1,1,78,0,0.0694971,"\int \frac{\cos ^4(x)}{i+\cot (x)} \, dx","Int[Cos[x]^4/(I + Cot[x]),x]","-\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}","-\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,"(-I/16)*x + 1/(32*(I - Cot[x])^2) + (I/8)/(I - Cot[x]) - (I/24)/(I + Cot[x])^3 + 5/(32*(I + Cot[x])^2) + ((3*I)/16)/(I + Cot[x])","A",5,4,13,0.3077,1,"{3516, 848, 88, 203}"
2,1,29,0,0.1607352,"\int \frac{\cos ^3(x)}{i+\cot (x)} \, dx","Int[Cos[x]^3/(I + Cot[x]),x]","\frac{1}{5} i \sin ^5(x)-\frac{1}{3} i \sin ^3(x)-\frac{1}{5} \cos ^5(x)","\frac{1}{5} i \sin ^5(x)-\frac{1}{3} i \sin ^3(x)-\frac{1}{5} \cos ^5(x)",1,"-Cos[x]^5/5 - (I/3)*Sin[x]^3 + (I/5)*Sin[x]^5","A",9,7,13,0.5385,1,"{3518, 3108, 3107, 2565, 30, 2564, 14}"
3,1,50,0,0.0604951,"\int \frac{\cos ^2(x)}{i+\cot (x)} \, dx","Int[Cos[x]^2/(I + Cot[x]),x]","-\frac{i x}{8}+\frac{i}{8 (-\cot (x)+i)}+\frac{i}{4 (\cot (x)+i)}+\frac{1}{8 (\cot (x)+i)^2}","-\frac{i x}{8}+\frac{i}{8 (-\cot (x)+i)}+\frac{i}{4 (\cot (x)+i)}+\frac{1}{8 (\cot (x)+i)^2}",1,"(-I/8)*x + (I/8)/(I - Cot[x]) + 1/(8*(I + Cot[x])^2) + (I/4)/(I + Cot[x])","A",5,4,13,0.3077,1,"{3516, 848, 88, 203}"
4,1,19,0,0.1002026,"\int \frac{\cos (x)}{i+\cot (x)} \, dx","Int[Cos[x]/(I + Cot[x]),x]","-\frac{1}{3} \cos ^3(x)-\frac{1}{3} i \sin ^3(x)","-\frac{1}{3} \cos ^3(x)-\frac{1}{3} i \sin ^3(x)",1,"-Cos[x]^3/3 - (I/3)*Sin[x]^3","A",8,6,11,0.5455,1,"{3518, 3108, 3107, 2565, 30, 2564}"
5,1,18,0,0.1016743,"\int \frac{\sec (x)}{i+\cot (x)} \, dx","Int[Sec[x]/(I + Cot[x]),x]","i \sin (x)-\cos (x)-i \tanh ^{-1}(\sin (x))","i \sin (x)-\cos (x)-i \tanh ^{-1}(\sin (x))",1,"(-I)*ArcTanh[Sin[x]] - Cos[x] + I*Sin[x]","A",8,7,11,0.6364,1,"{3518, 3108, 3107, 2638, 2592, 321, 206}"
6,1,20,0,0.0411211,"\int \frac{\sec ^2(x)}{i+\cot (x)} \, dx","Int[Sec[x]^2/(I + Cot[x]),x]","i x-i \tan (x)-\log (\sin (x))+\log (\tan (x))","i x-i \tan (x)-\log (\sin (x))+\log (\tan (x))",1,"I*x - Log[Sin[x]] + Log[Tan[x]] - I*Tan[x]","A",3,2,13,0.1538,1,"{3516, 44}"
7,1,22,0,0.1482426,"\int \frac{\sec ^3(x)}{i+\cot (x)} \, dx","Int[Sec[x]^3/(I + Cot[x]),x]","\sec (x)+\frac{1}{2} i \tanh ^{-1}(\sin (x))-\frac{1}{2} i \tan (x) \sec (x)","\sec (x)+\frac{1}{2} i \tanh ^{-1}(\sin (x))-\frac{1}{2} i \tan (x) \sec (x)",1,"(I/2)*ArcTanh[Sin[x]] + Sec[x] - (I/2)*Sec[x]*Tan[x]","A",8,7,13,0.5385,1,"{3518, 3108, 3107, 2606, 8, 2611, 3770}"
8,1,19,0,0.0447802,"\int \frac{\sec ^4(x)}{i+\cot (x)} \, dx","Int[Sec[x]^4/(I + Cot[x]),x]","\frac{\tan ^2(x)}{2}-\frac{1}{3} i \tan ^3(x)","\frac{\tan ^2(x)}{2}-\frac{1}{3} i \tan ^3(x)",1,"Tan[x]^2/2 - (I/3)*Tan[x]^3","A",4,3,13,0.2308,1,"{3516, 848, 43}"
