1,1,67,51,0.0089694,"\int x^5 \coth ^{-1}(a x) \, dx","Integrate[x^5*ArcCoth[a*x],x]","\frac{\log (1-a x)}{12 a^6}-\frac{\log (a x+1)}{12 a^6}+\frac{x}{6 a^5}+\frac{x^3}{18 a^3}+\frac{1}{6} x^6 \coth ^{-1}(a x)+\frac{x^5}{30 a}","-\frac{\tanh ^{-1}(a x)}{6 a^6}+\frac{x}{6 a^5}+\frac{x^3}{18 a^3}+\frac{1}{6} x^6 \coth ^{-1}(a x)+\frac{x^5}{30 a}",1,"x/(6*a^5) + x^3/(18*a^3) + x^5/(30*a) + (x^6*ArcCoth[a*x])/6 + Log[1 - a*x]/(12*a^6) - Log[1 + a*x]/(12*a^6)","A",1
2,1,50,50,0.0089108,"\int x^4 \coth ^{-1}(a x) \, dx","Integrate[x^4*ArcCoth[a*x],x]","\frac{x^2}{10 a^3}+\frac{\log \left(1-a^2 x^2\right)}{10 a^5}+\frac{1}{5} x^5 \coth ^{-1}(a x)+\frac{x^4}{20 a}","\frac{x^2}{10 a^3}+\frac{\log \left(1-a^2 x^2\right)}{10 a^5}+\frac{1}{5} x^5 \coth ^{-1}(a x)+\frac{x^4}{20 a}",1,"x^2/(10*a^3) + x^4/(20*a) + (x^5*ArcCoth[a*x])/5 + Log[1 - a^2*x^2]/(10*a^5)","A",1
3,1,57,41,0.0085174,"\int x^3 \coth ^{-1}(a x) \, dx","Integrate[x^3*ArcCoth[a*x],x]","\frac{\log (1-a x)}{8 a^4}-\frac{\log (a x+1)}{8 a^4}+\frac{x}{4 a^3}+\frac{1}{4} x^4 \coth ^{-1}(a x)+\frac{x^3}{12 a}","-\frac{\tanh ^{-1}(a x)}{4 a^4}+\frac{x}{4 a^3}+\frac{1}{4} x^4 \coth ^{-1}(a x)+\frac{x^3}{12 a}",1,"x/(4*a^3) + x^3/(12*a) + (x^4*ArcCoth[a*x])/4 + Log[1 - a*x]/(8*a^4) - Log[1 + a*x]/(8*a^4)","A",1
4,1,40,40,0.0078892,"\int x^2 \coth ^{-1}(a x) \, dx","Integrate[x^2*ArcCoth[a*x],x]","\frac{\log \left(1-a^2 x^2\right)}{6 a^3}+\frac{1}{3} x^3 \coth ^{-1}(a x)+\frac{x^2}{6 a}","\frac{\log \left(1-a^2 x^2\right)}{6 a^3}+\frac{1}{3} x^3 \coth ^{-1}(a x)+\frac{x^2}{6 a}",1,"x^2/(6*a) + (x^3*ArcCoth[a*x])/3 + Log[1 - a^2*x^2]/(6*a^3)","A",1
5,1,47,31,0.0071474,"\int x \coth ^{-1}(a x) \, dx","Integrate[x*ArcCoth[a*x],x]","\frac{\log (1-a x)}{4 a^2}-\frac{\log (a x+1)}{4 a^2}+\frac{1}{2} x^2 \coth ^{-1}(a x)+\frac{x}{2 a}","-\frac{\tanh ^{-1}(a x)}{2 a^2}+\frac{1}{2} x^2 \coth ^{-1}(a x)+\frac{x}{2 a}",1,"x/(2*a) + (x^2*ArcCoth[a*x])/2 + Log[1 - a*x]/(4*a^2) - Log[1 + a*x]/(4*a^2)","A",1
6,1,25,25,0.0031356,"\int \coth ^{-1}(a x) \, dx","Integrate[ArcCoth[a*x],x]","\frac{\log \left(1-a^2 x^2\right)}{2 a}+x \coth ^{-1}(a x)","\frac{\log \left(1-a^2 x^2\right)}{2 a}+x \coth ^{-1}(a x)",1,"x*ArcCoth[a*x] + Log[1 - a^2*x^2]/(2*a)","A",1
7,1,26,28,0.008461,"\int \frac{\coth ^{-1}(a x)}{x} \, dx","Integrate[ArcCoth[a*x]/x,x]","\frac{1}{2} \left(\text{Li}_2\left(-\frac{1}{a x}\right)-\text{Li}_2\left(\frac{1}{a x}\right)\right)","\frac{1}{2} \text{Li}_2\left(-\frac{1}{a x}\right)-\frac{1}{2} \text{Li}_2\left(\frac{1}{a x}\right)",1,"(PolyLog[2, -(1/(a*x))] - PolyLog[2, 1/(a*x)])/2","A",1
8,1,30,30,0.007705,"\int \frac{\coth ^{-1}(a x)}{x^2} \, dx","Integrate[ArcCoth[a*x]/x^2,x]","-\frac{1}{2} a \log \left(1-a^2 x^2\right)+a \log (x)-\frac{\coth ^{-1}(a x)}{x}","-\frac{1}{2} a \log \left(1-a^2 x^2\right)+a \log (x)-\frac{\coth ^{-1}(a x)}{x}",1,"-(ArcCoth[a*x]/x) + a*Log[x] - (a*Log[1 - a^2*x^2])/2","A",1
9,1,47,31,0.0082397,"\int \frac{\coth ^{-1}(a x)}{x^3} \, dx","Integrate[ArcCoth[a*x]/x^3,x]","-\frac{1}{4} a^2 \log (1-a x)+\frac{1}{4} a^2 \log (a x+1)-\frac{\coth ^{-1}(a x)}{2 x^2}-\frac{a}{2 x}","\frac{1}{2} a^2 \tanh ^{-1}(a x)-\frac{\coth ^{-1}(a x)}{2 x^2}-\frac{a}{2 x}",1,"-1/2*a/x - ArcCoth[a*x]/(2*x^2) - (a^2*Log[1 - a*x])/4 + (a^2*Log[1 + a*x])/4","A",1
10,1,47,47,0.0092326,"\int \frac{\coth ^{-1}(a x)}{x^4} \, dx","Integrate[ArcCoth[a*x]/x^4,x]","\frac{1}{3} a^3 \log (x)-\frac{1}{6} a^3 \log \left(1-a^2 x^2\right)-\frac{\coth ^{-1}(a x)}{3 x^3}-\frac{a}{6 x^2}","\frac{1}{3} a^3 \log (x)-\frac{1}{6} a^3 \log \left(1-a^2 x^2\right)-\frac{\coth ^{-1}(a x)}{3 x^3}-\frac{a}{6 x^2}",1,"-1/6*a/x^2 - ArcCoth[a*x]/(3*x^3) + (a^3*Log[x])/3 - (a^3*Log[1 - a^2*x^2])/6","A",1
11,1,57,41,0.0088603,"\int \frac{\coth ^{-1}(a x)}{x^5} \, dx","Integrate[ArcCoth[a*x]/x^5,x]","-\frac{1}{8} a^4 \log (1-a x)+\frac{1}{8} a^4 \log (a x+1)-\frac{a^3}{4 x}-\frac{\coth ^{-1}(a x)}{4 x^4}-\frac{a}{12 x^3}","\frac{1}{4} a^4 \tanh ^{-1}(a x)-\frac{a^3}{4 x}-\frac{\coth ^{-1}(a x)}{4 x^4}-\frac{a}{12 x^3}",1,"-1/12*a/x^3 - a^3/(4*x) - ArcCoth[a*x]/(4*x^4) - (a^4*Log[1 - a*x])/8 + (a^4*Log[1 + a*x])/8","A",1
12,1,80,105,0.0218522,"\int x^5 \coth ^{-1}(a x)^2 \, dx","Integrate[x^5*ArcCoth[a*x]^2,x]","\frac{30 \left(a^6 x^6-1\right) \coth ^{-1}(a x)^2+3 a^4 x^4+16 a^2 x^2+46 \log \left(1-a^2 x^2\right)+4 a x \left(3 a^4 x^4+5 a^2 x^2+15\right) \coth ^{-1}(a x)}{180 a^6}","-\frac{\coth ^{-1}(a x)^2}{6 a^6}+\frac{x \coth ^{-1}(a x)}{3 a^5}+\frac{4 x^2}{45 a^4}+\frac{x^3 \coth ^{-1}(a x)}{9 a^3}+\frac{x^4}{60 a^2}+\frac{23 \log \left(1-a^2 x^2\right)}{90 a^6}+\frac{1}{6} x^6 \coth ^{-1}(a x)^2+\frac{x^5 \coth ^{-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)*ArcCoth[a*x] + 30*(-1 + a^6*x^6)*ArcCoth[a*x]^2 + 46*Log[1 - a^2*x^2])/(180*a^6)","A",1
13,1,87,127,0.4510225,"\int x^4 \coth ^{-1}(a x)^2 \, dx","Integrate[x^4*ArcCoth[a*x]^2,x]","\frac{6 \left(a^5 x^5-1\right) \coth ^{-1}(a x)^2+a x \left(a^2 x^2+9\right)+3 \coth ^{-1}(a x) \left(a^4 x^4+2 a^2 x^2-4 \log \left(1-e^{-2 \coth ^{-1}(a x)}\right)-3\right)+6 \text{Li}_2\left(e^{-2 \coth ^{-1}(a x)}\right)}{30 a^5}","-\frac{\text{Li}_2\left(1-\frac{2}{1-a x}\right)}{5 a^5}-\frac{3 \tanh ^{-1}(a x)}{10 a^5}+\frac{\coth ^{-1}(a x)^2}{5 a^5}-\frac{2 \log \left(\frac{2}{1-a x}\right) \coth ^{-1}(a x)}{5 a^5}+\frac{3 x}{10 a^4}+\frac{x^2 \coth ^{-1}(a x)}{5 a^3}+\frac{x^3}{30 a^2}+\frac{1}{5} x^5 \coth ^{-1}(a x)^2+\frac{x^4 \coth ^{-1}(a x)}{10 a}",1,"(a*x*(9 + a^2*x^2) + 6*(-1 + a^5*x^5)*ArcCoth[a*x]^2 + 3*ArcCoth[a*x]*(-3 + 2*a^2*x^2 + a^4*x^4 - 4*Log[1 - E^(-2*ArcCoth[a*x])]) + 6*PolyLog[2, E^(-2*ArcCoth[a*x])])/(30*a^5)","A",0
14,1,62,81,0.0190986,"\int x^3 \coth ^{-1}(a x)^2 \, dx","Integrate[x^3*ArcCoth[a*x]^2,x]","\frac{3 \left(a^4 x^4-1\right) \coth ^{-1}(a x)^2+a^2 x^2+4 \log \left(1-a^2 x^2\right)+2 a x \left(a^2 x^2+3\right) \coth ^{-1}(a x)}{12 a^4}","-\frac{\coth ^{-1}(a x)^2}{4 a^4}+\frac{x \coth ^{-1}(a x)}{2 a^3}+\frac{x^2}{12 a^2}+\frac{\log \left(1-a^2 x^2\right)}{3 a^4}+\frac{1}{4} x^4 \coth ^{-1}(a x)^2+\frac{x^3 \coth ^{-1}(a x)}{6 a}",1,"(a^2*x^2 + 2*a*x*(3 + a^2*x^2)*ArcCoth[a*x] + 3*(-1 + a^4*x^4)*ArcCoth[a*x]^2 + 4*Log[1 - a^2*x^2])/(12*a^4)","A",1
15,1,66,103,0.2444965,"\int x^2 \coth ^{-1}(a x)^2 \, dx","Integrate[x^2*ArcCoth[a*x]^2,x]","\frac{\left(a^3 x^3-1\right) \coth ^{-1}(a x)^2+\coth ^{-1}(a x) \left(a^2 x^2-2 \log \left(1-e^{-2 \coth ^{-1}(a x)}\right)-1\right)+\text{Li}_2\left(e^{-2 \coth ^{-1}(a x)}\right)+a x}{3 a^3}","-\frac{\text{Li}_2\left(1-\frac{2}{1-a x}\right)}{3 a^3}-\frac{\tanh ^{-1}(a x)}{3 a^3}+\frac{\coth ^{-1}(a x)^2}{3 a^3}-\frac{2 \log \left(\frac{2}{1-a x}\right) \coth ^{-1}(a x)}{3 a^3}+\frac{x}{3 a^2}+\frac{1}{3} x^3 \coth ^{-1}(a x)^2+\frac{x^2 \coth ^{-1}(a x)}{3 a}",1,"(a*x + (-1 + a^3*x^3)*ArcCoth[a*x]^2 + ArcCoth[a*x]*(-1 + a^2*x^2 - 2*Log[1 - E^(-2*ArcCoth[a*x])]) + PolyLog[2, E^(-2*ArcCoth[a*x])])/(3*a^3)","A",0
16,1,43,54,0.0115454,"\int x \coth ^{-1}(a x)^2 \, dx","Integrate[x*ArcCoth[a*x]^2,x]","\frac{\log \left(1-a^2 x^2\right)+\left(a^2 x^2-1\right) \coth ^{-1}(a x)^2+2 a x \coth ^{-1}(a x)}{2 a^2}","\frac{\log \left(1-a^2 x^2\right)}{2 a^2}-\frac{\coth ^{-1}(a x)^2}{2 a^2}+\frac{1}{2} x^2 \coth ^{-1}(a x)^2+\frac{x \coth ^{-1}(a x)}{a}",1,"(2*a*x*ArcCoth[a*x] + (-1 + a^2*x^2)*ArcCoth[a*x]^2 + Log[1 - a^2*x^2])/(2*a^2)","A",1
17,1,46,58,0.0835209,"\int \coth ^{-1}(a x)^2 \, dx","Integrate[ArcCoth[a*x]^2,x]","\frac{\text{Li}_2\left(e^{-2 \coth ^{-1}(a x)}\right)+\coth ^{-1}(a x) \left((a x-1) \coth ^{-1}(a x)-2 \log \left(1-e^{-2 \coth ^{-1}(a x)}\right)\right)}{a}","-\frac{\text{Li}_2\left(1-\frac{2}{1-a x}\right)}{a}+x \coth ^{-1}(a x)^2+\frac{\coth ^{-1}(a x)^2}{a}-\frac{2 \log \left(\frac{2}{1-a x}\right) \coth ^{-1}(a x)}{a}",1,"(ArcCoth[a*x]*((-1 + a*x)*ArcCoth[a*x] - 2*Log[1 - E^(-2*ArcCoth[a*x])]) + PolyLog[2, E^(-2*ArcCoth[a*x])])/a","A",0
18,1,114,97,0.0583342,"\int \frac{\coth ^{-1}(a x)^2}{x} \, dx","Integrate[ArcCoth[a*x]^2/x,x]","-\coth ^{-1}(a x) \text{Li}_2\left(-e^{-2 \coth ^{-1}(a x)}\right)-\coth ^{-1}(a x) \text{Li}_2\left(e^{2 \coth ^{-1}(a x)}\right)-\frac{1}{2} \text{Li}_3\left(-e^{-2 \coth ^{-1}(a x)}\right)+\frac{1}{2} \text{Li}_3\left(e^{2 \coth ^{-1}(a x)}\right)+\frac{2}{3} \coth ^{-1}(a x)^3+\coth ^{-1}(a x)^2 \log \left(e^{-2 \coth ^{-1}(a x)}+1\right)-\coth ^{-1}(a x)^2 \log \left(1-e^{2 \coth ^{-1}(a x)}\right)","\frac{1}{2} \text{Li}_3\left(1-\frac{2}{a x+1}\right)-\frac{1}{2} \text{Li}_3\left(1-\frac{2 a x}{a x+1}\right)+\text{Li}_2\left(1-\frac{2}{a x+1}\right) \coth ^{-1}(a x)-\text{Li}_2\left(1-\frac{2 a x}{a x+1}\right) \coth ^{-1}(a x)+2 \coth ^{-1}\left(1-\frac{2}{1-a x}\right) \coth ^{-1}(a x)^2",1,"(2*ArcCoth[a*x]^3)/3 + ArcCoth[a*x]^2*Log[1 + E^(-2*ArcCoth[a*x])] - ArcCoth[a*x]^2*Log[1 - E^(2*ArcCoth[a*x])] - ArcCoth[a*x]*PolyLog[2, -E^(-2*ArcCoth[a*x])] - ArcCoth[a*x]*PolyLog[2, E^(2*ArcCoth[a*x])] - PolyLog[3, -E^(-2*ArcCoth[a*x])]/2 + PolyLog[3, E^(2*ArcCoth[a*x])]/2","A",0
19,1,49,55,0.1013836,"\int \frac{\coth ^{-1}(a x)^2}{x^2} \, dx","Integrate[ArcCoth[a*x]^2/x^2,x]","-a \text{Li}_2\left(-e^{-2 \coth ^{-1}(a x)}\right)+\frac{(a x-1) \coth ^{-1}(a x)^2}{x}+2 a \coth ^{-1}(a x) \log \left(e^{-2 \coth ^{-1}(a x)}+1\right)","-a \text{Li}_2\left(\frac{2}{a x+1}-1\right)+a \coth ^{-1}(a x)^2-\frac{\coth ^{-1}(a x)^2}{x}+2 a \log \left(2-\frac{2}{a x+1}\right) \coth ^{-1}(a x)",1,"((-1 + a*x)*ArcCoth[a*x]^2)/x + 2*a*ArcCoth[a*x]*Log[1 + E^(-2*ArcCoth[a*x])] - a*PolyLog[2, -E^(-2*ArcCoth[a*x])]","A",0
20,1,57,61,0.0161491,"\int \frac{\coth ^{-1}(a x)^2}{x^3} \, dx","Integrate[ArcCoth[a*x]^2/x^3,x]","-\frac{1}{2} a^2 \log \left(1-a^2 x^2\right)+\frac{\left(a^2 x^2-1\right) \coth ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-\frac{a \coth ^{-1}(a x)}{x}","-\frac{1}{2} a^2 \log \left(1-a^2 x^2\right)+a^2 \log (x)+\frac{1}{2} a^2 \coth ^{-1}(a x)^2-\frac{\coth ^{-1}(a x)^2}{2 x^2}-\frac{a \coth ^{-1}(a x)}{x}",1,"-((a*ArcCoth[a*x])/x) + ((-1 + a^2*x^2)*ArcCoth[a*x]^2)/(2*x^2) + a^2*Log[x] - (a^2*Log[1 - a^2*x^2])/2","A",1
21,1,87,103,0.2115224,"\int \frac{\coth ^{-1}(a x)^2}{x^4} \, dx","Integrate[ArcCoth[a*x]^2/x^4,x]","\frac{-a^3 x^3 \text{Li}_2\left(-e^{-2 \coth ^{-1}(a x)}\right)+\left(a^3 x^3-1\right) \coth ^{-1}(a x)^2-a^2 x^2+a x \coth ^{-1}(a x) \left(a^2 x^2+2 a^2 x^2 \log \left(e^{-2 \coth ^{-1}(a x)}+1\right)-1\right)}{3 x^3}","-\frac{1}{3} a^3 \text{Li}_2\left(\frac{2}{a x+1}-1\right)+\frac{1}{3} a^3 \tanh ^{-1}(a x)+\frac{1}{3} a^3 \coth ^{-1}(a x)^2+\frac{2}{3} a^3 \log \left(2-\frac{2}{a x+1}\right) \coth ^{-1}(a x)-\frac{a^2}{3 x}-\frac{\coth ^{-1}(a x)^2}{3 x^3}-\frac{a \coth ^{-1}(a x)}{3 x^2}",1,"(-(a^2*x^2) + (-1 + a^3*x^3)*ArcCoth[a*x]^2 + a*x*ArcCoth[a*x]*(-1 + a^2*x^2 + 2*a^2*x^2*Log[1 + E^(-2*ArcCoth[a*x])]) - a^3*x^3*PolyLog[2, -E^(-2*ArcCoth[a*x])])/(3*x^3)","A",0
22,1,82,90,0.0208529,"\int \frac{\coth ^{-1}(a x)^2}{x^5} \, dx","Integrate[ArcCoth[a*x]^2/x^5,x]","\frac{\left(a^4 x^4-1\right) \coth ^{-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) \coth ^{-1}(a x)}{6 x^3}-\frac{1}{3} a^4 \log \left(1-a^2 x^2\right)","\frac{2}{3} a^4 \log (x)+\frac{1}{4} a^4 \coth ^{-1}(a x)^2-\frac{a^3 \coth ^{-1}(a x)}{2 x}-\frac{a^2}{12 x^2}-\frac{1}{3} a^4 \log \left(1-a^2 x^2\right)-\frac{\coth ^{-1}(a x)^2}{4 x^4}-\frac{a \coth ^{-1}(a x)}{6 x^3}",1,"-1/12*a^2/x^2 - (a*(1 + 3*a^2*x^2)*ArcCoth[a*x])/(6*x^3) + ((-1 + a^4*x^4)*ArcCoth[a*x]^2)/(4*x^4) + (2*a^4*Log[x])/3 - (a^4*Log[1 - a^2*x^2])/3","A",1
23,1,117,186,0.5350606,"\int x^5 \coth ^{-1}(a x)^3 \, dx","Integrate[x^5*ArcCoth[a*x]^3,x]","\frac{10 \left(a^6 x^6-1\right) \coth ^{-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\right) \coth ^{-1}(a x)^2+\coth ^{-1}(a x) \left(3 a^4 x^4+16 a^2 x^2-92 \log \left(1-e^{-2 \coth ^{-1}(a x)}\right)-19\right)+46 \text{Li}_2\left(e^{-2 \coth ^{-1}(a x)}\right)}{60 a^6}","-\frac{23 \text{Li}_2\left(1-\frac{2}{1-a x}\right)}{30 a^6}-\frac{19 \tanh ^{-1}(a x)}{60 a^6}-\frac{\coth ^{-1}(a x)^3}{6 a^6}+\frac{23 \coth ^{-1}(a x)^2}{30 a^6}-\frac{23 \log \left(\frac{2}{1-a x}\right) \coth ^{-1}(a x)}{15 a^6}+\frac{19 x}{60 a^5}+\frac{x \coth ^{-1}(a x)^2}{2 a^5}+\frac{4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac{x^3}{60 a^3}+\frac{x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac{x^4 \coth ^{-1}(a x)}{20 a^2}+\frac{1}{6} x^6 \coth ^{-1}(a x)^3+\frac{x^5 \coth ^{-1}(a x)^2}{10 a}",1,"(a*x*(19 + a^2*x^2) + 2*(-23 + 15*a*x + 5*a^3*x^3 + 3*a^5*x^5)*ArcCoth[a*x]^2 + 10*(-1 + a^6*x^6)*ArcCoth[a*x]^3 + ArcCoth[a*x]*(-19 + 16*a^2*x^2 + 3*a^4*x^4 - 92*Log[1 - E^(-2*ArcCoth[a*x])]) + 46*PolyLog[2, E^(-2*ArcCoth[a*x])])/(60*a^6)","A",0
24,1,175,196,0.5876196,"\int x^4 \coth ^{-1}(a x)^3 \, dx","Integrate[x^4*ArcCoth[a*x]^3,x]","\frac{8 a^5 x^5 \coth ^{-1}(a x)^3+6 a^4 x^4 \coth ^{-1}(a x)^2+4 a^3 x^3 \coth ^{-1}(a x)+2 a^2 x^2-40 \log \left(\frac{1}{a x \sqrt{1-\frac{1}{a^2 x^2}}}\right)+12 a^2 x^2 \coth ^{-1}(a x)^2-24 \coth ^{-1}(a x) \text{Li}_2\left(e^{2 \coth ^{-1}(a x)}\right)+12 \text{Li}_3\left(e^{2 \coth ^{-1}(a x)}\right)+36 a x \coth ^{-1}(a x)+8 \coth ^{-1}(a x)^3-18 \coth ^{-1}(a x)^2-24 \coth ^{-1}(a x)^2 \log \left(1-e^{2 \coth ^{-1}(a x)}\right)-i \pi ^3-2}{40 a^5}","\frac{3 \text{Li}_3\left(1-\frac{2}{1-a x}\right)}{10 a^5}-\frac{3 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \coth ^{-1}(a x)}{5 a^5}+\frac{\coth ^{-1}(a x)^3}{5 a^5}-\frac{9 \coth ^{-1}(a x)^2}{20 a^5}-\frac{3 \log \left(\frac{2}{1-a x}\right) \coth ^{-1}(a x)^2}{5 a^5}+\frac{9 x \coth ^{-1}(a x)}{10 a^4}+\frac{x^2}{20 a^3}+\frac{3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac{x^3 \coth ^{-1}(a x)}{10 a^2}+\frac{\log \left(1-a^2 x^2\right)}{2 a^5}+\frac{1}{5} x^5 \coth ^{-1}(a x)^3+\frac{3 x^4 \coth ^{-1}(a x)^2}{20 a}",1,"(-2 - I*Pi^3 + 2*a^2*x^2 + 36*a*x*ArcCoth[a*x] + 4*a^3*x^3*ArcCoth[a*x] - 18*ArcCoth[a*x]^2 + 12*a^2*x^2*ArcCoth[a*x]^2 + 6*a^4*x^4*ArcCoth[a*x]^2 + 8*ArcCoth[a*x]^3 + 8*a^5*x^5*ArcCoth[a*x]^3 - 24*ArcCoth[a*x]^2*Log[1 - E^(2*ArcCoth[a*x])] - 40*Log[1/(a*Sqrt[1 - 1/(a^2*x^2)]*x)] - 24*ArcCoth[a*x]*PolyLog[2, E^(2*ArcCoth[a*x])] + 12*PolyLog[3, E^(2*ArcCoth[a*x])])/(40*a^5)","C",0
25,1,88,139,0.3083203,"\int x^3 \coth ^{-1}(a x)^3 \, dx","Integrate[x^3*ArcCoth[a*x]^3,x]","\frac{\left(a^4 x^4-1\right) \coth ^{-1}(a x)^3+\left(a^3 x^3+3 a x-4\right) \coth ^{-1}(a x)^2+\coth ^{-1}(a x) \left(a^2 x^2-8 \log \left(1-e^{-2 \coth ^{-1}(a x)}\right)-1\right)+4 \text{Li}_2\left(e^{-2 \coth ^{-1}(a x)}\right)+a x}{4 a^4}","-\frac{\text{Li}_2\left(1-\frac{2}{1-a x}\right)}{a^4}-\frac{\tanh ^{-1}(a x)}{4 a^4}-\frac{\coth ^{-1}(a x)^3}{4 a^4}+\frac{\coth ^{-1}(a x)^2}{a^4}-\frac{2 \log \left(\frac{2}{1-a x}\right) \coth ^{-1}(a x)}{a^4}+\frac{x}{4 a^3}+\frac{3 x \coth ^{-1}(a x)^2}{4 a^3}+\frac{x^2 \coth ^{-1}(a x)}{4 a^2}+\frac{1}{4} x^4 \coth ^{-1}(a x)^3+\frac{x^3 \coth ^{-1}(a x)^2}{4 a}",1,"(a*x + (-4 + 3*a*x + a^3*x^3)*ArcCoth[a*x]^2 + (-1 + a^4*x^4)*ArcCoth[a*x]^3 + ArcCoth[a*x]*(-1 + a^2*x^2 - 8*Log[1 - E^(-2*ArcCoth[a*x])]) + 4*PolyLog[2, E^(-2*ArcCoth[a*x])])/(4*a^4)","A",0
26,1,140,149,0.3818337,"\int x^2 \coth ^{-1}(a x)^3 \, dx","Integrate[x^2*ArcCoth[a*x]^3,x]","\frac{8 a^3 x^3 \coth ^{-1}(a x)^3-24 \log \left(\frac{1}{a x \sqrt{1-\frac{1}{a^2 x^2}}}\right)+12 a^2 x^2 \coth ^{-1}(a x)^2-24 \coth ^{-1}(a x) \text{Li}_2\left(e^{2 \coth ^{-1}(a x)}\right)+12 \text{Li}_3\left(e^{2 \coth ^{-1}(a x)}\right)+8 \coth ^{-1}(a x)^3-12 \coth ^{-1}(a x)^2+24 a x \coth ^{-1}(a x)-24 \coth ^{-1}(a x)^2 \log \left(1-e^{2 \coth ^{-1}(a x)}\right)-i \pi ^3}{24 a^3}","\frac{\text{Li}_3\left(1-\frac{2}{1-a x}\right)}{2 a^3}-\frac{\text{Li}_2\left(1-\frac{2}{1-a x}\right) \coth ^{-1}(a x)}{a^3}+\frac{\coth ^{-1}(a x)^3}{3 a^3}-\frac{\coth ^{-1}(a x)^2}{2 a^3}-\frac{\log \left(\frac{2}{1-a x}\right) \coth ^{-1}(a x)^2}{a^3}+\frac{x \coth ^{-1}(a x)}{a^2}+\frac{\log \left(1-a^2 x^2\right)}{2 a^3}+\frac{1}{3} x^3 \coth ^{-1}(a x)^3+\frac{x^2 \coth ^{-1}(a x)^2}{2 a}",1,"((-I)*Pi^3 + 24*a*x*ArcCoth[a*x] - 12*ArcCoth[a*x]^2 + 12*a^2*x^2*ArcCoth[a*x]^2 + 8*ArcCoth[a*x]^3 + 8*a^3*x^3*ArcCoth[a*x]^3 - 24*ArcCoth[a*x]^2*Log[1 - E^(2*ArcCoth[a*x])] - 24*Log[1/(a*Sqrt[1 - 1/(a^2*x^2)]*x)] - 24*ArcCoth[a*x]*PolyLog[2, E^(2*ArcCoth[a*x])] + 12*PolyLog[3, E^(2*ArcCoth[a*x])])/(24*a^3)","C",0
27,1,68,95,0.138893,"\int x \coth ^{-1}(a x)^3 \, dx","Integrate[x*ArcCoth[a*x]^3,x]","\frac{\coth ^{-1}(a x) \left(\left(a^2 x^2-1\right) \coth ^{-1}(a x)^2+3 (a x-1) \coth ^{-1}(a x)-6 \log \left(1-e^{-2 \coth ^{-1}(a x)}\right)\right)+3 \text{Li}_2\left(e^{-2 \coth ^{-1}(a x)}\right)}{2 a^2}","-\frac{3 \text{Li}_2\left(1-\frac{2}{1-a x}\right)}{2 a^2}-\frac{\coth ^{-1}(a x)^3}{2 a^2}+\frac{3 \coth ^{-1}(a x)^2}{2 a^2}-\frac{3 \log \left(\frac{2}{1-a x}\right) \coth ^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \coth ^{-1}(a x)^3+\frac{3 x \coth ^{-1}(a x)^2}{2 a}",1,"(ArcCoth[a*x]*(3*(-1 + a*x)*ArcCoth[a*x] + (-1 + a^2*x^2)*ArcCoth[a*x]^2 - 6*Log[1 - E^(-2*ArcCoth[a*x])]) + 3*PolyLog[2, E^(-2*ArcCoth[a*x])])/(2*a^2)","A",0
28,1,79,85,0.1004607,"\int \coth ^{-1}(a x)^3 \, dx","Integrate[ArcCoth[a*x]^3,x]","-\frac{3 \coth ^{-1}(a x) \text{Li}_2\left(e^{2 \coth ^{-1}(a x)}\right)}{a}+\frac{3 \text{Li}_3\left(e^{2 \coth ^{-1}(a x)}\right)}{2 a}+x \coth ^{-1}(a x)^3+\frac{\coth ^{-1}(a x)^3}{a}-\frac{3 \coth ^{-1}(a x)^2 \log \left(1-e^{2 \coth ^{-1}(a x)}\right)}{a}","\frac{3 \text{Li}_3\left(1-\frac{2}{1-a x}\right)}{2 a}-\frac{3 \text{Li}_2\left(1-\frac{2}{1-a x}\right) \coth ^{-1}(a x)}{a}+x \coth ^{-1}(a x)^3+\frac{\coth ^{-1}(a x)^3}{a}-\frac{3 \log \left(\frac{2}{1-a x}\right) \coth ^{-1}(a x)^2}{a}",1,"ArcCoth[a*x]^3/a + x*ArcCoth[a*x]^3 - (3*ArcCoth[a*x]^2*Log[1 - E^(2*ArcCoth[a*x])])/a - (3*ArcCoth[a*x]*PolyLog[2, E^(2*ArcCoth[a*x])])/a + (3*PolyLog[3, E^(2*ArcCoth[a*x])])/(2*a)","A",0
29,1,156,150,0.0755184,"\int \frac{\coth ^{-1}(a x)^3}{x} \, dx","Integrate[ArcCoth[a*x]^3/x,x]","\frac{1}{64} \left(-96 \coth ^{-1}(a x)^2 \text{Li}_2\left(-e^{-2 \coth ^{-1}(a x)}\right)-96 \coth ^{-1}(a x)^2 \text{Li}_2\left(e^{2 \coth ^{-1}(a x)}\right)-96 \coth ^{-1}(a x) \text{Li}_3\left(-e^{-2 \coth ^{-1}(a x)}\right)+96 \coth ^{-1}(a x) \text{Li}_3\left(e^{2 \coth ^{-1}(a x)}\right)-48 \text{Li}_4\left(-e^{-2 \coth ^{-1}(a x)}\right)-48 \text{Li}_4\left(e^{2 \coth ^{-1}(a x)}\right)+32 \coth ^{-1}(a x)^4+64 \coth ^{-1}(a x)^3 \log \left(e^{-2 \coth ^{-1}(a x)}+1\right)-64 \coth ^{-1}(a x)^3 \log \left(1-e^{2 \coth ^{-1}(a x)}\right)-\pi ^4\right)","\frac{3}{4} \text{Li}_4\left(1-\frac{2}{a x+1}\right)-\frac{3}{4} \text{Li}_4\left(1-\frac{2 a x}{a x+1}\right)+\frac{3}{2} \text{Li}_2\left(1-\frac{2}{a x+1}\right) \coth ^{-1}(a x)^2-\frac{3}{2} \text{Li}_2\left(1-\frac{2 a x}{a x+1}\right) \coth ^{-1}(a x)^2+\frac{3}{2} \text{Li}_3\left(1-\frac{2}{a x+1}\right) \coth ^{-1}(a x)-\frac{3}{2} \text{Li}_3\left(1-\frac{2 a x}{a x+1}\right) \coth ^{-1}(a x)+2 \coth ^{-1}\left(1-\frac{2}{1-a x}\right) \coth ^{-1}(a x)^3",1,"(-Pi^4 + 32*ArcCoth[a*x]^4 + 64*ArcCoth[a*x]^3*Log[1 + E^(-2*ArcCoth[a*x])] - 64*ArcCoth[a*x]^3*Log[1 - E^(2*ArcCoth[a*x])] - 96*ArcCoth[a*x]^2*PolyLog[2, -E^(-2*ArcCoth[a*x])] - 96*ArcCoth[a*x]^2*PolyLog[2, E^(2*ArcCoth[a*x])] - 96*ArcCoth[a*x]*PolyLog[3, -E^(-2*ArcCoth[a*x])] + 96*ArcCoth[a*x]*PolyLog[3, E^(2*ArcCoth[a*x])] - 48*PolyLog[4, -E^(-2*ArcCoth[a*x])] - 48*PolyLog[4, E^(2*ArcCoth[a*x])])/64","A",0
30,1,72,79,0.1314963,"\int \frac{\coth ^{-1}(a x)^3}{x^2} \, dx","Integrate[ArcCoth[a*x]^3/x^2,x]","-3 a \coth ^{-1}(a x) \text{Li}_2\left(-e^{-2 \coth ^{-1}(a x)}\right)-\frac{3}{2} a \text{Li}_3\left(-e^{-2 \coth ^{-1}(a x)}\right)+\frac{(a x-1) \coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left(e^{-2 \coth ^{-1}(a x)}+1\right)","-\frac{3}{2} a \text{Li}_3\left(\frac{2}{a x+1}-1\right)-3 a \text{Li}_2\left(\frac{2}{a x+1}-1\right) \coth ^{-1}(a x)+a \coth ^{-1}(a x)^3-\frac{\coth ^{-1}(a x)^3}{x}+3 a \log \left(2-\frac{2}{a x+1}\right) \coth ^{-1}(a x)^2",1,"((-1 + a*x)*ArcCoth[a*x]^3)/x + 3*a*ArcCoth[a*x]^2*Log[1 + E^(-2*ArcCoth[a*x])] - 3*a*ArcCoth[a*x]*PolyLog[2, -E^(-2*ArcCoth[a*x])] - (3*a*PolyLog[3, -E^(-2*ArcCoth[a*x])])/2","A",0
31,1,79,95,0.1682537,"\int \frac{\coth ^{-1}(a x)^3}{x^3} \, dx","Integrate[ArcCoth[a*x]^3/x^3,x]","\frac{1}{2} \left(\frac{\coth ^{-1}(a x) \left(\left(a^2 x^2-1\right) \coth ^{-1}(a x)^2+6 a^2 x^2 \log \left(e^{-2 \coth ^{-1}(a x)}+1\right)+3 a x (a x-1) \coth ^{-1}(a x)\right)}{x^2}-3 a^2 \text{Li}_2\left(-e^{-2 \coth ^{-1}(a x)}\right)\right)","-\frac{3}{2} a^2 \text{Li}_2\left(\frac{2}{a x+1}-1\right)+\frac{1}{2} a^2 \coth ^{-1}(a x)^3+\frac{3}{2} a^2 \coth ^{-1}(a x)^2+3 a^2 \log \left(2-\frac{2}{a x+1}\right) \coth ^{-1}(a x)-\frac{\coth ^{-1}(a x)^3}{2 x^2}-\frac{3 a \coth ^{-1}(a x)^2}{2 x}",1,"((ArcCoth[a*x]*(3*a*x*(-1 + a*x)*ArcCoth[a*x] + (-1 + a^2*x^2)*ArcCoth[a*x]^2 + 6*a^2*x^2*Log[1 + E^(-2*ArcCoth[a*x])]))/x^2 - 3*a^2*PolyLog[2, -E^(-2*ArcCoth[a*x])])/2","A",0
32,1,142,154,0.2174341,"\int \frac{\coth ^{-1}(a x)^3}{x^4} \, dx","Integrate[ArcCoth[a*x]^3/x^4,x]","\frac{1}{6} \left(-6 a^3 \coth ^{-1}(a x) \text{Li}_2\left(-e^{-2 \coth ^{-1}(a x)}\right)-3 a^3 \text{Li}_3\left(-e^{-2 \coth ^{-1}(a x)}\right)+2 a^3 \coth ^{-1}(a x)^3+3 a^3 \coth ^{-1}(a x)^2+6 a^3 \coth ^{-1}(a x)^2 \log \left(e^{-2 \coth ^{-1}(a x)}+1\right)-\frac{6 a^2 \coth ^{-1}(a x)}{x}+6 a^3 \log \left(\frac{1}{\sqrt{1-\frac{1}{a^2 x^2}}}\right)-\frac{2 \coth ^{-1}(a x)^3}{x^3}-\frac{3 a \coth ^{-1}(a x)^2}{x^2}\right)","-\frac{1}{2} a^3 \text{Li}_3\left(\frac{2}{a x+1}-1\right)-a^3 \text{Li}_2\left(\frac{2}{a x+1}-1\right) \coth ^{-1}(a x)+a^3 \log (x)+\frac{1}{3} a^3 \coth ^{-1}(a x)^3+\frac{1}{2} a^3 \coth ^{-1}(a x)^2+a^3 \log \left(2-\frac{2}{a x+1}\right) \coth ^{-1}(a x)^2-\frac{a^2 \coth ^{-1}(a x)}{x}-\frac{1}{2} a^3 \log \left(1-a^2 x^2\right)-\frac{\coth ^{-1}(a x)^3}{3 x^3}-\frac{a \coth ^{-1}(a x)^2}{2 x^2}",1,"((-6*a^2*ArcCoth[a*x])/x + 3*a^3*ArcCoth[a*x]^2 - (3*a*ArcCoth[a*x]^2)/x^2 + 2*a^3*ArcCoth[a*x]^3 - (2*ArcCoth[a*x]^3)/x^3 + 6*a^3*ArcCoth[a*x]^2*Log[1 + E^(-2*ArcCoth[a*x])] + 6*a^3*Log[1/Sqrt[1 - 1/(a^2*x^2)]] - 6*a^3*ArcCoth[a*x]*PolyLog[2, -E^(-2*ArcCoth[a*x])] - 3*a^3*PolyLog[3, -E^(-2*ArcCoth[a*x])])/6","A",0
33,1,118,141,0.2376863,"\int \frac{\coth ^{-1}(a x)^3}{x^5} \, dx","Integrate[ArcCoth[a*x]^3/x^5,x]","\frac{-4 a^4 x^4 \text{Li}_2\left(-e^{-2 \coth ^{-1}(a x)}\right)+\left(a^4 x^4-1\right) \coth ^{-1}(a x)^3-a^3 x^3+a^2 x^2 \coth ^{-1}(a x) \left(a^2 x^2+8 a^2 x^2 \log \left(e^{-2 \coth ^{-1}(a x)}+1\right)-1\right)+a x \left(4 a^3 x^3-3 a^2 x^2-1\right) \coth ^{-1}(a x)^2}{4 x^4}","-a^4 \text{Li}_2\left(\frac{2}{a x+1}-1\right)+\frac{1}{4} a^4 \tanh ^{-1}(a x)+\frac{1}{4} a^4 \coth ^{-1}(a x)^3+a^4 \coth ^{-1}(a x)^2+2 a^4 \log \left(2-\frac{2}{a x+1}\right) \coth ^{-1}(a x)-\frac{a^3}{4 x}-\frac{3 a^3 \coth ^{-1}(a x)^2}{4 x}-\frac{a^2 \coth ^{-1}(a x)}{4 x^2}-\frac{\coth ^{-1}(a x)^3}{4 x^4}-\frac{a \coth ^{-1}(a x)^2}{4 x^3}",1,"(-(a^3*x^3) + a*x*(-1 - 3*a^2*x^2 + 4*a^3*x^3)*ArcCoth[a*x]^2 + (-1 + a^4*x^4)*ArcCoth[a*x]^3 + a^2*x^2*ArcCoth[a*x]*(-1 + a^2*x^2 + 8*a^2*x^2*Log[1 + E^(-2*ArcCoth[a*x])]) - 4*a^4*x^4*PolyLog[2, -E^(-2*ArcCoth[a*x])])/(4*x^4)","A",0
34,1,565,164,7.8041956,"\int \frac{\coth ^{-1}(c x)^2}{d+e x} \, dx","Integrate[ArcCoth[c*x]^2/(d + e*x),x]","\frac{\frac{24 (e-c d) (c d+e) \left(2 c d \sqrt{1-\frac{e^2}{c^2 d^2}} \coth ^{-1}(c x)^3 e^{-\tanh ^{-1}\left(\frac{e}{c d}\right)}+3 e \coth ^{-1}(c x)^2 \log \left(\frac{d+e x}{x \sqrt{1-\frac{1}{c^2 x^2}}}\right)-3 i \pi  e \log \left(\frac{1}{\sqrt{1-\frac{1}{c^2 x^2}}}\right) \coth ^{-1}(c x)-6 e \coth ^{-1}(c x) \text{Li}_2\left(-e^{\coth ^{-1}(c x)+\tanh ^{-1}\left(\frac{e}{c d}\right)}\right)-6 e \coth ^{-1}(c x) \text{Li}_2\left(e^{\coth ^{-1}(c x)+\tanh ^{-1}\left(\frac{e}{c d}\right)}\right)+6 e \text{Li}_3\left(-e^{\coth ^{-1}(c x)+\tanh ^{-1}\left(\frac{e}{c d}\right)}\right)+6 e \text{Li}_3\left(e^{\coth ^{-1}(c x)+\tanh ^{-1}\left(\frac{e}{c d}\right)}\right)-3 e \coth ^{-1}(c x)^2 \log \left(\frac{1}{2} e^{-\coth ^{-1}(c x)} \left(c d \left(e^{2 \coth ^{-1}(c x)}-1\right)+e \left(e^{2 \coth ^{-1}(c x)}+1\right)\right)\right)-3 e \coth ^{-1}(c x)^2 \log \left(1-e^{\tanh ^{-1}\left(\frac{e}{c d}\right)+\coth ^{-1}(c x)}\right)-3 e \coth ^{-1}(c x)^2 \log \left(e^{\tanh ^{-1}\left(\frac{e}{c d}\right)+\coth ^{-1}(c x)}+1\right)-6 e \coth ^{-1}(c x) \tanh ^{-1}\left(\frac{e}{c d}\right) \log \left(\frac{1}{2} i e^{-\tanh ^{-1}\left(\frac{e}{c d}\right)-\coth ^{-1}(c x)} \left(e^{2 \left(\tanh ^{-1}\left(\frac{e}{c d}\right)+\coth ^{-1}(c x)\right)}-1\right)\right)+6 e \coth ^{-1}(c x) \tanh ^{-1}\left(\frac{e}{c d}\right) \log \left(i \sinh \left(\tanh ^{-1}\left(\frac{e}{c d}\right)+\coth ^{-1}(c x)\right)\right)-c d \coth ^{-1}(c x)^3+3 e \coth ^{-1}(c x)^3+3 i \pi  e \coth ^{-1}(c x) \log \left(\frac{1}{2} \left(e^{-\coth ^{-1}(c x)}+e^{\coth ^{-1}(c x)}\right)\right)\right)}{3 c^2 d^2-3 e^2}+8 c d \coth ^{-1}(c x)^3-24 e \coth ^{-1}(c x) \text{Li}_2\left(e^{2 \coth ^{-1}(c x)}\right)+12 e \text{Li}_3\left(e^{2 \coth ^{-1}(c x)}\right)+8 e \coth ^{-1}(c x)^3-24 e \coth ^{-1}(c x)^2 \log \left(1-e^{2 \coth ^{-1}(c x)}\right)-i \pi ^3 e}{24 e^2}","-\frac{\text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{2 e}-\frac{\coth ^{-1}(c x) \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right)}{e}+\frac{\coth ^{-1}(c x)^2 \log \left(\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right)}{e}+\frac{\text{Li}_3\left(1-\frac{2}{c x+1}\right)}{2 e}+\frac{\text{Li}_2\left(1-\frac{2}{c x+1}\right) \coth ^{-1}(c x)}{e}-\frac{\log \left(\frac{2}{c x+1}\right) \coth ^{-1}(c x)^2}{e}",1,"((-I)*e*Pi^3 + 8*c*d*ArcCoth[c*x]^3 + 8*e*ArcCoth[c*x]^3 - 24*e*ArcCoth[c*x]^2*Log[1 - E^(2*ArcCoth[c*x])] - 24*e*ArcCoth[c*x]*PolyLog[2, E^(2*ArcCoth[c*x])] + 12*e*PolyLog[3, E^(2*ArcCoth[c*x])] + (24*(-(c*d) + e)*(c*d + e)*(-(c*d*ArcCoth[c*x]^3) + 3*e*ArcCoth[c*x]^3 + (2*c*d*Sqrt[1 - e^2/(c^2*d^2)]*ArcCoth[c*x]^3)/E^ArcTanh[e/(c*d)] + (3*I)*e*Pi*ArcCoth[c*x]*Log[(E^(-ArcCoth[c*x]) + E^ArcCoth[c*x])/2] - 3*e*ArcCoth[c*x]^2*Log[1 - E^(ArcCoth[c*x] + ArcTanh[e/(c*d)])] - 3*e*ArcCoth[c*x]^2*Log[1 + E^(ArcCoth[c*x] + ArcTanh[e/(c*d)])] - 6*e*ArcCoth[c*x]*ArcTanh[e/(c*d)]*Log[(I/2)*E^(-ArcCoth[c*x] - ArcTanh[e/(c*d)])*(-1 + E^(2*(ArcCoth[c*x] + ArcTanh[e/(c*d)])))] - 3*e*ArcCoth[c*x]^2*Log[(c*d*(-1 + E^(2*ArcCoth[c*x])) + e*(1 + E^(2*ArcCoth[c*x])))/(2*E^ArcCoth[c*x])] - (3*I)*e*Pi*ArcCoth[c*x]*Log[1/Sqrt[1 - 1/(c^2*x^2)]] + 3*e*ArcCoth[c*x]^2*Log[(d + e*x)/(Sqrt[1 - 1/(c^2*x^2)]*x)] + 6*e*ArcCoth[c*x]*ArcTanh[e/(c*d)]*Log[I*Sinh[ArcCoth[c*x] + ArcTanh[e/(c*d)]]] - 6*e*ArcCoth[c*x]*PolyLog[2, -E^(ArcCoth[c*x] + ArcTanh[e/(c*d)])] - 6*e*ArcCoth[c*x]*PolyLog[2, E^(ArcCoth[c*x] + ArcTanh[e/(c*d)])] + 6*e*PolyLog[3, -E^(ArcCoth[c*x] + ArcTanh[e/(c*d)])] + 6*e*PolyLog[3, E^(ArcCoth[c*x] + ArcTanh[e/(c*d)])]))/(3*c^2*d^2 - 3*e^2))/(24*e^2)","C",0
35,1,213,245,0.1161812,"\int \left(c+d x^2\right)^4 \coth ^{-1}(a x) \, dx","Integrate[(c + d*x^2)^4*ArcCoth[a*x],x]","\frac{24 a^9 x \coth ^{-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(1-a^2 x^2\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(1-a^2 x^2\right)}{630 a^9}+c^4 x \coth ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \coth ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \coth ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \coth ^{-1}(a x)+\frac{1}{9} d^4 x^9 \coth ^{-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)*ArcCoth[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
36,1,150,169,0.0833664,"\int \left(c+d x^2\right)^3 \coth ^{-1}(a x) \, dx","Integrate[(c + d*x^2)^3*ArcCoth[a*x],x]","\frac{12 a^7 x \coth ^{-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(1-a^2 x^2\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(1-a^2 x^2\right)}{70 a^7}+c^3 x \coth ^{-1}(a x)+c^2 d x^3 \coth ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \coth ^{-1}(a x)+\frac{1}{7} d^3 x^7 \coth ^{-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)*ArcCoth[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
37,1,98,110,0.052011,"\int \left(c+d x^2\right)^2 \coth ^{-1}(a x) \, dx","Integrate[(c + d*x^2)^2*ArcCoth[a*x],x]","\frac{4 a^5 x \coth ^{-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(1-a^2 x^2\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(1-a^2 x^2\right)}{30 a^5}+c^2 x \coth ^{-1}(a x)+\frac{2}{3} c d x^3 \coth ^{-1}(a x)+\frac{1}{5} d^2 x^5 \coth ^{-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)*ArcCoth[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
38,1,69,57,0.0106656,"\int \left(c+d x^2\right) \coth ^{-1}(a x) \, dx","Integrate[(c + d*x^2)*ArcCoth[a*x],x]","\frac{c \log \left(1-a^2 x^2\right)}{2 a}+\frac{d \log \left(1-a^2 x^2\right)}{6 a^3}+c x \coth ^{-1}(a x)+\frac{1}{3} d x^3 \coth ^{-1}(a x)+\frac{d x^2}{6 a}","\frac{\left(3 a^2 c+d\right) \log \left(1-a^2 x^2\right)}{6 a^3}+c x \coth ^{-1}(a x)+\frac{1}{3} d x^3 \coth ^{-1}(a x)+\frac{d x^2}{6 a}",1,"(d*x^2)/(6*a) + c*x*ArcCoth[a*x] + (d*x^3*ArcCoth[a*x])/3 + (c*Log[1 - a^2*x^2])/(2*a) + (d*Log[1 - a^2*x^2])/(6*a^3)","A",1
39,1,671,390,1.3910123,"\int \frac{\coth ^{-1}(a x)}{c+d x^2} \, dx","Integrate[ArcCoth[a*x]/(c + d*x^2),x]","\frac{a \left(i \left(\text{Li}_2\left(\frac{\left(c a^2-d+2 i \sqrt{a^2 c d}\right) \left(i a d x+\sqrt{a^2 c d}\right)}{\left(c a^2+d\right) \left(\sqrt{a^2 c d}-i a d x\right)}\right)-\text{Li}_2\left(\frac{\left(c a^2-d-2 i \sqrt{a^2 c d}\right) \left(i a d x+\sqrt{a^2 c d}\right)}{\left(c a^2+d\right) \left(\sqrt{a^2 c d}-i a d x\right)}\right)\right)-2 i \cos ^{-1}\left(\frac{a^2 c-d}{a^2 c+d}\right) \tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)+4 \coth ^{-1}(a x) \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)-\log \left(\frac{2 d (a x-1) \left(a^2 c-i \sqrt{a^2 c d}\right)}{\left(a^2 c+d\right) \left(a d x+i \sqrt{a^2 c d}\right)}\right) \left(2 \tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)+\cos ^{-1}\left(\frac{a^2 c-d}{a^2 c+d}\right)\right)-\log \left(\frac{2 d (a x+1) \left(a^2 c+i \sqrt{a^2 c d}\right)}{\left(a^2 c+d\right) \left(a d x+i \sqrt{a^2 c d}\right)}\right) \left(\cos ^{-1}\left(\frac{a^2 c-d}{a^2 c+d}\right)-2 \tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)\right)+\left(2 \left(\tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)+\tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)+\cos ^{-1}\left(\frac{a^2 c-d}{a^2 c+d}\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{-\coth ^{-1}(a x)}}{\sqrt{a^2 c+d} \sqrt{\left(a^2 c+d\right) \cosh \left(2 \coth ^{-1}(a x)\right)+a^2 (-c)+d}}\right)+\left(\cos ^{-1}\left(\frac{a^2 c-d}{a^2 c+d}\right)-2 \left(\tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)+\tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{\coth ^{-1}(a x)}}{\sqrt{a^2 c+d} \sqrt{\left(a^2 c+d\right) \cosh \left(2 \coth ^{-1}(a x)\right)+a^2 (-c)+d}}\right)\right)}{4 \sqrt{a^2 c d}}","-\frac{i \text{Li}_2\left(\frac{2 \sqrt{c} \sqrt{d} (1-a x)}{\left(i a \sqrt{c}-\sqrt{d}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}+1\right)}{4 \sqrt{c} \sqrt{d}}+\frac{i \text{Li}_2\left(1-\frac{2 \sqrt{c} \sqrt{d} (a x+1)}{\left(i \sqrt{c} a+\sqrt{d}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}\right)}{4 \sqrt{c} \sqrt{d}}-\frac{\log \left(1-\frac{1}{a x}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} \sqrt{d}}+\frac{\log \left(\frac{1}{a x}+1\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 \sqrt{c} \sqrt{d}}+\frac{\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right) \log \left(-\frac{2 \sqrt{c} \sqrt{d} (1-a x)}{\left(-\sqrt{d}+i a \sqrt{c}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}\right)}{2 \sqrt{c} \sqrt{d}}-\frac{\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right) \log \left(\frac{2 \sqrt{c} \sqrt{d} (a x+1)}{\left(\sqrt{d}+i a \sqrt{c}\right) \left(\sqrt{c}-i \sqrt{d} x\right)}\right)}{2 \sqrt{c} \sqrt{d}}",1,"(a*((-2*I)*ArcCos[(a^2*c - d)/(a^2*c + d)]*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + 4*ArcCoth[a*x]*ArcTan[(a*d*x)/Sqrt[a^2*c*d]] - (ArcCos[(a^2*c - d)/(a^2*c + d)] + 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[(2*d*(a^2*c - I*Sqrt[a^2*c*d])*(-1 + a*x))/((a^2*c + d)*(I*Sqrt[a^2*c*d] + a*d*x))] - (ArcCos[(a^2*c - d)/(a^2*c + d)] - 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[(2*d*(a^2*c + I*Sqrt[a^2*c*d])*(1 + a*x))/((a^2*c + d)*(I*Sqrt[a^2*c*d] + a*d*x))] + (ArcCos[(a^2*c - d)/(a^2*c + d)] + 2*(ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d])/(Sqrt[a^2*c + d]*E^ArcCoth[a*x]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]]])] + (ArcCos[(a^2*c - d)/(a^2*c + d)] - 2*(ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d]*E^ArcCoth[a*x])/(Sqrt[a^2*c + d]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]]])] + I*(-PolyLog[2, ((a^2*c - d - (2*I)*Sqrt[a^2*c*d])*(Sqrt[a^2*c*d] + I*a*d*x))/((a^2*c + d)*(Sqrt[a^2*c*d] - I*a*d*x))] + PolyLog[2, ((a^2*c - d + (2*I)*Sqrt[a^2*c*d])*(Sqrt[a^2*c*d] + I*a*d*x))/((a^2*c + d)*(Sqrt[a^2*c*d] - I*a*d*x))])))/(4*Sqrt[a^2*c*d])","A",1
40,1,755,590,7.7825863,"\int \frac{\coth ^{-1}(a x)}{\left(c+d x^2\right)^2} \, dx","Integrate[ArcCoth[a*x]/(c + d*x^2)^2,x]","-\frac{a \left(\frac{i \left(\text{Li}_2\left(\frac{\left(c a^2-d-2 i \sqrt{a^2 c d}\right) \left(i a d x+\sqrt{a^2 c d}\right)}{\left(c a^2+d\right) \left(\sqrt{a^2 c d}-i a d x\right)}\right)-\text{Li}_2\left(\frac{\left(c a^2-d+2 i \sqrt{a^2 c d}\right) \left(i a d x+\sqrt{a^2 c d}\right)}{\left(c a^2+d\right) \left(\sqrt{a^2 c d}-i a d x\right)}\right)\right)+2 i \cos ^{-1}\left(\frac{a^2 c-d}{a^2 c+d}\right) \tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)-4 \coth ^{-1}(a x) \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)+\log \left(\frac{2 d (a x-1) \left(a^2 c-i \sqrt{a^2 c d}\right)}{\left(a^2 c+d\right) \left(a d x+i \sqrt{a^2 c d}\right)}\right) \left(2 \tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)+\cos ^{-1}\left(\frac{a^2 c-d}{a^2 c+d}\right)\right)+\log \left(\frac{2 d (a x+1) \left(a^2 c+i \sqrt{a^2 c d}\right)}{\left(a^2 c+d\right) \left(a d x+i \sqrt{a^2 c d}\right)}\right) \left(\cos ^{-1}\left(\frac{a^2 c-d}{a^2 c+d}\right)-2 \tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)\right)-\left(2 \left(\tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)+\tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)+\cos ^{-1}\left(\frac{a^2 c-d}{a^2 c+d}\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{-\coth ^{-1}(a x)}}{\sqrt{a^2 c+d} \sqrt{\left(a^2 c+d\right) \cosh \left(2 \coth ^{-1}(a x)\right)+a^2 (-c)+d}}\right)-\left(\cos ^{-1}\left(\frac{a^2 c-d}{a^2 c+d}\right)-2 \left(\tan ^{-1}\left(\frac{a c}{x \sqrt{a^2 c d}}\right)+\tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{\coth ^{-1}(a x)}}{\sqrt{a^2 c+d} \sqrt{\left(a^2 c+d\right) \cosh \left(2 \coth ^{-1}(a x)\right)+a^2 (-c)+d}}\right)}{\sqrt{a^2 c d}}+\frac{2 \log \left(1-\frac{\left(a^2 c+d\right) \cosh \left(2 \coth ^{-1}(a x)\right)}{a^2 c-d}\right)}{a^2 c+d}-\frac{4 \coth ^{-1}(a x) \sinh \left(2 \coth ^{-1}(a x)\right)}{\left(a^2 c+d\right) \cosh \left(2 \coth ^{-1}(a x)\right)+a^2 (-c)+d}\right)}{8 c}","\frac{a \log \left(1-a^2 x^2\right)}{4 c \left(a^2 c+d\right)}-\frac{a \log \left(c+d x^2\right)}{4 c \left(a^2 c+d\right)}+\frac{i \text{Li}_2\left(\frac{a \left(\sqrt{c}-i \sqrt{d} x\right)}{a \sqrt{c}-i \sqrt{d}}\right)}{8 c^{3/2} \sqrt{d}}-\frac{i \text{Li}_2\left(\frac{a \left(\sqrt{c}-i \sqrt{d} x\right)}{\sqrt{c} a+i \sqrt{d}}\right)}{8 c^{3/2} \sqrt{d}}+\frac{i \text{Li}_2\left(\frac{a \left(i \sqrt{d} x+\sqrt{c}\right)}{a \sqrt{c}-i \sqrt{d}}\right)}{8 c^{3/2} \sqrt{d}}-\frac{i \text{Li}_2\left(\frac{a \left(i \sqrt{d} x+\sqrt{c}\right)}{\sqrt{c} a+i \sqrt{d}}\right)}{8 c^{3/2} \sqrt{d}}+\frac{i \log \left(1-\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(\frac{\sqrt{d} (1-a x)}{\sqrt{d}+i a \sqrt{c}}\right)}{8 c^{3/2} \sqrt{d}}-\frac{i \log \left(1-\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(-\frac{\sqrt{d} (a x+1)}{-\sqrt{d}+i a \sqrt{c}}\right)}{8 c^{3/2} \sqrt{d}}-\frac{i \log \left(1+\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(-\frac{\sqrt{d} (1-a x)}{-\sqrt{d}+i a \sqrt{c}}\right)}{8 c^{3/2} \sqrt{d}}+\frac{i \log \left(1+\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(\frac{\sqrt{d} (a x+1)}{\sqrt{d}+i a \sqrt{c}}\right)}{8 c^{3/2} \sqrt{d}}+\frac{\coth ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{2 c^{3/2} \sqrt{d}}+\frac{x \coth ^{-1}(a x)}{2 c \left(c+d x^2\right)}",1,"-1/8*(a*((2*Log[1 - ((a^2*c + d)*Cosh[2*ArcCoth[a*x]])/(a^2*c - d)])/(a^2*c + d) + ((2*I)*ArcCos[(a^2*c - d)/(a^2*c + d)]*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] - 4*ArcCoth[a*x]*ArcTan[(a*d*x)/Sqrt[a^2*c*d]] + (ArcCos[(a^2*c - d)/(a^2*c + d)] + 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[(2*d*(a^2*c - I*Sqrt[a^2*c*d])*(-1 + a*x))/((a^2*c + d)*(I*Sqrt[a^2*c*d] + a*d*x))] + (ArcCos[(a^2*c - d)/(a^2*c + d)] - 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[(2*d*(a^2*c + I*Sqrt[a^2*c*d])*(1 + a*x))/((a^2*c + d)*(I*Sqrt[a^2*c*d] + a*d*x))] - (ArcCos[(a^2*c - d)/(a^2*c + d)] + 2*(ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d])/(Sqrt[a^2*c + d]*E^ArcCoth[a*x]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]]])] - (ArcCos[(a^2*c - d)/(a^2*c + d)] - 2*(ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d]*E^ArcCoth[a*x])/(Sqrt[a^2*c + d]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]]])] + I*(PolyLog[2, ((a^2*c - d - (2*I)*Sqrt[a^2*c*d])*(Sqrt[a^2*c*d] + I*a*d*x))/((a^2*c + d)*(Sqrt[a^2*c*d] - I*a*d*x))] - PolyLog[2, ((a^2*c - d + (2*I)*Sqrt[a^2*c*d])*(Sqrt[a^2*c*d] + I*a*d*x))/((a^2*c + d)*(Sqrt[a^2*c*d] - I*a*d*x))]))/Sqrt[a^2*c*d] - (4*ArcCoth[a*x]*Sinh[2*ArcCoth[a*x]])/(-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]])))/c","A",0
41,1,1846,657,12.9470094,"\int \frac{\coth ^{-1}(a x)}{\left(c+d x^2\right)^3} \, dx","Integrate[ArcCoth[a*x]/(c + d*x^2)^3,x]","a^5 \left(-\frac{3 d \log \left(1-\frac{\left(-c a^2-d\right) \cosh \left(2 \coth ^{-1}(a x)\right)}{d-a^2 c}\right)}{16 a^4 c^2 \left(c a^2+d\right)^2}-\frac{5 \log \left(1-\frac{\left(-c a^2-d\right) \cosh \left(2 \coth ^{-1}(a x)\right)}{d-a^2 c}\right)}{16 a^2 c \left(c a^2+d\right)^2}+\frac{3 d \left(-2 i \cos ^{-1}\left(\frac{a^2 c-d}{c a^2+d}\right) \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right)+4 \coth ^{-1}(a x) \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)-\left(\cos ^{-1}\left(\frac{a^2 c-d}{c a^2+d}\right)-2 \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right)\right) \log \left(1-\frac{\left(-c a^2+d-2 i \sqrt{a^2 c d}\right) \left(2 d-\frac{2 i \sqrt{a^2 c d}}{a x}\right)}{\left(c a^2+d\right) \left(2 d+\frac{2 i \sqrt{a^2 c d}}{a x}\right)}\right)+\left(-\cos ^{-1}\left(\frac{a^2 c-d}{c a^2+d}\right)-2 \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right)\right) \log \left(1-\frac{\left(-c a^2+d+2 i \sqrt{a^2 c d}\right) \left(2 d-\frac{2 i \sqrt{a^2 c d}}{a x}\right)}{\left(c a^2+d\right) \left(2 d+\frac{2 i \sqrt{a^2 c d}}{a x}\right)}\right)+\left(\cos ^{-1}\left(\frac{a^2 c-d}{c a^2+d}\right)+2 i \left(-i \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right)-i \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{-\coth ^{-1}(a x)}}{\sqrt{c a^2+d} \sqrt{-c a^2+d+\left(c a^2+d\right) \cosh \left(2 \coth ^{-1}(a x)\right)}}\right)+\left(\cos ^{-1}\left(\frac{a^2 c-d}{c a^2+d}\right)-2 i \left(-i \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right)-i \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{\coth ^{-1}(a x)}}{\sqrt{c a^2+d} \sqrt{-c a^2+d+\left(c a^2+d\right) \cosh \left(2 \coth ^{-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(2 d-\frac{2 i \sqrt{a^2 c d}}{a x}\right)}{\left(c a^2+d\right) \left(2 d+\frac{2 i \sqrt{a^2 c d}}{a x}\right)}\right)-\text{Li}_2\left(\frac{\left(-c a^2+d+2 i \sqrt{a^2 c d}\right) \left(2 d-\frac{2 i \sqrt{a^2 c d}}{a x}\right)}{\left(c a^2+d\right) \left(2 d+\frac{2 i \sqrt{a^2 c d}}{a x}\right)}\right)\right)\right)}{32 a^4 c^2 \sqrt{a^2 c d} \left(c a^2+d\right)}+\frac{3 \left(-2 i \cos ^{-1}\left(\frac{a^2 c-d}{c a^2+d}\right) \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right)+4 \coth ^{-1}(a x) \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)-\left(\cos ^{-1}\left(\frac{a^2 c-d}{c a^2+d}\right)-2 \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right)\right) \log \left(1-\frac{\left(-c a^2+d-2 i \sqrt{a^2 c d}\right) \left(2 d-\frac{2 i \sqrt{a^2 c d}}{a x}\right)}{\left(c a^2+d\right) \left(2 d+\frac{2 i \sqrt{a^2 c d}}{a x}\right)}\right)+\left(-\cos ^{-1}\left(\frac{a^2 c-d}{c a^2+d}\right)-2 \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right)\right) \log \left(1-\frac{\left(-c a^2+d+2 i \sqrt{a^2 c d}\right) \left(2 d-\frac{2 i \sqrt{a^2 c d}}{a x}\right)}{\left(c a^2+d\right) \left(2 d+\frac{2 i \sqrt{a^2 c d}}{a x}\right)}\right)+\left(\cos ^{-1}\left(\frac{a^2 c-d}{c a^2+d}\right)+2 i \left(-i \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right)-i \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{-\coth ^{-1}(a x)}}{\sqrt{c a^2+d} \sqrt{-c a^2+d+\left(c a^2+d\right) \cosh \left(2 \coth ^{-1}(a x)\right)}}\right)+\left(\cos ^{-1}\left(\frac{a^2 c-d}{c a^2+d}\right)-2 i \left(-i \tan ^{-1}\left(\frac{a c}{\sqrt{a^2 c d} x}\right)-i \tan ^{-1}\left(\frac{a d x}{\sqrt{a^2 c d}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{a^2 c d} e^{\coth ^{-1}(a x)}}{\sqrt{c a^2+d} \sqrt{-c a^2+d+\left(c a^2+d\right) \cosh \left(2 \coth ^{-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(2 d-\frac{2 i \sqrt{a^2 c d}}{a x}\right)}{\left(c a^2+d\right) \left(2 d+\frac{2 i \sqrt{a^2 c d}}{a x}\right)}\right)-\text{Li}_2\left(\frac{\left(-c a^2+d+2 i \sqrt{a^2 c d}\right) \left(2 d-\frac{2 i \sqrt{a^2 c d}}{a x}\right)}{\left(c a^2+d\right) \left(2 d+\frac{2 i \sqrt{a^2 c d}}{a x}\right)}\right)\right)\right)}{32 a^2 c \sqrt{a^2 c d} \left(c a^2+d\right)}-\frac{d \coth ^{-1}(a x) \sinh \left(2 \coth ^{-1}(a x)\right)}{2 a^2 c \left(c a^2+d\right) \left(-c a^2+c \cosh \left(2 \coth ^{-1}(a x)\right) a^2+d+d \cosh \left(2 \coth ^{-1}(a x)\right)\right)^2}-\frac{-5 c^2 \coth ^{-1}(a x) \sinh \left(2 \coth ^{-1}(a x)\right) a^4+2 c d a^2-8 c d \coth ^{-1}(a x) \sinh \left(2 \coth ^{-1}(a x)\right) a^2-3 d^2 \coth ^{-1}(a x) \sinh \left(2 \coth ^{-1}(a x)\right)}{8 a^4 c^2 \left(c a^2+d\right)^2 \left(-c a^2+c \cosh \left(2 \coth ^{-1}(a x)\right) a^2+d+d \cosh \left(2 \coth ^{-1}(a x)\right)\right)}\right)","\frac{a \left(5 a^2 c+3 d\right) \log \left(1-a^2 x^2\right)}{16 c^2 \left(a^2 c+d\right)^2}-\frac{a \left(5 a^2 c+3 d\right) \log \left(c+d x^2\right)}{16 c^2 \left(a^2 c+d\right)^2}+\frac{a}{8 c \left(a^2 c+d\right) \left(c+d x^2\right)}+\frac{3 i \text{Li}_2\left(\frac{a \left(\sqrt{c}-i \sqrt{d} x\right)}{a \sqrt{c}-i \sqrt{d}}\right)}{32 c^{5/2} \sqrt{d}}-\frac{3 i \text{Li}_2\left(\frac{a \left(\sqrt{c}-i \sqrt{d} x\right)}{\sqrt{c} a+i \sqrt{d}}\right)}{32 c^{5/2} \sqrt{d}}+\frac{3 i \text{Li}_2\left(\frac{a \left(i \sqrt{d} x+\sqrt{c}\right)}{a \sqrt{c}-i \sqrt{d}}\right)}{32 c^{5/2} \sqrt{d}}-\frac{3 i \text{Li}_2\left(\frac{a \left(i \sqrt{d} x+\sqrt{c}\right)}{\sqrt{c} a+i \sqrt{d}}\right)}{32 c^{5/2} \sqrt{d}}+\frac{3 i \log \left(1-\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(\frac{\sqrt{d} (1-a x)}{\sqrt{d}+i a \sqrt{c}}\right)}{32 c^{5/2} \sqrt{d}}-\frac{3 i \log \left(1-\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(-\frac{\sqrt{d} (a x+1)}{-\sqrt{d}+i a \sqrt{c}}\right)}{32 c^{5/2} \sqrt{d}}-\frac{3 i \log \left(1+\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(-\frac{\sqrt{d} (1-a x)}{-\sqrt{d}+i a \sqrt{c}}\right)}{32 c^{5/2} \sqrt{d}}+\frac{3 i \log \left(1+\frac{i \sqrt{d} x}{\sqrt{c}}\right) \log \left(\frac{\sqrt{d} (a x+1)}{\sqrt{d}+i a \sqrt{c}}\right)}{32 c^{5/2} \sqrt{d}}+\frac{3 \coth ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{8 c^{5/2} \sqrt{d}}+\frac{3 x \coth ^{-1}(a x)}{8 c^2 \left(c+d x^2\right)}+\frac{x \coth ^{-1}(a x)}{4 c \left(c+d x^2\right)^2}",1,"a^5*((-5*Log[1 - ((-(a^2*c) - d)*Cosh[2*ArcCoth[a*x]])/(-(a^2*c) + d)])/(16*a^2*c*(a^2*c + d)^2) - (3*d*Log[1 - ((-(a^2*c) - d)*Cosh[2*ArcCoth[a*x]])/(-(a^2*c) + d)])/(16*a^4*c^2*(a^2*c + d)^2) + (3*((-2*I)*ArcCos[(a^2*c - d)/(a^2*c + d)]*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + 4*ArcCoth[a*x]*ArcTan[(a*d*x)/Sqrt[a^2*c*d]] - (ArcCos[(a^2*c - d)/(a^2*c + d)] - 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[1 - ((-(a^2*c) + d - (2*I)*Sqrt[a^2*c*d])*(2*d - ((2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/(a*x)))] + (-ArcCos[(a^2*c - d)/(a^2*c + d)] - 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[1 - ((-(a^2*c) + d + (2*I)*Sqrt[a^2*c*d])*(2*d - ((2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/(a*x)))] + (ArcCos[(a^2*c - d)/(a^2*c + d)] + (2*I)*((-I)*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] - I*ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d])/(Sqrt[a^2*c + d]*E^ArcCoth[a*x]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]]])] + (ArcCos[(a^2*c - d)/(a^2*c + d)] - (2*I)*((-I)*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] - I*ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d]*E^ArcCoth[a*x])/(Sqrt[a^2*c + d]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]]])] + I*(PolyLog[2, ((-(a^2*c) + d - (2*I)*Sqrt[a^2*c*d])*(2*d - ((2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/(a*x)))] - PolyLog[2, ((-(a^2*c) + d + (2*I)*Sqrt[a^2*c*d])*(2*d - ((2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/(a*x)))])))/(32*a^2*c*Sqrt[a^2*c*d]*(a^2*c + d)) + (3*d*((-2*I)*ArcCos[(a^2*c - d)/(a^2*c + d)]*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] + 4*ArcCoth[a*x]*ArcTan[(a*d*x)/Sqrt[a^2*c*d]] - (ArcCos[(a^2*c - d)/(a^2*c + d)] - 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[1 - ((-(a^2*c) + d - (2*I)*Sqrt[a^2*c*d])*(2*d - ((2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/(a*x)))] + (-ArcCos[(a^2*c - d)/(a^2*c + d)] - 2*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)])*Log[1 - ((-(a^2*c) + d + (2*I)*Sqrt[a^2*c*d])*(2*d - ((2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/(a*x)))] + (ArcCos[(a^2*c - d)/(a^2*c + d)] + (2*I)*((-I)*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] - I*ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d])/(Sqrt[a^2*c + d]*E^ArcCoth[a*x]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]]])] + (ArcCos[(a^2*c - d)/(a^2*c + d)] - (2*I)*((-I)*ArcTan[(a*c)/(Sqrt[a^2*c*d]*x)] - I*ArcTan[(a*d*x)/Sqrt[a^2*c*d]]))*Log[(Sqrt[2]*Sqrt[a^2*c*d]*E^ArcCoth[a*x])/(Sqrt[a^2*c + d]*Sqrt[-(a^2*c) + d + (a^2*c + d)*Cosh[2*ArcCoth[a*x]]])] + I*(PolyLog[2, ((-(a^2*c) + d - (2*I)*Sqrt[a^2*c*d])*(2*d - ((2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/(a*x)))] - PolyLog[2, ((-(a^2*c) + d + (2*I)*Sqrt[a^2*c*d])*(2*d - ((2*I)*Sqrt[a^2*c*d])/(a*x)))/((a^2*c + d)*(2*d + ((2*I)*Sqrt[a^2*c*d])/(a*x)))])))/(32*a^4*c^2*Sqrt[a^2*c*d]*(a^2*c + d)) - (d*ArcCoth[a*x]*Sinh[2*ArcCoth[a*x]])/(2*a^2*c*(a^2*c + d)*(-(a^2*c) + d + a^2*c*Cosh[2*ArcCoth[a*x]] + d*Cosh[2*ArcCoth[a*x]])^2) - (2*a^2*c*d - 5*a^4*c^2*ArcCoth[a*x]*Sinh[2*ArcCoth[a*x]] - 8*a^2*c*d*ArcCoth[a*x]*Sinh[2*ArcCoth[a*x]] - 3*d^2*ArcCoth[a*x]*Sinh[2*ArcCoth[a*x]])/(8*a^4*c^2*(a^2*c + d)^2*(-(a^2*c) + d + a^2*c*Cosh[2*ArcCoth[a*x]] + d*Cosh[2*ArcCoth[a*x]])))","B",0
