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

Optimal. Leaf size=23 \[ \text{Unintegrable}\left (\frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.03318, 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 = {} \[ \int \frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.0724, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.559, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-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(1/(-a^2*x^2+1)^(1/2)/arctanh(a*x)^3,x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-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(1/(-a^2*x^2+1)^(1/2)/arctanh(a*x)^3,x, algorithm="giac")

[Out]

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