1,1,51,51,0.0027949,"\int x^5 \cot ^{-1}(a x) \, dx","Integrate[x^5*ArcCot[a*x],x]","-\frac{\tan ^{-1}(a x)}{6 a^6}+\frac{x}{6 a^5}-\frac{x^3}{18 a^3}+\frac{1}{6} x^6 \cot ^{-1}(a x)+\frac{x^5}{30 a}","-\frac{\tan ^{-1}(a x)}{6 a^6}+\frac{x}{6 a^5}-\frac{x^3}{18 a^3}+\frac{1}{6} x^6 \cot ^{-1}(a x)+\frac{x^5}{30 a}",1,"x/(6*a^5) - x^3/(18*a^3) + x^5/(30*a) + (x^6*ArcCot[a*x])/6 - ArcTan[a*x]/(6*a^6)","A",1
2,1,49,49,0.0155448,"\int x^4 \cot ^{-1}(a x) \, dx","Integrate[x^4*ArcCot[a*x],x]","-\frac{x^2}{10 a^3}+\frac{\log \left(a^2 x^2+1\right)}{10 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)+\frac{x^4}{20 a}","-\frac{x^2}{10 a^3}+\frac{\log \left(a^2 x^2+1\right)}{10 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)+\frac{x^4}{20 a}",1,"-1/10*x^2/a^3 + x^4/(20*a) + (x^5*ArcCot[a*x])/5 + Log[1 + a^2*x^2]/(10*a^5)","A",1
3,1,41,41,0.0026133,"\int x^3 \cot ^{-1}(a x) \, dx","Integrate[x^3*ArcCot[a*x],x]","\frac{\tan ^{-1}(a x)}{4 a^4}-\frac{x}{4 a^3}+\frac{1}{4} x^4 \cot ^{-1}(a x)+\frac{x^3}{12 a}","\frac{\tan ^{-1}(a x)}{4 a^4}-\frac{x}{4 a^3}+\frac{1}{4} x^4 \cot ^{-1}(a x)+\frac{x^3}{12 a}",1,"-1/4*x/a^3 + x^3/(12*a) + (x^4*ArcCot[a*x])/4 + ArcTan[a*x]/(4*a^4)","A",1
4,1,39,39,0.0080967,"\int x^2 \cot ^{-1}(a x) \, dx","Integrate[x^2*ArcCot[a*x],x]","-\frac{\log \left(a^2 x^2+1\right)}{6 a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)+\frac{x^2}{6 a}","-\frac{\log \left(a^2 x^2+1\right)}{6 a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)+\frac{x^2}{6 a}",1,"x^2/(6*a) + (x^3*ArcCot[a*x])/3 - Log[1 + a^2*x^2]/(6*a^3)","A",1
5,1,31,31,0.0019396,"\int x \cot ^{-1}(a x) \, dx","Integrate[x*ArcCot[a*x],x]","-\frac{\tan ^{-1}(a x)}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)+\frac{x}{2 a}","-\frac{\tan ^{-1}(a x)}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)+\frac{x}{2 a}",1,"x/(2*a) + (x^2*ArcCot[a*x])/2 - ArcTan[a*x]/(2*a^2)","A",1
6,1,24,24,0.0027599,"\int \cot ^{-1}(a x) \, dx","Integrate[ArcCot[a*x],x]","\frac{\log \left(a^2 x^2+1\right)}{2 a}+x \cot ^{-1}(a x)","\frac{\log \left(a^2 x^2+1\right)}{2 a}+x \cot ^{-1}(a x)",1,"x*ArcCot[a*x] + Log[1 + a^2*x^2]/(2*a)","A",1
7,1,37,37,0.0032888,"\int \frac{\cot ^{-1}(a x)}{x} \, dx","Integrate[ArcCot[a*x]/x,x]","\frac{1}{2} i \text{Li}_2\left(\frac{i}{a x}\right)-\frac{1}{2} i \text{Li}_2\left(-\frac{i}{a x}\right)","\frac{1}{2} i \text{Li}_2\left(\frac{i}{a x}\right)-\frac{1}{2} i \text{Li}_2\left(-\frac{i}{a x}\right)",1,"(-1/2*I)*PolyLog[2, (-I)/(a*x)] + (I/2)*PolyLog[2, I/(a*x)]","A",1
8,1,30,30,0.0025242,"\int \frac{\cot ^{-1}(a x)}{x^2} \, dx","Integrate[ArcCot[a*x]/x^2,x]","\frac{1}{2} a \log \left(a^2 x^2+1\right)-a \log (x)-\frac{\cot ^{-1}(a x)}{x}","\frac{1}{2} a \log \left(a^2 x^2+1\right)-a \log (x)-\frac{\cot ^{-1}(a x)}{x}",1,"-(ArcCot[a*x]/x) - a*Log[x] + (a*Log[1 + a^2*x^2])/2","A",1
9,1,36,31,0.0027754,"\int \frac{\cot ^{-1}(a x)}{x^3} \, dx","Integrate[ArcCot[a*x]/x^3,x]","\frac{a \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-a^2 x^2\right)}{2 x}-\frac{\cot ^{-1}(a x)}{2 x^2}","\frac{1}{2} a^2 \tan ^{-1}(a x)-\frac{\cot ^{-1}(a x)}{2 x^2}+\frac{a}{2 x}",1,"-1/2*ArcCot[a*x]/x^2 + (a*Hypergeometric2F1[-1/2, 1, 1/2, -(a^2*x^2)])/(2*x)","C",1
10,1,44,46,0.0119263,"\int \frac{\cot ^{-1}(a x)}{x^4} \, dx","Integrate[ArcCot[a*x]/x^4,x]","-\frac{1}{6} a \left(a^2 \log \left(a^2 x^2+1\right)-2 a^2 \log (x)-\frac{1}{x^2}\right)-\frac{\cot ^{-1}(a x)}{3 x^3}","\frac{1}{3} a^3 \log (x)-\frac{1}{6} a^3 \log \left(a^2 x^2+1\right)-\frac{\cot ^{-1}(a x)}{3 x^3}+\frac{a}{6 x^2}",1,"-1/3*ArcCot[a*x]/x^3 - (a*(-x^(-2) - 2*a^2*Log[x] + a^2*Log[1 + a^2*x^2]))/6","A",1
11,1,36,41,0.0028217,"\int \frac{\cot ^{-1}(a x)}{x^5} \, dx","Integrate[ArcCot[a*x]/x^5,x]","\frac{a \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-a^2 x^2\right)}{12 x^3}-\frac{\cot ^{-1}(a x)}{4 x^4}","-\frac{1}{4} a^4 \tan ^{-1}(a x)-\frac{a^3}{4 x}-\frac{\cot ^{-1}(a x)}{4 x^4}+\frac{a}{12 x^3}",1,"-1/4*ArcCot[a*x]/x^4 + (a*Hypergeometric2F1[-3/2, 1, -1/2, -(a^2*x^2)])/(12*x^3)","C",1
12,1,79,104,0.0252747,"\int x^5 \cot ^{-1}(a x)^2 \, dx","Integrate[x^5*ArcCot[a*x]^2,x]","\frac{30 \left(a^6 x^6+1\right) \cot ^{-1}(a x)^2+3 a^4 x^4-16 a^2 x^2+46 \log \left(a^2 x^2+1\right)+4 a x \left(3 a^4 x^4-5 a^2 x^2+15\right) \cot ^{-1}(a x)}{180 a^6}","\frac{\cot ^{-1}(a x)^2}{6 a^6}+\frac{x \cot ^{-1}(a x)}{3 a^5}-\frac{4 x^2}{45 a^4}-\frac{x^3 \cot ^{-1}(a x)}{9 a^3}+\frac{x^4}{60 a^2}+\frac{23 \log \left(a^2 x^2+1\right)}{90 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^2+\frac{x^5 \cot ^{-1}(a x)}{15 a}",1,"(-16*a^2*x^2 + 3*a^4*x^4 + 4*a*x*(15 - 5*a^2*x^2 + 3*a^4*x^4)*ArcCot[a*x] + 30*(1 + a^6*x^6)*ArcCot[a*x]^2 + 46*Log[1 + a^2*x^2])/(180*a^6)","A",1
13,1,95,135,0.5481635,"\int x^4 \cot ^{-1}(a x)^2 \, dx","Integrate[x^4*ArcCot[a*x]^2,x]","\frac{6 \left(a^5 x^5+i\right) \cot ^{-1}(a x)^2+a x \left(a^2 x^2-9\right)+3 \cot ^{-1}(a x) \left(a^4 x^4-2 a^2 x^2-4 \log \left(1-e^{2 i \cot ^{-1}(a x)}\right)-3\right)+6 i \text{Li}_2\left(e^{2 i \cot ^{-1}(a x)}\right)}{30 a^5}","\frac{i \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{5 a^5}+\frac{3 \tan ^{-1}(a x)}{10 a^5}+\frac{i \cot ^{-1}(a x)^2}{5 a^5}-\frac{2 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{5 a^5}-\frac{3 x}{10 a^4}-\frac{x^2 \cot ^{-1}(a x)}{5 a^3}+\frac{x^3}{30 a^2}+\frac{1}{5} x^5 \cot ^{-1}(a x)^2+\frac{x^4 \cot ^{-1}(a x)}{10 a}",1,"(a*x*(-9 + a^2*x^2) + 6*(I + a^5*x^5)*ArcCot[a*x]^2 + 3*ArcCot[a*x]*(-3 - 2*a^2*x^2 + a^4*x^4 - 4*Log[1 - E^((2*I)*ArcCot[a*x])]) + (6*I)*PolyLog[2, E^((2*I)*ArcCot[a*x])])/(30*a^5)","A",0
14,1,61,80,0.0212965,"\int x^3 \cot ^{-1}(a x)^2 \, dx","Integrate[x^3*ArcCot[a*x]^2,x]","\frac{3 \left(a^4 x^4-1\right) \cot ^{-1}(a x)^2+a^2 x^2-4 \log \left(a^2 x^2+1\right)+2 a x \left(a^2 x^2-3\right) \cot ^{-1}(a x)}{12 a^4}","-\frac{\cot ^{-1}(a x)^2}{4 a^4}-\frac{x \cot ^{-1}(a x)}{2 a^3}+\frac{x^2}{12 a^2}-\frac{\log \left(a^2 x^2+1\right)}{3 a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^2+\frac{x^3 \cot ^{-1}(a x)}{6 a}",1,"(a^2*x^2 + 2*a*x*(-3 + a^2*x^2)*ArcCot[a*x] + 3*(-1 + a^4*x^4)*ArcCot[a*x]^2 - 4*Log[1 + a^2*x^2])/(12*a^4)","A",1
15,1,76,111,0.2949683,"\int x^2 \cot ^{-1}(a x)^2 \, dx","Integrate[x^2*ArcCot[a*x]^2,x]","\frac{\left(a^3 x^3-i\right) \cot ^{-1}(a x)^2+\cot ^{-1}(a x) \left(a^2 x^2+2 \log \left(1-e^{2 i \cot ^{-1}(a x)}\right)+1\right)-i \text{Li}_2\left(e^{2 i \cot ^{-1}(a x)}\right)+a x}{3 a^3}","-\frac{i \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{3 a^3}-\frac{\tan ^{-1}(a x)}{3 a^3}-\frac{i \cot ^{-1}(a x)^2}{3 a^3}+\frac{2 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{3 a^3}+\frac{x}{3 a^2}+\frac{1}{3} x^3 \cot ^{-1}(a x)^2+\frac{x^2 \cot ^{-1}(a x)}{3 a}",1,"(a*x + (-I + a^3*x^3)*ArcCot[a*x]^2 + ArcCot[a*x]*(1 + a^2*x^2 + 2*Log[1 - E^((2*I)*ArcCot[a*x])]) - I*PolyLog[2, E^((2*I)*ArcCot[a*x])])/(3*a^3)","A",0
16,1,42,53,0.0141506,"\int x \cot ^{-1}(a x)^2 \, dx","Integrate[x*ArcCot[a*x]^2,x]","\frac{\log \left(a^2 x^2+1\right)+\left(a^2 x^2+1\right) \cot ^{-1}(a x)^2+2 a x \cot ^{-1}(a x)}{2 a^2}","\frac{\log \left(a^2 x^2+1\right)}{2 a^2}+\frac{\cot ^{-1}(a x)^2}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)^2+\frac{x \cot ^{-1}(a x)}{a}",1,"(2*a*x*ArcCot[a*x] + (1 + a^2*x^2)*ArcCot[a*x]^2 + Log[1 + a^2*x^2])/(2*a^2)","A",1
17,1,56,67,0.0848877,"\int \cot ^{-1}(a x)^2 \, dx","Integrate[ArcCot[a*x]^2,x]","\frac{i \text{Li}_2\left(e^{2 i \cot ^{-1}(a x)}\right)+\cot ^{-1}(a x) \left((a x+i) \cot ^{-1}(a x)-2 \log \left(1-e^{2 i \cot ^{-1}(a x)}\right)\right)}{a}","\frac{i \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{a}+x \cot ^{-1}(a x)^2+\frac{i \cot ^{-1}(a x)^2}{a}-\frac{2 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{a}",1,"(ArcCot[a*x]*((I + a*x)*ArcCot[a*x] - 2*Log[1 - E^((2*I)*ArcCot[a*x])]) + I*PolyLog[2, E^((2*I)*ArcCot[a*x])])/a","A",0
18,1,132,116,0.063563,"\int \frac{\cot ^{-1}(a x)^2}{x} \, dx","Integrate[ArcCot[a*x]^2/x,x]","-i \cot ^{-1}(a x) \text{Li}_2\left(e^{-2 i \cot ^{-1}(a x)}\right)-i \cot ^{-1}(a x) \text{Li}_2\left(-e^{2 i \cot ^{-1}(a x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{-2 i \cot ^{-1}(a x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{2 i \cot ^{-1}(a x)}\right)-\frac{2}{3} i \cot ^{-1}(a x)^3-\cot ^{-1}(a x)^2 \log \left(1-e^{-2 i \cot ^{-1}(a x)}\right)+\cot ^{-1}(a x)^2 \log \left(1+e^{2 i \cot ^{-1}(a x)}\right)","-\frac{1}{2} \text{Li}_3\left(1-\frac{2 i}{a x+i}\right)+\frac{1}{2} \text{Li}_3\left(1-\frac{2 a x}{a x+i}\right)-i \text{Li}_2\left(1-\frac{2 i}{a x+i}\right) \cot ^{-1}(a x)+i \text{Li}_2\left(1-\frac{2 a x}{a x+i}\right) \cot ^{-1}(a x)+2 \cot ^{-1}(a x)^2 \coth ^{-1}\left(1-\frac{2}{1+i a x}\right)",1,"((-2*I)/3)*ArcCot[a*x]^3 - ArcCot[a*x]^2*Log[1 - E^((-2*I)*ArcCot[a*x])] + ArcCot[a*x]^2*Log[1 + E^((2*I)*ArcCot[a*x])] - I*ArcCot[a*x]*PolyLog[2, E^((-2*I)*ArcCot[a*x])] - I*ArcCot[a*x]*PolyLog[2, -E^((2*I)*ArcCot[a*x])] - PolyLog[3, E^((-2*I)*ArcCot[a*x])]/2 + PolyLog[3, -E^((2*I)*ArcCot[a*x])]/2","A",0
19,1,64,66,0.0491651,"\int \frac{\cot ^{-1}(a x)^2}{x^2} \, dx","Integrate[ArcCot[a*x]^2/x^2,x]","a \left(i \text{Li}_2\left(-e^{2 i \cot ^{-1}(a x)}\right)-\frac{\cot ^{-1}(a x)^2}{a x}+i \cot ^{-1}(a x)^2-2 \cot ^{-1}(a x) \log \left(1+e^{2 i \cot ^{-1}(a x)}\right)\right)","-i a \text{Li}_2\left(\frac{2}{1-i a x}-1\right)-i a \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{x}-2 a \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)",1,"a*(I*ArcCot[a*x]^2 - ArcCot[a*x]^2/(a*x) - 2*ArcCot[a*x]*Log[1 + E^((2*I)*ArcCot[a*x])] + I*PolyLog[2, -E^((2*I)*ArcCot[a*x])])","A",0
20,1,56,59,0.0182686,"\int \frac{\cot ^{-1}(a x)^2}{x^3} \, dx","Integrate[ArcCot[a*x]^2/x^3,x]","-\frac{1}{2} a^2 \log \left(a^2 x^2+1\right)+\frac{\left(-a^2 x^2-1\right) \cot ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)+\frac{a \cot ^{-1}(a x)}{x}","-\frac{1}{2} a^2 \log \left(a^2 x^2+1\right)+a^2 \log (x)-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+\frac{a \cot ^{-1}(a x)}{x}",1,"(a*ArcCot[a*x])/x + ((-1 - a^2*x^2)*ArcCot[a*x]^2)/(2*x^2) + a^2*Log[x] - (a^2*Log[1 + a^2*x^2])/2","A",1
21,1,96,113,0.2496053,"\int \frac{\cot ^{-1}(a x)^2}{x^4} \, dx","Integrate[ArcCot[a*x]^2/x^4,x]","\frac{-i a^3 x^3 \text{Li}_2\left(-e^{2 i \cot ^{-1}(a x)}\right)+\left(-1-i a^3 x^3\right) \cot ^{-1}(a x)^2-a^2 x^2+a x \cot ^{-1}(a x) \left(a^2 x^2+2 a^2 x^2 \log \left(1+e^{2 i \cot ^{-1}(a x)}\right)+1\right)}{3 x^3}","\frac{1}{3} i a^3 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)-\frac{1}{3} a^3 \tan ^{-1}(a x)+\frac{1}{3} i a^3 \cot ^{-1}(a x)^2+\frac{2}{3} a^3 \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)-\frac{a^2}{3 x}-\frac{\cot ^{-1}(a x)^2}{3 x^3}+\frac{a \cot ^{-1}(a x)}{3 x^2}",1,"(-(a^2*x^2) + (-1 - I*a^3*x^3)*ArcCot[a*x]^2 + a*x*ArcCot[a*x]*(1 + a^2*x^2 + 2*a^2*x^2*Log[1 + E^((2*I)*ArcCot[a*x])]) - I*a^3*x^3*PolyLog[2, -E^((2*I)*ArcCot[a*x])])/(3*x^3)","A",0
22,1,81,89,0.0208537,"\int \frac{\cot ^{-1}(a x)^2}{x^5} \, dx","Integrate[ArcCot[a*x]^2/x^5,x]","\frac{\left(a^4 x^4-1\right) \cot ^{-1}(a x)^2}{4 x^4}-\frac{2}{3} a^4 \log (x)-\frac{a^2}{12 x^2}-\frac{a \left(3 a^2 x^2-1\right) \cot ^{-1}(a x)}{6 x^3}+\frac{1}{3} a^4 \log \left(a^2 x^2+1\right)","-\frac{2}{3} a^4 \log (x)+\frac{1}{4} a^4 \cot ^{-1}(a x)^2-\frac{a^3 \cot ^{-1}(a x)}{2 x}-\frac{a^2}{12 x^2}+\frac{1}{3} a^4 \log \left(a^2 x^2+1\right)-\frac{\cot ^{-1}(a x)^2}{4 x^4}+\frac{a \cot ^{-1}(a x)}{6 x^3}",1,"-1/12*a^2/x^2 - (a*(-1 + 3*a^2*x^2)*ArcCot[a*x])/(6*x^3) + ((-1 + a^4*x^4)*ArcCot[a*x]^2)/(4*x^4) - (2*a^4*Log[x])/3 + (a^4*Log[1 + a^2*x^2])/3","A",1
23,1,125,194,0.6869945,"\int x^5 \cot ^{-1}(a x)^3 \, dx","Integrate[x^5*ArcCot[a*x]^3,x]","\frac{10 \left(a^6 x^6+1\right) \cot ^{-1}(a x)^3+a x \left(a^2 x^2-19\right)+2 \left(3 a^5 x^5-5 a^3 x^3+15 a x+23 i\right) \cot ^{-1}(a x)^2+\cot ^{-1}(a x) \left(3 a^4 x^4-16 a^2 x^2-92 \log \left(1-e^{2 i \cot ^{-1}(a x)}\right)-19\right)+46 i \text{Li}_2\left(e^{2 i \cot ^{-1}(a x)}\right)}{60 a^6}","\frac{23 i \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{30 a^6}+\frac{19 \tan ^{-1}(a x)}{60 a^6}+\frac{\cot ^{-1}(a x)^3}{6 a^6}+\frac{23 i \cot ^{-1}(a x)^2}{30 a^6}-\frac{23 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{15 a^6}-\frac{19 x}{60 a^5}+\frac{x \cot ^{-1}(a x)^2}{2 a^5}-\frac{4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac{x^3}{60 a^3}-\frac{x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac{x^4 \cot ^{-1}(a x)}{20 a^2}+\frac{1}{6} x^6 \cot ^{-1}(a x)^3+\frac{x^5 \cot ^{-1}(a x)^2}{10 a}",1,"(a*x*(-19 + a^2*x^2) + 2*(23*I + 15*a*x - 5*a^3*x^3 + 3*a^5*x^5)*ArcCot[a*x]^2 + 10*(1 + a^6*x^6)*ArcCot[a*x]^3 + ArcCot[a*x]*(-19 - 16*a^2*x^2 + 3*a^4*x^4 - 92*Log[1 - E^((2*I)*ArcCot[a*x])]) + (46*I)*PolyLog[2, E^((2*I)*ArcCot[a*x])])/(60*a^6)","A",0
24,1,184,205,0.6420392,"\int x^4 \cot ^{-1}(a x)^3 \, dx","Integrate[x^4*ArcCot[a*x]^3,x]","\frac{8 a^5 x^5 \cot ^{-1}(a x)^3+6 a^4 x^4 \cot ^{-1}(a x)^2+4 a^3 x^3 \cot ^{-1}(a x)+2 a^2 x^2+40 \log \left(\frac{1}{a x \sqrt{\frac{1}{a^2 x^2}+1}}\right)-12 a^2 x^2 \cot ^{-1}(a x)^2-24 i \cot ^{-1}(a x) \text{Li}_2\left(e^{-2 i \cot ^{-1}(a x)}\right)-12 \text{Li}_3\left(e^{-2 i \cot ^{-1}(a x)}\right)-36 a x \cot ^{-1}(a x)-8 i \cot ^{-1}(a x)^3-18 \cot ^{-1}(a x)^2-24 \cot ^{-1}(a x)^2 \log \left(1-e^{-2 i \cot ^{-1}(a x)}\right)+i \pi ^3+2}{40 a^5}","-\frac{3 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)}{10 a^5}+\frac{3 i \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \cot ^{-1}(a x)}{5 a^5}+\frac{i \cot ^{-1}(a x)^3}{5 a^5}-\frac{9 \cot ^{-1}(a x)^2}{20 a^5}-\frac{3 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)^2}{5 a^5}-\frac{9 x \cot ^{-1}(a x)}{10 a^4}+\frac{x^2}{20 a^3}-\frac{3 x^2 \cot ^{-1}(a x)^2}{10 a^3}+\frac{x^3 \cot ^{-1}(a x)}{10 a^2}-\frac{\log \left(a^2 x^2+1\right)}{2 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^3+\frac{3 x^4 \cot ^{-1}(a x)^2}{20 a}",1,"(2 + I*Pi^3 + 2*a^2*x^2 - 36*a*x*ArcCot[a*x] + 4*a^3*x^3*ArcCot[a*x] - 18*ArcCot[a*x]^2 - 12*a^2*x^2*ArcCot[a*x]^2 + 6*a^4*x^4*ArcCot[a*x]^2 - (8*I)*ArcCot[a*x]^3 + 8*a^5*x^5*ArcCot[a*x]^3 - 24*ArcCot[a*x]^2*Log[1 - E^((-2*I)*ArcCot[a*x])] + 40*Log[1/(a*Sqrt[1 + 1/(a^2*x^2)]*x)] - (24*I)*ArcCot[a*x]*PolyLog[2, E^((-2*I)*ArcCot[a*x])] - 12*PolyLog[3, E^((-2*I)*ArcCot[a*x])])/(40*a^5)","A",0
25,1,96,148,0.3801501,"\int x^3 \cot ^{-1}(a x)^3 \, dx","Integrate[x^3*ArcCot[a*x]^3,x]","\frac{\left(a^4 x^4-1\right) \cot ^{-1}(a x)^3+\left(a^3 x^3-3 a x-4 i\right) \cot ^{-1}(a x)^2+\cot ^{-1}(a x) \left(a^2 x^2+8 \log \left(1-e^{2 i \cot ^{-1}(a x)}\right)+1\right)-4 i \text{Li}_2\left(e^{2 i \cot ^{-1}(a x)}\right)+a x}{4 a^4}","-\frac{i \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{a^4}-\frac{\tan ^{-1}(a x)}{4 a^4}-\frac{\cot ^{-1}(a x)^3}{4 a^4}-\frac{i \cot ^{-1}(a x)^2}{a^4}+\frac{2 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{a^4}+\frac{x}{4 a^3}-\frac{3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac{x^2 \cot ^{-1}(a x)}{4 a^2}+\frac{1}{4} x^4 \cot ^{-1}(a x)^3+\frac{x^3 \cot ^{-1}(a x)^2}{4 a}",1,"(a*x + (-4*I - 3*a*x + a^3*x^3)*ArcCot[a*x]^2 + (-1 + a^4*x^4)*ArcCot[a*x]^3 + ArcCot[a*x]*(1 + a^2*x^2 + 8*Log[1 - E^((2*I)*ArcCot[a*x])]) - (4*I)*PolyLog[2, E^((2*I)*ArcCot[a*x])])/(4*a^4)","A",0
26,1,149,157,0.3537751,"\int x^2 \cot ^{-1}(a x)^3 \, dx","Integrate[x^2*ArcCot[a*x]^3,x]","\frac{8 a^3 x^3 \cot ^{-1}(a x)^3-24 \log \left(\frac{1}{a x \sqrt{\frac{1}{a^2 x^2}+1}}\right)+12 a^2 x^2 \cot ^{-1}(a x)^2+24 i \cot ^{-1}(a x) \text{Li}_2\left(e^{-2 i \cot ^{-1}(a x)}\right)+12 \text{Li}_3\left(e^{-2 i \cot ^{-1}(a x)}\right)+8 i \cot ^{-1}(a x)^3+12 \cot ^{-1}(a x)^2+24 a x \cot ^{-1}(a x)+24 \cot ^{-1}(a x)^2 \log \left(1-e^{-2 i \cot ^{-1}(a x)}\right)-i \pi ^3}{24 a^3}","\frac{\text{Li}_3\left(1-\frac{2}{i a x+1}\right)}{2 a^3}-\frac{i \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \cot ^{-1}(a x)}{a^3}-\frac{i \cot ^{-1}(a x)^3}{3 a^3}+\frac{\cot ^{-1}(a x)^2}{2 a^3}+\frac{\log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)^2}{a^3}+\frac{x \cot ^{-1}(a x)}{a^2}+\frac{\log \left(a^2 x^2+1\right)}{2 a^3}+\frac{1}{3} x^3 \cot ^{-1}(a x)^3+\frac{x^2 \cot ^{-1}(a x)^2}{2 a}",1,"((-I)*Pi^3 + 24*a*x*ArcCot[a*x] + 12*ArcCot[a*x]^2 + 12*a^2*x^2*ArcCot[a*x]^2 + (8*I)*ArcCot[a*x]^3 + 8*a^3*x^3*ArcCot[a*x]^3 + 24*ArcCot[a*x]^2*Log[1 - E^((-2*I)*ArcCot[a*x])] - 24*Log[1/(a*Sqrt[1 + 1/(a^2*x^2)]*x)] + (24*I)*ArcCot[a*x]*PolyLog[2, E^((-2*I)*ArcCot[a*x])] + 12*PolyLog[3, E^((-2*I)*ArcCot[a*x])])/(24*a^3)","A",0
27,1,76,103,0.1005603,"\int x \cot ^{-1}(a x)^3 \, dx","Integrate[x*ArcCot[a*x]^3,x]","\frac{\cot ^{-1}(a x) \left(\left(a^2 x^2+1\right) \cot ^{-1}(a x)^2+3 (a x+i) \cot ^{-1}(a x)-6 \log \left(1-e^{2 i \cot ^{-1}(a x)}\right)\right)+3 i \text{Li}_2\left(e^{2 i \cot ^{-1}(a x)}\right)}{2 a^2}","\frac{3 i \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{2 a^2}+\frac{\cot ^{-1}(a x)^3}{2 a^2}+\frac{3 i \cot ^{-1}(a x)^2}{2 a^2}-\frac{3 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)^3+\frac{3 x \cot ^{-1}(a x)^2}{2 a}",1,"(ArcCot[a*x]*(3*(I + a*x)*ArcCot[a*x] + (1 + a^2*x^2)*ArcCot[a*x]^2 - 6*Log[1 - E^((2*I)*ArcCot[a*x])]) + (3*I)*PolyLog[2, E^((2*I)*ArcCot[a*x])])/(2*a^2)","A",0
28,1,90,96,0.1155088,"\int \cot ^{-1}(a x)^3 \, dx","Integrate[ArcCot[a*x]^3,x]","-\frac{3 i \cot ^{-1}(a x) \text{Li}_2\left(e^{-2 i \cot ^{-1}(a x)}\right)}{a}-\frac{3 \text{Li}_3\left(e^{-2 i \cot ^{-1}(a x)}\right)}{2 a}+x \cot ^{-1}(a x)^3-\frac{i \cot ^{-1}(a x)^3}{a}-\frac{3 \cot ^{-1}(a x)^2 \log \left(1-e^{-2 i \cot ^{-1}(a x)}\right)}{a}","-\frac{3 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)}{2 a}+\frac{3 i \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \cot ^{-1}(a x)}{a}+x \cot ^{-1}(a x)^3+\frac{i \cot ^{-1}(a x)^3}{a}-\frac{3 \log \left(\frac{2}{1+i a x}\right) \cot ^{-1}(a x)^2}{a}",1,"((-I)*ArcCot[a*x]^3)/a + x*ArcCot[a*x]^3 - (3*ArcCot[a*x]^2*Log[1 - E^((-2*I)*ArcCot[a*x])])/a - ((3*I)*ArcCot[a*x]*PolyLog[2, E^((-2*I)*ArcCot[a*x])])/a - (3*PolyLog[3, E^((-2*I)*ArcCot[a*x])])/(2*a)","A",0
29,1,180,178,0.0840071,"\int \frac{\cot ^{-1}(a x)^3}{x} \, dx","Integrate[ArcCot[a*x]^3/x,x]","\frac{1}{64} i \left(-96 \cot ^{-1}(a x)^2 \text{Li}_2\left(e^{-2 i \cot ^{-1}(a x)}\right)-96 \cot ^{-1}(a x)^2 \text{Li}_2\left(-e^{2 i \cot ^{-1}(a x)}\right)+96 i \cot ^{-1}(a x) \text{Li}_3\left(e^{-2 i \cot ^{-1}(a x)}\right)-96 i \cot ^{-1}(a x) \text{Li}_3\left(-e^{2 i \cot ^{-1}(a x)}\right)+48 \text{Li}_4\left(e^{-2 i \cot ^{-1}(a x)}\right)+48 \text{Li}_4\left(-e^{2 i \cot ^{-1}(a x)}\right)-32 \cot ^{-1}(a x)^4+64 i \cot ^{-1}(a x)^3 \log \left(1-e^{-2 i \cot ^{-1}(a x)}\right)-64 i \cot ^{-1}(a x)^3 \log \left(1+e^{2 i \cot ^{-1}(a x)}\right)+\pi ^4\right)","\frac{3}{4} i \text{Li}_4\left(1-\frac{2 i}{a x+i}\right)-\frac{3}{4} i \text{Li}_4\left(1-\frac{2 a x}{a x+i}\right)-\frac{3}{2} i \text{Li}_2\left(1-\frac{2 i}{a x+i}\right) \cot ^{-1}(a x)^2+\frac{3}{2} i \text{Li}_2\left(1-\frac{2 a x}{a x+i}\right) \cot ^{-1}(a x)^2-\frac{3}{2} \text{Li}_3\left(1-\frac{2 i}{a x+i}\right) \cot ^{-1}(a x)+\frac{3}{2} \text{Li}_3\left(1-\frac{2 a x}{a x+i}\right) \cot ^{-1}(a x)+2 \cot ^{-1}(a x)^3 \coth ^{-1}\left(1-\frac{2}{1+i a x}\right)",1,"(I/64)*(Pi^4 - 32*ArcCot[a*x]^4 + (64*I)*ArcCot[a*x]^3*Log[1 - E^((-2*I)*ArcCot[a*x])] - (64*I)*ArcCot[a*x]^3*Log[1 + E^((2*I)*ArcCot[a*x])] - 96*ArcCot[a*x]^2*PolyLog[2, E^((-2*I)*ArcCot[a*x])] - 96*ArcCot[a*x]^2*PolyLog[2, -E^((2*I)*ArcCot[a*x])] + (96*I)*ArcCot[a*x]*PolyLog[3, E^((-2*I)*ArcCot[a*x])] - (96*I)*ArcCot[a*x]*PolyLog[3, -E^((2*I)*ArcCot[a*x])] + 48*PolyLog[4, E^((-2*I)*ArcCot[a*x])] + 48*PolyLog[4, -E^((2*I)*ArcCot[a*x])])","A",0
30,1,83,93,0.0832102,"\int \frac{\cot ^{-1}(a x)^3}{x^2} \, dx","Integrate[ArcCot[a*x]^3/x^2,x]","3 i a \cot ^{-1}(a x) \text{Li}_2\left(-e^{2 i \cot ^{-1}(a x)}\right)-\frac{3}{2} a \text{Li}_3\left(-e^{2 i \cot ^{-1}(a x)}\right)+\frac{(-1+i a x) \cot ^{-1}(a x)^3}{x}-3 a \cot ^{-1}(a x)^2 \log \left(1+e^{2 i \cot ^{-1}(a x)}\right)","-\frac{3}{2} a \text{Li}_3\left(\frac{2}{1-i a x}-1\right)-3 i a \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \cot ^{-1}(a x)-i a \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{x}-3 a \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)^2",1,"((-1 + I*a*x)*ArcCot[a*x]^3)/x - 3*a*ArcCot[a*x]^2*Log[1 + E^((2*I)*ArcCot[a*x])] + (3*I)*a*ArcCot[a*x]*PolyLog[2, -E^((2*I)*ArcCot[a*x])] - (3*a*PolyLog[3, -E^((2*I)*ArcCot[a*x])])/2","A",0
31,1,90,105,0.1765587,"\int \frac{\cot ^{-1}(a x)^3}{x^3} \, dx","Integrate[ArcCot[a*x]^3/x^3,x]","-\frac{3}{2} i a^2 \text{Li}_2\left(-e^{2 i \cot ^{-1}(a x)}\right)-\frac{\cot ^{-1}(a x) \left(\left(a^2 x^2+1\right) \cot ^{-1}(a x)^2-6 a^2 x^2 \log \left(1+e^{2 i \cot ^{-1}(a x)}\right)+3 i a x (a x+i) \cot ^{-1}(a x)\right)}{2 x^2}","\frac{3}{2} i a^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)-\frac{1}{2} a^2 \cot ^{-1}(a x)^3+\frac{3}{2} i a^2 \cot ^{-1}(a x)^2+3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)-\frac{\cot ^{-1}(a x)^3}{2 x^2}+\frac{3 a \cot ^{-1}(a x)^2}{2 x}",1,"-1/2*(ArcCot[a*x]*((3*I)*a*x*(I + a*x)*ArcCot[a*x] + (1 + a^2*x^2)*ArcCot[a*x]^2 - 6*a^2*x^2*Log[1 + E^((2*I)*ArcCot[a*x])]))/x^2 - ((3*I)/2)*a^2*PolyLog[2, -E^((2*I)*ArcCot[a*x])]","A",0
32,1,151,167,0.2588723,"\int \frac{\cot ^{-1}(a x)^3}{x^4} \, dx","Integrate[ArcCot[a*x]^3/x^4,x]","\frac{1}{6} \left(-6 i a^3 \cot ^{-1}(a x) \text{Li}_2\left(-e^{2 i \cot ^{-1}(a x)}\right)+3 a^3 \text{Li}_3\left(-e^{2 i \cot ^{-1}(a x)}\right)-2 i a^3 \cot ^{-1}(a x)^3+3 a^3 \cot ^{-1}(a x)^2+6 a^3 \cot ^{-1}(a x)^2 \log \left(1+e^{2 i \cot ^{-1}(a x)}\right)-\frac{6 a^2 \cot ^{-1}(a x)}{x}-6 a^3 \log \left(\frac{1}{\sqrt{\frac{1}{a^2 x^2}+1}}\right)-\frac{2 \cot ^{-1}(a x)^3}{x^3}+\frac{3 a \cot ^{-1}(a x)^2}{x^2}\right)","\frac{1}{2} a^3 \text{Li}_3\left(\frac{2}{1-i a x}-1\right)+i a^3 \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \cot ^{-1}(a x)-a^3 \log (x)+\frac{1}{3} i a^3 \cot ^{-1}(a x)^3+\frac{1}{2} a^3 \cot ^{-1}(a x)^2+a^3 \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)^2-\frac{a^2 \cot ^{-1}(a x)}{x}+\frac{1}{2} a^3 \log \left(a^2 x^2+1\right)-\frac{\cot ^{-1}(a x)^3}{3 x^3}+\frac{a \cot ^{-1}(a x)^2}{2 x^2}",1,"((-6*a^2*ArcCot[a*x])/x + 3*a^3*ArcCot[a*x]^2 + (3*a*ArcCot[a*x]^2)/x^2 - (2*I)*a^3*ArcCot[a*x]^3 - (2*ArcCot[a*x]^3)/x^3 + 6*a^3*ArcCot[a*x]^2*Log[1 + E^((2*I)*ArcCot[a*x])] - 6*a^3*Log[1/Sqrt[1 + 1/(a^2*x^2)]] - (6*I)*a^3*ArcCot[a*x]*PolyLog[2, -E^((2*I)*ArcCot[a*x])] + 3*a^3*PolyLog[3, -E^((2*I)*ArcCot[a*x])])/6","A",0
33,1,126,152,0.2827716,"\int \frac{\cot ^{-1}(a x)^3}{x^5} \, dx","Integrate[ArcCot[a*x]^3/x^5,x]","\frac{4 i a^4 x^4 \text{Li}_2\left(-e^{2 i \cot ^{-1}(a x)}\right)+\left(a^4 x^4-1\right) \cot ^{-1}(a x)^3+a^3 x^3-a^2 x^2 \cot ^{-1}(a x) \left(a^2 x^2+8 a^2 x^2 \log \left(1+e^{2 i \cot ^{-1}(a x)}\right)+1\right)+\left(4 i a^4 x^4-3 a^3 x^3+a x\right) \cot ^{-1}(a x)^2}{4 x^4}","-i a^4 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)+\frac{1}{4} a^4 \tan ^{-1}(a x)+\frac{1}{4} a^4 \cot ^{-1}(a x)^3-i a^4 \cot ^{-1}(a x)^2-2 a^4 \log \left(2-\frac{2}{1-i a x}\right) \cot ^{-1}(a x)+\frac{a^3}{4 x}-\frac{3 a^3 \cot ^{-1}(a x)^2}{4 x}-\frac{a^2 \cot ^{-1}(a x)}{4 x^2}-\frac{\cot ^{-1}(a x)^3}{4 x^4}+\frac{a \cot ^{-1}(a x)^2}{4 x^3}",1,"(a^3*x^3 + (a*x - 3*a^3*x^3 + (4*I)*a^4*x^4)*ArcCot[a*x]^2 + (-1 + a^4*x^4)*ArcCot[a*x]^3 - a^2*x^2*ArcCot[a*x]*(1 + a^2*x^2 + 8*a^2*x^2*Log[1 + E^((2*I)*ArcCot[a*x])]) + (4*I)*a^4*x^4*PolyLog[2, -E^((2*I)*ArcCot[a*x])])/(4*x^4)","A",0
34,0,0,13,0.9350077,"\int x^m \cot ^{-1}(a x)^3 \, dx","Integrate[x^m*ArcCot[a*x]^3,x]","\int x^m \cot ^{-1}(a x)^3 \, dx","\text{Int}\left(x^m \cot ^{-1}(a x)^3,x\right)",0,"Integrate[x^m*ArcCot[a*x]^3, x]","A",-1
35,0,0,13,0.9310016,"\int x^m \cot ^{-1}(a x)^2 \, dx","Integrate[x^m*ArcCot[a*x]^2,x]","\int x^m \cot ^{-1}(a x)^2 \, dx","\text{Int}\left(x^m \cot ^{-1}(a x)^2,x\right)",0,"Integrate[x^m*ArcCot[a*x]^2, x]","A",-1
36,1,52,57,0.0236534,"\int x^m \cot ^{-1}(a x) \, dx","Integrate[x^m*ArcCot[a*x],x]","\frac{x^{m+1} \left(a x \, _2F_1\left(1,\frac{m}{2}+1;\frac{m}{2}+2;-a^2 x^2\right)+(m+2) \cot ^{-1}(a x)\right)}{(m+1) (m+2)}","\frac{a x^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m^2+3 m+2}+\frac{x^{m+1} \cot ^{-1}(a x)}{m+1}",1,"(x^(1 + m)*((2 + m)*ArcCot[a*x] + a*x*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, -(a^2*x^2)]))/((1 + m)*(2 + m))","A",1
37,1,32,40,0.0294764,"\int \frac{x^4 \cot ^{-1}(x)}{1+x^2} \, dx","Integrate[(x^4*ArcCot[x])/(1 + x^2),x]","\frac{1}{6} \left(x^2-4 \log \left(x^2+1\right)+2 \left(x^2-3\right) x \cot ^{-1}(x)-3 \cot ^{-1}(x)^2\right)","\frac{1}{3} x^3 \cot ^{-1}(x)+\frac{x^2}{6}-\frac{2}{3} \log \left(x^2+1\right)-x \cot ^{-1}(x)-\frac{1}{2} \cot ^{-1}(x)^2",1,"(x^2 + 2*x*(-3 + x^2)*ArcCot[x] - 3*ArcCot[x]^2 - 4*Log[1 + x^2])/6","A",1
38,1,241,67,0.0657285,"\int \frac{x^3 \cot ^{-1}(x)}{1+x^2} \, dx","Integrate[(x^3*ArcCot[x])/(1 + x^2),x]","-\frac{1}{4} i \text{Li}_2\left(-\frac{1}{2} i (i-x)\right)+\frac{1}{4} i \text{Li}_2\left(-\frac{1}{2} i (x+i)\right)+\frac{1}{2} x^2 \cot ^{-1}(x)+\frac{x}{2}+\frac{1}{8} i \log ^2(-x+i)-\frac{1}{8} i \log ^2(x+i)-\frac{1}{4} i \log (-x+i) \log \left(-\frac{-x+i}{x}\right)-\frac{1}{4} i \log (-x+i) \log \left(-\frac{1}{2} i (x+i)\right)+\frac{1}{4} i \log \left(-\frac{1}{2} i (-x+i)\right) \log (x+i)-\frac{1}{4} i \log \left(-\frac{-x+i}{x}\right) \log (x+i)+\frac{1}{4} i \log (-x+i) \log \left(\frac{x+i}{x}\right)+\frac{1}{4} i \log (x+i) \log \left(\frac{x+i}{x}\right)-\frac{1}{2} \tan ^{-1}(x)","-\frac{1}{2} i \text{Li}_2\left(1-\frac{2}{i x+1}\right)+\frac{1}{2} x^2 \cot ^{-1}(x)+\frac{x}{2}-\frac{1}{2} \tan ^{-1}(x)-\frac{1}{2} i \cot ^{-1}(x)^2+\log \left(\frac{2}{1+i x}\right) \cot ^{-1}(x)",1,"x/2 + (x^2*ArcCot[x])/2 - ArcTan[x]/2 + (I/8)*Log[I - x]^2 - (I/4)*Log[I - x]*Log[-((I - x)/x)] - (I/4)*Log[I - x]*Log[(-1/2*I)*(I + x)] + (I/4)*Log[(-1/2*I)*(I - x)]*Log[I + x] - (I/4)*Log[-((I - x)/x)]*Log[I + x] - (I/8)*Log[I + x]^2 + (I/4)*Log[I - x]*Log[(I + x)/x] + (I/4)*Log[I + x]*Log[(I + x)/x] - (I/4)*PolyLog[2, (-1/2*I)*(I - x)] + (I/4)*PolyLog[2, (-1/2*I)*(I + x)]","B",0
39,1,23,23,0.0155807,"\int \frac{x^2 \cot ^{-1}(x)}{1+x^2} \, dx","Integrate[(x^2*ArcCot[x])/(1 + x^2),x]","\frac{1}{2} \log \left(x^2+1\right)+\frac{1}{2} \cot ^{-1}(x)^2+x \cot ^{-1}(x)","\frac{1}{2} \log \left(x^2+1\right)+\frac{1}{2} \cot ^{-1}(x)^2+x \cot ^{-1}(x)",1,"x*ArcCot[x] + ArcCot[x]^2/2 + Log[1 + x^2]/2","A",1
40,1,221,48,0.051267,"\int \frac{x \cot ^{-1}(x)}{1+x^2} \, dx","Integrate[(x*ArcCot[x])/(1 + x^2),x]","\frac{1}{4} i \text{Li}_2\left(-\frac{1}{2} i (i-x)\right)-\frac{1}{4} i \text{Li}_2\left(-\frac{1}{2} i (x+i)\right)-\frac{1}{8} i \log ^2(-x+i)+\frac{1}{8} i \log ^2(x+i)+\frac{1}{4} i \log \left(-\frac{-x+i}{x}\right) \log (-x+i)+\frac{1}{4} i \log \left(-\frac{1}{2} i (x+i)\right) \log (-x+i)-\frac{1}{4} i \log \left(\frac{x+i}{x}\right) \log (-x+i)-\frac{1}{4} i \log \left(-\frac{1}{2} i (-x+i)\right) \log (x+i)+\frac{1}{4} i \log \left(-\frac{-x+i}{x}\right) \log (x+i)-\frac{1}{4} i \log (x+i) \log \left(\frac{x+i}{x}\right)","\frac{1}{2} i \text{Li}_2\left(1-\frac{2}{i x+1}\right)+\frac{1}{2} i \cot ^{-1}(x)^2-\log \left(\frac{2}{1+i x}\right) \cot ^{-1}(x)",1,"(-1/8*I)*Log[I - x]^2 + (I/4)*Log[I - x]*Log[-((I - x)/x)] + (I/4)*Log[I - x]*Log[(-1/2*I)*(I + x)] - (I/4)*Log[(-1/2*I)*(I - x)]*Log[I + x] + (I/4)*Log[-((I - x)/x)]*Log[I + x] + (I/8)*Log[I + x]^2 - (I/4)*Log[I - x]*Log[(I + x)/x] - (I/4)*Log[I + x]*Log[(I + x)/x] + (I/4)*PolyLog[2, (-1/2*I)*(I - x)] - (I/4)*PolyLog[2, (-1/2*I)*(I + x)]","B",0
41,1,8,8,0.0030653,"\int \frac{\cot ^{-1}(x)}{1+x^2} \, dx","Integrate[ArcCot[x]/(1 + x^2),x]","-\frac{1}{2} \cot ^{-1}(x)^2","-\frac{1}{2} \cot ^{-1}(x)^2",1,"-1/2*ArcCot[x]^2","A",1
42,1,251,49,0.0610745,"\int \frac{\cot ^{-1}(x)}{x \left(1+x^2\right)} \, dx","Integrate[ArcCot[x]/(x*(1 + x^2)),x]","-\frac{1}{4} i \text{Li}_2\left(-\frac{1}{2} i (i-x)\right)-\frac{1}{2} i \text{Li}_2\left(-\frac{i}{x}\right)+\frac{1}{2} i \text{Li}_2\left(\frac{i}{x}\right)+\frac{1}{4} i \text{Li}_2\left(-\frac{1}{2} i (x+i)\right)+\frac{1}{8} i \log ^2(-x+i)-\frac{1}{8} i \log ^2(x+i)-\frac{1}{4} i \log \left(-\frac{-x+i}{x}\right) \log (-x+i)-\frac{1}{4} i \log \left(-\frac{1}{2} i (x+i)\right) \log (-x+i)+\frac{1}{4} i \log \left(\frac{x+i}{x}\right) \log (-x+i)+\frac{1}{4} i \log \left(-\frac{1}{2} i (-x+i)\right) \log (x+i)-\frac{1}{4} i \log \left(-\frac{-x+i}{x}\right) \log (x+i)+\frac{1}{4} i \log (x+i) \log \left(\frac{x+i}{x}\right)","\frac{1}{2} i \text{Li}_2\left(\frac{2}{1-i x}-1\right)+\frac{1}{2} i \cot ^{-1}(x)^2+\log \left(2-\frac{2}{1-i x}\right) \cot ^{-1}(x)",1,"(I/8)*Log[I - x]^2 - (I/4)*Log[I - x]*Log[-((I - x)/x)] - (I/4)*Log[I - x]*Log[(-1/2*I)*(I + x)] + (I/4)*Log[(-1/2*I)*(I - x)]*Log[I + x] - (I/4)*Log[-((I - x)/x)]*Log[I + x] - (I/8)*Log[I + x]^2 + (I/4)*Log[I - x]*Log[(I + x)/x] + (I/4)*Log[I + x]*Log[(I + x)/x] - (I/4)*PolyLog[2, (-1/2*I)*(I - x)] - (I/2)*PolyLog[2, (-I)/x] + (I/2)*PolyLog[2, I/x] + (I/4)*PolyLog[2, (-1/2*I)*(I + x)]","B",0
43,1,30,30,0.0148405,"\int \frac{\cot ^{-1}(x)}{x^2 \left(1+x^2\right)} \, dx","Integrate[ArcCot[x]/(x^2*(1 + x^2)),x]","\frac{1}{2} \log \left(x^2+1\right)-\log (x)+\frac{1}{2} \cot ^{-1}(x)^2-\frac{\cot ^{-1}(x)}{x}","\frac{1}{2} \log \left(x^2+1\right)-\log (x)+\frac{1}{2} \cot ^{-1}(x)^2-\frac{\cot ^{-1}(x)}{x}",1,"-(ArcCot[x]/x) + ArcCot[x]^2/2 - Log[x] + Log[1 + x^2]/2","A",1
44,1,280,72,0.0665495,"\int \frac{\cot ^{-1}(x)}{x^3 \left(1+x^2\right)} \, dx","Integrate[ArcCot[x]/(x^3*(1 + x^2)),x]","\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-x^2\right)}{2 x}+\frac{1}{4} i \text{Li}_2\left(-\frac{1}{2} i (i-x)\right)+\frac{1}{2} i \text{Li}_2\left(-\frac{i}{x}\right)-\frac{1}{2} i \text{Li}_2\left(\frac{i}{x}\right)-\frac{1}{4} i \text{Li}_2\left(-\frac{1}{2} i (x+i)\right)-\frac{\cot ^{-1}(x)}{2 x^2}-\frac{1}{8} i \log ^2(-x+i)+\frac{1}{8} i \log ^2(x+i)+\frac{1}{4} i \log \left(-\frac{-x+i}{x}\right) \log (-x+i)+\frac{1}{4} i \log \left(-\frac{1}{2} i (x+i)\right) \log (-x+i)-\frac{1}{4} i \log \left(\frac{x+i}{x}\right) \log (-x+i)-\frac{1}{4} i \log \left(-\frac{1}{2} i (-x+i)\right) \log (x+i)+\frac{1}{4} i \log \left(-\frac{-x+i}{x}\right) \log (x+i)-\frac{1}{4} i \log (x+i) \log \left(\frac{x+i}{x}\right)","-\frac{1}{2} i \text{Li}_2\left(\frac{2}{1-i x}-1\right)-\frac{\cot ^{-1}(x)}{2 x^2}+\frac{1}{2 x}+\frac{1}{2} \tan ^{-1}(x)-\frac{1}{2} i \cot ^{-1}(x)^2-\log \left(2-\frac{2}{1-i x}\right) \cot ^{-1}(x)",1,"-1/2*ArcCot[x]/x^2 + Hypergeometric2F1[-1/2, 1, 1/2, -x^2]/(2*x) - (I/8)*Log[I - x]^2 + (I/4)*Log[I - x]*Log[-((I - x)/x)] + (I/4)*Log[I - x]*Log[(-1/2*I)*(I + x)] - (I/4)*Log[(-1/2*I)*(I - x)]*Log[I + x] + (I/4)*Log[-((I - x)/x)]*Log[I + x] + (I/8)*Log[I + x]^2 - (I/4)*Log[I - x]*Log[(I + x)/x] - (I/4)*Log[I + x]*Log[(I + x)/x] + (I/4)*PolyLog[2, (-1/2*I)*(I - x)] + (I/2)*PolyLog[2, (-I)/x] - (I/2)*PolyLog[2, I/x] - (I/4)*PolyLog[2, (-1/2*I)*(I + x)]","C",0
45,1,47,47,0.0212355,"\int \frac{\cot ^{-1}(x)}{x^4 \left(1+x^2\right)} \, dx","Integrate[ArcCot[x]/(x^4*(1 + x^2)),x]","-\frac{\cot ^{-1}(x)}{3 x^3}+\frac{1}{6 x^2}-\frac{2}{3} \log \left(x^2+1\right)+\frac{4 \log (x)}{3}-\frac{1}{2} \cot ^{-1}(x)^2+\frac{\cot ^{-1}(x)}{x}","-\frac{\cot ^{-1}(x)}{3 x^3}+\frac{1}{6 x^2}-\frac{2}{3} \log \left(x^2+1\right)+\frac{4 \log (x)}{3}-\frac{1}{2} \cot ^{-1}(x)^2+\frac{\cot ^{-1}(x)}{x}",1,"1/(6*x^2) - ArcCot[x]/(3*x^3) + ArcCot[x]/x - ArcCot[x]^2/2 + (4*Log[x])/3 - (2*Log[1 + x^2])/3","A",1
46,1,626,206,1.7662445,"\int \frac{x^2 \cot ^{-1}(c x)}{1+x^2} \, dx","Integrate[(x^2*ArcCot[c*x])/(1 + x^2),x]","\frac{\frac{1}{4} \sqrt{-c^2} \left(i \left(\text{Li}_2\left(\frac{\left(c^2+2 i \sqrt{-c^2}+1\right) \left(c x+\sqrt{-c^2}\right)}{\left(c^2-1\right) \left(\sqrt{-c^2}-c x\right)}\right)-\text{Li}_2\left(\frac{\left(c^2-2 i \sqrt{-c^2}+1\right) \left(c x+\sqrt{-c^2}\right)}{\left(c^2-1\right) \left(\sqrt{-c^2}-c x\right)}\right)\right)+2 \cos ^{-1}\left(\frac{c^2+1}{c^2-1}\right) \tanh ^{-1}\left(\frac{\sqrt{-c^2}}{c x}\right)-4 \cot ^{-1}(c x) \tanh ^{-1}\left(\frac{c x}{\sqrt{-c^2}}\right)-\log \left(-\frac{2 \left(c^2+i \sqrt{-c^2}\right) (c x-i)}{\left(c^2-1\right) \left(\sqrt{-c^2}-c x\right)}\right) \left(\cos ^{-1}\left(\frac{c^2+1}{c^2-1}\right)-2 i \tanh ^{-1}\left(\frac{\sqrt{-c^2}}{c x}\right)\right)-\log \left(\frac{2 i \left(\sqrt{-c^2}+i c^2\right) (c x+i)}{\left(c^2-1\right) \left(\sqrt{-c^2}-c x\right)}\right) \left(\cos ^{-1}\left(\frac{c^2+1}{c^2-1}\right)+2 i \tanh ^{-1}\left(\frac{\sqrt{-c^2}}{c x}\right)\right)+\left(-2 i \tanh ^{-1}\left(\frac{\sqrt{-c^2}}{c x}\right)+2 i \tanh ^{-1}\left(\frac{c x}{\sqrt{-c^2}}\right)+\cos ^{-1}\left(\frac{c^2+1}{c^2-1}\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2} e^{-i \cot ^{-1}(c x)}}{\sqrt{c^2-1} \sqrt{\left(c^2-1\right) \cos \left(2 \cot ^{-1}(c x)\right)-c^2-1}}\right)+\left(2 i \tanh ^{-1}\left(\frac{\sqrt{-c^2}}{c x}\right)-2 i \tanh ^{-1}\left(\frac{c x}{\sqrt{-c^2}}\right)+\cos ^{-1}\left(\frac{c^2+1}{c^2-1}\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2} e^{i \cot ^{-1}(c x)}}{\sqrt{c^2-1} \sqrt{\left(c^2-1\right) \cos \left(2 \cot ^{-1}(c x)\right)-c^2-1}}\right)\right)-\log \left(\frac{1}{c x \sqrt{\frac{1}{c^2 x^2}+1}}\right)+c x \cot ^{-1}(c x)}{c}","\frac{\log \left(c^2 x^2+1\right)}{2 c}+\frac{1}{4} \text{Li}_2\left(\frac{2 i (i-c x)}{(1-c) (1-i x)}+1\right)-\frac{1}{4} \text{Li}_2\left(\frac{2 i (c x+i)}{(c+1) (1-i x)}+1\right)+x \cot ^{-1}(c x)-\frac{1}{2} i \tan ^{-1}(x) \log \left(1-\frac{i}{c x}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(1+\frac{i}{c x}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)-\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)",1,"(c*x*ArcCot[c*x] - Log[1/(c*Sqrt[1 + 1/(c^2*x^2)]*x)] + (Sqrt[-c^2]*(2*ArcCos[(1 + c^2)/(-1 + c^2)]*ArcTanh[Sqrt[-c^2]/(c*x)] - 4*ArcCot[c*x]*ArcTanh[(c*x)/Sqrt[-c^2]] - (ArcCos[(1 + c^2)/(-1 + c^2)] - (2*I)*ArcTanh[Sqrt[-c^2]/(c*x)])*Log[(-2*(c^2 + I*Sqrt[-c^2])*(-I + c*x))/((-1 + c^2)*(Sqrt[-c^2] - c*x))] - (ArcCos[(1 + c^2)/(-1 + c^2)] + (2*I)*ArcTanh[Sqrt[-c^2]/(c*x)])*Log[((2*I)*(I*c^2 + Sqrt[-c^2])*(I + c*x))/((-1 + c^2)*(Sqrt[-c^2] - c*x))] + (ArcCos[(1 + c^2)/(-1 + c^2)] - (2*I)*ArcTanh[Sqrt[-c^2]/(c*x)] + (2*I)*ArcTanh[(c*x)/Sqrt[-c^2]])*Log[(Sqrt[2]*Sqrt[-c^2])/(Sqrt[-1 + c^2]*E^(I*ArcCot[c*x])*Sqrt[-1 - c^2 + (-1 + c^2)*Cos[2*ArcCot[c*x]]])] + (ArcCos[(1 + c^2)/(-1 + c^2)] + (2*I)*ArcTanh[Sqrt[-c^2]/(c*x)] - (2*I)*ArcTanh[(c*x)/Sqrt[-c^2]])*Log[(Sqrt[2]*Sqrt[-c^2]*E^(I*ArcCot[c*x]))/(Sqrt[-1 + c^2]*Sqrt[-1 - c^2 + (-1 + c^2)*Cos[2*ArcCot[c*x]]])] + I*(-PolyLog[2, ((1 + c^2 - (2*I)*Sqrt[-c^2])*(Sqrt[-c^2] + c*x))/((-1 + c^2)*(Sqrt[-c^2] - c*x))] + PolyLog[2, ((1 + c^2 + (2*I)*Sqrt[-c^2])*(Sqrt[-c^2] + c*x))/((-1 + c^2)*(Sqrt[-c^2] - c*x))])))/4)/c","B",1
