1,1,184,101,0.7137129,"\int x^3 \cot (a+b x) \, dx","Integrate[x^3*Cot[a + b*x],x]","\frac{4 b^3 x^3 \log \left(1-e^{-i (a+b x)}\right)+4 b^3 x^3 \log \left(1+e^{-i (a+b x)}\right)+12 i b^2 x^2 \text{Li}_2\left(-e^{-i (a+b x)}\right)+12 i b^2 x^2 \text{Li}_2\left(e^{-i (a+b x)}\right)+24 b x \text{Li}_3\left(-e^{-i (a+b x)}\right)+24 b x \text{Li}_3\left(e^{-i (a+b x)}\right)-24 i \text{Li}_4\left(-e^{-i (a+b x)}\right)-24 i \text{Li}_4\left(e^{-i (a+b x)}\right)+i b^4 x^4}{4 b^4}","\frac{3 i \text{Li}_4\left(e^{2 i (a+b x)}\right)}{4 b^4}+\frac{3 x \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}-\frac{3 i x^2 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}+\frac{x^3 \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{i x^4}{4}",1,"(I*b^4*x^4 + 4*b^3*x^3*Log[1 - E^((-I)*(a + b*x))] + 4*b^3*x^3*Log[1 + E^((-I)*(a + b*x))] + (12*I)*b^2*x^2*PolyLog[2, -E^((-I)*(a + b*x))] + (12*I)*b^2*x^2*PolyLog[2, E^((-I)*(a + b*x))] + 24*b*x*PolyLog[3, -E^((-I)*(a + b*x))] + 24*b*x*PolyLog[3, E^((-I)*(a + b*x))] - (24*I)*PolyLog[4, -E^((-I)*(a + b*x))] - (24*I)*PolyLog[4, E^((-I)*(a + b*x))])/(4*b^4)","A",1
2,1,136,74,0.4635859,"\int x^2 \cot (a+b x) \, dx","Integrate[x^2*Cot[a + b*x],x]","\frac{3 b^2 x^2 \log \left(1-e^{-i (a+b x)}\right)+3 b^2 x^2 \log \left(1+e^{-i (a+b x)}\right)+6 i b x \text{Li}_2\left(-e^{-i (a+b x)}\right)+6 i b x \text{Li}_2\left(e^{-i (a+b x)}\right)+6 \text{Li}_3\left(-e^{-i (a+b x)}\right)+6 \text{Li}_3\left(e^{-i (a+b x)}\right)+i b^3 x^3}{3 b^3}","\frac{\text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}-\frac{i x \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}+\frac{x^2 \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{i x^3}{3}",1,"(I*b^3*x^3 + 3*b^2*x^2*Log[1 - E^((-I)*(a + b*x))] + 3*b^2*x^2*Log[1 + E^((-I)*(a + b*x))] + (6*I)*b*x*PolyLog[2, -E^((-I)*(a + b*x))] + (6*I)*b*x*PolyLog[2, E^((-I)*(a + b*x))] + 6*PolyLog[3, -E^((-I)*(a + b*x))] + 6*PolyLog[3, E^((-I)*(a + b*x))])/(3*b^3)","A",1
3,1,135,53,3.7278016,"\int x \cot (a+b x) \, dx","Integrate[x*Cot[a + b*x],x]","\frac{1}{2} \left(-\frac{i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)-i b x \left(\pi -2 \tan ^{-1}(\tan (a))\right)-2 \left(\tan ^{-1}(\tan (a))+b x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))}{b^2}+x^2 \cot (a)-x^2 e^{i \tan ^{-1}(\tan (a))} \cot (a) \sqrt{\sec ^2(a)}\right)","-\frac{i \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}+\frac{x \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{i x^2}{2}",1,"(x^2*Cot[a] - ((-I)*b*x*(Pi - 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])/b^2 - E^(I*ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^2])/2","B",0
4,0,0,13,1.8984662,"\int \frac{\cot (a+b x)}{x} \, dx","Integrate[Cot[a + b*x]/x,x]","\int \frac{\cot (a+b x)}{x} \, dx","\text{Int}\left(\frac{\cot (a+b x)}{x},x\right)",0,"Integrate[Cot[a + b*x]/x, x]","A",-1
5,0,0,13,3.3249053,"\int \frac{\cot (a+b x)}{x^2} \, dx","Integrate[Cot[a + b*x]/x^2,x]","\int \frac{\cot (a+b x)}{x^2} \, dx","\text{Int}\left(\frac{\cot (a+b x)}{x^2},x\right)",0,"Integrate[Cot[a + b*x]/x^2, x]","A",-1
6,1,171,97,0.9181042,"\int x^3 \cot ^2(a+b x) \, dx","Integrate[x^3*Cot[a + b*x]^2,x]","\frac{-\frac{2 i b^3 x^3}{-1+e^{2 i a}}+3 b^2 x^2 \log \left(1-e^{-i (a+b x)}\right)+3 b^2 x^2 \log \left(1+e^{-i (a+b x)}\right)+6 i b x \text{Li}_2\left(-e^{-i (a+b x)}\right)+6 i b x \text{Li}_2\left(e^{-i (a+b x)}\right)+6 \text{Li}_3\left(-e^{-i (a+b x)}\right)+6 \text{Li}_3\left(e^{-i (a+b x)}\right)}{b^4}+\frac{x^3 \csc (a) \sin (b x) \csc (a+b x)}{b}-\frac{x^4}{4}","\frac{3 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^4}-\frac{3 i x \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^3}+\frac{3 x^2 \log \left(1-e^{2 i (a+b x)}\right)}{b^2}-\frac{x^3 \cot (a+b x)}{b}-\frac{i x^3}{b}-\frac{x^4}{4}",1,"-1/4*x^4 + (((-2*I)*b^3*x^3)/(-1 + E^((2*I)*a)) + 3*b^2*x^2*Log[1 - E^((-I)*(a + b*x))] + 3*b^2*x^2*Log[1 + E^((-I)*(a + b*x))] + (6*I)*b*x*PolyLog[2, -E^((-I)*(a + b*x))] + (6*I)*b*x*PolyLog[2, E^((-I)*(a + b*x))] + 6*PolyLog[3, -E^((-I)*(a + b*x))] + 6*PolyLog[3, E^((-I)*(a + b*x))])/b^4 + (x^3*Csc[a]*Csc[a + b*x]*Sin[b*x])/b","A",1
7,1,153,74,5.3547612,"\int x^2 \cot ^2(a+b x) \, dx","Integrate[x^2*Cot[a + b*x]^2,x]","\frac{-b^2 x^2 e^{i \tan ^{-1}(\tan (a))} \cot (a) \sqrt{\sec ^2(a)}-i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+i b x \left(\pi -2 \tan ^{-1}(\tan (a))\right)+2 \left(\tan ^{-1}(\tan (a))+b x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)-2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)+\pi  \log \left(1+e^{-2 i b x}\right)-\pi  \log (\cos (b x))}{b^3}+\frac{x^2 \csc (a) \sin (b x) \csc (a+b x)}{b}-\frac{x^3}{3}","-\frac{i \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^3}+\frac{2 x \log \left(1-e^{2 i (a+b x)}\right)}{b^2}-\frac{x^2 \cot (a+b x)}{b}-\frac{i x^2}{b}-\frac{x^3}{3}",1,"-1/3*x^3 + (I*b*x*(Pi - 2*ArcTan[Tan[a]]) + Pi*Log[1 + E^((-2*I)*b*x)] + 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] - Pi*Log[Cos[b*x]] - 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] - I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))] - b^2*E^(I*ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^2])/b^3 + (x^2*Csc[a]*Csc[a + b*x]*Sin[b*x])/b","B",0
8,1,44,31,0.2190311,"\int x \cot ^2(a+b x) \, dx","Integrate[x*Cot[a + b*x]^2,x]","\frac{\log (\sin (a+b x))}{b^2}-\frac{x \cot (a)}{b}+\frac{x \csc (a) \sin (b x) \csc (a+b x)}{b}-\frac{x^2}{2}","\frac{\log (\sin (a+b x))}{b^2}-\frac{x \cot (a+b x)}{b}-\frac{x^2}{2}",1,"-1/2*x^2 - (x*Cot[a])/b + Log[Sin[a + b*x]]/b^2 + (x*Csc[a]*Csc[a + b*x]*Sin[b*x])/b","A",1
9,0,0,15,3.816477,"\int \frac{\cot ^2(a+b x)}{x} \, dx","Integrate[Cot[a + b*x]^2/x,x]","\int \frac{\cot ^2(a+b x)}{x} \, dx","\text{Int}\left(\frac{\cot ^2(a+b x)}{x},x\right)",0,"Integrate[Cot[a + b*x]^2/x, x]","A",-1
10,0,0,15,3.4289827,"\int \frac{\cot ^2(a+b x)}{x^2} \, dx","Integrate[Cot[a + b*x]^2/x^2,x]","\int \frac{\cot ^2(a+b x)}{x^2} \, dx","\text{Int}\left(\frac{\cot ^2(a+b x)}{x^2},x\right)",0,"Integrate[Cot[a + b*x]^2/x^2, x]","A",-1
11,1,461,202,6.8072331,"\int x^3 \cot ^3(a+b x) \, dx","Integrate[x^3*Cot[a + b*x]^3,x]","\frac{3 x^2 \csc (a) \sin (b x) \csc (a+b x)}{2 b^2}-\frac{3 \csc (a) \sec (a) \left(b^2 x^2 e^{i \tan ^{-1}(\tan (a))}+\frac{\tan (a) \left(i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-2 \left(\tan ^{-1}(\tan (a))+b x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\tan ^2(a)+1}}\right)}{2 b^4 \sqrt{\sec ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}+\frac{e^{i a} \csc (a) \left(e^{-2 i a} b^4 x^4+2 i \left(1-e^{-2 i a}\right) b^3 x^3 \log \left(1-e^{-i (a+b x)}\right)+2 i \left(1-e^{-2 i a}\right) b^3 x^3 \log \left(1+e^{-i (a+b x)}\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^2 x^2 \text{Li}_2\left(-e^{-i (a+b x)}\right)-2 i b x \text{Li}_3\left(-e^{-i (a+b x)}\right)-2 \text{Li}_4\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^2 x^2 \text{Li}_2\left(e^{-i (a+b x)}\right)-2 i b x \text{Li}_3\left(e^{-i (a+b x)}\right)-2 \text{Li}_4\left(e^{-i (a+b x)}\right)\right)\right)}{4 b^4}-\frac{x^3 \csc ^2(a+b x)}{2 b}-\frac{1}{4} x^4 \cot (a)","-\frac{3 i \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^4}-\frac{3 i \text{Li}_4\left(e^{2 i (a+b x)}\right)}{4 b^4}-\frac{3 x \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}+\frac{3 x \log \left(1-e^{2 i (a+b x)}\right)}{b^3}+\frac{3 i x^2 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}-\frac{3 x^2 \cot (a+b x)}{2 b^2}-\frac{x^3 \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{x^3 \cot ^2(a+b x)}{2 b}-\frac{3 i x^2}{2 b^2}-\frac{x^3}{2 b}+\frac{i x^4}{4}",1,"-1/4*(x^4*Cot[a]) - (x^3*Csc[a + b*x]^2)/(2*b) + (E^(I*a)*Csc[a]*((b^4*x^4)/E^((2*I)*a) + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 - E^((-I)*(a + b*x))] + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, -E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-I)*(a + b*x))] - 2*PolyLog[4, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, E^((-I)*(a + b*x))] - 2*PolyLog[4, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/(4*b^4) + (3*x^2*Csc[a]*Csc[a + b*x]*Sin[b*x])/(2*b^2) - (3*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(2*b^4*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
12,1,221,126,5.4851907,"\int x^2 \cot ^3(a+b x) \, dx","Integrate[x^2*Cot[a + b*x]^3,x]","-\frac{2 b^3 x^3 \cot (a)+3 b^2 x^2 \csc ^2(a+b x)+2 e^{-i a} \sin (a) (\cot (a)+i) \left(-b^3 x^3 \cot (a)+3 b^2 x^2 \log \left(1-e^{-i (a+b x)}\right)+3 b^2 x^2 \log \left(1+e^{-i (a+b x)}\right)+6 i b x \text{Li}_2\left(-e^{-i (a+b x)}\right)+6 i b x \text{Li}_2\left(e^{-i (a+b x)}\right)+6 \text{Li}_3\left(-e^{-i (a+b x)}\right)+6 \text{Li}_3\left(e^{-i (a+b x)}\right)+i b^3 x^3\right)+6 b x \cot (a)-6 \log (\sin (a+b x))-6 b x \csc (a) \sin (b x) \csc (a+b x)}{6 b^3}","-\frac{\text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}+\frac{\log (\sin (a+b x))}{b^3}+\frac{i x \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}-\frac{x \cot (a+b x)}{b^2}-\frac{x^2 \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{x^2 \cot ^2(a+b x)}{2 b}-\frac{x^2}{2 b}+\frac{i x^3}{3}",1,"-1/6*(6*b*x*Cot[a] + 2*b^3*x^3*Cot[a] + 3*b^2*x^2*Csc[a + b*x]^2 - 6*Log[Sin[a + b*x]] + (2*(I + Cot[a])*(I*b^3*x^3 - b^3*x^3*Cot[a] + 3*b^2*x^2*Log[1 - E^((-I)*(a + b*x))] + 3*b^2*x^2*Log[1 + E^((-I)*(a + b*x))] + (6*I)*b*x*PolyLog[2, -E^((-I)*(a + b*x))] + (6*I)*b*x*PolyLog[2, E^((-I)*(a + b*x))] + 6*PolyLog[3, -E^((-I)*(a + b*x))] + 6*PolyLog[3, E^((-I)*(a + b*x))])*Sin[a])/E^(I*a) - 6*b*x*Csc[a]*Csc[a + b*x]*Sin[b*x])/b^3","A",1
13,1,179,91,4.2950525,"\int x \cot ^3(a+b x) \, dx","Integrate[x*Cot[a + b*x]^3,x]","\frac{-b^2 x^2 \cot (a)+b^2 x^2 e^{i \tan ^{-1}(\tan (a))} \cot (a) \sqrt{\sec ^2(a)}+i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)-b x \csc ^2(a+b x)-2 b x \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)+\csc (a) \sin (b x) \csc (a+b x)+2 \tan ^{-1}(\tan (a)) \left(-\log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)+\log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)+i b x\right)-i \pi  b x-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))}{2 b^2}","\frac{i \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}-\frac{\cot (a+b x)}{2 b^2}-\frac{x \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{x \cot ^2(a+b x)}{2 b}-\frac{x}{2 b}+\frac{i x^2}{2}",1,"((-I)*b*Pi*x - b^2*x^2*Cot[a] - b*x*Csc[a + b*x]^2 - Pi*Log[1 + E^((-2*I)*b*x)] - 2*b*x*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*(I*b*x - Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Log[Sin[b*x + ArcTan[Tan[a]]]]) + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))] + b^2*E^(I*ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^2] + Csc[a]*Csc[a + b*x]*Sin[b*x])/(2*b^2)","A",0
14,0,0,15,6.5055369,"\int \frac{\cot ^3(a+b x)}{x} \, dx","Integrate[Cot[a + b*x]^3/x,x]","\int \frac{\cot ^3(a+b x)}{x} \, dx","\text{Int}\left(\frac{\cot ^3(a+b x)}{x},x\right)",0,"Integrate[Cot[a + b*x]^3/x, x]","A",-1
15,0,0,15,5.2701638,"\int \frac{\cot ^3(a+b x)}{x^2} \, dx","Integrate[Cot[a + b*x]^3/x^2,x]","\int \frac{\cot ^3(a+b x)}{x^2} \, dx","\text{Int}\left(\frac{\cot ^3(a+b x)}{x^2},x\right)",0,"Integrate[Cot[a + b*x]^3/x^2, x]","A",-1
16,1,246,189,0.6280513,"\int \frac{(c+d x)^3}{a+i a \cot (e+f x)} \, dx","Integrate[(c + d*x)^3/(a + I*a*Cot[e + f*x]),x]","\frac{i (\cos (2 e)+i \sin (2 e)) \cos (2 f x) \left(4 c^3 f^3+6 c^2 d f^2 (2 f x+i)+6 c d^2 f \left(2 f^2 x^2+2 i f x-1\right)+d^3 \left(4 f^3 x^3+6 i f^2 x^2-6 f x-3 i\right)\right)-(\cos (2 e)+i \sin (2 e)) \sin (2 f x) \left(4 c^3 f^3+6 c^2 d f^2 (2 f x+i)+6 c d^2 f \left(2 f^2 x^2+2 i f x-1\right)+d^3 \left(4 f^3 x^3+6 i f^2 x^2-6 f x-3 i\right)\right)+2 f^4 x \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)}{16 a f^4}","\frac{3 i d^2 (c+d x)}{4 f^3 (a+i a \cot (e+f x))}+\frac{3 d (c+d x)^2}{4 f^2 (a+i a \cot (e+f x))}-\frac{i (c+d x)^3}{2 f (a+i a \cot (e+f x))}-\frac{3 d (c+d x)^2}{8 a f^2}+\frac{i (c+d x)^3}{4 a f}+\frac{(c+d x)^4}{8 a d}-\frac{3 d^3}{8 f^4 (a+i a \cot (e+f x))}-\frac{3 i d^3 x}{8 a f^3}",1,"(2*f^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + I*(4*c^3*f^3 + 6*c^2*d*f^2*(I + 2*f*x) + 6*c*d^2*f*(-1 + (2*I)*f*x + 2*f^2*x^2) + d^3*(-3*I - 6*f*x + (6*I)*f^2*x^2 + 4*f^3*x^3))*Cos[2*f*x]*(Cos[2*e] + I*Sin[2*e]) - (4*c^3*f^3 + 6*c^2*d*f^2*(I + 2*f*x) + 6*c*d^2*f*(-1 + (2*I)*f*x + 2*f^2*x^2) + d^3*(-3*I - 6*f*x + (6*I)*f^2*x^2 + 4*f^3*x^3))*(Cos[2*e] + I*Sin[2*e])*Sin[2*f*x])/(16*a*f^4)","A",1
