1,1,51,78,0.1231156,"\int \frac{\sin ^4(x)}{i+\cot (x)} \, dx","Integrate[Sin[x]^4/(I + Cot[x]),x]","\frac{1}{192} (-15 \cos (2 x)+6 \cos (4 x)-i (60 x-45 \sin (2 x)+9 \sin (4 x)-\sin (6 x)-i \cos (6 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}",1,"(-15*Cos[2*x] + 6*Cos[4*x] - I*(60*x - I*Cos[6*x] - 45*Sin[2*x] + 9*Sin[4*x] - Sin[6*x]))/192","A",1
2,1,46,37,0.0982578,"\int \frac{\sin ^3(x)}{i+\cot (x)} \, dx","Integrate[Sin[x]^3/(I + Cot[x]),x]","\frac{\csc (x) (-40 \sin (2 x)+4 \sin (4 x)+20 i \cos (2 x)-i \cos (4 x)+45 i)}{120 (\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,"(Csc[x]*(45*I + (20*I)*Cos[2*x] - I*Cos[4*x] - 40*Sin[2*x] + 4*Sin[4*x]))/(120*(I + Cot[x]))","A",1
3,1,36,50,0.0534601,"\int \frac{\sin ^2(x)}{i+\cot (x)} \, dx","Integrate[Sin[x]^2/(I + Cot[x]),x]","-\frac{1}{32} i (12 x-8 \sin (2 x)+\sin (4 x)-4 i \cos (2 x)+i \cos (4 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}",1,"(-1/32*I)*(12*x - (4*I)*Cos[2*x] + I*Cos[4*x] - 8*Sin[2*x] + Sin[4*x])","A",1
4,1,27,25,0.0457281,"\int \frac{\sin (x)}{i+\cot (x)} \, dx","Integrate[Sin[x]/(I + Cot[x]),x]","\frac{1}{6} (\sin (x)+i \cos (x)) (2 i \sin (2 x)+\cos (2 x)+3)","\frac{2}{3} i \cos (x)+\frac{i \sin (x)}{3 (\cot (x)+i)}",1,"((I*Cos[x] + Sin[x])*(3 + Cos[2*x] + (2*I)*Sin[2*x]))/6","A",1
5,1,9,14,0.0035987,"\int \frac{\csc (x)}{i+\cot (x)} \, dx","Integrate[Csc[x]/(I + Cot[x]),x]","\sin (x)+i \cos (x)","\frac{i \csc (x)}{\cot (x)+i}",1,"I*Cos[x] + Sin[x]","A",1
6,1,9,9,0.0033801,"\int \frac{\csc ^2(x)}{i+\cot (x)} \, dx","Integrate[Csc[x]^2/(I + Cot[x]),x]","\log (\sin (x))-i x","\log (\sin (x))-i x",1,"(-I)*x + Log[Sin[x]]","A",1
7,1,26,12,0.0481863,"\int \frac{\csc ^3(x)}{i+\cot (x)} \, dx","Integrate[Csc[x]^3/(I + Cot[x]),x]","-\csc (x)+i \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)","-\csc (x)+i \tanh ^{-1}(\cos (x))",1,"-Csc[x] + I*(Log[Cos[x/2]] - Log[Sin[x/2]])","B",1
8,1,15,15,0.0273455,"\int \frac{\csc ^4(x)}{i+\cot (x)} \, dx","Integrate[Csc[x]^4/(I + Cot[x]),x]","-\frac{\csc ^2(x)}{2}+i \cot (x)","-\frac{\cot ^2(x)}{2}+i \cot (x)",1,"I*Cot[x] - Csc[x]^2/2","A",1
9,1,67,28,0.1006149,"\int \frac{\csc ^5(x)}{i+\cot (x)} \, dx","Integrate[Csc[x]^5/(I + Cot[x]),x]","\frac{1}{24} i \csc ^3(x) \left(6 \sin (2 x)+3 \sin (3 x) \log \left(\sin \left(\frac{x}{2}\right)\right)+9 \sin (x) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)-3 \sin (3 x) \log \left(\cos \left(\frac{x}{2}\right)\right)+8 i\right)","-\frac{1}{3} \csc ^3(x)+\frac{1}{2} i \tanh ^{-1}(\cos (x))+\frac{1}{2} i \cot (x) \csc (x)",1,"(I/24)*Csc[x]^3*(8*I + 9*(Log[Cos[x/2]] - Log[Sin[x/2]])*Sin[x] + 6*Sin[2*x] - 3*Log[Cos[x/2]]*Sin[3*x] + 3*Log[Sin[x/2]]*Sin[3*x])","B",1