47,1,343,188,0.0918885,"\int \frac{x \cot ^{-1}(c x)}{1+x^2} \, dx","Integrate[(x*ArcCot[c*x])/(1 + x^2),x]","-\frac{1}{4} i \text{Li}_2\left(\frac{i c (i-x)}{1-c}\right)+\frac{1}{4} i \text{Li}_2\left(-\frac{i c (i-x)}{c+1}\right)+\frac{1}{4} i \text{Li}_2\left(\frac{i c (x+i)}{1-c}\right)-\frac{1}{4} i \text{Li}_2\left(-\frac{i c (x+i)}{c+1}\right)-\frac{1}{4} i \log (-x+i) \log \left(-\frac{i (-c x+i)}{1-c}\right)-\frac{1}{4} i \log (x+i) \log \left(-\frac{i (-c x+i)}{c+1}\right)+\frac{1}{4} i \log (-x+i) \log \left(-\frac{-c x+i}{c x}\right)+\frac{1}{4} i \log (x+i) \log \left(-\frac{-c x+i}{c x}\right)+\frac{1}{4} i \log (x+i) \log \left(-\frac{i (c x+i)}{1-c}\right)+\frac{1}{4} i \log (-x+i) \log \left(-\frac{i (c x+i)}{c+1}\right)-\frac{1}{4} i \log (-x+i) \log \left(\frac{c x+i}{c x}\right)-\frac{1}{4} i \log (x+i) \log \left(\frac{c x+i}{c x}\right)","-\frac{1}{2} i \text{Li}_2\left(1-\frac{2}{1-i c x}\right)+\frac{1}{4} i \text{Li}_2\left(1-\frac{2 i c (i-x)}{(1-c) (1-i c x)}\right)+\frac{1}{4} i \text{Li}_2\left(\frac{2 i c (x+i)}{(c+1) (1-i c x)}+1\right)+\log \left(\frac{2}{1-i c x}\right) \left(-\cot ^{-1}(c x)\right)+\frac{1}{2} \log \left(\frac{2 i c (-x+i)}{(1-c) (1-i c x)}\right) \cot ^{-1}(c x)+\frac{1}{2} \log \left(-\frac{2 i c (x+i)}{(c+1) (1-i c x)}\right) \cot ^{-1}(c x)",1,"(-1/4*I)*Log[I - x]*Log[((-I)*(I - c*x))/(1 - c)] - (I/4)*Log[I + x]*Log[((-I)*(I - c*x))/(1 + c)] + (I/4)*Log[I - x]*Log[-((I - c*x)/(c*x))] + (I/4)*Log[I + x]*Log[-((I - c*x)/(c*x))] + (I/4)*Log[I + x]*Log[((-I)*(I + c*x))/(1 - c)] + (I/4)*Log[I - x]*Log[((-I)*(I + c*x))/(1 + c)] - (I/4)*Log[I - x]*Log[(I + c*x)/(c*x)] - (I/4)*Log[I + x]*Log[(I + c*x)/(c*x)] - (I/4)*PolyLog[2, (I*c*(I - x))/(1 - c)] + (I/4)*PolyLog[2, ((-I)*c*(I - x))/(1 + c)] + (I/4)*PolyLog[2, (I*c*(I + x))/(1 - c)] - (I/4)*PolyLog[2, ((-I)*c*(I + x))/(1 + c)]","A",0
48,1,319,183,0.0888242,"\int \frac{\cot ^{-1}(c x)}{1+x^2} \, dx","Integrate[ArcCot[c*x]/(1 + x^2),x]","-\frac{1}{4} \text{Li}_2\left(\frac{i c (i-x)}{1-c}\right)+\frac{1}{4} \text{Li}_2\left(-\frac{i c (i-x)}{c+1}\right)-\frac{1}{4} \text{Li}_2\left(\frac{i c (x+i)}{1-c}\right)+\frac{1}{4} \text{Li}_2\left(-\frac{i c (x+i)}{c+1}\right)-\frac{1}{4} \log (-x+i) \log \left(-\frac{i (-c x+i)}{1-c}\right)+\frac{1}{4} \log (x+i) \log \left(-\frac{i (-c x+i)}{c+1}\right)+\frac{1}{4} \log (-x+i) \log \left(-\frac{-c x+i}{c x}\right)-\frac{1}{4} \log (x+i) \log \left(-\frac{-c x+i}{c x}\right)-\frac{1}{4} \log (x+i) \log \left(-\frac{i (c x+i)}{1-c}\right)+\frac{1}{4} \log (-x+i) \log \left(-\frac{i (c x+i)}{c+1}\right)-\frac{1}{4} \log (-x+i) \log \left(\frac{c x+i}{c x}\right)+\frac{1}{4} \log (x+i) \log \left(\frac{c x+i}{c x}\right)","-\frac{1}{4} \text{Li}_2\left(\frac{2 i (i-c x)}{(1-c) (1-i x)}+1\right)+\frac{1}{4} \text{Li}_2\left(\frac{2 i (c x+i)}{(c+1) (1-i x)}+1\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(1-\frac{i}{c x}\right)-\frac{1}{2} i \tan ^{-1}(x) \log \left(1+\frac{i}{c x}\right)-\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)",1,"-1/4*(Log[I - x]*Log[((-I)*(I - c*x))/(1 - c)]) + (Log[I + x]*Log[((-I)*(I - c*x))/(1 + c)])/4 + (Log[I - x]*Log[-((I - c*x)/(c*x))])/4 - (Log[I + x]*Log[-((I - c*x)/(c*x))])/4 - (Log[I + x]*Log[((-I)*(I + c*x))/(1 - c)])/4 + (Log[I - x]*Log[((-I)*(I + c*x))/(1 + c)])/4 - (Log[I - x]*Log[(I + c*x)/(c*x)])/4 + (Log[I + x]*Log[(I + c*x)/(c*x)])/4 - PolyLog[2, (I*c*(I - x))/(1 - c)]/4 + PolyLog[2, ((-I)*c*(I - x))/(1 + c)]/4 - PolyLog[2, (I*c*(I + x))/(1 - c)]/4 + PolyLog[2, ((-I)*c*(I + x))/(1 + c)]/4","A",0
49,1,379,223,0.1034086,"\int \frac{\cot ^{-1}(c x)}{x \left(1+x^2\right)} \, dx","Integrate[ArcCot[c*x]/(x*(1 + x^2)),x]","\frac{1}{4} i \text{Li}_2\left(\frac{i c (i-x)}{1-c}\right)-\frac{1}{4} i \text{Li}_2\left(-\frac{i c (i-x)}{c+1}\right)-\frac{1}{2} i \text{Li}_2\left(-\frac{i}{c x}\right)+\frac{1}{2} i \text{Li}_2\left(\frac{i}{c x}\right)-\frac{1}{4} i \text{Li}_2\left(\frac{i c (x+i)}{1-c}\right)+\frac{1}{4} i \text{Li}_2\left(-\frac{i c (x+i)}{c+1}\right)+\frac{1}{4} i \log (-x+i) \log \left(-\frac{i (-c x+i)}{1-c}\right)+\frac{1}{4} i \log (x+i) \log \left(-\frac{i (-c x+i)}{c+1}\right)-\frac{1}{4} i \log (-x+i) \log \left(-\frac{-c x+i}{c x}\right)-\frac{1}{4} i \log (x+i) \log \left(-\frac{-c x+i}{c x}\right)-\frac{1}{4} i \log (x+i) \log \left(-\frac{i (c x+i)}{1-c}\right)-\frac{1}{4} i \log (-x+i) \log \left(-\frac{i (c x+i)}{c+1}\right)+\frac{1}{4} i \log (-x+i) \log \left(\frac{c x+i}{c x}\right)+\frac{1}{4} i \log (x+i) \log \left(\frac{c x+i}{c x}\right)","-\frac{1}{2} i \text{Li}_2\left(-\frac{i}{c x}\right)+\frac{1}{2} i \text{Li}_2\left(\frac{i}{c x}\right)+\frac{1}{2} i \text{Li}_2\left(1-\frac{2}{1-i c x}\right)-\frac{1}{4} i \text{Li}_2\left(1-\frac{2 i c (i-x)}{(1-c) (1-i c x)}\right)-\frac{1}{4} i \text{Li}_2\left(\frac{2 i c (x+i)}{(c+1) (1-i c x)}+1\right)+\log \left(\frac{2}{1-i c x}\right) \cot ^{-1}(c x)-\frac{1}{2} \log \left(\frac{2 i c (-x+i)}{(1-c) (1-i c x)}\right) \cot ^{-1}(c x)-\frac{1}{2} \log \left(-\frac{2 i c (x+i)}{(c+1) (1-i c x)}\right) \cot ^{-1}(c x)",1,"(I/4)*Log[I - x]*Log[((-I)*(I - c*x))/(1 - c)] + (I/4)*Log[I + x]*Log[((-I)*(I - c*x))/(1 + c)] - (I/4)*Log[I - x]*Log[-((I - c*x)/(c*x))] - (I/4)*Log[I + x]*Log[-((I - c*x)/(c*x))] - (I/4)*Log[I + x]*Log[((-I)*(I + c*x))/(1 - c)] - (I/4)*Log[I - x]*Log[((-I)*(I + c*x))/(1 + c)] + (I/4)*Log[I - x]*Log[(I + c*x)/(c*x)] + (I/4)*Log[I + x]*Log[(I + c*x)/(c*x)] + (I/4)*PolyLog[2, (I*c*(I - x))/(1 - c)] - (I/4)*PolyLog[2, ((-I)*c*(I - x))/(1 + c)] - (I/2)*PolyLog[2, (-I)/(c*x)] + (I/2)*PolyLog[2, I/(c*x)] - (I/4)*PolyLog[2, (I*c*(I + x))/(1 - c)] + (I/4)*PolyLog[2, ((-I)*c*(I + x))/(1 + c)]","A",0
50,1,348,212,0.1043372,"\int \frac{\cot ^{-1}(c x)}{x^2 \left(1+x^2\right)} \, dx","Integrate[ArcCot[c*x]/(x^2*(1 + x^2)),x]","\frac{1}{2} c \log \left(c^2 x^2+1\right)+\frac{1}{4} \text{Li}_2\left(\frac{i c (i-x)}{1-c}\right)-\frac{1}{4} \text{Li}_2\left(-\frac{i c (i-x)}{c+1}\right)+\frac{1}{4} \text{Li}_2\left(\frac{i c (x+i)}{1-c}\right)-\frac{1}{4} \text{Li}_2\left(-\frac{i c (x+i)}{c+1}\right)-c \log (x)+\frac{1}{4} \log (-x+i) \log \left(-\frac{i (-c x+i)}{1-c}\right)-\frac{1}{4} \log (x+i) \log \left(-\frac{i (-c x+i)}{c+1}\right)-\frac{1}{4} \log (-x+i) \log \left(-\frac{-c x+i}{c x}\right)+\frac{1}{4} \log (x+i) \log \left(-\frac{-c x+i}{c x}\right)+\frac{1}{4} \log (x+i) \log \left(-\frac{i (c x+i)}{1-c}\right)-\frac{1}{4} \log (-x+i) \log \left(-\frac{i (c x+i)}{c+1}\right)+\frac{1}{4} \log (-x+i) \log \left(\frac{c x+i}{c x}\right)-\frac{1}{4} \log (x+i) \log \left(\frac{c x+i}{c x}\right)-\frac{\cot ^{-1}(c x)}{x}","\frac{1}{2} c \log \left(c^2 x^2+1\right)+\frac{1}{4} \text{Li}_2\left(\frac{2 i (i-c x)}{(1-c) (1-i x)}+1\right)-\frac{1}{4} \text{Li}_2\left(\frac{2 i (c x+i)}{(c+1) (1-i x)}+1\right)-c \log (x)-\frac{\cot ^{-1}(c x)}{x}-\frac{1}{2} i \tan ^{-1}(x) \log \left(1-\frac{i}{c x}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(1+\frac{i}{c x}\right)+\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right)-\frac{1}{2} i \tan ^{-1}(x) \log \left(-\frac{2 i (c x+i)}{(c+1) (1-i x)}\right)",1,"-(ArcCot[c*x]/x) - c*Log[x] + (Log[I - x]*Log[((-I)*(I - c*x))/(1 - c)])/4 - (Log[I + x]*Log[((-I)*(I - c*x))/(1 + c)])/4 - (Log[I - x]*Log[-((I - c*x)/(c*x))])/4 + (Log[I + x]*Log[-((I - c*x)/(c*x))])/4 + (Log[I + x]*Log[((-I)*(I + c*x))/(1 - c)])/4 - (Log[I - x]*Log[((-I)*(I + c*x))/(1 + c)])/4 + (Log[I - x]*Log[(I + c*x)/(c*x)])/4 - (Log[I + x]*Log[(I + c*x)/(c*x)])/4 + (c*Log[1 + c^2*x^2])/2 + PolyLog[2, (I*c*(I - x))/(1 - c)]/4 - PolyLog[2, ((-I)*c*(I - x))/(1 + c)]/4 + PolyLog[2, (I*c*(I + x))/(1 - c)]/4 - PolyLog[2, ((-I)*c*(I + x))/(1 + c)]/4","A",0
51,1,5,5,0.0250934,"\int \frac{1}{\left(1+x^2\right) \cot ^{-1}(x)} \, dx","Integrate[1/((1 + x^2)*ArcCot[x]),x]","-\log \left(\cot ^{-1}(x)\right)","-\log \left(\cot ^{-1}(x)\right)",1,"-Log[ArcCot[x]]","A",1
52,1,13,13,0.0083683,"\int \frac{\cot ^{-1}(x)^n}{1+x^2} \, dx","Integrate[ArcCot[x]^n/(1 + x^2),x]","-\frac{\cot ^{-1}(x)^{n+1}}{n+1}","-\frac{\cot ^{-1}(x)^{n+1}}{n+1}",1,"-(ArcCot[x]^(1 + n)/(1 + n))","A",1
53,1,212,244,0.189911,"\int \left(c+d x^2\right)^4 \cot ^{-1}(a x) \, dx","Integrate[(c + d*x^2)^4*ArcCot[a*x],x]","\frac{24 a^9 x \cot ^{-1}(a x) \left(315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right)+a^2 d x^2 \left(3 a^6 \left(1680 c^3+756 c^2 d x^2+240 c d^2 x^4+35 d^3 x^6\right)-4 a^4 d \left(1134 c^2+270 c d x^2+35 d^2 x^4\right)+30 a^2 d^2 \left(72 c+7 d x^2\right)-420 d^3\right)+12 \left(315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right) \log \left(a^2 x^2+1\right)}{7560 a^9}","\frac{d^3 x^6 \left(36 a^2 c-7 d\right)}{378 a^3}+\frac{d^2 x^4 \left(378 a^4 c^2-180 a^2 c d+35 d^2\right)}{1260 a^5}+\frac{d x^2 \left(420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right)}{630 a^7}+\frac{\left(315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right) \log \left(a^2 x^2+1\right)}{630 a^9}+c^4 x \cot ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cot ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cot ^{-1}(a x)+\frac{d^4 x^8}{72 a}",1,"(a^2*d*x^2*(-420*d^3 + 30*a^2*d^2*(72*c + 7*d*x^2) - 4*a^4*d*(1134*c^2 + 270*c*d*x^2 + 35*d^2*x^4) + 3*a^6*(1680*c^3 + 756*c^2*d*x^2 + 240*c*d^2*x^4 + 35*d^3*x^6)) + 24*a^9*x*(315*c^4 + 420*c^3*d*x^2 + 378*c^2*d^2*x^4 + 180*c*d^3*x^6 + 35*d^4*x^8)*ArcCot[a*x] + 12*(315*a^8*c^4 - 420*a^6*c^3*d + 378*a^4*c^2*d^2 - 180*a^2*c*d^3 + 35*d^4)*Log[1 + a^2*x^2])/(7560*a^9)","A",1
54,1,149,168,0.1177248,"\int \left(c+d x^2\right)^3 \cot ^{-1}(a x) \, dx","Integrate[(c + d*x^2)^3*ArcCot[a*x],x]","\frac{12 a^7 x \cot ^{-1}(a x) \left(35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right)+a^2 d x^2 \left(a^4 \left(210 c^2+63 c d x^2+10 d^2 x^4\right)-3 a^2 d \left(42 c+5 d x^2\right)+30 d^2\right)+6 \left(35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right) \log \left(a^2 x^2+1\right)}{420 a^7}","\frac{d^2 x^4 \left(21 a^2 c-5 d\right)}{140 a^3}+\frac{d x^2 \left(35 a^4 c^2-21 a^2 c d+5 d^2\right)}{70 a^5}+\frac{\left(35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right) \log \left(a^2 x^2+1\right)}{70 a^7}+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x)+\frac{d^3 x^6}{42 a}",1,"(a^2*d*x^2*(30*d^2 - 3*a^2*d*(42*c + 5*d*x^2) + a^4*(210*c^2 + 63*c*d*x^2 + 10*d^2*x^4)) + 12*a^7*x*(35*c^3 + 35*c^2*d*x^2 + 21*c*d^2*x^4 + 5*d^3*x^6)*ArcCot[a*x] + 6*(35*a^6*c^3 - 35*a^4*c^2*d + 21*a^2*c*d^2 - 5*d^3)*Log[1 + a^2*x^2])/(420*a^7)","A",1
55,1,97,109,0.076137,"\int \left(c+d x^2\right)^2 \cot ^{-1}(a x) \, dx","Integrate[(c + d*x^2)^2*ArcCot[a*x],x]","\frac{4 a^5 x \cot ^{-1}(a x) \left(15 c^2+10 c d x^2+3 d^2 x^4\right)+a^2 d x^2 \left(a^2 \left(20 c+3 d x^2\right)-6 d\right)+\left(30 a^4 c^2-20 a^2 c d+6 d^2\right) \log \left(a^2 x^2+1\right)}{60 a^5}","\frac{d x^2 \left(10 a^2 c-3 d\right)}{30 a^3}+\frac{\left(15 a^4 c^2-10 a^2 c d+3 d^2\right) \log \left(a^2 x^2+1\right)}{30 a^5}+c^2 x \cot ^{-1}(a x)+\frac{2}{3} c d x^3 \cot ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cot ^{-1}(a x)+\frac{d^2 x^4}{20 a}",1,"(a^2*d*x^2*(-6*d + a^2*(20*c + 3*d*x^2)) + 4*a^5*x*(15*c^2 + 10*c*d*x^2 + 3*d^2*x^4)*ArcCot[a*x] + (30*a^4*c^2 - 20*a^2*c*d + 6*d^2)*Log[1 + a^2*x^2])/(60*a^5)","A",1
56,1,67,58,0.0113122,"\int \left(c+d x^2\right) \cot ^{-1}(a x) \, dx","Integrate[(c + d*x^2)*ArcCot[a*x],x]","\frac{c \log \left(a^2 x^2+1\right)}{2 a}-\frac{d \log \left(a^2 x^2+1\right)}{6 a^3}+c x \cot ^{-1}(a x)+\frac{1}{3} d x^3 \cot ^{-1}(a x)+\frac{d x^2}{6 a}","\frac{\left(3 a^2 c-d\right) \log \left(a^2 x^2+1\right)}{6 a^3}+c x \cot ^{-1}(a x)+\frac{1}{3} d x^3 \cot ^{-1}(a x)+\frac{d x^2}{6 a}",1,"(d*x^2)/(6*a) + c*x*ArcCot[a*x] + (d*x^3*ArcCot[a*x])/3 + (c*Log[1 + a^2*x^2])/(2*a) - (d*Log[1 + a^2*x^2])/(6*a^3)","A",1
57,1,523,403,0.3262219,"\int \frac{\cot ^{-1}(a x)}{c+d x^2} \, dx","Integrate[ArcCot[a*x]/(c + d*x^2),x]","\frac{i \left(-\text{Li}_2\left(\frac{a \left(\sqrt{-c}-\sqrt{d} x\right)}{a \sqrt{-c}-i \sqrt{d}}\right)+\text{Li}_2\left(\frac{a \left(\sqrt{-c}-\sqrt{d} x\right)}{\sqrt{-c} a+i \sqrt{d}}\right)-\text{Li}_2\left(\frac{a \left(\sqrt{d} x+\sqrt{-c}\right)}{a \sqrt{-c}-i \sqrt{d}}\right)+\text{Li}_2\left(\frac{a \left(\sqrt{d} x+\sqrt{-c}\right)}{\sqrt{-c} a+i \sqrt{d}}\right)+\log \left(1-\frac{i}{a x}\right) \log \left(\sqrt{-c}-\sqrt{d} x\right)-\log \left(1+\frac{i}{a x}\right) \log \left(\sqrt{-c}-\sqrt{d} x\right)-\log \left(\sqrt{-c}-\sqrt{d} x\right) \log \left(\frac{\sqrt{d} (a x-i)}{a \sqrt{-c}-i \sqrt{d}}\right)+\log \left(\sqrt{-c}-\sqrt{d} x\right) \log \left(\frac{\sqrt{d} (a x+i)}{a \sqrt{-c}+i \sqrt{d}}\right)-\log \left(1-\frac{i}{a x}\right) \log \left(\sqrt{-c}+\sqrt{d} x\right)+\log \left(1+\frac{i}{a x}\right) \log \left(\sqrt{-c}+\sqrt{d} x\right)+\log \left(\sqrt{-c}+\sqrt{d} x\right) \log \left(\frac{\sqrt{d} (-a x+i)}{a \sqrt{-c}+i \sqrt{d}}\right)-\log \left(\sqrt{-c}+\sqrt{d} x\right) \log \left(-\frac{\sqrt{d} (a x+i)}{a \sqrt{-c}-i \sqrt{d}}\right)\right)}{4 \sqrt{-c} \sqrt{d}}","-\frac{\text{Li}_2\left(1-\frac{2 i \sqrt{c} \sqrt{d} (i-a x)}{\left(a \sqrt{c}-\sqrt{d}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}\right)}{4 \sqrt{c} \sqrt{d}}+\frac{\text{Li}_2\left(\frac{2 i \sqrt{c} \sqrt{d} (a x+i)}{\left(\sqrt{c} a+\sqrt{d}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}+1\right)}{4 \sqrt{c} \sqrt{d}}+\frac{i \log \left(1-\frac{i}{a x}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} \sqrt{d}}-\frac{i \log \left(1+\frac{i}{a x}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} \sqrt{d}}-\frac{i \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right) \log \left(\frac{2 i \sqrt{c} \sqrt{d} (-a x+i)}{\left(a \sqrt{c}-\sqrt{d}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}\right)}{2 \sqrt{c} \sqrt{d}}+\frac{i \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right) \log \left(-\frac{2 i \sqrt{c} \sqrt{d} (a x+i)}{\left(a \sqrt{c}+\sqrt{d}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}\right)}{2 \sqrt{c} \sqrt{d}}",1,"((I/4)*(Log[1 - I/(a*x)]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[1 + I/(a*x)]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[(Sqrt[d]*(-I + a*x))/(a*Sqrt[-c] - I*Sqrt[d])]*Log[Sqrt[-c] - Sqrt[d]*x] + Log[(Sqrt[d]*(I + a*x))/(a*Sqrt[-c] + I*Sqrt[d])]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[1 - I/(a*x)]*Log[Sqrt[-c] + Sqrt[d]*x] + Log[1 + I/(a*x)]*Log[Sqrt[-c] + Sqrt[d]*x] + Log[(Sqrt[d]*(I - a*x))/(a*Sqrt[-c] + I*Sqrt[d])]*Log[Sqrt[-c] + Sqrt[d]*x] - Log[-((Sqrt[d]*(I + a*x))/(a*Sqrt[-c] - I*Sqrt[d]))]*Log[Sqrt[-c] + Sqrt[d]*x] - PolyLog[2, (a*(Sqrt[-c] - Sqrt[d]*x))/(a*Sqrt[-c] - I*Sqrt[d])] + PolyLog[2, (a*(Sqrt[-c] - Sqrt[d]*x))/(a*Sqrt[-c] + I*Sqrt[d])] - PolyLog[2, (a*(Sqrt[-c] + Sqrt[d]*x))/(a*Sqrt[-c] - I*Sqrt[d])] + PolyLog[2, (a*(Sqrt[-c] + Sqrt[d]*x))/(a*Sqrt[-c] + I*Sqrt[d])]))/(Sqrt[-c]*Sqrt[d])","A",0
58,1,806,801,8.2560522,"\int \frac{\cot ^{-1}(a x)}{\left(c+d x^2\right)^2} \, dx","Integrate[ArcCot[a*x]/(c + d*x^2)^2,x]","-\frac{a \left(\frac{2 \log \left(\frac{c a^2+d+\left(d-a^2 c\right) \cos \left(2 \cot ^{-1}(a x)\right)}{c a^2+d}\right)}{a^2 c-d}+\frac{2 \cos ^{-1}\left(\frac{c a^2+d}{a^2 c-d}\right) \tanh ^{-1}\left(\frac{a c}{\sqrt{-a^2 c d} x}\right)+4 \cot ^{-1}(a x) \tanh ^{-1}\left(\frac{a d x}{\sqrt{-a^2 c d}}\right)+\left(\cos ^{-1}\left(\frac{c a^2+d}{a^2 c-d}\right)-2 i \tanh ^{-1}\left(\frac{a c}{\sqrt{-a^2 c d} x}\right)\right) \log \left(\frac{2 i d \left(i c a^2+\sqrt{-a^2 c d}\right) (a x+i)}{\left(a^2 c-d\right) \left(\sqrt{-a^2 c d}-a d x\right)}\right)+\left(\cos ^{-1}\left(\frac{c a^2+d}{a^2 c-d}\right)+2 i \tanh ^{-1}\left(\frac{a c}{\sqrt{-a^2 c d} x}\right)\right) \log \left(\frac{2 d \left(c a^2+i \sqrt{-a^2 c d}\right) (a x-i)}{\left(a^2 c-d\right) \left(a d x-\sqrt{-a^2 c d}\right)}\right)-\left(\cos ^{-1}\left(\frac{c a^2+d}{a^2 c-d}\right)+2 i \tanh ^{-1}\left(\frac{a c}{\sqrt{-a^2 c d} x}\right)+2 i \tanh ^{-1}\left(\frac{a d x}{\sqrt{-a^2 c d}}\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-a^2 c d} e^{-i \cot ^{-1}(a x)}}{\sqrt{a^2 c-d} \sqrt{-c a^2-d+\left(a^2 c-d\right) \cos \left(2 \cot ^{-1}(a x)\right)}}\right)-\left(\cos ^{-1}\left(\frac{c a^2+d}{a^2 c-d}\right)-2 i \tanh ^{-1}\left(\frac{a c}{\sqrt{-a^2 c d} x}\right)-2 i \tanh ^{-1}\left(\frac{a d x}{\sqrt{-a^2 c d}}\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-a^2 c d} e^{i \cot ^{-1}(a x)}}{\sqrt{a^2 c-d} \sqrt{-c a^2-d+\left(a^2 c-d\right) \cos \left(2 \cot ^{-1}(a x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(c a^2+d-2 i \sqrt{-a^2 c d}\right) \left(a d x+\sqrt{-a^2 c d}\right)}{\left(a^2 c-d\right) \left(\sqrt{-a^2 c d}-a d x\right)}\right)-\text{Li}_2\left(\frac{\left(c a^2+d+2 i \sqrt{-a^2 c d}\right) \left(a d x+\sqrt{-a^2 c d}\right)}{\left(a^2 c-d\right) \left(\sqrt{-a^2 c d}-a d x\right)}\right)\right)}{\sqrt{-a^2 c d}}-\frac{4 \cot ^{-1}(a x) \sin \left(2 \cot ^{-1}(a x)\right)}{c a^2+d+\left(d-a^2 c\right) \cos \left(2 \cot ^{-1}(a x)\right)}\right)}{8 c}","\frac{\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right) \cot ^{-1}(a x)}{2 c^{3/2} \sqrt{d}}+\frac{x \cot ^{-1}(a x)}{2 c \left(d x^2+c\right)}-\frac{i a \log \left(\frac{\sqrt{d} \left(1-\sqrt{-a^2} x\right)}{i \sqrt{-a^2} \sqrt{c}+\sqrt{d}}\right) \log \left(1-\frac{i \sqrt{d} x}{\sqrt{c}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{i a \log \left(-\frac{\sqrt{d} \left(\sqrt{-a^2} x+1\right)}{i \sqrt{-a^2} \sqrt{c}-\sqrt{d}}\right) \log \left(1-\frac{i \sqrt{d} x}{\sqrt{c}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{i a \log \left(-\frac{\sqrt{d} \left(1-\sqrt{-a^2} x\right)}{i \sqrt{-a^2} \sqrt{c}-\sqrt{d}}\right) \log \left(\frac{i \sqrt{d} x}{\sqrt{c}}+1\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}-\frac{i a \log \left(\frac{\sqrt{d} \left(\sqrt{-a^2} x+1\right)}{i \sqrt{-a^2} \sqrt{c}+\sqrt{d}}\right) \log \left(\frac{i \sqrt{d} x}{\sqrt{c}}+1\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{a \log \left(a^2 x^2+1\right)}{4 c \left(a^2 c-d\right)}-\frac{a \log \left(d x^2+c\right)}{4 c \left(a^2 c-d\right)}-\frac{i a \text{Li}_2\left(\frac{\sqrt{-a^2} \left(\sqrt{c}-i \sqrt{d} x\right)}{\sqrt{-a^2} \sqrt{c}-i \sqrt{d}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{i a \text{Li}_2\left(\frac{\sqrt{-a^2} \left(\sqrt{c}-i \sqrt{d} x\right)}{\sqrt{-a^2} \sqrt{c}+i \sqrt{d}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}-\frac{i a \text{Li}_2\left(\frac{\sqrt{-a^2} \left(i \sqrt{d} x+\sqrt{c}\right)}{\sqrt{-a^2} \sqrt{c}-i \sqrt{d}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}+\frac{i a \text{Li}_2\left(\frac{\sqrt{-a^2} \left(i \sqrt{d} x+\sqrt{c}\right)}{\sqrt{-a^2} \sqrt{c}+i \sqrt{d}}\right)}{8 \sqrt{-a^2} c^{3/2} \sqrt{d}}",1,"-1/8*(a*((2*Log[(a^2*c + d + (-(a^2*c) + d)*Cos[2*ArcCot[a*x]])/(a^2*c + d)])/(a^2*c - d) + (2*ArcCos[(a^2*c + d)/(a^2*c - d)]*ArcTanh[(a*c)/(Sqrt[-(a^2*c*d)]*x)] + 4*ArcCot[a*x]*ArcTanh[(a*d*x)/Sqrt[-(a^2*c*d)]] + (ArcCos[(a^2*c + d)/(a^2*c - d)] - (2*I)*ArcTanh[(a*c)/(Sqrt[-(a^2*c*d)]*x)])*Log[((2*I)*d*(I*a^2*c + Sqrt[-(a^2*c*d)])*(I + a*x))/((a^2*c - d)*(Sqrt[-(a^2*c*d)] - a*d*x))] + (ArcCos[(a^2*c + d)/(a^2*c - d)] + (2*I)*ArcTanh[(a*c)/(Sqrt[-(a^2*c*d)]*x)])*Log[(2*d*(a^2*c + I*Sqrt[-(a^2*c*d)])*(-I + a*x))/((a^2*c - d)*(-Sqrt[-(a^2*c*d)] + a*d*x))] - (ArcCos[(a^2*c + d)/(a^2*c - d)] + (2*I)*ArcTanh[(a*c)/(Sqrt[-(a^2*c*d)]*x)] + (2*I)*ArcTanh[(a*d*x)/Sqrt[-(a^2*c*d)]])*Log[(Sqrt[2]*Sqrt[-(a^2*c*d)])/(Sqrt[a^2*c - d]*E^(I*ArcCot[a*x])*Sqrt[-(a^2*c) - d + (a^2*c - d)*Cos[2*ArcCot[a*x]]])] - (ArcCos[(a^2*c + d)/(a^2*c - d)] - (2*I)*ArcTanh[(a*c)/(Sqrt[-(a^2*c*d)]*x)] - (2*I)*ArcTanh[(a*d*x)/Sqrt[-(a^2*c*d)]])*Log[(Sqrt[2]*Sqrt[-(a^2*c*d)]*E^(I*ArcCot[a*x]))/(Sqrt[a^2*c - d]*Sqrt[-(a^2*c) - d + (a^2*c - d)*Cos[2*ArcCot[a*x]]])] + I*(PolyLog[2, ((a^2*c + d - (2*I)*Sqrt[-(a^2*c*d)])*(Sqrt[-(a^2*c*d)] + a*d*x))/((a^2*c - d)*(Sqrt[-(a^2*c*d)] - a*d*x))] - PolyLog[2, ((a^2*c + d + (2*I)*Sqrt[-(a^2*c*d)])*(Sqrt[-(a^2*c*d)] + a*d*x))/((a^2*c - d)*(Sqrt[-(a^2*c*d)] - a*d*x))]))/Sqrt[-(a^2*c*d)] - (4*ArcCot[a*x]*Sin[2*ArcCot[a*x]])/(a^2*c + d + (-(a^2*c) + d)*Cos[2*ArcCot[a*x]])))/c","A",0
59,0,0,19,6.843784,"\int \sqrt{c+d x^2} \cot ^{-1}(a x) \, dx","Integrate[Sqrt[c + d*x^2]*ArcCot[a*x],x]","\int \sqrt{c+d x^2} \cot ^{-1}(a x) \, dx","\text{Int}\left(\cot ^{-1}(a x) \sqrt{c+d x^2},x\right)",0,"Integrate[Sqrt[c + d*x^2]*ArcCot[a*x], x]","A",-1
60,0,0,19,4.234085,"\int \frac{\cot ^{-1}(a x)}{\sqrt{c+d x^2}} \, dx","Integrate[ArcCot[a*x]/Sqrt[c + d*x^2],x]","\int \frac{\cot ^{-1}(a x)}{\sqrt{c+d x^2}} \, dx","\text{Int}\left(\frac{\cot ^{-1}(a x)}{\sqrt{c+d x^2}},x\right)",0,"Integrate[ArcCot[a*x]/Sqrt[c + d*x^2], x]","A",-1
61,1,169,66,0.2659357,"\int \frac{\cot ^{-1}(a x)}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[ArcCot[a*x]/(c + d*x^2)^(3/2),x]","\frac{\frac{2 x \cot ^{-1}(a x)}{\sqrt{c+d x^2}}+\frac{-\log \left(\frac{4 a c \left(\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c-i d x\right)}{(a x+i) \sqrt{a^2 c-d}}\right)-\log \left(\frac{4 a c \left(\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c+i d x\right)}{(a x-i) \sqrt{a^2 c-d}}\right)}{\sqrt{a^2 c-d}}}{2 c}","\frac{x \cot ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{\tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{c \sqrt{a^2 c-d}}",1,"((2*x*ArcCot[a*x])/Sqrt[c + d*x^2] + (-Log[(4*a*c*(a*c - I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/(Sqrt[a^2*c - d]*(I + a*x))] - Log[(4*a*c*(a*c + I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/(Sqrt[a^2*c - d]*(-I + a*x))])/Sqrt[a^2*c - d])/(2*c)","C",1
62,1,262,134,0.7314059,"\int \frac{\cot ^{-1}(a x)}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[ArcCot[a*x]/(c + d*x^2)^(5/2),x]","-\frac{\frac{\left(3 a^2 c-2 d\right) \log \left(\frac{12 a c^2 \sqrt{a^2 c-d} \left(\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c-i d x\right)}{(a x+i) \left(3 a^2 c-2 d\right)}\right)}{\left(a^2 c-d\right)^{3/2}}+\frac{\left(3 a^2 c-2 d\right) \log \left(\frac{12 a c^2 \sqrt{a^2 c-d} \left(\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c+i d x\right)}{(a x-i) \left(3 a^2 c-2 d\right)}\right)}{\left(a^2 c-d\right)^{3/2}}-\frac{2 a c}{\left(a^2 c-d\right) \sqrt{c+d x^2}}-\frac{2 x \cot ^{-1}(a x) \left(3 c+2 d x^2\right)}{\left(c+d x^2\right)^{3/2}}}{6 c^2}","-\frac{\left(3 a^2 c-2 d\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{3 c^2 \left(a^2 c-d\right)^{3/2}}+\frac{a}{3 c \left(a^2 c-d\right) \sqrt{c+d x^2}}+\frac{2 x \cot ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{x \cot ^{-1}(a x)}{3 c \left(c+d x^2\right)^{3/2}}",1,"-1/6*((-2*a*c)/((a^2*c - d)*Sqrt[c + d*x^2]) - (2*x*(3*c + 2*d*x^2)*ArcCot[a*x])/(c + d*x^2)^(3/2) + ((3*a^2*c - 2*d)*Log[(12*a*c^2*Sqrt[a^2*c - d]*(a*c - I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/((3*a^2*c - 2*d)*(I + a*x))])/(a^2*c - d)^(3/2) + ((3*a^2*c - 2*d)*Log[(12*a*c^2*Sqrt[a^2*c - d]*(a*c + I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/((3*a^2*c - 2*d)*(-I + a*x))])/(a^2*c - d)^(3/2))/c^2","C",1
63,1,345,208,1.1357894,"\int \frac{\cot ^{-1}(a x)}{\left(c+d x^2\right)^{7/2}} \, dx","Integrate[ArcCot[a*x]/(c + d*x^2)^(7/2),x]","-\frac{-\frac{2 a c \left(a^2 c \left(8 c+7 d x^2\right)-d \left(5 c+4 d x^2\right)\right)}{\left(d-a^2 c\right)^2 \left(c+d x^2\right)^{3/2}}+\frac{\left(15 a^4 c^2-20 a^2 c d+8 d^2\right) \log \left(\frac{60 a c^3 \left(a^2 c-d\right)^{3/2} \left(\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c-i d x\right)}{(a x+i) \left(15 a^4 c^2-20 a^2 c d+8 d^2\right)}\right)}{\left(a^2 c-d\right)^{5/2}}+\frac{\left(15 a^4 c^2-20 a^2 c d+8 d^2\right) \log \left(\frac{60 a c^3 \left(a^2 c-d\right)^{3/2} \left(\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c+i d x\right)}{(a x-i) \left(15 a^4 c^2-20 a^2 c d+8 d^2\right)}\right)}{\left(a^2 c-d\right)^{5/2}}-\frac{2 x \cot ^{-1}(a x) \left(15 c^2+20 c d x^2+8 d^2 x^4\right)}{\left(c+d x^2\right)^{5/2}}}{30 c^3}","\frac{a \left(7 a^2 c-4 d\right)}{15 c^2 \left(a^2 c-d\right)^2 \sqrt{c+d x^2}}+\frac{a}{15 c \left(a^2 c-d\right) \left(c+d x^2\right)^{3/2}}-\frac{\left(15 a^4 c^2-20 a^2 c d+8 d^2\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{15 c^3 \left(a^2 c-d\right)^{5/2}}+\frac{8 x \cot ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}+\frac{4 x \cot ^{-1}(a x)}{15 c^2 \left(c+d x^2\right)^{3/2}}+\frac{x \cot ^{-1}(a x)}{5 c \left(c+d x^2\right)^{5/2}}",1,"-1/30*((-2*a*c*(-(d*(5*c + 4*d*x^2)) + a^2*c*(8*c + 7*d*x^2)))/((-(a^2*c) + d)^2*(c + d*x^2)^(3/2)) - (2*x*(15*c^2 + 20*c*d*x^2 + 8*d^2*x^4)*ArcCot[a*x])/(c + d*x^2)^(5/2) + ((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*Log[(60*a*c^3*(a^2*c - d)^(3/2)*(a*c - I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*(I + a*x))])/(a^2*c - d)^(5/2) + ((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*Log[(60*a*c^3*(a^2*c - d)^(3/2)*(a*c + I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*(-I + a*x))])/(a^2*c - d)^(5/2))/c^3","C",1
64,1,450,293,1.6743965,"\int \frac{\cot ^{-1}(a x)}{\left(c+d x^2\right)^{9/2}} \, dx","Integrate[ArcCot[a*x]/(c + d*x^2)^(9/2),x]","\frac{\frac{2 a c \left(3 c^2 \left(d-a^2 c\right)^2+c \left(11 a^2 c-6 d\right) \left(a^2 c-d\right) \left(c+d x^2\right)+3 \left(19 a^4 c^2-22 a^2 c d+8 d^2\right) \left(c+d x^2\right)^2\right)}{\left(a^2 c-d\right)^3 \left(c+d x^2\right)^{5/2}}-\frac{3 \left(35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right) \log \left(\frac{140 a c^4 \left(a^2 c-d\right)^{5/2} \left(\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c-i d x\right)}{(a x+i) \left(35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right)}\right)}{\left(a^2 c-d\right)^{7/2}}-\frac{3 \left(35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right) \log \left(\frac{140 a c^4 \left(a^2 c-d\right)^{5/2} \left(\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c+i d x\right)}{(a x-i) \left(35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right)}\right)}{\left(a^2 c-d\right)^{7/2}}+\frac{6 x \cot ^{-1}(a x) \left(35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right)}{\left(c+d x^2\right)^{7/2}}}{210 c^4}","\frac{a \left(11 a^2 c-6 d\right)}{105 c^2 \left(a^2 c-d\right)^2 \left(c+d x^2\right)^{3/2}}+\frac{a}{35 c \left(a^2 c-d\right) \left(c+d x^2\right)^{5/2}}+\frac{a \left(19 a^4 c^2-22 a^2 c d+8 d^2\right)}{35 c^3 \left(a^2 c-d\right)^3 \sqrt{c+d x^2}}-\frac{\left(35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{35 c^4 \left(a^2 c-d\right)^{7/2}}+\frac{16 x \cot ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \cot ^{-1}(a x)}{35 c^3 \left(c+d x^2\right)^{3/2}}+\frac{6 x \cot ^{-1}(a x)}{35 c^2 \left(c+d x^2\right)^{5/2}}+\frac{x \cot ^{-1}(a x)}{7 c \left(c+d x^2\right)^{7/2}}",1,"((2*a*c*(3*c^2*(-(a^2*c) + d)^2 + c*(11*a^2*c - 6*d)*(a^2*c - d)*(c + d*x^2) + 3*(19*a^4*c^2 - 22*a^2*c*d + 8*d^2)*(c + d*x^2)^2))/((a^2*c - d)^3*(c + d*x^2)^(5/2)) + (6*x*(35*c^3 + 70*c^2*d*x^2 + 56*c*d^2*x^4 + 16*d^3*x^6)*ArcCot[a*x])/(c + d*x^2)^(7/2) - (3*(35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*Log[(140*a*c^4*(a^2*c - d)^(5/2)*(a*c - I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/((35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*(I + a*x))])/(a^2*c - d)^(7/2) - (3*(35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*Log[(140*a*c^4*(a^2*c - d)^(5/2)*(a*c + I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/((35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*(-I + a*x))])/(a^2*c - d)^(7/2))/(210*c^4)","C",1
65,1,136,195,1.2467288,"\int \sqrt{a+a x^2} \cot ^{-1}(x) \, dx","Integrate[Sqrt[a + a*x^2]*ArcCot[x],x]","-\frac{\left(a \left(x^2+1\right)\right)^{3/2} \left(4 i \text{Li}_2\left(-e^{i \cot ^{-1}(x)}\right)-4 i \text{Li}_2\left(e^{i \cot ^{-1}(x)}\right)-2 \cot \left(\frac{1}{2} \cot ^{-1}(x)\right)+4 \cot ^{-1}(x) \log \left(1-e^{i \cot ^{-1}(x)}\right)-4 \cot ^{-1}(x) \log \left(1+e^{i \cot ^{-1}(x)}\right)-2 \tan \left(\frac{1}{2} \cot ^{-1}(x)\right)-\cot ^{-1}(x) \csc ^2\left(\frac{1}{2} \cot ^{-1}(x)\right)+\cot ^{-1}(x) \sec ^2\left(\frac{1}{2} \cot ^{-1}(x)\right)\right)}{8 a \left(\frac{1}{x^2}+1\right)^{3/2} x^3}","-\frac{i a \sqrt{x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i x+1}}{\sqrt{1-i x}}\right)}{2 \sqrt{a x^2+a}}+\frac{i a \sqrt{x^2+1} \text{Li}_2\left(\frac{i \sqrt{i x+1}}{\sqrt{1-i x}}\right)}{2 \sqrt{a x^2+a}}+\frac{1}{2} \sqrt{a x^2+a}+\frac{1}{2} x \sqrt{a x^2+a} \cot ^{-1}(x)-\frac{i a \sqrt{x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i x}}{\sqrt{1-i x}}\right) \cot ^{-1}(x)}{\sqrt{a x^2+a}}",1,"-1/8*((a*(1 + x^2))^(3/2)*(-2*Cot[ArcCot[x]/2] - ArcCot[x]*Csc[ArcCot[x]/2]^2 + 4*ArcCot[x]*Log[1 - E^(I*ArcCot[x])] - 4*ArcCot[x]*Log[1 + E^(I*ArcCot[x])] + (4*I)*PolyLog[2, -E^(I*ArcCot[x])] - (4*I)*PolyLog[2, E^(I*ArcCot[x])] + ArcCot[x]*Sec[ArcCot[x]/2]^2 - 2*Tan[ArcCot[x]/2]))/(a*(1 + x^(-2))^(3/2)*x^3)","A",0
66,1,89,155,0.1162683,"\int \frac{\cot ^{-1}(x)}{\sqrt{a+a x^2}} \, dx","Integrate[ArcCot[x]/Sqrt[a + a*x^2],x]","-\frac{\sqrt{a \left(x^2+1\right)} \left(i \text{Li}_2\left(-e^{i \cot ^{-1}(x)}\right)-i \text{Li}_2\left(e^{i \cot ^{-1}(x)}\right)+\cot ^{-1}(x) \left(\log \left(1-e^{i \cot ^{-1}(x)}\right)-\log \left(1+e^{i \cot ^{-1}(x)}\right)\right)\right)}{a \sqrt{\frac{1}{x^2}+1} x}","-\frac{i \sqrt{x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i x+1}}{\sqrt{1-i x}}\right)}{\sqrt{a x^2+a}}+\frac{i \sqrt{x^2+1} \text{Li}_2\left(\frac{i \sqrt{i x+1}}{\sqrt{1-i x}}\right)}{\sqrt{a x^2+a}}-\frac{2 i \sqrt{x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i x}}{\sqrt{1-i x}}\right) \cot ^{-1}(x)}{\sqrt{a x^2+a}}",1,"-((Sqrt[a*(1 + x^2)]*(ArcCot[x]*(Log[1 - E^(I*ArcCot[x])] - Log[1 + E^(I*ArcCot[x])]) + I*PolyLog[2, -E^(I*ArcCot[x])] - I*PolyLog[2, E^(I*ArcCot[x])]))/(a*Sqrt[1 + x^(-2)]*x))","A",0
67,1,21,35,0.0253086,"\int \frac{\cot ^{-1}(x)}{\left(a+a x^2\right)^{3/2}} \, dx","Integrate[ArcCot[x]/(a + a*x^2)^(3/2),x]","\frac{x \cot ^{-1}(x)-1}{a \sqrt{a \left(x^2+1\right)}}","\frac{x \cot ^{-1}(x)}{a \sqrt{a x^2+a}}-\frac{1}{a \sqrt{a x^2+a}}",1,"(-1 + x*ArcCot[x])/(a*Sqrt[a*(1 + x^2)])","A",1
68,1,37,79,0.0347137,"\int \frac{\cot ^{-1}(x)}{\left(a+a x^2\right)^{5/2}} \, dx","Integrate[ArcCot[x]/(a + a*x^2)^(5/2),x]","\frac{\left(6 x^3+9 x\right) \cot ^{-1}(x)-6 x^2-7}{9 a \left(a \left(x^2+1\right)\right)^{3/2}}","-\frac{2}{3 a^2 \sqrt{a x^2+a}}+\frac{2 x \cot ^{-1}(x)}{3 a^2 \sqrt{a x^2+a}}-\frac{1}{9 a \left(a x^2+a\right)^{3/2}}+\frac{x \cot ^{-1}(x)}{3 a \left(a x^2+a\right)^{3/2}}",1,"(-7 - 6*x^2 + (9*x + 6*x^3)*ArcCot[x])/(9*a*(a*(1 + x^2))^(3/2))","A",1
69,1,47,118,0.0441029,"\int \frac{\cot ^{-1}(x)}{\left(a+a x^2\right)^{7/2}} \, dx","Integrate[ArcCot[x]/(a + a*x^2)^(7/2),x]","\frac{-120 x^4-260 x^2+15 \left(8 x^4+20 x^2+15\right) x \cot ^{-1}(x)-149}{225 a \left(a \left(x^2+1\right)\right)^{5/2}}","-\frac{8}{15 a^3 \sqrt{a x^2+a}}+\frac{8 x \cot ^{-1}(x)}{15 a^3 \sqrt{a x^2+a}}-\frac{4}{45 a^2 \left(a x^2+a\right)^{3/2}}+\frac{4 x \cot ^{-1}(x)}{15 a^2 \left(a x^2+a\right)^{3/2}}-\frac{1}{25 a \left(a x^2+a\right)^{5/2}}+\frac{x \cot ^{-1}(x)}{5 a \left(a x^2+a\right)^{5/2}}",1,"(-149 - 260*x^2 - 120*x^4 + 15*x*(15 + 20*x^2 + 8*x^4)*ArcCot[x])/(225*a*(a*(1 + x^2))^(5/2))","A",1
70,1,25,32,0.0223224,"\int \frac{x \cot ^{-1}(x)}{\left(1+x^2\right)^2} \, dx","Integrate[(x*ArcCot[x])/(1 + x^2)^2,x]","-\frac{x^2 \tan ^{-1}(x)+x+\tan ^{-1}(x)+2 \cot ^{-1}(x)}{4 x^2+4}","-\frac{x}{4 \left(x^2+1\right)}-\frac{\cot ^{-1}(x)}{2 \left(x^2+1\right)}-\frac{1}{4} \tan ^{-1}(x)",1,"-((x + 2*ArcCot[x] + ArcTan[x] + x^2*ArcTan[x])/(4 + 4*x^2))","A",1
71,1,36,44,0.0250625,"\int \frac{x \cot ^{-1}(x)}{\left(1+x^2\right)^3} \, dx","Integrate[(x*ArcCot[x])/(1 + x^2)^3,x]","-\frac{x \left(3 x^2+5\right)+3 \left(x^2+1\right)^2 \tan ^{-1}(x)+8 \cot ^{-1}(x)}{32 \left(x^2+1\right)^2}","-\frac{3 x}{32 \left(x^2+1\right)}-\frac{x}{16 \left(x^2+1\right)^2}-\frac{\cot ^{-1}(x)}{4 \left(x^2+1\right)^2}-\frac{3}{32} \tan ^{-1}(x)",1,"-1/32*(x*(5 + 3*x^2) + 8*ArcCot[x] + 3*(1 + x^2)^2*ArcTan[x])/(1 + x^2)^2","A",1
