \left(a^3 \cot (c+d x)+a^3\right)}{3 d e (e \cot (c+d x))^{3/2}}",1,"(-2*Sqrt[2]*a^3*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*e^(5/2)) + (16*a^3)/(3*d*e^2*Sqrt[e*Cot[c + d*x]]) + (2*(a^3 + a^3*Cot[c + d*x]))/(3*d*e*(e*Cot[c + d*x])^(3/2))","A",4,4,25,0.1600,1,"{3565, 3628, 3532, 208}"
21,1,141,0,0.2287583,"\int \frac{(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx","Int[(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(7/2),x]","\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 \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{2 \left(a^3 \cot (c+d x)+a^3\right)}{5 d e (e \cot (c+d x))^{5/2}}","\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 \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{2 \left(a^3 \cot (c+d x)+a^3\right)}{5 d e (e \cot (c+d x))^{5/2}}",1,"(-2*Sqrt[2]*a^3*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*e^(7/2)) + (8*a^3)/(5*d*e^2*(e*Cot[c + d*x])^(3/2)) + (4*a^3)/(d*e^3*Sqrt[e*Cot[c + d*x]]) + (2*(a^3 + a^3*Cot[c + d*x]))/(5*d*e*(e*Cot[c + d*x])^(5/2))","A",5,5,25,0.2000,1,"{3565, 3628, 3529, 3532, 205}"
22,1,165,0,0.2958865,"\int \frac{(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx","Int[(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(9/2),x]","-\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 \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{2 \left(a^3 \cot (c+d x)+a^3\right)}{7 d e (e \cot (c+d x))^{7/2}}","-\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 \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{2 \left(a^3 \cot (c+d x)+a^3\right)}{7 d e (e \cot (c+d x))^{7/2}}",1,"(2*Sqrt[2]*a^3*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*e^(9/2)) + (32*a^3)/(35*d*e^2*(e*Cot[c + d*x])^(5/2)) + (4*a^3)/(3*d*e^3*(e*Cot[c + d*x])^(3/2)) - (4*a^3)/(d*e^4*Sqrt[e*Cot[c + d*x]]) + (2*(a^3 + a^3*Cot[c + d*x]))/(7*d*e*(e*Cot[c + d*x])^(7/2))","A",6,5,25,0.2000,1,"{3565, 3628, 3529, 3532, 208}"
23,1,111,0,0.4514039,"\int \frac{(e \cot (c+d x))^{5/2}}{a+a \cot (c+d x)} \, dx","Int[(e*Cot[c + d*x])^(5/2)/(a + a*Cot[c + d*x]),x]","-\frac{2 e^2 \sqrt{e \cot (c+d x)}}{a d}+\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}+\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}",1,"(e^(5/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(a*d) - (e^(5/2)*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(Sqrt[2]*a*d) - (2*e^2*Sqrt[e*Cot[c + d*x]])/(a*d)","A",7,6,25,0.2400,1,"{3566, 3653, 3532, 205, 3634, 63}"
24,1,87,0,0.239068,"\int \frac{(e \cot (c+d x))^{3/2}}{a+a \cot (c+d x)} \, dx","Int[(e*Cot[c + d*x])^(3/2)/(a + a*Cot[c + d*x]),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}","\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,"-((e^(3/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(a*d)) + (e^(3/2)*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(Sqrt[2]*a*d)","A",6,6,25,0.2400,1,"{3573, 3532, 208, 3634, 63, 205}"
25,1,87,0,0.2181336,"\int \frac{\sqrt{e \cot (c+d x)}}{a+a \cot (c+d x)} \, dx","Int[Sqrt[e*Cot[c + d*x]]/(a + a*Cot[c + d*x]),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}","\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[e]*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(a*d) + (Sqrt[e]*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(Sqrt[2]*a*d)","A",6,5,25,0.2000,1,"{3572, 3532, 205, 3634, 63}"
26,1,83,0,0.218891,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx","Int[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])),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}}","-\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,"-(ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]]/(a*d*Sqrt[e])) - ArcTanh[(Sqrt[e]*(1 + Cot[c + d*x]))/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])]/(Sqrt[2]*a*d*Sqrt[e])","A",6,6,25,0.2400,1,"{3574, 3532, 208, 3634, 63, 205}"
27,1,111,0,0.4517722,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx","Int[1/((e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])),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)}}","\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,"ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]]/(a*d*e^(3/2)) - ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])]/(Sqrt[2]*a*d*e^(3/2)) + 2/(a*d*e*Sqrt[e*Cot[c + d*x]])","A",7,6,25,0.2400,1,"{3569, 3653, 3532, 205, 3634, 63}"
28,1,135,0,0.5383331,"\int \frac{1}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))} \, dx","Int[1/((e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])),x]","-\frac{2}{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}{3 a d e (e \cot (c+d x))^{3/2}}","-\frac{2}{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}{3 a d e (e \cot (c+d x))^{3/2}}",1,"-(ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]]/(a*d*e^(5/2))) + ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])]/(Sqrt[2]*a*d*e^(5/2)) + 2/(3*a*d*e*(e*Cot[c + d*x])^(3/2)) - 2/(a*d*e^2*Sqrt[e*Cot[c + d*x]])","A",10,10,25,0.4000,1,"{3569, 3649, 12, 16, 3573, 3532, 208, 3634, 63, 205}"
29,1,281,0,0.5440043,"\int \frac{(e \cot (c+d x))^{5/2}}{(a+a \cot (c+d x))^2} \, dx","Int[(e*Cot[c + d*x])^(5/2)/(a + a*Cot[c + d*x])^2,x]","\frac{e^2 \sqrt{e \cot (c+d x)}}{2 d \left(a^2 \cot (c+d x)+a^2\right)}-\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)}-\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}",1,"(-3*e^(5/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(2*a^2*d) - (e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(2*Sqrt[2]*a^2*d) + (e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(2*Sqrt[2]*a^2*d) + (e^2*Sqrt[e*Cot[c + d*x]])/(2*d*(a^2 + a^2*Cot[c + d*x])) - (e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(4*Sqrt[2]*a^2*d) + (e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(4*Sqrt[2]*a^2*d)","A",17,14,25,0.5600,1,"{3565, 3653, 12, 3476, 329, 211, 1165, 628, 1162, 617, 204, 3634, 63, 205}"
30,1,279,0,0.5636294,"\int \frac{(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^2} \, dx","Int[(e*Cot[c + d*x])^(3/2)/(a + a*Cot[c + d*x])^2,x]","-\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)}","-\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,"(e^(3/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(2*a^2*d) + (e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(2*Sqrt[2]*a^2*d) - (e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(2*Sqrt[2]*a^2*d) - (e*Sqrt[e*Cot[c + d*x]])/(2*d*(a^2 + a^2*Cot[c + d*x])) - (e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(4*Sqrt[2]*a^2*d) + (e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(4*Sqrt[2]*a^2*d)","A",18,15,25,0.6000,1,"{3567, 3653, 12, 16, 3476, 329, 297, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
31,1,278,0,0.531331,"\int \frac{\sqrt{e \cot (c+d x)}}{(a+a \cot (c+d x))^2} \, dx","Int[Sqrt[e*Cot[c + d*x]]/(a + a*Cot[c + d*x])^2,x]","\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}","\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]*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(2*a^2*d) + (Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(2*Sqrt[2]*a^2*d) - (Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(2*Sqrt[2]*a^2*d) + Sqrt[e*Cot[c + d*x]]/(2*d*(a^2 + a^2*Cot[c + d*x])) + (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(4*Sqrt[2]*a^2*d) - (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(4*Sqrt[2]*a^2*d)","A",17,14,25,0.5600,1,"{3568, 3653, 12, 3476, 329, 211, 1165, 628, 1162, 617, 204, 3634, 63, 205}"
32,1,281,0,0.5658054,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^2} \, dx","Int[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^2),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}}","-\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,"(-3*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(2*a^2*d*Sqrt[e]) - ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(2*Sqrt[2]*a^2*d*Sqrt[e]) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(2*Sqrt[2]*a^2*d*Sqrt[e]) - Sqrt[e*Cot[c + d*x]]/(2*d*e*(a^2 + a^2*Cot[c + d*x])) + Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]]/(4*Sqrt[2]*a^2*d*Sqrt[e]) - Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]]/(4*Sqrt[2]*a^2*d*Sqrt[e])","A",18,15,25,0.6000,1,"{3569, 3653, 12, 16, 3476, 329, 297, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
