1,1,187,263,1.2597626,"\int x^3 \tanh ^{-1}(a+b x)^2 \, dx","Integrate[x^3*ArcTanh[a + b*x]^2,x]","-\frac{12 \left(a^3+a\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a+b x)}\right)+36 a^2 \log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right)+11 a^2-2 \tanh ^{-1}(a+b x) \left(12 \left(a^3+a\right) \log \left(e^{-2 \tanh ^{-1}(a+b x)}+1\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) \tanh ^{-1}(a+b x)^2+10 a b x+8 \log \left(\frac{1}{\sqrt{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) \tanh ^{-1}(a+b x)^2}{b^4}+\frac{\left(6 a^2+1\right) (a+b x) \tanh ^{-1}(a+b x)}{2 b^4}+\frac{2 a \left(a^2+1\right) \log \left(\frac{2}{-a-b x+1}\right) \tanh ^{-1}(a+b x)}{b^4}-\frac{\left(a^4+6 a^2+1\right) \tanh ^{-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+b x)^3 \tanh ^{-1}(a+b x)}{6 b^4}-\frac{a (a+b x)^2 \tanh ^{-1}(a+b x)}{b^4}+\frac{a \tanh ^{-1}(a+b x)}{b^4}-\frac{a x}{b^3}+\frac{1}{4} x^4 \tanh ^{-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)*ArcTanh[a + b*x]^2 - 2*ArcTanh[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*ArcTanh[a + b*x])]) + 8*Log[1/Sqrt[1 - (a + b*x)^2]] + 36*a^2*Log[1/Sqrt[1 - (a + b*x)^2]] + 12*(a + a^3)*PolyLog[2, -E^(-2*ArcTanh[a + b*x])])/b^4","A",0
2,1,463,204,2.3590004,"\int x^2 \tanh ^{-1}(a+b x)^2 \, dx","Integrate[x^2*ArcTanh[a + b*x]^2,x]","-\frac{\left(1-(a+b x)^2\right)^{3/2} \left(-\frac{4 \left(3 a^2+1\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a+b x)}\right)}{\left(1-(a+b x)^2\right)^{3/2}}-\frac{3 a^2 (a+b x) \tanh ^{-1}(a+b x)^2}{\sqrt{1-(a+b x)^2}}+\frac{3 \left(3 a^2-4 a+1\right) \tanh ^{-1}(a+b x)^2+2 \tanh ^{-1}(a+b x) \left(\left(9 a^2+3\right) \log \left(e^{-2 \tanh ^{-1}(a+b x)}+1\right)+2\right)-18 a \log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right)}{\sqrt{1-(a+b x)^2}}-3 a^2 \tanh ^{-1}(a+b x)^2 \sinh \left(3 \tanh ^{-1}(a+b x)\right)+3 a^2 \tanh ^{-1}(a+b x)^2 \cosh \left(3 \tanh ^{-1}(a+b x)\right)+6 a^2 \tanh ^{-1}(a+b x) \log \left(e^{-2 \tanh ^{-1}(a+b x)}+1\right) \cosh \left(3 \tanh ^{-1}(a+b x)\right)-\frac{a+b x}{\sqrt{1-(a+b x)^2}}+\frac{6 a (a+b x) \tanh ^{-1}(a+b x)}{\sqrt{1-(a+b x)^2}}+\frac{3 (a+b x) \tanh ^{-1}(a+b x)^2}{\sqrt{1-(a+b x)^2}}+6 a \tanh ^{-1}(a+b x) \sinh \left(3 \tanh ^{-1}(a+b x)\right)-\tanh ^{-1}(a+b x)^2 \sinh \left(3 \tanh ^{-1}(a+b x)\right)-\sinh \left(3 \tanh ^{-1}(a+b x)\right)+\tanh ^{-1}(a+b x)^2 \cosh \left(3 \tanh ^{-1}(a+b x)\right)-6 a \log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right) \cosh \left(3 \tanh ^{-1}(a+b x)\right)+2 \tanh ^{-1}(a+b x) \log \left(e^{-2 \tanh ^{-1}(a+b x)}+1\right) \cosh \left(3 \tanh ^{-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) \tanh ^{-1}(a+b x)^2}{3 b^3}+\frac{\left(3 a^2+1\right) \tanh ^{-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) \tanh ^{-1}(a+b x)}{3 b^3}-\frac{a \log \left(1-(a+b x)^2\right)}{b^3}+\frac{(a+b x)^2 \tanh ^{-1}(a+b x)}{3 b^3}-\frac{2 a (a+b x) \tanh ^{-1}(a+b x)}{b^3}-\frac{\tanh ^{-1}(a+b x)}{3 b^3}+\frac{1}{3} x^3 \tanh ^{-1}(a+b x)^2+\frac{x}{3 b^2}",1,"-1/12*((1 - (a + b*x)^2)^(3/2)*(-((a + b*x)/Sqrt[1 - (a + b*x)^2]) + (6*a*(a + b*x)*ArcTanh[a + b*x])/Sqrt[1 - (a + b*x)^2] + (3*(a + b*x)*ArcTanh[a + b*x]^2)/Sqrt[1 - (a + b*x)^2] - (3*a^2*(a + b*x)*ArcTanh[a + b*x]^2)/Sqrt[1 - (a + b*x)^2] + ArcTanh[a + b*x]^2*Cosh[3*ArcTanh[a + b*x]] + 3*a^2*ArcTanh[a + b*x]^2*Cosh[3*ArcTanh[a + b*x]] + 2*ArcTanh[a + b*x]*Cosh[3*ArcTanh[a + b*x]]*Log[1 + E^(-2*ArcTanh[a + b*x])] + 6*a^2*ArcTanh[a + b*x]*Cosh[3*ArcTanh[a + b*x]]*Log[1 + E^(-2*ArcTanh[a + b*x])] - 6*a*Cosh[3*ArcTanh[a + b*x]]*Log[1/Sqrt[1 - (a + b*x)^2]] + (3*(1 - 4*a + 3*a^2)*ArcTanh[a + b*x]^2 + 2*ArcTanh[a + b*x]*(2 + (3 + 9*a^2)*Log[1 + E^(-2*ArcTanh[a + b*x])]) - 18*a*Log[1/Sqrt[1 - (a + b*x)^2]])/Sqrt[1 - (a + b*x)^2] - (4*(1 + 3*a^2)*PolyLog[2, -E^(-2*ArcTanh[a + b*x])])/(1 - (a + b*x)^2)^(3/2) - Sinh[3*ArcTanh[a + b*x]] + 6*a*ArcTanh[a + b*x]*Sinh[3*ArcTanh[a + b*x]] - ArcTanh[a + b*x]^2*Sinh[3*ArcTanh[a + b*x]] - 3*a^2*ArcTanh[a + b*x]^2*Sinh[3*ArcTanh[a + b*x]]))/b^3","B",0
3,1,98,136,0.2812527,"\int x \tanh ^{-1}(a+b x)^2 \, dx","Integrate[x*ArcTanh[a + b*x]^2,x]","\frac{\left(-a^2+2 a+b^2 x^2-1\right) \tanh ^{-1}(a+b x)^2-2 a \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a+b x)}\right)-2 \log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right)+2 \tanh ^{-1}(a+b x) \left(2 a \log \left(e^{-2 \tanh ^{-1}(a+b x)}+1\right)+a+b x\right)}{2 b^2}","-\frac{\left(a^2+1\right) \tanh ^{-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 \tanh ^{-1}(a+b x)^2}{b^2}+\frac{(a+b x) \tanh ^{-1}(a+b x)}{b^2}+\frac{2 a \log \left(\frac{2}{-a-b x+1}\right) \tanh ^{-1}(a+b x)}{b^2}+\frac{1}{2} x^2 \tanh ^{-1}(a+b x)^2",1,"((-1 + 2*a - a^2 + b^2*x^2)*ArcTanh[a + b*x]^2 + 2*ArcTanh[a + b*x]*(a + b*x + 2*a*Log[1 + E^(-2*ArcTanh[a + b*x])]) - 2*Log[1/Sqrt[1 - (a + b*x)^2]] - 2*a*PolyLog[2, -E^(-2*ArcTanh[a + b*x])])/(2*b^2)","A",0
4,1,55,81,0.068417,"\int \tanh ^{-1}(a+b x)^2 \, dx","Integrate[ArcTanh[a + b*x]^2,x]","\frac{\text{Li}_2\left(-e^{-2 \tanh ^{-1}(a+b x)}\right)+\tanh ^{-1}(a+b x) \left((a+b x-1) \tanh ^{-1}(a+b x)-2 \log \left(e^{-2 \tanh ^{-1}(a+b x)}+1\right)\right)}{b}","-\frac{\text{Li}_2\left(-\frac{a+b x+1}{-a-b x+1}\right)}{b}+\frac{(a+b x) \tanh ^{-1}(a+b x)^2}{b}+\frac{\tanh ^{-1}(a+b x)^2}{b}-\frac{2 \log \left(\frac{2}{-a-b x+1}\right) \tanh ^{-1}(a+b x)}{b}",1,"(ArcTanh[a + b*x]*((-1 + a + b*x)*ArcTanh[a + b*x] - 2*Log[1 + E^(-2*ArcTanh[a + b*x])]) + PolyLog[2, -E^(-2*ArcTanh[a + b*x])])/b","A",0
5,1,472,148,2.6449777,"\int \frac{\tanh ^{-1}(a+b x)^2}{x} \, dx","Integrate[ArcTanh[a + b*x]^2/x,x]","\frac{2 \sqrt{1-a^2} e^{\tanh ^{-1}(a)} \tanh ^{-1}(a+b x)^3}{3 a}+\tanh ^{-1}(a+b x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a+b x)}\right)+2 \tanh ^{-1}(a+b x) \text{Li}_2\left(-e^{\tanh ^{-1}(a+b x)-\tanh ^{-1}(a)}\right)+2 \tanh ^{-1}(a+b x) \text{Li}_2\left(e^{\tanh ^{-1}(a+b x)-\tanh ^{-1}(a)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(a+b x)}\right)-2 \text{Li}_3\left(-e^{\tanh ^{-1}(a+b x)-\tanh ^{-1}(a)}\right)-2 \text{Li}_3\left(e^{\tanh ^{-1}(a+b x)-\tanh ^{-1}(a)}\right)-\frac{2 \tanh ^{-1}(a+b x)^3}{3 a}-\frac{4}{3} \tanh ^{-1}(a+b x)^3-\tanh ^{-1}(a+b x)^2 \log \left(e^{-2 \tanh ^{-1}(a+b x)}+1\right)+\tanh ^{-1}(a+b x)^2 \log \left(\frac{1}{2} e^{-\tanh ^{-1}(a+b x)} \left(a e^{2 \tanh ^{-1}(a+b x)}-e^{2 \tanh ^{-1}(a+b x)}+a+1\right)\right)+\tanh ^{-1}(a+b x)^2 \log \left(1-e^{\tanh ^{-1}(a+b x)-\tanh ^{-1}(a)}\right)+\tanh ^{-1}(a+b x)^2 \log \left(e^{\tanh ^{-1}(a+b x)-\tanh ^{-1}(a)}+1\right)-\log \left(-\frac{b x}{\sqrt{1-(a+b x)^2}}\right) \tanh ^{-1}(a+b x)^2-i \pi  \tanh ^{-1}(a+b x) \log \left(\frac{1}{2} \left(e^{-\tanh ^{-1}(a+b x)}+e^{\tanh ^{-1}(a+b x)}\right)\right)-2 \tanh ^{-1}(a) \tanh ^{-1}(a+b x) \log \left(\frac{1}{2} i \left(e^{\tanh ^{-1}(a+b x)-\tanh ^{-1}(a)}-e^{\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)}\right)\right)+i \pi  \log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right) \tanh ^{-1}(a+b x)+2 \tanh ^{-1}(a) \tanh ^{-1}(a+b x) \log \left(-i \sinh \left(\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)\right)\right)","\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) \tanh ^{-1}(a+b x)-\text{Li}_2\left(1-\frac{2 b x}{(1-a) (a+b x+1)}\right) \tanh ^{-1}(a+b x)-\log \left(\frac{2}{a+b x+1}\right) \tanh ^{-1}(a+b x)^2+\log \left(\frac{2 b x}{(1-a) (a+b x+1)}\right) \tanh ^{-1}(a+b x)^2",1,"(-4*ArcTanh[a + b*x]^3)/3 - (2*ArcTanh[a + b*x]^3)/(3*a) + (2*Sqrt[1 - a^2]*E^ArcTanh[a]*ArcTanh[a + b*x]^3)/(3*a) - ArcTanh[a + b*x]^2*Log[1 + E^(-2*ArcTanh[a + b*x])] - I*Pi*ArcTanh[a + b*x]*Log[(E^(-ArcTanh[a + b*x]) + E^ArcTanh[a + b*x])/2] + ArcTanh[a + b*x]^2*Log[(1 + a - E^(2*ArcTanh[a + b*x]) + a*E^(2*ArcTanh[a + b*x]))/(2*E^ArcTanh[a + b*x])] + ArcTanh[a + b*x]^2*Log[1 - E^(-ArcTanh[a] + ArcTanh[a + b*x])] + ArcTanh[a + b*x]^2*Log[1 + E^(-ArcTanh[a] + ArcTanh[a + b*x])] - 2*ArcTanh[a]*ArcTanh[a + b*x]*Log[(I/2)*(-E^(ArcTanh[a] - ArcTanh[a + b*x]) + E^(-ArcTanh[a] + ArcTanh[a + b*x]))] + I*Pi*ArcTanh[a + b*x]*Log[1/Sqrt[1 - (a + b*x)^2]] - ArcTanh[a + b*x]^2*Log[-((b*x)/Sqrt[1 - (a + b*x)^2])] + 2*ArcTanh[a]*ArcTanh[a + b*x]*Log[(-I)*Sinh[ArcTanh[a] - ArcTanh[a + b*x]]] + ArcTanh[a + b*x]*PolyLog[2, -E^(-2*ArcTanh[a + b*x])] + 2*ArcTanh[a + b*x]*PolyLog[2, -E^(-ArcTanh[a] + ArcTanh[a + b*x])] + 2*ArcTanh[a + b*x]*PolyLog[2, E^(-ArcTanh[a] + ArcTanh[a + b*x])] + PolyLog[3, -E^(-2*ArcTanh[a + b*x])]/2 - 2*PolyLog[3, -E^(-ArcTanh[a] + ArcTanh[a + b*x])] - 2*PolyLog[3, E^(-ArcTanh[a] + ArcTanh[a + b*x])]","C",0
6,1,208,251,1.4370466,"\int \frac{\tanh ^{-1}(a+b x)^2}{x^2} \, dx","Integrate[ArcTanh[a + b*x]^2/x^2,x]","\frac{-\left(\left(a^3+a^2 b x+b x \left(\sqrt{1-a^2} e^{\tanh ^{-1}(a)}-1\right)-a\right) \tanh ^{-1}(a+b x)^2\right)+a b x \text{Li}_2\left(e^{2 \tanh ^{-1}(a)-2 \tanh ^{-1}(a+b x)}\right)+a b x \tanh ^{-1}(a+b x) \left(-2 \log \left(1-e^{2 \tanh ^{-1}(a)-2 \tanh ^{-1}(a+b x)}\right)+2 \tanh ^{-1}(a)-i \pi \right)+a b x \left(i \pi  \left(\log \left(e^{2 \tanh ^{-1}(a+b x)}+1\right)-\log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right)\right)+2 \tanh ^{-1}(a) \left(\log \left(1-e^{2 \tanh ^{-1}(a)-2 \tanh ^{-1}(a+b x)}\right)-\log \left(-i \sinh \left(\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)\right)\right)\right)\right)}{a \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) \tanh ^{-1}(a+b x)}{1-a^2}+\frac{2 b \log \left(\frac{2 b x}{(1-a) (a+b x+1)}\right) \tanh ^{-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{\tanh ^{-1}(a+b x)^2}{x}+\frac{b \log \left(\frac{2}{-a-b x+1}\right) \tanh ^{-1}(a+b x)}{1-a}+\frac{b \log \left(\frac{2}{a+b x+1}\right) \tanh ^{-1}(a+b x)}{a+1}",1,"(-((-a + a^3 + a^2*b*x + b*(-1 + Sqrt[1 - a^2]*E^ArcTanh[a])*x)*ArcTanh[a + b*x]^2) + a*b*x*ArcTanh[a + b*x]*((-I)*Pi + 2*ArcTanh[a] - 2*Log[1 - E^(2*ArcTanh[a] - 2*ArcTanh[a + b*x])]) + a*b*x*(I*Pi*(Log[1 + E^(2*ArcTanh[a + b*x])] - Log[1/Sqrt[1 - (a + b*x)^2]]) + 2*ArcTanh[a]*(Log[1 - E^(2*ArcTanh[a] - 2*ArcTanh[a + b*x])] - Log[(-I)*Sinh[ArcTanh[a] - ArcTanh[a + b*x]]])) + a*b*x*PolyLog[2, E^(2*ArcTanh[a] - 2*ArcTanh[a + b*x])])/(a*(-1 + a^2)*x)","C",0
7,1,271,370,2.3495575,"\int \frac{\tanh ^{-1}(a+b x)^2}{x^3} \, dx","Integrate[ArcTanh[a + b*x]^2/x^3,x]","\frac{2 b x \tanh ^{-1}(a+b x) \left(a^2+a b x+i \pi  a b x-2 a b x \tanh ^{-1}(a)+2 a b x \log \left(1-e^{2 \tanh ^{-1}(a)-2 \tanh ^{-1}(a+b x)}\right)-1\right)-\left(a^4-a^2 \left(b^2 x^2+2\right)-b^2 x^2 \left(2 \sqrt{1-a^2} e^{\tanh ^{-1}(a)}-1\right)+1\right) \tanh ^{-1}(a+b x)^2-2 a b^2 x^2 \text{Li}_2\left(e^{2 \tanh ^{-1}(a)-2 \tanh ^{-1}(a+b x)}\right)+2 b^2 x^2 \left(i \pi  a \log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right)+\log \left(-\frac{b x}{\sqrt{1-(a+b x)^2}}\right)-i \pi  a \log \left(e^{2 \tanh ^{-1}(a+b x)}+1\right)-2 a \tanh ^{-1}(a) \left(\log \left(1-e^{2 \tanh ^{-1}(a)-2 \tanh ^{-1}(a+b x)}\right)-\log \left(-i \sinh \left(\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)\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) \tanh ^{-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) \tanh ^{-1}(a+b x)}{\left(1-a^2\right)^2}-\frac{b \tanh ^{-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) \tanh ^{-1}(a+b x)}{2 (1-a)^2}-\frac{b^2 \log \left(\frac{2}{a+b x+1}\right) \tanh ^{-1}(a+b x)}{2 (a+1)^2}-\frac{\tanh ^{-1}(a+b x)^2}{2 x^2}",1,"(-((1 + a^4 - b^2*(-1 + 2*Sqrt[1 - a^2]*E^ArcTanh[a])*x^2 - a^2*(2 + b^2*x^2))*ArcTanh[a + b*x]^2) + 2*b*x*ArcTanh[a + b*x]*(-1 + a^2 + a*b*x + I*a*b*Pi*x - 2*a*b*x*ArcTanh[a] + 2*a*b*x*Log[1 - E^(2*ArcTanh[a] - 2*ArcTanh[a + b*x])]) + 2*b^2*x^2*((-I)*a*Pi*Log[1 + E^(2*ArcTanh[a + b*x])] + I*a*Pi*Log[1/Sqrt[1 - (a + b*x)^2]] + Log[-((b*x)/Sqrt[1 - (a + b*x)^2])] - 2*a*ArcTanh[a]*(Log[1 - E^(2*ArcTanh[a] - 2*ArcTanh[a + b*x])] - Log[(-I)*Sinh[ArcTanh[a] - ArcTanh[a + b*x]]])) - 2*a*b^2*x^2*PolyLog[2, E^(2*ArcTanh[a] - 2*ArcTanh[a + b*x])])/(2*(-1 + a^2)^2*x^2)","C",0
8,1,75,56,0.0904647,"\int \frac{\tanh ^{-1}(1+b x)^2}{x} \, dx","Integrate[ArcTanh[1 + b*x]^2/x,x]","\tanh ^{-1}(b x+1) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(b x+1)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(b x+1)}\right)-\frac{2}{3} \tanh ^{-1}(b x+1)^3-\tanh ^{-1}(b x+1)^2 \log \left(e^{-2 \tanh ^{-1}(b x+1)}+1\right)","\frac{1}{2} \text{Li}_3\left(1+\frac{2}{b x}\right)-\text{Li}_2\left(1+\frac{2}{b x}\right) \tanh ^{-1}(b x+1)-\log \left(-\frac{2}{b x}\right) \tanh ^{-1}(b x+1)^2",1,"(-2*ArcTanh[1 + b*x]^3)/3 - ArcTanh[1 + b*x]^2*Log[1 + E^(-2*ArcTanh[1 + b*x])] + ArcTanh[1 + b*x]*PolyLog[2, -E^(-2*ArcTanh[1 + b*x])] + PolyLog[3, -E^(-2*ArcTanh[1 + b*x])]/2","A",0
9,1,78,72,0.1044739,"\int (c e+d e x)^3 \left(a+b \tanh ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcTanh[c + d*x]),x]","\frac{e^3 \left(6 a (c+d x)^4+2 b (c+d x)^3+6 b (c+d x)+3 b \log (-c-d x+1)-3 b \log (c+d x+1)+6 b (c+d x)^4 \tanh ^{-1}(c+d x)\right)}{24 d}","\frac{e^3 (c+d x)^4 \left(a+b \tanh ^{-1}(c+d x)\right)}{4 d}+\frac{b e^3 (c+d x)^3}{12 d}-\frac{b e^3 \tanh ^{-1}(c+d x)}{4 d}+\frac{1}{4} b e^3 x",1,"(e^3*(6*b*(c + d*x) + 2*b*(c + d*x)^3 + 6*a*(c + d*x)^4 + 6*b*(c + d*x)^4*ArcTanh[c + d*x] + 3*b*Log[1 - c - d*x] - 3*b*Log[1 + c + d*x]))/(24*d)","A",1
10,1,59,69,0.0754448,"\int (c e+d e x)^2 \left(a+b \tanh ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcTanh[c + d*x]),x]","\frac{e^2 \left((c+d x)^2 (2 a (c+d x)+b)+b \log \left(1-(c+d x)^2\right)+2 b (c+d x)^3 \tanh ^{-1}(c+d x)\right)}{6 d}","\frac{e^2 (c+d x)^3 \left(a+b \tanh ^{-1}(c+d x)\right)}{3 d}+\frac{b e^2 (c+d x)^2}{6 d}+\frac{b e^2 \log \left(1-(c+d x)^2\right)}{6 d}",1,"(e^2*((c + d*x)^2*(b + 2*a*(c + d*x)) + 2*b*(c + d*x)^3*ArcTanh[c + d*x] + b*Log[1 - (c + d*x)^2]))/(6*d)","A",1
11,1,77,48,0.0494028,"\int (c e+d e x) \left(a+b \tanh ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcTanh[c + d*x]),x]","\frac{e \left(2 a c^2+4 a c d x+2 a d^2 x^2+b \log (-c-d x+1)-b \log (c+d x+1)+2 b (c+d x)^2 \tanh ^{-1}(c+d x)+2 b c+2 b d x\right)}{4 d}","\frac{e (c+d x)^2 \left(a+b \tanh ^{-1}(c+d x)\right)}{2 d}-\frac{b e \tanh ^{-1}(c+d x)}{2 d}+\frac{b e x}{2}",1,"(e*(2*b*c + 2*a*c^2 + 2*b*d*x + 4*a*c*d*x + 2*a*d^2*x^2 + 2*b*(c + d*x)^2*ArcTanh[c + d*x] + b*Log[1 - c - d*x] - b*Log[1 + c + d*x]))/(4*d)","A",1
12,1,54,54,0.0250525,"\int \frac{a+b \tanh ^{-1}(c+d x)}{c e+d e x} \, dx","Integrate[(a + b*ArcTanh[c + d*x])/(c*e + d*e*x),x]","\frac{a \log (c+d x)}{d e}-\frac{b \text{Li}_2(-c-d x)}{2 d e}+\frac{b \text{Li}_2(c+d x)}{2 d e}","\frac{a \log (c+d x)}{d e}-\frac{b \text{Li}_2(-c-d x)}{2 d e}+\frac{b \text{Li}_2(c+d x)}{2 d e}",1,"(a*Log[c + d*x])/(d*e) - (b*PolyLog[2, -c - d*x])/(2*d*e) + (b*PolyLog[2, c + d*x])/(2*d*e)","A",1
13,1,69,63,0.0705161,"\int \frac{a+b \tanh ^{-1}(c+d x)}{(c e+d e x)^2} \, dx","Integrate[(a + b*ArcTanh[c + d*x])/(c*e + d*e*x)^2,x]","-\frac{\frac{2 a}{c+d x}+b \log \left(-c^2-2 c d x-d^2 x^2+1\right)-2 b \log (c+d x)+\frac{2 b \tanh ^{-1}(c+d x)}{c+d x}}{2 d e^2}","-\frac{a+b \tanh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac{b \log (c+d x)}{d e^2}-\frac{b \log \left(1-(c+d x)^2\right)}{2 d e^2}",1,"-1/2*((2*a)/(c + d*x) + (2*b*ArcTanh[c + d*x])/(c + d*x) - 2*b*Log[c + d*x] + b*Log[1 - c^2 - 2*c*d*x - d^2*x^2])/(d*e^2)","A",1
14,1,100,63,0.0627084,"\int \frac{a+b \tanh ^{-1}(c+d x)}{(c e+d e x)^3} \, dx","Integrate[(a + b*ArcTanh[c + d*x])/(c*e + d*e*x)^3,x]","-\frac{a}{2 d e^3 (c+d x)^2}-\frac{b}{2 d e^3 (c+d x)}-\frac{b \log (-c-d x+1)}{4 d e^3}+\frac{b \log (c+d x+1)}{4 d e^3}-\frac{b \tanh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}","-\frac{a+b \tanh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac{b}{2 d e^3 (c+d x)}+\frac{b \tanh ^{-1}(c+d x)}{2 d e^3}",1,"-1/2*a/(d*e^3*(c + d*x)^2) - b/(2*d*e^3*(c + d*x)) - (b*ArcTanh[c + d*x])/(2*d*e^3*(c + d*x)^2) - (b*Log[1 - c - d*x])/(4*d*e^3) + (b*Log[1 + c + d*x])/(4*d*e^3)","A",1
15,1,148,159,0.1922188,"\int (c e+d e x)^3 \left(a+b \tanh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcTanh[c + d*x])^2,x]","\frac{e^3 \left(3 a^2 (c+d x)^4+2 a b (c+d x)^3+6 a b (c+d x)+b (3 a+4 b) \log (-c-d x+1)+b (4 b-3 a) \log (c+d x+1)+2 b (c+d x) \tanh ^{-1}(c+d x) \left(3 a (c+d x)^3+b (c+d x)^2+3 b\right)+b^2 (c+d x)^2+3 b^2 \left((c+d x)^4-1\right) \tanh ^{-1}(c+d x)^2\right)}{12 d}","\frac{e^3 (c+d x)^4 \left(a+b \tanh ^{-1}(c+d x)\right)^2}{4 d}+\frac{b e^3 (c+d x)^3 \left(a+b \tanh ^{-1}(c+d x)\right)}{6 d}-\frac{e^3 \left(a+b \tanh ^{-1}(c+d x)\right)^2}{4 d}+\frac{1}{2} a b e^3 x+\frac{b^2 e^3 (c+d x)^2}{12 d}+\frac{b^2 e^3 \log \left(1-(c+d x)^2\right)}{3 d}+\frac{b^2 e^3 (c+d x) \tanh ^{-1}(c+d x)}{2 d}",1,"(e^3*(6*a*b*(c + d*x) + b^2*(c + d*x)^2 + 2*a*b*(c + d*x)^3 + 3*a^2*(c + d*x)^4 + 2*b*(c + d*x)*(3*b + b*(c + d*x)^2 + 3*a*(c + d*x)^3)*ArcTanh[c + d*x] + 3*b^2*(-1 + (c + d*x)^4)*ArcTanh[c + d*x]^2 + b*(3*a + 4*b)*Log[1 - c - d*x] + b*(-3*a + 4*b)*Log[1 + c + d*x]))/(12*d)","A",1