72,1,28,34,0.0151123,"\int \frac{\cot ^{-1}(x)}{\left(1+x^2\right)^2} \, dx","Integrate[ArcCot[x]/(1 + x^2)^2,x]","-\frac{\left(x^2+1\right) \cot ^{-1}(x)^2-2 x \cot ^{-1}(x)+1}{4 \left(x^2+1\right)}","-\frac{1}{4 \left(x^2+1\right)}+\frac{x \cot ^{-1}(x)}{2 \left(x^2+1\right)}-\frac{1}{4} \cot ^{-1}(x)^2",1,"-1/4*(1 - 2*x*ArcCot[x] + (1 + x^2)*ArcCot[x]^2)/(1 + x^2)","A",1
73,1,46,56,0.027109,"\int \frac{\cot ^{-1}(x)^2}{\left(1+x^2\right)^2} \, dx","Integrate[ArcCot[x]^2/(1 + x^2)^2,x]","-\frac{3 \left(\left(x^2+1\right) \tan ^{-1}(x)+x\right)+2 \left(x^2+1\right) \cot ^{-1}(x)^3-6 x \cot ^{-1}(x)^2+6 \cot ^{-1}(x)}{12 \left(x^2+1\right)}","-\frac{x}{4 \left(x^2+1\right)}+\frac{x \cot ^{-1}(x)^2}{2 \left(x^2+1\right)}-\frac{\cot ^{-1}(x)}{2 \left(x^2+1\right)}-\frac{1}{4} \tan ^{-1}(x)-\frac{1}{6} \cot ^{-1}(x)^3",1,"-1/12*(6*ArcCot[x] - 6*x*ArcCot[x]^2 + 2*(1 + x^2)*ArcCot[x]^3 + 3*(x + (1 + x^2)*ArcTan[x]))/(1 + x^2)","A",1
74,1,41,41,0.0163149,"\int x^5 \cot ^{-1}\left(a x^2\right) \, dx","Integrate[x^5*ArcCot[a*x^2],x]","-\frac{\log \left(a^2 x^4+1\right)}{12 a^3}+\frac{x^4}{12 a}+\frac{1}{6} x^6 \cot ^{-1}\left(a x^2\right)","-\frac{\log \left(a^2 x^4+1\right)}{12 a^3}+\frac{x^4}{12 a}+\frac{1}{6} x^6 \cot ^{-1}\left(a x^2\right)",1,"x^4/(12*a) + (x^6*ArcCot[a*x^2])/6 - Log[1 + a^2*x^4]/(12*a^3)","A",1
75,1,37,37,0.0062193,"\int x^3 \cot ^{-1}\left(a x^2\right) \, dx","Integrate[x^3*ArcCot[a*x^2],x]","-\frac{\tan ^{-1}\left(a x^2\right)}{4 a^2}+\frac{x^2}{4 a}+\frac{1}{4} x^4 \cot ^{-1}\left(a x^2\right)","-\frac{\tan ^{-1}\left(a x^2\right)}{4 a^2}+\frac{x^2}{4 a}+\frac{1}{4} x^4 \cot ^{-1}\left(a x^2\right)",1,"x^2/(4*a) + (x^4*ArcCot[a*x^2])/4 - ArcTan[a*x^2]/(4*a^2)","A",1
76,1,31,31,0.0066604,"\int x \cot ^{-1}\left(a x^2\right) \, dx","Integrate[x*ArcCot[a*x^2],x]","\frac{\log \left(a^2 x^4+1\right)}{4 a}+\frac{1}{2} x^2 \cot ^{-1}\left(a x^2\right)","\frac{\log \left(a^2 x^4+1\right)}{4 a}+\frac{1}{2} x^2 \cot ^{-1}\left(a x^2\right)",1,"(x^2*ArcCot[a*x^2])/2 + Log[1 + a^2*x^4]/(4*a)","A",1
77,1,37,37,0.0064111,"\int \frac{\cot ^{-1}\left(a x^2\right)}{x} \, dx","Integrate[ArcCot[a*x^2]/x,x]","\frac{1}{4} i \text{Li}_2\left(\frac{i}{a x^2}\right)-\frac{1}{4} i \text{Li}_2\left(-\frac{i}{a x^2}\right)","\frac{1}{4} i \text{Li}_2\left(\frac{i}{a x^2}\right)-\frac{1}{4} i \text{Li}_2\left(-\frac{i}{a x^2}\right)",1,"(-1/4*I)*PolyLog[2, (-I)/(a*x^2)] + (I/4)*PolyLog[2, I/(a*x^2)]","A",1
78,1,34,34,0.0074272,"\int \frac{\cot ^{-1}\left(a x^2\right)}{x^3} \, dx","Integrate[ArcCot[a*x^2]/x^3,x]","\frac{1}{4} a \log \left(a^2 x^4+1\right)-\frac{\cot ^{-1}\left(a x^2\right)}{2 x^2}-a \log (x)","\frac{1}{4} a \log \left(a^2 x^4+1\right)-\frac{\cot ^{-1}\left(a x^2\right)}{2 x^2}-a \log (x)",1,"-1/2*ArcCot[a*x^2]/x^2 - a*Log[x] + (a*Log[1 + a^2*x^4])/4","A",1
79,1,38,35,0.0070134,"\int \frac{\cot ^{-1}\left(a x^2\right)}{x^5} \, dx","Integrate[ArcCot[a*x^2]/x^5,x]","\frac{a \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-a^2 x^4\right)}{4 x^2}-\frac{\cot ^{-1}\left(a x^2\right)}{4 x^4}","\frac{1}{4} a^2 \tan ^{-1}\left(a x^2\right)+\frac{a}{4 x^2}-\frac{\cot ^{-1}\left(a x^2\right)}{4 x^4}",1,"-1/4*ArcCot[a*x^2]/x^4 + (a*Hypergeometric2F1[-1/2, 1, 1/2, -(a^2*x^4)])/(4*x^2)","C",1
80,1,136,152,0.0499417,"\int x^4 \cot ^{-1}\left(a x^2\right) \, dx","Integrate[x^4*ArcCot[a*x^2],x]","\frac{8 a^{3/2} x^3+12 a^{5/2} x^5 \cot ^{-1}\left(a x^2\right)-3 \sqrt{2} \log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)+3 \sqrt{2} \log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)+6 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)-6 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{60 a^{5/2}}","-\frac{\log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{10 \sqrt{2} a^{5/2}}+\frac{\log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{10 \sqrt{2} a^{5/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{5 \sqrt{2} a^{5/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{5 \sqrt{2} a^{5/2}}+\frac{2 x^3}{15 a}+\frac{1}{5} x^5 \cot ^{-1}\left(a x^2\right)",1,"(8*a^(3/2)*x^3 + 12*a^(5/2)*x^5*ArcCot[a*x^2] + 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[a]*x] - 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[a]*x] - 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2] + 3*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2])/(60*a^(5/2))","A",1
81,1,133,150,0.0331729,"\int x^2 \cot ^{-1}\left(a x^2\right) \, dx","Integrate[x^2*ArcCot[a*x^2],x]","\frac{4 a^{3/2} x^3 \cot ^{-1}\left(a x^2\right)+\sqrt{2} \log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)-\sqrt{2} \log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)+8 \sqrt{a} x+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{12 a^{3/2}}","\frac{\log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{6 \sqrt{2} a^{3/2}}-\frac{\log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{6 \sqrt{2} a^{3/2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{3 \sqrt{2} a^{3/2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{3 \sqrt{2} a^{3/2}}+\frac{1}{3} x^3 \cot ^{-1}\left(a x^2\right)+\frac{2 x}{3 a}",1,"(8*Sqrt[a]*x + 4*a^(3/2)*x^3*ArcCot[a*x^2] + 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[a]*x] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[a]*x] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2])/(12*a^(3/2))","A",1
82,1,102,132,0.0384351,"\int \cot ^{-1}\left(a x^2\right) \, dx","Integrate[ArcCot[a*x^2],x]","x \cot ^{-1}\left(a x^2\right)+\frac{\log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)-\log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)-2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)+2 \tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{2 \sqrt{2} \sqrt{a}}","\frac{\log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{2 \sqrt{2} \sqrt{a}}-\frac{\log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{2 \sqrt{2} \sqrt{a}}+x \cot ^{-1}\left(a x^2\right)-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{\sqrt{2} \sqrt{a}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{\sqrt{2} \sqrt{a}}",1,"x*ArcCot[a*x^2] + (-2*ArcTan[1 - Sqrt[2]*Sqrt[a]*x] + 2*ArcTan[1 + Sqrt[2]*Sqrt[a]*x] + Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2] - Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2])/(2*Sqrt[2]*Sqrt[a])","A",1
83,1,105,135,0.0472124,"\int \frac{\cot ^{-1}\left(a x^2\right)}{x^2} \, dx","Integrate[ArcCot[a*x^2]/x^2,x]","\frac{\sqrt{a} \left(\log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)-\log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)\right)}{2 \sqrt{2}}-\frac{\cot ^{-1}\left(a x^2\right)}{x}","\frac{\sqrt{a} \log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{2 \sqrt{2}}-\frac{\sqrt{a} \log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{2 \sqrt{2}}-\frac{\cot ^{-1}\left(a x^2\right)}{x}+\frac{\sqrt{a} \tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{\sqrt{2}}-\frac{\sqrt{a} \tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{\sqrt{2}}",1,"-(ArcCot[a*x^2]/x) + (Sqrt[a]*(2*ArcTan[1 - Sqrt[2]*Sqrt[a]*x] - 2*ArcTan[1 + Sqrt[2]*Sqrt[a]*x] + Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2] - Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2]))/(2*Sqrt[2])","A",1
84,1,146,150,0.0558255,"\int \frac{\cot ^{-1}\left(a x^2\right)}{x^4} \, dx","Integrate[ArcCot[a*x^2]/x^4,x]","\frac{a x^2 \left(\sqrt{2} \sqrt{a} x \log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)-\sqrt{2} \sqrt{a} x \log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)-2 \sqrt{2} \sqrt{a} x \tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)+2 \sqrt{2} \sqrt{a} x \tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)+8\right)-4 \cot ^{-1}\left(a x^2\right)}{12 x^3}","\frac{a^{3/2} \log \left(a x^2-\sqrt{2} \sqrt{a} x+1\right)}{6 \sqrt{2}}-\frac{a^{3/2} \log \left(a x^2+\sqrt{2} \sqrt{a} x+1\right)}{6 \sqrt{2}}-\frac{a^{3/2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{a} x\right)}{3 \sqrt{2}}+\frac{a^{3/2} \tan ^{-1}\left(\sqrt{2} \sqrt{a} x+1\right)}{3 \sqrt{2}}-\frac{\cot ^{-1}\left(a x^2\right)}{3 x^3}+\frac{2 a}{3 x}",1,"(-4*ArcCot[a*x^2] + a*x^2*(8 - 2*Sqrt[2]*Sqrt[a]*x*ArcTan[1 - Sqrt[2]*Sqrt[a]*x] + 2*Sqrt[2]*Sqrt[a]*x*ArcTan[1 + Sqrt[2]*Sqrt[a]*x] + Sqrt[2]*Sqrt[a]*x*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2] - Sqrt[2]*Sqrt[a]*x*Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2]))/(12*x^3)","A",1
85,1,40,51,0.0169176,"\int x^2 \cot ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x^2*ArcCot[Sqrt[x]],x]","\frac{1}{45} \left(15 x^3 \cot ^{-1}\left(\sqrt{x}\right)+\left(3 x^2-5 x+15\right) \sqrt{x}-15 \tan ^{-1}\left(\sqrt{x}\right)\right)","\frac{x^{5/2}}{15}-\frac{x^{3/2}}{9}+\frac{1}{3} x^3 \cot ^{-1}\left(\sqrt{x}\right)+\frac{\sqrt{x}}{3}-\frac{1}{3} \tan ^{-1}\left(\sqrt{x}\right)",1,"(Sqrt[x]*(15 - 5*x + 3*x^2) + 15*x^3*ArcCot[Sqrt[x]] - 15*ArcTan[Sqrt[x]])/45","A",1
86,1,33,42,0.0126701,"\int x \cot ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x*ArcCot[Sqrt[x]],x]","\frac{1}{6} \left(3 x^2 \cot ^{-1}\left(\sqrt{x}\right)+(x-3) \sqrt{x}+3 \tan ^{-1}\left(\sqrt{x}\right)\right)","\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \cot ^{-1}\left(\sqrt{x}\right)-\frac{\sqrt{x}}{2}+\frac{1}{2} \tan ^{-1}\left(\sqrt{x}\right)",1,"((-3 + x)*Sqrt[x] + 3*x^2*ArcCot[Sqrt[x]] + 3*ArcTan[Sqrt[x]])/6","A",1
87,1,22,22,0.0069154,"\int \cot ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[ArcCot[Sqrt[x]],x]","\sqrt{x}-\tan ^{-1}\left(\sqrt{x}\right)+x \cot ^{-1}\left(\sqrt{x}\right)","\sqrt{x}-\tan ^{-1}\left(\sqrt{x}\right)+x \cot ^{-1}\left(\sqrt{x}\right)",1,"Sqrt[x] + x*ArcCot[Sqrt[x]] - ArcTan[Sqrt[x]]","A",1
88,1,31,31,0.0062115,"\int \frac{\cot ^{-1}\left(\sqrt{x}\right)}{x} \, dx","Integrate[ArcCot[Sqrt[x]]/x,x]","i \text{Li}_2\left(\frac{i}{\sqrt{x}}\right)-i \text{Li}_2\left(-\frac{i}{\sqrt{x}}\right)","i \text{Li}_2\left(\frac{i}{\sqrt{x}}\right)-i \text{Li}_2\left(-\frac{i}{\sqrt{x}}\right)",1,"(-I)*PolyLog[2, (-I)/Sqrt[x]] + I*PolyLog[2, I/Sqrt[x]]","A",1
89,1,29,23,0.0100709,"\int \frac{\cot ^{-1}\left(\sqrt{x}\right)}{x^2} \, dx","Integrate[ArcCot[Sqrt[x]]/x^2,x]","\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-x\right)}{\sqrt{x}}-\frac{\cot ^{-1}\left(\sqrt{x}\right)}{x}","\frac{1}{\sqrt{x}}+\tan ^{-1}\left(\sqrt{x}\right)-\frac{\cot ^{-1}\left(\sqrt{x}\right)}{x}",1,"-(ArcCot[Sqrt[x]]/x) + Hypergeometric2F1[-1/2, 1, 1/2, -x]/Sqrt[x]","C",1
90,1,34,42,0.0108713,"\int \frac{\cot ^{-1}\left(\sqrt{x}\right)}{x^3} \, dx","Integrate[ArcCot[Sqrt[x]]/x^3,x]","\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-x\right)}{6 x^{3/2}}-\frac{\cot ^{-1}\left(\sqrt{x}\right)}{2 x^2}","\frac{1}{6 x^{3/2}}-\frac{\cot ^{-1}\left(\sqrt{x}\right)}{2 x^2}-\frac{1}{2 \sqrt{x}}-\frac{1}{2} \tan ^{-1}\left(\sqrt{x}\right)",1,"-1/2*ArcCot[Sqrt[x]]/x^2 + Hypergeometric2F1[-3/2, 1, -1/2, -x]/(6*x^(3/2))","C",1
91,1,29,36,0.0167102,"\int x^{3/2} \cot ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x^(3/2)*ArcCot[Sqrt[x]],x]","\frac{1}{10} \left(4 x^{5/2} \cot ^{-1}\left(\sqrt{x}\right)+(x-2) x+2 \log (x+1)\right)","\frac{2}{5} x^{5/2} \cot ^{-1}\left(\sqrt{x}\right)+\frac{x^2}{10}-\frac{x}{5}+\frac{1}{5} \log (x+1)",1,"((-2 + x)*x + 4*x^(5/2)*ArcCot[Sqrt[x]] + 2*Log[1 + x])/10","A",1
92,1,25,29,0.0108914,"\int \sqrt{x} \cot ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[Sqrt[x]*ArcCot[Sqrt[x]],x]","\frac{1}{3} \left(2 x^{3/2} \cot ^{-1}\left(\sqrt{x}\right)+x-\log (x+1)\right)","\frac{2}{3} x^{3/2} \cot ^{-1}\left(\sqrt{x}\right)+\frac{x}{3}-\frac{1}{3} \log (x+1)",1,"(x + 2*x^(3/2)*ArcCot[Sqrt[x]] - Log[1 + x])/3","A",1
93,1,18,18,0.0079056,"\int \frac{\cot ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Integrate[ArcCot[Sqrt[x]]/Sqrt[x],x]","\log (x+1)+2 \sqrt{x} \cot ^{-1}\left(\sqrt{x}\right)","\log (x+1)+2 \sqrt{x} \cot ^{-1}\left(\sqrt{x}\right)",1,"2*Sqrt[x]*ArcCot[Sqrt[x]] + Log[1 + x]","A",1
94,1,22,22,0.0129493,"\int \frac{\cot ^{-1}\left(\sqrt{x}\right)}{x^{3/2}} \, dx","Integrate[ArcCot[Sqrt[x]]/x^(3/2),x]","-\log (x)+\log (x+1)-\frac{2 \cot ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}}","-\log (x)+\log (x+1)-\frac{2 \cot ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}}",1,"(-2*ArcCot[Sqrt[x]])/Sqrt[x] - Log[x] + Log[1 + x]","A",1
95,1,29,37,0.0200165,"\int \frac{\cot ^{-1}\left(\sqrt{x}\right)}{x^{5/2}} \, dx","Integrate[ArcCot[Sqrt[x]]/x^(5/2),x]","\frac{1}{3} \left(-\frac{2 \cot ^{-1}\left(\sqrt{x}\right)}{x^{3/2}}+\frac{1}{x}+\log (x)-\log (x+1)\right)","-\frac{2 \cot ^{-1}\left(\sqrt{x}\right)}{3 x^{3/2}}+\frac{1}{3 x}+\frac{\log (x)}{3}-\frac{1}{3} \log (x+1)",1,"(x^(-1) - (2*ArcCot[Sqrt[x]])/x^(3/2) + Log[x] - Log[1 + x])/3","A",1
96,1,17,17,0.0020288,"\int \cot ^{-1}\left(\frac{1}{x}\right) \, dx","Integrate[ArcCot[x^(-1)],x]","x \cot ^{-1}\left(\frac{1}{x}\right)-\frac{1}{2} \log \left(x^2+1\right)","x \cot ^{-1}\left(\frac{1}{x}\right)-\frac{1}{2} \log \left(x^2+1\right)",1,"x*ArcCot[x^(-1)] - Log[1 + x^2]/2","A",1
97,1,40,47,0.0178199,"\int \frac{\cot ^{-1}\left(a x^n\right)}{x} \, dx","Integrate[ArcCot[a*x^n]/x,x]","-\frac{i \left(\text{Li}_2\left(-\frac{i x^{-n}}{a}\right)-\text{Li}_2\left(\frac{i x^{-n}}{a}\right)\right)}{2 n}","\frac{i \text{Li}_2\left(\frac{i x^{-n}}{a}\right)}{2 n}-\frac{i \text{Li}_2\left(-\frac{i x^{-n}}{a}\right)}{2 n}",1,"((-1/2*I)*(PolyLog[2, (-I)/(a*x^n)] - PolyLog[2, I/(a*x^n)]))/n","A",1
98,1,37,37,0.0074324,"\int \frac{\cot ^{-1}\left(a x^5\right)}{x} \, dx","Integrate[ArcCot[a*x^5]/x,x]","\frac{1}{10} i \text{Li}_2\left(\frac{i}{a x^5}\right)-\frac{1}{10} i \text{Li}_2\left(-\frac{i}{a x^5}\right)","\frac{1}{10} i \text{Li}_2\left(\frac{i}{a x^5}\right)-\frac{1}{10} i \text{Li}_2\left(-\frac{i}{a x^5}\right)",1,"(-1/10*I)*PolyLog[2, (-I)/(a*x^5)] + (I/10)*PolyLog[2, I/(a*x^5)]","A",1
99,1,95,106,0.0766337,"\int x^3 \cot ^{-1}(a+b x) \, dx","Integrate[x^3*ArcCot[a + b*x],x]","\frac{6 \left(6 a^2-1\right) b x+6 b^4 x^4 \cot ^{-1}(a+b x)+2 (a+b x)^3-12 a (a+b x)^2-3 i (a-i)^4 \log (-a-b x+i)+3 i (a+i)^4 \log (a+b x+i)}{24 b^4}","\frac{a \left(1-a^2\right) \log \left((a+b x)^2+1\right)}{2 b^4}-\frac{\left(1-6 a^2\right) x}{4 b^3}+\frac{\left(a^4-6 a^2+1\right) \tan ^{-1}(a+b x)}{4 b^4}+\frac{(a+b x)^3}{12 b^4}-\frac{a (a+b x)^2}{2 b^4}+\frac{1}{4} x^4 \cot ^{-1}(a+b x)",1,"(6*(-1 + 6*a^2)*b*x - 12*a*(a + b*x)^2 + 2*(a + b*x)^3 + 6*b^4*x^4*ArcCot[a + b*x] - (3*I)*(-I + a)^4*Log[I - a - b*x] + (3*I)*(I + a)^4*Log[I + a + b*x])/(24*b^4)","C",1
100,1,114,80,0.0474362,"\int x^2 \cot ^{-1}(a+b x) \, dx","Integrate[x^2*ArcCot[a + b*x],x]","\frac{\frac{1}{3} b \left(\frac{a+b x}{b}-\frac{a}{b}\right)^3 \cot ^{-1}(a+b x)+\frac{1}{3} b \left(\frac{(a+b x)^2}{2 b^3}-\frac{(1-i a)^3 \log (a+b x+i)}{2 b^3}-\frac{(1+i a)^3 \log (-a-b x+i)}{2 b^3}-\frac{3 a x}{b^2}\right)}{b}","-\frac{\left(1-3 a^2\right) \log \left((a+b x)^2+1\right)}{6 b^3}+\frac{a \left(3-a^2\right) \tan ^{-1}(a+b x)}{3 b^3}+\frac{(a+b x)^2}{6 b^3}-\frac{a x}{b^2}+\frac{1}{3} x^3 \cot ^{-1}(a+b x)",1,"((b*(-(a/b) + (a + b*x)/b)^3*ArcCot[a + b*x])/3 + (b*((-3*a*x)/b^2 + (a + b*x)^2/(2*b^3) - ((1 + I*a)^3*Log[I - a - b*x])/(2*b^3) - ((1 - I*a)^3*Log[I + a + b*x])/(2*b^3)))/3)/b","C",1
101,1,90,60,0.0375015,"\int x \cot ^{-1}(a+b x) \, dx","Integrate[x*ArcCot[a + b*x],x]","\frac{i a^2 \log (a+b x+i)+2 b^2 x^2 \cot ^{-1}(a+b x)-2 a \log (a+b x+i)-i (a-i)^2 \log (-a-b x+i)-i \log (a+b x+i)+2 b x}{4 b^2}","-\frac{\left(1-a^2\right) \tan ^{-1}(a+b x)}{2 b^2}-\frac{a \log \left((a+b x)^2+1\right)}{2 b^2}+\frac{1}{2} x^2 \cot ^{-1}(a+b x)+\frac{x}{2 b}",1,"(2*b*x + 2*b^2*x^2*ArcCot[a + b*x] - I*(-I + a)^2*Log[I - a - b*x] - I*Log[I + a + b*x] - 2*a*Log[I + a + b*x] + I*a^2*Log[I + a + b*x])/(4*b^2)","C",1
102,1,44,33,0.0150735,"\int \cot ^{-1}(a+b x) \, dx","Integrate[ArcCot[a + b*x],x]","\frac{\log \left(a^2+2 a b x+b^2 x^2+1\right)-2 a \tan ^{-1}(a+b x)}{2 b}+x \cot ^{-1}(a+b x)","\frac{\log \left((a+b x)^2+1\right)}{2 b}+\frac{(a+b x) \cot ^{-1}(a+b x)}{b}",1,"x*ArcCot[a + b*x] + (-2*a*ArcTan[a + b*x] + Log[1 + a^2 + 2*a*b*x + b^2*x^2])/(2*b)","A",1
103,1,251,120,0.0285204,"\int \frac{\cot ^{-1}(a+b x)}{x} \, dx","Integrate[ArcCot[a + b*x]/x,x]","-\frac{1}{2} i \text{Li}_2\left(-\frac{b \left(\frac{a+b x}{b}-\frac{a}{b}\right)}{a-i}\right)+\frac{1}{2} i \text{Li}_2\left(-\frac{b \left(\frac{a+b x}{b}-\frac{a}{b}\right)}{a+i}\right)-\frac{1}{2} i \log \left(\frac{a+b x-i}{b \left(\frac{a}{b}-\frac{i}{b}\right)}\right) \log \left(\frac{a+b x}{b}-\frac{a}{b}\right)+\frac{1}{2} i \log \left(\frac{a+b x-i}{a+b x}\right) \log \left(\frac{a+b x}{b}-\frac{a}{b}\right)+\frac{1}{2} i \log \left(\frac{a+b x+i}{b \left(\frac{a}{b}+\frac{i}{b}\right)}\right) \log \left(\frac{a+b x}{b}-\frac{a}{b}\right)-\frac{1}{2} i \log \left(\frac{a+b x+i}{a+b x}\right) \log \left(\frac{a+b x}{b}-\frac{a}{b}\right)","-\frac{1}{2} i \text{Li}_2\left(1-\frac{2}{1-i (a+b x)}\right)+\frac{1}{2} i \text{Li}_2\left(1-\frac{2 b x}{(i-a) (1-i (a+b x))}\right)+\log \left(\frac{2}{1-i (a+b x)}\right) \left(-\cot ^{-1}(a+b x)\right)+\log \left(\frac{2 b x}{(-a+i) (1-i (a+b x))}\right) \cot ^{-1}(a+b x)",1,"(-1/2*I)*Log[(-I + a + b*x)/(((-I)/b + a/b)*b)]*Log[-(a/b) + (a + b*x)/b] + (I/2)*Log[(-I + a + b*x)/(a + b*x)]*Log[-(a/b) + (a + b*x)/b] + (I/2)*Log[(I + a + b*x)/((I/b + a/b)*b)]*Log[-(a/b) + (a + b*x)/b] - (I/2)*Log[(I + a + b*x)/(a + b*x)]*Log[-(a/b) + (a + b*x)/b] - (I/2)*PolyLog[2, -((b*(-(a/b) + (a + b*x)/b))/(-I + a))] + (I/2)*PolyLog[2, -((b*(-(a/b) + (a + b*x)/b))/(I + a))]","B",0
104,1,66,62,0.0667311,"\int \frac{\cot ^{-1}(a+b x)}{x^2} \, dx","Integrate[ArcCot[a + b*x]/x^2,x]","-\frac{\cot ^{-1}(a+b x)}{x}+\frac{b ((1-i a) \log (-a-b x+i)+(1+i a) \log (a+b x+i)-2 \log (x))}{2 \left(a^2+1\right)}","-\frac{b \log (x)}{a^2+1}+\frac{b \log \left((a+b x)^2+1\right)}{2 \left(a^2+1\right)}+\frac{a b \tan ^{-1}(a+b x)}{a^2+1}-\frac{\cot ^{-1}(a+b x)}{x}",1,"-(ArcCot[a + b*x]/x) + (b*(-2*Log[x] + (1 - I*a)*Log[I - a - b*x] + (1 + I*a)*Log[I + a + b*x]))/(2*(1 + a^2))","C",1
105,1,92,95,0.1122382,"\int \frac{\cot ^{-1}(a+b x)}{x^3} \, dx","Integrate[ArcCot[a + b*x]/x^3,x]","\frac{-2 \cot ^{-1}(a+b x)+\frac{b x \left(i (a+i)^2 b x \log (-a-b x+i)+4 a b x \log (x)+(a-i) ((-1-i a) b x \log (a+b x+i)+2 (a+i))\right)}{\left(a^2+1\right)^2}}{4 x^2}","\frac{a b^2 \log (x)}{\left(a^2+1\right)^2}-\frac{a b^2 \log \left((a+b x)^2+1\right)}{2 \left(a^2+1\right)^2}+\frac{\left(1-a^2\right) b^2 \tan ^{-1}(a+b x)}{2 \left(a^2+1\right)^2}+\frac{b}{2 \left(a^2+1\right) x}-\frac{\cot ^{-1}(a+b x)}{2 x^2}",1,"(-2*ArcCot[a + b*x] + (b*x*(4*a*b*x*Log[x] + I*(I + a)^2*b*x*Log[I - a - b*x] + (-I + a)*(2*(I + a) + (-1 - I*a)*b*x*Log[I + a + b*x])))/(1 + a^2)^2)/(4*x^2)","C",1
106,1,126,129,0.1553928,"\int \frac{\cot ^{-1}(a+b x)}{x^4} \, dx","Integrate[ArcCot[a + b*x]/x^4,x]","\frac{2 \left(1-3 a^2\right) b^3 x^3 \log (x)+(a-i) b x \left((a+i) \left(a^2-4 a b x+1\right)+i (a-i)^2 b^2 x^2 \log (a+b x+i)\right)-2 \left(a^2+1\right)^3 \cot ^{-1}(a+b x)+(-1+i a)^3 b^3 x^3 \log (-a-b x+i)}{6 \left(a^2+1\right)^3 x^3}","\frac{\left(1-3 a^2\right) b^3 \log (x)}{3 \left(a^2+1\right)^3}-\frac{\left(1-3 a^2\right) b^3 \log \left((a+b x)^2+1\right)}{6 \left(a^2+1\right)^3}-\frac{a \left(3-a^2\right) b^3 \tan ^{-1}(a+b x)}{3 \left(a^2+1\right)^3}-\frac{2 a b^2}{3 \left(a^2+1\right)^2 x}+\frac{b}{6 \left(a^2+1\right) x^2}-\frac{\cot ^{-1}(a+b x)}{3 x^3}",1,"(-2*(1 + a^2)^3*ArcCot[a + b*x] + 2*(1 - 3*a^2)*b^3*x^3*Log[x] + (-1 + I*a)^3*b^3*x^3*Log[I - a - b*x] + (-I + a)*b*x*((I + a)*(1 + a^2 - 4*a*b*x) + I*(-I + a)^2*b^2*x^2*Log[I + a + b*x]))/(6*(1 + a^2)^3*x^3)","C",1
107,1,563,642,0.6020161,"\int \frac{\cot ^{-1}(a+b x)}{c+d x^2} \, dx","Integrate[ArcCot[a + b*x]/(c + d*x^2),x]","-\frac{i \left(\text{Li}_2\left(\frac{b \left(\sqrt{-c}-\sqrt{d} x\right)}{\sqrt{d} (a-i)+b \sqrt{-c}}\right)-\text{Li}_2\left(\frac{b \left(\sqrt{-c}-\sqrt{d} x\right)}{\sqrt{d} (a+i)+b \sqrt{-c}}\right)-\text{Li}_2\left(\frac{b \left(\sqrt{d} x+\sqrt{-c}\right)}{b \sqrt{-c}-(a-i) \sqrt{d}}\right)+\text{Li}_2\left(\frac{b \left(\sqrt{d} x+\sqrt{-c}\right)}{b \sqrt{-c}-(a+i) \sqrt{d}}\right)+\log \left(\sqrt{-c}-\sqrt{d} x\right) \log \left(\frac{\sqrt{d} (a+b x-i)}{b \sqrt{-c}+(a-i) \sqrt{d}}\right)-\log \left(\frac{a+b x-i}{a+b x}\right) \log \left(\sqrt{-c}-\sqrt{d} x\right)-\log \left(\sqrt{-c}-\sqrt{d} x\right) \log \left(\frac{\sqrt{d} (a+b x+i)}{b \sqrt{-c}+(a+i) \sqrt{d}}\right)+\log \left(\frac{a+b x+i}{a+b x}\right) \log \left(\sqrt{-c}-\sqrt{d} x\right)-\log \left(\sqrt{-c}+\sqrt{d} x\right) \log \left(-\frac{\sqrt{d} (a+b x-i)}{b \sqrt{-c}-(a-i) \sqrt{d}}\right)+\log \left(\frac{a+b x-i}{a+b x}\right) \log \left(\sqrt{-c}+\sqrt{d} x\right)+\log \left(\sqrt{-c}+\sqrt{d} x\right) \log \left(-\frac{\sqrt{d} (a+b x+i)}{b \sqrt{-c}-(a+i) \sqrt{d}}\right)-\log \left(\frac{a+b x+i}{a+b x}\right) \log \left(\sqrt{-c}+\sqrt{d} x\right)\right)}{4 \sqrt{-c} \sqrt{d}}","\frac{\text{Li}_2\left(-\frac{\left(b \sqrt{c}-i a \sqrt{d}\right) (-a-b x+i)}{\left(b \sqrt{c}-(i a+1) \sqrt{d}\right) (a+b x)}\right)}{4 \sqrt{c} \sqrt{d}}-\frac{\text{Li}_2\left(-\frac{\left(i \sqrt{d} a+b \sqrt{c}\right) (-a-b x+i)}{\left(\sqrt{d} (i a+1)+b \sqrt{c}\right) (a+b x)}\right)}{4 \sqrt{c} \sqrt{d}}-\frac{\text{Li}_2\left(\frac{\left(b \sqrt{c}-i a \sqrt{d}\right) (a+b x+i)}{\left(\sqrt{d} (1-i a)+b \sqrt{c}\right) (a+b x)}\right)}{4 \sqrt{c} \sqrt{d}}+\frac{\text{Li}_2\left(\frac{\left(i \sqrt{d} a+b \sqrt{c}\right) (a+b x+i)}{\left(i \sqrt{d} (a+i)+b \sqrt{c}\right) (a+b x)}\right)}{4 \sqrt{c} \sqrt{d}}-\frac{\log \left(\frac{a+b x+i}{a+b x}\right) \log \left(-\frac{b \left(-\sqrt{d} x+i \sqrt{c}\right)}{(a+b x) \left(b \sqrt{c}+(1-i a) \sqrt{d}\right)}\right)}{4 \sqrt{c} \sqrt{d}}+\frac{\log \left(-\frac{-a-b x+i}{a+b x}\right) \log \left(\frac{i b \left(\sqrt{c}+i \sqrt{d} x\right)}{(a+b x) \left(b \sqrt{c}-(1+i a) \sqrt{d}\right)}\right)}{4 \sqrt{c} \sqrt{d}}-\frac{\log \left(-\frac{-a-b x+i}{a+b x}\right) \log \left(\frac{b \left(\sqrt{d} x+i \sqrt{c}\right)}{(a+b x) \left(b \sqrt{c}+(1+i a) \sqrt{d}\right)}\right)}{4 \sqrt{c} \sqrt{d}}+\frac{\log \left(\frac{a+b x+i}{a+b x}\right) \log \left(-\frac{b \left(\sqrt{d} x+i \sqrt{c}\right)}{(a+b x) \left(b \sqrt{c}+i (a+i) \sqrt{d}\right)}\right)}{4 \sqrt{c} \sqrt{d}}",1,"((-1/4*I)*(Log[(Sqrt[d]*(-I + a + b*x))/(b*Sqrt[-c] + (-I + a)*Sqrt[d])]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[(-I + a + b*x)/(a + b*x)]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[(Sqrt[d]*(I + a + b*x))/(b*Sqrt[-c] + (I + a)*Sqrt[d])]*Log[Sqrt[-c] - Sqrt[d]*x] + Log[(I + a + b*x)/(a + b*x)]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[-((Sqrt[d]*(-I + a + b*x))/(b*Sqrt[-c] - (-I + a)*Sqrt[d]))]*Log[Sqrt[-c] + Sqrt[d]*x] + Log[(-I + a + b*x)/(a + b*x)]*Log[Sqrt[-c] + Sqrt[d]*x] + Log[-((Sqrt[d]*(I + a + b*x))/(b*Sqrt[-c] - (I + a)*Sqrt[d]))]*Log[Sqrt[-c] + Sqrt[d]*x] - Log[(I + a + b*x)/(a + b*x)]*Log[Sqrt[-c] + Sqrt[d]*x] + PolyLog[2, (b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (-I + a)*Sqrt[d])] - PolyLog[2, (b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (I + a)*Sqrt[d])] - PolyLog[2, (b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (-I + a)*Sqrt[d])] + PolyLog[2, (b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (I + a)*Sqrt[d])]))/(Sqrt[-c]*Sqrt[d])","A",0
108,1,345,152,0.0474418,"\int \frac{\cot ^{-1}(a+b x)}{c+d x} \, dx","Integrate[ArcCot[a + b*x]/(c + d*x),x]","\frac{i \text{Li}_2\left(\frac{b \left(\frac{b c-a d}{b}+\frac{d (a+b x)}{b}\right)}{b c-a d-i d}\right)}{2 d}-\frac{i \text{Li}_2\left(\frac{b \left(\frac{b c-a d}{b}+\frac{d (a+b x)}{b}\right)}{b c-a d+i d}\right)}{2 d}-\frac{i \log \left(\frac{d (a+b x-i)}{b \left(-\frac{b c-a d}{b}-\frac{i d}{b}\right)}\right) \log \left(\frac{b c-a d}{b}+\frac{d (a+b x)}{b}\right)}{2 d}+\frac{i \log \left(\frac{a+b x-i}{a+b x}\right) \log \left(\frac{b c-a d}{b}+\frac{d (a+b x)}{b}\right)}{2 d}+\frac{i \log \left(\frac{d (a+b x+i)}{b \left(-\frac{b c-a d}{b}+\frac{i d}{b}\right)}\right) \log \left(\frac{b c-a d}{b}+\frac{d (a+b x)}{b}\right)}{2 d}-\frac{i \log \left(\frac{a+b x+i}{a+b x}\right) \log \left(\frac{b c-a d}{b}+\frac{d (a+b x)}{b}\right)}{2 d}","\frac{i \text{Li}_2\left(1-\frac{2 b (c+d x)}{(b c-a d+i d) (1-i (a+b x))}\right)}{2 d}+\frac{\cot ^{-1}(a+b x) \log \left(\frac{2 b (c+d x)}{(1-i (a+b x)) (-a d+b c+i d)}\right)}{d}-\frac{i \text{Li}_2\left(1-\frac{2}{1-i (a+b x)}\right)}{2 d}-\frac{\log \left(\frac{2}{1-i (a+b x)}\right) \cot ^{-1}(a+b x)}{d}",1,"((-1/2*I)*Log[(d*(-I + a + b*x))/(b*(((-I)*d)/b - (b*c - a*d)/b))]*Log[(b*c - a*d)/b + (d*(a + b*x))/b])/d + ((I/2)*Log[(-I + a + b*x)/(a + b*x)]*Log[(b*c - a*d)/b + (d*(a + b*x))/b])/d + ((I/2)*Log[(d*(I + a + b*x))/(b*((I*d)/b - (b*c - a*d)/b))]*Log[(b*c - a*d)/b + (d*(a + b*x))/b])/d - ((I/2)*Log[(I + a + b*x)/(a + b*x)]*Log[(b*c - a*d)/b + (d*(a + b*x))/b])/d + ((I/2)*PolyLog[2, (b*((b*c - a*d)/b + (d*(a + b*x))/b))/(b*c - I*d - a*d)])/d - ((I/2)*PolyLog[2, (b*((b*c - a*d)/b + (d*(a + b*x))/b))/(b*c + I*d - a*d)])/d","B",0
109,1,602,338,11.1239757,"\int \frac{\cot ^{-1}(a+b x)}{c+\frac{d}{x}} \, dx","Integrate[ArcCot[a + b*x]/(c + d/x),x]","-\frac{\left((a+b x)^2+1\right) \left(a b c d \sqrt{\frac{\left(a^2+1\right) c^2-2 a b c d+b^2 d^2}{(a c-b d)^2}} \cot ^{-1}(a+b x)^2 e^{-i \tan ^{-1}\left(\frac{c}{a c-b d}\right)}-b^2 d^2 \sqrt{\frac{\left(a^2+1\right) c^2-2 a b c d+b^2 d^2}{(a c-b d)^2}} \cot ^{-1}(a+b x)^2 e^{-i \tan ^{-1}\left(\frac{c}{a c-b d}\right)}+b^2 d^2 \cot ^{-1}(a+b x)^2+2 c^2 \log \left(\frac{1}{(a+b x) \sqrt{\frac{1}{(a+b x)^2}+1}}\right)-2 c^2 (a+b x) \cot ^{-1}(a+b x)-i b c d \text{Li}_2\left(\exp \left(2 i \left(\cot ^{-1}(a+b x)-\tan ^{-1}\left(\frac{c}{a c-b d}\right)\right)\right)\right)+2 b c d \cot ^{-1}(a+b x) \log \left(1-\exp \left(2 i \left(\cot ^{-1}(a+b x)-\tan ^{-1}\left(\frac{c}{a c-b d}\right)\right)\right)\right)-2 b c d \tan ^{-1}\left(\frac{c}{a c-b d}\right) \log \left(1-\exp \left(2 i \left(\cot ^{-1}(a+b x)-\tan ^{-1}\left(\frac{c}{a c-b d}\right)\right)\right)\right)+i b c d \text{Li}_2\left(e^{2 i \cot ^{-1}(a+b x)}\right)-\pi  b c d \log \left(\frac{1}{\sqrt{\frac{1}{(a+b x)^2}+1}}\right)-a b c d \cot ^{-1}(a+b x)^2+i b c d \cot ^{-1}(a+b x)^2+i \pi  b c d \cot ^{-1}(a+b x)+\pi  b c d \log \left(1+e^{-2 i \cot ^{-1}(a+b x)}\right)-2 b c d \cot ^{-1}(a+b x) \log \left(1-e^{2 i \cot ^{-1}(a+b x)}\right)+2 i b c d \cot ^{-1}(a+b x) \tan ^{-1}\left(\frac{c}{a c-b d}\right)+2 b c d \tan ^{-1}\left(\frac{c}{a c-b d}\right) \log \left(\sin \left(\cot ^{-1}(a+b x)-\tan ^{-1}\left(\frac{c}{a c-b d}\right)\right)\right)\right)}{2 b c^3 (a+b x)^2 \sqrt{\frac{a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}} \sqrt{\frac{1}{(a+b x)^2}+1}}","-\frac{i d \text{Li}_2\left(-\frac{b (d+c x)}{(a+i) c-b d}\right)}{2 c^2}+\frac{i d \text{Li}_2\left(\frac{b (d+c x)}{-a c+i c+b d}\right)}{2 c^2}+\frac{i d \log (c x+d) \log \left(\frac{c (-a-b x+i)}{-a c+b d+i c}\right)}{2 c^2}-\frac{i d \log \left(-\frac{-a-b x+i}{a+b x}\right) \log (c x+d)}{2 c^2}-\frac{i d \log (c x+d) \log \left(\frac{c (a+b x+i)}{-b d+(a+i) c}\right)}{2 c^2}+\frac{i d \log \left(\frac{a+b x+i}{a+b x}\right) \log (c x+d)}{2 c^2}+\frac{\log (-a-b x+i)}{2 b c}+\frac{i (a+b x) \log \left(-\frac{-a-b x+i}{a+b x}\right)}{2 b c}+\frac{\log (a+b x+i)}{2 b c}-\frac{i (a+b x) \log \left(\frac{a+b x+i}{a+b x}\right)}{2 b c}",1,"-1/2*((1 + (a + b*x)^2)*(I*b*c*d*Pi*ArcCot[a + b*x] - 2*c^2*(a + b*x)*ArcCot[a + b*x] + I*b*c*d*ArcCot[a + b*x]^2 - a*b*c*d*ArcCot[a + b*x]^2 + b^2*d^2*ArcCot[a + b*x]^2 + (a*b*c*d*Sqrt[((1 + a^2)*c^2 - 2*a*b*c*d + b^2*d^2)/(a*c - b*d)^2]*ArcCot[a + b*x]^2)/E^(I*ArcTan[c/(a*c - b*d)]) - (b^2*d^2*Sqrt[((1 + a^2)*c^2 - 2*a*b*c*d + b^2*d^2)/(a*c - b*d)^2]*ArcCot[a + b*x]^2)/E^(I*ArcTan[c/(a*c - b*d)]) + (2*I)*b*c*d*ArcCot[a + b*x]*ArcTan[c/(a*c - b*d)] + b*c*d*Pi*Log[1 + E^((-2*I)*ArcCot[a + b*x])] - 2*b*c*d*ArcCot[a + b*x]*Log[1 - E^((2*I)*ArcCot[a + b*x])] + 2*b*c*d*ArcCot[a + b*x]*Log[1 - E^((2*I)*(ArcCot[a + b*x] - ArcTan[c/(a*c - b*d)]))] - 2*b*c*d*ArcTan[c/(a*c - b*d)]*Log[1 - E^((2*I)*(ArcCot[a + b*x] - ArcTan[c/(a*c - b*d)]))] - b*c*d*Pi*Log[1/Sqrt[1 + (a + b*x)^(-2)]] + 2*c^2*Log[1/((a + b*x)*Sqrt[1 + (a + b*x)^(-2)])] + 2*b*c*d*ArcTan[c/(a*c - b*d)]*Log[Sin[ArcCot[a + b*x] - ArcTan[c/(a*c - b*d)]]] + I*b*c*d*PolyLog[2, E^((2*I)*ArcCot[a + b*x])] - I*b*c*d*PolyLog[2, E^((2*I)*(ArcCot[a + b*x] - ArcTan[c/(a*c - b*d)]))]))/(b*c^3*(a + b*x)^2*Sqrt[(1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*Sqrt[1 + (a + b*x)^(-2)])","A",0
110,1,5764,735,36.5593284,"\int \frac{\cot ^{-1}(a+b x)}{c+\frac{d}{x^2}} \, dx","Integrate[ArcCot[a + b*x]/(c + d/x^2),x]","\text{Result too large to show}","-\frac{\sqrt{d} \text{Li}_2\left(\frac{b \left(\sqrt{d}-i \sqrt{c} x\right)}{\sqrt{c} (i a+1)+b \sqrt{d}}\right)}{4 c^{3/2}}+\frac{\sqrt{d} \text{Li}_2\left(\frac{b \left(\sqrt{d}-i \sqrt{c} x\right)}{i \sqrt{c} (a+i)+b \sqrt{d}}\right)}{4 c^{3/2}}+\frac{\sqrt{d} \text{Li}_2\left(-\frac{b \left(i \sqrt{c} x+\sqrt{d}\right)}{(i a+1) \sqrt{c}-b \sqrt{d}}\right)}{4 c^{3/2}}-\frac{\sqrt{d} \text{Li}_2\left(\frac{b \left(i \sqrt{c} x+\sqrt{d}\right)}{\sqrt{c} (1-i a)+b \sqrt{d}}\right)}{4 c^{3/2}}-\frac{\sqrt{d} \log \left(1-\frac{i \sqrt{c} x}{\sqrt{d}}\right) \log \left(\frac{\sqrt{c} (-a-b x+i)}{(-a+i) \sqrt{c}+i b \sqrt{d}}\right)}{4 c^{3/2}}+\frac{\sqrt{d} \log \left(1-\frac{i \sqrt{c} x}{\sqrt{d}}\right) \log \left(\frac{\sqrt{c} (a+b x+i)}{(a+i) \sqrt{c}-i b \sqrt{d}}\right)}{4 c^{3/2}}+\frac{\sqrt{d} \log \left(1+\frac{i \sqrt{c} x}{\sqrt{d}}\right) \log \left(\frac{\sqrt{c} (-a-b x+i)}{(-a+i) \sqrt{c}-i b \sqrt{d}}\right)}{4 c^{3/2}}-\frac{\sqrt{d} \log \left(1+\frac{i \sqrt{c} x}{\sqrt{d}}\right) \log \left(\frac{\sqrt{c} (a+b x+i)}{(a+i) \sqrt{c}+i b \sqrt{d}}\right)}{4 c^{3/2}}-\frac{i \sqrt{d} \log \left(-\frac{-a-b x+i}{a+b x}\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)}{2 c^{3/2}}+\frac{i \sqrt{d} \log \left(\frac{a+b x+i}{a+b x}\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)}{2 c^{3/2}}+\frac{\log (-a-b x+i)}{2 b c}+\frac{i (a+b x) \log \left(-\frac{-a-b x+i}{a+b x}\right)}{2 b c}+\frac{\log (a+b x+i)}{2 b c}-\frac{i (a+b x) \log \left(\frac{a+b x+i}{a+b x}\right)}{2 b c}",1,"Result too large to show","B",0
111,1,618,693,0.8069353,"\int \frac{\cot ^{-1}(a+b x)}{c+d \sqrt{x}} \, dx","Integrate[ArcCot[a + b*x]/(c + d*Sqrt[x]),x]","-\frac{i \left(c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{-a-i} d}\right)+c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{-a-i} d}\right)-c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{i-a} d}\right)-c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{i-a} d}\right)+c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{\sqrt{b} c+\sqrt{-a-i} d}\right)-c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{\sqrt{b} c+\sqrt{-a+i} d}\right)+c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{-\sqrt{b} c+\sqrt{-a-i} d}\right)-c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{-\sqrt{b} c+\sqrt{-a+i} d}\right)+c \log \left(\frac{a+b x-i}{a+b x}\right) \log \left(c+d \sqrt{x}\right)-c \log \left(\frac{a+b x+i}{a+b x}\right) \log \left(c+d \sqrt{x}\right)-d \sqrt{x} \log \left(\frac{a+b x-i}{a+b x}\right)+d \sqrt{x} \log \left(\frac{a+b x+i}{a+b x}\right)+\frac{2 \sqrt{a+i} d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right)}{\sqrt{b}}-\frac{2 \sqrt{-a+i} d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right)}{\sqrt{b}}\right)}{d^2}","-\frac{i c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{-a-i} d}\right)}{d^2}-\frac{i c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{-a-i} d}\right)}{d^2}+\frac{i c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{i-a} d}\right)}{d^2}+\frac{i c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{i-a} d}\right)}{d^2}-\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(-\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{\sqrt{b} c+\sqrt{-a-i} d}\right)}{d^2}+\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(-\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{\sqrt{b} c+\sqrt{-a+i} d}\right)}{d^2}-\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(-\frac{d \left(\sqrt{b} \sqrt{x}+\sqrt{-a-i}\right)}{\sqrt{b} c-\sqrt{-a-i} d}\right)}{d^2}+\frac{i c \log \left(c+d \sqrt{x}\right) \log \left(-\frac{d \left(\sqrt{b} \sqrt{x}+\sqrt{-a+i}\right)}{\sqrt{b} c-\sqrt{-a+i} d}\right)}{d^2}-\frac{i c \log \left(-\frac{-a-b x+i}{a+b x}\right) \log \left(c+d \sqrt{x}\right)}{d^2}+\frac{i c \log \left(\frac{a+b x+i}{a+b x}\right) \log \left(c+d \sqrt{x}\right)}{d^2}+\frac{i \sqrt{x} \log \left(-\frac{-a-b x+i}{a+b x}\right)}{d}-\frac{i \sqrt{x} \log \left(\frac{a+b x+i}{a+b x}\right)}{d}-\frac{2 i \sqrt{a+i} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right)}{\sqrt{b} d}+\frac{2 i \sqrt{-a+i} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{-a+i}}\right)}{\sqrt{b} d}",1,"((-I)*((2*Sqrt[I + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]])/Sqrt[b] - (2*Sqrt[I - a]*d*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[I - a]])/Sqrt[b] + c*Log[(d*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-I - a]*d)]*Log[c + d*Sqrt[x]] - c*Log[(d*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[I - a]*d)]*Log[c + d*Sqrt[x]] + c*Log[(d*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(-(Sqrt[b]*c) + Sqrt[-I - a]*d)]*Log[c + d*Sqrt[x]] - c*Log[(d*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(-(Sqrt[b]*c) + Sqrt[I - a]*d)]*Log[c + d*Sqrt[x]] - d*Sqrt[x]*Log[(-I + a + b*x)/(a + b*x)] + c*Log[c + d*Sqrt[x]]*Log[(-I + a + b*x)/(a + b*x)] + d*Sqrt[x]*Log[(I + a + b*x)/(a + b*x)] - c*Log[c + d*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)] + c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-I - a]*d)] + c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-I - a]*d)] - c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[I - a]*d)] - c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[I - a]*d)]))/d^2","A",1