33,1,306,0,0.7956351,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2} \, dx","Int[1/((e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])^2),x]","-\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)}}","-\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,"(5*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(2*a^2*d*e^(3/2)) - ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(2*Sqrt[2]*a^2*d*e^(3/2)) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(2*Sqrt[2]*a^2*d*e^(3/2)) + 5/(2*a^2*d*e*Sqrt[e*Cot[c + d*x]]) - 1/(2*d*e*Sqrt[e*Cot[c + d*x]]*(a^2 + a^2*Cot[c + d*x])) - Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]]/(4*Sqrt[2]*a^2*d*e^(3/2)) + Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]]/(4*Sqrt[2]*a^2*d*e^(3/2))","A",18,15,25,0.6000,1,"{3569, 3649, 3653, 12, 3476, 329, 211, 1165, 628, 1162, 617, 204, 3634, 63, 205}"
34,1,331,0,1.0753887,"\int \frac{1}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^2} \, dx","Int[1/((e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^2),x]","-\frac{9}{2 a^2 d e^2 \sqrt{e \cot (c+d x)}}-\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{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}}","-\frac{9}{2 a^2 d e^2 \sqrt{e \cot (c+d x)}}-\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{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,"(-7*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(2*a^2*d*e^(5/2)) + ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(2*Sqrt[2]*a^2*d*e^(5/2)) - ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(2*Sqrt[2]*a^2*d*e^(5/2)) + 7/(6*a^2*d*e*(e*Cot[c + d*x])^(3/2)) - 9/(2*a^2*d*e^2*Sqrt[e*Cot[c + d*x]]) - 1/(2*d*e*(e*Cot[c + d*x])^(3/2)*(a^2 + a^2*Cot[c + d*x])) - Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]]/(4*Sqrt[2]*a^2*d*e^(5/2)) + Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]]/(4*Sqrt[2]*a^2*d*e^(5/2))","A",20,16,25,0.6400,1,"{3569, 3649, 3653, 12, 16, 3476, 329, 297, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
35,1,164,0,0.6174292,"\int \frac{(e \cot (c+d x))^{5/2}}{(a+a \cot (c+d x))^3} \, dx","Int[(e*Cot[c + d*x])^(5/2)/(a + a*Cot[c + d*x])^3,x]","-\frac{5 e^2 \sqrt{e \cot (c+d x)}}{8 a^3 d (\cot (c+d x)+1)}-\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{e^2 \sqrt{e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}","-\frac{5 e^2 \sqrt{e \cot (c+d x)}}{8 a^3 d (\cot (c+d x)+1)}-\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{e^2 \sqrt{e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}",1,"-(e^(5/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d) + (e^(5/2)*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(2*Sqrt[2]*a^3*d) - (5*e^2*Sqrt[e*Cot[c + d*x]])/(8*a^3*d*(1 + Cot[c + d*x])) + (e^2*Sqrt[e*Cot[c + d*x]])/(4*a*d*(a + a*Cot[c + d*x])^2)","A",8,8,25,0.3200,1,"{3565, 3649, 3654, 3532, 208, 3634, 63, 205}"
36,1,164,0,0.6625918,"\int \frac{(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^3} \, dx","Int[(e*Cot[c + d*x])^(3/2)/(a + a*Cot[c + d*x])^3,x]","\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}","\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,"(5*e^(3/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d) + (e^(3/2)*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(2*Sqrt[2]*a^3*d) - (e*Sqrt[e*Cot[c + d*x]])/(4*a*d*(a + a*Cot[c + d*x])^2) + (e*Sqrt[e*Cot[c + d*x]])/(8*d*(a^3 + a^3*Cot[c + d*x]))","A",8,7,25,0.2800,1,"{3567, 3649, 3653, 3532, 205, 3634, 63}"
37,1,161,0,0.5923297,"\int \frac{\sqrt{e \cot (c+d x)}}{(a+a \cot (c+d x))^3} \, dx","Int[Sqrt[e*Cot[c + d*x]]/(a + a*Cot[c + d*x])^3,x]","\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}","\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,"-(Sqrt[e]*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d) - (Sqrt[e]*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(2*Sqrt[2]*a^3*d) + Sqrt[e*Cot[c + d*x]]/(4*a*d*(a + a*Cot[c + d*x])^2) + (3*Sqrt[e*Cot[c + d*x]])/(8*d*(a^3 + a^3*Cot[c + d*x]))","A",8,8,25,0.3200,1,"{3568, 3649, 3654, 3532, 208, 3634, 63, 205}"
38,1,165,0,0.6477558,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^3} \, dx","Int[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^3),x]","-\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}","-\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,"(-11*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d*Sqrt[e]) - ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])]/(2*Sqrt[2]*a^3*d*Sqrt[e]) - (7*Sqrt[e*Cot[c + d*x]])/(8*a^3*d*e*(1 + Cot[c + d*x])) - Sqrt[e*Cot[c + d*x]]/(4*a*d*e*(a + a*Cot[c + d*x])^2)","A",8,7,25,0.2800,1,"{3569, 3649, 3653, 3532, 205, 3634, 63}"
39,1,189,0,0.8633122,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3} \, dx","Int[1/((e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])^3),x]","\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)}}","\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,"(31*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d*e^(3/2)) + ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])]/(2*Sqrt[2]*a^3*d*e^(3/2)) + 27/(8*a^3*d*e*Sqrt[e*Cot[c + d*x]]) - 9/(8*a^3*d*e*Sqrt[e*Cot[c + d*x]]*(1 + Cot[c + d*x])) - 1/(4*a*d*e*Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^2)","A",9,8,25,0.3200,1,"{3569, 3649, 3654, 3532, 208, 3634, 63, 205}"
40,1,215,0,1.1045898,"\int \frac{1}{(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3} \, dx","Int[1/((e*Cot[c + d*x])^(5/2)*(a + a*Cot[c + d*x])^3),x]","-\frac{63}{8 a^3 d e^2 \sqrt{e \cot (c+d x)}}-\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{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}}","-\frac{63}{8 a^3 d e^2 \sqrt{e \cot (c+d x)}}-\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{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,"(-59*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d*e^(5/2)) + ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])]/(2*Sqrt[2]*a^3*d*e^(5/2)) + 55/(24*a^3*d*e*(e*Cot[c + d*x])^(3/2)) - 63/(8*a^3*d*e^2*Sqrt[e*Cot[c + d*x]]) - 11/(8*a^3*d*e*(e*Cot[c + d*x])^(3/2)*(1 + Cot[c + d*x])) - 1/(4*a*d*e*(e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])^2)","A",10,8,25,0.3200,1,"{3569, 3649, 3650, 3653, 3532, 205, 3634, 63}"
41,1,223,0,0.2664326,"\int \cot ^2(x) \sqrt{1+\cot (x)} \, dx","Int[Cot[x]^2*Sqrt[1 + Cot[x]],x]","-\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)","-\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,"-(Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]]) + Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] - (2*(1 + Cot[x])^(3/2))/3 + Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[2*(1 + Sqrt[2])]) - Log[1 + Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[2*(1 + Sqrt[2])])","A",12,9,13,0.6923,1,"{3543, 3485, 700, 1127, 1161, 618, 204, 1164, 628}"
42,1,135,0,0.2402905,"\int \cot (x) \sqrt{1+\cot (x)} \, dx","Int[Cot[x]*Sqrt[1 + Cot[x]],x]","-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)","-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 + Sqrt[2])/2]*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2])*Cot[x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Cot[x]])] + Sqrt[(1 + Sqrt[2])/2]*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt[2])*Cot[x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Cot[x]])] - 2*Sqrt[1 + Cot[x]]","A",6,5,11,0.4545,1,"{3528, 3536, 3535, 203, 207}"
43,1,139,0,0.2347658,"\int \cot ^2(x) (1+\cot (x))^{3/2} \, dx","Int[Cot[x]^2*(1 + Cot[x])^(3/2),x]","-\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)","-\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,"-(Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Cot[x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])]) - Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Cot[x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])] + 2*Sqrt[1 + Cot[x]] - (2*(1 + Cot[x])^(5/2))/5","A",8,7,13,0.5385,1,"{3543, 3482, 12, 3536, 3535, 203, 207}"
