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

Optimal. Leaf size=171 \[ 2 a \text{PolyLog}\left (2,-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-2 a \text{PolyLog}\left (2,\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )+\frac{2 a^2 x}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{2 a \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}-4 a \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.312958, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {6030, 6008, 6018, 5962, 191} \[ 2 a \text{PolyLog}\left (2,-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-2 a \text{PolyLog}\left (2,\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )+\frac{2 a^2 x}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{2 a \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}-4 a \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*x)/Sqrt[1 - a^2*x^2] - (2*a*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] + (a^2*x*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2]
- (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/x - 4*a*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + 2*a*PolyLog[2
, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - 2*a*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + 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 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 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)^2}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx &=a^2 \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\int \frac{\tanh ^{-1}(a x)^2}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 a \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}+(2 a) \int \frac{\tanh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \, dx+\left (2 a^2\right ) \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 a^2 x}{\sqrt{1-a^2 x^2}}-\frac{2 a \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}-4 a \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+2 a \text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-2 a \text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )\\ \end{align*}

Mathematica [A]  time = 1.07844, size = 215, normalized size = 1.26 \[ \frac{a \left (4 \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )-4 \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right )+4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \log \left (e^{-\tanh ^{-1}(a x)}+1\right )-\frac{2 \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^2 \sinh ^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )}{a x}+4 a x+2 a x \tanh ^{-1}(a x)^2-4 \tanh ^{-1}(a x)-\frac{1}{2} a x \tanh ^{-1}(a x)^2 \text{csch}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right )}{2 \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*(4*a*x - 4*ArcTanh[a*x] + 2*a*x*ArcTanh[a*x]^2 - (a*x*ArcTanh[a*x]^2*Csch[ArcTanh[a*x]/2]^2)/2 + 4*Sqrt[1 -
a^2*x^2]*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] - 4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])
] + 4*Sqrt[1 - a^2*x^2]*PolyLog[2, -E^(-ArcTanh[a*x])] - 4*Sqrt[1 - a^2*x^2]*PolyLog[2, E^(-ArcTanh[a*x])] - (
2*(-1 + a^2*x^2)*ArcTanh[a*x]^2*Sinh[ArcTanh[a*x]/2]^2)/(a*x)))/(2*Sqrt[1 - a^2*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.27, size = 207, normalized size = 1.2 \begin{align*} -{\frac{a \left ( \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}-2\,{\it Artanh} \left ( ax \right ) +2 \right ) }{2\,ax-2}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{ \left ( \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+2\,{\it Artanh} \left ( ax \right ) +2 \right ) a}{2\,ax+2}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{x}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-2\,a{\it Artanh} \left ( ax \right ) \ln \left ( 1+{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -2\,a{\it polylog} \left ( 2,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +2\,a{\it Artanh} \left ( ax \right ) \ln \left ( 1-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +2\,a{\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)^2/x^2/(-a^2*x^2+1)^(3/2),x)

[Out]

-1/2*a*(arctanh(a*x)^2-2*arctanh(a*x)+2)*(-(a*x-1)*(a*x+1))^(1/2)/(a*x-1)-1/2*(arctanh(a*x)^2+2*arctanh(a*x)+2
)*a*(-(a*x-1)*(a*x+1))^(1/2)/(a*x+1)-(-(a*x-1)*(a*x+1))^(1/2)*arctanh(a*x)^2/x-2*a*arctanh(a*x)*ln(1+(a*x+1)/(
-a^2*x^2+1)^(1/2))-2*a*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*a*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2)
)+2*a*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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}}{a^{4} x^{6} - 2 \, a^{2} x^{4} + x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-a^2*x^2 + 1)*arctanh(a*x)^2/(a^4*x^6 - 2*a^2*x^4 + x^2), x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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