17,1,149,137,0.3662233,"\int \frac{(c+d x)^2}{a+i a \cot (e+f x)} \, dx","Integrate[(c + d*x)^2/(a + I*a*Cot[e + f*x]),x]","\frac{4 f^3 x \left(3 c^2+3 c d x+d^2 x^2\right)+3 (\cos (2 e)+i \sin (2 e)) \cos (2 f x) ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d ((1+i) f x+i))+3 i (\cos (2 e)+i \sin (2 e)) \sin (2 f x) ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d ((1+i) f x+i))}{24 a f^3}","\frac{d (c+d x)}{2 f^2 (a+i a \cot (e+f x))}-\frac{i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac{i (c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}+\frac{i d^2}{4 f^3 (a+i a \cot (e+f x))}-\frac{d^2 x}{4 a f^2}",1,"(4*f^3*x*(3*c^2 + 3*c*d*x + d^2*x^2) + 3*((1 + I)*c*f + d*(-1 + (1 + I)*f*x))*((1 + I)*c*f + d*(I + (1 + I)*f*x))*Cos[2*f*x]*(Cos[2*e] + I*Sin[2*e]) + (3*I)*((1 + I)*c*f + d*(-1 + (1 + I)*f*x))*((1 + I)*c*f + d*(I + (1 + I)*f*x))*(Cos[2*e] + I*Sin[2*e])*Sin[2*f*x])/(24*a*f^3)","A",1
18,1,107,84,0.247206,"\int \frac{c+d x}{a+i a \cot (e+f x)} \, dx","Integrate[(c + d*x)/(a + I*a*Cot[e + f*x]),x]","\frac{(\cos (e+f x)+i \sin (e+f x)) \left(\left(2 c f (2 f x+i)+d \left(2 f^2 x^2+2 i f x-1\right)\right) \cos (e+f x)-i \left(2 c f (2 f x-i)+d \left(2 f^2 x^2-2 i f x+1\right)\right) \sin (e+f x)\right)}{8 a f^2}","-\frac{i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac{(c+d x)^2}{4 a d}+\frac{d}{4 f^2 (a+i a \cot (e+f x))}+\frac{i d x}{4 a f}",1,"((Cos[e + f*x] + I*Sin[e + f*x])*((2*c*f*(I + 2*f*x) + d*(-1 + (2*I)*f*x + 2*f^2*x^2))*Cos[e + f*x] - I*(2*c*f*(-I + 2*f*x) + d*(1 - (2*I)*f*x + 2*f^2*x^2))*Sin[e + f*x]))/(8*a*f^2)","A",1
19,1,77,161,0.284395,"\int \frac{1}{(c+d x) (a+i a \cot (e+f x))} \, dx","Integrate[1/((c + d*x)*(a + I*a*Cot[e + f*x])),x]","\frac{\log (c+d x)-\left(\text{Ci}\left(\frac{2 f (c+d x)}{d}\right)+i \text{Si}\left(\frac{2 f (c+d x)}{d}\right)\right) \left(\cos \left(2 e-\frac{2 c f}{d}\right)+i \sin \left(2 e-\frac{2 c f}{d}\right)\right)}{2 a d}","-\frac{i \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \sin \left(2 e-\frac{2 c f}{d}\right)}{2 a d}-\frac{\text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)}{2 a d}+\frac{\sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{2 a d}-\frac{i \cos \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{2 a d}+\frac{\log (c+d x)}{2 a d}",1,"(Log[c + d*x] - (Cos[2*e - (2*c*f)/d] + I*Sin[2*e - (2*c*f)/d])*(CosIntegral[(2*f*(c + d*x))/d] + I*SinIntegral[(2*f*(c + d*x))/d]))/(2*a*d)","A",1
20,1,215,166,1.2448458,"\int \frac{1}{(c+d x)^2 (a+i a \cot (e+f x))} \, dx","Integrate[1/((c + d*x)^2*(a + I*a*Cot[e + f*x])),x]","\frac{\left(\cos \left(f \left(x-\frac{c}{d}\right)+e\right)+i \sin \left(f \left(x-\frac{c}{d}\right)+e\right)\right) \left(2 f (c+d x) \text{Ci}\left(\frac{2 f (c+d x)}{d}\right) \left(\sin \left(e-\frac{f (c+d x)}{d}\right)-i \cos \left(e-\frac{f (c+d x)}{d}\right)\right)+2 f (c+d x) \text{Si}\left(\frac{2 f (c+d x)}{d}\right) \left(\cos \left(e-\frac{f (c+d x)}{d}\right)+i \sin \left(e-\frac{f (c+d x)}{d}\right)\right)+d \left(i \left(\sin \left(f \left(x-\frac{c}{d}\right)+e\right)+\sin \left(f \left(\frac{c}{d}+x\right)+e\right)\right)-\cos \left(f \left(x-\frac{c}{d}\right)+e\right)+\cos \left(f \left(\frac{c}{d}+x\right)+e\right)\right)\right)}{2 a d^2 (c+d x)}","\frac{f \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \sin \left(2 e-\frac{2 c f}{d}\right)}{a d^2}-\frac{i f \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)}{a d^2}+\frac{i f \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{a d^2}+\frac{f \cos \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{a d^2}-\frac{1}{d (c+d x) (a+i a \cot (e+f x))}",1,"((Cos[e + f*(-(c/d) + x)] + I*Sin[e + f*(-(c/d) + x)])*(d*(-Cos[e + f*(-(c/d) + x)] + Cos[e + f*(c/d + x)] + I*(Sin[e + f*(-(c/d) + x)] + Sin[e + f*(c/d + x)])) + 2*f*(c + d*x)*CosIntegral[(2*f*(c + d*x))/d]*((-I)*Cos[e - (f*(c + d*x))/d] + Sin[e - (f*(c + d*x))/d]) + 2*f*(c + d*x)*(Cos[e - (f*(c + d*x))/d] + I*Sin[e - (f*(c + d*x))/d])*SinIntegral[(2*f*(c + d*x))/d]))/(2*a*d^2*(c + d*x))","A",0
21,1,283,227,1.5920715,"\int \frac{1}{(c+d x)^3 (a+i a \cot (e+f x))} \, dx","Integrate[1/((c + d*x)^3*(a + I*a*Cot[e + f*x])),x]","\frac{\left(\cos \left(f \left(x-\frac{c}{d}\right)+e\right)+i \sin \left(f \left(x-\frac{c}{d}\right)+e\right)\right) \left(4 f^2 (c+d x)^2 \text{Ci}\left(\frac{2 f (c+d x)}{d}\right) \left(\cos \left(e-\frac{f (c+d x)}{d}\right)+i \sin \left(e-\frac{f (c+d x)}{d}\right)\right)+i \left(4 f^2 (c+d x)^2 \text{Si}\left(\frac{2 f (c+d x)}{d}\right) \left(\cos \left(e-\frac{f (c+d x)}{d}\right)+i \sin \left(e-\frac{f (c+d x)}{d}\right)\right)+d \left(d \sin \left(f \left(x-\frac{c}{d}\right)+e\right)+d \sin \left(f \left(\frac{c}{d}+x\right)+e\right)+2 i c f \sin \left(f \left(\frac{c}{d}+x\right)+e\right)+2 i d f x \sin \left(f \left(\frac{c}{d}+x\right)+e\right)+i d \cos \left(f \left(x-\frac{c}{d}\right)+e\right)+(2 c f+2 d f x-i d) \cos \left(f \left(\frac{c}{d}+x\right)+e\right)\right)\right)\right)}{4 a d^3 (c+d x)^2}","\frac{i f^2 \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \sin \left(2 e-\frac{2 c f}{d}\right)}{a d^3}+\frac{f^2 \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)}{a d^3}-\frac{f^2 \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{a d^3}+\frac{i f^2 \cos \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{a d^3}-\frac{i f}{d^2 (c+d x) (a+i a \cot (e+f x))}+\frac{i f}{2 a d^2 (c+d x)}-\frac{1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}",1,"((Cos[e + f*(-(c/d) + x)] + I*Sin[e + f*(-(c/d) + x)])*(4*f^2*(c + d*x)^2*CosIntegral[(2*f*(c + d*x))/d]*(Cos[e - (f*(c + d*x))/d] + I*Sin[e - (f*(c + d*x))/d]) + I*(d*(I*d*Cos[e + f*(-(c/d) + x)] + ((-I)*d + 2*c*f + 2*d*f*x)*Cos[e + f*(c/d + x)] + d*Sin[e + f*(-(c/d) + x)] + d*Sin[e + f*(c/d + x)] + (2*I)*c*f*Sin[e + f*(c/d + x)] + (2*I)*d*f*x*Sin[e + f*(c/d + x)]) + 4*f^2*(c + d*x)^2*(Cos[e - (f*(c + d*x))/d] + I*Sin[e - (f*(c + d*x))/d])*SinIntegral[(2*f*(c + d*x))/d])))/(4*a*d^3*(c + d*x)^2)","A",0
22,1,362,270,1.6053635,"\int \frac{(c+d x)^3}{(a+i a \cot (e+f x))^2} \, dx","Integrate[(c + d*x)^3/(a + I*a*Cot[e + f*x])^2,x]","\frac{(\cos (2 (e+f x))+i \sin (2 (e+f x))) \left(\left(32 c^3 f^3 (4 f x-i)+24 c^2 d f^2 \left(8 f^2 x^2-4 i f x+1\right)+4 c d^2 f \left(32 f^3 x^3-24 i f^2 x^2+12 f x+3 i\right)+d^3 \left(32 f^4 x^4-32 i f^3 x^3+24 f^2 x^2+12 i f x-3\right)\right) \cos (2 (e+f x))-i \left(\left(32 c^3 f^3 (4 f x+i)+24 c^2 d f^2 \left(8 f^2 x^2+4 i f x-1\right)+4 c d^2 f \left(32 f^3 x^3+24 i f^2 x^2-12 f x-3 i\right)+d^3 \left(32 f^4 x^4+32 i f^3 x^3-24 f^2 x^2-12 i f x+3\right)\right) \sin (2 (e+f x))-32 \left(4 c^3 f^3+6 c^2 d f^2 (2 f x+i)+6 c d^2 f \left(2 f^2 x^2+2 i f x-1\right)+d^3 \left(4 f^3 x^3+6 i f^2 x^2-6 f x-3 i\right)\right)\right)\right)}{512 a^2 f^4}","-\frac{3 i d^2 (c+d x) e^{2 i e+2 i f x}}{8 a^2 f^3}+\frac{3 i d^2 (c+d x) e^{4 i e+4 i f x}}{128 a^2 f^3}-\frac{3 d (c+d x)^2 e^{2 i e+2 i f x}}{8 a^2 f^2}+\frac{3 d (c+d x)^2 e^{4 i e+4 i f x}}{64 a^2 f^2}+\frac{i (c+d x)^3 e^{2 i e+2 i f x}}{4 a^2 f}-\frac{i (c+d x)^3 e^{4 i e+4 i f x}}{16 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}+\frac{3 d^3 e^{2 i e+2 i f x}}{16 a^2 f^4}-\frac{3 d^3 e^{4 i e+4 i f x}}{512 a^2 f^4}",1,"((Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*((32*c^3*f^3*(-I + 4*f*x) + 24*c^2*d*f^2*(1 - (4*I)*f*x + 8*f^2*x^2) + 4*c*d^2*f*(3*I + 12*f*x - (24*I)*f^2*x^2 + 32*f^3*x^3) + d^3*(-3 + (12*I)*f*x + 24*f^2*x^2 - (32*I)*f^3*x^3 + 32*f^4*x^4))*Cos[2*(e + f*x)] - I*(-32*(4*c^3*f^3 + 6*c^2*d*f^2*(I + 2*f*x) + 6*c*d^2*f*(-1 + (2*I)*f*x + 2*f^2*x^2) + d^3*(-3*I - 6*f*x + (6*I)*f^2*x^2 + 4*f^3*x^3)) + (32*c^3*f^3*(I + 4*f*x) + 24*c^2*d*f^2*(-1 + (4*I)*f*x + 8*f^2*x^2) + 4*c*d^2*f*(-3*I - 12*f*x + (24*I)*f^2*x^2 + 32*f^3*x^3) + d^3*(3 - (12*I)*f*x - 24*f^2*x^2 + (32*I)*f^3*x^3 + 32*f^4*x^4))*Sin[2*(e + f*x)])))/(512*a^2*f^4)","A",1
23,1,255,202,0.737178,"\int \frac{(c+d x)^2}{(a+i a \cot (e+f x))^2} \, dx","Integrate[(c + d*x)^2/(a + I*a*Cot[e + f*x])^2,x]","\frac{32 f^3 x \left(3 c^2+3 c d x+d^2 x^2\right)+48 (\cos (2 e)+i \sin (2 e)) \cos (2 f x) ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d ((1+i) f x+i))-3 (\cos (4 e)+i \sin (4 e)) \cos (4 f x) ((2+2 i) c f+d (-1+(2+2 i) f x)) ((2+2 i) c f+d ((2+2 i) f x+i))+48 i (\cos (2 e)+i \sin (2 e)) \sin (2 f x) ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d ((1+i) f x+i))-3 (\cos (4 e)+i \sin (4 e)) \sin (4 f x) (-(2+2 i) c f+(-2-2 i) d f x+d) ((2-2 i) c f+(2-2 i) d f x+d)}{384 a^2 f^3}","-\frac{d (c+d x) e^{2 i e+2 i f x}}{4 a^2 f^2}+\frac{d (c+d x) e^{4 i e+4 i f x}}{32 a^2 f^2}+\frac{i (c+d x)^2 e^{2 i e+2 i f x}}{4 a^2 f}-\frac{i (c+d x)^2 e^{4 i e+4 i f x}}{16 a^2 f}+\frac{(c+d x)^3}{12 a^2 d}-\frac{i d^2 e^{2 i e+2 i f x}}{8 a^2 f^3}+\frac{i d^2 e^{4 i e+4 i f x}}{128 a^2 f^3}",1,"(32*f^3*x*(3*c^2 + 3*c*d*x + d^2*x^2) + 48*((1 + I)*c*f + d*(-1 + (1 + I)*f*x))*((1 + I)*c*f + d*(I + (1 + I)*f*x))*Cos[2*f*x]*(Cos[2*e] + I*Sin[2*e]) - 3*((2 + 2*I)*c*f + d*(-1 + (2 + 2*I)*f*x))*((2 + 2*I)*c*f + d*(I + (2 + 2*I)*f*x))*Cos[4*f*x]*(Cos[4*e] + I*Sin[4*e]) + (48*I)*((1 + I)*c*f + d*(-1 + (1 + I)*f*x))*((1 + I)*c*f + d*(I + (1 + I)*f*x))*(Cos[2*e] + I*Sin[2*e])*Sin[2*f*x] - 3*(d - (2 + 2*I)*c*f - (2 + 2*I)*d*f*x)*(d + (2 - 2*I)*c*f + (2 - 2*I)*d*f*x)*(Cos[4*e] + I*Sin[4*e])*Sin[4*f*x])/(384*a^2*f^3)","A",1
24,1,165,151,0.6130301,"\int \frac{c+d x}{(a+i a \cot (e+f x))^2} \, dx","Integrate[(c + d*x)/(a + I*a*Cot[e + f*x])^2,x]","\frac{8 i (2 c f+d (2 f x+i)) \cos (2 (e+f x))+(-4 i c f-4 i d f x+d) \cos (4 (e+f x))-16 c f \sin (2 (e+f x))+4 c f \sin (4 (e+f x))+16 c e f+16 c f^2 x-8 d e^2-8 i d \sin (2 (e+f x))-16 d f x \sin (2 (e+f x))+i d \sin (4 (e+f x))+4 d f x \sin (4 (e+f x))+8 d f^2 x^2}{64 a^2 f^2}","-\frac{i (c+d x)}{4 f \left(a^2+i a^2 \cot (e+f x)\right)}+\frac{x (c+d x)}{4 a^2}+\frac{3 d}{16 f^2 \left(a^2+i a^2 \cot (e+f x)\right)}+\frac{3 i d x}{16 a^2 f}-\frac{d x^2}{8 a^2}-\frac{i (c+d x)}{4 f (a+i a \cot (e+f x))^2}+\frac{d}{16 f^2 (a+i a \cot (e+f x))^2}",1,"(-8*d*e^2 + 16*c*e*f + 16*c*f^2*x + 8*d*f^2*x^2 + (8*I)*(2*c*f + d*(I + 2*f*x))*Cos[2*(e + f*x)] + (d - (4*I)*c*f - (4*I)*d*f*x)*Cos[4*(e + f*x)] - (8*I)*d*Sin[2*(e + f*x)] - 16*c*f*Sin[2*(e + f*x)] - 16*d*f*x*Sin[2*(e + f*x)] + I*d*Sin[4*(e + f*x)] + 4*c*f*Sin[4*(e + f*x)] + 4*d*f*x*Sin[4*(e + f*x)])/(64*a^2*f^2)","A",1
25,1,136,305,0.6093475,"\int \frac{1}{(c+d x) (a+i a \cot (e+f x))^2} \, dx","Integrate[1/((c + d*x)*(a + I*a*Cot[e + f*x])^2),x]","\frac{-2 \left(\text{Ci}\left(\frac{2 f (c+d x)}{d}\right)+i \text{Si}\left(\frac{2 f (c+d x)}{d}\right)\right) \left(\cos \left(2 e-\frac{2 c f}{d}\right)+i \sin \left(2 e-\frac{2 c f}{d}\right)\right)+\left(\text{Ci}\left(\frac{4 f (c+d x)}{d}\right)+i \text{Si}\left(\frac{4 f (c+d x)}{d}\right)\right) \left(\cos \left(4 e-\frac{4 c f}{d}\right)+i \sin \left(4 e-\frac{4 c f}{d}\right)\right)+\log (c+d x)}{4 a^2 d}","-\frac{i \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \sin \left(2 e-\frac{2 c f}{d}\right)}{2 a^2 d}+\frac{i \text{Ci}\left(4 x f+\frac{4 c f}{d}\right) \sin \left(4 e-\frac{4 c f}{d}\right)}{4 a^2 d}-\frac{\text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)}{2 a^2 d}+\frac{\text{Ci}\left(4 x f+\frac{4 c f}{d}\right) \cos \left(4 e-\frac{4 c f}{d}\right)}{4 a^2 d}+\frac{\sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{2 a^2 d}-\frac{\sin \left(4 e-\frac{4 c f}{d}\right) \text{Si}\left(4 x f+\frac{4 c f}{d}\right)}{4 a^2 d}-\frac{i \cos \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{2 a^2 d}+\frac{i \cos \left(4 e-\frac{4 c f}{d}\right) \text{Si}\left(4 x f+\frac{4 c f}{d}\right)}{4 a^2 d}+\frac{\log (c+d x)}{4 a^2 d}",1,"(Log[c + d*x] - 2*(Cos[2*e - (2*c*f)/d] + I*Sin[2*e - (2*c*f)/d])*(CosIntegral[(2*f*(c + d*x))/d] + I*SinIntegral[(2*f*(c + d*x))/d]) + (Cos[4*e - (4*c*f)/d] + I*Sin[4*e - (4*c*f)/d])*(CosIntegral[(4*f*(c + d*x))/d] + I*SinIntegral[(4*f*(c + d*x))/d]))/(4*a^2*d)","A",1