44,1,221,0,0.2086224,"\int \cot (x) (1+\cot (x))^{3/2} \, dx","Int[Cot[x]*(1 + Cot[x])^(3/2),x]","-\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)","-\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,"-(Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]]) + Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] - 2*Sqrt[1 + Cot[x]] - (2*(1 + Cot[x])^(3/2))/3 - Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[1 + Sqrt[2]]) + Log[1 + Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[1 + Sqrt[2]])","A",14,9,11,0.8182,1,"{3528, 12, 3485, 708, 1094, 634, 618, 204, 628}"
45,1,214,0,0.1850294,"\int \frac{\cot ^2(x)}{\sqrt{1+\cot (x)}} \, dx","Int[Cot[x]^2/Sqrt[1 + Cot[x]],x]","-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)","-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,"-(Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]])/2 + (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]])/2 - 2*Sqrt[1 + Cot[x]] - Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(4*Sqrt[1 + Sqrt[2]]) + Log[1 + Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(4*Sqrt[1 + Sqrt[2]])","A",12,8,13,0.6154,1,"{3543, 3485, 708, 1094, 634, 618, 204, 628}"
46,1,121,0,0.127488,"\int \frac{\cot (x)}{\sqrt{1+\cot (x)}} \, dx","Int[Cot[x]/Sqrt[1 + Cot[x]],x]","\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)","\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,"(Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Cot[x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])])/2 + (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Cot[x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])])/2","A",5,4,11,0.3636,1,"{3536, 3535, 203, 207}"
47,1,139,0,0.1933981,"\int \frac{\cot ^2(x)}{(1+\cot (x))^{3/2}} \, dx","Int[Cot[x]^2/(1 + Cot[x])^(3/2),x]","\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)","\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 + Sqrt[2])/2]*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2])*Cot[x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Cot[x]])])/2 + (Sqrt[(1 + Sqrt[2])/2]*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt[2])*Cot[x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Cot[x]])])/2 + 1/Sqrt[1 + Cot[x]]","A",6,5,13,0.3846,1,"{3542, 3536, 3535, 203, 207}"
48,1,226,0,0.193162,"\int \frac{\cot (x)}{(1+\cot (x))^{3/2}} \, dx","Int[Cot[x]/(1 + Cot[x])^(3/2),x]","-\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)","-\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,"(Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]])/2 - (Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]])/2 - 1/Sqrt[1 + Cot[x]] - Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(4*Sqrt[2*(1 + Sqrt[2])]) + Log[1 + Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(4*Sqrt[2*(1 + Sqrt[2])])","A",13,10,11,0.9091,1,"{3529, 21, 3485, 700, 1127, 1161, 618, 204, 1164, 628}"
49,1,143,0,0.2045846,"\int \frac{\cot ^2(x)}{(1+\cot (x))^{5/2}} \, dx","Int[Cot[x]^2/(1 + Cot[x])^(5/2),x]","-\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)","-\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,"(Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Cot[x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])])/4 + (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Cot[x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])])/4 + 1/(3*(1 + Cot[x])^(3/2)) - 1/Sqrt[1 + Cot[x]]","A",8,7,13,0.5385,1,"{3542, 3529, 12, 3536, 3535, 203, 207}"
50,1,216,0,0.176363,"\int \frac{\cot (x)}{(1+\cot (x))^{5/2}} \, dx","Int[Cot[x]/(1 + Cot[x])^(5/2),x]","-\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)","-\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,"(Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]])/4 - (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]])/4 - 1/(3*(1 + Cot[x])^(3/2)) + Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(8*Sqrt[1 + Sqrt[2]]) - Log[1 + Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(8*Sqrt[1 + Sqrt[2]])","A",13,9,11,0.8182,1,"{3529, 21, 3485, 708, 1094, 634, 618, 204, 628}"
51,1,247,0,0.2071189,"\int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x)) \, dx","Int[(e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x]),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}+\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}","-\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,"-(((a + b)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d)) + ((a + b)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (2*a*e*Sqrt[e*Cot[c + d*x]])/d - (2*b*(e*Cot[c + d*x])^(3/2))/(3*d) - ((a - b)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d) + ((a - b)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d)","A",12,8,23,0.3478,1,"{3528, 3534, 1168, 1162, 617, 204, 1165, 628}"
52,1,226,0,0.1722158,"\int \sqrt{e \cot (c+d x)} (a+b \cot (c+d x)) \, dx","Int[Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x]),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}+\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}","-\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,"((a - b)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - ((a - b)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (2*b*Sqrt[e*Cot[c + d*x]])/d - ((a + b)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d) + ((a + b)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d)","A",11,8,23,0.3478,1,"{3528, 3534, 1168, 1162, 617, 204, 1165, 628}"
53,1,208,0,0.1416262,"\int \frac{a+b \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx","Int[(a + b*Cot[c + d*x])/Sqrt[e*Cot[c + d*x]],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}}","\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,"((a + b)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - ((a + b)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) + ((a - b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) - ((a - b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e])","A",10,7,23,0.3043,1,"{3534, 1168, 1162, 617, 204, 1165, 628}"
54,1,229,0,0.1937903,"\int \frac{a+b \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx","Int[(a + b*Cot[c + d*x])/(e*Cot[c + d*x])^(3/2),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}}-\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)}}","\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,"-(((a - b)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2))) + ((a - b)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) + (2*a)/(d*e*Sqrt[e*Cot[c + d*x]]) + ((a + b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2)) - ((a + b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2))","A",11,8,23,0.3478,1,"{3529, 3534, 1168, 1162, 617, 204, 1165, 628}"
55,1,252,0,0.2581929,"\int \frac{a+b \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx","Int[(a + b*Cot[c + d*x])/(e*Cot[c + d*x])^(5/2),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^{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)}}","-\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,"-(((a + b)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2))) + ((a + b)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2)) + (2*a)/(3*d*e*(e*Cot[c + d*x])^(3/2)) + (2*b)/(d*e^2*Sqrt[e*Cot[c + d*x]]) - ((a - b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2)) + ((a - b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2))","A",12,8,23,0.3478,1,"{3529, 3534, 1168, 1162, 617, 204, 1165, 628}"
56,1,317,0,0.3336633,"\int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2 \, dx","Int[(e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x])^2,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}","-\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,"-(((a^2 + 2*a*b - b^2)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d)) + ((a^2 + 2*a*b - b^2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (2*(a^2 - b^2)*e*Sqrt[e*Cot[c + d*x]])/d - (4*a*b*(e*Cot[c + d*x])^(3/2))/(3*d) - (2*b^2*(e*Cot[c + d*x])^(5/2))/(5*d*e) - ((a^2 - 2*a*b - b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d) + ((a^2 - 2*a*b - b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d)","A",13,9,25,0.3600,1,"{3543, 3528, 3534, 1168, 1162, 617, 204, 1165, 628}"
