Integrand size = 10, antiderivative size = 150 \[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )+\frac {3}{2} \coth ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+a x}\right )-\frac {3}{2} \coth ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2 a x}{1+a x}\right )+\frac {3}{2} \coth ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+a x}\right )-\frac {3}{2} \coth ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2 a x}{1+a x}\right )+\frac {3}{4} \operatorname {PolyLog}\left (4,1-\frac {2}{1+a x}\right )-\frac {3}{4} \operatorname {PolyLog}\left (4,1-\frac {2 a x}{1+a x}\right ) \]
2*arccoth(a*x)^3*arccoth(1-2/(-a*x+1))+3/2*arccoth(a*x)^2*polylog(2,1-2/(a *x+1))-3/2*arccoth(a*x)^2*polylog(2,1-2*a*x/(a*x+1))+3/2*arccoth(a*x)*poly log(3,1-2/(a*x+1))-3/2*arccoth(a*x)*polylog(3,1-2*a*x/(a*x+1))+3/4*polylog (4,1-2/(a*x+1))-3/4*polylog(4,1-2*a*x/(a*x+1))
Time = 0.06 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.04 \[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=\frac {1}{64} \left (-\pi ^4+32 \coth ^{-1}(a x)^4+64 \coth ^{-1}(a x)^3 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )-64 \coth ^{-1}(a x)^3 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )-96 \coth ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )-96 \coth ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(a x)}\right )-96 \coth ^{-1}(a x) \operatorname {PolyLog}\left (3,-e^{-2 \coth ^{-1}(a x)}\right )+96 \coth ^{-1}(a x) \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(a x)}\right )-48 \operatorname {PolyLog}\left (4,-e^{-2 \coth ^{-1}(a x)}\right )-48 \operatorname {PolyLog}\left (4,e^{2 \coth ^{-1}(a x)}\right )\right ) \]
(-Pi^4 + 32*ArcCoth[a*x]^4 + 64*ArcCoth[a*x]^3*Log[1 + E^(-2*ArcCoth[a*x]) ] - 64*ArcCoth[a*x]^3*Log[1 - E^(2*ArcCoth[a*x])] - 96*ArcCoth[a*x]^2*Poly Log[2, -E^(-2*ArcCoth[a*x])] - 96*ArcCoth[a*x]^2*PolyLog[2, E^(2*ArcCoth[a *x])] - 96*ArcCoth[a*x]*PolyLog[3, -E^(-2*ArcCoth[a*x])] + 96*ArcCoth[a*x] *PolyLog[3, E^(2*ArcCoth[a*x])] - 48*PolyLog[4, -E^(-2*ArcCoth[a*x])] - 48 *PolyLog[4, E^(2*ArcCoth[a*x])])/64
Time = 0.98 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.21, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6449, 6615, 6619, 6623, 7164}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\coth ^{-1}(a x)^3}{x} \, dx\) |
| \(\Big \downarrow \) 6449 |
| \(\displaystyle 2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )-6 a \int \frac {\coth ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\) |
| \(\Big \downarrow \) 6615 |
| \(\displaystyle 2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )-6 a \left (\frac {1}{2} \int \frac {\coth ^{-1}(a x)^2 \log \left (\frac {2 a x}{a x+1}\right )}{1-a^2 x^2}dx-\frac {1}{2} \int \frac {\coth ^{-1}(a x)^2 \log \left (\frac {2}{a x+1}\right )}{1-a^2 x^2}dx\right )\) |
| \(\Big \downarrow \) 6619 |
| \(\displaystyle 2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )-6 a \left (\frac {1}{2} \left (\int \frac {\coth ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)^2}{2 a}\right )+\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+1}\right ) \coth ^{-1}(a x)^2}{2 a}-\int \frac {\coth ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+1}\right )}{1-a^2 x^2}dx\right )\right )\) |
| \(\Big \downarrow \) 6623 |
| \(\displaystyle 2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )-6 a \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)^2}{2 a}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)}{2 a}\right )+\frac {1}{2} \left (-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2 a x}{a x+1}\right )}{1-a^2 x^2}dx+\frac {\operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+1}\right ) \coth ^{-1}(a x)^2}{2 a}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2 a x}{a x+1}\right ) \coth ^{-1}(a x)}{2 a}\right )\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle 2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )-6 a \left (\frac {1}{2} \left (-\frac {\operatorname {PolyLog}\left (4,1-\frac {2}{a x+1}\right )}{4 a}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)^2}{2 a}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)}{2 a}\right )+\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (4,1-\frac {2 a x}{a x+1}\right )}{4 a}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+1}\right ) \coth ^{-1}(a x)^2}{2 a}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2 a x}{a x+1}\right ) \coth ^{-1}(a x)}{2 a}\right )\right )\) |