26,1,203,434,0.5874296,"\int \frac{1}{(c+d x)^2 (a+i a \cot (e+f x))^2} \, dx","Integrate[1/((c + d*x)^2*(a + I*a*Cot[e + f*x])^2),x]","\frac{4 f (c+d x) \left(\text{Ci}\left(\frac{2 f (c+d x)}{d}\right)+i \text{Si}\left(\frac{2 f (c+d x)}{d}\right)\right) \left(\sin \left(2 e-\frac{2 c f}{d}\right)-i \cos \left(2 e-\frac{2 c f}{d}\right)\right)+(c+d x) \left(\text{Ci}\left(\frac{4 f (c+d x)}{d}\right)+i \text{Si}\left(\frac{4 f (c+d x)}{d}\right)\right) \left(-4 f \sin \left(4 e-\frac{4 c f}{d}\right)+4 i f \cos \left(4 e-\frac{4 c f}{d}\right)\right)+2 d (\cos (2 (e+f x))+i \sin (2 (e+f x)))-d (\cos (4 (e+f x))+i \sin (4 (e+f x)))-d}{4 a^2 d^2 (c+d x)}","-\frac{f \text{Ci}\left(4 x f+\frac{4 c f}{d}\right) \sin \left(4 e-\frac{4 c f}{d}\right)}{a^2 d^2}+\frac{f \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \sin \left(2 e-\frac{2 c f}{d}\right)}{a^2 d^2}-\frac{i f \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)}{a^2 d^2}+\frac{i f \text{Ci}\left(4 x f+\frac{4 c f}{d}\right) \cos \left(4 e-\frac{4 c f}{d}\right)}{a^2 d^2}+\frac{i f \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{a^2 d^2}-\frac{i f \sin \left(4 e-\frac{4 c f}{d}\right) \text{Si}\left(4 x f+\frac{4 c f}{d}\right)}{a^2 d^2}+\frac{f \cos \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{a^2 d^2}-\frac{f \cos \left(4 e-\frac{4 c f}{d}\right) \text{Si}\left(4 x f+\frac{4 c f}{d}\right)}{a^2 d^2}+\frac{\sin ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac{i \sin (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac{i \sin (4 e+4 f x)}{4 a^2 d (c+d x)}-\frac{\cos ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac{\cos (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac{1}{4 a^2 d (c+d x)}",1,"(-d + 2*d*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]) - d*(Cos[4*(e + f*x)] + I*Sin[4*(e + f*x)]) + 4*f*(c + d*x)*((-I)*Cos[2*e - (2*c*f)/d] + Sin[2*e - (2*c*f)/d])*(CosIntegral[(2*f*(c + d*x))/d] + I*SinIntegral[(2*f*(c + d*x))/d]) + (c + d*x)*((4*I)*f*Cos[4*e - (4*c*f)/d] - 4*f*Sin[4*e - (4*c*f)/d])*(CosIntegral[(4*f*(c + d*x))/d] + I*SinIntegral[(4*f*(c + d*x))/d]))/(4*a^2*d^2*(c + d*x))","A",1
27,1,664,396,2.8497874,"\int \frac{(c+d x)^3}{(a+i a \cot (e+f x))^3} \, dx","Integrate[(c + d*x)^3/(a + I*a*Cot[e + f*x])^3,x]","\frac{(\cos (3 (e+f x))+i \sin (3 (e+f x))) \left(-3456 i c^3 f^4 x \sin (3 (e+f x))+7776 c^3 f^3 \sin (e+f x)-576 c^3 f^3 \sin (3 (e+f x))-5184 i c^2 d f^4 x^2 \sin (3 (e+f x))+23328 c^2 d f^3 x \sin (e+f x)-1728 c^2 d f^3 x \sin (3 (e+f x))+9720 i c^2 d f^2 \sin (e+f x)-288 i c^2 d f^2 \sin (3 (e+f x))+81 i \left(32 c^3 f^3+24 c^2 d f^2 (4 f x+3 i)+12 c d^2 f \left(8 f^2 x^2+12 i f x-7\right)+d^3 \left(32 f^3 x^3+72 i f^2 x^2-84 f x-45 i\right)\right) \cos (e+f x)+16 \left(36 c^3 f^3 (6 f x+i)+18 c^2 d f^2 \left(18 f^2 x^2+6 i f x-1\right)+6 c d^2 f \left(36 f^3 x^3+18 i f^2 x^2-6 f x-i\right)+d^3 \left(54 f^4 x^4+36 i f^3 x^3-18 f^2 x^2-6 i f x+1\right)\right) \cos (3 (e+f x))-3456 i c d^2 f^4 x^3 \sin (3 (e+f x))+23328 c d^2 f^3 x^2 \sin (e+f x)-1728 c d^2 f^3 x^2 \sin (3 (e+f x))+19440 i c d^2 f^2 x \sin (e+f x)-576 i c d^2 f^2 x \sin (3 (e+f x))-8748 c d^2 f \sin (e+f x)+96 c d^2 f \sin (3 (e+f x))-864 i d^3 f^4 x^4 \sin (3 (e+f x))+7776 d^3 f^3 x^3 \sin (e+f x)-576 d^3 f^3 x^3 \sin (3 (e+f x))+9720 i d^3 f^2 x^2 \sin (e+f x)-288 i d^3 f^2 x^2 \sin (3 (e+f x))-8748 d^3 f x \sin (e+f x)+96 d^3 f x \sin (3 (e+f x))-4131 i d^3 \sin (e+f x)+16 i d^3 \sin (3 (e+f x))\right)}{27648 a^3 f^4}","-\frac{9 i d^2 (c+d x) e^{2 i e+2 i f x}}{32 a^3 f^3}+\frac{9 i d^2 (c+d x) e^{4 i e+4 i f x}}{256 a^3 f^3}-\frac{i d^2 (c+d x) e^{6 i e+6 i f x}}{288 a^3 f^3}-\frac{9 d (c+d x)^2 e^{2 i e+2 i f x}}{32 a^3 f^2}+\frac{9 d (c+d x)^2 e^{4 i e+4 i f x}}{128 a^3 f^2}-\frac{d (c+d x)^2 e^{6 i e+6 i f x}}{96 a^3 f^2}+\frac{3 i (c+d x)^3 e^{2 i e+2 i f x}}{16 a^3 f}-\frac{3 i (c+d x)^3 e^{4 i e+4 i f x}}{32 a^3 f}+\frac{i (c+d x)^3 e^{6 i e+6 i f x}}{48 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}+\frac{9 d^3 e^{2 i e+2 i f x}}{64 a^3 f^4}-\frac{9 d^3 e^{4 i e+4 i f x}}{1024 a^3 f^4}+\frac{d^3 e^{6 i e+6 i f x}}{1728 a^3 f^4}",1,"((Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*((81*I)*(32*c^3*f^3 + 24*c^2*d*f^2*(3*I + 4*f*x) + 12*c*d^2*f*(-7 + (12*I)*f*x + 8*f^2*x^2) + d^3*(-45*I - 84*f*x + (72*I)*f^2*x^2 + 32*f^3*x^3))*Cos[e + f*x] + 16*(36*c^3*f^3*(I + 6*f*x) + 18*c^2*d*f^2*(-1 + (6*I)*f*x + 18*f^2*x^2) + 6*c*d^2*f*(-I - 6*f*x + (18*I)*f^2*x^2 + 36*f^3*x^3) + d^3*(1 - (6*I)*f*x - 18*f^2*x^2 + (36*I)*f^3*x^3 + 54*f^4*x^4))*Cos[3*(e + f*x)] - (4131*I)*d^3*Sin[e + f*x] - 8748*c*d^2*f*Sin[e + f*x] + (9720*I)*c^2*d*f^2*Sin[e + f*x] + 7776*c^3*f^3*Sin[e + f*x] - 8748*d^3*f*x*Sin[e + f*x] + (19440*I)*c*d^2*f^2*x*Sin[e + f*x] + 23328*c^2*d*f^3*x*Sin[e + f*x] + (9720*I)*d^3*f^2*x^2*Sin[e + f*x] + 23328*c*d^2*f^3*x^2*Sin[e + f*x] + 7776*d^3*f^3*x^3*Sin[e + f*x] + (16*I)*d^3*Sin[3*(e + f*x)] + 96*c*d^2*f*Sin[3*(e + f*x)] - (288*I)*c^2*d*f^2*Sin[3*(e + f*x)] - 576*c^3*f^3*Sin[3*(e + f*x)] + 96*d^3*f*x*Sin[3*(e + f*x)] - (576*I)*c*d^2*f^2*x*Sin[3*(e + f*x)] - 1728*c^2*d*f^3*x*Sin[3*(e + f*x)] - (3456*I)*c^3*f^4*x*Sin[3*(e + f*x)] - (288*I)*d^3*f^2*x^2*Sin[3*(e + f*x)] - 1728*c*d^2*f^3*x^2*Sin[3*(e + f*x)] - (5184*I)*c^2*d*f^4*x^2*Sin[3*(e + f*x)] - 576*d^3*f^3*x^3*Sin[3*(e + f*x)] - (3456*I)*c*d^2*f^4*x^3*Sin[3*(e + f*x)] - (864*I)*d^3*f^4*x^4*Sin[3*(e + f*x)]))/(27648*a^3*f^4)","A",1
28,1,369,294,0.8115688,"\int \frac{(c+d x)^2}{(a+i a \cot (e+f x))^3} \, dx","Integrate[(c + d*x)^2/(a + I*a*Cot[e + f*x])^3,x]","\frac{288 f^3 x \left(3 c^2+3 c d x+d^2 x^2\right)+648 (\cos (2 e)+i \sin (2 e)) \cos (2 f x) ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d ((1+i) f x+i))-81 (\cos (4 e)+i \sin (4 e)) \cos (4 f x) ((2+2 i) c f+d (-1+(2+2 i) f x)) ((2+2 i) c f+d ((2+2 i) f x+i))+8 (\cos (6 e)+i \sin (6 e)) \cos (6 f x) ((3+3 i) c f+d (-1+(3+3 i) f x)) ((3+3 i) c f+d ((3+3 i) f x+i))+648 i (\cos (2 e)+i \sin (2 e)) \sin (2 f x) ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d ((1+i) f x+i))-81 (\cos (4 e)+i \sin (4 e)) \sin (4 f x) (-(2+2 i) c f+(-2-2 i) d f x+d) ((2-2 i) c f+(2-2 i) d f x+d)+8 i (\cos (6 e)+i \sin (6 e)) \sin (6 f x) ((3+3 i) c f+d (-1+(3+3 i) f x)) ((3+3 i) c f+d ((3+3 i) f x+i))}{6912 a^3 f^3}","-\frac{3 d (c+d x) e^{2 i e+2 i f x}}{16 a^3 f^2}+\frac{3 d (c+d x) e^{4 i e+4 i f x}}{64 a^3 f^2}-\frac{d (c+d x) e^{6 i e+6 i f x}}{144 a^3 f^2}+\frac{3 i (c+d x)^2 e^{2 i e+2 i f x}}{16 a^3 f}-\frac{3 i (c+d x)^2 e^{4 i e+4 i f x}}{32 a^3 f}+\frac{i (c+d x)^2 e^{6 i e+6 i f x}}{48 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}-\frac{3 i d^2 e^{2 i e+2 i f x}}{32 a^3 f^3}+\frac{3 i d^2 e^{4 i e+4 i f x}}{256 a^3 f^3}-\frac{i d^2 e^{6 i e+6 i f x}}{864 a^3 f^3}",1,"(288*f^3*x*(3*c^2 + 3*c*d*x + d^2*x^2) + 648*((1 + I)*c*f + d*(-1 + (1 + I)*f*x))*((1 + I)*c*f + d*(I + (1 + I)*f*x))*Cos[2*f*x]*(Cos[2*e] + I*Sin[2*e]) - 81*((2 + 2*I)*c*f + d*(-1 + (2 + 2*I)*f*x))*((2 + 2*I)*c*f + d*(I + (2 + 2*I)*f*x))*Cos[4*f*x]*(Cos[4*e] + I*Sin[4*e]) + 8*((3 + 3*I)*c*f + d*(-1 + (3 + 3*I)*f*x))*((3 + 3*I)*c*f + d*(I + (3 + 3*I)*f*x))*Cos[6*f*x]*(Cos[6*e] + I*Sin[6*e]) + (648*I)*((1 + I)*c*f + d*(-1 + (1 + I)*f*x))*((1 + I)*c*f + d*(I + (1 + I)*f*x))*(Cos[2*e] + I*Sin[2*e])*Sin[2*f*x] - 81*(d - (2 + 2*I)*c*f - (2 + 2*I)*d*f*x)*(d + (2 - 2*I)*c*f + (2 - 2*I)*d*f*x)*(Cos[4*e] + I*Sin[4*e])*Sin[4*f*x] + (8*I)*((3 + 3*I)*c*f + d*(-1 + (3 + 3*I)*f*x))*((3 + 3*I)*c*f + d*(I + (3 + 3*I)*f*x))*(Cos[6*e] + I*Sin[6*e])*Sin[6*f*x])/(6912*a^3*f^3)","A",1
29,1,244,209,0.6633656,"\int \frac{c+d x}{(a+i a \cot (e+f x))^3} \, dx","Integrate[(c + d*x)/(a + I*a*Cot[e + f*x])^3,x]","\frac{108 i (2 c f+d (2 f x+i)) \cos (2 (e+f x))+27 (-4 i c f-4 i d f x+d) \cos (4 (e+f x))-216 c f \sin (2 (e+f x))+108 c f \sin (4 (e+f x))-24 c f \sin (6 (e+f x))+24 i c f \cos (6 (e+f x))+144 c e f+144 c f^2 x-72 d e^2-108 i d \sin (2 (e+f x))-216 d f x \sin (2 (e+f x))+27 i d \sin (4 (e+f x))+108 d f x \sin (4 (e+f x))-4 i d \sin (6 (e+f x))-24 d f x \sin (6 (e+f x))-4 d \cos (6 (e+f x))+24 i d f x \cos (6 (e+f x))+72 d f^2 x^2}{1152 a^3 f^2}","-\frac{i (c+d x)}{8 f \left(a^3+i a^3 \cot (e+f x)\right)}+\frac{x (c+d x)}{8 a^3}+\frac{11 d}{96 f^2 \left(a^3+i a^3 \cot (e+f x)\right)}+\frac{11 i d x}{96 a^3 f}-\frac{d x^2}{16 a^3}-\frac{i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}-\frac{i (c+d x)}{6 f (a+i a \cot (e+f x))^3}+\frac{5 d}{96 a f^2 (a+i a \cot (e+f x))^2}+\frac{d}{36 f^2 (a+i a \cot (e+f x))^3}",1,"(-72*d*e^2 + 144*c*e*f + 144*c*f^2*x + 72*d*f^2*x^2 + (108*I)*(2*c*f + d*(I + 2*f*x))*Cos[2*(e + f*x)] + 27*(d - (4*I)*c*f - (4*I)*d*f*x)*Cos[4*(e + f*x)] - 4*d*Cos[6*(e + f*x)] + (24*I)*c*f*Cos[6*(e + f*x)] + (24*I)*d*f*x*Cos[6*(e + f*x)] - (108*I)*d*Sin[2*(e + f*x)] - 216*c*f*Sin[2*(e + f*x)] - 216*d*f*x*Sin[2*(e + f*x)] + (27*I)*d*Sin[4*(e + f*x)] + 108*c*f*Sin[4*(e + f*x)] + 108*d*f*x*Sin[4*(e + f*x)] - (4*I)*d*Sin[6*(e + f*x)] - 24*c*f*Sin[6*(e + f*x)] - 24*d*f*x*Sin[6*(e + f*x)])/(1152*a^3*f^2)","A",1
30,1,197,449,0.5783841,"\int \frac{1}{(c+d x) (a+i a \cot (e+f x))^3} \, dx","Integrate[1/((c + d*x)*(a + I*a*Cot[e + f*x])^3),x]","\frac{-3 \left(\text{Ci}\left(\frac{2 f (c+d x)}{d}\right)+i \text{Si}\left(\frac{2 f (c+d x)}{d}\right)\right) \left(\cos \left(2 e-\frac{2 c f}{d}\right)+i \sin \left(2 e-\frac{2 c f}{d}\right)\right)+3 \left(\text{Ci}\left(\frac{4 f (c+d x)}{d}\right)+i \text{Si}\left(\frac{4 f (c+d x)}{d}\right)\right) \left(\cos \left(4 e-\frac{4 c f}{d}\right)+i \sin \left(4 e-\frac{4 c f}{d}\right)\right)-\left(\text{Ci}\left(\frac{6 f (c+d x)}{d}\right)+i \text{Si}\left(\frac{6 f (c+d x)}{d}\right)\right) \left(\cos \left(6 e-\frac{6 c f}{d}\right)+i \sin \left(6 e-\frac{6 c f}{d}\right)\right)+\log (c+d x)}{8 a^3 d}","-\frac{3 i \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \sin \left(2 e-\frac{2 c f}{d}\right)}{8 a^3 d}-\frac{i \text{Ci}\left(6 x f+\frac{6 c f}{d}\right) \sin \left(6 e-\frac{6 c f}{d}\right)}{8 a^3 d}+\frac{3 i \text{Ci}\left(4 x f+\frac{4 c f}{d}\right) \sin \left(4 e-\frac{4 c f}{d}\right)}{8 a^3 d}-\frac{3 \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)}{8 a^3 d}+\frac{3 \text{Ci}\left(4 x f+\frac{4 c f}{d}\right) \cos \left(4 e-\frac{4 c f}{d}\right)}{8 a^3 d}-\frac{\text{Ci}\left(6 x f+\frac{6 c f}{d}\right) \cos \left(6 e-\frac{6 c f}{d}\right)}{8 a^3 d}+\frac{3 \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{8 a^3 d}-\frac{3 \sin \left(4 e-\frac{4 c f}{d}\right) \text{Si}\left(4 x f+\frac{4 c f}{d}\right)}{8 a^3 d}+\frac{\sin \left(6 e-\frac{6 c f}{d}\right) \text{Si}\left(6 x f+\frac{6 c f}{d}\right)}{8 a^3 d}-\frac{3 i \cos \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{8 a^3 d}+\frac{3 i \cos \left(4 e-\frac{4 c f}{d}\right) \text{Si}\left(4 x f+\frac{4 c f}{d}\right)}{8 a^3 d}-\frac{i \cos \left(6 e-\frac{6 c f}{d}\right) \text{Si}\left(6 x f+\frac{6 c f}{d}\right)}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d}",1,"(Log[c + d*x] - 3*(Cos[2*e - (2*c*f)/d] + I*Sin[2*e - (2*c*f)/d])*(CosIntegral[(2*f*(c + d*x))/d] + I*SinIntegral[(2*f*(c + d*x))/d]) + 3*(Cos[4*e - (4*c*f)/d] + I*Sin[4*e - (4*c*f)/d])*(CosIntegral[(4*f*(c + d*x))/d] + I*SinIntegral[(4*f*(c + d*x))/d]) - (Cos[6*e - (6*c*f)/d] + I*Sin[6*e - (6*c*f)/d])*(CosIntegral[(6*f*(c + d*x))/d] + I*SinIntegral[(6*f*(c + d*x))/d]))/(8*a^3*d)","A",1
