1,1,78,0,0.0639931,"\int \frac{\sin ^4(x)}{i+\cot (x)} \, dx","Int[Sin[x]^4/(I + Cot[x]),x]","-\frac{5 i x}{16}-\frac{i}{8 (-\cot (x)+i)}+\frac{3 i}{16 (\cot (x)+i)}+\frac{1}{32 (-\cot (x)+i)^2}-\frac{3}{32 (\cot (x)+i)^2}-\frac{i}{24 (\cot (x)+i)^3}","-\frac{5 i x}{16}-\frac{i}{8 (-\cot (x)+i)}+\frac{3 i}{16 (\cot (x)+i)}+\frac{1}{32 (-\cot (x)+i)^2}-\frac{3}{32 (\cot (x)+i)^2}-\frac{i}{24 (\cot (x)+i)^3}",1,"((-5*I)/16)*x + 1/(32*(I - Cot[x])^2) - (I/8)/(I - Cot[x]) - (I/24)/(I + Cot[x])^3 - 3/(32*(I + Cot[x])^2) + ((3*I)/16)/(I + Cot[x])","A",4,3,13,0.2308,1,"{3487, 44, 203}"
2,1,37,0,0.0399543,"\int \frac{\sin ^3(x)}{i+\cot (x)} \, dx","Int[Sin[x]^3/(I + Cot[x]),x]","-\frac{4}{15} i \cos ^3(x)+\frac{4}{5} i \cos (x)+\frac{i \sin ^3(x)}{5 (\cot (x)+i)}","-\frac{4}{15} i \cos ^3(x)+\frac{4}{5} i \cos (x)+\frac{i \sin ^3(x)}{5 (\cot (x)+i)}",1,"((4*I)/5)*Cos[x] - ((4*I)/15)*Cos[x]^3 + ((I/5)*Sin[x]^3)/(I + Cot[x])","A",3,2,13,0.1538,1,"{3502, 2633}"
3,1,50,0,0.050595,"\int \frac{\sin ^2(x)}{i+\cot (x)} \, dx","Int[Sin[x]^2/(I + Cot[x]),x]","-\frac{3 i x}{8}-\frac{i}{8 (-\cot (x)+i)}+\frac{i}{4 (\cot (x)+i)}-\frac{1}{8 (\cot (x)+i)^2}","-\frac{3 i x}{8}-\frac{i}{8 (-\cot (x)+i)}+\frac{i}{4 (\cot (x)+i)}-\frac{1}{8 (\cot (x)+i)^2}",1,"((-3*I)/8)*x - (I/8)/(I - Cot[x]) - 1/(8*(I + Cot[x])^2) + (I/4)/(I + Cot[x])","A",4,3,13,0.2308,1,"{3487, 44, 203}"
4,1,25,0,0.0299948,"\int \frac{\sin (x)}{i+\cot (x)} \, dx","Int[Sin[x]/(I + Cot[x]),x]","\frac{2}{3} i \cos (x)+\frac{i \sin (x)}{3 (\cot (x)+i)}","\frac{2}{3} i \cos (x)+\frac{i \sin (x)}{3 (\cot (x)+i)}",1,"((2*I)/3)*Cos[x] + ((I/3)*Sin[x])/(I + Cot[x])","A",2,2,11,0.1818,1,"{3502, 2638}"
5,1,14,0,0.0187808,"\int \frac{\csc (x)}{i+\cot (x)} \, dx","Int[Csc[x]/(I + Cot[x]),x]","\frac{i \csc (x)}{\cot (x)+i}","\frac{i \csc (x)}{\cot (x)+i}",1,"(I*Csc[x])/(I + Cot[x])","A",1,1,11,0.09091,1,"{3488}"
6,1,9,0,0.032013,"\int \frac{\csc ^2(x)}{i+\cot (x)} \, dx","Int[Csc[x]^2/(I + Cot[x]),x]","\log (\sin (x))-i x","\log (\sin (x))-i x",1,"(-I)*x + Log[Sin[x]]","A",2,2,13,0.1538,1,"{3487, 31}"