42,0,0,19,26.7415387,"\int \sqrt{c+d x^2} \coth ^{-1}(a x) \, dx","Integrate[Sqrt[c + d*x^2]*ArcCoth[a*x],x]","\int \sqrt{c+d x^2} \coth ^{-1}(a x) \, dx","\text{Int}\left(\coth ^{-1}(a x) \sqrt{c+d x^2},x\right)",0,"Integrate[Sqrt[c + d*x^2]*ArcCoth[a*x], x]","A",-1
43,0,0,19,4.3837072,"\int \frac{\coth ^{-1}(a x)}{\sqrt{c+d x^2}} \, dx","Integrate[ArcCoth[a*x]/Sqrt[c + d*x^2],x]","\int \frac{\coth ^{-1}(a x)}{\sqrt{c+d x^2}} \, dx","\text{Int}\left(\frac{\coth ^{-1}(a x)}{\sqrt{c+d x^2}},x\right)",0,"Integrate[ArcCoth[a*x]/Sqrt[c + d*x^2], x]","A",-1
44,1,119,62,0.111787,"\int \frac{\coth ^{-1}(a x)}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[ArcCoth[a*x]/(c + d*x^2)^(3/2),x]","\frac{\frac{-\log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c-d x\right)-\log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c+d x\right)+\log (1-a x)+\log (a x+1)}{\sqrt{a^2 c+d}}+\frac{2 x \coth ^{-1}(a x)}{\sqrt{c+d x^2}}}{2 c}","\frac{x \coth ^{-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*ArcCoth[a*x])/Sqrt[c + d*x^2] + (Log[1 - a*x] + Log[1 + a*x] - Log[a*c - d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]] - Log[a*c + d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]])/Sqrt[a^2*c + d])/(2*c)","A",1
45,1,226,128,0.2970418,"\int \frac{\coth ^{-1}(a x)}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[ArcCoth[a*x]/(c + d*x^2)^(5/2),x]","\frac{\frac{2 a c}{\left(a^2 c+d\right) \sqrt{c+d x^2}}-\frac{\left(3 a^2 c+2 d\right) \log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c-d x\right)}{\left(a^2 c+d\right)^{3/2}}-\frac{\left(3 a^2 c+2 d\right) \log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c+d x\right)}{\left(a^2 c+d\right)^{3/2}}+\frac{\left(3 a^2 c+2 d\right) \log (1-a x)}{\left(a^2 c+d\right)^{3/2}}+\frac{\left(3 a^2 c+2 d\right) \log (a x+1)}{\left(a^2 c+d\right)^{3/2}}+\frac{2 x \coth ^{-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 \coth ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{x \coth ^{-1}(a x)}{3 c \left(c+d x^2\right)^{3/2}}",1,"((2*a*c)/((a^2*c + d)*Sqrt[c + d*x^2]) + (2*x*(3*c + 2*d*x^2)*ArcCoth[a*x])/(c + d*x^2)^(3/2) + ((3*a^2*c + 2*d)*Log[1 - a*x])/(a^2*c + d)^(3/2) + ((3*a^2*c + 2*d)*Log[1 + a*x])/(a^2*c + d)^(3/2) - ((3*a^2*c + 2*d)*Log[a*c - d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]])/(a^2*c + d)^(3/2) - ((3*a^2*c + 2*d)*Log[a*c + d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]])/(a^2*c + d)^(3/2))/(6*c^2)","A",1
46,1,329,200,0.566723,"\int \frac{\coth ^{-1}(a x)}{\left(c+d x^2\right)^{7/2}} \, dx","Integrate[ArcCoth[a*x]/(c + d*x^2)^(7/2),x]","\frac{2 x \left(a^2 c+d\right)^{5/2} \coth ^{-1}(a x) \left(15 c^2+20 c d x^2+8 d^2 x^4\right)+2 a c \sqrt{a^2 c+d} \left(c+d x^2\right) \left(a^2 c \left(8 c+7 d x^2\right)+d \left(5 c+4 d x^2\right)\right)+\left(15 a^4 c^2+20 a^2 c d+8 d^2\right) \log (1-a x) \left(c+d x^2\right)^{5/2}+\left(15 a^4 c^2+20 a^2 c d+8 d^2\right) \log (a x+1) \left(c+d x^2\right)^{5/2}-\left(15 a^4 c^2+20 a^2 c d+8 d^2\right) \left(c+d x^2\right)^{5/2} \log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c-d x\right)-\left(15 a^4 c^2+20 a^2 c d+8 d^2\right) \left(c+d x^2\right)^{5/2} \log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c+d x\right)}{30 c^3 \left(a^2 c+d\right)^{5/2} \left(c+d x^2\right)^{5/2}}","\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 \coth ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}+\frac{4 x \coth ^{-1}(a x)}{15 c^2 \left(c+d x^2\right)^{3/2}}+\frac{x \coth ^{-1}(a x)}{5 c \left(c+d x^2\right)^{5/2}}",1,"(2*a*c*Sqrt[a^2*c + d]*(c + d*x^2)*(d*(5*c + 4*d*x^2) + a^2*c*(8*c + 7*d*x^2)) + 2*(a^2*c + d)^(5/2)*x*(15*c^2 + 20*c*d*x^2 + 8*d^2*x^4)*ArcCoth[a*x] + (15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*(c + d*x^2)^(5/2)*Log[1 - a*x] + (15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*(c + d*x^2)^(5/2)*Log[1 + a*x] - (15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*(c + d*x^2)^(5/2)*Log[a*c - 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)*(c + d*x^2)^(5/2)*Log[a*c + d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]])/(30*c^3*(a^2*c + d)^(5/2)*(c + d*x^2)^(5/2))","A",1
47,1,431,283,0.9373744,"\int \frac{\coth ^{-1}(a x)}{\left(c+d x^2\right)^{9/2}} \, dx","Integrate[ArcCoth[a*x]/(c + d*x^2)^(9/2),x]","\frac{6 x \left(a^2 c+d\right)^{7/2} \coth ^{-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)+2 a c \sqrt{a^2 c+d} \left(c+d x^2\right) \left(3 c^2 \left(a^2 c+d\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)+3 \left(35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right) \log (1-a x) \left(c+d x^2\right)^{7/2}+3 \left(35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right) \log (a x+1) \left(c+d x^2\right)^{7/2}-3 \left(35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right) \left(c+d x^2\right)^{7/2} \log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c-d x\right)-3 \left(35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right) \left(c+d x^2\right)^{7/2} \log \left(\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c+d x\right)}{210 c^4 \left(a^2 c+d\right)^{7/2} \left(c+d x^2\right)^{7/2}}","\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 \coth ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \coth ^{-1}(a x)}{35 c^3 \left(c+d x^2\right)^{3/2}}+\frac{6 x \coth ^{-1}(a x)}{35 c^2 \left(c+d x^2\right)^{5/2}}+\frac{x \coth ^{-1}(a x)}{7 c \left(c+d x^2\right)^{7/2}}",1,"(2*a*c*Sqrt[a^2*c + d]*(c + d*x^2)*(3*c^2*(a^2*c + d)^2 + c*(a^2*c + d)*(11*a^2*c + 6*d)*(c + d*x^2) + 3*(19*a^4*c^2 + 22*a^2*c*d + 8*d^2)*(c + d*x^2)^2) + 6*(a^2*c + d)^(7/2)*x*(35*c^3 + 70*c^2*d*x^2 + 56*c*d^2*x^4 + 16*d^3*x^6)*ArcCoth[a*x] + 3*(35*a^6*c^3 + 70*a^4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*(c + d*x^2)^(7/2)*Log[1 - a*x] + 3*(35*a^6*c^3 + 70*a^4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*(c + d*x^2)^(7/2)*Log[1 + a*x] - 3*(35*a^6*c^3 + 70*a^4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*(c + d*x^2)^(7/2)*Log[a*c - d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]] - 3*(35*a^6*c^3 + 70*a^4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*(c + d*x^2)^(7/2)*Log[a*c + d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]])/(210*c^4*(a^2*c + d)^(7/2)*(c + d*x^2)^(7/2))","A",1
48,1,125,186,0.970056,"\int \sqrt{a-a x^2} \coth ^{-1}(x) \, dx","Integrate[Sqrt[a - a*x^2]*ArcCoth[x],x]","-\frac{\sqrt{a-a x^2} \left(-4 \text{Li}_2\left(-e^{-\coth ^{-1}(x)}\right)+4 \text{Li}_2\left(e^{-\coth ^{-1}(x)}\right)-2 \coth \left(\frac{1}{2} \coth ^{-1}(x)\right)-4 \coth ^{-1}(x) \log \left(1-e^{-\coth ^{-1}(x)}\right)+4 \coth ^{-1}(x) \log \left(e^{-\coth ^{-1}(x)}+1\right)+2 \tanh \left(\frac{1}{2} \coth ^{-1}(x)\right)-\coth ^{-1}(x) \text{csch}^2\left(\frac{1}{2} \coth ^{-1}(x)\right)-\coth ^{-1}(x) \text{sech}^2\left(\frac{1}{2} \coth ^{-1}(x)\right)\right)}{8 \sqrt{1-\frac{1}{x^2}} x}","-\frac{i a \sqrt{1-x^2} \text{Li}_2\left(-\frac{i \sqrt{1-x}}{\sqrt{x+1}}\right)}{2 \sqrt{a-a x^2}}+\frac{i a \sqrt{1-x^2} \text{Li}_2\left(\frac{i \sqrt{1-x}}{\sqrt{x+1}}\right)}{2 \sqrt{a-a x^2}}+\frac{1}{2} \sqrt{a-a x^2}+\frac{1}{2} x \sqrt{a-a x^2} \coth ^{-1}(x)-\frac{a \sqrt{1-x^2} \tan ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{x+1}}\right) \coth ^{-1}(x)}{\sqrt{a-a x^2}}",1,"-1/8*(Sqrt[a - a*x^2]*(-2*Coth[ArcCoth[x]/2] - ArcCoth[x]*Csch[ArcCoth[x]/2]^2 - 4*ArcCoth[x]*Log[1 - E^(-ArcCoth[x])] + 4*ArcCoth[x]*Log[1 + E^(-ArcCoth[x])] - 4*PolyLog[2, -E^(-ArcCoth[x])] + 4*PolyLog[2, E^(-ArcCoth[x])] - ArcCoth[x]*Sech[ArcCoth[x]/2]^2 + 2*Tanh[ArcCoth[x]/2]))/(Sqrt[1 - x^(-2)]*x)","A",0
49,1,77,144,0.118601,"\int \frac{\coth ^{-1}(x)}{\sqrt{a-a x^2}} \, dx","Integrate[ArcCoth[x]/Sqrt[a - a*x^2],x]","\frac{\sqrt{a-a x^2} \left(\text{Li}_2\left(-e^{-\coth ^{-1}(x)}\right)-\text{Li}_2\left(e^{-\coth ^{-1}(x)}\right)+\coth ^{-1}(x) \left(\log \left(1-e^{-\coth ^{-1}(x)}\right)-\log \left(e^{-\coth ^{-1}(x)}+1\right)\right)\right)}{a \sqrt{1-\frac{1}{x^2}} x}","-\frac{i \sqrt{1-x^2} \text{Li}_2\left(-\frac{i \sqrt{1-x}}{\sqrt{x+1}}\right)}{\sqrt{a-a x^2}}+\frac{i \sqrt{1-x^2} \text{Li}_2\left(\frac{i \sqrt{1-x}}{\sqrt{x+1}}\right)}{\sqrt{a-a x^2}}-\frac{2 \sqrt{1-x^2} \tan ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{x+1}}\right) \coth ^{-1}(x)}{\sqrt{a-a x^2}}",1,"(Sqrt[a - a*x^2]*(ArcCoth[x]*(Log[1 - E^(-ArcCoth[x])] - Log[1 + E^(-ArcCoth[x])]) + PolyLog[2, -E^(-ArcCoth[x])] - PolyLog[2, E^(-ArcCoth[x])]))/(a*Sqrt[1 - x^(-2)]*x)","A",0
50,1,30,37,0.0465847,"\int \frac{\coth ^{-1}(x)}{\left(a-a x^2\right)^{3/2}} \, dx","Integrate[ArcCoth[x]/(a - a*x^2)^(3/2),x]","\frac{\sqrt{a-a x^2} \left(1-x \coth ^{-1}(x)\right)}{a^2 \left(x^2-1\right)}","\frac{x \coth ^{-1}(x)}{a \sqrt{a-a x^2}}-\frac{1}{a \sqrt{a-a x^2}}",1,"(Sqrt[a - a*x^2]*(1 - x*ArcCoth[x]))/(a^2*(-1 + x^2))","A",1
51,1,45,83,0.0520613,"\int \frac{\coth ^{-1}(x)}{\left(a-a x^2\right)^{5/2}} \, dx","Integrate[ArcCoth[x]/(a - a*x^2)^(5/2),x]","-\frac{\sqrt{a-a x^2} \left(\left(6 x^3-9 x\right) \coth ^{-1}(x)-6 x^2+7\right)}{9 a^3 \left(x^2-1\right)^2}","-\frac{2}{3 a^2 \sqrt{a-a x^2}}+\frac{2 x \coth ^{-1}(x)}{3 a^2 \sqrt{a-a x^2}}-\frac{1}{9 a \left(a-a x^2\right)^{3/2}}+\frac{x \coth ^{-1}(x)}{3 a \left(a-a x^2\right)^{3/2}}",1,"-1/9*(Sqrt[a - a*x^2]*(7 - 6*x^2 + (-9*x + 6*x^3)*ArcCoth[x]))/(a^3*(-1 + x^2)^2)","A",1
52,1,55,124,0.0626545,"\int \frac{\coth ^{-1}(x)}{\left(a-a x^2\right)^{7/2}} \, dx","Integrate[ArcCoth[x]/(a - a*x^2)^(7/2),x]","\frac{\sqrt{a-a x^2} \left(120 x^4-260 x^2-15 \left(8 x^4-20 x^2+15\right) x \coth ^{-1}(x)+149\right)}{225 a^4 \left(x^2-1\right)^3}","-\frac{8}{15 a^3 \sqrt{a-a x^2}}+\frac{8 x \coth ^{-1}(x)}{15 a^3 \sqrt{a-a x^2}}-\frac{4}{45 a^2 \left(a-a x^2\right)^{3/2}}+\frac{4 x \coth ^{-1}(x)}{15 a^2 \left(a-a x^2\right)^{3/2}}-\frac{1}{25 a \left(a-a x^2\right)^{5/2}}+\frac{x \coth ^{-1}(x)}{5 a \left(a-a x^2\right)^{5/2}}",1,"(Sqrt[a - a*x^2]*(149 - 260*x^2 + 120*x^4 - 15*x*(15 - 20*x^2 + 8*x^4)*ArcCoth[x]))/(225*a^4*(-1 + x^2)^3)","A",1
53,1,3,3,0.025442,"\int \frac{1}{\left(1-x^2\right) \coth ^{-1}(x)} \, dx","Integrate[1/((1 - x^2)*ArcCoth[x]),x]","\log \left(\coth ^{-1}(x)\right)","\log \left(\coth ^{-1}(x)\right)",1,"Log[ArcCoth[x]]","A",1
54,1,12,12,0.0091785,"\int \frac{\coth ^{-1}(x)^n}{1-x^2} \, dx","Integrate[ArcCoth[x]^n/(1 - x^2),x]","\frac{\coth ^{-1}(x)^{n+1}}{n+1}","\frac{\coth ^{-1}(x)^{n+1}}{n+1}",1,"ArcCoth[x]^(1 + n)/(1 + n)","A",1
55,1,61,62,0.0660651,"\int \frac{\coth ^{-1}(x)^2}{\left(1-x^2\right)^2} \, dx","Integrate[ArcCoth[x]^2/(1 - x^2)^2,x]","\frac{-3 \left(x^2-1\right) \log (1-x)+3 \left(x^2-1\right) \log (x+1)+4 \left(x^2-1\right) \coth ^{-1}(x)^3-6 x-12 x \coth ^{-1}(x)^2+12 \coth ^{-1}(x)}{24 \left(x^2-1\right)}","\frac{x}{4 \left(1-x^2\right)}+\frac{x \coth ^{-1}(x)^2}{2 \left(1-x^2\right)}-\frac{\coth ^{-1}(x)}{2 \left(1-x^2\right)}+\frac{1}{4} \tanh ^{-1}(x)+\frac{1}{6} \coth ^{-1}(x)^3",1,"(-6*x + 12*ArcCoth[x] - 12*x*ArcCoth[x]^2 + 4*(-1 + x^2)*ArcCoth[x]^3 - 3*(-1 + x^2)*Log[1 - x] + 3*(-1 + x^2)*Log[1 + x])/(24*(-1 + x^2))","A",1
56,1,34,37,0.0506053,"\int \frac{x \coth ^{-1}(x)}{1-x^2} \, dx","Integrate[(x*ArcCoth[x])/(1 - x^2),x]","\frac{1}{2} \left(\coth ^{-1}(x) \left(\coth ^{-1}(x)+2 \log \left(1-e^{-2 \coth ^{-1}(x)}\right)\right)-\text{Li}_2\left(e^{-2 \coth ^{-1}(x)}\right)\right)","\frac{1}{2} \text{Li}_2\left(\frac{x+1}{x-1}\right)-\frac{1}{2} \coth ^{-1}(x)^2+\log \left(\frac{2}{1-x}\right) \coth ^{-1}(x)",1,"(ArcCoth[x]*(ArcCoth[x] + 2*Log[1 - E^(-2*ArcCoth[x])]) - PolyLog[2, E^(-2*ArcCoth[x])])/2","A",0
57,1,8,8,0.0040185,"\int \frac{\coth ^{-1}(x)}{1-x^2} \, dx","Integrate[ArcCoth[x]/(1 - x^2),x]","\frac{1}{2} \coth ^{-1}(x)^2","\frac{1}{2} \coth ^{-1}(x)^2",1,"ArcCoth[x]^2/2","A",1
58,1,44,36,0.027542,"\int \frac{x \coth ^{-1}(x)}{\left(1-x^2\right)^2} \, dx","Integrate[(x*ArcCoth[x])/(1 - x^2)^2,x]","\frac{x}{4 \left(x^2-1\right)}-\frac{\coth ^{-1}(x)}{2 \left(x^2-1\right)}+\frac{1}{8} \log (1-x)-\frac{1}{8} \log (x+1)","-\frac{x}{4 \left(1-x^2\right)}+\frac{\coth ^{-1}(x)}{2 \left(1-x^2\right)}-\frac{1}{4} \tanh ^{-1}(x)",1,"x/(4*(-1 + x^2)) - ArcCoth[x]/(2*(-1 + x^2)) + Log[1 - x]/8 - Log[1 + x]/8","A",1
59,1,28,38,0.0325329,"\int \frac{\coth ^{-1}(x)}{\left(1-x^2\right)^2} \, dx","Integrate[ArcCoth[x]/(1 - x^2)^2,x]","\frac{\left(x^2-1\right) \coth ^{-1}(x)^2-2 x \coth ^{-1}(x)+1}{4 \left(x^2-1\right)}","-\frac{1}{4 \left(1-x^2\right)}+\frac{x \coth ^{-1}(x)}{2 \left(1-x^2\right)}+\frac{1}{4} \coth ^{-1}(x)^2",1,"(1 - 2*x*ArcCoth[x] + (-1 + x^2)*ArcCoth[x]^2)/(4*(-1 + x^2))","A",1
60,1,50,50,0.0527834,"\int \frac{x \coth ^{-1}(x)}{\left(1-x^2\right)^3} \, dx","Integrate[(x*ArcCoth[x])/(1 - x^2)^3,x]","\frac{1}{64} \left(\frac{6 x}{x^2-1}-\frac{4 x}{\left(x^2-1\right)^2}+\frac{16 \coth ^{-1}(x)}{\left(x^2-1\right)^2}+3 \log (1-x)-3 \log (x+1)\right)","-\frac{3 x}{32 \left(1-x^2\right)}-\frac{x}{16 \left(1-x^2\right)^2}+\frac{\coth ^{-1}(x)}{4 \left(1-x^2\right)^2}-\frac{3}{32} \tanh ^{-1}(x)",1,"((-4*x)/(-1 + x^2)^2 + (6*x)/(-1 + x^2) + (16*ArcCoth[x])/(-1 + x^2)^2 + 3*Log[1 - x] - 3*Log[1 + x])/64","A",1
61,1,43,67,0.0553888,"\int \frac{\coth ^{-1}(x)}{\left(1-x^2\right)^3} \, dx","Integrate[ArcCoth[x]/(1 - x^2)^3,x]","-\frac{-3 x^2+2 \left(3 x^2-5\right) x \coth ^{-1}(x)-3 \left(x^2-1\right)^2 \coth ^{-1}(x)^2+4}{16 \left(x^2-1\right)^2}","-\frac{3}{16 \left(1-x^2\right)}-\frac{1}{16 \left(1-x^2\right)^2}+\frac{3 x \coth ^{-1}(x)}{8 \left(1-x^2\right)}+\frac{x \coth ^{-1}(x)}{4 \left(1-x^2\right)^2}+\frac{3}{16} \coth ^{-1}(x)^2",1,"-1/16*(4 - 3*x^2 + 2*x*(-5 + 3*x^2)*ArcCoth[x] - 3*(-1 + x^2)^2*ArcCoth[x]^2)/(-1 + x^2)^2","A",1
62,1,81,101,0.0421286,"\int x^3 \coth ^{-1}(a+b x) \, dx","Integrate[x^3*ArcCoth[a + b*x],x]","\frac{6 \left(3 a^2+1\right) b x+6 b^4 x^4 \coth ^{-1}(a+b x)-6 a b^2 x^2+3 (a-1)^4 \log (-a-b x+1)-3 (a+1)^4 \log (a+b x+1)+2 b^3 x^3}{24 b^4}","\frac{\left(6 a^2+1\right) x}{4 b^3}+\frac{(a+b x)^3}{12 b^4}-\frac{a (a+b x)^2}{2 b^4}+\frac{(1-a)^4 \log (-a-b x+1)}{8 b^4}-\frac{(a+1)^4 \log (a+b x+1)}{8 b^4}+\frac{1}{4} x^4 \coth ^{-1}(a+b x)",1,"(6*(1 + 3*a^2)*b*x - 6*a*b^2*x^2 + 2*b^3*x^3 + 6*b^4*x^4*ArcCoth[a + b*x] + 3*(-1 + a)^4*Log[1 - a - b*x] - 3*(1 + a)^4*Log[1 + a + b*x])/(24*b^4)","A",1
63,1,92,78,0.0243655,"\int x^2 \coth ^{-1}(a+b x) \, dx","Integrate[x^2*ArcCoth[a + b*x],x]","\frac{\left(-a^3+3 a^2-3 a+1\right) \log (-a-b x+1)}{6 b^3}+\frac{\left(a^3+3 a^2+3 a+1\right) \log (a+b x+1)}{6 b^3}-\frac{2 a x}{3 b^2}+\frac{1}{3} x^3 \coth ^{-1}(a+b x)+\frac{x^2}{6 b}","\frac{(a+b x)^2}{6 b^3}+\frac{(1-a)^3 \log (-a-b x+1)}{6 b^3}+\frac{(a+1)^3 \log (a+b x+1)}{6 b^3}-\frac{a x}{b^2}+\frac{1}{3} x^3 \coth ^{-1}(a+b x)",1,"(-2*a*x)/(3*b^2) + x^2/(6*b) + (x^3*ArcCoth[a + b*x])/3 + ((1 - 3*a + 3*a^2 - a^3)*Log[1 - a - b*x])/(6*b^3) + ((1 + 3*a + 3*a^2 + a^3)*Log[1 + a + b*x])/(6*b^3)","A",1
64,1,56,65,0.023483,"\int x \coth ^{-1}(a+b x) \, dx","Integrate[x*ArcCoth[a + b*x],x]","\frac{2 b^2 x^2 \coth ^{-1}(a+b x)+(a-1)^2 \log (-a-b x+1)-(a+1)^2 \log (a+b x+1)+2 b x}{4 b^2}","\frac{(1-a)^2 \log (-a-b x+1)}{4 b^2}-\frac{(a+1)^2 \log (a+b x+1)}{4 b^2}+\frac{1}{2} x^2 \coth ^{-1}(a+b x)+\frac{x}{2 b}",1,"(2*b*x + 2*b^2*x^2*ArcCoth[a + b*x] + (-1 + a)^2*Log[1 - a - b*x] - (1 + a)^2*Log[1 + a + b*x])/(4*b^2)","A",1
65,1,43,35,0.0146604,"\int \coth ^{-1}(a+b x) \, dx","Integrate[ArcCoth[a + b*x],x]","\frac{(a+1) \log (a+b x+1)-(a-1) \log (-a-b x+1)}{2 b}+x \coth ^{-1}(a+b x)","\frac{\log \left(1-(a+b x)^2\right)}{2 b}+\frac{(a+b x) \coth ^{-1}(a+b x)}{b}",1,"x*ArcCoth[a + b*x] + (-((-1 + a)*Log[1 - a - b*x]) + (1 + a)*Log[1 + a + b*x])/(2*b)","A",1
66,1,259,92,0.1693257,"\int \frac{\coth ^{-1}(a+b x)}{x} \, dx","Integrate[ArcCoth[a + b*x]/x,x]","\frac{1}{8} \left(-4 \text{Li}_2\left(e^{2 \tanh ^{-1}(a)-2 \tanh ^{-1}(a+b x)}\right)-4 \text{Li}_2\left(-e^{2 \tanh ^{-1}(a+b x)}\right)+4 \left(\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)\right)^2-\left(\pi -2 i \tanh ^{-1}(a+b x)\right)^2-8 \left(\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)\right) \log \left(1-e^{2 \tanh ^{-1}(a)-2 \tanh ^{-1}(a+b x)}\right)-4 i \left(\pi -2 i \tanh ^{-1}(a+b x)\right) \log \left(e^{2 \tanh ^{-1}(a+b x)}+1\right)+4 \log \left(\frac{2}{\sqrt{1-(a+b x)^2}}\right) \left(2 \tanh ^{-1}(a+b x)+i \pi \right)+8 \left(\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)\right) \log \left(-2 i \sinh \left(\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)\right)\right)\right)+\tanh ^{-1}(a+b x) \left(-\log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right)+\log \left(-i \sinh \left(\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)\right)\right)\right)+\log (x) \left(\coth ^{-1}(a+b x)-\tanh ^{-1}(a+b x)\right)","\frac{1}{2} \text{Li}_2\left(1-\frac{2}{a+b x+1}\right)-\frac{1}{2} \text{Li}_2\left(1-\frac{2 b x}{(1-a) (a+b x+1)}\right)+\log \left(\frac{2}{a+b x+1}\right) \left(-\coth ^{-1}(a+b x)\right)+\log \left(\frac{2 b x}{(1-a) (a+b x+1)}\right) \coth ^{-1}(a+b x)",1,"(ArcCoth[a + b*x] - ArcTanh[a + b*x])*Log[x] + ArcTanh[a + b*x]*(-Log[1/Sqrt[1 - (a + b*x)^2]] + Log[(-I)*Sinh[ArcTanh[a] - ArcTanh[a + b*x]]]) + (4*(ArcTanh[a] - ArcTanh[a + b*x])^2 - (Pi - (2*I)*ArcTanh[a + b*x])^2 - 8*(ArcTanh[a] - ArcTanh[a + b*x])*Log[1 - E^(2*ArcTanh[a] - 2*ArcTanh[a + b*x])] - (4*I)*(Pi - (2*I)*ArcTanh[a + b*x])*Log[1 + E^(2*ArcTanh[a + b*x])] + 4*(I*Pi + 2*ArcTanh[a + b*x])*Log[2/Sqrt[1 - (a + b*x)^2]] + 8*(ArcTanh[a] - ArcTanh[a + b*x])*Log[(-2*I)*Sinh[ArcTanh[a] - ArcTanh[a + b*x]]] - 4*PolyLog[2, E^(2*ArcTanh[a] - 2*ArcTanh[a + b*x])] - 4*PolyLog[2, -E^(2*ArcTanh[a + b*x])])/8","C",0
67,1,55,64,0.0559458,"\int \frac{\coth ^{-1}(a+b x)}{x^2} \, dx","Integrate[ArcCoth[a + b*x]/x^2,x]","\frac{b ((a+1) \log (-a-b x+1)-(a-1) \log (a+b x+1)-2 \log (x))}{2 \left(a^2-1\right)}-\frac{\coth ^{-1}(a+b x)}{x}","\frac{b \log (x)}{1-a^2}-\frac{b \log (-a-b x+1)}{2 (1-a)}-\frac{b \log (a+b x+1)}{2 (a+1)}-\frac{\coth ^{-1}(a+b x)}{x}",1,"-(ArcCoth[a + b*x]/x) + (b*(-2*Log[x] + (1 + a)*Log[1 - a - b*x] - (-1 + a)*Log[1 + a + b*x]))/(2*(-1 + a^2))","A",1
68,1,76,90,0.1173433,"\int \frac{\coth ^{-1}(a+b x)}{x^3} \, dx","Integrate[ArcCoth[a + b*x]/x^3,x]","\frac{1}{4} \left(b \left(\frac{4 a b \log (x)}{\left(a^2-1\right)^2}+\frac{2}{\left(a^2-1\right) x}-\frac{b \log (-a-b x+1)}{(a-1)^2}+\frac{b \log (a+b x+1)}{(a+1)^2}\right)-\frac{2 \coth ^{-1}(a+b x)}{x^2}\right)","\frac{a b^2 \log (x)}{\left(1-a^2\right)^2}-\frac{b}{2 \left(1-a^2\right) x}-\frac{b^2 \log (-a-b x+1)}{4 (1-a)^2}+\frac{b^2 \log (a+b x+1)}{4 (a+1)^2}-\frac{\coth ^{-1}(a+b x)}{2 x^2}",1,"((-2*ArcCoth[a + b*x])/x^2 + b*(2/((-1 + a^2)*x) + (4*a*b*Log[x])/(-1 + a^2)^2 - (b*Log[1 - a - b*x])/(-1 + a)^2 + (b*Log[1 + a + b*x])/(1 + a)^2))/4","A",1
69,1,203,263,1.7027467,"\int x^3 \coth ^{-1}(a+b x)^2 \, dx","Integrate[x^3*ArcCoth[a + b*x]^2,x]","-\frac{12 \left(a^3+a\right) \text{Li}_2\left(e^{-2 \coth ^{-1}(a+b x)}\right)+36 a^2 \log \left(\frac{1}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}\right)+11 a^2-2 \coth ^{-1}(a+b x) \left(12 \left(a^3+a\right) \log \left(1-e^{-2 \coth ^{-1}(a+b x)}\right)+13 a^3+9 a^2 b x-3 a b^2 x^2+9 a+b^3 x^3+3 b x\right)+3 \left(a^4-4 a^3+6 a^2-4 a-b^4 x^4+1\right) \coth ^{-1}(a+b x)^2+10 a b x+8 \log \left(\frac{1}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}\right)-b^2 x^2+1}{12 b^4}","\frac{a \left(a^2+1\right) \text{Li}_2\left(-\frac{a+b x+1}{-a-b x+1}\right)}{b^4}+\frac{\left(6 a^2+1\right) \log \left(1-(a+b x)^2\right)}{4 b^4}-\frac{a \left(a^2+1\right) \coth ^{-1}(a+b x)^2}{b^4}+\frac{\left(6 a^2+1\right) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}+\frac{2 a \left(a^2+1\right) \log \left(\frac{2}{-a-b x+1}\right) \coth ^{-1}(a+b x)}{b^4}-\frac{\left(a^4+6 a^2+1\right) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac{(a+b x)^2}{12 b^4}+\frac{\log \left(1-(a+b x)^2\right)}{12 b^4}+\frac{a \tanh ^{-1}(a+b x)}{b^4}+\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac{a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}-\frac{a x}{b^3}+\frac{1}{4} x^4 \coth ^{-1}(a+b x)^2",1,"-1/12*(1 + 11*a^2 + 10*a*b*x - b^2*x^2 + 3*(1 - 4*a + 6*a^2 - 4*a^3 + a^4 - b^4*x^4)*ArcCoth[a + b*x]^2 - 2*ArcCoth[a + b*x]*(9*a + 13*a^3 + 3*b*x + 9*a^2*b*x - 3*a*b^2*x^2 + b^3*x^3 + 12*(a + a^3)*Log[1 - E^(-2*ArcCoth[a + b*x])]) + 8*Log[1/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)])] + 36*a^2*Log[1/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)])] + 12*(a + a^3)*PolyLog[2, E^(-2*ArcCoth[a + b*x])])/b^4","A",0
70,1,607,204,4.5391443,"\int x^2 \coth ^{-1}(a+b x)^2 \, dx","Integrate[x^2*ArcCoth[a + b*x]^2,x]","-\frac{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \left(1-(a+b x)^2\right) \left(\frac{4 \left(3 a^2+1\right) \text{Li}_2\left(e^{-2 \coth ^{-1}(a+b x)}\right)}{(a+b x)^3 \left(1-\frac{1}{(a+b x)^2}\right)^{3/2}}+\frac{9 a^2 \coth ^{-1}(a+b x)^2}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}+\frac{-3 \left(a^2-1\right) \coth ^{-1}(a+b x)^2+6 a \coth ^{-1}(a+b x)-1}{\sqrt{1-\frac{1}{(a+b x)^2}}}+3 a^2 \coth ^{-1}(a+b x)^2 \cosh \left(3 \coth ^{-1}(a+b x)\right)+\frac{18 a^2 \coth ^{-1}(a+b x) \log \left(1-e^{-2 \coth ^{-1}(a+b x)}\right)}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}-3 a^2 \coth ^{-1}(a+b x)^2 \sinh \left(3 \coth ^{-1}(a+b x)\right)-6 a^2 \coth ^{-1}(a+b x) \log \left(1-e^{-2 \coth ^{-1}(a+b x)}\right) \sinh \left(3 \coth ^{-1}(a+b x)\right)-\frac{18 a \log \left(\frac{1}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}\right)}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}-\frac{12 a \coth ^{-1}(a+b x)^2}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}+\frac{3 \coth ^{-1}(a+b x)^2}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}+\frac{4 \coth ^{-1}(a+b x)}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}-6 a \coth ^{-1}(a+b x) \cosh \left(3 \coth ^{-1}(a+b x)\right)+\coth ^{-1}(a+b x)^2 \cosh \left(3 \coth ^{-1}(a+b x)\right)+\cosh \left(3 \coth ^{-1}(a+b x)\right)+\frac{6 \coth ^{-1}(a+b x) \log \left(1-e^{-2 \coth ^{-1}(a+b x)}\right)}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}-\coth ^{-1}(a+b x)^2 \sinh \left(3 \coth ^{-1}(a+b x)\right)+6 a \log \left(\frac{1}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}\right) \sinh \left(3 \coth ^{-1}(a+b x)\right)-2 \coth ^{-1}(a+b x) \log \left(1-e^{-2 \coth ^{-1}(a+b x)}\right) \sinh \left(3 \coth ^{-1}(a+b x)\right)\right)}{12 b^3}","-\frac{\left(3 a^2+1\right) \text{Li}_2\left(-\frac{a+b x+1}{-a-b x+1}\right)}{3 b^3}+\frac{a \left(a^2+3\right) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac{\left(3 a^2+1\right) \coth ^{-1}(a+b x)^2}{3 b^3}-\frac{2 \left(3 a^2+1\right) \log \left(\frac{2}{-a-b x+1}\right) \coth ^{-1}(a+b x)}{3 b^3}-\frac{a \log \left(1-(a+b x)^2\right)}{b^3}-\frac{\tanh ^{-1}(a+b x)}{3 b^3}+\frac{(a+b x)^2 \coth ^{-1}(a+b x)}{3 b^3}-\frac{2 a (a+b x) \coth ^{-1}(a+b x)}{b^3}+\frac{1}{3} x^3 \coth ^{-1}(a+b x)^2+\frac{x}{3 b^2}",1,"-1/12*((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]*(1 - (a + b*x)^2)*((4*ArcCoth[a + b*x])/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) + (3*ArcCoth[a + b*x]^2)/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) - (12*a*ArcCoth[a + b*x]^2)/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) + (9*a^2*ArcCoth[a + b*x]^2)/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) + (-1 + 6*a*ArcCoth[a + b*x] - 3*(-1 + a^2)*ArcCoth[a + b*x]^2)/Sqrt[1 - (a + b*x)^(-2)] + Cosh[3*ArcCoth[a + b*x]] - 6*a*ArcCoth[a + b*x]*Cosh[3*ArcCoth[a + b*x]] + ArcCoth[a + b*x]^2*Cosh[3*ArcCoth[a + b*x]] + 3*a^2*ArcCoth[a + b*x]^2*Cosh[3*ArcCoth[a + b*x]] + (6*ArcCoth[a + b*x]*Log[1 - E^(-2*ArcCoth[a + b*x])])/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) + (18*a^2*ArcCoth[a + b*x]*Log[1 - E^(-2*ArcCoth[a + b*x])])/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) - (18*a*Log[1/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)])])/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) + (4*(1 + 3*a^2)*PolyLog[2, E^(-2*ArcCoth[a + b*x])])/((a + b*x)^3*(1 - (a + b*x)^(-2))^(3/2)) - ArcCoth[a + b*x]^2*Sinh[3*ArcCoth[a + b*x]] - 3*a^2*ArcCoth[a + b*x]^2*Sinh[3*ArcCoth[a + b*x]] - 2*ArcCoth[a + b*x]*Log[1 - E^(-2*ArcCoth[a + b*x])]*Sinh[3*ArcCoth[a + b*x]] - 6*a^2*ArcCoth[a + b*x]*Log[1 - E^(-2*ArcCoth[a + b*x])]*Sinh[3*ArcCoth[a + b*x]] + 6*a*Log[1/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)])]*Sinh[3*ArcCoth[a + b*x]]))/b^3","B",0
71,1,106,136,0.2600163,"\int x \coth ^{-1}(a+b x)^2 \, dx","Integrate[x*ArcCoth[a + b*x]^2,x]","\frac{\left(-a^2+2 a+b^2 x^2-1\right) \coth ^{-1}(a+b x)^2-2 a \text{Li}_2\left(e^{-2 \coth ^{-1}(a+b x)}\right)-2 \log \left(\frac{1}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}\right)+2 \coth ^{-1}(a+b x) \left(2 a \log \left(1-e^{-2 \coth ^{-1}(a+b x)}\right)+a+b x\right)}{2 b^2}","-\frac{\left(a^2+1\right) \coth ^{-1}(a+b x)^2}{2 b^2}+\frac{a \text{Li}_2\left(-\frac{a+b x+1}{-a-b x+1}\right)}{b^2}+\frac{\log \left(1-(a+b x)^2\right)}{2 b^2}-\frac{a \coth ^{-1}(a+b x)^2}{b^2}+\frac{(a+b x) \coth ^{-1}(a+b x)}{b^2}+\frac{2 a \log \left(\frac{2}{-a-b x+1}\right) \coth ^{-1}(a+b x)}{b^2}+\frac{1}{2} x^2 \coth ^{-1}(a+b x)^2",1,"((-1 + 2*a - a^2 + b^2*x^2)*ArcCoth[a + b*x]^2 + 2*ArcCoth[a + b*x]*(a + b*x + 2*a*Log[1 - E^(-2*ArcCoth[a + b*x])]) - 2*Log[1/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)])] - 2*a*PolyLog[2, E^(-2*ArcCoth[a + b*x])])/(2*b^2)","A",0
72,1,55,81,0.0752378,"\int \coth ^{-1}(a+b x)^2 \, dx","Integrate[ArcCoth[a + b*x]^2,x]","\frac{\text{Li}_2\left(e^{-2 \coth ^{-1}(a+b x)}\right)+\coth ^{-1}(a+b x) \left((a+b x-1) \coth ^{-1}(a+b x)-2 \log \left(1-e^{-2 \coth ^{-1}(a+b x)}\right)\right)}{b}","-\frac{\text{Li}_2\left(-\frac{a+b x+1}{-a-b x+1}\right)}{b}+\frac{(a+b x) \coth ^{-1}(a+b x)^2}{b}+\frac{\coth ^{-1}(a+b x)^2}{b}-\frac{2 \log \left(\frac{2}{-a-b x+1}\right) \coth ^{-1}(a+b x)}{b}",1,"(ArcCoth[a + b*x]*((-1 + a + b*x)*ArcCoth[a + b*x] - 2*Log[1 - E^(-2*ArcCoth[a + b*x])]) + PolyLog[2, E^(-2*ArcCoth[a + b*x])])/b","A",0
73,1,547,148,2.8705454,"\int \frac{\coth ^{-1}(a+b x)^2}{x} \, dx","Integrate[ArcCoth[a + b*x]^2/x,x]","\frac{2}{3} \sqrt{1-\frac{1}{a^2}} a e^{\tanh ^{-1}\left(\frac{1}{a}\right)} \coth ^{-1}(a+b x)^3+2 \coth ^{-1}(a+b x) \text{Li}_2\left(-\sqrt{\frac{a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right)+2 \coth ^{-1}(a+b x) \text{Li}_2\left(\sqrt{\frac{a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right)-\coth ^{-1}(a+b x) \text{Li}_2\left(e^{2 \coth ^{-1}(a+b x)}\right)-2 \text{Li}_3\left(-\sqrt{\frac{a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right)-2 \text{Li}_3\left(\sqrt{\frac{a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right)+\frac{1}{2} \text{Li}_3\left(e^{2 \coth ^{-1}(a+b x)}\right)-\frac{2}{3} a \coth ^{-1}(a+b x)^3-\frac{2}{3} \coth ^{-1}(a+b x)^3+\coth ^{-1}(a+b x)^2 \log \left(1-\sqrt{\frac{a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right)+\coth ^{-1}(a+b x)^2 \log \left(\sqrt{\frac{a-1}{a+1}} e^{\coth ^{-1}(a+b x)}+1\right)-\coth ^{-1}(a+b x)^2 \log \left(1-e^{2 \coth ^{-1}(a+b x)}\right)+\coth ^{-1}(a+b x)^2 \log \left(\frac{1}{2} e^{-\coth ^{-1}(a+b x)} \left(a \left(e^{2 \coth ^{-1}(a+b x)}-1\right)-e^{2 \coth ^{-1}(a+b x)}-1\right)\right)-\log \left(-\frac{b x}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}\right) \coth ^{-1}(a+b x)^2-i \pi  \coth ^{-1}(a+b x) \log \left(\frac{1}{2} \left(e^{-\coth ^{-1}(a+b x)}+e^{\coth ^{-1}(a+b x)}\right)\right)+i \pi  \log \left(\frac{1}{\sqrt{1-\frac{1}{(a+b x)^2}}}\right) \coth ^{-1}(a+b x)-2 \tanh ^{-1}\left(\frac{1}{a}\right) \coth ^{-1}(a+b x) \log \left(\frac{1}{2} i \left(e^{\coth ^{-1}(a+b x)-\tanh ^{-1}\left(\frac{1}{a}\right)}-e^{\tanh ^{-1}\left(\frac{1}{a}\right)-\coth ^{-1}(a+b x)}\right)\right)+2 \tanh ^{-1}\left(\frac{1}{a}\right) \coth ^{-1}(a+b x) \log \left(i \sinh \left(\coth ^{-1}(a+b x)-\tanh ^{-1}\left(\frac{1}{a}\right)\right)\right)-\frac{i \pi ^3}{24}","\frac{1}{2} \text{Li}_3\left(1-\frac{2}{a+b x+1}\right)-\frac{1}{2} \text{Li}_3\left(1-\frac{2 b x}{(1-a) (a+b x+1)}\right)+\text{Li}_2\left(1-\frac{2}{a+b x+1}\right) \coth ^{-1}(a+b x)-\text{Li}_2\left(1-\frac{2 b x}{(1-a) (a+b x+1)}\right) \coth ^{-1}(a+b x)-\log \left(\frac{2}{a+b x+1}\right) \coth ^{-1}(a+b x)^2+\log \left(\frac{2 b x}{(1-a) (a+b x+1)}\right) \coth ^{-1}(a+b x)^2",1,"(-1/24*I)*Pi^3 - (2*ArcCoth[a + b*x]^3)/3 - (2*a*ArcCoth[a + b*x]^3)/3 + (2*Sqrt[1 - a^(-2)]*a*E^ArcTanh[a^(-1)]*ArcCoth[a + b*x]^3)/3 - I*Pi*ArcCoth[a + b*x]*Log[(E^(-ArcCoth[a + b*x]) + E^ArcCoth[a + b*x])/2] + ArcCoth[a + b*x]^2*Log[1 - Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x]] + ArcCoth[a + b*x]^2*Log[1 + Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x]] - ArcCoth[a + b*x]^2*Log[1 - E^(2*ArcCoth[a + b*x])] - 2*ArcCoth[a + b*x]*ArcTanh[a^(-1)]*Log[(I/2)*(E^(ArcCoth[a + b*x] - ArcTanh[a^(-1)]) - E^(-ArcCoth[a + b*x] + ArcTanh[a^(-1)]))] + ArcCoth[a + b*x]^2*Log[(-1 - E^(2*ArcCoth[a + b*x]) + a*(-1 + E^(2*ArcCoth[a + b*x])))/(2*E^ArcCoth[a + b*x])] + I*Pi*ArcCoth[a + b*x]*Log[1/Sqrt[1 - (a + b*x)^(-2)]] - ArcCoth[a + b*x]^2*Log[-((b*x)/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]))] + 2*ArcCoth[a + b*x]*ArcTanh[a^(-1)]*Log[I*Sinh[ArcCoth[a + b*x] - ArcTanh[a^(-1)]]] + 2*ArcCoth[a + b*x]*PolyLog[2, -(Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x])] + 2*ArcCoth[a + b*x]*PolyLog[2, Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x]] - ArcCoth[a + b*x]*PolyLog[2, E^(2*ArcCoth[a + b*x])] - 2*PolyLog[3, -(Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x])] - 2*PolyLog[3, Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x]] + PolyLog[3, E^(2*ArcCoth[a + b*x])]/2","C",0
74,1,206,251,1.1240251,"\int \frac{\coth ^{-1}(a+b x)^2}{x^2} \, dx","Integrate[ArcCoth[a + b*x]^2/x^2,x]","\frac{-\left(\left(\sqrt{1-\frac{1}{a^2}} a b x e^{\tanh ^{-1}\left(\frac{1}{a}\right)}+a^2-1\right) \coth ^{-1}(a+b x)^2\right)+b x \text{Li}_2\left(e^{2 \tanh ^{-1}\left(\frac{1}{a}\right)-2 \coth ^{-1}(a+b x)}\right)+b x \coth ^{-1}(a+b x) \left(-2 \log \left(1-e^{2 \tanh ^{-1}\left(\frac{1}{a}\right)-2 \coth ^{-1}(a+b x)}\right)+2 \tanh ^{-1}\left(\frac{1}{a}\right)-i \pi \right)+b x \left(i \pi  \left(\log \left(e^{2 \coth ^{-1}(a+b x)}+1\right)-\log \left(\frac{1}{\sqrt{1-\frac{1}{(a+b x)^2}}}\right)\right)+2 \tanh ^{-1}\left(\frac{1}{a}\right) \left(\log \left(1-e^{2 \tanh ^{-1}\left(\frac{1}{a}\right)-2 \coth ^{-1}(a+b x)}\right)-\log \left(i \sinh \left(\coth ^{-1}(a+b x)-\tanh ^{-1}\left(\frac{1}{a}\right)\right)\right)\right)\right)}{\left(a^2-1\right) x}","\frac{b \text{Li}_2\left(1-\frac{2}{a+b x+1}\right)}{1-a^2}-\frac{b \text{Li}_2\left(1-\frac{2 b x}{(1-a) (a+b x+1)}\right)}{1-a^2}-\frac{2 b \log \left(\frac{2}{a+b x+1}\right) \coth ^{-1}(a+b x)}{1-a^2}+\frac{2 b \log \left(\frac{2 b x}{(1-a) (a+b x+1)}\right) \coth ^{-1}(a+b x)}{1-a^2}+\frac{b \text{Li}_2\left(-\frac{a+b x+1}{-a-b x+1}\right)}{2 (1-a)}-\frac{b \text{Li}_2\left(1-\frac{2}{a+b x+1}\right)}{2 (a+1)}-\frac{\coth ^{-1}(a+b x)^2}{x}+\frac{b \log \left(\frac{2}{-a-b x+1}\right) \coth ^{-1}(a+b x)}{1-a}+\frac{b \log \left(\frac{2}{a+b x+1}\right) \coth ^{-1}(a+b x)}{a+1}",1,"(-((-1 + a^2 + Sqrt[1 - a^(-2)]*a*b*E^ArcTanh[a^(-1)]*x)*ArcCoth[a + b*x]^2) + b*x*ArcCoth[a + b*x]*((-I)*Pi + 2*ArcTanh[a^(-1)] - 2*Log[1 - E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[a^(-1)])]) + b*x*(I*Pi*(Log[1 + E^(2*ArcCoth[a + b*x])] - Log[1/Sqrt[1 - (a + b*x)^(-2)]]) + 2*ArcTanh[a^(-1)]*(Log[1 - E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[a^(-1)])] - Log[I*Sinh[ArcCoth[a + b*x] - ArcTanh[a^(-1)]]])) + b*x*PolyLog[2, E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[a^(-1)])])/((-1 + a^2)*x)","C",0
75,1,291,370,2.3504696,"\int \frac{\coth ^{-1}(a+b x)^2}{x^3} \, dx","Integrate[ArcCoth[a + b*x]^2/x^3,x]","\frac{2 b x \coth ^{-1}(a+b x) \left(a^2+a b x+i \pi  a b x-2 a b x \tanh ^{-1}\left(\frac{1}{a}\right)+2 a b x \log \left(1-e^{2 \tanh ^{-1}\left(\frac{1}{a}\right)-2 \coth ^{-1}(a+b x)}\right)-1\right)+\left(-a^4+a^2 \left(b^2 x^2 \left(2 \sqrt{1-\frac{1}{a^2}} e^{\tanh ^{-1}\left(\frac{1}{a}\right)}-1\right)+2\right)+b^2 x^2-1\right) \coth ^{-1}(a+b x)^2-2 a b^2 x^2 \text{Li}_2\left(e^{2 \tanh ^{-1}\left(\frac{1}{a}\right)-2 \coth ^{-1}(a+b x)}\right)+2 b^2 x^2 \left(i \pi  a \log \left(\frac{1}{\sqrt{1-\frac{1}{(a+b x)^2}}}\right)+\log \left(-\frac{b x}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}\right)-i \pi  a \log \left(e^{2 \coth ^{-1}(a+b x)}+1\right)-2 a \tanh ^{-1}\left(\frac{1}{a}\right) \left(\log \left(1-e^{2 \tanh ^{-1}\left(\frac{1}{a}\right)-2 \coth ^{-1}(a+b x)}\right)-\log \left(i \sinh \left(\coth ^{-1}(a+b x)-\tanh ^{-1}\left(\frac{1}{a}\right)\right)\right)\right)\right)}{2 \left(a^2-1\right)^2 x^2}","\frac{a b^2 \text{Li}_2\left(1-\frac{2}{a+b x+1}\right)}{\left(1-a^2\right)^2}-\frac{a b^2 \text{Li}_2\left(1-\frac{2 b x}{(1-a) (a+b x+1)}\right)}{\left(1-a^2\right)^2}+\frac{b^2 \log (x)}{\left(1-a^2\right)^2}-\frac{2 a b^2 \log \left(\frac{2}{a+b x+1}\right) \coth ^{-1}(a+b x)}{\left(1-a^2\right)^2}+\frac{2 a b^2 \log \left(\frac{2 b x}{(1-a) (a+b x+1)}\right) \coth ^{-1}(a+b x)}{\left(1-a^2\right)^2}-\frac{b \coth ^{-1}(a+b x)}{\left(1-a^2\right) x}+\frac{b^2 \text{Li}_2\left(-\frac{a+b x+1}{-a-b x+1}\right)}{4 (1-a)^2}+\frac{b^2 \text{Li}_2\left(1-\frac{2}{a+b x+1}\right)}{4 (a+1)^2}-\frac{b^2 \log (-a-b x+1)}{2 (1-a)^2 (a+1)}-\frac{b^2 \log (a+b x+1)}{2 (1-a) (a+1)^2}+\frac{b^2 \log \left(\frac{2}{-a-b x+1}\right) \coth ^{-1}(a+b x)}{2 (1-a)^2}-\frac{b^2 \log \left(\frac{2}{a+b x+1}\right) \coth ^{-1}(a+b x)}{2 (a+1)^2}-\frac{\coth ^{-1}(a+b x)^2}{2 x^2}",1,"((-1 - a^4 + b^2*x^2 + a^2*(2 + b^2*(-1 + 2*Sqrt[1 - a^(-2)]*E^ArcTanh[a^(-1)])*x^2))*ArcCoth[a + b*x]^2 + 2*b*x*ArcCoth[a + b*x]*(-1 + a^2 + a*b*x + I*a*b*Pi*x - 2*a*b*x*ArcTanh[a^(-1)] + 2*a*b*x*Log[1 - E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[a^(-1)])]) + 2*b^2*x^2*((-I)*a*Pi*Log[1 + E^(2*ArcCoth[a + b*x])] + I*a*Pi*Log[1/Sqrt[1 - (a + b*x)^(-2)]] + Log[-((b*x)/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]))] - 2*a*ArcTanh[a^(-1)]*(Log[1 - E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[a^(-1)])] - Log[I*Sinh[ArcCoth[a + b*x] - ArcTanh[a^(-1)]]])) - 2*a*b^2*x^2*PolyLog[2, E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[a^(-1)])])/(2*(-1 + a^2)^2*x^2)","C",0
76,1,529,673,0.6195889,"\int \frac{\coth ^{-1}(a+b x)}{c+d x^2} \, dx","Integrate[ArcCoth[a + b*x]/(c + d*x^2),x]","\frac{\text{Li}_2\left(\frac{b \left(\sqrt{-c}-\sqrt{d} x\right)}{\sqrt{d} (a-1)+b \sqrt{-c}}\right)-\text{Li}_2\left(\frac{b \left(\sqrt{-c}-\sqrt{d} x\right)}{\sqrt{d} (a+1)+b \sqrt{-c}}\right)-\text{Li}_2\left(\frac{b \left(\sqrt{d} x+\sqrt{-c}\right)}{b \sqrt{-c}-(a-1) \sqrt{d}}\right)+\text{Li}_2\left(\frac{b \left(\sqrt{d} x+\sqrt{-c}\right)}{b \sqrt{-c}-(a+1) \sqrt{d}}\right)+\log \left(\sqrt{-c}-\sqrt{d} x\right) \log \left(\frac{\sqrt{d} (a+b x-1)}{(a-1) \sqrt{d}+b \sqrt{-c}}\right)-\log \left(\frac{a+b x-1}{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+1)}{(a+1) \sqrt{d}+b \sqrt{-c}}\right)+\log \left(\frac{a+b x+1}{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-1)}{b \sqrt{-c}-(a-1) \sqrt{d}}\right)+\log \left(\frac{a+b x-1}{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+1)}{b \sqrt{-c}-(a+1) \sqrt{d}}\right)-\log \left(\frac{a+b x+1}{a+b x}\right) \log \left(\sqrt{-c}+\sqrt{d} x\right)}{4 \sqrt{-c} \sqrt{d}}","\frac{\text{Li}_2\left(-\frac{\left(d a^2+b^2 c\right) (-a-b x+1)}{\left(c b^2-\sqrt{-c} \sqrt{d} b-(1-a) a d\right) (a+b x)}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{\text{Li}_2\left(-\frac{\left(d a^2+b^2 c\right) (-a-b x+1)}{\left(c b^2+\sqrt{-c} \sqrt{d} b-(1-a) a d\right) (a+b x)}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{Li}_2\left(\frac{\left(d a^2+b^2 c\right) (a+b x+1)}{\left(c b^2-\sqrt{-c} \sqrt{d} b+a (a+1) d\right) (a+b x)}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{\text{Li}_2\left(\frac{\left(d a^2+b^2 c\right) (a+b x+1)}{\left(c b^2+\sqrt{-c} \sqrt{d} b+a (a+1) d\right) (a+b x)}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{\log \left(-\frac{-a-b x+1}{a+b x}\right) \log \left(\frac{(-a-b x+1) \left(a^2 d+b^2 c\right)}{(a+b x) \left(-(1-a) a d+b^2 c-b \sqrt{-c} \sqrt{d}\right)}+1\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{\log \left(-\frac{-a-b x+1}{a+b x}\right) \log \left(\frac{(-a-b x+1) \left(a^2 d+b^2 c\right)}{(a+b x) \left(-(1-a) a d+b^2 c+b \sqrt{-c} \sqrt{d}\right)}+1\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{\log \left(\frac{a+b x+1}{a+b x}\right) \log \left(1-\frac{(a+b x+1) \left(a^2 d+b^2 c\right)}{(a+b x) \left(a (a+1) d+b^2 c-b \sqrt{-c} \sqrt{d}\right)}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{\log \left(\frac{a+b x+1}{a+b x}\right) \log \left(1-\frac{(a+b x+1) \left(a^2 d+b^2 c\right)}{(a+b x) \left(a (a+1) d+b^2 c+b \sqrt{-c} \sqrt{d}\right)}\right)}{4 \sqrt{-c} \sqrt{d}}",1,"(Log[(Sqrt[d]*(-1 + a + b*x))/(b*Sqrt[-c] + (-1 + a)*Sqrt[d])]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[(-1 + a + b*x)/(a + b*x)]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[(Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])]*Log[Sqrt[-c] - Sqrt[d]*x] + Log[(1 + a + b*x)/(a + b*x)]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[-((Sqrt[d]*(-1 + a + b*x))/(b*Sqrt[-c] - (-1 + a)*Sqrt[d]))]*Log[Sqrt[-c] + Sqrt[d]*x] + Log[(-1 + a + b*x)/(a + b*x)]*Log[Sqrt[-c] + Sqrt[d]*x] + Log[-((Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] - (1 + a)*Sqrt[d]))]*Log[Sqrt[-c] + Sqrt[d]*x] - Log[(1 + a + b*x)/(a + b*x)]*Log[Sqrt[-c] + Sqrt[d]*x] + PolyLog[2, (b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (-1 + a)*Sqrt[d])] - PolyLog[2, (b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])] - PolyLog[2, (b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (-1 + a)*Sqrt[d])] + PolyLog[2, (b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (1 + a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d])","A",0
77,1,185,120,0.0734929,"\int \frac{\coth ^{-1}(a+b x)}{c+d x} \, dx","Integrate[ArcCoth[a + b*x]/(c + d*x),x]","-\frac{\text{Li}_2\left(\frac{b (c+d x)}{b c-a d-d}\right)}{2 d}+\frac{\text{Li}_2\left(\frac{b (c+d x)}{b c-a d+d}\right)}{2 d}+\frac{\log (c+d x) \log \left(\frac{d (-a-b x+1)}{-a d+b c+d}\right)}{2 d}-\frac{\log \left(\frac{a+b x-1}{a+b x}\right) \log (c+d x)}{2 d}-\frac{\log (c+d x) \log \left(-\frac{d (a+b x+1)}{-a d+b c-d}\right)}{2 d}+\frac{\log \left(\frac{a+b x+1}{a+b x}\right) \log (c+d x)}{2 d}","-\frac{\text{Li}_2\left(1-\frac{2 b (c+d x)}{(b c-a d+d) (a+b x+1)}\right)}{2 d}+\frac{\coth ^{-1}(a+b x) \log \left(\frac{2 b (c+d x)}{(a+b x+1) (-a d+b c+d)}\right)}{d}+\frac{\text{Li}_2\left(1-\frac{2}{a+b x+1}\right)}{2 d}-\frac{\log \left(\frac{2}{a+b x+1}\right) \coth ^{-1}(a+b x)}{d}",1,"(Log[(d*(1 - a - b*x))/(b*c + d - a*d)]*Log[c + d*x])/(2*d) - (Log[(-1 + a + b*x)/(a + b*x)]*Log[c + d*x])/(2*d) - (Log[-((d*(1 + a + b*x))/(b*c - d - a*d))]*Log[c + d*x])/(2*d) + (Log[(1 + a + b*x)/(a + b*x)]*Log[c + d*x])/(2*d) - PolyLog[2, (b*(c + d*x))/(b*c - d - a*d)]/(2*d) + PolyLog[2, (b*(c + d*x))/(b*c + d - a*d)]/(2*d)","A",0
78,1,502,292,4.4936995,"\int \frac{\coth ^{-1}(a+b x)}{c+\frac{d}{x}} \, dx","Integrate[ArcCoth[a + b*x]/(c + d/x),x]","\frac{b^2 d^2 \sqrt{1-\frac{c^2}{(a c-b d)^2}} \coth ^{-1}(a+b x)^2 e^{\tanh ^{-1}\left(\frac{c}{a c-b d}\right)}-b^2 d^2 \coth ^{-1}(a+b x)^2-a b c d \sqrt{1-\frac{c^2}{(a c-b d)^2}} \coth ^{-1}(a+b x)^2 e^{\tanh ^{-1}\left(\frac{c}{a c-b d}\right)}-2 c^2 \log \left(\frac{1}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}\right)+2 a c^2 \coth ^{-1}(a+b x)+2 b c^2 x \coth ^{-1}(a+b x)+b c d \text{Li}_2\left(\exp \left(2 \tanh ^{-1}\left(\frac{c}{a c-b d}\right)-2 \coth ^{-1}(a+b x)\right)\right)-2 b c d \coth ^{-1}(a+b x) \log \left(1-\exp \left(2 \tanh ^{-1}\left(\frac{c}{a c-b d}\right)-2 \coth ^{-1}(a+b x)\right)\right)+2 b c d \tanh ^{-1}\left(\frac{c}{a c-b d}\right) \log \left(1-\exp \left(2 \tanh ^{-1}\left(\frac{c}{a c-b d}\right)-2 \coth ^{-1}(a+b x)\right)\right)-b c d \text{Li}_2\left(e^{-2 \coth ^{-1}(a+b x)}\right)-i \pi  b c d \log \left(\frac{1}{\sqrt{1-\frac{1}{(a+b x)^2}}}\right)+a b c d \coth ^{-1}(a+b x)^2+b c d \coth ^{-1}(a+b x)^2-i \pi  b c d \coth ^{-1}(a+b x)+2 b c d \coth ^{-1}(a+b x) \log \left(1-e^{-2 \coth ^{-1}(a+b x)}\right)+i \pi  b c d \log \left(e^{2 \coth ^{-1}(a+b x)}+1\right)+2 b c d \coth ^{-1}(a+b x) \tanh ^{-1}\left(\frac{c}{a c-b d}\right)-2 b c d \tanh ^{-1}\left(\frac{c}{a c-b d}\right) \log \left(i \sinh \left(\coth ^{-1}(a+b x)-\tanh ^{-1}\left(\frac{c}{a c-b d}\right)\right)\right)}{2 b c^3}","\frac{d \text{Li}_2\left(-\frac{b (d+c x)}{a c+c-b d}\right)}{2 c^2}-\frac{d \text{Li}_2\left(\frac{b (d+c x)}{-a c+c+b d}\right)}{2 c^2}+\frac{d \log \left(-\frac{-a-b x+1}{a+b x}\right) \log (c x+d)}{2 c^2}-\frac{d \log (c x+d) \log \left(\frac{c (-a-b x+1)}{-a c+b d+c}\right)}{2 c^2}+\frac{d \log (c x+d) \log \left(\frac{c (a+b x+1)}{a c-b d+c}\right)}{2 c^2}-\frac{d \log \left(\frac{a+b x+1}{a+b x}\right) \log (c x+d)}{2 c^2}+\frac{(-a-b x+1) \log \left(-\frac{-a-b x+1}{a+b x}\right)}{2 b c}+\frac{\log (a+b x)}{2 b c}+\frac{\log (a+b x+1)}{2 b c}+\frac{(a+b x) \log \left(\frac{a+b x+1}{a+b x}\right)}{2 b c}",1,"(2*a*c^2*ArcCoth[a + b*x] - I*b*c*d*Pi*ArcCoth[a + b*x] + 2*b*c^2*x*ArcCoth[a + b*x] + b*c*d*ArcCoth[a + b*x]^2 + a*b*c*d*ArcCoth[a + b*x]^2 - b^2*d^2*ArcCoth[a + b*x]^2 - a*b*c*d*Sqrt[1 - c^2/(a*c - b*d)^2]*E^ArcTanh[c/(a*c - b*d)]*ArcCoth[a + b*x]^2 + b^2*d^2*Sqrt[1 - c^2/(a*c - b*d)^2]*E^ArcTanh[c/(a*c - b*d)]*ArcCoth[a + b*x]^2 + 2*b*c*d*ArcCoth[a + b*x]*ArcTanh[c/(a*c - b*d)] + 2*b*c*d*ArcCoth[a + b*x]*Log[1 - E^(-2*ArcCoth[a + b*x])] + I*b*c*d*Pi*Log[1 + E^(2*ArcCoth[a + b*x])] - 2*b*c*d*ArcCoth[a + b*x]*Log[1 - E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[c/(a*c - b*d)])] + 2*b*c*d*ArcTanh[c/(a*c - b*d)]*Log[1 - E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[c/(a*c - b*d)])] - I*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*ArcTanh[c/(a*c - b*d)]*Log[I*Sinh[ArcCoth[a + b*x] - ArcTanh[c/(a*c - b*d)]]] - b*c*d*PolyLog[2, E^(-2*ArcCoth[a + b*x])] + b*c*d*PolyLog[2, E^(-2*ArcCoth[a + b*x] + 2*ArcTanh[c/(a*c - b*d)])])/(2*b*c^3)","C",0
79,1,5552,738,36.3463648,"\int \frac{\coth ^{-1}(a+b x)}{c+\frac{d}{x^2}} \, dx","Integrate[ArcCoth[a + b*x]/(c + d/x^2),x]","\text{Result too large to show}","-\frac{\sqrt{d} \left(\log (a+b x-1)-\log \left(-\frac{-a-b x+1}{a+b x}\right)-\log (a+b x)\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)}{2 c^{3/2}}-\frac{\sqrt{d} \left(\log (a+b x)-\log (a+b x+1)+\log \left(\frac{a+b x+1}{a+b x}\right)\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)}{2 c^{3/2}}+\frac{\sqrt{d} \text{Li}_2\left(\frac{\sqrt{-c} (-a-b x+1)}{(1-a) \sqrt{-c}-b \sqrt{d}}\right)}{4 (-c)^{3/2}}-\frac{\sqrt{d} \text{Li}_2\left(\frac{\sqrt{-c} (-a-b x+1)}{\sqrt{-c} (1-a)+b \sqrt{d}}\right)}{4 (-c)^{3/2}}+\frac{\sqrt{d} \text{Li}_2\left(\frac{\sqrt{-c} (a+b x+1)}{(a+1) \sqrt{-c}-b \sqrt{d}}\right)}{4 (-c)^{3/2}}-\frac{\sqrt{d} \text{Li}_2\left(\frac{\sqrt{-c} (a+b x+1)}{\sqrt{-c} (a+1)+b \sqrt{d}}\right)}{4 (-c)^{3/2}}+\frac{\sqrt{d} \log (a+b x-1) \log \left(-\frac{b \left(\sqrt{d}-\sqrt{-c} x\right)}{(1-a) \sqrt{-c}-b \sqrt{d}}\right)}{4 (-c)^{3/2}}-\frac{\sqrt{d} \log (a+b x-1) \log \left(\frac{b \left(\sqrt{-c} x+\sqrt{d}\right)}{(1-a) \sqrt{-c}+b \sqrt{d}}\right)}{4 (-c)^{3/2}}-\frac{\sqrt{d} \log (a+b x+1) \log \left(\frac{b \left(\sqrt{d}-\sqrt{-c} x\right)}{(a+1) \sqrt{-c}+b \sqrt{d}}\right)}{4 (-c)^{3/2}}+\frac{\sqrt{d} \log (a+b x+1) \log \left(-\frac{b \left(\sqrt{-c} x+\sqrt{d}\right)}{(a+1) \sqrt{-c}-b \sqrt{d}}\right)}{4 (-c)^{3/2}}+\frac{(-a-b x+1) \log (a+b x-1)}{2 b c}+\frac{x \left(\log (a+b x-1)-\log \left(-\frac{-a-b x+1}{a+b x}\right)-\log (a+b x)\right)}{2 c}+\frac{(a+b x+1) \log (a+b x+1)}{2 b c}+\frac{x \left(\log (a+b x)-\log (a+b x+1)+\log \left(\frac{a+b x+1}{a+b x}\right)\right)}{2 c}",1,"Result too large to show","C",0
