1,1,60,78,0.0945904,"\int \frac{\cos ^4(x)}{i+\cot (x)} \, dx","Integrate[Cos[x]^4/(I + Cot[x]),x]","-\frac{1}{192} i \left(12 x+3 \sin (2 x)-3 \sin (4 x)-\sin (6 x)-18 i \cos ^2(x)-6 i \cos (2 x)-6 i \cos (4 x)-i \cos (6 x)\right)","-\frac{i x}{16}+\frac{i}{8 (-\cot (x)+i)}+\frac{3 i}{16 (\cot (x)+i)}+\frac{1}{32 (-\cot (x)+i)^2}+\frac{5}{32 (\cot (x)+i)^2}-\frac{i}{24 (\cot (x)+i)^3}",1,"(-1/192*I)*(12*x - (18*I)*Cos[x]^2 - (6*I)*Cos[2*x] - (6*I)*Cos[4*x] - I*Cos[6*x] + 3*Sin[2*x] - 3*Sin[4*x] - Sin[6*x])","A",1
2,1,42,29,0.0967644,"\int \frac{\cos ^3(x)}{i+\cot (x)} \, dx","Integrate[Cos[x]^3/(I + Cot[x]),x]","-\frac{\csc (x) (i (10 \sin (2 x)+\sin (4 x))+20 \cos (2 x)+4 \cos (4 x))}{120 (\cot (x)+i)}","\frac{1}{5} i \sin ^5(x)-\frac{1}{3} i \sin ^3(x)-\frac{1}{5} \cos ^5(x)",1,"-1/120*(Csc[x]*(20*Cos[2*x] + 4*Cos[4*x] + I*(10*Sin[2*x] + Sin[4*x])))/(I + Cot[x])","A",1
3,1,32,50,0.0669672,"\int \frac{\cos ^2(x)}{i+\cot (x)} \, dx","Integrate[Cos[x]^2/(I + Cot[x]),x]","-\frac{1}{32} i (4 x-\sin (4 x)-4 i \cos (2 x)-i \cos (4 x))","-\frac{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)*(4*x - (4*I)*Cos[2*x] - I*Cos[4*x] - Sin[4*x])","A",1
4,1,19,19,0.0137156,"\int \frac{\cos (x)}{i+\cot (x)} \, dx","Integrate[Cos[x]/(I + Cot[x]),x]","\frac{1}{3} \left(-\cos ^3(x)-i \sin ^3(x)\right)","-\frac{1}{3} \cos ^3(x)-\frac{1}{3} i \sin ^3(x)",1,"(-Cos[x]^3 - I*Sin[x]^3)/3","A",1
5,1,44,18,0.0363138,"\int \frac{\sec (x)}{i+\cot (x)} \, dx","Integrate[Sec[x]/(I + Cot[x]),x]","-\cos (x)+i \left(\sin (x)+\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)","i \sin (x)-\cos (x)-i \tanh ^{-1}(\sin (x))",1,"-Cos[x] + I*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]] + Sin[x])","B",1
6,1,17,20,0.0362109,"\int \frac{\sec ^2(x)}{i+\cot (x)} \, dx","Integrate[Sec[x]^2/(I + Cot[x]),x]","i (x-\tan (x)+i \log (\cos (x)))","i x-i \tan (x)-\log (\sin (x))+\log (\tan (x))",1,"I*(x + I*Log[Cos[x]] - Tan[x])","A",1
7,1,48,22,0.1497025,"\int \frac{\sec ^3(x)}{i+\cot (x)} \, dx","Integrate[Sec[x]^3/(I + Cot[x]),x]","-\frac{1}{2} i \left((\tan (x)+2 i) \sec (x)+\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)","\sec (x)+\frac{1}{2} i \tanh ^{-1}(\sin (x))-\frac{1}{2} i \tan (x) \sec (x)",1,"(-1/2*I)*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]] + Sec[x]*(2*I + Tan[x]))","B",1
8,1,24,19,0.055145,"\int \frac{\sec ^4(x)}{i+\cot (x)} \, dx","Integrate[Sec[x]^4/(I + Cot[x]),x]","\frac{1}{6} \left(2 i \tan (x)+(3-2 i \tan (x)) \sec ^2(x)\right)","\frac{\tan ^2(x)}{2}-\frac{1}{3} i \tan ^3(x)",1,"(Sec[x]^2*(3 - (2*I)*Tan[x]) + (2*I)*Tan[x])/6","A",1
9,1,61,40,0.5959358,"\int \frac{\sec ^5(x)}{i+\cot (x)} \, dx","Integrate[Sec[x]^5/(I + Cot[x]),x]","-\frac{1}{48} i \left(6 \left(\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)+\sec ^3(x) (-3 (\cos (2 x)-3) \tan (x)+16 i)\right)","\frac{\sec ^3(x)}{3}+\frac{1}{8} i \tanh ^{-1}(\sin (x))-\frac{1}{4} i \tan (x) \sec ^3(x)+\frac{1}{8} i \tan (x) \sec (x)",1,"(-1/48*I)*(6*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) + Sec[x]^3*(16*I - 3*(-3 + Cos[2*x])*Tan[x]))","A",1
