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

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

[Out]

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

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Rubi [A]  time = 0.118475, 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 = {} \[ \int \frac{1}{x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.0374, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.286, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x{\it Artanh} \left ( ax \right ) } \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(1/x/(-a^2*x^2+1)^(3/2)/arctanh(a*x),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-a^2*x^2 + 1)/((a^4*x^5 - 2*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{1}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}} \operatorname{atanh}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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