57,1,288,0,0.2745582,"\int \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2 \, dx","Int[Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^2,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}","-\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,"((a^2 - 2*a*b - b^2)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - ((a^2 - 2*a*b - b^2)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (4*a*b*Sqrt[e*Cot[c + d*x]])/d - (2*b^2*(e*Cot[c + d*x])^(3/2))/(3*d*e) - ((a^2 + 2*a*b - b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d) + ((a^2 + 2*a*b - b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d)","A",12,9,25,0.3600,1,"{3543, 3528, 3534, 1168, 1162, 617, 204, 1165, 628}"
58,1,267,0,0.2493933,"\int \frac{(a+b \cot (c+d x))^2}{\sqrt{e \cot (c+d x)}} \, dx","Int[(a + b*Cot[c + d*x])^2/Sqrt[e*Cot[c + d*x]],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}","\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,"((a^2 + 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - ((a^2 + 2*a*b - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - (2*b^2*Sqrt[e*Cot[c + d*x]])/(d*e) + ((a^2 - 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) - ((a^2 - 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e])","A",11,8,25,0.3200,1,"{3543, 3534, 1168, 1162, 617, 204, 1165, 628}"
59,1,267,0,0.2576256,"\int \frac{(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx","Int[(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(3/2),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}}-\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)}}","\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,"-(((a^2 - 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2))) + ((a^2 - 2*a*b - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) + (2*a^2)/(d*e*Sqrt[e*Cot[c + d*x]]) + ((a^2 + 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2)) - ((a^2 + 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2))","A",11,8,25,0.3200,1,"{3542, 3534, 1168, 1162, 617, 204, 1165, 628}"
60,1,291,0,0.3343385,"\int \frac{(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx","Int[(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(5/2),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^{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)}}","-\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,"-(((a^2 + 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2))) + ((a^2 + 2*a*b - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2)) + (2*a^2)/(3*d*e*(e*Cot[c + d*x])^(3/2)) + (4*a*b)/(d*e^2*Sqrt[e*Cot[c + d*x]]) - ((a^2 - 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2)) + ((a^2 - 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2))","A",12,9,25,0.3600,1,"{3542, 3529, 3534, 1168, 1162, 617, 204, 1165, 628}"
61,1,322,0,0.4279517,"\int \frac{(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx","Int[(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(7/2),x]","-\frac{2 \left(a^2-b^2\right)}{d e^3 \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^{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 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}}","-\frac{2 \left(a^2-b^2\right)}{d e^3 \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^{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 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,"((a^2 - 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) - ((a^2 - 2*a*b - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) + (2*a^2)/(5*d*e*(e*Cot[c + d*x])^(5/2)) + (4*a*b)/(3*d*e^2*(e*Cot[c + d*x])^(3/2)) - (2*(a^2 - b^2))/(d*e^3*Sqrt[e*Cot[c + d*x]]) - ((a^2 + 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2)) + ((a^2 + 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2))","A",13,9,25,0.3600,1,"{3542, 3529, 3534, 1168, 1162, 617, 204, 1165, 628}"
62,1,372,0,0.5636925,"\int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3 \, dx","Int[(e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x])^3,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}","-\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,"-(((a - b)*(a^2 + 4*a*b + b^2)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (2*a*(a^2 - 3*b^2)*e*Sqrt[e*Cot[c + d*x]])/d - (2*b*(3*a^2 - b^2)*(e*Cot[c + d*x])^(3/2))/(3*d) - (32*a*b^2*(e*Cot[c + d*x])^(5/2))/(35*d*e) - (2*b^2*(e*Cot[c + d*x])^(5/2)*(a + b*Cot[c + d*x]))/(7*d*e) - ((a + b)*(a^2 - 4*a*b + b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d) + ((a + b)*(a^2 - 4*a*b + b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d)","A",14,10,25,0.4000,1,"{3566, 3630, 3528, 3534, 1168, 1162, 617, 204, 1165, 628}"
63,1,342,0,0.4769959,"\int \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^3 \, dx","Int[Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^3,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}","-\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,"((a + b)*(a^2 - 4*a*b + b^2)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (2*b*(3*a^2 - b^2)*Sqrt[e*Cot[c + d*x]])/d - (8*a*b^2*(e*Cot[c + d*x])^(3/2))/(5*d*e) - (2*b^2*(e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x]))/(5*d*e) - ((a - b)*(a^2 + 4*a*b + b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d)","A",13,10,25,0.4000,1,"{3566, 3630, 3528, 3534, 1168, 1162, 617, 204, 1165, 628}"
64,1,313,0,0.4253661,"\int \frac{(a+b \cot (c+d x))^3}{\sqrt{e \cot (c+d x)}} \, dx","Int[(a + b*Cot[c + d*x])^3/Sqrt[e*Cot[c + d*x]],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}","\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,"((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - (16*a*b^2*Sqrt[e*Cot[c + d*x]])/(3*d*e) - (2*b^2*Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x]))/(3*d*e) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e])","A",12,9,25,0.3600,1,"{3566, 3630, 3534, 1168, 1162, 617, 204, 1165, 628}"
65,1,313,0,0.4198928,"\int \frac{(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx","Int[(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(3/2),x]","-\frac{2 b \left(a^2+b^2\right) \sqrt{e \cot (c+d x)}}{d e^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) \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 a^2 (a+b \cot (c+d x))}{d e \sqrt{e \cot (c+d x)}}","-\frac{2 b \left(a^2+b^2\right) \sqrt{e \cot (c+d x)}}{d e^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) \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 a^2 (a+b \cot (c+d x))}{d e \sqrt{e \cot (c+d x)}}",1,"-(((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2))) + ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) - (2*b*(a^2 + b^2)*Sqrt[e*Cot[c + d*x]])/(d*e^2) + (2*a^2*(a + b*Cot[c + d*x]))/(d*e*Sqrt[e*Cot[c + d*x]]) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2)) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2))","A",12,9,25,0.3600,1,"{3565, 3630, 3534, 1168, 1162, 617, 204, 1165, 628}"
66,1,313,0,0.4575407,"\int \frac{(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{5/2}} \, dx","Int[(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(5/2),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}}","-\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,"-(((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2))) + ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2)) + (16*a^2*b)/(3*d*e^2*Sqrt[e*Cot[c + d*x]]) + (2*a^2*(a + b*Cot[c + d*x]))/(3*d*e*(e*Cot[c + d*x])^(3/2)) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2)) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2))","A",12,9,25,0.3600,1,"{3565, 3628, 3534, 1168, 1162, 617, 204, 1165, 628}"
