3.247 \(\int \frac{\tanh ^{-1}(a x)^3}{x^3 (1-a^2 x^2)} \, dx\)

Optimal. Leaf size=200 \[ -\frac{3}{2} a^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{3}{4} a^2 \text{PolyLog}\left (4,\frac{2}{a x+1}-1\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )+\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+\frac{1}{2} a^2 \tanh ^{-1}(a x)^3+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2+a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^3+3 a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^3}{2 x^2}-\frac{3 a \tanh ^{-1}(a x)^2}{2 x} \]

[Out]

(3*a^2*ArcTanh[a*x]^2)/2 - (3*a*ArcTanh[a*x]^2)/(2*x) + (a^2*ArcTanh[a*x]^3)/2 - ArcTanh[a*x]^3/(2*x^2) + (a^2
*ArcTanh[a*x]^4)/4 + 3*a^2*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] + a^2*ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)] - (3*a^
2*PolyLog[2, -1 + 2/(1 + a*x)])/2 - (3*a^2*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)])/2 - (3*a^2*ArcTanh[a*x
]*PolyLog[3, -1 + 2/(1 + a*x)])/2 - (3*a^2*PolyLog[4, -1 + 2/(1 + a*x)])/4

________________________________________________________________________________________

Rubi [A]  time = 0.49564, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {5982, 5916, 5988, 5932, 2447, 5948, 6056, 6060, 6610} \[ -\frac{3}{2} a^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{3}{4} a^2 \text{PolyLog}\left (4,\frac{2}{a x+1}-1\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )+\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+\frac{1}{2} a^2 \tanh ^{-1}(a x)^3+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2+a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^3+3 a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^3}{2 x^2}-\frac{3 a \tanh ^{-1}(a x)^2}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^3/(x^3*(1 - a^2*x^2)),x]

[Out]

(3*a^2*ArcTanh[a*x]^2)/2 - (3*a*ArcTanh[a*x]^2)/(2*x) + (a^2*ArcTanh[a*x]^3)/2 - ArcTanh[a*x]^3/(2*x^2) + (a^2
*ArcTanh[a*x]^4)/4 + 3*a^2*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] + a^2*ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)] - (3*a^
2*PolyLog[2, -1 + 2/(1 + a*x)])/2 - (3*a^2*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)])/2 - (3*a^2*ArcTanh[a*x
]*PolyLog[3, -1 + 2/(1 + a*x)])/2 - (3*a^2*PolyLog[4, -1 + 2/(1 + a*x)])/4

Rule 5982

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d
, Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x], x] - Dist[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]

Rule 5916

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcT
anh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 5988

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] + Dist[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]

Rule 5932

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] - Dist[(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]

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 5948

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> 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]

Rule 6056

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[((a + b*ArcTa
nh[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] - Dist[(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]

Rule 6060

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[((a
+ b*ArcTanh[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] + Dist[(b*p)/2, Int[((a + b*ArcTanh[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]

Rule 6610

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]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a x)^3}{x^3 \left (1-a^2 x^2\right )} \, dx &=a^2 \int \frac{\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx+\int \frac{\tanh ^{-1}(a x)^3}{x^3} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^3}{2 x^2}+\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+\frac{1}{2} (3 a) \int \frac{\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx+a^2 \int \frac{\tanh ^{-1}(a x)^3}{x (1+a x)} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^3}{2 x^2}+\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )+\frac{1}{2} (3 a) \int \frac{\tanh ^{-1}(a x)^2}{x^2} \, dx+\frac{1}{2} \left (3 a^3\right ) \int \frac{\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx-\left (3 a^3\right ) \int \frac{\tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{3 a \tanh ^{-1}(a x)^2}{2 x}+\frac{1}{2} a^2 \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{2 x^2}+\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )+\left (3 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx+\left (3 a^3\right ) \int \frac{\tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac{3}{2} a^2 \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{2 x}+\frac{1}{2} a^2 \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{2 x^2}+\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+a x}\right )+\left (3 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{x (1+a x)} \, dx+\frac{1}{2} \left (3 a^3\right ) \int \frac{\text{Li}_3\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac{3}{2} a^2 \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{2 x}+\frac{1}{2} a^2 \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{2 x^2}+\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+3 a^2 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+a x}\right )-\frac{3}{4} a^2 \text{Li}_4\left (-1+\frac{2}{1+a x}\right )-\left (3 a^3\right ) \int \frac{\log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac{3}{2} a^2 \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{2 x}+\frac{1}{2} a^2 \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{2 x^2}+\frac{1}{4} a^2 \tanh ^{-1}(a x)^4+3 a^2 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+a x}\right )-\frac{3}{4} a^2 \text{Li}_4\left (-1+\frac{2}{1+a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.444073, size = 165, normalized size = 0.82 \[ -\frac{1}{64} a^2 \left (-96 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )+96 \tanh ^{-1}(a x) \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )+96 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )-48 \text{PolyLog}\left (4,e^{2 \tanh ^{-1}(a x)}\right )+\frac{32 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}{a^2 x^2}+16 \tanh ^{-1}(a x)^4+\frac{96 \tanh ^{-1}(a x)^2}{a x}-96 \tanh ^{-1}(a x)^2-64 \tanh ^{-1}(a x)^3 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-192 \tanh ^{-1}(a x) \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )-\pi ^4\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTanh[a*x]^3/(x^3*(1 - a^2*x^2)),x]