16,1,150,179,0.4701031,"\int (c e+d e x)^2 \left(a+b \tanh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcTanh[c + d*x])^2,x]","\frac{e^2 \left(a^2 (c+d x)^3+a b \left((c+d x)^2+\log \left((c+d x)^2-1\right)+2 (c+d x)^3 \tanh ^{-1}(c+d x)\right)+b^2 \left(\text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+(c+d x)^3 \tanh ^{-1}(c+d x)^2+(c+d x)^2 \tanh ^{-1}(c+d x)-\tanh ^{-1}(c+d x)^2-\tanh ^{-1}(c+d x)-2 \tanh ^{-1}(c+d x) \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)+c+d x\right)\right)}{3 d}","\frac{e^2 (c+d x)^3 \left(a+b \tanh ^{-1}(c+d x)\right)^2}{3 d}+\frac{b e^2 (c+d x)^2 \left(a+b \tanh ^{-1}(c+d x)\right)}{3 d}+\frac{e^2 \left(a+b \tanh ^{-1}(c+d x)\right)^2}{3 d}-\frac{2 b e^2 \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{3 d}-\frac{b^2 e^2 \text{Li}_2\left(-\frac{c+d x+1}{-c-d x+1}\right)}{3 d}-\frac{b^2 e^2 \tanh ^{-1}(c+d x)}{3 d}+\frac{1}{3} b^2 e^2 x",1,"(e^2*(a^2*(c + d*x)^3 + a*b*((c + d*x)^2 + 2*(c + d*x)^3*ArcTanh[c + d*x] + Log[-1 + (c + d*x)^2]) + b^2*(c + d*x - ArcTanh[c + d*x] + (c + d*x)^2*ArcTanh[c + d*x] - ArcTanh[c + d*x]^2 + (c + d*x)^3*ArcTanh[c + d*x]^2 - 2*ArcTanh[c + d*x]*Log[1 + E^(-2*ArcTanh[c + d*x])] + PolyLog[2, -E^(-2*ArcTanh[c + d*x])])))/(3*d)","A",0
17,1,134,95,0.102795,"\int (c e+d e x) \left(a+b \tanh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcTanh[c + d*x])^2,x]","e \left(\frac{a^2 (c+d x)^2}{2 d}+\frac{\left(a b+b^2\right) \log (-c-d x+1)}{2 d}+\frac{\left(b^2-a b\right) \log (c+d x+1)}{2 d}+\frac{a b (c+d x)}{d}+\frac{b (c+d x) \tanh ^{-1}(c+d x) (a (c+d x)+b)}{d}+\frac{\left(b^2 (c+d x)^2-b^2\right) \tanh ^{-1}(c+d x)^2}{2 d}\right)","\frac{e (c+d x)^2 \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d}-\frac{e \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d}+a b e x+\frac{b^2 e \log \left(1-(c+d x)^2\right)}{2 d}+\frac{b^2 e (c+d x) \tanh ^{-1}(c+d x)}{d}",1,"e*((a*b*(c + d*x))/d + (a^2*(c + d*x)^2)/(2*d) + (b*(c + d*x)*(b + a*(c + d*x))*ArcTanh[c + d*x])/d + ((-b^2 + b^2*(c + d*x)^2)*ArcTanh[c + d*x]^2)/(2*d) + ((a*b + b^2)*Log[1 - c - d*x])/(2*d) + ((-(a*b) + b^2)*Log[1 + c + d*x])/(2*d))","A",1
18,1,424,168,0.392575,"\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{c e+d e x} \, dx","Integrate[(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x),x]","\frac{a^2 \log (c+d x)-\frac{1}{4} a b \left(4 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c+d x)}\right)+4 \text{Li}_2\left(-e^{2 \tanh ^{-1}(c+d x)}\right)-4 i \pi  \log \left(\frac{2}{\sqrt{1-(c+d x)^2}}\right)-8 \tanh ^{-1}(c+d x)^2-4 i \pi  \tanh ^{-1}(c+d x)-8 \tanh ^{-1}(c+d x) \log \left(1-e^{-2 \tanh ^{-1}(c+d x)}\right)+8 \tanh ^{-1}(c+d x) \log \left(e^{2 \tanh ^{-1}(c+d x)}+1\right)-8 \log \left(\frac{2}{\sqrt{1-(c+d x)^2}}\right) \tanh ^{-1}(c+d x)+8 \log \left(\frac{2 i (c+d x)}{\sqrt{1-(c+d x)^2}}\right) \tanh ^{-1}(c+d x)+4 i \pi  \log \left(e^{2 \tanh ^{-1}(c+d x)}+1\right)+\pi ^2\right)+2 a b \left(-\log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)+\log \left(\frac{i (c+d x)}{\sqrt{1-(c+d x)^2}}\right)\right) \tanh ^{-1}(c+d x)+b^2 \left(\tanh ^{-1}(c+d x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+\tanh ^{-1}(c+d x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c+d x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(c+d x)}\right)-\frac{2}{3} \tanh ^{-1}(c+d x)^3-\tanh ^{-1}(c+d x)^2 \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)+\tanh ^{-1}(c+d x)^2 \log \left(1-e^{2 \tanh ^{-1}(c+d x)}\right)+\frac{i \pi ^3}{24}\right)}{d e}","-\frac{b \text{Li}_2\left(1-\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{d e}+\frac{b \text{Li}_2\left(\frac{2}{-c-d x+1}-1\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{d e}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)^2}{d e}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{-c-d x+1}\right)}{2 d e}-\frac{b^2 \text{Li}_3\left(\frac{2}{-c-d x+1}-1\right)}{2 d e}",1,"(a^2*Log[c + d*x] + 2*a*b*ArcTanh[c + d*x]*(-Log[1/Sqrt[1 - (c + d*x)^2]] + Log[(I*(c + d*x))/Sqrt[1 - (c + d*x)^2]]) - (a*b*(Pi^2 - (4*I)*Pi*ArcTanh[c + d*x] - 8*ArcTanh[c + d*x]^2 - 8*ArcTanh[c + d*x]*Log[1 - E^(-2*ArcTanh[c + d*x])] + (4*I)*Pi*Log[1 + E^(2*ArcTanh[c + d*x])] + 8*ArcTanh[c + d*x]*Log[1 + E^(2*ArcTanh[c + d*x])] - (4*I)*Pi*Log[2/Sqrt[1 - (c + d*x)^2]] - 8*ArcTanh[c + d*x]*Log[2/Sqrt[1 - (c + d*x)^2]] + 8*ArcTanh[c + d*x]*Log[((2*I)*(c + d*x))/Sqrt[1 - (c + d*x)^2]] + 4*PolyLog[2, E^(-2*ArcTanh[c + d*x])] + 4*PolyLog[2, -E^(2*ArcTanh[c + d*x])]))/4 + b^2*((I/24)*Pi^3 - (2*ArcTanh[c + d*x]^3)/3 - ArcTanh[c + d*x]^2*Log[1 + E^(-2*ArcTanh[c + d*x])] + ArcTanh[c + d*x]^2*Log[1 - E^(2*ArcTanh[c + d*x])] + ArcTanh[c + d*x]*PolyLog[2, -E^(-2*ArcTanh[c + d*x])] + ArcTanh[c + d*x]*PolyLog[2, E^(2*ArcTanh[c + d*x])] + PolyLog[3, -E^(-2*ArcTanh[c + d*x])]/2 - PolyLog[3, E^(2*ArcTanh[c + d*x])]/2))/(d*e)","C",0
19,1,126,104,0.2540683,"\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{(c e+d e x)^2} \, dx","Integrate[(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^2,x]","\frac{a \left(2 b (c+d x) \log \left(\frac{c+d x}{\sqrt{1-(c+d x)^2}}\right)-a\right)+2 b \tanh ^{-1}(c+d x) \left(b (c+d x) \log \left(1-e^{-2 \tanh ^{-1}(c+d x)}\right)-a\right)-b^2 (c+d x) \text{Li}_2\left(e^{-2 \tanh ^{-1}(c+d x)}\right)+b^2 (c+d x-1) \tanh ^{-1}(c+d x)^2}{d e^2 (c+d x)}","-\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{d e^2 (c+d x)}+\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{d e^2}+\frac{2 b \log \left(2-\frac{2}{c+d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{d e^2}-\frac{b^2 \text{Li}_2\left(\frac{2}{c+d x+1}-1\right)}{d e^2}",1,"(b^2*(-1 + c + d*x)*ArcTanh[c + d*x]^2 + 2*b*ArcTanh[c + d*x]*(-a + b*(c + d*x)*Log[1 - E^(-2*ArcTanh[c + d*x])]) + a*(-a + 2*b*(c + d*x)*Log[(c + d*x)/Sqrt[1 - (c + d*x)^2]]) - b^2*(c + d*x)*PolyLog[2, E^(-2*ArcTanh[c + d*x])])/(d*e^2*(c + d*x))","A",0
20,1,136,119,0.1678193,"\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{(c e+d e x)^3} \, dx","Integrate[(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^3,x]","\frac{-\frac{a^2}{(c+d x)^2}-\frac{2 a b}{c+d x}-b (a+b) \log (-c-d x+1)+b (a-b) \log (c+d x+1)-\frac{2 b \tanh ^{-1}(c+d x) (a+b (c+d x))}{(c+d x)^2}+\frac{b^2 \left(c^2+2 c d x+d^2 x^2-1\right) \tanh ^{-1}(c+d x)^2}{(c+d x)^2}+2 b^2 \log (c+d x)}{2 d e^3}","-\frac{b \left(a+b \tanh ^{-1}(c+d x)\right)}{d e^3 (c+d x)}-\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d e^3 (c+d x)^2}+\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d e^3}+\frac{b^2 \log (c+d x)}{d e^3}-\frac{b^2 \log \left(1-(c+d x)^2\right)}{2 d e^3}",1,"(-(a^2/(c + d*x)^2) - (2*a*b)/(c + d*x) - (2*b*(a + b*(c + d*x))*ArcTanh[c + d*x])/(c + d*x)^2 + (b^2*(-1 + c^2 + 2*c*d*x + d^2*x^2)*ArcTanh[c + d*x]^2)/(c + d*x)^2 - b*(a + b)*Log[1 - c - d*x] + 2*b^2*Log[c + d*x] + (a - b)*b*Log[1 + c + d*x])/(2*d*e^3)","A",1
21,1,218,180,0.5380027,"\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{(c e+d e x)^4} \, dx","Integrate[(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^4,x]","-\frac{a^2-a b \left((c+d x) \left(c^2+2 c d x+2 (c+d x)^2 \log \left(\frac{c+d x}{\sqrt{1-(c+d x)^2}}\right)+d^2 x^2-1\right)-2 \tanh ^{-1}(c+d x)\right)+b^2 \left((c+d x)^3 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c+d x)}\right)+(c+d x)^2+(c+d x)^2 \tanh ^{-1}(c+d x)^2+\left(1-(c+d x)^2\right) \tanh ^{-1}(c+d x)^2+(c+d x) \tanh ^{-1}(c+d x) \left(-(c+d x)^2+(c+d x)^2 \left(-\tanh ^{-1}(c+d x)\right)-2 (c+d x)^2 \log \left(1-e^{-2 \tanh ^{-1}(c+d x)}\right)+1\right)\right)}{3 d e^4 (c+d x)^3}","-\frac{b \left(a+b \tanh ^{-1}(c+d x)\right)}{3 d e^4 (c+d x)^2}-\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{3 d e^4 (c+d x)^3}+\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{3 d e^4}+\frac{2 b \log \left(2-\frac{2}{c+d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{3 d e^4}-\frac{b^2 \text{Li}_2\left(\frac{2}{c+d x+1}-1\right)}{3 d e^4}-\frac{b^2}{3 d e^4 (c+d x)}+\frac{b^2 \tanh ^{-1}(c+d x)}{3 d e^4}",1,"-1/3*(a^2 - a*b*(-2*ArcTanh[c + d*x] + (c + d*x)*(-1 + c^2 + 2*c*d*x + d^2*x^2 + 2*(c + d*x)^2*Log[(c + d*x)/Sqrt[1 - (c + d*x)^2]])) + b^2*((c + d*x)^2 + (c + d*x)^2*ArcTanh[c + d*x]^2 + (1 - (c + d*x)^2)*ArcTanh[c + d*x]^2 + (c + d*x)*ArcTanh[c + d*x]*(1 - (c + d*x)^2 - (c + d*x)^2*ArcTanh[c + d*x] - 2*(c + d*x)^2*Log[1 - E^(-2*ArcTanh[c + d*x])]) + (c + d*x)^3*PolyLog[2, E^(-2*ArcTanh[c + d*x])]))/(d*e^4*(c + d*x)^3)","A",0
22,1,218,172,0.2634934,"\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{(c e+d e x)^5} \, dx","Integrate[(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^5,x]","-\frac{\frac{3 a^2}{(c+d x)^4}+\frac{2 b \tanh ^{-1}(c+d x) \left(3 a+b \left(3 c^3+9 c^2 d x+9 c d^2 x^2+c+3 d^3 x^3+d x\right)\right)}{(c+d x)^4}+\frac{6 a b}{c+d x}+\frac{2 a b}{(c+d x)^3}+b (3 a+4 b) \log (-c-d x+1)-b (3 a-4 b) \log (c+d x+1)-\frac{3 b^2 \left(c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4-1\right) \tanh ^{-1}(c+d x)^2}{(c+d x)^4}+\frac{b^2}{(c+d x)^2}-8 b^2 \log (c+d x)}{12 d e^5}","-\frac{b \left(a+b \tanh ^{-1}(c+d x)\right)}{2 d e^5 (c+d x)}-\frac{b \left(a+b \tanh ^{-1}(c+d x)\right)}{6 d e^5 (c+d x)^3}-\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{4 d e^5 (c+d x)^4}+\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{4 d e^5}-\frac{b^2}{12 d e^5 (c+d x)^2}+\frac{2 b^2 \log (c+d x)}{3 d e^5}-\frac{b^2 \log \left(1-(c+d x)^2\right)}{3 d e^5}",1,"-1/12*((3*a^2)/(c + d*x)^4 + (2*a*b)/(c + d*x)^3 + b^2/(c + d*x)^2 + (6*a*b)/(c + d*x) + (2*b*(3*a + b*(c + 3*c^3 + d*x + 9*c^2*d*x + 9*c*d^2*x^2 + 3*d^3*x^3))*ArcTanh[c + d*x])/(c + d*x)^4 - (3*b^2*(-1 + c^4 + 4*c^3*d*x + 6*c^2*d^2*x^2 + 4*c*d^3*x^3 + d^4*x^4)*ArcTanh[c + d*x]^2)/(c + d*x)^4 + b*(3*a + 4*b)*Log[1 - c - d*x] - 8*b^2*Log[c + d*x] - (3*a - 4*b)*b*Log[1 + c + d*x])/(d*e^5)","A",1
23,1,336,263,0.7428272,"\int (c e+d e x)^2 \left(a+b \tanh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcTanh[c + d*x])^3,x]","\frac{e^2 \left(2 a^3 (c+d x)^3+3 a^2 b (c+d x)^2+3 a^2 b \log \left(1-(c+d x)^2\right)+6 a^2 b (c+d x)^3 \tanh ^{-1}(c+d x)+6 a b^2 \left(\text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+(c+d x)^3 \tanh ^{-1}(c+d x)^2+(c+d x)^2 \tanh ^{-1}(c+d x)-\tanh ^{-1}(c+d x)^2-\tanh ^{-1}(c+d x)-2 \tanh ^{-1}(c+d x) \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)+c+d x\right)+b^3 \left(6 \tanh ^{-1}(c+d x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+3 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)-6 \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)+2 (c+d x) \tanh ^{-1}(c+d x)^3-2 (c+d x) \left(1-(c+d x)^2\right) \tanh ^{-1}(c+d x)^3-2 \tanh ^{-1}(c+d x)^3-3 \left(1-(c+d x)^2\right) \tanh ^{-1}(c+d x)^2+6 (c+d x) \tanh ^{-1}(c+d x)-6 \tanh ^{-1}(c+d x)^2 \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)\right)\right)}{6 d}","-\frac{b^2 e^2 \text{Li}_2\left(1-\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{d}+a b^2 e^2 x-\frac{b e^2 \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d}+\frac{b e^2 (c+d x)^2 \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d}+\frac{e^2 (c+d x)^3 \left(a+b \tanh ^{-1}(c+d x)\right)^3}{3 d}+\frac{e^2 \left(a+b \tanh ^{-1}(c+d x)\right)^3}{3 d}-\frac{b e^2 \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)^2}{d}+\frac{b^3 e^2 \text{Li}_3\left(1-\frac{2}{-c-d x+1}\right)}{2 d}+\frac{b^3 e^2 \log \left(1-(c+d x)^2\right)}{2 d}+\frac{b^3 e^2 (c+d x) \tanh ^{-1}(c+d x)}{d}",1,"(e^2*(3*a^2*b*(c + d*x)^2 + 2*a^3*(c + d*x)^3 + 6*a^2*b*(c + d*x)^3*ArcTanh[c + d*x] + 3*a^2*b*Log[1 - (c + d*x)^2] + 6*a*b^2*(c + d*x - ArcTanh[c + d*x] + (c + d*x)^2*ArcTanh[c + d*x] - ArcTanh[c + d*x]^2 + (c + d*x)^3*ArcTanh[c + d*x]^2 - 2*ArcTanh[c + d*x]*Log[1 + E^(-2*ArcTanh[c + d*x])] + PolyLog[2, -E^(-2*ArcTanh[c + d*x])]) + b^3*(6*(c + d*x)*ArcTanh[c + d*x] - 3*(1 - (c + d*x)^2)*ArcTanh[c + d*x]^2 - 2*ArcTanh[c + d*x]^3 + 2*(c + d*x)*ArcTanh[c + d*x]^3 - 2*(c + d*x)*(1 - (c + d*x)^2)*ArcTanh[c + d*x]^3 - 6*ArcTanh[c + d*x]^2*Log[1 + E^(-2*ArcTanh[c + d*x])] - 6*Log[1/Sqrt[1 - (c + d*x)^2]] + 6*ArcTanh[c + d*x]*PolyLog[2, -E^(-2*ArcTanh[c + d*x])] + 3*PolyLog[3, -E^(-2*ArcTanh[c + d*x])])))/(6*d)","A",0
24,1,213,160,1.1347193,"\int (c e+d e x) \left(a+b \tanh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcTanh[c + d*x])^3,x]","\frac{e \left(a \left(-3 a b \left(c^2-1\right) \log (-c-d x+1)+3 a b \left(c^2-1\right) \log (c+d x+1)+2 a d x (2 a c+a d x+3 b)-12 b^2 \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)\right)+6 b^2 (c+d x-1) \tanh ^{-1}(c+d x)^2 (a (c+d x+1)+b)+6 b \tanh ^{-1}(c+d x) \left(a (a d x (2 c+d x)+2 b (c+d x))-2 b^2 \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)\right)+2 b^3 \left(c^2+2 c d x+d^2 x^2-1\right) \tanh ^{-1}(c+d x)^3+6 b^3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)\right)}{4 d}","-\frac{3 b^2 e \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{d}+\frac{3 b e \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d}+\frac{3 b e (c+d x) \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d}+\frac{e (c+d x)^2 \left(a+b \tanh ^{-1}(c+d x)\right)^3}{2 d}-\frac{e \left(a+b \tanh ^{-1}(c+d x)\right)^3}{2 d}-\frac{3 b^3 e \text{Li}_2\left(-\frac{c+d x+1}{-c-d x+1}\right)}{2 d}",1,"(e*(6*b^2*(-1 + c + d*x)*(b + a*(1 + c + d*x))*ArcTanh[c + d*x]^2 + 2*b^3*(-1 + c^2 + 2*c*d*x + d^2*x^2)*ArcTanh[c + d*x]^3 + 6*b*ArcTanh[c + d*x]*(a*(2*b*(c + d*x) + a*d*x*(2*c + d*x)) - 2*b^2*Log[1 + E^(-2*ArcTanh[c + d*x])]) + a*(2*a*d*x*(3*b + 2*a*c + a*d*x) - 3*a*b*(-1 + c^2)*Log[1 - c - d*x] + 3*a*b*(-1 + c^2)*Log[1 + c + d*x] - 12*b^2*Log[1/Sqrt[1 - (c + d*x)^2]]) + 6*b^3*PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/(4*d)","A",0
25,1,599,257,0.5114166,"\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^3}{c e+d e x} \, dx","Integrate[(a + b*ArcTanh[c + d*x])^3/(c*e + d*e*x),x]","\frac{64 a^3 \log (c+d x)-96 i a^2 b \left(-i \text{Li}_2\left(e^{-2 \tanh ^{-1}(c+d x)}\right)-i \text{Li}_2\left(-e^{2 \tanh ^{-1}(c+d x)}\right)-\frac{1}{4} i \left(\pi -2 i \tanh ^{-1}(c+d x)\right)^2+i \tanh ^{-1}(c+d x)^2+\left(\pi -2 i \tanh ^{-1}(c+d x)\right) \log \left(e^{2 \tanh ^{-1}(c+d x)}+1\right)-\log \left(\frac{2}{\sqrt{1-(c+d x)^2}}\right) \left(\pi -2 i \tanh ^{-1}(c+d x)\right)+2 i \tanh ^{-1}(c+d x) \log \left(1-e^{-2 \tanh ^{-1}(c+d x)}\right)-2 i \log \left(\frac{2 i (c+d x)}{\sqrt{1-(c+d x)^2}}\right) \tanh ^{-1}(c+d x)\right)+192 a^2 b \left(-\log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)+\log \left(\frac{i (c+d x)}{\sqrt{1-(c+d x)^2}}\right)\right) \tanh ^{-1}(c+d x)+8 a b^2 \left(24 \tanh ^{-1}(c+d x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+24 \tanh ^{-1}(c+d x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c+d x)}\right)+12 \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)-12 \text{Li}_3\left(e^{2 \tanh ^{-1}(c+d x)}\right)-16 \tanh ^{-1}(c+d x)^3-24 \tanh ^{-1}(c+d x)^2 \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)+24 \tanh ^{-1}(c+d x)^2 \log \left(1-e^{2 \tanh ^{-1}(c+d x)}\right)+i \pi ^3\right)+b^3 \left(96 \tanh ^{-1}(c+d x)^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+96 \tanh ^{-1}(c+d x)^2 \text{Li}_2\left(e^{2 \tanh ^{-1}(c+d x)}\right)+96 \tanh ^{-1}(c+d x) \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)-96 \tanh ^{-1}(c+d x) \text{Li}_3\left(e^{2 \tanh ^{-1}(c+d x)}\right)+48 \text{Li}_4\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+48 \text{Li}_4\left(e^{2 \tanh ^{-1}(c+d x)}\right)-32 \tanh ^{-1}(c+d x)^4-64 \tanh ^{-1}(c+d x)^3 \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)+64 \tanh ^{-1}(c+d x)^3 \log \left(1-e^{2 \tanh ^{-1}(c+d x)}\right)+\pi ^4\right)}{64 d e}","\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{2 d e}-\frac{3 b^2 \text{Li}_3\left(\frac{2}{-c-d x+1}-1\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{2 d e}-\frac{3 b \text{Li}_2\left(1-\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d e}+\frac{3 b \text{Li}_2\left(\frac{2}{-c-d x+1}-1\right) \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d e}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)^3}{d e}-\frac{3 b^3 \text{Li}_4\left(1-\frac{2}{-c-d x+1}\right)}{4 d e}+\frac{3 b^3 \text{Li}_4\left(\frac{2}{-c-d x+1}-1\right)}{4 d e}",1,"(64*a^3*Log[c + d*x] + 192*a^2*b*ArcTanh[c + d*x]*(-Log[1/Sqrt[1 - (c + d*x)^2]] + Log[(I*(c + d*x))/Sqrt[1 - (c + d*x)^2]]) - (96*I)*a^2*b*((-1/4*I)*(Pi - (2*I)*ArcTanh[c + d*x])^2 + I*ArcTanh[c + d*x]^2 + (2*I)*ArcTanh[c + d*x]*Log[1 - E^(-2*ArcTanh[c + d*x])] + (Pi - (2*I)*ArcTanh[c + d*x])*Log[1 + E^(2*ArcTanh[c + d*x])] - (Pi - (2*I)*ArcTanh[c + d*x])*Log[2/Sqrt[1 - (c + d*x)^2]] - (2*I)*ArcTanh[c + d*x]*Log[((2*I)*(c + d*x))/Sqrt[1 - (c + d*x)^2]] - I*PolyLog[2, E^(-2*ArcTanh[c + d*x])] - I*PolyLog[2, -E^(2*ArcTanh[c + d*x])]) + 8*a*b^2*(I*Pi^3 - 16*ArcTanh[c + d*x]^3 - 24*ArcTanh[c + d*x]^2*Log[1 + E^(-2*ArcTanh[c + d*x])] + 24*ArcTanh[c + d*x]^2*Log[1 - E^(2*ArcTanh[c + d*x])] + 24*ArcTanh[c + d*x]*PolyLog[2, -E^(-2*ArcTanh[c + d*x])] + 24*ArcTanh[c + d*x]*PolyLog[2, E^(2*ArcTanh[c + d*x])] + 12*PolyLog[3, -E^(-2*ArcTanh[c + d*x])] - 12*PolyLog[3, E^(2*ArcTanh[c + d*x])]) + b^3*(Pi^4 - 32*ArcTanh[c + d*x]^4 - 64*ArcTanh[c + d*x]^3*Log[1 + E^(-2*ArcTanh[c + d*x])] + 64*ArcTanh[c + d*x]^3*Log[1 - E^(2*ArcTanh[c + d*x])] + 96*ArcTanh[c + d*x]^2*PolyLog[2, -E^(-2*ArcTanh[c + d*x])] + 96*ArcTanh[c + d*x]^2*PolyLog[2, E^(2*ArcTanh[c + d*x])] + 96*ArcTanh[c + d*x]*PolyLog[3, -E^(-2*ArcTanh[c + d*x])] - 96*ArcTanh[c + d*x]*PolyLog[3, E^(2*ArcTanh[c + d*x])] + 48*PolyLog[4, -E^(-2*ArcTanh[c + d*x])] + 48*PolyLog[4, E^(2*ArcTanh[c + d*x])]))/(64*d*e)","C",0