67,1,343,0,0.5614048,"\int \frac{(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx","Int[(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(7/2),x]","-\frac{2 a \left(a^2-3 b^2\right)}{d e^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^{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{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}}","-\frac{2 a \left(a^2-3 b^2\right)}{d e^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^{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{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,"((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) + (8*a^2*b)/(5*d*e^2*(e*Cot[c + d*x])^(3/2)) - (2*a*(a^2 - 3*b^2))/(d*e^3*Sqrt[e*Cot[c + d*x]]) + (2*a^2*(a + b*Cot[c + d*x]))/(5*d*e*(e*Cot[c + d*x])^(5/2)) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2)) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2))","A",13,10,25,0.4000,1,"{3565, 3628, 3529, 3534, 1168, 1162, 617, 204, 1165, 628}"
68,1,377,0,0.6623372,"\int \frac{(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx","Int[(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(9/2),x]","-\frac{2 a \left(a^2-3 b^2\right)}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac{2 b \left(3 a^2-b^2\right)}{d e^4 \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^{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{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}}","-\frac{2 a \left(a^2-3 b^2\right)}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac{2 b \left(3 a^2-b^2\right)}{d e^4 \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^{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{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,"((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(9/2)) - ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(9/2)) + (32*a^2*b)/(35*d*e^2*(e*Cot[c + d*x])^(5/2)) - (2*a*(a^2 - 3*b^2))/(3*d*e^3*(e*Cot[c + d*x])^(3/2)) - (2*b*(3*a^2 - b^2))/(d*e^4*Sqrt[e*Cot[c + d*x]]) + (2*a^2*(a + b*Cot[c + d*x]))/(7*d*e*(e*Cot[c + d*x])^(7/2)) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(9/2)) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(9/2))","A",14,10,25,0.4000,1,"{3565, 3628, 3529, 3534, 1168, 1162, 617, 204, 1165, 628}"
69,1,325,0,0.6603185,"\int \frac{(e \cot (c+d x))^{5/2}}{a+b \cot (c+d x)} \, dx","Int[(e*Cot[c + d*x])^(5/2)/(a + b*Cot[c + d*x]),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{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{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 e^2 \sqrt{e \cot (c+d x)}}{b d}","\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{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{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 e^2 \sqrt{e \cot (c+d x)}}{b d}",1,"(2*a^(5/2)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(b^(3/2)*(a^2 + b^2)*d) - ((a + b)*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d) + ((a + b)*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*e^2*Sqrt[e*Cot[c + d*x]])/(b*d) + ((a - b)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d)","A",15,12,25,0.4800,1,"{3566, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
70,1,302,0,0.3787099,"\int \frac{(e \cot (c+d x))^{3/2}}{a+b \cot (c+d x)} \, dx","Int[(e*Cot[c + d*x])^(3/2)/(a + b*Cot[c + d*x]),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{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)}-\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{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{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)}-\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)}",1,"(-2*a^(3/2)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(Sqrt[b]*(a^2 + b^2)*d) - ((a - b)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d) + ((a - b)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a + b)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d)","A",14,11,25,0.4400,1,"{3573, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
71,1,302,0,0.3756145,"\int \frac{\sqrt{e \cot (c+d x)}}{a+b \cot (c+d x)} \, dx","Int[Sqrt[e*Cot[c + d*x]]/(a + b*Cot[c + d*x]),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)}","-\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,"(2*Sqrt[a]*Sqrt[b]*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/((a^2 + b^2)*d) + ((a + b)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a - b)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d)","A",14,11,25,0.4400,1,"{3572, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
72,1,302,0,0.369796,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))} \, dx","Int[1/(Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])),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{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)}+\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{(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{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)}+\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)}",1,"(-2*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(Sqrt[a]*(a^2 + b^2)*d*Sqrt[e]) + ((a - b)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d*Sqrt[e]) - ((a - b)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d*Sqrt[e]) + ((a + b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d*Sqrt[e]) - ((a + b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d*Sqrt[e])","A",14,11,25,0.4400,1,"{3574, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
73,1,325,0,0.6596984,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx","Int[1/((e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x])),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{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{(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}{a d e \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{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{(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}{a d e \sqrt{e \cot (c+d x)}}",1,"(2*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(a^(3/2)*(a^2 + b^2)*d*e^(3/2)) - ((a + b)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d*e^(3/2)) + ((a + b)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d*e^(3/2)) + 2/(a*d*e*Sqrt[e*Cot[c + d*x]]) + ((a - b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d*e^(3/2)) - ((a - b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d*e^(3/2))","A",15,12,25,0.4800,1,"{3569, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
74,1,351,0,0.962181,"\int \frac{1}{(e \cot (c+d x))^{5/2} (a+b \cot (c+d x))} \, dx","Int[1/((e*Cot[c + d*x])^(5/2)*(a + b*Cot[c + d*x])),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^{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{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{(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}{3 a d e (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{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{(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}{3 a d e (e \cot (c+d x))^{3/2}}",1,"(-2*b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(a^(5/2)*(a^2 + b^2)*d*e^(5/2)) - ((a - b)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d*e^(5/2)) + ((a - b)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d*e^(5/2)) + 2/(3*a*d*e*(e*Cot[c + d*x])^(3/2)) - (2*b)/(a^2*d*e^2*Sqrt[e*Cot[c + d*x]]) - ((a + b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d*e^(5/2)) + ((a + b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d*e^(5/2))","A",16,13,25,0.5200,1,"{3569, 3649, 3654, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