9,1,40,0,0.1655043,"\int \frac{\sec ^5(x)}{i+\cot (x)} \, dx","Int[Sec[x]^5/(I + Cot[x]),x]","\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)","\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,"(I/8)*ArcTanh[Sin[x]] + Sec[x]^3/3 + (I/8)*Sec[x]*Tan[x] - (I/4)*Sec[x]^3*Tan[x]","A",9,8,13,0.6154,1,"{3518, 3108, 3107, 2606, 30, 2611, 3768, 3770}"
10,1,37,0,0.053519,"\int \frac{\sec ^6(x)}{i+\cot (x)} \, dx","Int[Sec[x]^6/(I + Cot[x]),x]","-\frac{1}{5} i \tan ^5(x)+\frac{\tan ^4(x)}{4}-\frac{1}{3} i \tan ^3(x)+\frac{\tan ^2(x)}{2}","-\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,"Tan[x]^2/2 - (I/3)*Tan[x]^3 + Tan[x]^4/4 - (I/5)*Tan[x]^5","A",4,3,13,0.2308,1,"{3516, 848, 75}"
11,1,126,0,0.3031966,"\int \frac{\cos ^4(x)}{a+b \cot (x)} \, dx","Int[Cos[x]^4/(a + b*Cot[x]),x]","\frac{a x \left(-6 a^2 b^2+3 a^4-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(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(-6 a^2 b^2+3 a^4-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(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}",1,"(a*(3*a^4 - 6*a^2*b^2 - b^4)*x)/(8*(a^2 + b^2)^3) - (a^4*b*Log[b*Cos[x] + a*Sin[x]])/(a^2 + b^2)^3 + ((4*b*(2*a^2 + b^2) + a*(5*a^2 + 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",8,6,13,0.4615,1,"{3516, 1647, 801, 635, 203, 260}"
12,1,123,0,0.2081976,"\int \frac{\cos ^3(x)}{a+b \cot (x)} \, dx","Int[Cos[x]^3/(a + b*Cot[x]),x]","-\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}}","-\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,"(a^3*b*ArcTanh[(a*Cos[x] - b*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (a^2*b*Cos[x])/(a^2 + b^2)^2 - (b*Cos[x]^3)/(3*(a^2 + b^2)) - (a*b^2*Sin[x])/(a^2 + b^2)^2 + (a*Sin[x])/(a^2 + b^2) - (a*Sin[x]^3)/(3*(a^2 + b^2))","A",10,9,13,0.6923,1,"{3518, 3109, 2633, 2565, 30, 3100, 2637, 3074, 206}"
13,1,73,0,0.1609312,"\int \frac{\cos ^2(x)}{a+b \cot (x)} \, dx","Int[Cos[x]^2/(a + b*Cot[x]),x]","\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}","\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,"(a*(a^2 - b^2)*x)/(2*(a^2 + b^2)^2) - (a^2*b*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,"{3516, 1647, 801, 635, 203, 260}"
14,1,65,0,0.0888049,"\int \frac{\cos (x)}{a+b \cot (x)} \, dx","Int[Cos[x]/(a + b*Cot[x]),x]","\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}}","\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,"(a*b*ArcTanh[(a*Cos[x] - b*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (b*Cos[x])/(a^2 + b^2) + (a*Sin[x])/(a^2 + b^2)","A",6,6,11,0.5455,1,"{3518, 3109, 2637, 2638, 3074, 206}"
