Integrand size = 22, antiderivative size = 135 \[ \int \frac {x^3 \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=-\frac {x \text {arctanh}(a x)}{a^3}+\frac {\text {arctanh}(a x)^2}{2 a^4}-\frac {x^2 \text {arctanh}(a x)^2}{2 a^2}-\frac {\text {arctanh}(a x)^3}{3 a^4}+\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {\log \left (1-a^2 x^2\right )}{2 a^4}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^4}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^4} \]
-x*arctanh(a*x)/a^3+1/2*arctanh(a*x)^2/a^4-1/2*x^2*arctanh(a*x)^2/a^2-1/3* arctanh(a*x)^3/a^4+arctanh(a*x)^2*ln(2/(-a*x+1))/a^4-1/2*ln(-a^2*x^2+1)/a^ 4+arctanh(a*x)*polylog(2,1-2/(-a*x+1))/a^4-1/2*polylog(3,1-2/(-a*x+1))/a^4
Time = 0.16 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=-\frac {a x \text {arctanh}(a x)-\frac {1}{2} \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2-\frac {1}{3} \text {arctanh}(a x)^3-\text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )-\log \left (\frac {1}{\sqrt {1-a^2 x^2}}\right )+\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )}{a^4} \]
-((a*x*ArcTanh[a*x] - ((1 - a^2*x^2)*ArcTanh[a*x]^2)/2 - ArcTanh[a*x]^3/3 - ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] - Log[1/Sqrt[1 - a^2*x^2]] + ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] + PolyLog[3, -E^(-2*ArcTanh [a*x])]/2)/a^4)
Time = 1.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.19, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6542, 6452, 6542, 6436, 240, 6510, 6546, 6470, 6620, 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.
| \(\displaystyle \int \frac {x^3 \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {\int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\int x \text {arctanh}(a x)^2dx}{a^2}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {\int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)^2-a \int \frac {x^2 \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {\int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)^2-a \left (\frac {\int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int \text {arctanh}(a x)dx}{a^2}\right )}{a^2}\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle \frac {\int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)^2-a \left (\frac {\int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {x \text {arctanh}(a x)-a \int \frac {x}{1-a^2 x^2}dx}{a^2}\right )}{a^2}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {\int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)^2-a \left (\frac {\int \frac {\text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)}{a^2}\right )}{a^2}\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {\int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)^2-a \left (\frac {\text {arctanh}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)}{a^2}\right )}{a^2}\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {\frac {\int \frac {\text {arctanh}(a x)^2}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)^2-a \left (\frac {\text {arctanh}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)}{a^2}\right )}{a^2}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \int \frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)^2-a \left (\frac {\text {arctanh}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)}{a^2}\right )}{a^2}\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle \frac {\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)^2-a \left (\frac {\text {arctanh}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)}{a^2}\right )}{a^2}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{4 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)^2-a \left (\frac {\text {arctanh}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)}{a^2}\right )}{a^2}\) |
-(((x^2*ArcTanh[a*x]^2)/2 - a*(ArcTanh[a*x]^2/(2*a^3) - (x*ArcTanh[a*x] + Log[1 - a^2*x^2]/(2*a))/a^2))/a^2) + (-1/3*ArcTanh[a*x]^3/a^2 + ((ArcTanh[ a*x]^2*Log[2/(1 - a*x)])/a - 2*(-1/2*(ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a *x)])/a + PolyLog[3, 1 - 2/(1 - a*x)]/(4*a)))/a)/a^2
3.3.34.3.1 Defintions of rubi rules used
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTanh[c*x^n]) ^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
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_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[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[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTanh[c* x])^p, x], x] - Simp[d*(f^2/e) 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] && GtQ[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*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[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 = 2.49 (sec) , antiderivative size = 728, normalized size of antiderivative = 5.39