75,1,437,0,1.1083293,"\int \frac{(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^2} \, dx","Int[(e*Cot[c + d*x])^(7/2)/(a + b*Cot[c + d*x])^2,x]","-\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{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{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}+\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{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{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}+\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}",1,"(a^(5/2)*(3*a^2 + 7*b^2)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(b^(5/2)*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*e^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*e^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((3*a^2 + 2*b^2)*e^3*Sqrt[e*Cot[c + d*x]])/(b^2*(a^2 + b^2)*d) + (a^2*e^2*(e*Cot[c + d*x])^(3/2))/(b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])) + ((a^2 + 2*a*b - b^2)*e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)","A",16,13,25,0.5200,1,"{3565, 3647, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
76,1,393,0,0.7395336,"\int \frac{(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^2} \, dx","Int[(e*Cot[c + d*x])^(5/2)/(a + b*Cot[c + d*x])^2,x]","\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{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{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}-\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{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{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}-\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}",1,"-((a^(3/2)*(a^2 + 5*b^2)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(b^(3/2)*(a^2 + b^2)^2*d)) - ((a^2 + 2*a*b - b^2)*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (a^2*e^2*Sqrt[e*Cot[c + d*x]])/(b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])) + ((a^2 - 2*a*b - b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)","A",15,12,25,0.4800,1,"{3565, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
77,1,387,0,0.676897,"\int \frac{(e \cot (c+d x))^{3/2}}{(a+b \cot (c+d x))^2} \, dx","Int[(e*Cot[c + d*x])^(3/2)/(a + b*Cot[c + d*x])^2,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))}","-\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,"-((Sqrt[a]*(a^2 - 3*b^2)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(Sqrt[b]*(a^2 + b^2)^2*d)) - ((a^2 - 2*a*b - b^2)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (a*e*Sqrt[e*Cot[c + d*x]])/((a^2 + b^2)*d*(a + b*Cot[c + d*x])) - ((a^2 + 2*a*b - b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)","A",15,12,25,0.4800,1,"{3567, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
78,1,386,0,0.6459523,"\int \frac{\sqrt{e \cot (c+d x)}}{(a+b \cot (c+d x))^2} \, dx","Int[Sqrt[e*Cot[c + d*x]]/(a + b*Cot[c + d*x])^2,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}","\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[b]*(3*a^2 - b^2)*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(Sqrt[a]*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (b*Sqrt[e*Cot[c + d*x]])/((a^2 + b^2)*d*(a + b*Cot[c + d*x])) - ((a^2 - 2*a*b - b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)","A",15,12,25,0.4800,1,"{3568, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
79,1,394,0,0.7380444,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2} \, dx","Int[1/(Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^2),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{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}+\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^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{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}+\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}",1,"-((b^(3/2)*(5*a^2 + b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(a^(3/2)*(a^2 + b^2)^2*d*Sqrt[e])) + ((a^2 - 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d*Sqrt[e]) - ((a^2 - 2*a*b - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d*Sqrt[e]) - (b^2*Sqrt[e*Cot[c + d*x]])/(a*(a^2 + b^2)*d*e*(a + b*Cot[c + d*x])) + ((a^2 + 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d*Sqrt[e]) - ((a^2 + 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d*Sqrt[e])","A",15,12,25,0.4800,1,"{3569, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
80,1,437,0,1.0949583,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2} \, dx","Int[1/((e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x])^2),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{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}-\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{\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{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}-\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)}}",1,"(b^(5/2)*(7*a^2 + 3*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(a^(5/2)*(a^2 + b^2)^2*d*e^(3/2)) - ((a^2 + 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d*e^(3/2)) + ((a^2 + 2*a*b - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d*e^(3/2)) + (2*a^2 + 3*b^2)/(a^2*(a^2 + b^2)*d*e*Sqrt[e*Cot[c + d*x]]) - b^2/(a*(a^2 + b^2)*d*e*Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])) + ((a^2 - 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d*e^(3/2)) - ((a^2 - 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d*e^(3/2))","A",16,13,25,0.5200,1,"{3569, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
81,1,529,0,1.6305611,"\int \frac{(e \cot (c+d x))^{9/2}}{(a+b \cot (c+d x))^3} \, dx","Int[(e*Cot[c + d*x])^(9/2)/(a + b*Cot[c + d*x])^3,x]","-\frac{e^4 \left(31 a^2 b^2+15 a^4+8 b^4\right) \sqrt{e \cot (c+d x)}}{4 b^3 d \left(a^2+b^2\right)^2}+\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^{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{a^{5/2} e^{9/2} \left(46 a^2 b^2+15 a^4+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}+\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{e^4 \left(31 a^2 b^2+15 a^4+8 b^4\right) \sqrt{e \cot (c+d x)}}{4 b^3 d \left(a^2+b^2\right)^2}+\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^{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{a^{5/2} e^{9/2} \left(46 a^2 b^2+15 a^4+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}+\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}",1,"(a^(5/2)*(15*a^4 + 46*a^2*b^2 + 63*b^4)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*b^(7/2)*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*e^(9/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*e^(9/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((15*a^4 + 31*a^2*b^2 + 8*b^4)*e^4*Sqrt[e*Cot[c + d*x]])/(4*b^3*(a^2 + b^2)^2*d) + (a^2*e^2*(e*Cot[c + d*x])^(5/2))/(2*b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) + (a^2*(5*a^2 + 13*b^2)*e^3*(e*Cot[c + d*x])^(3/2))/(4*b^2*(a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) - ((a + b)*(a^2 - 4*a*b + b^2)*e^(9/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*e^(9/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)","A",17,14,25,0.5600,1,"{3565, 3645, 3647, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
82,1,476,0,1.2304229,"\int \frac{(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^3} \, dx","Int[(e*Cot[c + d*x])^(7/2)/(a + b*Cot[c + d*x])^3,x]","\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{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{a^{3/2} e^{7/2} \left(6 a^2 b^2+3 a^4+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}+\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{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{a^{3/2} e^{7/2} \left(6 a^2 b^2+3 a^4+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}+\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}",1,"-(a^(3/2)*(3*a^4 + 6*a^2*b^2 + 35*b^4)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*b^(5/2)*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*e^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*e^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (a^2*e^2*(e*Cot[c + d*x])^(3/2))/(2*b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) + (a^2*(3*a^2 + 11*b^2)*e^3*Sqrt[e*Cot[c + d*x]])/(4*b^2*(a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) + ((a - b)*(a^2 + 4*a*b + b^2)*e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)","A",16,13,25,0.5200,1,"{3565, 3645, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