7,1,12,0,0.0345983,"\int \frac{\csc ^3(x)}{i+\cot (x)} \, dx","Int[Csc[x]^3/(I + Cot[x]),x]","-\csc (x)+i \tanh ^{-1}(\cos (x))","-\csc (x)+i \tanh ^{-1}(\cos (x))",1,"I*ArcTanh[Cos[x]] - Csc[x]","A",2,2,13,0.1538,1,"{3501, 3770}"
8,1,15,0,0.0327547,"\int \frac{\csc ^4(x)}{i+\cot (x)} \, dx","Int[Csc[x]^4/(I + Cot[x]),x]","-\frac{\cot ^2(x)}{2}+i \cot (x)","-\frac{\cot ^2(x)}{2}+i \cot (x)",1,"I*Cot[x] - Cot[x]^2/2","A",2,1,13,0.07692,1,"{3487}"
9,1,28,0,0.0404402,"\int \frac{\csc ^5(x)}{i+\cot (x)} \, dx","Int[Csc[x]^5/(I + Cot[x]),x]","-\frac{1}{3} \csc ^3(x)+\frac{1}{2} i \tanh ^{-1}(\cos (x))+\frac{1}{2} i \cot (x) \csc (x)","-\frac{1}{3} \csc ^3(x)+\frac{1}{2} i \tanh ^{-1}(\cos (x))+\frac{1}{2} i \cot (x) \csc (x)",1,"(I/2)*ArcTanh[Cos[x]] + (I/2)*Cot[x]*Csc[x] - Csc[x]^3/3","A",3,3,13,0.2308,1,"{3501, 3768, 3770}"
10,1,33,0,0.0391877,"\int \frac{\csc ^6(x)}{i+\cot (x)} \, dx","Int[Csc[x]^6/(I + Cot[x]),x]","-\frac{1}{4} \cot ^4(x)+\frac{1}{3} i \cot ^3(x)-\frac{\cot ^2(x)}{2}+i \cot (x)","-\frac{1}{4} \cot ^4(x)+\frac{1}{3} i \cot ^3(x)-\frac{\cot ^2(x)}{2}+i \cot (x)",1,"I*Cot[x] - Cot[x]^2/2 + (I/3)*Cot[x]^3 - Cot[x]^4/4","A",3,2,13,0.1538,1,"{3487, 43}"
11,1,40,0,0.0453658,"\int \frac{\csc ^7(x)}{i+\cot (x)} \, dx","Int[Csc[x]^7/(I + Cot[x]),x]","-\frac{1}{5} \csc ^5(x)+\frac{3}{8} i \tanh ^{-1}(\cos (x))+\frac{1}{4} i \cot (x) \csc ^3(x)+\frac{3}{8} i \cot (x) \csc (x)","-\frac{1}{5} \csc ^5(x)+\frac{3}{8} i \tanh ^{-1}(\cos (x))+\frac{1}{4} i \cot (x) \csc ^3(x)+\frac{3}{8} i \cot (x) \csc (x)",1,"((3*I)/8)*ArcTanh[Cos[x]] + ((3*I)/8)*Cot[x]*Csc[x] + (I/4)*Cot[x]*Csc[x]^3 - Csc[x]^5/5","A",4,3,13,0.2308,1,"{3501, 3768, 3770}"
12,1,81,0,0.0996165,"\int \frac{\csc ^6(x)}{a+b \cot (x)} \, dx","Int[Csc[x]^6/(a + b*Cot[x]),x]","-\frac{\left(a^2+2 b^2\right) \cot ^2(x)}{2 b^3}+\frac{a \left(a^2+2 b^2\right) \cot (x)}{b^4}-\frac{\left(a^2+b^2\right)^2 \log (a+b \cot (x))}{b^5}+\frac{a \cot ^3(x)}{3 b^2}-\frac{\cot ^4(x)}{4 b}","-\frac{\left(a^2+2 b^2\right) \cot ^2(x)}{2 b^3}+\frac{a \left(a^2+2 b^2\right) \cot (x)}{b^4}-\frac{\left(a^2+b^2\right)^2 \log (a+b \cot (x))}{b^5}+\frac{a \cot ^3(x)}{3 b^2}-\frac{\cot ^4(x)}{4 b}",1,"(a*(a^2 + 2*b^2)*Cot[x])/b^4 - ((a^2 + 2*b^2)*Cot[x]^2)/(2*b^3) + (a*Cot[x]^3)/(3*b^2) - Cot[x]^4/(4*b) - ((a^2 + b^2)^2*Log[a + b*Cot[x]])/b^5","A",3,2,13,0.1538,1,"{3506, 697}"
