Integrand size = 22, antiderivative size = 103 \[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=-\frac {\text {arctanh}(a x)^3}{a^3}-\frac {x \text {arctanh}(a x)^3}{a^2}+\frac {\text {arctanh}(a x)^4}{4 a^3}+\frac {3 \text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3}+\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3}-\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^3} \]
-arctanh(a*x)^3/a^3-x*arctanh(a*x)^3/a^2+1/4*arctanh(a*x)^4/a^3+3*arctanh( a*x)^2*ln(2/(-a*x+1))/a^3+3*arctanh(a*x)*polylog(2,1-2/(-a*x+1))/a^3-3/2*p olylog(3,1-2/(-a*x+1))/a^3
Time = 0.24 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.76 \[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=\frac {\text {arctanh}(a x)^2 \left ((4-4 a x) \text {arctanh}(a x)+\text {arctanh}(a x)^2+12 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )\right )-12 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )-6 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )}{4 a^3} \]
(ArcTanh[a*x]^2*((4 - 4*a*x)*ArcTanh[a*x] + ArcTanh[a*x]^2 + 12*Log[1 + E^ (-2*ArcTanh[a*x])]) - 12*ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] - 6 *PolyLog[3, -E^(-2*ArcTanh[a*x])])/(4*a^3)
Time = 0.95 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6542, 6436, 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.
\(\displaystyle \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx\) |
\(\Big \downarrow \) 6542 |
\(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^3}{1-a^2 x^2}dx}{a^2}-\frac {\int \text {arctanh}(a x)^3dx}{a^2}\) |
\(\Big \downarrow \) 6436 |
\(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^3}{1-a^2 x^2}dx}{a^2}-\frac {x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a^2}\) |
\(\Big \downarrow \) 6510 |
\(\displaystyle \frac {\text {arctanh}(a x)^4}{4 a^3}-\frac {x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a^2}\) |
\(\Big \downarrow \) 6546 |
\(\displaystyle \frac {\text {arctanh}(a x)^4}{4 a^3}-\frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\int \frac {\text {arctanh}(a x)^2}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a^2}\) |
\(\Big \downarrow \) 6470 |
\(\displaystyle \frac {\text {arctanh}(a x)^4}{4 a^3}-\frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \int \frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a^2}\) |
\(\Big \downarrow \) 6620 |
\(\displaystyle \frac {\text {arctanh}(a x)^4}{4 a^3}-\frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a^2}\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle \frac {\text {arctanh}(a x)^4}{4 a^3}-\frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{4 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a^2}\) |
ArcTanh[a*x]^4/(4*a^3) - (x*ArcTanh[a*x]^3 - 3*a*(-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
3.3.42.3.1 Defintions of rubi rules used
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_)]*(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 = 0.44 (sec) , antiderivative size = 736, normalized size of antiderivative = 7.15
method | result | size |
derivativedivides | \(\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}+\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 {\operatorname {arctanh}\left (a x \right )^{4}}{4}+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+\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}-\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}-\operatorname {arctanh}\left (a x \right )^{3}-\frac {3 \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{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 )}{\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}+\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}{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 {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}}{\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 {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}}{a^{3}}\) | \(736\) |
default | \(\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}+\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 {\operatorname {arctanh}\left (a x \right )^{4}}{4}+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+\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}-\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}-\operatorname {arctanh}\left (a x \right )^{3}-\frac {3 \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{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 )}{\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}+\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}{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 {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}}{\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 {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}}{a^{3}}\) | \(736\) |
parts | \(-\frac {x \operatorname {arctanh}\left (a x \right )^{3}}{a^{2}}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{2 a^{3}}-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x -1\right )}{2 a^{3}}-\frac {3 a \left (\frac {2 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{3 a^{4}}-\frac {\operatorname {arctanh}\left (a x \right )^{4}}{6 a^{4}}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{3}}{3 a^{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}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{6 a^{4}}+\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}}{3 a^{4}}-\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}}{3 a^{4}}-\frac {2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{4}}-\frac {2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{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}}{6 a^{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}}{6 a^{4}}-\frac {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}}{6 a^{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 )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{6 a^{4}}-\frac {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}}{6 a^{4}}+\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}}{3 a^{4}}+\frac {2 \operatorname {arctanh}\left (a x \right )^{3}}{3 a^{4}}+\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{4}}\right )}{2}\) | \(791\) |
1/a^3*(-arctanh(a*x)^3*a*x-1/2*arctanh(a*x)^3*ln(a*x-1)+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))+1/4*arctanh(a*x)^ 4+3*arctanh(a*x)^2*ln(1+(a*x+1)^2/(-a^2*x^2+1))+3*arctanh(a*x)*polylog(2,- (a*x+1)^2/(-a^2*x^2+1))+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)^3-1/4*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*arctanh(a*x)^ 3-arctanh(a*x)^3-3/2*polylog(3,-(a*x+1)^2/(-a^2*x^2+1))-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)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)) *arctanh(a*x)^3+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/(1-(a*x+1)^2/(a^2*x^2-1)))^2*arctanh(a*x)^3+1/2*I*Pi*arc tanh(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*csgn(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)^3-1/4*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)
\[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=\int { -\frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a^{2} x^{2} - 1} \,d x } \]
\[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=- \int \frac {x^{2} \operatorname {atanh}^{3}{\left (a x \right )}}{a^{2} x^{2} - 1}\, dx \]
\[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=\int { -\frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a^{2} x^{2} - 1} \,d x } \]
1/64*(4*(2*a*x - log(a*x + 1) - 2)*log(-a*x + 1)^3 + log(-a*x + 1)^4 - 6*( 4*(a*x + 1)*log(a*x + 1) - log(a*x + 1)^2)*log(-a*x + 1)^2)/a^3 + 1/8*inte grate(-1/2*(2*a^2*x^2*log(a*x + 1)^3 - 3*((2*a^2*x^2 - a*x - 1)*log(a*x + 1)^2 + 4*(a^2*x^2 + 2*a*x + 1)*log(a*x + 1))*log(-a*x + 1))/(a^4*x^2 - a^2 ), x)
\[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=\int { -\frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a^{2} x^{2} - 1} \,d x } \]
Timed out. \[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=-\int \frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^3}{a^2\,x^2-1} \,d x \]