83,1,470,0,1.294836,"\int \frac{(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^3} \, dx","Int[(e*Cot[c + d*x])^(5/2)/(a + b*Cot[c + d*x])^3,x]","-\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{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{\sqrt{a} e^{5/2} \left(18 a^2 b^2+a^4-15 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}-\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{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{\sqrt{a} e^{5/2} \left(18 a^2 b^2+a^4-15 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}-\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}",1,"-(Sqrt[a]*(a^4 + 18*a^2*b^2 - 15*b^4)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*b^(3/2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (a^2*e^2*Sqrt[e*Cot[c + d*x]])/(2*b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) - (a*(a^2 + 9*b^2)*e^2*Sqrt[e*Cot[c + d*x]])/(4*b*(a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) + ((a + b)*(a^2 - 4*a*b + b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)","A",16,13,25,0.5200,1,"{3565, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
84,1,461,0,1.2341284,"\int \frac{(e \cot (c+d x))^{3/2}}{(a+b \cot (c+d x))^3} \, dx","Int[(e*Cot[c + d*x])^(3/2)/(a + b*Cot[c + d*x])^3,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} \left(-26 a^2 b^2+3 a^4+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}-\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} (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} \left(-26 a^2 b^2+3 a^4+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}-\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}",1,"-((3*a^4 - 26*a^2*b^2 + 3*b^4)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*Sqrt[a]*Sqrt[b]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (a*e*Sqrt[e*Cot[c + d*x]])/(2*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) - ((3*a^2 - 5*b^2)*e*Sqrt[e*Cot[c + d*x]])/(4*(a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) - ((a - b)*(a^2 + 4*a*b + b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)","A",16,13,25,0.5200,1,"{3567, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
85,1,463,0,1.1468676,"\int \frac{\sqrt{e \cot (c+d x)}}{(a+b \cot (c+d x))^3} \, dx","Int[Sqrt[e*Cot[c + d*x]]/(a + b*Cot[c + d*x])^3,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{b} \sqrt{e} \left(-18 a^2 b^2+15 a^4-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}+\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{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{b} \sqrt{e} \left(-18 a^2 b^2+15 a^4-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}+\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}",1,"(Sqrt[b]*(15*a^4 - 18*a^2*b^2 - b^4)*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*a^(3/2)*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (b*Sqrt[e*Cot[c + d*x]])/(2*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) + (b*(7*a^2 - b^2)*Sqrt[e*Cot[c + d*x]])/(4*a*(a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) - ((a + b)*(a^2 - 4*a*b + b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)","A",16,13,25,0.5200,1,"{3568, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
86,1,476,0,1.2420618,"\int \frac{1}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^3} \, dx","Int[1/(Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^3),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{b^{3/2} \left(6 a^2 b^2+35 a^4+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}+\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^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{b^{3/2} \left(6 a^2 b^2+35 a^4+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}+\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}",1,"-(b^(3/2)*(35*a^4 + 6*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*a^(5/2)*(a^2 + b^2)^3*d*Sqrt[e]) + ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d*Sqrt[e]) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d*Sqrt[e]) - (b^2*Sqrt[e*Cot[c + d*x]])/(2*a*(a^2 + b^2)*d*e*(a + b*Cot[c + d*x])^2) - (b^2*(11*a^2 + 3*b^2)*Sqrt[e*Cot[c + d*x]])/(4*a^2*(a^2 + b^2)^2*d*e*(a + b*Cot[c + d*x])) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d*Sqrt[e]) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d*Sqrt[e])","A",16,13,25,0.5200,1,"{3569, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
87,1,529,0,1.65768,"\int \frac{1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3} \, dx","Int[1/((e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x])^3),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{b^{5/2} \left(46 a^2 b^2+63 a^4+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{(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{31 a^2 b^2+8 a^4+15 b^4}{4 a^3 d e \left(a^2+b^2\right)^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^{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{b^{5/2} \left(46 a^2 b^2+63 a^4+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{(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{31 a^2 b^2+8 a^4+15 b^4}{4 a^3 d e \left(a^2+b^2\right)^2 \sqrt{e \cot (c+d x)}}",1,"(b^(5/2)*(63*a^4 + 46*a^2*b^2 + 15*b^4)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*a^(7/2)*(a^2 + b^2)^3*d*e^(3/2)) - ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d*e^(3/2)) + ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d*e^(3/2)) + (8*a^4 + 31*a^2*b^2 + 15*b^4)/(4*a^3*(a^2 + b^2)^2*d*e*Sqrt[e*Cot[c + d*x]]) - b^2/(2*a*(a^2 + b^2)*d*e*Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^2) - (b^2*(13*a^2 + 5*b^2))/(4*a^2*(a^2 + b^2)^2*d*e*Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d*e^(3/2)) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d*e^(3/2))","A",17,13,25,0.5200,1,"{3569, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}"
88,1,167,0,0.2434868,"\int (a+b \cot (c+d x))^n \, dx","Int[(a + b*Cot[c + d*x])^n,x]","\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)}","\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,"-(b*(a + b*Cot[c + d*x])^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Cot[c + d*x])/(a - Sqrt[-b^2])])/(2*Sqrt[-b^2]*(a - Sqrt[-b^2])*d*(1 + n)) + (b*(a + b*Cot[c + d*x])^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Cot[c + d*x])/(a + Sqrt[-b^2])])/(2*Sqrt[-b^2]*(a + Sqrt[-b^2])*d*(1 + n))","A",5,3,12,0.2500,1,"{3485, 712, 68}"
89,1,193,0,0.2719939,"\int (a+b \cot (e+f x))^m (d \tan (e+f x))^n \, dx","Int[(a + b*Cot[e + f*x])^m*(d*Tan[e + f*x])^n,x]","-\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)}","-\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,"-(AppellF1[1 - n, -m, 1, 2 - n, -((b*Cot[e + f*x])/a), (-I)*Cot[e + f*x]]*Cot[e + f*x]*(a + b*Cot[e + f*x])^m*(d*Tan[e + f*x])^n)/(2*f*(1 - n)*(1 + (b*Cot[e + f*x])/a)^m) - (AppellF1[1 - n, -m, 1, 2 - n, -((b*Cot[e + f*x])/a), I*Cot[e + f*x]]*Cot[e + f*x]*(a + b*Cot[e + f*x])^m*(d*Tan[e + f*x])^n)/(2*f*(1 - n)*(1 + (b*Cot[e + f*x])/a)^m)","A",8,5,23,0.2174,1,"{4242, 3575, 912, 135, 133}"
90,1,45,0,0.0652656,"\int \frac{1+i \cot (c+d x)}{\sqrt{a+b \cot (c+d x)}} \, dx","Int[(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",3,3,27,0.1111,1,"{3537, 63, 208}"
91,1,45,0,0.0587035,"\int \frac{1-i \cot (c+d x)}{\sqrt{a+b \cot (c+d x)}} \, dx","Int[(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",3,3,27,0.1111,1,"{3537, 63, 208}"
92,1,59,0,0.0778469,"\int \frac{A+B \cot (c+d x)}{a+b \cot (c+d x)} \, dx","Int[(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x]),x]","\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)}","\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,"((a*A + b*B)*x)/(a^2 + b^2) - ((A*b - a*B)*Log[b*Cos[c + d*x] + a*Sin[c + d*x]])/((a^2 + b^2)*d)","A",2,2,23,0.08696,1,"{3531, 3530}"
93,1,111,0,0.1494542,"\int \frac{A+B \cot (c+d x)}{(a+b \cot (c+d x))^2} \, dx","Int[(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^2,x]","\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}","\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,"((a^2*A - A*b^2 + 2*a*b*B)*x)/(a^2 + b^2)^2 + (A*b - a*B)/((a^2 + b^2)*d*(a + b*Cot[c + d*x])) - ((2*a*A*b - a^2*B + b^2*B)*Log[b*Cos[c + d*x] + a*Sin[c + d*x]])/((a^2 + b^2)^2*d)","A",3,3,23,0.1304,1,"{3529, 3531, 3530}"