13,1,38,0,0.0648538,"\int \frac{\csc ^4(x)}{a+b \cot (x)} \, dx","Int[Csc[x]^4/(a + b*Cot[x]),x]","-\frac{\left(a^2+b^2\right) \log (a+b \cot (x))}{b^3}+\frac{a \cot (x)}{b^2}-\frac{\cot ^2(x)}{2 b}","-\frac{\left(a^2+b^2\right) \log (a+b \cot (x))}{b^3}+\frac{a \cot (x)}{b^2}-\frac{\cot ^2(x)}{2 b}",1,"(a*Cot[x])/b^2 - Cot[x]^2/(2*b) - ((a^2 + b^2)*Log[a + b*Cot[x]])/b^3","A",3,2,13,0.1538,1,"{3506, 697}"
14,1,12,0,0.0418816,"\int \frac{\csc ^2(x)}{a+b \cot (x)} \, dx","Int[Csc[x]^2/(a + b*Cot[x]),x]","-\frac{\log (a+b \cot (x))}{b}","-\frac{\log (a+b \cot (x))}{b}",1,"-(Log[a + b*Cot[x]]/b)","A",2,2,13,0.1538,1,"{3506, 31}"
15,1,72,0,0.1297044,"\int \frac{\sin ^2(x)}{a+b \cot (x)} \, dx","Int[Sin[x]^2/(a + b*Cot[x]),x]","\frac{a x \left(a^2+3 b^2\right)}{2 \left(a^2+b^2\right)^2}-\frac{\sin ^2(x) (a \cot (x)+b)}{2 \left(a^2+b^2\right)}-\frac{b^3 \log (a \sin (x)+b \cos (x))}{\left(a^2+b^2\right)^2}","\frac{a x \left(a^2+3 b^2\right)}{2 \left(a^2+b^2\right)^2}-\frac{\sin ^2(x) (a \cot (x)+b)}{2 \left(a^2+b^2\right)}-\frac{b^3 \log (a \sin (x)+b \cos (x))}{\left(a^2+b^2\right)^2}",1,"(a*(a^2 + 3*b^2)*x)/(2*(a^2 + b^2)^2) - (b^3*Log[b*Cos[x] + a*Sin[x]])/(a^2 + b^2)^2 - ((b + a*Cot[x])*Sin[x]^2)/(2*(a^2 + b^2))","A",7,6,13,0.4615,1,"{3506, 741, 801, 635, 203, 260}"
16,1,120,0,0.1930158,"\int \frac{\sin ^4(x)}{a+b \cot (x)} \, dx","Int[Sin[x]^4/(a + b*Cot[x]),x]","\frac{a x \left(10 a^2 b^2+3 a^4+15 b^4\right)}{8 \left(a^2+b^2\right)^3}-\frac{\sin ^4(x) (a \cot (x)+b)}{4 \left(a^2+b^2\right)}-\frac{\sin ^2(x) \left(a \left(3 a^2+7 b^2\right) \cot (x)+4 b^3\right)}{8 \left(a^2+b^2\right)^2}-\frac{b^5 \log (a \sin (x)+b \cos (x))}{\left(a^2+b^2\right)^3}","\frac{a x \left(10 a^2 b^2+3 a^4+15 b^4\right)}{8 \left(a^2+b^2\right)^3}-\frac{\sin ^4(x) (a \cot (x)+b)}{4 \left(a^2+b^2\right)}-\frac{\sin ^2(x) \left(a \left(3 a^2+7 b^2\right) \cot (x)+4 b^3\right)}{8 \left(a^2+b^2\right)^2}-\frac{b^5 \log (a \sin (x)+b \cos (x))}{\left(a^2+b^2\right)^3}",1,"(a*(3*a^4 + 10*a^2*b^2 + 15*b^4)*x)/(8*(a^2 + b^2)^3) - (b^5*Log[b*Cos[x] + a*Sin[x]])/(a^2 + b^2)^3 - ((4*b^3 + a*(3*a^2 + 7*b^2)*Cot[x])*Sin[x]^2)/(8*(a^2 + b^2)^2) - ((b + a*Cot[x])*Sin[x]^4)/(4*(a^2 + b^2))","A",8,7,13,0.5385,1,"{3506, 741, 823, 801, 635, 203, 260}"