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right ) \left (-6 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 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \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 {arctanh}\left (a x \right )-3 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 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 )-6 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 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 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \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 {arctanh}\left (a x \right )-3 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 )+6 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 )-6 i \pi \,\operatorname {arctanh}\left (a x \right )+4 \operatorname {arctanh}\left (a x \right )^{2}-12 \ln \left (2\right ) \operatorname {arctanh}\left (a x \right )-6 \,\operatorname {arctanh}\left (a x \right )+12 a x +12\right )}{12}+\ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{4}}\) | \(728\) |
| default | \(\frac {-\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right ) \left (-6 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 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \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 {arctanh}\left (a x \right )-3 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 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 )-6 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 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 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \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 {arctanh}\left (a x \right )-3 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 )+6 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 )-6 i \pi \,\operatorname {arctanh}\left (a x \right )+4 \operatorname {arctanh}\left (a x \right )^{2}-12 \ln \left (2\right ) \operatorname {arctanh}\left (a x \right )-6 \,\operatorname {arctanh}\left (a x \right )+12 a x +12\right )}{12}+\ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{4}}\) | \(728\) |
| parts | \(\text {Expression too large to display}\) | \(886\) |
1/a^4*(-1/2*a^2*x^2*arctanh(a*x)^2-1/2*arctanh(a*x)^2*ln(a*x-1)-1/2*arctan h(a*x)^2*ln(a*x+1)+arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))+arctanh(a *x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))-1/2*polylog(3,-(a*x+1)^2/(-a^2*x^2+ 1))-1/12*arctanh(a*x)*(-6*I*arctanh(a*x)*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1))) ^3*Pi+3*I*arctanh(a*x)*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))*csgn(I*(a*x+1)^2/ (a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))*Pi-3* I*arctanh(a*x)*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))*csgn(I*(a*x+1)^2/(a^2*x^2 -1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2*Pi-3*I*arctanh(a*x)*csgn(I*(a*x+1)/(-a^2* x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*Pi-6*I*arctanh(a*x)*csgn(I*( a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*Pi-3*I*arctanh( a*x)*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*Pi+3*I*arctanh(a*x)*csgn(I*(a*x+1)^2/ (a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2*Pi- 3*I*arctanh(a*x)*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^3 *Pi+6*I*arctanh(a*x)*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))^2*Pi-6*I*arctanh(a* x)*Pi+4*arctanh(a*x)^2-12*ln(2)*arctanh(a*x)-6*arctanh(a*x)+12*a*x+12)+ln( 1+(a*x+1)^2/(-a^2*x^2+1)))
\[ \int \frac {x^3 \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=\int { -\frac {x^{3} \operatorname {artanh}\left (a x\right )^{2}}{a^{2} x^{2} - 1} \,d x } \]
\[ \int \frac {x^3 \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=- \int \frac {x^{3} \operatorname {atanh}^{2}{\left (a x \right )}}{a^{2} x^{2} - 1}\, dx \]
\[ \int \frac {x^3 \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=\int { -\frac {x^{3} \operatorname {artanh}\left (a x\right )^{2}}{a^{2} x^{2} - 1} \,d x } \]
-1/24*(3*(a^2*x^2 + log(a*x + 1))*log(-a*x + 1)^2 + log(-a*x + 1)^3)/a^4 + 1/4*integrate(-(a^3*x^3*log(a*x + 1)^2 - (a^3*x^3 + a^2*x^2 + (2*a^3*x^3 + a*x + 1)*log(a*x + 1))*log(-a*x + 1))/(a^5*x^2 - a^3), x)
\[ \int \frac {x^3 \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=\int { -\frac {x^{3} \operatorname {artanh}\left (a x\right )^{2}}{a^{2} x^{2} - 1} \,d x } \]
Timed out. \[ \int \frac {x^3 \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=-\int \frac {x^3\,{\mathrm {atanh}\left (a\,x\right )}^2}{a^2\,x^2-1} \,d x \]