2*ArcCoth[a*x]^3*ArcCoth[1 - 2/(1 - a*x)] - 6*a*((-1/2*(ArcCoth[a*x]^2*Pol yLog[2, 1 - 2/(1 + a*x)])/a - (ArcCoth[a*x]*PolyLog[3, 1 - 2/(1 + a*x)])/( 2*a) - PolyLog[4, 1 - 2/(1 + a*x)]/(4*a))/2 + ((ArcCoth[a*x]^2*PolyLog[2, 1 - (2*a*x)/(1 + a*x)])/(2*a) + (ArcCoth[a*x]*PolyLog[3, 1 - (2*a*x)/(1 + a*x)])/(2*a) + PolyLog[4, 1 - (2*a*x)/(1 + a*x)]/(4*a))/2)
3.1.29.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcCoth[c*x])^p*ArcCoth[1 - 2/(1 - c*x)], x] - Simp[2*b*c*p Int[(a + b *ArcCoth[c*x])^(p - 1)*(ArcCoth[1 - 2/(1 - c*x)]/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
Int[(ArcCoth[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*( x_)^2), x_Symbol] :> Simp[1/2 Int[Log[SimplifyIntegrand[1 + 1/u, x]]*((a + b*ArcCoth[c*x])^p/(d + e*x^2)), x], x] - Simp[1/2 Int[Log[SimplifyInteg rand[1 - 1/u, x]]*((a + b*ArcCoth[c*x])^p/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 2), x_Symbol] :> Simp[(a + b*ArcCoth[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x ] - Simp[b*(p/2) Int[(a + b*ArcCoth[c*x])^(p - 1)*(PolyLog[2, 1 - u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]
Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_ .)*(x_)^2), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^p)*(PolyLog[k + 1, u]/ (2*c*d)), x] + Simp[b*(p/2) Int[(a + b*ArcCoth[c*x])^(p - 1)*(PolyLog[k + 1, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] & & EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.67 (sec) , antiderivative size = 536, normalized size of antiderivative = 3.57
| method | result | size |
| derivativedivides | \(\ln \left (a x \right ) \operatorname {arccoth}\left (a x \right )^{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right ) \left (\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right )-\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )-\operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right )+\operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{2}\right ) \operatorname {arccoth}\left (a x \right )^{3}}{2}+\operatorname {arccoth}\left (a x \right )^{3} \ln \left (\frac {a x +1}{a x -1}-1\right )-\operatorname {arccoth}\left (a x \right )^{3} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-3 \operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \operatorname {polylog}\left (4, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-\operatorname {arccoth}\left (a x \right )^{3} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-3 \operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \operatorname {polylog}\left (4, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{a x -1}\right )}{2}-\frac {3 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{a x -1}\right )}{2}+\frac {3 \operatorname {polylog}\left (4, -\frac {a x +1}{a x -1}\right )}{4}\) | \(536\) |
