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

Optimal. Leaf size=42 \[ \frac{\text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^4}-\frac{\text{Unintegrable}\left (\frac{x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)},x\right )}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0629171, 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 = {} \[ \int \frac{x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx &=\int \frac{x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ \end{align*}

Mathematica [A]  time = 1.59783, size = 0, normalized size = 0. \[ \int \frac{x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.237, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ( -{a}^{2}{x}^{2}+1 \right ) ^{2}{\it Artanh} \left ( ax \right ) }}\, 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),x)

[Out]

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

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Maxima [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 )}\,{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),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

Integral(x**3/((a*x - 1)**2*(a*x + 1)**2*atanh(a*x)), 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 )}\,{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),x, algorithm="giac")

[Out]

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