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

Optimal. Leaf size=305 \[ -\frac{3 i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a^3}+\frac{3 i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a^3}-\frac{3 i \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{2 a^3}+\frac{3 i \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{2 a^3}+\frac{3 i \tanh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac{3 i \tanh ^{-1}(a x) \text{PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac{3 i \text{PolyLog}\left (4,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac{3 i \text{PolyLog}\left (4,i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 a^2}-\frac{3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a^3}+\frac{\tanh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac{6 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{a^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.339874, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6016, 5994, 5950, 5952, 4180, 2531, 6609, 2282, 6589} \[ -\frac{3 i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a^3}+\frac{3 i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a^3}-\frac{3 i \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{2 a^3}+\frac{3 i \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{2 a^3}+\frac{3 i \tanh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac{3 i \tanh ^{-1}(a x) \text{PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac{3 i \text{PolyLog}\left (4,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac{3 i \text{PolyLog}\left (4,i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 a^2}-\frac{3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a^3}+\frac{\tanh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac{6 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6016

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

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 5950

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

Rule 5952

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

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && 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]

Rubi steps

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

Mathematica [A]  time = 4.46736, size = 570, normalized size = 1.87 \[ -\frac{i \left (192 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )+192 i \pi \tanh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )+384 \tanh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{-\tanh ^{-1}(a x)}\right )-384 \tanh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )-48 \left (-4 i \pi \tanh ^{-1}(a x)-4 \left (\tanh ^{-1}(a x)^2+2\right )+\pi ^2\right ) \text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-384 \text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )-48 \pi ^2 \text{PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )+192 i \pi \text{PolyLog}\left (3,-i e^{-\tanh ^{-1}(a x)}\right )-192 i \pi \text{PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )+384 \text{PolyLog}\left (4,-i e^{-\tanh ^{-1}(a x)}\right )+384 \text{PolyLog}\left (4,-i e^{\tanh ^{-1}(a x)}\right )-64 i a x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3-192 i \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2-16 \tanh ^{-1}(a x)^4-32 i \pi \tanh ^{-1}(a x)^3+24 \pi ^2 \tanh ^{-1}(a x)^2+8 i \pi ^3 \tanh ^{-1}(a x)-64 \tanh ^{-1}(a x)^3 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+64 \tanh ^{-1}(a x)^3 \log \left (1+i e^{\tanh ^{-1}(a x)}\right )-96 i \pi \tanh ^{-1}(a x)^2 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+96 i \pi \tanh ^{-1}(a x)^2 \log \left (1-i e^{\tanh ^{-1}(a x)}\right )+384 \tanh ^{-1}(a x) \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )+48 \pi ^2 \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-384 \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-48 \pi ^2 \tanh ^{-1}(a x) \log \left (1-i e^{\tanh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\tanh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac{1}{4} \left (\pi +2 i \tanh ^{-1}(a x)\right )\right )\right )+7 \pi ^4\right )}{128 a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I/128)*(7*Pi^4 + (8*I)*Pi^3*ArcTanh[a*x] + 24*Pi^2*ArcTanh[a*x]^2 - (192*I)*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^
2 - (32*I)*Pi*ArcTanh[a*x]^3 - (64*I)*a*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3 - 16*ArcTanh[a*x]^4 + 384*ArcTanh[a
*x]*Log[1 - I/E^ArcTanh[a*x]] + (8*I)*Pi^3*Log[1 + I/E^ArcTanh[a*x]] - 384*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*
x]] + 48*Pi^2*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] - (96*I)*Pi*ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] - 64
*ArcTanh[a*x]^3*Log[1 + I/E^ArcTanh[a*x]] - 48*Pi^2*ArcTanh[a*x]*Log[1 - I*E^ArcTanh[a*x]] + (96*I)*Pi*ArcTanh
[a*x]^2*Log[1 - I*E^ArcTanh[a*x]] - (8*I)*Pi^3*Log[1 + I*E^ArcTanh[a*x]] + 64*ArcTanh[a*x]^3*Log[1 + I*E^ArcTa
nh[a*x]] + (8*I)*Pi^3*Log[Tan[(Pi + (2*I)*ArcTanh[a*x])/4]] - 48*(Pi^2 - (4*I)*Pi*ArcTanh[a*x] - 4*(2 + ArcTan
h[a*x]^2))*PolyLog[2, (-I)/E^ArcTanh[a*x]] - 384*PolyLog[2, I/E^ArcTanh[a*x]] + 192*ArcTanh[a*x]^2*PolyLog[2,
(-I)*E^ArcTanh[a*x]] - 48*Pi^2*PolyLog[2, I*E^ArcTanh[a*x]] + (192*I)*Pi*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a
*x]] + (192*I)*Pi*PolyLog[3, (-I)/E^ArcTanh[a*x]] + 384*ArcTanh[a*x]*PolyLog[3, (-I)/E^ArcTanh[a*x]] - 384*Arc
Tanh[a*x]*PolyLog[3, (-I)*E^ArcTanh[a*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcTanh[a*x]] + 384*PolyLog[4, (-I)/E^Ar
cTanh[a*x]] + 384*PolyLog[4, (-I)*E^ArcTanh[a*x]]))/a^3

________________________________________________________________________________________

Maple [F]  time = 0.285, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{atanh}^{3}{\left (a x \right )}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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