26,1,248,143,0.6050654,"\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^3}{(c e+d e x)^2} \, dx","Integrate[(a + b*ArcTanh[c + d*x])^3/(c*e + d*e*x)^2,x]","\frac{-\frac{2 a^3}{c+d x}-3 a^2 b \log \left(-c^2-2 c d x-d^2 x^2+1\right)+6 a^2 b \log (c+d x)-\frac{6 a^2 b \tanh ^{-1}(c+d x)}{c+d x}+6 a b^2 \left(\tanh ^{-1}(c+d x) \left(\left(1-\frac{1}{c+d x}\right) \tanh ^{-1}(c+d x)+2 \log \left(1-e^{-2 \tanh ^{-1}(c+d x)}\right)\right)-\text{Li}_2\left(e^{-2 \tanh ^{-1}(c+d x)}\right)\right)+2 b^3 \left(3 \tanh ^{-1}(c+d x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c+d x)}\right)-\frac{3}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(c+d x)}\right)-\frac{\tanh ^{-1}(c+d x)^3}{c+d x}-\tanh ^{-1}(c+d x)^3+3 \tanh ^{-1}(c+d x)^2 \log \left(1-e^{2 \tanh ^{-1}(c+d x)}\right)+\frac{i \pi ^3}{8}\right)}{2 d e^2}","-\frac{3 b^2 \text{Li}_2\left(\frac{2}{c+d x+1}-1\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{d e^2}-\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^3}{d e^2 (c+d x)}+\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^3}{d e^2}+\frac{3 b \log \left(2-\frac{2}{c+d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)^2}{d e^2}-\frac{3 b^3 \text{Li}_3\left(\frac{2}{c+d x+1}-1\right)}{2 d e^2}",1,"((-2*a^3)/(c + d*x) - (6*a^2*b*ArcTanh[c + d*x])/(c + d*x) + 6*a^2*b*Log[c + d*x] - 3*a^2*b*Log[1 - c^2 - 2*c*d*x - d^2*x^2] + 6*a*b^2*(ArcTanh[c + d*x]*((1 - (c + d*x)^(-1))*ArcTanh[c + d*x] + 2*Log[1 - E^(-2*ArcTanh[c + d*x])]) - PolyLog[2, E^(-2*ArcTanh[c + d*x])]) + 2*b^3*((I/8)*Pi^3 - ArcTanh[c + d*x]^3 - ArcTanh[c + d*x]^3/(c + d*x) + 3*ArcTanh[c + d*x]^2*Log[1 - E^(2*ArcTanh[c + d*x])] + 3*ArcTanh[c + d*x]*PolyLog[2, E^(2*ArcTanh[c + d*x])] - (3*PolyLog[3, E^(2*ArcTanh[c + d*x])])/2))/(2*d*e^2)","C",0
27,1,335,166,1.178447,"\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^3}{(c e+d e x)^3} \, dx","Integrate[(a + b*ArcTanh[c + d*x])^3/(c*e + d*e*x)^3,x]","\frac{-4 a^3-12 a^2 b c-12 a^2 b d x+12 b \tanh ^{-1}(c+d x) \left(a \left(a \left(c^2+2 c d x+d^2 x^2-1\right)-2 b (c+d x)\right)+2 b^2 (c+d x)^2 \log \left(1-e^{-2 \tanh ^{-1}(c+d x)}\right)\right)+24 a b^2 c^2 \log \left(\frac{c+d x}{\sqrt{1-(c+d x)^2}}\right)+24 a b^2 d^2 x^2 \log \left(\frac{c+d x}{\sqrt{1-(c+d x)^2}}\right)+48 a b^2 c d x \log \left(\frac{c+d x}{\sqrt{1-(c+d x)^2}}\right)+12 b^2 (c+d x-1) \tanh ^{-1}(c+d x)^2 (a (c+d x+1)+b (c+d x))+i \pi ^3 b^3 c^3+4 b^3 \left(c^2+2 c d x+d^2 x^2-1\right) \tanh ^{-1}(c+d x)^3+2 i \pi ^3 b^3 c^2 d x+i \pi ^3 b^3 c d^2 x^2-12 b^3 (c+d x)^2 \text{Li}_2\left(e^{-2 \tanh ^{-1}(c+d x)}\right)}{8 d e^3 (c+d x)^2}","\frac{3 b^2 \log \left(2-\frac{2}{c+d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{d e^3}-\frac{3 b \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d e^3 (c+d x)}+\frac{3 b \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d e^3}-\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^3}{2 d e^3 (c+d x)^2}+\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^3}{2 d e^3}-\frac{3 b^3 \text{Li}_2\left(\frac{2}{c+d x+1}-1\right)}{2 d e^3}",1,"(-4*a^3 - 12*a^2*b*c + I*b^3*c^3*Pi^3 - 12*a^2*b*d*x + (2*I)*b^3*c^2*d*Pi^3*x + I*b^3*c*d^2*Pi^3*x^2 + 12*b^2*(-1 + c + d*x)*(b*(c + d*x) + a*(1 + c + d*x))*ArcTanh[c + d*x]^2 + 4*b^3*(-1 + c^2 + 2*c*d*x + d^2*x^2)*ArcTanh[c + d*x]^3 + 12*b*ArcTanh[c + d*x]*(a*(-2*b*(c + d*x) + a*(-1 + c^2 + 2*c*d*x + d^2*x^2)) + 2*b^2*(c + d*x)^2*Log[1 - E^(-2*ArcTanh[c + d*x])]) + 24*a*b^2*c^2*Log[(c + d*x)/Sqrt[1 - (c + d*x)^2]] + 48*a*b^2*c*d*x*Log[(c + d*x)/Sqrt[1 - (c + d*x)^2]] + 24*a*b^2*d^2*x^2*Log[(c + d*x)/Sqrt[1 - (c + d*x)^2]] - 12*b^3*(c + d*x)^2*PolyLog[2, E^(-2*ArcTanh[c + d*x])])/(8*d*e^3*(c + d*x)^2)","C",0
28,1,393,269,1.3344533,"\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^3}{(c e+d e x)^4} \, dx","Integrate[(a + b*ArcTanh[c + d*x])^3/(c*e + d*e*x)^4,x]","\frac{-\frac{2 a^3}{(c+d x)^3}-3 a^2 b \log \left(-c^2-2 c d x-d^2 x^2+1\right)-\frac{3 a^2 b}{(c+d x)^2}+6 a^2 b \log (c+d x)-\frac{6 a^2 b \tanh ^{-1}(c+d x)}{(c+d x)^3}+6 a b^2 \left(-\text{Li}_2\left(e^{-2 \tanh ^{-1}(c+d x)}\right)-\frac{(c+d x)^2+\tanh ^{-1}(c+d x)^2}{(c+d x)^3}+\tanh ^{-1}(c+d x) \left(-\frac{1-(c+d x)^2}{(c+d x)^2}+\tanh ^{-1}(c+d x)+2 \log \left(1-e^{-2 \tanh ^{-1}(c+d x)}\right)\right)\right)+6 b^3 \left(\tanh ^{-1}(c+d x) \text{Li}_2\left(e^{2 \tanh ^{-1}(c+d x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{2 \tanh ^{-1}(c+d x)}\right)+\log \left(\frac{c+d x}{\sqrt{1-(c+d x)^2}}\right)-\frac{\left(1-(c+d x)^2\right) \tanh ^{-1}(c+d x)^3}{3 (c+d x)^3}-\frac{\tanh ^{-1}(c+d x)^3}{3 (c+d x)}-\frac{1}{3} \tanh ^{-1}(c+d x)^3-\frac{\left(1-(c+d x)^2\right) \tanh ^{-1}(c+d x)^2}{2 (c+d x)^2}-\frac{\tanh ^{-1}(c+d x)}{c+d x}+\tanh ^{-1}(c+d x)^2 \log \left(1-e^{2 \tanh ^{-1}(c+d x)}\right)+\frac{i \pi ^3}{24}\right)}{6 d e^4}","-\frac{b^2 \text{Li}_2\left(\frac{2}{c+d x+1}-1\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{d e^4}-\frac{b^2 \left(a+b \tanh ^{-1}(c+d x)\right)}{d e^4 (c+d x)}-\frac{b \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d e^4 (c+d x)^2}+\frac{b \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d e^4}-\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^3}{3 d e^4 (c+d x)^3}+\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^3}{3 d e^4}+\frac{b \log \left(2-\frac{2}{c+d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)^2}{d e^4}-\frac{b^3 \text{Li}_3\left(\frac{2}{c+d x+1}-1\right)}{2 d e^4}+\frac{b^3 \log (c+d x)}{d e^4}-\frac{b^3 \log \left(1-(c+d x)^2\right)}{2 d e^4}",1,"((-2*a^3)/(c + d*x)^3 - (3*a^2*b)/(c + d*x)^2 - (6*a^2*b*ArcTanh[c + d*x])/(c + d*x)^3 + 6*a^2*b*Log[c + d*x] - 3*a^2*b*Log[1 - c^2 - 2*c*d*x - d^2*x^2] + 6*a*b^2*(-(((c + d*x)^2 + ArcTanh[c + d*x]^2)/(c + d*x)^3) + ArcTanh[c + d*x]*(-((1 - (c + d*x)^2)/(c + d*x)^2) + ArcTanh[c + d*x] + 2*Log[1 - E^(-2*ArcTanh[c + d*x])]) - PolyLog[2, E^(-2*ArcTanh[c + d*x])]) + 6*b^3*((I/24)*Pi^3 - ArcTanh[c + d*x]/(c + d*x) - ((1 - (c + d*x)^2)*ArcTanh[c + d*x]^2)/(2*(c + d*x)^2) - ArcTanh[c + d*x]^3/3 - ArcTanh[c + d*x]^3/(3*(c + d*x)) - ((1 - (c + d*x)^2)*ArcTanh[c + d*x]^3)/(3*(c + d*x)^3) + ArcTanh[c + d*x]^2*Log[1 - E^(2*ArcTanh[c + d*x])] + Log[(c + d*x)/Sqrt[1 - (c + d*x)^2]] + ArcTanh[c + d*x]*PolyLog[2, E^(2*ArcTanh[c + d*x])] - PolyLog[3, E^(2*ArcTanh[c + d*x])]/2))/(6*d*e^4)","C",0
29,1,31,21,0.0034606,"\int \frac{\tanh ^{-1}(1+x)}{2+2 x} \, dx","Integrate[ArcTanh[1 + x]/(2 + 2*x),x]","\frac{1}{4} \text{Li}_2\left(\frac{1}{2} (2 x+2)\right)-\frac{1}{4} \text{Li}_2\left(\frac{1}{2} (-2 x-2)\right)","\frac{\text{Li}_2(x+1)}{4}-\frac{\text{Li}_2(-x-1)}{4}",1,"-1/4*PolyLog[2, (-2 - 2*x)/2] + PolyLog[2, (2 + 2*x)/2]/4","A",1
30,1,52,32,0.0063543,"\int \frac{\tanh ^{-1}(a+b x)}{\frac{a d}{b}+d x} \, dx","Integrate[ArcTanh[a + b*x]/((a*d)/b + d*x),x]","b \left(\frac{\text{Li}_2\left(\frac{a d+b x d}{d}\right)}{2 b d}-\frac{\text{Li}_2\left(-\frac{a d+b x d}{d}\right)}{2 b d}\right)","\frac{\text{Li}_2(a+b x)}{2 d}-\frac{\text{Li}_2(-a-b x)}{2 d}",1,"b*(-1/2*PolyLog[2, -((a*d + b*d*x)/d)]/(b*d) + PolyLog[2, (a*d + b*d*x)/d]/(2*b*d))","A",1
31,1,270,168,0.2814563,"\int (e+f x)^3 \left(a+b \tanh ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)^3*(a + b*ArcTanh[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) \tanh ^{-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 \tanh ^{-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)*ArcTanh[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
32,1,174,120,0.1679669,"\int (e+f x)^2 \left(a+b \tanh ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)^2*(a + b*ArcTanh[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) \tanh ^{-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 \tanh ^{-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)*ArcTanh[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
33,1,138,97,0.0451803,"\int (e+f x) \left(a+b \tanh ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)*(a + b*ArcTanh[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 \tanh ^{-1}(c+d x)+\frac{1}{2} b f x^2 \tanh ^{-1}(c+d x)+\frac{b f x}{2 d}","\frac{(e+f x)^2 \left(a+b \tanh ^{-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*ArcTanh[c + d*x] + (b*f*x^2*ArcTanh[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
34,1,48,40,0.0158744,"\int \left(a+b \tanh ^{-1}(c+d x)\right) \, dx","Integrate[a + b*ArcTanh[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 \tanh ^{-1}(c+d x)","a x+\frac{b \log \left(1-(c+d x)^2\right)}{2 d}+\frac{b (c+d x) \tanh ^{-1}(c+d x)}{d}",1,"a*x + b*x*ArcTanh[c + d*x] + (b*(-((-1 + c)*Log[1 - c - d*x]) + (1 + c)*Log[1 + c + d*x]))/(2*d)","A",1
35,1,126,130,0.0761463,"\int \frac{a+b \tanh ^{-1}(c+d x)}{e+f x} \, dx","Integrate[(a + b*ArcTanh[c + d*x])/(e + f*x),x]","\frac{2 a \log (e+f x)-b \text{Li}_2\left(\frac{f (c+d x-1)}{(c-1) f-d e}\right)+b \text{Li}_2\left(\frac{f (c+d x+1)}{-d e+c f+f}\right)-b \log (-c-d x+1) \log \left(\frac{d (e+f x)}{-c f+d e+f}\right)+b \log (c+d x+1) \log \left(\frac{d (e+f x)}{d e-(c+1) f}\right)}{2 f}","\frac{\left(a+b \tanh ^{-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 \tanh ^{-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,"(2*a*Log[e + f*x] - b*Log[1 - c - d*x]*Log[(d*(e + f*x))/(d*e + f - c*f)] + b*Log[1 + c + d*x]*Log[(d*(e + f*x))/(d*e - (1 + c)*f)] - b*PolyLog[2, (f*(-1 + c + d*x))/(-(d*e) + (-1 + c)*f)] + b*PolyLog[2, (f*(1 + c + d*x))/(-(d*e) + f + c*f)])/(2*f)","A",0
36,1,125,115,0.1814029,"\int \frac{a+b \tanh ^{-1}(c+d x)}{(e+f x)^2} \, dx","Integrate[(a + b*ArcTanh[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 \tanh ^{-1}(c+d x)}{f (e+f x)}\right)","-\frac{a+b \tanh ^{-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*ArcTanh[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
37,1,174,167,0.3199817,"\int \frac{a+b \tanh ^{-1}(c+d x)}{(e+f x)^3} \, dx","Integrate[(a + b*ArcTanh[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 \tanh ^{-1}(c+d x)}{f (e+f x)^2}\right)","-\frac{a+b \tanh ^{-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*ArcTanh[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
38,1,1082,562,7.4811021,"\int (e+f x)^3 \left(a+b \tanh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(e + f*x)^3*(a + b*ArcTanh[c + d*x])^2,x]","\frac{1}{12} \left(3 a^2 f^3 x^4+12 a^2 e f^2 x^3+18 a^2 e^2 f x^2+12 a^2 e^3 x+a b \left(6 x \left(4 e^3+6 f x e^2+4 f^2 x^2 e+f^3 x^3\right) \tanh ^{-1}(c+d x)-\frac{-2 d f x \left(\left(18 e^2+6 f x e+f^2 x^2\right) d^2-3 c f (8 e+f x) d+3 \left(3 c^2+1\right) f^2\right)+3 (c-1) \left(4 d^3 e^3-6 (c-1) d^2 f e^2+4 (c-1)^2 d f^2 e-(c-1)^3 f^3\right) \log (-c-d x+1)+3 (c+1) \left(-4 d^3 e^3+6 (c+1) d^2 f e^2-4 (c+1)^2 d f^2 e+(c+1)^3 f^3\right) \log (c+d x+1)}{d^4}\right)+\frac{12 b^2 e^3 \left(\tanh ^{-1}(c+d x) \left((c+d x-1) \tanh ^{-1}(c+d x)-2 \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right)\right)+\text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)\right)}{d}-\frac{18 b^2 e^2 f \left(\left(c^2-2 c-d^2 x^2+1\right) \tanh ^{-1}(c+d x)^2-2 \left(2 \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right) c+c+d x\right) \tanh ^{-1}(c+d x)+2 \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)+2 c \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)\right)}{d^2}+\frac{b^2 f^3 \left(-36 \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right) c^2-11 c^2-10 d x c+d^2 x^2-3 \left(c^4-4 c^3+6 c^2-4 c-d^4 x^4+1\right) \tanh ^{-1}(c+d x)^2+2 \tanh ^{-1}(c+d x) \left(13 c^3+9 d x c^2-3 d^2 x^2 c+9 c+d^3 x^3+3 d x+12 \left(c^3+c\right) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right)\right)-8 \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)-12 \left(c^3+c\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)-1\right)}{d^4}-\frac{3 b^2 e f^2 \left(1-(c+d x)^2\right)^{3/2} \left(-\frac{3 (c+d x) \tanh ^{-1}(c+d x)^2 c^2}{\sqrt{1-(c+d x)^2}}+3 \tanh ^{-1}(c+d x)^2 \cosh \left(3 \tanh ^{-1}(c+d x)\right) c^2+6 \tanh ^{-1}(c+d x) \cosh \left(3 \tanh ^{-1}(c+d x)\right) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right) c^2-3 \tanh ^{-1}(c+d x)^2 \sinh \left(3 \tanh ^{-1}(c+d x)\right) c^2+\frac{6 (c+d x) \tanh ^{-1}(c+d x) c}{\sqrt{1-(c+d x)^2}}-6 \cosh \left(3 \tanh ^{-1}(c+d x)\right) \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right) c+6 \tanh ^{-1}(c+d x) \sinh \left(3 \tanh ^{-1}(c+d x)\right) c+\frac{3 (c+d x) \tanh ^{-1}(c+d x)^2}{\sqrt{1-(c+d x)^2}}+\tanh ^{-1}(c+d x)^2 \cosh \left(3 \tanh ^{-1}(c+d x)\right)+2 \tanh ^{-1}(c+d x) \cosh \left(3 \tanh ^{-1}(c+d x)\right) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right)+\frac{3 \left(3 c^2-4 c+1\right) \tanh ^{-1}(c+d x)^2+2 \left(\left(9 c^2+3\right) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right)+2\right) \tanh ^{-1}(c+d x)-18 c \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)}{\sqrt{1-(c+d x)^2}}-\frac{4 \left(3 c^2+1\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)}{\left(1-(c+d x)^2\right)^{3/2}}-\tanh ^{-1}(c+d x)^2 \sinh \left(3 \tanh ^{-1}(c+d x)\right)-\sinh \left(3 \tanh ^{-1}(c+d x)\right)-\frac{c+d x}{\sqrt{1-(c+d x)^2}}\right)}{d^3}\right)","\frac{(d e-c f) \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right) \left(a+b \tanh ^{-1}(c+d x)\right)^2}{d^4}-\frac{2 b (d e-c f) \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right) \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{d^4}+\frac{a b f x \left(\left(6 c^2+1\right) f^2-12 c d e f+6 d^2 e^2\right)}{2 d^3}-\frac{\left(6 \left(c^2+1\right) d^2 e^2 f^2-4 c \left(c^2+3\right) d e f^3+\left(c^4+6 c^2+1\right) f^4-4 c d^3 e^3 f+d^4 e^4\right) \left(a+b \tanh ^{-1}(c+d x)\right)^2}{4 d^4 f}+\frac{b f^2 (c+d x)^2 (d e-c f) \left(a+b \tanh ^{-1}(c+d x)\right)}{d^4}+\frac{b f^3 (c+d x)^3 \left(a+b \tanh ^{-1}(c+d x)\right)}{6 d^4}+\frac{(e+f x)^4 \left(a+b \tanh ^{-1}(c+d x)\right)^2}{4 f}-\frac{b^2 (d e-c f) \left(\left(c^2+1\right) f^2-2 c d e f+d^2 e^2\right) \text{Li}_2\left(-\frac{c+d x+1}{-c-d x+1}\right)}{d^4}+\frac{b^2 f \left(\left(6 c^2+1\right) f^2-12 c d e f+6 d^2 e^2\right) \log \left(1-(c+d x)^2\right)}{4 d^4}+\frac{b^2 f (c+d x) \left(\left(6 c^2+1\right) f^2-12 c d e f+6 d^2 e^2\right) \tanh ^{-1}(c+d x)}{2 d^4}-\frac{b^2 f^2 (d e-c f) \tanh ^{-1}(c+d x)}{d^4}+\frac{b^2 f^3 (c+d x)^2}{12 d^4}+\frac{b^2 f^3 \log \left(1-(c+d x)^2\right)}{12 d^4}+\frac{b^2 f^2 x (d e-c f)}{d^3}",1,"(12*a^2*e^3*x + 18*a^2*e^2*f*x^2 + 12*a^2*e*f^2*x^3 + 3*a^2*f^3*x^4 + a*b*(6*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)*ArcTanh[c + d*x] - (-2*d*f*x*(3*(1 + 3*c^2)*f^2 - 3*c*d*f*(8*e + f*x) + d^2*(18*e^2 + 6*e*f*x + f^2*x^2)) + 3*(-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*(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])/d^4) + (12*b^2*e^3*(ArcTanh[c + d*x]*((-1 + c + d*x)*ArcTanh[c + d*x] - 2*Log[1 + E^(-2*ArcTanh[c + d*x])]) + PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/d - (18*b^2*e^2*f*((1 - 2*c + c^2 - d^2*x^2)*ArcTanh[c + d*x]^2 - 2*ArcTanh[c + d*x]*(c + d*x + 2*c*Log[1 + E^(-2*ArcTanh[c + d*x])]) + 2*Log[1/Sqrt[1 - (c + d*x)^2]] + 2*c*PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/d^2 + (b^2*f^3*(-1 - 11*c^2 - 10*c*d*x + d^2*x^2 - 3*(1 - 4*c + 6*c^2 - 4*c^3 + c^4 - d^4*x^4)*ArcTanh[c + d*x]^2 + 2*ArcTanh[c + d*x]*(9*c + 13*c^3 + 3*d*x + 9*c^2*d*x - 3*c*d^2*x^2 + d^3*x^3 + 12*(c + c^3)*Log[1 + E^(-2*ArcTanh[c + d*x])]) - 8*Log[1/Sqrt[1 - (c + d*x)^2]] - 36*c^2*Log[1/Sqrt[1 - (c + d*x)^2]] - 12*(c + c^3)*PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/d^4 - (3*b^2*e*f^2*(1 - (c + d*x)^2)^(3/2)*(-((c + d*x)/Sqrt[1 - (c + d*x)^2]) + (6*c*(c + d*x)*ArcTanh[c + d*x])/Sqrt[1 - (c + d*x)^2] + (3*(c + d*x)*ArcTanh[c + d*x]^2)/Sqrt[1 - (c + d*x)^2] - (3*c^2*(c + d*x)*ArcTanh[c + d*x]^2)/Sqrt[1 - (c + d*x)^2] + ArcTanh[c + d*x]^2*Cosh[3*ArcTanh[c + d*x]] + 3*c^2*ArcTanh[c + d*x]^2*Cosh[3*ArcTanh[c + d*x]] + 2*ArcTanh[c + d*x]*Cosh[3*ArcTanh[c + d*x]]*Log[1 + E^(-2*ArcTanh[c + d*x])] + 6*c^2*ArcTanh[c + d*x]*Cosh[3*ArcTanh[c + d*x]]*Log[1 + E^(-2*ArcTanh[c + d*x])] - 6*c*Cosh[3*ArcTanh[c + d*x]]*Log[1/Sqrt[1 - (c + d*x)^2]] + (3*(1 - 4*c + 3*c^2)*ArcTanh[c + d*x]^2 + 2*ArcTanh[c + d*x]*(2 + (3 + 9*c^2)*Log[1 + E^(-2*ArcTanh[c + d*x])]) - 18*c*Log[1/Sqrt[1 - (c + d*x)^2]])/Sqrt[1 - (c + d*x)^2] - (4*(1 + 3*c^2)*PolyLog[2, -E^(-2*ArcTanh[c + d*x])])/(1 - (c + d*x)^2)^(3/2) - Sinh[3*ArcTanh[c + d*x]] + 6*c*ArcTanh[c + d*x]*Sinh[3*ArcTanh[c + d*x]] - ArcTanh[c + d*x]^2*Sinh[3*ArcTanh[c + d*x]] - 3*c^2*ArcTanh[c + d*x]^2*Sinh[3*ArcTanh[c + d*x]]))/d^3)/12","A",0