31,1,292,712,0.6433562,"\int \frac{1}{(c+d x)^2 (a+i a \cot (e+f x))^3} \, dx","Integrate[1/((c + d*x)^2*(a + I*a*Cot[e + f*x])^3),x]","\frac{6 f (c+d x) \left(\text{Ci}\left(\frac{2 f (c+d x)}{d}\right)+i \text{Si}\left(\frac{2 f (c+d x)}{d}\right)\right) \left(\sin \left(2 e-\frac{2 c f}{d}\right)-i \cos \left(2 e-\frac{2 c f}{d}\right)\right)+12 i f (c+d x) \left(\text{Ci}\left(\frac{4 f (c+d x)}{d}\right)+i \text{Si}\left(\frac{4 f (c+d x)}{d}\right)\right) \left(\cos \left(4 e-\frac{4 c f}{d}\right)+i \sin \left(4 e-\frac{4 c f}{d}\right)\right)+6 f (c+d x) \left(\text{Ci}\left(\frac{6 f (c+d x)}{d}\right)+i \text{Si}\left(\frac{6 f (c+d x)}{d}\right)\right) \left(\sin \left(6 e-\frac{6 c f}{d}\right)-i \cos \left(6 e-\frac{6 c f}{d}\right)\right)+3 d (\cos (2 (e+f x))+i \sin (2 (e+f x)))-3 d (\cos (4 (e+f x))+i \sin (4 (e+f x)))+d (\cos (6 (e+f x))+i \sin (6 (e+f x)))-d}{8 a^3 d^2 (c+d x)}","\frac{3 f \text{Ci}\left(6 x f+\frac{6 c f}{d}\right) \sin \left(6 e-\frac{6 c f}{d}\right)}{4 a^3 d^2}-\frac{3 f \text{Ci}\left(4 x f+\frac{4 c f}{d}\right) \sin \left(4 e-\frac{4 c f}{d}\right)}{2 a^3 d^2}+\frac{3 f \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \sin \left(2 e-\frac{2 c f}{d}\right)}{4 a^3 d^2}-\frac{3 i f \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)}{4 a^3 d^2}+\frac{3 i f \text{Ci}\left(4 x f+\frac{4 c f}{d}\right) \cos \left(4 e-\frac{4 c f}{d}\right)}{2 a^3 d^2}-\frac{3 i f \text{Ci}\left(6 x f+\frac{6 c f}{d}\right) \cos \left(6 e-\frac{6 c f}{d}\right)}{4 a^3 d^2}+\frac{3 i f \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{4 a^3 d^2}-\frac{3 i f \sin \left(4 e-\frac{4 c f}{d}\right) \text{Si}\left(4 x f+\frac{4 c f}{d}\right)}{2 a^3 d^2}+\frac{3 i f \sin \left(6 e-\frac{6 c f}{d}\right) \text{Si}\left(6 x f+\frac{6 c f}{d}\right)}{4 a^3 d^2}+\frac{3 f \cos \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{4 a^3 d^2}-\frac{3 f \cos \left(4 e-\frac{4 c f}{d}\right) \text{Si}\left(4 x f+\frac{4 c f}{d}\right)}{2 a^3 d^2}+\frac{3 f \cos \left(6 e-\frac{6 c f}{d}\right) \text{Si}\left(6 x f+\frac{6 c f}{d}\right)}{4 a^3 d^2}-\frac{i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{15 i \sin (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac{3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac{3 i \sin (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac{\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{9 \cos (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac{3 \cos (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{1}{8 a^3 d (c+d x)}",1,"(-d + 3*d*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]) - 3*d*(Cos[4*(e + f*x)] + I*Sin[4*(e + f*x)]) + d*(Cos[6*(e + f*x)] + I*Sin[6*(e + f*x)]) + 6*f*(c + d*x)*((-I)*Cos[2*e - (2*c*f)/d] + Sin[2*e - (2*c*f)/d])*(CosIntegral[(2*f*(c + d*x))/d] + I*SinIntegral[(2*f*(c + d*x))/d]) + (12*I)*f*(c + d*x)*(Cos[4*e - (4*c*f)/d] + I*Sin[4*e - (4*c*f)/d])*(CosIntegral[(4*f*(c + d*x))/d] + I*SinIntegral[(4*f*(c + d*x))/d]) + 6*f*(c + d*x)*((-I)*Cos[6*e - (6*c*f)/d] + Sin[6*e - (6*c*f)/d])*(CosIntegral[(6*f*(c + d*x))/d] + I*SinIntegral[(6*f*(c + d*x))/d]))/(8*a^3*d^2*(c + d*x))","A",1
32,0,0,26,10.3778153,"\int (c+d x)^m (a+i a \cot (e+f x))^2 \, dx","Integrate[(c + d*x)^m*(a + I*a*Cot[e + f*x])^2,x]","\int (c+d x)^m (a+i a \cot (e+f x))^2 \, dx","\text{Int}\left((c+d x)^m (a+i a \cot (e+f x))^2,x\right)",0,"Integrate[(c + d*x)^m*(a + I*a*Cot[e + f*x])^2, x]","A",-1
33,0,0,24,5.7383665,"\int (c+d x)^m (a+i a \cot (e+f x)) \, dx","Integrate[(c + d*x)^m*(a + I*a*Cot[e + f*x]),x]","\int (c+d x)^m (a+i a \cot (e+f x)) \, dx","\text{Int}\left((c+d x)^m (a+i a \cot (e+f x)),x\right)",0,"Integrate[(c + d*x)^m*(a + I*a*Cot[e + f*x]), x]","A",-1
34,1,190,98,1.2919181,"\int \frac{(c+d x)^m}{a+i a \cot (e+f x)} \, dx","Integrate[(c + d*x)^m/(a + I*a*Cot[e + f*x]),x]","\frac{2^{-m-2} (c+d x)^m \left(\frac{i f (c+d x)}{d}\right)^m \left(\frac{f^2 (c+d x)^2}{d^2}\right)^{-m} \left(\cos \left(e-\frac{c f}{d}\right)+i \sin \left(e-\frac{c f}{d}\right)\right) \left(f 2^{m+1} (c+d x) \left(-\frac{i f (c+d x)}{d}\right)^m \left(\cos \left(e-\frac{c f}{d}\right)-i \sin \left(e-\frac{c f}{d}\right)\right)+i d (m+1) \left(\cos \left(e-\frac{c f}{d}\right)+i \sin \left(e-\frac{c f}{d}\right)\right) \Gamma \left(m+1,-\frac{2 i f (c+d x)}{d}\right)\right)}{a d f (m+1)}","\frac{(c+d x)^{m+1}}{2 a d (m+1)}+\frac{i 2^{-m-2} e^{2 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i f (c+d x)}{d}\right)}{a f}",1,"(2^(-2 - m)*(c + d*x)^m*((I*f*(c + d*x))/d)^m*(2^(1 + m)*f*(c + d*x)*(((-I)*f*(c + d*x))/d)^m*(Cos[e - (c*f)/d] - I*Sin[e - (c*f)/d]) + I*d*(1 + m)*Gamma[1 + m, ((-2*I)*f*(c + d*x))/d]*(Cos[e - (c*f)/d] + I*Sin[e - (c*f)/d]))*(Cos[e - (c*f)/d] + I*Sin[e - (c*f)/d]))/(a*d*f*(1 + m)*((f^2*(c + d*x)^2)/d^2)^m)","A",1
35,1,152,171,2.832075,"\int \frac{(c+d x)^m}{(a+i a \cot (e+f x))^2} \, dx","Integrate[(c + d*x)^m/(a + I*a*Cot[e + f*x])^2,x]","\frac{(c+d x)^m \left(i 2^{2-m} e^{2 i \left(e-\frac{c f}{d}\right)} \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i f (c+d x)}{d}\right)-i 4^{-m} e^{4 i \left(e-\frac{c f}{d}\right)} \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{4 i f (c+d x)}{d}\right)+\frac{4 f (c+d x)}{d (m+1)}\right)}{16 a^2 f}","\frac{i 2^{-m-2} e^{2 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i f (c+d x)}{d}\right)}{a^2 f}-\frac{i 4^{-m-2} e^{4 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{4 i f (c+d x)}{d}\right)}{a^2 f}+\frac{(c+d x)^{m+1}}{4 a^2 d (m+1)}",1,"((c + d*x)^m*((4*f*(c + d*x))/(d*(1 + m)) + (I*2^(2 - m)*E^((2*I)*(e - (c*f)/d))*Gamma[1 + m, ((-2*I)*f*(c + d*x))/d])/(((-I)*f*(c + d*x))/d)^m - (I*E^((4*I)*(e - (c*f)/d))*Gamma[1 + m, ((-4*I)*f*(c + d*x))/d])/(4^m*(((-I)*f*(c + d*x))/d)^m)))/(16*a^2*f)","A",1
36,1,238,251,4.0993796,"\int \frac{(c+d x)^m}{(a+i a \cot (e+f x))^3} \, dx","Integrate[(c + d*x)^m/(a + I*a*Cot[e + f*x])^3,x]","\frac{2^{-2 m-5} 3^{-m-1} e^{-\frac{6 i c f}{d}} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \left(i d 2^{m+1} 3^{m+2} (m+1) e^{\frac{4 i c f}{d}+2 i e} \Gamma \left(m+1,-\frac{2 i f (c+d x)}{d}\right)-i d 3^{m+2} (m+1) e^{2 i \left(\frac{c f}{d}+2 e\right)} \Gamma \left(m+1,-\frac{4 i f (c+d x)}{d}\right)+i d e^{6 i e} 2^{m+1} (m+1) \Gamma \left(m+1,-\frac{6 i f (c+d x)}{d}\right)+f 12^{m+1} e^{\frac{6 i c f}{d}} (c+d x) \left(-\frac{i f (c+d x)}{d}\right)^m\right)}{a^3 d f (m+1)}","\frac{3 i 2^{-m-4} e^{2 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i f (c+d x)}{d}\right)}{a^3 f}-\frac{3 i 2^{-2 m-5} e^{4 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{4 i f (c+d x)}{d}\right)}{a^3 f}+\frac{i 2^{-m-4} 3^{-m-1} e^{6 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{6 i f (c+d x)}{d}\right)}{a^3 f}+\frac{(c+d x)^{m+1}}{8 a^3 d (m+1)}",1,"(2^(-5 - 2*m)*3^(-1 - m)*(c + d*x)^m*(12^(1 + m)*E^(((6*I)*c*f)/d)*f*(c + d*x)*(((-I)*f*(c + d*x))/d)^m + I*2^(1 + m)*3^(2 + m)*d*E^((2*I)*e + ((4*I)*c*f)/d)*(1 + m)*Gamma[1 + m, ((-2*I)*f*(c + d*x))/d] - I*3^(2 + m)*d*E^((2*I)*(2*e + (c*f)/d))*(1 + m)*Gamma[1 + m, ((-4*I)*f*(c + d*x))/d] + I*2^(1 + m)*d*E^((6*I)*e)*(1 + m)*Gamma[1 + m, ((-6*I)*f*(c + d*x))/d]))/(a^3*d*E^(((6*I)*c*f)/d)*f*(1 + m)*(((-I)*f*(c + d*x))/d)^m)","A",1
37,1,730,147,7.5029846,"\int (c+d x)^3 (a+b \cot (e+f x)) \, dx","Integrate[(c + d*x)^3*(a + b*Cot[e + f*x]),x]","\frac{1}{4} x \csc (e) \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right) (a \sin (e)+b \cos (e))+\frac{b c^3 \csc (e) (\sin (e) \log (\sin (e) \cos (f x)+\cos (e) \sin (f x))-f x \cos (e))}{f \left(\sin ^2(e)+\cos ^2(e)\right)}-\frac{3 b c^2 d \csc (e) \sec (e) \left(f^2 x^2 e^{i \tan ^{-1}(\tan (e))}+\frac{\tan (e) \left(i \text{Li}_2\left(e^{2 i \left(f x+\tan ^{-1}(\tan (e))\right)}\right)+i f x \left(2 \tan ^{-1}(\tan (e))-\pi \right)-2 \left(\tan ^{-1}(\tan (e))+f x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (e))+f x\right)}\right)+2 \tan ^{-1}(\tan (e)) \log \left(\sin \left(\tan ^{-1}(\tan (e))+f x\right)\right)-\pi  \log \left(1+e^{-2 i f x}\right)+\pi  \log (\cos (f x))\right)}{\sqrt{\tan ^2(e)+1}}\right)}{2 f^2 \sqrt{\sec ^2(e) \left(\sin ^2(e)+\cos ^2(e)\right)}}-\frac{b c d^2 e^{i e} \csc (e) \left(2 e^{-2 i e} f^3 x^3+3 i \left(1-e^{-2 i e}\right) f^2 x^2 \log \left(1-e^{-i (e+f x)}\right)+3 i \left(1-e^{-2 i e}\right) f^2 x^2 \log \left(1+e^{-i (e+f x)}\right)-6 e^{-2 i e} \left(-1+e^{2 i e}\right) \left(f x \text{Li}_2\left(-e^{-i (e+f x)}\right)-i \text{Li}_3\left(-e^{-i (e+f x)}\right)\right)-6 e^{-2 i e} \left(-1+e^{2 i e}\right) \left(f x \text{Li}_2\left(e^{-i (e+f x)}\right)-i \text{Li}_3\left(e^{-i (e+f x)}\right)\right)\right)}{2 f^3}-\frac{b d^3 e^{i e} \csc (e) \left(e^{-2 i e} f^4 x^4+2 i \left(1-e^{-2 i e}\right) f^3 x^3 \log \left(1-e^{-i (e+f x)}\right)+2 i \left(1-e^{-2 i e}\right) f^3 x^3 \log \left(1+e^{-i (e+f x)}\right)-6 e^{-2 i e} \left(-1+e^{2 i e}\right) \left(f^2 x^2 \text{Li}_2\left(-e^{-i (e+f x)}\right)-2 i f x \text{Li}_3\left(-e^{-i (e+f x)}\right)-2 \text{Li}_4\left(-e^{-i (e+f x)}\right)\right)-6 e^{-2 i e} \left(-1+e^{2 i e}\right) \left(f^2 x^2 \text{Li}_2\left(e^{-i (e+f x)}\right)-2 i f x \text{Li}_3\left(e^{-i (e+f x)}\right)-2 \text{Li}_4\left(e^{-i (e+f x)}\right)\right)\right)}{4 f^4}","\frac{a (c+d x)^4}{4 d}+\frac{3 b d^2 (c+d x) \text{Li}_3\left(e^{2 i (e+f x)}\right)}{2 f^3}-\frac{3 i b d (c+d x)^2 \text{Li}_2\left(e^{2 i (e+f x)}\right)}{2 f^2}+\frac{b (c+d x)^3 \log \left(1-e^{2 i (e+f x)}\right)}{f}-\frac{i b (c+d x)^4}{4 d}+\frac{3 i b d^3 \text{Li}_4\left(e^{2 i (e+f x)}\right)}{4 f^4}",1,"-1/2*(b*c*d^2*E^(I*e)*Csc[e]*((2*f^3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 - E^((-I)*(e + f*x))] + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, -E^((-I)*(e + f*x))] - I*PolyLog[3, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, E^((-I)*(e + f*x))] - I*PolyLog[3, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/f^3 - (b*d^3*E^(I*e)*Csc[e]*((f^4*x^4)/E^((2*I)*e) + (2*I)*(1 - E^((-2*I)*e))*f^3*x^3*Log[1 - E^((-I)*(e + f*x))] + (2*I)*(1 - E^((-2*I)*e))*f^3*x^3*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f^2*x^2*PolyLog[2, -E^((-I)*(e + f*x))] - (2*I)*f*x*PolyLog[3, -E^((-I)*(e + f*x))] - 2*PolyLog[4, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f^2*x^2*PolyLog[2, E^((-I)*(e + f*x))] - (2*I)*f*x*PolyLog[3, E^((-I)*(e + f*x))] - 2*PolyLog[4, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/(4*f^4) + (x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Csc[e]*(b*Cos[e] + a*Sin[e]))/4 + (b*c^3*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) - (3*b*c^2*d*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(2*f^2*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)])","B",0
38,1,406,112,2.5711261,"\int (c+d x)^2 (a+b \cot (e+f x)) \, dx","Integrate[(c + d*x)^2*(a + b*Cot[e + f*x]),x]","\frac{3 a c^2 f^3 x+3 a c d f^3 x^2+a d^2 f^3 x^3+3 b c^2 f^2 \log (\sin (e+f x))+3 b c d f^3 x^2 \cot (e)-3 b c d f^3 x^2 e^{i \tan ^{-1}(\tan (e))} \cot (e) \sqrt{\sec ^2(e)}-6 i b c d f^2 x \tan ^{-1}(\tan (e))+6 b c d f^2 x \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (e))+f x\right)}\right)-3 i b c d f \text{Li}_2\left(e^{2 i \left(f x+\tan ^{-1}(\tan (e))\right)}\right)+6 b c d f \tan ^{-1}(\tan (e)) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (e))+f x\right)}\right)-6 b c d f \tan ^{-1}(\tan (e)) \log \left(\sin \left(\tan ^{-1}(\tan (e))+f x\right)\right)+3 i \pi  b c d f^2 x+3 \pi  b c d f \log \left(1+e^{-2 i f x}\right)-3 \pi  b c d f \log (\cos (f x))+3 b d^2 f^2 x^2 \log \left(1-e^{-i (e+f x)}\right)+3 b d^2 f^2 x^2 \log \left(1+e^{-i (e+f x)}\right)+6 i b d^2 f x \text{Li}_2\left(-e^{-i (e+f x)}\right)+6 i b d^2 f x \text{Li}_2\left(e^{-i (e+f x)}\right)+6 b d^2 \text{Li}_3\left(-e^{-i (e+f x)}\right)+6 b d^2 \text{Li}_3\left(e^{-i (e+f x)}\right)+i b d^2 f^3 x^3}{3 f^3}","\frac{a (c+d x)^3}{3 d}-\frac{i b d (c+d x) \text{Li}_2\left(e^{2 i (e+f x)}\right)}{f^2}+\frac{b (c+d x)^2 \log \left(1-e^{2 i (e+f x)}\right)}{f}-\frac{i b (c+d x)^3}{3 d}+\frac{b d^2 \text{Li}_3\left(e^{2 i (e+f x)}\right)}{2 f^3}",1,"(3*a*c^2*f^3*x + (3*I)*b*c*d*f^2*Pi*x + 3*a*c*d*f^3*x^2 + a*d^2*f^3*x^3 + I*b*d^2*f^3*x^3 - (6*I)*b*c*d*f^2*x*ArcTan[Tan[e]] + 3*b*c*d*f^3*x^2*Cot[e] + 3*b*c*d*f*Pi*Log[1 + E^((-2*I)*f*x)] + 3*b*d^2*f^2*x^2*Log[1 - E^((-I)*(e + f*x))] + 3*b*d^2*f^2*x^2*Log[1 + E^((-I)*(e + f*x))] + 6*b*c*d*f^2*x*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + 6*b*c*d*f*ArcTan[Tan[e]]*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] - 3*b*c*d*f*Pi*Log[Cos[f*x]] + 3*b*c^2*f^2*Log[Sin[e + f*x]] - 6*b*c*d*f*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + (6*I)*b*d^2*f*x*PolyLog[2, -E^((-I)*(e + f*x))] + (6*I)*b*d^2*f*x*PolyLog[2, E^((-I)*(e + f*x))] - (3*I)*b*c*d*f*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))] + 6*b*d^2*PolyLog[3, -E^((-I)*(e + f*x))] + 6*b*d^2*PolyLog[3, E^((-I)*(e + f*x))] - 3*b*c*d*E^(I*ArcTan[Tan[e]])*f^3*x^2*Cot[e]*Sqrt[Sec[e]^2])/(3*f^3)","B",0
