Integrand size = 24, antiderivative size = 187 \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {6 a}{\sqrt {1-a^2 x^2}}+\frac {6 a^2 x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {3 a \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}-6 a \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2+\frac {a^2 x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}-6 a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )+6 a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )+6 a \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-6 a \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right ) \]
-6*a*arctanh((a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)^2-6*a*arctanh(a*x)*p olylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*a*arctanh(a*x)*polylog(2,(a*x+1)/( -a^2*x^2+1)^(1/2))+6*a*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-6*a*polylog( 3,(a*x+1)/(-a^2*x^2+1)^(1/2))-6*a/(-a^2*x^2+1)^(1/2)+6*a^2*x*arctanh(a*x)/ (-a^2*x^2+1)^(1/2)-3*a*arctanh(a*x)^2/(-a^2*x^2+1)^(1/2)+a^2*x*arctanh(a*x )^3/(-a^2*x^2+1)^(1/2)-arctanh(a*x)^3*(-a^2*x^2+1)^(1/2)/x
Time = 1.41 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.44 \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {6 a}{\sqrt {1-a^2 x^2}}+\frac {6 a^2 x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {3 a \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}+\frac {a^2 x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {a^2 x \text {arctanh}(a x)^3 \text {csch}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )}{4 \sqrt {1-a^2 x^2}}+3 a \text {arctanh}(a x)^2 \log \left (1-e^{-\text {arctanh}(a x)}\right )-3 a \text {arctanh}(a x)^2 \log \left (1+e^{-\text {arctanh}(a x)}\right )+6 a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )-6 a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )+6 a \operatorname {PolyLog}\left (3,-e^{-\text {arctanh}(a x)}\right )-6 a \operatorname {PolyLog}\left (3,e^{-\text {arctanh}(a x)}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3 \sinh ^2\left (\frac {1}{2} \text {arctanh}(a x)\right )}{x} \]
(-6*a)/Sqrt[1 - a^2*x^2] + (6*a^2*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (3*a *ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2] + (a^2*x*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x ^2] - (a^2*x*ArcTanh[a*x]^3*Csch[ArcTanh[a*x]/2]^2)/(4*Sqrt[1 - a^2*x^2]) + 3*a*ArcTanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x])] - 3*a*ArcTanh[a*x]^2*Log[1 + E^(-ArcTanh[a*x])] + 6*a*ArcTanh[a*x]*PolyLog[2, -E^(-ArcTanh[a*x])] - 6*a*ArcTanh[a*x]*PolyLog[2, E^(-ArcTanh[a*x])] + 6*a*PolyLog[3, -E^(-ArcTa nh[a*x])] - 6*a*PolyLog[3, E^(-ArcTanh[a*x])] + (Sqrt[1 - a^2*x^2]*ArcTanh [a*x]^3*Sinh[ArcTanh[a*x]/2]^2)/x
Result contains complex when optimal does not.
Time = 1.41 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6592, 6524, 6520, 6570, 6582, 3042, 26, 4670, 3011, 2720, 7143}
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.
\(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6592 |
\(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx+\int \frac {\text {arctanh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 6524 |
\(\displaystyle a^2 \left (6 \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}\right )+\int \frac {\text {arctanh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 6520 |
\(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}}dx+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\) |
\(\Big \downarrow \) 6570 |
\(\displaystyle 3 a \int \frac {\text {arctanh}(a x)^2}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\) |
\(\Big \downarrow \) 6582 |
\(\displaystyle 3 a \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a x}d\text {arctanh}(a x)-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 a \int i \text {arctanh}(a x)^2 \csc (i \text {arctanh}(a x))d\text {arctanh}(a x)-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle 3 i a \int \text {arctanh}(a x)^2 \csc (i \text {arctanh}(a x))d\text {arctanh}(a x)-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle 3 i a \left (2 i \int \text {arctanh}(a x) \log \left (1-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-2 i \int \text {arctanh}(a x) \log \left (1+e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle 3 i a \left (-2 i \left (\int \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+2 i \left (\int \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle 3 i a \left (-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )+3 i a \left (-2 i \left (\operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+2 i \left (\operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )\) |
-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/x) + a^2*((-3*ArcTanh[a*x]^2)/(a*Sqrt [1 - a^2*x^2]) + (x*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2] + 6*(-(1/(a*Sqrt[1 - a^2*x^2])) + (x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2])) + (3*I)*a*((2*I)*ArcTan h[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 - (2*I)*(-(ArcTanh[a*x]*PolyLog[2, -E^Arc Tanh[a*x]]) + PolyLog[3, -E^ArcTanh[a*x]]) + (2*I)*(-(ArcTanh[a*x]*PolyLog [2, E^ArcTanh[a*x]]) + PolyLog[3, E^ArcTanh[a*x]]))
3.5.9.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symb ol] :> Simp[-b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTanh[c*x])/(d* Sqrt[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[(-b)*p*((a + b*ArcTanh[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) ), x] + (Simp[x*((a + b*ArcTanh[c*x])^p/(d*Sqrt[d + e*x^2])), x] + Simp[b^2 *p*(p - 1) Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(d*(m + 1))), x] - Simp[b*c*(p/(m + 1)) Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2 ]), x_Symbol] :> Simp[1/Sqrt[d] Subst[Int[(a + b*x)^p*Csch[x], x], x, Arc Tanh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[1/d Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh [c*x])^p, x], x] - Simp[e/d Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh[c* x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Integers Q[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(235)=470\).
Time = 0.19 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.61
method | result | size |
default | \(-\frac {2 \operatorname {arctanh}\left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a x +3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}-3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a x -6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a x +6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}-6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a x +6 \,\operatorname {arctanh}\left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-3 a x \operatorname {arctanh}\left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+6 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}-6 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a x -6 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}+6 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a x -\sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right )^{3}-6 a x \sqrt {-a^{2} x^{2}+1}}{x \left (a^{2} x^{2}-1\right )}\) | \(488\) |
-(2*arctanh(a*x)^3*(-a^2*x^2+1)^(1/2)*a^2*x^2-3*arctanh(a*x)^2*ln(1-(a*x+1 )/(-a^2*x^2+1)^(1/2))*a^3*x^3+3*arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^( 1/2))*a*x+3*arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))*a^3*x^3-3*arct anh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))*a*x-6*arctanh(a*x)*polylog(2,( a*x+1)/(-a^2*x^2+1)^(1/2))*a^3*x^3+6*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2* x^2+1)^(1/2))*a*x+6*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))*a^ 3*x^3-6*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))*a*x+6*arctanh( a*x)*(-a^2*x^2+1)^(1/2)*a^2*x^2-3*a*x*arctanh(a*x)^2*(-a^2*x^2+1)^(1/2)+6* polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))*a^3*x^3-6*polylog(3,(a*x+1)/(-a^2*x^ 2+1)^(1/2))*a*x-6*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))*a^3*x^3+6*polylog (3,-(a*x+1)/(-a^2*x^2+1)^(1/2))*a*x-(-a^2*x^2+1)^(1/2)*arctanh(a*x)^3-6*a* x*(-a^2*x^2+1)^(1/2))/x/(a^2*x^2-1)
\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{x^{2} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]