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

Optimal. Leaf size=151 \[ a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )-a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )-a^2 \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )+a^2 \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}+a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2 \]

[Out]

-((a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(2*x^2) + a^2*ArcTanh[E^ArcTanh[a
*x]]*ArcTanh[a*x]^2 - a^2*ArcTanh[Sqrt[1 - a^2*x^2]] + a^2*ArcTanh[a*x]*PolyLog[2, -E^ArcTanh[a*x]] - a^2*ArcT
anh[a*x]*PolyLog[2, E^ArcTanh[a*x]] - a^2*PolyLog[3, -E^ArcTanh[a*x]] + a^2*PolyLog[3, E^ArcTanh[a*x]]

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Rubi [A]  time = 0.544467, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {6014, 6026, 6008, 266, 63, 208, 6020, 4182, 2531, 2282, 6589} \[ a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )-a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )-a^2 \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )+a^2 \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}+a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(2*x^2) + a^2*ArcTanh[E^ArcTanh[a
*x]]*ArcTanh[a*x]^2 - a^2*ArcTanh[Sqrt[1 - a^2*x^2]] + a^2*ArcTanh[a*x]*PolyLog[2, -E^ArcTanh[a*x]] - a^2*ArcT
anh[a*x]*PolyLog[2, E^ArcTanh[a*x]] - a^2*PolyLog[3, -E^ArcTanh[a*x]] + a^2*PolyLog[3, E^ArcTanh[a*x]]

Rule 6014

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

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

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

Mathematica [A]  time = 1.2173, size = 188, normalized size = 1.25 \[ \frac{1}{8} a^2 \left (-8 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )+8 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right )-8 \text{PolyLog}\left (3,-e^{-\tanh ^{-1}(a x)}\right )+8 \text{PolyLog}\left (3,e^{-\tanh ^{-1}(a x)}\right )+4 \tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right ) \tanh ^{-1}(a x)-4 \tanh ^{-1}(a x)^2 \log \left (1-e^{-\tanh ^{-1}(a x)}\right )+4 \tanh ^{-1}(a x)^2 \log \left (e^{-\tanh ^{-1}(a x)}+1\right )+8 \log \left (\tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right )-4 \tanh ^{-1}(a x) \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right )+\tanh ^{-1}(a x)^2 \left (-\text{csch}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right )-\tanh ^{-1}(a x)^2 \text{sech}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*(-4*ArcTanh[a*x]*Coth[ArcTanh[a*x]/2] - ArcTanh[a*x]^2*Csch[ArcTanh[a*x]/2]^2 - 4*ArcTanh[a*x]^2*Log[1 -
E^(-ArcTanh[a*x])] + 4*ArcTanh[a*x]^2*Log[1 + E^(-ArcTanh[a*x])] + 8*Log[Tanh[ArcTanh[a*x]/2]] - 8*ArcTanh[a*x
]*PolyLog[2, -E^(-ArcTanh[a*x])] + 8*ArcTanh[a*x]*PolyLog[2, E^(-ArcTanh[a*x])] - 8*PolyLog[3, -E^(-ArcTanh[a*
x])] + 8*PolyLog[3, E^(-ArcTanh[a*x])] - ArcTanh[a*x]^2*Sech[ArcTanh[a*x]/2]^2 + 4*ArcTanh[a*x]*Tanh[ArcTanh[a
*x]/2]))/8

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Maple [A]  time = 0.289, size = 231, normalized size = 1.5 \begin{align*} -{\frac{{\it Artanh} \left ( ax \right ) \left ( 2\,ax+{\it Artanh} \left ( ax \right ) \right ) }{2\,{x}^{2}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2}\ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{a}^{2}{\it Artanh} \left ( ax \right ){\it polylog} \left ( 2,-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) -{a}^{2}{\it polylog} \left ( 3,-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) -{\frac{{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2}\ln \left ( 1-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{a}^{2}{\it Artanh} \left ( ax \right ){\it polylog} \left ( 2,{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) +{a}^{2}{\it polylog} \left ( 3,{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) -2\,{a}^{2}{\it Artanh} \left ({\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)^2*(-a^2*x^2+1)^(1/2)/x^3,x)

[Out]

-1/2*(-(a*x-1)*(a*x+1))^(1/2)*arctanh(a*x)*(2*a*x+arctanh(a*x))/x^2+1/2*a^2*arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*
x^2+1)^(1/2))+a^2*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-a^2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2
))-1/2*a^2*arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-a^2*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/
2))+a^2*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))-2*a^2*arctanh((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{\sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )^{2}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-a^2*x^2 + 1)*arctanh(a*x)^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 )^{2}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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