80,1,575,619,0.7069678,"\int \frac{\coth ^{-1}(a+b x)}{c+d \sqrt{x}} \, dx","Integrate[ArcCoth[a + b*x]/(c + d*Sqrt[x]),x]","\frac{c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{-a-1} d}\right)+c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{-a-1} d}\right)-c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{1-a} d}\right)-c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{1-a} d}\right)+c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-1} d+\sqrt{b} c}\right)-c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(\sqrt{1-a}-\sqrt{b} \sqrt{x}\right)}{\sqrt{1-a} d+\sqrt{b} c}\right)+c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-1} d-\sqrt{b} c}\right)-c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(\sqrt{1-a}+\sqrt{b} \sqrt{x}\right)}{\sqrt{1-a} d-\sqrt{b} c}\right)+c \log \left(\frac{a+b x-1}{a+b x}\right) \log \left(c+d \sqrt{x}\right)-c \log \left(\frac{a+b x+1}{a+b x}\right) \log \left(c+d \sqrt{x}\right)-d \sqrt{x} \log \left(\frac{a+b x-1}{a+b x}\right)+d \sqrt{x} \log \left(\frac{a+b x+1}{a+b x}\right)+\frac{2 \sqrt{a+1} d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right)}{\sqrt{b}}-\frac{2 \sqrt{1-a} d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right)}{\sqrt{b}}}{d^2}","\frac{c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{-a-1} d}\right)}{d^2}+\frac{c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{-a-1} d}\right)}{d^2}-\frac{c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c-\sqrt{1-a} d}\right)}{d^2}-\frac{c \text{Li}_2\left(\frac{\sqrt{b} \left(c+d \sqrt{x}\right)}{\sqrt{b} c+\sqrt{1-a} d}\right)}{d^2}+\frac{c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-1} d+\sqrt{b} c}\right)}{d^2}-\frac{c \log \left(c+d \sqrt{x}\right) \log \left(\frac{d \left(\sqrt{1-a}-\sqrt{b} \sqrt{x}\right)}{\sqrt{1-a} d+\sqrt{b} c}\right)}{d^2}+\frac{c \log \left(c+d \sqrt{x}\right) \log \left(-\frac{d \left(\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right)}{\sqrt{b} c-\sqrt{-a-1} d}\right)}{d^2}-\frac{c \log \left(c+d \sqrt{x}\right) \log \left(-\frac{d \left(\sqrt{1-a}+\sqrt{b} \sqrt{x}\right)}{\sqrt{b} c-\sqrt{1-a} d}\right)}{d^2}+\frac{c \log \left(-\frac{-a-b x+1}{a+b x}\right) \log \left(c+d \sqrt{x}\right)}{d^2}-\frac{c \log \left(\frac{a+b x+1}{a+b x}\right) \log \left(c+d \sqrt{x}\right)}{d^2}-\frac{\sqrt{x} \log \left(-\frac{-a-b x+1}{a+b x}\right)}{d}+\frac{\sqrt{x} \log \left(\frac{a+b x+1}{a+b x}\right)}{d}+\frac{2 \sqrt{a+1} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right)}{\sqrt{b} d}-\frac{2 \sqrt{1-a} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right)}{\sqrt{b} d}",1,"((2*Sqrt[1 + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]])/Sqrt[b] - (2*Sqrt[1 - a]*d*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[1 - a]])/Sqrt[b] + c*Log[(d*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-1 - a]*d)]*Log[c + d*Sqrt[x]] - c*Log[(d*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[1 - a]*d)]*Log[c + d*Sqrt[x]] + c*Log[(d*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(-(Sqrt[b]*c) + Sqrt[-1 - a]*d)]*Log[c + d*Sqrt[x]] - c*Log[(d*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(-(Sqrt[b]*c) + Sqrt[1 - a]*d)]*Log[c + d*Sqrt[x]] - d*Sqrt[x]*Log[(-1 + a + b*x)/(a + b*x)] + c*Log[c + d*Sqrt[x]]*Log[(-1 + a + b*x)/(a + b*x)] + d*Sqrt[x]*Log[(1 + a + b*x)/(a + b*x)] - c*Log[c + d*Sqrt[x]]*Log[(1 + a + b*x)/(a + b*x)] + c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-1 - a]*d)] + c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-1 - a]*d)] - c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[1 - a]*d)] - c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[1 - a]*d)])/d^2","A",1
81,1,719,738,0.744733,"\int \frac{\coth ^{-1}(a+b x)}{c+\frac{d}{\sqrt{x}}} \, dx","Integrate[ArcCoth[a + b*x]/(c + d/Sqrt[x]),x]","\frac{-a c^2 \log (-a-b x+1)+c^2 \log (-a-b x+1)-b c^2 x \log \left(\frac{a+b x-1}{a+b x}\right)+a c^2 \log (a+b x+1)+c^2 \log (a+b x+1)+b c^2 x \log \left(\frac{a+b x+1}{a+b x}\right)-2 b d^2 \text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{b} d-\sqrt{-a-1} c}\right)-2 b d^2 \text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{-a-1} c+\sqrt{b} d}\right)+2 b d^2 \text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{b} d-\sqrt{1-a} c}\right)+2 b d^2 \text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{1-a} c+\sqrt{b} d}\right)-2 b d^2 \log \left(c \sqrt{x}+d\right) \log \left(\frac{c \left(\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-1} c+\sqrt{b} d}\right)+2 b d^2 \log \left(c \sqrt{x}+d\right) \log \left(\frac{c \left(\sqrt{1-a}-\sqrt{b} \sqrt{x}\right)}{\sqrt{1-a} c+\sqrt{b} d}\right)-2 b d^2 \log \left(c \sqrt{x}+d\right) \log \left(\frac{c \left(\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-1} c-\sqrt{b} d}\right)+2 b d^2 \log \left(c \sqrt{x}+d\right) \log \left(\frac{c \left(\sqrt{1-a}+\sqrt{b} \sqrt{x}\right)}{\sqrt{1-a} c-\sqrt{b} d}\right)-2 b d^2 \log \left(\frac{a+b x-1}{a+b x}\right) \log \left(c \sqrt{x}+d\right)+2 b d^2 \log \left(\frac{a+b x+1}{a+b x}\right) \log \left(c \sqrt{x}+d\right)+2 b c d \sqrt{x} \log \left(\frac{a+b x-1}{a+b x}\right)-2 b c d \sqrt{x} \log \left(\frac{a+b x+1}{a+b x}\right)-4 \sqrt{a+1} \sqrt{b} c d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right)+4 \sqrt{1-a} \sqrt{b} c d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right)}{2 b c^3}","-\frac{d^2 \text{Li}_2\left(-\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{-a-1} c-\sqrt{b} d}\right)}{c^3}+\frac{d^2 \text{Li}_2\left(-\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{1-a} c-\sqrt{b} d}\right)}{c^3}-\frac{d^2 \text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{-a-1} c+\sqrt{b} d}\right)}{c^3}+\frac{d^2 \text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{1-a} c+\sqrt{b} d}\right)}{c^3}-\frac{d^2 \log \left(c \sqrt{x}+d\right) \log \left(\frac{c \left(\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-1} c+\sqrt{b} d}\right)}{c^3}+\frac{d^2 \log \left(c \sqrt{x}+d\right) \log \left(\frac{c \left(\sqrt{1-a}-\sqrt{b} \sqrt{x}\right)}{\sqrt{1-a} c+\sqrt{b} d}\right)}{c^3}-\frac{d^2 \log \left(c \sqrt{x}+d\right) \log \left(\frac{c \left(\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-1} c-\sqrt{b} d}\right)}{c^3}+\frac{d^2 \log \left(c \sqrt{x}+d\right) \log \left(\frac{c \left(\sqrt{1-a}+\sqrt{b} \sqrt{x}\right)}{\sqrt{1-a} c-\sqrt{b} d}\right)}{c^3}-\frac{d^2 \log \left(-\frac{-a-b x+1}{a+b x}\right) \log \left(c \sqrt{x}+d\right)}{c^3}+\frac{d^2 \log \left(\frac{a+b x+1}{a+b x}\right) \log \left(c \sqrt{x}+d\right)}{c^3}+\frac{d \sqrt{x} \log \left(-\frac{-a-b x+1}{a+b x}\right)}{c^2}-\frac{d \sqrt{x} \log \left(\frac{a+b x+1}{a+b x}\right)}{c^2}-\frac{2 \sqrt{a+1} d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right)}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right)}{\sqrt{b} c^2}+\frac{(1-a) \log (-a-b x+1)}{2 b c}-\frac{x \log \left(-\frac{-a-b x+1}{a+b x}\right)}{2 c}+\frac{(a+1) \log (a+b x+1)}{2 b c}+\frac{x \log \left(\frac{a+b x+1}{a+b x}\right)}{2 c}",1,"(-4*Sqrt[1 + a]*Sqrt[b]*c*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]] + 4*Sqrt[1 - a]*Sqrt[b]*c*d*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[1 - a]] - 2*b*d^2*Log[(c*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]] + 2*b*d^2*Log[(c*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]] - 2*b*d^2*Log[(c*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]] + 2*b*d^2*Log[(c*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]] + c^2*Log[1 - a - b*x] - a*c^2*Log[1 - a - b*x] + 2*b*c*d*Sqrt[x]*Log[(-1 + a + b*x)/(a + b*x)] - b*c^2*x*Log[(-1 + a + b*x)/(a + b*x)] - 2*b*d^2*Log[d + c*Sqrt[x]]*Log[(-1 + a + b*x)/(a + b*x)] + c^2*Log[1 + a + b*x] + a*c^2*Log[1 + a + b*x] - 2*b*c*d*Sqrt[x]*Log[(1 + a + b*x)/(a + b*x)] + b*c^2*x*Log[(1 + a + b*x)/(a + b*x)] + 2*b*d^2*Log[d + c*Sqrt[x]]*Log[(1 + a + b*x)/(a + b*x)] - 2*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[-1 - a]*c) + Sqrt[b]*d)] - 2*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)] + 2*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[1 - a]*c) + Sqrt[b]*d)] + 2*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a]*c + Sqrt[b]*d)])/(2*b*c^3)","A",1
82,1,596,335,0.9620416,"\int \frac{\coth ^{-1}(d+e x)}{a+b x+c x^2} \, dx","Integrate[ArcCoth[d + e*x]/(a + b*x + c*x^2),x]","\frac{-\text{Li}_2\left(\frac{e \left(-b-2 c x+\sqrt{b^2-4 a c}\right)}{2 c (d+1)+\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 d c+2 c+b e-\sqrt{b^2-4 a c} 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-1)}\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+1)}\right)+\log \left(-\sqrt{b^2-4 a c}+b+2 c x\right) \log \left(\frac{2 c (d+e x-1)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c (d-1)}\right)-\log \left(\sqrt{b^2-4 a c}+b+2 c x\right) \log \left(\frac{2 c (d+e x-1)}{2 c (d-1)-e \left(\sqrt{b^2-4 a c}+b\right)}\right)-\log \left(\frac{d+e x-1}{d+e x}\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x\right)+\log \left(\frac{d+e x-1}{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+1)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c (d+1)}\right)+\log \left(\sqrt{b^2-4 a c}+b+2 c x\right) \log \left(\frac{2 c (d+e x+1)}{2 c (d+1)-e \left(\sqrt{b^2-4 a c}+b\right)}\right)+\log \left(\frac{d+e x+1}{d+e x}\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c x\right)-\log \left(\frac{d+e x+1}{d+e x}\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{2 \sqrt{b^2-4 a c}}","-\frac{\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 c+b e-\sqrt{b^2-4 a c} e\right) (d+e x+1)}+1\right)}{2 \sqrt{b^2-4 a c}}+\frac{\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 (1-d)+\left(b+\sqrt{b^2-4 a c}\right) e\right) (d+e x+1)}+1\right)}{2 \sqrt{b^2-4 a c}}+\frac{\coth ^{-1}(d+e x) \log \left(\frac{2 e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{(d+e x+1) \left(e \left(b-\sqrt{b^2-4 a c}\right)+2 c (1-d)\right)}\right)}{\sqrt{b^2-4 a c}}-\frac{\coth ^{-1}(d+e x) \log \left(\frac{2 e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{(d+e x+1) \left(e \left(\sqrt{b^2-4 a c}+b\right)+2 c (1-d)\right)}\right)}{\sqrt{b^2-4 a c}}",1,"(Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(-1 + d + e*x))/(2*c*(-1 + d) + (-b + Sqrt[b^2 - 4*a*c])*e)] - Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(-1 + d + e*x))/(2*c*(-1 + d) - (b + Sqrt[b^2 - 4*a*c])*e)] - Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(-1 + d + e*x)/(d + e*x)] + Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(-1 + d + e*x)/(d + e*x)] - Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(1 + d + e*x))/(2*c*(1 + d) + (-b + Sqrt[b^2 - 4*a*c])*e)] + Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(1 + d + e*x))/(2*c*(1 + d) - (b + Sqrt[b^2 - 4*a*c])*e)] + Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(1 + d + e*x)/(d + e*x)] - Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(1 + d + e*x)/(d + e*x)] - PolyLog[2, (e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*(1 + d) + (-b + Sqrt[b^2 - 4*a*c])*e)] + PolyLog[2, (e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c - 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e)] - PolyLog[2, (e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*(-1 + d) + (b + Sqrt[b^2 - 4*a*c])*e)] + PolyLog[2, (e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*(1 + d) + (b + Sqrt[b^2 - 4*a*c])*e)])/(2*Sqrt[b^2 - 4*a*c])","A",0
83,1,59,51,0.0186846,"\int x^2 \coth ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x^2*ArcCoth[Sqrt[x]],x]","\frac{1}{90} \left(6 x^{5/2}+10 x^{3/2}+30 x^3 \coth ^{-1}\left(\sqrt{x}\right)+30 \sqrt{x}+15 \log \left(1-\sqrt{x}\right)-15 \log \left(\sqrt{x}+1\right)\right)","\frac{x^{5/2}}{15}+\frac{x^{3/2}}{9}+\frac{1}{3} x^3 \coth ^{-1}\left(\sqrt{x}\right)+\frac{\sqrt{x}}{3}-\frac{1}{3} \tanh ^{-1}\left(\sqrt{x}\right)",1,"(30*Sqrt[x] + 10*x^(3/2) + 6*x^(5/2) + 30*x^3*ArcCoth[Sqrt[x]] + 15*Log[1 - Sqrt[x]] - 15*Log[1 + Sqrt[x]])/90","A",1
84,1,52,42,0.0138348,"\int x \coth ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x*ArcCoth[Sqrt[x]],x]","\frac{1}{12} \left(2 x^{3/2}+6 x^2 \coth ^{-1}\left(\sqrt{x}\right)+6 \sqrt{x}+3 \log \left(1-\sqrt{x}\right)-3 \log \left(\sqrt{x}+1\right)\right)","\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \coth ^{-1}\left(\sqrt{x}\right)+\frac{\sqrt{x}}{2}-\frac{1}{2} \tanh ^{-1}\left(\sqrt{x}\right)",1,"(6*Sqrt[x] + 2*x^(3/2) + 6*x^2*ArcCoth[Sqrt[x]] + 3*Log[1 - Sqrt[x]] - 3*Log[1 + Sqrt[x]])/12","A",1
85,1,22,22,0.0070272,"\int \coth ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[ArcCoth[Sqrt[x]],x]","\sqrt{x}-\tanh ^{-1}\left(\sqrt{x}\right)+x \coth ^{-1}\left(\sqrt{x}\right)","\sqrt{x}-\tanh ^{-1}\left(\sqrt{x}\right)+x \coth ^{-1}\left(\sqrt{x}\right)",1,"Sqrt[x] + x*ArcCoth[Sqrt[x]] - ArcTanh[Sqrt[x]]","A",1
86,1,19,19,0.0101593,"\int \frac{\coth ^{-1}\left(\sqrt{x}\right)}{x} \, dx","Integrate[ArcCoth[Sqrt[x]]/x,x]","\text{Li}_2\left(-\frac{1}{\sqrt{x}}\right)-\text{Li}_2\left(\frac{1}{\sqrt{x}}\right)","\text{Li}_2\left(-\frac{1}{\sqrt{x}}\right)-\text{Li}_2\left(\frac{1}{\sqrt{x}}\right)",1,"PolyLog[2, -(1/Sqrt[x])] - PolyLog[2, 1/Sqrt[x]]","A",1
87,1,45,25,0.0206899,"\int \frac{\coth ^{-1}\left(\sqrt{x}\right)}{x^2} \, dx","Integrate[ArcCoth[Sqrt[x]]/x^2,x]","-\frac{1}{\sqrt{x}}-\frac{1}{2} \log \left(1-\sqrt{x}\right)+\frac{1}{2} \log \left(\sqrt{x}+1\right)-\frac{\coth ^{-1}\left(\sqrt{x}\right)}{x}","-\frac{1}{\sqrt{x}}+\tanh ^{-1}\left(\sqrt{x}\right)-\frac{\coth ^{-1}\left(\sqrt{x}\right)}{x}",1,"-(1/Sqrt[x]) - ArcCoth[Sqrt[x]]/x - Log[1 - Sqrt[x]]/2 + Log[1 + Sqrt[x]]/2","A",1
88,1,58,42,0.0223082,"\int \frac{\coth ^{-1}\left(\sqrt{x}\right)}{x^3} \, dx","Integrate[ArcCoth[Sqrt[x]]/x^3,x]","-\frac{1}{6 x^{3/2}}-\frac{\coth ^{-1}\left(\sqrt{x}\right)}{2 x^2}-\frac{1}{2 \sqrt{x}}-\frac{1}{4} \log \left(1-\sqrt{x}\right)+\frac{1}{4} \log \left(\sqrt{x}+1\right)","-\frac{1}{6 x^{3/2}}-\frac{\coth ^{-1}\left(\sqrt{x}\right)}{2 x^2}-\frac{1}{2 \sqrt{x}}+\frac{1}{2} \tanh ^{-1}\left(\sqrt{x}\right)",1,"-1/6*1/x^(3/2) - 1/(2*Sqrt[x]) - ArcCoth[Sqrt[x]]/(2*x^2) - Log[1 - Sqrt[x]]/4 + Log[1 + Sqrt[x]]/4","A",1
89,1,31,38,0.0163179,"\int x^{3/2} \coth ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x^(3/2)*ArcCoth[Sqrt[x]],x]","\frac{1}{10} \left(4 x^{5/2} \coth ^{-1}\left(\sqrt{x}\right)+(x+2) x+2 \log (1-x)\right)","\frac{2}{5} x^{5/2} \coth ^{-1}\left(\sqrt{x}\right)+\frac{x^2}{10}+\frac{x}{5}+\frac{1}{5} \log (1-x)",1,"(x*(2 + x) + 4*x^(5/2)*ArcCoth[Sqrt[x]] + 2*Log[1 - x])/10","A",1
90,1,25,31,0.01269,"\int \sqrt{x} \coth ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[Sqrt[x]*ArcCoth[Sqrt[x]],x]","\frac{1}{3} \left(2 x^{3/2} \coth ^{-1}\left(\sqrt{x}\right)+x+\log (1-x)\right)","\frac{2}{3} x^{3/2} \coth ^{-1}\left(\sqrt{x}\right)+\frac{x}{3}+\frac{1}{3} \log (1-x)",1,"(x + 2*x^(3/2)*ArcCoth[Sqrt[x]] + Log[1 - x])/3","A",1
91,1,20,20,0.0095633,"\int \frac{\coth ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Integrate[ArcCoth[Sqrt[x]]/Sqrt[x],x]","\log (1-x)+2 \sqrt{x} \coth ^{-1}\left(\sqrt{x}\right)","\log (1-x)+2 \sqrt{x} \coth ^{-1}\left(\sqrt{x}\right)",1,"2*Sqrt[x]*ArcCoth[Sqrt[x]] + Log[1 - x]","A",1
92,1,24,24,0.0215591,"\int \frac{\coth ^{-1}\left(\sqrt{x}\right)}{x^{3/2}} \, dx","Integrate[ArcCoth[Sqrt[x]]/x^(3/2),x]","-\log (1-x)+\log (x)-\frac{2 \coth ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}}","-\log (1-x)+\log (x)-\frac{2 \coth ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}}",1,"(-2*ArcCoth[Sqrt[x]])/Sqrt[x] - Log[1 - x] + Log[x]","A",1
93,1,26,28,0.0149604,"\int \frac{\coth ^{-1}\left(a x^5\right)}{x} \, dx","Integrate[ArcCoth[a*x^5]/x,x]","\frac{1}{10} \left(\text{Li}_2\left(-\frac{1}{a x^5}\right)-\text{Li}_2\left(\frac{1}{a x^5}\right)\right)","\frac{1}{10} \text{Li}_2\left(-\frac{1}{a x^5}\right)-\frac{1}{10} \text{Li}_2\left(\frac{1}{a x^5}\right)",1,"(PolyLog[2, -(1/(a*x^5))] - PolyLog[2, 1/(a*x^5)])/10","A",1
94,1,19,19,0.0020576,"\int \coth ^{-1}\left(\frac{1}{x}\right) \, dx","Integrate[ArcCoth[x^(-1)],x]","\frac{1}{2} \log \left(1-x^2\right)+x \coth ^{-1}\left(\frac{1}{x}\right)","\frac{1}{2} \log \left(1-x^2\right)+x \coth ^{-1}\left(\frac{1}{x}\right)",1,"x*ArcCoth[x^(-1)] + Log[1 - x^2]/2","A",1
95,1,97,38,0.0543515,"\int \frac{\coth ^{-1}\left(a x^n\right)}{x} \, dx","Integrate[ArcCoth[a*x^n]/x,x]","\frac{-\text{Li}_2\left(1-a x^n\right)+\text{Li}_2\left(a x^n+1\right)+n \log (x) \log \left(a x^n-1\right)-n \log (x) \log \left(a x^n+1\right)-\log \left(a x^n\right) \log \left(a x^n-1\right)+\log \left(-a x^n\right) \log \left(a x^n+1\right)+2 n \log (x) \coth ^{-1}\left(a x^n\right)}{2 n}","\frac{\text{Li}_2\left(-\frac{x^{-n}}{a}\right)}{2 n}-\frac{\text{Li}_2\left(\frac{x^{-n}}{a}\right)}{2 n}",1,"(2*n*ArcCoth[a*x^n]*Log[x] + n*Log[x]*Log[-1 + a*x^n] - Log[a*x^n]*Log[-1 + a*x^n] - n*Log[x]*Log[1 + a*x^n] + Log[-(a*x^n)]*Log[1 + a*x^n] - PolyLog[2, 1 - a*x^n] + PolyLog[2, 1 + a*x^n])/(2*n)","B",1
96,1,66,39,0.0334712,"\int (a+b x) \coth ^{-1}(a+b x) \, dx","Integrate[(a + b*x)*ArcCoth[a + b*x],x]","\frac{a^2 \log (a+b x+1)-\left(a^2-1\right) \log (-a-b x+1)-\log (a+b x+1)+2 b x (2 a+b x) \coth ^{-1}(a+b x)+2 b x}{4 b}","-\frac{\tanh ^{-1}(a+b x)}{2 b}+\frac{(a+b x)^2 \coth ^{-1}(a+b x)}{2 b}+\frac{x}{2}",1,"(2*b*x + 2*b*x*(2*a + b*x)*ArcCoth[a + b*x] - (-1 + a^2)*Log[1 - a - b*x] - Log[1 + a + b*x] + a^2*Log[1 + a + b*x])/(4*b)","A",1
97,1,42,54,0.0410854,"\int (a+b x)^2 \coth ^{-1}(a+b x) \, dx","Integrate[(a + b*x)^2*ArcCoth[a + b*x],x]","\frac{(a+b x)^2+\log \left(1-(a+b x)^2\right)+2 (a+b x)^3 \coth ^{-1}(a+b x)}{6 b}","\frac{(a+b x)^2}{6 b}+\frac{\log \left(1-(a+b x)^2\right)}{6 b}+\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}",1,"((a + b*x)^2 + 2*(a + b*x)^3*ArcCoth[a + b*x] + Log[1 - (a + b*x)^2])/(6*b)","A",1
98,1,286,35,0.02563,"\int \frac{\coth ^{-1}(a+b x)}{a+b x} \, dx","Integrate[ArcCoth[a + b*x]/(a + b*x),x]","-\frac{\text{Li}_2(-a-b x)}{2 b}+\frac{\text{Li}_2(a+b x)}{2 b}-\frac{\log ^2\left(\frac{a b-(a-1) b}{b (a+b x)}\right)}{4 b}+\frac{\log ^2\left(\frac{a b-(a+1) b}{b (a+b x)}\right)}{4 b}-\frac{\log \left(\frac{b (a+b x-1)}{(a-1) b-a b}\right) \log \left(\frac{a b-(a-1) b}{b (a+b x)}\right)}{2 b}+\frac{\log \left(\frac{a+b x-1}{a+b x}\right) \log \left(\frac{a b-(a-1) b}{b (a+b x)}\right)}{2 b}+\frac{\log \left(\frac{b (-a-b x-1)}{(-a-1) b+a b}\right) \log \left(\frac{a b-(a+1) b}{b (a+b x)}\right)}{2 b}-\frac{\log \left(\frac{a b-(a+1) b}{b (a+b x)}\right) \log \left(\frac{a+b x+1}{a+b x}\right)}{2 b}","\frac{\text{Li}_2\left(-\frac{1}{a+b x}\right)}{2 b}-\frac{\text{Li}_2\left(\frac{1}{a+b x}\right)}{2 b}",1,"-1/2*(Log[(b*(-1 + a + b*x))/((-1 + a)*b - a*b)]*Log[(-((-1 + a)*b) + a*b)/(b*(a + b*x))])/b - Log[(-((-1 + a)*b) + a*b)/(b*(a + b*x))]^2/(4*b) + (Log[(b*(-1 - a - b*x))/((-1 - a)*b + a*b)]*Log[(a*b - (1 + a)*b)/(b*(a + b*x))])/(2*b) + Log[(a*b - (1 + a)*b)/(b*(a + b*x))]^2/(4*b) + (Log[(-((-1 + a)*b) + a*b)/(b*(a + b*x))]*Log[(-1 + a + b*x)/(a + b*x)])/(2*b) - (Log[(a*b - (1 + a)*b)/(b*(a + b*x))]*Log[(1 + a + b*x)/(a + b*x)])/(2*b) - PolyLog[2, -a - b*x]/(2*b) + PolyLog[2, a + b*x]/(2*b)","B",0
99,1,43,48,0.0287355,"\int \frac{\coth ^{-1}(a+b x)}{(a+b x)^2} \, dx","Integrate[ArcCoth[a + b*x]/(a + b*x)^2,x]","-\frac{-2 \log (a+b x)+\log \left(1-(a+b x)^2\right)+\frac{2 \coth ^{-1}(a+b x)}{a+b x}}{2 b}","\frac{\log (a+b x)}{b}-\frac{\log \left(1-(a+b x)^2\right)}{2 b}-\frac{\coth ^{-1}(a+b x)}{b (a+b x)}",1,"-1/2*((2*ArcCoth[a + b*x])/(a + b*x) - 2*Log[a + b*x] + Log[1 - (a + b*x)^2])/b","A",1
100,1,117,25,0.0152774,"\int \frac{\coth ^{-1}(1+x)}{2+2 x} \, dx","Integrate[ArcCoth[1 + x]/(2 + 2*x),x]","-\frac{\text{Li}_2(-x-1)}{4}+\frac{\text{Li}_2(x+1)}{4}+\frac{1}{8} \log ^2\left(-\frac{1}{x+1}\right)-\frac{1}{8} \log ^2\left(\frac{1}{x+1}\right)+\frac{1}{4} \log (x+2) \log \left(-\frac{1}{x+1}\right)-\frac{1}{4} \log \left(\frac{x+2}{x+1}\right) \log \left(-\frac{1}{x+1}\right)-\frac{1}{4} \log (-x) \log \left(\frac{1}{x+1}\right)+\frac{1}{4} \log \left(\frac{1}{x+1}\right) \log \left(\frac{x}{x+1}\right)","\frac{1}{4} \text{Li}_2\left(-\frac{1}{x+1}\right)-\frac{1}{4} \text{Li}_2\left(\frac{1}{x+1}\right)",1,"Log[-(1 + x)^(-1)]^2/8 - (Log[-x]*Log[(1 + x)^(-1)])/4 - Log[(1 + x)^(-1)]^2/8 + (Log[(1 + x)^(-1)]*Log[x/(1 + x)])/4 + (Log[-(1 + x)^(-1)]*Log[2 + x])/4 - (Log[-(1 + x)^(-1)]*Log[(2 + x)/(1 + x)])/4 - PolyLog[2, -1 - x]/4 + PolyLog[2, 1 + x]/4","B",0
101,1,312,35,0.0279262,"\int \frac{\coth ^{-1}(a+b x)}{\frac{a d}{b}+d x} \, dx","Integrate[ArcCoth[a + b*x]/((a*d)/b + d*x),x]","b \left(-\frac{\text{Li}_2(-a-b x)}{2 b d}+\frac{\text{Li}_2(a+b x)}{2 b d}-\frac{\log ^2\left(\frac{a b-(a-1) b}{b (a+b x)}\right)}{4 b d}+\frac{\log ^2\left(\frac{a b-(a+1) b}{b (a+b x)}\right)}{4 b d}-\frac{\log \left(\frac{b (a+b x-1)}{(a-1) b-a b}\right) \log \left(\frac{a b-(a-1) b}{b (a+b x)}\right)}{2 b d}+\frac{\log \left(\frac{a+b x-1}{a+b x}\right) \log \left(\frac{a b-(a-1) b}{b (a+b x)}\right)}{2 b d}+\frac{\log \left(\frac{b (-a-b x-1)}{(-a-1) b+a b}\right) \log \left(\frac{a b-(a+1) b}{b (a+b x)}\right)}{2 b d}-\frac{\log \left(\frac{a b-(a+1) b}{b (a+b x)}\right) \log \left(\frac{a+b x+1}{a+b x}\right)}{2 b d}\right)","\frac{\text{Li}_2\left(-\frac{1}{a+b x}\right)}{2 d}-\frac{\text{Li}_2\left(\frac{1}{a+b x}\right)}{2 d}",1,"b*(-1/2*(Log[(b*(-1 + a + b*x))/((-1 + a)*b - a*b)]*Log[(-((-1 + a)*b) + a*b)/(b*(a + b*x))])/(b*d) - Log[(-((-1 + a)*b) + a*b)/(b*(a + b*x))]^2/(4*b*d) + (Log[(b*(-1 - a - b*x))/((-1 - a)*b + a*b)]*Log[(a*b - (1 + a)*b)/(b*(a + b*x))])/(2*b*d) + Log[(a*b - (1 + a)*b)/(b*(a + b*x))]^2/(4*b*d) + (Log[(-((-1 + a)*b) + a*b)/(b*(a + b*x))]*Log[(-1 + a + b*x)/(a + b*x)])/(2*b*d) - (Log[(a*b - (1 + a)*b)/(b*(a + b*x))]*Log[(1 + a + b*x)/(a + b*x)])/(2*b*d) - PolyLog[2, -a - b*x]/(2*b*d) + PolyLog[2, a + b*x]/(2*b*d))","B",0
102,1,270,168,0.3019811,"\int (e+f x)^3 \left(a+b \coth ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)^3*(a + b*ArcCoth[c + d*x]),x]","\frac{6 d x \left(4 a d^3 e^3+b f \left(\left(3 c^2+1\right) f^2-8 c d e f+6 d^2 e^2\right)\right)+6 d^2 f x^2 \left(6 a d^2 e^2+b f (2 d e-c f)\right)+2 d^3 f^2 x^3 (12 a d e+b f)+6 a d^4 f^3 x^4+6 b d^4 x \left(4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right) \coth ^{-1}(c+d x)-3 b (c-1) \left(-6 (c-1) d^2 e^2 f+4 (c-1)^2 d e f^2-(c-1)^3 f^3+4 d^3 e^3\right) \log (-c-d x+1)-3 b (c+1) \left(6 (c+1) d^2 e^2 f-4 (c+1)^2 d e f^2+(c+1)^3 f^3-4 d^3 e^3\right) \log (c+d x+1)}{24 d^4}","\frac{(e+f x)^4 \left(a+b \coth ^{-1}(c+d x)\right)}{4 f}+\frac{b f x \left(\left(6 c^2+1\right) f^2-12 c d e f+6 d^2 e^2\right)}{4 d^3}+\frac{b f^2 (c+d x)^2 (d e-c f)}{2 d^4}-\frac{b (-c f+d e-f)^4 \log (c+d x+1)}{8 d^4 f}+\frac{b (-c f+d e+f)^4 \log (-c-d x+1)}{8 d^4 f}+\frac{b f^3 (c+d x)^3}{12 d^4}",1,"(6*d*(4*a*d^3*e^3 + b*f*(6*d^2*e^2 - 8*c*d*e*f + (1 + 3*c^2)*f^2))*x + 6*d^2*f*(6*a*d^2*e^2 + b*f*(2*d*e - c*f))*x^2 + 2*d^3*f^2*(12*a*d*e + b*f)*x^3 + 6*a*d^4*f^3*x^4 + 6*b*d^4*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)*ArcCoth[c + d*x] - 3*b*(-1 + c)*(4*d^3*e^3 - 6*(-1 + c)*d^2*e^2*f + 4*(-1 + c)^2*d*e*f^2 - (-1 + c)^3*f^3)*Log[1 - c - d*x] - 3*b*(1 + c)*(-4*d^3*e^3 + 6*(1 + c)*d^2*e^2*f - 4*(1 + c)^2*d*e*f^2 + (1 + c)^3*f^3)*Log[1 + c + d*x])/(24*d^4)","A",1
103,1,174,120,0.1705623,"\int (e+f x)^2 \left(a+b \coth ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)^2*(a + b*ArcCoth[c + d*x]),x]","\frac{2 d x \left(3 a d^2 e^2+b f (3 d e-2 c f)\right)+d^2 f x^2 (6 a d e+b f)+2 a d^3 f^2 x^3+2 b d^3 x \left(3 e^2+3 e f x+f^2 x^2\right) \coth ^{-1}(c+d x)-b (c-1) \left(-3 (c-1) d e f+(c-1)^2 f^2+3 d^2 e^2\right) \log (-c-d x+1)+b (c+1) \left(-3 (c+1) d e f+(c+1)^2 f^2+3 d^2 e^2\right) \log (c+d x+1)}{6 d^3}","\frac{(e+f x)^3 \left(a+b \coth ^{-1}(c+d x)\right)}{3 f}+\frac{b (-c f+d e+f)^3 \log (-c-d x+1)}{6 d^3 f}-\frac{b (d e-(c+1) f)^3 \log (c+d x+1)}{6 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,"(2*d*(3*a*d^2*e^2 + b*f*(3*d*e - 2*c*f))*x + d^2*f*(6*a*d*e + b*f)*x^2 + 2*a*d^3*f^2*x^3 + 2*b*d^3*x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCoth[c + d*x] - b*(-1 + c)*(3*d^2*e^2 - 3*(-1 + c)*d*e*f + (-1 + c)^2*f^2)*Log[1 - c - d*x] + b*(1 + c)*(3*d^2*e^2 - 3*(1 + c)*d*e*f + (1 + c)^2*f^2)*Log[1 + c + d*x])/(6*d^3)","A",1
104,1,138,97,0.0476516,"\int (e+f x) \left(a+b \coth ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)*(a + b*ArcCoth[c + d*x]),x]","a e x+\frac{1}{2} a f x^2+\frac{b \left(c^2-2 c+1\right) f \log (-c-d x+1)}{4 d^2}+\frac{b \left(-c^2-2 c-1\right) f \log (c+d x+1)}{4 d^2}+\frac{b e ((c+1) \log (c+d x+1)-(c-1) \log (-c-d x+1))}{2 d}+b e x \coth ^{-1}(c+d x)+\frac{1}{2} b f x^2 \coth ^{-1}(c+d x)+\frac{b f x}{2 d}","\frac{(e+f x)^2 \left(a+b \coth ^{-1}(c+d x)\right)}{2 f}+\frac{b (-c f+d e+f)^2 \log (-c-d x+1)}{4 d^2 f}-\frac{b (d e-(c+1) f)^2 \log (c+d x+1)}{4 d^2 f}+\frac{b f x}{2 d}",1,"a*e*x + (b*f*x)/(2*d) + (a*f*x^2)/2 + b*e*x*ArcCoth[c + d*x] + (b*f*x^2*ArcCoth[c + d*x])/2 + (b*(1 - 2*c + c^2)*f*Log[1 - c - d*x])/(4*d^2) + (b*(-1 - 2*c - c^2)*f*Log[1 + c + d*x])/(4*d^2) + (b*e*(-((-1 + c)*Log[1 - c - d*x]) + (1 + c)*Log[1 + c + d*x]))/(2*d)","A",1
105,1,48,40,0.0168066,"\int \left(a+b \coth ^{-1}(c+d x)\right) \, dx","Integrate[a + b*ArcCoth[c + d*x],x]","a x+\frac{b ((c+1) \log (c+d x+1)-(c-1) \log (-c-d x+1))}{2 d}+b x \coth ^{-1}(c+d x)","a x+\frac{b \log \left(1-(c+d x)^2\right)}{2 d}+\frac{b (c+d x) \coth ^{-1}(c+d x)}{d}",1,"a*x + b*x*ArcCoth[c + d*x] + (b*(-((-1 + c)*Log[1 - c - d*x]) + (1 + c)*Log[1 + c + d*x]))/(2*d)","A",1
106,1,206,130,0.1173704,"\int \frac{a+b \coth ^{-1}(c+d x)}{e+f x} \, dx","Integrate[(a + b*ArcCoth[c + d*x])/(e + f*x),x]","\frac{a \log (e+f x)}{f}-\frac{b \text{Li}_2\left(\frac{d (e+f x)}{d e-c f-f}\right)}{2 f}+\frac{b \text{Li}_2\left(\frac{d (e+f x)}{d e-c f+f}\right)}{2 f}+\frac{b \log (e+f x) \log \left(\frac{f (-c-d x+1)}{-c f+d e+f}\right)}{2 f}-\frac{b \log \left(-\frac{-c-d x+1}{c+d x}\right) \log (e+f x)}{2 f}-\frac{b \log (e+f x) \log \left(-\frac{f (c+d x+1)}{-c f+d e-f}\right)}{2 f}+\frac{b \log \left(\frac{c+d x+1}{c+d x}\right) \log (e+f x)}{2 f}","\frac{\left(a+b \coth ^{-1}(c+d x)\right) \log \left(\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right)}{f}-\frac{\log \left(\frac{2}{c+d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)}{f}-\frac{b \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right)}{2 f}+\frac{b \text{Li}_2\left(1-\frac{2}{c+d x+1}\right)}{2 f}",1,"(a*Log[e + f*x])/f + (b*Log[(f*(1 - c - d*x))/(d*e + f - c*f)]*Log[e + f*x])/(2*f) - (b*Log[-((1 - c - d*x)/(c + d*x))]*Log[e + f*x])/(2*f) - (b*Log[-((f*(1 + c + d*x))/(d*e - f - c*f))]*Log[e + f*x])/(2*f) + (b*Log[(1 + c + d*x)/(c + d*x)]*Log[e + f*x])/(2*f) - (b*PolyLog[2, (d*(e + f*x))/(d*e - f - c*f)])/(2*f) + (b*PolyLog[2, (d*(e + f*x))/(d*e + f - c*f)])/(2*f)","A",0
107,1,125,115,0.1927964,"\int \frac{a+b \coth ^{-1}(c+d x)}{(e+f x)^2} \, dx","Integrate[(a + b*ArcCoth[c + d*x])/(e + f*x)^2,x]","\frac{1}{2} \left(-\frac{2 a}{f (e+f x)}-\frac{2 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 \log (-c-d x+1)}{f ((c-1) f-d e)}-\frac{b d \log (c+d x+1)}{f (c f-d e+f)}-\frac{2 b \coth ^{-1}(c+d x)}{f (e+f x)}\right)","-\frac{a+b \coth ^{-1}(c+d x)}{f (e+f x)}-\frac{b d \log (-c-d x+1)}{2 f (-c f+d e+f)}+\frac{b d \log (c+d x+1)}{2 f (-c f+d e-f)}-\frac{b d \log (e+f x)}{(-c f+d e+f) (d e-(c+1) f)}",1,"((-2*a)/(f*(e + f*x)) - (2*b*ArcCoth[c + d*x])/(f*(e + f*x)) + (b*d*Log[1 - c - d*x])/(f*(-(d*e) + (-1 + c)*f)) - (b*d*Log[1 + c + d*x])/(f*(-(d*e) + f + c*f)) - (2*b*d*Log[e + f*x])/(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2))/2","A",1
108,1,174,167,0.3311451,"\int \frac{a+b \coth ^{-1}(c+d x)}{(e+f x)^3} \, dx","Integrate[(a + b*ArcCoth[c + d*x])/(e + f*x)^3,x]","\frac{1}{4} \left(-\frac{2 a}{f (e+f x)^2}+\frac{2 b d}{(e+f x) \left(\left(c^2-1\right) f^2-2 c d e f+d^2 e^2\right)}-\frac{4 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 \log (-c-d x+1)}{f (-c f+d e+f)^2}+\frac{b d^2 \log (c+d x+1)}{f (c f-d e+f)^2}-\frac{2 b \coth ^{-1}(c+d x)}{f (e+f x)^2}\right)","-\frac{a+b \coth ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac{b d^2 \log (-c-d x+1)}{4 f (-c f+d e+f)^2}+\frac{b d^2 \log (c+d x+1)}{4 f (-c f+d e-f)^2}-\frac{b d^2 (d e-c f) \log (e+f x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac{b d}{2 (e+f x) (-c f+d e+f) (d e-(c+1) f)}",1,"((-2*a)/(f*(e + f*x)^2) + (2*b*d)/((d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)*(e + f*x)) - (2*b*ArcCoth[c + d*x])/(f*(e + f*x)^2) - (b*d^2*Log[1 - c - d*x])/(f*(d*e + f - c*f)^2) + (b*d^2*Log[1 + c + d*x])/(f*(-(d*e) + f + c*f)^2) - (4*b*d^2*(d*e - c*f)*Log[e + f*x])/(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)^2)/4","A",1
109,1,1054,374,7.3619423,"\int (e+f x)^2 \left(a+b \coth ^{-1}(c+d x)\right)^2 \, dx","Integrate[(e + f*x)^2*(a + b*ArcCoth[c + d*x])^2,x]","\frac{1}{3} a^2 f^2 x^3+a^2 e f x^2+a^2 e^2 x+\frac{1}{3} a b \left(2 x \left(3 e^2+3 f x e+f^2 x^2\right) \coth ^{-1}(c+d x)+\frac{d f x (6 d e-4 c f+d f x)-(c-1) \left(3 d^2 e^2-3 (c-1) d f e+(c-1)^2 f^2\right) \log (-c-d x+1)+(c+1) \left(3 d^2 e^2-3 (c+1) d f e+(c+1)^2 f^2\right) \log (c+d x+1)}{d^3}\right)+\frac{b^2 e^2 \left(1-(c+d x)^2\right) \left(\coth ^{-1}(c+d x) \left(-\left((c+d x) \coth ^{-1}(c+d x)\right)+\coth ^{-1}(c+d x)+2 \log \left(1-e^{-2 \coth ^{-1}(c+d x)}\right)\right)-\text{Li}_2\left(e^{-2 \coth ^{-1}(c+d x)}\right)\right)}{d (c+d x)^2 \left(1-\frac{1}{(c+d x)^2}\right)}-\frac{b^2 e f \left(1-(c+d x)^2\right) \left(2 c \coth ^{-1}(c+d x)^2+(c+d x)^2 \left(1-\frac{1}{(c+d x)^2}\right) \coth ^{-1}(c+d x)^2-2 (c+d x) \left(c \coth ^{-1}(c+d x)-1\right) \coth ^{-1}(c+d x)+4 c \log \left(1-e^{-2 \coth ^{-1}(c+d x)}\right) \coth ^{-1}(c+d x)-2 \log \left(\frac{1}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}\right)-2 c \text{Li}_2\left(e^{-2 \coth ^{-1}(c+d x)}\right)\right)}{d^2 (c+d x)^2 \left(1-\frac{1}{(c+d x)^2}\right)}-\frac{b^2 f^2 (c+d x) \sqrt{1-\frac{1}{(c+d x)^2}} \left(1-(c+d x)^2\right) \left(\frac{9 \coth ^{-1}(c+d x)^2 c^2}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}+3 \coth ^{-1}(c+d x)^2 \cosh \left(3 \coth ^{-1}(c+d x)\right) c^2+\frac{18 \coth ^{-1}(c+d x) \log \left(1-e^{-2 \coth ^{-1}(c+d x)}\right) c^2}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}-3 \coth ^{-1}(c+d x)^2 \sinh \left(3 \coth ^{-1}(c+d x)\right) c^2-6 \coth ^{-1}(c+d x) \log \left(1-e^{-2 \coth ^{-1}(c+d x)}\right) \sinh \left(3 \coth ^{-1}(c+d x)\right) c^2-\frac{12 \coth ^{-1}(c+d x)^2 c}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}-6 \coth ^{-1}(c+d x) \cosh \left(3 \coth ^{-1}(c+d x)\right) c-\frac{18 \log \left(\frac{1}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}\right) c}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}+6 \log \left(\frac{1}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}\right) \sinh \left(3 \coth ^{-1}(c+d x)\right) c+\frac{3 \coth ^{-1}(c+d x)^2}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}+\frac{4 \coth ^{-1}(c+d x)}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}+\frac{-3 c^2 \coth ^{-1}(c+d x)^2+3 \coth ^{-1}(c+d x)^2+6 c \coth ^{-1}(c+d x)-1}{\sqrt{1-\frac{1}{(c+d x)^2}}}+\coth ^{-1}(c+d x)^2 \cosh \left(3 \coth ^{-1}(c+d x)\right)+\cosh \left(3 \coth ^{-1}(c+d x)\right)+\frac{6 \coth ^{-1}(c+d x) \log \left(1-e^{-2 \coth ^{-1}(c+d x)}\right)}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}+\frac{4 \left(3 c^2+1\right) \text{Li}_2\left(e^{-2 \coth ^{-1}(c+d x)}\right)}{(c+d x)^3 \left(1-\frac{1}{(c+d x)^2}\right)^{3/2}}-\coth ^{-1}(c+d x)^2 \sinh \left(3 \coth ^{-1}(c+d x)\right)-2 \coth ^{-1}(c+d x) \log \left(1-e^{-2 \coth ^{-1}(c+d x)}\right) \sinh \left(3 \coth ^{-1}(c+d x)\right)\right)}{12 d^3}","-\frac{(d e-c f) \left(\left(c^2+3\right) f^2-2 c d e f+d^2 e^2\right) \left(a+b \coth ^{-1}(c+d x)\right)^2}{3 d^3 f}+\frac{\left(\left(3 c^2+1\right) f^2-6 c d e f+3 d^2 e^2\right) \left(a+b \coth ^{-1}(c+d x)\right)^2}{3 d^3}-\frac{2 b \left(\left(3 c^2+1\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)}{3 d^3}+\frac{b f^2 (c+d x)^2 \left(a+b \coth ^{-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 \coth ^{-1}(c+d x)\right)^2}{3 f}-\frac{b^2 \left(\left(3 c^2+1\right) f^2-6 c d e f+3 d^2 e^2\right) \text{Li}_2\left(-\frac{c+d x+1}{-c-d x+1}\right)}{3 d^3}+\frac{b^2 f (d e-c f) \log \left(1-(c+d x)^2\right)}{d^3}+\frac{2 b^2 f (c+d x) (d e-c f) \coth ^{-1}(c+d x)}{d^3}-\frac{b^2 f^2 \tanh ^{-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*(2*x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCoth[c + d*x] + (d*f*x*(6*d*e - 4*c*f + d*f*x) - (-1 + c)*(3*d^2*e^2 - 3*(-1 + c)*d*e*f + (-1 + c)^2*f^2)*Log[1 - c - d*x] + (1 + c)*(3*d^2*e^2 - 3*(1 + c)*d*e*f + (1 + c)^2*f^2)*Log[1 + c + d*x])/d^3))/3 + (b^2*e^2*(1 - (c + d*x)^2)*(ArcCoth[c + d*x]*(ArcCoth[c + d*x] - (c + d*x)*ArcCoth[c + d*x] + 2*Log[1 - E^(-2*ArcCoth[c + d*x])]) - PolyLog[2, E^(-2*ArcCoth[c + d*x])]))/(d*(c + d*x)^2*(1 - (c + d*x)^(-2))) - (b^2*e*f*(1 - (c + d*x)^2)*(2*c*ArcCoth[c + d*x]^2 + (c + d*x)^2*(1 - (c + d*x)^(-2))*ArcCoth[c + d*x]^2 - 2*(c + d*x)*ArcCoth[c + d*x]*(-1 + c*ArcCoth[c + d*x]) + 4*c*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])] - 2*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])] - 2*c*PolyLog[2, E^(-2*ArcCoth[c + d*x])]))/(d^2*(c + d*x)^2*(1 - (c + d*x)^(-2))) - (b^2*f^2*(c + d*x)*Sqrt[1 - (c + d*x)^(-2)]*(1 - (c + d*x)^2)*((4*ArcCoth[c + d*x])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (3*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (12*c*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (9*c^2*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (-1 + 6*c*ArcCoth[c + d*x] + 3*ArcCoth[c + d*x]^2 - 3*c^2*ArcCoth[c + d*x]^2)/Sqrt[1 - (c + d*x)^(-2)] + Cosh[3*ArcCoth[c + d*x]] - 6*c*ArcCoth[c + d*x]*Cosh[3*ArcCoth[c + d*x]] + ArcCoth[c + d*x]^2*Cosh[3*ArcCoth[c + d*x]] + 3*c^2*ArcCoth[c + d*x]^2*Cosh[3*ArcCoth[c + d*x]] + (6*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (18*c^2*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (18*c*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (4*(1 + 3*c^2)*PolyLog[2, E^(-2*ArcCoth[c + d*x])])/((c + d*x)^3*(1 - (c + d*x)^(-2))^(3/2)) - ArcCoth[c + d*x]^2*Sinh[3*ArcCoth[c + d*x]] - 3*c^2*ArcCoth[c + d*x]^2*Sinh[3*ArcCoth[c + d*x]] - 2*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])]*Sinh[3*ArcCoth[c + d*x]] - 6*c^2*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])]*Sinh[3*ArcCoth[c + d*x]] + 6*c*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])]*Sinh[3*ArcCoth[c + d*x]]))/(12*d^3)","B",0
110,1,295,221,0.6085126,"\int (e+f x) \left(a+b \coth ^{-1}(c+d x)\right)^2 \, dx","Integrate[(e + f*x)*(a + b*ArcCoth[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 \coth ^{-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 \coth ^{-1}(c+d x)}\right)\right)-4 a b d e \log \left(\frac{1}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}\right)+a b f \log (-c-d x+1)-a b f \log (c+d x+1)+4 a b c f \log \left(\frac{1}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}\right)+2 a b c f+2 a b d f x+2 b^2 (d e-c f) \text{Li}_2\left(e^{-2 \coth ^{-1}(c+d x)}\right)+b^2 (c+d x-1) \coth ^{-1}(c+d x)^2 (-c f+2 d e+d f x+f)-2 b^2 f \log \left(\frac{1}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}\right)}{2 d^2}","-\frac{\left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right) \left(a+b \coth ^{-1}(c+d x)\right)^2}{2 d^2 f}+\frac{(d e-c f) \left(a+b \coth ^{-1}(c+d x)\right)^2}{d^2}-\frac{2 b (d e-c f) \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)}{d^2}+\frac{(e+f x)^2 \left(a+b \coth ^{-1}(c+d x)\right)^2}{2 f}+\frac{a b f x}{d}-\frac{b^2 (d e-c f) \text{Li}_2\left(-\frac{c+d x+1}{-c-d x+1}\right)}{d^2}+\frac{b^2 f \log \left(1-(c+d x)^2\right)}{2 d^2}+\frac{b^2 f (c+d x) \coth ^{-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*(-1 + c + d*x)*(2*d*e + f - c*f + d*f*x)*ArcCoth[c + d*x]^2 + 2*b*ArcCoth[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*ArcCoth[c + d*x])]) + a*b*f*Log[1 - c - d*x] - a*b*f*Log[1 + 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*b^2*(d*e - c*f)*PolyLog[2, E^(-2*ArcCoth[c + d*x])])/(2*d^2)","A",0
111,1,111,97,0.1658941,"\int \left(a+b \coth ^{-1}(c+d x)\right)^2 \, dx","Integrate[(a + b*ArcCoth[c + d*x])^2,x]","\frac{a \left(a c+a d x-2 b \log \left(\frac{1}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}\right)\right)+2 b \coth ^{-1}(c+d x) \left(a c+a d x-b \log \left(1-e^{-2 \coth ^{-1}(c+d x)}\right)\right)+b^2 \text{Li}_2\left(e^{-2 \coth ^{-1}(c+d x)}\right)+b^2 (c+d x-1) \coth ^{-1}(c+d x)^2}{d}","\frac{(c+d x) \left(a+b \coth ^{-1}(c+d x)\right)^2}{d}+\frac{\left(a+b \coth ^{-1}(c+d x)\right)^2}{d}-\frac{2 b \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)}{d}-\frac{b^2 \text{Li}_2\left(-\frac{c+d x+1}{-c-d x+1}\right)}{d}",1,"(b^2*(-1 + c + d*x)*ArcCoth[c + d*x]^2 + 2*b*ArcCoth[c + d*x]*(a*c + a*d*x - b*Log[1 - E^(-2*ArcCoth[c + d*x])]) + a*(a*c + a*d*x - 2*b*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])]) + b^2*PolyLog[2, E^(-2*ArcCoth[c + d*x])])/d","A",0
112,1,3759,214,31.0590737,"\int \frac{\left(a+b \coth ^{-1}(c+d x)\right)^2}{e+f x} \, dx","Integrate[(a + b*ArcCoth[c + d*x])^2/(e + f*x),x]","\text{Result too large to show}","-\frac{b \left(a+b \coth ^{-1}(c+d x)\right) \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right)}{f}+\frac{\left(a+b \coth ^{-1}(c+d x)\right)^2 \log \left(\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right)}{f}+\frac{b \text{Li}_2\left(1-\frac{2}{c+d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)}{f}-\frac{\log \left(\frac{2}{c+d x+1}\right) \left(a+b \coth ^{-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+f) (c+d x+1)}\right)}{2 f}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{c+d x+1}\right)}{2 f}",1,"(a^2*Log[e + f*x])/f + 2*a*b*(((ArcCoth[c + d*x] - ArcTanh[c + d*x])*Log[e + f*x])/f - (I*(I*ArcTanh[c + d*x]*(-Log[1/Sqrt[1 - (c + d*x)^2]] + Log[I*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]]) + ((-I)*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTanh[c + d*x])^2 - (I/4)*(Pi - (2*I)*ArcTanh[c + d*x])^2 + 2*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTanh[c + d*x])*Log[1 - E^((2*I)*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTanh[c + d*x]))] + (Pi - (2*I)*ArcTanh[c + d*x])*Log[1 - E^(I*(Pi - (2*I)*ArcTanh[c + d*x]))] - (Pi - (2*I)*ArcTanh[c + d*x])*Log[2*Sin[(Pi - (2*I)*ArcTanh[c + d*x])/2]] - 2*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTanh[c + d*x])*Log[(2*I)*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]] - I*PolyLog[2, E^((2*I)*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTanh[c + d*x]))] - I*PolyLog[2, E^(I*(Pi - (2*I)*ArcTanh[c + d*x]))])/2))/f) - (b^2*(d*e - c*f + f*(c + d*x))*(1 - (c + d*x)^2)*(-1/24*(I*f*Pi^3 - 8*d*e*ArcCoth[c + d*x]^3 - 8*f*ArcCoth[c + d*x]^3 + 8*c*f*ArcCoth[c + d*x]^3 + 24*f*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])] + 24*f*ArcCoth[c + d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])] - 12*f*PolyLog[3, E^(2*ArcCoth[c + d*x])])/f^2 + ((-(d*e) - f + c*f)*(-(d*e) + f + c*f)*(2*d^2*e^2*ArcCoth[c + d*x]^3 - 8*d*e*f*ArcCoth[c + d*x]^3 - 4*c*d*e*f*ArcCoth[c + d*x]^3 + 4*d*e*E^(2*ArcTanh[f/(d*e - c*f)])*f*ArcCoth[c + d*x]^3 - 10*f^2*ArcCoth[c + d*x]^3 + 8*c*f^2*ArcCoth[c + d*x]^3 + 2*c^2*f^2*ArcCoth[c + d*x]^3 - 4*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*ArcCoth[c + d*x]^3 - 4*c*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*ArcCoth[c + d*x]^3 - (4*d^2*e^2*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] - (4*d*e*f*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] + (8*c*d*e*f*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] + (4*c*f^2*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] - (4*c^2*f^2*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] + (6*I)*d*e*f*Pi*ArcCoth[c + d*x]*Log[2] + (6*I)*f^2*Pi*ArcCoth[c + d*x]*Log[2] - (6*I)*c*f^2*Pi*ArcCoth[c + d*x]*Log[2] - d*e*f*ArcCoth[c + d*x]^2*Log[64] - f^2*ArcCoth[c + d*x]^2*Log[64] + c*f^2*ArcCoth[c + d*x]^2*Log[64] - (6*I)*d*e*f*Pi*ArcCoth[c + d*x]*Log[E^(-ArcCoth[c + d*x]) + E^ArcCoth[c + d*x]] - (6*I)*f^2*Pi*ArcCoth[c + d*x]*Log[E^(-ArcCoth[c + d*x]) + E^ArcCoth[c + d*x]] + (6*I)*c*f^2*Pi*ArcCoth[c + d*x]*Log[E^(-ArcCoth[c + d*x]) + E^ArcCoth[c + d*x]] + 6*d*e*f*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 6*d*e*E^(2*ArcTanh[f/(d*e - c*f)])*f*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 6*f^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 6*c*f^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 6*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 6*c*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 12*d*e*f*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[(I/2)*E^(-ArcCoth[c + d*x] - ArcTanh[f/(d*e - c*f)])*(-1 + E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])))] + 12*f^2*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[(I/2)*E^(-ArcCoth[c + d*x] - ArcTanh[f/(d*e - c*f)])*(-1 + E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])))] - 12*c*f^2*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[(I/2)*E^(-ArcCoth[c + d*x] - ArcTanh[f/(d*e - c*f)])*(-1 + E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])))] + 6*d*e*f*ArcCoth[c + d*x]^2*Log[-((d*e*(-1 + E^(2*ArcCoth[c + d*x])) + (1 + c + E^(2*ArcCoth[c + d*x]) - c*E^(2*ArcCoth[c + d*x]))*f)/E^ArcCoth[c + d*x])] + 6*f^2*ArcCoth[c + d*x]^2*Log[-((d*e*(-1 + E^(2*ArcCoth[c + d*x])) + (1 + c + E^(2*ArcCoth[c + d*x]) - c*E^(2*ArcCoth[c + d*x]))*f)/E^ArcCoth[c + d*x])] - 6*c*f^2*ArcCoth[c + d*x]^2*Log[-((d*e*(-1 + E^(2*ArcCoth[c + d*x])) + (1 + c + E^(2*ArcCoth[c + d*x]) - c*E^(2*ArcCoth[c + d*x]))*f)/E^ArcCoth[c + d*x])] + 6*d*e*f*ArcCoth[c + d*x]^2*Log[1 - (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + 6*f^2*ArcCoth[c + d*x]^2*Log[1 - (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 6*c*f^2*ArcCoth[c + d*x]^2*Log[1 - (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + 6*d*e*f*ArcCoth[c + d*x]^2*Log[1 + (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + 6*f^2*ArcCoth[c + d*x]^2*Log[1 + (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 6*c*f^2*ArcCoth[c + d*x]^2*Log[1 + (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + (6*I)*d*e*f*Pi*ArcCoth[c + d*x]*Log[1/Sqrt[1 - (c + d*x)^(-2)]] + (6*I)*f^2*Pi*ArcCoth[c + d*x]*Log[1/Sqrt[1 - (c + d*x)^(-2)]] - (6*I)*c*f^2*Pi*ArcCoth[c + d*x]*Log[1/Sqrt[1 - (c + d*x)^(-2)]] - 6*d*e*f*ArcCoth[c + d*x]^2*Log[-(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)])] - 6*f^2*ArcCoth[c + d*x]^2*Log[-(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)])] + 6*c*f^2*ArcCoth[c + d*x]^2*Log[-(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)])] - 12*d*e*f*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[I*Sinh[ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]] - 12*f^2*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[I*Sinh[ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]] + 12*c*f^2*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[I*Sinh[ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]] + 6*f*(-(d*e*(-1 + E^(2*ArcTanh[f/(d*e - c*f)]))) + (1 + E^(2*ArcTanh[f/(d*e - c*f)]) + c*(-1 + E^(2*ArcTanh[f/(d*e - c*f)])))*f)*ArcCoth[c + d*x]*PolyLog[2, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 12*f*(d*e + f - c*f)*ArcCoth[c + d*x]*PolyLog[2, -((E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f])] + 12*d*e*f*ArcCoth[c + d*x]*PolyLog[2, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + 12*f^2*ArcCoth[c + d*x]*PolyLog[2, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 12*c*f^2*ArcCoth[c + d*x]*PolyLog[2, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 3*d*e*f*PolyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 3*d*e*E^(2*ArcTanh[f/(d*e - c*f)])*f*PolyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 3*f^2*PolyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 3*c*f^2*PolyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 3*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*PolyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 3*c*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*PolyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 12*d*e*f*PolyLog[3, -((E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f])] - 12*f^2*PolyLog[3, -((E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f])] + 12*c*f^2*PolyLog[3, -((E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f])] - 12*d*e*f*PolyLog[3, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 12*f^2*PolyLog[3, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + 12*c*f^2*PolyLog[3, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]]))/(6*f^2*(d*e + f - c*f)^2*(d*e - (1 + c)*f))))/(d*(c + d*x)^2*(e + f*x)*(1 - (c + d*x)^(-2)))","C",0