[Out]

-(a^2*(-Pi^4 - 96*ArcTanh[a*x]^2 + (96*ArcTanh[a*x]^2)/(a*x) + (32*(1 - a^2*x^2)*ArcTanh[a*x]^3)/(a^2*x^2) + 1
6*ArcTanh[a*x]^4 - 192*ArcTanh[a*x]*Log[1 - E^(-2*ArcTanh[a*x])] - 64*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x]
)] + 96*PolyLog[2, E^(-2*ArcTanh[a*x])] - 96*ArcTanh[a*x]^2*PolyLog[2, E^(2*ArcTanh[a*x])] + 96*ArcTanh[a*x]*P
olyLog[3, E^(2*ArcTanh[a*x])] - 48*PolyLog[4, E^(2*ArcTanh[a*x])]))/64

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Maple [B]  time = 1.251, size = 406, normalized size = 2. \begin{align*} -{\frac{{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}}{4}}+{\frac{{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{2}}-{\frac{3\,{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2}}-{\frac{3\,a \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2\,x}}-{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{2\,{x}^{2}}}+{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}\ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) +3\,{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -6\,{a}^{2}{\it Artanh} \left ( ax \right ){\it polylog} \left ( 3,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +6\,{a}^{2}{\it polylog} \left ( 4,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}\ln \left ( 1-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) +3\,{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -6\,{a}^{2}{\it Artanh} \left ( ax \right ){\it polylog} \left ( 3,{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +6\,{a}^{2}{\it polylog} \left ( 4,{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +3\,{a}^{2}{\it Artanh} \left ( ax \right ) \ln \left ( 1+{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +3\,{a}^{2}{\it polylog} \left ( 2,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +3\,{a}^{2}{\it Artanh} \left ( ax \right ) \ln \left ( 1-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +3\,{a}^{2}{\it polylog} \left ( 2,{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)^3/x^3/(-a^2*x^2+1),x)

[Out]

-1/4*a^2*arctanh(a*x)^4+1/2*a^2*arctanh(a*x)^3-3/2*a^2*arctanh(a*x)^2-3/2*a*arctanh(a*x)^2/x-1/2*arctanh(a*x)^
3/x^2+a^2*arctanh(a*x)^3*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*arctanh(a*x)^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)
^(1/2))-6*a^2*arctanh(a*x)*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*a^2*polylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))
+a^2*arctanh(a*x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*arctanh(a*x)^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2)
)-6*a^2*arctanh(a*x)*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+6*a^2*polylog(4,(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*a
rctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*arctanh(a*x)*l
n(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{2} x^{2} \log \left (-a x + 1\right )^{4} + 4 \,{\left (a^{2} x^{2} \log \left (a x + 1\right ) + 1\right )} \log \left (-a x + 1\right )^{3}}{64 \, x^{2}} - \frac{1}{8} \, \int \frac{2 \, \log \left (a x + 1\right )^{3} - 6 \, \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right ) + 3 \,{\left (a^{2} x^{2} + a x +{\left (a^{4} x^{4} + a^{3} x^{3} + 2\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{2}}{2 \,{\left (a^{2} x^{5} - x^{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^3/x^3/(-a^2*x^2+1),x, algorithm="maxima")

[Out]

1/64*(a^2*x^2*log(-a*x + 1)^4 + 4*(a^2*x^2*log(a*x + 1) + 1)*log(-a*x + 1)^3)/x^2 - 1/8*integrate(1/2*(2*log(a
*x + 1)^3 - 6*log(a*x + 1)^2*log(-a*x + 1) + 3*(a^2*x^2 + a*x + (a^4*x^4 + a^3*x^3 + 2)*log(a*x + 1))*log(-a*x
+ 1)^2)/(a^2*x^5 - x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\operatorname{artanh}\left (a x\right )^{3}}{a^{2} x^{5} - x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^3/x^3/(-a^2*x^2+1),x, algorithm="fricas")

[Out]

integral(-arctanh(a*x)^3/(a^2*x^5 - x^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\operatorname{atanh}^{3}{\left (a x \right )}}{a^{2} x^{5} - x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)**3/x**3/(-a**2*x**2+1),x)

[Out]

-Integral(atanh(a*x)**3/(a**2*x**5 - x**3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\operatorname{artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^3/x^3/(-a^2*x^2+1),x, algorithm="giac")

[Out]

integrate(-arctanh(a*x)^3/((a^2*x^2 - 1)*x^3), x)