112,1,809,830,0.7396645,"\int \frac{\cot ^{-1}(a+b x)}{c+\frac{d}{\sqrt{x}}} \, dx","Integrate[ArcCot[a + b*x]/(c + d/Sqrt[x]),x]","\frac{i a \log (-a-b x+i) c^2+\log (-a-b x+i) c^2+i b x \log \left(\frac{a+b x-i}{a+b x}\right) c^2-i a \log (a+b x+i) c^2+\log (a+b x+i) c^2-i b x \log \left(\frac{a+b x+i}{a+b x}\right) c^2+4 i \sqrt{a+i} \sqrt{b} d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right) c-4 i \sqrt{i-a} \sqrt{b} d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{i-a}}\right) c-2 i b d \sqrt{x} \log \left(\frac{a+b x-i}{a+b x}\right) c+2 i b d \sqrt{x} \log \left(\frac{a+b x+i}{a+b x}\right) c+2 i b d^2 \log \left(\frac{c \left(\sqrt{-a-i}-\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-i} c+\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right)-2 i b d^2 \log \left(\frac{c \left(\sqrt{i-a}-\sqrt{b} \sqrt{x}\right)}{\sqrt{i-a} c+\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right)+2 i b d^2 \log \left(\frac{c \left(\sqrt{-a-i}+\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-i} c-\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right)-2 i b d^2 \log \left(\frac{c \left(\sqrt{i-a}+\sqrt{b} \sqrt{x}\right)}{\sqrt{i-a} c-\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right)+2 i b d^2 \log \left(\sqrt{x} c+d\right) \log \left(\frac{a+b x-i}{a+b x}\right)-2 i b d^2 \log \left(\sqrt{x} c+d\right) \log \left(\frac{a+b x+i}{a+b x}\right)+2 i b d^2 \text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{b} d-\sqrt{-a-i} c}\right)+2 i b d^2 \text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{-a-i} c+\sqrt{b} d}\right)-2 i b d^2 \text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{b} d-\sqrt{i-a} c}\right)-2 i b d^2 \text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{i-a} c+\sqrt{b} d}\right)}{2 b c^3}","\frac{i \log \left(\frac{c \left(\sqrt{-a-i}-\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-i} c+\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right) d^2}{c^3}-\frac{i \log \left(\frac{c \left(\sqrt{i-a}-\sqrt{b} \sqrt{x}\right)}{\sqrt{i-a} c+\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right) d^2}{c^3}+\frac{i \log \left(\frac{c \left(\sqrt{-a-i}+\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-i} c-\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right) d^2}{c^3}-\frac{i \log \left(\frac{c \left(\sqrt{i-a}+\sqrt{b} \sqrt{x}\right)}{\sqrt{i-a} c-\sqrt{b} d}\right) \log \left(\sqrt{x} c+d\right) d^2}{c^3}+\frac{i \log \left(\sqrt{x} c+d\right) \log \left(-\frac{-a-b x+i}{a+b x}\right) d^2}{c^3}-\frac{i \log \left(\sqrt{x} c+d\right) \log \left(\frac{a+b x+i}{a+b x}\right) d^2}{c^3}+\frac{i \text{Li}_2\left(-\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{-a-i} c-\sqrt{b} d}\right) d^2}{c^3}-\frac{i \text{Li}_2\left(-\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{i-a} c-\sqrt{b} d}\right) d^2}{c^3}+\frac{i \text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{-a-i} c+\sqrt{b} d}\right) d^2}{c^3}-\frac{i \text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{i-a} c+\sqrt{b} d}\right) d^2}{c^3}+\frac{2 i \sqrt{a+i} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+i}}\right) d}{\sqrt{b} c^2}-\frac{2 i \sqrt{i-a} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{i-a}}\right) d}{\sqrt{b} c^2}-\frac{i \sqrt{x} \log \left(-\frac{-a-b x+i}{a+b x}\right) d}{c^2}+\frac{i \sqrt{x} \log \left(\frac{a+b x+i}{a+b x}\right) d}{c^2}+\frac{(i a+1) \log (-a-b x+i)}{2 b c}+\frac{i x \log \left(-\frac{-a-b x+i}{a+b x}\right)}{2 c}+\frac{(1-i a) \log (a+b x+i)}{2 b c}-\frac{i x \log \left(\frac{a+b x+i}{a+b x}\right)}{2 c}",1,"((4*I)*Sqrt[I + a]*Sqrt[b]*c*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]] - (4*I)*Sqrt[I - a]*Sqrt[b]*c*d*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[I - a]] + (2*I)*b*d^2*Log[(c*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]] - (2*I)*b*d^2*Log[(c*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]] + (2*I)*b*d^2*Log[(c*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]] - (2*I)*b*d^2*Log[(c*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]] + c^2*Log[I - a - b*x] + I*a*c^2*Log[I - a - b*x] - (2*I)*b*c*d*Sqrt[x]*Log[(-I + a + b*x)/(a + b*x)] + I*b*c^2*x*Log[(-I + a + b*x)/(a + b*x)] + (2*I)*b*d^2*Log[d + c*Sqrt[x]]*Log[(-I + a + b*x)/(a + b*x)] + c^2*Log[I + a + b*x] - I*a*c^2*Log[I + a + b*x] + (2*I)*b*c*d*Sqrt[x]*Log[(I + a + b*x)/(a + b*x)] - I*b*c^2*x*Log[(I + a + b*x)/(a + b*x)] - (2*I)*b*d^2*Log[d + c*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)] + (2*I)*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[-I - a]*c) + Sqrt[b]*d)] + (2*I)*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)] - (2*I)*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[I - a]*c) + Sqrt[b]*d)] - (2*I)*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)])/(2*b*c^3)","A",1
113,1,629,367,0.4950479,"\int \frac{\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx","Integrate[ArcCot[d + e*x]/(a + b*x + c*x^2),x]","-\frac{i \left(\text{Li}_2\left(\frac{e \left(-b-2 c x+\sqrt{b^2-4 a c}\right)}{2 c (d-i)+\left(\sqrt{b^2-4 a c}-b\right) e}\right)-\text{Li}_2\left(\frac{e \left(-b-2 c x+\sqrt{b^2-4 a c}\right)}{2 c (d+i)+\left(\sqrt{b^2-4 a c}-b\right) e}\right)-\text{Li}_2\left(\frac{e \left(b+2 c x+\sqrt{b^2-4 a c}\right)}{\left(b+\sqrt{b^2-4 a c}\right) e-2 c (d-i)}\right)+\text{Li}_2\left(\frac{e \left(b+2 c x+\sqrt{b^2-4 a c}\right)}{\left(b+\sqrt{b^2-4 a c}\right) e-2 c (d+i)}\right)+\log \left(-\sqrt{b^2-4 a c}+b+2 c x\right) \log \left(\frac{2 c (d+e x-i)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c (d-i)}\right)-\log \left(\sqrt{b^2-4 a c}+b+2 c x\right) \log \left(\frac{2 c (d+e x-i)}{-e \left(\sqrt{b^2-4 a c}+b\right)+2 c (d-i)}\right)-\log \left(\frac{d+e x-i}{d+e x}\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x\right)+\log \left(\frac{d+e x-i}{d+e x}\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x\right)-\log \left(-\sqrt{b^2-4 a c}+b+2 c x\right) \log \left(\frac{2 c (d+e x+i)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c (d+i)}\right)+\log \left(\sqrt{b^2-4 a c}+b+2 c x\right) \log \left(\frac{2 c (d+e x+i)}{-e \left(\sqrt{b^2-4 a c}+b\right)+2 c (d+i)}\right)+\log \left(\frac{d+e x+i}{d+e x}\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x\right)-\log \left(\frac{d+e x+i}{d+e x}\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x\right)\right)}{2 \sqrt{b^2-4 a c}}","\frac{i \text{Li}_2\left(\frac{2 \left(2 c d-\left(b-\sqrt{b^2-4 a c}\right) e-2 c (d+e x)\right)}{\left(-2 d c+2 i c+b e-\sqrt{b^2-4 a c} e\right) (1-i (d+e x))}+1\right)}{2 \sqrt{b^2-4 a c}}-\frac{i \text{Li}_2\left(\frac{2 \left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e-2 c (d+e x)\right)}{\left(2 c (i-d)+\left(b+\sqrt{b^2-4 a c}\right) e\right) (1-i (d+e x))}+1\right)}{2 \sqrt{b^2-4 a c}}+\frac{\cot ^{-1}(d+e x) \log \left(\frac{2 e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{(1-i (d+e x)) \left(e \left(b-\sqrt{b^2-4 a c}\right)+2 c (-d+i)\right)}\right)}{\sqrt{b^2-4 a c}}-\frac{\cot ^{-1}(d+e x) \log \left(\frac{2 e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{(1-i (d+e x)) \left(e \left(\sqrt{b^2-4 a c}+b\right)+2 c (-d+i)\right)}\right)}{\sqrt{b^2-4 a c}}",1,"((-1/2*I)*(Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(-I + d + e*x))/(2*c*(-I + d) + (-b + Sqrt[b^2 - 4*a*c])*e)] - Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(-I + d + e*x))/(2*c*(-I + d) - (b + Sqrt[b^2 - 4*a*c])*e)] - Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(-I + d + e*x)/(d + e*x)] + Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(-I + d + e*x)/(d + e*x)] - Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(I + d + e*x))/(2*c*(I + d) + (-b + Sqrt[b^2 - 4*a*c])*e)] + Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(I + d + e*x))/(2*c*(I + d) - (b + Sqrt[b^2 - 4*a*c])*e)] + Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(I + d + e*x)/(d + e*x)] - Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(I + d + e*x)/(d + e*x)] + PolyLog[2, (e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*(-I + d) + (-b + Sqrt[b^2 - 4*a*c])*e)] - PolyLog[2, (e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*(I + d) + (-b + Sqrt[b^2 - 4*a*c])*e)] - PolyLog[2, (e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*(-I + d) + (b + Sqrt[b^2 - 4*a*c])*e)] + PolyLog[2, (e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*(I + d) + (b + Sqrt[b^2 - 4*a*c])*e)]))/Sqrt[b^2 - 4*a*c]","A",0
114,1,127,132,0.1600895,"\int \frac{\cot ^{-1}(a+b x)}{\sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx","Integrate[ArcCot[a + b*x]/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2],x]","-\frac{\sqrt{a^2+2 a b x+b^2 x^2+1} \left(i \text{Li}_2\left(-e^{i \cot ^{-1}(a+b x)}\right)-i \text{Li}_2\left(e^{i \cot ^{-1}(a+b x)}\right)+\cot ^{-1}(a+b x) \left(\log \left(1-e^{i \cot ^{-1}(a+b x)}\right)-\log \left(1+e^{i \cot ^{-1}(a+b x)}\right)\right)\right)}{b (a+b x) \sqrt{\frac{1}{(a+b x)^2}+1}}","-\frac{i \text{Li}_2\left(-\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{b}+\frac{i \text{Li}_2\left(\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{b}-\frac{2 i \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right) \cot ^{-1}(a+b x)}{b}",1,"-((Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(ArcCot[a + b*x]*(Log[1 - E^(I*ArcCot[a + b*x])] - Log[1 + E^(I*ArcCot[a + b*x])]) + I*PolyLog[2, -E^(I*ArcCot[a + b*x])] - I*PolyLog[2, E^(I*ArcCot[a + b*x])]))/(b*(a + b*x)*Sqrt[1 + (a + b*x)^(-2)]))","A",0
115,1,138,216,0.0915899,"\int \frac{\cot ^{-1}(a+b x)}{\sqrt{\left(1+a^2\right) c+2 a b c x+b^2 c x^2}} \, dx","Integrate[ArcCot[a + b*x]/Sqrt[(1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2],x]","-\frac{\left((a+b x)^2+1\right) \left(i \text{Li}_2\left(-e^{i \cot ^{-1}(a+b x)}\right)-i \text{Li}_2\left(e^{i \cot ^{-1}(a+b x)}\right)+\cot ^{-1}(a+b x) \left(\log \left(1-e^{i \cot ^{-1}(a+b x)}\right)-\log \left(1+e^{i \cot ^{-1}(a+b x)}\right)\right)\right)}{b (a+b x) \sqrt{\frac{1}{(a+b x)^2}+1} \sqrt{c \left(a^2+2 a b x+b^2 x^2+1\right)}}","-\frac{i \sqrt{(a+b x)^2+1} \text{Li}_2\left(-\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{b \sqrt{c (a+b x)^2+c}}+\frac{i \sqrt{(a+b x)^2+1} \text{Li}_2\left(\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{b \sqrt{c (a+b x)^2+c}}-\frac{2 i \sqrt{(a+b x)^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right) \cot ^{-1}(a+b x)}{b \sqrt{c (a+b x)^2+c}}",1,"-(((1 + (a + b*x)^2)*(ArcCot[a + b*x]*(Log[1 - E^(I*ArcCot[a + b*x])] - Log[1 + E^(I*ArcCot[a + b*x])]) + I*PolyLog[2, -E^(I*ArcCot[a + b*x])] - I*PolyLog[2, E^(I*ArcCot[a + b*x])]))/(b*(a + b*x)*Sqrt[c*(1 + a^2 + 2*a*b*x + b^2*x^2)]*Sqrt[1 + (a + b*x)^(-2)]))","A",0
116,1,177,23,0.4103404,"\int \frac{\cot ^{-1}(a+b x)}{\sqrt[3]{1+a^2+2 a b x+b^2 x^2}} \, dx","Integrate[ArcCot[a + b*x]/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3),x]","\frac{6 \Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \left(4 (a+b x) \, _2F_1\left(1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right) \cot ^{-1}(a+b x)+5 \left(a^2+2 a b x+b^2 x^2+1\right) \left(2 (a+b x) \cot ^{-1}(a+b x)-3\right)\right)-5 \sqrt[3]{2} \sqrt{\pi } \Gamma \left(\frac{5}{3}\right) \, _3F_2\left(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right)}{20 b \Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \left(a^2+2 a b x+b^2 x^2+1\right)^{4/3}}","\text{Int}\left(\frac{\cot ^{-1}(a+b x)}{\sqrt[3]{(a+b x)^2+1}},x\right)",0,"(6*Gamma[11/6]*Gamma[7/3]*(5*(1 + a^2 + 2*a*b*x + b^2*x^2)*(-3 + 2*(a + b*x)*ArcCot[a + b*x]) + 4*(a + b*x)*ArcCot[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) - 5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)])/(20*b*(1 + a^2 + 2*a*b*x + b^2*x^2)^(4/3)*Gamma[11/6]*Gamma[7/3])","B",0
117,1,180,25,0.1121065,"\int \frac{\cot ^{-1}(a+b x)}{\sqrt[3]{\left(1+a^2\right) c+2 a b c x+b^2 c x^2}} \, dx","Integrate[ArcCot[a + b*x]/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]","\frac{c \left(6 \Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \left(4 (a+b x) \, _2F_1\left(1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right) \cot ^{-1}(a+b x)+5 \left(a^2+2 a b x+b^2 x^2+1\right) \left(2 (a+b x) \cot ^{-1}(a+b x)-3\right)\right)-5 \sqrt[3]{2} \sqrt{\pi } \Gamma \left(\frac{5}{3}\right) \, _3F_2\left(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right)\right)}{20 b \Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \left(c \left(a^2+2 a b x+b^2 x^2+1\right)\right)^{4/3}}","\text{Int}\left(\frac{\cot ^{-1}(a+b x)}{\sqrt[3]{c (a+b x)^2+c}},x\right)",0,"(c*(6*Gamma[11/6]*Gamma[7/3]*(5*(1 + a^2 + 2*a*b*x + b^2*x^2)*(-3 + 2*(a + b*x)*ArcCot[a + b*x]) + 4*(a + b*x)*ArcCot[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) - 5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]))/(20*b*(c*(1 + a^2 + 2*a*b*x + b^2*x^2))^(4/3)*Gamma[11/6]*Gamma[7/3])","B",0
118,1,202,187,1.5461648,"\int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx","Integrate[((a + b*x)^2*ArcCot[a + b*x])/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2],x]","-\frac{\sqrt{(a+b x)^2 \left(\frac{1}{(a+b x)^2}+1\right)} \left(-4 i \text{Li}_2\left(-e^{i \cot ^{-1}(a+b x)}\right)+4 i \text{Li}_2\left(e^{i \cot ^{-1}(a+b x)}\right)-2 \cot \left(\frac{1}{2} \cot ^{-1}(a+b x)\right)-4 \cot ^{-1}(a+b x) \log \left(1-e^{i \cot ^{-1}(a+b x)}\right)+4 \cot ^{-1}(a+b x) \log \left(1+e^{i \cot ^{-1}(a+b x)}\right)-2 \tan \left(\frac{1}{2} \cot ^{-1}(a+b x)\right)-\cot ^{-1}(a+b x) \csc ^2\left(\frac{1}{2} \cot ^{-1}(a+b x)\right)+\cot ^{-1}(a+b x) \sec ^2\left(\frac{1}{2} \cot ^{-1}(a+b x)\right)\right)}{8 b (a+b x) \sqrt{\frac{1}{(a+b x)^2}+1}}","\frac{i \text{Li}_2\left(-\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{2 b}-\frac{i \text{Li}_2\left(\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{2 b}+\frac{\sqrt{(a+b x)^2+1}}{2 b}+\frac{(a+b x) \sqrt{(a+b x)^2+1} \cot ^{-1}(a+b x)}{2 b}+\frac{i \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right) \cot ^{-1}(a+b x)}{b}",1,"-1/8*(Sqrt[(a + b*x)^2*(1 + (a + b*x)^(-2))]*(-2*Cot[ArcCot[a + b*x]/2] - ArcCot[a + b*x]*Csc[ArcCot[a + b*x]/2]^2 - 4*ArcCot[a + b*x]*Log[1 - E^(I*ArcCot[a + b*x])] + 4*ArcCot[a + b*x]*Log[1 + E^(I*ArcCot[a + b*x])] - (4*I)*PolyLog[2, -E^(I*ArcCot[a + b*x])] + (4*I)*PolyLog[2, E^(I*ArcCot[a + b*x])] + ArcCot[a + b*x]*Sec[ArcCot[a + b*x]/2]^2 - 2*Tan[ArcCot[a + b*x]/2]))/(b*(a + b*x)*Sqrt[1 + (a + b*x)^(-2)])","A",0
119,1,207,281,0.9276807,"\int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt{\left(1+a^2\right) c+2 a b c x+b^2 c x^2}} \, dx","Integrate[((a + b*x)^2*ArcCot[a + b*x])/Sqrt[(1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2],x]","-\frac{\sqrt{c \left(a^2+2 a b x+b^2 x^2+1\right)} \left(-4 i \text{Li}_2\left(-e^{i \cot ^{-1}(a+b x)}\right)+4 i \text{Li}_2\left(e^{i \cot ^{-1}(a+b x)}\right)-2 \cot \left(\frac{1}{2} \cot ^{-1}(a+b x)\right)-4 \cot ^{-1}(a+b x) \log \left(1-e^{i \cot ^{-1}(a+b x)}\right)+4 \cot ^{-1}(a+b x) \log \left(1+e^{i \cot ^{-1}(a+b x)}\right)-2 \tan \left(\frac{1}{2} \cot ^{-1}(a+b x)\right)-\cot ^{-1}(a+b x) \csc ^2\left(\frac{1}{2} \cot ^{-1}(a+b x)\right)+\cot ^{-1}(a+b x) \sec ^2\left(\frac{1}{2} \cot ^{-1}(a+b x)\right)\right)}{8 b c (a+b x) \sqrt{\frac{1}{(a+b x)^2}+1}}","\frac{i \sqrt{(a+b x)^2+1} \text{Li}_2\left(-\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{2 b \sqrt{c (a+b x)^2+c}}-\frac{i \sqrt{(a+b x)^2+1} \text{Li}_2\left(\frac{i \sqrt{i (a+b x)+1}}{\sqrt{1-i (a+b x)}}\right)}{2 b \sqrt{c (a+b x)^2+c}}+\frac{\sqrt{c (a+b x)^2+c}}{2 b c}+\frac{(a+b x) \sqrt{c (a+b x)^2+c} \cot ^{-1}(a+b x)}{2 b c}+\frac{i \sqrt{(a+b x)^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right) \cot ^{-1}(a+b x)}{b \sqrt{c (a+b x)^2+c}}",1,"-1/8*(Sqrt[c*(1 + a^2 + 2*a*b*x + b^2*x^2)]*(-2*Cot[ArcCot[a + b*x]/2] - ArcCot[a + b*x]*Csc[ArcCot[a + b*x]/2]^2 - 4*ArcCot[a + b*x]*Log[1 - E^(I*ArcCot[a + b*x])] + 4*ArcCot[a + b*x]*Log[1 + E^(I*ArcCot[a + b*x])] - (4*I)*PolyLog[2, -E^(I*ArcCot[a + b*x])] + (4*I)*PolyLog[2, E^(I*ArcCot[a + b*x])] + ArcCot[a + b*x]*Sec[ArcCot[a + b*x]/2]^2 - 2*Tan[ArcCot[a + b*x]/2]))/(b*c*(a + b*x)*Sqrt[1 + (a + b*x)^(-2)])","A",0
120,1,198,30,1.0031921,"\int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt[3]{1+a^2+2 a b x+b^2 x^2}} \, dx","Integrate[((a + b*x)^2*ArcCot[a + b*x])/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3),x]","\frac{3 \left(5 \sqrt[3]{2} \sqrt{\pi } \Gamma \left(\frac{5}{3}\right) \, _3F_2\left(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right)+\Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \left(5 \left((a+b x)^2+1\right) \left(3 \left((a+b x)^2+7\right)+4 (a+b x) \left((a+b x)^2-2\right) \cot ^{-1}(a+b x)\right)-24 (a+b x) \, _2F_1\left(1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right) \cot ^{-1}(a+b x)\right)\right)}{140 b \Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \sqrt[3]{a^2+2 a b x+b^2 x^2+1} \left((a+b x)^2+1\right)}","\text{Int}\left(\frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt[3]{(a+b x)^2+1}},x\right)",0,"(3*(Gamma[11/6]*Gamma[7/3]*(5*(1 + (a + b*x)^2)*(3*(7 + (a + b*x)^2) + 4*(a + b*x)*(-2 + (a + b*x)^2)*ArcCot[a + b*x]) - 24*(a + b*x)*ArcCot[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) + 5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]))/(140*b*(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3)*(1 + (a + b*x)^2)*Gamma[11/6]*Gamma[7/3])","B",0
121,1,200,32,0.2247707,"\int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt[3]{\left(1+a^2\right) c+2 a b c x+b^2 c x^2}} \, dx","Integrate[((a + b*x)^2*ArcCot[a + b*x])/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]","\frac{3 \left(5 \sqrt[3]{2} \sqrt{\pi } \Gamma \left(\frac{5}{3}\right) \, _3F_2\left(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right)+\Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \left(5 \left((a+b x)^2+1\right) \left(3 \left((a+b x)^2+7\right)+4 (a+b x) \left((a+b x)^2-2\right) \cot ^{-1}(a+b x)\right)-24 (a+b x) \, _2F_1\left(1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right) \cot ^{-1}(a+b x)\right)\right)}{140 b \Gamma \left(\frac{11}{6}\right) \Gamma \left(\frac{7}{3}\right) \left((a+b x)^2+1\right) \sqrt[3]{c \left(a^2+2 a b x+b^2 x^2+1\right)}}","\text{Int}\left(\frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt[3]{c (a+b x)^2+c}},x\right)",0,"(3*(Gamma[11/6]*Gamma[7/3]*(5*(1 + (a + b*x)^2)*(3*(7 + (a + b*x)^2) + 4*(a + b*x)*(-2 + (a + b*x)^2)*ArcCot[a + b*x]) - 24*(a + b*x)*ArcCot[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) + 5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]))/(140*b*(c*(1 + a^2 + 2*a*b*x + b^2*x^2))^(1/3)*(1 + (a + b*x)^2)*Gamma[11/6]*Gamma[7/3])","B",0
122,1,42,52,0.0147075,"\int (a+b x)^2 \cot ^{-1}(a+b x) \, dx","Integrate[(a + b*x)^2*ArcCot[a + b*x],x]","\frac{(a+b x)^2-\log \left((a+b x)^2+1\right)+2 (a+b x)^3 \cot ^{-1}(a+b x)}{6 b}","\frac{(a+b x)^2}{6 b}-\frac{\log \left((a+b x)^2+1\right)}{6 b}+\frac{(a+b x)^3 \cot ^{-1}(a+b x)}{3 b}",1,"((a + b*x)^2 + 2*(a + b*x)^3*ArcCot[a + b*x] - Log[1 + (a + b*x)^2])/(6*b)","A",1
123,1,141,39,0.0592707,"\int (a+b x) \cot ^{-1}(a+b x) \, dx","Integrate[(a + b*x)*ArcCot[a + b*x],x]","\frac{a \left(\log \left(a^2+2 a b x+b^2 x^2+1\right)-2 a \tan ^{-1}(a+b x)\right)}{2 b}+\frac{1}{2} b \left(-\frac{i (-a+i)^2 \log (-a-b x+i)}{2 b^2}+\frac{i (a+i)^2 \log (a+b x+i)}{2 b^2}+\frac{x}{b}\right)+\frac{1}{2} b \left(\frac{a+b x}{b}-\frac{a}{b}\right)^2 \cot ^{-1}(a+b x)+a x \cot ^{-1}(a+b x)","-\frac{\tan ^{-1}(a+b x)}{2 b}+\frac{(a+b x)^2 \cot ^{-1}(a+b x)}{2 b}+\frac{x}{2}",1,"a*x*ArcCot[a + b*x] + (b*(-(a/b) + (a + b*x)/b)^2*ArcCot[a + b*x])/2 + (b*(x/b - ((I/2)*(I - a)^2*Log[I - a - b*x])/b^2 + ((I/2)*(I + a)^2*Log[I + a + b*x])/b^2))/2 + (a*(-2*a*ArcTan[a + b*x] + Log[1 + a^2 + 2*a*b*x + b^2*x^2]))/(2*b)","C",1
124,1,38,45,0.0083503,"\int \frac{\cot ^{-1}(a+b x)}{a+b x} \, dx","Integrate[ArcCot[a + b*x]/(a + b*x),x]","-\frac{i \left(\text{Li}_2\left(-\frac{i}{a+b x}\right)-\text{Li}_2\left(\frac{i}{a+b x}\right)\right)}{2 b}","\frac{i \text{Li}_2\left(\frac{i}{a+b x}\right)}{2 b}-\frac{i \text{Li}_2\left(-\frac{i}{a+b x}\right)}{2 b}",1,"((-1/2*I)*(PolyLog[2, (-I)/(a + b*x)] - PolyLog[2, I/(a + b*x)]))/b","A",1
125,1,40,47,0.0209799,"\int \frac{\cot ^{-1}(a+b x)}{(a+b x)^2} \, dx","Integrate[ArcCot[a + b*x]/(a + b*x)^2,x]","-\frac{\log (a+b x)-\frac{1}{2} \log \left((a+b x)^2+1\right)+\frac{\cot ^{-1}(a+b x)}{a+b x}}{b}","-\frac{\log (a+b x)}{b}+\frac{\log \left((a+b x)^2+1\right)}{2 b}-\frac{\cot ^{-1}(a+b x)}{b (a+b x)}",1,"-((ArcCot[a + b*x]/(a + b*x) + Log[a + b*x] - Log[1 + (a + b*x)^2]/2)/b)","A",1
126,1,35,35,0.005346,"\int \frac{\cot ^{-1}(1+x)}{2+2 x} \, dx","Integrate[ArcCot[1 + x]/(2 + 2*x),x]","\frac{1}{4} i \text{Li}_2\left(\frac{i}{x+1}\right)-\frac{1}{4} i \text{Li}_2\left(-\frac{i}{x+1}\right)","\frac{1}{4} i \text{Li}_2\left(\frac{i}{x+1}\right)-\frac{1}{4} i \text{Li}_2\left(-\frac{i}{x+1}\right)",1,"(-1/4*I)*PolyLog[2, (-I)/(1 + x)] + (I/4)*PolyLog[2, I/(1 + x)]","A",1
127,1,38,45,0.0081495,"\int \frac{\cot ^{-1}(a+b x)}{\frac{a d}{b}+d x} \, dx","Integrate[ArcCot[a + b*x]/((a*d)/b + d*x),x]","-\frac{i \left(\text{Li}_2\left(-\frac{i}{a+b x}\right)-\text{Li}_2\left(\frac{i}{a+b x}\right)\right)}{2 d}","\frac{i \text{Li}_2\left(\frac{i}{a+b x}\right)}{2 d}-\frac{i \text{Li}_2\left(-\frac{i}{a+b x}\right)}{2 d}",1,"((-1/2*I)*(PolyLog[2, (-I)/(a + b*x)] - PolyLog[2, I/(a + b*x)]))/d","A",1
128,0,0,21,9.2581674,"\int (a+b x)^2 \sqrt{\cot ^{-1}(a+b x)} \, dx","Integrate[(a + b*x)^2*Sqrt[ArcCot[a + b*x]],x]","\int (a+b x)^2 \sqrt{\cot ^{-1}(a+b x)} \, dx","\text{Int}\left((a+b x)^2 \sqrt{\cot ^{-1}(a+b x)},x\right)",0,"Integrate[(a + b*x)^2*Sqrt[ArcCot[a + b*x]], x]","A",-1
129,1,157,233,0.291145,"\int (e+f x)^3 \left(a+b \cot ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)^3*(a + b*ArcCot[c + d*x]),x]","\frac{(e+f x)^4 \left(a+b \cot ^{-1}(c+d x)\right)+\frac{b \left(6 d f^2 x \left(\left(6 c^2-1\right) f^2-12 c d e f+6 d^2 e^2\right)+12 f^3 (c+d x)^2 (d e-c f)-3 i (d e-(c-i) f)^4 \log (-c-d x+i)+3 i (d e-(c+i) f)^4 \log (c+d x+i)+2 f^4 (c+d x)^3\right)}{6 d^4}}{4 f}","\frac{(e+f x)^4 \left(a+b \cot ^{-1}(c+d x)\right)}{4 f}+\frac{b f x \left(-\left(1-6 c^2\right) f^2-12 c d e f+6 d^2 e^2\right)}{4 d^3}+\frac{b \left(-6 \left(1-c^2\right) d^2 e^2 f^2+4 c \left(3-c^2\right) d e f^3+\left(c^4-6 c^2+1\right) f^4-4 c d^3 e^3 f+d^4 e^4\right) \tan ^{-1}(c+d x)}{4 d^4 f}+\frac{b f^2 (c+d x)^2 (d e-c f)}{2 d^4}+\frac{b (d e-c f) (-c f+d e+f) (d e-(c+1) f) \log \left((c+d x)^2+1\right)}{2 d^4}+\frac{b f^3 (c+d x)^3}{12 d^4}",1,"((e + f*x)^4*(a + b*ArcCot[c + d*x]) + (b*(6*d*f^2*(6*d^2*e^2 - 12*c*d*e*f + (-1 + 6*c^2)*f^2)*x + 12*f^3*(d*e - c*f)*(c + d*x)^2 + 2*f^4*(c + d*x)^3 - (3*I)*(d*e - (-I + c)*f)^4*Log[I - c - d*x] + (3*I)*(d*e - (I + c)*f)^4*Log[I + c + d*x]))/(6*d^4))/(4*f)","C",1
130,1,118,154,0.1575194,"\int (e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)^2*(a + b*ArcCot[c + d*x]),x]","\frac{(e+f x)^3 \left(a+b \cot ^{-1}(c+d x)\right)+\frac{b \left(6 d f^2 x (d e-c f)-i (d e-(c-i) f)^3 \log (-c-d x+i)+i (d e-(c+i) f)^3 \log (c+d x+i)+f^3 (c+d x)^2\right)}{2 d^3}}{3 f}","\frac{(e+f x)^3 \left(a+b \cot ^{-1}(c+d x)\right)}{3 f}+\frac{b \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left((c+d x)^2+1\right)}{6 d^3}+\frac{b (d e-c f) \left(-\left(3-c^2\right) f^2-2 c d e f+d^2 e^2\right) \tan ^{-1}(c+d x)}{3 d^3 f}+\frac{b f^2 (c+d x)^2}{6 d^3}+\frac{b f x (d e-c f)}{d^2}",1,"((e + f*x)^3*(a + b*ArcCot[c + d*x]) + (b*(6*d*f^2*(d*e - c*f)*x + f^3*(c + d*x)^2 - I*(d*e - (-I + c)*f)^3*Log[I - c - d*x] + I*(d*e - (I + c)*f)^3*Log[I + c + d*x]))/(2*d^3))/(3*f)","C",1
131,1,163,97,0.0935155,"\int (e+f x) \left(a+b \cot ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)*(a + b*ArcCot[c + d*x]),x]","a e x+\frac{1}{2} a f x^2+\frac{b e \left(\log \left(c^2+2 c d x+d^2 x^2+1\right)-2 c \tan ^{-1}(c+d x)\right)}{2 d}+\frac{b f \left(\frac{1}{2} d \left(\frac{c+d x}{d}-\frac{c}{d}\right)^2 \cot ^{-1}(c+d x)+\frac{1}{2} d \left(-\frac{i (-c+i)^2 \log (-c-d x+i)}{2 d^2}+\frac{i (c+i)^2 \log (c+d x+i)}{2 d^2}+\frac{x}{d}\right)\right)}{d}+b e x \cot ^{-1}(c+d x)","\frac{(e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right)}{2 f}+\frac{b (d e-c f) \log \left((c+d x)^2+1\right)}{2 d^2}+\frac{b (-c f+d e+f) (d e-(c+1) f) \tan ^{-1}(c+d x)}{2 d^2 f}+\frac{b f x}{2 d}",1,"a*e*x + (a*f*x^2)/2 + b*e*x*ArcCot[c + d*x] + (b*f*((d*(-(c/d) + (c + d*x)/d)^2*ArcCot[c + d*x])/2 + (d*(x/d - ((I/2)*(I - c)^2*Log[I - c - d*x])/d^2 + ((I/2)*(I + c)^2*Log[I + c + d*x])/d^2))/2))/d + (b*e*(-2*c*ArcTan[c + d*x] + Log[1 + c^2 + 2*c*d*x + d^2*x^2]))/(2*d)","C",1
132,1,49,38,0.0138335,"\int \left(a+b \cot ^{-1}(c+d x)\right) \, dx","Integrate[a + b*ArcCot[c + d*x],x]","a x+\frac{b \left(\log \left(c^2+2 c d x+d^2 x^2+1\right)-2 c \tan ^{-1}(c+d x)\right)}{2 d}+b x \cot ^{-1}(c+d x)","a x+\frac{b \log \left((c+d x)^2+1\right)}{2 d}+\frac{b (c+d x) \cot ^{-1}(c+d x)}{d}",1,"a*x + b*x*ArcCot[c + d*x] + (b*(-2*c*ArcTan[c + d*x] + Log[1 + c^2 + 2*c*d*x + d^2*x^2]))/(2*d)","A",1
133,1,304,162,0.1085256,"\int \frac{a+b \cot ^{-1}(c+d x)}{e+f x} \, dx","Integrate[(a + b*ArcCot[c + d*x])/(e + f*x),x]","\frac{a \log (f (c+d x)-c f+d e)}{f}-\frac{i b \text{Li}_2\left(\frac{d e-c f+f (c+d x)}{d e+(i-c) f}\right)}{2 f}+\frac{i b \text{Li}_2\left(\frac{d e-c f+f (c+d x)}{d e-(c+i) f}\right)}{2 f}-\frac{i b \log \left(\frac{f (-c-d x+i)}{d e+(-c+i) f}\right) \log (f (c+d x)-c f+d e)}{2 f}+\frac{i b \log \left(-\frac{-c-d x+i}{c+d x}\right) \log (f (c+d x)-c f+d e)}{2 f}+\frac{i b \log \left(-\frac{f (c+d x+i)}{d e-(c+i) f}\right) \log (f (c+d x)-c f+d e)}{2 f}-\frac{i b \log \left(\frac{c+d x+i}{c+d x}\right) \log (f (c+d x)-c f+d e)}{2 f}","\frac{\left(a+b \cot ^{-1}(c+d x)\right) \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{f}-\frac{\log \left(\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{f}+\frac{i b \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right)}{2 f}-\frac{i b \text{Li}_2\left(1-\frac{2}{1-i (c+d x)}\right)}{2 f}",1,"(a*Log[d*e - c*f + f*(c + d*x)])/f - ((I/2)*b*Log[(f*(I - c - d*x))/(d*e + (I - c)*f)]*Log[d*e - c*f + f*(c + d*x)])/f + ((I/2)*b*Log[-((I - c - d*x)/(c + d*x))]*Log[d*e - c*f + f*(c + d*x)])/f + ((I/2)*b*Log[-((f*(I + c + d*x))/(d*e - (I + c)*f))]*Log[d*e - c*f + f*(c + d*x)])/f - ((I/2)*b*Log[(I + c + d*x)/(c + d*x)]*Log[d*e - c*f + f*(c + d*x)])/f - ((I/2)*b*PolyLog[2, (d*e - c*f + f*(c + d*x))/(d*e + (I - c)*f)])/f + ((I/2)*b*PolyLog[2, (d*e - c*f + f*(c + d*x))/(d*e - (I + c)*f)])/f","A",0
134,1,118,153,0.1987312,"\int \frac{a+b \cot ^{-1}(c+d x)}{(e+f x)^2} \, dx","Integrate[(a + b*ArcCot[c + d*x])/(e + f*x)^2,x]","\frac{-\frac{a+b \cot ^{-1}(c+d x)}{e+f x}+\frac{b d ((-i c f+i d e+f) \log (-c-d x+i)+(i c f-i d e+f) \log (c+d x+i)-2 f \log (d (e+f x)))}{2 \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}}{f}","-\frac{a+b \cot ^{-1}(c+d x)}{f (e+f x)}+\frac{b d \log \left(c^2+2 c d x+d^2 x^2+1\right)}{2 \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}-\frac{b d \log (e+f x)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{b d (d e-c f) \tan ^{-1}(c+d x)}{f \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}",1,"(-((a + b*ArcCot[c + d*x])/(e + f*x)) + (b*d*((I*d*e + f - I*c*f)*Log[I - c - d*x] + ((-I)*d*e + f + I*c*f)*Log[I + c + d*x] - 2*f*Log[d*(e + f*x)]))/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)))/f","C",1
135,1,180,228,0.6188177,"\int \frac{a+b \cot ^{-1}(c+d x)}{(e+f x)^3} \, dx","Integrate[(a + b*ArcCot[c + d*x])/(e + f*x)^3,x]","\frac{-\frac{a+b \cot ^{-1}(c+d x)}{(e+f x)^2}+\frac{b d f}{(e+f x) \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}-\frac{2 b d^2 f (d e-c f) \log (d (e+f x))}{\left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)^2}+\frac{i b d^2 \log (-c-d x+i)}{2 (d e-(c-i) f)^2}-\frac{i b d^2 \log (c+d x+i)}{2 (d e-(c+i) f)^2}}{2 f}","-\frac{a+b \cot ^{-1}(c+d x)}{2 f (e+f x)^2}+\frac{b d^2 (d e-c f) \log \left(c^2+2 c d x+d^2 x^2+1\right)}{2 \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)^2}+\frac{b d}{2 (e+f x) \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}-\frac{b d^2 (d e-c f) \log (e+f x)}{\left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)^2}-\frac{b d^2 (-c f+d e+f) (d e-(c+1) f) \tan ^{-1}(c+d x)}{2 f \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)^2}",1,"((b*d*f)/((d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(e + f*x)) - (a + b*ArcCot[c + d*x])/(e + f*x)^2 + ((I/2)*b*d^2*Log[I - c - d*x])/(d*e - (-I + c)*f)^2 - ((I/2)*b*d^2*Log[I + c + d*x])/(d*e - (I + c)*f)^2 - (2*b*d^2*f*(d*e - c*f)*Log[d*(e + f*x)])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^2)/(2*f)","C",1
136,1,665,382,5.3700071,"\int (e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right)^2 \, dx","Integrate[(e + f*x)^2*(a + b*ArcCot[c + d*x])^2,x]","a^2 e^2 x+a^2 e f x^2+\frac{1}{3} a^2 f^2 x^3+\frac{a b \left(\left(\left(3 c^2-1\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left(c^2+2 c d x+d^2 x^2+1\right)-2 \left(c^3 f^2-3 c^2 d e f+3 c d^2 e^2-3 c f^2+3 d e f\right) \tan ^{-1}(c+d x)+2 d^3 x \left(3 e^2+3 e f x+f^2 x^2\right) \cot ^{-1}(c+d x)+d f x (-4 c f+6 d e+d f x)\right)}{3 d^3}+\frac{b^2 f^2 \left(4 i \left(3 c^2-1\right) \text{Li}_2\left(e^{2 i \cot ^{-1}(c+d x)}\right)+(c+d x) \left((c+d x)^2+1\right) \left(3 \left(c^2+1\right) \cot ^{-1}(c+d x)^2-6 c \cot ^{-1}(c+d x)+1\right)-(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} \left((c+d x)^2+1\right) \left(\left(3 c^2-1\right) \cot ^{-1}(c+d x)^2-6 c \cot ^{-1}(c+d x)+1\right) \cos \left(3 \cot ^{-1}(c+d x)\right)+2 \left((c+d x)^2+1\right) \left(-i \cot ^{-1}(c+d x)^2 \left(\left(3 c^2-1\right) \cos \left(2 \cot ^{-1}(c+d x)\right)-3 c^2-6 i c+1\right)+2 \cot ^{-1}(c+d x) \left(\left(1-3 c^2\right) \log \left(1-e^{2 i \cot ^{-1}(c+d x)}\right)+\left(3 c^2-1\right) \cos \left(2 \cot ^{-1}(c+d x)\right) \log \left(1-e^{2 i \cot ^{-1}(c+d x)}\right)+1\right)-6 c \log \left(\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right) \left(\cos \left(2 \cot ^{-1}(c+d x)\right)-1\right)\right)\right)}{12 d^3}+\frac{b^2 e f \left(\left(-c^2-2 i c+d^2 x^2+1\right) \cot ^{-1}(c+d x)^2-2 i c \text{Li}_2\left(e^{2 i \cot ^{-1}(c+d x)}\right)-2 \log \left(\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right)+2 \cot ^{-1}(c+d x) \left(2 c \log \left(1-e^{2 i \cot ^{-1}(c+d x)}\right)+c+d x\right)\right)}{d^2}+\frac{b^2 e^2 \left(i \text{Li}_2\left(e^{2 i \cot ^{-1}(c+d x)}\right)+\cot ^{-1}(c+d x) \left((c+d x+i) \cot ^{-1}(c+d x)-2 \log \left(1-e^{2 i \cot ^{-1}(c+d x)}\right)\right)\right)}{d}","\frac{i \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{3 d^3}-\frac{(d e-c f) \left(-\left(3-c^2\right) f^2-2 c d e f+d^2 e^2\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{3 d^3 f}-\frac{2 b \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{3 d^3}+\frac{b f^2 (c+d x)^2 \left(a+b \cot ^{-1}(c+d x)\right)}{3 d^3}+\frac{2 a b f x (d e-c f)}{d^2}+\frac{(e+f x)^3 \left(a+b \cot ^{-1}(c+d x)\right)^2}{3 f}+\frac{i b^2 \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right)}{3 d^3}+\frac{b^2 f (d e-c f) \log \left((c+d x)^2+1\right)}{d^3}+\frac{2 b^2 f (c+d x) (d e-c f) \cot ^{-1}(c+d x)}{d^3}-\frac{b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}+\frac{b^2 f^2 x}{3 d^2}",1,"a^2*e^2*x + a^2*e*f*x^2 + (a^2*f^2*x^3)/3 + (a*b*(d*f*x*(6*d*e - 4*c*f + d*f*x) + 2*d^3*x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCot[c + d*x] - 2*(3*c*d^2*e^2 + 3*d*e*f - 3*c^2*d*e*f - 3*c*f^2 + c^3*f^2)*ArcTan[c + d*x] + (3*d^2*e^2 - 6*c*d*e*f + (-1 + 3*c^2)*f^2)*Log[1 + c^2 + 2*c*d*x + d^2*x^2]))/(3*d^3) + (b^2*e^2*(ArcCot[c + d*x]*((I + c + d*x)*ArcCot[c + d*x] - 2*Log[1 - E^((2*I)*ArcCot[c + d*x])]) + I*PolyLog[2, E^((2*I)*ArcCot[c + d*x])]))/d + (b^2*e*f*((1 - (2*I)*c - c^2 + d^2*x^2)*ArcCot[c + d*x]^2 + 2*ArcCot[c + d*x]*(c + d*x + 2*c*Log[1 - E^((2*I)*ArcCot[c + d*x])]) - 2*Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])] - (2*I)*c*PolyLog[2, E^((2*I)*ArcCot[c + d*x])]))/d^2 + (b^2*f^2*((c + d*x)*(1 + (c + d*x)^2)*(1 - 6*c*ArcCot[c + d*x] + 3*(1 + c^2)*ArcCot[c + d*x]^2) - (c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(1 + (c + d*x)^2)*(1 - 6*c*ArcCot[c + d*x] + (-1 + 3*c^2)*ArcCot[c + d*x]^2)*Cos[3*ArcCot[c + d*x]] + 2*(1 + (c + d*x)^2)*((-I)*ArcCot[c + d*x]^2*(1 - (6*I)*c - 3*c^2 + (-1 + 3*c^2)*Cos[2*ArcCot[c + d*x]]) + 2*ArcCot[c + d*x]*(1 + (1 - 3*c^2)*Log[1 - E^((2*I)*ArcCot[c + d*x])] + (-1 + 3*c^2)*Cos[2*ArcCot[c + d*x]]*Log[1 - E^((2*I)*ArcCot[c + d*x])]) - 6*c*(-1 + Cos[2*ArcCot[c + d*x]])*Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])]) + (4*I)*(-1 + 3*c^2)*PolyLog[2, E^((2*I)*ArcCot[c + d*x])]))/(12*d^3)","A",0