39,1,795,374,3.8089121,"\int (e+f x)^2 \left(a+b \tanh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(e + f*x)^2*(a + b*ArcTanh[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) \tanh ^{-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(\tanh ^{-1}(c+d x) \left((c+d x-1) \tanh ^{-1}(c+d x)-2 \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right)\right)+\text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)\right)}{d}+\frac{b^2 e f \left(\left(-c^2+2 c+d^2 x^2-1\right) \tanh ^{-1}(c+d x)^2+2 \left(2 \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right) c+c+d x\right) \tanh ^{-1}(c+d x)-2 \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)-2 c \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)\right)}{d^2}-\frac{b^2 f^2 \left(1-(c+d x)^2\right)^{3/2} \left(-\frac{3 (c+d x) \tanh ^{-1}(c+d x)^2 c^2}{\sqrt{1-(c+d x)^2}}+3 \tanh ^{-1}(c+d x)^2 \cosh \left(3 \tanh ^{-1}(c+d x)\right) c^2+6 \tanh ^{-1}(c+d x) \cosh \left(3 \tanh ^{-1}(c+d x)\right) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right) c^2-3 \tanh ^{-1}(c+d x)^2 \sinh \left(3 \tanh ^{-1}(c+d x)\right) c^2+\frac{6 (c+d x) \tanh ^{-1}(c+d x) c}{\sqrt{1-(c+d x)^2}}-6 \cosh \left(3 \tanh ^{-1}(c+d x)\right) \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right) c+6 \tanh ^{-1}(c+d x) \sinh \left(3 \tanh ^{-1}(c+d x)\right) c+\frac{3 (c+d x) \tanh ^{-1}(c+d x)^2}{\sqrt{1-(c+d x)^2}}+\tanh ^{-1}(c+d x)^2 \cosh \left(3 \tanh ^{-1}(c+d x)\right)+2 \tanh ^{-1}(c+d x) \cosh \left(3 \tanh ^{-1}(c+d x)\right) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right)+\frac{3 \left(3 c^2-4 c+1\right) \tanh ^{-1}(c+d x)^2+2 \left(\left(9 c^2+3\right) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right)+2\right) \tanh ^{-1}(c+d x)-18 c \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)}{\sqrt{1-(c+d x)^2}}-\frac{4 \left(3 c^2+1\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)}{\left(1-(c+d x)^2\right)^{3/2}}-\tanh ^{-1}(c+d x)^2 \sinh \left(3 \tanh ^{-1}(c+d x)\right)-\sinh \left(3 \tanh ^{-1}(c+d x)\right)-\frac{c+d x}{\sqrt{1-(c+d x)^2}}\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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-1}(c+d x)\right)}{3 d^3}+\frac{b f^2 (c+d x)^2 \left(a+b \tanh ^{-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 \tanh ^{-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) \tanh ^{-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)*ArcTanh[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*(ArcTanh[c + d*x]*((-1 + c + d*x)*ArcTanh[c + d*x] - 2*Log[1 + E^(-2*ArcTanh[c + d*x])]) + PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/d + (b^2*e*f*((-1 + 2*c - c^2 + d^2*x^2)*ArcTanh[c + d*x]^2 + 2*ArcTanh[c + d*x]*(c + d*x + 2*c*Log[1 + E^(-2*ArcTanh[c + d*x])]) - 2*Log[1/Sqrt[1 - (c + d*x)^2]] - 2*c*PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/d^2 - (b^2*f^2*(1 - (c + d*x)^2)^(3/2)*(-((c + d*x)/Sqrt[1 - (c + d*x)^2]) + (6*c*(c + d*x)*ArcTanh[c + d*x])/Sqrt[1 - (c + d*x)^2] + (3*(c + d*x)*ArcTanh[c + d*x]^2)/Sqrt[1 - (c + d*x)^2] - (3*c^2*(c + d*x)*ArcTanh[c + d*x]^2)/Sqrt[1 - (c + d*x)^2] + ArcTanh[c + d*x]^2*Cosh[3*ArcTanh[c + d*x]] + 3*c^2*ArcTanh[c + d*x]^2*Cosh[3*ArcTanh[c + d*x]] + 2*ArcTanh[c + d*x]*Cosh[3*ArcTanh[c + d*x]]*Log[1 + E^(-2*ArcTanh[c + d*x])] + 6*c^2*ArcTanh[c + d*x]*Cosh[3*ArcTanh[c + d*x]]*Log[1 + E^(-2*ArcTanh[c + d*x])] - 6*c*Cosh[3*ArcTanh[c + d*x]]*Log[1/Sqrt[1 - (c + d*x)^2]] + (3*(1 - 4*c + 3*c^2)*ArcTanh[c + d*x]^2 + 2*ArcTanh[c + d*x]*(2 + (3 + 9*c^2)*Log[1 + E^(-2*ArcTanh[c + d*x])]) - 18*c*Log[1/Sqrt[1 - (c + d*x)^2]])/Sqrt[1 - (c + d*x)^2] - (4*(1 + 3*c^2)*PolyLog[2, -E^(-2*ArcTanh[c + d*x])])/(1 - (c + d*x)^2)^(3/2) - Sinh[3*ArcTanh[c + d*x]] + 6*c*ArcTanh[c + d*x]*Sinh[3*ArcTanh[c + d*x]] - ArcTanh[c + d*x]^2*Sinh[3*ArcTanh[c + d*x]] - 3*c^2*ArcTanh[c + d*x]^2*Sinh[3*ArcTanh[c + d*x]]))/(12*d^3)","B",0
40,1,271,221,0.5409062,"\int (e+f x) \left(a+b \tanh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(e + f*x)*(a + b*ArcTanh[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 \tanh ^{-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(e^{-2 \tanh ^{-1}(c+d x)}+1\right)\right)-4 a b d e \log \left(\frac{1}{\sqrt{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}{\sqrt{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 \tanh ^{-1}(c+d x)}\right)+b^2 (c+d x-1) \tanh ^{-1}(c+d x)^2 (-c f+2 d e+d f x+f)-2 b^2 f \log \left(\frac{1}{\sqrt{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 \tanh ^{-1}(c+d x)\right)^2}{2 d^2 f}+\frac{(d e-c f) \left(a+b \tanh ^{-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 \tanh ^{-1}(c+d x)\right)}{d^2}+\frac{(e+f x)^2 \left(a+b \tanh ^{-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) \tanh ^{-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)*ArcTanh[c + d*x]^2 + 2*b*ArcTanh[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*ArcTanh[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/Sqrt[1 - (c + d*x)^2]] - 2*b^2*f*Log[1/Sqrt[1 - (c + d*x)^2]] + 4*a*b*c*f*Log[1/Sqrt[1 - (c + d*x)^2]] + 2*b^2*(d*e - c*f)*PolyLog[2, -E^(-2*ArcTanh[c + d*x])])/(2*d^2)","A",0
41,1,107,97,0.1889525,"\int \left(a+b \tanh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(a + b*ArcTanh[c + d*x])^2,x]","\frac{a (a d x+(b-b c) \log (-c-d x+1)+b (c+1) \log (c+d x+1))+2 b \tanh ^{-1}(c+d x) \left(a d x-b \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)\right)+b^2 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+b^2 (c+d x-1) \tanh ^{-1}(c+d x)^2}{d}","\frac{(c+d x) \left(a+b \tanh ^{-1}(c+d x)\right)^2}{d}+\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{d}-\frac{2 b \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-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)*ArcTanh[c + d*x]^2 + 2*b*ArcTanh[c + d*x]*(a*d*x - b*Log[1 + E^(-2*ArcTanh[c + d*x])]) + a*(a*d*x + (b - b*c)*Log[1 - c - d*x] + b*(1 + c)*Log[1 + c + d*x]) + b^2*PolyLog[2, -E^(-2*ArcTanh[c + d*x])])/d","A",0
42,1,1757,214,24.2759521,"\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{e+f x} \, dx","Integrate[(a + b*ArcTanh[c + d*x])^2/(e + f*x),x]","\frac{\log (e+f x) a^2}{f}-\frac{2 i b \left(i \tanh ^{-1}(c+d x) \left(\log \left(i \sinh \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)\right)-\log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)\right)+\frac{1}{2} \left(-i \left(i \tanh ^{-1}\left(\frac{d e-c f}{f}\right)+i \tanh ^{-1}(c+d x)\right)^2+2 \log \left(1-e^{2 i \left(i \tanh ^{-1}\left(\frac{d e-c f}{f}\right)+i \tanh ^{-1}(c+d x)\right)}\right) \left(i \tanh ^{-1}\left(\frac{d e-c f}{f}\right)+i \tanh ^{-1}(c+d x)\right)-2 \log \left(2 i \sinh \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)\right) \left(i \tanh ^{-1}\left(\frac{d e-c f}{f}\right)+i \tanh ^{-1}(c+d x)\right)-\frac{1}{4} i \left(\pi -2 i \tanh ^{-1}(c+d x)\right)^2+\left(\pi -2 i \tanh ^{-1}(c+d x)\right) \log \left(1-e^{i \left(\pi -2 i \tanh ^{-1}(c+d x)\right)}\right)-\left(\pi -2 i \tanh ^{-1}(c+d x)\right) \log \left(2 \sin \left(\frac{1}{2} \left(\pi -2 i \tanh ^{-1}(c+d x)\right)\right)\right)-i \text{Li}_2\left(e^{2 i \left(i \tanh ^{-1}\left(\frac{d e-c f}{f}\right)+i \tanh ^{-1}(c+d x)\right)}\right)-i \text{Li}_2\left(e^{i \left(\pi -2 i \tanh ^{-1}(c+d x)\right)}\right)\right)\right) a}{f}+\frac{b^2 (d e-c f+f (c+d x)) \left(\frac{2 \left(d e \tanh ^{-1}(c+d x)-(c+1) f \tanh ^{-1}(c+d x)+3 (d e-c f) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right)\right) \tanh ^{-1}(c+d x)^2+\left(6 c f \tanh ^{-1}(c+d x)-6 d e \tanh ^{-1}(c+d x)\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+(3 c f-3 d e) \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)}{6 f (c f-d e)}-\frac{(-d e+c f-f) (-d e+c f+f) \left(-3 d e \tanh ^{-1}(c+d x)^3+3 c f \tanh ^{-1}(c+d x)^3-2 e^{-\tanh ^{-1}\left(\frac{d e-c f}{f}\right)} \sqrt{-c^2+\frac{2 d e c}{f}-\frac{d^2 e^2}{f^2}+1} f \tanh ^{-1}(c+d x)^3+f \tanh ^{-1}(c+d x)^3+3 d e \log \left(1-e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)^2-3 c f \log \left(1-e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)^2+3 d e \log \left(1+e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)^2-3 c f \log \left(1+e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)^2+3 d e \log \left(\frac{1}{2} e^{-\tanh ^{-1}(c+d x)} \left(d e \left(1+e^{2 \tanh ^{-1}(c+d x)}\right)-\left(e^{2 \tanh ^{-1}(c+d x)} c+c-e^{2 \tanh ^{-1}(c+d x)}+1\right) f\right)\right) \tanh ^{-1}(c+d x)^2-3 c f \log \left(\frac{1}{2} e^{-\tanh ^{-1}(c+d x)} \left(d e \left(1+e^{2 \tanh ^{-1}(c+d x)}\right)-\left(e^{2 \tanh ^{-1}(c+d x)} c+c-e^{2 \tanh ^{-1}(c+d x)}+1\right) f\right)\right) \tanh ^{-1}(c+d x)^2-3 d e \log \left(\frac{d e}{\sqrt{1-(c+d x)^2}}-\frac{c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right) \tanh ^{-1}(c+d x)^2+3 c f \log \left(\frac{d e}{\sqrt{1-(c+d x)^2}}-\frac{c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right) \tanh ^{-1}(c+d x)^2-3 i d e \pi  \log \left(\frac{1}{2} e^{-\tanh ^{-1}(c+d x)} \left(1+e^{2 \tanh ^{-1}(c+d x)}\right)\right) \tanh ^{-1}(c+d x)+3 i c f \pi  \log \left(\frac{1}{2} e^{-\tanh ^{-1}(c+d x)} \left(1+e^{2 \tanh ^{-1}(c+d x)}\right)\right) \tanh ^{-1}(c+d x)+6 d e \tanh ^{-1}\left(\frac{d e-c f}{f}\right) \log \left(\frac{1}{2} i e^{-\tanh ^{-1}\left(\frac{d e-c f}{f}\right)-\tanh ^{-1}(c+d x)} \left(-1+e^{2 \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)}\right)\right) \tanh ^{-1}(c+d x)-6 c f \tanh ^{-1}\left(\frac{d e-c f}{f}\right) \log \left(\frac{1}{2} i e^{-\tanh ^{-1}\left(\frac{d e-c f}{f}\right)-\tanh ^{-1}(c+d x)} \left(-1+e^{2 \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)}\right)\right) \tanh ^{-1}(c+d x)+3 i d e \pi  \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right) \tanh ^{-1}(c+d x)-3 i c f \pi  \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right) \tanh ^{-1}(c+d x)-6 d e \tanh ^{-1}\left(\frac{d e-c f}{f}\right) \log \left(i \sinh \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)\right) \tanh ^{-1}(c+d x)+6 c f \tanh ^{-1}\left(\frac{d e-c f}{f}\right) \log \left(i \sinh \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)\right) \tanh ^{-1}(c+d x)+6 (d e-c f) \text{Li}_2\left(-e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)+6 (d e-c f) \text{Li}_2\left(e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)-6 d e \text{Li}_3\left(-e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right)+6 c f \text{Li}_3\left(-e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right)-6 d e \text{Li}_3\left(e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right)+6 c f \text{Li}_3\left(e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right)\right)}{3 f (c f-d e) \left(d^2 e^2-2 c d f e+\left(c^2-1\right) f^2\right)}\right)}{d (e+f x)}","-\frac{b \left(a+b \tanh ^{-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 \tanh ^{-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 \tanh ^{-1}(c+d x)\right)}{f}-\frac{\log \left(\frac{2}{c+d x+1}\right) \left(a+b \tanh ^{-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*I)*a*b*(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))*((2*ArcTanh[c + d*x]^2*(d*e*ArcTanh[c + d*x] - (1 + c)*f*ArcTanh[c + d*x] + 3*(d*e - c*f)*Log[1 + E^(-2*ArcTanh[c + d*x])]) + (-6*d*e*ArcTanh[c + d*x] + 6*c*f*ArcTanh[c + d*x])*PolyLog[2, -E^(-2*ArcTanh[c + d*x])] + (-3*d*e + 3*c*f)*PolyLog[3, -E^(-2*ArcTanh[c + d*x])])/(6*f*(-(d*e) + c*f)) - ((-(d*e) - f + c*f)*(-(d*e) + f + c*f)*(-3*d*e*ArcTanh[c + d*x]^3 + f*ArcTanh[c + d*x]^3 + 3*c*f*ArcTanh[c + d*x]^3 - (2*Sqrt[1 - c^2 - (d^2*e^2)/f^2 + (2*c*d*e)/f]*f*ArcTanh[c + d*x]^3)/E^ArcTanh[(d*e - c*f)/f] - (3*I)*d*e*Pi*ArcTanh[c + d*x]*Log[(1 + E^(2*ArcTanh[c + d*x]))/(2*E^ArcTanh[c + d*x])] + (3*I)*c*f*Pi*ArcTanh[c + d*x]*Log[(1 + E^(2*ArcTanh[c + d*x]))/(2*E^ArcTanh[c + d*x])] + 3*d*e*ArcTanh[c + d*x]^2*Log[1 - E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 3*c*f*ArcTanh[c + d*x]^2*Log[1 - E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 3*d*e*ArcTanh[c + d*x]^2*Log[1 + E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 3*c*f*ArcTanh[c + d*x]^2*Log[1 + E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*d*e*ArcTanh[(d*e - c*f)/f]*ArcTanh[c + d*x]*Log[(I/2)*E^(-ArcTanh[(d*e - c*f)/f] - ArcTanh[c + d*x])*(-1 + E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])))] - 6*c*f*ArcTanh[(d*e - c*f)/f]*ArcTanh[c + d*x]*Log[(I/2)*E^(-ArcTanh[(d*e - c*f)/f] - ArcTanh[c + d*x])*(-1 + E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])))] + 3*d*e*ArcTanh[c + d*x]^2*Log[(d*e*(1 + E^(2*ArcTanh[c + d*x])) - (1 + c - E^(2*ArcTanh[c + d*x]) + c*E^(2*ArcTanh[c + d*x]))*f)/(2*E^ArcTanh[c + d*x])] - 3*c*f*ArcTanh[c + d*x]^2*Log[(d*e*(1 + E^(2*ArcTanh[c + d*x])) - (1 + c - E^(2*ArcTanh[c + d*x]) + c*E^(2*ArcTanh[c + d*x]))*f)/(2*E^ArcTanh[c + d*x])] + (3*I)*d*e*Pi*ArcTanh[c + d*x]*Log[1/Sqrt[1 - (c + d*x)^2]] - (3*I)*c*f*Pi*ArcTanh[c + d*x]*Log[1/Sqrt[1 - (c + d*x)^2]] - 3*d*e*ArcTanh[c + d*x]^2*Log[(d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2]] + 3*c*f*ArcTanh[c + d*x]^2*Log[(d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2]] - 6*d*e*ArcTanh[(d*e - c*f)/f]*ArcTanh[c + d*x]*Log[I*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]] + 6*c*f*ArcTanh[(d*e - c*f)/f]*ArcTanh[c + d*x]*Log[I*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]] + 6*(d*e - c*f)*ArcTanh[c + d*x]*PolyLog[2, -E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*(d*e - c*f)*ArcTanh[c + d*x]*PolyLog[2, E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 6*d*e*PolyLog[3, -E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*c*f*PolyLog[3, -E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 6*d*e*PolyLog[3, E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*c*f*PolyLog[3, E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])]))/(3*f*(-(d*e) + c*f)*(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2))))/(d*(e + f*x))","C",0
