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

Optimal. Leaf size=123 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2},x\right )}{2 a}-\frac{a x}{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac{1}{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}-\frac{\sqrt{1-a^2 x^2}}{2 a x \tanh ^{-1}(a x)^2}+\frac{1}{2} \text{Shi}\left (\tanh ^{-1}(a x)\right ) \]

[Out]

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

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Rubi [A]  time = 0.484825, 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 = {} \[ \int \frac{1}{x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 18.0729, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.311, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \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(1/x/(-a^2*x^2+1)^(3/2)/arctanh(a*x)^3,x)

[Out]

int(1/x/(-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{1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x \operatorname{artanh}\left (a x\right )^{3}}\,{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)^3,x, algorithm="maxima")

[Out]

integrate(1/((-a^2*x^2 + 1)^(3/2)*x*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^{4} x^{5} - 2 \, a^{2} x^{3} + x\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/x/(-a^2*x^2+1)^(3/2)/arctanh(a*x)^3,x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*x^2 + 1)/((a^4*x^5 - 2*a^2*x^3 + x)*arctanh(a*x)^3), 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}^{3}{\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)**3,x)

[Out]

Integral(1/(x*(-(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{1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x \operatorname{artanh}\left (a x\right )^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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