15,1,47,0,0.1139742,"\int \frac{\sec (x)}{a+b \cot (x)} \, dx","Int[Sec[x]/(a + b*Cot[x]),x]","\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}","\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,"ArcTanh[Sin[x]]/a + (b*ArcTanh[(a*Cos[x] - b*Sin[x])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2])","A",6,5,11,0.4545,1,"{3518, 3110, 3770, 3074, 206}"
16,1,29,0,0.055597,"\int \frac{\sec ^2(x)}{a+b \cot (x)} \, dx","Int[Sec[x]^2/(a + b*Cot[x]),x]","-\frac{b \log (\tan (x))}{a^2}-\frac{b \log (a+b \cot (x))}{a^2}+\frac{\tan (x)}{a}","-\frac{b \log (\tan (x))}{a^2}-\frac{b \log (a+b \cot (x))}{a^2}+\frac{\tan (x)}{a}",1,"-((b*Log[a + b*Cot[x]])/a^2) - (b*Log[Tan[x]])/a^2 + Tan[x]/a","A",3,2,13,0.1538,1,"{3516, 44}"
17,1,79,0,0.2007493,"\int \frac{\sec ^3(x)}{a+b \cot (x)} \, dx","Int[Sec[x]^3/(a + b*Cot[x]),x]","\frac{b^2 \tanh ^{-1}(\sin (x))}{a^3}+\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{b \sec (x)}{a^2}+\frac{\tanh ^{-1}(\sin (x))}{2 a}+\frac{\tan (x) \sec (x)}{2 a}","\frac{b^2 \tanh ^{-1}(\sin (x))}{a^3}+\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{b \sec (x)}{a^2}+\frac{\tanh ^{-1}(\sin (x))}{2 a}+\frac{\tan (x) \sec (x)}{2 a}",1,"ArcTanh[Sin[x]]/(2*a) + (b^2*ArcTanh[Sin[x]])/a^3 + (b*Sqrt[a^2 + b^2]*ArcTanh[(a*Cos[x] - b*Sin[x])/Sqrt[a^2 + b^2]])/a^3 - (b*Sec[x])/a^2 + (Sec[x]*Tan[x])/(2*a)","A",9,7,13,0.5385,1,"{3518, 3110, 3768, 3770, 3104, 3074, 206}"
18,1,73,0,0.0980419,"\int \frac{\sec ^4(x)}{a+b \cot (x)} \, dx","Int[Sec[x]^4/(a + b*Cot[x]),x]","\frac{\left(a^2+b^2\right) \tan (x)}{a^3}-\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{b \tan ^2(x)}{2 a^2}+\frac{\tan ^3(x)}{3 a}","\frac{\left(a^2+b^2\right) \tan (x)}{a^3}-\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{b \tan ^2(x)}{2 a^2}+\frac{\tan ^3(x)}{3 a}",1,"-((b*(a^2 + b^2)*Log[a + b*Cot[x]])/a^4) - (b*(a^2 + b^2)*Log[Tan[x]])/a^4 + ((a^2 + b^2)*Tan[x])/a^3 - (b*Tan[x]^2)/(2*a^2) + Tan[x]^3/(3*a)","A",3,2,13,0.1538,1,"{3516, 894}"
19,1,25,0,0.0903716,"\int \frac{\sec (x)}{1+2 \cot (x)} \, dx","Int[Sec[x]/(1 + 2*Cot[x]),x]","\tanh ^{-1}(\sin (x))+\frac{2 \tanh ^{-1}\left(\frac{\cos (x)-2 \sin (x)}{\sqrt{5}}\right)}{\sqrt{5}}","\tanh ^{-1}(\sin (x))+\frac{2 \tanh ^{-1}\left(\frac{\cos (x)-2 \sin (x)}{\sqrt{5}}\right)}{\sqrt{5}}",1,"(2*ArcTanh[(Cos[x] - 2*Sin[x])/Sqrt[5]])/Sqrt[5] + ArcTanh[Sin[x]]","A",6,5,11,0.4545,1,"{3518, 3110, 3770, 3074, 206}"