113,1,470,480,8.8989296,"\int \frac{\left(a+b \coth ^{-1}(c+d x)\right)^2}{(e+f x)^2} \, dx","Integrate[(a + b*ArcCoth[c + d*x])^2/(e + f*x)^2,x]","\frac{-\frac{a^2}{f}+\frac{2 a b \left(\coth ^{-1}(c+d x) \left(c^2 (-f)+c d (e-f x)+d^2 e x+f\right)-d (e+f x) \log \left(-\frac{d (e+f x)}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}\right)\right)}{(-c f+d e+f) (d e-(c+1) f)}+\frac{b^2 d \left(1-(c+d x)^2\right) (e+f x) \left(\frac{f \left(-\text{Li}_2\left(\exp \left(-2 \left(\coth ^{-1}(c+d x)+\tanh ^{-1}\left(\frac{f}{d e-c f}\right)\right)\right)\right)-2 \tanh ^{-1}\left(\frac{f}{c f-d e}\right) \log \left(1-\exp \left(-2 \left(\tanh ^{-1}\left(\frac{f}{d e-c f}\right)+\coth ^{-1}(c+d x)\right)\right)\right)+\coth ^{-1}(c+d x) \left(2 \log \left(1-\exp \left(-2 \left(\tanh ^{-1}\left(\frac{f}{d e-c f}\right)+\coth ^{-1}(c+d x)\right)\right)\right)+2 \tanh ^{-1}\left(\frac{f}{d e-c f}\right)+i \pi \right)+2 \tanh ^{-1}\left(\frac{f}{c f-d e}\right) \log \left(i \sinh \left(\tanh ^{-1}\left(\frac{f}{d e-c f}\right)+\coth ^{-1}(c+d x)\right)\right)+i \pi  \log \left(\frac{1}{\sqrt{1-\frac{1}{(c+d x)^2}}}\right)-i \pi  \log \left(e^{2 \coth ^{-1}(c+d x)}+1\right)\right)}{\left(c^2-1\right) f^2-2 c d e f+d^2 e^2}+\frac{\coth ^{-1}(c+d x)^2 e^{\tanh ^{-1}\left(\frac{f}{c f-d e}\right)}}{(c f-d e) \sqrt{1-\frac{f^2}{(d e-c f)^2}}}+\frac{\coth ^{-1}(c+d x)^2}{d e+d f x}\right)}{(c+d x)^2 \left(f-\frac{f}{(c+d x)^2}\right)}}{e+f x}","\frac{2 a b d \log (e+f x)}{f^2-(d e-c f)^2}-\frac{a b d \log (-c-d x+1)}{f (-c f+d e+f)}+\frac{a b d \log (c+d x+1)}{f (-c f+d e-f)}-\frac{\left(a+b \coth ^{-1}(c+d x)\right)^2}{f (e+f x)}+\frac{b^2 d \text{Li}_2\left(-\frac{c+d x+1}{-c-d x+1}\right)}{2 f (-c f+d e+f)}+\frac{b^2 d \text{Li}_2\left(1-\frac{2}{c+d x+1}\right)}{2 f (-c f+d e-f)}-\frac{b^2 d \text{Li}_2\left(1-\frac{2}{c+d x+1}\right)}{(-c f+d e+f) (d e-(c+1) f)}+\frac{b^2 d \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right)}{(-c f+d e+f) (d e-(c+1) f)}+\frac{b^2 d \log \left(\frac{2}{-c-d x+1}\right) \coth ^{-1}(c+d x)}{f (-c f+d e+f)}-\frac{b^2 d \log \left(\frac{2}{c+d x+1}\right) \coth ^{-1}(c+d x)}{f (-c f+d e-f)}+\frac{2 b^2 d \log \left(\frac{2}{c+d x+1}\right) \coth ^{-1}(c+d x)}{(-c f+d e+f) (d e-(c+1) f)}-\frac{2 b^2 d \coth ^{-1}(c+d x) \log \left(\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right)}{(-c f+d e+f) (d e-(c+1) f)}",1,"(-(a^2/f) + (2*a*b*((f - c^2*f + d^2*e*x + c*d*(e - f*x))*ArcCoth[c + d*x] - d*(e + f*x)*Log[-((d*(e + f*x))/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]))]))/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (b^2*d*(e + f*x)*(1 - (c + d*x)^2)*((E^ArcTanh[f/(-(d*e) + c*f)]*ArcCoth[c + d*x]^2)/((-(d*e) + c*f)*Sqrt[1 - f^2/(d*e - c*f)^2]) + ArcCoth[c + d*x]^2/(d*e + d*f*x) + (f*((-I)*Pi*Log[1 + E^(2*ArcCoth[c + d*x])] - 2*ArcTanh[f/(-(d*e) + c*f)]*Log[1 - E^(-2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + ArcCoth[c + d*x]*(I*Pi + 2*ArcTanh[f/(d*e - c*f)] + 2*Log[1 - E^(-2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))]) + I*Pi*Log[1/Sqrt[1 - (c + d*x)^(-2)]] + 2*ArcTanh[f/(-(d*e) + c*f)]*Log[I*Sinh[ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]] - PolyLog[2, E^(-2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))]))/(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)))/((c + d*x)^2*(f - f/(c + d*x)^2)))/(e + f*x)","C",0
114,1,2594,546,10.5983568,"\int (e+f x)^2 \left(a+b \coth ^{-1}(c+d x)\right)^3 \, dx","Integrate[(e + f*x)^2*(a + b*ArcCoth[c + d*x])^3,x]","\text{Result too large to show}","-\frac{b^2 \left(\left(3 c^2+1\right) f^2-6 c d e f+3 d^2 e^2\right) \text{Li}_2\left(1-\frac{2}{-c-d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)}{d^3}-\frac{6 b^2 f (d e-c f) \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)}{d^3}+\frac{a b^2 f^2 x}{d^2}-\frac{(d e-c f) \left(\left(c^2+3\right) f^2-2 c d e f+d^2 e^2\right) \left(a+b \coth ^{-1}(c+d x)\right)^3}{3 d^3 f}+\frac{\left(\left(3 c^2+1\right) f^2-6 c d e f+3 d^2 e^2\right) \left(a+b \coth ^{-1}(c+d x)\right)^3}{3 d^3}-\frac{b \left(\left(3 c^2+1\right) f^2-6 c d e f+3 d^2 e^2\right) \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)^2}{d^3}+\frac{3 b f (d e-c f) \left(a+b \coth ^{-1}(c+d x)\right)^2}{d^3}+\frac{3 b f (c+d x) (d e-c f) \left(a+b \coth ^{-1}(c+d x)\right)^2}{d^3}-\frac{b f^2 \left(a+b \coth ^{-1}(c+d x)\right)^2}{2 d^3}+\frac{b f^2 (c+d x)^2 \left(a+b \coth ^{-1}(c+d x)\right)^2}{2 d^3}+\frac{(e+f x)^3 \left(a+b \coth ^{-1}(c+d x)\right)^3}{3 f}+\frac{b^3 \left(\left(3 c^2+1\right) f^2-6 c d e f+3 d^2 e^2\right) \text{Li}_3\left(1-\frac{2}{-c-d x+1}\right)}{2 d^3}-\frac{3 b^3 f (d e-c f) \text{Li}_2\left(-\frac{c+d x+1}{-c-d x+1}\right)}{d^3}+\frac{b^3 f^2 \log \left(1-(c+d x)^2\right)}{2 d^3}+\frac{b^3 f^2 (c+d x) \coth ^{-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)*ArcCoth[c + d*x] + ((3*a^2*b*d^2*e^2 - 3*a^2*b*c*d^2*e^2 + 3*a^2*b*d*e*f - 6*a^2*b*c*d*e*f + 3*a^2*b*c^2*d*e*f + a^2*b*f^2 - 3*a^2*b*c*f^2 + 3*a^2*b*c^2*f^2 - a^2*b*c^3*f^2)*Log[1 - c - d*x])/(2*d^3) + ((3*a^2*b*d^2*e^2 + 3*a^2*b*c*d^2*e^2 - 3*a^2*b*d*e*f - 6*a^2*b*c*d*e*f - 3*a^2*b*c^2*d*e*f + a^2*b*f^2 + 3*a^2*b*c*f^2 + 3*a^2*b*c^2*f^2 + a^2*b*c^3*f^2)*Log[1 + c + d*x])/(2*d^3) + (3*a*b^2*e^2*(1 - (c + d*x)^2)*(ArcCoth[c + d*x]*(ArcCoth[c + d*x] - (c + d*x)*ArcCoth[c + d*x] + 2*Log[1 - E^(-2*ArcCoth[c + d*x])]) - PolyLog[2, E^(-2*ArcCoth[c + d*x])]))/(d*(c + d*x)^2*(1 - (c + d*x)^(-2))) - (3*a*b^2*e*f*(1 - (c + d*x)^2)*(2*c*ArcCoth[c + d*x]^2 + (c + d*x)^2*(1 - (c + d*x)^(-2))*ArcCoth[c + d*x]^2 - 2*(c + d*x)*ArcCoth[c + d*x]*(-1 + c*ArcCoth[c + d*x]) + 4*c*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])] - 2*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])] - 2*c*PolyLog[2, E^(-2*ArcCoth[c + d*x])]))/(d^2*(c + d*x)^2*(1 - (c + d*x)^(-2))) + (b^3*e^2*(1 - (c + d*x)^2)*((I/8)*Pi^3 - ArcCoth[c + d*x]^3 - (c + d*x)*ArcCoth[c + d*x]^3 + 3*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])] + 3*ArcCoth[c + d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])] - (3*PolyLog[3, E^(2*ArcCoth[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*ArcCoth[c + d*x]^2 + 12*(c + d*x)*ArcCoth[c + d*x]^2 - 8*c*ArcCoth[c + d*x]^3 - 8*c*(c + d*x)*ArcCoth[c + d*x]^3 + 4*(c + d*x)^2*(1 - (c + d*x)^(-2))*ArcCoth[c + d*x]^3 - 24*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])] + 24*c*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])] + 12*PolyLog[2, E^(-2*ArcCoth[c + d*x])] + 24*c*ArcCoth[c + d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])] - 12*c*PolyLog[3, E^(2*ArcCoth[c + d*x])]))/(4*d^2*(c + d*x)^2*(1 - (c + d*x)^(-2))) - (a*b^2*f^2*(c + d*x)*Sqrt[1 - (c + d*x)^(-2)]*(1 - (c + d*x)^2)*((4*ArcCoth[c + d*x])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (3*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (12*c*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (9*c^2*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (-1 + 6*c*ArcCoth[c + d*x] + 3*ArcCoth[c + d*x]^2 - 3*c^2*ArcCoth[c + d*x]^2)/Sqrt[1 - (c + d*x)^(-2)] + Cosh[3*ArcCoth[c + d*x]] - 6*c*ArcCoth[c + d*x]*Cosh[3*ArcCoth[c + d*x]] + ArcCoth[c + d*x]^2*Cosh[3*ArcCoth[c + d*x]] + 3*c^2*ArcCoth[c + d*x]^2*Cosh[3*ArcCoth[c + d*x]] + (6*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (18*c^2*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (18*c*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (4*(1 + 3*c^2)*PolyLog[2, E^(-2*ArcCoth[c + d*x])])/((c + d*x)^3*(1 - (c + d*x)^(-2))^(3/2)) - ArcCoth[c + d*x]^2*Sinh[3*ArcCoth[c + d*x]] - 3*c^2*ArcCoth[c + d*x]^2*Sinh[3*ArcCoth[c + d*x]] - 2*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])]*Sinh[3*ArcCoth[c + d*x]] - 6*c^2*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])]*Sinh[3*ArcCoth[c + d*x]] + 6*c*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])]*Sinh[3*ArcCoth[c + d*x]]))/(4*d^3) + (b^3*f^2*(1 - (c + d*x)^2)*(3*c*PolyLog[2, E^(-2*ArcCoth[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*ArcCoth[c + d*x])/Sqrt[1 - (c + d*x)^(-2)] - (72*c*ArcCoth[c + d*x]^2)/Sqrt[1 - (c + d*x)^(-2)] - (48*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (216*c*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (24*ArcCoth[c + d*x]^3)/Sqrt[1 - (c + d*x)^(-2)] + (24*c^2*ArcCoth[c + d*x]^3)/Sqrt[1 - (c + d*x)^(-2)] + (24*ArcCoth[c + d*x]^3)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (96*c*ArcCoth[c + d*x]^3)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (72*c^2*ArcCoth[c + d*x]^3)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - 24*ArcCoth[c + d*x]*Cosh[3*ArcCoth[c + d*x]] + 72*c*ArcCoth[c + d*x]^2*Cosh[3*ArcCoth[c + d*x]] - 8*ArcCoth[c + d*x]^3*Cosh[3*ArcCoth[c + d*x]] - 24*c^2*ArcCoth[c + d*x]^3*Cosh[3*ArcCoth[c + d*x]] + (432*c*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (72*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (216*c^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[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)]) + (96*(1 + 3*c^2)*ArcCoth[c + d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])])/((c + d*x)^3*(1 - (c + d*x)^(-2))^(3/2)) - (48*(1 + 3*c^2)*PolyLog[3, E^(2*ArcCoth[c + d*x])])/((c + d*x)^3*(1 - (c + d*x)^(-2))^(3/2)) + I*Pi^3*Sinh[3*ArcCoth[c + d*x]] + (3*I)*c^2*Pi^3*Sinh[3*ArcCoth[c + d*x]] - 72*c*ArcCoth[c + d*x]^2*Sinh[3*ArcCoth[c + d*x]] - 8*ArcCoth[c + d*x]^3*Sinh[3*ArcCoth[c + d*x]] - 24*c^2*ArcCoth[c + d*x]^3*Sinh[3*ArcCoth[c + d*x]] - 144*c*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])]*Sinh[3*ArcCoth[c + d*x]] + 24*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])]*Sinh[3*ArcCoth[c + d*x]] + 72*c^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])]*Sinh[3*ArcCoth[c + d*x]] + 24*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])]*Sinh[3*ArcCoth[c + d*x]]))/96))/(d^3*(c + d*x)^2*(1 - (c + d*x)^(-2)))","C",0
115,1,600,326,1.4567479,"\int (e+f x) \left(a+b \coth ^{-1}(c+d x)\right)^3 \, dx","Integrate[(e + f*x)*(a + b*ArcCoth[c + d*x])^3,x]","\frac{2 a^3 f (c+d x)^2+2 a^2 (c+d x) (-2 a c f+2 a d e+3 b f)+3 a^2 b (-2 c f+2 d e+f) \log (-c-d x+1)+3 a^2 b (2 d e-(2 c+1) f) \log (c+d x+1)-6 a^2 b (c+d x) \coth ^{-1}(c+d x) (c f-d (2 e+f x))+12 a b^2 d e \left(\text{Li}_2\left(e^{-2 \coth ^{-1}(c+d x)}\right)+\coth ^{-1}(c+d x) \left((c+d x-1) \coth ^{-1}(c+d x)-2 \log \left(1-e^{-2 \coth ^{-1}(c+d x)}\right)\right)\right)-12 a b^2 c f \left(\text{Li}_2\left(e^{-2 \coth ^{-1}(c+d x)}\right)+\coth ^{-1}(c+d x) \left((c+d x-1) \coth ^{-1}(c+d x)-2 \log \left(1-e^{-2 \coth ^{-1}(c+d x)}\right)\right)\right)+12 a b^2 f \left(-\log \left(\frac{1}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}\right)+\frac{1}{2} \left((c+d x)^2-1\right) \coth ^{-1}(c+d x)^2+(c+d x) \coth ^{-1}(c+d x)\right)+2 b^3 f \left(\coth ^{-1}(c+d x) \left(\left(c^2+2 c d x+d^2 x^2-1\right) \coth ^{-1}(c+d x)^2+3 (c+d x-1) \coth ^{-1}(c+d x)-6 \log \left(1-e^{-2 \coth ^{-1}(c+d x)}\right)\right)+3 \text{Li}_2\left(e^{-2 \coth ^{-1}(c+d x)}\right)\right)+4 b^3 d e \left(-3 \coth ^{-1}(c+d x) \text{Li}_2\left(e^{2 \coth ^{-1}(c+d x)}\right)+\frac{3}{2} \text{Li}_3\left(e^{2 \coth ^{-1}(c+d x)}\right)+(c+d x) \coth ^{-1}(c+d x)^3+\coth ^{-1}(c+d x)^3-3 \coth ^{-1}(c+d x)^2 \log \left(1-e^{2 \coth ^{-1}(c+d x)}\right)-\frac{i \pi ^3}{8}\right)-4 b^3 c f \left(-3 \coth ^{-1}(c+d x) \text{Li}_2\left(e^{2 \coth ^{-1}(c+d x)}\right)+\frac{3}{2} \text{Li}_3\left(e^{2 \coth ^{-1}(c+d x)}\right)+(c+d x) \coth ^{-1}(c+d x)^3+\coth ^{-1}(c+d x)^3-3 \coth ^{-1}(c+d x)^2 \log \left(1-e^{2 \coth ^{-1}(c+d x)}\right)-\frac{i \pi ^3}{8}\right)}{4 d^2}","-\frac{3 b^2 (d e-c f) \text{Li}_2\left(1-\frac{2}{-c-d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)}{d^2}-\frac{3 b^2 f \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)}{d^2}-\frac{\left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right) \left(a+b \coth ^{-1}(c+d x)\right)^3}{2 d^2 f}+\frac{(d e-c f) \left(a+b \coth ^{-1}(c+d x)\right)^3}{d^2}-\frac{3 b (d e-c f) \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)^2}{d^2}+\frac{3 b f \left(a+b \coth ^{-1}(c+d x)\right)^2}{2 d^2}+\frac{3 b f (c+d x) \left(a+b \coth ^{-1}(c+d x)\right)^2}{2 d^2}+\frac{(e+f x)^2 \left(a+b \coth ^{-1}(c+d x)\right)^3}{2 f}+\frac{3 b^3 (d e-c f) \text{Li}_3\left(1-\frac{2}{-c-d x+1}\right)}{2 d^2}-\frac{3 b^3 f \text{Li}_2\left(-\frac{c+d x+1}{-c-d x+1}\right)}{2 d^2}",1,"(2*a^2*(2*a*d*e + 3*b*f - 2*a*c*f)*(c + d*x) + 2*a^3*f*(c + d*x)^2 - 6*a^2*b*(c + d*x)*(c*f - d*(2*e + f*x))*ArcCoth[c + d*x] + 3*a^2*b*(2*d*e + f - 2*c*f)*Log[1 - c - d*x] + 3*a^2*b*(2*d*e - (1 + 2*c)*f)*Log[1 + c + d*x] + 12*a*b^2*f*((c + d*x)*ArcCoth[c + d*x] + ((-1 + (c + d*x)^2)*ArcCoth[c + d*x]^2)/2 - Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])]) + 12*a*b^2*d*e*(ArcCoth[c + d*x]*((-1 + c + d*x)*ArcCoth[c + d*x] - 2*Log[1 - E^(-2*ArcCoth[c + d*x])]) + PolyLog[2, E^(-2*ArcCoth[c + d*x])]) - 12*a*b^2*c*f*(ArcCoth[c + d*x]*((-1 + c + d*x)*ArcCoth[c + d*x] - 2*Log[1 - E^(-2*ArcCoth[c + d*x])]) + PolyLog[2, E^(-2*ArcCoth[c + d*x])]) + 2*b^3*f*(ArcCoth[c + d*x]*(3*(-1 + c + d*x)*ArcCoth[c + d*x] + (-1 + c^2 + 2*c*d*x + d^2*x^2)*ArcCoth[c + d*x]^2 - 6*Log[1 - E^(-2*ArcCoth[c + d*x])]) + 3*PolyLog[2, E^(-2*ArcCoth[c + d*x])]) + 4*b^3*d*e*((-1/8*I)*Pi^3 + ArcCoth[c + d*x]^3 + (c + d*x)*ArcCoth[c + d*x]^3 - 3*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])] - 3*ArcCoth[c + d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])] + (3*PolyLog[3, E^(2*ArcCoth[c + d*x])])/2) - 4*b^3*c*f*((-1/8*I)*Pi^3 + ArcCoth[c + d*x]^3 + (c + d*x)*ArcCoth[c + d*x]^3 - 3*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])] - 3*ArcCoth[c + d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])] + (3*PolyLog[3, E^(2*ArcCoth[c + d*x])])/2))/(4*d^2)","C",0
116,1,208,132,0.3302313,"\int \left(a+b \coth ^{-1}(c+d x)\right)^3 \, dx","Integrate[(a + b*ArcCoth[c + d*x])^3,x]","\frac{2 a^3 (c+d x)+3 a^2 b \log \left(1-(c+d x)^2\right)+6 a^2 b (c+d x) \coth ^{-1}(c+d x)+6 a b^2 \left(\text{Li}_2\left(e^{-2 \coth ^{-1}(c+d x)}\right)+\coth ^{-1}(c+d x) \left((c+d x-1) \coth ^{-1}(c+d x)-2 \log \left(1-e^{-2 \coth ^{-1}(c+d x)}\right)\right)\right)+2 b^3 \left(-3 \coth ^{-1}(c+d x) \text{Li}_2\left(e^{2 \coth ^{-1}(c+d x)}\right)+\frac{3}{2} \text{Li}_3\left(e^{2 \coth ^{-1}(c+d x)}\right)+(c+d x) \coth ^{-1}(c+d x)^3+\coth ^{-1}(c+d x)^3-3 \coth ^{-1}(c+d x)^2 \log \left(1-e^{2 \coth ^{-1}(c+d x)}\right)-\frac{i \pi ^3}{8}\right)}{2 d}","-\frac{3 b^2 \text{Li}_2\left(1-\frac{2}{-c-d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)}{d}+\frac{(c+d x) \left(a+b \coth ^{-1}(c+d x)\right)^3}{d}+\frac{\left(a+b \coth ^{-1}(c+d x)\right)^3}{d}-\frac{3 b \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)^2}{d}+\frac{3 b^3 \text{Li}_3\left(1-\frac{2}{-c-d x+1}\right)}{2 d}",1,"(2*a^3*(c + d*x) + 6*a^2*b*(c + d*x)*ArcCoth[c + d*x] + 3*a^2*b*Log[1 - (c + d*x)^2] + 6*a*b^2*(ArcCoth[c + d*x]*((-1 + c + d*x)*ArcCoth[c + d*x] - 2*Log[1 - E^(-2*ArcCoth[c + d*x])]) + PolyLog[2, E^(-2*ArcCoth[c + d*x])]) + 2*b^3*((-1/8*I)*Pi^3 + ArcCoth[c + d*x]^3 + (c + d*x)*ArcCoth[c + d*x]^3 - 3*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])] - 3*ArcCoth[c + d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])] + (3*PolyLog[3, E^(2*ArcCoth[c + d*x])])/2))/(2*d)","C",0
117,0,0,308,28.5714184,"\int \frac{\left(a+b \coth ^{-1}(c+d x)\right)^3}{e+f x} \, dx","Integrate[(a + b*ArcCoth[c + d*x])^3/(e + f*x),x]","\int \frac{\left(a+b \coth ^{-1}(c+d x)\right)^3}{e+f x} \, dx","-\frac{3 b^2 \left(a+b \coth ^{-1}(c+d x)\right) \text{Li}_3\left(1-\frac{2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right)}{2 f}+\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{c+d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)}{2 f}-\frac{3 b \left(a+b \coth ^{-1}(c+d x)\right)^2 \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right)}{2 f}+\frac{\left(a+b \coth ^{-1}(c+d x)\right)^3 \log \left(\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right)}{f}+\frac{3 b \text{Li}_2\left(1-\frac{2}{c+d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)^2}{2 f}-\frac{\log \left(\frac{2}{c+d x+1}\right) \left(a+b \coth ^{-1}(c+d x)\right)^3}{f}-\frac{3 b^3 \text{Li}_4\left(1-\frac{2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right)}{4 f}+\frac{3 b^3 \text{Li}_4\left(1-\frac{2}{c+d x+1}\right)}{4 f}",1,"Integrate[(a + b*ArcCoth[c + d*x])^3/(e + f*x), x]","F",-1
118,1,3937,1089,31.312186,"\int \frac{\left(a+b \coth ^{-1}(c+d x)\right)^3}{(e+f x)^2} \, dx","Integrate[(a + b*ArcCoth[c + d*x])^3/(e + f*x)^2,x]","\text{Result too large to show}","\frac{3 d \coth ^{-1}(c+d x)^2 \log \left(\frac{2}{-c-d x+1}\right) b^3}{2 f (d e-c f+f)}-\frac{3 d \coth ^{-1}(c+d x)^2 \log \left(\frac{2}{c+d x+1}\right) b^3}{2 f (d e-c f-f)}+\frac{3 d \coth ^{-1}(c+d x)^2 \log \left(\frac{2}{c+d x+1}\right) b^3}{(d e-c f+f) (d e-(c+1) f)}-\frac{3 d \coth ^{-1}(c+d x)^2 \log \left(\frac{2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right) b^3}{(d e-c f+f) (d e-(c+1) f)}+\frac{3 d \coth ^{-1}(c+d x) \text{Li}_2\left(1-\frac{2}{-c-d x+1}\right) b^3}{2 f (d e-c f+f)}+\frac{3 d \coth ^{-1}(c+d x) \text{Li}_2\left(1-\frac{2}{c+d x+1}\right) b^3}{2 f (d e-c f-f)}-\frac{3 d \coth ^{-1}(c+d x) \text{Li}_2\left(1-\frac{2}{c+d x+1}\right) b^3}{(d e-c f+f) (d e-(c+1) f)}+\frac{3 d \coth ^{-1}(c+d x) \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right) b^3}{(d e-c f+f) (d e-(c+1) f)}-\frac{3 d \text{Li}_3\left(1-\frac{2}{-c-d x+1}\right) b^3}{4 f (d e-c f+f)}+\frac{3 d \text{Li}_3\left(1-\frac{2}{c+d x+1}\right) b^3}{4 f (d e-c f-f)}-\frac{3 d \text{Li}_3\left(1-\frac{2}{c+d x+1}\right) b^3}{2 (d e-c f+f) (d e-(c+1) f)}+\frac{3 d \text{Li}_3\left(1-\frac{2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right) b^3}{2 (d e-c f+f) (d e-(c+1) f)}+\frac{3 a d \coth ^{-1}(c+d x) \log \left(\frac{2}{-c-d x+1}\right) b^2}{f (d e-c f+f)}-\frac{3 a d \coth ^{-1}(c+d x) \log \left(\frac{2}{c+d x+1}\right) b^2}{f (d e-c f-f)}+\frac{6 a d \coth ^{-1}(c+d x) \log \left(\frac{2}{c+d x+1}\right) b^2}{(d e-c f+f) (d e-(c+1) f)}-\frac{6 a d \coth ^{-1}(c+d x) \log \left(\frac{2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right) b^2}{(d e-c f+f) (d e-(c+1) f)}+\frac{3 a d \text{Li}_2\left(-\frac{c+d x+1}{-c-d x+1}\right) b^2}{2 f (d e-c f+f)}+\frac{3 a d \text{Li}_2\left(1-\frac{2}{c+d x+1}\right) b^2}{2 f (d e-c f-f)}-\frac{3 a d \text{Li}_2\left(1-\frac{2}{c+d x+1}\right) b^2}{(d e-c f+f) (d e-(c+1) f)}+\frac{3 a d \text{Li}_2\left(1-\frac{2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right) b^2}{(d e-c f+f) (d e-(c+1) f)}-\frac{3 a^2 d \log (-c-d x+1) b}{2 f (d e-c f+f)}+\frac{3 a^2 d \log (c+d x+1) b}{2 f (d e-c f-f)}+\frac{3 a^2 d \log (e+f x) b}{f^2-(d e-c f)^2}-\frac{\left(a+b \coth ^{-1}(c+d x)\right)^3}{f (e+f x)}",1,"-(a^3/(f*(e + f*x))) - (3*a^2*b*ArcCoth[c + d*x])/(f*(e + f*x)) + (3*a^2*b*d*Log[1 - c - d*x])/(2*f*(-(d*e) - f + c*f)) - (3*a^2*b*d*Log[1 + c + d*x])/(2*f*(-(d*e) + f + c*f)) - (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*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^ArcTanh[f/(-(d*e) + c*f)]*ArcCoth[c + d*x]^2)/((-(d*e) + c*f)*Sqrt[1 - f^2/(d*e - c*f)^2]) + ArcCoth[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*ArcCoth[c + d*x] + 2*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)] - I*Pi*Log[1 + E^(2*ArcCoth[c + d*x])] + 2*ArcCoth[c + d*x]*Log[1 - E^(-2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 2*ArcTanh[f/(-(d*e) + c*f)]*Log[1 - E^(-2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + I*Pi*Log[1/Sqrt[1 - (c + d*x)^(-2)]] + 2*ArcTanh[f/(-(d*e) + c*f)]*Log[I*Sinh[ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]] - PolyLog[2, E^(-2*(ArcCoth[c + d*x] + ArcTanh[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*ArcCoth[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*(2*d^2*e^2*ArcCoth[c + d*x]^3 - 8*d*e*f*ArcCoth[c + d*x]^3 - 4*c*d*e*f*ArcCoth[c + d*x]^3 + 4*d*e*E^(2*ArcTanh[f/(d*e - c*f)])*f*ArcCoth[c + d*x]^3 - 10*f^2*ArcCoth[c + d*x]^3 + 8*c*f^2*ArcCoth[c + d*x]^3 + 2*c^2*f^2*ArcCoth[c + d*x]^3 - 4*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*ArcCoth[c + d*x]^3 - 4*c*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*ArcCoth[c + d*x]^3 - (4*d^2*e^2*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] - (4*d*e*f*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] + (8*c*d*e*f*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] + (4*c*f^2*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] - (4*c^2*f^2*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] + (6*I)*d*e*f*Pi*ArcCoth[c + d*x]*Log[2] + (6*I)*f^2*Pi*ArcCoth[c + d*x]*Log[2] - (6*I)*c*f^2*Pi*ArcCoth[c + d*x]*Log[2] - d*e*f*ArcCoth[c + d*x]^2*Log[64] - f^2*ArcCoth[c + d*x]^2*Log[64] + c*f^2*ArcCoth[c + d*x]^2*Log[64] - (6*I)*d*e*f*Pi*ArcCoth[c + d*x]*Log[E^(-ArcCoth[c + d*x]) + E^ArcCoth[c + d*x]] - (6*I)*f^2*Pi*ArcCoth[c + d*x]*Log[E^(-ArcCoth[c + d*x]) + E^ArcCoth[c + d*x]] + (6*I)*c*f^2*Pi*ArcCoth[c + d*x]*Log[E^(-ArcCoth[c + d*x]) + E^ArcCoth[c + d*x]] + 6*d*e*f*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 6*d*e*E^(2*ArcTanh[f/(d*e - c*f)])*f*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 6*f^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 6*c*f^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 6*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 6*c*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 12*d*e*f*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[(I/2)*E^(-ArcCoth[c + d*x] - ArcTanh[f/(d*e - c*f)])*(-1 + E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])))] + 12*f^2*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[(I/2)*E^(-ArcCoth[c + d*x] - ArcTanh[f/(d*e - c*f)])*(-1 + E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])))] - 12*c*f^2*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[(I/2)*E^(-ArcCoth[c + d*x] - ArcTanh[f/(d*e - c*f)])*(-1 + E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])))] + 6*d*e*f*ArcCoth[c + d*x]^2*Log[-((d*e*(-1 + E^(2*ArcCoth[c + d*x])) + (1 + c + E^(2*ArcCoth[c + d*x]) - c*E^(2*ArcCoth[c + d*x]))*f)/E^ArcCoth[c + d*x])] + 6*f^2*ArcCoth[c + d*x]^2*Log[-((d*e*(-1 + E^(2*ArcCoth[c + d*x])) + (1 + c + E^(2*ArcCoth[c + d*x]) - c*E^(2*ArcCoth[c + d*x]))*f)/E^ArcCoth[c + d*x])] - 6*c*f^2*ArcCoth[c + d*x]^2*Log[-((d*e*(-1 + E^(2*ArcCoth[c + d*x])) + (1 + c + E^(2*ArcCoth[c + d*x]) - c*E^(2*ArcCoth[c + d*x]))*f)/E^ArcCoth[c + d*x])] + 6*d*e*f*ArcCoth[c + d*x]^2*Log[1 - (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + 6*f^2*ArcCoth[c + d*x]^2*Log[1 - (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 6*c*f^2*ArcCoth[c + d*x]^2*Log[1 - (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + 6*d*e*f*ArcCoth[c + d*x]^2*Log[1 + (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + 6*f^2*ArcCoth[c + d*x]^2*Log[1 + (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 6*c*f^2*ArcCoth[c + d*x]^2*Log[1 + (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + (6*I)*d*e*f*Pi*ArcCoth[c + d*x]*Log[1/Sqrt[1 - (c + d*x)^(-2)]] + (6*I)*f^2*Pi*ArcCoth[c + d*x]*Log[1/Sqrt[1 - (c + d*x)^(-2)]] - (6*I)*c*f^2*Pi*ArcCoth[c + d*x]*Log[1/Sqrt[1 - (c + d*x)^(-2)]] - 6*d*e*f*ArcCoth[c + d*x]^2*Log[-(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)])] - 6*f^2*ArcCoth[c + d*x]^2*Log[-(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)])] + 6*c*f^2*ArcCoth[c + d*x]^2*Log[-(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)])] - 12*d*e*f*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[I*Sinh[ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]] - 12*f^2*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[I*Sinh[ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]] + 12*c*f^2*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[I*Sinh[ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]] + 6*f*(-(d*e*(-1 + E^(2*ArcTanh[f/(d*e - c*f)]))) + (1 + E^(2*ArcTanh[f/(d*e - c*f)]) + c*(-1 + E^(2*ArcTanh[f/(d*e - c*f)])))*f)*ArcCoth[c + d*x]*PolyLog[2, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 12*f*(d*e + f - c*f)*ArcCoth[c + d*x]*PolyLog[2, -((E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f])] + 12*d*e*f*ArcCoth[c + d*x]*PolyLog[2, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + 12*f^2*ArcCoth[c + d*x]*PolyLog[2, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 12*c*f^2*ArcCoth[c + d*x]*PolyLog[2, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 3*d*e*f*PolyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 3*d*e*E^(2*ArcTanh[f/(d*e - c*f)])*f*PolyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 3*f^2*PolyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 3*c*f^2*PolyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 3*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*PolyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 3*c*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*PolyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 12*d*e*f*PolyLog[3, -((E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f])] - 12*f^2*PolyLog[3, -((E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f])] + 12*c*f^2*PolyLog[3, -((E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f])] - 12*d*e*f*PolyLog[3, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 12*f^2*PolyLog[3, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + 12*c*f^2*PolyLog[3, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]]))/(2*f*(d*e + f - c*f)^2*(d*e - (1 + c)*f))))/(d^2*(e + f*x)^2)","C",0
119,0,0,162,2.5282216,"\int (e+f x)^m \left(a+b \coth ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)^m*(a + b*ArcCoth[c + d*x]),x]","\int (e+f x)^m \left(a+b \coth ^{-1}(c+d x)\right) \, dx","\frac{(e+f x)^{m+1} \left(a+b \coth ^{-1}(c+d x)\right)}{f (m+1)}+\frac{b d (e+f x)^{m+2} \, _2F_1\left(1,m+2;m+3;\frac{d (e+f x)}{d e-c f-f}\right)}{2 f (m+1) (m+2) (d e-(c+1) f)}-\frac{b d (e+f x)^{m+2} \, _2F_1\left(1,m+2;m+3;\frac{d (e+f x)}{d e-c f+f}\right)}{2 f (m+1) (m+2) (-c f+d e+f)}",1,"Integrate[(e + f*x)^m*(a + b*ArcCoth[c + d*x]), x]","F",-1
120,0,0,23,2.6518029,"\int (e+f x)^m \left(a+b \coth ^{-1}(c+d x)\right)^2 \, dx","Integrate[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^2,x]","\int (e+f x)^m \left(a+b \coth ^{-1}(c+d x)\right)^2 \, dx","\text{Int}\left((e+f x)^m \left(a+b \coth ^{-1}(c+d x)\right)^2,x\right)",0,"Integrate[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^2, x]","A",-1
121,0,0,23,0.3513853,"\int (e+f x)^m \left(a+b \coth ^{-1}(c+d x)\right)^3 \, dx","Integrate[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^3,x]","\int (e+f x)^m \left(a+b \coth ^{-1}(c+d x)\right)^3 \, dx","\text{Int}\left((e+f x)^m \left(a+b \coth ^{-1}(c+d x)\right)^3,x\right)",0,"Integrate[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^3, x]","A",-1
122,0,0,43,0.0970266,"\int \frac{\left(a+b \coth ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^n}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2),x]","\int \frac{\left(a+b \coth ^{-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 \coth ^{-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*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]","A",-1
123,0,0,460,0.2885098,"\int \frac{\left(a+b \coth ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^3}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2),x]","\int \frac{\left(a+b \coth ^{-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}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1}\right) \left(a+b \coth ^{-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}}+1\right)}\right) \left(a+b \coth ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{2 c}-\frac{3 b \text{Li}_2\left(1-\frac{2}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1}\right) \left(a+b \coth ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 c}+\frac{3 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}}+1\right)}\right) \left(a+b \coth ^{-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{\sqrt{1-c x}}{\sqrt{c x+1}}}\right) \left(a+b \coth ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^3}{c}-\frac{3 b^3 \text{Li}_4\left(1-\frac{2}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1}\right)}{4 c}+\frac{3 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}}+1\right)}\right)}{4 c}",1,"Integrate[(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2), x]","F",-1
124,0,0,302,0.5407378,"\int \frac{\left(a+b \coth ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2),x]","\int \frac{\left(a+b \coth ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2}{1-c^2 x^2} \, dx","-\frac{b \text{Li}_2\left(1-\frac{2}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1}\right) \left(a+b \coth ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}+\frac{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}}+1\right)}\right) \left(a+b \coth ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}-\frac{2 \coth ^{-1}\left(1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right) \left(a+b \coth ^{-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}{\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+1}\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}}+1\right)}\right)}{2 c}",1,"Integrate[(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2), x]","F",-1
125,1,98,89,0.4007553,"\int \frac{a+b \coth ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(1 - c^2*x^2),x]","\frac{a \tanh ^{-1}(c x)}{c}+\frac{b \left(\text{Li}_2\left(-e^{-\tanh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\tanh ^{-1}(c x)}\right)+\tanh ^{-1}(c x) \left(2 \coth ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)+\log \left(1-e^{-\tanh ^{-1}(c x)}\right)-\log \left(e^{-\tanh ^{-1}(c x)}+1\right)\right)\right)}{2 c}","-\frac{a \log \left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}{c}-\frac{b \text{Li}_2\left(-\frac{\sqrt{c x+1}}{\sqrt{1-c x}}\right)}{2 c}+\frac{b \text{Li}_2\left(\frac{\sqrt{c x+1}}{\sqrt{1-c x}}\right)}{2 c}",1,"(a*ArcTanh[c*x])/c + (b*(ArcTanh[c*x]*(2*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]] + Log[1 - E^(-ArcTanh[c*x])] - Log[1 + E^(-ArcTanh[c*x])]) + PolyLog[2, -E^(-ArcTanh[c*x])] - PolyLog[2, E^(-ArcTanh[c*x])]))/(2*c)","A",0
126,0,0,43,0.0952829,"\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \coth ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)} \, dx","Integrate[1/((1 - c^2*x^2)*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])),x]","\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \coth ^{-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 \coth ^{-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*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]","A",-1
127,0,0,43,0.7918461,"\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \coth ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2} \, dx","Integrate[1/((1 - c^2*x^2)*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2),x]","\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \coth ^{-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 \coth ^{-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*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]","A",-1
128,1,34,37,0.0646717,"\int x^m \coth ^{-1}(\tanh (a+b x)) \, dx","Integrate[x^m*ArcCoth[Tanh[a + b*x]],x]","x^m \left(\frac{x \left(\coth ^{-1}(\tanh (a+b x))-b x\right)}{m+1}+\frac{b x^2}{m+2}\right)","\frac{x^{m+1} \coth ^{-1}(\tanh (a+b x))}{m+1}-\frac{b x^{m+2}}{m^2+3 m+2}",1,"x^m*((b*x^2)/(2 + m) + (x*(-(b*x) + ArcCoth[Tanh[a + b*x]]))/(1 + m))","A",1
129,1,20,23,0.0173743,"\int x^2 \coth ^{-1}(\tanh (a+b x)) \, dx","Integrate[x^2*ArcCoth[Tanh[a + b*x]],x]","-\frac{1}{12} x^3 \left(b x-4 \coth ^{-1}(\tanh (a+b x))\right)","\frac{1}{3} x^3 \coth ^{-1}(\tanh (a+b x))-\frac{b x^4}{12}",1,"-1/12*(x^3*(b*x - 4*ArcCoth[Tanh[a + b*x]]))","A",1
130,1,20,23,0.0150958,"\int x \coth ^{-1}(\tanh (a+b x)) \, dx","Integrate[x*ArcCoth[Tanh[a + b*x]],x]","-\frac{1}{6} x^2 \left(b x-3 \coth ^{-1}(\tanh (a+b x))\right)","\frac{1}{2} x^2 \coth ^{-1}(\tanh (a+b x))-\frac{b x^3}{6}",1,"-1/6*(x^2*(b*x - 3*ArcCoth[Tanh[a + b*x]]))","A",1
131,1,18,16,0.0077599,"\int \coth ^{-1}(\tanh (a+b x)) \, dx","Integrate[ArcCoth[Tanh[a + b*x]],x]","x \coth ^{-1}(\tanh (a+b x))-\frac{b x^2}{2}","\frac{\coth ^{-1}(\tanh (a+b x))^2}{2 b}",1,"-1/2*(b*x^2) + x*ArcCoth[Tanh[a + b*x]]","A",1
132,1,19,21,0.0142107,"\int \frac{\coth ^{-1}(\tanh (a+b x))}{x} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]/x,x]","\log (x) \left(\coth ^{-1}(\tanh (a+b x))-b x\right)+b x","b x-\log (x) \left(b x-\coth ^{-1}(\tanh (a+b x))\right)",1,"b*x + (-(b*x) + ArcCoth[Tanh[a + b*x]])*Log[x]","A",1
133,1,18,17,0.0163617,"\int \frac{\coth ^{-1}(\tanh (a+b x))}{x^2} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]/x^2,x]","-\frac{\coth ^{-1}(\tanh (a+b x))}{x}+b \log (x)+b","b \log (x)-\frac{\coth ^{-1}(\tanh (a+b x))}{x}",1,"b - ArcCoth[Tanh[a + b*x]]/x + b*Log[x]","A",1
134,1,18,23,0.0140086,"\int \frac{\coth ^{-1}(\tanh (a+b x))}{x^3} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]/x^3,x]","-\frac{\coth ^{-1}(\tanh (a+b x))+b x}{2 x^2}","-\frac{\coth ^{-1}(\tanh (a+b x))}{2 x^2}-\frac{b}{2 x}",1,"-1/2*(b*x + ArcCoth[Tanh[a + b*x]])/x^2","A",1
135,1,20,23,0.0158046,"\int \frac{\coth ^{-1}(\tanh (a+b x))}{x^4} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]/x^4,x]","-\frac{2 \coth ^{-1}(\tanh (a+b x))+b x}{6 x^3}","-\frac{\coth ^{-1}(\tanh (a+b x))}{3 x^3}-\frac{b}{6 x^2}",1,"-1/6*(b*x + 2*ArcCoth[Tanh[a + b*x]])/x^3","A",1
136,1,62,71,0.1681749,"\int x^m \coth ^{-1}(\tanh (a+b x))^2 \, dx","Integrate[x^m*ArcCoth[Tanh[a + b*x]]^2,x]","\frac{x^{m+1} \left(\left(m^2+5 m+6\right) \coth ^{-1}(\tanh (a+b x))^2-2 b (m+3) x \coth ^{-1}(\tanh (a+b x))+2 b^2 x^2\right)}{(m+1) (m+2) (m+3)}","-\frac{2 b x^{m+2} \coth ^{-1}(\tanh (a+b x))}{m^2+3 m+2}+\frac{x^{m+1} \coth ^{-1}(\tanh (a+b x))^2}{m+1}+\frac{2 b^2 x^{m+3}}{m^3+6 m^2+11 m+6}",1,"(x^(1 + m)*(2*b^2*x^2 - 2*b*(3 + m)*x*ArcCoth[Tanh[a + b*x]] + (6 + 5*m + m^2)*ArcCoth[Tanh[a + b*x]]^2))/((1 + m)*(2 + m)*(3 + m))","A",1
137,1,37,42,0.033809,"\int x^3 \coth ^{-1}(\tanh (a+b x))^2 \, dx","Integrate[x^3*ArcCoth[Tanh[a + b*x]]^2,x]","\frac{1}{60} x^4 \left(-6 b x \coth ^{-1}(\tanh (a+b x))+15 \coth ^{-1}(\tanh (a+b x))^2+b^2 x^2\right)","-\frac{1}{10} b x^5 \coth ^{-1}(\tanh (a+b x))+\frac{1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^2+\frac{b^2 x^6}{60}",1,"(x^4*(b^2*x^2 - 6*b*x*ArcCoth[Tanh[a + b*x]] + 15*ArcCoth[Tanh[a + b*x]]^2))/60","A",1
138,1,37,42,0.0549092,"\int x^2 \coth ^{-1}(\tanh (a+b x))^2 \, dx","Integrate[x^2*ArcCoth[Tanh[a + b*x]]^2,x]","\frac{1}{30} x^3 \left(-5 b x \coth ^{-1}(\tanh (a+b x))+10 \coth ^{-1}(\tanh (a+b x))^2+b^2 x^2\right)","-\frac{1}{6} b x^4 \coth ^{-1}(\tanh (a+b x))+\frac{1}{3} x^3 \coth ^{-1}(\tanh (a+b x))^2+\frac{b^2 x^5}{30}",1,"(x^3*(b^2*x^2 - 5*b*x*ArcCoth[Tanh[a + b*x]] + 10*ArcCoth[Tanh[a + b*x]]^2))/30","A",1
139,1,74,34,0.0756628,"\int x \coth ^{-1}(\tanh (a+b x))^2 \, dx","Integrate[x*ArcCoth[Tanh[a + b*x]]^2,x]","\frac{(a+b x) \left(4 \left(2 a^2+a b x-b^2 x^2\right) \coth ^{-1}(\tanh (a+b x))-\left((3 a-b x) (a+b x)^2\right)-6 (a-b x) \coth ^{-1}(\tanh (a+b x))^2\right)}{12 b^2}","\frac{x \coth ^{-1}(\tanh (a+b x))^3}{3 b}-\frac{\coth ^{-1}(\tanh (a+b x))^4}{12 b^2}",1,"((a + b*x)*(-((3*a - b*x)*(a + b*x)^2) + 4*(2*a^2 + a*b*x - b^2*x^2)*ArcCoth[Tanh[a + b*x]] - 6*(a - b*x)*ArcCoth[Tanh[a + b*x]]^2))/(12*b^2)","B",1
140,1,16,16,0.0059973,"\int \coth ^{-1}(\tanh (a+b x))^2 \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^2,x]","\frac{\coth ^{-1}(\tanh (a+b x))^3}{3 b}","\frac{\coth ^{-1}(\tanh (a+b x))^3}{3 b}",1,"ArcCoth[Tanh[a + b*x]]^3/(3*b)","A",1
141,1,53,49,0.0638309,"\int \frac{\coth ^{-1}(\tanh (a+b x))^2}{x} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^2/x,x]","\frac{1}{2} (a+b x)^2-(a+b x) \left(-2 \coth ^{-1}(\tanh (a+b x))+a+2 b x\right)+\log (b x) \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^2","-b x \left(b x-\coth ^{-1}(\tanh (a+b x))\right)+\frac{1}{2} \coth ^{-1}(\tanh (a+b x))^2+\log (x) \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2",1,"(a + b*x)^2/2 - (a + b*x)*(a + 2*b*x - 2*ArcCoth[Tanh[a + b*x]]) + (-(b*x) + ArcCoth[Tanh[a + b*x]])^2*Log[b*x]","A",1
142,1,37,39,0.0494609,"\int \frac{\coth ^{-1}(\tanh (a+b x))^2}{x^2} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^2/x^2,x]","-\frac{\coth ^{-1}(\tanh (a+b x))^2}{x}+2 b (\log (x)+1) \coth ^{-1}(\tanh (a+b x))-2 b^2 x \log (x)","-\frac{\coth ^{-1}(\tanh (a+b x))^2}{x}-2 b \log (x) \left(b x-\coth ^{-1}(\tanh (a+b x))\right)+2 b^2 x",1,"-(ArcCoth[Tanh[a + b*x]]^2/x) - 2*b^2*x*Log[x] + 2*b*ArcCoth[Tanh[a + b*x]]*(1 + Log[x])","A",1
143,1,42,36,0.0368268,"\int \frac{\coth ^{-1}(\tanh (a+b x))^2}{x^3} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^2/x^3,x]","-\frac{2 b x \coth ^{-1}(\tanh (a+b x))+\coth ^{-1}(\tanh (a+b x))^2-b^2 x^2 (2 \log (x)+3)}{2 x^2}","-\frac{\coth ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac{b \coth ^{-1}(\tanh (a+b x))}{x}+b^2 \log (x)",1,"-1/2*(2*b*x*ArcCoth[Tanh[a + b*x]] + ArcCoth[Tanh[a + b*x]]^2 - b^2*x^2*(3 + 2*Log[x]))/x^2","A",1
144,1,34,31,0.0455981,"\int \frac{\coth ^{-1}(\tanh (a+b x))^2}{x^4} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^2/x^4,x]","-\frac{b x \coth ^{-1}(\tanh (a+b x))+\coth ^{-1}(\tanh (a+b x))^2+b^2 x^2}{3 x^3}","\frac{\coth ^{-1}(\tanh (a+b x))^3}{3 x^3 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}",1,"-1/3*(b^2*x^2 + b*x*ArcCoth[Tanh[a + b*x]] + ArcCoth[Tanh[a + b*x]]^2)/x^3","A",1
145,1,37,64,0.0312422,"\int \frac{\coth ^{-1}(\tanh (a+b x))^2}{x^5} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^2/x^5,x]","-\frac{2 b x \coth ^{-1}(\tanh (a+b x))+3 \coth ^{-1}(\tanh (a+b x))^2+b^2 x^2}{12 x^4}","\frac{\coth ^{-1}(\tanh (a+b x))^3}{4 x^4 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}+\frac{b \coth ^{-1}(\tanh (a+b x))^3}{12 x^3 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2}",1,"-1/12*(b^2*x^2 + 2*b*x*ArcCoth[Tanh[a + b*x]] + 3*ArcCoth[Tanh[a + b*x]]^2)/x^4","A",1
146,1,97,110,0.2762198,"\int x^m \coth ^{-1}(\tanh (a+b x))^3 \, dx","Integrate[x^m*ArcCoth[Tanh[a + b*x]]^3,x]","\frac{x^{m+1} \left(6 b^2 (m+4) x^2 \coth ^{-1}(\tanh (a+b x))-3 b \left(m^2+7 m+12\right) x \coth ^{-1}(\tanh (a+b x))^2+\left(m^3+9 m^2+26 m+24\right) \coth ^{-1}(\tanh (a+b x))^3-6 b^3 x^3\right)}{(m+1) (m+2) (m+3) (m+4)}","\frac{6 b^2 x^{m+3} \coth ^{-1}(\tanh (a+b x))}{m^3+6 m^2+11 m+6}-\frac{3 b x^{m+2} \coth ^{-1}(\tanh (a+b x))^2}{m^2+3 m+2}+\frac{x^{m+1} \coth ^{-1}(\tanh (a+b x))^3}{m+1}-\frac{6 b^3 x^{m+4}}{(m+1) \left(m^3+9 m^2+26 m+24\right)}",1,"(x^(1 + m)*(-6*b^3*x^3 + 6*b^2*(4 + m)*x^2*ArcCoth[Tanh[a + b*x]] - 3*b*(12 + 7*m + m^2)*x*ArcCoth[Tanh[a + b*x]]^2 + (24 + 26*m + 9*m^2 + m^3)*ArcCoth[Tanh[a + b*x]]^3))/((1 + m)*(2 + m)*(3 + m)*(4 + m))","A",1
147,1,54,61,0.034833,"\int x^4 \coth ^{-1}(\tanh (a+b x))^3 \, dx","Integrate[x^4*ArcCoth[Tanh[a + b*x]]^3,x]","-\frac{1}{280} x^5 \left(-8 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+28 b x \coth ^{-1}(\tanh (a+b x))^2-56 \coth ^{-1}(\tanh (a+b x))^3+b^3 x^3\right)","\frac{1}{35} b^2 x^7 \coth ^{-1}(\tanh (a+b x))-\frac{1}{10} b x^6 \coth ^{-1}(\tanh (a+b x))^2+\frac{1}{5} x^5 \coth ^{-1}(\tanh (a+b x))^3-\frac{1}{280} b^3 x^8",1,"-1/280*(x^5*(b^3*x^3 - 8*b^2*x^2*ArcCoth[Tanh[a + b*x]] + 28*b*x*ArcCoth[Tanh[a + b*x]]^2 - 56*ArcCoth[Tanh[a + b*x]]^3))","A",1
148,1,54,61,0.026176,"\int x^3 \coth ^{-1}(\tanh (a+b x))^3 \, dx","Integrate[x^3*ArcCoth[Tanh[a + b*x]]^3,x]","-\frac{1}{140} x^4 \left(-7 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+21 b x \coth ^{-1}(\tanh (a+b x))^2-35 \coth ^{-1}(\tanh (a+b x))^3+b^3 x^3\right)","\frac{1}{20} b^2 x^6 \coth ^{-1}(\tanh (a+b x))-\frac{3}{20} b x^5 \coth ^{-1}(\tanh (a+b x))^2+\frac{1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^3-\frac{1}{140} b^3 x^7",1,"-1/140*(x^4*(b^3*x^3 - 7*b^2*x^2*ArcCoth[Tanh[a + b*x]] + 21*b*x*ArcCoth[Tanh[a + b*x]]^2 - 35*ArcCoth[Tanh[a + b*x]]^3))","A",1
149,1,54,53,0.0249378,"\int x^2 \coth ^{-1}(\tanh (a+b x))^3 \, dx","Integrate[x^2*ArcCoth[Tanh[a + b*x]]^3,x]","-\frac{1}{60} x^3 \left(-6 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+15 b x \coth ^{-1}(\tanh (a+b x))^2-20 \coth ^{-1}(\tanh (a+b x))^3+b^3 x^3\right)","\frac{\coth ^{-1}(\tanh (a+b x))^6}{60 b^3}-\frac{x \coth ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac{x^2 \coth ^{-1}(\tanh (a+b x))^4}{4 b}",1,"-1/60*(x^3*(b^3*x^3 - 6*b^2*x^2*ArcCoth[Tanh[a + b*x]] + 15*b*x*ArcCoth[Tanh[a + b*x]]^2 - 20*ArcCoth[Tanh[a + b*x]]^3))","A",1
150,1,99,34,0.0740608,"\int x \coth ^{-1}(\tanh (a+b x))^3 \, dx","Integrate[x*ArcCoth[Tanh[a + b*x]]^3,x]","\frac{(a+b x) \left(10 \left(2 a^2+a b x-b^2 x^2\right) \coth ^{-1}(\tanh (a+b x))^2+(4 a-b x) (a+b x)^3-5 (3 a-b x) (a+b x)^2 \coth ^{-1}(\tanh (a+b x))-10 (a-b x) \coth ^{-1}(\tanh (a+b x))^3\right)}{20 b^2}","\frac{x \coth ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{\coth ^{-1}(\tanh (a+b x))^5}{20 b^2}",1,"((a + b*x)*((4*a - b*x)*(a + b*x)^3 - 5*(3*a - b*x)*(a + b*x)^2*ArcCoth[Tanh[a + b*x]] + 10*(2*a^2 + a*b*x - b^2*x^2)*ArcCoth[Tanh[a + b*x]]^2 - 10*(a - b*x)*ArcCoth[Tanh[a + b*x]]^3))/(20*b^2)","B",1
151,1,16,16,0.0075427,"\int \coth ^{-1}(\tanh (a+b x))^3 \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^3,x]","\frac{\coth ^{-1}(\tanh (a+b x))^4}{4 b}","\frac{\coth ^{-1}(\tanh (a+b x))^4}{4 b}",1,"ArcCoth[Tanh[a + b*x]]^4/(4*b)","A",1
152,1,104,77,0.0965439,"\int \frac{\coth ^{-1}(\tanh (a+b x))^3}{x} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^3/x,x]","(a+b x) \left(a^2-3 a \left(-\coth ^{-1}(\tanh (a+b x))+a+b x\right)+3 \left(-\coth ^{-1}(\tanh (a+b x))+a+b x\right)^2\right)+\frac{1}{3} (a+b x)^3-\frac{1}{2} (a+b x)^2 \left(-3 \coth ^{-1}(\tanh (a+b x))+2 a+3 b x\right)+\log (b x) \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^3","b x \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2-\frac{1}{2} \coth ^{-1}(\tanh (a+b x))^2 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)+\frac{1}{3} \coth ^{-1}(\tanh (a+b x))^3-\log (x) \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^3",1,"(a + b*x)^3/3 + (a + b*x)*(a^2 - 3*a*(a + b*x - ArcCoth[Tanh[a + b*x]]) + 3*(a + b*x - ArcCoth[Tanh[a + b*x]])^2) - ((a + b*x)^2*(2*a + 3*b*x - 3*ArcCoth[Tanh[a + b*x]]))/2 + (-(b*x) + ArcCoth[Tanh[a + b*x]])^3*Log[b*x]","A",1
153,1,62,68,0.0419341,"\int \frac{\coth ^{-1}(\tanh (a+b x))^3}{x^2} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^3/x^2,x]","-6 b^2 x \log (x) \coth ^{-1}(\tanh (a+b x))-\frac{\coth ^{-1}(\tanh (a+b x))^3}{x}+3 b (\log (x)+1) \coth ^{-1}(\tanh (a+b x))^2+\frac{3}{2} b^3 x^2 (2 \log (x)-1)","-3 b^2 x \left(b x-\coth ^{-1}(\tanh (a+b x))\right)-\frac{\coth ^{-1}(\tanh (a+b x))^3}{x}+\frac{3}{2} b \coth ^{-1}(\tanh (a+b x))^2+3 b \log (x) \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2",1,"-(ArcCoth[Tanh[a + b*x]]^3/x) - 6*b^2*x*ArcCoth[Tanh[a + b*x]]*Log[x] + 3*b*ArcCoth[Tanh[a + b*x]]^2*(1 + Log[x]) + (3*b^3*x^2*(-1 + 2*Log[x]))/2","A",1
154,1,66,60,0.0376369,"\int \frac{\coth ^{-1}(\tanh (a+b x))^3}{x^3} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^3/x^3,x]","3 b^2 \log (x) \left(\coth ^{-1}(\tanh (a+b x))-b x\right)-\frac{\left(\coth ^{-1}(\tanh (a+b x))-b x\right)^3}{2 x^2}-\frac{3 b \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^2}{x}+b^3 x","-3 b^2 \log (x) \left(b x-\coth ^{-1}(\tanh (a+b x))\right)-\frac{\coth ^{-1}(\tanh (a+b x))^3}{2 x^2}-\frac{3 b \coth ^{-1}(\tanh (a+b x))^2}{2 x}+3 b^3 x",1,"b^3*x - (3*b*(-(b*x) + ArcCoth[Tanh[a + b*x]])^2)/x - (-(b*x) + ArcCoth[Tanh[a + b*x]])^3/(2*x^2) + 3*b^2*(-(b*x) + ArcCoth[Tanh[a + b*x]])*Log[x]","A",1
155,1,60,55,0.0256958,"\int \frac{\coth ^{-1}(\tanh (a+b x))^3}{x^4} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^3/x^4,x]","\frac{-6 b^2 x^2 \coth ^{-1}(\tanh (a+b x))-3 b x \coth ^{-1}(\tanh (a+b x))^2-2 \coth ^{-1}(\tanh (a+b x))^3+b^3 x^3 (6 \log (x)+11)}{6 x^3}","-\frac{b^2 \coth ^{-1}(\tanh (a+b x))}{x}-\frac{\coth ^{-1}(\tanh (a+b x))^3}{3 x^3}-\frac{b \coth ^{-1}(\tanh (a+b x))^2}{2 x^2}+b^3 \log (x)",1,"(-6*b^2*x^2*ArcCoth[Tanh[a + b*x]] - 3*b*x*ArcCoth[Tanh[a + b*x]]^2 - 2*ArcCoth[Tanh[a + b*x]]^3 + b^3*x^3*(11 + 6*Log[x]))/(6*x^3)","A",1
156,1,50,31,0.0241513,"\int \frac{\coth ^{-1}(\tanh (a+b x))^3}{x^5} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^3/x^5,x]","-\frac{b^2 x^2 \coth ^{-1}(\tanh (a+b x))+b x \coth ^{-1}(\tanh (a+b x))^2+\coth ^{-1}(\tanh (a+b x))^3+b^3 x^3}{4 x^4}","\frac{\coth ^{-1}(\tanh (a+b x))^4}{4 x^4 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}",1,"-1/4*(b^3*x^3 + b^2*x^2*ArcCoth[Tanh[a + b*x]] + b*x*ArcCoth[Tanh[a + b*x]]^2 + ArcCoth[Tanh[a + b*x]]^3)/x^4","A",1
157,1,54,64,0.0365885,"\int \frac{\coth ^{-1}(\tanh (a+b x))^3}{x^6} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^3/x^6,x]","-\frac{2 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+3 b x \coth ^{-1}(\tanh (a+b x))^2+4 \coth ^{-1}(\tanh (a+b x))^3+b^3 x^3}{20 x^5}","\frac{\coth ^{-1}(\tanh (a+b x))^4}{5 x^5 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}+\frac{b \coth ^{-1}(\tanh (a+b x))^4}{20 x^4 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2}",1,"-1/20*(b^3*x^3 + 2*b^2*x^2*ArcCoth[Tanh[a + b*x]] + 3*b*x*ArcCoth[Tanh[a + b*x]]^2 + 4*ArcCoth[Tanh[a + b*x]]^3)/x^5","A",1
158,1,51,53,0.0934846,"\int \frac{x^m}{\coth ^{-1}(\tanh (a+b x))} \, dx","Integrate[x^m/ArcCoth[Tanh[a + b*x]],x]","\frac{x^{m+1} \, _2F_1\left(1,m+1;m+2;-\frac{b x}{\coth ^{-1}(\tanh (a+b x))-b x}\right)}{(m+1) \left(\coth ^{-1}(\tanh (a+b x))-b x\right)}","-\frac{x^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right)}{(m+1) \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}",1,"(x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/(-(b*x) + ArcCoth[Tanh[a + b*x]]))])/((1 + m)*(-(b*x) + ArcCoth[Tanh[a + b*x]]))","A",1