43,1,425,480,7.5260274,"\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{(e+f x)^2} \, dx","Integrate[(a + b*ArcTanh[c + d*x])^2/(e + f*x)^2,x]","\frac{-\frac{a^2}{f}+\frac{2 a b \left(\tanh ^{-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)}{\sqrt{1-(c+d x)^2}}\right)\right)}{(-c f+d e+f) (d e-(c+1) f)}+\frac{b^2 d (e+f x) \left(\frac{(d e-c f) \left(\text{Li}_2\left(\exp \left(-2 \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)\right)\right)-2 \tanh ^{-1}(c+d x) \log \left(1-\exp \left(-2 \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)\right)\right)-2 \tanh ^{-1}\left(\frac{d e-c f}{f}\right) \left(\log \left(1-\exp \left(-2 \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)\right)\right)-\log \left(i \sinh \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)\right)+\tanh ^{-1}(c+d x)\right)+i \pi  \log \left(e^{2 \tanh ^{-1}(c+d x)}+1\right)-i \pi  \left(\log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)+\tanh ^{-1}(c+d x)\right)\right)}{\left(c^2-1\right) f^2-2 c d e f+d^2 e^2}-\frac{\tanh ^{-1}(c+d x)^2 e^{-\tanh ^{-1}\left(\frac{d e-c f}{f}\right)}}{f \sqrt{1-\frac{(d e-c f)^2}{f^2}}}+\frac{(c+d x) \tanh ^{-1}(c+d x)^2}{d (e+f x)}\right)}{d e-c f}}{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 \tanh ^{-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) \tanh ^{-1}(c+d x)}{f (-c f+d e+f)}-\frac{b^2 d \log \left(\frac{2}{c+d x+1}\right) \tanh ^{-1}(c+d x)}{f (-c f+d e-f)}+\frac{2 b^2 d \log \left(\frac{2}{c+d x+1}\right) \tanh ^{-1}(c+d x)}{(-c f+d e+f) (d e-(c+1) f)}-\frac{2 b^2 d \tanh ^{-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))*ArcTanh[c + d*x] - d*(e + f*x)*Log[(d*(e + f*x))/Sqrt[1 - (c + d*x)^2]]))/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (b^2*d*(e + f*x)*(-(ArcTanh[c + d*x]^2/(E^ArcTanh[(d*e - c*f)/f]*f*Sqrt[1 - (d*e - c*f)^2/f^2])) + ((c + d*x)*ArcTanh[c + d*x]^2)/(d*(e + f*x)) + ((d*e - c*f)*(I*Pi*Log[1 + E^(2*ArcTanh[c + d*x])] - 2*ArcTanh[c + d*x]*Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] - I*Pi*(ArcTanh[c + d*x] + Log[1/Sqrt[1 - (c + d*x)^2]]) - 2*ArcTanh[(d*e - c*f)/f]*(ArcTanh[c + d*x] + Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] - Log[I*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]]) + PolyLog[2, E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))]))/(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)))/(d*e - c*f))/(e + f*x)","C",0
44,1,1968,750,14.7887376,"\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{(e+f x)^3} \, dx","Integrate[(a + b*ArcTanh[c + d*x])^2/(e + f*x)^3,x]","\frac{a b \left(\frac{f \left(\frac{(d e-c f+f) (d e-(c+1) f)}{\left(\frac{d e-c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)^2}+2\right) \tanh ^{-1}(c+d x)}{(d e-c f+f)^2 (-d e+c f+f)^2}-\frac{(c+d x) \left(2 c \tanh ^{-1}(c+d x) f+f-2 d e \tanh ^{-1}(c+d x)\right)}{(d e-c f) (d e-c f+f) (d e-(c+1) f) \sqrt{1-(c+d x)^2} \left(\frac{d e-c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)}-\frac{2 (d e-c f) \log \left(\frac{d e}{\sqrt{1-(c+d x)^2}}-\frac{c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)}{\left(d^2 e^2-2 c d f e+\left(c^2-1\right) f^2\right)^2}\right) (d e-c f+f (c+d x))^3}{d (e+f x)^3}+\frac{b^2 \left(\frac{f \left(1-(c+d x)^2\right)^{3/2} \tanh ^{-1}(c+d x)^2 \left(\frac{d e}{\sqrt{1-(c+d x)^2}}-\frac{c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)^3}{2 (d e-c f-f) (d e-c f+f) (d e-c f+f (c+d x))^3 \left(-\frac{d e}{\sqrt{1-(c+d x)^2}}+\frac{c f}{\sqrt{1-(c+d x)^2}}-\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)^2}+\frac{\left(1-(c+d x)^2\right)^{3/2} \left(-\frac{d e (c+d x) \tanh ^{-1}(c+d x)^2}{\sqrt{1-(c+d x)^2}}+\frac{c f (c+d x) \tanh ^{-1}(c+d x)^2}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x) \tanh ^{-1}(c+d x)}{\sqrt{1-(c+d x)^2}}\right) \left(\frac{d e}{\sqrt{1-(c+d x)^2}}-\frac{c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)^3}{(d e-c f) (d e-c f-f) (d e-c f+f) (d e-c f+f (c+d x))^3 \left(-\frac{d e}{\sqrt{1-(c+d x)^2}}+\frac{c f}{\sqrt{1-(c+d x)^2}}-\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)}+\frac{f \left(1-(c+d x)^2\right)^{3/2} \left((d e-c f) \log \left(\frac{d e-c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)-f \tanh ^{-1}(c+d x)\right) \left(\frac{d e}{\sqrt{1-(c+d x)^2}}-\frac{c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)^3}{(d e-c f) (d e-c f-f) (d e-c f+f) \left((d e-c f)^2-f^2\right) (d e-c f+f (c+d x))^3}+\frac{c \left(1-(c+d x)^2\right)^{3/2} \left(e^{-\tanh ^{-1}\left(\frac{d e-c f}{f}\right)} \tanh ^{-1}(c+d x)^2-\frac{i (d e-c f) \left(-\left(\left(2 i \tanh ^{-1}\left(\frac{d e-c f}{f}\right)-\pi \right) \tanh ^{-1}(c+d x)\right)-2 \left(i \tanh ^{-1}\left(\frac{d e-c f}{f}\right)+i \tanh ^{-1}(c+d x)\right) \log \left(1-e^{2 i \left(i \tanh ^{-1}\left(\frac{d e-c f}{f}\right)+i \tanh ^{-1}(c+d x)\right)}\right)-\pi  \log \left(1+e^{2 \tanh ^{-1}(c+d x)}\right)+\pi  \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)+2 i \tanh ^{-1}\left(\frac{d e-c f}{f}\right) \log \left(i \sinh \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)\right)+i \text{Li}_2\left(e^{2 i \left(i \tanh ^{-1}\left(\frac{d e-c f}{f}\right)+i \tanh ^{-1}(c+d x)\right)}\right)\right)}{f \sqrt{1-\frac{(d e-c f)^2}{f^2}}}\right) \left(\frac{d e}{\sqrt{1-(c+d x)^2}}-\frac{c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)^3}{(d e-c f) (d e-c f-f) (d e-c f+f) \sqrt{\frac{f^2-(d e-c f)^2}{f^2}} (d e-c f+f (c+d x))^3}-\frac{d e \left(1-(c+d x)^2\right)^{3/2} \left(e^{-\tanh ^{-1}\left(\frac{d e-c f}{f}\right)} \tanh ^{-1}(c+d x)^2-\frac{i (d e-c f) \left(-\left(\left(2 i \tanh ^{-1}\left(\frac{d e-c f}{f}\right)-\pi \right) \tanh ^{-1}(c+d x)\right)-2 \left(i \tanh ^{-1}\left(\frac{d e-c f}{f}\right)+i \tanh ^{-1}(c+d x)\right) \log \left(1-e^{2 i \left(i \tanh ^{-1}\left(\frac{d e-c f}{f}\right)+i \tanh ^{-1}(c+d x)\right)}\right)-\pi  \log \left(1+e^{2 \tanh ^{-1}(c+d x)}\right)+\pi  \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)+2 i \tanh ^{-1}\left(\frac{d e-c f}{f}\right) \log \left(i \sinh \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)\right)+i \text{Li}_2\left(e^{2 i \left(i \tanh ^{-1}\left(\frac{d e-c f}{f}\right)+i \tanh ^{-1}(c+d x)\right)}\right)\right)}{f \sqrt{1-\frac{(d e-c f)^2}{f^2}}}\right) \left(\frac{d e}{\sqrt{1-(c+d x)^2}}-\frac{c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)^3}{f (d e-c f) (d e-c f-f) (d e-c f+f) \sqrt{\frac{f^2-(d e-c f)^2}{f^2}} (d e-c f+f (c+d x))^3}\right) (d e-c f+f (c+d x))^3}{d (e+f x)^3}-\frac{a^2}{2 f (e+f x)^2}","-\frac{a b d^2 \log (-c-d x+1)}{2 f (-c f+d e+f)^2}+\frac{a b d^2 \log (c+d x+1)}{2 f (-c f+d e-f)^2}-\frac{2 a b d^2 (d e-c f) \log (e+f x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}-\frac{a b d}{(e+f x) \left(f^2-(d e-c f)^2\right)}-\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 f (e+f x)^2}+\frac{b^2 d^2 \text{Li}_2\left(-\frac{c+d x+1}{-c-d x+1}\right)}{4 f (-c f+d e+f)^2}+\frac{b^2 d^2 \text{Li}_2\left(1-\frac{2}{c+d x+1}\right)}{4 f (-c f+d e-f)^2}-\frac{b^2 d^2 (d e-c f) \text{Li}_2\left(1-\frac{2}{c+d x+1}\right)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac{b^2 d^2 (d e-c f) \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)^2 (d e-(c+1) f)^2}+\frac{b^2 d^2 \log (-c-d x+1)}{2 (-c f+d e+f)^2 (d e-(c+1) f)}-\frac{b^2 d^2 \log (c+d x+1)}{2 (-c f+d e+f) (d e-(c+1) f)^2}+\frac{b^2 d^2 f \log (e+f x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac{b^2 d^2 \log \left(\frac{2}{-c-d x+1}\right) \tanh ^{-1}(c+d x)}{2 f (-c f+d e+f)^2}-\frac{b^2 d^2 \log \left(\frac{2}{c+d x+1}\right) \tanh ^{-1}(c+d x)}{2 f (-c f+d e-f)^2}+\frac{2 b^2 d^2 (d e-c f) \log \left(\frac{2}{c+d x+1}\right) \tanh ^{-1}(c+d x)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}-\frac{2 b^2 d^2 (d e-c f) \tanh ^{-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)^2 (d e-(c+1) f)^2}+\frac{b^2 d \tanh ^{-1}(c+d x)}{(e+f x) (-c f+d e+f) (d e-(c+1) f)}",1,"-1/2*a^2/(f*(e + f*x)^2) + (a*b*(d*e - c*f + f*(c + d*x))^3*((f*(2 + ((d*e + f - c*f)*(d*e - (1 + c)*f))/((d*e - c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])^2)*ArcTanh[c + d*x])/((d*e + f - c*f)^2*(-(d*e) + f + c*f)^2) - ((c + d*x)*(f - 2*d*e*ArcTanh[c + d*x] + 2*c*f*ArcTanh[c + d*x]))/((d*e - c*f)*(d*e + f - c*f)*(d*e - (1 + c)*f)*Sqrt[1 - (c + d*x)^2]*((d*e - c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])) - (2*(d*e - c*f)*Log[(d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2]])/(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)^2))/(d*(e + f*x)^3) + (b^2*(d*e - c*f + f*(c + d*x))^3*((f*(1 - (c + d*x)^2)^(3/2)*((d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])^3*ArcTanh[c + d*x]^2)/(2*(d*e - f - c*f)*(d*e + f - c*f)*(d*e - c*f + f*(c + d*x))^3*(-((d*e)/Sqrt[1 - (c + d*x)^2]) + (c*f)/Sqrt[1 - (c + d*x)^2] - (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])^2) + ((1 - (c + d*x)^2)^(3/2)*((d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])^3*((f*(c + d*x)*ArcTanh[c + d*x])/Sqrt[1 - (c + d*x)^2] - (d*e*(c + d*x)*ArcTanh[c + d*x]^2)/Sqrt[1 - (c + d*x)^2] + (c*f*(c + d*x)*ArcTanh[c + d*x]^2)/Sqrt[1 - (c + d*x)^2]))/((d*e - c*f)*(d*e - f - c*f)*(d*e + f - c*f)*(d*e - c*f + f*(c + d*x))^3*(-((d*e)/Sqrt[1 - (c + d*x)^2]) + (c*f)/Sqrt[1 - (c + d*x)^2] - (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])) + (f*(1 - (c + d*x)^2)^(3/2)*((d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])^3*(-(f*ArcTanh[c + d*x]) + (d*e - c*f)*Log[(d*e - c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2]]))/((d*e - c*f)*(d*e - f - c*f)*(d*e + f - c*f)*(-f^2 + (d*e - c*f)^2)*(d*e - c*f + f*(c + d*x))^3) + (c*(1 - (c + d*x)^2)^(3/2)*((d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])^3*(ArcTanh[c + d*x]^2/E^ArcTanh[(d*e - c*f)/f] - (I*(d*e - c*f)*(-((-Pi + (2*I)*ArcTanh[(d*e - c*f)/f])*ArcTanh[c + d*x]) - 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*Log[1 + E^(2*ArcTanh[c + d*x])] + Pi*Log[1/Sqrt[1 - (c + d*x)^2]] + (2*I)*ArcTanh[(d*e - c*f)/f]*Log[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]))]))/(f*Sqrt[1 - (d*e - c*f)^2/f^2])))/((d*e - c*f)*(d*e - f - c*f)*(d*e + f - c*f)*Sqrt[(f^2 - (d*e - c*f)^2)/f^2]*(d*e - c*f + f*(c + d*x))^3) - (d*e*(1 - (c + d*x)^2)^(3/2)*((d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])^3*(ArcTanh[c + d*x]^2/E^ArcTanh[(d*e - c*f)/f] - (I*(d*e - c*f)*(-((-Pi + (2*I)*ArcTanh[(d*e - c*f)/f])*ArcTanh[c + d*x]) - 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*Log[1 + E^(2*ArcTanh[c + d*x])] + Pi*Log[1/Sqrt[1 - (c + d*x)^2]] + (2*I)*ArcTanh[(d*e - c*f)/f]*Log[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]))]))/(f*Sqrt[1 - (d*e - c*f)^2/f^2])))/(f*(d*e - c*f)*(d*e - f - c*f)*(d*e + f - c*f)*Sqrt[(f^2 - (d*e - c*f)^2)/f^2]*(d*e - c*f + f*(c + d*x))^3)))/(d*(e + f*x)^3)","C",0
45,1,1868,546,10.1605447,"\int (e+f x)^2 \left(a+b \tanh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(e + f*x)^2*(a + b*ArcTanh[c + d*x])^3,x]","\frac{e^2 \left(\left((c+d x) \tanh ^{-1}(c+d x)-\tanh ^{-1}(c+d x)-3 \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right)\right) \tanh ^{-1}(c+d x)^2+3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)+\frac{3}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)\right) b^3}{d}+\frac{e f \left(-\tanh ^{-1}(c+d x) \left(-2 c \tanh ^{-1}(c+d x)^2+\left(1-(c+d x)^2\right) \tanh ^{-1}(c+d x)^2+(c+d x) \left(2 c \tanh ^{-1}(c+d x)-3\right) \tanh ^{-1}(c+d x)-6 c \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)+3 \tanh ^{-1}(c+d x)+6 \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right)\right)+\left(3-6 c \tanh ^{-1}(c+d x)\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)-3 c \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)\right) b^3}{d^2}+\frac{f^2 \left(\left(3 \tanh ^{-1}(c+d x) c^2-3 c+\tanh ^{-1}(c+d x)\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)-\frac{1}{12} \left(1-(c+d x)^2\right)^{3/2} \left(3 c^2 \cosh \left(3 \tanh ^{-1}(c+d x)\right) \tanh ^{-1}(c+d x)^3+\cosh \left(3 \tanh ^{-1}(c+d x)\right) \tanh ^{-1}(c+d x)^3-3 c^2 \sinh \left(3 \tanh ^{-1}(c+d x)\right) \tanh ^{-1}(c+d x)^3-\sinh \left(3 \tanh ^{-1}(c+d x)\right) \tanh ^{-1}(c+d x)^3-\frac{3 c^2 (c+d x) \tanh ^{-1}(c+d x)^3}{\sqrt{1-(c+d x)^2}}+\frac{3 (c+d x) \tanh ^{-1}(c+d x)^3}{\sqrt{1-(c+d x)^2}}-9 c \cosh \left(3 \tanh ^{-1}(c+d x)\right) \tanh ^{-1}(c+d x)^2+9 c^2 \cosh \left(3 \tanh ^{-1}(c+d x)\right) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)^2+3 \cosh \left(3 \tanh ^{-1}(c+d x)\right) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)^2+9 c \sinh \left(3 \tanh ^{-1}(c+d x)\right) \tanh ^{-1}(c+d x)^2+\frac{9 c (c+d x) \tanh ^{-1}(c+d x)^2}{\sqrt{1-(c+d x)^2}}-18 c \cosh \left(3 \tanh ^{-1}(c+d x)\right) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)-3 \sinh \left(3 \tanh ^{-1}(c+d x)\right) \tanh ^{-1}(c+d x)-\frac{3 (c+d x) \tanh ^{-1}(c+d x)}{\sqrt{1-(c+d x)^2}}+3 \cosh \left(3 \tanh ^{-1}(c+d x)\right) \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)+\frac{3 \left(\left(3 \tanh ^{-1}(c+d x) c^2-4 \tanh ^{-1}(c+d x) c-9 c+\tanh ^{-1}(c+d x)+2\right) \tanh ^{-1}(c+d x)^2+3 \left(3 \tanh ^{-1}(c+d x) c^2-6 c+\tanh ^{-1}(c+d x)\right) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)+3 \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)\right)}{\sqrt{1-(c+d x)^2}}-\frac{6 \left(3 c^2+1\right) \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)}{\left(1-(c+d x)^2\right)^{3/2}}\right)\right) b^3}{d^3}+\frac{3 a e^2 \left(\tanh ^{-1}(c+d x) \left((c+d x) \tanh ^{-1}(c+d x)-\tanh ^{-1}(c+d x)-2 \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right)\right)+\text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)\right) b^2}{d}-\frac{3 a e f \left(\left(1-(c+d x)^2\right) \tanh ^{-1}(c+d x)^2+2 \left(-c \tanh ^{-1}(c+d x)^2+c (c+d x) \tanh ^{-1}(c+d x)^2-(c+d x) \tanh ^{-1}(c+d x)-2 c \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)+\log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)\right)+2 c \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)\right) b^2}{d^2}-\frac{a f^2 \left(1-(c+d x)^2\right)^{3/2} \left(-\frac{3 (c+d x) \tanh ^{-1}(c+d x)^2 c^2}{\sqrt{1-(c+d x)^2}}+3 \tanh ^{-1}(c+d x)^2 \cosh \left(3 \tanh ^{-1}(c+d x)\right) c^2+6 \tanh ^{-1}(c+d x) \cosh \left(3 \tanh ^{-1}(c+d x)\right) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right) c^2-3 \tanh ^{-1}(c+d x)^2 \sinh \left(3 \tanh ^{-1}(c+d x)\right) c^2+\frac{6 (c+d x) \tanh ^{-1}(c+d x) c}{\sqrt{1-(c+d x)^2}}-6 \cosh \left(3 \tanh ^{-1}(c+d x)\right) \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right) c+6 \tanh ^{-1}(c+d x) \sinh \left(3 \tanh ^{-1}(c+d x)\right) c+\frac{3 (c+d x) \tanh ^{-1}(c+d x)^2}{\sqrt{1-(c+d x)^2}}+\tanh ^{-1}(c+d x)^2 \cosh \left(3 \tanh ^{-1}(c+d x)\right)+2 \tanh ^{-1}(c+d x) \cosh \left(3 \tanh ^{-1}(c+d x)\right) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right)+\frac{\tanh ^{-1}(c+d x) \left(3 \left(3 c^2-4 c+1\right) \tanh ^{-1}(c+d x)+4\right)+6 \left(3 \tanh ^{-1}(c+d x) c^2+\tanh ^{-1}(c+d x)\right) \log \left(1+e^{-2 \tanh ^{-1}(c+d x)}\right)-18 c \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)}{\sqrt{1-(c+d x)^2}}-\frac{4 \left(3 c^2+1\right) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)}{\left(1-(c+d x)^2\right)^{3/2}}-\tanh ^{-1}(c+d x)^2 \sinh \left(3 \tanh ^{-1}(c+d x)\right)-\sinh \left(3 \tanh ^{-1}(c+d x)\right)-\frac{c+d x}{\sqrt{1-(c+d x)^2}}\right) b^2}{4 d^3}+a^2 x \left(3 e^2+3 f x e+f^2 x^2\right) \tanh ^{-1}(c+d x) b+\frac{1}{3} a^3 f^2 x^3+\frac{a^2 f (2 a d e+b f) x^2}{2 d}+\frac{a^2 \left(a d^2 e^2+3 b d f e-2 b c f^2\right) x}{d^2}+\frac{\left(-a^2 b f^2 c^3+3 a^2 b f^2 c^2+3 a^2 b d e f c^2-3 a^2 b d^2 e^2 c-3 a^2 b f^2 c-6 a^2 b d e f c+3 a^2 b d^2 e^2+a^2 b f^2+3 a^2 b d e f\right) \log (-c-d x+1)}{2 d^3}+\frac{\left(a^2 b f^2 c^3+3 a^2 b f^2 c^2-3 a^2 b d e f c^2+3 a^2 b d^2 e^2 c+3 a^2 b f^2 c-6 a^2 b d e f c+3 a^2 b d^2 e^2+a^2 b f^2-3 a^2 b d e f\right) \log (c+d x+1)}{2 d^3}","-\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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-1}(c+d x)\right)^2}{d^3}+\frac{3 b f (d e-c f) \left(a+b \tanh ^{-1}(c+d x)\right)^2}{d^3}+\frac{3 b f (c+d x) (d e-c f) \left(a+b \tanh ^{-1}(c+d x)\right)^2}{d^3}-\frac{b f^2 \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d^3}+\frac{b f^2 (c+d x)^2 \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d^3}+\frac{(e+f x)^3 \left(a+b \tanh ^{-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) \tanh ^{-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)*ArcTanh[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*(ArcTanh[c + d*x]*(-ArcTanh[c + d*x] + (c + d*x)*ArcTanh[c + d*x] - 2*Log[1 + E^(-2*ArcTanh[c + d*x])]) + PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/d - (3*a*b^2*e*f*((1 - (c + d*x)^2)*ArcTanh[c + d*x]^2 + 2*(-((c + d*x)*ArcTanh[c + d*x]) - c*ArcTanh[c + d*x]^2 + c*(c + d*x)*ArcTanh[c + d*x]^2 - 2*c*ArcTanh[c + d*x]*Log[1 + E^(-2*ArcTanh[c + d*x])] + Log[1/Sqrt[1 - (c + d*x)^2]]) + 2*c*PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/d^2 + (b^3*e^2*(ArcTanh[c + d*x]^2*(-ArcTanh[c + d*x] + (c + d*x)*ArcTanh[c + d*x] - 3*Log[1 + E^(-2*ArcTanh[c + d*x])]) + 3*ArcTanh[c + d*x]*PolyLog[2, -E^(-2*ArcTanh[c + d*x])] + (3*PolyLog[3, -E^(-2*ArcTanh[c + d*x])])/2))/d + (b^3*e*f*(-(ArcTanh[c + d*x]*(3*ArcTanh[c + d*x] - 2*c*ArcTanh[c + d*x]^2 + (1 - (c + d*x)^2)*ArcTanh[c + d*x]^2 + (c + d*x)*ArcTanh[c + d*x]*(-3 + 2*c*ArcTanh[c + d*x]) + 6*Log[1 + E^(-2*ArcTanh[c + d*x])] - 6*c*ArcTanh[c + d*x]*Log[1 + E^(-2*ArcTanh[c + d*x])])) + (3 - 6*c*ArcTanh[c + d*x])*PolyLog[2, -E^(-2*ArcTanh[c + d*x])] - 3*c*PolyLog[3, -E^(-2*ArcTanh[c + d*x])]))/d^2 - (a*b^2*f^2*(1 - (c + d*x)^2)^(3/2)*(-((c + d*x)/Sqrt[1 - (c + d*x)^2]) + (6*c*(c + d*x)*ArcTanh[c + d*x])/Sqrt[1 - (c + d*x)^2] + (3*(c + d*x)*ArcTanh[c + d*x]^2)/Sqrt[1 - (c + d*x)^2] - (3*c^2*(c + d*x)*ArcTanh[c + d*x]^2)/Sqrt[1 - (c + d*x)^2] + ArcTanh[c + d*x]^2*Cosh[3*ArcTanh[c + d*x]] + 3*c^2*ArcTanh[c + d*x]^2*Cosh[3*ArcTanh[c + d*x]] + 2*ArcTanh[c + d*x]*Cosh[3*ArcTanh[c + d*x]]*Log[1 + E^(-2*ArcTanh[c + d*x])] + 6*c^2*ArcTanh[c + d*x]*Cosh[3*ArcTanh[c + d*x]]*Log[1 + E^(-2*ArcTanh[c + d*x])] - 6*c*Cosh[3*ArcTanh[c + d*x]]*Log[1/Sqrt[1 - (c + d*x)^2]] + (ArcTanh[c + d*x]*(4 + 3*(1 - 4*c + 3*c^2)*ArcTanh[c + d*x]) + 6*(ArcTanh[c + d*x] + 3*c^2*ArcTanh[c + d*x])*Log[1 + E^(-2*ArcTanh[c + d*x])] - 18*c*Log[1/Sqrt[1 - (c + d*x)^2]])/Sqrt[1 - (c + d*x)^2] - (4*(1 + 3*c^2)*PolyLog[2, -E^(-2*ArcTanh[c + d*x])])/(1 - (c + d*x)^2)^(3/2) - Sinh[3*ArcTanh[c + d*x]] + 6*c*ArcTanh[c + d*x]*Sinh[3*ArcTanh[c + d*x]] - ArcTanh[c + d*x]^2*Sinh[3*ArcTanh[c + d*x]] - 3*c^2*ArcTanh[c + d*x]^2*Sinh[3*ArcTanh[c + d*x]]))/(4*d^3) + (b^3*f^2*((-3*c + ArcTanh[c + d*x] + 3*c^2*ArcTanh[c + d*x])*PolyLog[2, -E^(-2*ArcTanh[c + d*x])] - ((1 - (c + d*x)^2)^(3/2)*((-3*(c + d*x)*ArcTanh[c + d*x])/Sqrt[1 - (c + d*x)^2] + (9*c*(c + d*x)*ArcTanh[c + d*x]^2)/Sqrt[1 - (c + d*x)^2] + (3*(c + d*x)*ArcTanh[c + d*x]^3)/Sqrt[1 - (c + d*x)^2] - (3*c^2*(c + d*x)*ArcTanh[c + d*x]^3)/Sqrt[1 - (c + d*x)^2] - 9*c*ArcTanh[c + d*x]^2*Cosh[3*ArcTanh[c + d*x]] + ArcTanh[c + d*x]^3*Cosh[3*ArcTanh[c + d*x]] + 3*c^2*ArcTanh[c + d*x]^3*Cosh[3*ArcTanh[c + d*x]] - 18*c*ArcTanh[c + d*x]*Cosh[3*ArcTanh[c + d*x]]*Log[1 + E^(-2*ArcTanh[c + d*x])] + 3*ArcTanh[c + d*x]^2*Cosh[3*ArcTanh[c + d*x]]*Log[1 + E^(-2*ArcTanh[c + d*x])] + 9*c^2*ArcTanh[c + d*x]^2*Cosh[3*ArcTanh[c + d*x]]*Log[1 + E^(-2*ArcTanh[c + d*x])] + 3*Cosh[3*ArcTanh[c + d*x]]*Log[1/Sqrt[1 - (c + d*x)^2]] + (3*(ArcTanh[c + d*x]^2*(2 - 9*c + ArcTanh[c + d*x] - 4*c*ArcTanh[c + d*x] + 3*c^2*ArcTanh[c + d*x]) + 3*ArcTanh[c + d*x]*(-6*c + ArcTanh[c + d*x] + 3*c^2*ArcTanh[c + d*x])*Log[1 + E^(-2*ArcTanh[c + d*x])] + 3*Log[1/Sqrt[1 - (c + d*x)^2]]))/Sqrt[1 - (c + d*x)^2] - (6*(1 + 3*c^2)*PolyLog[3, -E^(-2*ArcTanh[c + d*x])])/(1 - (c + d*x)^2)^(3/2) - 3*ArcTanh[c + d*x]*Sinh[3*ArcTanh[c + d*x]] + 9*c*ArcTanh[c + d*x]^2*Sinh[3*ArcTanh[c + d*x]] - ArcTanh[c + d*x]^3*Sinh[3*ArcTanh[c + d*x]] - 3*c^2*ArcTanh[c + d*x]^3*Sinh[3*ArcTanh[c + d*x]]))/12))/d^3","B",0
46,1,566,326,1.1549063,"\int (e+f x) \left(a+b \tanh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(e + f*x)*(a + b*ArcTanh[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) \tanh ^{-1}(c+d x) (c f-d (2 e+f x))+12 a b^2 d e \left(\text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+\tanh ^{-1}(c+d x) \left((c+d x-1) \tanh ^{-1}(c+d x)-2 \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)\right)\right)-12 a b^2 c f \left(\text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+\tanh ^{-1}(c+d x) \left((c+d x-1) \tanh ^{-1}(c+d x)-2 \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)\right)\right)+12 a b^2 f \left(-\log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)-\frac{1}{2} \left(1-(c+d x)^2\right) \tanh ^{-1}(c+d x)^2+(c+d x) \tanh ^{-1}(c+d x)\right)+2 b^3 f \left(\tanh ^{-1}(c+d x) \left(\left(c^2+2 c d x+d^2 x^2-1\right) \tanh ^{-1}(c+d x)^2+3 (c+d x-1) \tanh ^{-1}(c+d x)-6 \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)\right)+3 \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)\right)+4 b^3 d e \left(3 \tanh ^{-1}(c+d x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+\frac{3}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+\tanh ^{-1}(c+d x)^2 \left((c+d x-1) \tanh ^{-1}(c+d x)-3 \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)\right)\right)-4 b^3 c f \left(3 \tanh ^{-1}(c+d x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+\frac{3}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+\tanh ^{-1}(c+d x)^2 \left((c+d x-1) \tanh ^{-1}(c+d x)-3 \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)\right)\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 \tanh ^{-1}(c+d x)\right)}{d^2}-\frac{3 b^2 f \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-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 \tanh ^{-1}(c+d x)\right)^3}{2 d^2 f}+\frac{(d e-c f) \left(a+b \tanh ^{-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 \tanh ^{-1}(c+d x)\right)^2}{d^2}+\frac{3 b f \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d^2}+\frac{3 b f (c+d x) \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 d^2}+\frac{(e+f x)^2 \left(a+b \tanh ^{-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))*ArcTanh[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)*ArcTanh[c + d*x] - ((1 - (c + d*x)^2)*ArcTanh[c + d*x]^2)/2 - Log[1/Sqrt[1 - (c + d*x)^2]]) + 12*a*b^2*d*e*(ArcTanh[c + d*x]*((-1 + c + d*x)*ArcTanh[c + d*x] - 2*Log[1 + E^(-2*ArcTanh[c + d*x])]) + PolyLog[2, -E^(-2*ArcTanh[c + d*x])]) - 12*a*b^2*c*f*(ArcTanh[c + d*x]*((-1 + c + d*x)*ArcTanh[c + d*x] - 2*Log[1 + E^(-2*ArcTanh[c + d*x])]) + PolyLog[2, -E^(-2*ArcTanh[c + d*x])]) + 2*b^3*f*(ArcTanh[c + d*x]*(3*(-1 + c + d*x)*ArcTanh[c + d*x] + (-1 + c^2 + 2*c*d*x + d^2*x^2)*ArcTanh[c + d*x]^2 - 6*Log[1 + E^(-2*ArcTanh[c + d*x])]) + 3*PolyLog[2, -E^(-2*ArcTanh[c + d*x])]) + 4*b^3*d*e*(ArcTanh[c + d*x]^2*((-1 + c + d*x)*ArcTanh[c + d*x] - 3*Log[1 + E^(-2*ArcTanh[c + d*x])]) + 3*ArcTanh[c + d*x]*PolyLog[2, -E^(-2*ArcTanh[c + d*x])] + (3*PolyLog[3, -E^(-2*ArcTanh[c + d*x])])/2) - 4*b^3*c*f*(ArcTanh[c + d*x]^2*((-1 + c + d*x)*ArcTanh[c + d*x] - 3*Log[1 + E^(-2*ArcTanh[c + d*x])]) + 3*ArcTanh[c + d*x]*PolyLog[2, -E^(-2*ArcTanh[c + d*x])] + (3*PolyLog[3, -E^(-2*ArcTanh[c + d*x])])/2))/(4*d^2)","A",0
47,1,205,132,0.2721803,"\int \left(a+b \tanh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(a + b*ArcTanh[c + d*x])^3,x]","\frac{2 a^3 d x-3 a^2 b (c-1) \log (-c-d x+1)+3 a^2 b (c+1) \log (c+d x+1)+6 a^2 b d x \tanh ^{-1}(c+d x)+6 a b^2 \left(\text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+\tanh ^{-1}(c+d x) \left((c+d x-1) \tanh ^{-1}(c+d x)-2 \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)\right)\right)+2 b^3 \left(3 \tanh ^{-1}(c+d x) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+\frac{3}{2} \text{Li}_3\left(-e^{-2 \tanh ^{-1}(c+d x)}\right)+\tanh ^{-1}(c+d x)^2 \left((c+d x-1) \tanh ^{-1}(c+d x)-3 \log \left(e^{-2 \tanh ^{-1}(c+d x)}+1\right)\right)\right)}{2 d}","-\frac{3 b^2 \text{Li}_2\left(1-\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{d}+\frac{(c+d x) \left(a+b \tanh ^{-1}(c+d x)\right)^3}{d}+\frac{\left(a+b \tanh ^{-1}(c+d x)\right)^3}{d}-\frac{3 b \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-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*d*x + 6*a^2*b*d*x*ArcTanh[c + d*x] - 3*a^2*b*(-1 + c)*Log[1 - c - d*x] + 3*a^2*b*(1 + c)*Log[1 + c + d*x] + 6*a*b^2*(ArcTanh[c + d*x]*((-1 + c + d*x)*ArcTanh[c + d*x] - 2*Log[1 + E^(-2*ArcTanh[c + d*x])]) + PolyLog[2, -E^(-2*ArcTanh[c + d*x])]) + 2*b^3*(ArcTanh[c + d*x]^2*((-1 + c + d*x)*ArcTanh[c + d*x] - 3*Log[1 + E^(-2*ArcTanh[c + d*x])]) + 3*ArcTanh[c + d*x]*PolyLog[2, -E^(-2*ArcTanh[c + d*x])] + (3*PolyLog[3, -E^(-2*ArcTanh[c + d*x])])/2))/(2*d)","A",0
48,0,0,308,22.9116697,"\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^3}{e+f x} \, dx","Integrate[(a + b*ArcTanh[c + d*x])^3/(e + f*x),x]","\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^3}{e+f x} \, dx","-\frac{3 b^2 \left(a+b \tanh ^{-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 \tanh ^{-1}(c+d x)\right)}{2 f}-\frac{3 b \left(a+b \tanh ^{-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 \tanh ^{-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 \tanh ^{-1}(c+d x)\right)^2}{2 f}-\frac{\log \left(\frac{2}{c+d x+1}\right) \left(a+b \tanh ^{-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*ArcTanh[c + d*x])^3/(e + f*x), x]","F",-1
49,1,1946,1089,23.2105047,"\int \frac{\left(a+b \tanh ^{-1}(c+d x)\right)^3}{(e+f x)^2} \, dx","Integrate[(a + b*ArcTanh[c + d*x])^3/(e + f*x)^2,x]","-\frac{a^3}{f (e+f x)}-\frac{3 b \tanh ^{-1}(c+d x) a^2}{f (e+f x)}+\frac{3 b d \log (-c-d x+1) a^2}{2 f (-d e+c f-f)}-\frac{3 b d \log (c+d x+1) a^2}{2 f (-d e+c f+f)}-\frac{3 b d \log (e+f x) a^2}{d^2 e^2-2 c d f e+c^2 f^2-f^2}+\frac{3 b^2 \left(1-(c+d x)^2\right) \left(\frac{d e-c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)^2 \left(-\frac{e^{-\tanh ^{-1}\left(\frac{d e-c f}{f}\right)} \tanh ^{-1}(c+d x)^2}{f \sqrt{1-\frac{(d e-c f)^2}{f^2}}}+\frac{(c+d x) \tanh ^{-1}(c+d x)^2}{\sqrt{1-(c+d x)^2} \left(\frac{d e}{\sqrt{1-(c+d x)^2}}-\frac{c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)}+\frac{(d e-c f) \left(i \pi  \log \left(1+e^{2 \tanh ^{-1}(c+d x)}\right)-2 \tanh ^{-1}(c+d x) \log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)}\right)-i \pi  \left(\tanh ^{-1}(c+d x)+\log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right)\right)-2 \tanh ^{-1}\left(\frac{d e-c f}{f}\right) \left(\tanh ^{-1}(c+d x)+\log \left(1-e^{-2 \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)}\right)-\log \left(i \sinh \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)\right)\right)+\text{Li}_2\left(e^{-2 \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)}\right)\right)}{d^2 e^2-2 c d f e+\left(c^2-1\right) f^2}\right) a}{d (d e-c f) (e+f x)^2}+\frac{b^3 \left(1-(c+d x)^2\right) \left(\frac{d e-c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)^2 \left(\frac{d (c+d x) \tanh ^{-1}(c+d x)^3}{(d e-c f) \sqrt{1-(c+d x)^2} \left(\frac{d e}{\sqrt{1-(c+d x)^2}}-\frac{c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right)}-\frac{d \left(-3 d e \tanh ^{-1}(c+d x)^3+3 c f \tanh ^{-1}(c+d x)^3-2 e^{-\tanh ^{-1}\left(\frac{d e-c f}{f}\right)} \sqrt{-c^2+\frac{2 d e c}{f}-\frac{d^2 e^2}{f^2}+1} f \tanh ^{-1}(c+d x)^3+f \tanh ^{-1}(c+d x)^3+3 d e \log \left(1-e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)^2-3 c f \log \left(1-e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)^2+3 d e \log \left(1+e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)^2-3 c f \log \left(1+e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)^2+3 d e \log \left(\frac{1}{2} e^{-\tanh ^{-1}(c+d x)} \left(d e \left(1+e^{2 \tanh ^{-1}(c+d x)}\right)-\left(e^{2 \tanh ^{-1}(c+d x)} c+c-e^{2 \tanh ^{-1}(c+d x)}+1\right) f\right)\right) \tanh ^{-1}(c+d x)^2-3 c f \log \left(\frac{1}{2} e^{-\tanh ^{-1}(c+d x)} \left(d e \left(1+e^{2 \tanh ^{-1}(c+d x)}\right)-\left(e^{2 \tanh ^{-1}(c+d x)} c+c-e^{2 \tanh ^{-1}(c+d x)}+1\right) f\right)\right) \tanh ^{-1}(c+d x)^2-3 d e \log \left(\frac{d e}{\sqrt{1-(c+d x)^2}}-\frac{c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right) \tanh ^{-1}(c+d x)^2+3 c f \log \left(\frac{d e}{\sqrt{1-(c+d x)^2}}-\frac{c f}{\sqrt{1-(c+d x)^2}}+\frac{f (c+d x)}{\sqrt{1-(c+d x)^2}}\right) \tanh ^{-1}(c+d x)^2-3 i d e \pi  \log \left(\frac{1}{2} e^{-\tanh ^{-1}(c+d x)} \left(1+e^{2 \tanh ^{-1}(c+d x)}\right)\right) \tanh ^{-1}(c+d x)+3 i c f \pi  \log \left(\frac{1}{2} e^{-\tanh ^{-1}(c+d x)} \left(1+e^{2 \tanh ^{-1}(c+d x)}\right)\right) \tanh ^{-1}(c+d x)+6 d e \tanh ^{-1}\left(\frac{d e-c f}{f}\right) \log \left(\frac{1}{2} i e^{-\tanh ^{-1}\left(\frac{d e-c f}{f}\right)-\tanh ^{-1}(c+d x)} \left(-1+e^{2 \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)}\right)\right) \tanh ^{-1}(c+d x)-6 c f \tanh ^{-1}\left(\frac{d e-c f}{f}\right) \log \left(\frac{1}{2} i e^{-\tanh ^{-1}\left(\frac{d e-c f}{f}\right)-\tanh ^{-1}(c+d x)} \left(-1+e^{2 \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)}\right)\right) \tanh ^{-1}(c+d x)+3 i d e \pi  \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right) \tanh ^{-1}(c+d x)-3 i c f \pi  \log \left(\frac{1}{\sqrt{1-(c+d x)^2}}\right) \tanh ^{-1}(c+d x)-6 d e \tanh ^{-1}\left(\frac{d e-c f}{f}\right) \log \left(i \sinh \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)\right) \tanh ^{-1}(c+d x)+6 c f \tanh ^{-1}\left(\frac{d e-c f}{f}\right) \log \left(i \sinh \left(\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)\right)\right) \tanh ^{-1}(c+d x)+6 (d e-c f) \text{Li}_2\left(-e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)+6 (d e-c f) \text{Li}_2\left(e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right) \tanh ^{-1}(c+d x)-6 d e \text{Li}_3\left(-e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right)+6 c f \text{Li}_3\left(-e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right)-6 d e \text{Li}_3\left(e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right)+6 c f \text{Li}_3\left(e^{\tanh ^{-1}\left(\frac{d e-c f}{f}\right)+\tanh ^{-1}(c+d x)}\right)\right)}{(d e-c f) \left(d^2 e^2-2 c d f e+\left(c^2-1\right) f^2\right)}\right)}{d^2 (e+f x)^2}","\frac{3 d \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-1}(c+d x)\right)^3}{f (e+f x)}",1,"-(a^3/(f*(e + f*x))) - (3*a^2*b*ArcTanh[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)*((d*e - c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])^2*(-(ArcTanh[c + d*x]^2/(E^ArcTanh[(d*e - c*f)/f]*f*Sqrt[1 - (d*e - c*f)^2/f^2])) + ((c + d*x)*ArcTanh[c + d*x]^2)/(Sqrt[1 - (c + d*x)^2]*((d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])) + ((d*e - c*f)*(I*Pi*Log[1 + E^(2*ArcTanh[c + d*x])] - 2*ArcTanh[c + d*x]*Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] - I*Pi*(ArcTanh[c + d*x] + Log[1/Sqrt[1 - (c + d*x)^2]]) - 2*ArcTanh[(d*e - c*f)/f]*(ArcTanh[c + d*x] + Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] - Log[I*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]]) + PolyLog[2, E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))]))/(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)))/(d*(d*e - c*f)*(e + f*x)^2) + (b^3*(1 - (c + d*x)^2)*((d*e - c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])^2*((d*(c + d*x)*ArcTanh[c + d*x]^3)/((d*e - c*f)*Sqrt[1 - (c + d*x)^2]*((d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])) - (d*(-3*d*e*ArcTanh[c + d*x]^3 + f*ArcTanh[c + d*x]^3 + 3*c*f*ArcTanh[c + d*x]^3 - (2*Sqrt[1 - c^2 - (d^2*e^2)/f^2 + (2*c*d*e)/f]*f*ArcTanh[c + d*x]^3)/E^ArcTanh[(d*e - c*f)/f] - (3*I)*d*e*Pi*ArcTanh[c + d*x]*Log[(1 + E^(2*ArcTanh[c + d*x]))/(2*E^ArcTanh[c + d*x])] + (3*I)*c*f*Pi*ArcTanh[c + d*x]*Log[(1 + E^(2*ArcTanh[c + d*x]))/(2*E^ArcTanh[c + d*x])] + 3*d*e*ArcTanh[c + d*x]^2*Log[1 - E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 3*c*f*ArcTanh[c + d*x]^2*Log[1 - E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 3*d*e*ArcTanh[c + d*x]^2*Log[1 + E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 3*c*f*ArcTanh[c + d*x]^2*Log[1 + E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*d*e*ArcTanh[(d*e - c*f)/f]*ArcTanh[c + d*x]*Log[(I/2)*E^(-ArcTanh[(d*e - c*f)/f] - ArcTanh[c + d*x])*(-1 + E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])))] - 6*c*f*ArcTanh[(d*e - c*f)/f]*ArcTanh[c + d*x]*Log[(I/2)*E^(-ArcTanh[(d*e - c*f)/f] - ArcTanh[c + d*x])*(-1 + E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])))] + 3*d*e*ArcTanh[c + d*x]^2*Log[(d*e*(1 + E^(2*ArcTanh[c + d*x])) - (1 + c - E^(2*ArcTanh[c + d*x]) + c*E^(2*ArcTanh[c + d*x]))*f)/(2*E^ArcTanh[c + d*x])] - 3*c*f*ArcTanh[c + d*x]^2*Log[(d*e*(1 + E^(2*ArcTanh[c + d*x])) - (1 + c - E^(2*ArcTanh[c + d*x]) + c*E^(2*ArcTanh[c + d*x]))*f)/(2*E^ArcTanh[c + d*x])] + (3*I)*d*e*Pi*ArcTanh[c + d*x]*Log[1/Sqrt[1 - (c + d*x)^2]] - (3*I)*c*f*Pi*ArcTanh[c + d*x]*Log[1/Sqrt[1 - (c + d*x)^2]] - 3*d*e*ArcTanh[c + d*x]^2*Log[(d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2]] + 3*c*f*ArcTanh[c + d*x]^2*Log[(d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2]] - 6*d*e*ArcTanh[(d*e - c*f)/f]*ArcTanh[c + d*x]*Log[I*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]] + 6*c*f*ArcTanh[(d*e - c*f)/f]*ArcTanh[c + d*x]*Log[I*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]] + 6*(d*e - c*f)*ArcTanh[c + d*x]*PolyLog[2, -E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*(d*e - c*f)*ArcTanh[c + d*x]*PolyLog[2, E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 6*d*e*PolyLog[3, -E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*c*f*PolyLog[3, -E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 6*d*e*PolyLog[3, E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*c*f*PolyLog[3, E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])]))/((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2))))/(d^2*(e + f*x)^2)","C",0