10,1,26,37,0.0890018,"\int \frac{\sec ^6(x)}{i+\cot (x)} \, dx","Integrate[Sec[x]^6/(I + Cot[x]),x]","\frac{1}{60} \sec ^4(x) \left(15-4 i \sin ^2(x) (\cos (2 x)+4) \tan (x)\right)","-\frac{1}{5} i \tan ^5(x)+\frac{\tan ^4(x)}{4}-\frac{1}{3} i \tan ^3(x)+\frac{\tan ^2(x)}{2}",1,"(Sec[x]^4*(15 - (4*I)*(4 + Cos[2*x])*Sin[x]^2*Tan[x]))/60","A",1
11,1,179,126,0.6117979,"\int \frac{\cos ^4(x)}{a+b \cot (x)} \, dx","Integrate[Cos[x]^4/(a + b*Cot[x]),x]","\frac{12 a^5 x+8 a^5 \sin (2 x)+a^5 \sin (4 x)-32 i a^4 b x-a^4 b \cos (4 x)+32 i a^4 b \tan ^{-1}(\tan (x))-16 a^4 b \log \left((a \sin (x)+b \cos (x))^2\right)-24 a^3 b^2 x+8 a^3 b^2 \sin (2 x)+2 a^3 b^2 \sin (4 x)-2 a^2 b^3 \cos (4 x)-4 b \left(3 a^4+4 a^2 b^2+b^4\right) \cos (2 x)-4 a b^4 x+a b^4 \sin (4 x)-b^5 \cos (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{\sin ^2(x) \left(a \left(5 a^2+b^2\right) \cot (x)+4 b \left(2 a^2+b^2\right)\right)}{8 \left(a^2+b^2\right)^2}-\frac{a^4 b \log (a \sin (x)+b \cos (x))}{\left(a^2+b^2\right)^3}+\frac{a x \left(3 a^4-6 a^2 b^2-b^4\right)}{8 \left(a^2+b^2\right)^3}",1,"(12*a^5*x - (32*I)*a^4*b*x - 24*a^3*b^2*x - 4*a*b^4*x + (32*I)*a^4*b*ArcTan[Tan[x]] - 4*b*(3*a^4 + 4*a^2*b^2 + b^4)*Cos[2*x] - a^4*b*Cos[4*x] - 2*a^2*b^3*Cos[4*x] - b^5*Cos[4*x] - 16*a^4*b*Log[(b*Cos[x] + a*Sin[x])^2] + 8*a^5*Sin[2*x] + 8*a^3*b^2*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)","C",1
12,1,112,123,1.1109738,"\int \frac{\cos ^3(x)}{a+b \cot (x)} \, dx","Integrate[Cos[x]^3/(a + b*Cot[x]),x]","-\frac{3 b \left(5 a^2+b^2\right) \cos (x)+b \left(a^2+b^2\right) \cos (3 x)-2 a \sin (x) \left(\left(a^2+b^2\right) \cos (2 x)+5 a^2-b^2\right)}{12 \left(a^2+b^2\right)^2}-\frac{2 a^3 b \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{a \sin ^3(x)}{3 \left(a^2+b^2\right)}+\frac{a \sin (x)}{a^2+b^2}-\frac{a b^2 \sin (x)}{\left(a^2+b^2\right)^2}-\frac{b \cos ^3(x)}{3 \left(a^2+b^2\right)}-\frac{a^2 b \cos (x)}{\left(a^2+b^2\right)^2}+\frac{a^3 b \tanh ^{-1}\left(\frac{a \cos (x)-b \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}",1,"(-2*a^3*b*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (3*b*(5*a^2 + b^2)*Cos[x] + b*(a^2 + b^2)*Cos[3*x] - 2*a*(5*a^2 - b^2 + (a^2 + b^2)*Cos[2*x])*Sin[x])/(12*(a^2 + b^2)^2)","A",1
13,1,82,73,0.3231422,"\int \frac{\cos ^2(x)}{a+b \cot (x)} \, dx","Integrate[Cos[x]^2/(a + b*Cot[x]),x]","\frac{-b \left(a^2+b^2\right) \cos (2 x)+a \left(\left(a^2+b^2\right) \sin (2 x)+2 x (a-i b)^2-2 a b \log \left((a \sin (x)+b \cos (x))^2\right)\right)+4 i a^2 b \tan ^{-1}(\tan (x))}{4 \left(a^2+b^2\right)^2}","\frac{a x \left(a^2-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{a^2 b \log (a \sin (x)+b \cos (x))}{\left(a^2+b^2\right)^2}",1,"((4*I)*a^2*b*ArcTan[Tan[x]] - b*(a^2 + b^2)*Cos[2*x] + a*(2*(a - I*b)^2*x - 2*a*b*Log[(b*Cos[x] + a*Sin[x])^2] + (a^2 + b^2)*Sin[2*x]))/(4*(a^2 + b^2)^2)","C",1
14,1,61,65,0.2024347,"\int \frac{\cos (x)}{a+b \cot (x)} \, dx","Integrate[Cos[x]/(a + b*Cot[x]),x]","\frac{a \sin (x)-b \cos (x)}{a^2+b^2}-\frac{2 a b \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 \sin (x)}{a^2+b^2}-\frac{b \cos (x)}{a^2+b^2}+\frac{a b \tanh ^{-1}\left(\frac{a \cos (x)-b \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{3/2}}",1,"(-2*a*b*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) + (-(b*Cos[x]) + a*Sin[x])/(a^2 + b^2)","A",1