159,1,79,81,0.0469343,"\int \frac{x^3}{\coth ^{-1}(\tanh (a+b x))} \, dx","Integrate[x^3/ArcCoth[Tanh[a + b*x]],x]","-\frac{\left(\coth ^{-1}(\tanh (a+b x))-b x\right)^3 \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^4}+\frac{x \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^2}{b^3}-\frac{x^2 \left(\coth ^{-1}(\tanh (a+b x))-b x\right)}{2 b^2}+\frac{x^3}{3 b}","\frac{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^3 \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^4}+\frac{x \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2}{b^3}+\frac{x^2 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}{2 b^2}+\frac{x^3}{3 b}",1,"x^3/(3*b) - (x^2*(-(b*x) + ArcCoth[Tanh[a + b*x]]))/(2*b^2) + (x*(-(b*x) + ArcCoth[Tanh[a + b*x]])^2)/b^3 - ((-(b*x) + ArcCoth[Tanh[a + b*x]])^3*Log[ArcCoth[Tanh[a + b*x]]])/b^4","A",1
160,1,55,56,0.0390453,"\int \frac{x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx","Integrate[x^2/ArcCoth[Tanh[a + b*x]],x]","\frac{\left(\coth ^{-1}(\tanh (a+b x))-b x\right)^2 \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^3}-\frac{x \left(\coth ^{-1}(\tanh (a+b x))-b x\right)}{b^2}+\frac{x^2}{2 b}","\frac{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2 \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^3}+\frac{x \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}{b^2}+\frac{x^2}{2 b}",1,"x^2/(2*b) - (x*(-(b*x) + ArcCoth[Tanh[a + b*x]]))/b^2 + ((-(b*x) + ArcCoth[Tanh[a + b*x]])^2*Log[ArcCoth[Tanh[a + b*x]]])/b^3","A",1
161,1,31,31,0.0255879,"\int \frac{x}{\coth ^{-1}(\tanh (a+b x))} \, dx","Integrate[x/ArcCoth[Tanh[a + b*x]],x]","\frac{x}{b}-\frac{\left(\coth ^{-1}(\tanh (a+b x))-b x\right) \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^2}","\frac{\left(b x-\coth ^{-1}(\tanh (a+b x))\right) \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^2}+\frac{x}{b}",1,"x/b - ((-(b*x) + ArcCoth[Tanh[a + b*x]])*Log[ArcCoth[Tanh[a + b*x]]])/b^2","A",1
162,1,12,12,0.0485988,"\int \frac{1}{\coth ^{-1}(\tanh (a+b x))} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^(-1),x]","\frac{\log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b}","\frac{\log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b}",1,"Log[ArcCoth[Tanh[a + b*x]]]/b","A",1
163,1,29,44,0.0282453,"\int \frac{1}{x \coth ^{-1}(\tanh (a+b x))} \, dx","Integrate[1/(x*ArcCoth[Tanh[a + b*x]]),x]","\frac{\log \left(\coth ^{-1}(\tanh (a+b x))\right)-\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))}","\frac{\log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac{\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))}",1,"(-Log[x] + Log[ArcCoth[Tanh[a + b*x]]])/(b*x - ArcCoth[Tanh[a + b*x]])","A",1
164,1,45,65,0.0250038,"\int \frac{1}{x^2 \coth ^{-1}(\tanh (a+b x))} \, dx","Integrate[1/(x^2*ArcCoth[Tanh[a + b*x]]),x]","\frac{b x \left(\log \left(\coth ^{-1}(\tanh (a+b x))\right)-\log (x)+1\right)-\coth ^{-1}(\tanh (a+b x))}{x \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^2}","\frac{1}{x \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}-\frac{b \log (x)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2}+\frac{b \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2}",1,"(-ArcCoth[Tanh[a + b*x]] + b*x*(1 - Log[x] + Log[ArcCoth[Tanh[a + b*x]]]))/(x*(-(b*x) + ArcCoth[Tanh[a + b*x]])^2)","A",1
165,1,66,92,0.0270171,"\int \frac{1}{x^3 \coth ^{-1}(\tanh (a+b x))} \, dx","Integrate[1/(x^3*ArcCoth[Tanh[a + b*x]]),x]","\frac{b^2 x^2 \left(2 \log \left(\coth ^{-1}(\tanh (a+b x))\right)-2 \log (x)+3\right)-4 b x \coth ^{-1}(\tanh (a+b x))+\coth ^{-1}(\tanh (a+b x))^2}{2 x^2 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^3}","-\frac{b^2 \log (x)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^3}+\frac{b^2 \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^3}+\frac{1}{2 x^2 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}+\frac{b}{x \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2}",1,"(-4*b*x*ArcCoth[Tanh[a + b*x]] + ArcCoth[Tanh[a + b*x]]^2 + b^2*x^2*(3 - 2*Log[x] + 2*Log[ArcCoth[Tanh[a + b*x]]]))/(2*x^2*(b*x - ArcCoth[Tanh[a + b*x]])^3)","A",1
166,1,51,65,0.5382366,"\int \frac{x^m}{\coth ^{-1}(\tanh (a+b x))^2} \, dx","Integrate[x^m/ArcCoth[Tanh[a + b*x]]^2,x]","\frac{x^{m+1} \, _2F_1\left(2,m+1;m+2;-\frac{b x}{\coth ^{-1}(\tanh (a+b x))-b x}\right)}{(m+1) \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^2}","-\frac{x^m \, _2F_1\left(1,m;m+1;\frac{b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right)}{b \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}-\frac{x^m}{b \coth ^{-1}(\tanh (a+b x))}",1,"(x^(1 + m)*Hypergeometric2F1[2, 1 + m, 2 + m, -((b*x)/(-(b*x) + ArcCoth[Tanh[a + b*x]]))])/((1 + m)*(-(b*x) + ArcCoth[Tanh[a + b*x]])^2)","A",1
167,1,106,98,0.0913687,"\int \frac{x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx","Integrate[x^4/ArcCoth[Tanh[a + b*x]]^2,x]","-\frac{\left(\coth ^{-1}(\tanh (a+b x))-b x\right)^4}{b^5 \coth ^{-1}(\tanh (a+b x))}-\frac{4 \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^3 \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^5}+\frac{3 x \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^2}{b^4}-\frac{x^2 \left(\coth ^{-1}(\tanh (a+b x))-b x\right)}{b^3}+\frac{x^3}{3 b^2}","\frac{4 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^3 \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^5}+\frac{4 x \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2}{b^4}+\frac{2 x^2 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}{b^3}-\frac{x^4}{b \coth ^{-1}(\tanh (a+b x))}+\frac{4 x^3}{3 b^2}",1,"x^3/(3*b^2) - (x^2*(-(b*x) + ArcCoth[Tanh[a + b*x]]))/b^3 + (3*x*(-(b*x) + ArcCoth[Tanh[a + b*x]])^2)/b^4 - (-(b*x) + ArcCoth[Tanh[a + b*x]])^4/(b^5*ArcCoth[Tanh[a + b*x]]) - (4*(-(b*x) + ArcCoth[Tanh[a + b*x]])^3*Log[ArcCoth[Tanh[a + b*x]]])/b^5","A",1
168,1,83,75,0.0542784,"\int \frac{x^3}{\coth ^{-1}(\tanh (a+b x))^2} \, dx","Integrate[x^3/ArcCoth[Tanh[a + b*x]]^2,x]","\frac{\left(\coth ^{-1}(\tanh (a+b x))-b x\right)^3}{b^4 \coth ^{-1}(\tanh (a+b x))}+\frac{3 \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^2 \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^4}-\frac{2 x \left(\coth ^{-1}(\tanh (a+b x))-b x\right)}{b^3}+\frac{x^2}{2 b^2}","\frac{3 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2 \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^4}+\frac{3 x \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}{b^3}-\frac{x^3}{b \coth ^{-1}(\tanh (a+b x))}+\frac{3 x^2}{2 b^2}",1,"x^2/(2*b^2) - (2*x*(-(b*x) + ArcCoth[Tanh[a + b*x]]))/b^3 + (-(b*x) + ArcCoth[Tanh[a + b*x]])^3/(b^4*ArcCoth[Tanh[a + b*x]]) + (3*(-(b*x) + ArcCoth[Tanh[a + b*x]])^2*Log[ArcCoth[Tanh[a + b*x]]])/b^4","A",1
169,1,56,50,0.0704652,"\int \frac{x^2}{\coth ^{-1}(\tanh (a+b x))^2} \, dx","Integrate[x^2/ArcCoth[Tanh[a + b*x]]^2,x]","\frac{-\frac{\left(\coth ^{-1}(\tanh (a+b x))-b x\right)^2}{\coth ^{-1}(\tanh (a+b x))}+2 \left(b x-\coth ^{-1}(\tanh (a+b x))\right) \log \left(\coth ^{-1}(\tanh (a+b x))\right)+b x}{b^3}","\frac{2 \left(b x-\coth ^{-1}(\tanh (a+b x))\right) \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^3}-\frac{x^2}{b \coth ^{-1}(\tanh (a+b x))}+\frac{2 x}{b^2}",1,"(b*x - (-(b*x) + ArcCoth[Tanh[a + b*x]])^2/ArcCoth[Tanh[a + b*x]] + 2*(b*x - ArcCoth[Tanh[a + b*x]])*Log[ArcCoth[Tanh[a + b*x]]])/b^3","A",1
170,1,27,28,0.0554583,"\int \frac{x}{\coth ^{-1}(\tanh (a+b x))^2} \, dx","Integrate[x/ArcCoth[Tanh[a + b*x]]^2,x]","\frac{-\frac{b x}{\coth ^{-1}(\tanh (a+b x))}+\log \left(\coth ^{-1}(\tanh (a+b x))\right)+1}{b^2}","\frac{\log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^2}-\frac{x}{b \coth ^{-1}(\tanh (a+b x))}",1,"(1 - (b*x)/ArcCoth[Tanh[a + b*x]] + Log[ArcCoth[Tanh[a + b*x]]])/b^2","A",1
171,1,14,14,0.0062153,"\int \frac{1}{\coth ^{-1}(\tanh (a+b x))^2} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^(-2),x]","-\frac{1}{b \coth ^{-1}(\tanh (a+b x))}","-\frac{1}{b \coth ^{-1}(\tanh (a+b x))}",1,"-(1/(b*ArcCoth[Tanh[a + b*x]]))","A",1
172,1,53,70,0.0735846,"\int \frac{1}{x \coth ^{-1}(\tanh (a+b x))^2} \, dx","Integrate[1/(x*ArcCoth[Tanh[a + b*x]]^2),x]","\frac{\coth ^{-1}(\tanh (a+b x)) \left(-\log \left(\coth ^{-1}(\tanh (a+b x))\right)+\log (b x)+1\right)-b x}{\coth ^{-1}(\tanh (a+b x)) \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^2}","-\frac{1}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right) \coth ^{-1}(\tanh (a+b x))}+\frac{\log (x)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2}-\frac{\log \left(\coth ^{-1}(\tanh (a+b x))\right)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2}",1,"(-(b*x) + ArcCoth[Tanh[a + b*x]]*(1 + Log[b*x] - Log[ArcCoth[Tanh[a + b*x]]]))/(ArcCoth[Tanh[a + b*x]]*(-(b*x) + ArcCoth[Tanh[a + b*x]])^2)","A",1
173,1,70,102,0.0609925,"\int \frac{1}{x^2 \coth ^{-1}(\tanh (a+b x))^2} \, dx","Integrate[1/(x^2*ArcCoth[Tanh[a + b*x]]^2),x]","\frac{\coth ^{-1}(\tanh (a+b x))^2+2 b x \coth ^{-1}(\tanh (a+b x)) \left(\log (x)-\log \left(\coth ^{-1}(\tanh (a+b x))\right)\right)-b^2 x^2}{x \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^3 \coth ^{-1}(\tanh (a+b x))}","-\frac{2 b}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2 \coth ^{-1}(\tanh (a+b x))}+\frac{1}{x \left(b x-\coth ^{-1}(\tanh (a+b x))\right) \coth ^{-1}(\tanh (a+b x))}+\frac{2 b \log (x)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^3}-\frac{2 b \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^3}",1,"(-(b^2*x^2) + ArcCoth[Tanh[a + b*x]]^2 + 2*b*x*ArcCoth[Tanh[a + b*x]]*(Log[x] - Log[ArcCoth[Tanh[a + b*x]]]))/(x*(b*x - ArcCoth[Tanh[a + b*x]])^3*ArcCoth[Tanh[a + b*x]])","A",1
174,1,92,143,0.0445815,"\int \frac{1}{x^3 \coth ^{-1}(\tanh (a+b x))^2} \, dx","Integrate[1/(x^3*ArcCoth[Tanh[a + b*x]]^2),x]","-\frac{-3 b^2 x^2 \coth ^{-1}(\tanh (a+b x)) \left(-2 \log \left(\coth ^{-1}(\tanh (a+b x))\right)+2 \log (x)-1\right)-6 b x \coth ^{-1}(\tanh (a+b x))^2+\coth ^{-1}(\tanh (a+b x))^3+2 b^3 x^3}{2 x^2 \coth ^{-1}(\tanh (a+b x)) \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^4}","-\frac{3 b^2}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^3 \coth ^{-1}(\tanh (a+b x))}+\frac{3 b^2 \log (x)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^4}-\frac{3 b^2 \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^4}+\frac{1}{2 x^2 \left(b x-\coth ^{-1}(\tanh (a+b x))\right) \coth ^{-1}(\tanh (a+b x))}+\frac{3 b}{2 x \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2 \coth ^{-1}(\tanh (a+b x))}",1,"-1/2*(2*b^3*x^3 - 6*b*x*ArcCoth[Tanh[a + b*x]]^2 + ArcCoth[Tanh[a + b*x]]^3 - 3*b^2*x^2*ArcCoth[Tanh[a + b*x]]*(-1 + 2*Log[x] - 2*Log[ArcCoth[Tanh[a + b*x]]]))/(x^2*ArcCoth[Tanh[a + b*x]]*(-(b*x) + ArcCoth[Tanh[a + b*x]])^4)","A",1
175,1,51,94,0.5427389,"\int \frac{x^m}{\coth ^{-1}(\tanh (a+b x))^3} \, dx","Integrate[x^m/ArcCoth[Tanh[a + b*x]]^3,x]","\frac{x^{m+1} \, _2F_1\left(3,m+1;m+2;-\frac{b x}{\coth ^{-1}(\tanh (a+b x))-b x}\right)}{(m+1) \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^3}","-\frac{m x^{m-1} \, _2F_1\left(1,m-1;m;\frac{b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right)}{2 b^2 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}-\frac{m x^{m-1}}{2 b^2 \coth ^{-1}(\tanh (a+b x))}-\frac{x^m}{2 b \coth ^{-1}(\tanh (a+b x))^2}",1,"(x^(1 + m)*Hypergeometric2F1[3, 1 + m, 2 + m, -((b*x)/(-(b*x) + ArcCoth[Tanh[a + b*x]]))])/((1 + m)*(-(b*x) + ArcCoth[Tanh[a + b*x]])^3)","A",1
176,1,114,92,0.0434358,"\int \frac{x^4}{\coth ^{-1}(\tanh (a+b x))^3} \, dx","Integrate[x^4/ArcCoth[Tanh[a + b*x]]^3,x]","-\frac{\left(\coth ^{-1}(\tanh (a+b x))-b x\right)^4}{2 b^5 \coth ^{-1}(\tanh (a+b x))^2}+\frac{4 \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^3}{b^5 \coth ^{-1}(\tanh (a+b x))}+\frac{6 \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^2 \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^5}-\frac{3 x \left(\coth ^{-1}(\tanh (a+b x))-b x\right)}{b^4}+\frac{x^2}{2 b^3}","\frac{6 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2 \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^5}+\frac{6 x \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}{b^4}-\frac{2 x^3}{b^2 \coth ^{-1}(\tanh (a+b x))}-\frac{x^4}{2 b \coth ^{-1}(\tanh (a+b x))^2}+\frac{3 x^2}{b^3}",1,"x^2/(2*b^3) - (3*x*(-(b*x) + ArcCoth[Tanh[a + b*x]]))/b^4 + (4*(-(b*x) + ArcCoth[Tanh[a + b*x]])^3)/(b^5*ArcCoth[Tanh[a + b*x]]) - (-(b*x) + ArcCoth[Tanh[a + b*x]])^4/(2*b^5*ArcCoth[Tanh[a + b*x]]^2) + (6*(-(b*x) + ArcCoth[Tanh[a + b*x]])^2*Log[ArcCoth[Tanh[a + b*x]]])/b^5","A",1
177,1,86,71,0.0456004,"\int \frac{x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx","Integrate[x^3/ArcCoth[Tanh[a + b*x]]^3,x]","-\frac{3 b^2 x^2 \coth ^{-1}(\tanh (a+b x))-b x \coth ^{-1}(\tanh (a+b x))^2 \left(6 \log \left(\coth ^{-1}(\tanh (a+b x))\right)+11\right)+\coth ^{-1}(\tanh (a+b x))^3 \left(6 \log \left(\coth ^{-1}(\tanh (a+b x))\right)+5\right)+b^3 x^3}{2 b^4 \coth ^{-1}(\tanh (a+b x))^2}","\frac{3 \left(b x-\coth ^{-1}(\tanh (a+b x))\right) \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^4}-\frac{3 x^2}{2 b^2 \coth ^{-1}(\tanh (a+b x))}-\frac{x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}+\frac{3 x}{b^3}",1,"-1/2*(b^3*x^3 + 3*b^2*x^2*ArcCoth[Tanh[a + b*x]] + ArcCoth[Tanh[a + b*x]]^3*(5 + 6*Log[ArcCoth[Tanh[a + b*x]]]) - b*x*ArcCoth[Tanh[a + b*x]]^2*(11 + 6*Log[ArcCoth[Tanh[a + b*x]]]))/(b^4*ArcCoth[Tanh[a + b*x]]^2)","A",1
178,1,49,47,0.0350448,"\int \frac{x^2}{\coth ^{-1}(\tanh (a+b x))^3} \, dx","Integrate[x^2/ArcCoth[Tanh[a + b*x]]^3,x]","\frac{-\frac{b^2 x^2}{\coth ^{-1}(\tanh (a+b x))^2}-\frac{2 b x}{\coth ^{-1}(\tanh (a+b x))}+2 \log \left(\coth ^{-1}(\tanh (a+b x))\right)+3}{2 b^3}","\frac{\log \left(\coth ^{-1}(\tanh (a+b x))\right)}{b^3}-\frac{x}{b^2 \coth ^{-1}(\tanh (a+b x))}-\frac{x^2}{2 b \coth ^{-1}(\tanh (a+b x))^2}",1,"(3 - (b^2*x^2)/ArcCoth[Tanh[a + b*x]]^2 - (2*b*x)/ArcCoth[Tanh[a + b*x]] + 2*Log[ArcCoth[Tanh[a + b*x]]])/(2*b^3)","A",1
179,1,27,34,0.0495662,"\int \frac{x}{\coth ^{-1}(\tanh (a+b x))^3} \, dx","Integrate[x/ArcCoth[Tanh[a + b*x]]^3,x]","-\frac{\coth ^{-1}(\tanh (a+b x))+b x}{2 b^2 \coth ^{-1}(\tanh (a+b x))^2}","-\frac{1}{2 b^2 \coth ^{-1}(\tanh (a+b x))}-\frac{x}{2 b \coth ^{-1}(\tanh (a+b x))^2}",1,"-1/2*(b*x + ArcCoth[Tanh[a + b*x]])/(b^2*ArcCoth[Tanh[a + b*x]]^2)","A",1
180,1,16,16,0.0060519,"\int \frac{1}{\coth ^{-1}(\tanh (a+b x))^3} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^(-3),x]","-\frac{1}{2 b \coth ^{-1}(\tanh (a+b x))^2}","-\frac{1}{2 b \coth ^{-1}(\tanh (a+b x))^2}",1,"-1/2*1/(b*ArcCoth[Tanh[a + b*x]]^2)","A",1
181,1,74,97,0.1102982,"\int \frac{1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx","Integrate[1/(x*ArcCoth[Tanh[a + b*x]]^3),x]","\frac{-4 b x \coth ^{-1}(\tanh (a+b x))+\coth ^{-1}(\tanh (a+b x))^2 \left(-2 \log \left(\coth ^{-1}(\tanh (a+b x))\right)+2 \log (b x)+3\right)+b^2 x^2}{2 \coth ^{-1}(\tanh (a+b x))^2 \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^3}","\frac{1}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2 \coth ^{-1}(\tanh (a+b x))}-\frac{1}{2 \left(b x-\coth ^{-1}(\tanh (a+b x))\right) \coth ^{-1}(\tanh (a+b x))^2}-\frac{\log (x)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^3}+\frac{\log \left(\coth ^{-1}(\tanh (a+b x))\right)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^3}",1,"(b^2*x^2 - 4*b*x*ArcCoth[Tanh[a + b*x]] + ArcCoth[Tanh[a + b*x]]^2*(3 + 2*Log[b*x] - 2*Log[ArcCoth[Tanh[a + b*x]]]))/(2*ArcCoth[Tanh[a + b*x]]^2*(-(b*x) + ArcCoth[Tanh[a + b*x]])^3)","A",1
182,1,93,131,0.0454165,"\int \frac{1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx","Integrate[1/(x^2*ArcCoth[Tanh[a + b*x]]^3),x]","-\frac{-6 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+2 \coth ^{-1}(\tanh (a+b x))^3+3 b x \coth ^{-1}(\tanh (a+b x))^2 \left(-2 \log \left(\coth ^{-1}(\tanh (a+b x))\right)+2 \log (x)+1\right)+b^3 x^3}{2 x \coth ^{-1}(\tanh (a+b x))^2 \left(\coth ^{-1}(\tanh (a+b x))-b x\right)^4}","\frac{3 b}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^3 \coth ^{-1}(\tanh (a+b x))}-\frac{3 b}{2 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac{1}{x \left(b x-\coth ^{-1}(\tanh (a+b x))\right) \coth ^{-1}(\tanh (a+b x))^2}-\frac{3 b \log (x)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^4}+\frac{3 b \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^4}",1,"-1/2*(b^3*x^3 - 6*b^2*x^2*ArcCoth[Tanh[a + b*x]] + 2*ArcCoth[Tanh[a + b*x]]^3 + 3*b*x*ArcCoth[Tanh[a + b*x]]^2*(1 + 2*Log[x] - 2*Log[ArcCoth[Tanh[a + b*x]]]))/(x*ArcCoth[Tanh[a + b*x]]^2*(-(b*x) + ArcCoth[Tanh[a + b*x]])^4)","A",1
183,1,107,170,0.0456774,"\int \frac{1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx","Integrate[1/(x^3*ArcCoth[Tanh[a + b*x]]^3),x]","\frac{8 b^3 x^3 \coth ^{-1}(\tanh (a+b x))-12 b^2 x^2 \coth ^{-1}(\tanh (a+b x))^2 \left(\log (x)-\log \left(\coth ^{-1}(\tanh (a+b x))\right)\right)-8 b x \coth ^{-1}(\tanh (a+b x))^3+\coth ^{-1}(\tanh (a+b x))^4-b^4 x^4}{2 x^2 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^5 \coth ^{-1}(\tanh (a+b x))^2}","\frac{6 b^2}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^4 \coth ^{-1}(\tanh (a+b x))}-\frac{3 b^2}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^3 \coth ^{-1}(\tanh (a+b x))^2}-\frac{6 b^2 \log (x)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^5}+\frac{6 b^2 \log \left(\coth ^{-1}(\tanh (a+b x))\right)}{\left(b x-\coth ^{-1}(\tanh (a+b x))\right)^5}+\frac{1}{2 x^2 \left(b x-\coth ^{-1}(\tanh (a+b x))\right) \coth ^{-1}(\tanh (a+b x))^2}+\frac{2 b}{x \left(b x-\coth ^{-1}(\tanh (a+b x))\right)^2 \coth ^{-1}(\tanh (a+b x))^2}",1,"(-(b^4*x^4) + 8*b^3*x^3*ArcCoth[Tanh[a + b*x]] - 8*b*x*ArcCoth[Tanh[a + b*x]]^3 + ArcCoth[Tanh[a + b*x]]^4 - 12*b^2*x^2*ArcCoth[Tanh[a + b*x]]^2*(Log[x] - Log[ArcCoth[Tanh[a + b*x]]]))/(2*x^2*(b*x - ArcCoth[Tanh[a + b*x]])^5*ArcCoth[Tanh[a + b*x]]^2)","A",1
184,1,71,79,0.1402617,"\int x^m \coth ^{-1}(\tanh (a+b x))^n \, dx","Integrate[x^m*ArcCoth[Tanh[a + b*x]]^n,x]","\frac{x^{m+1} \coth ^{-1}(\tanh (a+b x))^n \left(\frac{b x}{\coth ^{-1}(\tanh (a+b x))-b x}+1\right)^{-n} \, _2F_1\left(m+1,-n;m+2;-\frac{b x}{\coth ^{-1}(\tanh (a+b x))-b x}\right)}{m+1}","\frac{x^m \left(\frac{b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right)^{-m} \coth ^{-1}(\tanh (a+b x))^{n+1} \, _2F_1\left(-m,n+1;n+2;-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right)}{b (n+1)}",1,"(x^(1 + m)*ArcCoth[Tanh[a + b*x]]^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((b*x)/(-(b*x) + ArcCoth[Tanh[a + b*x]]))])/((1 + m)*(1 + (b*x)/(-(b*x) + ArcCoth[Tanh[a + b*x]]))^n)","A",1
185,1,146,165,0.108376,"\int x^4 \coth ^{-1}(\tanh (a+b x))^n \, dx","Integrate[x^4*ArcCoth[Tanh[a + b*x]]^n,x]","\frac{\coth ^{-1}(\tanh (a+b x))^{n+1} \left(-4 b^3 \left(n^3+12 n^2+47 n+60\right) x^3 \coth ^{-1}(\tanh (a+b x))+12 b^2 \left(n^2+9 n+20\right) x^2 \coth ^{-1}(\tanh (a+b x))^2-24 b (n+5) x \coth ^{-1}(\tanh (a+b x))^3+24 \coth ^{-1}(\tanh (a+b x))^4+b^4 \left(n^4+14 n^3+71 n^2+154 n+120\right) x^4\right)}{b^5 (n+1) (n+2) (n+3) (n+4) (n+5)}","\frac{24 \coth ^{-1}(\tanh (a+b x))^{n+5}}{b^5 (n+1) (n+2) (n+3) (n+4) (n+5)}-\frac{24 x \coth ^{-1}(\tanh (a+b x))^{n+4}}{b^4 (n+1) (n+2) (n+3) (n+4)}+\frac{12 x^2 \coth ^{-1}(\tanh (a+b x))^{n+3}}{b^3 (n+1) (n+2) (n+3)}-\frac{4 x^3 \coth ^{-1}(\tanh (a+b x))^{n+2}}{b^2 (n+1) (n+2)}+\frac{x^4 \coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)}",1,"(ArcCoth[Tanh[a + b*x]]^(1 + n)*(b^4*(120 + 154*n + 71*n^2 + 14*n^3 + n^4)*x^4 - 4*b^3*(60 + 47*n + 12*n^2 + n^3)*x^3*ArcCoth[Tanh[a + b*x]] + 12*b^2*(20 + 9*n + n^2)*x^2*ArcCoth[Tanh[a + b*x]]^2 - 24*b*(5 + n)*x*ArcCoth[Tanh[a + b*x]]^3 + 24*ArcCoth[Tanh[a + b*x]]^4))/(b^5*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n))","A",1
186,1,106,121,0.0781488,"\int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx","Integrate[x^3*ArcCoth[Tanh[a + b*x]]^n,x]","\frac{\coth ^{-1}(\tanh (a+b x))^{n+1} \left(-3 b^2 \left(n^2+7 n+12\right) x^2 \coth ^{-1}(\tanh (a+b x))+6 b (n+4) x \coth ^{-1}(\tanh (a+b x))^2-6 \coth ^{-1}(\tanh (a+b x))^3+b^3 \left(n^3+9 n^2+26 n+24\right) x^3\right)}{b^4 (n+1) (n+2) (n+3) (n+4)}","-\frac{6 \coth ^{-1}(\tanh (a+b x))^{n+4}}{b^4 (n+1) (n+2) (n+3) (n+4)}+\frac{6 x \coth ^{-1}(\tanh (a+b x))^{n+3}}{b^3 (n+1) (n+2) (n+3)}-\frac{3 x^2 \coth ^{-1}(\tanh (a+b x))^{n+2}}{b^2 (n+1) (n+2)}+\frac{x^3 \coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)}",1,"(ArcCoth[Tanh[a + b*x]]^(1 + n)*(b^3*(24 + 26*n + 9*n^2 + n^3)*x^3 - 3*b^2*(12 + 7*n + n^2)*x^2*ArcCoth[Tanh[a + b*x]] + 6*b*(4 + n)*x*ArcCoth[Tanh[a + b*x]]^2 - 6*ArcCoth[Tanh[a + b*x]]^3))/(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n))","A",1
187,1,71,82,0.0650307,"\int x^2 \coth ^{-1}(\tanh (a+b x))^n \, dx","Integrate[x^2*ArcCoth[Tanh[a + b*x]]^n,x]","\frac{\coth ^{-1}(\tanh (a+b x))^{n+1} \left(-2 b (n+3) x \coth ^{-1}(\tanh (a+b x))+2 \coth ^{-1}(\tanh (a+b x))^2+b^2 \left(n^2+5 n+6\right) x^2\right)}{b^3 (n+1) (n+2) (n+3)}","\frac{2 \coth ^{-1}(\tanh (a+b x))^{n+3}}{b^3 (n+1) (n+2) (n+3)}-\frac{2 x \coth ^{-1}(\tanh (a+b x))^{n+2}}{b^2 (n+1) (n+2)}+\frac{x^2 \coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)}",1,"(ArcCoth[Tanh[a + b*x]]^(1 + n)*(b^2*(6 + 5*n + n^2)*x^2 - 2*b*(3 + n)*x*ArcCoth[Tanh[a + b*x]] + 2*ArcCoth[Tanh[a + b*x]]^2))/(b^3*(1 + n)*(2 + n)*(3 + n))","A",1
188,1,41,48,0.0447787,"\int x \coth ^{-1}(\tanh (a+b x))^n \, dx","Integrate[x*ArcCoth[Tanh[a + b*x]]^n,x]","\frac{\left(b (n+2) x-\coth ^{-1}(\tanh (a+b x))\right) \coth ^{-1}(\tanh (a+b x))^{n+1}}{b^2 (n+1) (n+2)}","\frac{x \coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)}-\frac{\coth ^{-1}(\tanh (a+b x))^{n+2}}{b^2 (n+1) (n+2)}",1,"((b*(2 + n)*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]]^(1 + n))/(b^2*(1 + n)*(2 + n))","A",1
189,1,20,20,0.015888,"\int \coth ^{-1}(\tanh (a+b x))^n \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^n,x]","\frac{\coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)}","\frac{\coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)}",1,"ArcCoth[Tanh[a + b*x]]^(1 + n)/(b*(1 + n))","A",1
190,1,60,64,0.0890879,"\int \frac{\coth ^{-1}(\tanh (a+b x))^n}{x} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^n/x,x]","\frac{\coth ^{-1}(\tanh (a+b x))^n \left(\frac{\coth ^{-1}(\tanh (a+b x))}{b x}\right)^{-n} \, _2F_1\left(-n,-n;1-n;1-\frac{\coth ^{-1}(\tanh (a+b x))}{b x}\right)}{n}","\frac{\coth ^{-1}(\tanh (a+b x))^{n+1} \, _2F_1\left(1,n+1;n+2;-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right)}{(n+1) \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}",1,"(ArcCoth[Tanh[a + b*x]]^n*Hypergeometric2F1[-n, -n, 1 - n, 1 - ArcCoth[Tanh[a + b*x]]/(b*x)])/(n*(ArcCoth[Tanh[a + b*x]]/(b*x))^n)","A",1
191,1,67,71,0.0412613,"\int \frac{\coth ^{-1}(\tanh (a+b x))^n}{x^2} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^n/x^2,x]","\frac{\coth ^{-1}(\tanh (a+b x))^n \left(\frac{\coth ^{-1}(\tanh (a+b x))}{b x}\right)^{-n} \, _2F_1\left(1-n,-n;2-n;1-\frac{\coth ^{-1}(\tanh (a+b x))}{b x}\right)}{(n-1) x}","\frac{b \coth ^{-1}(\tanh (a+b x))^n \, _2F_1\left(1,n;n+1;-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right)}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac{\coth ^{-1}(\tanh (a+b x))^n}{x}",1,"(ArcCoth[Tanh[a + b*x]]^n*Hypergeometric2F1[1 - n, -n, 2 - n, 1 - ArcCoth[Tanh[a + b*x]]/(b*x)])/((-1 + n)*x*(ArcCoth[Tanh[a + b*x]]/(b*x))^n)","A",1
192,1,67,101,0.0437849,"\int \frac{\coth ^{-1}(\tanh (a+b x))^n}{x^3} \, dx","Integrate[ArcCoth[Tanh[a + b*x]]^n/x^3,x]","\frac{\coth ^{-1}(\tanh (a+b x))^n \left(\frac{\coth ^{-1}(\tanh (a+b x))}{b x}\right)^{-n} \, _2F_1\left(2-n,-n;3-n;1-\frac{\coth ^{-1}(\tanh (a+b x))}{b x}\right)}{(n-2) x^2}","\frac{b^2 n \coth ^{-1}(\tanh (a+b x))^{n-1} \, _2F_1\left(1,n-1;n;-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right)}{2 \left(b x-\coth ^{-1}(\tanh (a+b x))\right)}-\frac{\coth ^{-1}(\tanh (a+b x))^n}{2 x^2}-\frac{b n \coth ^{-1}(\tanh (a+b x))^{n-1}}{2 x}",1,"(ArcCoth[Tanh[a + b*x]]^n*Hypergeometric2F1[2 - n, -n, 3 - n, 1 - ArcCoth[Tanh[a + b*x]]/(b*x)])/((-2 + n)*x^2*(ArcCoth[Tanh[a + b*x]]/(b*x))^n)","A",1
193,1,34,37,0.0353969,"\int x^m \coth ^{-1}(\tanh (a+b x)) \, dx","Integrate[x^m*ArcCoth[Tanh[a + b*x]],x]","x^m \left(\frac{x \left(\coth ^{-1}(\tanh (a+b x))-b x\right)}{m+1}+\frac{b x^2}{m+2}\right)","\frac{x^{m+1} \coth ^{-1}(\tanh (a+b x))}{m+1}-\frac{b x^{m+2}}{m^2+3 m+2}",1,"x^m*((b*x^2)/(2 + m) + (x*(-(b*x) + ArcCoth[Tanh[a + b*x]]))/(1 + m))","A",1
194,1,20,23,0.0384803,"\int x^2 \coth ^{-1}(\coth (a+b x)) \, dx","Integrate[x^2*ArcCoth[Coth[a + b*x]],x]","-\frac{1}{12} x^3 \left(b x-4 \coth ^{-1}(\coth (a+b x))\right)","\frac{1}{3} x^3 \coth ^{-1}(\coth (a+b x))-\frac{b x^4}{12}",1,"-1/12*(x^3*(b*x - 4*ArcCoth[Coth[a + b*x]]))","A",1
195,1,20,23,0.0165409,"\int x \coth ^{-1}(\coth (a+b x)) \, dx","Integrate[x*ArcCoth[Coth[a + b*x]],x]","-\frac{1}{6} x^2 \left(b x-3 \coth ^{-1}(\coth (a+b x))\right)","\frac{1}{2} x^2 \coth ^{-1}(\coth (a+b x))-\frac{b x^3}{6}",1,"-1/6*(x^2*(b*x - 3*ArcCoth[Coth[a + b*x]]))","A",1
196,1,18,16,0.0069187,"\int \coth ^{-1}(\coth (a+b x)) \, dx","Integrate[ArcCoth[Coth[a + b*x]],x]","x \coth ^{-1}(\coth (a+b x))-\frac{b x^2}{2}","\frac{\coth ^{-1}(\coth (a+b x))^2}{2 b}",1,"-1/2*(b*x^2) + x*ArcCoth[Coth[a + b*x]]","A",1
197,1,19,21,0.0180747,"\int \frac{\coth ^{-1}(\coth (a+b x))}{x} \, dx","Integrate[ArcCoth[Coth[a + b*x]]/x,x]","\log (x) \left(\coth ^{-1}(\coth (a+b x))-b x\right)+b x","b x-\log (x) \left(b x-\coth ^{-1}(\coth (a+b x))\right)",1,"b*x + (-(b*x) + ArcCoth[Coth[a + b*x]])*Log[x]","A",1
198,1,18,17,0.017443,"\int \frac{\coth ^{-1}(\coth (a+b x))}{x^2} \, dx","Integrate[ArcCoth[Coth[a + b*x]]/x^2,x]","-\frac{\coth ^{-1}(\coth (a+b x))}{x}+b \log (x)+b","b \log (x)-\frac{\coth ^{-1}(\coth (a+b x))}{x}",1,"b - ArcCoth[Coth[a + b*x]]/x + b*Log[x]","A",1
199,1,18,23,0.0154167,"\int \frac{\coth ^{-1}(\coth (a+b x))}{x^3} \, dx","Integrate[ArcCoth[Coth[a + b*x]]/x^3,x]","-\frac{\coth ^{-1}(\coth (a+b x))+b x}{2 x^2}","-\frac{\coth ^{-1}(\coth (a+b x))}{2 x^2}-\frac{b}{2 x}",1,"-1/2*(b*x + ArcCoth[Coth[a + b*x]])/x^2","A",1
200,1,47,27,0.0215532,"\int \coth ^{-1}(\cosh (x)) \, dx","Integrate[ArcCoth[Cosh[x]],x]","\text{Li}_2\left(-e^{-x}\right)-\text{Li}_2\left(e^{-x}\right)+x \left(\log \left(1-e^{-x}\right)-\log \left(e^{-x}+1\right)\right)+x \coth ^{-1}(\cosh (x))","-\text{Li}_2\left(-e^x\right)+\text{Li}_2\left(e^x\right)-2 x \tanh ^{-1}\left(e^x\right)+x \coth ^{-1}(\cosh (x))",1,"x*ArcCoth[Cosh[x]] + x*(Log[1 - E^(-x)] - Log[1 + E^(-x)]) + PolyLog[2, -E^(-x)] - PolyLog[2, E^(-x)]","A",1
201,1,81,51,0.0151988,"\int x \coth ^{-1}(\cosh (x)) \, dx","Integrate[x*ArcCoth[Cosh[x]],x]","\frac{1}{2} \left(2 x \text{Li}_2\left(-e^{-x}\right)-2 x \text{Li}_2\left(e^{-x}\right)+2 \text{Li}_3\left(-e^{-x}\right)-2 \text{Li}_3\left(e^{-x}\right)+x^2 \log \left(1-e^{-x}\right)-x^2 \log \left(e^{-x}+1\right)+x^2 \coth ^{-1}(\cosh (x))\right)","-x \text{Li}_2\left(-e^x\right)+x \text{Li}_2\left(e^x\right)+\text{Li}_3\left(-e^x\right)-\text{Li}_3\left(e^x\right)-x^2 \tanh ^{-1}\left(e^x\right)+\frac{1}{2} x^2 \coth ^{-1}(\cosh (x))",1,"(x^2*ArcCoth[Cosh[x]] + x^2*Log[1 - E^(-x)] - x^2*Log[1 + E^(-x)] + 2*x*PolyLog[2, -E^(-x)] - 2*x*PolyLog[2, E^(-x)] + 2*PolyLog[3, -E^(-x)] - 2*PolyLog[3, E^(-x)])/2","A",1
202,1,109,77,0.0339804,"\int x^2 \coth ^{-1}(\cosh (x)) \, dx","Integrate[x^2*ArcCoth[Cosh[x]],x]","\frac{1}{24} \left(24 x^2 \text{Li}_2\left(-e^{-x}\right)+24 x^2 \text{Li}_2\left(e^x\right)+48 x \text{Li}_3\left(-e^{-x}\right)-48 x \text{Li}_3\left(e^x\right)+48 \text{Li}_4\left(-e^{-x}\right)+48 \text{Li}_4\left(e^x\right)-2 x^4-8 x^3 \log \left(e^{-x}+1\right)+8 x^3 \log \left(1-e^x\right)+8 x^3 \coth ^{-1}(\cosh (x))+\pi ^4\right)","-x^2 \text{Li}_2\left(-e^x\right)+x^2 \text{Li}_2\left(e^x\right)+2 x \text{Li}_3\left(-e^x\right)-2 x \text{Li}_3\left(e^x\right)-2 \text{Li}_4\left(-e^x\right)+2 \text{Li}_4\left(e^x\right)-\frac{2}{3} x^3 \tanh ^{-1}\left(e^x\right)+\frac{1}{3} x^3 \coth ^{-1}(\cosh (x))",1,"(Pi^4 - 2*x^4 + 8*x^3*ArcCoth[Cosh[x]] - 8*x^3*Log[1 + E^(-x)] + 8*x^3*Log[1 - E^x] + 24*x^2*PolyLog[2, -E^(-x)] + 24*x^2*PolyLog[2, E^x] + 48*x*PolyLog[3, -E^(-x)] - 48*x*PolyLog[3, E^x] + 48*PolyLog[4, -E^(-x)] + 48*PolyLog[4, E^x])/24","A",1
203,1,345,307,0.5039203,"\int x^2 \coth ^{-1}(c+d \tanh (a+b x)) \, dx","Integrate[x^2*ArcCoth[c + d*Tanh[a + b*x]],x]","\frac{4 b^3 x^3 \log \left(\frac{(c-d-1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d-1}+1\right)-4 b^3 x^3 \log \left(\frac{(c-d+1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d+1}+1\right)-6 b^2 x^2 \text{Li}_2\left(\frac{(c-d-1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d-1}\right)+6 b^2 x^2 \text{Li}_2\left(\frac{(c-d+1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d+1}\right)-6 b x \text{Li}_3\left(\frac{(c-d-1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d-1}\right)+6 b x \text{Li}_3\left(\frac{(c-d+1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d+1}\right)-3 \text{Li}_4\left(\frac{(c-d-1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d-1}\right)+3 \text{Li}_4\left(\frac{(c-d+1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d+1}\right)}{24 b^3}+\frac{1}{3} x^3 \coth ^{-1}(d \tanh (a+b x)+c)","\frac{\text{Li}_4\left(-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)}{8 b^3}-\frac{\text{Li}_4\left(-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)}{8 b^3}-\frac{x \text{Li}_3\left(-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)}{4 b^2}+\frac{x \text{Li}_3\left(-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)}{4 b^2}+\frac{x^2 \text{Li}_2\left(-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)}{4 b}-\frac{x^2 \text{Li}_2\left(-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)}{4 b}+\frac{1}{6} x^3 \log \left(\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}+1\right)-\frac{1}{6} x^3 \log \left(\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}+1\right)+\frac{1}{3} x^3 \coth ^{-1}(d \tanh (a+b x)+c)",1,"(x^3*ArcCoth[c + d*Tanh[a + b*x]])/3 + (4*b^3*x^3*Log[1 + ((-1 + c - d)*(Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]))/(-1 + c + d)] - 4*b^3*x^3*Log[1 + ((1 + c - d)*(Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]))/(1 + c + d)] - 6*b^2*x^2*PolyLog[2, ((-1 + c - d)*(-Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]))/(-1 + c + d)] + 6*b^2*x^2*PolyLog[2, ((1 + c - d)*(-Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]))/(1 + c + d)] - 6*b*x*PolyLog[3, ((-1 + c - d)*(-Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]))/(-1 + c + d)] + 6*b*x*PolyLog[3, ((1 + c - d)*(-Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]))/(1 + c + d)] - 3*PolyLog[4, ((-1 + c - d)*(-Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]))/(-1 + c + d)] + 3*PolyLog[4, ((1 + c - d)*(-Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]))/(1 + c + d)])/(24*b^3)","A",1
204,1,259,231,0.1813526,"\int x \coth ^{-1}(c+d \tanh (a+b x)) \, dx","Integrate[x*ArcCoth[c + d*Tanh[a + b*x]],x]","\frac{2 b^2 x^2 \log \left(\frac{(c-d-1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d-1}+1\right)-2 b^2 x^2 \log \left(\frac{(c-d+1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d+1}+1\right)-2 b x \text{Li}_2\left(\frac{(c-d-1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d-1}\right)+2 b x \text{Li}_2\left(\frac{(c-d+1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d+1}\right)-\text{Li}_3\left(\frac{(c-d-1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d-1}\right)+\text{Li}_3\left(\frac{(c-d+1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d+1}\right)}{8 b^2}+\frac{1}{2} x^2 \coth ^{-1}(d \tanh (a+b x)+c)","-\frac{\text{Li}_3\left(-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)}{8 b^2}+\frac{\text{Li}_3\left(-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)}{8 b^2}+\frac{x \text{Li}_2\left(-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)}{4 b}-\frac{x \text{Li}_2\left(-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)}{4 b}+\frac{1}{4} x^2 \log \left(\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}+1\right)-\frac{1}{4} x^2 \log \left(\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}+1\right)+\frac{1}{2} x^2 \coth ^{-1}(d \tanh (a+b x)+c)",1,"(x^2*ArcCoth[c + d*Tanh[a + b*x]])/2 + (2*b^2*x^2*Log[1 + ((-1 + c - d)*(Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]))/(-1 + c + d)] - 2*b^2*x^2*Log[1 + ((1 + c - d)*(Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]))/(1 + c + d)] - 2*b*x*PolyLog[2, ((-1 + c - d)*(-Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]))/(-1 + c + d)] + 2*b*x*PolyLog[2, ((1 + c - d)*(-Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]))/(1 + c + d)] - PolyLog[3, ((-1 + c - d)*(-Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]))/(-1 + c + d)] + PolyLog[3, ((1 + c - d)*(-Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]))/(1 + c + d)])/(8*b^2)","A",1
205,1,131,150,1.4232297,"\int \coth ^{-1}(c+d \tanh (a+b x)) \, dx","Integrate[ArcCoth[c + d*Tanh[a + b*x]],x]","\frac{\text{Li}_2\left(-\frac{(c+d-1) e^{2 (a+b x)}}{c-d-1}\right)-\text{Li}_2\left(-\frac{(c+d+1) e^{2 (a+b x)}}{c-d+1}\right)+2 b x \left(\log \left(\frac{(c+d-1) e^{2 (a+b x)}}{c-d-1}+1\right)-\log \left(\frac{(c+d+1) e^{2 (a+b x)}}{c-d+1}+1\right)\right)}{4 b}+x \coth ^{-1}(d \tanh (a+b x)+c)","\frac{\text{Li}_2\left(-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)}{4 b}-\frac{\text{Li}_2\left(-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)}{4 b}+\frac{1}{2} x \log \left(\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}+1\right)-\frac{1}{2} x \log \left(\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}+1\right)+x \coth ^{-1}(d \tanh (a+b x)+c)",1,"x*ArcCoth[c + d*Tanh[a + b*x]] + (2*b*x*(Log[1 + ((-1 + c + d)*E^(2*(a + b*x)))/(-1 + c - d)] - Log[1 + ((1 + c + d)*E^(2*(a + b*x)))/(1 + c - d)]) + PolyLog[2, -(((-1 + c + d)*E^(2*(a + b*x)))/(-1 + c - d))] - PolyLog[2, -(((1 + c + d)*E^(2*(a + b*x)))/(1 + c - d))])/(4*b)","A",1
206,0,0,18,9.5365609,"\int \frac{\coth ^{-1}(c+d \tanh (a+b x))}{x} \, dx","Integrate[ArcCoth[c + d*Tanh[a + b*x]]/x,x]","\int \frac{\coth ^{-1}(c+d \tanh (a+b x))}{x} \, dx","\text{Int}\left(\frac{\coth ^{-1}(d \tanh (a+b x)+c)}{x},x\right)",0,"Integrate[ArcCoth[c + d*Tanh[a + b*x]]/x, x]","A",-1
207,1,144,155,0.2050615,"\int x^3 \coth ^{-1}(1+d+d \tanh (a+b x)) \, dx","Integrate[x^3*ArcCoth[1 + d + d*Tanh[a + b*x]],x]","\frac{1}{16} \left(\frac{3 \text{Li}_5\left(-\frac{e^{-2 (a+b x)}}{d+1}\right)}{b^4}+\frac{6 x \text{Li}_4\left(-\frac{e^{-2 (a+b x)}}{d+1}\right)}{b^3}+\frac{6 x^2 \text{Li}_3\left(-\frac{e^{-2 (a+b x)}}{d+1}\right)}{b^2}+\frac{4 x^3 \text{Li}_2\left(-\frac{e^{-2 (a+b x)}}{d+1}\right)}{b}-2 x^4 \log \left(\frac{e^{-2 (a+b x)}}{d+1}+1\right)+4 x^4 \coth ^{-1}(d \tanh (a+b x)+d+1)\right)","\frac{3 \text{Li}_5\left(-\left((d+1) e^{2 a+2 b x}\right)\right)}{16 b^4}-\frac{3 x \text{Li}_4\left(-\left((d+1) e^{2 a+2 b x}\right)\right)}{8 b^3}+\frac{3 x^2 \text{Li}_3\left(-\left((d+1) e^{2 a+2 b x}\right)\right)}{8 b^2}-\frac{x^3 \text{Li}_2\left(-\left((d+1) e^{2 a+2 b x}\right)\right)}{4 b}-\frac{1}{8} x^4 \log \left((d+1) e^{2 a+2 b x}+1\right)+\frac{1}{4} x^4 \coth ^{-1}(d \tanh (a+b x)+d+1)+\frac{b x^5}{20}",1,"(4*x^4*ArcCoth[1 + d + d*Tanh[a + b*x]] - 2*x^4*Log[1 + 1/((1 + d)*E^(2*(a + b*x)))] + (4*x^3*PolyLog[2, -(1/((1 + d)*E^(2*(a + b*x))))])/b + (6*x^2*PolyLog[3, -(1/((1 + d)*E^(2*(a + b*x))))])/b^2 + (6*x*PolyLog[4, -(1/((1 + d)*E^(2*(a + b*x))))])/b^3 + (3*PolyLog[5, -(1/((1 + d)*E^(2*(a + b*x))))])/b^4)/16","A",1
208,1,118,128,0.1193786,"\int x^2 \coth ^{-1}(1+d+d \tanh (a+b x)) \, dx","Integrate[x^2*ArcCoth[1 + d + d*Tanh[a + b*x]],x]","\frac{1}{24} \left(\frac{3 \text{Li}_4\left(-\frac{e^{-2 (a+b x)}}{d+1}\right)}{b^3}+\frac{6 x \text{Li}_3\left(-\frac{e^{-2 (a+b x)}}{d+1}\right)}{b^2}+\frac{6 x^2 \text{Li}_2\left(-\frac{e^{-2 (a+b x)}}{d+1}\right)}{b}-4 x^3 \log \left(\frac{e^{-2 (a+b x)}}{d+1}+1\right)+8 x^3 \coth ^{-1}(d \tanh (a+b x)+d+1)\right)","-\frac{\text{Li}_4\left(-\left((d+1) e^{2 a+2 b x}\right)\right)}{8 b^3}+\frac{x \text{Li}_3\left(-\left((d+1) e^{2 a+2 b x}\right)\right)}{4 b^2}-\frac{x^2 \text{Li}_2\left(-\left((d+1) e^{2 a+2 b x}\right)\right)}{4 b}-\frac{1}{6} x^3 \log \left((d+1) e^{2 a+2 b x}+1\right)+\frac{1}{3} x^3 \coth ^{-1}(d \tanh (a+b x)+d+1)+\frac{b x^4}{12}",1,"(8*x^3*ArcCoth[1 + d + d*Tanh[a + b*x]] - 4*x^3*Log[1 + 1/((1 + d)*E^(2*(a + b*x)))] + (6*x^2*PolyLog[2, -(1/((1 + d)*E^(2*(a + b*x))))])/b + (6*x*PolyLog[3, -(1/((1 + d)*E^(2*(a + b*x))))])/b^2 + (3*PolyLog[4, -(1/((1 + d)*E^(2*(a + b*x))))])/b^3)/24","A",1
209,1,91,101,0.1015087,"\int x \coth ^{-1}(1+d+d \tanh (a+b x)) \, dx","Integrate[x*ArcCoth[1 + d + d*Tanh[a + b*x]],x]","\frac{2 b^2 x^2 \left(2 \coth ^{-1}(d \tanh (a+b x)+d+1)-\log \left(\frac{e^{-2 (a+b x)}}{d+1}+1\right)\right)+2 b x \text{Li}_2\left(-\frac{e^{-2 (a+b x)}}{d+1}\right)+\text{Li}_3\left(-\frac{e^{-2 (a+b x)}}{d+1}\right)}{8 b^2}","\frac{\text{Li}_3\left(-\left((d+1) e^{2 a+2 b x}\right)\right)}{8 b^2}-\frac{x \text{Li}_2\left(-\left((d+1) e^{2 a+2 b x}\right)\right)}{4 b}-\frac{1}{4} x^2 \log \left((d+1) e^{2 a+2 b x}+1\right)+\frac{1}{2} x^2 \coth ^{-1}(d \tanh (a+b x)+d+1)+\frac{b x^3}{6}",1,"(2*b^2*x^2*(2*ArcCoth[1 + d + d*Tanh[a + b*x]] - Log[1 + 1/((1 + d)*E^(2*(a + b*x)))]) + 2*b*x*PolyLog[2, -(1/((1 + d)*E^(2*(a + b*x))))] + PolyLog[3, -(1/((1 + d)*E^(2*(a + b*x))))])/(8*b^2)","A",1
210,1,201,69,0.9131635,"\int \coth ^{-1}(1+d+d \tanh (a+b x)) \, dx","Integrate[ArcCoth[1 + d + d*Tanh[a + b*x]],x]","\frac{-2 \text{Li}_2\left(-\sqrt{-d-1} e^{a+b x}\right)-2 \text{Li}_2\left(\sqrt{-d-1} e^{a+b x}\right)-2 \log \left(e^{a+b x}\right) \log \left(1-\sqrt{-d-1} e^{a+b x}\right)-2 \log \left(e^{a+b x}\right) \log \left(\sqrt{-d-1} e^{a+b x}+1\right)+2 \log \left(e^{a+b x}\right) \log \left((d+1) e^{a+b x}+e^{-a-b x}\right)-2 b x \log (d \sinh (a+b x)+(d+2) \cosh (a+b x))+\log ^2\left(e^{a+b x}\right)+b^2 x^2}{4 b}+x \coth ^{-1}(d \tanh (a+b x)+d+1)","-\frac{\text{Li}_2\left(-\left((d+1) e^{2 a+2 b x}\right)\right)}{4 b}-\frac{1}{2} x \log \left((d+1) e^{2 a+2 b x}+1\right)+x \coth ^{-1}(d \tanh (a+b x)+d+1)+\frac{b x^2}{2}",1,"x*ArcCoth[1 + d + d*Tanh[a + b*x]] + (b^2*x^2 + Log[E^(a + b*x)]^2 - 2*Log[E^(a + b*x)]*Log[1 - Sqrt[-1 - d]*E^(a + b*x)] - 2*Log[E^(a + b*x)]*Log[1 + Sqrt[-1 - d]*E^(a + b*x)] + 2*Log[E^(a + b*x)]*Log[E^(-a - b*x) + (1 + d)*E^(a + b*x)] - 2*b*x*Log[(2 + d)*Cosh[a + b*x] + d*Sinh[a + b*x]] - 2*PolyLog[2, -(Sqrt[-1 - d]*E^(a + b*x))] - 2*PolyLog[2, Sqrt[-1 - d]*E^(a + b*x)])/(4*b)","B",0
211,0,0,19,3.3510707,"\int \frac{\coth ^{-1}(1+d+d \tanh (a+b x))}{x} \, dx","Integrate[ArcCoth[1 + d + d*Tanh[a + b*x]]/x,x]","\int \frac{\coth ^{-1}(1+d+d \tanh (a+b x))}{x} \, dx","\text{Int}\left(\frac{\coth ^{-1}(d \tanh (a+b x)+d+1)}{x},x\right)",0,"Integrate[ArcCoth[1 + d + d*Tanh[a + b*x]]/x, x]","A",-1
212,1,144,168,0.2177601,"\int x^3 \coth ^{-1}(1-d-d \tanh (a+b x)) \, dx","Integrate[x^3*ArcCoth[1 - d - d*Tanh[a + b*x]],x]","\frac{1}{16} \left(\frac{3 \text{Li}_5\left(\frac{e^{-2 (a+b x)}}{d-1}\right)}{b^4}+\frac{6 x \text{Li}_4\left(\frac{e^{-2 (a+b x)}}{d-1}\right)}{b^3}+\frac{6 x^2 \text{Li}_3\left(\frac{e^{-2 (a+b x)}}{d-1}\right)}{b^2}+\frac{4 x^3 \text{Li}_2\left(\frac{e^{-2 (a+b x)}}{d-1}\right)}{b}-2 x^4 \log \left(1-\frac{e^{-2 (a+b x)}}{d-1}\right)+4 x^4 \coth ^{-1}(d (-\tanh (a+b x))-d+1)\right)","\frac{3 \text{Li}_5\left(-\left((1-d) e^{2 a+2 b x}\right)\right)}{16 b^4}-\frac{3 x \text{Li}_4\left(-\left((1-d) e^{2 a+2 b x}\right)\right)}{8 b^3}+\frac{3 x^2 \text{Li}_3\left(-\left((1-d) e^{2 a+2 b x}\right)\right)}{8 b^2}-\frac{x^3 \text{Li}_2\left(-\left((1-d) e^{2 a+2 b x}\right)\right)}{4 b}-\frac{1}{8} x^4 \log \left((1-d) e^{2 a+2 b x}+1\right)+\frac{1}{4} x^4 \coth ^{-1}(d (-\tanh (a+b x))-d+1)+\frac{b x^5}{20}",1,"(4*x^4*ArcCoth[1 - d - d*Tanh[a + b*x]] - 2*x^4*Log[1 - 1/((-1 + d)*E^(2*(a + b*x)))] + (4*x^3*PolyLog[2, 1/((-1 + d)*E^(2*(a + b*x)))])/b + (6*x^2*PolyLog[3, 1/((-1 + d)*E^(2*(a + b*x)))])/b^2 + (6*x*PolyLog[4, 1/((-1 + d)*E^(2*(a + b*x)))])/b^3 + (3*PolyLog[5, 1/((-1 + d)*E^(2*(a + b*x)))])/b^4)/16","A",1
213,1,119,139,0.1151351,"\int x^2 \coth ^{-1}(1-d-d \tanh (a+b x)) \, dx","Integrate[x^2*ArcCoth[1 - d - d*Tanh[a + b*x]],x]","\frac{1}{24} \left(\frac{3 \text{Li}_4\left(\frac{e^{-2 (a+b x)}}{d-1}\right)}{b^3}+\frac{6 x \text{Li}_3\left(\frac{e^{-2 (a+b x)}}{d-1}\right)}{b^2}+\frac{6 x^2 \text{Li}_2\left(\frac{e^{-2 (a+b x)}}{d-1}\right)}{b}-4 x^3 \log \left(1-\frac{e^{-2 (a+b x)}}{d-1}\right)+8 x^3 \coth ^{-1}(d (-\tanh (a+b x))-d+1)\right)","-\frac{\text{Li}_4\left(-\left((1-d) e^{2 a+2 b x}\right)\right)}{8 b^3}+\frac{x \text{Li}_3\left(-\left((1-d) e^{2 a+2 b x}\right)\right)}{4 b^2}-\frac{x^2 \text{Li}_2\left(-\left((1-d) e^{2 a+2 b x}\right)\right)}{4 b}-\frac{1}{6} x^3 \log \left((1-d) e^{2 a+2 b x}+1\right)+\frac{1}{3} x^3 \coth ^{-1}(d (-\tanh (a+b x))-d+1)+\frac{b x^4}{12}",1,"(8*x^3*ArcCoth[1 - d - d*Tanh[a + b*x]] - 4*x^3*Log[1 - 1/((-1 + d)*E^(2*(a + b*x)))] + (6*x^2*PolyLog[2, 1/((-1 + d)*E^(2*(a + b*x)))])/b + (6*x*PolyLog[3, 1/((-1 + d)*E^(2*(a + b*x)))])/b^2 + (3*PolyLog[4, 1/((-1 + d)*E^(2*(a + b*x)))])/b^3)/24","A",1
214,1,93,110,0.1007993,"\int x \coth ^{-1}(1-d-d \tanh (a+b x)) \, dx","Integrate[x*ArcCoth[1 - d - d*Tanh[a + b*x]],x]","\frac{2 b^2 x^2 \left(2 \coth ^{-1}(d (-\tanh (a+b x))-d+1)-\log \left(1-\frac{e^{-2 (a+b x)}}{d-1}\right)\right)+2 b x \text{Li}_2\left(\frac{e^{-2 (a+b x)}}{d-1}\right)+\text{Li}_3\left(\frac{e^{-2 (a+b x)}}{d-1}\right)}{8 b^2}","\frac{\text{Li}_3\left(-\left((1-d) e^{2 a+2 b x}\right)\right)}{8 b^2}-\frac{x \text{Li}_2\left(-\left((1-d) e^{2 a+2 b x}\right)\right)}{4 b}-\frac{1}{4} x^2 \log \left((1-d) e^{2 a+2 b x}+1\right)+\frac{1}{2} x^2 \coth ^{-1}(d (-\tanh (a+b x))-d+1)+\frac{b x^3}{6}",1,"(2*b^2*x^2*(2*ArcCoth[1 - d - d*Tanh[a + b*x]] - Log[1 - 1/((-1 + d)*E^(2*(a + b*x)))]) + 2*b*x*PolyLog[2, 1/((-1 + d)*E^(2*(a + b*x)))] + PolyLog[3, 1/((-1 + d)*E^(2*(a + b*x)))])/(8*b^2)","A",1
215,1,200,76,0.9847082,"\int \coth ^{-1}(1-d-d \tanh (a+b x)) \, dx","Integrate[ArcCoth[1 - d - d*Tanh[a + b*x]],x]","\frac{-2 \text{Li}_2\left(-\sqrt{d-1} e^{a+b x}\right)-2 \text{Li}_2\left(\sqrt{d-1} e^{a+b x}\right)-2 \log \left(e^{a+b x}\right) \log \left(1-\sqrt{d-1} e^{a+b x}\right)-2 \log \left(e^{a+b x}\right) \log \left(\sqrt{d-1} e^{a+b x}+1\right)+2 \log \left(e^{a+b x}\right) \log \left(e^{-a-b x} \left((d-1) e^{2 (a+b x)}-1\right)\right)-2 b x \log (d \sinh (a+b x)+(d-2) \cosh (a+b x))+\log ^2\left(e^{a+b x}\right)+b^2 x^2}{4 b}+x \coth ^{-1}(d (-\tanh (a+b x))-d+1)","-\frac{\text{Li}_2\left(-\left((1-d) e^{2 a+2 b x}\right)\right)}{4 b}-\frac{1}{2} x \log \left((1-d) e^{2 a+2 b x}+1\right)+x \coth ^{-1}(d (-\tanh (a+b x))-d+1)+\frac{b x^2}{2}",1,"x*ArcCoth[1 - d - d*Tanh[a + b*x]] + (b^2*x^2 + Log[E^(a + b*x)]^2 - 2*Log[E^(a + b*x)]*Log[1 - Sqrt[-1 + d]*E^(a + b*x)] - 2*Log[E^(a + b*x)]*Log[1 + Sqrt[-1 + d]*E^(a + b*x)] + 2*Log[E^(a + b*x)]*Log[E^(-a - b*x)*(-1 + (-1 + d)*E^(2*(a + b*x)))] - 2*b*x*Log[(-2 + d)*Cosh[a + b*x] + d*Sinh[a + b*x]] - 2*PolyLog[2, -(Sqrt[-1 + d]*E^(a + b*x))] - 2*PolyLog[2, Sqrt[-1 + d]*E^(a + b*x)])/(4*b)","B",0
216,0,0,22,3.3911413,"\int \frac{\coth ^{-1}(1-d-d \tanh (a+b x))}{x} \, dx","Integrate[ArcCoth[1 - d - d*Tanh[a + b*x]]/x,x]","\int \frac{\coth ^{-1}(1-d-d \tanh (a+b x))}{x} \, dx","\text{Int}\left(\frac{\coth ^{-1}(d (-\tanh (a+b x))-d+1)}{x},x\right)",0,"Integrate[ArcCoth[1 - d - d*Tanh[a + b*x]]/x, x]","A",-1
217,1,353,303,0.4486055,"\int x^2 \coth ^{-1}(c+d \coth (a+b x)) \, dx","Integrate[x^2*ArcCoth[c + d*Coth[a + b*x]],x]","\frac{4 b^3 x^3 \log \left(\frac{2 (\cosh (a+b x)-\sinh (a+b x)) ((c-1) \sinh (a+b x)+d \cosh (a+b x))}{c+d-1}\right)-4 b^3 x^3 \log \left(\frac{(c-d+1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d+1}+1\right)-6 b^2 x^2 \text{Li}_2\left(\frac{(c-d-1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d-1}\right)+6 b^2 x^2 \text{Li}_2\left(\frac{(c-d+1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d+1}\right)-6 b x \text{Li}_3\left(\frac{(c-d-1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d-1}\right)+6 b x \text{Li}_3\left(\frac{(c-d+1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d+1}\right)-3 \text{Li}_4\left(\frac{(c-d-1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d-1}\right)+3 \text{Li}_4\left(\frac{(c-d+1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d+1}\right)}{24 b^3}+\frac{1}{3} x^3 \coth ^{-1}(d \coth (a+b x)+c)","\frac{\text{Li}_4\left(\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)}{8 b^3}-\frac{\text{Li}_4\left(\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)}{8 b^3}-\frac{x \text{Li}_3\left(\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)}{4 b^2}+\frac{x \text{Li}_3\left(\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)}{4 b^2}+\frac{x^2 \text{Li}_2\left(\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)}{4 b}-\frac{x^2 \text{Li}_2\left(\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)}{4 b}+\frac{1}{6} x^3 \log \left(1-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)-\frac{1}{6} x^3 \log \left(1-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)+\frac{1}{3} x^3 \coth ^{-1}(d \coth (a+b x)+c)",1,"(x^3*ArcCoth[c + d*Coth[a + b*x]])/3 + (4*b^3*x^3*Log[(2*(Cosh[a + b*x] - Sinh[a + b*x])*(d*Cosh[a + b*x] + (-1 + c)*Sinh[a + b*x]))/(-1 + c + d)] - 4*b^3*x^3*Log[1 + ((1 + c - d)*(-Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]))/(1 + c + d)] - 6*b^2*x^2*PolyLog[2, ((-1 + c - d)*(Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]))/(-1 + c + d)] + 6*b^2*x^2*PolyLog[2, ((1 + c - d)*(Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]))/(1 + c + d)] - 6*b*x*PolyLog[3, ((-1 + c - d)*(Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]))/(-1 + c + d)] + 6*b*x*PolyLog[3, ((1 + c - d)*(Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]))/(1 + c + d)] - 3*PolyLog[4, ((-1 + c - d)*(Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]))/(-1 + c + d)] + 3*PolyLog[4, ((1 + c - d)*(Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]))/(1 + c + d)])/(24*b^3)","A",0
