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

Optimal. Leaf size=360 \[ 3 a^2 \text{PolyLog}\left (2,-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-3 a^2 \text{PolyLog}\left (2,\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{9}{2} a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+\frac{9}{2} a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+9 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-9 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-9 a^2 \text{PolyLog}\left (4,-e^{\tanh ^{-1}(a x)}\right )+9 a^2 \text{PolyLog}\left (4,e^{\tanh ^{-1}(a x)}\right )-\frac{6 a^3 x}{\sqrt{1-a^2 x^2}}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{a^2 \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \]

[Out]

(-6*a^3*x)/Sqrt[1 - a^2*x^2] + (6*a^2*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (3*a^3*x*ArcTanh[a*x]^2)/Sqrt[1 - a^2*
x^2] - (3*a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(2*x) + (a^2*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2] - (Sqrt[1 - a^2*x
^2]*ArcTanh[a*x]^3)/(2*x^2) - 3*a^2*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^3 - 6*a^2*ArcTanh[a*x]*ArcTanh[Sqrt[1
- a*x]/Sqrt[1 + a*x]] - (9*a^2*ArcTanh[a*x]^2*PolyLog[2, -E^ArcTanh[a*x]])/2 + (9*a^2*ArcTanh[a*x]^2*PolyLog[
2, E^ArcTanh[a*x]])/2 + 3*a^2*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - 3*a^2*PolyLog[2, Sqrt[1 - a*x]/Sqrt
[1 + a*x]] + 9*a^2*ArcTanh[a*x]*PolyLog[3, -E^ArcTanh[a*x]] - 9*a^2*ArcTanh[a*x]*PolyLog[3, E^ArcTanh[a*x]] -
9*a^2*PolyLog[4, -E^ArcTanh[a*x]] + 9*a^2*PolyLog[4, E^ArcTanh[a*x]]

________________________________________________________________________________________

Rubi [A]  time = 0.982383, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 13, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.542, Rules used = {6030, 6026, 6008, 6018, 6020, 4182, 2531, 6609, 2282, 6589, 5994, 5962, 191} \[ 3 a^2 \text{PolyLog}\left (2,-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-3 a^2 \text{PolyLog}\left (2,\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{9}{2} a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+\frac{9}{2} a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+9 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-9 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-9 a^2 \text{PolyLog}\left (4,-e^{\tanh ^{-1}(a x)}\right )+9 a^2 \text{PolyLog}\left (4,e^{\tanh ^{-1}(a x)}\right )-\frac{6 a^3 x}{\sqrt{1-a^2 x^2}}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{a^2 \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*a^3*x)/Sqrt[1 - a^2*x^2] + (6*a^2*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (3*a^3*x*ArcTanh[a*x]^2)/Sqrt[1 - a^2*
x^2] - (3*a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(2*x) + (a^2*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2] - (Sqrt[1 - a^2*x
^2]*ArcTanh[a*x]^3)/(2*x^2) - 3*a^2*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^3 - 6*a^2*ArcTanh[a*x]*ArcTanh[Sqrt[1
- a*x]/Sqrt[1 + a*x]] - (9*a^2*ArcTanh[a*x]^2*PolyLog[2, -E^ArcTanh[a*x]])/2 + (9*a^2*ArcTanh[a*x]^2*PolyLog[
2, E^ArcTanh[a*x]])/2 + 3*a^2*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - 3*a^2*PolyLog[2, Sqrt[1 - a*x]/Sqrt
[1 + a*x]] + 9*a^2*ArcTanh[a*x]*PolyLog[3, -E^ArcTanh[a*x]] - 9*a^2*ArcTanh[a*x]*PolyLog[3, E^ArcTanh[a*x]] -
9*a^2*PolyLog[4, -E^ArcTanh[a*x]] + 9*a^2*PolyLog[4, E^ArcTanh[a*x]]

Rule 6030

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/d, Int
[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[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] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[
m, 0] && NeQ[p, -1]