137,1,286,220,0.5846713,"\int (e+f x) \left(a+b \cot ^{-1}(c+d x)\right)^2 \, dx","Integrate[(e + f*x)*(a + b*ArcCot[c + d*x])^2,x]","\frac{-a^2 c^2 f+2 a^2 c d e+2 a^2 d^2 e x+a^2 d^2 f x^2+2 b \cot ^{-1}(c+d x) \left(-((c+d x) (a c f-a d (2 e+f x)-b f))-2 b (d e-c f) \log \left(1-e^{2 i \cot ^{-1}(c+d x)}\right)\right)-4 a b d e \log \left(\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right)+4 a b c f \log \left(\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right)-2 a b f \tan ^{-1}(c+d x)+2 a b c f+2 a b d f x+2 i b^2 (d e-c f) \text{Li}_2\left(e^{2 i \cot ^{-1}(c+d x)}\right)+b^2 (c+d x+i) \cot ^{-1}(c+d x)^2 (d (2 e+f x)-(c+i) f)-2 b^2 f \log \left(\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right)}{2 d^2}","\frac{i (d e-c f) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^2}-\frac{(-c f+d e+f) (d e-(c+1) f) \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^2 f}-\frac{2 b (d e-c f) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^2}+\frac{(e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 f}+\frac{a b f x}{d}+\frac{i b^2 (d e-c f) \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right)}{d^2}+\frac{b^2 f \log \left((c+d x)^2+1\right)}{2 d^2}+\frac{b^2 f (c+d x) \cot ^{-1}(c+d x)}{d^2}",1,"(2*a^2*c*d*e + 2*a*b*c*f - a^2*c^2*f + 2*a^2*d^2*e*x + 2*a*b*d*f*x + a^2*d^2*f*x^2 + b^2*(I + c + d*x)*(-((I + c)*f) + d*(2*e + f*x))*ArcCot[c + d*x]^2 - 2*a*b*f*ArcTan[c + d*x] + 2*b*ArcCot[c + d*x]*(-((c + d*x)*(-(b*f) + a*c*f - a*d*(2*e + f*x))) - 2*b*(d*e - c*f)*Log[1 - E^((2*I)*ArcCot[c + d*x])]) - 4*a*b*d*e*Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])] - 2*b^2*f*Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])] + 4*a*b*c*f*Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])] + (2*I)*b^2*(d*e - c*f)*PolyLog[2, E^((2*I)*ArcCot[c + d*x])])/(2*d^2)","A",0
138,1,118,102,0.1698604,"\int \left(a+b \cot ^{-1}(c+d x)\right)^2 \, dx","Integrate[(a + b*ArcCot[c + d*x])^2,x]","\frac{a \left(a c+a d x-2 b \log \left(\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right)\right)+2 b \cot ^{-1}(c+d x) \left(a c+a d x-b \log \left(1-e^{2 i \cot ^{-1}(c+d x)}\right)\right)+i b^2 \text{Li}_2\left(e^{2 i \cot ^{-1}(c+d x)}\right)+b^2 (c+d x+i) \cot ^{-1}(c+d x)^2}{d}","\frac{(c+d x) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d}+\frac{i \left(a+b \cot ^{-1}(c+d x)\right)^2}{d}-\frac{2 b \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d}+\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right)}{d}",1,"(b^2*(I + c + d*x)*ArcCot[c + d*x]^2 + 2*b*ArcCot[c + d*x]*(a*c + a*d*x - b*Log[1 - E^((2*I)*ArcCot[c + d*x])]) + a*(a*c + a*d*x - 2*b*Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])]) + I*b^2*PolyLog[2, E^((2*I)*ArcCot[c + d*x])])/d","A",0
139,0,0,261,7.0370822,"\int \frac{\left(a+b \cot ^{-1}(c+d x)\right)^2}{e+f x} \, dx","Integrate[(a + b*ArcCot[c + d*x])^2/(e + f*x),x]","\int \frac{\left(a+b \cot ^{-1}(c+d x)\right)^2}{e+f x} \, dx","\frac{i b \left(a+b \cot ^{-1}(c+d x)\right) \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right)}{f}+\frac{\left(a+b \cot ^{-1}(c+d x)\right)^2 \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{f}-\frac{i b \text{Li}_2\left(1-\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{f}-\frac{\log \left(\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{f}+\frac{b^2 \text{Li}_3\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right)}{2 f}-\frac{b^2 \text{Li}_3\left(1-\frac{2}{1-i (c+d x)}\right)}{2 f}",1,"Integrate[(a + b*ArcCot[c + d*x])^2/(e + f*x), x]","F",-1
140,1,454,567,9.5037156,"\int \frac{\left(a+b \cot ^{-1}(c+d x)\right)^2}{(e+f x)^2} \, dx","Integrate[(a + b*ArcCot[c + d*x])^2/(e + f*x)^2,x]","-\frac{a^2+\frac{2 a b f \left(\cot ^{-1}(c+d x) \left(c^2 f-c d e+c d f x-d^2 e x+f\right)+d (e+f x) \log \left(-\frac{d (e+f x)}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right)\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{b^2 d \left((c+d x)^2+1\right) (e+f x) \left(\frac{f \left(-i \text{Li}_2\left(\exp \left(2 i \left(\cot ^{-1}(c+d x)+\tan ^{-1}\left(\frac{f}{d e-c f}\right)\right)\right)\right)+2 \cot ^{-1}(c+d x) \log \left(1-\exp \left(2 i \left(\tan ^{-1}\left(\frac{f}{d e-c f}\right)+\cot ^{-1}(c+d x)\right)\right)\right)+2 \tan ^{-1}\left(\frac{f}{c f-d e}\right) \left(-\log \left(1-\exp \left(2 i \left(\tan ^{-1}\left(\frac{f}{d e-c f}\right)+\cot ^{-1}(c+d x)\right)\right)\right)+\log \left(\sin \left(\tan ^{-1}\left(\frac{f}{d e-c f}\right)+\cot ^{-1}(c+d x)\right)\right)+i \cot ^{-1}(c+d x)\right)-\pi  \log \left(\frac{1}{\sqrt{\frac{1}{(c+d x)^2}+1}}\right)+i \pi  \cot ^{-1}(c+d x)+\pi  \log \left(1+e^{-2 i \cot ^{-1}(c+d x)}\right)\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{\cot ^{-1}(c+d x)^2 e^{i \tan ^{-1}\left(\frac{f}{d e-c f}\right)}}{(c f-d e) \sqrt{\frac{f^2}{(d e-c f)^2}+1}}+\frac{\cot ^{-1}(c+d x)^2}{d e+d f x}\right)}{(c+d x)^2 \left(\frac{1}{(c+d x)^2}+1\right)}}{f (e+f x)}","-\frac{2 a b d \log (e+f x)}{(d e-c f)^2+f^2}+\frac{a b d \log \left((c+d x)^2+1\right)}{(d e-c f)^2+f^2}-\frac{2 a b d (d e-c f) \tan ^{-1}(c+d x)}{f \left((d e-c f)^2+f^2\right)}-\frac{\left(a+b \cot ^{-1}(c+d x)\right)^2}{f (e+f x)}+\frac{i b^2 d \text{Li}_2\left(1-\frac{2}{1-i (c+d x)}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{i b^2 d \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{i b^2 d \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{i b^2 d \cot ^{-1}(c+d x)^2}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}+\frac{b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right)}+\frac{2 b^2 d \log \left(\frac{2}{1-i (c+d x)}\right) \cot ^{-1}(c+d x)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{2 b^2 d \cot ^{-1}(c+d x) \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}-\frac{2 b^2 d \log \left(\frac{2}{1+i (c+d x)}\right) \cot ^{-1}(c+d x)}{\left(c^2+1\right) f^2-2 c d e f+d^2 e^2}",1,"-((a^2 + (2*a*b*f*((-(c*d*e) + f + c^2*f - d^2*e*x + c*d*f*x)*ArcCot[c + d*x] + d*(e + f*x)*Log[-((d*(e + f*x))/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]))]))/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (b^2*d*(e + f*x)*(1 + (c + d*x)^2)*((E^(I*ArcTan[f/(d*e - c*f)])*ArcCot[c + d*x]^2)/((-(d*e) + c*f)*Sqrt[1 + f^2/(d*e - c*f)^2]) + ArcCot[c + d*x]^2/(d*e + d*f*x) + (f*(I*Pi*ArcCot[c + d*x] + Pi*Log[1 + E^((-2*I)*ArcCot[c + d*x])] + 2*ArcCot[c + d*x]*Log[1 - E^((2*I)*(ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]))] - Pi*Log[1/Sqrt[1 + (c + d*x)^(-2)]] + 2*ArcTan[f/(-(d*e) + c*f)]*(I*ArcCot[c + d*x] - Log[1 - E^((2*I)*(ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]))] + Log[Sin[ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]]]) - I*PolyLog[2, E^((2*I)*(ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]))]))/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)))/((c + d*x)^2*(1 + (c + d*x)^(-2))))/(f*(e + f*x)))","A",0
141,1,2336,565,11.4286601,"\int (e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right)^3 \, dx","Integrate[(e + f*x)^2*(a + b*ArcCot[c + d*x])^3,x]","\text{Result too large to show}","\frac{i b^2 \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^3}-\frac{6 b^2 f (d e-c f) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^3}+\frac{a b^2 f^2 x}{d^2}+\frac{i \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \left(a+b \cot ^{-1}(c+d x)\right)^3}{3 d^3}-\frac{(d e-c f) \left(-\left(3-c^2\right) f^2-2 c d e f+d^2 e^2\right) \left(a+b \cot ^{-1}(c+d x)\right)^3}{3 d^3 f}-\frac{b \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^3}+\frac{3 i b f (d e-c f) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^3}+\frac{3 b f (c+d x) (d e-c f) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^3}+\frac{b f^2 \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^3}+\frac{b f^2 (c+d x)^2 \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^3}+\frac{(e+f x)^3 \left(a+b \cot ^{-1}(c+d x)\right)^3}{3 f}-\frac{b^3 \left(-\left(1-3 c^2\right) f^2-6 c d e f+3 d^2 e^2\right) \text{Li}_3\left(1-\frac{2}{i (c+d x)+1}\right)}{2 d^3}+\frac{3 i b^3 f (d e-c f) \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right)}{d^3}+\frac{b^3 f^2 \log \left((c+d x)^2+1\right)}{2 d^3}+\frac{b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3}",1,"(a^2*(a*d^2*e^2 + 3*b*d*e*f - 2*b*c*f^2)*x)/d^2 + (a^2*f*(2*a*d*e + b*f)*x^2)/(2*d) + (a^3*f^2*x^3)/3 + a^2*b*x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCot[c + d*x] + ((-3*a^2*b*c*d^2*e^2 - 3*a^2*b*d*e*f + 3*a^2*b*c^2*d*e*f + 3*a^2*b*c*f^2 - a^2*b*c^3*f^2)*ArcTan[c + d*x])/d^3 + ((3*a^2*b*d^2*e^2 - 6*a^2*b*c*d*e*f - a^2*b*f^2 + 3*a^2*b*c^2*f^2)*Log[1 + c^2 + 2*c*d*x + d^2*x^2])/(2*d^3) + (a*b^2*f^2*x^2*(1 + (c + d*x)^2)*((c + d*x)*(1 - 6*c*ArcCot[c + d*x] + 3*ArcCot[c + d*x]^2 + 3*c^2*ArcCot[c + d*x]^2) - (c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(1 - 6*c*ArcCot[c + d*x] - ArcCot[c + d*x]^2 + 3*c^2*ArcCot[c + d*x]^2)*Cos[3*ArcCot[c + d*x]] - 2*(-2*ArcCot[c + d*x] + I*ArcCot[c + d*x]^2 + 6*c*ArcCot[c + d*x]^2 - (3*I)*c^2*ArcCot[c + d*x]^2 + 2*(-1 + 3*c^2)*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] - 6*c*Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])] + Cos[2*ArcCot[c + d*x]]*(I*(-1 + 3*c^2)*ArcCot[c + d*x]^2 + (2 - 6*c^2)*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] + 6*c*Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])])) + ((4*I)*(-1 + 3*c^2)*PolyLog[2, E^((2*I)*ArcCot[c + d*x])])/((c + d*x)^2*(1 + (c + d*x)^(-2)))))/(4*d*(c + d*x)^2*(1 + (c + d*x)^(-2))*(1/Sqrt[1 + (c + d*x)^(-2)] - c/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]))^2) - (3*a*b^2*e^2*(1 + (c + d*x)^2)*(-((c + d*x)*ArcCot[c + d*x]^2) + 2*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] - I*(ArcCot[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcCot[c + d*x])])))/(d*(c + d*x)^2*(1 + (c + d*x)^(-2))) + (6*a*b^2*e*f*(1 + (c + d*x)^2)*(((c + d*x)*ArcCot[c + d*x])/d^2 - (c*(c + d*x)*ArcCot[c + d*x]^2)/d^2 + ((c + d*x)^2*(1 + (c + d*x)^(-2))*ArcCot[c + d*x]^2)/(2*d^2) - Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])]/d^2 + (2*c*(ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] - (I/2)*(ArcCot[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcCot[c + d*x])])))/d^2))/((c + d*x)^2*(1 + (c + d*x)^(-2))) - (b^3*e^2*(1 + (c + d*x)^2)*((-1/8*I)*Pi^3 + I*ArcCot[c + d*x]^3 - (c + d*x)*ArcCot[c + d*x]^3 + 3*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])] + (3*I)*ArcCot[c + d*x]*PolyLog[2, E^((-2*I)*ArcCot[c + d*x])] + (3*PolyLog[3, E^((-2*I)*ArcCot[c + d*x])])/2))/(d*(c + d*x)^2*(1 + (c + d*x)^(-2))) + (b^3*e*f*(1 + (c + d*x)^2)*((-I)*c*Pi^3 + (12*I)*ArcCot[c + d*x]^2 + 12*(c + d*x)*ArcCot[c + d*x]^2 + (8*I)*c*ArcCot[c + d*x]^3 - 8*c*(c + d*x)*ArcCot[c + d*x]^3 + 4*(c + d*x)^2*(1 + (c + d*x)^(-2))*ArcCot[c + d*x]^3 + 24*c*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])] - 24*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] + (24*I)*c*ArcCot[c + d*x]*PolyLog[2, E^((-2*I)*ArcCot[c + d*x])] + (12*I)*PolyLog[2, E^((2*I)*ArcCot[c + d*x])] + 12*c*PolyLog[3, E^((-2*I)*ArcCot[c + d*x])]))/(4*d^2*(c + d*x)^2*(1 + (c + d*x)^(-2))) - (b^3*f^2*(1 + (c + d*x)^2)*(I*(-1 + 3*c^2)*ArcCot[c + d*x]*PolyLog[2, E^((-2*I)*ArcCot[c + d*x])] + ((c + d*x)^3*(1 + (c + d*x)^(-2))^(3/2)*(((3*I)*Pi^3)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) - ((9*I)*c^2*Pi^3)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) - (24*ArcCot[c + d*x])/Sqrt[1 + (c + d*x)^(-2)] + (72*c*ArcCot[c + d*x]^2)/Sqrt[1 + (c + d*x)^(-2)] - (48*ArcCot[c + d*x]^2)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + ((216*I)*c*ArcCot[c + d*x]^2)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) - (24*ArcCot[c + d*x]^3)/Sqrt[1 + (c + d*x)^(-2)] - (24*c^2*ArcCot[c + d*x]^3)/Sqrt[1 + (c + d*x)^(-2)] - ((24*I)*ArcCot[c + d*x]^3)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + (96*c*ArcCot[c + d*x]^3)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + ((72*I)*c^2*ArcCot[c + d*x]^3)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + 24*ArcCot[c + d*x]*Cos[3*ArcCot[c + d*x]] - 72*c*ArcCot[c + d*x]^2*Cos[3*ArcCot[c + d*x]] - 8*ArcCot[c + d*x]^3*Cos[3*ArcCot[c + d*x]] + 24*c^2*ArcCot[c + d*x]^3*Cos[3*ArcCot[c + d*x]] - (72*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])])/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + (216*c^2*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])])/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) - (432*c*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])])/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + (72*Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])])/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + ((288*I)*c*PolyLog[2, E^((2*I)*ArcCot[c + d*x])])/((c + d*x)^3*(1 + (c + d*x)^(-2))^(3/2)) + (48*(-1 + 3*c^2)*PolyLog[3, E^((-2*I)*ArcCot[c + d*x])])/((c + d*x)^3*(1 + (c + d*x)^(-2))^(3/2)) - I*Pi^3*Sin[3*ArcCot[c + d*x]] + (3*I)*c^2*Pi^3*Sin[3*ArcCot[c + d*x]] - (72*I)*c*ArcCot[c + d*x]^2*Sin[3*ArcCot[c + d*x]] + (8*I)*ArcCot[c + d*x]^3*Sin[3*ArcCot[c + d*x]] - (24*I)*c^2*ArcCot[c + d*x]^3*Sin[3*ArcCot[c + d*x]] + 24*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])]*Sin[3*ArcCot[c + d*x]] - 72*c^2*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])]*Sin[3*ArcCot[c + d*x]] + 144*c*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])]*Sin[3*ArcCot[c + d*x]] - 24*Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])]*Sin[3*ArcCot[c + d*x]]))/96))/(d^3*(c + d*x)^2*(1 + (c + d*x)^(-2)))","B",0
142,1,630,337,1.2696275,"\int (e+f x) \left(a+b \cot ^{-1}(c+d x)\right)^3 \, dx","Integrate[(e + f*x)*(a + b*ArcCot[c + d*x])^3,x]","\frac{a^3 f (c+d x)^2+a^2 (c+d x) (-2 a c f+2 a d e+3 b f)+3 a^2 b (d e-c f) \log \left((c+d x)^2+1\right)-3 a^2 b (c+d x) \cot ^{-1}(c+d x) (c f-d (2 e+f x))-3 a^2 b f \tan ^{-1}(c+d x)+6 a b^2 d e \left(i \text{Li}_2\left(e^{2 i \cot ^{-1}(c+d x)}\right)+\cot ^{-1}(c+d x) \left((c+d x+i) \cot ^{-1}(c+d x)-2 \log \left(1-e^{2 i \cot ^{-1}(c+d x)}\right)\right)\right)-6 a b^2 c f \left(i \text{Li}_2\left(e^{2 i \cot ^{-1}(c+d x)}\right)+\cot ^{-1}(c+d x) \left((c+d x+i) \cot ^{-1}(c+d x)-2 \log \left(1-e^{2 i \cot ^{-1}(c+d x)}\right)\right)\right)+6 a b^2 f \left(-\log \left(\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right)+\frac{1}{2} \left((c+d x)^2+1\right) \cot ^{-1}(c+d x)^2+(c+d x) \cot ^{-1}(c+d x)\right)+2 b^3 d e \left(-3 i \cot ^{-1}(c+d x) \text{Li}_2\left(e^{-2 i \cot ^{-1}(c+d x)}\right)-\frac{3}{2} \text{Li}_3\left(e^{-2 i \cot ^{-1}(c+d x)}\right)+(c+d x) \cot ^{-1}(c+d x)^3-i \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left(1-e^{-2 i \cot ^{-1}(c+d x)}\right)+\frac{i \pi ^3}{8}\right)+b^3 f \left(3 i \left(\cot ^{-1}(c+d x)^2+\text{Li}_2\left(e^{2 i \cot ^{-1}(c+d x)}\right)\right)+\left((c+d x)^2+1\right) \cot ^{-1}(c+d x)^3+3 (c+d x) \cot ^{-1}(c+d x)^2-6 \cot ^{-1}(c+d x) \log \left(1-e^{2 i \cot ^{-1}(c+d x)}\right)\right)-2 b^3 c f \left(-3 i \cot ^{-1}(c+d x) \text{Li}_2\left(e^{-2 i \cot ^{-1}(c+d x)}\right)-\frac{3}{2} \text{Li}_3\left(e^{-2 i \cot ^{-1}(c+d x)}\right)+(c+d x) \cot ^{-1}(c+d x)^3-i \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left(1-e^{-2 i \cot ^{-1}(c+d x)}\right)+\frac{i \pi ^3}{8}\right)}{2 d^2}","\frac{3 i b^2 (d e-c f) \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^2}-\frac{3 b^2 f \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d^2}+\frac{i (d e-c f) \left(a+b \cot ^{-1}(c+d x)\right)^3}{d^2}-\frac{(-c f+d e+f) (d e-(c+1) f) \left(a+b \cot ^{-1}(c+d x)\right)^3}{2 d^2 f}-\frac{3 b (d e-c f) \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d^2}+\frac{3 i b f \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^2}+\frac{3 b f (c+d x) \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 d^2}+\frac{(e+f x)^2 \left(a+b \cot ^{-1}(c+d x)\right)^3}{2 f}-\frac{3 b^3 (d e-c f) \text{Li}_3\left(1-\frac{2}{i (c+d x)+1}\right)}{2 d^2}+\frac{3 i b^3 f \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right)}{2 d^2}",1,"(a^2*(2*a*d*e + 3*b*f - 2*a*c*f)*(c + d*x) + a^3*f*(c + d*x)^2 - 3*a^2*b*(c + d*x)*(c*f - d*(2*e + f*x))*ArcCot[c + d*x] - 3*a^2*b*f*ArcTan[c + d*x] + 6*a*b^2*f*((c + d*x)*ArcCot[c + d*x] + ((1 + (c + d*x)^2)*ArcCot[c + d*x]^2)/2 - Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])]) + 3*a^2*b*(d*e - c*f)*Log[1 + (c + d*x)^2] + 6*a*b^2*d*e*(ArcCot[c + d*x]*((I + c + d*x)*ArcCot[c + d*x] - 2*Log[1 - E^((2*I)*ArcCot[c + d*x])]) + I*PolyLog[2, E^((2*I)*ArcCot[c + d*x])]) - 6*a*b^2*c*f*(ArcCot[c + d*x]*((I + c + d*x)*ArcCot[c + d*x] - 2*Log[1 - E^((2*I)*ArcCot[c + d*x])]) + I*PolyLog[2, E^((2*I)*ArcCot[c + d*x])]) + b^3*f*(3*(c + d*x)*ArcCot[c + d*x]^2 + (1 + (c + d*x)^2)*ArcCot[c + d*x]^3 - 6*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] + (3*I)*(ArcCot[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcCot[c + d*x])])) + 2*b^3*d*e*((I/8)*Pi^3 - I*ArcCot[c + d*x]^3 + (c + d*x)*ArcCot[c + d*x]^3 - 3*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])] - (3*I)*ArcCot[c + d*x]*PolyLog[2, E^((-2*I)*ArcCot[c + d*x])] - (3*PolyLog[3, E^((-2*I)*ArcCot[c + d*x])])/2) - 2*b^3*c*f*((I/8)*Pi^3 - I*ArcCot[c + d*x]^3 + (c + d*x)*ArcCot[c + d*x]^3 - 3*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])] - (3*I)*ArcCot[c + d*x]*PolyLog[2, E^((-2*I)*ArcCot[c + d*x])] - (3*PolyLog[3, E^((-2*I)*ArcCot[c + d*x])])/2))/(2*d^2)","A",0
143,1,228,143,0.3524714,"\int \left(a+b \cot ^{-1}(c+d x)\right)^3 \, dx","Integrate[(a + b*ArcCot[c + d*x])^3,x]","\frac{2 a^3 (c+d x)+3 a^2 b \log \left((c+d x)^2+1\right)+6 a^2 b (c+d x) \cot ^{-1}(c+d x)+6 a b^2 \left(i \text{Li}_2\left(e^{2 i \cot ^{-1}(c+d x)}\right)+\cot ^{-1}(c+d x) \left((c+d x+i) \cot ^{-1}(c+d x)-2 \log \left(1-e^{2 i \cot ^{-1}(c+d x)}\right)\right)\right)+2 b^3 \left(-3 i \cot ^{-1}(c+d x) \text{Li}_2\left(e^{-2 i \cot ^{-1}(c+d x)}\right)-\frac{3}{2} \text{Li}_3\left(e^{-2 i \cot ^{-1}(c+d x)}\right)+(c+d x) \cot ^{-1}(c+d x)^3-i \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left(1-e^{-2 i \cot ^{-1}(c+d x)}\right)+\frac{i \pi ^3}{8}\right)}{2 d}","\frac{3 i b^2 \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{d}+\frac{(c+d x) \left(a+b \cot ^{-1}(c+d x)\right)^3}{d}+\frac{i \left(a+b \cot ^{-1}(c+d x)\right)^3}{d}-\frac{3 b \log \left(\frac{2}{1+i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{d}-\frac{3 b^3 \text{Li}_3\left(1-\frac{2}{i (c+d x)+1}\right)}{2 d}",1,"(2*a^3*(c + d*x) + 6*a^2*b*(c + d*x)*ArcCot[c + d*x] + 3*a^2*b*Log[1 + (c + d*x)^2] + 6*a*b^2*(ArcCot[c + d*x]*((I + c + d*x)*ArcCot[c + d*x] - 2*Log[1 - E^((2*I)*ArcCot[c + d*x])]) + I*PolyLog[2, E^((2*I)*ArcCot[c + d*x])]) + 2*b^3*((I/8)*Pi^3 - I*ArcCot[c + d*x]^3 + (c + d*x)*ArcCot[c + d*x]^3 - 3*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])] - (3*I)*ArcCot[c + d*x]*PolyLog[2, E^((-2*I)*ArcCot[c + d*x])] - (3*PolyLog[3, E^((-2*I)*ArcCot[c + d*x])])/2))/(2*d)","A",0
144,0,0,372,45.8631725,"\int \frac{\left(a+b \cot ^{-1}(c+d x)\right)^3}{e+f x} \, dx","Integrate[(a + b*ArcCot[c + d*x])^3/(e + f*x),x]","\int \frac{\left(a+b \cot ^{-1}(c+d x)\right)^3}{e+f x} \, dx","\frac{3 b^2 \left(a+b \cot ^{-1}(c+d x)\right) \text{Li}_3\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right)}{2 f}-\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)}{2 f}+\frac{3 i b \left(a+b \cot ^{-1}(c+d x)\right)^2 \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right)}{2 f}+\frac{\left(a+b \cot ^{-1}(c+d x)\right)^3 \log \left(\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right)}{f}-\frac{3 i b \text{Li}_2\left(1-\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^2}{2 f}-\frac{\log \left(\frac{2}{1-i (c+d x)}\right) \left(a+b \cot ^{-1}(c+d x)\right)^3}{f}-\frac{3 i b^3 \text{Li}_4\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right)}{4 f}+\frac{3 i b^3 \text{Li}_4\left(1-\frac{2}{1-i (c+d x)}\right)}{4 f}",1,"Integrate[(a + b*ArcCot[c + d*x])^3/(e + f*x), x]","F",-1
145,1,1584,1233,38.7592958,"\int \frac{\left(a+b \cot ^{-1}(c+d x)\right)^3}{(e+f x)^2} \, dx","Integrate[(a + b*ArcCot[c + d*x])^3/(e + f*x)^2,x]","-\frac{a^3}{f (e+f x)}-\frac{3 b \cot ^{-1}(c+d x) a^2}{f (e+f x)}+\frac{3 b d (c f-d e) \tan ^{-1}(c+d x) a^2}{f \left(d^2 e^2-2 c d f e+c^2 f^2+f^2\right)}-\frac{3 b d \log (e+f x) a^2}{d^2 e^2-2 c d f e+c^2 f^2+f^2}+\frac{3 b d \log \left(c^2+2 d x c+d^2 x^2+1\right) a^2}{2 \left(d^2 e^2-2 c d f e+c^2 f^2+f^2\right)}-\frac{3 b^2 \left((c+d x)^2+1\right) \left(\frac{f}{\sqrt{1+\frac{1}{(c+d x)^2}}}+\frac{d e-c f}{(c+d x) \sqrt{1+\frac{1}{(c+d x)^2}}}\right)^2 \left(\frac{e^{i \tan ^{-1}\left(\frac{f}{d e-c f}\right)} \cot ^{-1}(c+d x)^2}{(c f-d e) \sqrt{\frac{f^2}{(d e-c f)^2}+1}}+\frac{\cot ^{-1}(c+d x)^2}{(c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left(\frac{f}{\sqrt{1+\frac{1}{(c+d x)^2}}}+\frac{d e-c f}{(c+d x) \sqrt{1+\frac{1}{(c+d x)^2}}}\right)}+\frac{f \left(2 \log \left(1-e^{2 i \left(\cot ^{-1}(c+d x)+\tan ^{-1}\left(\frac{f}{d e-c f}\right)\right)}\right) \cot ^{-1}(c+d x)+i \pi  \cot ^{-1}(c+d x)+\pi  \log \left(1+e^{-2 i \cot ^{-1}(c+d x)}\right)-\pi  \log \left(\frac{1}{\sqrt{1+\frac{1}{(c+d x)^2}}}\right)+2 \tan ^{-1}\left(\frac{f}{c f-d e}\right) \left(i \cot ^{-1}(c+d x)-\log \left(1-e^{2 i \left(\cot ^{-1}(c+d x)+\tan ^{-1}\left(\frac{f}{d e-c f}\right)\right)}\right)+\log \left(\sin \left(\cot ^{-1}(c+d x)+\tan ^{-1}\left(\frac{f}{d e-c f}\right)\right)\right)\right)-i \text{Li}_2\left(e^{2 i \left(\cot ^{-1}(c+d x)+\tan ^{-1}\left(\frac{f}{d e-c f}\right)\right)}\right)\right)}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}\right) a}{d f (e+f x)^2}+\frac{b^3 \left((c+d x)^2+1\right) \left(\frac{f}{\sqrt{1+\frac{1}{(c+d x)^2}}}+\frac{d e-c f}{(c+d x) \sqrt{1+\frac{1}{(c+d x)^2}}}\right)^2 \left(\frac{d \cot ^{-1}(c+d x)^3}{f (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left(-\frac{d e}{(c+d x) \sqrt{1+\frac{1}{(c+d x)^2}}}-\frac{f}{\sqrt{1+\frac{1}{(c+d x)^2}}}+\frac{c f}{(c+d x) \sqrt{1+\frac{1}{(c+d x)^2}}}\right)}+\frac{d \left(\cot ^{-1}(c+d x) \left(-2 d e \cot ^{-1}(c+d x)^2+2 c f \cot ^{-1}(c+d x)^2-2 i f \cot ^{-1}(c+d x)^2+4 d e e^{i \tan ^{-1}\left(\frac{f}{d e-c f}\right)} \sqrt{\frac{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}{(d e-c f)^2}} \cot ^{-1}(c+d x)^2-4 c e^{i \tan ^{-1}\left(\frac{f}{d e-c f}\right)} f \sqrt{\frac{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}{(d e-c f)^2}} \cot ^{-1}(c+d x)^2-12 f \log \left(1-e^{2 i \left(\cot ^{-1}(c+d x)+\tan ^{-1}\left(\frac{f}{d e-c f}\right)\right)}\right) \cot ^{-1}(c+d x)+6 f \log \left(-\left(\left(\frac{i}{(c+d x) \sqrt{1+\frac{1}{(c+d x)^2}}}+\frac{1}{\sqrt{1+\frac{1}{(c+d x)^2}}}\right) \left(\frac{-1+e^{2 i \tan ^{-1}\left(\frac{f}{d e-c f}\right)}}{\sqrt{1+\frac{1}{(c+d x)^2}}}+\frac{i \left(1+e^{2 i \tan ^{-1}\left(\frac{f}{d e-c f}\right)}\right)}{(c+d x) \sqrt{1+\frac{1}{(c+d x)^2}}}\right)\right)\right) \cot ^{-1}(c+d x)-6 f \pi  \log \left(1+e^{-2 i \cot ^{-1}(c+d x)}\right)+12 f \tan ^{-1}\left(\frac{f}{c f-d e}\right) \log \left(1-e^{2 i \left(\cot ^{-1}(c+d x)+\tan ^{-1}\left(\frac{f}{d e-c f}\right)\right)}\right)+6 f \pi  \log \left(\frac{\frac{1}{\sqrt{1+\frac{1}{(c+d x)^2}}}-\frac{i}{(c+d x) \sqrt{1+\frac{1}{(c+d x)^2}}}}{\sqrt{1+\frac{1}{(c+d x)^2}}}\right)-12 f \tan ^{-1}\left(\frac{f}{d e-c f}\right) \left(\log \left(\frac{1}{2} e^{-i \tan ^{-1}\left(\frac{f}{d e-c f}\right)} \left(\frac{1+e^{2 i \tan ^{-1}\left(\frac{f}{d e-c f}\right)}}{(c+d x) \sqrt{1+\frac{1}{(c+d x)^2}}}-\frac{i \left(-1+e^{2 i \tan ^{-1}\left(\frac{f}{d e-c f}\right)}\right)}{\sqrt{1+\frac{1}{(c+d x)^2}}}\right)\right)-\log \left(-e^{2 i \tan ^{-1}\left(\frac{f}{d e-c f}\right)} \cos \left(2 \cot ^{-1}(c+d x)\right)-i e^{2 i \tan ^{-1}\left(\frac{f}{d e-c f}\right)} \sin \left(2 \cot ^{-1}(c+d x)\right)+1\right)-\log \left(\sin \left(\cot ^{-1}(c+d x)+\tan ^{-1}\left(\frac{f}{d e-c f}\right)\right)\right)\right)+f \pi  \log (64)\right)+6 i f \cot ^{-1}(c+d x) \text{Li}_2\left(e^{2 i \left(\cot ^{-1}(c+d x)+\tan ^{-1}\left(\frac{f}{d e-c f}\right)\right)}\right)-3 f \text{Li}_3\left(e^{2 i \left(\cot ^{-1}(c+d x)+\tan ^{-1}\left(\frac{f}{d e-c f}\right)\right)}\right)\right)}{2 f \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}\right)}{d^2 (e+f x)^2}","\frac{i d \cot ^{-1}(c+d x)^3 b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{d (d e-c f) \cot ^{-1}(c+d x)^3 b^3}{f \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}+\frac{3 d \cot ^{-1}(c+d x)^2 \log \left(\frac{2}{1-i (c+d x)}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 d \cot ^{-1}(c+d x)^2 \log \left(\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 d \cot ^{-1}(c+d x)^2 \log \left(\frac{2}{i (c+d x)+1}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i d \cot ^{-1}(c+d x) \text{Li}_2\left(1-\frac{2}{1-i (c+d x)}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 i d \cot ^{-1}(c+d x) \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i d \cot ^{-1}(c+d x) \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right) b^3}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 d \text{Li}_3\left(1-\frac{2}{1-i (c+d x)}\right) b^3}{2 \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}-\frac{3 d \text{Li}_3\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^3}{2 \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}-\frac{3 d \text{Li}_3\left(1-\frac{2}{i (c+d x)+1}\right) b^3}{2 \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}+\frac{3 i a d \cot ^{-1}(c+d x)^2 b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 a d (d e-c f) \cot ^{-1}(c+d x)^2 b^2}{f \left(d^2 e^2-2 c d f e+\left(c^2+1\right) f^2\right)}+\frac{6 a d \cot ^{-1}(c+d x) \log \left(\frac{2}{1-i (c+d x)}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{6 a d \cot ^{-1}(c+d x) \log \left(\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{6 a d \cot ^{-1}(c+d x) \log \left(\frac{2}{i (c+d x)+1}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i a d \text{Li}_2\left(1-\frac{2}{1-i (c+d x)}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 i a d \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}+\frac{3 i a d \text{Li}_2\left(1-\frac{2}{i (c+d x)+1}\right) b^2}{d^2 e^2-2 c d f e+\left(c^2+1\right) f^2}-\frac{3 a^2 d (d e-c f) \tan ^{-1}(c+d x) b}{f \left(f^2+(d e-c f)^2\right)}-\frac{3 a^2 d \log (e+f x) b}{f^2+(d e-c f)^2}+\frac{3 a^2 d \log \left((c+d x)^2+1\right) b}{2 \left(f^2+(d e-c f)^2\right)}-\frac{\left(a+b \cot ^{-1}(c+d x)\right)^3}{f (e+f x)}",1,"-(a^3/(f*(e + f*x))) - (3*a^2*b*ArcCot[c + d*x])/(f*(e + f*x)) + (3*a^2*b*d*(-(d*e) + c*f)*ArcTan[c + d*x])/(f*(d^2*e^2 - 2*c*d*e*f + f^2 + c^2*f^2)) - (3*a^2*b*d*Log[e + f*x])/(d^2*e^2 - 2*c*d*e*f + f^2 + c^2*f^2) + (3*a^2*b*d*Log[1 + c^2 + 2*c*d*x + d^2*x^2])/(2*(d^2*e^2 - 2*c*d*e*f + f^2 + c^2*f^2)) - (3*a*b^2*(1 + (c + d*x)^2)*(f/Sqrt[1 + (c + d*x)^(-2)] + (d*e - c*f)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]))^2*((E^(I*ArcTan[f/(d*e - c*f)])*ArcCot[c + d*x]^2)/((-(d*e) + c*f)*Sqrt[1 + f^2/(d*e - c*f)^2]) + ArcCot[c + d*x]^2/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(f/Sqrt[1 + (c + d*x)^(-2)] + (d*e - c*f)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]))) + (f*(I*Pi*ArcCot[c + d*x] + Pi*Log[1 + E^((-2*I)*ArcCot[c + d*x])] + 2*ArcCot[c + d*x]*Log[1 - E^((2*I)*(ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]))] - Pi*Log[1/Sqrt[1 + (c + d*x)^(-2)]] + 2*ArcTan[f/(-(d*e) + c*f)]*(I*ArcCot[c + d*x] - Log[1 - E^((2*I)*(ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]))] + Log[Sin[ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]]]) - I*PolyLog[2, E^((2*I)*(ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]))]))/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)))/(d*f*(e + f*x)^2) + (b^3*(1 + (c + d*x)^2)*(f/Sqrt[1 + (c + d*x)^(-2)] + (d*e - c*f)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]))^2*((d*ArcCot[c + d*x]^3)/(f*(c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(-(f/Sqrt[1 + (c + d*x)^(-2)]) - (d*e)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + (c*f)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]))) + (d*(ArcCot[c + d*x]*(-2*d*e*ArcCot[c + d*x]^2 - (2*I)*f*ArcCot[c + d*x]^2 + 2*c*f*ArcCot[c + d*x]^2 + 4*d*e*E^(I*ArcTan[f/(d*e - c*f)])*Sqrt[(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCot[c + d*x]^2 - 4*c*E^(I*ArcTan[f/(d*e - c*f)])*f*Sqrt[(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCot[c + d*x]^2 + f*Pi*Log[64] - 6*f*Pi*Log[1 + E^((-2*I)*ArcCot[c + d*x])] - 12*f*ArcCot[c + d*x]*Log[1 - E^((2*I)*(ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]))] + 12*f*ArcTan[f/(-(d*e) + c*f)]*Log[1 - E^((2*I)*(ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]))] + 6*f*Pi*Log[(1/Sqrt[1 + (c + d*x)^(-2)] - I/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]))/Sqrt[1 + (c + d*x)^(-2)]] + 6*f*ArcCot[c + d*x]*Log[-((1/Sqrt[1 + (c + d*x)^(-2)] + I/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]))*((-1 + E^((2*I)*ArcTan[f/(d*e - c*f)]))/Sqrt[1 + (c + d*x)^(-2)] + (I*(1 + E^((2*I)*ArcTan[f/(d*e - c*f)])))/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])))] - 12*f*ArcTan[f/(d*e - c*f)]*(Log[(((-I)*(-1 + E^((2*I)*ArcTan[f/(d*e - c*f)])))/Sqrt[1 + (c + d*x)^(-2)] + (1 + E^((2*I)*ArcTan[f/(d*e - c*f)]))/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]))/(2*E^(I*ArcTan[f/(d*e - c*f)]))] - Log[1 - E^((2*I)*ArcTan[f/(d*e - c*f)])*Cos[2*ArcCot[c + d*x]] - I*E^((2*I)*ArcTan[f/(d*e - c*f)])*Sin[2*ArcCot[c + d*x]]] - Log[Sin[ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]]])) + (6*I)*f*ArcCot[c + d*x]*PolyLog[2, E^((2*I)*(ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]))] - 3*f*PolyLog[3, E^((2*I)*(ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]))]))/(2*f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2))))/(d^2*(e + f*x)^2)","A",0
146,1,162,177,0.3603023,"\int (e+f x)^m \left(a+b \cot ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)^m*(a + b*ArcCot[c + d*x]),x]","\frac{(e+f x)^{m+1} \left(2 \left(a+b \cot ^{-1}(c+d x)\right)+\frac{b d (e+f x) \left((d e-(c+i) f) \, _2F_1\left(1,m+2;m+3;\frac{d (e+f x)}{d e-(c-i) f}\right)+(-d e+(c-i) f) \, _2F_1\left(1,m+2;m+3;\frac{d (e+f x)}{d e-(c+i) f}\right)\right)}{(m+2) (i c f-i d e+f) (d e-(c+i) f)}\right)}{2 f (m+1)}","\frac{(e+f x)^{m+1} \left(a+b \cot ^{-1}(c+d x)\right)}{f (m+1)}+\frac{i b d (e+f x)^{m+2} \, _2F_1\left(1,m+2;m+3;\frac{d (e+f x)}{d e-c f+i f}\right)}{2 f (m+1) (m+2) (d e+(-c+i) f)}-\frac{i b d (e+f x)^{m+2} \, _2F_1\left(1,m+2;m+3;\frac{d (e+f x)}{d e-(c+i) f}\right)}{2 f (m+1) (m+2) (d e-(c+i) f)}",1,"((e + f*x)^(1 + m)*(2*(a + b*ArcCot[c + d*x]) + (b*d*(e + f*x)*((d*e - (I + c)*f)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e - (-I + c)*f)] + (-(d*e) + (-I + c)*f)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e - (I + c)*f)]))/(((-I)*d*e + f + I*c*f)*(d*e - (I + c)*f)*(2 + m))))/(2*f*(1 + m))","A",1
147,0,0,23,5.4411476,"\int (e+f x)^m \left(a+b \cot ^{-1}(c+d x)\right)^2 \, dx","Integrate[(e + f*x)^m*(a + b*ArcCot[c + d*x])^2,x]","\int (e+f x)^m \left(a+b \cot ^{-1}(c+d x)\right)^2 \, dx","\text{Int}\left((e+f x)^m \left(a+b \cot ^{-1}(c+d x)\right)^2,x\right)",0,"Integrate[(e + f*x)^m*(a + b*ArcCot[c + d*x])^2, x]","A",-1
148,0,0,23,0.4853998,"\int (e+f x)^m \left(a+b \cot ^{-1}(c+d x)\right)^3 \, dx","Integrate[(e + f*x)^m*(a + b*ArcCot[c + d*x])^3,x]","\int (e+f x)^m \left(a+b \cot ^{-1}(c+d x)\right)^3 \, dx","\text{Int}\left((e+f x)^m \left(a+b \cot ^{-1}(c+d x)\right)^3,x\right)",0,"Integrate[(e + f*x)^m*(a + b*ArcCot[c + d*x])^3, x]","A",-1
149,1,37,42,0.0160137,"\int x^3 \cot ^{-1}\left(a+b x^4\right) \, dx","Integrate[x^3*ArcCot[a + b*x^4],x]","\frac{\log \left(\left(a+b x^4\right)^2+1\right)+2 \left(a+b x^4\right) \cot ^{-1}\left(a+b x^4\right)}{8 b}","\frac{\log \left(\left(a+b x^4\right)^2+1\right)}{8 b}+\frac{\left(a+b x^4\right) \cot ^{-1}\left(a+b x^4\right)}{4 b}",1,"(2*(a + b*x^4)*ArcCot[a + b*x^4] + Log[1 + (a + b*x^4)^2])/(8*b)","A",1
150,1,40,45,0.0361626,"\int x^{-1+n} \cot ^{-1}\left(a+b x^n\right) \, dx","Integrate[x^(-1 + n)*ArcCot[a + b*x^n],x]","\frac{\log \left(\left(a+b x^n\right)^2+1\right)+2 \left(a+b x^n\right) \cot ^{-1}\left(a+b x^n\right)}{2 b n}","\frac{\log \left(\left(a+b x^n\right)^2+1\right)}{2 b n}+\frac{\left(a+b x^n\right) \cot ^{-1}\left(a+b x^n\right)}{b n}",1,"(2*(a + b*x^n)*ArcCot[a + b*x^n] + Log[1 + (a + b*x^n)^2])/(2*b*n)","A",1
151,0,0,43,0.091085,"\int \frac{\left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^n}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2),x]","\int \frac{\left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^n}{1-c^2 x^2} \, dx","\text{Int}\left(\frac{\left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^n}{1-c^2 x^2},x\right)",0,"Integrate[(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]","A",-1