218,1,267,229,0.1646537,"\int x \coth ^{-1}(c+d \coth (a+b x)) \, dx","Integrate[x*ArcCoth[c + d*Coth[a + b*x]],x]","\frac{2 b^2 x^2 \log \left(\frac{2 (\cosh (a+b x)-\sinh (a+b x)) ((c-1) \sinh (a+b x)+d \cosh (a+b x))}{c+d-1}\right)-2 b^2 x^2 \log \left(\frac{(c-d+1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d+1}+1\right)-2 b x \text{Li}_2\left(\frac{(c-d-1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d-1}\right)+2 b x \text{Li}_2\left(\frac{(c-d+1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d+1}\right)-\text{Li}_3\left(\frac{(c-d-1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d-1}\right)+\text{Li}_3\left(\frac{(c-d+1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d+1}\right)}{8 b^2}+\frac{1}{2} x^2 \coth ^{-1}(d \coth (a+b x)+c)","-\frac{\text{Li}_3\left(\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)}{8 b^2}+\frac{\text{Li}_3\left(\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)}{8 b^2}+\frac{x \text{Li}_2\left(\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)}{4 b}-\frac{x \text{Li}_2\left(\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)}{4 b}+\frac{1}{4} x^2 \log \left(1-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)-\frac{1}{4} x^2 \log \left(1-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)+\frac{1}{2} x^2 \coth ^{-1}(d \coth (a+b x)+c)",1,"(x^2*ArcCoth[c + d*Coth[a + b*x]])/2 + (2*b^2*x^2*Log[(2*(Cosh[a + b*x] - Sinh[a + b*x])*(d*Cosh[a + b*x] + (-1 + c)*Sinh[a + b*x]))/(-1 + c + d)] - 2*b^2*x^2*Log[1 + ((1 + c - d)*(-Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]))/(1 + c + d)] - 2*b*x*PolyLog[2, ((-1 + c - d)*(Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]))/(-1 + c + d)] + 2*b*x*PolyLog[2, ((1 + c - d)*(Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]))/(1 + c + d)] - PolyLog[3, ((-1 + c - d)*(Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]))/(-1 + c + d)] + PolyLog[3, ((1 + c - d)*(Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]))/(1 + c + d)])/(8*b^2)","A",0
219,1,131,150,1.2627346,"\int \coth ^{-1}(c+d \coth (a+b x)) \, dx","Integrate[ArcCoth[c + d*Coth[a + b*x]],x]","x \coth ^{-1}(d \coth (a+b x)+c)-\frac{-\text{Li}_2\left(\frac{(c+d-1) e^{2 (a+b x)}}{c-d-1}\right)+\text{Li}_2\left(\frac{(c+d+1) e^{2 (a+b x)}}{c-d+1}\right)-2 b x \left(\log \left(1-\frac{(c+d-1) e^{2 (a+b x)}}{c-d-1}\right)-\log \left(1-\frac{(c+d+1) e^{2 (a+b x)}}{c-d+1}\right)\right)}{4 b}","\frac{\text{Li}_2\left(\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)}{4 b}-\frac{\text{Li}_2\left(\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)}{4 b}+\frac{1}{2} x \log \left(1-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right)-\frac{1}{2} x \log \left(1-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right)+x \coth ^{-1}(d \coth (a+b x)+c)",1,"x*ArcCoth[c + d*Coth[a + b*x]] - (-2*b*x*(Log[1 - ((-1 + c + d)*E^(2*(a + b*x)))/(-1 + c - d)] - Log[1 - ((1 + c + d)*E^(2*(a + b*x)))/(1 + c - d)]) - PolyLog[2, ((-1 + c + d)*E^(2*(a + b*x)))/(-1 + c - d)] + PolyLog[2, ((1 + c + d)*E^(2*(a + b*x)))/(1 + c - d)])/(4*b)","A",1
220,0,0,18,6.2276136,"\int \frac{\coth ^{-1}(c+d \coth (a+b x))}{x} \, dx","Integrate[ArcCoth[c + d*Coth[a + b*x]]/x,x]","\int \frac{\coth ^{-1}(c+d \coth (a+b x))}{x} \, dx","\text{Int}\left(\frac{\coth ^{-1}(d \coth (a+b x)+c)}{x},x\right)",0,"Integrate[ArcCoth[c + d*Coth[a + b*x]]/x, x]","A",-1
221,1,141,152,0.2007627,"\int x^3 \coth ^{-1}(1+d+d \coth (a+b x)) \, dx","Integrate[x^3*ArcCoth[1 + d + d*Coth[a + b*x]],x]","\frac{1}{16} \left(\frac{3 \text{Li}_5\left(\frac{e^{-2 (a+b x)}}{d+1}\right)}{b^4}+\frac{6 x \text{Li}_4\left(\frac{e^{-2 (a+b x)}}{d+1}\right)}{b^3}+\frac{6 x^2 \text{Li}_3\left(\frac{e^{-2 (a+b x)}}{d+1}\right)}{b^2}+\frac{4 x^3 \text{Li}_2\left(\frac{e^{-2 (a+b x)}}{d+1}\right)}{b}-2 x^4 \log \left(1-\frac{e^{-2 (a+b x)}}{d+1}\right)+4 x^4 \coth ^{-1}(d \coth (a+b x)+d+1)\right)","\frac{3 \text{Li}_5\left((d+1) e^{2 a+2 b x}\right)}{16 b^4}-\frac{3 x \text{Li}_4\left((d+1) e^{2 a+2 b x}\right)}{8 b^3}+\frac{3 x^2 \text{Li}_3\left((d+1) e^{2 a+2 b x}\right)}{8 b^2}-\frac{x^3 \text{Li}_2\left((d+1) e^{2 a+2 b x}\right)}{4 b}-\frac{1}{8} x^4 \log \left(1-(d+1) e^{2 a+2 b x}\right)+\frac{1}{4} x^4 \coth ^{-1}(d \coth (a+b x)+d+1)+\frac{b x^5}{20}",1,"(4*x^4*ArcCoth[1 + d + d*Coth[a + b*x]] - 2*x^4*Log[1 - 1/((1 + d)*E^(2*(a + b*x)))] + (4*x^3*PolyLog[2, 1/((1 + d)*E^(2*(a + b*x)))])/b + (6*x^2*PolyLog[3, 1/((1 + d)*E^(2*(a + b*x)))])/b^2 + (6*x*PolyLog[4, 1/((1 + d)*E^(2*(a + b*x)))])/b^3 + (3*PolyLog[5, 1/((1 + d)*E^(2*(a + b*x)))])/b^4)/16","A",1
222,1,116,126,0.11472,"\int x^2 \coth ^{-1}(1+d+d \coth (a+b x)) \, dx","Integrate[x^2*ArcCoth[1 + d + d*Coth[a + b*x]],x]","\frac{1}{24} \left(\frac{3 \text{Li}_4\left(\frac{e^{-2 (a+b x)}}{d+1}\right)}{b^3}+\frac{6 x \text{Li}_3\left(\frac{e^{-2 (a+b x)}}{d+1}\right)}{b^2}+\frac{6 x^2 \text{Li}_2\left(\frac{e^{-2 (a+b x)}}{d+1}\right)}{b}-4 x^3 \log \left(1-\frac{e^{-2 (a+b x)}}{d+1}\right)+8 x^3 \coth ^{-1}(d \coth (a+b x)+d+1)\right)","-\frac{\text{Li}_4\left((d+1) e^{2 a+2 b x}\right)}{8 b^3}+\frac{x \text{Li}_3\left((d+1) e^{2 a+2 b x}\right)}{4 b^2}-\frac{x^2 \text{Li}_2\left((d+1) e^{2 a+2 b x}\right)}{4 b}-\frac{1}{6} x^3 \log \left(1-(d+1) e^{2 a+2 b x}\right)+\frac{1}{3} x^3 \coth ^{-1}(d \coth (a+b x)+d+1)+\frac{b x^4}{12}",1,"(8*x^3*ArcCoth[1 + d + d*Coth[a + b*x]] - 4*x^3*Log[1 - 1/((1 + d)*E^(2*(a + b*x)))] + (6*x^2*PolyLog[2, 1/((1 + d)*E^(2*(a + b*x)))])/b + (6*x*PolyLog[3, 1/((1 + d)*E^(2*(a + b*x)))])/b^2 + (3*PolyLog[4, 1/((1 + d)*E^(2*(a + b*x)))])/b^3)/24","A",1
223,1,90,100,0.1028887,"\int x \coth ^{-1}(1+d+d \coth (a+b x)) \, dx","Integrate[x*ArcCoth[1 + d + d*Coth[a + b*x]],x]","\frac{2 b^2 x^2 \left(2 \coth ^{-1}(d \coth (a+b x)+d+1)-\log \left(1-\frac{e^{-2 (a+b x)}}{d+1}\right)\right)+2 b x \text{Li}_2\left(\frac{e^{-2 (a+b x)}}{d+1}\right)+\text{Li}_3\left(\frac{e^{-2 (a+b x)}}{d+1}\right)}{8 b^2}","\frac{\text{Li}_3\left((d+1) e^{2 a+2 b x}\right)}{8 b^2}-\frac{x \text{Li}_2\left((d+1) e^{2 a+2 b x}\right)}{4 b}-\frac{1}{4} x^2 \log \left(1-(d+1) e^{2 a+2 b x}\right)+\frac{1}{2} x^2 \coth ^{-1}(d \coth (a+b x)+d+1)+\frac{b x^3}{6}",1,"(2*b^2*x^2*(2*ArcCoth[1 + d + d*Coth[a + b*x]] - Log[1 - 1/((1 + d)*E^(2*(a + b*x)))]) + 2*b*x*PolyLog[2, 1/((1 + d)*E^(2*(a + b*x)))] + PolyLog[3, 1/((1 + d)*E^(2*(a + b*x)))])/(8*b^2)","A",1
224,1,197,69,0.8688868,"\int \coth ^{-1}(1+d+d \coth (a+b x)) \, dx","Integrate[ArcCoth[1 + d + d*Coth[a + b*x]],x]","\frac{-2 \text{Li}_2\left(-\sqrt{d+1} e^{a+b x}\right)-2 \text{Li}_2\left(\sqrt{d+1} e^{a+b x}\right)-2 \log \left(e^{a+b x}\right) \log \left(1-\sqrt{d+1} e^{a+b x}\right)-2 \log \left(e^{a+b x}\right) \log \left(\sqrt{d+1} e^{a+b x}+1\right)+2 \log \left(e^{a+b x}\right) \log \left(e^{-a-b x} \left((d+1) e^{2 (a+b x)}-1\right)\right)-2 b x \log ((d+2) \sinh (a+b x)+d \cosh (a+b x))+\log ^2\left(e^{a+b x}\right)+b^2 x^2}{4 b}+x \coth ^{-1}(d \coth (a+b x)+d+1)","-\frac{\text{Li}_2\left((d+1) e^{2 a+2 b x}\right)}{4 b}-\frac{1}{2} x \log \left(1-(d+1) e^{2 a+2 b x}\right)+x \coth ^{-1}(d \coth (a+b x)+d+1)+\frac{b x^2}{2}",1,"x*ArcCoth[1 + d + d*Coth[a + b*x]] + (b^2*x^2 + Log[E^(a + b*x)]^2 - 2*Log[E^(a + b*x)]*Log[1 - Sqrt[1 + d]*E^(a + b*x)] - 2*Log[E^(a + b*x)]*Log[1 + Sqrt[1 + d]*E^(a + b*x)] + 2*Log[E^(a + b*x)]*Log[E^(-a - b*x)*(-1 + (1 + d)*E^(2*(a + b*x)))] - 2*b*x*Log[d*Cosh[a + b*x] + (2 + d)*Sinh[a + b*x]] - 2*PolyLog[2, -(Sqrt[1 + d]*E^(a + b*x))] - 2*PolyLog[2, Sqrt[1 + d]*E^(a + b*x)])/(4*b)","B",0
225,0,0,19,3.5048432,"\int \frac{\coth ^{-1}(1+d+d \coth (a+b x))}{x} \, dx","Integrate[ArcCoth[1 + d + d*Coth[a + b*x]]/x,x]","\int \frac{\coth ^{-1}(1+d+d \coth (a+b x))}{x} \, dx","\text{Int}\left(\frac{\coth ^{-1}(d \coth (a+b x)+d+1)}{x},x\right)",0,"Integrate[ArcCoth[1 + d + d*Coth[a + b*x]]/x, x]","A",-1
226,1,147,165,0.2149656,"\int x^3 \coth ^{-1}(1-d-d \coth (a+b x)) \, dx","Integrate[x^3*ArcCoth[1 - d - d*Coth[a + b*x]],x]","\frac{1}{16} \left(\frac{3 \text{Li}_5\left(-\frac{e^{-2 (a+b x)}}{d-1}\right)}{b^4}+\frac{6 x \text{Li}_4\left(-\frac{e^{-2 (a+b x)}}{d-1}\right)}{b^3}+\frac{6 x^2 \text{Li}_3\left(-\frac{e^{-2 (a+b x)}}{d-1}\right)}{b^2}+\frac{4 x^3 \text{Li}_2\left(-\frac{e^{-2 (a+b x)}}{d-1}\right)}{b}-2 x^4 \log \left(\frac{e^{-2 (a+b x)}}{d-1}+1\right)+4 x^4 \coth ^{-1}(d (-\coth (a+b x))-d+1)\right)","\frac{3 \text{Li}_5\left((1-d) e^{2 a+2 b x}\right)}{16 b^4}-\frac{3 x \text{Li}_4\left((1-d) e^{2 a+2 b x}\right)}{8 b^3}+\frac{3 x^2 \text{Li}_3\left((1-d) e^{2 a+2 b x}\right)}{8 b^2}-\frac{x^3 \text{Li}_2\left((1-d) e^{2 a+2 b x}\right)}{4 b}-\frac{1}{8} x^4 \log \left(1-(1-d) e^{2 a+2 b x}\right)+\frac{1}{4} x^4 \coth ^{-1}(d (-\coth (a+b x))-d+1)+\frac{b x^5}{20}",1,"(4*x^4*ArcCoth[1 - d - d*Coth[a + b*x]] - 2*x^4*Log[1 + 1/((-1 + d)*E^(2*(a + b*x)))] + (4*x^3*PolyLog[2, -(1/((-1 + d)*E^(2*(a + b*x))))])/b + (6*x^2*PolyLog[3, -(1/((-1 + d)*E^(2*(a + b*x))))])/b^2 + (6*x*PolyLog[4, -(1/((-1 + d)*E^(2*(a + b*x))))])/b^3 + (3*PolyLog[5, -(1/((-1 + d)*E^(2*(a + b*x))))])/b^4)/16","A",1
227,1,121,137,0.1216971,"\int x^2 \coth ^{-1}(1-d-d \coth (a+b x)) \, dx","Integrate[x^2*ArcCoth[1 - d - d*Coth[a + b*x]],x]","\frac{1}{24} \left(\frac{3 \text{Li}_4\left(-\frac{e^{-2 (a+b x)}}{d-1}\right)}{b^3}+\frac{6 x \text{Li}_3\left(-\frac{e^{-2 (a+b x)}}{d-1}\right)}{b^2}+\frac{6 x^2 \text{Li}_2\left(-\frac{e^{-2 (a+b x)}}{d-1}\right)}{b}-4 x^3 \log \left(\frac{e^{-2 (a+b x)}}{d-1}+1\right)+8 x^3 \coth ^{-1}(d (-\coth (a+b x))-d+1)\right)","-\frac{\text{Li}_4\left((1-d) e^{2 a+2 b x}\right)}{8 b^3}+\frac{x \text{Li}_3\left((1-d) e^{2 a+2 b x}\right)}{4 b^2}-\frac{x^2 \text{Li}_2\left((1-d) e^{2 a+2 b x}\right)}{4 b}-\frac{1}{6} x^3 \log \left(1-(1-d) e^{2 a+2 b x}\right)+\frac{1}{3} x^3 \coth ^{-1}(d (-\coth (a+b x))-d+1)+\frac{b x^4}{12}",1,"(8*x^3*ArcCoth[1 - d - d*Coth[a + b*x]] - 4*x^3*Log[1 + 1/((-1 + d)*E^(2*(a + b*x)))] + (6*x^2*PolyLog[2, -(1/((-1 + d)*E^(2*(a + b*x))))])/b + (6*x*PolyLog[3, -(1/((-1 + d)*E^(2*(a + b*x))))])/b^2 + (3*PolyLog[4, -(1/((-1 + d)*E^(2*(a + b*x))))])/b^3)/24","A",1
228,1,94,109,0.1188164,"\int x \coth ^{-1}(1-d-d \coth (a+b x)) \, dx","Integrate[x*ArcCoth[1 - d - d*Coth[a + b*x]],x]","\frac{2 b^2 x^2 \left(2 \coth ^{-1}(d (-\coth (a+b x))-d+1)-\log \left(\frac{e^{-2 (a+b x)}}{d-1}+1\right)\right)+2 b x \text{Li}_2\left(-\frac{e^{-2 (a+b x)}}{d-1}\right)+\text{Li}_3\left(-\frac{e^{-2 (a+b x)}}{d-1}\right)}{8 b^2}","\frac{\text{Li}_3\left((1-d) e^{2 a+2 b x}\right)}{8 b^2}-\frac{x \text{Li}_2\left((1-d) e^{2 a+2 b x}\right)}{4 b}-\frac{1}{4} x^2 \log \left(1-(1-d) e^{2 a+2 b x}\right)+\frac{1}{2} x^2 \coth ^{-1}(d (-\coth (a+b x))-d+1)+\frac{b x^3}{6}",1,"(2*b^2*x^2*(2*ArcCoth[1 - d - d*Coth[a + b*x]] - Log[1 + 1/((-1 + d)*E^(2*(a + b*x)))]) + 2*b*x*PolyLog[2, -(1/((-1 + d)*E^(2*(a + b*x))))] + PolyLog[3, -(1/((-1 + d)*E^(2*(a + b*x))))])/(8*b^2)","A",1
229,1,208,76,0.7905008,"\int \coth ^{-1}(1-d-d \coth (a+b x)) \, dx","Integrate[ArcCoth[1 - d - d*Coth[a + b*x]],x]","\frac{-2 \text{Li}_2\left(-\sqrt{1-d} e^{a+b x}\right)-2 \text{Li}_2\left(\sqrt{1-d} e^{a+b x}\right)-2 \log \left(e^{a+b x}\right) \log \left(1-\sqrt{1-d} e^{a+b x}\right)-2 \log \left(e^{a+b x}\right) \log \left(\sqrt{1-d} e^{a+b x}+1\right)+2 \log \left(e^{a+b x}\right) \log \left(e^{-a-b x} \left((d-1) e^{2 (a+b x)}+1\right)\right)-2 b x \log ((d-2) \sinh (a+b x)+d \cosh (a+b x))+\log ^2\left(e^{a+b x}\right)+b^2 x^2}{4 b}+x \coth ^{-1}(d (-\coth (a+b x))-d+1)","-\frac{\text{Li}_2\left((1-d) e^{2 a+2 b x}\right)}{4 b}-\frac{1}{2} x \log \left(1-(1-d) e^{2 a+2 b x}\right)+x \coth ^{-1}(d (-\coth (a+b x))-d+1)+\frac{b x^2}{2}",1,"x*ArcCoth[1 - d - d*Coth[a + b*x]] + (b^2*x^2 + Log[E^(a + b*x)]^2 - 2*Log[E^(a + b*x)]*Log[1 - Sqrt[1 - d]*E^(a + b*x)] - 2*Log[E^(a + b*x)]*Log[1 + Sqrt[1 - d]*E^(a + b*x)] + 2*Log[E^(a + b*x)]*Log[E^(-a - b*x)*(1 + (-1 + d)*E^(2*(a + b*x)))] - 2*b*x*Log[d*Cosh[a + b*x] + (-2 + d)*Sinh[a + b*x]] - 2*PolyLog[2, -(Sqrt[1 - d]*E^(a + b*x))] - 2*PolyLog[2, Sqrt[1 - d]*E^(a + b*x)])/(4*b)","B",0
230,0,0,22,3.6333667,"\int \frac{\coth ^{-1}(1-d-d \coth (a+b x))}{x} \, dx","Integrate[ArcCoth[1 - d - d*Coth[a + b*x]]/x,x]","\int \frac{\coth ^{-1}(1-d-d \coth (a+b x))}{x} \, dx","\text{Int}\left(\frac{\coth ^{-1}(d (-\coth (a+b x))-d+1)}{x},x\right)",0,"Integrate[ArcCoth[1 - d - d*Coth[a + b*x]]/x, x]","A",-1
231,1,654,302,0.3508992,"\int (e+f x)^3 \coth ^{-1}(\tan (a+b x)) \, dx","Integrate[(e + f*x)^3*ArcCoth[Tan[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) \coth ^{-1}(\tan (a+b x))+\frac{-8 b^4 e^3 x \log \left(1-i e^{2 i (a+b x)}\right)+8 b^4 e^3 x \log \left(1+i e^{2 i (a+b x)}\right)-12 b^4 e^2 f x^2 \log \left(1-i e^{2 i (a+b x)}\right)+12 b^4 e^2 f x^2 \log \left(1+i e^{2 i (a+b x)}\right)-8 b^4 e f^2 x^3 \log \left(1-i e^{2 i (a+b x)}\right)+8 b^4 e f^2 x^3 \log \left(1+i e^{2 i (a+b x)}\right)-2 b^4 f^3 x^4 \log \left(1-i e^{2 i (a+b x)}\right)+2 b^4 f^3 x^4 \log \left(1+i e^{2 i (a+b x)}\right)-4 i b^3 (e+f x)^3 \text{Li}_2\left(-i e^{2 i (a+b x)}\right)+4 i b^3 (e+f x)^3 \text{Li}_2\left(i e^{2 i (a+b x)}\right)+6 b^2 e^2 f \text{Li}_3\left(-i e^{2 i (a+b x)}\right)-6 b^2 e^2 f \text{Li}_3\left(i e^{2 i (a+b x)}\right)+12 b^2 e f^2 x \text{Li}_3\left(-i e^{2 i (a+b x)}\right)-12 b^2 e f^2 x \text{Li}_3\left(i e^{2 i (a+b x)}\right)+6 b^2 f^3 x^2 \text{Li}_3\left(-i e^{2 i (a+b x)}\right)-6 b^2 f^3 x^2 \text{Li}_3\left(i e^{2 i (a+b x)}\right)+6 i b e f^2 \text{Li}_4\left(-i e^{2 i (a+b x)}\right)-6 i b e f^2 \text{Li}_4\left(i e^{2 i (a+b x)}\right)+6 i b f^3 x \text{Li}_4\left(-i e^{2 i (a+b x)}\right)-6 i b f^3 x \text{Li}_4\left(i e^{2 i (a+b x)}\right)-3 f^3 \text{Li}_5\left(-i e^{2 i (a+b x)}\right)+3 f^3 \text{Li}_5\left(i e^{2 i (a+b x)}\right)}{16 b^4}","-\frac{3 f^3 \text{Li}_5\left(-i e^{2 i (a+b x)}\right)}{16 b^4}+\frac{3 f^3 \text{Li}_5\left(i e^{2 i (a+b x)}\right)}{16 b^4}+\frac{3 i f^2 (e+f x) \text{Li}_4\left(-i e^{2 i (a+b x)}\right)}{8 b^3}-\frac{3 i f^2 (e+f x) \text{Li}_4\left(i e^{2 i (a+b x)}\right)}{8 b^3}+\frac{3 f (e+f x)^2 \text{Li}_3\left(-i e^{2 i (a+b x)}\right)}{8 b^2}-\frac{3 f (e+f x)^2 \text{Li}_3\left(i e^{2 i (a+b x)}\right)}{8 b^2}-\frac{i (e+f x)^3 \text{Li}_2\left(-i e^{2 i (a+b x)}\right)}{4 b}+\frac{i (e+f x)^3 \text{Li}_2\left(i e^{2 i (a+b x)}\right)}{4 b}+\frac{i (e+f x)^4 \tan ^{-1}\left(e^{2 i (a+b x)}\right)}{4 f}+\frac{(e+f x)^4 \coth ^{-1}(\tan (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)*ArcCoth[Tan[a + b*x]])/4 + (-8*b^4*e^3*x*Log[1 - I*E^((2*I)*(a + b*x))] - 12*b^4*e^2*f*x^2*Log[1 - I*E^((2*I)*(a + b*x))] - 8*b^4*e*f^2*x^3*Log[1 - I*E^((2*I)*(a + b*x))] - 2*b^4*f^3*x^4*Log[1 - I*E^((2*I)*(a + b*x))] + 8*b^4*e^3*x*Log[1 + I*E^((2*I)*(a + b*x))] + 12*b^4*e^2*f*x^2*Log[1 + I*E^((2*I)*(a + b*x))] + 8*b^4*e*f^2*x^3*Log[1 + I*E^((2*I)*(a + b*x))] + 2*b^4*f^3*x^4*Log[1 + I*E^((2*I)*(a + b*x))] - (4*I)*b^3*(e + f*x)^3*PolyLog[2, (-I)*E^((2*I)*(a + b*x))] + (4*I)*b^3*(e + f*x)^3*PolyLog[2, I*E^((2*I)*(a + b*x))] + 6*b^2*e^2*f*PolyLog[3, (-I)*E^((2*I)*(a + b*x))] + 12*b^2*e*f^2*x*PolyLog[3, (-I)*E^((2*I)*(a + b*x))] + 6*b^2*f^3*x^2*PolyLog[3, (-I)*E^((2*I)*(a + b*x))] - 6*b^2*e^2*f*PolyLog[3, I*E^((2*I)*(a + b*x))] - 12*b^2*e*f^2*x*PolyLog[3, I*E^((2*I)*(a + b*x))] - 6*b^2*f^3*x^2*PolyLog[3, I*E^((2*I)*(a + b*x))] + (6*I)*b*e*f^2*PolyLog[4, (-I)*E^((2*I)*(a + b*x))] + (6*I)*b*f^3*x*PolyLog[4, (-I)*E^((2*I)*(a + b*x))] - (6*I)*b*e*f^2*PolyLog[4, I*E^((2*I)*(a + b*x))] - (6*I)*b*f^3*x*PolyLog[4, I*E^((2*I)*(a + b*x))] - 3*f^3*PolyLog[5, (-I)*E^((2*I)*(a + b*x))] + 3*f^3*PolyLog[5, I*E^((2*I)*(a + b*x))])/(16*b^4)","B",1
232,1,409,234,0.1958685,"\int (e+f x)^2 \coth ^{-1}(\tan (a+b x)) \, dx","Integrate[(e + f*x)^2*ArcCoth[Tan[a + b*x]],x]","\frac{1}{3} x \left(3 e^2+3 e f x+f^2 x^2\right) \coth ^{-1}(\tan (a+b x))+\frac{-12 b^3 e^2 x \log \left(1-i e^{2 i (a+b x)}\right)+12 b^3 e^2 x \log \left(1+i e^{2 i (a+b x)}\right)-12 b^3 e f x^2 \log \left(1-i e^{2 i (a+b x)}\right)+12 b^3 e f x^2 \log \left(1+i e^{2 i (a+b x)}\right)-4 b^3 f^2 x^3 \log \left(1-i e^{2 i (a+b x)}\right)+4 b^3 f^2 x^3 \log \left(1+i e^{2 i (a+b x)}\right)-6 i b^2 (e+f x)^2 \text{Li}_2\left(-i e^{2 i (a+b x)}\right)+6 i b^2 (e+f x)^2 \text{Li}_2\left(i e^{2 i (a+b x)}\right)+6 b e f \text{Li}_3\left(-i e^{2 i (a+b x)}\right)-6 b e f \text{Li}_3\left(i e^{2 i (a+b x)}\right)+6 b f^2 x \text{Li}_3\left(-i e^{2 i (a+b x)}\right)-6 b f^2 x \text{Li}_3\left(i e^{2 i (a+b x)}\right)+3 i f^2 \text{Li}_4\left(-i e^{2 i (a+b x)}\right)-3 i f^2 \text{Li}_4\left(i e^{2 i (a+b x)}\right)}{24 b^3}","\frac{i f^2 \text{Li}_4\left(-i e^{2 i (a+b x)}\right)}{8 b^3}-\frac{i f^2 \text{Li}_4\left(i e^{2 i (a+b x)}\right)}{8 b^3}+\frac{f (e+f x) \text{Li}_3\left(-i e^{2 i (a+b x)}\right)}{4 b^2}-\frac{f (e+f x) \text{Li}_3\left(i e^{2 i (a+b x)}\right)}{4 b^2}-\frac{i (e+f x)^2 \text{Li}_2\left(-i e^{2 i (a+b x)}\right)}{4 b}+\frac{i (e+f x)^2 \text{Li}_2\left(i e^{2 i (a+b x)}\right)}{4 b}+\frac{i (e+f x)^3 \tan ^{-1}\left(e^{2 i (a+b x)}\right)}{3 f}+\frac{(e+f x)^3 \coth ^{-1}(\tan (a+b x))}{3 f}",1,"(x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCoth[Tan[a + b*x]])/3 + (-12*b^3*e^2*x*Log[1 - I*E^((2*I)*(a + b*x))] - 12*b^3*e*f*x^2*Log[1 - I*E^((2*I)*(a + b*x))] - 4*b^3*f^2*x^3*Log[1 - I*E^((2*I)*(a + b*x))] + 12*b^3*e^2*x*Log[1 + I*E^((2*I)*(a + b*x))] + 12*b^3*e*f*x^2*Log[1 + I*E^((2*I)*(a + b*x))] + 4*b^3*f^2*x^3*Log[1 + I*E^((2*I)*(a + b*x))] - (6*I)*b^2*(e + f*x)^2*PolyLog[2, (-I)*E^((2*I)*(a + b*x))] + (6*I)*b^2*(e + f*x)^2*PolyLog[2, I*E^((2*I)*(a + b*x))] + 6*b*e*f*PolyLog[3, (-I)*E^((2*I)*(a + b*x))] + 6*b*f^2*x*PolyLog[3, (-I)*E^((2*I)*(a + b*x))] - 6*b*e*f*PolyLog[3, I*E^((2*I)*(a + b*x))] - 6*b*f^2*x*PolyLog[3, I*E^((2*I)*(a + b*x))] + (3*I)*f^2*PolyLog[4, (-I)*E^((2*I)*(a + b*x))] - (3*I)*f^2*PolyLog[4, I*E^((2*I)*(a + b*x))])/(24*b^3)","A",1
233,1,263,162,0.1385395,"\int (e+f x) \coth ^{-1}(\tan (a+b x)) \, dx","Integrate[(e + f*x)*ArcCoth[Tan[a + b*x]],x]","-b e \left(\frac{i \text{Li}_2\left(-i e^{i (2 a+2 b x)}\right)}{4 b^2}-\frac{i \text{Li}_2\left(i e^{i (2 a+2 b x)}\right)}{4 b^2}-\frac{i x \tan ^{-1}\left(e^{2 i a+2 i b x}\right)}{b}\right)+\frac{f \left(4 i b^2 x^2 \tan ^{-1}(\cos (2 (a+b x))+i \sin (2 (a+b x)))+2 i b x \text{Li}_2(i \cos (2 (a+b x))-\sin (2 (a+b x)))-2 i b x \text{Li}_2(\sin (2 (a+b x))-i \cos (2 (a+b x)))-\text{Li}_3(i \cos (2 (a+b x))-\sin (2 (a+b x)))+\text{Li}_3(\sin (2 (a+b x))-i \cos (2 (a+b x)))\right)}{8 b^2}+e x \coth ^{-1}(\tan (a+b x))+\frac{1}{2} f x^2 \coth ^{-1}(\tan (a+b x))","\frac{f \text{Li}_3\left(-i e^{2 i (a+b x)}\right)}{8 b^2}-\frac{f \text{Li}_3\left(i e^{2 i (a+b x)}\right)}{8 b^2}-\frac{i (e+f x) \text{Li}_2\left(-i e^{2 i (a+b x)}\right)}{4 b}+\frac{i (e+f x) \text{Li}_2\left(i e^{2 i (a+b x)}\right)}{4 b}+\frac{i (e+f x)^2 \tan ^{-1}\left(e^{2 i (a+b x)}\right)}{2 f}+\frac{(e+f x)^2 \coth ^{-1}(\tan (a+b x))}{2 f}",1,"e*x*ArcCoth[Tan[a + b*x]] + (f*x^2*ArcCoth[Tan[a + b*x]])/2 - b*e*(((-I)*x*ArcTan[E^((2*I)*a + (2*I)*b*x)])/b + ((I/4)*PolyLog[2, (-I)*E^(I*(2*a + 2*b*x))])/b^2 - ((I/4)*PolyLog[2, I*E^(I*(2*a + 2*b*x))])/b^2) + (f*((4*I)*b^2*x^2*ArcTan[Cos[2*(a + b*x)] + I*Sin[2*(a + b*x)]] + (2*I)*b*x*PolyLog[2, I*Cos[2*(a + b*x)] - Sin[2*(a + b*x)]] - (2*I)*b*x*PolyLog[2, (-I)*Cos[2*(a + b*x)] + Sin[2*(a + b*x)]] - PolyLog[3, I*Cos[2*(a + b*x)] - Sin[2*(a + b*x)]] + PolyLog[3, (-I)*Cos[2*(a + b*x)] + Sin[2*(a + b*x)]]))/(8*b^2)","A",0
234,1,78,79,0.0223144,"\int \coth ^{-1}(\tan (a+b x)) \, dx","Integrate[ArcCoth[Tan[a + b*x]],x]","\frac{-i \text{Li}_2\left(-i e^{2 i (a+b x)}\right)+i \text{Li}_2\left(i e^{2 i (a+b x)}\right)+4 b x \left(\coth ^{-1}(\tan (a+b x))+i \tan ^{-1}\left(e^{2 i (a+b x)}\right)\right)}{4 b}","-\frac{i \text{Li}_2\left(-i e^{2 i (a+b x)}\right)}{4 b}+\frac{i \text{Li}_2\left(i e^{2 i (a+b x)}\right)}{4 b}+i x \tan ^{-1}\left(e^{2 i (a+b x)}\right)+x \coth ^{-1}(\tan (a+b x))",1,"(4*b*x*(ArcCoth[Tan[a + b*x]] + I*ArcTan[E^((2*I)*(a + b*x))]) - I*PolyLog[2, (-I)*E^((2*I)*(a + b*x))] + I*PolyLog[2, I*E^((2*I)*(a + b*x))])/(4*b)","A",1
235,0,0,18,0.9895595,"\int \frac{\coth ^{-1}(\tan (a+b x))}{e+f x} \, dx","Integrate[ArcCoth[Tan[a + b*x]]/(e + f*x),x]","\int \frac{\coth ^{-1}(\tan (a+b x))}{e+f x} \, dx","\text{Int}\left(\frac{\coth ^{-1}(\tan (a+b x))}{e+f x},x\right)",0,"Integrate[ArcCoth[Tan[a + b*x]]/(e + f*x), x]","A",-1
236,1,346,395,0.3395756,"\int x^2 \coth ^{-1}(c+d \tan (a+b x)) \, dx","Integrate[x^2*ArcCoth[c + d*Tan[a + b*x]],x]","\frac{1}{3} x^3 \coth ^{-1}(d \tan (a+b x)+c)+\frac{4 b^3 x^3 \log \left(1+\frac{(c-i d-1) e^{2 i (a+b x)}}{c+i d-1}\right)-4 b^3 x^3 \log \left(1+\frac{(c-i d+1) e^{2 i (a+b x)}}{c+i d+1}\right)-6 i 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 i 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 x \text{Li}_3\left(\frac{(-c+i d+1) e^{2 i (a+b x)}}{c+i d-1}\right)-6 b x \text{Li}_3\left(-\frac{(c-i d+1) e^{2 i (a+b x)}}{c+i d+1}\right)+3 i \text{Li}_4\left(\frac{(-c+i d+1) e^{2 i (a+b x)}}{c+i d-1}\right)-3 i \text{Li}_4\left(-\frac{(c-i d+1) e^{2 i (a+b x)}}{c+i d+1}\right)}{24 b^3}","\frac{i \text{Li}_4\left(-\frac{(-c+i d+1) e^{2 i a+2 i b x}}{-c-i d+1}\right)}{8 b^3}-\frac{i \text{Li}_4\left(-\frac{(c-i d+1) e^{2 i a+2 i b x}}{c+i d+1}\right)}{8 b^3}+\frac{x \text{Li}_3\left(-\frac{(-c+i d+1) e^{2 i a+2 i b x}}{-c-i d+1}\right)}{4 b^2}-\frac{x \text{Li}_3\left(-\frac{(c-i d+1) e^{2 i a+2 i b x}}{c+i d+1}\right)}{4 b^2}-\frac{i x^2 \text{Li}_2\left(-\frac{(-c+i d+1) e^{2 i a+2 i b x}}{-c-i d+1}\right)}{4 b}+\frac{i x^2 \text{Li}_2\left(-\frac{(c-i d+1) e^{2 i a+2 i b x}}{c+i d+1}\right)}{4 b}+\frac{1}{6} x^3 \log \left(1+\frac{(-c+i d+1) e^{2 i a+2 i b x}}{-c-i d+1}\right)-\frac{1}{6} x^3 \log \left(1+\frac{(c-i d+1) e^{2 i a+2 i b x}}{c+i d+1}\right)+\frac{1}{3} x^3 \coth ^{-1}(d \tan (a+b x)+c)",1,"(x^3*ArcCoth[c + d*Tan[a + b*x]])/3 + (4*b^3*x^3*Log[1 + ((-1 + c - I*d)*E^((2*I)*(a + b*x)))/(-1 + c + I*d)] - 4*b^3*x^3*Log[1 + ((1 + c - I*d)*E^((2*I)*(a + b*x)))/(1 + c + I*d)] - (6*I)*b^2*x^2*PolyLog[2, ((1 - c + I*d)*E^((2*I)*(a + b*x)))/(-1 + c + I*d)] + (6*I)*b^2*x^2*PolyLog[2, -(((1 + c - I*d)*E^((2*I)*(a + b*x)))/(1 + c + I*d))] + 6*b*x*PolyLog[3, ((1 - c + I*d)*E^((2*I)*(a + b*x)))/(-1 + c + I*d)] - 6*b*x*PolyLog[3, -(((1 + c - I*d)*E^((2*I)*(a + b*x)))/(1 + c + I*d))] + (3*I)*PolyLog[4, ((1 - c + I*d)*E^((2*I)*(a + b*x)))/(-1 + c + I*d)] - (3*I)*PolyLog[4, -(((1 + c - I*d)*E^((2*I)*(a + b*x)))/(1 + c + I*d))])/(24*b^3)","A",1
237,1,257,295,0.1555918,"\int x \coth ^{-1}(c+d \tan (a+b x)) \, dx","Integrate[x*ArcCoth[c + d*Tan[a + b*x]],x]","\frac{1}{2} x^2 \coth ^{-1}(d \tan (a+b x)+c)+\frac{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+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+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-1}\right)-\text{Li}_3\left(-\frac{(c-i d+1) e^{2 i (a+b x)}}{c+i d+1}\right)}{8 b^2}","\frac{\text{Li}_3\left(-\frac{(-c+i d+1) e^{2 i a+2 i b x}}{-c-i d+1}\right)}{8 b^2}-\frac{\text{Li}_3\left(-\frac{(c-i d+1) e^{2 i a+2 i b x}}{c+i d+1}\right)}{8 b^2}-\frac{i x \text{Li}_2\left(-\frac{(-c+i d+1) e^{2 i a+2 i b x}}{-c-i d+1}\right)}{4 b}+\frac{i x \text{Li}_2\left(-\frac{(c-i d+1) e^{2 i a+2 i b x}}{c+i d+1}\right)}{4 b}+\frac{1}{4} x^2 \log \left(1+\frac{(-c+i d+1) e^{2 i a+2 i b x}}{-c-i d+1}\right)-\frac{1}{4} x^2 \log \left(1+\frac{(c-i d+1) e^{2 i a+2 i b x}}{c+i d+1}\right)+\frac{1}{2} x^2 \coth ^{-1}(d \tan (a+b x)+c)",1,"(x^2*ArcCoth[c + d*Tan[a + b*x]])/2 + (2*b^2*x^2*Log[1 + ((-1 + c - I*d)*E^((2*I)*(a + b*x)))/(-1 + c + I*d)] - 2*b^2*x^2*Log[1 + ((1 + c - I*d)*E^((2*I)*(a + b*x)))/(1 + c + I*d)] - (2*I)*b*x*PolyLog[2, ((1 - c + I*d)*E^((2*I)*(a + b*x)))/(-1 + c + I*d)] + (2*I)*b*x*PolyLog[2, -(((1 + c - I*d)*E^((2*I)*(a + b*x)))/(1 + c + I*d))] + PolyLog[3, ((1 - c + I*d)*E^((2*I)*(a + b*x)))/(-1 + c + I*d)] - PolyLog[3, -(((1 + c - I*d)*E^((2*I)*(a + b*x)))/(1 + c + I*d))])/(8*b^2)","A",1
238,1,4654,194,13.414994,"\int \coth ^{-1}(c+d \tan (a+b x)) \, dx","Integrate[ArcCoth[c + d*Tan[a + b*x]],x]","\text{Result too large to show}","-\frac{i \text{Li}_2\left(-\frac{(-c+i d+1) e^{2 i a+2 i b x}}{-c-i d+1}\right)}{4 b}+\frac{i \text{Li}_2\left(-\frac{(c-i d+1) e^{2 i a+2 i b x}}{c+i d+1}\right)}{4 b}+\frac{1}{2} x \log \left(1+\frac{(-c+i d+1) e^{2 i a+2 i b x}}{-c-i d+1}\right)-\frac{1}{2} x \log \left(1+\frac{(c-i d+1) e^{2 i a+2 i b x}}{c+i d+1}\right)+x \coth ^{-1}(d \tan (a+b x)+c)",1,"x*ArcCoth[c + d*Tan[a + b*x]] + (d*(-(a*Log[-(Sec[(a + b*x)/2]^2*((-1 + c)*Cos[a + b*x] + d*Sin[a + b*x]))]) + a*Log[Sec[(a + b*x)/2]^2*(Cos[a + b*x] + c*Cos[a + b*x] + d*Sin[a + b*x])] + (a + b*x)*Log[(-d + Sqrt[1 - 2*c + c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2]] + I*Log[((-1 + c)*(1 + I*Tan[(a + b*x)/2]))/(-1 + c + I*d - I*Sqrt[1 - 2*c + c^2 + d^2])]*Log[(-d + Sqrt[1 - 2*c + c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2]] - I*Log[-(((-1 + c)*(I + Tan[(a + b*x)/2]))/(I - I*c - d + Sqrt[1 - 2*c + c^2 + d^2]))]*Log[(-d + Sqrt[1 - 2*c + c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2]] + (a + b*x)*Log[(d + Sqrt[1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2]] + I*Log[((-1 + c)*(-I + Tan[(a + b*x)/2]))/(I - I*c + d + Sqrt[1 - 2*c + c^2 + d^2])]*Log[(d + Sqrt[1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2]] - I*Log[((-1 + c)*(I + Tan[(a + b*x)/2]))/(-I + I*c + d + Sqrt[1 - 2*c + c^2 + d^2])]*Log[(d + Sqrt[1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2]] - (a + b*x)*Log[-((d + Sqrt[1 + 2*c + c^2 + d^2])/(1 + c)) + Tan[(a + b*x)/2]] - I*Log[((1 + c)*(-I + Tan[(a + b*x)/2]))/(-I - I*c + d + Sqrt[1 + 2*c + c^2 + d^2])]*Log[-((d + Sqrt[1 + 2*c + c^2 + d^2])/(1 + c)) + Tan[(a + b*x)/2]] + I*Log[((1 + c)*(I + Tan[(a + b*x)/2]))/(I + I*c + d + Sqrt[1 + 2*c + c^2 + d^2])]*Log[-((d + Sqrt[1 + 2*c + c^2 + d^2])/(1 + c)) + Tan[(a + b*x)/2]] - (a + b*x)*Log[(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])/(1 + c)] + I*Log[((1 + c)*(1 - I*Tan[(a + b*x)/2]))/(1 + c - I*d + I*Sqrt[1 + 2*c + c^2 + d^2])]*Log[(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])/(1 + c)] - I*Log[((1 + c)*(1 + I*Tan[(a + b*x)/2]))/(1 + c + I*d - I*Sqrt[1 + 2*c + c^2 + d^2])]*Log[(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])/(1 + c)] + I*PolyLog[2, (d + Sqrt[1 - 2*c + c^2 + d^2] - (-1 + c)*Tan[(a + b*x)/2])/(I - I*c + d + Sqrt[1 - 2*c + c^2 + d^2])] - I*PolyLog[2, (d + Sqrt[1 - 2*c + c^2 + d^2] - (-1 + c)*Tan[(a + b*x)/2])/(-I + I*c + d + Sqrt[1 - 2*c + c^2 + d^2])] - I*PolyLog[2, (-d + Sqrt[1 - 2*c + c^2 + d^2] + (-1 + c)*Tan[(a + b*x)/2])/(I - I*c - d + Sqrt[1 - 2*c + c^2 + d^2])] + I*PolyLog[2, (-d + Sqrt[1 - 2*c + c^2 + d^2] + (-1 + c)*Tan[(a + b*x)/2])/(-I + I*c - d + Sqrt[1 - 2*c + c^2 + d^2])] - I*PolyLog[2, (d + Sqrt[1 + 2*c + c^2 + d^2] - (1 + c)*Tan[(a + b*x)/2])/(-I - I*c + d + Sqrt[1 + 2*c + c^2 + d^2])] + I*PolyLog[2, (d + Sqrt[1 + 2*c + c^2 + d^2] - (1 + c)*Tan[(a + b*x)/2])/(I + I*c + d + Sqrt[1 + 2*c + c^2 + d^2])] + I*PolyLog[2, (-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])/(-I - I*c - d + Sqrt[1 + 2*c + c^2 + d^2])] - I*PolyLog[2, (-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])/(I + I*c - d + Sqrt[1 + 2*c + c^2 + 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)]))))/(Log[(-d + Sqrt[1 - 2*c + c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2]] + Log[(d + Sqrt[1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2]] - Log[-((d + Sqrt[1 + 2*c + c^2 + d^2])/(1 + c)) + Tan[(a + b*x)/2]] - Log[(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])/(1 + c)] + (Log[(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])/(1 + c)]*Sec[(a + b*x)/2]^2)/(2*(1 - I*Tan[(a + b*x)/2])) - (Log[(-d + Sqrt[1 - 2*c + c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(2*(1 + I*Tan[(a + b*x)/2])) + (Log[(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])/(1 + c)]*Sec[(a + b*x)/2]^2)/(2*(1 + I*Tan[(a + b*x)/2])) + ((I/2)*Log[(d + Sqrt[1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(-I + Tan[(a + b*x)/2]) - ((I/2)*Log[-((d + Sqrt[1 + 2*c + c^2 + d^2])/(1 + c)) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(-I + Tan[(a + b*x)/2]) - ((I/2)*Log[(-d + Sqrt[1 - 2*c + c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(I + Tan[(a + b*x)/2]) - ((I/2)*Log[(d + Sqrt[1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(I + Tan[(a + b*x)/2]) + ((I/2)*Log[-((d + Sqrt[1 + 2*c + c^2 + d^2])/(1 + c)) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(I + Tan[(a + b*x)/2]) + ((a + b*x)*Sec[(a + b*x)/2]^2)/(2*((-d + Sqrt[1 - 2*c + c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2])) + ((I/2)*Log[((-1 + c)*(1 + I*Tan[(a + b*x)/2]))/(-1 + c + I*d - I*Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/((-d + Sqrt[1 - 2*c + c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2]) - ((I/2)*Log[-(((-1 + c)*(I + Tan[(a + b*x)/2]))/(I - I*c - d + Sqrt[1 - 2*c + c^2 + d^2]))]*Sec[(a + b*x)/2]^2)/((-d + Sqrt[1 - 2*c + c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2]) + ((a + b*x)*Sec[(a + b*x)/2]^2)/(2*((d + Sqrt[1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2])) + ((I/2)*Log[((-1 + c)*(-I + Tan[(a + b*x)/2]))/(I - I*c + d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/((d + Sqrt[1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2]) - ((I/2)*Log[((-1 + c)*(I + Tan[(a + b*x)/2]))/(-I + I*c + d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/((d + Sqrt[1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2]) - ((a + b*x)*Sec[(a + b*x)/2]^2)/(2*(-((d + Sqrt[1 + 2*c + c^2 + d^2])/(1 + c)) + Tan[(a + b*x)/2])) - ((I/2)*Log[((1 + c)*(-I + Tan[(a + b*x)/2]))/(-I - I*c + d + Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-((d + Sqrt[1 + 2*c + c^2 + d^2])/(1 + c)) + Tan[(a + b*x)/2]) + ((I/2)*Log[((1 + c)*(I + Tan[(a + b*x)/2]))/(I + I*c + d + Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-((d + Sqrt[1 + 2*c + c^2 + d^2])/(1 + c)) + Tan[(a + b*x)/2]) + ((I/2)*(-1 + c)*Log[1 - (d + Sqrt[1 - 2*c + c^2 + d^2] - (-1 + c)*Tan[(a + b*x)/2])/(I - I*c + d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(d + Sqrt[1 - 2*c + c^2 + d^2] - (-1 + c)*Tan[(a + b*x)/2]) - ((I/2)*(-1 + c)*Log[1 - (d + Sqrt[1 - 2*c + c^2 + d^2] - (-1 + c)*Tan[(a + b*x)/2])/(-I + I*c + d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(d + Sqrt[1 - 2*c + c^2 + d^2] - (-1 + c)*Tan[(a + b*x)/2]) + ((I/2)*(-1 + c)*Log[1 - (-d + Sqrt[1 - 2*c + c^2 + d^2] + (-1 + c)*Tan[(a + b*x)/2])/(I - I*c - d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-d + Sqrt[1 - 2*c + c^2 + d^2] + (-1 + c)*Tan[(a + b*x)/2]) - ((I/2)*(-1 + c)*Log[1 - (-d + Sqrt[1 - 2*c + c^2 + d^2] + (-1 + c)*Tan[(a + b*x)/2])/(-I + I*c - d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-d + Sqrt[1 - 2*c + c^2 + d^2] + (-1 + c)*Tan[(a + b*x)/2]) - ((I/2)*(1 + c)*Log[1 - (d + Sqrt[1 + 2*c + c^2 + d^2] - (1 + c)*Tan[(a + b*x)/2])/(-I - I*c + d + Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(d + Sqrt[1 + 2*c + c^2 + d^2] - (1 + c)*Tan[(a + b*x)/2]) + ((I/2)*(1 + c)*Log[1 - (d + Sqrt[1 + 2*c + c^2 + d^2] - (1 + c)*Tan[(a + b*x)/2])/(I + I*c + d + Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(d + Sqrt[1 + 2*c + c^2 + d^2] - (1 + c)*Tan[(a + b*x)/2]) - ((1 + c)*(a + b*x)*Sec[(a + b*x)/2]^2)/(2*(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])) + ((I/2)*(1 + c)*Log[((1 + c)*(1 - I*Tan[(a + b*x)/2]))/(1 + c - I*d + I*Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2]) - ((I/2)*(1 + c)*Log[((1 + c)*(1 + I*Tan[(a + b*x)/2]))/(1 + c + I*d - I*Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2]) - ((I/2)*(1 + c)*Log[1 - (-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])/(-I - I*c - d + Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2]) + ((I/2)*(1 + c)*Log[1 - (-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])/(I + I*c - d + Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2]) + (a*Cos[(a + b*x)/2]^2*(-(Sec[(a + b*x)/2]^2*(d*Cos[a + b*x] - (-1 + c)*Sin[a + b*x])) - Sec[(a + b*x)/2]^2*((-1 + c)*Cos[a + b*x] + d*Sin[a + b*x])*Tan[(a + b*x)/2]))/((-1 + c)*Cos[a + b*x] + d*Sin[a + b*x]) + (a*Cos[(a + b*x)/2]^2*(Sec[(a + b*x)/2]^2*(d*Cos[a + b*x] - Sin[a + b*x] - c*Sin[a + b*x]) + Sec[(a + b*x)/2]^2*(Cos[a + b*x] + c*Cos[a + b*x] + d*Sin[a + b*x])*Tan[(a + b*x)/2]))/(Cos[a + b*x] + c*Cos[a + b*x] + d*Sin[a + b*x]))","B",0
239,0,0,18,0.3928409,"\int \frac{\coth ^{-1}(c+d \tan (a+b x))}{x} \, dx","Integrate[ArcCoth[c + d*Tan[a + b*x]]/x,x]","\int \frac{\coth ^{-1}(c+d \tan (a+b x))}{x} \, dx","\text{Int}\left(\frac{\coth ^{-1}(d \tan (a+b x)+c)}{x},x\right)",0,"Integrate[ArcCoth[c + d*Tan[a + b*x]]/x, x]","A",-1
240,1,155,170,0.2236154,"\int x^2 \coth ^{-1}(1-i d+d \tan (a+b x)) \, dx","Integrate[x^2*ArcCoth[1 - I*d + d*Tan[a + b*x]],x]","\frac{1}{3} x^3 \coth ^{-1}(d \tan (a+b x)-i d+1)-\frac{4 b^3 x^3 \log \left(1+\frac{i e^{-2 i (a+b x)}}{d+i}\right)+6 i b^2 x^2 \text{Li}_2\left(-\frac{i e^{-2 i (a+b x)}}{d+i}\right)+6 b x \text{Li}_3\left(-\frac{i e^{-2 i (a+b x)}}{d+i}\right)-3 i \text{Li}_4\left(-\frac{i e^{-2 i (a+b x)}}{d+i}\right)}{24 b^3}","-\frac{i \text{Li}_4\left(-\left((1-i d) e^{2 i a+2 i b x}\right)\right)}{8 b^3}-\frac{x \text{Li}_3\left(-\left((1-i d) e^{2 i a+2 i b x}\right)\right)}{4 b^2}+\frac{i x^2 \text{Li}_2\left(-\left((1-i d) e^{2 i a+2 i b x}\right)\right)}{4 b}-\frac{1}{6} x^3 \log \left(1+(1-i d) e^{2 i a+2 i b x}\right)+\frac{1}{3} x^3 \coth ^{-1}(d \tan (a+b x)-i d+1)+\frac{1}{12} i b x^4",1,"(x^3*ArcCoth[1 - I*d + d*Tan[a + b*x]])/3 - (4*b^3*x^3*Log[1 + I/((I + d)*E^((2*I)*(a + b*x)))] + (6*I)*b^2*x^2*PolyLog[2, (-I)/((I + d)*E^((2*I)*(a + b*x)))] + 6*b*x*PolyLog[3, (-I)/((I + d)*E^((2*I)*(a + b*x)))] - (3*I)*PolyLog[4, (-I)/((I + d)*E^((2*I)*(a + b*x)))])/(24*b^3)","A",1
241,1,119,133,0.1167063,"\int x \coth ^{-1}(1-i d+d \tan (a+b x)) \, dx","Integrate[x*ArcCoth[1 - I*d + d*Tan[a + b*x]],x]","\frac{1}{2} x^2 \coth ^{-1}(d \tan (a+b x)-i d+1)-\frac{2 b^2 x^2 \log \left(1+\frac{i e^{-2 i (a+b x)}}{d+i}\right)+2 i b x \text{Li}_2\left(-\frac{i e^{-2 i (a+b x)}}{d+i}\right)+\text{Li}_3\left(-\frac{i e^{-2 i (a+b x)}}{d+i}\right)}{8 b^2}","-\frac{\text{Li}_3\left(-\left((1-i d) e^{2 i a+2 i b x}\right)\right)}{8 b^2}+\frac{i x \text{Li}_2\left(-\left((1-i d) e^{2 i a+2 i b x}\right)\right)}{4 b}-\frac{1}{4} x^2 \log \left(1+(1-i d) e^{2 i a+2 i b x}\right)+\frac{1}{2} x^2 \coth ^{-1}(d \tan (a+b x)-i d+1)+\frac{1}{6} i b x^3",1,"(x^2*ArcCoth[1 - I*d + d*Tan[a + b*x]])/2 - (2*b^2*x^2*Log[1 + I/((I + d)*E^((2*I)*(a + b*x)))] + (2*I)*b*x*PolyLog[2, (-I)/((I + d)*E^((2*I)*(a + b*x)))] + PolyLog[3, (-I)/((I + d)*E^((2*I)*(a + b*x)))])/(8*b^2)","A",1
242,1,766,93,3.8511483,"\int \coth ^{-1}(1-i d+d \tan (a+b x)) \, dx","Integrate[ArcCoth[1 - I*d + d*Tan[a + b*x]],x]","\frac{x \sec ^2(a+b x) (\cos (b x)+i \sin (b x)) (\sin (b x)+i \cos (b x)) (d \sin (a+b x)+(2-i d) \cos (a+b x)) \left(\text{Li}_2\left(-\frac{1}{2} (\cos (a)+i \sin (a)) (d \cos (a)+i (d+2 i) \sin (a)) (\tan (b x)-i)\right)-\text{Li}_2\left(\frac{\sec (b x) (d \cos (a)+i (d+2 i) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 (d+i)}\right)-\log (1-i \tan (b x)) \log \left(\frac{(\cos (a)-i \sin (a)) \sec (b x) (i d \sin (a+b x)+(d+2 i) \cos (a+b x))}{2 (d+i)}\right)+\log (1+i \tan (b x)) \log \left(\frac{\sec (b x) (d \sin (a+b x)+(2-i d) \cos (a+b x))}{2 \cos (a)-2 i \sin (a)}\right)-\text{Li}_2(i \sin (2 b x)-\cos (2 b x))+2 i b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))\right)}{(\tan (a+b x)-i) (d \tan (a+b x)-i d+2) (i d \sin (a+b x)+(d+2 i) \cos (a+b x)) \left(\frac{\sec ^2(b x) \log \left(\frac{\sec (b x) (d \sin (a+b x)+(2-i d) \cos (a+b x))}{2 \cos (a)-2 i \sin (a)}\right)}{\tan (b x)-i}-\frac{\sec ^2(b x) \log \left(1+\frac{1}{2} (\cos (a)+i \sin (a)) (\tan (b x)-i) (d \cos (a)+i (d+2 i) \sin (a))\right)}{\tan (b x)-i}-\frac{\sec ^2(b x) \log \left(\frac{(\cos (a)-i \sin (a)) \sec (b x) (i d \sin (a+b x)+(d+2 i) \cos (a+b x))}{2 (d+i)}\right)}{\tan (b x)+i}+\frac{i \sec (b x) (d \cos (a)+i (d+2 i) \sin (a)) \log (1+i \tan (b x))}{i d \sin (a+b x)+(d+2 i) \cos (a+b x)}+\frac{\sec (b x) ((d+2 i) \sin (a)-i d \cos (a)) \log (1-i \tan (b x))}{i d \sin (a+b x)+(d+2 i) \cos (a+b x)}+(\tan (b x)-i) \log \left(\frac{(\cos (a)-i \sin (a)) \sec (b x) (i d \sin (a+b x)+(d+2 i) \cos (a+b x))}{2 (d+i)}\right)+2 b x (1-i \tan (b x))\right)}+x \coth ^{-1}(d \tan (a+b x)-i d+1)","\frac{i \text{Li}_2\left(-\left((1-i d) e^{2 i a+2 i b x}\right)\right)}{4 b}-\frac{1}{2} x \log \left(1+(1-i d) e^{2 i a+2 i b x}\right)+x \coth ^{-1}(d \tan (a+b x)-i d+1)+\frac{1}{2} i b x^2",1,"x*ArcCoth[1 - I*d + d*Tan[a + b*x]] + (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])*((2*I + d)*Cos[a + b*x] + I*d*Sin[a + b*x]))/(2*(I + d))]*Log[1 - I*Tan[b*x]] + Log[(Sec[b*x]*((2 - I*d)*Cos[a + b*x] + d*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]*(d*Cos[a] + I*(2*I + d)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*(I + d))] + PolyLog[2, -1/2*((Cos[a] + I*Sin[a])*(d*Cos[a] + I*(2*I + d)*Sin[a])*(-I + Tan[b*x]))])*Sec[a + b*x]^2*(Cos[b*x] + I*Sin[b*x])*(I*Cos[b*x] + Sin[b*x])*((2 - I*d)*Cos[a + b*x] + d*Sin[a + b*x]))/(((2*I + d)*Cos[a + b*x] + I*d*Sin[a + b*x])*((I*Log[1 + I*Tan[b*x]]*Sec[b*x]*(d*Cos[a] + I*(2*I + d)*Sin[a]))/((2*I + d)*Cos[a + b*x] + I*d*Sin[a + b*x]) + (Log[1 - I*Tan[b*x]]*Sec[b*x]*((-I)*d*Cos[a] + (2*I + d)*Sin[a]))/((2*I + d)*Cos[a + b*x] + I*d*Sin[a + b*x]) + 2*b*x*(1 - I*Tan[b*x]) + (Log[(Sec[b*x]*((2 - I*d)*Cos[a + b*x] + d*Sin[a + b*x]))/(2*Cos[a] - (2*I)*Sin[a])]*Sec[b*x]^2)/(-I + Tan[b*x]) - (Log[1 + ((Cos[a] + I*Sin[a])*(d*Cos[a] + I*(2*I + d)*Sin[a])*(-I + Tan[b*x]))/2]*Sec[b*x]^2)/(-I + Tan[b*x]) + Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((2*I + d)*Cos[a + b*x] + I*d*Sin[a + b*x]))/(2*(I + d))]*(-I + Tan[b*x]) - (Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((2*I + d)*Cos[a + b*x] + I*d*Sin[a + b*x]))/(2*(I + d))]*Sec[b*x]^2)/(I + Tan[b*x]))*(-I + Tan[a + b*x])*(2 - I*d + d*Tan[a + b*x]))","B",0
243,0,0,23,0.8114917,"\int \frac{\coth ^{-1}(1-i d+d \tan (a+b x))}{x} \, dx","Integrate[ArcCoth[1 - I*d + d*Tan[a + b*x]]/x,x]","\int \frac{\coth ^{-1}(1-i d+d \tan (a+b x))}{x} \, dx","\text{Int}\left(\frac{\coth ^{-1}(d \tan (a+b x)-i d+1)}{x},x\right)",0,"Integrate[ArcCoth[1 - I*d + d*Tan[a + b*x]]/x, x]","A",-1
244,1,156,171,0.2137584,"\int x^2 \coth ^{-1}(1+i d-d \tan (a+b x)) \, dx","Integrate[x^2*ArcCoth[1 + I*d - d*Tan[a + b*x]],x]","\frac{1}{3} x^3 \coth ^{-1}(d (-\tan (a+b x))+i d+1)-\frac{4 b^3 x^3 \log \left(1-\frac{i e^{-2 i (a+b x)}}{d-i}\right)+6 i b^2 x^2 \text{Li}_2\left(\frac{i e^{-2 i (a+b x)}}{d-i}\right)+6 b x \text{Li}_3\left(\frac{i e^{-2 i (a+b x)}}{d-i}\right)-3 i \text{Li}_4\left(\frac{i e^{-2 i (a+b x)}}{d-i}\right)}{24 b^3}","-\frac{i \text{Li}_4\left(-\left((i d+1) e^{2 i a+2 i b x}\right)\right)}{8 b^3}-\frac{x \text{Li}_3\left(-\left((i d+1) e^{2 i a+2 i b x}\right)\right)}{4 b^2}+\frac{i x^2 \text{Li}_2\left(-\left((i d+1) e^{2 i a+2 i b x}\right)\right)}{4 b}-\frac{1}{6} x^3 \log \left(1+(1+i d) e^{2 i a+2 i b x}\right)+\frac{1}{3} x^3 \coth ^{-1}(d (-\tan (a+b x))+i d+1)+\frac{1}{12} i b x^4",1,"(x^3*ArcCoth[1 + I*d - d*Tan[a + b*x]])/3 - (4*b^3*x^3*Log[1 - I/((-I + d)*E^((2*I)*(a + b*x)))] + (6*I)*b^2*x^2*PolyLog[2, I/((-I + d)*E^((2*I)*(a + b*x)))] + 6*b*x*PolyLog[3, I/((-I + d)*E^((2*I)*(a + b*x)))] - (3*I)*PolyLog[4, I/((-I + d)*E^((2*I)*(a + b*x)))])/(24*b^3)","A",1
245,1,120,134,0.1194166,"\int x \coth ^{-1}(1+i d-d \tan (a+b x)) \, dx","Integrate[x*ArcCoth[1 + I*d - d*Tan[a + b*x]],x]","\frac{1}{2} x^2 \coth ^{-1}(d (-\tan (a+b x))+i d+1)-\frac{2 b^2 x^2 \log \left(1-\frac{i e^{-2 i (a+b x)}}{d-i}\right)+2 i b x \text{Li}_2\left(\frac{i e^{-2 i (a+b x)}}{d-i}\right)+\text{Li}_3\left(\frac{i e^{-2 i (a+b x)}}{d-i}\right)}{8 b^2}","-\frac{\text{Li}_3\left(-\left((i d+1) e^{2 i a+2 i b x}\right)\right)}{8 b^2}+\frac{i x \text{Li}_2\left(-\left((i d+1) e^{2 i a+2 i b x}\right)\right)}{4 b}-\frac{1}{4} x^2 \log \left(1+(1+i d) e^{2 i a+2 i b x}\right)+\frac{1}{2} x^2 \coth ^{-1}(d (-\tan (a+b x))+i d+1)+\frac{1}{6} i b x^3",1,"(x^2*ArcCoth[1 + I*d - d*Tan[a + b*x]])/2 - (2*b^2*x^2*Log[1 - I/((-I + d)*E^((2*I)*(a + b*x)))] + (2*I)*b*x*PolyLog[2, I/((-I + d)*E^((2*I)*(a + b*x)))] + PolyLog[3, I/((-I + d)*E^((2*I)*(a + b*x)))])/(8*b^2)","A",1
246,1,723,94,3.2387155,"\int \coth ^{-1}(1+i d-d \tan (a+b x)) \, dx","Integrate[ArcCoth[1 + I*d - d*Tan[a + b*x]],x]","x \coth ^{-1}(d (-\tan (a+b x))+i d+1)-\frac{x \sec (a+b x) (\cos (b x)+i \sin (b x)) (\sin (b x)+i \cos (b x)) \left(-\text{Li}_2\left(\frac{1}{2} (\cos (a)+i \sin (a)) (d \cos (a)+(i d+2) \sin (a)) (\tan (b x)-i)\right)+\text{Li}_2\left(\frac{\sec (b x) (d \cos (a)+(i d+2) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 (d-i)}\right)+\log (1-i \tan (b x)) \log \left(\frac{(\cos (a)-i \sin (a)) \sec (b x) (i d \sin (a+b x)+(d-2 i) \cos (a+b x))}{2 (d-i)}\right)-\log (1+i \tan (b x)) \log \left(\frac{\sec (b x) (-d \sin (a+b x)+(2+i d) \cos (a+b x))}{2 \cos (a)-2 i \sin (a)}\right)+\text{Li}_2(i \sin (2 b x)-\cos (2 b x))-2 i b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))\right)}{(\tan (a+b x)-i) (i d \sin (a+b x)+(d-2 i) \cos (a+b x)) \left(-\frac{\sec ^2(b x) \log \left(\frac{\sec (b x) (-d \sin (a+b x)+(2+i d) \cos (a+b x))}{2 \cos (a)-2 i \sin (a)}\right)}{\tan (b x)-i}+\frac{\sec ^2(b x) \log \left(1-\frac{1}{2} (\cos (a)+i \sin (a)) (\tan (b x)-i) (d \cos (a)+(2+i d) \sin (a))\right)}{\tan (b x)-i}+\frac{\sec ^2(b x) \log \left(\frac{(\cos (a)-i \sin (a)) \sec (b x) (i d \sin (a+b x)+(d-2 i) \cos (a+b x))}{2 (d-i)}\right)}{\tan (b x)+i}+\frac{i \sec (b x) (d \cos (a)+(2+i d) \sin (a)) \log (1-i \tan (b x))}{i d \sin (a+b x)+(d-2 i) \cos (a+b x)}+\frac{\sec (b x) ((d-2 i) \sin (a)-i d \cos (a)) \log (1+i \tan (b x))}{i d \sin (a+b x)+(d-2 i) \cos (a+b x)}-(\tan (b x)-i) \log \left(1-\frac{\sec (b x) (d \cos (a)+(2+i d) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 (d-i)}\right)+2 i b x (\tan (b x)+i)\right)}","\frac{i \text{Li}_2\left(-\left((i d+1) e^{2 i a+2 i b x}\right)\right)}{4 b}-\frac{1}{2} x \log \left(1+(1+i d) e^{2 i a+2 i b x}\right)+x \coth ^{-1}(d (-\tan (a+b x))+i d+1)+\frac{1}{2} i b x^2",1,"x*ArcCoth[1 + I*d - d*Tan[a + b*x]] - (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])*((-2*I + d)*Cos[a + b*x] + I*d*Sin[a + b*x]))/(2*(-I + d))]*Log[1 - I*Tan[b*x]] - Log[(Sec[b*x]*((2 + I*d)*Cos[a + b*x] - d*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]*(d*Cos[a] + (2 + I*d)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*(-I + d))] - PolyLog[2, ((Cos[a] + I*Sin[a])*(d*Cos[a] + (2 + I*d)*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]))/(((-2*I + d)*Cos[a + b*x] + I*d*Sin[a + b*x])*((I*Log[1 - I*Tan[b*x]]*Sec[b*x]*(d*Cos[a] + (2 + I*d)*Sin[a]))/((-2*I + d)*Cos[a + b*x] + I*d*Sin[a + b*x]) + (Log[1 + I*Tan[b*x]]*Sec[b*x]*((-I)*d*Cos[a] + (-2*I + d)*Sin[a]))/((-2*I + d)*Cos[a + b*x] + I*d*Sin[a + b*x]) - (Log[(Sec[b*x]*((2 + I*d)*Cos[a + b*x] - d*Sin[a + b*x]))/(2*Cos[a] - (2*I)*Sin[a])]*Sec[b*x]^2)/(-I + Tan[b*x]) + (Log[1 - ((Cos[a] + I*Sin[a])*(d*Cos[a] + (2 + I*d)*Sin[a])*(-I + Tan[b*x]))/2]*Sec[b*x]^2)/(-I + Tan[b*x]) - Log[1 - (Sec[b*x]*(d*Cos[a] + (2 + I*d)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*(-I + d))]*(-I + Tan[b*x]) + (Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((-2*I + d)*Cos[a + b*x] + I*d*Sin[a + b*x]))/(2*(-I + d))]*Sec[b*x]^2)/(I + Tan[b*x]) + (2*I)*b*x*(I + Tan[b*x]))*(-I + Tan[a + b*x]))","B",0
247,0,0,24,0.8025789,"\int \frac{\coth ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx","Integrate[ArcCoth[1 + I*d - d*Tan[a + b*x]]/x,x]","\int \frac{\coth ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx","\text{Int}\left(\frac{\coth ^{-1}(d (-\tan (a+b x))+i d+1)}{x},x\right)",0,"Integrate[ArcCoth[1 + I*d - d*Tan[a + b*x]]/x, x]","A",-1
