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

Optimal. Leaf size=101 \[ -\frac{\text{Unintegrable}\left (\frac{1}{\tanh ^{-1}(a x)^2},x\right )}{2 a^3}+\frac{\text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{a^4}-\frac{x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{a^2 x^2+1}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{x}{2 a^3 \tanh ^{-1}(a x)^2} \]

[Out]

x/(2*a^3*ArcTanh[a*x]^2) - x/(2*a^3*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (1 + a^2*x^2)/(2*a^4*(1 - a^2*x^2)*ArcTanh
[a*x]) + SinhIntegral[2*ArcTanh[a*x]]/a^4 - Unintegrable[ArcTanh[a*x]^(-2), x]/(2*a^3)

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Rubi [A]  time = 0.22487, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

x/(2*a^3*ArcTanh[a*x]^2) - x/(2*a^3*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (1 + a^2*x^2)/(2*a^4*(1 - a^2*x^2)*ArcTanh
[a*x]) + SinhIntegral[2*ArcTanh[a*x]]/a^4 - Defer[Int][ArcTanh[a*x]^(-2), x]/(2*a^3)

Rubi steps

\begin{align*} \int \frac{x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx &=\frac{\int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx}{a^2}-\frac{\int \frac{x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3} \, dx}{a^2}\\ &=\frac{x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac{x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{\int \frac{1}{\tanh ^{-1}(a x)^2} \, dx}{2 a^3}+\frac{2 \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx}{a^2}\\ &=\frac{x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac{x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}-\frac{\int \frac{1}{\tanh ^{-1}(a x)^2} \, dx}{2 a^3}\\ &=\frac{x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac{x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}-\frac{\int \frac{1}{\tanh ^{-1}(a x)^2} \, dx}{2 a^3}\\ &=\frac{x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac{x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}-\frac{\int \frac{1}{\tanh ^{-1}(a x)^2} \, dx}{2 a^3}\\ &=\frac{x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac{x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{a^4}-\frac{\int \frac{1}{\tanh ^{-1}(a x)^2} \, dx}{2 a^3}\\ \end{align*}

Mathematica [A]  time = 10.8905, size = 0, normalized size = 0. \[ \int \frac{x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.227, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ( -{a}^{2}{x}^{2}+1 \right ) ^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, a x^{3} -{\left (a^{2} x^{4} - 3 \, x^{2}\right )} \log \left (a x + 1\right ) +{\left (a^{2} x^{4} - 3 \, x^{2}\right )} \log \left (-a x + 1\right )}{{\left (a^{4} x^{2} - a^{2}\right )} \log \left (a x + 1\right )^{2} - 2 \,{\left (a^{4} x^{2} - a^{2}\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right ) +{\left (a^{4} x^{2} - a^{2}\right )} \log \left (-a x + 1\right )^{2}} - \int -\frac{2 \,{\left (a^{4} x^{5} - 2 \, a^{2} x^{3} + 3 \, x\right )}}{{\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (a x + 1\right ) -{\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (-a x + 1\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*x^3 - (a^2*x^4 - 3*x^2)*log(a*x + 1) + (a^2*x^4 - 3*x^2)*log(-a*x + 1))/((a^4*x^2 - a^2)*log(a*x + 1)^2 -
2*(a^4*x^2 - a^2)*log(a*x + 1)*log(-a*x + 1) + (a^4*x^2 - a^2)*log(-a*x + 1)^2) - integrate(-2*(a^4*x^5 - 2*a
^2*x^3 + 3*x)/((a^6*x^4 - 2*a^4*x^2 + a^2)*log(a*x + 1) - (a^6*x^4 - 2*a^4*x^2 + a^2)*log(-a*x + 1)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3}}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname{artanh}\left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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