39,1,204,83,5.036305,"\int (c+d x) (a+b \cot (e+f x)) \, dx","Integrate[(c + d*x)*(a + b*Cot[e + f*x]),x]","a c x+\frac{1}{2} a d x^2+\frac{b c (\log (\tan (e+f x))+\log (\cos (e+f x)))}{f}-\frac{b d \csc (e) \sec (e) \left(f^2 x^2 e^{i \tan ^{-1}(\tan (e))}+\frac{\tan (e) \left(i \text{Li}_2\left(e^{2 i \left(f x+\tan ^{-1}(\tan (e))\right)}\right)+i f x \left(2 \tan ^{-1}(\tan (e))-\pi \right)-2 \left(\tan ^{-1}(\tan (e))+f x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (e))+f x\right)}\right)+2 \tan ^{-1}(\tan (e)) \log \left(\sin \left(\tan ^{-1}(\tan (e))+f x\right)\right)-\pi  \log \left(1+e^{-2 i f x}\right)+\pi  \log (\cos (f x))\right)}{\sqrt{\tan ^2(e)+1}}\right)}{2 f^2 \sqrt{\sec ^2(e) \left(\sin ^2(e)+\cos ^2(e)\right)}}+\frac{1}{2} b d x^2 \cot (e)","\frac{a (c+d x)^2}{2 d}+\frac{b (c+d x) \log \left(1-e^{2 i (e+f x)}\right)}{f}-\frac{i b (c+d x)^2}{2 d}-\frac{i b d \text{Li}_2\left(e^{2 i (e+f x)}\right)}{2 f^2}",1,"a*c*x + (a*d*x^2)/2 + (b*d*x^2*Cot[e])/2 + (b*c*(Log[Cos[e + f*x]] + Log[Tan[e + f*x]]))/f - (b*d*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(2*f^2*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)])","B",0
40,0,0,21,2.2273385,"\int \frac{a+b \cot (e+f x)}{c+d x} \, dx","Integrate[(a + b*Cot[e + f*x])/(c + d*x),x]","\int \frac{a+b \cot (e+f x)}{c+d x} \, dx","\text{Int}\left(\frac{a+b \cot (e+f x)}{c+d x},x\right)",0,"Integrate[(a + b*Cot[e + f*x])/(c + d*x), x]","A",-1
41,0,0,21,8.1915118,"\int \frac{a+b \cot (e+f x)}{(c+d x)^2} \, dx","Integrate[(a + b*Cot[e + f*x])/(c + d*x)^2,x]","\int \frac{a+b \cot (e+f x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{a+b \cot (e+f x)}{(c+d x)^2},x\right)",0,"Integrate[(a + b*Cot[e + f*x])/(c + d*x)^2, x]","A",-1
42,1,1611,295,7.4329327,"\int (c+d x)^3 (a+b \cot (e+f x))^2 \, dx","Integrate[(c + d*x)^3*(a + b*Cot[e + f*x])^2,x]","\frac{2 a b \csc (e) (\log (\cos (f x) \sin (e)+\cos (e) \sin (f x)) \sin (e)-f x \cos (e)) c^3}{f \left(\cos ^2(e)+\sin ^2(e)\right)}-\frac{3 a b d \csc (e) \sec (e) \left(e^{i \tan ^{-1}(\tan (e))} f^2 x^2+\frac{\left(i f x \left(2 \tan ^{-1}(\tan (e))-\pi \right)-\pi  \log \left(1+e^{-2 i f x}\right)-2 \left(f x+\tan ^{-1}(\tan (e))\right) \log \left(1-e^{2 i \left(f x+\tan ^{-1}(\tan (e))\right)}\right)+\pi  \log (\cos (f x))+2 \tan ^{-1}(\tan (e)) \log \left(\sin \left(f x+\tan ^{-1}(\tan (e))\right)\right)+i \text{Li}_2\left(e^{2 i \left(f x+\tan ^{-1}(\tan (e))\right)}\right)\right) \tan (e)}{\sqrt{\tan ^2(e)+1}}\right) c^2}{f^2 \sqrt{\sec ^2(e) \left(\cos ^2(e)+\sin ^2(e)\right)}}+\frac{3 b^2 d \csc (e) (\log (\cos (f x) \sin (e)+\cos (e) \sin (f x)) \sin (e)-f x \cos (e)) c^2}{f^2 \left(\cos ^2(e)+\sin ^2(e)\right)}-\frac{a b d^2 e^{i e} \csc (e) \left(2 e^{-2 i e} f^3 x^3+3 i \left(1-e^{-2 i e}\right) f^2 \log \left(1-e^{-i (e+f x)}\right) x^2+3 i \left(1-e^{-2 i e}\right) f^2 \log \left(1+e^{-i (e+f x)}\right) x^2-6 e^{-2 i e} \left(-1+e^{2 i e}\right) \left(f x \text{Li}_2\left(-e^{-i (e+f x)}\right)-i \text{Li}_3\left(-e^{-i (e+f x)}\right)\right)-6 e^{-2 i e} \left(-1+e^{2 i e}\right) \left(f x \text{Li}_2\left(e^{-i (e+f x)}\right)-i \text{Li}_3\left(e^{-i (e+f x)}\right)\right)\right) c}{f^3}-\frac{3 b^2 d^2 \csc (e) \sec (e) \left(e^{i \tan ^{-1}(\tan (e))} f^2 x^2+\frac{\left(i f x \left(2 \tan ^{-1}(\tan (e))-\pi \right)-\pi  \log \left(1+e^{-2 i f x}\right)-2 \left(f x+\tan ^{-1}(\tan (e))\right) \log \left(1-e^{2 i \left(f x+\tan ^{-1}(\tan (e))\right)}\right)+\pi  \log (\cos (f x))+2 \tan ^{-1}(\tan (e)) \log \left(\sin \left(f x+\tan ^{-1}(\tan (e))\right)\right)+i \text{Li}_2\left(e^{2 i \left(f x+\tan ^{-1}(\tan (e))\right)}\right)\right) \tan (e)}{\sqrt{\tan ^2(e)+1}}\right) c}{f^3 \sqrt{\sec ^2(e) \left(\cos ^2(e)+\sin ^2(e)\right)}}-\frac{b^2 d^3 e^{i e} \csc (e) \left(2 e^{-2 i e} f^3 x^3+3 i \left(1-e^{-2 i e}\right) f^2 \log \left(1-e^{-i (e+f x)}\right) x^2+3 i \left(1-e^{-2 i e}\right) f^2 \log \left(1+e^{-i (e+f x)}\right) x^2-6 e^{-2 i e} \left(-1+e^{2 i e}\right) \left(f x \text{Li}_2\left(-e^{-i (e+f x)}\right)-i \text{Li}_3\left(-e^{-i (e+f x)}\right)\right)-6 e^{-2 i e} \left(-1+e^{2 i e}\right) \left(f x \text{Li}_2\left(e^{-i (e+f x)}\right)-i \text{Li}_3\left(e^{-i (e+f x)}\right)\right)\right)}{2 f^4}-\frac{a b d^3 e^{i e} \csc (e) \left(e^{-2 i e} f^4 x^4+2 i \left(1-e^{-2 i e}\right) f^3 \log \left(1-e^{-i (e+f x)}\right) x^3+2 i \left(1-e^{-2 i e}\right) f^3 \log \left(1+e^{-i (e+f x)}\right) x^3-6 e^{-2 i e} \left(-1+e^{2 i e}\right) \left(f^2 \text{Li}_2\left(-e^{-i (e+f x)}\right) x^2-2 i f \text{Li}_3\left(-e^{-i (e+f x)}\right) x-2 \text{Li}_4\left(-e^{-i (e+f x)}\right)\right)-6 e^{-2 i e} \left(-1+e^{2 i e}\right) \left(f^2 \text{Li}_2\left(e^{-i (e+f x)}\right) x^2-2 i f \text{Li}_3\left(e^{-i (e+f x)}\right) x-2 \text{Li}_4\left(e^{-i (e+f x)}\right)\right)\right)}{2 f^4}+\frac{\csc (e) \csc (e+f x) \left(a^2 d^3 f \cos (f x) x^4-b^2 d^3 f \cos (f x) x^4-a^2 d^3 f \cos (2 e+f x) x^4+b^2 d^3 f \cos (2 e+f x) x^4+2 a b d^3 f \sin (f x) x^4+2 a b d^3 f \sin (2 e+f x) x^4+4 a^2 c d^2 f \cos (f x) x^3-4 b^2 c d^2 f \cos (f x) x^3-4 a^2 c d^2 f \cos (2 e+f x) x^3+4 b^2 c d^2 f \cos (2 e+f x) x^3+8 b^2 d^3 \sin (f x) x^3+8 a b c d^2 f \sin (f x) x^3+8 a b c d^2 f \sin (2 e+f x) x^3+6 a^2 c^2 d f \cos (f x) x^2-6 b^2 c^2 d f \cos (f x) x^2-6 a^2 c^2 d f \cos (2 e+f x) x^2+6 b^2 c^2 d f \cos (2 e+f x) x^2+24 b^2 c d^2 \sin (f x) x^2+12 a b c^2 d f \sin (f x) x^2+12 a b c^2 d f \sin (2 e+f x) x^2+4 a^2 c^3 f \cos (f x) x-4 b^2 c^3 f \cos (f x) x-4 a^2 c^3 f \cos (2 e+f x) x+4 b^2 c^3 f \cos (2 e+f x) x+24 b^2 c^2 d \sin (f x) x+8 a b c^3 f \sin (f x) x+8 a b c^3 f \sin (2 e+f x) x+8 b^2 c^3 \sin (f x)\right)}{8 f}","\frac{a^2 (c+d x)^4}{4 d}+\frac{3 a b d^2 (c+d x) \text{Li}_3\left(e^{2 i (e+f x)}\right)}{f^3}-\frac{3 i a b d (c+d x)^2 \text{Li}_2\left(e^{2 i (e+f x)}\right)}{f^2}+\frac{2 a b (c+d x)^3 \log \left(1-e^{2 i (e+f x)}\right)}{f}-\frac{i a b (c+d x)^4}{2 d}+\frac{3 i a b d^3 \text{Li}_4\left(e^{2 i (e+f x)}\right)}{2 f^4}-\frac{3 i b^2 d^2 (c+d x) \text{Li}_2\left(e^{2 i (e+f x)}\right)}{f^3}+\frac{3 b^2 d (c+d x)^2 \log \left(1-e^{2 i (e+f x)}\right)}{f^2}-\frac{b^2 (c+d x)^3 \cot (e+f x)}{f}-\frac{i b^2 (c+d x)^3}{f}-\frac{b^2 (c+d x)^4}{4 d}+\frac{3 b^2 d^3 \text{Li}_3\left(e^{2 i (e+f x)}\right)}{2 f^4}",1,"-1/2*(b^2*d^3*E^(I*e)*Csc[e]*((2*f^3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 - E^((-I)*(e + f*x))] + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, -E^((-I)*(e + f*x))] - I*PolyLog[3, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, E^((-I)*(e + f*x))] - I*PolyLog[3, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/f^4 - (a*b*c*d^2*E^(I*e)*Csc[e]*((2*f^3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 - E^((-I)*(e + f*x))] + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, -E^((-I)*(e + f*x))] - I*PolyLog[3, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, E^((-I)*(e + f*x))] - I*PolyLog[3, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/f^3 - (a*b*d^3*E^(I*e)*Csc[e]*((f^4*x^4)/E^((2*I)*e) + (2*I)*(1 - E^((-2*I)*e))*f^3*x^3*Log[1 - E^((-I)*(e + f*x))] + (2*I)*(1 - E^((-2*I)*e))*f^3*x^3*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f^2*x^2*PolyLog[2, -E^((-I)*(e + f*x))] - (2*I)*f*x*PolyLog[3, -E^((-I)*(e + f*x))] - 2*PolyLog[4, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f^2*x^2*PolyLog[2, E^((-I)*(e + f*x))] - (2*I)*f*x*PolyLog[3, E^((-I)*(e + f*x))] - 2*PolyLog[4, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/(2*f^4) + (3*b^2*c^2*d*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f^2*(Cos[e]^2 + Sin[e]^2)) + (2*a*b*c^3*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) + (Csc[e]*Csc[e + f*x]*(4*a^2*c^3*f*x*Cos[f*x] - 4*b^2*c^3*f*x*Cos[f*x] + 6*a^2*c^2*d*f*x^2*Cos[f*x] - 6*b^2*c^2*d*f*x^2*Cos[f*x] + 4*a^2*c*d^2*f*x^3*Cos[f*x] - 4*b^2*c*d^2*f*x^3*Cos[f*x] + a^2*d^3*f*x^4*Cos[f*x] - b^2*d^3*f*x^4*Cos[f*x] - 4*a^2*c^3*f*x*Cos[2*e + f*x] + 4*b^2*c^3*f*x*Cos[2*e + f*x] - 6*a^2*c^2*d*f*x^2*Cos[2*e + f*x] + 6*b^2*c^2*d*f*x^2*Cos[2*e + f*x] - 4*a^2*c*d^2*f*x^3*Cos[2*e + f*x] + 4*b^2*c*d^2*f*x^3*Cos[2*e + f*x] - a^2*d^3*f*x^4*Cos[2*e + f*x] + b^2*d^3*f*x^4*Cos[2*e + f*x] + 8*b^2*c^3*Sin[f*x] + 24*b^2*c^2*d*x*Sin[f*x] + 8*a*b*c^3*f*x*Sin[f*x] + 24*b^2*c*d^2*x^2*Sin[f*x] + 12*a*b*c^2*d*f*x^2*Sin[f*x] + 8*b^2*d^3*x^3*Sin[f*x] + 8*a*b*c*d^2*f*x^3*Sin[f*x] + 2*a*b*d^3*f*x^4*Sin[f*x] + 8*a*b*c^3*f*x*Sin[2*e + f*x] + 12*a*b*c^2*d*f*x^2*Sin[2*e + f*x] + 8*a*b*c*d^2*f*x^3*Sin[2*e + f*x] + 2*a*b*d^3*f*x^4*Sin[2*e + f*x]))/(8*f) - (3*b^2*c*d^2*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^3*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (3*a*b*c^2*d*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^2*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)])","B",0
43,1,729,227,7.2242327,"\int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx","Integrate[(c + d*x)^2*(a + b*Cot[e + f*x])^2,x]","\frac{1}{3} x \csc (e) \left(3 c^2+3 c d x+d^2 x^2\right) \left(a^2 \sin (e)+2 a b \cos (e)-b^2 \sin (e)\right)+\frac{2 a b c^2 \csc (e) (\sin (e) \log (\sin (e) \cos (f x)+\cos (e) \sin (f x))-f x \cos (e))}{f \left(\sin ^2(e)+\cos ^2(e)\right)}-\frac{2 a b c d \csc (e) \sec (e) \left(f^2 x^2 e^{i \tan ^{-1}(\tan (e))}+\frac{\tan (e) \left(i \text{Li}_2\left(e^{2 i \left(f x+\tan ^{-1}(\tan (e))\right)}\right)+i f x \left(2 \tan ^{-1}(\tan (e))-\pi \right)-2 \left(\tan ^{-1}(\tan (e))+f x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (e))+f x\right)}\right)+2 \tan ^{-1}(\tan (e)) \log \left(\sin \left(\tan ^{-1}(\tan (e))+f x\right)\right)-\pi  \log \left(1+e^{-2 i f x}\right)+\pi  \log (\cos (f x))\right)}{\sqrt{\tan ^2(e)+1}}\right)}{f^2 \sqrt{\sec ^2(e) \left(\sin ^2(e)+\cos ^2(e)\right)}}-\frac{a b d^2 e^{i e} \csc (e) \left(2 e^{-2 i e} f^3 x^3+3 i \left(1-e^{-2 i e}\right) f^2 x^2 \log \left(1-e^{-i (e+f x)}\right)+3 i \left(1-e^{-2 i e}\right) f^2 x^2 \log \left(1+e^{-i (e+f x)}\right)-6 e^{-2 i e} \left(-1+e^{2 i e}\right) \left(f x \text{Li}_2\left(-e^{-i (e+f x)}\right)-i \text{Li}_3\left(-e^{-i (e+f x)}\right)\right)-6 e^{-2 i e} \left(-1+e^{2 i e}\right) \left(f x \text{Li}_2\left(e^{-i (e+f x)}\right)-i \text{Li}_3\left(e^{-i (e+f x)}\right)\right)\right)}{3 f^3}+\frac{\csc (e) \csc (e+f x) \left(b^2 c^2 \sin (f x)+2 b^2 c d x \sin (f x)+b^2 d^2 x^2 \sin (f x)\right)}{f}+\frac{2 b^2 c d \csc (e) (\sin (e) \log (\sin (e) \cos (f x)+\cos (e) \sin (f x))-f x \cos (e))}{f^2 \left(\sin ^2(e)+\cos ^2(e)\right)}-\frac{b^2 d^2 \csc (e) \sec (e) \left(f^2 x^2 e^{i \tan ^{-1}(\tan (e))}+\frac{\tan (e) \left(i \text{Li}_2\left(e^{2 i \left(f x+\tan ^{-1}(\tan (e))\right)}\right)+i f x \left(2 \tan ^{-1}(\tan (e))-\pi \right)-2 \left(\tan ^{-1}(\tan (e))+f x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (e))+f x\right)}\right)+2 \tan ^{-1}(\tan (e)) \log \left(\sin \left(\tan ^{-1}(\tan (e))+f x\right)\right)-\pi  \log \left(1+e^{-2 i f x}\right)+\pi  \log (\cos (f x))\right)}{\sqrt{\tan ^2(e)+1}}\right)}{f^3 \sqrt{\sec ^2(e) \left(\sin ^2(e)+\cos ^2(e)\right)}}","\frac{a^2 (c+d x)^3}{3 d}-\frac{2 i a b d (c+d x) \text{Li}_2\left(e^{2 i (e+f x)}\right)}{f^2}+\frac{2 a b (c+d x)^2 \log \left(1-e^{2 i (e+f x)}\right)}{f}-\frac{2 i a b (c+d x)^3}{3 d}+\frac{a b d^2 \text{Li}_3\left(e^{2 i (e+f x)}\right)}{f^3}+\frac{2 b^2 d (c+d x) \log \left(1-e^{2 i (e+f x)}\right)}{f^2}-\frac{b^2 (c+d x)^2 \cot (e+f x)}{f}-\frac{i b^2 (c+d x)^2}{f}-\frac{b^2 (c+d x)^3}{3 d}-\frac{i b^2 d^2 \text{Li}_2\left(e^{2 i (e+f x)}\right)}{f^3}",1,"-1/3*(a*b*d^2*E^(I*e)*Csc[e]*((2*f^3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 - E^((-I)*(e + f*x))] + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, -E^((-I)*(e + f*x))] - I*PolyLog[3, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, E^((-I)*(e + f*x))] - I*PolyLog[3, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/f^3 + (x*(3*c^2 + 3*c*d*x + d^2*x^2)*Csc[e]*(2*a*b*Cos[e] + a^2*Sin[e] - b^2*Sin[e]))/3 + (2*b^2*c*d*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f^2*(Cos[e]^2 + Sin[e]^2)) + (2*a*b*c^2*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) + (Csc[e]*Csc[e + f*x]*(b^2*c^2*Sin[f*x] + 2*b^2*c*d*x*Sin[f*x] + b^2*d^2*x^2*Sin[f*x]))/f - (b^2*d^2*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^3*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (2*a*b*c*d*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^2*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)])","B",0