248,1,654,302,0.3108132,"\int (e+f x)^3 \coth ^{-1}(\cot (a+b x)) \, dx","Integrate[(e + f*x)^3*ArcCoth[Cot[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) \coth ^{-1}(\cot (a+b x))+\frac{-8 b^4 e^3 x \log \left(1-i e^{2 i (a+b x)}\right)+8 b^4 e^3 x \log \left(1+i e^{2 i (a+b x)}\right)-12 b^4 e^2 f x^2 \log \left(1-i e^{2 i (a+b x)}\right)+12 b^4 e^2 f x^2 \log \left(1+i e^{2 i (a+b x)}\right)-8 b^4 e f^2 x^3 \log \left(1-i e^{2 i (a+b x)}\right)+8 b^4 e f^2 x^3 \log \left(1+i e^{2 i (a+b x)}\right)-2 b^4 f^3 x^4 \log \left(1-i e^{2 i (a+b x)}\right)+2 b^4 f^3 x^4 \log \left(1+i e^{2 i (a+b x)}\right)-4 i b^3 (e+f x)^3 \text{Li}_2\left(-i e^{2 i (a+b x)}\right)+4 i b^3 (e+f x)^3 \text{Li}_2\left(i e^{2 i (a+b x)}\right)+6 b^2 e^2 f \text{Li}_3\left(-i e^{2 i (a+b x)}\right)-6 b^2 e^2 f \text{Li}_3\left(i e^{2 i (a+b x)}\right)+12 b^2 e f^2 x \text{Li}_3\left(-i e^{2 i (a+b x)}\right)-12 b^2 e f^2 x \text{Li}_3\left(i e^{2 i (a+b x)}\right)+6 b^2 f^3 x^2 \text{Li}_3\left(-i e^{2 i (a+b x)}\right)-6 b^2 f^3 x^2 \text{Li}_3\left(i e^{2 i (a+b x)}\right)+6 i b e f^2 \text{Li}_4\left(-i e^{2 i (a+b x)}\right)-6 i b e f^2 \text{Li}_4\left(i e^{2 i (a+b x)}\right)+6 i b f^3 x \text{Li}_4\left(-i e^{2 i (a+b x)}\right)-6 i b f^3 x \text{Li}_4\left(i e^{2 i (a+b x)}\right)-3 f^3 \text{Li}_5\left(-i e^{2 i (a+b x)}\right)+3 f^3 \text{Li}_5\left(i e^{2 i (a+b x)}\right)}{16 b^4}","-\frac{3 f^3 \text{Li}_5\left(-i e^{2 i (a+b x)}\right)}{16 b^4}+\frac{3 f^3 \text{Li}_5\left(i e^{2 i (a+b x)}\right)}{16 b^4}+\frac{3 i f^2 (e+f x) \text{Li}_4\left(-i e^{2 i (a+b x)}\right)}{8 b^3}-\frac{3 i f^2 (e+f x) \text{Li}_4\left(i e^{2 i (a+b x)}\right)}{8 b^3}+\frac{3 f (e+f x)^2 \text{Li}_3\left(-i e^{2 i (a+b x)}\right)}{8 b^2}-\frac{3 f (e+f x)^2 \text{Li}_3\left(i e^{2 i (a+b x)}\right)}{8 b^2}-\frac{i (e+f x)^3 \text{Li}_2\left(-i e^{2 i (a+b x)}\right)}{4 b}+\frac{i (e+f x)^3 \text{Li}_2\left(i e^{2 i (a+b x)}\right)}{4 b}+\frac{i (e+f x)^4 \tan ^{-1}\left(e^{2 i (a+b x)}\right)}{4 f}+\frac{(e+f x)^4 \coth ^{-1}(\cot (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)*ArcCoth[Cot[a + b*x]])/4 + (-8*b^4*e^3*x*Log[1 - I*E^((2*I)*(a + b*x))] - 12*b^4*e^2*f*x^2*Log[1 - I*E^((2*I)*(a + b*x))] - 8*b^4*e*f^2*x^3*Log[1 - I*E^((2*I)*(a + b*x))] - 2*b^4*f^3*x^4*Log[1 - I*E^((2*I)*(a + b*x))] + 8*b^4*e^3*x*Log[1 + I*E^((2*I)*(a + b*x))] + 12*b^4*e^2*f*x^2*Log[1 + I*E^((2*I)*(a + b*x))] + 8*b^4*e*f^2*x^3*Log[1 + I*E^((2*I)*(a + b*x))] + 2*b^4*f^3*x^4*Log[1 + I*E^((2*I)*(a + b*x))] - (4*I)*b^3*(e + f*x)^3*PolyLog[2, (-I)*E^((2*I)*(a + b*x))] + (4*I)*b^3*(e + f*x)^3*PolyLog[2, I*E^((2*I)*(a + b*x))] + 6*b^2*e^2*f*PolyLog[3, (-I)*E^((2*I)*(a + b*x))] + 12*b^2*e*f^2*x*PolyLog[3, (-I)*E^((2*I)*(a + b*x))] + 6*b^2*f^3*x^2*PolyLog[3, (-I)*E^((2*I)*(a + b*x))] - 6*b^2*e^2*f*PolyLog[3, I*E^((2*I)*(a + b*x))] - 12*b^2*e*f^2*x*PolyLog[3, I*E^((2*I)*(a + b*x))] - 6*b^2*f^3*x^2*PolyLog[3, I*E^((2*I)*(a + b*x))] + (6*I)*b*e*f^2*PolyLog[4, (-I)*E^((2*I)*(a + b*x))] + (6*I)*b*f^3*x*PolyLog[4, (-I)*E^((2*I)*(a + b*x))] - (6*I)*b*e*f^2*PolyLog[4, I*E^((2*I)*(a + b*x))] - (6*I)*b*f^3*x*PolyLog[4, I*E^((2*I)*(a + b*x))] - 3*f^3*PolyLog[5, (-I)*E^((2*I)*(a + b*x))] + 3*f^3*PolyLog[5, I*E^((2*I)*(a + b*x))])/(16*b^4)","B",1
249,1,409,234,0.197558,"\int (e+f x)^2 \coth ^{-1}(\cot (a+b x)) \, dx","Integrate[(e + f*x)^2*ArcCoth[Cot[a + b*x]],x]","\frac{1}{3} x \left(3 e^2+3 e f x+f^2 x^2\right) \coth ^{-1}(\cot (a+b x))+\frac{-12 b^3 e^2 x \log \left(1-i e^{2 i (a+b x)}\right)+12 b^3 e^2 x \log \left(1+i e^{2 i (a+b x)}\right)-12 b^3 e f x^2 \log \left(1-i e^{2 i (a+b x)}\right)+12 b^3 e f x^2 \log \left(1+i e^{2 i (a+b x)}\right)-4 b^3 f^2 x^3 \log \left(1-i e^{2 i (a+b x)}\right)+4 b^3 f^2 x^3 \log \left(1+i e^{2 i (a+b x)}\right)-6 i b^2 (e+f x)^2 \text{Li}_2\left(-i e^{2 i (a+b x)}\right)+6 i b^2 (e+f x)^2 \text{Li}_2\left(i e^{2 i (a+b x)}\right)+6 b e f \text{Li}_3\left(-i e^{2 i (a+b x)}\right)-6 b e f \text{Li}_3\left(i e^{2 i (a+b x)}\right)+6 b f^2 x \text{Li}_3\left(-i e^{2 i (a+b x)}\right)-6 b f^2 x \text{Li}_3\left(i e^{2 i (a+b x)}\right)+3 i f^2 \text{Li}_4\left(-i e^{2 i (a+b x)}\right)-3 i f^2 \text{Li}_4\left(i e^{2 i (a+b x)}\right)}{24 b^3}","\frac{i f^2 \text{Li}_4\left(-i e^{2 i (a+b x)}\right)}{8 b^3}-\frac{i f^2 \text{Li}_4\left(i e^{2 i (a+b x)}\right)}{8 b^3}+\frac{f (e+f x) \text{Li}_3\left(-i e^{2 i (a+b x)}\right)}{4 b^2}-\frac{f (e+f x) \text{Li}_3\left(i e^{2 i (a+b x)}\right)}{4 b^2}-\frac{i (e+f x)^2 \text{Li}_2\left(-i e^{2 i (a+b x)}\right)}{4 b}+\frac{i (e+f x)^2 \text{Li}_2\left(i e^{2 i (a+b x)}\right)}{4 b}+\frac{i (e+f x)^3 \tan ^{-1}\left(e^{2 i (a+b x)}\right)}{3 f}+\frac{(e+f x)^3 \coth ^{-1}(\cot (a+b x))}{3 f}",1,"(x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCoth[Cot[a + b*x]])/3 + (-12*b^3*e^2*x*Log[1 - I*E^((2*I)*(a + b*x))] - 12*b^3*e*f*x^2*Log[1 - I*E^((2*I)*(a + b*x))] - 4*b^3*f^2*x^3*Log[1 - I*E^((2*I)*(a + b*x))] + 12*b^3*e^2*x*Log[1 + I*E^((2*I)*(a + b*x))] + 12*b^3*e*f*x^2*Log[1 + I*E^((2*I)*(a + b*x))] + 4*b^3*f^2*x^3*Log[1 + I*E^((2*I)*(a + b*x))] - (6*I)*b^2*(e + f*x)^2*PolyLog[2, (-I)*E^((2*I)*(a + b*x))] + (6*I)*b^2*(e + f*x)^2*PolyLog[2, I*E^((2*I)*(a + b*x))] + 6*b*e*f*PolyLog[3, (-I)*E^((2*I)*(a + b*x))] + 6*b*f^2*x*PolyLog[3, (-I)*E^((2*I)*(a + b*x))] - 6*b*e*f*PolyLog[3, I*E^((2*I)*(a + b*x))] - 6*b*f^2*x*PolyLog[3, I*E^((2*I)*(a + b*x))] + (3*I)*f^2*PolyLog[4, (-I)*E^((2*I)*(a + b*x))] - (3*I)*f^2*PolyLog[4, I*E^((2*I)*(a + b*x))])/(24*b^3)","A",1
250,1,263,162,0.1225563,"\int (e+f x) \coth ^{-1}(\cot (a+b x)) \, dx","Integrate[(e + f*x)*ArcCoth[Cot[a + b*x]],x]","-b e \left(\frac{i \text{Li}_2\left(-i e^{i (2 a+2 b x)}\right)}{4 b^2}-\frac{i \text{Li}_2\left(i e^{i (2 a+2 b x)}\right)}{4 b^2}-\frac{i x \tan ^{-1}\left(e^{2 i a+2 i b x}\right)}{b}\right)+\frac{f \left(4 i b^2 x^2 \tan ^{-1}(\cos (2 (a+b x))+i \sin (2 (a+b x)))+2 i b x \text{Li}_2(i \cos (2 (a+b x))-\sin (2 (a+b x)))-2 i b x \text{Li}_2(\sin (2 (a+b x))-i \cos (2 (a+b x)))-\text{Li}_3(i \cos (2 (a+b x))-\sin (2 (a+b x)))+\text{Li}_3(\sin (2 (a+b x))-i \cos (2 (a+b x)))\right)}{8 b^2}+e x \coth ^{-1}(\cot (a+b x))+\frac{1}{2} f x^2 \coth ^{-1}(\cot (a+b x))","\frac{f \text{Li}_3\left(-i e^{2 i (a+b x)}\right)}{8 b^2}-\frac{f \text{Li}_3\left(i e^{2 i (a+b x)}\right)}{8 b^2}-\frac{i (e+f x) \text{Li}_2\left(-i e^{2 i (a+b x)}\right)}{4 b}+\frac{i (e+f x) \text{Li}_2\left(i e^{2 i (a+b x)}\right)}{4 b}+\frac{i (e+f x)^2 \tan ^{-1}\left(e^{2 i (a+b x)}\right)}{2 f}+\frac{(e+f x)^2 \coth ^{-1}(\cot (a+b x))}{2 f}",1,"e*x*ArcCoth[Cot[a + b*x]] + (f*x^2*ArcCoth[Cot[a + b*x]])/2 - b*e*(((-I)*x*ArcTan[E^((2*I)*a + (2*I)*b*x)])/b + ((I/4)*PolyLog[2, (-I)*E^(I*(2*a + 2*b*x))])/b^2 - ((I/4)*PolyLog[2, I*E^(I*(2*a + 2*b*x))])/b^2) + (f*((4*I)*b^2*x^2*ArcTan[Cos[2*(a + b*x)] + I*Sin[2*(a + b*x)]] + (2*I)*b*x*PolyLog[2, I*Cos[2*(a + b*x)] - Sin[2*(a + b*x)]] - (2*I)*b*x*PolyLog[2, (-I)*Cos[2*(a + b*x)] + Sin[2*(a + b*x)]] - PolyLog[3, I*Cos[2*(a + b*x)] - Sin[2*(a + b*x)]] + PolyLog[3, (-I)*Cos[2*(a + b*x)] + Sin[2*(a + b*x)]]))/(8*b^2)","A",0
251,1,78,79,0.0333525,"\int \coth ^{-1}(\cot (a+b x)) \, dx","Integrate[ArcCoth[Cot[a + b*x]],x]","\frac{-i \text{Li}_2\left(-i e^{2 i (a+b x)}\right)+i \text{Li}_2\left(i e^{2 i (a+b x)}\right)+4 b x \left(\coth ^{-1}(\cot (a+b x))+i \tan ^{-1}\left(e^{2 i (a+b x)}\right)\right)}{4 b}","-\frac{i \text{Li}_2\left(-i e^{2 i (a+b x)}\right)}{4 b}+\frac{i \text{Li}_2\left(i e^{2 i (a+b x)}\right)}{4 b}+i x \tan ^{-1}\left(e^{2 i (a+b x)}\right)+x \coth ^{-1}(\cot (a+b x))",1,"(4*b*x*(ArcCoth[Cot[a + b*x]] + I*ArcTan[E^((2*I)*(a + b*x))]) - I*PolyLog[2, (-I)*E^((2*I)*(a + b*x))] + I*PolyLog[2, I*E^((2*I)*(a + b*x))])/(4*b)","A",1
252,0,0,18,0.1162383,"\int \frac{\coth ^{-1}(\cot (a+b x))}{e+f x} \, dx","Integrate[ArcCoth[Cot[a + b*x]]/(e + f*x),x]","\int \frac{\coth ^{-1}(\cot (a+b x))}{e+f x} \, dx","\text{Int}\left(\frac{\coth ^{-1}(\cot (a+b x))}{e+f x},x\right)",0,"Integrate[ArcCoth[Cot[a + b*x]]/(e + f*x), x]","A",-1
253,1,339,391,0.3331307,"\int x^2 \coth ^{-1}(c+d \cot (a+b x)) \, dx","Integrate[x^2*ArcCoth[c + d*Cot[a + b*x]],x]","\frac{1}{3} x^3 \coth ^{-1}(d \cot (a+b x)+c)+\frac{4 b^3 x^3 \log \left(1-\frac{(c+i d-1) e^{2 i (a+b x)}}{c-i d-1}\right)-4 b^3 x^3 \log \left(1-\frac{(c+i d+1) e^{2 i (a+b x)}}{c-i d+1}\right)-6 i 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 i 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 x \text{Li}_3\left(\frac{(c+i d-1) e^{2 i (a+b x)}}{c-i d-1}\right)-6 b x \text{Li}_3\left(\frac{(c+i d+1) e^{2 i (a+b x)}}{c-i d+1}\right)+3 i \text{Li}_4\left(\frac{(c+i d-1) e^{2 i (a+b x)}}{c-i d-1}\right)-3 i \text{Li}_4\left(\frac{(c+i d+1) e^{2 i (a+b x)}}{c-i d+1}\right)}{24 b^3}","\frac{i \text{Li}_4\left(\frac{(-c-i d+1) e^{2 i a+2 i b x}}{-c+i d+1}\right)}{8 b^3}-\frac{i \text{Li}_4\left(\frac{(c+i d+1) e^{2 i a+2 i b x}}{c-i d+1}\right)}{8 b^3}+\frac{x \text{Li}_3\left(\frac{(-c-i d+1) e^{2 i a+2 i b x}}{-c+i d+1}\right)}{4 b^2}-\frac{x \text{Li}_3\left(\frac{(c+i d+1) e^{2 i a+2 i b x}}{c-i d+1}\right)}{4 b^2}-\frac{i x^2 \text{Li}_2\left(\frac{(-c-i d+1) e^{2 i a+2 i b x}}{-c+i d+1}\right)}{4 b}+\frac{i x^2 \text{Li}_2\left(\frac{(c+i d+1) e^{2 i a+2 i b x}}{c-i d+1}\right)}{4 b}+\frac{1}{6} x^3 \log \left(1-\frac{(-c-i d+1) e^{2 i a+2 i b x}}{-c+i d+1}\right)-\frac{1}{6} x^3 \log \left(1-\frac{(c+i d+1) e^{2 i a+2 i b x}}{c-i d+1}\right)+\frac{1}{3} x^3 \coth ^{-1}(d \cot (a+b x)+c)",1,"(x^3*ArcCoth[c + d*Cot[a + b*x]])/3 + (4*b^3*x^3*Log[1 - ((-1 + c + I*d)*E^((2*I)*(a + b*x)))/(-1 + c - I*d)] - 4*b^3*x^3*Log[1 - ((1 + c + I*d)*E^((2*I)*(a + b*x)))/(1 + c - I*d)] - (6*I)*b^2*x^2*PolyLog[2, ((-1 + c + I*d)*E^((2*I)*(a + b*x)))/(-1 + c - I*d)] + (6*I)*b^2*x^2*PolyLog[2, ((1 + c + I*d)*E^((2*I)*(a + b*x)))/(1 + c - I*d)] + 6*b*x*PolyLog[3, ((-1 + c + I*d)*E^((2*I)*(a + b*x)))/(-1 + c - I*d)] - 6*b*x*PolyLog[3, ((1 + c + I*d)*E^((2*I)*(a + b*x)))/(1 + c - I*d)] + (3*I)*PolyLog[4, ((-1 + c + I*d)*E^((2*I)*(a + b*x)))/(-1 + c - I*d)] - (3*I)*PolyLog[4, ((1 + c + I*d)*E^((2*I)*(a + b*x)))/(1 + c - I*d)])/(24*b^3)","A",1
254,1,253,293,0.1283674,"\int x \coth ^{-1}(c+d \cot (a+b x)) \, dx","Integrate[x*ArcCoth[c + d*Cot[a + b*x]],x]","\frac{1}{2} x^2 \coth ^{-1}(d \cot (a+b x)+c)+\frac{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+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+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-1}\right)-\text{Li}_3\left(\frac{(c+i d+1) e^{2 i (a+b x)}}{c-i d+1}\right)}{8 b^2}","\frac{\text{Li}_3\left(\frac{(-c-i d+1) e^{2 i a+2 i b x}}{-c+i d+1}\right)}{8 b^2}-\frac{\text{Li}_3\left(\frac{(c+i d+1) e^{2 i a+2 i b x}}{c-i d+1}\right)}{8 b^2}-\frac{i x \text{Li}_2\left(\frac{(-c-i d+1) e^{2 i a+2 i b x}}{-c+i d+1}\right)}{4 b}+\frac{i x \text{Li}_2\left(\frac{(c+i d+1) e^{2 i a+2 i b x}}{c-i d+1}\right)}{4 b}+\frac{1}{4} x^2 \log \left(1-\frac{(-c-i d+1) e^{2 i a+2 i b x}}{-c+i d+1}\right)-\frac{1}{4} x^2 \log \left(1-\frac{(c+i d+1) e^{2 i a+2 i b x}}{c-i d+1}\right)+\frac{1}{2} x^2 \coth ^{-1}(d \cot (a+b x)+c)",1,"(x^2*ArcCoth[c + d*Cot[a + b*x]])/2 + (2*b^2*x^2*Log[1 - ((-1 + c + I*d)*E^((2*I)*(a + b*x)))/(-1 + c - I*d)] - 2*b^2*x^2*Log[1 - ((1 + c + I*d)*E^((2*I)*(a + b*x)))/(1 + c - I*d)] - (2*I)*b*x*PolyLog[2, ((-1 + c + I*d)*E^((2*I)*(a + b*x)))/(-1 + c - I*d)] + (2*I)*b*x*PolyLog[2, ((1 + c + I*d)*E^((2*I)*(a + b*x)))/(1 + c - I*d)] + PolyLog[3, ((-1 + c + I*d)*E^((2*I)*(a + b*x)))/(-1 + c - I*d)] - PolyLog[3, ((1 + c + I*d)*E^((2*I)*(a + b*x)))/(1 + c - I*d)])/(8*b^2)","A",1
255,1,4463,194,13.1000998,"\int \coth ^{-1}(c+d \cot (a+b x)) \, dx","Integrate[ArcCoth[c + d*Cot[a + b*x]],x]","\text{Result too large to show}","-\frac{i \text{Li}_2\left(\frac{(-c-i d+1) e^{2 i a+2 i b x}}{-c+i d+1}\right)}{4 b}+\frac{i \text{Li}_2\left(\frac{(c+i d+1) e^{2 i a+2 i b x}}{c-i d+1}\right)}{4 b}+\frac{1}{2} x \log \left(1-\frac{(-c-i d+1) e^{2 i a+2 i b x}}{-c+i d+1}\right)-\frac{1}{2} x \log \left(1-\frac{(c+i d+1) e^{2 i a+2 i b x}}{c-i d+1}\right)+x \coth ^{-1}(d \cot (a+b x)+c)",1,"x*ArcCoth[c + d*Cot[a + b*x]] - (d*(a*Log[-(Sec[(a + b*x)/2]^2*(d*Cos[a + b*x] + (-1 + c)*Sin[a + b*x]))] - a*Log[-(Sec[(a + b*x)/2]^2*(d*Cos[a + b*x] + Sin[a + b*x] + c*Sin[a + b*x]))] - (a + b*x)*Log[-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]] - I*Log[(d*(-I + Tan[(a + b*x)/2]))/(-1 + c - I*d + Sqrt[1 - 2*c + c^2 + d^2])]*Log[-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]] + I*Log[(d*(I + Tan[(a + b*x)/2]))/(-1 + c + I*d + Sqrt[1 - 2*c + c^2 + d^2])]*Log[-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]] + (a + b*x)*Log[-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]] + I*Log[(d*(-I + Tan[(a + b*x)/2]))/(1 + c - I*d + Sqrt[1 + 2*c + c^2 + d^2])]*Log[-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]] - I*Log[(d*(I + Tan[(a + b*x)/2]))/(1 + c + I*d + Sqrt[1 + 2*c + c^2 + d^2])]*Log[-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]] - (a + b*x)*Log[(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d] - I*Log[-((d*(-I + Tan[(a + b*x)/2]))/(1 - c + I*d + Sqrt[1 - 2*c + c^2 + d^2]))]*Log[(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d] + I*Log[-((d*(I + Tan[(a + b*x)/2]))/(1 - c - I*d + Sqrt[1 - 2*c + c^2 + d^2]))]*Log[(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d] + (a + b*x)*Log[(-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d] + I*Log[-((d*(-I + Tan[(a + b*x)/2]))/(-1 - c + I*d + Sqrt[1 + 2*c + c^2 + d^2]))]*Log[(-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d] - I*Log[-((d*(I + Tan[(a + b*x)/2]))/(-1 - c - I*d + Sqrt[1 + 2*c + c^2 + d^2]))]*Log[(-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d] - I*PolyLog[2, (-1 + c + Sqrt[1 - 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(-1 + c - I*d + Sqrt[1 - 2*c + c^2 + d^2])] + I*PolyLog[2, (-1 + c + Sqrt[1 - 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(-1 + c + I*d + Sqrt[1 - 2*c + c^2 + d^2])] - I*PolyLog[2, (1 + c - Sqrt[1 + 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(1 + c + I*d - Sqrt[1 + 2*c + c^2 + d^2])] + I*PolyLog[2, (1 + c + Sqrt[1 + 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(1 + c - I*d + Sqrt[1 + 2*c + c^2 + d^2])] - I*PolyLog[2, (1 + c + Sqrt[1 + 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(1 + c + I*d + Sqrt[1 + 2*c + c^2 + d^2])] + I*PolyLog[2, (1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/(1 - c - I*d + Sqrt[1 - 2*c + c^2 + d^2])] - I*PolyLog[2, (1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/(1 - c + I*d + Sqrt[1 - 2*c + c^2 + d^2])] + I*PolyLog[2, (-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/(-1 - c + I*d + Sqrt[1 + 2*c + c^2 + 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)]))))/(-Log[-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]] + Log[-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]] - Log[(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d] + Log[(-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d] - ((I/2)*Log[-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(-I + Tan[(a + b*x)/2]) + ((I/2)*Log[-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(-I + Tan[(a + b*x)/2]) - ((I/2)*Log[(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d]*Sec[(a + b*x)/2]^2)/(-I + Tan[(a + b*x)/2]) + ((I/2)*Log[(-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d]*Sec[(a + b*x)/2]^2)/(-I + Tan[(a + b*x)/2]) + ((I/2)*Log[-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(I + Tan[(a + b*x)/2]) - ((I/2)*Log[-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(I + Tan[(a + b*x)/2]) + ((I/2)*Log[(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d]*Sec[(a + b*x)/2]^2)/(I + Tan[(a + b*x)/2]) - ((I/2)*Log[(-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/d]*Sec[(a + b*x)/2]^2)/(I + Tan[(a + b*x)/2]) - ((a + b*x)*Sec[(a + b*x)/2]^2)/(2*(-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2])) - ((I/2)*Log[(d*(-I + Tan[(a + b*x)/2]))/(-1 + c - I*d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]) + ((I/2)*Log[(d*(I + Tan[(a + b*x)/2]))/(-1 + c + I*d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-((-1 + c + Sqrt[1 - 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]) + ((a + b*x)*Sec[(a + b*x)/2]^2)/(2*(-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2])) + ((I/2)*Log[(d*(-I + Tan[(a + b*x)/2]))/(1 + c - I*d + Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]) - ((I/2)*Log[(d*(I + Tan[(a + b*x)/2]))/(1 + c + I*d + Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-((1 + c + Sqrt[1 + 2*c + c^2 + d^2])/d) + Tan[(a + b*x)/2]) - ((I/2)*d*Log[1 - (-1 + c + Sqrt[1 - 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(-1 + c - I*d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-1 + c + Sqrt[1 - 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2]) + ((I/2)*d*Log[1 - (-1 + c + Sqrt[1 - 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(-1 + c + I*d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-1 + c + Sqrt[1 - 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2]) - ((I/2)*d*Log[1 - (1 + c - Sqrt[1 + 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(1 + c + I*d - Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(1 + c - Sqrt[1 + 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2]) + ((I/2)*d*Log[1 - (1 + c + Sqrt[1 + 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(1 + c - I*d + Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(1 + c + Sqrt[1 + 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2]) - ((I/2)*d*Log[1 - (1 + c + Sqrt[1 + 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2])/(1 + c + I*d + Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(1 + c + Sqrt[1 + 2*c + c^2 + d^2] - d*Tan[(a + b*x)/2]) - (d*(a + b*x)*Sec[(a + b*x)/2]^2)/(2*(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])) - ((I/2)*d*Log[-((d*(-I + Tan[(a + b*x)/2]))/(1 - c + I*d + Sqrt[1 - 2*c + c^2 + d^2]))]*Sec[(a + b*x)/2]^2)/(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2]) + ((I/2)*d*Log[-((d*(I + Tan[(a + b*x)/2]))/(1 - c - I*d + Sqrt[1 - 2*c + c^2 + d^2]))]*Sec[(a + b*x)/2]^2)/(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2]) - ((I/2)*d*Log[1 - (1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/(1 - c - I*d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2]) + ((I/2)*d*Log[1 - (1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/(1 - c + I*d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(1 - c + Sqrt[1 - 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2]) + (d*(a + b*x)*Sec[(a + b*x)/2]^2)/(2*(-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])) + ((I/2)*d*Log[-((d*(-I + Tan[(a + b*x)/2]))/(-1 - c + I*d + Sqrt[1 + 2*c + c^2 + d^2]))]*Sec[(a + b*x)/2]^2)/(-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2]) - ((I/2)*d*Log[-((d*(I + Tan[(a + b*x)/2]))/(-1 - c - I*d + Sqrt[1 + 2*c + c^2 + d^2]))]*Sec[(a + b*x)/2]^2)/(-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2]) - ((I/2)*d*Log[1 - (-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2])/(-1 - c + I*d + Sqrt[1 + 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/(-1 - c + Sqrt[1 + 2*c + c^2 + d^2] + d*Tan[(a + b*x)/2]) - (a*Cos[(a + b*x)/2]^2*(-(Sec[(a + b*x)/2]^2*((-1 + c)*Cos[a + b*x] - d*Sin[a + b*x])) - Sec[(a + b*x)/2]^2*(d*Cos[a + b*x] + (-1 + c)*Sin[a + b*x])*Tan[(a + b*x)/2]))/(d*Cos[a + b*x] + (-1 + c)*Sin[a + b*x]) + (a*Cos[(a + b*x)/2]^2*(-(Sec[(a + b*x)/2]^2*(Cos[a + b*x] + c*Cos[a + b*x] - d*Sin[a + b*x])) - Sec[(a + b*x)/2]^2*(d*Cos[a + b*x] + Sin[a + b*x] + c*Sin[a + b*x])*Tan[(a + b*x)/2]))/(d*Cos[a + b*x] + Sin[a + b*x] + c*Sin[a + b*x]))","B",0
256,0,0,18,0.4413864,"\int \frac{\coth ^{-1}(c+d \cot (a+b x))}{x} \, dx","Integrate[ArcCoth[c + d*Cot[a + b*x]]/x,x]","\int \frac{\coth ^{-1}(c+d \cot (a+b x))}{x} \, dx","\text{Int}\left(\frac{\coth ^{-1}(d \cot (a+b x)+c)}{x},x\right)",0,"Integrate[ArcCoth[c + d*Cot[a + b*x]]/x, x]","A",-1
257,1,155,168,0.2048912,"\int x^2 \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx","Integrate[x^2*ArcCoth[1 + I*d + d*Cot[a + b*x]],x]","\frac{1}{3} x^3 \coth ^{-1}(d \cot (a+b x)+i d+1)-\frac{4 b^3 x^3 \log \left(1+\frac{i e^{-2 i (a+b x)}}{d-i}\right)+6 i b^2 x^2 \text{Li}_2\left(-\frac{i e^{-2 i (a+b x)}}{d-i}\right)+6 b x \text{Li}_3\left(-\frac{i e^{-2 i (a+b x)}}{d-i}\right)-3 i \text{Li}_4\left(-\frac{i e^{-2 i (a+b x)}}{d-i}\right)}{24 b^3}","-\frac{i \text{Li}_4\left((i d+1) e^{2 i a+2 i b x}\right)}{8 b^3}-\frac{x \text{Li}_3\left((i d+1) e^{2 i a+2 i b x}\right)}{4 b^2}+\frac{i x^2 \text{Li}_2\left((i d+1) e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{6} x^3 \log \left(1-(1+i d) e^{2 i a+2 i b x}\right)+\frac{1}{3} x^3 \coth ^{-1}(d \cot (a+b x)+i d+1)+\frac{1}{12} i b x^4",1,"(x^3*ArcCoth[1 + I*d + d*Cot[a + b*x]])/3 - (4*b^3*x^3*Log[1 + I/((-I + d)*E^((2*I)*(a + b*x)))] + (6*I)*b^2*x^2*PolyLog[2, (-I)/((-I + d)*E^((2*I)*(a + b*x)))] + 6*b*x*PolyLog[3, (-I)/((-I + d)*E^((2*I)*(a + b*x)))] - (3*I)*PolyLog[4, (-I)/((-I + d)*E^((2*I)*(a + b*x)))])/(24*b^3)","A",1
258,1,119,132,0.1165894,"\int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx","Integrate[x*ArcCoth[1 + I*d + d*Cot[a + b*x]],x]","\frac{1}{2} x^2 \coth ^{-1}(d \cot (a+b x)+i d+1)-\frac{2 b^2 x^2 \log \left(1+\frac{i e^{-2 i (a+b x)}}{d-i}\right)+2 i b x \text{Li}_2\left(-\frac{i e^{-2 i (a+b x)}}{d-i}\right)+\text{Li}_3\left(-\frac{i e^{-2 i (a+b x)}}{d-i}\right)}{8 b^2}","-\frac{\text{Li}_3\left((i d+1) e^{2 i a+2 i b x}\right)}{8 b^2}+\frac{i x \text{Li}_2\left((i d+1) e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{4} x^2 \log \left(1-(1+i d) e^{2 i a+2 i b x}\right)+\frac{1}{2} x^2 \coth ^{-1}(d \cot (a+b x)+i d+1)+\frac{1}{6} i b x^3",1,"(x^2*ArcCoth[1 + I*d + d*Cot[a + b*x]])/2 - (2*b^2*x^2*Log[1 + I/((-I + d)*E^((2*I)*(a + b*x)))] + (2*I)*b*x*PolyLog[2, (-I)/((-I + d)*E^((2*I)*(a + b*x)))] + PolyLog[3, (-I)/((-I + d)*E^((2*I)*(a + b*x)))])/(8*b^2)","A",1
259,1,709,93,3.7852025,"\int \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx","Integrate[ArcCoth[1 + I*d + d*Cot[a + b*x]],x]","\frac{x \csc ^2(a+b x) (\cos (b x)-i \sin (b x)) (\cos (b x)+i \sin (b x)) \left(i \text{Li}_2\left(\frac{(\cos (a)-i \sin (a)) ((-i d-2) \cos (a)+d \sin (a)) (\tan (b x)+i)}{2 (d-i)}\right)-i \text{Li}_2\left(\frac{1}{2} \sec (b x) ((i d+2) \cos (a)-d \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right)+i \log (1-i \tan (b x)) \log \left(\frac{(\cos (a)-i \sin (a)) \sec (b x) (d \cos (a+b x)+(2+i d) \sin (a+b x))}{2 (d-i)}\right)-i \log (1+i \tan (b x)) \log \left(\frac{1}{2} (\sin (a)-i \cos (a)) \sec (b x) (d \cos (a+b x)+(2+i d) \sin (a+b x))\right)+i \text{Li}_2(i \sin (2 b x)-\cos (2 b x))+2 b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))\right)}{(\cot (a+b x)+i) (d \cot (a+b x)+i d+2) \left(\frac{(d-2 i) \cos (a+b x) (\log (1-i \tan (b x))-\log (1+i \tan (b x)))}{d \cos (a+b x)+(2+i d) \sin (a+b x)}+\frac{d \sin (a+b x) (\log (1-i \tan (b x))-\log (1+i \tan (b x)))}{(d-2 i) \sin (a+b x)-i d \cos (a+b x)}+\log \left(1+\frac{1}{2} \sec (b x) (d \sin (a)+(-2-i d) \cos (a)) (\cos (a+b x)+i \sin (a+b x))\right)-\log \left(\frac{1}{2} (\sin (a)-i \cos (a)) \sec (b x) (d \cos (a+b x)+(2+i d) \sin (a+b x))\right)-i \tan (b x) \log \left(1+\frac{1}{2} \sec (b x) (d \sin (a)+(-2-i d) \cos (a)) (\cos (a+b x)+i \sin (a+b x))\right)+i \tan (b x) \log \left(\frac{1}{2} (\sin (a)-i \cos (a)) \sec (b x) (d \cos (a+b x)+(2+i d) \sin (a+b x))\right)+2 i b x+2 b x \tan (b x)-i \tan (b x) \log (1-i \tan (b x))+i \tan (b x) \log (1+i \tan (b x))\right)}+x \coth ^{-1}(d \cot (a+b x)+i d+1)","\frac{i \text{Li}_2\left((i d+1) e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{2} x \log \left(1-(1+i d) e^{2 i a+2 i b x}\right)+x \coth ^{-1}(d \cot (a+b x)+i d+1)+\frac{1}{2} i b x^2",1,"x*ArcCoth[1 + I*d + d*Cot[a + b*x]] + (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])*(d*Cos[a + b*x] + (2 + I*d)*Sin[a + b*x]))/(2*(-I + d))]*Log[1 - I*Tan[b*x]] - I*Log[(Sec[b*x]*((-I)*Cos[a] + Sin[a])*(d*Cos[a + b*x] + (2 + I*d)*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]*((2 + I*d)*Cos[a] - d*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2] + I*PolyLog[2, ((Cos[a] - I*Sin[a])*((-2 - I*d)*Cos[a] + d*Sin[a])*(I + Tan[b*x]))/(2*(-I + d))])*(Cos[b*x] - I*Sin[b*x])*(Cos[b*x] + I*Sin[b*x]))/((I + Cot[a + b*x])*(2 + I*d + d*Cot[a + b*x])*((2*I)*b*x + Log[1 + (Sec[b*x]*((-2 - I*d)*Cos[a] + d*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2] - Log[(Sec[b*x]*((-I)*Cos[a] + Sin[a])*(d*Cos[a + b*x] + (2 + I*d)*Sin[a + b*x]))/2] + ((-2*I + d)*Cos[a + b*x]*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]]))/(d*Cos[a + b*x] + (2 + I*d)*Sin[a + b*x]) + (d*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]])*Sin[a + b*x])/((-I)*d*Cos[a + b*x] + (-2*I + d)*Sin[a + b*x]) + 2*b*x*Tan[b*x] - I*Log[1 + (Sec[b*x]*((-2 - I*d)*Cos[a] + d*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2]*Tan[b*x] + I*Log[(Sec[b*x]*((-I)*Cos[a] + Sin[a])*(d*Cos[a + b*x] + (2 + I*d)*Sin[a + b*x]))/2]*Tan[b*x] - I*Log[1 - I*Tan[b*x]]*Tan[b*x] + I*Log[1 + I*Tan[b*x]]*Tan[b*x]))","B",0
260,0,0,23,0.782663,"\int \frac{\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx","Integrate[ArcCoth[1 + I*d + d*Cot[a + b*x]]/x,x]","\int \frac{\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx","\text{Int}\left(\frac{\coth ^{-1}(d \cot (a+b x)+i d+1)}{x},x\right)",0,"Integrate[ArcCoth[1 + I*d + d*Cot[a + b*x]]/x, x]","A",-1
261,1,155,169,0.2133273,"\int x^2 \coth ^{-1}(1-i d-d \cot (a+b x)) \, dx","Integrate[x^2*ArcCoth[1 - I*d - d*Cot[a + b*x]],x]","\frac{1}{3} x^3 \coth ^{-1}(d (-\cot (a+b x))-i d+1)-\frac{4 b^3 x^3 \log \left(1+\frac{e^{-2 i (a+b x)}}{-1+i d}\right)+6 i b^2 x^2 \text{Li}_2\left(\frac{i e^{-2 i (a+b x)}}{d+i}\right)+6 b x \text{Li}_3\left(\frac{i e^{-2 i (a+b x)}}{d+i}\right)-3 i \text{Li}_4\left(\frac{i e^{-2 i (a+b x)}}{d+i}\right)}{24 b^3}","-\frac{i \text{Li}_4\left((1-i d) e^{2 i a+2 i b x}\right)}{8 b^3}-\frac{x \text{Li}_3\left((1-i d) e^{2 i a+2 i b x}\right)}{4 b^2}+\frac{i x^2 \text{Li}_2\left((1-i d) e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{6} x^3 \log \left(1-(1-i d) e^{2 i a+2 i b x}\right)+\frac{1}{3} x^3 \coth ^{-1}(d (-\cot (a+b x))-i d+1)+\frac{1}{12} i b x^4",1,"(x^3*ArcCoth[1 - I*d - d*Cot[a + b*x]])/3 - (4*b^3*x^3*Log[1 + 1/((-1 + I*d)*E^((2*I)*(a + b*x)))] + (6*I)*b^2*x^2*PolyLog[2, I/((I + d)*E^((2*I)*(a + b*x)))] + 6*b*x*PolyLog[3, I/((I + d)*E^((2*I)*(a + b*x)))] - (3*I)*PolyLog[4, I/((I + d)*E^((2*I)*(a + b*x)))])/(24*b^3)","A",1
262,1,119,133,0.1099997,"\int x \coth ^{-1}(1-i d-d \cot (a+b x)) \, dx","Integrate[x*ArcCoth[1 - I*d - d*Cot[a + b*x]],x]","\frac{1}{2} x^2 \coth ^{-1}(d (-\cot (a+b x))-i d+1)-\frac{2 b^2 x^2 \log \left(1+\frac{e^{-2 i (a+b x)}}{-1+i d}\right)+2 i b x \text{Li}_2\left(\frac{i e^{-2 i (a+b x)}}{d+i}\right)+\text{Li}_3\left(\frac{i e^{-2 i (a+b x)}}{d+i}\right)}{8 b^2}","-\frac{\text{Li}_3\left((1-i d) e^{2 i a+2 i b x}\right)}{8 b^2}+\frac{i x \text{Li}_2\left((1-i d) e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{4} x^2 \log \left(1-(1-i d) e^{2 i a+2 i b x}\right)+\frac{1}{2} x^2 \coth ^{-1}(d (-\cot (a+b x))-i d+1)+\frac{1}{6} i b x^3",1,"(x^2*ArcCoth[1 - I*d - d*Cot[a + b*x]])/2 - (2*b^2*x^2*Log[1 + 1/((-1 + I*d)*E^((2*I)*(a + b*x)))] + (2*I)*b*x*PolyLog[2, I/((I + d)*E^((2*I)*(a + b*x)))] + PolyLog[3, I/((I + d)*E^((2*I)*(a + b*x)))])/(8*b^2)","A",1
263,1,605,94,2.9423307,"\int \coth ^{-1}(1-i d-d \cot (a+b x)) \, dx","Integrate[ArcCoth[1 - I*d - d*Cot[a + b*x]],x]","\frac{x \csc ^2(a+b x) (\cos (b x)-i \sin (b x)) (\cos (b x)+i \sin (b x)) \left(i \text{Li}_2\left(\frac{(\cos (a)-i \sin (a)) ((2-i d) \cos (a)+d \sin (a)) (\tan (b x)+i)}{2 (d+i)}\right)-i \text{Li}_2\left(\frac{1}{2} \sec (b x) ((2-i d) \cos (a)+d \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right)+i \log (1-i \tan (b x)) \log \left(\frac{(\cos (a)-i \sin (a)) \sec (b x) (d \cos (a+b x)+i (d+2 i) \sin (a+b x))}{2 (d+i)}\right)-i \log (1+i \tan (b x)) \log \left(\frac{i \sec (b x) (d \cos (a+b x)+i (d+2 i) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\right)+i \text{Li}_2(i \sin (2 b x)-\cos (2 b x))+2 b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))\right)}{(\cot (a+b x)+i) (d \cot (a+b x)+i d-2) \left(\frac{\sec ^2(b x) \log \left(\frac{i \sec (b x) (d \cos (a+b x)+i (d+2 i) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\right)}{1+i \tan (b x)}-\frac{\sec (b x) (i d \sin (a)+(d+2 i) \cos (a)) \log (1-i \tan (b x))}{d \cos (a+b x)+i (d+2 i) \sin (a+b x)}+\frac{\sec (b x) (i d \sin (a)+(d+2 i) \cos (a)) \log (1+i \tan (b x))}{d \cos (a+b x)+i (d+2 i) \sin (a+b x)}+i (\tan (b x)+i) \log \left(1-\frac{1}{2} \sec (b x) (d \sin (a)+(2-i d) \cos (a)) (\cos (a+b x)+i \sin (a+b x))\right)-2 b x (\tan (b x)+i)\right)}+x \coth ^{-1}(d (-\cot (a+b x))-i d+1)","\frac{i \text{Li}_2\left((1-i d) e^{2 i a+2 i b x}\right)}{4 b}-\frac{1}{2} x \log \left(1-(1-i d) e^{2 i a+2 i b x}\right)+x \coth ^{-1}(d (-\cot (a+b x))-i d+1)+\frac{1}{2} i b x^2",1,"x*ArcCoth[1 - I*d - d*Cot[a + b*x]] + (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])*(d*Cos[a + b*x] + I*(2*I + d)*Sin[a + b*x]))/(2*(I + d))]*Log[1 - I*Tan[b*x]] - I*Log[(I*Sec[b*x]*(d*Cos[a + b*x] + I*(2*I + d)*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]*((2 - I*d)*Cos[a] + d*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2] + I*PolyLog[2, ((Cos[a] - I*Sin[a])*((2 - I*d)*Cos[a] + d*Sin[a])*(I + Tan[b*x]))/(2*(I + d))])*(Cos[b*x] - I*Sin[b*x])*(Cos[b*x] + I*Sin[b*x]))/((I + Cot[a + b*x])*(-2 + I*d + d*Cot[a + b*x])*(-((Log[1 - I*Tan[b*x]]*Sec[b*x]*((2*I + d)*Cos[a] + I*d*Sin[a]))/(d*Cos[a + b*x] + I*(2*I + d)*Sin[a + b*x])) + (Log[1 + I*Tan[b*x]]*Sec[b*x]*((2*I + d)*Cos[a] + I*d*Sin[a]))/(d*Cos[a + b*x] + I*(2*I + d)*Sin[a + b*x]) + (Log[(I*Sec[b*x]*(d*Cos[a + b*x] + I*(2*I + d)*Sin[a + b*x]))/(2*Cos[a] - (2*I)*Sin[a])]*Sec[b*x]^2)/(1 + I*Tan[b*x]) - 2*b*x*(I + Tan[b*x]) + I*Log[1 - (Sec[b*x]*((2 - I*d)*Cos[a] + d*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2]*(I + Tan[b*x])))","B",0
264,0,0,24,0.8285965,"\int \frac{\coth ^{-1}(1-i d-d \cot (a+b x))}{x} \, dx","Integrate[ArcCoth[1 - I*d - d*Cot[a + b*x]]/x,x]","\int \frac{\coth ^{-1}(1-i d-d \cot (a+b x))}{x} \, dx","\text{Int}\left(\frac{\coth ^{-1}(d (-\cot (a+b x))-i d+1)}{x},x\right)",0,"Integrate[ArcCoth[1 - I*d - d*Cot[a + b*x]]/x, x]","A",-1
265,1,131,160,0.302262,"\int \frac{\left(a+b \coth ^{-1}\left(c x^n\right)\right) \left(d+e \log \left(f x^m\right)\right)}{x} \, dx","Integrate[((a + b*ArcCoth[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 \tanh ^{-1}\left(c x^n\right)+b \coth ^{-1}\left(c x^n\right)\right)","a d \log (x)+\frac{a e \log ^2\left(f x^m\right)}{2 m}+\frac{b d \text{Li}_2\left(-\frac{x^{-n}}{c}\right)}{2 n}-\frac{b d \text{Li}_2\left(\frac{x^{-n}}{c}\right)}{2 n}+\frac{b e \text{Li}_2\left(-\frac{x^{-n}}{c}\right) \log \left(f x^m\right)}{2 n}-\frac{b e \text{Li}_2\left(\frac{x^{-n}}{c}\right) \log \left(f x^m\right)}{2 n}+\frac{b e m \text{Li}_3\left(-\frac{x^{-n}}{c}\right)}{2 n^2}-\frac{b e m \text{Li}_3\left(\frac{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*ArcCoth[c*x^n] - b*ArcTanh[c*x^n])*Log[x]*(e*m*Log[x] - 2*(d + e*Log[f*x^m])))/2","C",1
266,1,236,297,0.1747748,"\int x^5 \left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right) \, dx","Integrate[x^5*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]","\frac{20 e \log \left(1-c^2 x^2\right) \left(15 a c^6 x^6+15 b \left(c^6 x^6-1\right) \coth ^{-1}(c x)+b c x \left(3 c^4 x^4+5 c^2 x^2+15\right)\right)+15 \log (1-c x) (-20 a e+10 b d-49 b e)-15 \log (c x+1) (20 a e+10 b d-49 b e)+100 a c^6 x^6 (3 d-e)-150 a c^4 e x^4-300 a c^2 e x^2+4 b c^5 x^5 (15 d-11 e)+10 b c^3 x^3 (10 d-19 e)-50 b c^2 x^2 \coth ^{-1}(c x) \left(e \left(2 c^4 x^4+3 c^2 x^2+6\right)-6 c^4 d x^4\right)+30 b c x (10 d-49 e)}{1800 c^6}","-\frac{e x^2 \left(a+b \coth ^{-1}(c x)\right)}{6 c^4}+\frac{1}{6} x^6 \left(a+b \coth ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{e x^4 \left(a+b \coth ^{-1}(c x)\right)}{12 c^2}-\frac{e \log \left(1-c^2 x^2\right) \left(a+b \coth ^{-1}(c x)\right)}{6 c^6}-\frac{1}{18} e x^6 \left(a+b \coth ^{-1}(c x)\right)-\frac{b (3 d-e) \tanh ^{-1}(c x)}{18 c^6}+\frac{137 b e \tanh ^{-1}(c x)}{180 c^6}+\frac{b x (3 d-e)}{18 c^5}-\frac{137 b e x}{180 c^5}+\frac{b x^3 (3 d-e)}{54 c^3}-\frac{47 b e x^3}{540 c^3}+\frac{b e x^5 \log \left(1-c^2 x^2\right)}{30 c}+\frac{b e x \log \left(1-c^2 x^2\right)}{6 c^5}+\frac{b e x^3 \log \left(1-c^2 x^2\right)}{18 c^3}+\frac{b x^5 (3 d-e)}{90 c}-\frac{b e x^5}{75 c}",1,"(30*b*c*(10*d - 49*e)*x - 300*a*c^2*e*x^2 + 10*b*c^3*(10*d - 19*e)*x^3 - 150*a*c^4*e*x^4 + 4*b*c^5*(15*d - 11*e)*x^5 + 100*a*c^6*(3*d - e)*x^6 - 50*b*c^2*x^2*(-6*c^4*d*x^4 + e*(6 + 3*c^2*x^2 + 2*c^4*x^4))*ArcCoth[c*x] + 15*(10*b*d - 20*a*e - 49*b*e)*Log[1 - c*x] - 15*(10*b*d + 20*a*e - 49*b*e)*Log[1 + c*x] + 20*e*(15*a*c^6*x^6 + b*c*x*(15 + 5*c^2*x^2 + 3*c^4*x^4) + 15*b*(-1 + c^6*x^6)*ArcCoth[c*x])*Log[1 - c^2*x^2])/(1800*c^6)","A",1
267,1,192,225,0.1513969,"\int x^3 \left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right) \, dx","Integrate[x^3*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]","\frac{12 e \log \left(1-c^2 x^2\right) \left(3 a c^4 x^4+3 b \left(c^4 x^4-1\right) \coth ^{-1}(c x)+b c x \left(c^2 x^2+3\right)\right)+3 \log (1-c x) (-12 a e+6 b d-25 b e)-3 \log (c x+1) (12 a e+6 b d-25 b e)+18 a c^4 x^4 (2 d-e)-36 a c^2 e x^2+2 b c^3 x^3 (6 d-7 e)-18 b c^2 x^2 \coth ^{-1}(c x) \left(e \left(c^2 x^2+2\right)-2 c^2 d x^2\right)+6 b c x (6 d-25 e)}{144 c^4}","\frac{1}{4} x^4 \left(a+b \coth ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{e x^2 \left(a+b \coth ^{-1}(c x)\right)}{4 c^2}-\frac{e \log \left(1-c^2 x^2\right) \left(a+b \coth ^{-1}(c x)\right)}{4 c^4}-\frac{1}{8} e x^4 \left(a+b \coth ^{-1}(c x)\right)-\frac{b (2 d-3 e) \tanh ^{-1}(c x)}{8 c^4}+\frac{2 b e \tanh ^{-1}(c x)}{3 c^4}+\frac{b x (2 d-3 e)}{8 c^3}-\frac{2 b e x}{3 c^3}+\frac{b e x^3 \log \left(1-c^2 x^2\right)}{12 c}+\frac{b e x \log \left(1-c^2 x^2\right)}{4 c^3}+\frac{b x^3 (2 d-e)}{24 c}-\frac{b e x^3}{18 c}",1,"(6*b*c*(6*d - 25*e)*x - 36*a*c^2*e*x^2 + 2*b*c^3*(6*d - 7*e)*x^3 + 18*a*c^4*(2*d - e)*x^4 - 18*b*c^2*x^2*(-2*c^2*d*x^2 + e*(2 + c^2*x^2))*ArcCoth[c*x] + 3*(6*b*d - 12*a*e - 25*b*e)*Log[1 - c*x] - 3*(6*b*d + 12*a*e - 25*b*e)*Log[1 + c*x] + 12*e*(3*a*c^4*x^4 + b*c*x*(3 + c^2*x^2) + 3*b*(-1 + c^4*x^4)*ArcCoth[c*x])*Log[1 - c^2*x^2])/(144*c^4)","A",1
268,1,129,140,0.1102369,"\int x \left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right) \, dx","Integrate[x*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]","\frac{2 e \log \left(1-c^2 x^2\right) \left(c x (a c x+b)+b \left(c^2 x^2-1\right) \coth ^{-1}(c x)\right)+\log (1-c x) (b (d-3 e)-2 a e)-\log (c x+1) (2 a e+b (d-3 e))+2 a c^2 x^2 (d-e)+2 b c^2 x^2 (d-e) \coth ^{-1}(c x)+2 b c x (d-3 e)}{4 c^2}","-\frac{e \left(1-c^2 x^2\right) \log \left(1-c^2 x^2\right) \left(a+b \coth ^{-1}(c x)\right)}{2 c^2}+\frac{1}{2} d x^2 \left(a+b \coth ^{-1}(c x)\right)-\frac{1}{2} e x^2 \left(a+b \coth ^{-1}(c x)\right)-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{b e x \log \left(1-c^2 x^2\right)}{2 c}+\frac{b e \tanh ^{-1}(c x)}{c^2}+\frac{b x (d-e)}{2 c}-\frac{b e x}{c}",1,"(2*b*c*(d - 3*e)*x + 2*a*c^2*(d - e)*x^2 + 2*b*c^2*(d - e)*x^2*ArcCoth[c*x] + (b*(d - 3*e) - 2*a*e)*Log[1 - c*x] - (b*(d - 3*e) + 2*a*e)*Log[1 + c*x] + 2*e*(c*x*(b + a*c*x) + b*(-1 + c^2*x^2)*ArcCoth[c*x])*Log[1 - c^2*x^2])/(4*c^2)","A",1
269,0,0,381,0.2229374,"\int \frac{\left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x} \, dx","Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/x,x]","\int \frac{\left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x} \, dx","-\frac{1}{2} a e \text{Li}_2\left(c^2 x^2\right)+a d \log (x)+\frac{1}{2} b e \text{Li}_2\left(-\frac{1}{c x}\right) \log \left(-c^2 x^2\right)-\frac{1}{2} b e \text{Li}_2\left(-\frac{1}{c x}\right) \left(\log \left(-c^2 x^2\right)-\log \left(1-c^2 x^2\right)+\log \left(1-\frac{1}{c x}\right)+\log \left(\frac{1}{c x}+1\right)\right)-\frac{1}{2} b e \text{Li}_2\left(\frac{1}{c x}\right) \log \left(-c^2 x^2\right)+\frac{1}{2} b e \text{Li}_2\left(\frac{1}{c x}\right) \left(\log \left(-c^2 x^2\right)-\log \left(1-c^2 x^2\right)+\log \left(1-\frac{1}{c x}\right)+\log \left(\frac{1}{c x}+1\right)\right)+\frac{1}{2} b d \text{Li}_2\left(-\frac{1}{c x}\right)-\frac{1}{2} b d \text{Li}_2\left(\frac{1}{c x}\right)+b e \text{Li}_3\left(\frac{c+\frac{1}{x}}{c}\right)-b e \text{Li}_3\left(1-\frac{1}{c x}\right)+b e \text{Li}_3\left(-\frac{1}{c x}\right)-b e \text{Li}_3\left(\frac{1}{c x}\right)+b e \text{Li}_2\left(1-\frac{1}{c x}\right) \log \left(1-\frac{1}{c x}\right)-b e \text{Li}_2\left(\frac{c+\frac{1}{x}}{c}\right) \log \left(\frac{c+\frac{1}{x}}{c}\right)+\frac{1}{2} b e \log \left(\frac{1}{c x}\right) \log ^2\left(1-\frac{1}{c x}\right)-\frac{1}{2} b e \log ^2\left(\frac{1}{c x}+1\right) \log \left(-\frac{1}{c x}\right)",1,"Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/x, x]","F",-1
270,1,161,247,0.1588869,"\int \frac{\left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x^3} \, dx","Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/x^3,x]","\frac{1}{2} \left(-\frac{e \log \left(1-c^2 x^2\right) \left(a+\left(b-b c^2 x^2\right) \coth ^{-1}(c x)+b c x\right)}{x^2}+c^2 e (a+b) \log (1-c x)+c^2 e (a-b) \log (c x+1)-2 a c^2 e \log (x)-\frac{a d}{x^2}-b c^2 e \left(\text{Li}_2\left(-\frac{1}{c x}\right)-\text{Li}_2\left(\frac{1}{c x}\right)\right)-\frac{b d \left(c x (c x \log (1-c x)-c x \log (c x+1)+2)+2 \coth ^{-1}(c x)\right)}{2 x^2}\right)","-\frac{\left(a+b \coth ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{2 x^2}+\frac{1}{2} c^2 e (a+b) \log (1-c x)+\frac{1}{2} c^2 e (a-b) \log (c x+1)-a c^2 e \log (x)-\frac{b c \left(e \log \left(1-c^2 x^2\right)+d\right)}{2 x}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left(e \log \left(1-c^2 x^2\right)+d\right)+\frac{1}{2} b c^2 e \text{Li}_2\left(1-\frac{2}{1-c x}\right)+\frac{1}{2} b c^2 e \text{Li}_2\left(\frac{2}{c x+1}-1\right)-\frac{1}{2} b c^2 e \tanh ^{-1}(c x)^2-\frac{1}{2} b c^2 e \coth ^{-1}(c x)^2+b c^2 e \log \left(\frac{2}{1-c x}\right) \tanh ^{-1}(c x)-b c^2 e \log \left(2-\frac{2}{c x+1}\right) \coth ^{-1}(c x)",1,"(-((a*d)/x^2) - 2*a*c^2*e*Log[x] + (a + b)*c^2*e*Log[1 - c*x] + (a - b)*c^2*e*Log[1 + c*x] - (b*d*(2*ArcCoth[c*x] + c*x*(2 + c*x*Log[1 - c*x] - c*x*Log[1 + c*x])))/(2*x^2) - (e*(a + b*c*x + (b - b*c^2*x^2)*ArcCoth[c*x])*Log[1 - c^2*x^2])/x^2 - b*c^2*e*(PolyLog[2, -(1/(c*x))] - PolyLog[2, 1/(c*x)]))/2","A",0
271,1,307,339,0.1529599,"\int \frac{\left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x^5} \, dx","Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/x^5,x]","\frac{1}{12} \log (1-c x) \left(3 a c^4 e+4 b c^4 e\right)+\frac{1}{12} \log (c x+1) \left(3 a c^4 e-4 b c^4 e\right)+\frac{e \log \left(1-c^2 x^2\right) \left(-3 a+3 b c^4 x^4 \coth ^{-1}(c x)-3 b c^3 x^3-b c x-3 b \coth ^{-1}(c x)\right)}{12 x^4}-\frac{1}{2} a c^4 e \log (x)+\frac{a c^2 e}{4 x^2}-\frac{a d}{4 x^4}-\frac{1}{4} b c^4 e \left(\text{Li}_2\left(-\frac{1}{c x}\right)-\text{Li}_2\left(\frac{1}{c x}\right)\right)+\frac{b c^3 e}{6 x}+b c^4 d \left(\frac{1}{4} \left(-\frac{1}{3 c^3 x^3}-\frac{1}{c x}-\frac{1}{2} \log (1-c x)+\frac{1}{2} \log (c x+1)\right)-\frac{\coth ^{-1}(c x)}{4 c^4 x^4}\right)-\frac{1}{2} b c^4 e \left(\frac{1}{2} \left(-\frac{1}{c x}-\frac{1}{2} \log (1-c x)+\frac{1}{2} \log (c x+1)\right)-\frac{\coth ^{-1}(c x)}{2 c^2 x^2}\right)","\frac{1}{12} c^4 e (3 a+4 b) \log (1-c x)+\frac{1}{12} c^4 e (3 a-4 b) \log (c x+1)-\frac{\left(a+b \coth ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{4 x^4}-\frac{1}{2} a c^4 e \log (x)+\frac{a c^2 e}{4 x^2}+\frac{1}{4} b c^4 e \text{Li}_2\left(1-\frac{2}{1-c x}\right)+\frac{1}{4} b c^4 e \text{Li}_2\left(\frac{2}{c x+1}-1\right)-\frac{1}{4} b c^4 e \tanh ^{-1}(c x)^2-\frac{1}{4} b c^4 e \tanh ^{-1}(c x)-\frac{1}{4} b c^4 e \coth ^{-1}(c x)^2+\frac{1}{2} b c^4 e \log \left(\frac{2}{1-c x}\right) \tanh ^{-1}(c x)-\frac{1}{2} b c^4 e \log \left(2-\frac{2}{c x+1}\right) \coth ^{-1}(c x)+\frac{5 b c^3 e}{12 x}-\frac{b c \left(e \log \left(1-c^2 x^2\right)+d\right)}{12 x^3}+\frac{b c^2 e \coth ^{-1}(c x)}{4 x^2}+\frac{1}{4} b c^4 \tanh ^{-1}(c x) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{b c^3 \left(e \log \left(1-c^2 x^2\right)+d\right)}{4 x}",1,"-1/4*(a*d)/x^4 + (a*c^2*e)/(4*x^2) + (b*c^3*e)/(6*x) - (a*c^4*e*Log[x])/2 + ((3*a*c^4*e + 4*b*c^4*e)*Log[1 - c*x])/12 - (b*c^4*e*(-1/2*ArcCoth[c*x]/(c^2*x^2) + (-(1/(c*x)) - Log[1 - c*x]/2 + Log[1 + c*x]/2)/2))/2 + b*c^4*d*(-1/4*ArcCoth[c*x]/(c^4*x^4) + (-1/3*1/(c^3*x^3) - 1/(c*x) - Log[1 - c*x]/2 + Log[1 + c*x]/2)/4) + ((3*a*c^4*e - 4*b*c^4*e)*Log[1 + c*x])/12 + (e*(-3*a - b*c*x - 3*b*c^3*x^3 - 3*b*ArcCoth[c*x] + 3*b*c^4*x^4*ArcCoth[c*x])*Log[1 - c^2*x^2])/(12*x^4) - (b*c^4*e*(PolyLog[2, -(1/(c*x))] - PolyLog[2, 1/(c*x)]))/4","A",0
272,1,236,315,0.1590822,"\int x^4 \left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right) \, dx","Integrate[x^4*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]","\frac{30 c^2 e x^2 \log \left(1-c^2 x^2\right) \left(4 a c^3 x^3+4 b c^3 x^3 \coth ^{-1}(c x)+b \left(c^2 x^2+2\right)\right)+2 \log (1-c x) (-60 a e+30 b d-137 b e)+2 \log (c x+1) (60 a e+30 b d-137 b e)+24 a c^5 x^5 (5 d-2 e)-80 a c^3 e x^3-240 a c e x+3 b c^4 x^4 (10 d-9 e)+2 b c^2 x^2 (30 d-77 e)+30 b e \log ^2\left(1-c^2 x^2\right)-8 b c x \coth ^{-1}(c x) \left(2 e \left(3 c^4 x^4+5 c^2 x^2+15\right)-15 c^4 d x^4\right)+120 b e \coth ^{-1}(c x)^2}{600 c^5}","-\frac{e (4 a+3 b) \log (1-c x)}{20 c^5}+\frac{e (4 a-3 b) \log (c x+1)}{20 c^5}+\frac{1}{5} x^5 \left(a+b \coth ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{2 a e x}{5 c^4}-\frac{2 a e x^3}{15 c^2}-\frac{2}{25} a e x^5+\frac{b e \coth ^{-1}(c x)^2}{5 c^5}-\frac{2 b e x \coth ^{-1}(c x)}{5 c^4}-\frac{77 b e x^2}{300 c^3}+\frac{b x^4 \left(e \log \left(1-c^2 x^2\right)+d\right)}{20 c}-\frac{2 b e x^3 \coth ^{-1}(c x)}{15 c^2}+\frac{b \log \left(1-c^2 x^2\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{10 c^5}-\frac{b e \log ^2\left(1-c^2 x^2\right)}{20 c^5}-\frac{23 b e \log \left(1-c^2 x^2\right)}{75 c^5}+\frac{b x^2 \left(e \log \left(1-c^2 x^2\right)+d\right)}{10 c^3}-\frac{2}{25} b e x^5 \coth ^{-1}(c x)-\frac{9 b e x^4}{200 c}",1,"(-240*a*c*e*x + 2*b*c^2*(30*d - 77*e)*x^2 - 80*a*c^3*e*x^3 + 3*b*c^4*(10*d - 9*e)*x^4 + 24*a*c^5*(5*d - 2*e)*x^5 - 8*b*c*x*(-15*c^4*d*x^4 + 2*e*(15 + 5*c^2*x^2 + 3*c^4*x^4))*ArcCoth[c*x] + 120*b*e*ArcCoth[c*x]^2 + 2*(30*b*d - 60*a*e - 137*b*e)*Log[1 - c*x] + 2*(30*b*d + 60*a*e - 137*b*e)*Log[1 + c*x] + 30*c^2*e*x^2*(4*a*c^3*x^3 + b*(2 + c^2*x^2) + 4*b*c^3*x^3*ArcCoth[c*x])*Log[1 - c^2*x^2] + 30*b*e*Log[1 - c^2*x^2]^2)/(600*c^5)","A",1
273,1,183,247,0.1281789,"\int x^2 \left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right) \, dx","Integrate[x^2*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]","\frac{6 c^2 e x^2 \log \left(1-c^2 x^2\right) \left(2 a c x+2 b c x \coth ^{-1}(c x)+b\right)+2 \log (1-c x) (-6 a e+3 b d-11 b e)+2 \log (c x+1) (6 a e+3 b d-11 b e)+4 a c^3 x^3 (3 d-2 e)-24 a c e x+2 b c^2 x^2 (3 d-5 e)+4 b c x \coth ^{-1}(c x) \left(3 c^2 d x^2-2 e \left(c^2 x^2+3\right)\right)+3 b e \log ^2\left(1-c^2 x^2\right)+12 b e \coth ^{-1}(c x)^2}{36 c^3}","-\frac{e (2 a+b) \log (1-c x)}{6 c^3}+\frac{e (2 a-b) \log (c x+1)}{6 c^3}+\frac{1}{3} x^3 \left(a+b \coth ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{2 a e x}{3 c^2}-\frac{2}{9} a e x^3+\frac{b e \coth ^{-1}(c x)^2}{3 c^3}+\frac{b x^2 \left(e \log \left(1-c^2 x^2\right)+d\right)}{6 c}-\frac{2 b e x \coth ^{-1}(c x)}{3 c^2}+\frac{b \log \left(1-c^2 x^2\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{6 c^3}-\frac{b e \log ^2\left(1-c^2 x^2\right)}{12 c^3}-\frac{4 b e \log \left(1-c^2 x^2\right)}{9 c^3}-\frac{2}{9} b e x^3 \coth ^{-1}(c x)-\frac{5 b e x^2}{18 c}",1,"(-24*a*c*e*x + 2*b*c^2*(3*d - 5*e)*x^2 + 4*a*c^3*(3*d - 2*e)*x^3 + 4*b*c*x*(3*c^2*d*x^2 - 2*e*(3 + c^2*x^2))*ArcCoth[c*x] + 12*b*e*ArcCoth[c*x]^2 + 2*(3*b*d - 6*a*e - 11*b*e)*Log[1 - c*x] + 2*(3*b*d + 6*a*e - 11*b*e)*Log[1 + c*x] + 6*c^2*e*x^2*(b + 2*a*c*x + 2*b*c*x*ArcCoth[c*x])*Log[1 - c^2*x^2] + 3*b*e*Log[1 - c^2*x^2]^2)/(36*c^3)","A",1
274,1,144,104,0.0189436,"\int \left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right) \, dx","Integrate[(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]","a e x \log \left(1-c^2 x^2\right)+\frac{2 a e \tanh ^{-1}(c x)}{c}+a d x-2 a e x+\frac{b d \log \left(1-c^2 x^2\right)}{2 c}+\frac{b e \log ^2\left(1-c^2 x^2\right)}{4 c}-\frac{b e \log \left(1-c^2 x^2\right)}{c}+b e x \log \left(1-c^2 x^2\right) \coth ^{-1}(c x)+b d x \coth ^{-1}(c x)+\frac{b e \coth ^{-1}(c x)^2}{c}-2 b e x \coth ^{-1}(c x)","x \left(a+b \coth ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)+\frac{e \left(a+b \coth ^{-1}(c x)\right)^2}{b c}-2 a e x+\frac{b \left(e \log \left(1-c^2 x^2\right)+d\right)^2}{4 c e}-\frac{b e \log \left(1-c^2 x^2\right)}{c}-2 b e x \coth ^{-1}(c x)",1,"a*d*x - 2*a*e*x + b*d*x*ArcCoth[c*x] - 2*b*e*x*ArcCoth[c*x] + (b*e*ArcCoth[c*x]^2)/c + (2*a*e*ArcTanh[c*x])/c + (b*d*Log[1 - c^2*x^2])/(2*c) - (b*e*Log[1 - c^2*x^2])/c + a*e*x*Log[1 - c^2*x^2] + b*e*x*ArcCoth[c*x]*Log[1 - c^2*x^2] + (b*e*Log[1 - c^2*x^2]^2)/(4*c)","A",1
275,1,332,105,0.205714,"\int \frac{\left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x^2} \, dx","Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/x^2,x]","-\frac{4 a e \log \left(1-c^2 x^2\right)+8 a c e x \tanh ^{-1}(c x)+4 a d+2 b c d x \log \left(1-c^2 x^2\right)-4 b c e x \log (x) \log \left(1-c^2 x^2\right)+2 b c e x \log \left(x-\frac{1}{c}\right) \log \left(1-c^2 x^2\right)+2 b c e x \log \left(\frac{1}{c}+x\right) \log \left(1-c^2 x^2\right)+4 b e \log \left(1-c^2 x^2\right) \coth ^{-1}(c x)-4 b c d x \log (x)+4 b d \coth ^{-1}(c x)+4 b c e x \text{Li}_2(-c x)+4 b c e x \text{Li}_2(c x)-2 b c e x \text{Li}_2\left(\frac{1}{2}-\frac{c x}{2}\right)-2 b c e x \text{Li}_2\left(\frac{1}{2} (c x+1)\right)-b c e x \log ^2\left(x-\frac{1}{c}\right)-b c e x \log ^2\left(\frac{1}{c}+x\right)-2 b c e x \log \left(\frac{1}{c}+x\right) \log \left(\frac{1}{2} (1-c x)\right)+4 b c e x \log (x) \log (1-c x)-2 b c e x \log \left(x-\frac{1}{c}\right) \log \left(\frac{1}{2} (c x+1)\right)+4 b c e x \log (x) \log (c x+1)+4 b c e x \coth ^{-1}(c x)^2}{4 x}","-\frac{\left(a+b \coth ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{x}-\frac{c e \left(a+b \coth ^{-1}(c x)\right)^2}{b}+\frac{1}{2} b c \log \left(1-\frac{1}{1-c^2 x^2}\right) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{1}{2} b c e \text{Li}_2\left(\frac{1}{1-c^2 x^2}\right)",1,"-1/4*(4*a*d + 4*b*d*ArcCoth[c*x] + 4*b*c*e*x*ArcCoth[c*x]^2 + 8*a*c*e*x*ArcTanh[c*x] - 4*b*c*d*x*Log[x] - b*c*e*x*Log[-c^(-1) + x]^2 - b*c*e*x*Log[c^(-1) + x]^2 - 2*b*c*e*x*Log[c^(-1) + x]*Log[(1 - c*x)/2] + 4*b*c*e*x*Log[x]*Log[1 - c*x] - 2*b*c*e*x*Log[-c^(-1) + x]*Log[(1 + c*x)/2] + 4*b*c*e*x*Log[x]*Log[1 + c*x] + 4*a*e*Log[1 - c^2*x^2] + 2*b*c*d*x*Log[1 - c^2*x^2] + 4*b*e*ArcCoth[c*x]*Log[1 - c^2*x^2] - 4*b*c*e*x*Log[x]*Log[1 - c^2*x^2] + 2*b*c*e*x*Log[-c^(-1) + x]*Log[1 - c^2*x^2] + 2*b*c*e*x*Log[c^(-1) + x]*Log[1 - c^2*x^2] + 4*b*c*e*x*PolyLog[2, -(c*x)] + 4*b*c*e*x*PolyLog[2, c*x] - 2*b*c*e*x*PolyLog[2, 1/2 - (c*x)/2] - 2*b*c*e*x*PolyLog[2, (1 + c*x)/2])/x","B",1