Rule 6026

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])^p)/(d*f*(m + 1)), x] + (-Dist[(b*c*p)/(f*(m + 1)), Int[((
f*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1))/Sqrt[d + e*x^2], x], x] + Dist[(c^2*(m + 2))/(f^2*(m + 1)), Int[((f
*x)^(m + 2)*(a + b*ArcTanh[c*x])^p)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e,
0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]

Rule 6008

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

Rule 6018

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Simp[(-2*(a + b*ArcTanh
[c*x])*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/Sqrt[d], x] + (Simp[(b*PolyLog[2, -(Sqrt[1 - c*x]/Sqrt[1 + c*x])]
)/Sqrt[d], x] - Simp[(b*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]])/Sqrt[d], x]) /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[c^2*d + e, 0] && GtQ[d, 0]

Rule 6020

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Dist[1/Sqrt[d], Su
bst[Int[(a + b*x)^p*Csch[x], x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGt
Q[p, 0] && GtQ[d, 0]

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(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]

Rule 2531

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] + Dist[(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]

Rule 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

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

Rule 5994

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[((d + e*x^2)
^(q + 1)*(a + b*ArcTanh[c*x])^p)/(2*e*(q + 1)), x] + Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcTan
h[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]

Rule 5962

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> -Simp[(b*p*(a + b*ArcTa
nh[c*x])^(p - 1))/(c*d*Sqrt[d + e*x^2]), x] + (Dist[b^2*p*(p - 1), Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2
)^(3/2), x], x] + Simp[(x*(a + b*ArcTanh[c*x])^p)/(d*Sqrt[d + e*x^2]), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ
[c^2*d + e, 0] && GtQ[p, 1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a x)^3}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx &=a^2 \int \frac{\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx+\int \frac{\tanh ^{-1}(a x)^3}{x^3 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} (3 a) \int \frac{\tanh ^{-1}(a x)^2}{x^2 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{2} a^2 \int \frac{\tanh ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}} \, dx+a^2 \int \frac{\tanh ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}} \, dx+a^4 \int \frac{x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}+\frac{a^2 \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int x^3 \text{csch}(x) \, dx,x,\tanh ^{-1}(a x)\right )+a^2 \operatorname{Subst}\left (\int x^3 \text{csch}(x) \, dx,x,\tanh ^{-1}(a x)\right )+\left (3 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \, dx-\left (3 a^3\right ) \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{6 a^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}+\frac{a^2 \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+3 a^2 \text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-3 a^2 \text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{1}{2} \left (3 a^2\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+\frac{1}{2} \left (3 a^2\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\left (3 a^2\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\left (6 a^3\right ) \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{6 a^3 x}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}+\frac{a^2 \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{9}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+\frac{9}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-3 a^2 \text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\left (3 a^2\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+\left (6 a^2\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\left (6 a^2\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{6 a^3 x}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}+\frac{a^2 \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{9}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+\frac{9}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-3 a^2 \text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+9 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-9 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-\left (3 a^2\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\left (6 a^2\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+\left (6 a^2\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{6 a^3 x}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}+\frac{a^2 \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{9}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+\frac{9}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-3 a^2 \text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+9 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-9 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )-\left (6 a^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )+\left (6 a^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )\\ &=-\frac{6 a^3 x}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a^3 x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}+\frac{a^2 \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{9}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+\frac{9}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-3 a^2 \text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+9 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-9 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-9 a^2 \text{Li}_4\left (-e^{\tanh ^{-1}(a x)}\right )+9 a^2 \text{Li}_4\left (e^{\tanh ^{-1}(a x)}\right )\\ \end{align*}

