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

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{x \left (1-a^2 x^2\right )}{\tanh ^{-1}(a x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0250282, 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 = {} \[ \int \frac{x \left (1-a^2 x^2\right )}{\tanh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(x*(1 - a^2*x^2))/ArcTanh[a*x],x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.74538, size = 0, normalized size = 0. \[ \int \frac{x \left (1-a^2 x^2\right )}{\tanh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.252, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ( -{a}^{2}{x}^{2}+1 \right ) }{{\it Artanh} \left ( ax \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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