10,1,23,33,0.0542472,"\int \frac{\csc ^6(x)}{i+\cot (x)} \, dx","Integrate[Csc[x]^6/(I + Cot[x]),x]","-\frac{\csc ^4(x)}{4}+\frac{1}{3} i \cot (x) \left(\csc ^2(x)+2\right)","-\frac{1}{4} \cot ^4(x)+\frac{1}{3} i \cot ^3(x)-\frac{\cot ^2(x)}{2}+i \cot (x)",1,"-1/4*Csc[x]^4 + (I/3)*Cot[x]*(2 + Csc[x]^2)","A",1
11,1,99,40,0.1508553,"\int \frac{\csc ^7(x)}{i+\cot (x)} \, dx","Integrate[Csc[x]^7/(I + Cot[x]),x]","\frac{1}{640} i \csc ^5(x) \left(140 \sin (2 x)-30 \sin (4 x)+75 \sin (3 x) \log \left(\sin \left(\frac{x}{2}\right)\right)-15 \sin (5 x) \log \left(\sin \left(\frac{x}{2}\right)\right)+150 \sin (x) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)-75 \sin (3 x) \log \left(\cos \left(\frac{x}{2}\right)\right)+15 \sin (5 x) \log \left(\cos \left(\frac{x}{2}\right)\right)+128 i\right)","-\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,"(I/640)*Csc[x]^5*(128*I + 150*(Log[Cos[x/2]] - Log[Sin[x/2]])*Sin[x] + 140*Sin[2*x] - 75*Log[Cos[x/2]]*Sin[3*x] + 75*Log[Sin[x/2]]*Sin[3*x] - 30*Sin[4*x] + 15*Log[Cos[x/2]]*Sin[5*x] - 15*Log[Sin[x/2]]*Sin[5*x])","B",1
12,1,85,81,0.4131587,"\int \frac{\csc ^6(x)}{a+b \cot (x)} \, dx","Integrate[Csc[x]^6/(a + b*Cot[x]),x]","\frac{-6 b^2 \left(a^2+b^2\right) \csc ^2(x)+4 a b \cot (x) \left(3 a^2+b^2 \csc ^2(x)+5 b^2\right)+12 \left(a^2+b^2\right)^2 (\log (\sin (x))-\log (a \sin (x)+b \cos (x)))-3 b^4 \csc ^4(x)}{12 b^5}","-\frac{\left(a^2+b^2\right)^2 \log (a+b \cot (x))}{b^5}+\frac{a \left(a^2+2 b^2\right) \cot (x)}{b^4}-\frac{\left(a^2+2 b^2\right) \cot ^2(x)}{2 b^3}+\frac{a \cot ^3(x)}{3 b^2}-\frac{\cot ^4(x)}{4 b}",1,"(-6*b^2*(a^2 + b^2)*Csc[x]^2 - 3*b^4*Csc[x]^4 + 4*a*b*Cot[x]*(3*a^2 + 5*b^2 + b^2*Csc[x]^2) + 12*(a^2 + b^2)^2*(Log[Sin[x]] - Log[b*Cos[x] + a*Sin[x]]))/(12*b^5)","A",1
13,1,48,38,0.1555607,"\int \frac{\csc ^4(x)}{a+b \cot (x)} \, dx","Integrate[Csc[x]^4/(a + b*Cot[x]),x]","\frac{2 \left(a^2+b^2\right) (\log (\sin (x))-\log (a \sin (x)+b \cos (x)))+2 a b \cot (x)-b^2 \csc ^2(x)}{2 b^3}","-\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,"(2*a*b*Cot[x] - b^2*Csc[x]^2 + 2*(a^2 + b^2)*(Log[Sin[x]] - Log[b*Cos[x] + a*Sin[x]]))/(2*b^3)","A",1
14,1,20,12,0.0438288,"\int \frac{\csc ^2(x)}{a+b \cot (x)} \, dx","Integrate[Csc[x]^2/(a + b*Cot[x]),x]","\frac{\log (\sin (x))-\log (a \sin (x)+b \cos (x))}{b}","-\frac{\log (a+b \cot (x))}{b}",1,"(Log[Sin[x]] - Log[b*Cos[x] + a*Sin[x]])/b","A",1