152,0,0,488,0.2959742,"\int \frac{\left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^3}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2),x]","\int \frac{\left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^3}{1-c^2 x^2} \, dx","\frac{3 b^2 \text{Li}_3\left(1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{2 c}-\frac{3 b^2 \text{Li}_3\left(1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{2 c}+\frac{3 i b \text{Li}_2\left(1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 c}-\frac{3 i b \text{Li}_2\left(1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 c}-\frac{2 \coth ^{-1}\left(1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^3}{c}-\frac{3 i b^3 \text{Li}_4\left(1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right)}{4 c}+\frac{3 i b^3 \text{Li}_4\left(1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right)}{4 c}",1,"Integrate[(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2), x]","F",-1
153,0,0,321,0.5205907,"\int \frac{\left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2),x]","\int \frac{\left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2}{1-c^2 x^2} \, dx","\frac{i b \text{Li}_2\left(1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}-\frac{i b \text{Li}_2\left(1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}-\frac{2 \coth ^{-1}\left(1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{c}+\frac{b^2 \text{Li}_3\left(1-\frac{2 i}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right)}{2 c}-\frac{b^2 \text{Li}_3\left(1-\frac{2 \sqrt{1-c x}}{\sqrt{c x+1} \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i\right)}\right)}{2 c}",1,"Integrate[(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2), x]","F",-1
154,1,93,98,0.0352043,"\int \frac{a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(1 - c^2*x^2),x]","-\frac{a \log \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)-\frac{1}{2} i b \text{Li}_2\left(-\frac{i \sqrt{c x+1}}{\sqrt{1-c x}}\right)+\frac{1}{2} i b \text{Li}_2\left(\frac{i \sqrt{c x+1}}{\sqrt{1-c x}}\right)}{c}","-\frac{a \log \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}{c}+\frac{i b \text{Li}_2\left(-\frac{i \sqrt{c x+1}}{\sqrt{1-c x}}\right)}{2 c}-\frac{i b \text{Li}_2\left(\frac{i \sqrt{c x+1}}{\sqrt{1-c x}}\right)}{2 c}",1,"-((a*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]] - (I/2)*b*PolyLog[2, ((-I)*Sqrt[1 + c*x])/Sqrt[1 - c*x]] + (I/2)*b*PolyLog[2, (I*Sqrt[1 + c*x])/Sqrt[1 - c*x]])/c)","A",1
155,0,0,43,0.0912657,"\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)} \, dx","Integrate[1/((1 - c^2*x^2)*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])),x]","\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)} \, dx","\text{Int}\left(\frac{1}{\left(1-c^2 x^2\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)},x\right)",0,"Integrate[1/((1 - c^2*x^2)*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]","A",-1
156,0,0,43,0.8178846,"\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2} \, dx","Integrate[1/((1 - c^2*x^2)*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2),x]","\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(1-c^2 x^2\right) \left(a+b \cot ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2},x\right)",0,"Integrate[1/((1 - c^2*x^2)*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]","A",-1
157,1,18,16,0.0087835,"\int \cot ^{-1}(\tan (a+b x)) \, dx","Integrate[ArcCot[Tan[a + b*x]],x]","x \cot ^{-1}(\tan (a+b x))+\frac{b x^2}{2}","-\frac{\cot ^{-1}(\tan (a+b x))^2}{2 b}",1,"(b*x^2)/2 + x*ArcCot[Tan[a + b*x]]","A",1
158,1,363,403,0.8913474,"\int x^2 \cot ^{-1}(c+d \tan (a+b x)) \, dx","Integrate[x^2*ArcCot[c + d*Tan[a + b*x]],x]","\frac{1}{3} x^3 \cot ^{-1}(d \tan (a+b x)+c)+\frac{-4 i b^3 x^3 \log \left(1+\frac{(c-i (d+1)) e^{2 i (a+b x)}}{c+i (d-1)}\right)+4 i b^3 x^3 \log \left(1+\frac{(c-i d+i) e^{2 i (a+b x)}}{c+i (d+1)}\right)-6 b^2 x^2 \text{Li}_2\left(-\frac{(c-i (d+1)) e^{2 i (a+b x)}}{c+i (d-1)}\right)+6 b^2 x^2 \text{Li}_2\left(-\frac{(c-i d+i) e^{2 i (a+b x)}}{c+i (d+1)}\right)-6 i b x \text{Li}_3\left(-\frac{(c-i (d+1)) e^{2 i (a+b x)}}{c+i (d-1)}\right)+6 i b x \text{Li}_3\left(-\frac{(c-i d+i) e^{2 i (a+b x)}}{c+i (d+1)}\right)+3 \text{Li}_4\left(-\frac{(c-i (d+1)) e^{2 i (a+b x)}}{c+i (d-1)}\right)-3 \text{Li}_4\left(-\frac{(c-i d+i) e^{2 i (a+b x)}}{c+i (d+1)}\right)}{24 b^3}","\frac{\text{Li}_4\left(-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{8 b^3}-\frac{\text{Li}_4\left(-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{8 b^3}-\frac{i x \text{Li}_3\left(-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{4 b^2}+\frac{i x \text{Li}_3\left(-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{4 b^2}-\frac{x^2 \text{Li}_2\left(-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{4 b}+\frac{x^2 \text{Li}_2\left(-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{4 b}-\frac{1}{6} i x^3 \log \left(1+\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)+\frac{1}{6} i x^3 \log \left(1+\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)+\frac{1}{3} x^3 \cot ^{-1}(d \tan (a+b x)+c)",1,"(x^3*ArcCot[c + d*Tan[a + b*x]])/3 + ((-4*I)*b^3*x^3*Log[1 + ((c - I*(1 + d))*E^((2*I)*(a + b*x)))/(c + I*(-1 + d))] + (4*I)*b^3*x^3*Log[1 + ((I + c - I*d)*E^((2*I)*(a + b*x)))/(c + I*(1 + d))] - 6*b^2*x^2*PolyLog[2, -(((c - I*(1 + d))*E^((2*I)*(a + b*x)))/(c + I*(-1 + d)))] + 6*b^2*x^2*PolyLog[2, -(((I + c - I*d)*E^((2*I)*(a + b*x)))/(c + I*(1 + d)))] - (6*I)*b*x*PolyLog[3, -(((c - I*(1 + d))*E^((2*I)*(a + b*x)))/(c + I*(-1 + d)))] + (6*I)*b*x*PolyLog[3, -(((I + c - I*d)*E^((2*I)*(a + b*x)))/(c + I*(1 + d)))] + 3*PolyLog[4, -(((c - I*(1 + d))*E^((2*I)*(a + b*x)))/(c + I*(-1 + d)))] - 3*PolyLog[4, -(((I + c - I*d)*E^((2*I)*(a + b*x)))/(c + I*(1 + d)))])/(24*b^3)","A",1
159,1,272,305,0.6174952,"\int x \cot ^{-1}(c+d \tan (a+b x)) \, dx","Integrate[x*ArcCot[c + d*Tan[a + b*x]],x]","\frac{1}{2} x^2 \cot ^{-1}(d \tan (a+b x)+c)-\frac{i \left(2 b^2 x^2 \log \left(1+\frac{(c-i (d+1)) e^{2 i (a+b x)}}{c+i (d-1)}\right)-2 b^2 x^2 \log \left(1+\frac{(c-i d+i) e^{2 i (a+b x)}}{c+i (d+1)}\right)-2 i b x \text{Li}_2\left(-\frac{(c-i (d+1)) e^{2 i (a+b x)}}{c+i (d-1)}\right)+2 i b x \text{Li}_2\left(-\frac{(c-i d+i) e^{2 i (a+b x)}}{c+i (d+1)}\right)+\text{Li}_3\left(-\frac{(c-i (d+1)) e^{2 i (a+b x)}}{c+i (d-1)}\right)-\text{Li}_3\left(-\frac{(c-i d+i) e^{2 i (a+b x)}}{c+i (d+1)}\right)\right)}{8 b^2}","-\frac{i \text{Li}_3\left(-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{8 b^2}+\frac{i \text{Li}_3\left(-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{8 b^2}-\frac{x \text{Li}_2\left(-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{4 b}+\frac{x \text{Li}_2\left(-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{4 b}-\frac{1}{4} i x^2 \log \left(1+\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)+\frac{1}{4} i x^2 \log \left(1+\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)+\frac{1}{2} x^2 \cot ^{-1}(d \tan (a+b x)+c)",1,"(x^2*ArcCot[c + d*Tan[a + b*x]])/2 - ((I/8)*(2*b^2*x^2*Log[1 + ((c - I*(1 + d))*E^((2*I)*(a + b*x)))/(c + I*(-1 + d))] - 2*b^2*x^2*Log[1 + ((I + c - I*d)*E^((2*I)*(a + b*x)))/(c + I*(1 + d))] - (2*I)*b*x*PolyLog[2, -(((c - I*(1 + d))*E^((2*I)*(a + b*x)))/(c + I*(-1 + d)))] + (2*I)*b*x*PolyLog[2, -(((I + c - I*d)*E^((2*I)*(a + b*x)))/(c + I*(1 + d)))] + PolyLog[3, -(((c - I*(1 + d))*E^((2*I)*(a + b*x)))/(c + I*(-1 + d)))] - PolyLog[3, -(((I + c - I*d)*E^((2*I)*(a + b*x)))/(c + I*(1 + d)))]))/b^2","A",1
160,1,555,198,1.7632259,"\int \cot ^{-1}(c+d \tan (a+b x)) \, dx","Integrate[ArcCot[c + d*Tan[a + b*x]],x]","x \cot ^{-1}(d \tan (a+b x)+c)-\frac{x \left(-i \sqrt{-d^2} \left(\text{Li}_2\left(\frac{d^2 (1-i \tan (a+b x))}{d^2+i c d-i \sqrt{-d^2}}\right)+\log (1-i \tan (a+b x)) \log \left(\frac{d^2 (-\tan (a+b x))-c d+\sqrt{-d^2}}{-c d+i d^2+\sqrt{-d^2}}\right)\right)+i \sqrt{-d^2} \left(\text{Li}_2\left(\frac{d^2 (1-i \tan (a+b x))}{d^2+i c d+i \sqrt{-d^2}}\right)+\log (1-i \tan (a+b x)) \log \left(\frac{d^2 \tan (a+b x)+c d+\sqrt{-d^2}}{c d-i d^2+\sqrt{-d^2}}\right)\right)+i \sqrt{-d^2} \left(\text{Li}_2\left(\frac{d^2 (i \tan (a+b x)+1)}{d^2-i c d+i \sqrt{-d^2}}\right)+\log (1+i \tan (a+b x)) \log \left(\frac{d^2 \tan (a+b x)+c d-\sqrt{-d^2}}{c d+i d^2-\sqrt{-d^2}}\right)\right)-i \sqrt{-d^2} \left(\text{Li}_2\left(\frac{d^2 (i \tan (a+b x)+1)}{d^2-i \left(c d+\sqrt{-d^2}\right)}\right)+\log (1+i \tan (a+b x)) \log \left(\frac{d^2 \tan (a+b x)+c d+\sqrt{-d^2}}{c d+i d^2+\sqrt{-d^2}}\right)\right)-4 a d \tan ^{-1}(d \tan (a+b x)+c)\right)}{2 d (-i \log (1-i \tan (a+b x))+i \log (1+i \tan (a+b x))+2 a)}","-\frac{\text{Li}_2\left(-\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)}{4 b}+\frac{\text{Li}_2\left(-\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)}{4 b}-\frac{1}{2} i x \log \left(1+\frac{(i c+d+1) e^{2 i a+2 i b x}}{i c-d+1}\right)+\frac{1}{2} i x \log \left(1+\frac{(c+i (1-d)) e^{2 i a+2 i b x}}{c+i (d+1)}\right)+x \cot ^{-1}(d \tan (a+b x)+c)",1,"x*ArcCot[c + d*Tan[a + b*x]] - (x*(-4*a*d*ArcTan[c + d*Tan[a + b*x]] - I*Sqrt[-d^2]*(Log[1 - I*Tan[a + b*x]]*Log[(-(c*d) + Sqrt[-d^2] - d^2*Tan[a + b*x])/(-(c*d) + I*d^2 + Sqrt[-d^2])] + PolyLog[2, (d^2*(1 - I*Tan[a + b*x]))/(I*c*d + d^2 - I*Sqrt[-d^2])]) + I*Sqrt[-d^2]*(Log[1 - I*Tan[a + b*x]]*Log[(c*d + Sqrt[-d^2] + d^2*Tan[a + b*x])/(c*d - I*d^2 + Sqrt[-d^2])] + PolyLog[2, (d^2*(1 - I*Tan[a + b*x]))/(I*c*d + d^2 + I*Sqrt[-d^2])]) + I*Sqrt[-d^2]*(Log[1 + I*Tan[a + b*x]]*Log[(c*d - Sqrt[-d^2] + d^2*Tan[a + b*x])/(c*d + I*d^2 - Sqrt[-d^2])] + PolyLog[2, (d^2*(1 + I*Tan[a + b*x]))/((-I)*c*d + d^2 + I*Sqrt[-d^2])]) - I*Sqrt[-d^2]*(Log[1 + I*Tan[a + b*x]]*Log[(c*d + Sqrt[-d^2] + d^2*Tan[a + b*x])/(c*d + I*d^2 + Sqrt[-d^2])] + PolyLog[2, (d^2*(1 + I*Tan[a + b*x]))/(d^2 - I*(c*d + Sqrt[-d^2]))])))/(2*d*(2*a - I*Log[1 - I*Tan[a + b*x]] + I*Log[1 + I*Tan[a + b*x]]))","B",0
161,0,0,18,0.4131997,"\int \frac{\cot ^{-1}(c+d \tan (a+b x))}{x} \, dx","Integrate[ArcCot[c + d*Tan[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+d \tan (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(d \tan (a+b x)+c)}{x},x\right)",0,"Integrate[ArcCot[c + d*Tan[a + b*x]]/x, x]","A",-1
162,1,136,154,0.2614339,"\int x^2 \cot ^{-1}(c+(1+i c) \tan (a+b x)) \, dx","Integrate[x^2*ArcCot[c + (1 + I*c)*Tan[a + b*x]],x]","\frac{1}{24} \left(\frac{3 \text{Li}_4\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)}{b^3}+\frac{6 i x \text{Li}_3\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)}{b^2}-\frac{6 x^2 \text{Li}_2\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)}{b}+4 i x^3 \log \left(1+\frac{i e^{-2 i (a+b x)}}{c}\right)+8 x^3 \cot ^{-1}(c+(1+i c) \tan (a+b x))\right)","-\frac{\text{Li}_4\left(i c e^{2 i a+2 i b x}\right)}{8 b^3}+\frac{i x \text{Li}_3\left(i c e^{2 i a+2 i b x}\right)}{4 b^2}+\frac{x^2 \text{Li}_2\left(i c e^{2 i a+2 i b x}\right)}{4 b}+\frac{1}{6} i x^3 \log \left(1-i c e^{2 i a+2 i b x}\right)+\frac{1}{3} x^3 \cot ^{-1}(c+(1+i c) \tan (a+b x))+\frac{b x^4}{12}",1,"(8*x^3*ArcCot[c + (1 + I*c)*Tan[a + b*x]] + (4*I)*x^3*Log[1 + I/(c*E^((2*I)*(a + b*x)))] - (6*x^2*PolyLog[2, (-I)/(c*E^((2*I)*(a + b*x)))])/b + ((6*I)*x*PolyLog[3, (-I)/(c*E^((2*I)*(a + b*x)))])/b^2 + (3*PolyLog[4, (-I)/(c*E^((2*I)*(a + b*x)))])/b^3)/24","A",1
163,1,110,123,0.1085143,"\int x \cot ^{-1}(c+(1+i c) \tan (a+b x)) \, dx","Integrate[x*ArcCot[c + (1 + I*c)*Tan[a + b*x]],x]","\frac{i \left(2 b^2 x^2 \log \left(1+\frac{i e^{-2 i (a+b x)}}{c}\right)+2 i b x \text{Li}_2\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)+\text{Li}_3\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)\right)}{8 b^2}+\frac{1}{2} x^2 \cot ^{-1}(c+(1+i c) \tan (a+b x))","\frac{i \text{Li}_3\left(i c e^{2 i a+2 i b x}\right)}{8 b^2}+\frac{x \text{Li}_2\left(i c e^{2 i a+2 i b x}\right)}{4 b}+\frac{1}{4} i x^2 \log \left(1-i c e^{2 i a+2 i b x}\right)+\frac{1}{2} x^2 \cot ^{-1}(c+(1+i c) \tan (a+b x))+\frac{b x^3}{6}",1,"(x^2*ArcCot[c + (1 + I*c)*Tan[a + b*x]])/2 + ((I/8)*(2*b^2*x^2*Log[1 + I/(c*E^((2*I)*(a + b*x)))] + (2*I)*b*x*PolyLog[2, (-I)/(c*E^((2*I)*(a + b*x)))] + PolyLog[3, (-I)/(c*E^((2*I)*(a + b*x)))]))/b^2","A",1
164,1,967,85,5.5844143,"\int \cot ^{-1}(c+(1+i c) \tan (a+b x)) \, dx","Integrate[ArcCot[c + (1 + I*c)*Tan[a + b*x]],x]","x \cot ^{-1}(c+(i c+1) \tan (a+b x))-\frac{i x \left(2 i b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))-\log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c+i) \cos (a+b x)+(i c+1) \sin (a+b x))}{2 c}\right) \log (1-i \tan (b x))+\log \left(\frac{\sec (b x) ((1-i c) \cos (a+b x)+(c-i) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\right) \log (i \tan (b x)+1)-\text{Li}_2(i \sin (2 b x)-\cos (2 b x))-\text{Li}_2\left(\frac{\sec (b x) ((c-i) \cos (a)+i (c+i) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)+\text{Li}_2\left(\frac{1}{2} \sec (b x) ((i c+1) \cos (a)-(c+i) \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right)\right) \sec ^2(a+b x) (\cos (b x)+i \sin (b x)) (i \cos (b x)+\sin (b x)) ((1-i c) \cos (a+b x)+(c-i) \sin (a+b x))}{((c+i) \cos (a+b x)+(i c+1) \sin (a+b x)) \left(\log (i \tan (b x)+1) \tan (b x) \cos ^2(a)+2 b x-i \log \left(1-\frac{\sec (b x) ((c-i) \cos (a)+i (c+i) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)-i \log \left(\frac{1}{2} \sec (b x) ((-i c-1) \cos (a)+(c+i) \sin (a)) (\cos (a+b x)+i \sin (a+b x))+1\right)+\log (i \tan (b x)+1) \sin ^2(a) \tan (b x)-2 i b x \tan (b x)+\log \left(1-\frac{\sec (b x) ((c-i) \cos (a)+i (c+i) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right) \tan (b x)-\log \left(\frac{1}{2} \sec (b x) ((-i c-1) \cos (a)+(c+i) \sin (a)) (\cos (a+b x)+i \sin (a+b x))+1\right) \tan (b x)-\log (1-i \tan (b x)) \tan (b x)-\frac{i (c-i) \cos (a+b x) (\log (1-i \tan (b x))-\log (i \tan (b x)+1))}{(c+i) \cos (a+b x)+(i c+1) \sin (a+b x)}+\frac{(c+i) (\log (1-i \tan (b x))-\log (i \tan (b x)+1)) \sin (a+b x)}{(c+i) \cos (a+b x)+(i c+1) \sin (a+b x)}+\frac{\log \left(\frac{\sec (b x) ((1-i c) \cos (a+b x)+(c-i) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\right) \sec ^2(b x)}{\tan (b x)-i}-\frac{\log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c+i) \cos (a+b x)+(i c+1) \sin (a+b x))}{2 c}\right) \sec ^2(b x)}{\tan (b x)+i}\right) (\tan (a+b x)-i) (-i c+(c-i) \tan (a+b x)+1)}","\frac{\text{Li}_2\left(i c e^{2 i a+2 i b x}\right)}{4 b}+\frac{1}{2} i x \log \left(1-i c e^{2 i a+2 i b x}\right)+x \cot ^{-1}(c+(1+i c) \tan (a+b x))+\frac{b x^2}{2}",1,"x*ArcCot[c + (1 + I*c)*Tan[a + b*x]] - (I*x*((2*I)*b*x*Log[2*Cos[b*x]*(Cos[b*x] - I*Sin[b*x])] - Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/(2*c)]*Log[1 - I*Tan[b*x]] + Log[(Sec[b*x]*((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]))/(2*Cos[a] - (2*I)*Sin[a])]*Log[1 + I*Tan[b*x]] - PolyLog[2, -Cos[2*b*x] + I*Sin[2*b*x]] - PolyLog[2, (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] + PolyLog[2, (Sec[b*x]*((1 + I*c)*Cos[a] - (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2])*Sec[a + b*x]^2*(Cos[b*x] + I*Sin[b*x])*(I*Cos[b*x] + Sin[b*x])*((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]))/(((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x])*(2*b*x - I*Log[1 - (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] - I*Log[1 + (Sec[b*x]*((-1 - I*c)*Cos[a] + (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2] - (I*(-I + c)*Cos[a + b*x]*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]]))/((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]) + ((I + c)*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]])*Sin[a + b*x])/((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]) - (2*I)*b*x*Tan[b*x] + Log[1 - (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)]*Tan[b*x] - Log[1 + (Sec[b*x]*((-1 - I*c)*Cos[a] + (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2]*Tan[b*x] - Log[1 - I*Tan[b*x]]*Tan[b*x] + Cos[a]^2*Log[1 + I*Tan[b*x]]*Tan[b*x] + Log[1 + I*Tan[b*x]]*Sin[a]^2*Tan[b*x] + (Log[(Sec[b*x]*((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]))/(2*Cos[a] - (2*I)*Sin[a])]*Sec[b*x]^2)/(-I + Tan[b*x]) - (Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/(2*c)]*Sec[b*x]^2)/(I + Tan[b*x]))*(-I + Tan[a + b*x])*(1 - I*c + (-I + c)*Tan[a + b*x]))","B",0
165,0,0,24,0.3567476,"\int \frac{\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx","Integrate[ArcCot[c + (1 + I*c)*Tan[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x},x\right)",0,"Integrate[ArcCot[c + (1 + I*c)*Tan[a + b*x]]/x, x]","A",-1
166,1,141,155,0.2554755,"\int x^2 \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx","Integrate[x^2*ArcCot[c - (1 - I*c)*Tan[a + b*x]],x]","\frac{1}{3} x^3 \cot ^{-1}(c+i (c+i) \tan (a+b x))-\frac{4 i b^3 x^3 \log \left(1-\frac{i e^{-2 i (a+b x)}}{c}\right)-6 b^2 x^2 \text{Li}_2\left(\frac{i e^{-2 i (a+b x)}}{c}\right)+6 i b x \text{Li}_3\left(\frac{i e^{-2 i (a+b x)}}{c}\right)+3 \text{Li}_4\left(\frac{i e^{-2 i (a+b x)}}{c}\right)}{24 b^3}","\frac{\text{Li}_4\left(-i c e^{2 i a+2 i b x}\right)}{8 b^3}-\frac{i x \text{Li}_3\left(-i c e^{2 i a+2 i b x}\right)}{4 b^2}-\frac{x^2 \text{Li}_2\left(-i c e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{6} i x^3 \log \left(1+i c e^{2 i a+2 i b x}\right)+\frac{1}{3} x^3 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac{b x^4}{12}",1,"(x^3*ArcCot[c + I*(I + c)*Tan[a + b*x]])/3 - ((4*I)*b^3*x^3*Log[1 - I/(c*E^((2*I)*(a + b*x)))] - 6*b^2*x^2*PolyLog[2, I/(c*E^((2*I)*(a + b*x)))] + (6*I)*b*x*PolyLog[3, I/(c*E^((2*I)*(a + b*x)))] + 3*PolyLog[4, I/(c*E^((2*I)*(a + b*x)))])/(24*b^3)","A",1
167,1,111,124,0.0969542,"\int x \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx","Integrate[x*ArcCot[c - (1 - I*c)*Tan[a + b*x]],x]","\frac{1}{2} x^2 \cot ^{-1}(c+i (c+i) \tan (a+b x))-\frac{i \left(2 b^2 x^2 \log \left(1-\frac{i e^{-2 i (a+b x)}}{c}\right)+2 i b x \text{Li}_2\left(\frac{i e^{-2 i (a+b x)}}{c}\right)+\text{Li}_3\left(\frac{i e^{-2 i (a+b x)}}{c}\right)\right)}{8 b^2}","-\frac{i \text{Li}_3\left(-i c e^{2 i a+2 i b x}\right)}{8 b^2}-\frac{x \text{Li}_2\left(-i c e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{4} i x^2 \log \left(1+i c e^{2 i a+2 i b x}\right)+\frac{1}{2} x^2 \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac{b x^3}{6}",1,"(x^2*ArcCot[c + I*(I + c)*Tan[a + b*x]])/2 - ((I/8)*(2*b^2*x^2*Log[1 - I/(c*E^((2*I)*(a + b*x)))] + (2*I)*b*x*PolyLog[2, I/(c*E^((2*I)*(a + b*x)))] + PolyLog[3, I/(c*E^((2*I)*(a + b*x)))]))/b^2","A",1
168,1,847,86,2.9780923,"\int \cot ^{-1}(c-(1-i c) \tan (a+b x)) \, dx","Integrate[ArcCot[c - (1 - I*c)*Tan[a + b*x]],x]","x \cot ^{-1}(c+i (c+i) \tan (a+b x))-\frac{i x \left(-2 i b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))+\log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c-i) \cos (a+b x)+i (c+i) \sin (a+b x))}{2 c}\right) \log (1-i \tan (b x))-\log \left(\frac{1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((i c+1) \cos (a+b x)-(c+i) \sin (a+b x))\right) \log (i \tan (b x)+1)+\text{Li}_2(i \sin (2 b x)-\cos (2 b x))+\text{Li}_2\left(\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)-\text{Li}_2\left(\frac{1}{2} (\cos (a)+i \sin (a)) ((c+i) \cos (a)+(i c+1) \sin (a)) (\tan (b x)-i)\right)\right) \sec (a+b x) (\cos (b x)+i \sin (b x)) (i \cos (b x)+\sin (b x))}{((c-i) \cos (a+b x)+i (c+i) \sin (a+b x)) \left(-\frac{\log \left(\frac{1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((i c+1) \cos (a+b x)-(c+i) \sin (a+b x))\right) \sec ^2(b x)}{\tan (b x)-i}+\frac{\log \left(1-\frac{1}{2} (\cos (a)+i \sin (a)) ((c+i) \cos (a)+(i c+1) \sin (a)) (\tan (b x)-i)\right) \sec ^2(b x)}{\tan (b x)-i}+\frac{\log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c-i) \cos (a+b x)+i (c+i) \sin (a+b x))}{2 c}\right) \sec ^2(b x)}{\tan (b x)+i}-2 b x+i \log \left(1-\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)+2 i b x \tan (b x)-\log \left(1-\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right) \tan (b x)+\log (1-i \tan (b x)) \tan (b x)-\log (i \tan (b x)+1) \tan (b x)+\frac{i (c+i) \cos (a+b x) (\log (1-i \tan (b x))-\log (i \tan (b x)+1))}{(c-i) \cos (a+b x)+i (c+i) \sin (a+b x)}+\frac{(i c+1) (\log (1-i \tan (b x))-\log (i \tan (b x)+1)) \sin (a+b x)}{(-i c-1) \cos (a+b x)+(c+i) \sin (a+b x)}\right) (\tan (a+b x)-i)}","-\frac{\text{Li}_2\left(-i c e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{2} i x \log \left(1+i c e^{2 i a+2 i b x}\right)+x \cot ^{-1}(c-(1-i c) \tan (a+b x))-\frac{b x^2}{2}",1,"x*ArcCot[c + I*(I + c)*Tan[a + b*x]] - (I*x*((-2*I)*b*x*Log[2*Cos[b*x]*(Cos[b*x] - I*Sin[b*x])] + Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]))/(2*c)]*Log[1 - I*Tan[b*x]] - Log[(Sec[b*x]*(Cos[a] + I*Sin[a])*((1 + I*c)*Cos[a + b*x] - (I + c)*Sin[a + b*x]))/2]*Log[1 + I*Tan[b*x]] + PolyLog[2, -Cos[2*b*x] + I*Sin[2*b*x]] + PolyLog[2, (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] - PolyLog[2, ((Cos[a] + I*Sin[a])*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(-I + Tan[b*x]))/2])*Sec[a + b*x]*(Cos[b*x] + I*Sin[b*x])*(I*Cos[b*x] + Sin[b*x]))/(((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x])*(-2*b*x + I*Log[1 - (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] + (I*(I + c)*Cos[a + b*x]*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]]))/((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]) + ((1 + I*c)*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]])*Sin[a + b*x])/((-1 - I*c)*Cos[a + b*x] + (I + c)*Sin[a + b*x]) + (2*I)*b*x*Tan[b*x] - Log[1 - (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)]*Tan[b*x] + Log[1 - I*Tan[b*x]]*Tan[b*x] - Log[1 + I*Tan[b*x]]*Tan[b*x] - (Log[(Sec[b*x]*(Cos[a] + I*Sin[a])*((1 + I*c)*Cos[a + b*x] - (I + c)*Sin[a + b*x]))/2]*Sec[b*x]^2)/(-I + Tan[b*x]) + (Log[1 - ((Cos[a] + I*Sin[a])*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(-I + Tan[b*x]))/2]*Sec[b*x]^2)/(-I + Tan[b*x]) + (Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]))/(2*c)]*Sec[b*x]^2)/(I + Tan[b*x]))*(-I + Tan[a + b*x]))","B",0
169,0,0,25,0.8353893,"\int \frac{\cot ^{-1}(c-(1-i c) \tan (a+b x))}{x} \, dx","Integrate[ArcCot[c - (1 - I*c)*Tan[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c-(1-i c) \tan (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c-(1-i c) \tan (a+b x))}{x},x\right)",0,"Integrate[ArcCot[c - (1 - I*c)*Tan[a + b*x]]/x, x]","A",-1
170,1,18,16,0.0094309,"\int \cot ^{-1}(\cot (a+b x)) \, dx","Integrate[ArcCot[Cot[a + b*x]],x]","x \cot ^{-1}(\cot (a+b x))-\frac{b x^2}{2}","\frac{\cot ^{-1}(\cot (a+b x))^2}{2 b}",1,"-1/2*(b*x^2) + x*ArcCot[Cot[a + b*x]]","A",1
171,1,359,399,0.9085686,"\int x^2 \cot ^{-1}(c+d \cot (a+b x)) \, dx","Integrate[x^2*ArcCot[c + d*Cot[a + b*x]],x]","\frac{1}{3} x^3 \cot ^{-1}(d \cot (a+b x)+c)+\frac{-4 i b^3 x^3 \log \left(1-\frac{(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right)+4 i b^3 x^3 \log \left(1-\frac{(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right)-6 b^2 x^2 \text{Li}_2\left(\frac{(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right)+6 b^2 x^2 \text{Li}_2\left(\frac{(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right)-6 i b x \text{Li}_3\left(\frac{(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right)+6 i b x \text{Li}_3\left(\frac{(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right)+3 \text{Li}_4\left(\frac{(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right)-3 \text{Li}_4\left(\frac{(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right)}{24 b^3}","\frac{\text{Li}_4\left(\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{8 b^3}-\frac{\text{Li}_4\left(\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{8 b^3}-\frac{i x \text{Li}_3\left(\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{4 b^2}+\frac{i x \text{Li}_3\left(\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{4 b^2}-\frac{x^2 \text{Li}_2\left(\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{4 b}+\frac{x^2 \text{Li}_2\left(\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{4 b}-\frac{1}{6} i x^3 \log \left(1-\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)+\frac{1}{6} i x^3 \log \left(1-\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)+\frac{1}{3} x^3 \cot ^{-1}(d \cot (a+b x)+c)",1,"(x^3*ArcCot[c + d*Cot[a + b*x]])/3 + ((-4*I)*b^3*x^3*Log[1 - ((c + I*(-1 + d))*E^((2*I)*(a + b*x)))/(c - I*(1 + d))] + (4*I)*b^3*x^3*Log[1 - ((c + I*(1 + d))*E^((2*I)*(a + b*x)))/(I + c - I*d)] - 6*b^2*x^2*PolyLog[2, ((c + I*(-1 + d))*E^((2*I)*(a + b*x)))/(c - I*(1 + d))] + 6*b^2*x^2*PolyLog[2, ((c + I*(1 + d))*E^((2*I)*(a + b*x)))/(I + c - I*d)] - (6*I)*b*x*PolyLog[3, ((c + I*(-1 + d))*E^((2*I)*(a + b*x)))/(c - I*(1 + d))] + (6*I)*b*x*PolyLog[3, ((c + I*(1 + d))*E^((2*I)*(a + b*x)))/(I + c - I*d)] + 3*PolyLog[4, ((c + I*(-1 + d))*E^((2*I)*(a + b*x)))/(c - I*(1 + d))] - 3*PolyLog[4, ((c + I*(1 + d))*E^((2*I)*(a + b*x)))/(I + c - I*d)])/(24*b^3)","A",1
172,1,270,303,0.5962933,"\int x \cot ^{-1}(c+d \cot (a+b x)) \, dx","Integrate[x*ArcCot[c + d*Cot[a + b*x]],x]","\frac{1}{2} x^2 \cot ^{-1}(d \cot (a+b x)+c)-\frac{i \left(2 b^2 x^2 \log \left(1-\frac{(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right)-2 b^2 x^2 \log \left(1-\frac{(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right)-2 i b x \text{Li}_2\left(\frac{(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right)+2 i b x \text{Li}_2\left(\frac{(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right)+\text{Li}_3\left(\frac{(c+i (d-1)) e^{2 i (a+b x)}}{c-i (d+1)}\right)-\text{Li}_3\left(\frac{(c+i (d+1)) e^{2 i (a+b x)}}{c-i d+i}\right)\right)}{8 b^2}","-\frac{i \text{Li}_3\left(\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{8 b^2}+\frac{i \text{Li}_3\left(\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{8 b^2}-\frac{x \text{Li}_2\left(\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{4 b}+\frac{x \text{Li}_2\left(\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{4 b}-\frac{1}{4} i x^2 \log \left(1-\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)+\frac{1}{4} i x^2 \log \left(1-\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)+\frac{1}{2} x^2 \cot ^{-1}(d \cot (a+b x)+c)",1,"(x^2*ArcCot[c + d*Cot[a + b*x]])/2 - ((I/8)*(2*b^2*x^2*Log[1 - ((c + I*(-1 + d))*E^((2*I)*(a + b*x)))/(c - I*(1 + d))] - 2*b^2*x^2*Log[1 - ((c + I*(1 + d))*E^((2*I)*(a + b*x)))/(I + c - I*d)] - (2*I)*b*x*PolyLog[2, ((c + I*(-1 + d))*E^((2*I)*(a + b*x)))/(c - I*(1 + d))] + (2*I)*b*x*PolyLog[2, ((c + I*(1 + d))*E^((2*I)*(a + b*x)))/(I + c - I*d)] + PolyLog[3, ((c + I*(-1 + d))*E^((2*I)*(a + b*x)))/(c - I*(1 + d))] - PolyLog[3, ((c + I*(1 + d))*E^((2*I)*(a + b*x)))/(I + c - I*d)]))/b^2","A",1
173,1,1649,198,13.0483351,"\int \cot ^{-1}(c+d \cot (a+b x)) \, dx","Integrate[ArcCot[c + d*Cot[a + b*x]],x]","x \cot ^{-1}(c+d \cot (a+b x))-\frac{d \left(4 a \sqrt{-d^2} \tan ^{-1}\left(\frac{\tan (a+b x) c^2+d c+\tan (a+b x)}{d}\right)+i d \log (i \tan (a+b x)+1) \log \left(\frac{\tan (a+b x) c^2+d c+\tan (a+b x)-\sqrt{-d^2}}{i c^2+d c-\sqrt{-d^2}+i}\right)+i d \log (1-i \tan (a+b x)) \log \left(\frac{\tan (a+b x) c^2+d c+\tan (a+b x)+\sqrt{-d^2}}{-i c^2+d c+\sqrt{-d^2}-i}\right)-i d \log (i \tan (a+b x)+1) \log \left(\frac{\tan (a+b x) c^2+d c+\tan (a+b x)+\sqrt{-d^2}}{i c^2+d c+\sqrt{-d^2}+i}\right)-i d \log (1-i \tan (a+b x)) \log \left(\frac{-c d-\left(c^2+1\right) \tan (a+b x)+\sqrt{-d^2}}{i c^2-d c+\sqrt{-d^2}+i}\right)-i d \text{Li}_2\left(\frac{\left(c^2+1\right) (1-i \tan (a+b x))}{c^2+i d c-i \sqrt{-d^2}+1}\right)+i d \text{Li}_2\left(\frac{\left(c^2+1\right) (1-i \tan (a+b x))}{c^2+i d c+i \sqrt{-d^2}+1}\right)-i d \text{Li}_2\left(\frac{\left(c^2+1\right) (i \tan (a+b x)+1)}{c^2-i d c-i \sqrt{-d^2}+1}\right)+i d \text{Li}_2\left(\frac{\left(c^2+1\right) (i \tan (a+b x)+1)}{c^2-i d c+i \sqrt{-d^2}+1}\right)\right) \left(\frac{2 a}{b \left(\cos (2 (a+b x)) c^2-c^2-2 d \sin (2 (a+b x)) c-d^2-d^2 \cos (2 (a+b x))+\cos (2 (a+b x))-1\right)}-\frac{2 (a+b x)}{b \left(\cos (2 (a+b x)) c^2-c^2-2 d \sin (2 (a+b x)) c-d^2-d^2 \cos (2 (a+b x))+\cos (2 (a+b x))-1\right)}\right)}{\frac{d \log \left(1-\frac{\left(c^2+1\right) (1-i \tan (a+b x))}{c^2+i d c-i \sqrt{-d^2}+1}\right) \sec ^2(a+b x)}{1-i \tan (a+b x)}-\frac{d \log \left(1-\frac{\left(c^2+1\right) (1-i \tan (a+b x))}{c^2+i d c+i \sqrt{-d^2}+1}\right) \sec ^2(a+b x)}{1-i \tan (a+b x)}+\frac{d \log \left(\frac{\tan (a+b x) c^2+d c+\tan (a+b x)+\sqrt{-d^2}}{-i c^2+d c+\sqrt{-d^2}-i}\right) \sec ^2(a+b x)}{1-i \tan (a+b x)}-\frac{d \log \left(\frac{-c d-\left(c^2+1\right) \tan (a+b x)+\sqrt{-d^2}}{i c^2-d c+\sqrt{-d^2}+i}\right) \sec ^2(a+b x)}{1-i \tan (a+b x)}-\frac{d \log \left(1-\frac{\left(c^2+1\right) (i \tan (a+b x)+1)}{c^2-i d c-i \sqrt{-d^2}+1}\right) \sec ^2(a+b x)}{i \tan (a+b x)+1}+\frac{d \log \left(1-\frac{\left(c^2+1\right) (i \tan (a+b x)+1)}{c^2-i d c+i \sqrt{-d^2}+1}\right) \sec ^2(a+b x)}{i \tan (a+b x)+1}-\frac{d \log \left(\frac{\tan (a+b x) c^2+d c+\tan (a+b x)-\sqrt{-d^2}}{i c^2+d c-\sqrt{-d^2}+i}\right) \sec ^2(a+b x)}{i \tan (a+b x)+1}+\frac{d \log \left(\frac{\tan (a+b x) c^2+d c+\tan (a+b x)+\sqrt{-d^2}}{i c^2+d c+\sqrt{-d^2}+i}\right) \sec ^2(a+b x)}{i \tan (a+b x)+1}+\frac{i \left(c^2+1\right) d \log (1-i \tan (a+b x)) \sec ^2(a+b x)}{-c d-\left(c^2+1\right) \tan (a+b x)+\sqrt{-d^2}}+\frac{i d \log (i \tan (a+b x)+1) \left(c^2 \sec ^2(a+b x)+\sec ^2(a+b x)\right)}{\tan (a+b x) c^2+d c+\tan (a+b x)-\sqrt{-d^2}}+\frac{i d \log (1-i \tan (a+b x)) \left(c^2 \sec ^2(a+b x)+\sec ^2(a+b x)\right)}{\tan (a+b x) c^2+d c+\tan (a+b x)+\sqrt{-d^2}}-\frac{i d \log (i \tan (a+b x)+1) \left(c^2 \sec ^2(a+b x)+\sec ^2(a+b x)\right)}{\tan (a+b x) c^2+d c+\tan (a+b x)+\sqrt{-d^2}}+\frac{4 a \sqrt{-d^2} \left(c^2 \sec ^2(a+b x)+\sec ^2(a+b x)\right)}{d \left(\frac{\left(\tan (a+b x) c^2+d c+\tan (a+b x)\right)^2}{d^2}+1\right)}}","-\frac{\text{Li}_2\left(\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)}{4 b}+\frac{\text{Li}_2\left(\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)}{4 b}-\frac{1}{2} i x \log \left(1-\frac{(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right)+\frac{1}{2} i x \log \left(1-\frac{(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right)+x \cot ^{-1}(d \cot (a+b x)+c)",1,"x*ArcCot[c + d*Cot[a + b*x]] - (d*(4*a*Sqrt[-d^2]*ArcTan[(c*d + Tan[a + b*x] + c^2*Tan[a + b*x])/d] + I*d*Log[1 + I*Tan[a + b*x]]*Log[(c*d - Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(I + I*c^2 + c*d - Sqrt[-d^2])] + I*d*Log[1 - I*Tan[a + b*x]]*Log[(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(-I - I*c^2 + c*d + Sqrt[-d^2])] - I*d*Log[1 + I*Tan[a + b*x]]*Log[(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(I + I*c^2 + c*d + Sqrt[-d^2])] - I*d*Log[1 - I*Tan[a + b*x]]*Log[(-(c*d) + Sqrt[-d^2] - (1 + c^2)*Tan[a + b*x])/(I + I*c^2 - c*d + Sqrt[-d^2])] - I*d*PolyLog[2, ((1 + c^2)*(1 - I*Tan[a + b*x]))/(1 + c^2 + I*c*d - I*Sqrt[-d^2])] + I*d*PolyLog[2, ((1 + c^2)*(1 - I*Tan[a + b*x]))/(1 + c^2 + I*c*d + I*Sqrt[-d^2])] - I*d*PolyLog[2, ((1 + c^2)*(1 + I*Tan[a + b*x]))/(1 + c^2 - I*c*d - I*Sqrt[-d^2])] + I*d*PolyLog[2, ((1 + c^2)*(1 + I*Tan[a + b*x]))/(1 + c^2 - I*c*d + I*Sqrt[-d^2])])*((2*a)/(b*(-1 - c^2 - d^2 + Cos[2*(a + b*x)] + c^2*Cos[2*(a + b*x)] - d^2*Cos[2*(a + b*x)] - 2*c*d*Sin[2*(a + b*x)])) - (2*(a + b*x))/(b*(-1 - c^2 - d^2 + Cos[2*(a + b*x)] + c^2*Cos[2*(a + b*x)] - d^2*Cos[2*(a + b*x)] - 2*c*d*Sin[2*(a + b*x)]))))/((d*Log[1 - ((1 + c^2)*(1 - I*Tan[a + b*x]))/(1 + c^2 + I*c*d - I*Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 - I*Tan[a + b*x]) - (d*Log[1 - ((1 + c^2)*(1 - I*Tan[a + b*x]))/(1 + c^2 + I*c*d + I*Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 - I*Tan[a + b*x]) + (d*Log[(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(-I - I*c^2 + c*d + Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 - I*Tan[a + b*x]) - (d*Log[(-(c*d) + Sqrt[-d^2] - (1 + c^2)*Tan[a + b*x])/(I + I*c^2 - c*d + Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 - I*Tan[a + b*x]) - (d*Log[1 - ((1 + c^2)*(1 + I*Tan[a + b*x]))/(1 + c^2 - I*c*d - I*Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 + I*Tan[a + b*x]) + (d*Log[1 - ((1 + c^2)*(1 + I*Tan[a + b*x]))/(1 + c^2 - I*c*d + I*Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 + I*Tan[a + b*x]) - (d*Log[(c*d - Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(I + I*c^2 + c*d - Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 + I*Tan[a + b*x]) + (d*Log[(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(I + I*c^2 + c*d + Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 + I*Tan[a + b*x]) + (I*d*Log[1 + I*Tan[a + b*x]]*(Sec[a + b*x]^2 + c^2*Sec[a + b*x]^2))/(c*d - Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x]) + (I*d*Log[1 - I*Tan[a + b*x]]*(Sec[a + b*x]^2 + c^2*Sec[a + b*x]^2))/(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x]) - (I*d*Log[1 + I*Tan[a + b*x]]*(Sec[a + b*x]^2 + c^2*Sec[a + b*x]^2))/(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x]) + (I*(1 + c^2)*d*Log[1 - I*Tan[a + b*x]]*Sec[a + b*x]^2)/(-(c*d) + Sqrt[-d^2] - (1 + c^2)*Tan[a + b*x]) + (4*a*Sqrt[-d^2]*(Sec[a + b*x]^2 + c^2*Sec[a + b*x]^2))/(d*(1 + (c*d + Tan[a + b*x] + c^2*Tan[a + b*x])^2/d^2)))","B",0
174,0,0,18,0.4220729,"\int \frac{\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx","Integrate[ArcCot[c + d*Cot[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(d \cot (a+b x)+c)}{x},x\right)",0,"Integrate[ArcCot[c + d*Cot[a + b*x]]/x, x]","A",-1