| default | \(\ln \left (a x \right ) \operatorname {arccoth}\left (a x \right )^{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right ) \left (\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right )-\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )-\operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right )+\operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{2}\right ) \operatorname {arccoth}\left (a x \right )^{3}}{2}+\operatorname {arccoth}\left (a x \right )^{3} \ln \left (\frac {a x +1}{a x -1}-1\right )-\operatorname {arccoth}\left (a x \right )^{3} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-3 \operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \operatorname {polylog}\left (4, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-\operatorname {arccoth}\left (a x \right )^{3} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-3 \operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \operatorname {polylog}\left (4, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{a x -1}\right )}{2}-\frac {3 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{a x -1}\right )}{2}+\frac {3 \operatorname {polylog}\left (4, -\frac {a x +1}{a x -1}\right )}{4}\) | \(536\) |
| parts | \(\ln \left (x \right ) \operatorname {arccoth}\left (a x \right )^{3}+3 a \left (\frac {\left (i \pi \,\operatorname {csgn}\left (\frac {i}{a}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{a \left (\frac {a x +1}{a x -1}-1\right )}\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{a}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{a \left (\frac {a x +1}{a x -1}-1\right )}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{2}-i \pi \,\operatorname {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{a \left (\frac {a x +1}{a x -1}-1\right )}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{a \left (\frac {a x +1}{a x -1}-1\right )}\right )^{3}+2 \ln \left (a \right )\right ) \operatorname {arccoth}\left (a x \right )^{3}}{6 a}+\frac {\operatorname {arccoth}\left (a x \right )^{3} \ln \left (\frac {a x +1}{a x -1}-1\right )}{3 a}-\frac {\operatorname {arccoth}\left (a x \right )^{3} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{3 a}-\frac {\operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}+\frac {2 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}-\frac {2 \operatorname {polylog}\left (4, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}-\frac {\operatorname {arccoth}\left (a x \right )^{3} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{3 a}-\frac {\operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}+\frac {2 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}-\frac {2 \operatorname {polylog}\left (4, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}+\frac {\operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{a x -1}\right )}{2 a}-\frac {\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{a x -1}\right )}{2 a}+\frac {\operatorname {polylog}\left (4, -\frac {a x +1}{a x -1}\right )}{4 a}\right )\) | \(859\) |
ln(a*x)*arccoth(a*x)^3+1/2*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1)*(1+(a*x+1)/(a*x -1)))*(csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I*(1+(a*x+1)/(a*x-1)))-csgn(I/((a* x+1)/(a*x-1)-1))*csgn(I/((a*x+1)/(a*x-1)-1)*(1+(a*x+1)/(a*x-1)))-csgn(I/(( a*x+1)/(a*x-1)-1)*(1+(a*x+1)/(a*x-1)))*csgn(I*(1+(a*x+1)/(a*x-1)))+csgn(I/ ((a*x+1)/(a*x-1)-1)*(1+(a*x+1)/(a*x-1)))^2)*arccoth(a*x)^3+arccoth(a*x)^3* ln((a*x+1)/(a*x-1)-1)-arccoth(a*x)^3*ln(1+1/((a*x-1)/(a*x+1))^(1/2))-3*arc coth(a*x)^2*polylog(2,-1/((a*x-1)/(a*x+1))^(1/2))+6*arccoth(a*x)*polylog(3 ,-1/((a*x-1)/(a*x+1))^(1/2))-6*polylog(4,-1/((a*x-1)/(a*x+1))^(1/2))-arcco th(a*x)^3*ln(1-1/((a*x-1)/(a*x+1))^(1/2))-3*arccoth(a*x)^2*polylog(2,1/((a *x-1)/(a*x+1))^(1/2))+6*arccoth(a*x)*polylog(3,1/((a*x-1)/(a*x+1))^(1/2))- 6*polylog(4,1/((a*x-1)/(a*x+1))^(1/2))+3/2*arccoth(a*x)^2*polylog(2,-(a*x+ 1)/(a*x-1))-3/2*arccoth(a*x)*polylog(3,-(a*x+1)/(a*x-1))+3/4*polylog(4,-(a *x+1)/(a*x-1))
\[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x} \,d x } \]
\[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=\int \frac {\operatorname {acoth}^{3}{\left (a x \right )}}{x}\, dx \]
\[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x} \,d x } \]
\[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x} \,d x } \]
Timed out. \[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=\int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x} \,d x \]