Mathematica [A]  time = 7.72953, size = 555, normalized size = 1.54 \[ \frac{a^2 \left (72 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+144 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \text{PolyLog}\left (3,-e^{-\tanh ^{-1}(a x)}\right )-144 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )+24 \sqrt{1-a^2 x^2} \left (3 \tanh ^{-1}(a x)^2+2\right ) \text{PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )-48 \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right )+144 \sqrt{1-a^2 x^2} \text{PolyLog}\left (4,-e^{-\tanh ^{-1}(a x)}\right )+144 \sqrt{1-a^2 x^2} \text{PolyLog}\left (4,e^{\tanh ^{-1}(a x)}\right )+3 \pi ^4 \sqrt{1-a^2 x^2}-6 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^4+12 \sqrt{1-a^2 x^2} \tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right ) \tanh ^{-1}(a x)^2-24 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3 \log \left (e^{-\tanh ^{-1}(a x)}+1\right )+24 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3 \log \left (1-e^{\tanh ^{-1}(a x)}\right )+48 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-48 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \log \left (e^{-\tanh ^{-1}(a x)}+1\right )-12 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right )-2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3 \text{csch}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )-2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3 \text{sech}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )-96 a x+16 \tanh ^{-1}(a x)^3-48 a x \tanh ^{-1}(a x)^2+96 \tanh ^{-1}(a x)\right )}{16 \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*(-96*a*x + 3*Pi^4*Sqrt[1 - a^2*x^2] + 96*ArcTanh[a*x] - 48*a*x*ArcTanh[a*x]^2 + 16*ArcTanh[a*x]^3 - 6*Sqr
t[1 - a^2*x^2]*ArcTanh[a*x]^4 - 12*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2*Coth[ArcTanh[a*x]/2] - 2*Sqrt[1 - a^2*x^2]
*ArcTanh[a*x]^3*Csch[ArcTanh[a*x]/2]^2 + 48*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] - 48*Sqr
t[1 - a^2*x^2]*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] - 24*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3*Log[1 + E^(-ArcTa
nh[a*x])] + 24*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3*Log[1 - E^ArcTanh[a*x]] + 24*Sqrt[1 - a^2*x^2]*(2 + 3*ArcTanh[
a*x]^2)*PolyLog[2, -E^(-ArcTanh[a*x])] - 48*Sqrt[1 - a^2*x^2]*PolyLog[2, E^(-ArcTanh[a*x])] + 72*Sqrt[1 - a^2*
x^2]*ArcTanh[a*x]^2*PolyLog[2, E^ArcTanh[a*x]] + 144*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*PolyLog[3, -E^(-ArcTanh[a*
x])] - 144*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*PolyLog[3, E^ArcTanh[a*x]] + 144*Sqrt[1 - a^2*x^2]*PolyLog[4, -E^(-A
rcTanh[a*x])] + 144*Sqrt[1 - a^2*x^2]*PolyLog[4, E^ArcTanh[a*x]] - 2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3*Sech[Arc
Tanh[a*x]/2]^2 + 12*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2*Tanh[ArcTanh[a*x]/2]))/(16*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.346, size = 482, normalized size = 1.3 \begin{align*} \text{result too large to display} \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)^(3/2),x)

[Out]

-1/2*a^2*(arctanh(a*x)^3-3*arctanh(a*x)^2+6*arctanh(a*x)-6)*(-(a*x-1)*(a*x+1))^(1/2)/(a*x-1)+1/2*(arctanh(a*x)
^3+3*arctanh(a*x)^2+6*arctanh(a*x)+6)*a^2*(-(a*x-1)*(a*x+1))^(1/2)/(a*x+1)-1/2*(-(a*x-1)*(a*x+1))^(1/2)*arctan
h(a*x)^2*(3*a*x+arctanh(a*x))/x^2-3/2*a^2*arctanh(a*x)^3*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-9/2*a^2*arctanh(a*x)
^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+9*a^2*arctanh(a*x)*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-9*a^2*poly
log(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))+3/2*a^2*arctanh(a*x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+9/2*a^2*arctanh(a*x
)^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-9*a^2*arctanh(a*x)*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+9*a^2*polyl
og(4,(a*x+1)/(-a^2*x^2+1)^(1/2))-3*a^2*arctanh(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)*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))

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Maxima [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 )}^{\frac{3}{2}} 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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )^{3}}{a^{4} x^{7} - 2 \, 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)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*x^2 + 1)*arctanh(a*x)^3/(a^4*x^7 - 2*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 )}}{x^{3} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, 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)**(3/2),x)

[Out]

Integral(atanh(a*x)**3/(x**3*(-(a*x - 1)*(a*x + 1))**(3/2)), 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 )}^{\frac{3}{2}} 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)^(3/2),x, algorithm="giac")

[Out]

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