44,1,200,137,2.2507859,"\int (c+d x) (a+b \cot (e+f x))^2 \, dx","Integrate[(c + d*x)*(a + b*Cot[e + f*x])^2,x]","\frac{\sin (e+f x) (a+b \cot (e+f x))^2 \left(\sin (e+f x) \left(-\left((e+f x) \left(a^2 (-2 c f+d e-d f x)+2 i a b d (e+f x)+b^2 (2 c f-d e+d f x)\right)\right)+2 b \log (\sin (e+f x)) (2 a c f-2 a d e+b d)+4 a b d (e+f x) \log \left(1-e^{2 i (e+f x)}\right)\right)-2 i a b d \text{Li}_2\left(e^{2 i (e+f x)}\right) \sin (e+f x)-2 b^2 f (c+d x) \cos (e+f x)\right)}{2 f^2 (a \sin (e+f x)+b \cos (e+f x))^2}","\frac{a^2 (c+d x)^2}{2 d}+\frac{2 a b (c+d x) \log \left(1-e^{2 i (e+f x)}\right)}{f}-\frac{i a b (c+d x)^2}{d}-\frac{i a b d \text{Li}_2\left(e^{2 i (e+f x)}\right)}{f^2}-\frac{b^2 (c+d x) \cot (e+f x)}{f}-b^2 c x+\frac{b^2 d \log (\sin (e+f x))}{f^2}-\frac{1}{2} b^2 d x^2",1,"((a + b*Cot[e + f*x])^2*Sin[e + f*x]*(-2*b^2*f*(c + d*x)*Cos[e + f*x] + (-((e + f*x)*((2*I)*a*b*d*(e + f*x) + a^2*(d*e - 2*c*f - d*f*x) + b^2*(-(d*e) + 2*c*f + d*f*x))) + 4*a*b*d*(e + f*x)*Log[1 - E^((2*I)*(e + f*x))] + 2*b*(b*d - 2*a*d*e + 2*a*c*f)*Log[Sin[e + f*x]])*Sin[e + f*x] - (2*I)*a*b*d*PolyLog[2, E^((2*I)*(e + f*x))]*Sin[e + f*x]))/(2*f^2*(b*Cos[e + f*x] + a*Sin[e + f*x])^2)","A",1
45,0,0,23,19.8471662,"\int \frac{(a+b \cot (e+f x))^2}{c+d x} \, dx","Integrate[(a + b*Cot[e + f*x])^2/(c + d*x),x]","\int \frac{(a+b \cot (e+f x))^2}{c+d x} \, dx","\text{Int}\left(\frac{(a+b \cot (e+f x))^2}{c+d x},x\right)",0,"Integrate[(a + b*Cot[e + f*x])^2/(c + d*x), x]","A",-1
46,0,0,23,16.276539,"\int \frac{(a+b \cot (e+f x))^2}{(c+d x)^2} \, dx","Integrate[(a + b*Cot[e + f*x])^2/(c + d*x)^2,x]","\int \frac{(a+b \cot (e+f x))^2}{(c+d x)^2} \, dx","\text{Int}\left(\frac{(a+b \cot (e+f x))^2}{(c+d x)^2},x\right)",0,"Integrate[(a + b*Cot[e + f*x])^2/(c + d*x)^2, x]","A",-1
47,1,3045,603,8.5437068,"\int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx","Integrate[(c + d*x)^3*(a + b*Cot[e + f*x])^3,x]","\text{Result too large to show}","\frac{a^3 (c+d x)^4}{4 d}+\frac{9 a^2 b d^2 (c+d x) \text{Li}_3\left(e^{2 i (e+f x)}\right)}{2 f^3}-\frac{9 i a^2 b d (c+d x)^2 \text{Li}_2\left(e^{2 i (e+f x)}\right)}{2 f^2}+\frac{3 a^2 b (c+d x)^3 \log \left(1-e^{2 i (e+f x)}\right)}{f}-\frac{3 i a^2 b (c+d x)^4}{4 d}+\frac{9 i a^2 b d^3 \text{Li}_4\left(e^{2 i (e+f x)}\right)}{4 f^4}-\frac{9 i a b^2 d^2 (c+d x) \text{Li}_2\left(e^{2 i (e+f x)}\right)}{f^3}+\frac{9 a b^2 d (c+d x)^2 \log \left(1-e^{2 i (e+f x)}\right)}{f^2}-\frac{3 a b^2 (c+d x)^3 \cot (e+f x)}{f}-\frac{3 i a b^2 (c+d x)^3}{f}-\frac{3 a b^2 (c+d x)^4}{4 d}+\frac{9 a b^2 d^3 \text{Li}_3\left(e^{2 i (e+f x)}\right)}{2 f^4}-\frac{3 b^3 d^2 (c+d x) \text{Li}_3\left(e^{2 i (e+f x)}\right)}{2 f^3}+\frac{3 b^3 d^2 (c+d x) \log \left(1-e^{2 i (e+f x)}\right)}{f^3}+\frac{3 i b^3 d (c+d x)^2 \text{Li}_2\left(e^{2 i (e+f x)}\right)}{2 f^2}-\frac{3 b^3 d (c+d x)^2 \cot (e+f x)}{2 f^2}-\frac{b^3 (c+d x)^3 \log \left(1-e^{2 i (e+f x)}\right)}{f}-\frac{b^3 (c+d x)^3 \cot ^2(e+f x)}{2 f}-\frac{3 i b^3 d (c+d x)^2}{2 f^2}-\frac{b^3 (c+d x)^3}{2 f}+\frac{i b^3 (c+d x)^4}{4 d}-\frac{3 i b^3 d^3 \text{Li}_2\left(e^{2 i (e+f x)}\right)}{2 f^4}-\frac{3 i b^3 d^3 \text{Li}_4\left(e^{2 i (e+f x)}\right)}{4 f^4}",1,"((-(b^3*c^3) - 3*b^3*c^2*d*x - 3*b^3*c*d^2*x^2 - b^3*d^3*x^3)*Csc[e + f*x]^2)/(2*f) - (3*a*b^2*d^3*E^(I*e)*Csc[e]*((2*f^3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 - E^((-I)*(e + f*x))] + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, -E^((-I)*(e + f*x))] - I*PolyLog[3, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, E^((-I)*(e + f*x))] - I*PolyLog[3, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/(2*f^4) - (3*a^2*b*c*d^2*E^(I*e)*Csc[e]*((2*f^3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 - E^((-I)*(e + f*x))] + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, -E^((-I)*(e + f*x))] - I*PolyLog[3, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, E^((-I)*(e + f*x))] - I*PolyLog[3, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/(2*f^3) + (b^3*c*d^2*E^(I*e)*Csc[e]*((2*f^3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 - E^((-I)*(e + f*x))] + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, -E^((-I)*(e + f*x))] - I*PolyLog[3, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, E^((-I)*(e + f*x))] - I*PolyLog[3, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/(2*f^3) - (3*a^2*b*d^3*E^(I*e)*Csc[e]*((f^4*x^4)/E^((2*I)*e) + (2*I)*(1 - E^((-2*I)*e))*f^3*x^3*Log[1 - E^((-I)*(e + f*x))] + (2*I)*(1 - E^((-2*I)*e))*f^3*x^3*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f^2*x^2*PolyLog[2, -E^((-I)*(e + f*x))] - (2*I)*f*x*PolyLog[3, -E^((-I)*(e + f*x))] - 2*PolyLog[4, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f^2*x^2*PolyLog[2, E^((-I)*(e + f*x))] - (2*I)*f*x*PolyLog[3, E^((-I)*(e + f*x))] - 2*PolyLog[4, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/(4*f^4) + (b^3*d^3*E^(I*e)*Csc[e]*((f^4*x^4)/E^((2*I)*e) + (2*I)*(1 - E^((-2*I)*e))*f^3*x^3*Log[1 - E^((-I)*(e + f*x))] + (2*I)*(1 - E^((-2*I)*e))*f^3*x^3*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f^2*x^2*PolyLog[2, -E^((-I)*(e + f*x))] - (2*I)*f*x*PolyLog[3, -E^((-I)*(e + f*x))] - 2*PolyLog[4, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f^2*x^2*PolyLog[2, E^((-I)*(e + f*x))] - (2*I)*f*x*PolyLog[3, E^((-I)*(e + f*x))] - 2*PolyLog[4, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/(4*f^4) + (3*b^3*c*d^2*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f^3*(Cos[e]^2 + Sin[e]^2)) + (9*a*b^2*c^2*d*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f^2*(Cos[e]^2 + Sin[e]^2)) + (3*a^2*b*c^3*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) - (b^3*c^3*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) + (3*x^2*(-(a^3*c^2*d) + (3*I)*a^2*b*c^2*d + 3*a*b^2*c^2*d - I*b^3*c^2*d + a^3*c^2*d*Cos[2*e] + (3*I)*a^2*b*c^2*d*Cos[2*e] - 3*a*b^2*c^2*d*Cos[2*e] - I*b^3*c^2*d*Cos[2*e] + I*a^3*c^2*d*Sin[2*e] - 3*a^2*b*c^2*d*Sin[2*e] - (3*I)*a*b^2*c^2*d*Sin[2*e] + b^3*c^2*d*Sin[2*e]))/(2*(-1 + Cos[2*e] + I*Sin[2*e])) + (x^3*(-(a^3*c*d^2) + (3*I)*a^2*b*c*d^2 + 3*a*b^2*c*d^2 - I*b^3*c*d^2 + a^3*c*d^2*Cos[2*e] + (3*I)*a^2*b*c*d^2*Cos[2*e] - 3*a*b^2*c*d^2*Cos[2*e] - I*b^3*c*d^2*Cos[2*e] + I*a^3*c*d^2*Sin[2*e] - 3*a^2*b*c*d^2*Sin[2*e] - (3*I)*a*b^2*c*d^2*Sin[2*e] + b^3*c*d^2*Sin[2*e]))/(-1 + Cos[2*e] + I*Sin[2*e]) + (x^4*(-(a^3*d^3) + (3*I)*a^2*b*d^3 + 3*a*b^2*d^3 - I*b^3*d^3 + a^3*d^3*Cos[2*e] + (3*I)*a^2*b*d^3*Cos[2*e] - 3*a*b^2*d^3*Cos[2*e] - I*b^3*d^3*Cos[2*e] + I*a^3*d^3*Sin[2*e] - 3*a^2*b*d^3*Sin[2*e] - (3*I)*a*b^2*d^3*Sin[2*e] + b^3*d^3*Sin[2*e]))/(4*(-1 + Cos[2*e] + I*Sin[2*e])) + x*(a^3*c^3 - 3*a*b^2*c^3 + ((3*I)*a^2*b*c^3)/(-1 + Cos[2*e] + I*Sin[2*e]) + ((3*I)*a^2*b*c^3*Cos[2*e] - 3*a^2*b*c^3*Sin[2*e])/(-1 + Cos[2*e] + I*Sin[2*e]) + ((-2*I)*b^3*c^3*Cos[2*e] + 2*b^3*c^3*Sin[2*e])/((-1 + Cos[2*e] + I*Sin[2*e])*(1 + Cos[2*e] + Cos[4*e] + I*Sin[2*e] + I*Sin[4*e])) + ((-2*I)*b^3*c^3*Cos[4*e] + 2*b^3*c^3*Sin[4*e])/((-1 + Cos[2*e] + I*Sin[2*e])*(1 + Cos[2*e] + Cos[4*e] + I*Sin[2*e] + I*Sin[4*e])) - (I*b^3*c^3)/(-1 + Cos[6*e] + I*Sin[6*e]) + ((-I)*b^3*c^3*Cos[6*e] + b^3*c^3*Sin[6*e])/(-1 + Cos[6*e] + I*Sin[6*e])) + (3*Csc[e]*Csc[e + f*x]*(b^3*c^2*d*Sin[f*x] + 2*a*b^2*c^3*f*Sin[f*x] + 2*b^3*c*d^2*x*Sin[f*x] + 6*a*b^2*c^2*d*f*x*Sin[f*x] + b^3*d^3*x^2*Sin[f*x] + 6*a*b^2*c*d^2*f*x^2*Sin[f*x] + 2*a*b^2*d^3*f*x^3*Sin[f*x]))/(2*f^2) - (3*b^3*d^3*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(2*f^4*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (9*a*b^2*c*d^2*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^3*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (9*a^2*b*c^2*d*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(2*f^2*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)]) + (3*b^3*c^2*d*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(2*f^2*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)])","B",0
48,1,2013,433,7.8976709,"\int (c+d x)^2 (a+b \cot (e+f x))^3 \, dx","Integrate[(c + d*x)^2*(a + b*Cot[e + f*x])^3,x]","\text{Result too large to show}","\frac{a^3 (c+d x)^3}{3 d}-\frac{3 i a^2 b d (c+d x) \text{Li}_2\left(e^{2 i (e+f x)}\right)}{f^2}+\frac{3 a^2 b (c+d x)^2 \log \left(1-e^{2 i (e+f x)}\right)}{f}-\frac{i a^2 b (c+d x)^3}{d}+\frac{3 a^2 b d^2 \text{Li}_3\left(e^{2 i (e+f x)}\right)}{2 f^3}+\frac{6 a b^2 d (c+d x) \log \left(1-e^{2 i (e+f x)}\right)}{f^2}-\frac{3 a b^2 (c+d x)^2 \cot (e+f x)}{f}-\frac{3 i a b^2 (c+d x)^2}{f}-\frac{a b^2 (c+d x)^3}{d}-\frac{3 i a b^2 d^2 \text{Li}_2\left(e^{2 i (e+f x)}\right)}{f^3}+\frac{i b^3 d (c+d x) \text{Li}_2\left(e^{2 i (e+f x)}\right)}{f^2}-\frac{b^3 d (c+d x) \cot (e+f x)}{f^2}-\frac{b^3 (c+d x)^2 \log \left(1-e^{2 i (e+f x)}\right)}{f}-\frac{b^3 (c+d x)^2 \cot ^2(e+f x)}{2 f}-\frac{b^3 c d x}{f}+\frac{i b^3 (c+d x)^3}{3 d}-\frac{b^3 d^2 \text{Li}_3\left(e^{2 i (e+f x)}\right)}{2 f^3}+\frac{b^3 d^2 \log (\sin (e+f x))}{f^3}-\frac{b^3 d^2 x^2}{2 f}",1,"-1/2*(a^2*b*d^2*E^(I*e)*Csc[e]*((2*f^3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 - E^((-I)*(e + f*x))] + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, -E^((-I)*(e + f*x))] - I*PolyLog[3, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, E^((-I)*(e + f*x))] - I*PolyLog[3, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/f^3 + (b^3*d^2*E^(I*e)*Csc[e]*((2*f^3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 - E^((-I)*(e + f*x))] + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, -E^((-I)*(e + f*x))] - I*PolyLog[3, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, E^((-I)*(e + f*x))] - I*PolyLog[3, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/(6*f^3) + (b^3*d^2*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f^3*(Cos[e]^2 + Sin[e]^2)) + (6*a*b^2*c*d*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f^2*(Cos[e]^2 + Sin[e]^2)) + (3*a^2*b*c^2*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) - (b^3*c^2*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) + (Csc[e]*Csc[e + f*x]^2*(6*b^3*c*d*Cos[e] + 18*a*b^2*c^2*f*Cos[e] + 6*b^3*d^2*x*Cos[e] + 36*a*b^2*c*d*f*x*Cos[e] + 18*a^2*b*c^2*f^2*x*Cos[e] - 6*b^3*c^2*f^2*x*Cos[e] + 18*a*b^2*d^2*f*x^2*Cos[e] + 18*a^2*b*c*d*f^2*x^2*Cos[e] - 6*b^3*c*d*f^2*x^2*Cos[e] + 6*a^2*b*d^2*f^2*x^3*Cos[e] - 2*b^3*d^2*f^2*x^3*Cos[e] - 6*b^3*c*d*Cos[e + 2*f*x] - 18*a*b^2*c^2*f*Cos[e + 2*f*x] - 6*b^3*d^2*x*Cos[e + 2*f*x] - 36*a*b^2*c*d*f*x*Cos[e + 2*f*x] - 9*a^2*b*c^2*f^2*x*Cos[e + 2*f*x] + 3*b^3*c^2*f^2*x*Cos[e + 2*f*x] - 18*a*b^2*d^2*f*x^2*Cos[e + 2*f*x] - 9*a^2*b*c*d*f^2*x^2*Cos[e + 2*f*x] + 3*b^3*c*d*f^2*x^2*Cos[e + 2*f*x] - 3*a^2*b*d^2*f^2*x^3*Cos[e + 2*f*x] + b^3*d^2*f^2*x^3*Cos[e + 2*f*x] - 9*a^2*b*c^2*f^2*x*Cos[3*e + 2*f*x] + 3*b^3*c^2*f^2*x*Cos[3*e + 2*f*x] - 9*a^2*b*c*d*f^2*x^2*Cos[3*e + 2*f*x] + 3*b^3*c*d*f^2*x^2*Cos[3*e + 2*f*x] - 3*a^2*b*d^2*f^2*x^3*Cos[3*e + 2*f*x] + b^3*d^2*f^2*x^3*Cos[3*e + 2*f*x] - 6*b^3*c^2*f*Sin[e] - 12*b^3*c*d*f*x*Sin[e] + 6*a^3*c^2*f^2*x*Sin[e] - 18*a*b^2*c^2*f^2*x*Sin[e] - 6*b^3*d^2*f*x^2*Sin[e] + 6*a^3*c*d*f^2*x^2*Sin[e] - 18*a*b^2*c*d*f^2*x^2*Sin[e] + 2*a^3*d^2*f^2*x^3*Sin[e] - 6*a*b^2*d^2*f^2*x^3*Sin[e] + 3*a^3*c^2*f^2*x*Sin[e + 2*f*x] - 9*a*b^2*c^2*f^2*x*Sin[e + 2*f*x] + 3*a^3*c*d*f^2*x^2*Sin[e + 2*f*x] - 9*a*b^2*c*d*f^2*x^2*Sin[e + 2*f*x] + a^3*d^2*f^2*x^3*Sin[e + 2*f*x] - 3*a*b^2*d^2*f^2*x^3*Sin[e + 2*f*x] - 3*a^3*c^2*f^2*x*Sin[3*e + 2*f*x] + 9*a*b^2*c^2*f^2*x*Sin[3*e + 2*f*x] - 3*a^3*c*d*f^2*x^2*Sin[3*e + 2*f*x] + 9*a*b^2*c*d*f^2*x^2*Sin[3*e + 2*f*x] - a^3*d^2*f^2*x^3*Sin[3*e + 2*f*x] + 3*a*b^2*d^2*f^2*x^3*Sin[3*e + 2*f*x]))/(12*f^2) - (3*a*b^2*d^2*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^3*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (3*a^2*b*c*d*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^2*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)]) + (b^3*c*d*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^2*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)])","B",0