15,1,76,47,0.0927033,"\int \frac{\sec (x)}{a+b \cot (x)} \, dx","Integrate[Sec[x]/(a + b*Cot[x]),x]","\frac{-\frac{2 b \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-a}{\sqrt{a^2+b^2}}\right)}{\sqrt{a^2+b^2}}-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{a}","\frac{b \tanh ^{-1}\left(\frac{a \cos (x)-b \sin (x)}{\sqrt{a^2+b^2}}\right)}{a \sqrt{a^2+b^2}}+\frac{\tanh ^{-1}(\sin (x))}{a}",1,"((-2*b*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] - Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])/a","A",1
16,1,27,29,0.0845742,"\int \frac{\sec ^2(x)}{a+b \cot (x)} \, dx","Integrate[Sec[x]^2/(a + b*Cot[x]),x]","\frac{-b \log (a \sin (x)+b \cos (x))+a \tan (x)+b \log (\cos (x))}{a^2}","-\frac{b \log (\tan (x))}{a^2}-\frac{b \log (a+b \cot (x))}{a^2}+\frac{\tan (x)}{a}",1,"(b*Log[Cos[x]] - b*Log[b*Cos[x] + a*Sin[x]] + a*Tan[x])/a^2","A",1
17,1,192,79,0.46914,"\int \frac{\sec ^3(x)}{a+b \cot (x)} \, dx","Integrate[Sec[x]^3/(a + b*Cot[x]),x]","-\frac{8 b \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-a}{\sqrt{a^2+b^2}}\right)+\sec ^2(x) \left(\left(a^2+2 b^2\right) \cos (2 x) \left(\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)-2 a^2 \sin (x)+a^2 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-a^2 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)+4 a b \cos (x)+2 b^2 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-2 b^2 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)}{4 a^3}","\frac{b^2 \tanh ^{-1}(\sin (x))}{a^3}-\frac{b \sec (x)}{a^2}+\frac{b \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{a \cos (x)-b \sin (x)}{\sqrt{a^2+b^2}}\right)}{a^3}+\frac{\tanh ^{-1}(\sin (x))}{2 a}+\frac{\tan (x) \sec (x)}{2 a}",1,"-1/4*(8*b*Sqrt[a^2 + b^2]*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]] + Sec[x]^2*(4*a*b*Cos[x] + a^2*Log[Cos[x/2] - Sin[x/2]] + 2*b^2*Log[Cos[x/2] - Sin[x/2]] + (a^2 + 2*b^2)*Cos[2*x]*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) - a^2*Log[Cos[x/2] + Sin[x/2]] - 2*b^2*Log[Cos[x/2] + Sin[x/2]] - 2*a^2*Sin[x]))/a^3","B",1
18,1,66,73,0.2827237,"\int \frac{\sec ^4(x)}{a+b \cot (x)} \, dx","Integrate[Sec[x]^4/(a + b*Cot[x]),x]","\frac{\left(4 a^3+6 a b^2\right) \tan (x)+6 b \left(a^2+b^2\right) (\log (\cos (x))-\log (a \sin (x)+b \cos (x)))+a^2 \sec ^2(x) (2 a \tan (x)-3 b)}{6 a^4}","-\frac{b \tan ^2(x)}{2 a^2}-\frac{b \left(a^2+b^2\right) \log (\tan (x))}{a^4}-\frac{b \left(a^2+b^2\right) \log (a+b \cot (x))}{a^4}+\frac{\left(a^2+b^2\right) \tan (x)}{a^3}+\frac{\tan ^3(x)}{3 a}",1,"(6*b*(a^2 + b^2)*(Log[Cos[x]] - Log[b*Cos[x] + a*Sin[x]]) + (4*a^3 + 6*a*b^2)*Tan[x] + a^2*Sec[x]^2*(-3*b + 2*a*Tan[x]))/(6*a^4)","A",1
19,1,57,25,0.0528384,"\int \frac{\sec (x)}{1+2 \cot (x)} \, dx","Integrate[Sec[x]/(1 + 2*Cot[x]),x]","\frac{4 \tanh ^{-1}\left(\frac{1-2 \tan \left(\frac{x}{2}\right)}{\sqrt{5}}\right)}{\sqrt{5}}-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","\tanh ^{-1}(\sin (x))+\frac{2 \tanh ^{-1}\left(\frac{\cos (x)-2 \sin (x)}{\sqrt{5}}\right)}{\sqrt{5}}",1,"(4*ArcTanh[(1 - 2*Tan[x/2])/Sqrt[5]])/Sqrt[5] - Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]]","B",1