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

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{1-a^2 x^2}{x \tanh ^{-1}(a x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0338011, 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 = {} \[ \int \frac{1-a^2 x^2}{x \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.14553, size = 0, normalized size = 0. \[ \int \frac{1-a^2 x^2}{x \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.263, size = 0, normalized size = 0. \begin{align*} \int{\frac{-{a}^{2}{x}^{2}+1}{x \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(a^4*x^4 - 2*a^2*x^2 + 1)/(a*x*log(a*x + 1) - a*x*log(-a*x + 1)) - integrate(-2*(3*a^4*x^4 - 2*a^2*x^2 - 1)
/(a*x^2*log(a*x + 1) - a*x^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{a^{2} x^{2} - 1}{x \operatorname{artanh}\left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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