49,1,276,278,3.646816,"\int (c+d x) (a+b \cot (e+f x))^3 \, dx","Integrate[(c + d*x)*(a + b*Cot[e + f*x])^3,x]","\frac{\sin (e+f x) (a+b \cot (e+f x))^3 \left(i b d \left(b^2-3 a^2\right) \text{Li}_2\left(e^{2 i (e+f x)}\right) \sin ^2(e+f x)+\sin ^2(e+f x) \left(2 b \log (\sin (e+f x)) \left(a^2 (3 c f-3 d e)+3 a b d+b^2 (d e-c f)\right)-2 b d \left(b^2-3 a^2\right) (e+f x) \log \left(1-e^{2 i (e+f x)}\right)-\left((e+f x) \left(a^3 (d (e-f x)-2 c f)+3 i a^2 b d (e+f x)+3 a b^2 (2 c f-d e+d f x)-i b^3 d (e+f x)\right)\right)\right)-\frac{1}{2} b^2 (\sin (2 (e+f x)) (6 a f (c+d x)+b d)+2 b f (c+d x))\right)}{2 f^2 (a \sin (e+f x)+b \cos (e+f x))^3}","\frac{a^3 (c+d x)^2}{2 d}+\frac{3 a^2 b (c+d x) \log \left(1-e^{2 i (e+f x)}\right)}{f}-\frac{3 i a^2 b (c+d x)^2}{2 d}-\frac{3 i a^2 b d \text{Li}_2\left(e^{2 i (e+f x)}\right)}{2 f^2}-\frac{3 a b^2 (c+d x) \cot (e+f x)}{f}-3 a b^2 c x+\frac{3 a b^2 d \log (\sin (e+f x))}{f^2}-\frac{3}{2} a b^2 d x^2-\frac{b^3 (c+d x) \log \left(1-e^{2 i (e+f x)}\right)}{f}-\frac{b^3 (c+d x) \cot ^2(e+f x)}{2 f}+\frac{i b^3 (c+d x)^2}{2 d}+\frac{i b^3 d \text{Li}_2\left(e^{2 i (e+f x)}\right)}{2 f^2}-\frac{b^3 d \cot (e+f x)}{2 f^2}-\frac{b^3 d x}{2 f}",1,"((a + b*Cot[e + f*x])^3*Sin[e + f*x]*((-((e + f*x)*((3*I)*a^2*b*d*(e + f*x) - I*b^3*d*(e + f*x) + 3*a*b^2*(-(d*e) + 2*c*f + d*f*x) + a^3*(-2*c*f + d*(e - f*x)))) - 2*b*(-3*a^2 + b^2)*d*(e + f*x)*Log[1 - E^((2*I)*(e + f*x))] + 2*b*(3*a*b*d + b^2*(d*e - c*f) + a^2*(-3*d*e + 3*c*f))*Log[Sin[e + f*x]])*Sin[e + f*x]^2 + I*b*(-3*a^2 + b^2)*d*PolyLog[2, E^((2*I)*(e + f*x))]*Sin[e + f*x]^2 - (b^2*(2*b*f*(c + d*x) + (b*d + 6*a*f*(c + d*x))*Sin[2*(e + f*x)]))/2))/(2*f^2*(b*Cos[e + f*x] + a*Sin[e + f*x])^3)","A",1
50,0,0,23,9.5936796,"\int \frac{(a+b \cot (e+f x))^3}{c+d x} \, dx","Integrate[(a + b*Cot[e + f*x])^3/(c + d*x),x]","\int \frac{(a+b \cot (e+f x))^3}{c+d x} \, dx","\text{Int}\left(\frac{(a+b \cot (e+f x))^3}{c+d x},x\right)",0,"Integrate[(a + b*Cot[e + f*x])^3/(c + d*x), x]","A",-1
51,0,0,23,12.556834,"\int \frac{(a+b \cot (e+f x))^3}{(c+d x)^2} \, dx","Integrate[(a + b*Cot[e + f*x])^3/(c + d*x)^2,x]","\int \frac{(a+b \cot (e+f x))^3}{(c+d x)^2} \, dx","\text{Int}\left(\frac{(a+b \cot (e+f x))^3}{(c+d x)^2},x\right)",0,"Integrate[(a + b*Cot[e + f*x])^3/(c + d*x)^2, x]","A",-1
52,1,345,242,2.1351822,"\int \frac{(c+d x)^3}{a+b \cot (e+f x)} \, dx","Integrate[(c + d*x)^3/(a + b*Cot[e + f*x]),x]","\frac{x \sin (e) \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)}{4 (a \sin (e)+b \cos (e))}+\frac{b \left(\frac{3 d \left(b \left(1+e^{2 i e}\right)-i a \left(-1+e^{2 i e}\right)\right) \left(2 f^2 (c+d x)^2 \text{Li}_2\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right)+d \left(-2 i f (c+d x) \text{Li}_3\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right)-d \text{Li}_4\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right)\right)\right)}{f^4 \left(a^2+b^2\right)}-\frac{4 \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right) (c+d x)^3 \log \left(1+\frac{(-a+i b) e^{-2 i (e+f x)}}{a+i b}\right)}{f \left(a^2+b^2\right)}+\frac{2 i (c+d x)^4}{d (a+i b)}\right)}{4 \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right)}","-\frac{3 b d^2 (c+d x) \text{Li}_3\left(\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right)}{2 f^3 \left(a^2+b^2\right)}+\frac{3 i b d (c+d x)^2 \text{Li}_2\left(\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right)}{2 f^2 \left(a^2+b^2\right)}-\frac{b (c+d x)^3 \log \left(1-\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right)}{f \left(a^2+b^2\right)}-\frac{3 i b d^3 \text{Li}_4\left(\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right)}{4 f^4 \left(a^2+b^2\right)}+\frac{(c+d x)^4}{4 d (a-i b)}",1,"(b*(((2*I)*(c + d*x)^4)/((a + I*b)*d) - (4*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(c + d*x)^3*Log[1 + (-a + I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/((a^2 + b^2)*f) + (3*d*((-I)*a*(-1 + E^((2*I)*e)) + b*(1 + E^((2*I)*e)))*(2*f^2*(c + d*x)^2*PolyLog[2, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))] + d*((-2*I)*f*(c + d*x)*PolyLog[3, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))] - d*PolyLog[4, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])))/((a^2 + b^2)*f^4)))/(4*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))) + (x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Sin[e])/(4*(b*Cos[e] + a*Sin[e]))","A",1
53,1,289,181,1.5469378,"\int \frac{(c+d x)^2}{a+b \cot (e+f x)} \, dx","Integrate[(c + d*x)^2/(a + b*Cot[e + f*x]),x]","\frac{x \sin (e) \left(3 c^2+3 c d x+d^2 x^2\right)}{3 (a \sin (e)+b \cos (e))}+\frac{b \left(\frac{3 d \left(b \left(1+e^{2 i e}\right)-i a \left(-1+e^{2 i e}\right)\right) \left(2 f (c+d x) \text{Li}_2\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right)-i d \text{Li}_3\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right)\right)}{f^3 \left(a^2+b^2\right)}-\frac{6 \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right) (c+d x)^2 \log \left(1+\frac{(-a+i b) e^{-2 i (e+f x)}}{a+i b}\right)}{f \left(a^2+b^2\right)}+\frac{4 i (c+d x)^3}{d (a+i b)}\right)}{6 \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right)}","\frac{i b d (c+d x) \text{Li}_2\left(\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right)}{f^2 \left(a^2+b^2\right)}-\frac{b (c+d x)^2 \log \left(1-\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right)}{f \left(a^2+b^2\right)}-\frac{b d^2 \text{Li}_3\left(\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right)}{2 f^3 \left(a^2+b^2\right)}+\frac{(c+d x)^3}{3 d (a-i b)}",1,"(b*(((4*I)*(c + d*x)^3)/((a + I*b)*d) - (6*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(c + d*x)^2*Log[1 + (-a + I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/((a^2 + b^2)*f) + (3*d*((-I)*a*(-1 + E^((2*I)*e)) + b*(1 + E^((2*I)*e)))*(2*f*(c + d*x)*PolyLog[2, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))] - I*d*PolyLog[3, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))]))/((a^2 + b^2)*f^3)))/(6*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))) + (x*(3*c^2 + 3*c*d*x + d^2*x^2)*Sin[e])/(3*(b*Cos[e] + a*Sin[e]))","A",1
54,1,182,126,1.8567548,"\int \frac{c+d x}{a+b \cot (e+f x)} \, dx","Integrate[(c + d*x)/(a + b*Cot[e + f*x]),x]","\frac{x \sin (e) (2 c+d x)}{2 (a \sin (e)+b \cos (e))}+\frac{1}{2} b \left(-\frac{2 (c+d x) \log \left(1+\frac{(-a+i b) e^{-2 i (e+f x)}}{a+i b}\right)}{f \left(a^2+b^2\right)}-\frac{i d \text{Li}_2\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right)}{f^2 \left(a^2+b^2\right)}+\frac{2 i (c+d x)^2}{d (a+i b) \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right)}\right)","-\frac{b (c+d x) \log \left(1-\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right)}{f \left(a^2+b^2\right)}+\frac{i b d \text{Li}_2\left(\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right)}{2 f^2 \left(a^2+b^2\right)}+\frac{(c+d x)^2}{2 d (a-i b)}",1,"(b*(((2*I)*(c + d*x)^2)/((a + I*b)*d*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))) - (2*(c + d*x)*Log[1 + (-a + I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/((a^2 + b^2)*f) - (I*d*PolyLog[2, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/((a^2 + b^2)*f^2)))/2 + (x*(2*c + d*x)*Sin[e])/(2*(b*Cos[e] + a*Sin[e]))","A",1
55,0,0,23,1.9667449,"\int \frac{1}{(c+d x) (a+b \cot (e+f x))} \, dx","Integrate[1/((c + d*x)*(a + b*Cot[e + f*x])),x]","\int \frac{1}{(c+d x) (a+b \cot (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x) (a+b \cot (e+f x))},x\right)",0,"Integrate[1/((c + d*x)*(a + b*Cot[e + f*x])), x]","A",-1
56,0,0,23,3.9438824,"\int \frac{1}{(c+d x)^2 (a+b \cot (e+f x))} \, dx","Integrate[1/((c + d*x)^2*(a + b*Cot[e + f*x])),x]","\int \frac{1}{(c+d x)^2 (a+b \cot (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x)^2 (a+b \cot (e+f x))},x\right)",0,"Integrate[1/((c + d*x)^2*(a + b*Cot[e + f*x])), x]","A",-1
57,1,1682,839,10.6597893,"\int \frac{(c+d x)^3}{(a+b \cot (e+f x))^2} \, dx","Integrate[(c + d*x)^3/(a + b*Cot[e + f*x])^2,x]","\frac{\left(-a d^3-i b d^3+a \cos (2 e) d^3-i b \cos (2 e) d^3+i a \sin (2 e) d^3+b \sin (2 e) d^3\right) x^4}{4 (a-i b) (a+i b) (\cos (2 e) a+i \sin (2 e) a-a+i b+i b \cos (2 e)-b \sin (2 e))}+\frac{\left(-a c d^2-i b c d^2+a c \cos (2 e) d^2-i b c \cos (2 e) d^2+i a c \sin (2 e) d^2+b c \sin (2 e) d^2\right) x^3}{(a-i b) (a+i b) (\cos (2 e) a+i \sin (2 e) a-a+i b+i b \cos (2 e)-b \sin (2 e))}+\frac{3 \left(-a d c^2-i b d c^2+a d \cos (2 e) c^2-i b d \cos (2 e) c^2+i a d \sin (2 e) c^2+b d \sin (2 e) c^2\right) x^2}{2 (a-i b) (a+i b) (\cos (2 e) a+i \sin (2 e) a-a+i b+i b \cos (2 e)-b \sin (2 e))}+\left(\frac{c^3}{\cos (4 e) a^2+i \sin (4 e) a^2+a^2-2 i b a+2 i b \cos (4 e) a-2 b \sin (4 e) a-b^2-b^2 \cos (4 e)-i b^2 \sin (4 e)}+\frac{\cos (4 e) c^3+i \sin (4 e) c^3}{\cos (4 e) a^2+i \sin (4 e) a^2+a^2-2 i b a+2 i b \cos (4 e) a-2 b \sin (4 e) a-b^2-b^2 \cos (4 e)-i b^2 \sin (4 e)}+\frac{(-i \cos (2 e) a+\sin (2 e) a-i a-b+b \cos (2 e)+i b \sin (2 e)) \left(4 a b \cos (2 e) c^3+4 i a b \sin (2 e) c^3\right)}{(a-i b) (a+i b) (\cos (2 e) a+i \sin (2 e) a-a+i b+i b \cos (2 e)-b \sin (2 e)) \left(\cos (4 e) a^2+i \sin (4 e) a^2+a^2-2 i b a+2 i b \cos (4 e) a-2 b \sin (4 e) a-b^2-b^2 \cos (4 e)-i b^2 \sin (4 e)\right)}\right) x+\frac{b \left(\frac{2 a f (c+d x)^4}{(a+i b) d}-\frac{4 b (c+d x)^3}{a+i b}+\frac{4 a d^3 \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right) x^3 \log \left(\frac{e^{-2 i (e+f x)} (i b-a)}{a+i b}+1\right)}{(a-i b) (b-i a)}-\frac{6 d^2 \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right) (b d-2 a c f) x^2 \log \left(\frac{e^{-2 i (e+f x)} (i b-a)}{a+i b}+1\right)}{(a-i b) (b-i a) f}+\frac{12 c d \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right) (a c f-b d) x \log \left(\frac{e^{-2 i (e+f x)} (i b-a)}{a+i b}+1\right)}{(a-i b) (b-i a) f}+\frac{2 c^2 \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right) (2 a c f-3 b d) \left(2 f x+i \log \left(a-(a+i b) e^{2 i (e+f x)}-i b\right)\right)}{\left(a^2+b^2\right) f}-\frac{6 c d \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right) (a c f-b d) \text{Li}_2\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right)}{\left(a^2+b^2\right) f^2}+\frac{3 d^2 \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right) (b d-2 a c f) \left(2 f x \text{Li}_2\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right)-i \text{Li}_3\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right)\right)}{\left(a^2+b^2\right) f^3}-\frac{3 a d^3 \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right) \left(2 f^2 \text{Li}_2\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right) x^2-2 i f \text{Li}_3\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right) x-\text{Li}_4\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right)\right)}{\left(a^2+b^2\right) f^3}\right)}{2 (a-i b) (a+i b) \left(b \left(1+e^{2 i e}\right)-i a \left(-1+e^{2 i e}\right)\right) f}+\frac{b^2 \sin (f x) c^3+3 b^2 d x \sin (f x) c^2+3 b^2 d^2 x^2 \sin (f x) c+b^2 d^3 x^3 \sin (f x)}{(a-i b) (a+i b) f (b \cos (e)+a \sin (e)) (b \cos (e+f x)+a \sin (e+f x))}","-\frac{b (c+d x)^4}{(a+i b)^2 (i a+b) d}+\frac{(c+d x)^4}{4 (a+i b)^2 d}-\frac{b^2 (c+d x)^4}{\left(a^2+b^2\right)^2 d}-\frac{2 b \log \left(1-\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right) (c+d x)^3}{(a-i b) (a+i b)^2 f}-\frac{2 i b^2 \log \left(1-\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right) (c+d x)^3}{\left(a^2+b^2\right)^2 f}-\frac{2 b^2 (c+d x)^3}{(a-i b) (a+i b)^2 \left(i a-(i a-b) e^{2 i e+2 i f x}+b\right) f}-\frac{2 i b^2 (c+d x)^3}{\left(a^2+b^2\right)^2 f}+\frac{3 b^2 d \log \left(1-\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right) (c+d x)^2}{\left(a^2+b^2\right)^2 f^2}-\frac{3 b d \text{Li}_2\left(\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right) (c+d