175,1,140,154,0.2325272,"\int x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx","Integrate[x^2*ArcCot[c + (1 - I*c)*Cot[a + b*x]],x]","\frac{1}{3} x^3 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac{4 i b^3 x^3 \log \left(1+\frac{i e^{-2 i (a+b x)}}{c}\right)-6 b^2 x^2 \text{Li}_2\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)+6 i b x \text{Li}_3\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)+3 \text{Li}_4\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)}{24 b^3}","\frac{\text{Li}_4\left(i c e^{2 i a+2 i b x}\right)}{8 b^3}-\frac{i x \text{Li}_3\left(i c e^{2 i a+2 i b x}\right)}{4 b^2}-\frac{x^2 \text{Li}_2\left(i c e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{6} i x^3 \log \left(1-i c e^{2 i a+2 i b x}\right)+\frac{1}{3} x^3 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac{b x^4}{12}",1,"(x^3*ArcCot[c + (1 - I*c)*Cot[a + b*x]])/3 - ((4*I)*b^3*x^3*Log[1 + I/(c*E^((2*I)*(a + b*x)))] - 6*b^2*x^2*PolyLog[2, (-I)/(c*E^((2*I)*(a + b*x)))] + (6*I)*b*x*PolyLog[3, (-I)/(c*E^((2*I)*(a + b*x)))] + 3*PolyLog[4, (-I)/(c*E^((2*I)*(a + b*x)))])/(24*b^3)","A",1
176,1,110,123,0.112918,"\int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx","Integrate[x*ArcCot[c + (1 - I*c)*Cot[a + b*x]],x]","\frac{1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac{i \left(2 b^2 x^2 \log \left(1+\frac{i e^{-2 i (a+b x)}}{c}\right)+2 i b x \text{Li}_2\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)+\text{Li}_3\left(-\frac{i e^{-2 i (a+b x)}}{c}\right)\right)}{8 b^2}","-\frac{i \text{Li}_3\left(i c e^{2 i a+2 i b x}\right)}{8 b^2}-\frac{x \text{Li}_2\left(i c e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{4} i x^2 \log \left(1-i c e^{2 i a+2 i b x}\right)+\frac{1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac{b x^3}{6}",1,"(x^2*ArcCot[c + (1 - I*c)*Cot[a + b*x]])/2 - ((I/8)*(2*b^2*x^2*Log[1 + I/(c*E^((2*I)*(a + b*x)))] + (2*I)*b*x*PolyLog[2, (-I)/(c*E^((2*I)*(a + b*x)))] + PolyLog[3, (-I)/(c*E^((2*I)*(a + b*x)))]))/b^2","A",1
177,1,929,85,6.8833244,"\int \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx","Integrate[ArcCot[c + (1 - I*c)*Cot[a + b*x]],x]","\frac{i x \left(2 b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))+i \log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c+i) \cos (a+b x)+(i c+1) \sin (a+b x))}{2 c}\right) \log (1-i \tan (b x))-i \log \left(\frac{\sec (b x) ((1-i c) \cos (a+b x)+(c-i) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\right) \log (i \tan (b x)+1)+i \text{Li}_2(i \sin (2 b x)-\cos (2 b x))+i \text{Li}_2\left(\frac{\sec (b x) ((c-i) \cos (a)+i (c+i) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)-i \text{Li}_2\left(\frac{1}{2} \sec (b x) ((i c+1) \cos (a)-(c+i) \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right)\right) (\cos (b x)-i \sin (b x)) (\cos (b x)+i \sin (b x)) \csc ^2(a+b x)}{(\cot (a+b x)+i) (i c+(c+i) \cot (a+b x)+1) \left(i \log (i \tan (b x)+1) \tan (b x) \cos ^2(a)+2 i b x+\log \left(1-\frac{\sec (b x) ((c-i) \cos (a)+i (c+i) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)+\log \left(\frac{1}{2} \sec (b x) ((-i c-1) \cos (a)+(c+i) \sin (a)) (\cos (a+b x)+i \sin (a+b x))+1\right)+i \log (i \tan (b x)+1) \sin ^2(a) \tan (b x)+2 b x \tan (b x)+i \log \left(1-\frac{\sec (b x) ((c-i) \cos (a)+i (c+i) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right) \tan (b x)-i \log \left(\frac{1}{2} \sec (b x) ((-i c-1) \cos (a)+(c+i) \sin (a)) (\cos (a+b x)+i \sin (a+b x))+1\right) \tan (b x)-i \log (1-i \tan (b x)) \tan (b x)+\frac{(c-i) \cos (a+b x) (\log (1-i \tan (b x))-\log (i \tan (b x)+1))}{(c+i) \cos (a+b x)+(i c+1) \sin (a+b x)}+\frac{(c+i) (\log (1-i \tan (b x))-\log (i \tan (b x)+1)) \sin (a+b x)}{(1-i c) \cos (a+b x)+(c-i) \sin (a+b x)}+\frac{i \log \left(\frac{\sec (b x) ((1-i c) \cos (a+b x)+(c-i) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\right) \sec ^2(b x)}{\tan (b x)-i}-\frac{i \log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c+i) \cos (a+b x)+(i c+1) \sin (a+b x))}{2 c}\right) \sec ^2(b x)}{\tan (b x)+i}\right)}+x \cot ^{-1}(c+(1-i c) \cot (a+b x))","-\frac{\text{Li}_2\left(i c e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{2} i x \log \left(1-i c e^{2 i a+2 i b x}\right)+x \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac{b x^2}{2}",1,"x*ArcCot[c + (1 - I*c)*Cot[a + b*x]] + (I*x*Csc[a + b*x]^2*(2*b*x*Log[2*Cos[b*x]*(Cos[b*x] - I*Sin[b*x])] + I*Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/(2*c)]*Log[1 - I*Tan[b*x]] - I*Log[(Sec[b*x]*((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]))/(2*Cos[a] - (2*I)*Sin[a])]*Log[1 + I*Tan[b*x]] + I*PolyLog[2, -Cos[2*b*x] + I*Sin[2*b*x]] + I*PolyLog[2, (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] - I*PolyLog[2, (Sec[b*x]*((1 + I*c)*Cos[a] - (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2])*(Cos[b*x] - I*Sin[b*x])*(Cos[b*x] + I*Sin[b*x]))/((I + Cot[a + b*x])*(1 + I*c + (I + c)*Cot[a + b*x])*((2*I)*b*x + Log[1 - (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] + Log[1 + (Sec[b*x]*((-1 - I*c)*Cos[a] + (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2] + ((-I + c)*Cos[a + b*x]*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]]))/((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]) + ((I + c)*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]])*Sin[a + b*x])/((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]) + 2*b*x*Tan[b*x] + I*Log[1 - (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)]*Tan[b*x] - I*Log[1 + (Sec[b*x]*((-1 - I*c)*Cos[a] + (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2]*Tan[b*x] - I*Log[1 - I*Tan[b*x]]*Tan[b*x] + I*Cos[a]^2*Log[1 + I*Tan[b*x]]*Tan[b*x] + I*Log[1 + I*Tan[b*x]]*Sin[a]^2*Tan[b*x] + (I*Log[(Sec[b*x]*((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]))/(2*Cos[a] - (2*I)*Sin[a])]*Sec[b*x]^2)/(-I + Tan[b*x]) - (I*Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/(2*c)]*Sec[b*x]^2)/(I + Tan[b*x])))","B",0
178,0,0,24,0.4143444,"\int \frac{\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx","Integrate[ArcCot[c + (1 - I*c)*Cot[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c+(1-i c) \cot (a+b x))}{x},x\right)",0,"Integrate[ArcCot[c + (1 - I*c)*Cot[a + b*x]]/x, x]","A",-1
179,1,136,155,0.2299881,"\int x^2 \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx","Integrate[x^2*ArcCot[c - (1 + I*c)*Cot[a + b*x]],x]","\frac{1}{24} \left(\frac{3 \text{Li}_4\left(\frac{i e^{-2 i (a+b x)}}{c}\right)}{b^3}+\frac{6 i x \text{Li}_3\left(\frac{i e^{-2 i (a+b x)}}{c}\right)}{b^2}-\frac{6 x^2 \text{Li}_2\left(\frac{i e^{-2 i (a+b x)}}{c}\right)}{b}+4 i x^3 \log \left(1-\frac{i e^{-2 i (a+b x)}}{c}\right)+8 x^3 \cot ^{-1}(c+(-1-i c) \cot (a+b x))\right)","-\frac{\text{Li}_4\left(-i c e^{2 i a+2 i b x}\right)}{8 b^3}+\frac{i x \text{Li}_3\left(-i c e^{2 i a+2 i b x}\right)}{4 b^2}+\frac{x^2 \text{Li}_2\left(-i c e^{2 i a+2 i b x}\right)}{4 b}+\frac{1}{6} i x^3 \log \left(1+i c e^{2 i a+2 i b x}\right)+\frac{1}{3} x^3 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac{b x^4}{12}",1,"(8*x^3*ArcCot[c + (-1 - I*c)*Cot[a + b*x]] + (4*I)*x^3*Log[1 - I/(c*E^((2*I)*(a + b*x)))] - (6*x^2*PolyLog[2, I/(c*E^((2*I)*(a + b*x)))])/b + ((6*I)*x*PolyLog[3, I/(c*E^((2*I)*(a + b*x)))])/b^2 + (3*PolyLog[4, I/(c*E^((2*I)*(a + b*x)))])/b^3)/24","A",1
180,1,110,124,0.1344507,"\int x \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx","Integrate[x*ArcCot[c - (1 + I*c)*Cot[a + b*x]],x]","\frac{i \left(2 b^2 x^2 \log \left(1-\frac{i e^{-2 i (a+b x)}}{c}\right)+2 i b x \text{Li}_2\left(\frac{i e^{-2 i (a+b x)}}{c}\right)+\text{Li}_3\left(\frac{i e^{-2 i (a+b x)}}{c}\right)\right)}{8 b^2}+\frac{1}{2} x^2 \cot ^{-1}(c+(-1-i c) \cot (a+b x))","\frac{i \text{Li}_3\left(-i c e^{2 i a+2 i b x}\right)}{8 b^2}+\frac{x \text{Li}_2\left(-i c e^{2 i a+2 i b x}\right)}{4 b}+\frac{1}{4} i x^2 \log \left(1+i c e^{2 i a+2 i b x}\right)+\frac{1}{2} x^2 \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac{b x^3}{6}",1,"(x^2*ArcCot[c + (-1 - I*c)*Cot[a + b*x]])/2 + ((I/8)*(2*b^2*x^2*Log[1 - I/(c*E^((2*I)*(a + b*x)))] + (2*I)*b*x*PolyLog[2, I/(c*E^((2*I)*(a + b*x)))] + PolyLog[3, I/(c*E^((2*I)*(a + b*x)))]))/b^2","A",1
181,1,872,86,2.8437844,"\int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx","Integrate[ArcCot[c - (1 + I*c)*Cot[a + b*x]],x]","x \cot ^{-1}(c+(-i c-1) \cot (a+b x))-\frac{i x \csc (a+b x) \left(2 b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))+i \log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c-i) \cos (a+b x)+i (c+i) \sin (a+b x))}{2 c}\right) \log (1-i \tan (b x))-i \log \left(\frac{1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((i c+1) \cos (a+b x)-(c+i) \sin (a+b x))\right) \log (i \tan (b x)+1)+i \text{Li}_2(i \sin (2 b x)-\cos (2 b x))+i \text{Li}_2\left(\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)-i \text{Li}_2\left(\frac{1}{2} (\cos (a)+i \sin (a)) ((c+i) \cos (a)+(i c+1) \sin (a)) (\tan (b x)-i)\right)\right) (\cos (b x)-i \sin (b x)) (\cos (b x)+i \sin (b x))}{(\cot (a+b x)+i) ((c-i) \cos (a+b x)+i (c+i) \sin (a+b x)) \left(\frac{\log \left(\frac{1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((i c+1) \cos (a+b x)-(c+i) \sin (a+b x))\right) \sec ^2(b x)}{i \tan (b x)+1}+\frac{i \log \left(1-\frac{1}{2} (\cos (a)+i \sin (a)) ((c+i) \cos (a)+(i c+1) \sin (a)) (\tan (b x)-i)\right) \sec ^2(b x)}{\tan (b x)-i}+\frac{i \log \left(\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c-i) \cos (a+b x)+i (c+i) \sin (a+b x))}{2 c}\right) \sec ^2(b x)}{\tan (b x)+i}-2 i b x-\log \left(1-\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right)-2 b x \tan (b x)-i \log \left(1-\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right) \tan (b x)+i \log (1-i \tan (b x)) \tan (b x)-i \log (i \tan (b x)+1) \tan (b x)-\frac{\log (1-i \tan (b x)) ((c+i) \cos (a+b x)+(i c+1) \sin (a+b x))}{(c-i) \cos (a+b x)+i (c+i) \sin (a+b x)}+\frac{\log (i \tan (b x)+1) ((c+i) \cos (a+b x)+(i c+1) \sin (a+b x))}{(c-i) \cos (a+b x)+i (c+i) \sin (a+b x)}\right)}","\frac{\text{Li}_2\left(-i c e^{2 i a+2 i b x}\right)}{4 b}+\frac{1}{2} i x \log \left(1+i c e^{2 i a+2 i b x}\right)+x \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac{b x^2}{2}",1,"x*ArcCot[c + (-1 - I*c)*Cot[a + b*x]] - (I*x*Csc[a + b*x]*(2*b*x*Log[2*Cos[b*x]*(Cos[b*x] - I*Sin[b*x])] + I*Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]))/(2*c)]*Log[1 - I*Tan[b*x]] - I*Log[(Sec[b*x]*(Cos[a] + I*Sin[a])*((1 + I*c)*Cos[a + b*x] - (I + c)*Sin[a + b*x]))/2]*Log[1 + I*Tan[b*x]] + I*PolyLog[2, -Cos[2*b*x] + I*Sin[2*b*x]] + I*PolyLog[2, (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] - I*PolyLog[2, ((Cos[a] + I*Sin[a])*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(-I + Tan[b*x]))/2])*(Cos[b*x] - I*Sin[b*x])*(Cos[b*x] + I*Sin[b*x]))/((I + Cot[a + b*x])*((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x])*((-2*I)*b*x - Log[1 - (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] - (Log[1 - I*Tan[b*x]]*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]) + (Log[1 + I*Tan[b*x]]*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]) + (Log[(Sec[b*x]*(Cos[a] + I*Sin[a])*((1 + I*c)*Cos[a + b*x] - (I + c)*Sin[a + b*x]))/2]*Sec[b*x]^2)/(1 + I*Tan[b*x]) - 2*b*x*Tan[b*x] - I*Log[1 - (Sec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)]*Tan[b*x] + I*Log[1 - I*Tan[b*x]]*Tan[b*x] - I*Log[1 + I*Tan[b*x]]*Tan[b*x] + (I*Log[1 - ((Cos[a] + I*Sin[a])*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(-I + Tan[b*x]))/2]*Sec[b*x]^2)/(-I + Tan[b*x]) + (I*Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]))/(2*c)]*Sec[b*x]^2)/(I + Tan[b*x])))","B",0
182,0,0,25,0.4126332,"\int \frac{\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx","Integrate[ArcCot[c - (1 + I*c)*Cot[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x},x\right)",0,"Integrate[ArcCot[c - (1 + I*c)*Cot[a + b*x]]/x, x]","A",-1
183,1,600,299,0.3618201,"\int (e+f x)^3 \cot ^{-1}(\tanh (a+b x)) \, dx","Integrate[(e + f*x)^3*ArcCot[Tanh[a + b*x]],x]","\frac{1}{4} x \left(4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right) \cot ^{-1}(\tanh (a+b x))+\frac{i \left(8 b^4 e^3 x \log \left(1-i e^{2 (a+b x)}\right)-8 b^4 e^3 x \log \left(1+i e^{2 (a+b x)}\right)+12 b^4 e^2 f x^2 \log \left(1-i e^{2 (a+b x)}\right)-12 b^4 e^2 f x^2 \log \left(1+i e^{2 (a+b x)}\right)+8 b^4 e f^2 x^3 \log \left(1-i e^{2 (a+b x)}\right)-8 b^4 e f^2 x^3 \log \left(1+i e^{2 (a+b x)}\right)+2 b^4 f^3 x^4 \log \left(1-i e^{2 (a+b x)}\right)-2 b^4 f^3 x^4 \log \left(1+i e^{2 (a+b x)}\right)-4 b^3 (e+f x)^3 \text{Li}_2\left(-i e^{2 (a+b x)}\right)+4 b^3 (e+f x)^3 \text{Li}_2\left(i e^{2 (a+b x)}\right)+6 b^2 e^2 f \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b^2 e^2 f \text{Li}_3\left(i e^{2 (a+b x)}\right)+12 b^2 e f^2 x \text{Li}_3\left(-i e^{2 (a+b x)}\right)-12 b^2 e f^2 x \text{Li}_3\left(i e^{2 (a+b x)}\right)+6 b^2 f^3 x^2 \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b^2 f^3 x^2 \text{Li}_3\left(i e^{2 (a+b x)}\right)-6 b e f^2 \text{Li}_4\left(-i e^{2 (a+b x)}\right)+6 b e f^2 \text{Li}_4\left(i e^{2 (a+b x)}\right)-6 b f^3 x \text{Li}_4\left(-i e^{2 (a+b x)}\right)+6 b f^3 x \text{Li}_4\left(i e^{2 (a+b x)}\right)+3 f^3 \text{Li}_5\left(-i e^{2 (a+b x)}\right)-3 f^3 \text{Li}_5\left(i e^{2 (a+b x)}\right)\right)}{16 b^4}","\frac{3 i f^3 \text{Li}_5\left(-i e^{2 a+2 b x}\right)}{16 b^4}-\frac{3 i f^3 \text{Li}_5\left(i e^{2 a+2 b x}\right)}{16 b^4}-\frac{3 i f^2 (e+f x) \text{Li}_4\left(-i e^{2 a+2 b x}\right)}{8 b^3}+\frac{3 i f^2 (e+f x) \text{Li}_4\left(i e^{2 a+2 b x}\right)}{8 b^3}+\frac{3 i f (e+f x)^2 \text{Li}_3\left(-i e^{2 a+2 b x}\right)}{8 b^2}-\frac{3 i f (e+f x)^2 \text{Li}_3\left(i e^{2 a+2 b x}\right)}{8 b^2}-\frac{i (e+f x)^3 \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}+\frac{i (e+f x)^3 \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}+\frac{(e+f x)^4 \tan ^{-1}\left(e^{2 a+2 b x}\right)}{4 f}+\frac{(e+f x)^4 \cot ^{-1}(\tanh (a+b x))}{4 f}",1,"(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)*ArcCot[Tanh[a + b*x]])/4 + ((I/16)*(8*b^4*e^3*x*Log[1 - I*E^(2*(a + b*x))] + 12*b^4*e^2*f*x^2*Log[1 - I*E^(2*(a + b*x))] + 8*b^4*e*f^2*x^3*Log[1 - I*E^(2*(a + b*x))] + 2*b^4*f^3*x^4*Log[1 - I*E^(2*(a + b*x))] - 8*b^4*e^3*x*Log[1 + I*E^(2*(a + b*x))] - 12*b^4*e^2*f*x^2*Log[1 + I*E^(2*(a + b*x))] - 8*b^4*e*f^2*x^3*Log[1 + I*E^(2*(a + b*x))] - 2*b^4*f^3*x^4*Log[1 + I*E^(2*(a + b*x))] - 4*b^3*(e + f*x)^3*PolyLog[2, (-I)*E^(2*(a + b*x))] + 4*b^3*(e + f*x)^3*PolyLog[2, I*E^(2*(a + b*x))] + 6*b^2*e^2*f*PolyLog[3, (-I)*E^(2*(a + b*x))] + 12*b^2*e*f^2*x*PolyLog[3, (-I)*E^(2*(a + b*x))] + 6*b^2*f^3*x^2*PolyLog[3, (-I)*E^(2*(a + b*x))] - 6*b^2*e^2*f*PolyLog[3, I*E^(2*(a + b*x))] - 12*b^2*e*f^2*x*PolyLog[3, I*E^(2*(a + b*x))] - 6*b^2*f^3*x^2*PolyLog[3, I*E^(2*(a + b*x))] - 6*b*e*f^2*PolyLog[4, (-I)*E^(2*(a + b*x))] - 6*b*f^3*x*PolyLog[4, (-I)*E^(2*(a + b*x))] + 6*b*e*f^2*PolyLog[4, I*E^(2*(a + b*x))] + 6*b*f^3*x*PolyLog[4, I*E^(2*(a + b*x))] + 3*f^3*PolyLog[5, (-I)*E^(2*(a + b*x))] - 3*f^3*PolyLog[5, I*E^(2*(a + b*x))]))/b^4","B",1
184,1,375,229,0.2080571,"\int (e+f x)^2 \cot ^{-1}(\tanh (a+b x)) \, dx","Integrate[(e + f*x)^2*ArcCot[Tanh[a + b*x]],x]","\frac{1}{3} x \left(3 e^2+3 e f x+f^2 x^2\right) \cot ^{-1}(\tanh (a+b x))+\frac{i \left(12 b^3 e^2 x \log \left(1-i e^{2 (a+b x)}\right)-12 b^3 e^2 x \log \left(1+i e^{2 (a+b x)}\right)+12 b^3 e f x^2 \log \left(1-i e^{2 (a+b x)}\right)-12 b^3 e f x^2 \log \left(1+i e^{2 (a+b x)}\right)+4 b^3 f^2 x^3 \log \left(1-i e^{2 (a+b x)}\right)-4 b^3 f^2 x^3 \log \left(1+i e^{2 (a+b x)}\right)-6 b^2 (e+f x)^2 \text{Li}_2\left(-i e^{2 (a+b x)}\right)+6 b^2 (e+f x)^2 \text{Li}_2\left(i e^{2 (a+b x)}\right)+6 b e f \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b e f \text{Li}_3\left(i e^{2 (a+b x)}\right)+6 b f^2 x \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b f^2 x \text{Li}_3\left(i e^{2 (a+b x)}\right)-3 f^2 \text{Li}_4\left(-i e^{2 (a+b x)}\right)+3 f^2 \text{Li}_4\left(i e^{2 (a+b x)}\right)\right)}{24 b^3}","-\frac{i f^2 \text{Li}_4\left(-i e^{2 a+2 b x}\right)}{8 b^3}+\frac{i f^2 \text{Li}_4\left(i e^{2 a+2 b x}\right)}{8 b^3}+\frac{i f (e+f x) \text{Li}_3\left(-i e^{2 a+2 b x}\right)}{4 b^2}-\frac{i f (e+f x) \text{Li}_3\left(i e^{2 a+2 b x}\right)}{4 b^2}-\frac{i (e+f x)^2 \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}+\frac{i (e+f x)^2 \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}+\frac{(e+f x)^3 \tan ^{-1}\left(e^{2 a+2 b x}\right)}{3 f}+\frac{(e+f x)^3 \cot ^{-1}(\tanh (a+b x))}{3 f}",1,"(x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCot[Tanh[a + b*x]])/3 + ((I/24)*(12*b^3*e^2*x*Log[1 - I*E^(2*(a + b*x))] + 12*b^3*e*f*x^2*Log[1 - I*E^(2*(a + b*x))] + 4*b^3*f^2*x^3*Log[1 - I*E^(2*(a + b*x))] - 12*b^3*e^2*x*Log[1 + I*E^(2*(a + b*x))] - 12*b^3*e*f*x^2*Log[1 + I*E^(2*(a + b*x))] - 4*b^3*f^2*x^3*Log[1 + I*E^(2*(a + b*x))] - 6*b^2*(e + f*x)^2*PolyLog[2, (-I)*E^(2*(a + b*x))] + 6*b^2*(e + f*x)^2*PolyLog[2, I*E^(2*(a + b*x))] + 6*b*e*f*PolyLog[3, (-I)*E^(2*(a + b*x))] + 6*b*f^2*x*PolyLog[3, (-I)*E^(2*(a + b*x))] - 6*b*e*f*PolyLog[3, I*E^(2*(a + b*x))] - 6*b*f^2*x*PolyLog[3, I*E^(2*(a + b*x))] - 3*f^2*PolyLog[4, (-I)*E^(2*(a + b*x))] + 3*f^2*PolyLog[4, I*E^(2*(a + b*x))]))/b^3","A",1
185,1,278,159,0.2399672,"\int (e+f x) \cot ^{-1}(\tanh (a+b x)) \, dx","Integrate[(e + f*x)*ArcCot[Tanh[a + b*x]],x]","\frac{i f \left(2 b^2 x^2 \log \left(1-i e^{2 (a+b x)}\right)-2 b^2 x^2 \log \left(1+i e^{2 (a+b x)}\right)-2 b x \text{Li}_2\left(-i e^{2 (a+b x)}\right)+2 b x \text{Li}_2\left(i e^{2 (a+b x)}\right)+\text{Li}_3\left(-i e^{2 (a+b x)}\right)-\text{Li}_3\left(i e^{2 (a+b x)}\right)\right)}{8 b^2}+\frac{e \left(-2 i \left(\text{Li}_2\left(-i e^{2 (a+b x)}\right)-\text{Li}_2\left(i e^{2 (a+b x)}\right)\right)-\left((-4 i a-4 i b x+\pi ) \left(\log \left(1-i e^{2 (a+b x)}\right)-\log \left(1+i e^{2 (a+b x)}\right)\right)\right)+(\pi -4 i a) \log \left(\cot \left(\frac{1}{4} (4 i a+4 i b x+\pi )\right)\right)\right)}{8 b}+e x \cot ^{-1}(\tanh (a+b x))+\frac{1}{2} f x^2 \cot ^{-1}(\tanh (a+b x))","\frac{i f \text{Li}_3\left(-i e^{2 a+2 b x}\right)}{8 b^2}-\frac{i f \text{Li}_3\left(i e^{2 a+2 b x}\right)}{8 b^2}-\frac{i (e+f x) \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}+\frac{i (e+f x) \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}+\frac{(e+f x)^2 \tan ^{-1}\left(e^{2 a+2 b x}\right)}{2 f}+\frac{(e+f x)^2 \cot ^{-1}(\tanh (a+b x))}{2 f}",1,"e*x*ArcCot[Tanh[a + b*x]] + (f*x^2*ArcCot[Tanh[a + b*x]])/2 + (e*(-(((-4*I)*a + Pi - (4*I)*b*x)*(Log[1 - I*E^(2*(a + b*x))] - Log[1 + I*E^(2*(a + b*x))])) + ((-4*I)*a + Pi)*Log[Cot[((4*I)*a + Pi + (4*I)*b*x)/4]] - (2*I)*(PolyLog[2, (-I)*E^(2*(a + b*x))] - PolyLog[2, I*E^(2*(a + b*x))])))/(8*b) + ((I/8)*f*(2*b^2*x^2*Log[1 - I*E^(2*(a + b*x))] - 2*b^2*x^2*Log[1 + I*E^(2*(a + b*x))] - 2*b*x*PolyLog[2, (-I)*E^(2*(a + b*x))] + 2*b*x*PolyLog[2, I*E^(2*(a + b*x))] + PolyLog[3, (-I)*E^(2*(a + b*x))] - PolyLog[3, I*E^(2*(a + b*x))]))/b^2","A",1
186,1,132,73,0.0568687,"\int \cot ^{-1}(\tanh (a+b x)) \, dx","Integrate[ArcCot[Tanh[a + b*x]],x]","x \cot ^{-1}(\tanh (a+b x))+\frac{-2 i \left(\text{Li}_2\left(-i e^{2 (a+b x)}\right)-\text{Li}_2\left(i e^{2 (a+b x)}\right)\right)-\left((-4 i a-4 i b x+\pi ) \left(\log \left(1-i e^{2 (a+b x)}\right)-\log \left(1+i e^{2 (a+b x)}\right)\right)\right)+(\pi -4 i a) \log \left(\cot \left(\frac{1}{4} (4 i a+4 i b x+\pi )\right)\right)}{8 b}","-\frac{i \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}+\frac{i \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}+x \tan ^{-1}\left(e^{2 a+2 b x}\right)+x \cot ^{-1}(\tanh (a+b x))",1,"x*ArcCot[Tanh[a + b*x]] + (-(((-4*I)*a + Pi - (4*I)*b*x)*(Log[1 - I*E^(2*(a + b*x))] - Log[1 + I*E^(2*(a + b*x))])) + ((-4*I)*a + Pi)*Log[Cot[((4*I)*a + Pi + (4*I)*b*x)/4]] - (2*I)*(PolyLog[2, (-I)*E^(2*(a + b*x))] - PolyLog[2, I*E^(2*(a + b*x))]))/(8*b)","A",1
187,0,0,18,4.3066346,"\int \frac{\cot ^{-1}(\tanh (a+b x))}{e+f x} \, dx","Integrate[ArcCot[Tanh[a + b*x]]/(e + f*x),x]","\int \frac{\cot ^{-1}(\tanh (a+b x))}{e+f x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(\tanh (a+b x))}{e+f x},x\right)",0,"Integrate[ArcCot[Tanh[a + b*x]]/(e + f*x), x]","A",-1
188,1,305,355,5.4564926,"\int x^2 \cot ^{-1}(c+d \tanh (a+b x)) \, dx","Integrate[x^2*ArcCot[c + d*Tanh[a + b*x]],x]","\frac{1}{3} x^3 \cot ^{-1}(d \tanh (a+b x)+c)-\frac{i \left(4 b^3 x^3 \log \left(1+\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-4 b^3 x^3 \log \left(1+\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)+6 b^2 x^2 \text{Li}_2\left(-\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-6 b^2 x^2 \text{Li}_2\left(-\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)-6 b x \text{Li}_3\left(-\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)+6 b x \text{Li}_3\left(-\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)+3 \text{Li}_4\left(-\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-3 \text{Li}_4\left(-\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)\right)}{24 b^3}","-\frac{i \text{Li}_4\left(-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^3}+\frac{i \text{Li}_4\left(-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^3}+\frac{i x \text{Li}_3\left(-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b^2}-\frac{i x \text{Li}_3\left(-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b^2}-\frac{i x^2 \text{Li}_2\left(-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i x^2 \text{Li}_2\left(-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b}-\frac{1}{6} i x^3 \log \left(1+\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)+\frac{1}{6} i x^3 \log \left(1+\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)+\frac{1}{3} x^3 \cot ^{-1}(d \tanh (a+b x)+c)",1,"(x^3*ArcCot[c + d*Tanh[a + b*x]])/3 - ((I/24)*(4*b^3*x^3*Log[1 + ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] - 4*b^3*x^3*Log[1 + ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)] + 6*b^2*x^2*PolyLog[2, -(((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d))] - 6*b^2*x^2*PolyLog[2, -(((I + c + d)*E^(2*(a + b*x)))/(I + c - d))] - 6*b*x*PolyLog[3, -(((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d))] + 6*b*x*PolyLog[3, -(((I + c + d)*E^(2*(a + b*x)))/(I + c - d))] + 3*PolyLog[4, -(((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d))] - 3*PolyLog[4, -(((I + c + d)*E^(2*(a + b*x)))/(I + c - d))]))/b^3","A",1
189,1,229,267,4.196183,"\int x \cot ^{-1}(c+d \tanh (a+b x)) \, dx","Integrate[x*ArcCot[c + d*Tanh[a + b*x]],x]","\frac{1}{2} x^2 \cot ^{-1}(d \tanh (a+b x)+c)-\frac{i \left(2 b^2 x^2 \log \left(1+\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-2 b^2 x^2 \log \left(1+\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)+2 b x \text{Li}_2\left(-\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-2 b x \text{Li}_2\left(-\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)-\text{Li}_3\left(-\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)+\text{Li}_3\left(-\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)\right)}{8 b^2}","\frac{i \text{Li}_3\left(-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^2}-\frac{i \text{Li}_3\left(-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^2}-\frac{i x \text{Li}_2\left(-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i x \text{Li}_2\left(-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b}-\frac{1}{4} i x^2 \log \left(1+\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)+\frac{1}{4} i x^2 \log \left(1+\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)+\frac{1}{2} x^2 \cot ^{-1}(d \tanh (a+b x)+c)",1,"(x^2*ArcCot[c + d*Tanh[a + b*x]])/2 - ((I/8)*(2*b^2*x^2*Log[1 + ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] - 2*b^2*x^2*Log[1 + ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)] + 2*b*x*PolyLog[2, -(((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d))] - 2*b*x*PolyLog[2, -(((I + c + d)*E^(2*(a + b*x)))/(I + c - d))] - PolyLog[3, -(((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d))] + PolyLog[3, -(((I + c + d)*E^(2*(a + b*x)))/(I + c - d))]))/b^2","A",1
190,1,288,174,1.5105812,"\int \cot ^{-1}(c+d \tanh (a+b x)) \, dx","Integrate[ArcCot[c + d*Tanh[a + b*x]],x]","x \cot ^{-1}(d \tanh (a+b x)+c)-\frac{d \text{Li}_2\left(-\frac{\left(c^2+2 d c+d^2+1\right) e^{2 (a+b x)}}{c^2-d^2+2 \sqrt{-d^2}+1}\right)-d \text{Li}_2\left(\frac{\left(c^2+2 d c+d^2+1\right) e^{2 (a+b x)}}{-c^2+d^2+2 \sqrt{-d^2}-1}\right)-2 d (a+b x) \log \left(\frac{2 \left((c+d)^2+1\right) e^{2 (a+b x)}}{2 c^2-2 d^2-4 \sqrt{-d^2}+2}+1\right)+2 d (a+b x) \log \left(\frac{\left((c+d)^2+1\right) e^{2 (a+b x)}}{c^2-d^2+2 \sqrt{-d^2}+1}+1\right)+4 a \sqrt{-d^2} \tan ^{-1}\left(\frac{\left(c^2+2 c d+d^2+1\right) e^{2 (a+b x)}+c^2-d^2+1}{2 d}\right)}{4 b \sqrt{-d^2}}","-\frac{i \text{Li}_2\left(-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i \text{Li}_2\left(-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b}-\frac{1}{2} i x \log \left(1+\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)+\frac{1}{2} i x \log \left(1+\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)+x \cot ^{-1}(d \tanh (a+b x)+c)",1,"x*ArcCot[c + d*Tanh[a + b*x]] - (4*a*Sqrt[-d^2]*ArcTan[(1 + c^2 - d^2 + (1 + c^2 + 2*c*d + d^2)*E^(2*(a + b*x)))/(2*d)] - 2*d*(a + b*x)*Log[1 + (2*(1 + (c + d)^2)*E^(2*(a + b*x)))/(2 + 2*c^2 - 2*d^2 - 4*Sqrt[-d^2])] + 2*d*(a + b*x)*Log[1 + ((1 + (c + d)^2)*E^(2*(a + b*x)))/(1 + c^2 - d^2 + 2*Sqrt[-d^2])] + d*PolyLog[2, -(((1 + c^2 + 2*c*d + d^2)*E^(2*(a + b*x)))/(1 + c^2 - d^2 + 2*Sqrt[-d^2]))] - d*PolyLog[2, ((1 + c^2 + 2*c*d + d^2)*E^(2*(a + b*x)))/(-1 - c^2 + d^2 + 2*Sqrt[-d^2])])/(4*b*Sqrt[-d^2])","A",1
191,0,0,18,9.510515,"\int \frac{\cot ^{-1}(c+d \tanh (a+b x))}{x} \, dx","Integrate[ArcCot[c + d*Tanh[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+d \tanh (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(d \tanh (a+b x)+c)}{x},x\right)",0,"Integrate[ArcCot[c + d*Tanh[a + b*x]]/x, x]","A",-1
192,1,128,142,0.2212749,"\int x^2 \cot ^{-1}(c+(i+c) \tanh (a+b x)) \, dx","Integrate[x^2*ArcCot[c + (I + c)*Tanh[a + b*x]],x]","\frac{1}{3} x^3 \cot ^{-1}(c+(c+i) \tanh (a+b x))-\frac{i \left(4 b^3 x^3 \log \left(1-\frac{i e^{-2 (a+b x)}}{c}\right)-6 b^2 x^2 \text{Li}_2\left(\frac{i e^{-2 (a+b x)}}{c}\right)-6 b x \text{Li}_3\left(\frac{i e^{-2 (a+b x)}}{c}\right)-3 \text{Li}_4\left(\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{24 b^3}","-\frac{i \text{Li}_4\left(-i c e^{2 a+2 b x}\right)}{8 b^3}+\frac{i x \text{Li}_3\left(-i c e^{2 a+2 b x}\right)}{4 b^2}-\frac{i x^2 \text{Li}_2\left(-i c e^{2 a+2 b x}\right)}{4 b}-\frac{1}{6} i x^3 \log \left(1+i c e^{2 a+2 b x}\right)+\frac{1}{3} x^3 \cot ^{-1}(c+(c+i) \tanh (a+b x))+\frac{1}{12} i b x^4",1,"(x^3*ArcCot[c + (I + c)*Tanh[a + b*x]])/3 - ((I/24)*(4*b^3*x^3*Log[1 - I/(c*E^(2*(a + b*x)))] - 6*b^2*x^2*PolyLog[2, I/(c*E^(2*(a + b*x)))] - 6*b*x*PolyLog[3, I/(c*E^(2*(a + b*x)))] - 3*PolyLog[4, I/(c*E^(2*(a + b*x)))]))/b^3","A",1
193,1,102,113,0.0890977,"\int x \cot ^{-1}(c+(i+c) \tanh (a+b x)) \, dx","Integrate[x*ArcCot[c + (I + c)*Tanh[a + b*x]],x]","\frac{1}{2} x^2 \cot ^{-1}(c+(c+i) \tanh (a+b x))-\frac{i \left(2 b^2 x^2 \log \left(1-\frac{i e^{-2 (a+b x)}}{c}\right)-2 b x \text{Li}_2\left(\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_3\left(\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{8 b^2}","\frac{i \text{Li}_3\left(-i c e^{2 a+2 b x}\right)}{8 b^2}-\frac{i x \text{Li}_2\left(-i c e^{2 a+2 b x}\right)}{4 b}-\frac{1}{4} i x^2 \log \left(1+i c e^{2 a+2 b x}\right)+\frac{1}{2} x^2 \cot ^{-1}(c+(c+i) \tanh (a+b x))+\frac{1}{6} i b x^3",1,"(x^2*ArcCot[c + (I + c)*Tanh[a + b*x]])/2 - ((I/8)*(2*b^2*x^2*Log[1 - I/(c*E^(2*(a + b*x)))] - 2*b*x*PolyLog[2, I/(c*E^(2*(a + b*x)))] - PolyLog[3, I/(c*E^(2*(a + b*x)))]))/b^2","A",1
194,1,71,79,0.8737504,"\int \cot ^{-1}(c+(i+c) \tanh (a+b x)) \, dx","Integrate[ArcCot[c + (I + c)*Tanh[a + b*x]],x]","x \cot ^{-1}(c+(c+i) \tanh (a+b x))-\frac{i \left(2 b x \log \left(1-\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_2\left(\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{4 b}","-\frac{i \text{Li}_2\left(-i c e^{2 a+2 b x}\right)}{4 b}-\frac{1}{2} i x \log \left(1+i c e^{2 a+2 b x}\right)+x \cot ^{-1}(c+(c+i) \tanh (a+b x))+\frac{1}{2} i b x^2",1,"x*ArcCot[c + (I + c)*Tanh[a + b*x]] - ((I/4)*(2*b*x*Log[1 - I/(c*E^(2*(a + b*x)))] - PolyLog[2, I/(c*E^(2*(a + b*x)))]))/b","A",1
195,0,0,22,3.4811024,"\int \frac{\cot ^{-1}(c+(i+c) \tanh (a+b x))}{x} \, dx","Integrate[ArcCot[c + (I + c)*Tanh[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+(i+c) \tanh (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c+(c+i) \tanh (a+b x))}{x},x\right)",0,"Integrate[ArcCot[c + (I + c)*Tanh[a + b*x]]/x, x]","A",-1
196,1,128,145,0.2310388,"\int x^2 \cot ^{-1}(c-(i-c) \tanh (a+b x)) \, dx","Integrate[x^2*ArcCot[c - (I - c)*Tanh[a + b*x]],x]","\frac{i \left(4 b^3 x^3 \log \left(1+\frac{i e^{-2 (a+b x)}}{c}\right)-6 b^2 x^2 \text{Li}_2\left(-\frac{i e^{-2 (a+b x)}}{c}\right)-6 b x \text{Li}_3\left(-\frac{i e^{-2 (a+b x)}}{c}\right)-3 \text{Li}_4\left(-\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{24 b^3}+\frac{1}{3} x^3 \cot ^{-1}(c+(c-i) \tanh (a+b x))","\frac{i \text{Li}_4\left(i c e^{2 a+2 b x}\right)}{8 b^3}-\frac{i x \text{Li}_3\left(i c e^{2 a+2 b x}\right)}{4 b^2}+\frac{i x^2 \text{Li}_2\left(i c e^{2 a+2 b x}\right)}{4 b}+\frac{1}{6} i x^3 \log \left(1-i c e^{2 a+2 b x}\right)+\frac{1}{3} x^3 \cot ^{-1}(c-(-c+i) \tanh (a+b x))-\frac{1}{12} i b x^4",1,"(x^3*ArcCot[c + (-I + c)*Tanh[a + b*x]])/3 + ((I/24)*(4*b^3*x^3*Log[1 + I/(c*E^(2*(a + b*x)))] - 6*b^2*x^2*PolyLog[2, (-I)/(c*E^(2*(a + b*x)))] - 6*b*x*PolyLog[3, (-I)/(c*E^(2*(a + b*x)))] - 3*PolyLog[4, (-I)/(c*E^(2*(a + b*x)))]))/b^3","A",1
197,1,102,116,0.1183722,"\int x \cot ^{-1}(c-(i-c) \tanh (a+b x)) \, dx","Integrate[x*ArcCot[c - (I - c)*Tanh[a + b*x]],x]","\frac{i \left(2 b^2 x^2 \log \left(1+\frac{i e^{-2 (a+b x)}}{c}\right)-2 b x \text{Li}_2\left(-\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_3\left(-\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{8 b^2}+\frac{1}{2} x^2 \cot ^{-1}(c+(c-i) \tanh (a+b x))","-\frac{i \text{Li}_3\left(i c e^{2 a+2 b x}\right)}{8 b^2}+\frac{i x \text{Li}_2\left(i c e^{2 a+2 b x}\right)}{4 b}+\frac{1}{4} i x^2 \log \left(1-i c e^{2 a+2 b x}\right)+\frac{1}{2} x^2 \cot ^{-1}(c-(-c+i) \tanh (a+b x))-\frac{1}{6} i b x^3",1,"(x^2*ArcCot[c + (-I + c)*Tanh[a + b*x]])/2 + ((I/8)*(2*b^2*x^2*Log[1 + I/(c*E^(2*(a + b*x)))] - 2*b*x*PolyLog[2, (-I)/(c*E^(2*(a + b*x)))] - PolyLog[3, (-I)/(c*E^(2*(a + b*x)))]))/b^2","A",1
198,1,71,82,0.7757806,"\int \cot ^{-1}(c-(i-c) \tanh (a+b x)) \, dx","Integrate[ArcCot[c - (I - c)*Tanh[a + b*x]],x]","\frac{i \left(2 b x \log \left(1+\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_2\left(-\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{4 b}+x \cot ^{-1}(c+(c-i) \tanh (a+b x))","\frac{i \text{Li}_2\left(i c e^{2 a+2 b x}\right)}{4 b}+\frac{1}{2} i x \log \left(1-i c e^{2 a+2 b x}\right)+x \cot ^{-1}(c-(-c+i) \tanh (a+b x))-\frac{1}{2} i b x^2",1,"x*ArcCot[c + (-I + c)*Tanh[a + b*x]] + ((I/4)*(2*b*x*Log[1 + I/(c*E^(2*(a + b*x)))] - PolyLog[2, (-I)/(c*E^(2*(a + b*x)))]))/b","A",1
199,0,0,25,3.4697149,"\int \frac{\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx","Integrate[ArcCot[c - (I - c)*Tanh[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c-(-c+i) \tanh (a+b x))}{x},x\right)",0,"Integrate[ArcCot[c - (I - c)*Tanh[a + b*x]]/x, x]","A",-1
200,1,600,299,0.3710502,"\int (e+f x)^3 \cot ^{-1}(\coth (a+b x)) \, dx","Integrate[(e + f*x)^3*ArcCot[Coth[a + b*x]],x]","\frac{1}{4} x \left(4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right) \cot ^{-1}(\coth (a+b x))-\frac{i \left(8 b^4 e^3 x \log \left(1-i e^{2 (a+b x)}\right)-8 b^4 e^3 x \log \left(1+i e^{2 (a+b x)}\right)+12 b^4 e^2 f x^2 \log \left(1-i e^{2 (a+b x)}\right)-12 b^4 e^2 f x^2 \log \left(1+i e^{2 (a+b x)}\right)+8 b^4 e f^2 x^3 \log \left(1-i e^{2 (a+b x)}\right)-8 b^4 e f^2 x^3 \log \left(1+i e^{2 (a+b x)}\right)+2 b^4 f^3 x^4 \log \left(1-i e^{2 (a+b x)}\right)-2 b^4 f^3 x^4 \log \left(1+i e^{2 (a+b x)}\right)-4 b^3 (e+f x)^3 \text{Li}_2\left(-i e^{2 (a+b x)}\right)+4 b^3 (e+f x)^3 \text{Li}_2\left(i e^{2 (a+b x)}\right)+6 b^2 e^2 f \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b^2 e^2 f \text{Li}_3\left(i e^{2 (a+b x)}\right)+12 b^2 e f^2 x \text{Li}_3\left(-i e^{2 (a+b x)}\right)-12 b^2 e f^2 x \text{Li}_3\left(i e^{2 (a+b x)}\right)+6 b^2 f^3 x^2 \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b^2 f^3 x^2 \text{Li}_3\left(i e^{2 (a+b x)}\right)-6 b e f^2 \text{Li}_4\left(-i e^{2 (a+b x)}\right)+6 b e f^2 \text{Li}_4\left(i e^{2 (a+b x)}\right)-6 b f^3 x \text{Li}_4\left(-i e^{2 (a+b x)}\right)+6 b f^3 x \text{Li}_4\left(i e^{2 (a+b x)}\right)+3 f^3 \text{Li}_5\left(-i e^{2 (a+b x)}\right)-3 f^3 \text{Li}_5\left(i e^{2 (a+b x)}\right)\right)}{16 b^4}","-\frac{3 i f^3 \text{Li}_5\left(-i e^{2 a+2 b x}\right)}{16 b^4}+\frac{3 i f^3 \text{Li}_5\left(i e^{2 a+2 b x}\right)}{16 b^4}+\frac{3 i f^2 (e+f x) \text{Li}_4\left(-i e^{2 a+2 b x}\right)}{8 b^3}-\frac{3 i f^2 (e+f x) \text{Li}_4\left(i e^{2 a+2 b x}\right)}{8 b^3}-\frac{3 i f (e+f x)^2 \text{Li}_3\left(-i e^{2 a+2 b x}\right)}{8 b^2}+\frac{3 i f (e+f x)^2 \text{Li}_3\left(i e^{2 a+2 b x}\right)}{8 b^2}+\frac{i (e+f x)^3 \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}-\frac{i (e+f x)^3 \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}-\frac{(e+f x)^4 \tan ^{-1}\left(e^{2 a+2 b x}\right)}{4 f}+\frac{(e+f x)^4 \cot ^{-1}(\coth (a+b x))}{4 f}",1,"(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)*ArcCot[Coth[a + b*x]])/4 - ((I/16)*(8*b^4*e^3*x*Log[1 - I*E^(2*(a + b*x))] + 12*b^4*e^2*f*x^2*Log[1 - I*E^(2*(a + b*x))] + 8*b^4*e*f^2*x^3*Log[1 - I*E^(2*(a + b*x))] + 2*b^4*f^3*x^4*Log[1 - I*E^(2*(a + b*x))] - 8*b^4*e^3*x*Log[1 + I*E^(2*(a + b*x))] - 12*b^4*e^2*f*x^2*Log[1 + I*E^(2*(a + b*x))] - 8*b^4*e*f^2*x^3*Log[1 + I*E^(2*(a + b*x))] - 2*b^4*f^3*x^4*Log[1 + I*E^(2*(a + b*x))] - 4*b^3*(e + f*x)^3*PolyLog[2, (-I)*E^(2*(a + b*x))] + 4*b^3*(e + f*x)^3*PolyLog[2, I*E^(2*(a + b*x))] + 6*b^2*e^2*f*PolyLog[3, (-I)*E^(2*(a + b*x))] + 12*b^2*e*f^2*x*PolyLog[3, (-I)*E^(2*(a + b*x))] + 6*b^2*f^3*x^2*PolyLog[3, (-I)*E^(2*(a + b*x))] - 6*b^2*e^2*f*PolyLog[3, I*E^(2*(a + b*x))] - 12*b^2*e*f^2*x*PolyLog[3, I*E^(2*(a + b*x))] - 6*b^2*f^3*x^2*PolyLog[3, I*E^(2*(a + b*x))] - 6*b*e*f^2*PolyLog[4, (-I)*E^(2*(a + b*x))] - 6*b*f^3*x*PolyLog[4, (-I)*E^(2*(a + b*x))] + 6*b*e*f^2*PolyLog[4, I*E^(2*(a + b*x))] + 6*b*f^3*x*PolyLog[4, I*E^(2*(a + b*x))] + 3*f^3*PolyLog[5, (-I)*E^(2*(a + b*x))] - 3*f^3*PolyLog[5, I*E^(2*(a + b*x))]))/b^4","B",1