276,1,457,197,0.3667515,"\int \frac{\left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x^4} \, dx","Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/x^4,x]","\frac{1}{6} \left(-4 a c^3 e \tanh ^{-1}(c x)-\frac{2 a e \log \left(1-c^2 x^2\right)}{x^3}+\frac{4 a c^2 e}{x}-\frac{2 a d}{x^3}+2 b c^3 d \log (x)-2 b c^3 e \text{Li}_2(-c x)-2 b c^3 e \text{Li}_2(c x)+b c^3 e \text{Li}_2\left(\frac{1}{2}-\frac{c x}{2}\right)+b c^3 e \text{Li}_2\left(\frac{1}{2} (c x+1)\right)+\frac{1}{2} b c^3 e \log ^2\left(x-\frac{1}{c}\right)+\frac{1}{2} b c^3 e \log ^2\left(\frac{1}{c}+x\right)-2 b c^3 e \log (x)+b c^3 e \log \left(\frac{1}{c}+x\right) \log \left(\frac{1}{2} (1-c x)\right)-2 b c^3 e \log (x) \log (1-c x)+b c^3 e \log \left(x-\frac{1}{c}\right) \log \left(\frac{1}{2} (c x+1)\right)-2 b c^3 e \log (x) \log (c x+1)-2 b c^3 e \coth ^{-1}(c x)^2-\frac{b c e \log \left(1-c^2 x^2\right)}{x^2}-\frac{2 b e \log \left(1-c^2 x^2\right) \coth ^{-1}(c x)}{x^3}+\frac{4 b c^2 e \coth ^{-1}(c x)}{x}-b c^3 d \log \left(1-c^2 x^2\right)-4 b c^3 e \log \left(\frac{1}{\sqrt{1-\frac{1}{c^2 x^2}}}\right)+b c^3 e \log \left(1-c^2 x^2\right)+2 b c^3 e \log (x) \log \left(1-c^2 x^2\right)-b c^3 e \log \left(x-\frac{1}{c}\right) \log \left(1-c^2 x^2\right)-b c^3 e \log \left(\frac{1}{c}+x\right) \log \left(1-c^2 x^2\right)-\frac{2 b d \coth ^{-1}(c x)}{x^3}-\frac{b c d}{x^2}\right)","-\frac{c^3 e \left(a+b \coth ^{-1}(c x)\right)^2}{3 b}-\frac{\left(a+b \coth ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{3 x^3}+\frac{2 c^2 e \left(a+b \coth ^{-1}(c x)\right)}{3 x}-b c^3 e \log (x)-\frac{b c \left(1-c^2 x^2\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{6 x^2}+\frac{1}{6} b c^3 \log \left(1-\frac{1}{1-c^2 x^2}\right) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{1}{6} b c^3 e \text{Li}_2\left(\frac{1}{1-c^2 x^2}\right)+\frac{1}{3} b c^3 e \log \left(1-c^2 x^2\right)",1,"((-2*a*d)/x^3 - (b*c*d)/x^2 + (4*a*c^2*e)/x - (2*b*d*ArcCoth[c*x])/x^3 + (4*b*c^2*e*ArcCoth[c*x])/x - 2*b*c^3*e*ArcCoth[c*x]^2 - 4*a*c^3*e*ArcTanh[c*x] - 4*b*c^3*e*Log[1/Sqrt[1 - 1/(c^2*x^2)]] + 2*b*c^3*d*Log[x] - 2*b*c^3*e*Log[x] + (b*c^3*e*Log[-c^(-1) + x]^2)/2 + (b*c^3*e*Log[c^(-1) + x]^2)/2 + b*c^3*e*Log[c^(-1) + x]*Log[(1 - c*x)/2] - 2*b*c^3*e*Log[x]*Log[1 - c*x] + b*c^3*e*Log[-c^(-1) + x]*Log[(1 + c*x)/2] - 2*b*c^3*e*Log[x]*Log[1 + c*x] - b*c^3*d*Log[1 - c^2*x^2] + b*c^3*e*Log[1 - c^2*x^2] - (2*a*e*Log[1 - c^2*x^2])/x^3 - (b*c*e*Log[1 - c^2*x^2])/x^2 - (2*b*e*ArcCoth[c*x]*Log[1 - c^2*x^2])/x^3 + 2*b*c^3*e*Log[x]*Log[1 - c^2*x^2] - b*c^3*e*Log[-c^(-1) + x]*Log[1 - c^2*x^2] - b*c^3*e*Log[c^(-1) + x]*Log[1 - c^2*x^2] - 2*b*c^3*e*PolyLog[2, -(c*x)] - 2*b*c^3*e*PolyLog[2, c*x] + b*c^3*e*PolyLog[2, 1/2 - (c*x)/2] + b*c^3*e*PolyLog[2, (1 + c*x)/2])/6","B",1
277,0,0,256,0.2971042,"\int \frac{\left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x^6} \, dx","Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/x^6,x]","\int \frac{\left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(1-c^2 x^2\right)\right)}{x^6} \, dx","-\frac{c^5 e \left(a+b \coth ^{-1}(c x)\right)^2}{5 b}+\frac{2 c^4 e \left(a+b \coth ^{-1}(c x)\right)}{5 x}-\frac{\left(a+b \coth ^{-1}(c x)\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{5 x^5}+\frac{2 c^2 e \left(a+b \coth ^{-1}(c x)\right)}{15 x^3}-\frac{5}{6} b c^5 e \log (x)+\frac{7 b c^3 e}{60 x^2}-\frac{b c \left(e \log \left(1-c^2 x^2\right)+d\right)}{20 x^4}+\frac{1}{10} b c^5 \log \left(1-\frac{1}{1-c^2 x^2}\right) \left(e \log \left(1-c^2 x^2\right)+d\right)-\frac{1}{10} b c^5 e \text{Li}_2\left(\frac{1}{1-c^2 x^2}\right)+\frac{19}{60} b c^5 e \log \left(1-c^2 x^2\right)-\frac{b c^3 \left(1-c^2 x^2\right) \left(e \log \left(1-c^2 x^2\right)+d\right)}{10 x^2}",1,"Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/x^6, x]","F",-1
278,1,677,512,4.4335415,"\int x \left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right) \, dx","Integrate[x*(a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]),x]","\frac{2 a c^2 d g x^2+2 a c^2 e g x^2 \log \left(f+g x^2\right)+2 a c^2 e f \log \left(f+g x^2\right)-2 a c^2 e g x^2+2 b c^2 d g x^2 \coth ^{-1}(c x)+b c^2 e f \text{Li}_2\left(\frac{e^{2 \coth ^{-1}(c x)} \left(f c^2+g\right)}{f c^2+2 \sqrt{-f} \sqrt{g} c-g}\right)+2 b e \left(c^2 f+g\right) \text{Li}_2\left(e^{-2 \coth ^{-1}(c x)}\right)+b e \left(c^2 f+g\right) \text{Li}_2\left(\frac{e^{2 \coth ^{-1}(c x)} \left(f c^2+g\right)}{f c^2-2 \sqrt{-f} \sqrt{g} c-g}\right)+b e g \text{Li}_2\left(\frac{e^{2 \coth ^{-1}(c x)} \left(f c^2+g\right)}{f c^2+2 \sqrt{-f} \sqrt{g} c-g}\right)+2 b c^2 e g x^2 \coth ^{-1}(c x) \log \left(f+g x^2\right)+2 b c^2 e f \coth ^{-1}(c x) \log \left(\frac{\left(c^2 f+g\right) e^{2 \coth ^{-1}(c x)}}{c^2 (-f)-2 c \sqrt{-f} \sqrt{g}+g}+1\right)+2 b c^2 e f \coth ^{-1}(c x) \log \left(\frac{\left(c^2 f+g\right) e^{2 \coth ^{-1}(c x)}}{c^2 (-f)+2 c \sqrt{-f} \sqrt{g}+g}+1\right)+2 b e g \coth ^{-1}(c x) \log \left(\frac{\left(c^2 f+g\right) e^{2 \coth ^{-1}(c x)}}{c^2 (-f)-2 c \sqrt{-f} \sqrt{g}+g}+1\right)+2 b e g \coth ^{-1}(c x) \log \left(\frac{\left(c^2 f+g\right) e^{2 \coth ^{-1}(c x)}}{c^2 (-f)+2 c \sqrt{-f} \sqrt{g}+g}+1\right)-4 b c^2 e f \coth ^{-1}(c x)^2-4 b c^2 e f \coth ^{-1}(c x) \log \left(1-e^{-2 \coth ^{-1}(c x)}\right)-2 b c^2 e g x^2 \coth ^{-1}(c x)+2 b c d g x-2 b d g \coth ^{-1}(c x)+2 b c e g x \log \left(f+g x^2\right)-2 b e g \coth ^{-1}(c x) \log \left(f+g x^2\right)+4 b c e \sqrt{f} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)-6 b c e g x-4 b e g \coth ^{-1}(c x)^2+2 b e g \coth ^{-1}(c x)-4 b e g \coth ^{-1}(c x) \log \left(1-e^{-2 \coth ^{-1}(c x)}\right)}{4 c^2 g}","\frac{1}{2} d x^2 \left(a+b \coth ^{-1}(c x)\right)+\frac{e \left(f+g x^2\right) \log \left(f+g x^2\right) \left(a+b \coth ^{-1}(c x)\right)}{2 g}-\frac{1}{2} e x^2 \left(a+b \coth ^{-1}(c x)\right)-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{b e \left(c^2 f+g\right) \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 c^2 g}-\frac{b e \left(c^2 f+g\right) \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{\left(c \sqrt{-f}-\sqrt{g}\right) (c x+1)}\right)}{4 c^2 g}-\frac{b e \left(c^2 f+g\right) \text{Li}_2\left(1-\frac{2 c \left(\sqrt{g} x+\sqrt{-f}\right)}{\left(\sqrt{-f} c+\sqrt{g}\right) (c x+1)}\right)}{4 c^2 g}-\frac{b e \left(c^2 f+g\right) \tanh ^{-1}(c x) \log \left(f+g x^2\right)}{2 c^2 g}-\frac{b e \left(c^2 f+g\right) \log \left(\frac{2}{c x+1}\right) \tanh ^{-1}(c x)}{c^2 g}+\frac{b e \left(c^2 f+g\right) \tanh ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{(c x+1) \left(c \sqrt{-f}-\sqrt{g}\right)}\right)}{2 c^2 g}+\frac{b e \left(c^2 f+g\right) \tanh ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}+\sqrt{g} x\right)}{(c x+1) \left(c \sqrt{-f}+\sqrt{g}\right)}\right)}{2 c^2 g}+\frac{b x (d-e)}{2 c}+\frac{b e x \log \left(f+g x^2\right)}{2 c}+\frac{b e \sqrt{f} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{c \sqrt{g}}-\frac{b e x}{c}",1,"(2*b*c*d*g*x - 6*b*c*e*g*x + 2*a*c^2*d*g*x^2 - 2*a*c^2*e*g*x^2 - 2*b*d*g*ArcCoth[c*x] + 2*b*e*g*ArcCoth[c*x] + 2*b*c^2*d*g*x^2*ArcCoth[c*x] - 2*b*c^2*e*g*x^2*ArcCoth[c*x] - 4*b*c^2*e*f*ArcCoth[c*x]^2 - 4*b*e*g*ArcCoth[c*x]^2 + 4*b*c*e*Sqrt[f]*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]] - 4*b*c^2*e*f*ArcCoth[c*x]*Log[1 - E^(-2*ArcCoth[c*x])] - 4*b*e*g*ArcCoth[c*x]*Log[1 - E^(-2*ArcCoth[c*x])] + 2*b*c^2*e*f*ArcCoth[c*x]*Log[1 + (E^(2*ArcCoth[c*x])*(c^2*f + g))/(-(c^2*f) - 2*c*Sqrt[-f]*Sqrt[g] + g)] + 2*b*e*g*ArcCoth[c*x]*Log[1 + (E^(2*ArcCoth[c*x])*(c^2*f + g))/(-(c^2*f) - 2*c*Sqrt[-f]*Sqrt[g] + g)] + 2*b*c^2*e*f*ArcCoth[c*x]*Log[1 + (E^(2*ArcCoth[c*x])*(c^2*f + g))/(-(c^2*f) + 2*c*Sqrt[-f]*Sqrt[g] + g)] + 2*b*e*g*ArcCoth[c*x]*Log[1 + (E^(2*ArcCoth[c*x])*(c^2*f + g))/(-(c^2*f) + 2*c*Sqrt[-f]*Sqrt[g] + g)] + 2*a*c^2*e*f*Log[f + g*x^2] + 2*b*c*e*g*x*Log[f + g*x^2] + 2*a*c^2*e*g*x^2*Log[f + g*x^2] - 2*b*e*g*ArcCoth[c*x]*Log[f + g*x^2] + 2*b*c^2*e*g*x^2*ArcCoth[c*x]*Log[f + g*x^2] + 2*b*e*(c^2*f + g)*PolyLog[2, E^(-2*ArcCoth[c*x])] + b*e*(c^2*f + g)*PolyLog[2, (E^(2*ArcCoth[c*x])*(c^2*f + g))/(c^2*f - 2*c*Sqrt[-f]*Sqrt[g] - g)] + b*c^2*e*f*PolyLog[2, (E^(2*ArcCoth[c*x])*(c^2*f + g))/(c^2*f + 2*c*Sqrt[-f]*Sqrt[g] - g)] + b*e*g*PolyLog[2, (E^(2*ArcCoth[c*x])*(c^2*f + g))/(c^2*f + 2*c*Sqrt[-f]*Sqrt[g] - g)])/(4*c^2*g)","A",0
279,1,1287,546,3.2229471,"\int \left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right) \, dx","Integrate[(a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]),x]","a d x-2 a e x+b d \coth ^{-1}(c x) x+a e \log \left(g x^2+f\right) x+\frac{2 a e \sqrt{f} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{g}}+\frac{b d \log \left(1-c^2 x^2\right)}{2 c}+b e \left(x \coth ^{-1}(c x)+\frac{\log \left(1-c^2 x^2\right)}{2 c}\right) \log \left(g x^2+f\right)+\frac{b e \left(-4 c x \coth ^{-1}(c x)+4 \log \left(\frac{1}{c \sqrt{1-\frac{1}{c^2 x^2}} x}\right)+\frac{\sqrt{c^2 f g} \left(-2 i \cos ^{-1}\left(\frac{c^2 f-g}{f c^2+g}\right) \tan ^{-1}\left(\frac{\sqrt{c^2 f g}}{c g x}\right)+4 \coth ^{-1}(c x) \tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)-\left(\cos ^{-1}\left(\frac{c^2 f-g}{f c^2+g}\right)+2 \tan ^{-1}\left(\frac{\sqrt{c^2 f g}}{c g x}\right)\right) \log \left(\frac{2 i g \left(i f c^2+\sqrt{c^2 f g}\right) \left(\frac{1}{c x}-1\right)}{\left(f c^2+g\right) \left(g+\frac{i \sqrt{c^2 f g}}{c x}\right)}\right)-\left(\cos ^{-1}\left(\frac{c^2 f-g}{f c^2+g}\right)-2 \tan ^{-1}\left(\frac{\sqrt{c^2 f g}}{c g x}\right)\right) \log \left(\frac{2 g \left(f c^2+i \sqrt{c^2 f g}\right) \left(1+\frac{1}{c x}\right)}{\left(f c^2+g\right) \left(g+\frac{i \sqrt{c^2 f g}}{c x}\right)}\right)+\left(\cos ^{-1}\left(\frac{c^2 f-g}{f c^2+g}\right)+2 \left(\tan ^{-1}\left(\frac{\sqrt{c^2 f g}}{c g x}\right)+\tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)\right)\right) \log \left(\frac{\sqrt{2} e^{-\coth ^{-1}(c x)} \sqrt{c^2 f g}}{\sqrt{f c^2+g} \sqrt{-f c^2+g+\left(f c^2+g\right) \cosh \left(2 \coth ^{-1}(c x)\right)}}\right)+\left(\cos ^{-1}\left(\frac{c^2 f-g}{f c^2+g}\right)-2 \left(\tan ^{-1}\left(\frac{\sqrt{c^2 f g}}{c g x}\right)+\tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)\right)\right) \log \left(\frac{\sqrt{2} e^{\coth ^{-1}(c x)} \sqrt{c^2 f g}}{\sqrt{f c^2+g} \sqrt{-f c^2+g+\left(f c^2+g\right) \cosh \left(2 \coth ^{-1}(c x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(f c^2-g+2 i \sqrt{c^2 f g}\right) \left(i g+\frac{\sqrt{c^2 f g}}{c x}\right)}{\left(f c^2+g\right) \left(\frac{\sqrt{c^2 f g}}{c x}-i g\right)}\right)-\text{Li}_2\left(\frac{\left(-f c^2+g+2 i \sqrt{c^2 f g}\right) \left(g-\frac{i \sqrt{c^2 f g}}{c x}\right)}{\left(f c^2+g\right) \left(g+\frac{i \sqrt{c^2 f g}}{c x}\right)}\right)\right)\right)}{g}\right)}{2 c}-\frac{b e g \left(\frac{\left(-\log \left(x-\frac{1}{c}\right)-\log \left(x+\frac{1}{c}\right)+\log \left(1-c^2 x^2\right)\right) \log \left(g x^2+f\right)}{2 g}+\frac{\log \left(x-\frac{1}{c}\right) \log \left(1-\frac{\sqrt{g} \left(x-\frac{1}{c}\right)}{-i \sqrt{f}-\frac{\sqrt{g}}{c}}\right)+\text{Li}_2\left(\frac{\sqrt{g} \left(x-\frac{1}{c}\right)}{-i \sqrt{f}-\frac{\sqrt{g}}{c}}\right)}{2 g}+\frac{\log \left(x-\frac{1}{c}\right) \log \left(1-\frac{\sqrt{g} \left(x-\frac{1}{c}\right)}{i \sqrt{f}-\frac{\sqrt{g}}{c}}\right)+\text{Li}_2\left(\frac{\sqrt{g} \left(x-\frac{1}{c}\right)}{i \sqrt{f}-\frac{\sqrt{g}}{c}}\right)}{2 g}+\frac{\log \left(x+\frac{1}{c}\right) \log \left(1-\frac{\sqrt{g} \left(x+\frac{1}{c}\right)}{\frac{\sqrt{g}}{c}-i \sqrt{f}}\right)+\text{Li}_2\left(\frac{\sqrt{g} \left(x+\frac{1}{c}\right)}{\frac{\sqrt{g}}{c}-i \sqrt{f}}\right)}{2 g}+\frac{\log \left(x+\frac{1}{c}\right) \log \left(1-\frac{\sqrt{g} \left(x+\frac{1}{c}\right)}{i \sqrt{f}+\frac{\sqrt{g}}{c}}\right)+\text{Li}_2\left(\frac{\sqrt{g} \left(x+\frac{1}{c}\right)}{i \sqrt{f}+\frac{\sqrt{g}}{c}}\right)}{2 g}\right)}{c}","x \left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)+\frac{2 a e \sqrt{f} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{g}}-2 a e x+\frac{b \log \left(\frac{g \left(1-c^2 x^2\right)}{c^2 f+g}\right) \left(d+e \log \left(f+g x^2\right)\right)}{2 c}+\frac{b e \text{Li}_2\left(\frac{c^2 \left(g x^2+f\right)}{f c^2+g}\right)}{2 c}-\frac{b e \log \left(1-c^2 x^2\right)}{c}-\frac{i b e \sqrt{f} \text{Li}_2\left(\frac{2 \sqrt{f} \sqrt{g} (1-c x)}{\left(i c \sqrt{f}-\sqrt{g}\right) \left(\sqrt{f}-i \sqrt{g} x\right)}+1\right)}{2 \sqrt{g}}+\frac{i b e \sqrt{f} \text{Li}_2\left(1-\frac{2 \sqrt{f} \sqrt{g} (c x+1)}{\left(i \sqrt{f} c+\sqrt{g}\right) \left(\sqrt{f}-i \sqrt{g} x\right)}\right)}{2 \sqrt{g}}-\frac{b e \sqrt{f} \log \left(1-\frac{1}{c x}\right) \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{g}}+\frac{b e \sqrt{f} \log \left(\frac{1}{c x}+1\right) \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{g}}+\frac{b e \sqrt{f} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right) \log \left(-\frac{2 \sqrt{f} \sqrt{g} (1-c x)}{\left(-\sqrt{g}+i c \sqrt{f}\right) \left(\sqrt{f}-i \sqrt{g} x\right)}\right)}{\sqrt{g}}-\frac{b e \sqrt{f} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right) \log \left(\frac{2 \sqrt{f} \sqrt{g} (c x+1)}{\left(\sqrt{g}+i c \sqrt{f}\right) \left(\sqrt{f}-i \sqrt{g} x\right)}\right)}{\sqrt{g}}-2 b e x \coth ^{-1}(c x)",1,"a*d*x - 2*a*e*x + b*d*x*ArcCoth[c*x] + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] + (b*d*Log[1 - c^2*x^2])/(2*c) + a*e*x*Log[f + g*x^2] + b*e*(x*ArcCoth[c*x] + Log[1 - c^2*x^2]/(2*c))*Log[f + g*x^2] + (b*e*(-4*c*x*ArcCoth[c*x] + 4*Log[1/(c*Sqrt[1 - 1/(c^2*x^2)]*x)] + (Sqrt[c^2*f*g]*((-2*I)*ArcCos[(c^2*f - g)/(c^2*f + g)]*ArcTan[Sqrt[c^2*f*g]/(c*g*x)] + 4*ArcCoth[c*x]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] - (ArcCos[(c^2*f - g)/(c^2*f + g)] + 2*ArcTan[Sqrt[c^2*f*g]/(c*g*x)])*Log[((2*I)*g*(I*c^2*f + Sqrt[c^2*f*g])*(-1 + 1/(c*x)))/((c^2*f + g)*(g + (I*Sqrt[c^2*f*g])/(c*x)))] - (ArcCos[(c^2*f - g)/(c^2*f + g)] - 2*ArcTan[Sqrt[c^2*f*g]/(c*g*x)])*Log[(2*g*(c^2*f + I*Sqrt[c^2*f*g])*(1 + 1/(c*x)))/((c^2*f + g)*(g + (I*Sqrt[c^2*f*g])/(c*x)))] + (ArcCos[(c^2*f - g)/(c^2*f + g)] + 2*(ArcTan[Sqrt[c^2*f*g]/(c*g*x)] + ArcTan[(c*g*x)/Sqrt[c^2*f*g]]))*Log[(Sqrt[2]*Sqrt[c^2*f*g])/(E^ArcCoth[c*x]*Sqrt[c^2*f + g]*Sqrt[-(c^2*f) + g + (c^2*f + g)*Cosh[2*ArcCoth[c*x]]])] + (ArcCos[(c^2*f - g)/(c^2*f + g)] - 2*(ArcTan[Sqrt[c^2*f*g]/(c*g*x)] + ArcTan[(c*g*x)/Sqrt[c^2*f*g]]))*Log[(Sqrt[2]*E^ArcCoth[c*x]*Sqrt[c^2*f*g])/(Sqrt[c^2*f + g]*Sqrt[-(c^2*f) + g + (c^2*f + g)*Cosh[2*ArcCoth[c*x]]])] + I*(-PolyLog[2, ((-(c^2*f) + g + (2*I)*Sqrt[c^2*f*g])*(g - (I*Sqrt[c^2*f*g])/(c*x)))/((c^2*f + g)*(g + (I*Sqrt[c^2*f*g])/(c*x)))] + PolyLog[2, ((c^2*f - g + (2*I)*Sqrt[c^2*f*g])*(I*g + Sqrt[c^2*f*g]/(c*x)))/((c^2*f + g)*((-I)*g + Sqrt[c^2*f*g]/(c*x)))])))/g))/(2*c) - (b*e*g*(((-Log[-c^(-1) + x] - Log[c^(-1) + x] + Log[1 - c^2*x^2])*Log[f + g*x^2])/(2*g) + (Log[-c^(-1) + x]*Log[1 - (Sqrt[g]*(-c^(-1) + x))/((-I)*Sqrt[f] - Sqrt[g]/c)] + PolyLog[2, (Sqrt[g]*(-c^(-1) + x))/((-I)*Sqrt[f] - Sqrt[g]/c)])/(2*g) + (Log[-c^(-1) + x]*Log[1 - (Sqrt[g]*(-c^(-1) + x))/(I*Sqrt[f] - Sqrt[g]/c)] + PolyLog[2, (Sqrt[g]*(-c^(-1) + x))/(I*Sqrt[f] - Sqrt[g]/c)])/(2*g) + (Log[c^(-1) + x]*Log[1 - (Sqrt[g]*(c^(-1) + x))/((-I)*Sqrt[f] + Sqrt[g]/c)] + PolyLog[2, (Sqrt[g]*(c^(-1) + x))/((-I)*Sqrt[f] + Sqrt[g]/c)])/(2*g) + (Log[c^(-1) + x]*Log[1 - (Sqrt[g]*(c^(-1) + x))/(I*Sqrt[f] + Sqrt[g]/c)] + PolyLog[2, (Sqrt[g]*(c^(-1) + x))/(I*Sqrt[f] + Sqrt[g]/c)])/(2*g)))/c","B",1
280,0,0,101,0.2404704,"\int \frac{\left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x} \, dx","Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]))/x,x]","\int \frac{\left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x} \, dx","b e \text{Int}\left(\frac{\coth ^{-1}(c x) \log \left(f+g x^2\right)}{x},x\right)+a d \log (x)+\frac{1}{2} a e \text{Li}_2\left(\frac{g x^2}{f}+1\right)+\frac{1}{2} a e \log \left(-\frac{g x^2}{f}\right) \log \left(f+g x^2\right)+\frac{1}{2} b d \text{Li}_2\left(-\frac{1}{c x}\right)-\frac{1}{2} b d \text{Li}_2\left(\frac{1}{c x}\right)",0,"Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]))/x, x]","A",-1
281,1,1236,560,3.5863436,"\int \frac{\left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x^2} \, dx","Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]))/x^2,x]","-\frac{b \coth ^{-1}(c x) d}{x}+b c \log (x) d-\frac{1}{2} b c \log \left(1-c^2 x^2\right) d-\frac{a d}{x}+a e \left(\frac{2 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{f}}-\frac{\log \left(g x^2+f\right)}{x}\right)+\frac{1}{2} b e \left(-\frac{\left(2 \coth ^{-1}(c x)+c x \left(\log \left(1-c^2 x^2\right)-2 \log (x)\right)\right) \log \left(g x^2+f\right)}{x}-2 c \left(\log (x) \left(\log \left(1-\frac{i \sqrt{g} x}{\sqrt{f}}\right)+\log \left(\frac{i \sqrt{g} x}{\sqrt{f}}+1\right)\right)+\text{Li}_2\left(-\frac{i \sqrt{g} x}{\sqrt{f}}\right)+\text{Li}_2\left(\frac{i \sqrt{g} x}{\sqrt{f}}\right)\right)+c \left(\log \left(x-\frac{1}{c}\right) \log \left(\frac{c \left(\sqrt{f}-i \sqrt{g} x\right)}{c \sqrt{f}-i \sqrt{g}}\right)+\log \left(x+\frac{1}{c}\right) \log \left(\frac{c \left(\sqrt{f}-i \sqrt{g} x\right)}{\sqrt{f} c+i \sqrt{g}}\right)+\log \left(x-\frac{1}{c}\right) \log \left(\frac{c \left(i \sqrt{g} x+\sqrt{f}\right)}{\sqrt{f} c+i \sqrt{g}}\right)-\left(\log \left(x-\frac{1}{c}\right)+\log \left(x+\frac{1}{c}\right)-\log \left(1-c^2 x^2\right)\right) \log \left(g x^2+f\right)+\log \left(x+\frac{1}{c}\right) \log \left(1-\frac{\sqrt{g} (c x+1)}{i \sqrt{f} c+\sqrt{g}}\right)+\text{Li}_2\left(\frac{c \sqrt{g} \left(x+\frac{1}{c}\right)}{i \sqrt{f} c+\sqrt{g}}\right)+\text{Li}_2\left(\frac{i \sqrt{g} (c x-1)}{c \sqrt{f}-i \sqrt{g}}\right)+\text{Li}_2\left(-\frac{i \sqrt{g} (c x-1)}{\sqrt{f} c+i \sqrt{g}}\right)+\text{Li}_2\left(\frac{i \sqrt{g} (c x+1)}{\sqrt{f} c+i \sqrt{g}}\right)\right)-\frac{c g \left(2 i \cos ^{-1}\left(\frac{c^2 f-g}{f c^2+g}\right) \tan ^{-1}\left(\frac{c f}{\sqrt{c^2 f g} x}\right)-4 \coth ^{-1}(c x) \tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)+\left(\cos ^{-1}\left(\frac{c^2 f-g}{f c^2+g}\right)+2 \tan ^{-1}\left(\frac{c f}{\sqrt{c^2 f g} x}\right)\right) \log \left(\frac{2 g \left(c^2 f-i \sqrt{c^2 f g}\right) (c x-1)}{\left(f c^2+g\right) \left(c g x+i \sqrt{c^2 f g}\right)}\right)+\left(\cos ^{-1}\left(\frac{c^2 f-g}{f c^2+g}\right)-2 \tan ^{-1}\left(\frac{c f}{\sqrt{c^2 f g} x}\right)\right) \log \left(\frac{2 g \left(f c^2+i \sqrt{c^2 f g}\right) (c x+1)}{\left(f c^2+g\right) \left(c g x+i \sqrt{c^2 f g}\right)}\right)-\left(\cos ^{-1}\left(\frac{c^2 f-g}{f c^2+g}\right)+2 \left(\tan ^{-1}\left(\frac{c f}{\sqrt{c^2 f g} x}\right)+\tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)\right)\right) \log \left(\frac{\sqrt{2} e^{-\coth ^{-1}(c x)} \sqrt{c^2 f g}}{\sqrt{f c^2+g} \sqrt{-f c^2+g+\left(f c^2+g\right) \cosh \left(2 \coth ^{-1}(c x)\right)}}\right)-\left(\cos ^{-1}\left(\frac{c^2 f-g}{f c^2+g}\right)-2 \left(\tan ^{-1}\left(\frac{c f}{\sqrt{c^2 f g} x}\right)+\tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)\right)\right) \log \left(\frac{\sqrt{2} e^{\coth ^{-1}(c x)} \sqrt{c^2 f g}}{\sqrt{f c^2+g} \sqrt{-f c^2+g+\left(f c^2+g\right) \cosh \left(2 \coth ^{-1}(c x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(f c^2-g-2 i \sqrt{c^2 f g}\right) \left(i c g x+\sqrt{c^2 f g}\right)}{\left(f c^2+g\right) \left(\sqrt{c^2 f g}-i c g x\right)}\right)-\text{Li}_2\left(\frac{\left(f c^2-g+2 i \sqrt{c^2 f g}\right) \left(i c g x+\sqrt{c^2 f g}\right)}{\left(f c^2+g\right) \left(\sqrt{c^2 f g}-i c g x\right)}\right)\right)\right)}{\sqrt{c^2 f g}}\right)","-\frac{\left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x}+\frac{2 a e \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{f}}-\frac{1}{2} b c \log \left(\frac{g \left(1-c^2 x^2\right)}{c^2 f+g}\right) \left(d+e \log \left(f+g x^2\right)\right)-\frac{1}{2} b c e \text{Li}_2\left(\frac{c^2 \left(g x^2+f\right)}{f c^2+g}\right)+\frac{1}{2} b c \log \left(-\frac{g x^2}{f}\right) \left(d+e \log \left(f+g x^2\right)\right)+\frac{1}{2} b c e \text{Li}_2\left(\frac{g x^2}{f}+1\right)-\frac{i b e \sqrt{g} \text{Li}_2\left(\frac{2 \sqrt{f} \sqrt{g} (1-c x)}{\left(i c \sqrt{f}-\sqrt{g}\right) \left(\sqrt{f}-i \sqrt{g} x\right)}+1\right)}{2 \sqrt{f}}+\frac{i b e \sqrt{g} \text{Li}_2\left(1-\frac{2 \sqrt{f} \sqrt{g} (c x+1)}{\left(i \sqrt{f} c+\sqrt{g}\right) \left(\sqrt{f}-i \sqrt{g} x\right)}\right)}{2 \sqrt{f}}-\frac{b e \sqrt{g} \log \left(1-\frac{1}{c x}\right) \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{f}}+\frac{b e \sqrt{g} \log \left(\frac{1}{c x}+1\right) \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{f}}+\frac{b e \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right) \log \left(-\frac{2 \sqrt{f} \sqrt{g} (1-c x)}{\left(-\sqrt{g}+i c \sqrt{f}\right) \left(\sqrt{f}-i \sqrt{g} x\right)}\right)}{\sqrt{f}}-\frac{b e \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right) \log \left(\frac{2 \sqrt{f} \sqrt{g} (c x+1)}{\left(\sqrt{g}+i c \sqrt{f}\right) \left(\sqrt{f}-i \sqrt{g} x\right)}\right)}{\sqrt{f}}",1,"-((a*d)/x) - (b*d*ArcCoth[c*x])/x + b*c*d*Log[x] - (b*c*d*Log[1 - c^2*x^2])/2 + a*e*((2*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] - Log[f + g*x^2]/x) + (b*e*(-(((2*ArcCoth[c*x] + c*x*(-2*Log[x] + Log[1 - c^2*x^2]))*Log[f + g*x^2])/x) - 2*c*(Log[x]*(Log[1 - (I*Sqrt[g]*x)/Sqrt[f]] + Log[1 + (I*Sqrt[g]*x)/Sqrt[f]]) + PolyLog[2, ((-I)*Sqrt[g]*x)/Sqrt[f]] + PolyLog[2, (I*Sqrt[g]*x)/Sqrt[f]]) + c*(Log[-c^(-1) + x]*Log[(c*(Sqrt[f] - I*Sqrt[g]*x))/(c*Sqrt[f] - I*Sqrt[g])] + Log[c^(-1) + x]*Log[(c*(Sqrt[f] - I*Sqrt[g]*x))/(c*Sqrt[f] + I*Sqrt[g])] + Log[-c^(-1) + x]*Log[(c*(Sqrt[f] + I*Sqrt[g]*x))/(c*Sqrt[f] + I*Sqrt[g])] - (Log[-c^(-1) + x] + Log[c^(-1) + x] - Log[1 - c^2*x^2])*Log[f + g*x^2] + Log[c^(-1) + x]*Log[1 - (Sqrt[g]*(1 + c*x))/(I*c*Sqrt[f] + Sqrt[g])] + PolyLog[2, (c*Sqrt[g]*(c^(-1) + x))/(I*c*Sqrt[f] + Sqrt[g])] + PolyLog[2, (I*Sqrt[g]*(-1 + c*x))/(c*Sqrt[f] - I*Sqrt[g])] + PolyLog[2, ((-I)*Sqrt[g]*(-1 + c*x))/(c*Sqrt[f] + I*Sqrt[g])] + PolyLog[2, (I*Sqrt[g]*(1 + c*x))/(c*Sqrt[f] + I*Sqrt[g])]) - (c*g*((2*I)*ArcCos[(c^2*f - g)/(c^2*f + g)]*ArcTan[(c*f)/(Sqrt[c^2*f*g]*x)] - 4*ArcCoth[c*x]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] + (ArcCos[(c^2*f - g)/(c^2*f + g)] + 2*ArcTan[(c*f)/(Sqrt[c^2*f*g]*x)])*Log[(2*g*(c^2*f - I*Sqrt[c^2*f*g])*(-1 + c*x))/((c^2*f + g)*(I*Sqrt[c^2*f*g] + c*g*x))] + (ArcCos[(c^2*f - g)/(c^2*f + g)] - 2*ArcTan[(c*f)/(Sqrt[c^2*f*g]*x)])*Log[(2*g*(c^2*f + I*Sqrt[c^2*f*g])*(1 + c*x))/((c^2*f + g)*(I*Sqrt[c^2*f*g] + c*g*x))] - (ArcCos[(c^2*f - g)/(c^2*f + g)] + 2*(ArcTan[(c*f)/(Sqrt[c^2*f*g]*x)] + ArcTan[(c*g*x)/Sqrt[c^2*f*g]]))*Log[(Sqrt[2]*Sqrt[c^2*f*g])/(E^ArcCoth[c*x]*Sqrt[c^2*f + g]*Sqrt[-(c^2*f) + g + (c^2*f + g)*Cosh[2*ArcCoth[c*x]]])] - (ArcCos[(c^2*f - g)/(c^2*f + g)] - 2*(ArcTan[(c*f)/(Sqrt[c^2*f*g]*x)] + ArcTan[(c*g*x)/Sqrt[c^2*f*g]]))*Log[(Sqrt[2]*E^ArcCoth[c*x]*Sqrt[c^2*f*g])/(Sqrt[c^2*f + g]*Sqrt[-(c^2*f) + g + (c^2*f + g)*Cosh[2*ArcCoth[c*x]]])] + I*(PolyLog[2, ((c^2*f - g - (2*I)*Sqrt[c^2*f*g])*(Sqrt[c^2*f*g] + I*c*g*x))/((c^2*f + g)*(Sqrt[c^2*f*g] - I*c*g*x))] - PolyLog[2, ((c^2*f - g + (2*I)*Sqrt[c^2*f*g])*(Sqrt[c^2*f*g] + I*c*g*x))/((c^2*f + g)*(Sqrt[c^2*f*g] - I*c*g*x))])))/Sqrt[c^2*f*g]))/2","B",0
282,1,1318,712,5.9036342,"\int \frac{\left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x^3} \, dx","Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]))/x^3,x]","-\frac{-4 b c e \sqrt{f} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right) x^2-4 a e g \log (x) x^2+2 a e g \log \left(g x^2+f\right) x^2+b c^2 e f \left(-4 \coth ^{-1}(c x)^2-4 \log \left(1-e^{-2 \coth ^{-1}(c x)}\right) \coth ^{-1}(c x)+2 \log \left(\frac{e^{2 \coth ^{-1}(c x)} \left(f c^2+g\right)}{-f c^2-2 \sqrt{-f} \sqrt{g} c+g}+1\right) \coth ^{-1}(c x)+2 \log \left(\frac{e^{2 \coth ^{-1}(c x)} \left(f c^2+g\right)}{-f c^2+2 \sqrt{-f} \sqrt{g} c+g}+1\right) \coth ^{-1}(c x)+2 \text{Li}_2\left(e^{-2 \coth ^{-1}(c x)}\right)+\text{Li}_2\left(\frac{e^{2 \coth ^{-1}(c x)} \left(f c^2+g\right)}{f c^2-2 \sqrt{-f} \sqrt{g} c-g}\right)+\text{Li}_2\left(\frac{e^{2 \coth ^{-1}(c x)} \left(f c^2+g\right)}{f c^2+2 \sqrt{-f} \sqrt{g} c-g}\right)\right) x^2+b d g \left(2 \coth ^{-1}(c x)^2-2 \left(\coth ^{-1}(c x)+2 \log \left(1+e^{-2 \coth ^{-1}(c x)}\right)\right) \coth ^{-1}(c x)-4 i \sin ^{-1}\left(\sqrt{\frac{g}{f c^2+g}}\right) \tanh ^{-1}\left(\frac{c f}{\sqrt{-c^2 f g} x}\right)+2 \left(\coth ^{-1}(c x)-i \sin ^{-1}\left(\sqrt{\frac{g}{f c^2+g}}\right)\right) \log \left(\frac{e^{-2 \coth ^{-1}(c x)} \left(\left(-1+e^{2 \coth ^{-1}(c x)}\right) f c^2+e^{2 \coth ^{-1}(c x)} g+g-2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)+2 \left(\coth ^{-1}(c x)+i \sin ^{-1}\left(\sqrt{\frac{g}{f c^2+g}}\right)\right) \log \left(\frac{e^{-2 \coth ^{-1}(c x)} \left(\left(-1+e^{2 \coth ^{-1}(c x)}\right) f c^2+e^{2 \coth ^{-1}(c x)} g+g+2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)+2 \text{Li}_2\left(-e^{-2 \coth ^{-1}(c x)}\right)-\text{Li}_2\left(\frac{e^{-2 \coth ^{-1}(c x)} \left(f c^2-g+2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)-\text{Li}_2\left(-\frac{e^{-2 \coth ^{-1}(c x)} \left(-f c^2+g+2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)\right) x^2+b e g \left(2 \coth ^{-1}(c x)^2-2 \left(\coth ^{-1}(c x)+2 \log \left(1+e^{-2 \coth ^{-1}(c x)}\right)\right) \coth ^{-1}(c x)-4 i \sin ^{-1}\left(\sqrt{\frac{g}{f c^2+g}}\right) \tanh ^{-1}\left(\frac{c f}{\sqrt{-c^2 f g} x}\right)+2 \left(\coth ^{-1}(c x)-i \sin ^{-1}\left(\sqrt{\frac{g}{f c^2+g}}\right)\right) \log \left(\frac{e^{-2 \coth ^{-1}(c x)} \left(\left(-1+e^{2 \coth ^{-1}(c x)}\right) f c^2+e^{2 \coth ^{-1}(c x)} g+g-2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)+2 \left(\coth ^{-1}(c x)+i \sin ^{-1}\left(\sqrt{\frac{g}{f c^2+g}}\right)\right) \log \left(\frac{e^{-2 \coth ^{-1}(c x)} \left(\left(-1+e^{2 \coth ^{-1}(c x)}\right) f c^2+e^{2 \coth ^{-1}(c x)} g+g+2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)+2 \text{Li}_2\left(-e^{-2 \coth ^{-1}(c x)}\right)-\text{Li}_2\left(\frac{e^{-2 \coth ^{-1}(c x)} \left(f c^2-g+2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)-\text{Li}_2\left(-\frac{e^{-2 \coth ^{-1}(c x)} \left(-f c^2+g+2 \sqrt{-c^2 f g}\right)}{f c^2+g}\right)\right) x^2+2 a d f+2 e f \left(a+b c x+\left(b-b c^2 x^2\right) \coth ^{-1}(c x)\right) \log \left(g x^2+f\right)-b d \left(g x^2 \left(2 \coth ^{-1}(c x) \left(-\coth ^{-1}(c x)+\log \left(\frac{e^{2 \coth ^{-1}(c x)} \left(f c^2+g\right)}{-f c^2-2 \sqrt{-f} \sqrt{g} c+g}+1\right)+\log \left(\frac{e^{2 \coth ^{-1}(c x)} \left(f c^2+g\right)}{-f c^2+2 \sqrt{-f} \sqrt{g} c+g}+1\right)\right)+\text{Li}_2\left(\frac{e^{2 \coth ^{-1}(c x)} \left(f c^2+g\right)}{f c^2-2 \sqrt{-f} \sqrt{g} c-g}\right)+\text{Li}_2\left(\frac{e^{2 \coth ^{-1}(c x)} \left(f c^2+g\right)}{f c^2+2 \sqrt{-f} \sqrt{g} c-g}\right)\right)-2 \left(g \coth ^{-1}(c x)^2 x^2-g \text{Li}_2\left(-e^{-2 \coth ^{-1}(c x)}\right) x^2+c f x+\coth ^{-1}(c x) \left(-c^2 f x^2+2 g \log \left(1+e^{-2 \coth ^{-1}(c x)}\right) x^2+f\right)\right)\right)}{4 f x^2}","-\frac{\left(a+b \coth ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{2 x^2}-\frac{a e g \log \left(f+g x^2\right)}{2 f}+\frac{a e g \log (x)}{f}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left(d+e \log \left(f+g x^2\right)\right)+\frac{1}{4} b c^2 e \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{\left(c \sqrt{-f}-\sqrt{g}\right) (c x+1)}\right)+\frac{1}{4} b c^2 e \text{Li}_2\left(1-\frac{2 c \left(\sqrt{g} x+\sqrt{-f}\right)}{\left(\sqrt{-f} c+\sqrt{g}\right) (c x+1)}\right)-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{(c x+1) \left(c \sqrt{-f}-\sqrt{g}\right)}\right)-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}+\sqrt{g} x\right)}{(c x+1) \left(c \sqrt{-f}+\sqrt{g}\right)}\right)-\frac{1}{2} b c^2 e \text{Li}_2\left(1-\frac{2}{c x+1}\right)+b c^2 e \log \left(\frac{2}{c x+1}\right) \tanh ^{-1}(c x)-\frac{b c \left(d+e \log \left(f+g x^2\right)\right)}{2 x}+\frac{b e g \text{Li}_2\left(-\frac{1}{c x}\right)}{2 f}-\frac{b e g \text{Li}_2\left(\frac{1}{c x}\right)}{2 f}-\frac{b e g \text{Li}_2\left(1-\frac{2}{c x+1}\right)}{2 f}+\frac{b e g \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{\left(c \sqrt{-f}-\sqrt{g}\right) (c x+1)}\right)}{4 f}+\frac{b e g \text{Li}_2\left(1-\frac{2 c \left(\sqrt{g} x+\sqrt{-f}\right)}{\left(\sqrt{-f} c+\sqrt{g}\right) (c x+1)}\right)}{4 f}+\frac{b c e \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{f}}+\frac{b e g \log \left(\frac{2}{c x+1}\right) \coth ^{-1}(c x)}{f}-\frac{b e g \coth ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{(c x+1) \left(c \sqrt{-f}-\sqrt{g}\right)}\right)}{2 f}-\frac{b e g \coth ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}+\sqrt{g} x\right)}{(c x+1) \left(c \sqrt{-f}+\sqrt{g}\right)}\right)}{2 f}",1,"-1/4*(2*a*d*f - 4*b*c*e*Sqrt[f]*Sqrt[g]*x^2*ArcTan[(Sqrt[g]*x)/Sqrt[f]] - 4*a*e*g*x^2*Log[x] + 2*a*e*g*x^2*Log[f + g*x^2] + 2*e*f*(a + b*c*x + (b - b*c^2*x^2)*ArcCoth[c*x])*Log[f + g*x^2] + b*c^2*e*f*x^2*(-4*ArcCoth[c*x]^2 - 4*ArcCoth[c*x]*Log[1 - E^(-2*ArcCoth[c*x])] + 2*ArcCoth[c*x]*Log[1 + (E^(2*ArcCoth[c*x])*(c^2*f + g))/(-(c^2*f) - 2*c*Sqrt[-f]*Sqrt[g] + g)] + 2*ArcCoth[c*x]*Log[1 + (E^(2*ArcCoth[c*x])*(c^2*f + g))/(-(c^2*f) + 2*c*Sqrt[-f]*Sqrt[g] + g)] + 2*PolyLog[2, E^(-2*ArcCoth[c*x])] + PolyLog[2, (E^(2*ArcCoth[c*x])*(c^2*f + g))/(c^2*f - 2*c*Sqrt[-f]*Sqrt[g] - g)] + PolyLog[2, (E^(2*ArcCoth[c*x])*(c^2*f + g))/(c^2*f + 2*c*Sqrt[-f]*Sqrt[g] - g)]) - b*d*(-2*(c*f*x + g*x^2*ArcCoth[c*x]^2 + ArcCoth[c*x]*(f - c^2*f*x^2 + 2*g*x^2*Log[1 + E^(-2*ArcCoth[c*x])]) - g*x^2*PolyLog[2, -E^(-2*ArcCoth[c*x])]) + g*x^2*(2*ArcCoth[c*x]*(-ArcCoth[c*x] + Log[1 + (E^(2*ArcCoth[c*x])*(c^2*f + g))/(-(c^2*f) - 2*c*Sqrt[-f]*Sqrt[g] + g)] + Log[1 + (E^(2*ArcCoth[c*x])*(c^2*f + g))/(-(c^2*f) + 2*c*Sqrt[-f]*Sqrt[g] + g)]) + PolyLog[2, (E^(2*ArcCoth[c*x])*(c^2*f + g))/(c^2*f - 2*c*Sqrt[-f]*Sqrt[g] - g)] + PolyLog[2, (E^(2*ArcCoth[c*x])*(c^2*f + g))/(c^2*f + 2*c*Sqrt[-f]*Sqrt[g] - g)])) + b*d*g*x^2*(2*ArcCoth[c*x]^2 - (4*I)*ArcSin[Sqrt[g/(c^2*f + g)]]*ArcTanh[(c*f)/(Sqrt[-(c^2*f*g)]*x)] - 2*ArcCoth[c*x]*(ArcCoth[c*x] + 2*Log[1 + E^(-2*ArcCoth[c*x])]) + 2*(ArcCoth[c*x] - I*ArcSin[Sqrt[g/(c^2*f + g)]])*Log[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] + 2*(ArcCoth[c*x] + I*ArcSin[Sqrt[g/(c^2*f + g)]])*Log[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] + 2*PolyLog[2, -E^(-2*ArcCoth[c*x])] - PolyLog[2, (c^2*f - g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] - PolyLog[2, -((-(c^2*f) + g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g)))]) + b*e*g*x^2*(2*ArcCoth[c*x]^2 - (4*I)*ArcSin[Sqrt[g/(c^2*f + g)]]*ArcTanh[(c*f)/(Sqrt[-(c^2*f*g)]*x)] - 2*ArcCoth[c*x]*(ArcCoth[c*x] + 2*Log[1 + E^(-2*ArcCoth[c*x])]) + 2*(ArcCoth[c*x] - I*ArcSin[Sqrt[g/(c^2*f + g)]])*Log[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] + 2*(ArcCoth[c*x] + I*ArcSin[Sqrt[g/(c^2*f + g)]])*Log[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] + 2*PolyLog[2, -E^(-2*ArcCoth[c*x])] - PolyLog[2, (c^2*f - g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] - PolyLog[2, -((-(c^2*f) + g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g)))]))/(f*x^2)","C",0
283,1,51,25,0.0352852,"\int \coth ^{-1}\left(e^x\right) \, dx","Integrate[ArcCoth[E^x],x]","-\frac{\text{Li}_2\left(-e^x\right)}{2}+\frac{\text{Li}_2\left(e^x\right)}{2}+\frac{1}{2} x \log \left(1-e^x\right)-\frac{1}{2} x \log \left(e^x+1\right)+x \coth ^{-1}\left(e^x\right)","\frac{\text{Li}_2\left(-e^{-x}\right)}{2}-\frac{\text{Li}_2\left(e^{-x}\right)}{2}",1,"x*ArcCoth[E^x] + (x*Log[1 - E^x])/2 - (x*Log[1 + E^x])/2 - PolyLog[2, -E^x]/2 + PolyLog[2, E^x]/2","B",1
284,1,71,51,0.0253883,"\int x \coth ^{-1}\left(e^x\right) \, dx","Integrate[x*ArcCoth[E^x],x]","\frac{1}{4} \left(-2 x \text{Li}_2\left(-e^x\right)+2 x \text{Li}_2\left(e^x\right)+2 \text{Li}_3\left(-e^x\right)-2 \text{Li}_3\left(e^x\right)+x^2 \log \left(1-e^x\right)-x^2 \log \left(e^x+1\right)+2 x^2 \coth ^{-1}\left(e^x\right)\right)","\frac{1}{2} x \text{Li}_2\left(-e^{-x}\right)-\frac{1}{2} x \text{Li}_2\left(e^{-x}\right)+\frac{\text{Li}_3\left(-e^{-x}\right)}{2}-\frac{\text{Li}_3\left(e^{-x}\right)}{2}",1,"(2*x^2*ArcCoth[E^x] + x^2*Log[1 - E^x] - x^2*Log[1 + E^x] - 2*x*PolyLog[2, -E^x] + 2*x*PolyLog[2, E^x] + 2*PolyLog[3, -E^x] - 2*PolyLog[3, E^x])/4","A",1
285,1,93,70,0.0258168,"\int x^2 \coth ^{-1}\left(e^x\right) \, dx","Integrate[x^2*ArcCoth[E^x],x]","\frac{1}{6} \left(-3 x^2 \text{Li}_2\left(-e^x\right)+3 x^2 \text{Li}_2\left(e^x\right)+6 x \text{Li}_3\left(-e^x\right)-6 x \text{Li}_3\left(e^x\right)-6 \text{Li}_4\left(-e^x\right)+6 \text{Li}_4\left(e^x\right)+x^3 \log \left(1-e^x\right)-x^3 \log \left(e^x+1\right)+2 x^3 \coth ^{-1}\left(e^x\right)\right)","\frac{1}{2} x^2 \text{Li}_2\left(-e^{-x}\right)-\frac{1}{2} x^2 \text{Li}_2\left(e^{-x}\right)+x \text{Li}_3\left(-e^{-x}\right)-x \text{Li}_3\left(e^{-x}\right)+\text{Li}_4\left(-e^{-x}\right)-\text{Li}_4\left(e^{-x}\right)",1,"(2*x^3*ArcCoth[E^x] + x^3*Log[1 - E^x] - x^3*Log[1 + E^x] - 3*x^2*PolyLog[2, -E^x] + 3*x^2*PolyLog[2, E^x] + 6*x*PolyLog[3, -E^x] - 6*x*PolyLog[3, E^x] - 6*PolyLog[4, -E^x] + 6*PolyLog[4, E^x])/6","A",1
286,1,68,41,0.0825535,"\int \coth ^{-1}\left(e^{a+b x}\right) \, dx","Integrate[ArcCoth[E^(a + b*x)],x]","\frac{-\text{Li}_2\left(-e^{a+b x}\right)+\text{Li}_2\left(e^{a+b x}\right)+b x \left(\log \left(1-e^{a+b x}\right)-\log \left(e^{a+b x}+1\right)+2 \coth ^{-1}\left(e^{a+b x}\right)\right)}{2 b}","\frac{\text{Li}_2\left(-e^{-a-b x}\right)}{2 b}-\frac{\text{Li}_2\left(e^{-a-b x}\right)}{2 b}",1,"(b*x*(2*ArcCoth[E^(a + b*x)] + Log[1 - E^(a + b*x)] - Log[1 + E^(a + b*x)]) - PolyLog[2, -E^(a + b*x)] + PolyLog[2, E^(a + b*x)])/(2*b)","A",1
287,1,113,83,0.0453973,"\int x \coth ^{-1}\left(e^{a+b x}\right) \, dx","Integrate[x*ArcCoth[E^(a + b*x)],x]","\frac{b^2 x^2 \log \left(1-e^{a+b x}\right)-b^2 x^2 \log \left(e^{a+b x}+1\right)+2 b^2 x^2 \coth ^{-1}\left(e^{a+b x}\right)-2 b x \text{Li}_2\left(-e^{a+b x}\right)+2 b x \text{Li}_2\left(e^{a+b x}\right)+2 \text{Li}_3\left(-e^{a+b x}\right)-2 \text{Li}_3\left(e^{a+b x}\right)}{4 b^2}","\frac{\text{Li}_3\left(-e^{-a-b x}\right)}{2 b^2}-\frac{\text{Li}_3\left(e^{-a-b x}\right)}{2 b^2}+\frac{x \text{Li}_2\left(-e^{-a-b x}\right)}{2 b}-\frac{x \text{Li}_2\left(e^{-a-b x}\right)}{2 b}",1,"(2*b^2*x^2*ArcCoth[E^(a + b*x)] + b^2*x^2*Log[1 - E^(a + b*x)] - b^2*x^2*Log[1 + E^(a + b*x)] - 2*b*x*PolyLog[2, -E^(a + b*x)] + 2*b*x*PolyLog[2, E^(a + b*x)] + 2*PolyLog[3, -E^(a + b*x)] - 2*PolyLog[3, E^(a + b*x)])/(4*b^2)","A",1
288,1,149,119,0.0417623,"\int x^2 \coth ^{-1}\left(e^{a+b x}\right) \, dx","Integrate[x^2*ArcCoth[E^(a + b*x)],x]","\frac{b^3 x^3 \log \left(1-e^{a+b x}\right)-b^3 x^3 \log \left(e^{a+b x}+1\right)+2 b^3 x^3 \coth ^{-1}\left(e^{a+b x}\right)-3 b^2 x^2 \text{Li}_2\left(-e^{a+b x}\right)+3 b^2 x^2 \text{Li}_2\left(e^{a+b x}\right)+6 b x \text{Li}_3\left(-e^{a+b x}\right)-6 b x \text{Li}_3\left(e^{a+b x}\right)-6 \text{Li}_4\left(-e^{a+b x}\right)+6 \text{Li}_4\left(e^{a+b x}\right)}{6 b^3}","\frac{\text{Li}_4\left(-e^{-a-b x}\right)}{b^3}-\frac{\text{Li}_4\left(e^{-a-b x}\right)}{b^3}+\frac{x \text{Li}_3\left(-e^{-a-b x}\right)}{b^2}-\frac{x \text{Li}_3\left(e^{-a-b x}\right)}{b^2}+\frac{x^2 \text{Li}_2\left(-e^{-a-b x}\right)}{2 b}-\frac{x^2 \text{Li}_2\left(e^{-a-b x}\right)}{2 b}",1,"(2*b^3*x^3*ArcCoth[E^(a + b*x)] + b^3*x^3*Log[1 - E^(a + b*x)] - b^3*x^3*Log[1 + E^(a + b*x)] - 3*b^2*x^2*PolyLog[2, -E^(a + b*x)] + 3*b^2*x^2*PolyLog[2, E^(a + b*x)] + 6*b*x*PolyLog[3, -E^(a + b*x)] - 6*b*x*PolyLog[3, E^(a + b*x)] - 6*PolyLog[4, -E^(a + b*x)] + 6*PolyLog[4, E^(a + b*x)])/(6*b^3)","A",1
289,1,108,168,0.0858727,"\int \coth ^{-1}\left(a+b f^{c+d x}\right) \, dx","Integrate[ArcCoth[a + b*f^(c + d*x)],x]","\frac{\text{Li}_2\left(-\frac{b f^{c+d x}}{a-1}\right)-\text{Li}_2\left(-\frac{b f^{c+d x}}{a+1}\right)+d x \log (f) \left(\log \left(\frac{a+b f^{c+d x}-1}{a-1}\right)-\log \left(\frac{a+b f^{c+d x}+1}{a+1}\right)+2 \coth ^{-1}\left(a+b f^{c+d x}\right)\right)}{2 d \log (f)}","\frac{\text{Li}_2\left(1-\frac{2}{b f^{c+d x}+a+1}\right)}{2 d \log (f)}-\frac{\text{Li}_2\left(1-\frac{2 b f^{c+d x}}{(1-a) \left(b f^{c+d x}+a+1\right)}\right)}{2 d \log (f)}-\frac{\log \left(\frac{2}{a+b f^{c+d x}+1}\right) \coth ^{-1}\left(a+b f^{c+d x}\right)}{d \log (f)}+\frac{\log \left(\frac{2 b f^{c+d x}}{(1-a) \left(a+b f^{c+d x}+1\right)}\right) \coth ^{-1}\left(a+b f^{c+d x}\right)}{d \log (f)}",1,"(d*x*Log[f]*(2*ArcCoth[a + b*f^(c + d*x)] + Log[(-1 + a + b*f^(c + d*x))/(-1 + a)] - Log[(1 + a + b*f^(c + d*x))/(1 + a)]) + PolyLog[2, -((b*f^(c + d*x))/(-1 + a))] - PolyLog[2, -((b*f^(c + d*x))/(1 + a))])/(2*d*Log[f])","A",1
290,1,177,216,0.1121973,"\int x \coth ^{-1}\left(a+b f^{c+d x}\right) \, dx","Integrate[x*ArcCoth[a + b*f^(c + d*x)],x]","\frac{d^2 x^2 \log ^2(f) \log \left(\frac{b f^{c+d x}}{a-1}+1\right)-d^2 x^2 \log ^2(f) \log \left(\frac{b f^{c+d x}}{a+1}+1\right)+2 d^2 x^2 \log ^2(f) \coth ^{-1}\left(a+b f^{c+d x}\right)-2 \text{Li}_3\left(-\frac{b f^{c+d x}}{a-1}\right)+2 \text{Li}_3\left(-\frac{b f^{c+d x}}{a+1}\right)+2 d x \log (f) \text{Li}_2\left(-\frac{b f^{c+d x}}{a-1}\right)-2 d x \log (f) \text{Li}_2\left(-\frac{b f^{c+d x}}{a+1}\right)}{4 d^2 \log ^2(f)}","-\frac{\text{Li}_3\left(\frac{b f^{c+d x}}{1-a}\right)}{2 d^2 \log ^2(f)}+\frac{\text{Li}_3\left(-\frac{b f^{c+d x}}{a+1}\right)}{2 d^2 \log ^2(f)}+\frac{x \text{Li}_2\left(\frac{b f^{c+d x}}{1-a}\right)}{2 d \log (f)}-\frac{x \text{Li}_2\left(-\frac{b f^{c+d x}}{a+1}\right)}{2 d \log (f)}+\frac{1}{4} x^2 \log \left(1-\frac{b f^{c+d x}}{1-a}\right)-\frac{1}{4} x^2 \log \left(\frac{b f^{c+d x}}{a+1}+1\right)-\frac{1}{4} x^2 \log \left(1-\frac{1}{a+b f^{c+d x}}\right)+\frac{1}{4} x^2 \log \left(\frac{1}{a+b f^{c+d x}}+1\right)",1,"(2*d^2*x^2*ArcCoth[a + b*f^(c + d*x)]*Log[f]^2 + d^2*x^2*Log[f]^2*Log[1 + (b*f^(c + d*x))/(-1 + a)] - d^2*x^2*Log[f]^2*Log[1 + (b*f^(c + d*x))/(1 + a)] + 2*d*x*Log[f]*PolyLog[2, -((b*f^(c + d*x))/(-1 + a))] - 2*d*x*Log[f]*PolyLog[2, -((b*f^(c + d*x))/(1 + a))] - 2*PolyLog[3, -((b*f^(c + d*x))/(-1 + a))] + 2*PolyLog[3, -((b*f^(c + d*x))/(1 + a))])/(4*d^2*Log[f]^2)","A",1
291,1,235,269,0.0762844,"\int x^2 \coth ^{-1}\left(a+b f^{c+d x}\right) \, dx","Integrate[x^2*ArcCoth[a + b*f^(c + d*x)],x]","\frac{d^3 x^3 \log ^3(f) \log \left(\frac{b f^{c+d x}}{a-1}+1\right)-d^3 x^3 \log ^3(f) \log \left(\frac{b f^{c+d x}}{a+1}+1\right)+2 d^3 x^3 \log ^3(f) \coth ^{-1}\left(a+b f^{c+d x}\right)+3 d^2 x^2 \log ^2(f) \text{Li}_2\left(-\frac{b f^{c+d x}}{a-1}\right)-3 d^2 x^2 \log ^2(f) \text{Li}_2\left(-\frac{b f^{c+d x}}{a+1}\right)+6 \text{Li}_4\left(-\frac{b f^{c+d x}}{a-1}\right)-6 \text{Li}_4\left(-\frac{b f^{c+d x}}{a+1}\right)-6 d x \log (f) \text{Li}_3\left(-\frac{b f^{c+d x}}{a-1}\right)+6 d x \log (f) \text{Li}_3\left(-\frac{b f^{c+d x}}{a+1}\right)}{6 d^3 \log ^3(f)}","\frac{\text{Li}_4\left(\frac{b f^{c+d x}}{1-a}\right)}{d^3 \log ^3(f)}-\frac{\text{Li}_4\left(-\frac{b f^{c+d x}}{a+1}\right)}{d^3 \log ^3(f)}-\frac{x \text{Li}_3\left(\frac{b f^{c+d x}}{1-a}\right)}{d^2 \log ^2(f)}+\frac{x \text{Li}_3\left(-\frac{b f^{c+d x}}{a+1}\right)}{d^2 \log ^2(f)}+\frac{x^2 \text{Li}_2\left(\frac{b f^{c+d x}}{1-a}\right)}{2 d \log (f)}-\frac{x^2 \text{Li}_2\left(-\frac{b f^{c+d x}}{a+1}\right)}{2 d \log (f)}+\frac{1}{6} x^3 \log \left(1-\frac{b f^{c+d x}}{1-a}\right)-\frac{1}{6} x^3 \log \left(\frac{b f^{c+d x}}{a+1}+1\right)-\frac{1}{6} x^3 \log \left(1-\frac{1}{a+b f^{c+d x}}\right)+\frac{1}{6} x^3 \log \left(\frac{1}{a+b f^{c+d x}}+1\right)",1,"(2*d^3*x^3*ArcCoth[a + b*f^(c + d*x)]*Log[f]^3 + d^3*x^3*Log[f]^3*Log[1 + (b*f^(c + d*x))/(-1 + a)] - d^3*x^3*Log[f]^3*Log[1 + (b*f^(c + d*x))/(1 + a)] + 3*d^2*x^2*Log[f]^2*PolyLog[2, -((b*f^(c + d*x))/(-1 + a))] - 3*d^2*x^2*Log[f]^2*PolyLog[2, -((b*f^(c + d*x))/(1 + a))] - 6*d*x*Log[f]*PolyLog[3, -((b*f^(c + d*x))/(-1 + a))] + 6*d*x*Log[f]*PolyLog[3, -((b*f^(c + d*x))/(1 + a))] + 6*PolyLog[4, -((b*f^(c + d*x))/(-1 + a))] - 6*PolyLog[4, -((b*f^(c + d*x))/(1 + a))])/(6*d^3*Log[f]^3)","A",1
292,1,17,17,0.057689,"\int \frac{1}{\left(a-a x^2\right) \left(b-2 b \coth ^{-1}(x)\right)} \, dx","Integrate[1/((a - a*x^2)*(b - 2*b*ArcCoth[x])),x]","-\frac{\log \left(2 \coth ^{-1}(x)-1\right)}{2 a b}","-\frac{\log \left(1-2 \coth ^{-1}(x)\right)}{2 a b}",1,"-1/2*Log[-1 + 2*ArcCoth[x]]/(a*b)","A",1
293,1,39,44,0.0180338,"\int x^3 \coth ^{-1}\left(a+b x^4\right) \, dx","Integrate[x^3*ArcCoth[a + b*x^4],x]","\frac{\log \left(1-\left(a+b x^4\right)^2\right)+2 \left(a+b x^4\right) \coth ^{-1}\left(a+b x^4\right)}{8 b}","\frac{\log \left(1-\left(a+b x^4\right)^2\right)}{8 b}+\frac{\left(a+b x^4\right) \coth ^{-1}\left(a+b x^4\right)}{4 b}",1,"(2*(a + b*x^4)*ArcCoth[a + b*x^4] + Log[1 - (a + b*x^4)^2])/(8*b)","A",1
294,1,42,47,0.0381062,"\int x^{-1+n} \coth ^{-1}\left(a+b x^n\right) \, dx","Integrate[x^(-1 + n)*ArcCoth[a + b*x^n],x]","\frac{\log \left(1-\left(a+b x^n\right)^2\right)+2 \left(a+b x^n\right) \coth ^{-1}\left(a+b x^n\right)}{2 b n}","\frac{\log \left(1-\left(a+b x^n\right)^2\right)}{2 b n}+\frac{\left(a+b x^n\right) \coth ^{-1}\left(a+b x^n\right)}{b n}",1,"(2*(a + b*x^n)*ArcCoth[a + b*x^n] + Log[1 - (a + b*x^n)^2])/(2*b*n)","A",1
295,1,153,107,0.1809351,"\int e^{c (a+b x)} \coth ^{-1}(\sinh (a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcCoth[Sinh[a*c + b*c*x]],x]","\frac{\log \left(-2 e^{c (a+b x)}-e^{2 c (a+b x)}+1\right)+\log \left(2 e^{c (a+b x)}-e^{2 c (a+b x)}+1\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{e^{c (a+b x)}-1}{\sqrt{2}}\right)+2 \sqrt{2} \tanh ^{-1}\left(\frac{e^{c (a+b x)}+1}{\sqrt{2}}\right)-2 e^{c (a+b x)} \coth ^{-1}\left(\frac{1}{2} e^{-c (a+b x)}-\frac{1}{2} e^{c (a+b x)}\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} \coth ^{-1}(\sinh (c (a+b x)))}{b c}",1,"(-2*E^(c*(a + b*x))*ArcCoth[1/(2*E^(c*(a + b*x))) - E^(c*(a + b*x))/2] - 2*Sqrt[2]*ArcTanh[(-1 + E^(c*(a + b*x)))/Sqrt[2]] + 2*Sqrt[2]*ArcTanh[(1 + E^(c*(a + b*x)))/Sqrt[2]] + Log[1 - 2*E^(c*(a + b*x)) - E^(2*c*(a + b*x))] + Log[1 + 2*E^(c*(a + b*x)) - E^(2*c*(a + b*x))])/(2*b*c)","A",0
296,1,60,49,0.0926675,"\int e^{c (a+b x)} \coth ^{-1}(\cosh (a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcCoth[Cosh[a*c + b*c*x]],x]","\frac{\log \left(1-e^{2 c (a+b x)}\right)+e^{c (a+b x)} \coth ^{-1}\left(\frac{1}{2} e^{-c (a+b x)} \left(e^{2 c (a+b x)}+1\right)\right)}{b c}","\frac{\log \left(1-e^{2 c (a+b x)}\right)}{b c}+\frac{e^{a c+b c x} \coth ^{-1}(\cosh (c (a+b x)))}{b c}",1,"(E^(c*(a + b*x))*ArcCoth[(1 + E^(2*c*(a + b*x)))/(2*E^(c*(a + b*x)))] + Log[1 - E^(2*c*(a + b*x))])/(b*c)","A",0
297,1,46,45,0.095342,"\int e^{c (a+b x)} \coth ^{-1}(\tanh (a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcCoth[Tanh[a*c + b*c*x]],x]","\frac{e^{c (a+b x)} \left(\coth ^{-1}\left(\frac{e^{2 c (a+b x)}-1}{e^{2 c (a+b x)}+1}\right)-1\right)}{b c}","\frac{e^{a c+b c x} \coth ^{-1}(\tanh (c (a+b x)))}{b c}-\frac{e^{a c+b c x}}{b c}",1,"(E^(c*(a + b*x))*(-1 + ArcCoth[(-1 + E^(2*c*(a + b*x)))/(1 + E^(2*c*(a + b*x)))]))/(b*c)","A",0
298,1,46,45,0.0921045,"\int e^{c (a+b x)} \coth ^{-1}(\coth (a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcCoth[Coth[a*c + b*c*x]],x]","\frac{e^{c (a+b x)} \left(\coth ^{-1}\left(\frac{e^{2 c (a+b x)}+1}{e^{2 c (a+b x)}-1}\right)-1\right)}{b c}","\frac{e^{a c+b c x} \coth ^{-1}(\coth (c (a+b x)))}{b c}-\frac{e^{a c+b c x}}{b c}",1,"(E^(c*(a + b*x))*(-1 + ArcCoth[(1 + E^(2*c*(a + b*x)))/(-1 + E^(2*c*(a + b*x)))]))/(b*c)","A",0
299,1,59,49,0.0891451,"\int e^{c (a+b x)} \coth ^{-1}(\text{sech}(a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcCoth[Sech[a*c + b*c*x]],x]","\frac{\log \left(1-e^{2 c (a+b x)}\right)+e^{c (a+b x)} \coth ^{-1}\left(\frac{2 e^{c (a+b x)}}{e^{2 c (a+b x)}+1}\right)}{b c}","\frac{\log \left(1-e^{2 c (a+b x)}\right)}{b c}+\frac{e^{a c+b c x} \coth ^{-1}(\text{sech}(c (a+b x)))}{b c}",1,"(E^(c*(a + b*x))*ArcCoth[(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
300,1,150,107,0.166529,"\int e^{c (a+b x)} \coth ^{-1}(\text{csch}(a c+b c x)) \, dx","Integrate[E^(c*(a + b*x))*ArcCoth[Csch[a*c + b*c*x]],x]","\frac{\log \left(-2 e^{c (a+b x)}-e^{2 c (a+b x)}+1\right)+\log \left(2 e^{c (a+b x)}-e^{2 c (a+b x)}+1\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{e^{c (a+b x)}-1}{\sqrt{2}}\right)+2 \sqrt{2} \tanh ^{-1}\left(\frac{e^{c (a+b x)}+1}{\sqrt{2}}\right)+2 e^{c (a+b x)} \coth ^{-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} \coth ^{-1}(\text{csch}(c (a+b x)))}{b c}",1,"(2*E^(c*(a + b*x))*ArcCoth[(2*E^(c*(a + b*x)))/(-1 + E^(2*c*(a + b*x)))] - 2*Sqrt[2]*ArcTanh[(-1 + E^(c*(a + b*x)))/Sqrt[2]] + 2*Sqrt[2]*ArcTanh[(1 + E^(c*(a + b*x)))/Sqrt[2]] + Log[1 - 2*E^(c*(a + b*x)) - E^(2*c*(a + b*x))] + Log[1 + 2*E^(c*(a + b*x)) - E^(2*c*(a + b*x))])/(2*b*c)","A",0