94,1,175,0,0.2756091,"\int \frac{A+B \cot (c+d x)}{(a+b \cot (c+d x))^3} \, dx","Int[(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^3,x]","\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(3 a^2 A b+a^3 (-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}","\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(3 a^2 A b+a^3 (-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,"((a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*x)/(a^2 + b^2)^3 + (A*b - a*B)/(2*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) + (2*a*A*b - a^2*B + b^2*B)/((a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) - ((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*Log[b*Cos[c + d*x] + a*Sin[c + d*x]])/((a^2 + b^2)^3*d)","A",4,3,23,0.1304,1,"{3529, 3531, 3530}"
95,1,188,0,0.4527091,"\int (a+b \cot (c+d x))^{5/2} (A+B \cot (c+d x)) \, dx","Int[(a + b*Cot[c + d*x])^(5/2)*(A + B*Cot[c + d*x]),x]","-\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}","-\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,"((a - I*b)^(5/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(5/2)*(I*A - B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/d - (2*(2*a*A*b + a^2*B - b^2*B)*Sqrt[a + b*Cot[c + d*x]])/d - (2*(A*b + a*B)*(a + b*Cot[c + d*x])^(3/2))/(3*d) - (2*B*(a + b*Cot[c + d*x])^(5/2))/(5*d)","A",10,5,25,0.2000,1,"{3528, 3539, 3537, 63, 208}"
96,1,150,0,0.3331591,"\int (a+b \cot (c+d x))^{3/2} (A+B \cot (c+d x)) \, dx","Int[(a + b*Cot[c + d*x])^(3/2)*(A + B*Cot[c + d*x]),x]","-\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}","-\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,"((a - I*b)^(3/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(3/2)*(I*A - B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/d - (2*(A*b + a*B)*Sqrt[a + b*Cot[c + d*x]])/d - (2*B*(a + b*Cot[c + d*x])^(3/2))/(3*d)","A",9,5,25,0.2000,1,"{3528, 3539, 3537, 63, 208}"
97,1,122,0,0.2617527,"\int \sqrt{a+b \cot (c+d x)} (A+B \cot (c+d x)) \, dx","Int[Sqrt[a + b*Cot[c + d*x]]*(A + B*Cot[c + d*x]),x]","\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}","\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,"(Sqrt[a - I*b]*(I*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/d - (Sqrt[a + I*b]*(I*A - B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/d - (2*B*Sqrt[a + b*Cot[c + d*x]])/d","A",8,5,25,0.2000,1,"{3528, 3539, 3537, 63, 208}"
98,1,151,0,0.2777049,"\int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{5/2} \, dx","Int[(-a + b*Cot[c + d*x])*(a + b*Cot[c + d*x])^(5/2),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}","\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,"-(((I*a - b)*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/d) + ((a + I*b)^(5/2)*(I*a + b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/d + (2*b*(a^2 + b^2)*Sqrt[a + b*Cot[c + d*x]])/d - (2*b*(a + b*Cot[c + d*x])^(5/2))/(5*d)","A",10,7,27,0.2593,1,"{3528, 12, 3482, 3539, 3537, 63, 208}"
99,1,408,0,0.5108739,"\int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx","Int[(-a + b*Cot[c + d*x])*(a + b*Cot[c + d*x])^(3/2),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}","\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,"(b*(a^2 + b^2)*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Cot[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*(a^2 + b^2)*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Cot[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (2*b*(a + b*Cot[c + d*x])^(3/2))/(3*d) + (b*(a^2 + b^2)*Log[a + Sqrt[a^2 + b^2] + b*Cot[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Cot[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (b*(a^2 + b^2)*Log[a + Sqrt[a^2 + b^2] + b*Cot[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Cot[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d)","A",13,9,27,0.3333,1,"{3528, 12, 3485, 700, 1129, 634, 618, 206, 628}"
100,1,422,0,0.4177571,"\int (-a+b \cot (c+d x)) \sqrt{a+b \cot (c+d x)} \, dx","Int[(-a + b*Cot[c + d*x])*Sqrt[a + b*Cot[c + d*x]],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}","-\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,"(b*Sqrt[a^2 + b^2]*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Cot[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*Sqrt[a^2 + b^2]*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Cot[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (2*b*Sqrt[a + b*Cot[c + d*x]])/d - (b*Sqrt[a^2 + b^2]*Log[a + Sqrt[a^2 + b^2] + b*Cot[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Cot[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (b*Sqrt[a^2 + b^2]*Log[a + Sqrt[a^2 + b^2] + b*Cot[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Cot[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d)","A",13,9,27,0.3333,1,"{3528, 12, 3485, 708, 1094, 634, 618, 206, 628}"
101,1,102,0,0.1602419,"\int \frac{A+B \cot (c+d x)}{\sqrt{a+b \cot (c+d x)}} \, dx","Int[(A + B*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]],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}}","\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,"((I*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) - ((I*A - B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)","A",7,4,25,0.1600,1,"{3539, 3537, 63, 208}"
102,1,138,0,0.2629597,"\int \frac{A+B \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx","Int[(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^(3/2),x]","\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}}","\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,"((I*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) - ((I*A - B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) + (2*(A*b - a*B))/((a^2 + b^2)*d*Sqrt[a + b*Cot[c + d*x]])","A",8,5,25,0.2000,1,"{3529, 3539, 3537, 63, 208}"
103,1,185,0,0.4020749,"\int \frac{A+B \cot (c+d x)}{(a+b \cot (c+d x))^{5/2}} \, dx","Int[(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^(5/2),x]","\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}}","\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,"((I*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d) - ((I*A - B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) + (2*(A*b - a*B))/(3*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^(3/2)) + (2*(2*a*A*b - a^2*B + b^2*B))/((a^2 + b^2)^2*d*Sqrt[a + b*Cot[c + d*x]])","A",9,5,25,0.2000,1,"{3529, 3539, 3537, 63, 208}"
104,1,102,0,0.1615773,"\int \frac{-a+b \cot (c+d x)}{\sqrt{a+b \cot (c+d x)}} \, dx","Int[(-a + b*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]],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}}","\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,"-(((I*a - b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)) + ((I*a + b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)","A",7,4,27,0.1481,1,"{3539, 3537, 63, 208}"
105,1,132,0,0.248112,"\int \frac{-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx","Int[(-a + b*Cot[c + d*x])/(a + b*Cot[c + d*x])^(3/2),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}}","-\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,"-(((I*a - b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d)) + ((I*a + b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (4*a*b)/((a^2 + b^2)*d*Sqrt[a + b*Cot[c + d*x]])","A",8,5,27,0.1852,1,"{3529, 3539, 3537, 63, 208}"
106,1,174,0,0.3818084,"\int \frac{-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{5/2}} \, dx","Int[(-a + b*Cot[c + d*x])/(a + b*Cot[c + d*x])^(5/2),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}}","-\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,"-(((I*a - b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d)) + ((I*a + b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) - (4*a*b)/(3*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^(3/2)) - (2*b*(3*a^2 - b^2))/((a^2 + b^2)^2*d*Sqrt[a + b*Cot[c + d*x]])","A",9,5,27,0.1852,1,"{3529, 3539, 3537, 63, 208}"