201,1,375,229,0.2184195,"\int (e+f x)^2 \cot ^{-1}(\coth (a+b x)) \, dx","Integrate[(e + f*x)^2*ArcCot[Coth[a + b*x]],x]","\frac{1}{3} x \left(3 e^2+3 e f x+f^2 x^2\right) \cot ^{-1}(\coth (a+b x))-\frac{i \left(12 b^3 e^2 x \log \left(1-i e^{2 (a+b x)}\right)-12 b^3 e^2 x \log \left(1+i e^{2 (a+b x)}\right)+12 b^3 e f x^2 \log \left(1-i e^{2 (a+b x)}\right)-12 b^3 e f x^2 \log \left(1+i e^{2 (a+b x)}\right)+4 b^3 f^2 x^3 \log \left(1-i e^{2 (a+b x)}\right)-4 b^3 f^2 x^3 \log \left(1+i e^{2 (a+b x)}\right)-6 b^2 (e+f x)^2 \text{Li}_2\left(-i e^{2 (a+b x)}\right)+6 b^2 (e+f x)^2 \text{Li}_2\left(i e^{2 (a+b x)}\right)+6 b e f \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b e f \text{Li}_3\left(i e^{2 (a+b x)}\right)+6 b f^2 x \text{Li}_3\left(-i e^{2 (a+b x)}\right)-6 b f^2 x \text{Li}_3\left(i e^{2 (a+b x)}\right)-3 f^2 \text{Li}_4\left(-i e^{2 (a+b x)}\right)+3 f^2 \text{Li}_4\left(i e^{2 (a+b x)}\right)\right)}{24 b^3}","\frac{i f^2 \text{Li}_4\left(-i e^{2 a+2 b x}\right)}{8 b^3}-\frac{i f^2 \text{Li}_4\left(i e^{2 a+2 b x}\right)}{8 b^3}-\frac{i f (e+f x) \text{Li}_3\left(-i e^{2 a+2 b x}\right)}{4 b^2}+\frac{i f (e+f x) \text{Li}_3\left(i e^{2 a+2 b x}\right)}{4 b^2}+\frac{i (e+f x)^2 \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}-\frac{i (e+f x)^2 \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}-\frac{(e+f x)^3 \tan ^{-1}\left(e^{2 a+2 b x}\right)}{3 f}+\frac{(e+f x)^3 \cot ^{-1}(\coth (a+b x))}{3 f}",1,"(x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCot[Coth[a + b*x]])/3 - ((I/24)*(12*b^3*e^2*x*Log[1 - I*E^(2*(a + b*x))] + 12*b^3*e*f*x^2*Log[1 - I*E^(2*(a + b*x))] + 4*b^3*f^2*x^3*Log[1 - I*E^(2*(a + b*x))] - 12*b^3*e^2*x*Log[1 + I*E^(2*(a + b*x))] - 12*b^3*e*f*x^2*Log[1 + I*E^(2*(a + b*x))] - 4*b^3*f^2*x^3*Log[1 + I*E^(2*(a + b*x))] - 6*b^2*(e + f*x)^2*PolyLog[2, (-I)*E^(2*(a + b*x))] + 6*b^2*(e + f*x)^2*PolyLog[2, I*E^(2*(a + b*x))] + 6*b*e*f*PolyLog[3, (-I)*E^(2*(a + b*x))] + 6*b*f^2*x*PolyLog[3, (-I)*E^(2*(a + b*x))] - 6*b*e*f*PolyLog[3, I*E^(2*(a + b*x))] - 6*b*f^2*x*PolyLog[3, I*E^(2*(a + b*x))] - 3*f^2*PolyLog[4, (-I)*E^(2*(a + b*x))] + 3*f^2*PolyLog[4, I*E^(2*(a + b*x))]))/b^3","A",1
202,1,278,159,0.2776619,"\int (e+f x) \cot ^{-1}(\coth (a+b x)) \, dx","Integrate[(e + f*x)*ArcCot[Coth[a + b*x]],x]","-\frac{i f \left(2 b^2 x^2 \log \left(1-i e^{2 (a+b x)}\right)-2 b^2 x^2 \log \left(1+i e^{2 (a+b x)}\right)-2 b x \text{Li}_2\left(-i e^{2 (a+b x)}\right)+2 b x \text{Li}_2\left(i e^{2 (a+b x)}\right)+\text{Li}_3\left(-i e^{2 (a+b x)}\right)-\text{Li}_3\left(i e^{2 (a+b x)}\right)\right)}{8 b^2}-\frac{e \left(-2 i \left(\text{Li}_2\left(-i e^{2 (a+b x)}\right)-\text{Li}_2\left(i e^{2 (a+b x)}\right)\right)-\left((-4 i a-4 i b x+\pi ) \left(\log \left(1-i e^{2 (a+b x)}\right)-\log \left(1+i e^{2 (a+b x)}\right)\right)\right)+(\pi -4 i a) \log \left(\cot \left(\frac{1}{4} (4 i a+4 i b x+\pi )\right)\right)\right)}{8 b}+e x \cot ^{-1}(\coth (a+b x))+\frac{1}{2} f x^2 \cot ^{-1}(\coth (a+b x))","-\frac{i f \text{Li}_3\left(-i e^{2 a+2 b x}\right)}{8 b^2}+\frac{i f \text{Li}_3\left(i e^{2 a+2 b x}\right)}{8 b^2}+\frac{i (e+f x) \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}-\frac{i (e+f x) \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}-\frac{(e+f x)^2 \tan ^{-1}\left(e^{2 a+2 b x}\right)}{2 f}+\frac{(e+f x)^2 \cot ^{-1}(\coth (a+b x))}{2 f}",1,"e*x*ArcCot[Coth[a + b*x]] + (f*x^2*ArcCot[Coth[a + b*x]])/2 - (e*(-(((-4*I)*a + Pi - (4*I)*b*x)*(Log[1 - I*E^(2*(a + b*x))] - Log[1 + I*E^(2*(a + b*x))])) + ((-4*I)*a + Pi)*Log[Cot[((4*I)*a + Pi + (4*I)*b*x)/4]] - (2*I)*(PolyLog[2, (-I)*E^(2*(a + b*x))] - PolyLog[2, I*E^(2*(a + b*x))])))/(8*b) - ((I/8)*f*(2*b^2*x^2*Log[1 - I*E^(2*(a + b*x))] - 2*b^2*x^2*Log[1 + I*E^(2*(a + b*x))] - 2*b*x*PolyLog[2, (-I)*E^(2*(a + b*x))] + 2*b*x*PolyLog[2, I*E^(2*(a + b*x))] + PolyLog[3, (-I)*E^(2*(a + b*x))] - PolyLog[3, I*E^(2*(a + b*x))]))/b^2","A",1
203,1,132,74,0.0498752,"\int \cot ^{-1}(\coth (a+b x)) \, dx","Integrate[ArcCot[Coth[a + b*x]],x]","x \cot ^{-1}(\coth (a+b x))-\frac{-2 i \left(\text{Li}_2\left(-i e^{2 (a+b x)}\right)-\text{Li}_2\left(i e^{2 (a+b x)}\right)\right)-\left((-4 i a-4 i b x+\pi ) \left(\log \left(1-i e^{2 (a+b x)}\right)-\log \left(1+i e^{2 (a+b x)}\right)\right)\right)+(\pi -4 i a) \log \left(\cot \left(\frac{1}{4} (4 i a+4 i b x+\pi )\right)\right)}{8 b}","\frac{i \text{Li}_2\left(-i e^{2 a+2 b x}\right)}{4 b}-\frac{i \text{Li}_2\left(i e^{2 a+2 b x}\right)}{4 b}-x \tan ^{-1}\left(e^{2 a+2 b x}\right)+x \cot ^{-1}(\coth (a+b x))",1,"x*ArcCot[Coth[a + b*x]] - (-(((-4*I)*a + Pi - (4*I)*b*x)*(Log[1 - I*E^(2*(a + b*x))] - Log[1 + I*E^(2*(a + b*x))])) + ((-4*I)*a + Pi)*Log[Cot[((4*I)*a + Pi + (4*I)*b*x)/4]] - (2*I)*(PolyLog[2, (-I)*E^(2*(a + b*x))] - PolyLog[2, I*E^(2*(a + b*x))]))/(8*b)","A",1
204,0,0,18,0.8448488,"\int \frac{\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx","Integrate[ArcCot[Coth[a + b*x]]/(e + f*x),x]","\int \frac{\cot ^{-1}(\coth (a+b x))}{e+f x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(\coth (a+b x))}{e+f x},x\right)",0,"Integrate[ArcCot[Coth[a + b*x]]/(e + f*x), x]","A",-1
205,1,299,351,5.7699715,"\int x^2 \cot ^{-1}(c+d \coth (a+b x)) \, dx","Integrate[x^2*ArcCot[c + d*Coth[a + b*x]],x]","\frac{1}{3} x^3 \cot ^{-1}(d \coth (a+b x)+c)-\frac{i \left(4 b^3 x^3 \log \left(1+\frac{(c+d-i) e^{2 (a+b x)}}{-c+d+i}\right)-4 b^3 x^3 \log \left(1+\frac{(c+d+i) e^{2 (a+b x)}}{-c+d-i}\right)+6 b^2 x^2 \text{Li}_2\left(\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-6 b^2 x^2 \text{Li}_2\left(\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)-6 b x \text{Li}_3\left(\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)+6 b x \text{Li}_3\left(\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)+3 \text{Li}_4\left(\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-3 \text{Li}_4\left(\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)\right)}{24 b^3}","-\frac{i \text{Li}_4\left(\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^3}+\frac{i \text{Li}_4\left(\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^3}+\frac{i x \text{Li}_3\left(\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b^2}-\frac{i x \text{Li}_3\left(\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b^2}-\frac{i x^2 \text{Li}_2\left(\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i x^2 \text{Li}_2\left(\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b}-\frac{1}{6} i x^3 \log \left(1-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)+\frac{1}{6} i x^3 \log \left(1-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)+\frac{1}{3} x^3 \cot ^{-1}(d \coth (a+b x)+c)",1,"(x^3*ArcCot[c + d*Coth[a + b*x]])/3 - ((I/24)*(4*b^3*x^3*Log[1 + ((-I + c + d)*E^(2*(a + b*x)))/(I - c + d)] - 4*b^3*x^3*Log[1 + ((I + c + d)*E^(2*(a + b*x)))/(-I - c + d)] + 6*b^2*x^2*PolyLog[2, ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] - 6*b^2*x^2*PolyLog[2, ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)] - 6*b*x*PolyLog[3, ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] + 6*b*x*PolyLog[3, ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)] + 3*PolyLog[4, ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] - 3*PolyLog[4, ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)]))/b^3","A",1
206,1,225,265,4.5125496,"\int x \cot ^{-1}(c+d \coth (a+b x)) \, dx","Integrate[x*ArcCot[c + d*Coth[a + b*x]],x]","\frac{1}{2} x^2 \cot ^{-1}(d \coth (a+b x)+c)-\frac{i \left(2 b^2 x^2 \log \left(1+\frac{(c+d-i) e^{2 (a+b x)}}{-c+d+i}\right)-2 b^2 x^2 \log \left(1+\frac{(c+d+i) e^{2 (a+b x)}}{-c+d-i}\right)+2 b x \text{Li}_2\left(\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)-2 b x \text{Li}_2\left(\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)-\text{Li}_3\left(\frac{(c+d-i) e^{2 (a+b x)}}{c-d-i}\right)+\text{Li}_3\left(\frac{(c+d+i) e^{2 (a+b x)}}{c-d+i}\right)\right)}{8 b^2}","\frac{i \text{Li}_3\left(\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{8 b^2}-\frac{i \text{Li}_3\left(\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{8 b^2}-\frac{i x \text{Li}_2\left(\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i x \text{Li}_2\left(\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b}-\frac{1}{4} i x^2 \log \left(1-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)+\frac{1}{4} i x^2 \log \left(1-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)+\frac{1}{2} x^2 \cot ^{-1}(d \coth (a+b x)+c)",1,"(x^2*ArcCot[c + d*Coth[a + b*x]])/2 - ((I/8)*(2*b^2*x^2*Log[1 + ((-I + c + d)*E^(2*(a + b*x)))/(I - c + d)] - 2*b^2*x^2*Log[1 + ((I + c + d)*E^(2*(a + b*x)))/(-I - c + d)] + 2*b*x*PolyLog[2, ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] - 2*b*x*PolyLog[2, ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)] - PolyLog[3, ((-I + c + d)*E^(2*(a + b*x)))/(-I + c - d)] + PolyLog[3, ((I + c + d)*E^(2*(a + b*x)))/(I + c - d)]))/b^2","A",1
207,1,287,174,1.3096948,"\int \cot ^{-1}(c+d \coth (a+b x)) \, dx","Integrate[ArcCot[c + d*Coth[a + b*x]],x]","x \cot ^{-1}(d \coth (a+b x)+c)-\frac{d \text{Li}_2\left(\frac{\left(c^2+2 d c+d^2+1\right) e^{2 (a+b x)}}{c^2-d^2+2 \sqrt{-d^2}+1}\right)-d \text{Li}_2\left(-\frac{\left(c^2+2 d c+d^2+1\right) e^{2 (a+b x)}}{-c^2+d^2+2 \sqrt{-d^2}-1}\right)+2 d (a+b x) \log \left(1-\frac{\left((c+d)^2+1\right) e^{2 (a+b x)}}{c^2-d^2+2 \sqrt{-d^2}+1}\right)-2 d (a+b x) \log \left(\frac{\left((c+d)^2+1\right) e^{2 (a+b x)}}{-c^2+d^2+2 \sqrt{-d^2}-1}+1\right)+4 a \sqrt{-d^2} \tan ^{-1}\left(\frac{-\left(c^2+2 c d+d^2+1\right) e^{2 (a+b x)}+c^2-d^2+1}{2 d}\right)}{4 b \sqrt{-d^2}}","-\frac{i \text{Li}_2\left(\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)}{4 b}+\frac{i \text{Li}_2\left(\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)}{4 b}-\frac{1}{2} i x \log \left(1-\frac{(-c-d+i) e^{2 a+2 b x}}{-c+d+i}\right)+\frac{1}{2} i x \log \left(1-\frac{(c+d+i) e^{2 a+2 b x}}{c-d+i}\right)+x \cot ^{-1}(d \coth (a+b x)+c)",1,"x*ArcCot[c + d*Coth[a + b*x]] - (4*a*Sqrt[-d^2]*ArcTan[(1 + c^2 - d^2 - (1 + c^2 + 2*c*d + d^2)*E^(2*(a + b*x)))/(2*d)] + 2*d*(a + b*x)*Log[1 - ((1 + (c + d)^2)*E^(2*(a + b*x)))/(1 + c^2 - d^2 + 2*Sqrt[-d^2])] - 2*d*(a + b*x)*Log[1 + ((1 + (c + d)^2)*E^(2*(a + b*x)))/(-1 - c^2 + d^2 + 2*Sqrt[-d^2])] + d*PolyLog[2, ((1 + c^2 + 2*c*d + d^2)*E^(2*(a + b*x)))/(1 + c^2 - d^2 + 2*Sqrt[-d^2])] - d*PolyLog[2, -(((1 + c^2 + 2*c*d + d^2)*E^(2*(a + b*x)))/(-1 - c^2 + d^2 + 2*Sqrt[-d^2]))])/(4*b*Sqrt[-d^2])","A",1
208,0,0,18,9.3159079,"\int \frac{\cot ^{-1}(c+d \coth (a+b x))}{x} \, dx","Integrate[ArcCot[c + d*Coth[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+d \coth (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(d \coth (a+b x)+c)}{x},x\right)",0,"Integrate[ArcCot[c + d*Coth[a + b*x]]/x, x]","A",-1
209,1,128,142,0.2185598,"\int x^2 \cot ^{-1}(c+(i+c) \coth (a+b x)) \, dx","Integrate[x^2*ArcCot[c + (I + c)*Coth[a + b*x]],x]","\frac{1}{3} x^3 \cot ^{-1}(c+(c+i) \coth (a+b x))-\frac{i \left(4 b^3 x^3 \log \left(1+\frac{i e^{-2 (a+b x)}}{c}\right)-6 b^2 x^2 \text{Li}_2\left(-\frac{i e^{-2 (a+b x)}}{c}\right)-6 b x \text{Li}_3\left(-\frac{i e^{-2 (a+b x)}}{c}\right)-3 \text{Li}_4\left(-\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{24 b^3}","-\frac{i \text{Li}_4\left(i c e^{2 a+2 b x}\right)}{8 b^3}+\frac{i x \text{Li}_3\left(i c e^{2 a+2 b x}\right)}{4 b^2}-\frac{i x^2 \text{Li}_2\left(i c e^{2 a+2 b x}\right)}{4 b}-\frac{1}{6} i x^3 \log \left(1-i c e^{2 a+2 b x}\right)+\frac{1}{3} x^3 \cot ^{-1}(c+(c+i) \coth (a+b x))+\frac{1}{12} i b x^4",1,"(x^3*ArcCot[c + (I + c)*Coth[a + b*x]])/3 - ((I/24)*(4*b^3*x^3*Log[1 + I/(c*E^(2*(a + b*x)))] - 6*b^2*x^2*PolyLog[2, (-I)/(c*E^(2*(a + b*x)))] - 6*b*x*PolyLog[3, (-I)/(c*E^(2*(a + b*x)))] - 3*PolyLog[4, (-I)/(c*E^(2*(a + b*x)))]))/b^3","A",1
210,1,102,113,0.0957747,"\int x \cot ^{-1}(c+(i+c) \coth (a+b x)) \, dx","Integrate[x*ArcCot[c + (I + c)*Coth[a + b*x]],x]","\frac{1}{2} x^2 \cot ^{-1}(c+(c+i) \coth (a+b x))-\frac{i \left(2 b^2 x^2 \log \left(1+\frac{i e^{-2 (a+b x)}}{c}\right)-2 b x \text{Li}_2\left(-\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_3\left(-\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{8 b^2}","\frac{i \text{Li}_3\left(i c e^{2 a+2 b x}\right)}{8 b^2}-\frac{i x \text{Li}_2\left(i c e^{2 a+2 b x}\right)}{4 b}-\frac{1}{4} i x^2 \log \left(1-i c e^{2 a+2 b x}\right)+\frac{1}{2} x^2 \cot ^{-1}(c+(c+i) \coth (a+b x))+\frac{1}{6} i b x^3",1,"(x^2*ArcCot[c + (I + c)*Coth[a + b*x]])/2 - ((I/8)*(2*b^2*x^2*Log[1 + I/(c*E^(2*(a + b*x)))] - 2*b*x*PolyLog[2, (-I)/(c*E^(2*(a + b*x)))] - PolyLog[3, (-I)/(c*E^(2*(a + b*x)))]))/b^2","A",1
211,1,71,79,0.7136099,"\int \cot ^{-1}(c+(i+c) \coth (a+b x)) \, dx","Integrate[ArcCot[c + (I + c)*Coth[a + b*x]],x]","x \cot ^{-1}(c+(c+i) \coth (a+b x))-\frac{i \left(2 b x \log \left(1+\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_2\left(-\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{4 b}","-\frac{i \text{Li}_2\left(i c e^{2 a+2 b x}\right)}{4 b}-\frac{1}{2} i x \log \left(1-i c e^{2 a+2 b x}\right)+x \cot ^{-1}(c+(c+i) \coth (a+b x))+\frac{1}{2} i b x^2",1,"x*ArcCot[c + (I + c)*Coth[a + b*x]] - ((I/4)*(2*b*x*Log[1 + I/(c*E^(2*(a + b*x)))] - PolyLog[2, (-I)/(c*E^(2*(a + b*x)))]))/b","A",1
212,0,0,22,3.4157689,"\int \frac{\cot ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx","Integrate[ArcCot[c + (I + c)*Coth[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c+(c+i) \coth (a+b x))}{x},x\right)",0,"Integrate[ArcCot[c + (I + c)*Coth[a + b*x]]/x, x]","A",-1
213,1,128,145,0.2351115,"\int x^2 \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx","Integrate[x^2*ArcCot[c - (I - c)*Coth[a + b*x]],x]","\frac{i \left(4 b^3 x^3 \log \left(1-\frac{i e^{-2 (a+b x)}}{c}\right)-6 b^2 x^2 \text{Li}_2\left(\frac{i e^{-2 (a+b x)}}{c}\right)-6 b x \text{Li}_3\left(\frac{i e^{-2 (a+b x)}}{c}\right)-3 \text{Li}_4\left(\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{24 b^3}+\frac{1}{3} x^3 \cot ^{-1}(c+(c-i) \coth (a+b x))","\frac{i \text{Li}_4\left(-i c e^{2 a+2 b x}\right)}{8 b^3}-\frac{i x \text{Li}_3\left(-i c e^{2 a+2 b x}\right)}{4 b^2}+\frac{i x^2 \text{Li}_2\left(-i c e^{2 a+2 b x}\right)}{4 b}+\frac{1}{6} i x^3 \log \left(1+i c e^{2 a+2 b x}\right)+\frac{1}{3} x^3 \cot ^{-1}(c-(-c+i) \coth (a+b x))-\frac{1}{12} i b x^4",1,"(x^3*ArcCot[c + (-I + c)*Coth[a + b*x]])/3 + ((I/24)*(4*b^3*x^3*Log[1 - I/(c*E^(2*(a + b*x)))] - 6*b^2*x^2*PolyLog[2, I/(c*E^(2*(a + b*x)))] - 6*b*x*PolyLog[3, I/(c*E^(2*(a + b*x)))] - 3*PolyLog[4, I/(c*E^(2*(a + b*x)))]))/b^3","A",1
214,1,102,116,0.1216846,"\int x \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx","Integrate[x*ArcCot[c - (I - c)*Coth[a + b*x]],x]","\frac{i \left(2 b^2 x^2 \log \left(1-\frac{i e^{-2 (a+b x)}}{c}\right)-2 b x \text{Li}_2\left(\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_3\left(\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{8 b^2}+\frac{1}{2} x^2 \cot ^{-1}(c+(c-i) \coth (a+b x))","-\frac{i \text{Li}_3\left(-i c e^{2 a+2 b x}\right)}{8 b^2}+\frac{i x \text{Li}_2\left(-i c e^{2 a+2 b x}\right)}{4 b}+\frac{1}{4} i x^2 \log \left(1+i c e^{2 a+2 b x}\right)+\frac{1}{2} x^2 \cot ^{-1}(c-(-c+i) \coth (a+b x))-\frac{1}{6} i b x^3",1,"(x^2*ArcCot[c + (-I + c)*Coth[a + b*x]])/2 + ((I/8)*(2*b^2*x^2*Log[1 - I/(c*E^(2*(a + b*x)))] - 2*b*x*PolyLog[2, I/(c*E^(2*(a + b*x)))] - PolyLog[3, I/(c*E^(2*(a + b*x)))]))/b^2","A",1
215,1,71,82,0.7155778,"\int \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx","Integrate[ArcCot[c - (I - c)*Coth[a + b*x]],x]","\frac{i \left(2 b x \log \left(1-\frac{i e^{-2 (a+b x)}}{c}\right)-\text{Li}_2\left(\frac{i e^{-2 (a+b x)}}{c}\right)\right)}{4 b}+x \cot ^{-1}(c+(c-i) \coth (a+b x))","\frac{i \text{Li}_2\left(-i c e^{2 a+2 b x}\right)}{4 b}+\frac{1}{2} i x \log \left(1+i c e^{2 a+2 b x}\right)+x \cot ^{-1}(c-(-c+i) \coth (a+b x))-\frac{1}{2} i b x^2",1,"x*ArcCot[c + (-I + c)*Coth[a + b*x]] + ((I/4)*(2*b*x*Log[1 - I/(c*E^(2*(a + b*x)))] - PolyLog[2, I/(c*E^(2*(a + b*x)))]))/b","A",1
216,0,0,25,3.5634622,"\int \frac{\cot ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx","Integrate[ArcCot[c - (I - c)*Coth[a + b*x]]/x,x]","\int \frac{\cot ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx","\text{Int}\left(\frac{\cot ^{-1}(c-(-c+i) \coth (a+b x))}{x},x\right)",0,"Integrate[ArcCot[c - (I - c)*Coth[a + b*x]]/x, x]","A",-1
217,1,132,187,0.3638547,"\int \frac{\left(a+b \cot ^{-1}\left(c x^n\right)\right) \left(d+e \log \left(f x^m\right)\right)}{x} \, dx","Integrate[((a + b*ArcCot[c*x^n])*(d + e*Log[f*x^m]))/x,x]","-\frac{b c x^n \left(d+e \log \left(f x^m\right)\right) \, _3F_2\left(\frac{1}{2},\frac{1}{2},1;\frac{3}{2},\frac{3}{2};-c^2 x^{2 n}\right)}{n}+\frac{b c e m x^n \, _4F_3\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},1;\frac{3}{2},\frac{3}{2},\frac{3}{2};-c^2 x^{2 n}\right)}{n^2}-\frac{1}{2} \log (x) \left(e m \log (x)-2 \left(d+e \log \left(f x^m\right)\right)\right) \left(a+b \tan ^{-1}\left(c x^n\right)+b \cot ^{-1}\left(c x^n\right)\right)","a d \log (x)+\frac{a e \log ^2\left(f x^m\right)}{2 m}-\frac{i b d \text{Li}_2\left(-\frac{i x^{-n}}{c}\right)}{2 n}+\frac{i b d \text{Li}_2\left(\frac{i x^{-n}}{c}\right)}{2 n}-\frac{i b e \text{Li}_2\left(-\frac{i x^{-n}}{c}\right) \log \left(f x^m\right)}{2 n}+\frac{i b e \text{Li}_2\left(\frac{i x^{-n}}{c}\right) \log \left(f x^m\right)}{2 n}-\frac{i b e m \text{Li}_3\left(-\frac{i x^{-n}}{c}\right)}{2 n^2}+\frac{i b e m \text{Li}_3\left(\frac{i x^{-n}}{c}\right)}{2 n^2}",1,"(b*c*e*m*x^n*HypergeometricPFQ[{1/2, 1/2, 1/2, 1}, {3/2, 3/2, 3/2}, -(c^2*x^(2*n))])/n^2 - (b*c*x^n*HypergeometricPFQ[{1/2, 1/2, 1}, {3/2, 3/2}, -(c^2*x^(2*n))]*(d + e*Log[f*x^m]))/n - ((a + b*ArcCot[c*x^n] + b*ArcTan[c*x^n])*Log[x]*(e*m*Log[x] - 2*(d + e*Log[f*x^m])))/2","C",1
218,1,59,35,0.0813612,"\int \cot ^{-1}\left(e^x\right) \, dx","Integrate[ArcCot[E^x],x]","x \cot ^{-1}\left(e^x\right)+\frac{1}{2} i \left(-\text{Li}_2\left(-i e^x\right)+\text{Li}_2\left(i e^x\right)+x \left(\log \left(1-i e^x\right)-\log \left(1+i e^x\right)\right)\right)","\frac{1}{2} i \text{Li}_2\left(i e^{-x}\right)-\frac{1}{2} i \text{Li}_2\left(-i e^{-x}\right)",1,"x*ArcCot[E^x] + (I/2)*(x*(Log[1 - I*E^x] - Log[1 + I*E^x]) - PolyLog[2, (-I)*E^x] + PolyLog[2, I*E^x])","A",1
219,1,58,71,0.0119627,"\int x \cot ^{-1}\left(e^x\right) \, dx","Integrate[x*ArcCot[E^x],x]","-\frac{1}{2} i \left(x \text{Li}_2\left(-i e^{-x}\right)-x \text{Li}_2\left(i e^{-x}\right)+\text{Li}_3\left(-i e^{-x}\right)-\text{Li}_3\left(i e^{-x}\right)\right)","-\frac{1}{2} i x \text{Li}_2\left(-i e^{-x}\right)+\frac{1}{2} i x \text{Li}_2\left(i e^{-x}\right)-\frac{1}{2} i \text{Li}_3\left(-i e^{-x}\right)+\frac{1}{2} i \text{Li}_3\left(i e^{-x}\right)",1,"(-1/2*I)*(x*PolyLog[2, (-I)/E^x] - x*PolyLog[2, I/E^x] + PolyLog[3, (-I)/E^x] - PolyLog[3, I/E^x])","A",1
220,1,103,103,0.0086083,"\int x^2 \cot ^{-1}\left(e^x\right) \, dx","Integrate[x^2*ArcCot[E^x],x]","-\frac{1}{2} i x^2 \text{Li}_2\left(-i e^{-x}\right)+\frac{1}{2} i x^2 \text{Li}_2\left(i e^{-x}\right)-i x \text{Li}_3\left(-i e^{-x}\right)+i x \text{Li}_3\left(i e^{-x}\right)-i \text{Li}_4\left(-i e^{-x}\right)+i \text{Li}_4\left(i e^{-x}\right)","-\frac{1}{2} i x^2 \text{Li}_2\left(-i e^{-x}\right)+\frac{1}{2} i x^2 \text{Li}_2\left(i e^{-x}\right)-i x \text{Li}_3\left(-i e^{-x}\right)+i x \text{Li}_3\left(i e^{-x}\right)-i \text{Li}_4\left(-i e^{-x}\right)+i \text{Li}_4\left(i e^{-x}\right)",1,"(-1/2*I)*x^2*PolyLog[2, (-I)/E^x] + (I/2)*x^2*PolyLog[2, I/E^x] - I*x*PolyLog[3, (-I)/E^x] + I*x*PolyLog[3, I/E^x] - I*PolyLog[4, (-I)/E^x] + I*PolyLog[4, I/E^x]","A",1
221,1,83,51,0.1796717,"\int \cot ^{-1}\left(e^{a+b x}\right) \, dx","Integrate[ArcCot[E^(a + b*x)],x]","x \cot ^{-1}\left(e^{a+b x}\right)+\frac{i \left(-\text{Li}_2\left(-i e^{a+b x}\right)+\text{Li}_2\left(i e^{a+b x}\right)+b x \left(\log \left(1-i e^{a+b x}\right)-\log \left(1+i e^{a+b x}\right)\right)\right)}{2 b}","\frac{i \text{Li}_2\left(i e^{-a-b x}\right)}{2 b}-\frac{i \text{Li}_2\left(-i e^{-a-b x}\right)}{2 b}",1,"x*ArcCot[E^(a + b*x)] + ((I/2)*(b*x*(Log[1 - I*E^(a + b*x)] - Log[1 + I*E^(a + b*x)]) - PolyLog[2, (-I)*E^(a + b*x)] + PolyLog[2, I*E^(a + b*x)]))/b","A",1
222,1,83,103,0.0164271,"\int x \cot ^{-1}\left(e^{a+b x}\right) \, dx","Integrate[x*ArcCot[E^(a + b*x)],x]","-\frac{i \left(b x \text{Li}_2\left(-i e^{-a-b x}\right)-b x \text{Li}_2\left(i e^{-a-b x}\right)+\text{Li}_3\left(-i e^{-a-b x}\right)-\text{Li}_3\left(i e^{-a-b x}\right)\right)}{2 b^2}","-\frac{i \text{Li}_3\left(-i e^{-a-b x}\right)}{2 b^2}+\frac{i \text{Li}_3\left(i e^{-a-b x}\right)}{2 b^2}-\frac{i x \text{Li}_2\left(-i e^{-a-b x}\right)}{2 b}+\frac{i x \text{Li}_2\left(i e^{-a-b x}\right)}{2 b}",1,"((-1/2*I)*(b*x*PolyLog[2, (-I)*E^(-a - b*x)] - b*x*PolyLog[2, I*E^(-a - b*x)] + PolyLog[3, (-I)*E^(-a - b*x)] - PolyLog[3, I*E^(-a - b*x)]))/b^2","A",1
223,1,151,151,0.0098873,"\int x^2 \cot ^{-1}\left(e^{a+b x}\right) \, dx","Integrate[x^2*ArcCot[E^(a + b*x)],x]","-\frac{i \text{Li}_4\left(-i e^{-a-b x}\right)}{b^3}+\frac{i \text{Li}_4\left(i e^{-a-b x}\right)}{b^3}-\frac{i x \text{Li}_3\left(-i e^{-a-b x}\right)}{b^2}+\frac{i x \text{Li}_3\left(i e^{-a-b x}\right)}{b^2}-\frac{i x^2 \text{Li}_2\left(-i e^{-a-b x}\right)}{2 b}+\frac{i x^2 \text{Li}_2\left(i e^{-a-b x}\right)}{2 b}","-\frac{i \text{Li}_4\left(-i e^{-a-b x}\right)}{b^3}+\frac{i \text{Li}_4\left(i e^{-a-b x}\right)}{b^3}-\frac{i x \text{Li}_3\left(-i e^{-a-b x}\right)}{b^2}+\frac{i x \text{Li}_3\left(i e^{-a-b x}\right)}{b^2}-\frac{i x^2 \text{Li}_2\left(-i e^{-a-b x}\right)}{2 b}+\frac{i x^2 \text{Li}_2\left(i e^{-a-b x}\right)}{2 b}",1,"((-1/2*I)*x^2*PolyLog[2, (-I)*E^(-a - b*x)])/b + ((I/2)*x^2*PolyLog[2, I*E^(-a - b*x)])/b - (I*x*PolyLog[3, (-I)*E^(-a - b*x)])/b^2 + (I*x*PolyLog[3, I*E^(-a - b*x)])/b^2 - (I*PolyLog[4, (-I)*E^(-a - b*x)])/b^3 + (I*PolyLog[4, I*E^(-a - b*x)])/b^3","A",1
224,1,167,196,0.187155,"\int \cot ^{-1}\left(a+b f^{c+d x}\right) \, dx","Integrate[ArcCot[a + b*f^(c + d*x)],x]","\frac{b \left(\text{Li}_2\left(-\frac{b^2 f^{c+d x}}{a b-\sqrt{-b^2}}\right)-\text{Li}_2\left(-\frac{b^2 f^{c+d x}}{a b+\sqrt{-b^2}}\right)+d x \log (f) \left(\log \left(\frac{b^2 f^{c+d x}}{a b-\sqrt{-b^2}}+1\right)-\log \left(\frac{b^2 f^{c+d x}}{a b+\sqrt{-b^2}}+1\right)\right)\right)}{2 \sqrt{-b^2} d \log (f)}+x \cot ^{-1}\left(a+b f^{c+d x}\right)","-\frac{i \text{Li}_2\left(1-\frac{2}{1-i \left(b f^{c+d x}+a\right)}\right)}{2 d \log (f)}+\frac{i \text{Li}_2\left(1-\frac{2 b f^{c+d x}}{(i-a) \left(1-i \left(b f^{c+d x}+a\right)\right)}\right)}{2 d \log (f)}-\frac{\log \left(\frac{2}{1-i \left(a+b f^{c+d x}\right)}\right) \cot ^{-1}\left(a+b f^{c+d x}\right)}{d \log (f)}+\frac{\log \left(\frac{2 b f^{c+d x}}{(-a+i) \left(1-i \left(a+b f^{c+d x}\right)\right)}\right) \cot ^{-1}\left(a+b f^{c+d x}\right)}{d \log (f)}",1,"x*ArcCot[a + b*f^(c + d*x)] + (b*(d*x*Log[f]*(Log[1 + (b^2*f^(c + d*x))/(a*b - Sqrt[-b^2])] - Log[1 + (b^2*f^(c + d*x))/(a*b + Sqrt[-b^2])]) + PolyLog[2, -((b^2*f^(c + d*x))/(a*b - Sqrt[-b^2]))] - PolyLog[2, -((b^2*f^(c + d*x))/(a*b + Sqrt[-b^2]))]))/(2*Sqrt[-b^2]*d*Log[f])","A",1
225,1,250,250,0.3072751,"\int x \cot ^{-1}\left(a+b f^{c+d x}\right) \, dx","Integrate[x*ArcCot[a + b*f^(c + d*x)],x]","\frac{i \text{Li}_3\left(\frac{b f^{c+d x}}{i-a}\right)}{2 d^2 \log ^2(f)}-\frac{i \text{Li}_3\left(-\frac{b f^{c+d x}}{a+i}\right)}{2 d^2 \log ^2(f)}-\frac{i x \text{Li}_2\left(\frac{b f^{c+d x}}{i-a}\right)}{2 d \log (f)}+\frac{i x \text{Li}_2\left(-\frac{b f^{c+d x}}{a+i}\right)}{2 d \log (f)}-\frac{1}{4} i x^2 \log \left(1-\frac{b f^{c+d x}}{-a+i}\right)+\frac{1}{4} i x^2 \log \left(1+\frac{b f^{c+d x}}{a+i}\right)+\frac{1}{4} i x^2 \log \left(1-\frac{i}{a+b f^{c+d x}}\right)-\frac{1}{4} i x^2 \log \left(1+\frac{i}{a+b f^{c+d x}}\right)","\frac{i \text{Li}_3\left(\frac{b f^{c+d x}}{i-a}\right)}{2 d^2 \log ^2(f)}-\frac{i \text{Li}_3\left(-\frac{b f^{c+d x}}{a+i}\right)}{2 d^2 \log ^2(f)}-\frac{i x \text{Li}_2\left(\frac{b f^{c+d x}}{i-a}\right)}{2 d \log (f)}+\frac{i x \text{Li}_2\left(-\frac{b f^{c+d x}}{a+i}\right)}{2 d \log (f)}-\frac{1}{4} i x^2 \log \left(1-\frac{b f^{c+d x}}{-a+i}\right)+\frac{1}{4} i x^2 \log \left(1+\frac{b f^{c+d x}}{a+i}\right)+\frac{1}{4} i x^2 \log \left(1-\frac{i}{a+b f^{c+d x}}\right)-\frac{1}{4} i x^2 \log \left(1+\frac{i}{a+b f^{c+d x}}\right)",1,"(-1/4*I)*x^2*Log[1 - (b*f^(c + d*x))/(I - a)] + (I/4)*x^2*Log[1 + (b*f^(c + d*x))/(I + a)] + (I/4)*x^2*Log[1 - I/(a + b*f^(c + d*x))] - (I/4)*x^2*Log[1 + I/(a + b*f^(c + d*x))] - ((I/2)*x*PolyLog[2, (b*f^(c + d*x))/(I - a)])/(d*Log[f]) + ((I/2)*x*PolyLog[2, -((b*f^(c + d*x))/(I + a))])/(d*Log[f]) + ((I/2)*PolyLog[3, (b*f^(c + d*x))/(I - a)])/(d^2*Log[f]^2) - ((I/2)*PolyLog[3, -((b*f^(c + d*x))/(I + a))])/(d^2*Log[f]^2)","A",1
226,1,313,313,0.2384255,"\int x^2 \cot ^{-1}\left(a+b f^{c+d x}\right) \, dx","Integrate[x^2*ArcCot[a + b*f^(c + d*x)],x]","-\frac{i \text{Li}_4\left(\frac{b f^{c+d x}}{i-a}\right)}{d^3 \log ^3(f)}+\frac{i \text{Li}_4\left(-\frac{b f^{c+d x}}{a+i}\right)}{d^3 \log ^3(f)}+\frac{i x \text{Li}_3\left(\frac{b f^{c+d x}}{i-a}\right)}{d^2 \log ^2(f)}-\frac{i x \text{Li}_3\left(-\frac{b f^{c+d x}}{a+i}\right)}{d^2 \log ^2(f)}-\frac{i x^2 \text{Li}_2\left(\frac{b f^{c+d x}}{i-a}\right)}{2 d \log (f)}+\frac{i x^2 \text{Li}_2\left(-\frac{b f^{c+d x}}{a+i}\right)}{2 d \log (f)}-\frac{1}{6} i x^3 \log \left(1-\frac{b f^{c+d x}}{-a+i}\right)+\frac{1}{6} i x^3 \log \left(1+\frac{b f^{c+d x}}{a+i}\right)+\frac{1}{6} i x^3 \log \left(1-\frac{i}{a+b f^{c+d x}}\right)-\frac{1}{6} i x^3 \log \left(1+\frac{i}{a+b f^{c+d x}}\right)","-\frac{i \text{Li}_4\left(\frac{b f^{c+d x}}{i-a}\right)}{d^3 \log ^3(f)}+\frac{i \text{Li}_4\left(-\frac{b f^{c+d x}}{a+i}\right)}{d^3 \log ^3(f)}+\frac{i x \text{Li}_3\left(\frac{b f^{c+d x}}{i-a}\right)}{d^2 \log ^2(f)}-\frac{i x \text{Li}_3\left(-\frac{b f^{c+d x}}{a+i}\right)}{d^2 \log ^2(f)}-\frac{i x^2 \text{Li}_2\left(\frac{b f^{c+d x}}{i-a}\right)}{2 d \log (f)}+\frac{i x^2 \text{Li}_2\left(-\frac{b f^{c+d x}}{a+i}\right)}{2 d \log (f)}-\frac{1}{6} i x^3 \log \left(1-\frac{b f^{c+d x}}{-a+i}\right)+\frac{1}{6} i x^3 \log \left(1+\frac{b f^{c+d x}}{a+i}\right)+\frac{1}{6} i x^3 \log \left(1-\frac{i}{a+b f^{c+d x}}\right)-\frac{1}{6} i x^3 \log \left(1+\frac{i}{a+b f^{c+d x}}\right)",1,"(-1/6*I)*x^3*Log[1 - (b*f^(c + d*x))/(I - a)] + (I/6)*x^3*Log[1 + (b*f^(c + d*x))/(I + a)] + (I/6)*x^3*Log[1 - I/(a + b*f^(c + d*x))] - (I/6)*x^3*Log[1 + I/(a + b*f^(c + d*x))] - ((I/2)*x^2*PolyLog[2, (b*f^(c + d*x))/(I - a)])/(d*Log[f]) + ((I/2)*x^2*PolyLog[2, -((b*f^(c + d*x))/(I + a))])/(d*Log[f]) + (I*x*PolyLog[3, (b*f^(c + d*x))/(I - a)])/(d^2*Log[f]^2) - (I*x*PolyLog[3, -((b*f^(c + d*x))/(I + a))])/(d^2*Log[f]^2) - (I*PolyLog[4, (b*f^(c + d*x))/(I - a)])/(d^3*Log[f]^3) + (I*PolyLog[4, -((b*f^(c + d*x))/(I + a))])/(d^3*Log[f]^3)","A",1
227,1,27,27,0.0225456,"\int e^{-x} \cot ^{-1}\left(e^x\right) \, dx","Integrate[ArcCot[E^x]/E^x,x]","-x+\frac{1}{2} \log \left(e^{2 x}+1\right)-e^{-x} \cot ^{-1}\left(e^x\right)","-x+\frac{1}{2} \log \left(e^{2 x}+1\right)-e^{-x} \cot ^{-1}\left(e^x\right)",1,"-x - ArcCot[E^x]/E^x + Log[1 + E^(2*x)]/2","A",1
228,1,17,17,0.0486987,"\int \frac{1}{\left(a+a x^2\right) \left(b-2 b \cot ^{-1}(x)\right)} \, dx","Integrate[1/((a + a*x^2)*(b - 2*b*ArcCot[x])),x]","\frac{\log \left(2 \cot ^{-1}(x)-1\right)}{2 a b}","\frac{\log \left(1-2 \cot ^{-1}(x)\right)}{2 a b}",1,"Log[-1 + 2*ArcCot[x]]/(2*a*b)","A",1
229,1,61,47,0.1115285,"\int e^{c (a+b x)} \cot ^{-1}(\sinh (a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcCot[Sinh[a*c + b*c*x]],x]","\frac{\log \left(e^{2 c (a+b x)}+1\right)-e^{c (a+b x)} \cot ^{-1}\left(\frac{1}{2} e^{-c (a+b x)}-\frac{1}{2} e^{c (a+b x)}\right)}{b c}","\frac{\log \left(e^{2 c (a+b x)}+1\right)}{b c}+\frac{e^{a c+b c x} \cot ^{-1}(\sinh (c (a+b x)))}{b c}",1,"(-(E^(c*(a + b*x))*ArcCot[1/(2*E^(c*(a + b*x))) - E^(c*(a + b*x))/2]) + Log[1 + E^(2*c*(a + b*x))])/(b*c)","A",0
230,1,146,103,0.1515913,"\int e^{c (a+b x)} \cot ^{-1}(\cosh (a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcCot[Cosh[a*c + b*c*x]],x]","\frac{\text{RootSum}\left[\text{$\#$1}^4+6 \text{$\#$1}^2+1\&,\frac{7 \text{$\#$1}^2 \log \left(e^{c (a+b x)}-\text{$\#$1}\right)-7 \text{$\#$1}^2 a c-7 \text{$\#$1}^2 b c x+\log \left(e^{c (a+b x)}-\text{$\#$1}\right)-a c-b c x}{3 \text{$\#$1}^2+1}\&\right]+4 c (a+b x)+2 e^{c (a+b x)} \cot ^{-1}\left(\frac{1}{2} e^{-c (a+b x)} \left(e^{2 c (a+b x)}+1\right)\right)}{2 b c}","\frac{\left(1-\sqrt{2}\right) \log \left(e^{2 c (a+b x)}+3-2 \sqrt{2}\right)}{2 b c}+\frac{\left(1+\sqrt{2}\right) \log \left(e^{2 c (a+b x)}+3+2 \sqrt{2}\right)}{2 b c}+\frac{e^{a c+b c x} \cot ^{-1}(\cosh (c (a+b x)))}{b c}",1,"(4*c*(a + b*x) + 2*E^(c*(a + b*x))*ArcCot[(1 + E^(2*c*(a + b*x)))/(2*E^(c*(a + b*x)))] + RootSum[1 + 6*#1^2 + #1^4 & , (-(a*c) - b*c*x + Log[E^(c*(a + b*x)) - #1] - 7*a*c*#1^2 - 7*b*c*x*#1^2 + 7*Log[E^(c*(a + b*x)) - #1]*#1^2)/(1 + 3*#1^2) & ])/(2*b*c)","C",0
231,1,89,180,0.1200607,"\int e^{c (a+b x)} \cot ^{-1}(\tanh (a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcCot[Tanh[a*c + b*c*x]],x]","\frac{\text{RootSum}\left[\text{$\#$1}^4+1\&,\frac{\log \left(e^{c (a+b x)}-\text{$\#$1}\right)-a c-b c x}{\text{$\#$1}}\&\right]+2 e^{c (a+b x)} \cot ^{-1}\left(\frac{e^{2 c (a+b x)}-1}{e^{2 c (a+b x)}+1}\right)}{2 b c}","\frac{\log \left(e^{2 c (a+b x)}-\sqrt{2} e^{a c+b c x}+1\right)}{2 \sqrt{2} b c}-\frac{\log \left(e^{2 c (a+b x)}+\sqrt{2} e^{a c+b c x}+1\right)}{2 \sqrt{2} b c}-\frac{\tan ^{-1}\left(1-\sqrt{2} e^{a c+b c x}\right)}{\sqrt{2} b c}+\frac{\tan ^{-1}\left(\sqrt{2} e^{a c+b c x}+1\right)}{\sqrt{2} b c}+\frac{e^{a c+b c x} \cot ^{-1}(\tanh (c (a+b x)))}{b c}",1,"(2*E^(c*(a + b*x))*ArcCot[(-1 + E^(2*c*(a + b*x)))/(1 + E^(2*c*(a + b*x)))] + RootSum[1 + #1^4 & , (-(a*c) - b*c*x + Log[E^(c*(a + b*x)) - #1])/#1 & ])/(2*b*c)","C",0
232,1,89,180,0.1199377,"\int e^{c (a+b x)} \cot ^{-1}(\coth (a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcCot[Coth[a*c + b*c*x]],x]","\frac{\text{RootSum}\left[\text{$\#$1}^4+1\&,\frac{-\log \left(e^{c (a+b x)}-\text{$\#$1}\right)+a c+b c x}{\text{$\#$1}}\&\right]+2 e^{c (a+b x)} \cot ^{-1}\left(\frac{e^{2 c (a+b x)}+1}{e^{2 c (a+b x)}-1}\right)}{2 b c}","-\frac{\log \left(e^{2 c (a+b x)}-\sqrt{2} e^{a c+b c x}+1\right)}{2 \sqrt{2} b c}+\frac{\log \left(e^{2 c (a+b x)}+\sqrt{2} e^{a c+b c x}+1\right)}{2 \sqrt{2} b c}+\frac{\tan ^{-1}\left(1-\sqrt{2} e^{a c+b c x}\right)}{\sqrt{2} b c}-\frac{\tan ^{-1}\left(\sqrt{2} e^{a c+b c x}+1\right)}{\sqrt{2} b c}+\frac{e^{a c+b c x} \cot ^{-1}(\coth (c (a+b x)))}{b c}",1,"(2*E^(c*(a + b*x))*ArcCot[(1 + E^(2*c*(a + b*x)))/(-1 + E^(2*c*(a + b*x)))] + RootSum[1 + #1^4 & , (a*c + b*c*x - Log[E^(c*(a + b*x)) - #1])/#1 & ])/(2*b*c)","C",0
233,1,145,103,0.1563477,"\int e^{c (a+b x)} \cot ^{-1}(\text{sech}(a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcCot[Sech[a*c + b*c*x]],x]","\frac{\text{RootSum}\left[\text{$\#$1}^4+6 \text{$\#$1}^2+1\&,\frac{-7 \text{$\#$1}^2 \log \left(e^{c (a+b x)}-\text{$\#$1}\right)+7 \text{$\#$1}^2 a c+7 \text{$\#$1}^2 b c x-\log \left(e^{c (a+b x)}-\text{$\#$1}\right)+a c+b c x}{3 \text{$\#$1}^2+1}\&\right]-4 c (a+b x)+2 e^{c (a+b x)} \cot ^{-1}\left(\frac{2 e^{c (a+b x)}}{e^{2 c (a+b x)}+1}\right)}{2 b c}","-\frac{\left(1-\sqrt{2}\right) \log \left(e^{2 c (a+b x)}+3-2 \sqrt{2}\right)}{2 b c}-\frac{\left(1+\sqrt{2}\right) \log \left(e^{2 c (a+b x)}+3+2 \sqrt{2}\right)}{2 b c}+\frac{e^{a c+b c x} \cot ^{-1}(\text{sech}(c (a+b x)))}{b c}",1,"(-4*c*(a + b*x) + 2*E^(c*(a + b*x))*ArcCot[(2*E^(c*(a + b*x)))/(1 + E^(2*c*(a + b*x)))] + RootSum[1 + 6*#1^2 + #1^4 & , (a*c + b*c*x - Log[E^(c*(a + b*x)) - #1] + 7*a*c*#1^2 + 7*b*c*x*#1^2 - 7*Log[E^(c*(a + b*x)) - #1]*#1^2)/(1 + 3*#1^2) & ])/(2*b*c)","C",0
234,1,59,48,0.1019429,"\int e^{c (a+b x)} \cot ^{-1}(\text{csch}(a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcCot[Csch[a*c + b*c*x]],x]","\frac{e^{c (a+b x)} \cot ^{-1}\left(\frac{2 e^{c (a+b x)}}{e^{2 c (a+b x)}-1}\right)-\log \left(e^{2 c (a+b x)}+1\right)}{b c}","\frac{e^{a c+b c x} \cot ^{-1}(\text{csch}(c (a+b x)))}{b c}-\frac{\log \left(e^{2 c (a+b x)}+1\right)}{b c}",1,"(E^(c*(a + b*x))*ArcCot[(2*E^(c*(a + b*x)))/(-1 + E^(2*c*(a + b*x)))] - Log[1 + E^(2*c*(a + b*x))])/(b*c)","A",0