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

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{x^m \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.100265, 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 = {} \[ \int \frac{x^m \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.475587, size = 0, normalized size = 0. \[ \int \frac{x^m \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.747, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ({\it Artanh} \left ( ax \right ) \right ) ^{3} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} x^{2} + 1} x^{m} \operatorname{artanh}\left (a x\right )^{3}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-a^2*x^2 + 1)*x^m*arctanh(a*x)^3/(a^4*x^4 - 2*a^2*x^2 + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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