50,0,0,23,4.5087198,"\int (e+f x)^m \left(a+b \tanh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(e + f*x)^m*(a + b*ArcTanh[c + d*x])^3,x]","\int (e+f x)^m \left(a+b \tanh ^{-1}(c+d x)\right)^3 \, dx","\text{Int}\left((e+f x)^m \left(a+b \tanh ^{-1}(c+d x)\right)^3,x\right)",0,"Integrate[(e + f*x)^m*(a + b*ArcTanh[c + d*x])^3, x]","A",-1
51,0,0,23,0.3026114,"\int (e+f x)^m \left(a+b \tanh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(e + f*x)^m*(a + b*ArcTanh[c + d*x])^2,x]","\int (e+f x)^m \left(a+b \tanh ^{-1}(c+d x)\right)^2 \, dx","\text{Int}\left((e+f x)^m \left(a+b \tanh ^{-1}(c+d x)\right)^2,x\right)",0,"Integrate[(e + f*x)^m*(a + b*ArcTanh[c + d*x])^2, x]","A",-1
52,0,0,162,0.0814683,"\int (e+f x)^m \left(a+b \tanh ^{-1}(c+d x)\right) \, dx","Integrate[(e + f*x)^m*(a + b*ArcTanh[c + d*x]),x]","\int (e+f x)^m \left(a+b \tanh ^{-1}(c+d x)\right) \, dx","\frac{(e+f x)^{m+1} \left(a+b \tanh ^{-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*ArcTanh[c + d*x]), x]","F",-1
53,1,623,780,0.7578574,"\int \frac{\tanh ^{-1}(a+b x)}{c+d x^3} \, dx","Integrate[ArcTanh[a + b*x]/(c + d*x^3),x]","\frac{-\text{Li}_2\left(-\frac{\sqrt[3]{d} (a+b x-1)}{b \sqrt[3]{c}-(a-1) \sqrt[3]{d}}\right)-(-1)^{2/3} \text{Li}_2\left(\frac{\sqrt[3]{-1} \sqrt[3]{d} (a+b x-1)}{\sqrt[3]{-1} \sqrt[3]{d} (a-1)+b \sqrt[3]{c}}\right)+\sqrt[3]{-1} \text{Li}_2\left(\frac{(-1)^{2/3} \sqrt[3]{d} (a+b x-1)}{(-1)^{2/3} (a-1) \sqrt[3]{d}-b \sqrt[3]{c}}\right)+\text{Li}_2\left(-\frac{\sqrt[3]{d} (a+b x+1)}{b \sqrt[3]{c}-(a+1) \sqrt[3]{d}}\right)+(-1)^{2/3} \text{Li}_2\left(\frac{\sqrt[3]{-1} \sqrt[3]{d} (a+b x+1)}{\sqrt[3]{-1} \sqrt[3]{d} (a+1)+b \sqrt[3]{c}}\right)-\sqrt[3]{-1} \text{Li}_2\left(\frac{(-1)^{2/3} \sqrt[3]{d} (a+b x+1)}{(-1)^{2/3} (a+1) \sqrt[3]{d}-b \sqrt[3]{c}}\right)-\log (-a-b x+1) \log \left(\frac{b \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{b \sqrt[3]{c}-(a-1) \sqrt[3]{d}}\right)+\log (a+b x+1) \log \left(\frac{b \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{b \sqrt[3]{c}-(a+1) \sqrt[3]{d}}\right)-(-1)^{2/3} \log (-a-b x+1) \log \left(\frac{b \left(\sqrt[3]{c}-\sqrt[3]{-1} \sqrt[3]{d} x\right)}{\sqrt[3]{-1} (a-1) \sqrt[3]{d}+b \sqrt[3]{c}}\right)+(-1)^{2/3} \log (a+b x+1) \log \left(\frac{b \left(\sqrt[3]{c}-\sqrt[3]{-1} \sqrt[3]{d} x\right)}{\sqrt[3]{-1} (a+1) \sqrt[3]{d}+b \sqrt[3]{c}}\right)+\sqrt[3]{-1} \log (-a-b x+1) \log \left(\frac{b \left(\sqrt[3]{c}+(-1)^{2/3} \sqrt[3]{d} x\right)}{b \sqrt[3]{c}-(-1)^{2/3} (a-1) \sqrt[3]{d}}\right)-\sqrt[3]{-1} \log (a+b x+1) \log \left(\frac{b \left(\sqrt[3]{c}+(-1)^{2/3} \sqrt[3]{d} x\right)}{b \sqrt[3]{c}-(-1)^{2/3} (a+1) \sqrt[3]{d}}\right)}{6 c^{2/3} \sqrt[3]{d}}","-\frac{\text{Li}_2\left(\frac{\sqrt[3]{d} (-a-b x+1)}{\sqrt[3]{d} (1-a)+b \sqrt[3]{c}}\right)}{6 c^{2/3} \sqrt[3]{d}}-\frac{(-1)^{2/3} \text{Li}_2\left(-\frac{\sqrt[3]{-1} \sqrt[3]{d} (-a-b x+1)}{b \sqrt[3]{c}-\sqrt[3]{-1} (1-a) \sqrt[3]{d}}\right)}{6 c^{2/3} \sqrt[3]{d}}+\frac{\sqrt[3]{-1} \text{Li}_2\left(\frac{(-1)^{2/3} \sqrt[3]{d} (-a-b x+1)}{(-1)^{2/3} \sqrt[3]{d} (1-a)+b \sqrt[3]{c}}\right)}{6 c^{2/3} \sqrt[3]{d}}+\frac{\text{Li}_2\left(-\frac{\sqrt[3]{d} (a+b x+1)}{b \sqrt[3]{c}-(a+1) \sqrt[3]{d}}\right)}{6 c^{2/3} \sqrt[3]{d}}+\frac{(-1)^{2/3} \text{Li}_2\left(\frac{\sqrt[3]{-1} \sqrt[3]{d} (a+b x+1)}{\sqrt[3]{-1} \sqrt[3]{d} (a+1)+b \sqrt[3]{c}}\right)}{6 c^{2/3} \sqrt[3]{d}}-\frac{\sqrt[3]{-1} \text{Li}_2\left(-\frac{(-1)^{2/3} \sqrt[3]{d} (a+b x+1)}{b \sqrt[3]{c}-(-1)^{2/3} (a+1) \sqrt[3]{d}}\right)}{6 c^{2/3} \sqrt[3]{d}}-\frac{\log (-a-b x+1) \log \left(\frac{b \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{(1-a) \sqrt[3]{d}+b \sqrt[3]{c}}\right)}{6 c^{2/3} \sqrt[3]{d}}+\frac{\log (a+b x+1) \log \left(\frac{b \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{b \sqrt[3]{c}-(a+1) \sqrt[3]{d}}\right)}{6 c^{2/3} \sqrt[3]{d}}-\frac{(-1)^{2/3} \log (-a-b x+1) \log \left(\frac{b \left(\sqrt[3]{c}-\sqrt[3]{-1} \sqrt[3]{d} x\right)}{b \sqrt[3]{c}-\sqrt[3]{-1} (1-a) \sqrt[3]{d}}\right)}{6 c^{2/3} \sqrt[3]{d}}+\frac{(-1)^{2/3} \log (a+b x+1) \log \left(\frac{b \left(\sqrt[3]{c}-\sqrt[3]{-1} \sqrt[3]{d} x\right)}{\sqrt[3]{-1} (a+1) \sqrt[3]{d}+b \sqrt[3]{c}}\right)}{6 c^{2/3} \sqrt[3]{d}}+\frac{\sqrt[3]{-1} \log (-a-b x+1) \log \left(\frac{b \left(\sqrt[3]{c}+(-1)^{2/3} \sqrt[3]{d} x\right)}{(-1)^{2/3} (1-a) \sqrt[3]{d}+b \sqrt[3]{c}}\right)}{6 c^{2/3} \sqrt[3]{d}}-\frac{\sqrt[3]{-1} \log (a+b x+1) \log \left(\frac{b \left(\sqrt[3]{c}+(-1)^{2/3} \sqrt[3]{d} x\right)}{b \sqrt[3]{c}-(-1)^{2/3} (a+1) \sqrt[3]{d}}\right)}{6 c^{2/3} \sqrt[3]{d}}",1,"(-(Log[1 - a - b*x]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) - (-1 + a)*d^(1/3))]) + Log[1 + a + b*x]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) - (1 + a)*d^(1/3))] - (-1)^(2/3)*Log[1 - a - b*x]*Log[(b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/3)*(-1 + a)*d^(1/3))] + (-1)^(2/3)*Log[1 + a + b*x]*Log[(b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/3)*(1 + a)*d^(1/3))] + (-1)^(1/3)*Log[1 - a - b*x]*Log[(b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) - (-1)^(2/3)*(-1 + a)*d^(1/3))] - (-1)^(1/3)*Log[1 + a + b*x]*Log[(b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) - (-1)^(2/3)*(1 + a)*d^(1/3))] - PolyLog[2, -((d^(1/3)*(-1 + a + b*x))/(b*c^(1/3) - (-1 + a)*d^(1/3)))] - (-1)^(2/3)*PolyLog[2, ((-1)^(1/3)*d^(1/3)*(-1 + a + b*x))/(b*c^(1/3) + (-1)^(1/3)*(-1 + a)*d^(1/3))] + (-1)^(1/3)*PolyLog[2, ((-1)^(2/3)*d^(1/3)*(-1 + a + b*x))/(-(b*c^(1/3)) + (-1)^(2/3)*(-1 + a)*d^(1/3))] + PolyLog[2, -((d^(1/3)*(1 + a + b*x))/(b*c^(1/3) - (1 + a)*d^(1/3)))] + (-1)^(2/3)*PolyLog[2, ((-1)^(1/3)*d^(1/3)*(1 + a + b*x))/(b*c^(1/3) + (-1)^(1/3)*(1 + a)*d^(1/3))] - (-1)^(1/3)*PolyLog[2, ((-1)^(2/3)*d^(1/3)*(1 + a + b*x))/(-(b*c^(1/3)) + (-1)^(2/3)*(1 + a)*d^(1/3))])/(6*c^(2/3)*d^(1/3))","A",1
54,1,365,481,0.3071175,"\int \frac{\tanh ^{-1}(a+b x)}{c+d x^2} \, dx","Integrate[ArcTanh[a + b*x]/(c + d*x^2),x]","\frac{\text{Li}_2\left(-\frac{\sqrt{d} (a+b x-1)}{b \sqrt{-c}-(a-1) \sqrt{d}}\right)-\text{Li}_2\left(\frac{\sqrt{d} (a+b x-1)}{\sqrt{d} (a-1)+b \sqrt{-c}}\right)-\text{Li}_2\left(-\frac{\sqrt{d} (a+b x+1)}{b \sqrt{-c}-(a+1) \sqrt{d}}\right)+\text{Li}_2\left(\frac{\sqrt{d} (a+b x+1)}{\sqrt{d} (a+1)+b \sqrt{-c}}\right)-\log (-a-b x+1) \log \left(\frac{b \left(\sqrt{-c}-\sqrt{d} x\right)}{(a-1) \sqrt{d}+b \sqrt{-c}}\right)+\log (a+b x+1) \log \left(\frac{b \left(\sqrt{-c}-\sqrt{d} x\right)}{(a+1) \sqrt{d}+b \sqrt{-c}}\right)+\log (-a-b x+1) \log \left(\frac{b \left(\sqrt{-c}+\sqrt{d} x\right)}{b \sqrt{-c}-(a-1) \sqrt{d}}\right)-\log (a+b x+1) \log \left(\frac{b \left(\sqrt{-c}+\sqrt{d} x\right)}{b \sqrt{-c}-(a+1) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}","-\frac{\text{Li}_2\left(-\frac{\sqrt{d} (-a-b x+1)}{b \sqrt{-c}-(1-a) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{Li}_2\left(\frac{\sqrt{d} (-a-b x+1)}{\sqrt{d} (1-a)+b \sqrt{-c}}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{\text{Li}_2\left(-\frac{\sqrt{d} (a+b x+1)}{b \sqrt{-c}-(a+1) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{\text{Li}_2\left(\frac{\sqrt{d} (a+b x+1)}{\sqrt{d} (a+1)+b \sqrt{-c}}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{\log (-a-b x+1) \log \left(\frac{b \left(\sqrt{-c}-\sqrt{d} x\right)}{b \sqrt{-c}-(1-a) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{\log (a+b x+1) \log \left(\frac{b \left(\sqrt{-c}-\sqrt{d} x\right)}{(a+1) \sqrt{d}+b \sqrt{-c}}\right)}{4 \sqrt{-c} \sqrt{d}}+\frac{\log (-a-b x+1) \log \left(\frac{b \left(\sqrt{-c}+\sqrt{d} x\right)}{(1-a) \sqrt{d}+b \sqrt{-c}}\right)}{4 \sqrt{-c} \sqrt{d}}-\frac{\log (a+b x+1) \log \left(\frac{b \left(\sqrt{-c}+\sqrt{d} x\right)}{b \sqrt{-c}-(a+1) \sqrt{d}}\right)}{4 \sqrt{-c} \sqrt{d}}",1,"(-(Log[1 - a - b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (-1 + a)*Sqrt[d])]) + Log[1 + a + b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])] + Log[1 - a - b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (-1 + a)*Sqrt[d])] - Log[1 + a + b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (1 + a)*Sqrt[d])] + PolyLog[2, -((Sqrt[d]*(-1 + a + b*x))/(b*Sqrt[-c] - (-1 + a)*Sqrt[d]))] - PolyLog[2, (Sqrt[d]*(-1 + a + b*x))/(b*Sqrt[-c] + (-1 + a)*Sqrt[d])] - PolyLog[2, -((Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] - (1 + a)*Sqrt[d]))] + PolyLog[2, (Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d])","A",1
55,1,138,120,0.013229,"\int \frac{\tanh ^{-1}(a+b x)}{c+d x} \, dx","Integrate[ArcTanh[a + b*x]/(c + d*x),x]","-\frac{\text{Li}_2\left(-\frac{d (-a-b x+1)}{-b c+a d-d}\right)}{2 d}+\frac{\text{Li}_2\left(\frac{d (a+b x+1)}{-b c+a d+d}\right)}{2 d}-\frac{\log (-a-b x+1) \log \left(-\frac{b (c+d x)}{-(1-a) d-b c}\right)}{2 d}+\frac{\log (a+b x+1) \log \left(\frac{b (c+d x)}{b c-(a+1) d}\right)}{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{\tanh ^{-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) \tanh ^{-1}(a+b x)}{d}",1,"-1/2*(Log[1 - a - b*x]*Log[-((b*(c + d*x))/(-(b*c) - (1 - a)*d))])/d + (Log[1 + a + b*x]*Log[(b*(c + d*x))/(b*c - (1 + a)*d)])/(2*d) - PolyLog[2, -((d*(1 - a - b*x))/(-(b*c) - d + a*d))]/(2*d) + PolyLog[2, (d*(1 + a + b*x))/(-(b*c) + d + a*d)]/(2*d)","A",0
56,1,759,186,4.14584,"\int \frac{\tanh ^{-1}(a+b x)}{c+\frac{d}{x}} \, dx","Integrate[ArcTanh[a + b*x]/(c + d/x),x]","\frac{b c d \sqrt{-a^2+\frac{2 a b d}{c}-\frac{b^2 d^2}{c^2}+1} \tanh ^{-1}(a+b x)^2 e^{\tanh ^{-1}\left(a-\frac{b d}{c}\right)}-2 a^2 c^2 \tanh ^{-1}(a+b x)+2 b^2 d^2 \tanh ^{-1}\left(a-\frac{b d}{c}\right) \log \left(1-\exp \left(2 \left(\tanh ^{-1}\left(a-\frac{b d}{c}\right)-\tanh ^{-1}(a+b x)\right)\right)\right)-2 b^2 d^2 \tanh ^{-1}(a+b x) \log \left(1-\exp \left(2 \left(\tanh ^{-1}\left(a-\frac{b d}{c}\right)-\tanh ^{-1}(a+b x)\right)\right)\right)+2 b^2 d^2 \tanh ^{-1}(a+b x) \tanh ^{-1}\left(a-\frac{b d}{c}\right)-2 b^2 d^2 \tanh ^{-1}\left(a-\frac{b d}{c}\right) \log \left(-i \sinh \left(\tanh ^{-1}\left(a-\frac{b d}{c}\right)-\tanh ^{-1}(a+b x)\right)\right)+2 b^2 c d x \tanh ^{-1}(a+b x)-i \pi  b^2 d^2 \log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right)+b^2 d^2 \tanh ^{-1}(a+b x)^2-i \pi  b^2 d^2 \tanh ^{-1}(a+b x)+2 b^2 d^2 \tanh ^{-1}(a+b x) \log \left(e^{-2 \tanh ^{-1}(a+b x)}+1\right)+i \pi  b^2 d^2 \log \left(e^{2 \tanh ^{-1}(a+b x)}+1\right)+2 a c^2 \log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right)-2 a b c^2 x \tanh ^{-1}(a+b x)+b d (b d-a c) \text{Li}_2\left(\exp \left(2 \left(\tanh ^{-1}\left(a-\frac{b d}{c}\right)-\tanh ^{-1}(a+b x)\right)\right)\right)-2 a b c d \tanh ^{-1}\left(a-\frac{b d}{c}\right) \log \left(1-\exp \left(2 \left(\tanh ^{-1}\left(a-\frac{b d}{c}\right)-\tanh ^{-1}(a+b x)\right)\right)\right)+2 a b c d \tanh ^{-1}(a+b x) \log \left(1-\exp \left(2 \left(\tanh ^{-1}\left(a-\frac{b d}{c}\right)-\tanh ^{-1}(a+b x)\right)\right)\right)+b d (a c-b d) \text{Li}_2\left(-e^{-2 \tanh ^{-1}(a+b x)}\right)-2 b c d \log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right)+i \pi  a b c d \log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right)-a b c d \tanh ^{-1}(a+b x)^2-b c d \tanh ^{-1}(a+b x)^2+2 a b c d \tanh ^{-1}(a+b x)-2 a b c d \tanh ^{-1}(a+b x) \tanh ^{-1}\left(a-\frac{b d}{c}\right)+i \pi  a b c d \tanh ^{-1}(a+b x)-2 a b c d \tanh ^{-1}(a+b x) \log \left(e^{-2 \tanh ^{-1}(a+b x)}+1\right)-i \pi  a b c d \log \left(e^{2 \tanh ^{-1}(a+b x)}+1\right)+2 a b c d \tanh ^{-1}\left(a-\frac{b d}{c}\right) \log \left(-i \sinh \left(\tanh ^{-1}\left(a-\frac{b d}{c}\right)-\tanh ^{-1}(a+b x)\right)\right)}{2 b c^2 (b d-a c)}","\frac{d \text{Li}_2\left(\frac{c (-a-b x+1)}{-a c+c+b d}\right)}{2 c^2}-\frac{d \text{Li}_2\left(\frac{c (a+b x+1)}{a c+c-b d}\right)}{2 c^2}+\frac{d \log (-a-b x+1) \log \left(\frac{b (c x+d)}{-a c+b d+c}\right)}{2 c^2}-\frac{d \log (a+b x+1) \log \left(-\frac{b (c x+d)}{a c-b d+c}\right)}{2 c^2}+\frac{(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac{(a+b x+1) \log (a+b x+1)}{2 b c}",1,"(-2*a^2*c^2*ArcTanh[a + b*x] + 2*a*b*c*d*ArcTanh[a + b*x] + I*a*b*c*d*Pi*ArcTanh[a + b*x] - I*b^2*d^2*Pi*ArcTanh[a + b*x] - 2*a*b*c^2*x*ArcTanh[a + b*x] + 2*b^2*c*d*x*ArcTanh[a + b*x] - 2*a*b*c*d*ArcTanh[a - (b*d)/c]*ArcTanh[a + b*x] + 2*b^2*d^2*ArcTanh[a - (b*d)/c]*ArcTanh[a + b*x] - b*c*d*ArcTanh[a + b*x]^2 - a*b*c*d*ArcTanh[a + b*x]^2 + b^2*d^2*ArcTanh[a + b*x]^2 + b*c*d*Sqrt[1 - a^2 + (2*a*b*d)/c - (b^2*d^2)/c^2]*E^ArcTanh[a - (b*d)/c]*ArcTanh[a + b*x]^2 - 2*a*b*c*d*ArcTanh[a - (b*d)/c]*Log[1 - E^(2*(ArcTanh[a - (b*d)/c] - ArcTanh[a + b*x]))] + 2*b^2*d^2*ArcTanh[a - (b*d)/c]*Log[1 - E^(2*(ArcTanh[a - (b*d)/c] - ArcTanh[a + b*x]))] + 2*a*b*c*d*ArcTanh[a + b*x]*Log[1 - E^(2*(ArcTanh[a - (b*d)/c] - ArcTanh[a + b*x]))] - 2*b^2*d^2*ArcTanh[a + b*x]*Log[1 - E^(2*(ArcTanh[a - (b*d)/c] - ArcTanh[a + b*x]))] - 2*a*b*c*d*ArcTanh[a + b*x]*Log[1 + E^(-2*ArcTanh[a + b*x])] + 2*b^2*d^2*ArcTanh[a + b*x]*Log[1 + E^(-2*ArcTanh[a + b*x])] - I*a*b*c*d*Pi*Log[1 + E^(2*ArcTanh[a + b*x])] + I*b^2*d^2*Pi*Log[1 + E^(2*ArcTanh[a + b*x])] + 2*a*c^2*Log[1/Sqrt[1 - (a + b*x)^2]] - 2*b*c*d*Log[1/Sqrt[1 - (a + b*x)^2]] + I*a*b*c*d*Pi*Log[1/Sqrt[1 - (a + b*x)^2]] - I*b^2*d^2*Pi*Log[1/Sqrt[1 - (a + b*x)^2]] + 2*a*b*c*d*ArcTanh[a - (b*d)/c]*Log[(-I)*Sinh[ArcTanh[a - (b*d)/c] - ArcTanh[a + b*x]]] - 2*b^2*d^2*ArcTanh[a - (b*d)/c]*Log[(-I)*Sinh[ArcTanh[a - (b*d)/c] - ArcTanh[a + b*x]]] + b*d*(-(a*c) + b*d)*PolyLog[2, E^(2*(ArcTanh[a - (b*d)/c] - ArcTanh[a + b*x]))] + b*d*(a*c - b*d)*PolyLog[2, -E^(-2*ArcTanh[a + b*x])])/(2*b*c^2*(-(a*c) + b*d))","C",0
57,1,1456,545,25.3215637,"\int \frac{\tanh ^{-1}(a+b x)}{c+\frac{d}{x^2}} \, dx","Integrate[ArcTanh[a + b*x]/(c + d/x^2),x]","\frac{(a+b x) \tanh ^{-1}(a+b x)-\log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right)}{b c}+\frac{\sqrt{d} \left(-2 i \sqrt{c} \tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) a^2+2 i \sqrt{c} \tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) a^2-2 \sqrt{c} \tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right) a^2-2 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right) a^2+2 \sqrt{c} \tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right) a^2+2 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right) a^2+2 \sqrt{c} \tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right) \log \left(-\sin \left(\tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)\right) a^2-2 \sqrt{c} \tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right) \log \left(-\sin \left(\tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)\right) a^2+b \sqrt{d} \sqrt{\frac{c (a-1)^2+b^2 d}{b^2 d}} e^{-i \tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right)} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)^2 a-b \sqrt{d} \sqrt{\frac{c (a+1)^2+b^2 d}{b^2 d}} e^{-i \tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right)} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)^2 a+b \sqrt{d} \sqrt{\frac{c (a-1)^2+b^2 d}{b^2 d}} e^{-i \tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right)} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)^2+b \sqrt{d} \sqrt{\frac{c (a+1)^2+b^2 d}{b^2 d}} e^{-i \tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right)} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)^2-2 b \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)^2+2 i \sqrt{c} \tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)-2 i \sqrt{c} \tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)-4 \left(a^2-1\right) \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) \tanh ^{-1}(a+b x)+2 \sqrt{c} \tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right)+2 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right)-2 \sqrt{c} \tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right)-2 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right) \log \left(1-e^{-2 i \left(\tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right)-2 \sqrt{c} \tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right) \log \left(-\sin \left(\tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)\right)+2 \sqrt{c} \tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right) \log \left(-\sin \left(\tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)\right)-i \left(a^2-1\right) \sqrt{c} \text{Li}_2\left(e^{-2 i \left(\tan ^{-1}\left(\frac{(a-1) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right)+i \left(a^2-1\right) \sqrt{c} \text{Li}_2\left(e^{-2 i \left(\tan ^{-1}\left(\frac{(a+1) \sqrt{c}}{b \sqrt{d}}\right)+\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}\right)\right)}{4 \left(a^2-1\right) c^2}","\frac{\sqrt{d} \text{Li}_2\left(\frac{\sqrt{-c} (-a-b x+1)}{-\sqrt{-c} 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{(a+b x+1) \log (a+b x+1)}{2 b c}",1,"((a + b*x)*ArcTanh[a + b*x] - Log[1/Sqrt[1 - (a + b*x)^2]])/(b*c) + (Sqrt[d]*((2*I)*Sqrt[c]*ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])]*ArcTan[(Sqrt[c]*x)/Sqrt[d]] - (2*I)*a^2*Sqrt[c]*ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])]*ArcTan[(Sqrt[c]*x)/Sqrt[d]] - (2*I)*Sqrt[c]*ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]*ArcTan[(Sqrt[c]*x)/Sqrt[d]] + (2*I)*a^2*Sqrt[c]*ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]*ArcTan[(Sqrt[c]*x)/Sqrt[d]] - 2*b*Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^2 + (b*Sqrt[d]*Sqrt[((-1 + a)^2*c + b^2*d)/(b^2*d)]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^2)/E^(I*ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])]) + (a*b*Sqrt[d]*Sqrt[((-1 + a)^2*c + b^2*d)/(b^2*d)]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^2)/E^(I*ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])]) + (b*Sqrt[d]*Sqrt[((1 + a)^2*c + b^2*d)/(b^2*d)]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^2)/E^(I*ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]) - (a*b*Sqrt[d]*Sqrt[((1 + a)^2*c + b^2*d)/(b^2*d)]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^2)/E^(I*ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]) - 4*(-1 + a^2)*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*ArcTanh[a + b*x] + 2*Sqrt[c]*ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])]*Log[1 - E^((-2*I)*(ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - 2*a^2*Sqrt[c]*ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])]*Log[1 - E^((-2*I)*(ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] + 2*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - 2*a^2*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - 2*Sqrt[c]*ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]*Log[1 - E^((-2*I)*(ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] + 2*a^2*Sqrt[c]*ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]*Log[1 - E^((-2*I)*(ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - 2*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] + 2*a^2*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - 2*Sqrt[c]*ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])]*Log[-Sin[ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]]] + 2*a^2*Sqrt[c]*ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])]*Log[-Sin[ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]]] + 2*Sqrt[c]*ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]*Log[-Sin[ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]]] - 2*a^2*Sqrt[c]*ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]*Log[-Sin[ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]]] - I*(-1 + a^2)*Sqrt[c]*PolyLog[2, E^((-2*I)*(ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] + I*(-1 + a^2)*Sqrt[c]*PolyLog[2, E^((-2*I)*(ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))]))/(4*(-1 + a^2)*c^2)","C",0
58,1,917,832,8.0630026,"\int \frac{\tanh ^{-1}(a+b x)}{c+\frac{d}{x^3}} \, dx","Integrate[ArcTanh[a + b*x]/(c + d/x^3),x]","-\frac{d \text{RootSum}\left[c \text{$\#$1}^3 a^3+3 c \text{$\#$1}^2 a^3+c a^3+3 c \text{$\#$1} a^3-3 c \text{$\#$1}^3 a^2-3 c \text{$\#$1}^2 a^2+3 c a^2+3 c \text{$\#$1} a^2+3 c \text{$\#$1}^3 a-3 c \text{$\#$1}^2 a+3 c a-3 c \text{$\#$1} a-c \text{$\#$1}^3-b^3 d \text{$\#$1}^3+3 c \text{$\#$1}^2-3 b^3 d \text{$\#$1}^2+c-b^3 d-3 c \text{$\#$1}-3 b^3 d \text{$\#$1}\&,\frac{-2 \text{$\#$1} \tanh ^{-1}(a+b x)^2+2 e^{-\tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right)} \sqrt{\frac{\text{$\#$1}}{(\text{$\#$1}+1)^2}} \tanh ^{-1}(a+b x)^2+2 e^{-\tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right)} \text{$\#$1}^2 \sqrt{\frac{\text{$\#$1}}{(\text{$\#$1}+1)^2}} \tanh ^{-1}(a+b x)^2+4 e^{-\tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right)} \text{$\#$1} \sqrt{\frac{\text{$\#$1}}{(\text{$\#$1}+1)^2}} \tanh ^{-1}(a+b x)^2-2 \tanh ^{-1}(a+b x)^2+2 \tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right) \text{$\#$1}^2 \tanh ^{-1}(a+b x)+2 \log \left(1-e^{-2 \left(\tanh ^{-1}(a+b x)+\tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right)\right)}\right) \text{$\#$1}^2 \tanh ^{-1}(a+b x)+i \pi  \text{$\#$1}^2 \tanh ^{-1}(a+b x)-2 \tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right) \tanh ^{-1}(a+b x)-2 \log \left(1-e^{-2 \left(\tanh ^{-1}(a+b x)+\tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right)\right)}\right) \tanh ^{-1}(a+b x)-i \pi  \tanh ^{-1}(a+b x)-i \pi  \log \left(1+e^{2 \tanh ^{-1}(a+b x)}\right) \text{$\#$1}^2+2 \tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right) \log \left(1-e^{-2 \left(\tanh ^{-1}(a+b x)+\tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right)\right)}\right) \text{$\#$1}^2+i \pi  \log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right) \text{$\#$1}^2-2 \tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right) \log \left(i \sinh \left(\tanh ^{-1}(a+b x)+\tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right)\right)\right) \text{$\#$1}^2-\text{Li}_2\left(e^{-2 \left(\tanh ^{-1}(a+b x)+\tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right)\right)}\right) \text{$\#$1}^2+i \pi  \log \left(1+e^{2 \tanh ^{-1}(a+b x)}\right)-2 \tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right) \log \left(1-e^{-2 \left(\tanh ^{-1}(a+b x)+\tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right)\right)}\right)-i \pi  \log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right)+2 \tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right) \log \left(i \sinh \left(\tanh ^{-1}(a+b x)+\tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right)\right)\right)+\text{Li}_2\left(e^{-2 \left(\tanh ^{-1}(a+b x)+\tanh ^{-1}\left(\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right)\right)}\right)}{c \text{$\#$1}^2 a^3+c a^3+2 c \text{$\#$1} a^3-2 c \text{$\#$1}^2 a^2+2 c a^2+c \text{$\#$1}^2 a+c a-2 c \text{$\#$1} a-b^3 d \text{$\#$1}^2-b^3 d-2 b^3 d \text{$\#$1}}\&\right] b^3-6 (a+b x) \tanh ^{-1}(a+b x)+6 \log \left(\frac{1}{\sqrt{1-(a+b x)^2}}\right)}{6 b c}","\frac{(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac{\sqrt[3]{d} \log \left(\frac{b \left(\sqrt[3]{c} x+\sqrt[3]{d}\right)}{\sqrt[3]{c} (1-a)+b \sqrt[3]{d}}\right) \log (-a-b x+1)}{6 c^{4/3}}+\frac{(-1)^{2/3} \sqrt[3]{d} \log \left(-\frac{b \left(\sqrt[3]{d}-\sqrt[3]{-1} \sqrt[3]{c} x\right)}{\sqrt[3]{-1} (1-a) \sqrt[3]{c}-b \sqrt[3]{d}}\right) \log (-a-b x+1)}{6 c^{4/3}}-\frac{\sqrt[3]{-1} \sqrt[3]{d} \log \left(\frac{b \left((-1)^{2/3} \sqrt[3]{c} x+\sqrt[3]{d}\right)}{(-1)^{2/3} \sqrt[3]{c} (1-a)+b \sqrt[3]{d}}\right) \log (-a-b x+1)}{6 c^{4/3}}+\frac{(a+b x+1) \log (a+b x+1)}{2 b c}-\frac{\sqrt[3]{d} \log (a+b x+1) \log \left(-\frac{b \left(\sqrt[3]{c} x+\sqrt[3]{d}\right)}{(a+1) \sqrt[3]{c}-b \sqrt[3]{d}}\right)}{6 c^{4/3}}-\frac{(-1)^{2/3} \sqrt[3]{d} \log (a+b x+1) \log \left(\frac{b \left(\sqrt[3]{d}-\sqrt[3]{-1} \sqrt[3]{c} x\right)}{\sqrt[3]{-1} \sqrt[3]{c} (a+1)+b \sqrt[3]{d}}\right)}{6 c^{4/3}}+\frac{\sqrt[3]{-1} \sqrt[3]{d} \log (a+b x+1) \log \left(-\frac{b \left((-1)^{2/3} \sqrt[3]{c} x+\sqrt[3]{d}\right)}{(-1)^{2/3} (a+1) \sqrt[3]{c}-b \sqrt[3]{d}}\right)}{6 c^{4/3}}+\frac{(-1)^{2/3} \sqrt[3]{d} \text{Li}_2\left(\frac{\sqrt[3]{-1} \sqrt[3]{c} (-a-b x+1)}{\sqrt[3]{-1} (1-a) \sqrt[3]{c}-b \sqrt[3]{d}}\right)}{6 c^{4/3}}+\frac{\sqrt[3]{d} \text{Li}_2\left(\frac{\sqrt[3]{c} (-a-b x+1)}{\sqrt[3]{c} (1-a)+b \sqrt[3]{d}}\right)}{6 c^{4/3}}-\frac{\sqrt[3]{-1} \sqrt[3]{d} \text{Li}_2\left(\frac{(-1)^{2/3} \sqrt[3]{c} (-a-b x+1)}{(-1)^{2/3} \sqrt[3]{c} (1-a)+b \sqrt[3]{d}}\right)}{6 c^{4/3}}-\frac{\sqrt[3]{d} \text{Li}_2\left(\frac{\sqrt[3]{c} (a+b x+1)}{(a+1) \sqrt[3]{c}-b \sqrt[3]{d}}\right)}{6 c^{4/3}}+\frac{\sqrt[3]{-1} \sqrt[3]{d} \text{Li}_2\left(\frac{(-1)^{2/3} \sqrt[3]{c} (a+b x+1)}{(-1)^{2/3} (a+1) \sqrt[3]{c}-b \sqrt[3]{d}}\right)}{6 c^{4/3}}-\frac{(-1)^{2/3} \sqrt[3]{d} \text{Li}_2\left(\frac{\sqrt[3]{-1} \sqrt[3]{c} (a+b x+1)}{\sqrt[3]{-1} \sqrt[3]{c} (a+1)+b \sqrt[3]{d}}\right)}{6 c^{4/3}}",1,"-1/6*(-6*(a + b*x)*ArcTanh[a + b*x] + 6*Log[1/Sqrt[1 - (a + b*x)^2]] + b^3*d*RootSum[c + 3*a*c + 3*a^2*c + a^3*c - b^3*d - 3*c*#1 - 3*a*c*#1 + 3*a^2*c*#1 + 3*a^3*c*#1 - 3*b^3*d*#1 + 3*c*#1^2 - 3*a*c*#1^2 - 3*a^2*c*#1^2 + 3*a^3*c*#1^2 - 3*b^3*d*#1^2 - c*#1^3 + 3*a*c*#1^3 - 3*a^2*c*#1^3 + a^3*c*#1^3 - b^3*d*#1^3 & , ((-I)*Pi*ArcTanh[a + b*x] - 2*ArcTanh[a + b*x]^2 - 2*ArcTanh[a + b*x]*ArcTanh[(1 - #1)/(1 + #1)] + I*Pi*Log[1 + E^(2*ArcTanh[a + b*x])] - 2*ArcTanh[a + b*x]*Log[1 - E^(-2*(ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]))] - 2*ArcTanh[(1 - #1)/(1 + #1)]*Log[1 - E^(-2*(ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]))] - I*Pi*Log[1/Sqrt[1 - (a + b*x)^2]] + 2*ArcTanh[(1 - #1)/(1 + #1)]*Log[I*Sinh[ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]]] + PolyLog[2, E^(-2*(ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]))] - 2*ArcTanh[a + b*x]^2*#1 + I*Pi*ArcTanh[a + b*x]*#1^2 + 2*ArcTanh[a + b*x]*ArcTanh[(1 - #1)/(1 + #1)]*#1^2 - I*Pi*Log[1 + E^(2*ArcTanh[a + b*x])]*#1^2 + 2*ArcTanh[a + b*x]*Log[1 - E^(-2*(ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]))]*#1^2 + 2*ArcTanh[(1 - #1)/(1 + #1)]*Log[1 - E^(-2*(ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]))]*#1^2 + I*Pi*Log[1/Sqrt[1 - (a + b*x)^2]]*#1^2 - 2*ArcTanh[(1 - #1)/(1 + #1)]*Log[I*Sinh[ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]]]*#1^2 - PolyLog[2, E^(-2*(ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]))]*#1^2 + (2*ArcTanh[a + b*x]^2*Sqrt[#1/(1 + #1)^2])/E^ArcTanh[(1 - #1)/(1 + #1)] + (4*ArcTanh[a + b*x]^2*#1*Sqrt[#1/(1 + #1)^2])/E^ArcTanh[(1 - #1)/(1 + #1)] + (2*ArcTanh[a + b*x]^2*#1^2*Sqrt[#1/(1 + #1)^2])/E^ArcTanh[(1 - #1)/(1 + #1)])/(a*c + 2*a^2*c + a^3*c - b^3*d - 2*a*c*#1 + 2*a^3*c*#1 - 2*b^3*d*#1 + a*c*#1^2 - 2*a^2*c*#1^2 + a^3*c*#1^2 - b^3*d*#1^2) & ])/(b*c)","C",0
59,1,549,585,0.5393285,"\int \frac{\tanh ^{-1}(a+b x)}{c+d \sqrt{x}} \, dx","Integrate[ArcTanh[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 (-a-b x+1) \log \left(c+d \sqrt{x}\right)-c \log (a+b x+1) \log \left(c+d \sqrt{x}\right)-d \sqrt{x} \log (-a-b x+1)+d \sqrt{x} \log (a+b x+1)+\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 (-a-b x+1) \log \left(c+d \sqrt{x}\right)}{d^2}-\frac{c \log (a+b x+1) \log \left(c+d \sqrt{x}\right)}{d^2}-\frac{\sqrt{x} \log (-a-b x+1)}{d}+\frac{\sqrt{x} \log (a+b x+1)}{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] + c*Log[c + d*Sqrt[x]]*Log[1 - a - b*x] + d*Sqrt[x]*Log[1 + a + b*x] - c*Log[c + d*Sqrt[x]]*Log[1 + 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
60,1,598,661,0.6384077,"\int \frac{\tanh ^{-1}(a+b x)}{c+\frac{d}{\sqrt{x}}} \, dx","Integrate[ArcTanh[a + b*x]/(c + d/Sqrt[x]),x]","\frac{-\frac{c^2 (a+b x-1) \log (-a-b x+1)}{b}+\frac{c^2 (a+b x+1) \log (a+b x+1)}{b}-2 d^2 \left(\text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{b} d-\sqrt{-a-1} c}\right)+\text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{-a-1} c+\sqrt{b} d}\right)+\log \left(c \sqrt{x}+d\right) \left(\log \left(\frac{c \left(\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-1} c+\sqrt{b} d}\right)+\log \left(\frac{c \left(\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right)}{\sqrt{-a-1} c-\sqrt{b} d}\right)\right)\right)+2 d^2 \left(\text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{b} d-\sqrt{1-a} c}\right)+\text{Li}_2\left(\frac{\sqrt{b} \left(\sqrt{x} c+d\right)}{\sqrt{1-a} c+\sqrt{b} d}\right)+\log \left(c \sqrt{x}+d\right) \left(\log \left(\frac{c \left(\sqrt{1-a}-\sqrt{b} \sqrt{x}\right)}{\sqrt{1-a} c+\sqrt{b} d}\right)+\log \left(\frac{c \left(\sqrt{1-a}+\sqrt{b} \sqrt{x}\right)}{\sqrt{1-a} c-\sqrt{b} d}\right)\right)\right)-2 d^2 \log (-a-b x+1) \log \left(c \sqrt{x}+d\right)+2 d^2 \log (a+b x+1) \log \left(c \sqrt{x}+d\right)+2 c d \sqrt{x} \log (-a-b x+1)-2 c d \sqrt{x} \log (a+b x+1)+4 c d \left(\sqrt{x}-\frac{\sqrt{a+1} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right)}{\sqrt{b}}\right)-4 c d \left(\sqrt{x}-\frac{\sqrt{1-a} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right)}{\sqrt{b}}\right)}{2 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 (-a-b x+1) \log \left(c \sqrt{x}+d\right)}{c^3}+\frac{d^2 \log (a+b x+1) \log \left(c \sqrt{x}+d\right)}{c^3}+\frac{d \sqrt{x} \log (-a-b x+1)}{c^2}-\frac{d \sqrt{x} \log (a+b x+1)}{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{(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac{(a+b x+1) \log (a+b x+1)}{2 b c}",1,"(4*c*d*(Sqrt[x] - (Sqrt[1 + a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]])/Sqrt[b]) - 4*c*d*(Sqrt[x] - (Sqrt[1 - a]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[1 - a]])/Sqrt[b]) + 2*c*d*Sqrt[x]*Log[1 - a - b*x] - (c^2*(-1 + a + b*x)*Log[1 - a - b*x])/b - 2*d^2*Log[d + c*Sqrt[x]]*Log[1 - a - b*x] - 2*c*d*Sqrt[x]*Log[1 + a + b*x] + (c^2*(1 + a + b*x)*Log[1 + a + b*x])/b + 2*d^2*Log[d + c*Sqrt[x]]*Log[1 + a + b*x] - 2*d^2*((Log[(c*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)] + Log[(c*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d)])*Log[d + c*Sqrt[x]] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[-1 - a]*c) + Sqrt[b]*d)] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)]) + 2*d^2*((Log[(c*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c + Sqrt[b]*d)] + Log[(c*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b]*d)])*Log[d + c*Sqrt[x]] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[1 - a]*c) + Sqrt[b]*d)] + PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a]*c + Sqrt[b]*d)]))/(2*c^3)","A",1
61,1,403,335,0.5722697,"\int \frac{\tanh ^{-1}(d+e x)}{a+b x+c x^2} \, dx","Integrate[ArcTanh[d + e*x]/(a + b*x + c*x^2),x]","\frac{-\text{Li}_2\left(\frac{2 c (d+e x-1)}{2 c (d-1)+\left(\sqrt{b^2-4 a c}-b\right) e}\right)+\text{Li}_2\left(\frac{2 c (d+e x-1)}{2 c (d-1)-\left(b+\sqrt{b^2-4 a c}\right) e}\right)+\text{Li}_2\left(\frac{2 c (d+e x+1)}{2 c (d+1)+\left(\sqrt{b^2-4 a c}-b\right) e}\right)-\text{Li}_2\left(\frac{2 c (d+e x+1)}{2 c (d+1)-\left(b+\sqrt{b^2-4 a c}\right) e}\right)+\log (-d-e x+1) \left(-\log \left(\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c (d-1)}\right)\right)+\log (-d-e x+1) \log \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c (d-1)}\right)+\log (d+e x+1) \log \left(\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c (d+1)}\right)-\log (d+e x+1) \log \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c (d+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 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{\tanh ^{-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{\tanh ^{-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[(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*(-1 + d) + (-b + Sqrt[b^2 - 4*a*c])*e)]*Log[1 - d - e*x]) + Log[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*(-1 + d) + (b + Sqrt[b^2 - 4*a*c])*e)]*Log[1 - d - e*x] + Log[(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*(1 + d) + (-b + Sqrt[b^2 - 4*a*c])*e)]*Log[1 + d + e*x] - Log[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*(1 + d) + (b + Sqrt[b^2 - 4*a*c])*e)]*Log[1 + d + e*x] - PolyLog[2, (2*c*(-1 + d + e*x))/(2*c*(-1 + d) + (-b + Sqrt[b^2 - 4*a*c])*e)] + PolyLog[2, (2*c*(-1 + d + e*x))/(2*c*(-1 + d) - (b + Sqrt[b^2 - 4*a*c])*e)] + PolyLog[2, (2*c*(1 + d + e*x))/(2*c*(1 + d) + (-b + Sqrt[b^2 - 4*a*c])*e)] - PolyLog[2, (2*c*(1 + d + e*x))/(2*c*(1 + d) - (b + Sqrt[b^2 - 4*a*c])*e)])/(2*Sqrt[b^2 - 4*a*c])","A",0
62,1,114,83,0.1210127,"\int \frac{(c e+d e x) \left(a+b \tanh ^{-1}(c+d x)\right)}{1-(c+d x)^2} \, dx","Integrate[((c*e + d*e*x)*(a + b*ArcTanh[c + d*x]))/(1 - (c + d*x)^2),x]","-\frac{e \left(4 a \log (-c-d x+1)+4 a \log (c+d x+1)-2 b \text{Li}_2\left(\frac{1}{2} (-c-d x+1)\right)+2 b \text{Li}_2\left(\frac{1}{2} (c+d x+1)\right)-b \log ^2(-c-d x+1)+b \log ^2(c+d x+1)+b \log (4) \log (c+d x-1)-b \log (4) \log (c+d x+1)\right)}{8 d}","-\frac{e \left(a+b \tanh ^{-1}(c+d x)\right)^2}{2 b d}+\frac{e \log \left(\frac{2}{-c-d x+1}\right) \left(a+b \tanh ^{-1}(c+d x)\right)}{d}+\frac{b e \text{Li}_2\left(-\frac{c+d x+1}{-c-d x+1}\right)}{2 d}",1,"-1/8*(e*(4*a*Log[1 - c - d*x] - b*Log[1 - c - d*x]^2 + b*Log[4]*Log[-1 + c + d*x] + 4*a*Log[1 + c + d*x] - b*Log[4]*Log[1 + c + d*x] + b*Log[1 + c + d*x]^2 - 2*b*PolyLog[2, (1 - c - d*x)/2] + 2*b*PolyLog[2, (1 + c + d*x)/2]))/d","A",0