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

Optimal. Leaf size=267 \[ 3 a^2 \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-3 a^2 \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {3}{2} a^2 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+\frac {3}{2} a^2 \tanh ^{-1}(a x)^2 \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+3 a^2 \tanh ^{-1}(a x) \text {Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-3 a^2 \tanh ^{-1}(a x) \text {Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-3 a^2 \text {Li}_4\left (-e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text {Li}_4\left (e^{\tanh ^{-1}(a x)}\right )-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-\frac {3 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.45, antiderivative size = 267, normalized size of antiderivative = 1.00, 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 = {6026, 6008, 6018, 6020, 4182, 2531, 6609, 2282, 6589} \[ 3 a^2 \text {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-3 a^2 \text {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {3}{2} a^2 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+\frac {3}{2} a^2 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+3 a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-3 a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-3 a^2 \text {PolyLog}\left (4,-e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text {PolyLog}\left (4,e^{\tanh ^{-1}(a x)}\right )-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-\frac {3 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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 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 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 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]

Rubi steps

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

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Mathematica [A]  time = 8.97, size = 416, normalized size = 1.56 \[ \frac {1}{16} a^2 \tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right ) \left (-\frac {4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{a x}-\frac {a x \tanh ^{-1}(a x)^3 \text {csch}^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}+24 \tanh ^{-1}(a x)^2 \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right ) \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )+48 \tanh ^{-1}(a x) \text {Li}_3\left (-e^{-\tanh ^{-1}(a x)}\right ) \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )-48 \tanh ^{-1}(a x) \text {Li}_3\left (e^{\tanh ^{-1}(a x)}\right ) \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )+24 \left (\tanh ^{-1}(a x)^2+2\right ) \text {Li}_2\left (-e^{-\tanh ^{-1}(a x)}\right ) \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )-48 \text {Li}_2\left (e^{-\tanh ^{-1}(a x)}\right ) \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )+48 \text {Li}_4\left (-e^{-\tanh ^{-1}(a x)}\right ) \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )+48 \text {Li}_4\left (e^{\tanh ^{-1}(a x)}\right ) \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )+12 \tanh ^{-1}(a x)^2-12 \tanh ^{-1}(a x)^2 \coth ^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )-2 \tanh ^{-1}(a x)^4 \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )+\pi ^4 \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )-8 \tanh ^{-1}(a x)^3 \log \left (e^{-\tanh ^{-1}(a x)}+1\right ) \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )+8 \tanh ^{-1}(a x)^3 \log \left (1-e^{\tanh ^{-1}(a x)}\right ) \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )+48 \tanh ^{-1}(a x) \log \left (1-e^{-\tanh ^{-1}(a x)}\right ) \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )-48 \tanh ^{-1}(a x) \log \left (e^{-\tanh ^{-1}(a x)}+1\right ) \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*(12*ArcTanh[a*x]^2 - (4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/(a*x) + Pi^4*Coth[ArcTanh[a*x]/2] - 2*ArcTanh[a
*x]^4*Coth[ArcTanh[a*x]/2] - 12*ArcTanh[a*x]^2*Coth[ArcTanh[a*x]/2]^2 - (a*x*ArcTanh[a*x]^3*Csch[ArcTanh[a*x]/
2]^4)/Sqrt[1 - a^2*x^2] + 48*ArcTanh[a*x]*Coth[ArcTanh[a*x]/2]*Log[1 - E^(-ArcTanh[a*x])] - 48*ArcTanh[a*x]*Co
th[ArcTanh[a*x]/2]*Log[1 + E^(-ArcTanh[a*x])] - 8*ArcTanh[a*x]^3*Coth[ArcTanh[a*x]/2]*Log[1 + E^(-ArcTanh[a*x]
)] + 8*ArcTanh[a*x]^3*Coth[ArcTanh[a*x]/2]*Log[1 - E^ArcTanh[a*x]] + 24*(2 + ArcTanh[a*x]^2)*Coth[ArcTanh[a*x]
/2]*PolyLog[2, -E^(-ArcTanh[a*x])] - 48*Coth[ArcTanh[a*x]/2]*PolyLog[2, E^(-ArcTanh[a*x])] + 24*ArcTanh[a*x]^2
*Coth[ArcTanh[a*x]/2]*PolyLog[2, E^ArcTanh[a*x]] + 48*ArcTanh[a*x]*Coth[ArcTanh[a*x]/2]*PolyLog[3, -E^(-ArcTan
h[a*x])] - 48*ArcTanh[a*x]*Coth[ArcTanh[a*x]/2]*PolyLog[3, E^ArcTanh[a*x]] + 48*Coth[ArcTanh[a*x]/2]*PolyLog[4
, -E^(-ArcTanh[a*x])] + 48*Coth[ArcTanh[a*x]/2]*PolyLog[4, E^ArcTanh[a*x]])*Tanh[ArcTanh[a*x]/2])/16

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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )^{3}}{a^{2} x^{5} - x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-a^2*x^2 + 1)*arctanh(a*x)^3/(a^2*x^5 - x^3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.50, size = 386, normalized size = 1.45 \[ -\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \arctanh \left (a x \right )^{2} \left (3 a x +\arctanh \left (a x \right )\right )}{2 x^{2}}-\frac {a^{2} \arctanh \left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {3 a^{2} \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+3 a^{2} \arctanh \left (a x \right ) \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 a^{2} \polylog \left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {a^{2} \arctanh \left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {3 a^{2} \arctanh \left (a x \right )^{2} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-3 a^{2} \arctanh \left (a x \right ) \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 a^{2} \polylog \left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 a^{2} \arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 a^{2} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 a^{2} \arctanh \left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 a^{2} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-(a*x-1)*(a*x+1))^(1/2)*arctanh(a*x)^2*(3*a*x+arctanh(a*x))/x^2-1/2*a^2*arctanh(a*x)^3*ln(1+(a*x+1)/(-a^
2*x^2+1)^(1/2))-3/2*a^2*arctanh(a*x)^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*arctanh(a*x)*polylog(3,-(a
*x+1)/(-a^2*x^2+1)^(1/2))-3*a^2*polylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*a^2*arctanh(a*x)^3*ln(1-(a*x+1)/(-a
^2*x^2+1)^(1/2))+3/2*a^2*arctanh(a*x)^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-3*a^2*arctanh(a*x)*polylog(3,(a*
x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*polylog(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x^3\,\sqrt {1-a^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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