x)^2}{(a+i b)^2 (i a+b) f^2}-\frac{3 b^2 d \text{Li}_2\left(\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right) (c+d x)^2}{\left(a^2+b^2\right)^2 f^2}-\frac{3 i b^2 d^2 \text{Li}_2\left(\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right) (c+d x)}{\left(a^2+b^2\right)^2 f^3}-\frac{3 b d^2 \text{Li}_3\left(\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right) (c+d x)}{(a-i b) (a+i b)^2 f^3}-\frac{3 i b^2 d^2 \text{Li}_3\left(\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right) (c+d x)}{\left(a^2+b^2\right)^2 f^3}+\frac{3 b^2 d^3 \text{Li}_3\left(\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right)}{2 \left(a^2+b^2\right)^2 f^4}+\frac{3 b d^3 \text{Li}_4\left(\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right)}{2 (a+i b)^2 (i a+b) f^4}+\frac{3 b^2 d^3 \text{Li}_4\left(\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right)}{2 \left(a^2+b^2\right)^2 f^4}",1,"(b*((-4*b*(c + d*x)^3)/(a + I*b) + (2*a*f*(c + d*x)^4)/((a + I*b)*d) + (12*c*d*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(-(b*d) + a*c*f)*x*Log[1 + (-a + I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/((a - I*b)*((-I)*a + b)*f) - (6*d^2*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(b*d - 2*a*c*f)*x^2*Log[1 + (-a + I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/((a - I*b)*((-I)*a + b)*f) + (4*a*d^3*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*x^3*Log[1 + (-a + I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/((a - I*b)*((-I)*a + b)) + (2*c^2*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(-3*b*d + 2*a*c*f)*(2*f*x + I*Log[a - I*b - (a + I*b)*E^((2*I)*(e + f*x))]))/((a^2 + b^2)*f) - (6*c*d*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(-(b*d) + a*c*f)*PolyLog[2, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/((a^2 + b^2)*f^2) + (3*d^2*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(b*d - 2*a*c*f)*(2*f*x*PolyLog[2, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))] - I*PolyLog[3, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))]))/((a^2 + b^2)*f^3) - (3*a*d^3*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(2*f^2*x^2*PolyLog[2, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))] - (2*I)*f*x*PolyLog[3, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))] - PolyLog[4, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))]))/((a^2 + b^2)*f^3)))/(2*(a - I*b)*(a + I*b)*((-I)*a*(-1 + E^((2*I)*e)) + b*(1 + E^((2*I)*e)))*f) + (3*x^2*(-(a*c^2*d) - I*b*c^2*d + a*c^2*d*Cos[2*e] - I*b*c^2*d*Cos[2*e] + I*a*c^2*d*Sin[2*e] + b*c^2*d*Sin[2*e]))/(2*(a - I*b)*(a + I*b)*(-a + I*b + a*Cos[2*e] + I*b*Cos[2*e] + I*a*Sin[2*e] - b*Sin[2*e])) + (x^3*(-(a*c*d^2) - I*b*c*d^2 + a*c*d^2*Cos[2*e] - I*b*c*d^2*Cos[2*e] + I*a*c*d^2*Sin[2*e] + b*c*d^2*Sin[2*e]))/((a - I*b)*(a + I*b)*(-a + I*b + a*Cos[2*e] + I*b*Cos[2*e] + I*a*Sin[2*e] - b*Sin[2*e])) + (x^4*(-(a*d^3) - I*b*d^3 + a*d^3*Cos[2*e] - I*b*d^3*Cos[2*e] + I*a*d^3*Sin[2*e] + b*d^3*Sin[2*e]))/(4*(a - I*b)*(a + I*b)*(-a + I*b + a*Cos[2*e] + I*b*Cos[2*e] + I*a*Sin[2*e] - b*Sin[2*e])) + x*(c^3/(a^2 - (2*I)*a*b - b^2 + a^2*Cos[4*e] + (2*I)*a*b*Cos[4*e] - b^2*Cos[4*e] + I*a^2*Sin[4*e] - 2*a*b*Sin[4*e] - I*b^2*Sin[4*e]) + (((-I)*a - b - I*a*Cos[2*e] + b*Cos[2*e] + a*Sin[2*e] + I*b*Sin[2*e])*(4*a*b*c^3*Cos[2*e] + (4*I)*a*b*c^3*Sin[2*e]))/((a - I*b)*(a + I*b)*(-a + I*b + a*Cos[2*e] + I*b*Cos[2*e] + I*a*Sin[2*e] - b*Sin[2*e])*(a^2 - (2*I)*a*b - b^2 + a^2*Cos[4*e] + (2*I)*a*b*Cos[4*e] - b^2*Cos[4*e] + I*a^2*Sin[4*e] - 2*a*b*Sin[4*e] - I*b^2*Sin[4*e])) + (c^3*Cos[4*e] + I*c^3*Sin[4*e])/(a^2 - (2*I)*a*b - b^2 + a^2*Cos[4*e] + (2*I)*a*b*Cos[4*e] - b^2*Cos[4*e] + I*a^2*Sin[4*e] - 2*a*b*Sin[4*e] - I*b^2*Sin[4*e])) + (b^2*c^3*Sin[f*x] + 3*b^2*c^2*d*x*Sin[f*x] + 3*b^2*c*d^2*x^2*Sin[f*x] + b^2*d^3*x^3*Sin[f*x])/((a - I*b)*(a + I*b)*f*(b*Cos[e] + a*Sin[e])*(b*Cos[e + f*x] + a*Sin[e + f*x]))","B",0
58,1,718,650,9.2125567,"\int \frac{(c+d x)^2}{(a+b \cot (e+f x))^2} \, dx","Integrate[(c + d*x)^2/(a + b*Cot[e + f*x])^2,x]","\frac{\frac{-f x \left(a^2+b^2\right) \left(3 c^2+3 c d x+d^2 x^2\right) \cos (2 e+f x)+f x \left(a^2-b^2\right) \left(3 c^2+3 c d x+d^2 x^2\right) \cos (f x)+2 b \sin (f x) \left(3 b (c+d x)^2-a f x \left(3 c^2+3 c d x+d^2 x^2\right)\right)}{(a \sin (e)+b \cos (e)) (a \sin (e+f x)+b \cos (e+f x))}+\frac{2 b \left(\frac{3 d \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right) (b d-2 a c f) \text{Li}_2\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right)}{f^2 \left(a^2+b^2\right)}+\frac{6 c \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right) (a c f-b d) \left(2 f x+i \log \left(-(a+i b) e^{2 i (e+f x)}+a-i b\right)\right)}{f \left(a^2+b^2\right)}-\frac{3 a d^2 \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right) \left(2 f x \text{Li}_2\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right)-i \text{Li}_3\left(\frac{(a-i b) e^{-2 i (e+f x)}}{a+i b}\right)\right)}{f^2 \left(a^2+b^2\right)}-\frac{6 d x \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right) (b d-2 a c f) \log \left(1+\frac{(-a+i b) e^{-2 i (e+f x)}}{a+i b}\right)}{f (a-i b) (b-i a)}+\frac{6 d x^2 (2 a c f-b d)}{a+i b}+\frac{12 c x (a c f-b d)}{a+i b}+\frac{6 a d^2 x^2 \left(a \left(-1+e^{2 i e}\right)+i b \left(1+e^{2 i e}\right)\right) \log \left(1+\frac{(-a+i b) e^{-2 i (e+f x)}}{a+i b}\right)}{(a-i b) (b-i a)}+\frac{4 a d^2 f x^3}{a+i b}\right)}{b \left(1+e^{2 i e}\right)-i a \left(-1+e^{2 i e}\right)}}{6 f \left(a^2+b^2\right)}","-\frac{2 b^2 d (c+d x) \text{Li}_2\left(\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right)}{f^2 \left(a^2+b^2\right)^2}+\frac{2 b^2 d (c+d x) \log \left(1-\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right)}{f^2 \left(a^2+b^2\right)^2}-\frac{2 i b^2 (c+d x)^2 \log \left(1-\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right)}{f \left(a^2+b^2\right)^2}-\frac{2 i b^2 (c+d x)^2}{f \left(a^2+b^2\right)^2}-\frac{4 b^2 (c+d x)^3}{3 d \left(a^2+b^2\right)^2}-\frac{i b^2 d^2 \text{Li}_2\left(\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right)}{f^3 \left(a^2+b^2\right)^2}-\frac{i b^2 d^2 \text{Li}_3\left(\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right)}{f^3 \left(a^2+b^2\right)^2}-\frac{2 b^2 (c+d x)^2}{f (a-i b) (a+i b)^2 \left(-(-b+i a) e^{2 i e+2 i f x}+i a+b\right)}-\frac{2 b d (c+d x) \text{Li}_2\left(\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right)}{f^2 (a+i b)^2 (b+i a)}-\frac{2 b (c+d x)^2 \log \left(1-\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right)}{f (a-i b) (a+i b)^2}-\frac{4 b (c+d x)^3}{3 d (a+i b)^2 (b+i a)}+\frac{(c+d x)^3}{3 d (a+i b)^2}-\frac{b d^2 \text{Li}_3\left(\frac{(a+i b) e^{2 i e+2 i f x}}{a-i b}\right)}{f^3 (a-i b) (a+i b)^2}",1,"((2*b*((12*c*(-(b*d) + a*c*f)*x)/(a + I*b) + (6*d*(-(b*d) + 2*a*c*f)*x^2)/(a + I*b) + (4*a*d^2*f*x^3)/(a + I*b) - (6*d*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(b*d - 2*a*c*f)*x*Log[1 + (-a + I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/((a - I*b)*((-I)*a + b)*f) + (6*a*d^2*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*x^2*Log[1 + (-a + I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/((a - I*b)*((-I)*a + b)) + (6*c*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(-(b*d) + a*c*f)*(2*f*x + I*Log[a - I*b - (a + I*b)*E^((2*I)*(e + f*x))]))/((a^2 + b^2)*f) + (3*d*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(b*d - 2*a*c*f)*PolyLog[2, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/((a^2 + b^2)*f^2) - (3*a*d^2*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(2*f*x*PolyLog[2, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))] - I*PolyLog[3, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))]))/((a^2 + b^2)*f^2)))/((-I)*a*(-1 + E^((2*I)*e)) + b*(1 + E^((2*I)*e))) + ((a^2 - b^2)*f*x*(3*c^2 + 3*c*d*x + d^2*x^2)*Cos[f*x] - (a^2 + b^2)*f*x*(3*c^2 + 3*c*d*x + d^2*x^2)*Cos[2*e + f*x] + 2*b*(3*b*(c + d*x)^2 - a*f*x*(3*c^2 + 3*c*d*x + d^2*x^2))*Sin[f*x])/((b*Cos[e] + a*Sin[e])*(b*Cos[e + f*x] + a*Sin[e + f*x])))/(6*(a^2 + b^2)*f)","A",1
59,1,730,213,7.0996533,"\int \frac{c+d x}{(a+b \cot (e+f x))^2} \, dx","Integrate[(c + d*x)/(a + b*Cot[e + f*x])^2,x]","-\frac{2 a c \csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x))^2 (b \log (a \sin (e+f x)+b \cos (e+f x))-a (e+f x))}{f (b-i a) (b+i a) \left(a^2+b^2\right) (a+b \cot (e+f x))^2}+\frac{d \csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x))^2 \left(\frac{b \left(i \text{Li}_2\left(e^{2 i \left(e+f x+\tan ^{-1}\left(\frac{b}{a}\right)\right)}\right)+i \left(2 \tan ^{-1}\left(\frac{b}{a}\right)-\pi \right) (e+f x)-2 \left(\tan ^{-1}\left(\frac{b}{a}\right)+e+f x\right) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{b}{a}\right)+e+f x\right)}\right)+2 \tan ^{-1}\left(\frac{b}{a}\right) \log \left(\sin \left(\tan ^{-1}\left(\frac{b}{a}\right)+e+f x\right)\right)-\pi  \log \left(1+e^{-2 i (e+f x)}\right)+\pi  \log (\cos (e+f x))\right)}{a \sqrt{\frac{b^2}{a^2}+1}}+e^{i \tan ^{-1}\left(\frac{b}{a}\right)} (e+f x)^2\right)}{f^2 (b-i a) (b+i a) \sqrt{\frac{a^2+b^2}{a^2}} (a+b \cot (e+f x))^2}+\frac{b d \csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x))^2 (b \log (a \sin (e+f x)+b \cos (e+f x))-a (e+f x))}{f^2 (b-i a) (b+i a) \left(a^2+b^2\right) (a+b \cot (e+f x))^2}+\frac{2 a d e \csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x))^2 (b \log (a \sin (e+f x)+b \cos (e+f x))-a (e+f x))}{f^2 (b-i a) (b+i a) \left(a^2+b^2\right) (a+b \cot (e+f x))^2}-\frac{(e+f x) \csc ^2(e+f x) (2 c f+d (e+f x)-2 d e) (a \sin (e+f x)+b \cos (e+f x))^2}{2 f^2 (b-i a) (b+i a) (a+b \cot (e+f x))^2}+\frac{\csc ^2(e+f x) (a \sin (e+f x)+b \cos (e+f x)) (b c f \sin (e+f x)-b d e \sin (e+f x)+b d (e+f x) \sin (e+f x))}{f^2 (b-i a) (b+i a) (a+b \cot (e+f x))^2}","\frac{b (-2 a c f-2 a d f x+b d) \log \left(1-\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right)}{f^2 \left(a^2+b^2\right)^2}+\frac{b (c+d x)}{f \left(a^2+b^2\right) (a+b \cot (e+f x))}-\frac{(c+d x)^2}{2 d \left(a^2+b^2\right)}+\frac{i a b d \text{Li}_2\left(\frac{(a+i b) e^{2 i (e+f x)}}{a-i b}\right)}{f^2 \left(a^2+b^2\right)^2}+\frac{(-2 a c f-2 a d f x+b d)^2}{4 a d f^2 (a-i b)^2 (a+i b)}",1,"-1/2*((e + f*x)*(-2*d*e + 2*c*f + d*(e + f*x))*Csc[e + f*x]^2*(b*Cos[e + f*x] + a*Sin[e + f*x])^2)/(((-I)*a + b)*(I*a + b)*f^2*(a + b*Cot[e + f*x])^2) + (b*d*Csc[e + f*x]^2*(-(a*(e + f*x)) + b*Log[b*Cos[e + f*x] + a*Sin[e + f*x]])*(b*Cos[e + f*x] + a*Sin[e + f*x])^2)/(((-I)*a + b)*(I*a + b)*(a^2 + b^2)*f^2*(a + b*Cot[e + f*x])^2) + (2*a*d*e*Csc[e + f*x]^2*(-(a*(e + f*x)) + b*Log[b*Cos[e + f*x] + a*Sin[e + f*x]])*(b*Cos[e + f*x] + a*Sin[e + f*x])^2)/(((-I)*a + b)*(I*a + b)*(a^2 + b^2)*f^2*(a + b*Cot[e + f*x])^2) - (2*a*c*Csc[e + f*x]^2*(-(a*(e + f*x)) + b*Log[b*Cos[e + f*x] + a*Sin[e + f*x]])*(b*Cos[e + f*x] + a*Sin[e + f*x])^2)/(((-I)*a + b)*(I*a + b)*(a^2 + b^2)*f*(a + b*Cot[e + f*x])^2) + (d*Csc[e + f*x]^2*(E^(I*ArcTan[b/a])*(e + f*x)^2 + (b*(I*(e + f*x)*(-Pi + 2*ArcTan[b/a]) - Pi*Log[1 + E^((-2*I)*(e + f*x))] - 2*(e + f*x + ArcTan[b/a])*Log[1 - E^((2*I)*(e + f*x + ArcTan[b/a]))] + Pi*Log[Cos[e + f*x]] + 2*ArcTan[b/a]*Log[Sin[e + f*x + ArcTan[b/a]]] + I*PolyLog[2, E^((2*I)*(e + f*x + ArcTan[b/a]))]))/(a*Sqrt[1 + b^2/a^2]))*(b*Cos[e + f*x] + a*Sin[e + f*x])^2)/(((-I)*a + b)*(I*a + b)*Sqrt[(a^2 + b^2)/a^2]*f^2*(a + b*Cot[e + f*x])^2) + (Csc[e + f*x]^2*(b*Cos[e + f*x] + a*Sin[e + f*x])*(-(b*d*e*Sin[e + f*x]) + b*c*f*Sin[e + f*x] + b*d*(e + f*x)*Sin[e + f*x]))/(((-I)*a + b)*(I*a + b)*f^2*(a + b*Cot[e + f*x])^2)","B",0
60,0,0,23,19.281314,"\int \frac{1}{(c+d x) (a+b \cot (e+f x))^2} \, dx","Integrate[1/((c + d*x)*(a + b*Cot[e + f*x])^2),x]","\int \frac{1}{(c+d x) (a+b \cot (e+f x))^2} \, dx","\text{Int}\left(\frac{1}{(c+d x) (a+b \cot (e+f x))^2},x\right)",0,"Integrate[1/((c + d*x)*(a + b*Cot[e + f*x])^2), x]","A",-1
61,0,0,23,17.1733084,"\int \frac{1}{(c+d x)^2 (a+b \cot (e+f x))^2} \, dx","Integrate[1/((c + d*x)^2*(a + b*Cot[e + f*x])^2),x]","\int \frac{1}{(c+d x)^2 (a+b \cot (e+f x))^2} \, dx","\text{Int}\left(\frac{1}{(c+d x)^2 (a+b \cot (e+f x))^2},x\right)",0,"Integrate[1/((c + d*x)^2*(a + b*Cot[e + f*x])^2), x]","A",-1