17,1,101,0,0.1669731,"\int \frac{\csc ^5(x)}{a+b \cot (x)} \, dx","Int[Csc[x]^5/(a + b*Cot[x]),x]","-\frac{\left(a^2+b^2\right) \csc (x)}{b^3}+\frac{a \left(a^2+b^2\right) \tanh ^{-1}(\cos (x))}{b^4}+\frac{\left(a^2+b^2\right)^{3/2} \tanh ^{-1}\left(\frac{\sin (x) (b-a \cot (x))}{\sqrt{a^2+b^2}}\right)}{b^4}+\frac{a \tanh ^{-1}(\cos (x))}{2 b^2}+\frac{a \cot (x) \csc (x)}{2 b^2}-\frac{\csc ^3(x)}{3 b}","-\frac{\left(a^2+b^2\right) \csc (x)}{b^3}+\frac{a \left(a^2+b^2\right) \tanh ^{-1}(\cos (x))}{b^4}+\frac{\left(a^2+b^2\right)^{3/2} \tanh ^{-1}\left(\frac{\sin (x) (b-a \cot (x))}{\sqrt{a^2+b^2}}\right)}{b^4}+\frac{a \tanh ^{-1}(\cos (x))}{2 b^2}+\frac{a \cot (x) \csc (x)}{2 b^2}-\frac{\csc ^3(x)}{3 b}",1,"(a*ArcTanh[Cos[x]])/(2*b^2) + (a*(a^2 + b^2)*ArcTanh[Cos[x]])/b^4 + ((a^2 + b^2)^(3/2)*ArcTanh[((b - a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]])/b^4 - ((a^2 + b^2)*Csc[x])/b^3 + (a*Cot[x]*Csc[x])/(2*b^2) - Csc[x]^3/(3*b)","A",9,6,13,0.4615,1,"{3510, 3486, 3768, 3770, 3509, 206}"
18,1,53,0,0.0848644,"\int \frac{\csc ^3(x)}{a+b \cot (x)} \, dx","Int[Csc[x]^3/(a + b*Cot[x]),x]","\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{\sin (x) (b-a \cot (x))}{\sqrt{a^2+b^2}}\right)}{b^2}+\frac{a \tanh ^{-1}(\cos (x))}{b^2}-\frac{\csc (x)}{b}","\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{\sin (x) (b-a \cot (x))}{\sqrt{a^2+b^2}}\right)}{b^2}+\frac{a \tanh ^{-1}(\cos (x))}{b^2}-\frac{\csc (x)}{b}",1,"(a*ArcTanh[Cos[x]])/b^2 + (Sqrt[a^2 + b^2]*ArcTanh[((b - a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]])/b^2 - Csc[x]/b","A",5,5,13,0.3846,1,"{3510, 3486, 3770, 3509, 206}"
19,1,36,0,0.0270206,"\int \frac{\csc (x)}{a+b \cot (x)} \, dx","Int[Csc[x]/(a + b*Cot[x]),x]","-\frac{\tanh ^{-1}\left(\frac{\sin (x) (a \cot (x)-b)}{\sqrt{a^2+b^2}}\right)}{\sqrt{a^2+b^2}}","-\frac{\tanh ^{-1}\left(\frac{\sin (x) (a \cot (x)-b)}{\sqrt{a^2+b^2}}\right)}{\sqrt{a^2+b^2}}",1,"-(ArcTanh[((-b + a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]]/Sqrt[a^2 + b^2])","A",2,2,11,0.1818,1,"{3509, 206}"