15,1,94,72,0.1883374,"\int \frac{\sin ^2(x)}{a+b \cot (x)} \, dx","Integrate[Sin[x]^2/(a + b*Cot[x]),x]","\frac{2 a^3 x-a^3 \sin (2 x)+b \left(a^2+b^2\right) \cos (2 x)-2 b^3 \log \left((a \sin (x)+b \cos (x))^2\right)+6 a b^2 x-a b^2 \sin (2 x)-4 i b^3 x+4 i b^3 \tan ^{-1}(\tan (x))}{4 \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,"(2*a^3*x + 6*a*b^2*x - (4*I)*b^3*x + (4*I)*b^3*ArcTan[Tan[x]] + b*(a^2 + b^2)*Cos[2*x] - 2*b^3*Log[(b*Cos[x] + a*Sin[x])^2] - a^3*Sin[2*x] - a*b^2*Sin[2*x])/(4*(a^2 + b^2)^2)","C",1
16,1,151,120,0.2628864,"\int \frac{\sin ^4(x)}{a+b \cot (x)} \, dx","Integrate[Sin[x]^4/(a + b*Cot[x]),x]","\frac{12 a^5 x-8 a^5 \sin (2 x)+a^5 \sin (4 x)+40 a^3 b^2 x-24 a^3 b^2 \sin (2 x)+2 a^3 b^2 \sin (4 x)-b \left(a^2+b^2\right)^2 \cos (4 x)+4 b \left(a^4+4 a^2 b^2+3 b^4\right) \cos (2 x)-32 b^5 \log (a \sin (x)+b \cos (x))+60 a b^4 x-16 a b^4 \sin (2 x)+a b^4 \sin (4 x)}{32 \left(a^2+b^2\right)^3}","-\frac{\sin ^4(x) (a \cot (x)+b)}{4 \left(a^2+b^2\right)}-\frac{b^5 \log (a \sin (x)+b \cos (x))}{\left(a^2+b^2\right)^3}-\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{a x \left(3 a^4+10 a^2 b^2+15 b^4\right)}{8 \left(a^2+b^2\right)^3}",1,"(12*a^5*x + 40*a^3*b^2*x + 60*a*b^4*x + 4*b*(a^4 + 4*a^2*b^2 + 3*b^4)*Cos[2*x] - b*(a^2 + b^2)^2*Cos[4*x] - 32*b^5*Log[b*Cos[x] + a*Sin[x]] - 8*a^5*Sin[2*x] - 24*a^3*b^2*Sin[2*x] - 16*a*b^4*Sin[2*x] + a^5*Sin[4*x] + 2*a^3*b^2*Sin[4*x] + a*b^4*Sin[4*x])/(32*(a^2 + b^2)^3)","A",1
17,1,198,101,1.2734034,"\int \frac{\csc ^5(x)}{a+b \cot (x)} \, dx","Integrate[Csc[x]^5/(a + b*Cot[x]),x]","-\frac{48 a^3 \log \left(\sin \left(\frac{x}{2}\right)\right)-48 a^3 \log \left(\cos \left(\frac{x}{2}\right)\right)+4 b \left(6 a^2+7 b^2\right) \cot \left(\frac{x}{2}\right)-96 \left(a^2+b^2\right)^{3/2} \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-a}{\sqrt{a^2+b^2}}\right)+24 a^2 b \tan \left(\frac{x}{2}\right)-6 a b^2 \csc ^2\left(\frac{x}{2}\right)+6 a b^2 \sec ^2\left(\frac{x}{2}\right)+72 a b^2 \log \left(\sin \left(\frac{x}{2}\right)\right)-72 a b^2 \log \left(\cos \left(\frac{x}{2}\right)\right)+28 b^3 \tan \left(\frac{x}{2}\right)+b^3 \sin (x) \csc ^4\left(\frac{x}{2}\right)+16 b^3 \sin ^4\left(\frac{x}{2}\right) \csc ^3(x)}{48 b^4}","\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{\left(a^2+b^2\right) \csc (x)}{b^3}+\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,"-1/48*(-96*(a^2 + b^2)^(3/2)*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]] + 4*b*(6*a^2 + 7*b^2)*Cot[x/2] - 6*a*b^2*Csc[x/2]^2 - 48*a^3*Log[Cos[x/2]] - 72*a*b^2*Log[Cos[x/2]] + 48*a^3*Log[Sin[x/2]] + 72*a*b^2*Log[Sin[x/2]] + 6*a*b^2*Sec[x/2]^2 + 16*b^3*Csc[x]^3*Sin[x/2]^4 + b^3*Csc[x/2]^4*Sin[x] + 24*a^2*b*Tan[x/2] + 28*b^3*Tan[x/2])/b^4","A",1
