3.281 \(\int \frac{x^4}{(1-a^2 x^2)^2 \tanh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=39 \[ \frac{\text{Unintegrable}\left (\frac{1}{\tanh ^{-1}(a x)},x\right )}{a^4}+\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^5}-\frac{3 \log \left (\tanh ^{-1}(a x)\right )}{2 a^5} \]

[Out]

CoshIntegral[2*ArcTanh[a*x]]/(2*a^5) - (3*Log[ArcTanh[a*x]])/(2*a^5) + Unintegrable[ArcTanh[a*x]^(-1), x]/a^4

________________________________________________________________________________________

Rubi [A]  time = 0.0635334, 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 = {} \[ \int \frac{x^4}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{x^4}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx &=\int \frac{x^4}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ \end{align*}

Mathematica [A]  time = 4.57327, size = 0, normalized size = 0. \[ \int \frac{x^4}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.191, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{4}}{ \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^4/(-a^2*x^2+1)^2/arctanh(a*x),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\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^4/(-a^2*x^2+1)^2/arctanh(a*x),x, algorithm="maxima")

[Out]

integrate(x^4/((a^2*x^2 - 1)^2*arctanh(a*x)), x)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{4}}{{\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^4/(-a^2*x^2+1)^2/arctanh(a*x),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\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**4/(-a**2*x**2+1)**2/atanh(a*x),x)

[Out]

Integral(x**4/((a*x - 1)**2*(a*x + 1)**2*atanh(a*x)), x)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\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^4/(-a^2*x^2+1)^2/arctanh(a*x),x, algorithm="giac")

[Out]

integrate(x^4/((a^2*x^2 - 1)^2*arctanh(a*x)), x)