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

Optimal. Leaf size=96 \[ -\frac{\text{Unintegrable}\left (\frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3},x\right )}{a^2}+\frac{\text{Chi}\left (\tanh ^{-1}(a x)\right )}{2 a^3}-\frac{x}{2 a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}-\frac{1}{2 a^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2} \]

[Out]

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

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Rubi [A]  time = 0.303998, 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 = {} \[ \int \frac{x^2}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{x^2}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^3} \, dx &=\frac{\int \frac{1}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^3} \, dx}{a^2}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3} \, dx}{a^2}\\ &=-\frac{1}{2 a^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3} \, dx}{a^2}+\frac{\int \frac{x}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ &=-\frac{1}{2 a^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac{x}{2 a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}+\frac{\int \frac{1}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)} \, dx}{2 a^2}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3} \, dx}{a^2}\\ &=-\frac{1}{2 a^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac{x}{2 a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^3}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3} \, dx}{a^2}\\ &=-\frac{1}{2 a^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac{x}{2 a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}+\frac{\text{Chi}\left (\tanh ^{-1}(a x)\right )}{2 a^3}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3} \, dx}{a^2}\\ \end{align*}

Mathematica [A]  time = 6.18917, size = 0, normalized size = 0. \[ \int \frac{x^2}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.273, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{ \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^2/(-a^2*x^2+1)^(3/2)/arctanh(a*x)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/((-a^2*x^2 + 1)^(3/2)*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} x^{2}}{{\left (a^{4} x^{4} - 2 \, 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(x^2/(-a^2*x^2+1)^(3/2)/arctanh(a*x)^3,x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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