Integrand size = 22, antiderivative size = 90 \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx=a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^3}{x}+\frac {1}{4} a \text {arctanh}(a x)^4+3 a \text {arctanh}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \]
a*arctanh(a*x)^3-arctanh(a*x)^3/x+1/4*a*arctanh(a*x)^4+3*a*arctanh(a*x)^2* ln(2-2/(a*x+1))-3*a*arctanh(a*x)*polylog(2,-1+2/(a*x+1))-3/2*a*polylog(3,- 1+2/(a*x+1))
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.03 \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx=-a \left (-\frac {i \pi ^3}{8}+\text {arctanh}(a x)^3+\frac {\text {arctanh}(a x)^3}{a x}-\frac {1}{4} \text {arctanh}(a x)^4-3 \text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )-3 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )\right ) \]
-(a*((-1/8*I)*Pi^3 + ArcTanh[a*x]^3 + ArcTanh[a*x]^3/(a*x) - ArcTanh[a*x]^ 4/4 - 3*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - 3*ArcTanh[a*x]*PolyLo g[2, E^(2*ArcTanh[a*x])] + (3*PolyLog[3, E^(2*ArcTanh[a*x])])/2))
Time = 0.95 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6544, 6452, 6510, 6550, 6494, 6618, 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.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{1-a^2 x^2}dx+\int \frac {\text {arctanh}(a x)^3}{x^2}dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 3 a \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )}dx+a^2 \int \frac {\text {arctanh}(a x)^3}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3}{x}\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle 3 a \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )}dx+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\) |
| \(\Big \downarrow \) 6550 |
| \(\displaystyle 3 a \left (\int \frac {\text {arctanh}(a x)^2}{x (a x+1)}dx+\frac {1}{3} \text {arctanh}(a x)^3\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle 3 a \left (-2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\) |
| \(\Big \downarrow \) 6618 |
| \(\displaystyle 3 a \left (-2 a \left (\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle 3 a \left (-2 a \left (\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{4 a}\right )+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\) |
-(ArcTanh[a*x]^3/x) + (a*ArcTanh[a*x]^4)/4 + 3*a*(ArcTanh[a*x]^3/3 + ArcTa nh[a*x]^2*Log[2 - 2/(1 + a*x)] - 2*a*((ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)])/(2*a) + PolyLog[3, -1 + 2/(1 + a*x)]/(4*a)))
3.3.46.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))] /(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c ^2*d^2 - e^2, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x ], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/(d + e*x ^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ d Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x ] - Simp[b*(p/2) Int[(a + b*ArcTanh[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[(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 = 0.38 (sec) , antiderivative size = 810, normalized size of antiderivative = 9.00
| method | result | size |
| derivativedivides | \(a \left (-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{a x}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{2}-\operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{3}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{4}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{4}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{3}}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{4}-\operatorname {arctanh}\left (a x \right )^{3}-6 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\operatorname {arctanh}\left (a x \right )^{4}}{4}+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{3}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{4}-6 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(810\) |
| default | \(a \left (-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{a x}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{2}-\operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{3}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{4}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{4}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{3}}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{4}-\operatorname {arctanh}\left (a x \right )^{3}-6 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\operatorname {arctanh}\left (a x \right )^{4}}{4}+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{3}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{4}-6 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(810\) |
| parts | \(\text {Expression too large to display}\) | \(811\) |
a*(-1/2*arctanh(a*x)^3*ln(a*x-1)-arctanh(a*x)^3/a/x+1/2*arctanh(a*x)^3*ln( a*x+1)-arctanh(a*x)^3*ln((a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*I*Pi*csgn(I/(1-(a *x+1)^2/(a^2*x^2-1)))^3*arctanh(a*x)^3-1/2*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1 )^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*arctanh(a*x)^3-1/4*I*Pi*csgn(I*(a *x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^3*arctanh(a*x)^3+1/4*I*Pi*c sgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2*csgn(I*(a*x+1)^2/ (a^2*x^2-1))*arctanh(a*x)^3+1/2*I*Pi*arctanh(a*x)^3+1/4*I*Pi*csgn(I*(a*x+1 )^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1) ))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*arctanh(a*x)^3-arctanh(a*x)^3-6*polylog(3 ,-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*arctanh(a*x)^4+6*arctanh(a*x)*polylog(2, -(a*x+1)/(-a^2*x^2+1)^(1/2))+3*arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1 /2))+6*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+3*arctanh(a*x)^2 *ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I*Pi*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1) ))^2*arctanh(a*x)^3-1/4*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^ 2*x^2-1)))^2*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))*arctanh(a*x)^3-1/4*I*Pi*csg n(I*(a*x+1)^2/(a^2*x^2-1))^3*arctanh(a*x)^3-1/4*I*Pi*csgn(I*(a*x+1)/(-a^2* x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*arctanh(a*x)^3-6*polylog(3,( a*x+1)/(-a^2*x^2+1)^(1/2)))
\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )} x^{2}} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx=- \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{a^{2} x^{4} - x^{2}}\, dx \]
\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )} x^{2}} \,d x } \]
1/64*(a*x*log(-a*x + 1)^4 - 4*(a*x*log(a*x + 1) + 2*a*x - 2)*log(-a*x + 1) ^3 + 6*(a*x*log(a*x + 1)^2 - 4*(a*x + 1)*log(a*x + 1))*log(-a*x + 1)^2)/x - 1/8*integrate(1/2*(2*log(a*x + 1)^3 + 3*((a^3*x^3 + a^2*x^2 - 2)*log(a*x + 1)^2 - 4*(a^3*x^3 + 2*a^2*x^2 + a*x)*log(a*x + 1))*log(-a*x + 1))/(a^2* x^4 - x^2), x)
\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx=-\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x^2\,\left (a^2\,x^2-1\right )} \,d x \]