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

Optimal. Leaf size=19 \[ \text{Unintegrable}\left (\frac{1-a^2 x^2}{\tanh ^{-1}(a x)^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0132488, 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 = {} \[ \int \frac{1-a^2 x^2}{\tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 - a^2*x^2)/ArcTanh[a*x]^3,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.14183, size = 0, normalized size = 0. \[ \int \frac{1-a^2 x^2}{\tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((-a^2*x^2+1)/arctanh(a*x)^3,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} - 2 \,{\left (a^{5} x^{5} - 2 \, a^{3} x^{3} + a x\right )} \log \left (a x + 1\right ) + 2 \,{\left (a^{5} x^{5} - 2 \, a^{3} x^{3} + a x\right )} \log \left (-a x + 1\right ) + 1\right )}}{a \log \left (a x + 1\right )^{2} - 2 \, a \log \left (a x + 1\right ) \log \left (-a x + 1\right ) + a \log \left (-a x + 1\right )^{2}} + \int -\frac{4 \,{\left (5 \, a^{4} x^{4} - 6 \, a^{2} x^{2} + 1\right )}}{\log \left (a x + 1\right ) - \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)/arctanh(a*x)^3,x, algorithm="maxima")

[Out]

-2*(a^4*x^4 - 2*a^2*x^2 - 2*(a^5*x^5 - 2*a^3*x^3 + a*x)*log(a*x + 1) + 2*(a^5*x^5 - 2*a^3*x^3 + a*x)*log(-a*x
+ 1) + 1)/(a*log(a*x + 1)^2 - 2*a*log(a*x + 1)*log(-a*x + 1) + a*log(-a*x + 1)^2) + integrate(-4*(5*a^4*x^4 -
6*a^2*x^2 + 1)/(log(a*x + 1) - 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}{\operatorname{artanh}\left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(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{a^{2} x^{2}}{\operatorname{atanh}^{3}{\left (a x \right )}}\, dx - \int - \frac{1}{\operatorname{atanh}^{3}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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