18,1,67,53,0.1261519,"\int \frac{\csc ^3(x)}{a+b \cot (x)} \, dx","Integrate[Csc[x]^3/(a + b*Cot[x]),x]","\frac{2 \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-a}{\sqrt{a^2+b^2}}\right)+a \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)-b \csc (x)}{b^2}","\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,"(2*Sqrt[a^2 + b^2]*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]] - b*Csc[x] + a*(Log[Cos[x/2]] - Log[Sin[x/2]]))/b^2","A",1
19,1,38,36,0.0305426,"\int \frac{\csc (x)}{a+b \cot (x)} \, dx","Integrate[Csc[x]/(a + b*Cot[x]),x]","\frac{2 \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-a}{\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,"(2*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2]","A",1
20,1,62,66,0.2393881,"\int \frac{\sin (x)}{a+b \cot (x)} \, dx","Integrate[Sin[x]/(a + b*Cot[x]),x]","\frac{2 b^2 \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-a}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{3/2}}-\frac{a \cos (x)+b \sin (x)}{a^2+b^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,"(2*b^2*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (a*Cos[x] + b*Sin[x])/(a^2 + b^2)","A",1
21,1,113,121,0.6064646,"\int \frac{\sin ^3(x)}{a+b \cot (x)} \, dx","Integrate[Sin[x]^3/(a + b*Cot[x]),x]","\frac{-3 a \left(3 a^2+7 b^2\right) \cos (x)+a \left(a^2+b^2\right) \cos (3 x)+2 b \sin (x) \left(\left(a^2+b^2\right) \cos (2 x)-a^2-7 b^2\right)}{12 \left(a^2+b^2\right)^2}+\frac{2 b^4 \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-a}{\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{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^3 \sin (x)}{\left(a^2+b^2\right)^2}",1,"(2*b^4*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) + (-3*a*(3*a^2 + 7*b^2)*Cos[x] + a*(a^2 + b^2)*Cos[3*x] + 2*b*(-a^2 - 7*b^2 + (a^2 + b^2)*Cos[2*x])*Sin[x])/(12*(a^2 + b^2)^2)","A",1
22,1,17,12,0.0292363,"\int \frac{\csc ^2(x)}{(a+b \cot (x))^2} \, dx","Integrate[Csc[x]^2/(a + b*Cot[x])^2,x]","\frac{\sin (x)}{b (a \sin (x)+b \cos (x))}","\frac{1}{b (a+b \cot (x))}",1,"Sin[x]/(b*(b*Cos[x] + a*Sin[x]))","A",1
23,1,19,20,0.2019194,"\int (a+b \cot (x))^n \csc ^2(x) \, dx","Integrate[(a + b*Cot[x])^n*Csc[x]^2,x]","-\frac{(a+b \cot (x))^{n+1}}{b n+b}","-\frac{(a+b \cot (x))^{n+1}}{b (n+1)}",1,"-((a + b*Cot[x])^(1 + n)/(b + b*n))","A",1