20,1,66,0,0.0868971,"\int \frac{\sin (x)}{a+b \cot (x)} \, dx","Int[Sin[x]/(a + b*Cot[x]),x]","-\frac{b \sin (x)}{a^2+b^2}-\frac{a \cos (x)}{a^2+b^2}+\frac{b^2 \tanh ^{-1}\left(\frac{\sin (x) (b-a \cot (x))}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{3/2}}","-\frac{b \sin (x)}{a^2+b^2}-\frac{a \cos (x)}{a^2+b^2}+\frac{b^2 \tanh ^{-1}\left(\frac{\sin (x) (b-a \cot (x))}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{3/2}}",1,"(b^2*ArcTanh[((b - a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (a*Cos[x])/(a^2 + b^2) - (b*Sin[x])/(a^2 + b^2)","A",5,5,11,0.4545,1,"{3511, 3486, 2638, 3509, 206}"
21,1,121,0,0.1588685,"\int \frac{\sin ^3(x)}{a+b \cot (x)} \, dx","Int[Sin[x]^3/(a + b*Cot[x]),x]","-\frac{b \sin ^3(x)}{3 \left(a^2+b^2\right)}-\frac{b^3 \sin (x)}{\left(a^2+b^2\right)^2}+\frac{a \cos ^3(x)}{3 \left(a^2+b^2\right)}-\frac{a b^2 \cos (x)}{\left(a^2+b^2\right)^2}-\frac{a \cos (x)}{a^2+b^2}+\frac{b^4 \tanh ^{-1}\left(\frac{\sin (x) (b-a \cot (x))}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}","-\frac{b \sin ^3(x)}{3 \left(a^2+b^2\right)}-\frac{b^3 \sin (x)}{\left(a^2+b^2\right)^2}+\frac{a \cos ^3(x)}{3 \left(a^2+b^2\right)}-\frac{a b^2 \cos (x)}{\left(a^2+b^2\right)^2}-\frac{a \cos (x)}{a^2+b^2}+\frac{b^4 \tanh ^{-1}\left(\frac{\sin (x) (b-a \cot (x))}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}",1,"(b^4*ArcTanh[((b - a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (a*b^2*Cos[x])/(a^2 + b^2)^2 - (a*Cos[x])/(a^2 + b^2) + (a*Cos[x]^3)/(3*(a^2 + b^2)) - (b^3*Sin[x])/(a^2 + b^2)^2 - (b*Sin[x]^3)/(3*(a^2 + b^2))","A",9,6,13,0.4615,1,"{3511, 3486, 2633, 2638, 3509, 206}"
22,1,12,0,0.0400216,"\int \frac{\csc ^2(x)}{(a+b \cot (x))^2} \, dx","Int[Csc[x]^2/(a + b*Cot[x])^2,x]","\frac{1}{b (a+b \cot (x))}","\frac{1}{b (a+b \cot (x))}",1,"1/(b*(a + b*Cot[x]))","A",2,2,13,0.1538,1,"{3506, 32}"
23,1,20,0,0.0427855,"\int (a+b \cot (x))^n \csc ^2(x) \, dx","Int[(a + b*Cot[x])^n*Csc[x]^2,x]","-\frac{(a+b \cot (x))^{n+1}}{b (n+1)}","-\frac{(a+b \cot (x))^{n+1}}{b (n+1)}",1,"-((a + b*Cot[x])^(1 + n)/(b*(1 + n)))","A",2,2,13,0.1538,1,"{3506, 32}"