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

Optimal. Leaf size=89 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)},x\right )}{a}-\frac{a x}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}-\frac{\sqrt{1-a^2 x^2}}{a x \tanh ^{-1}(a x)}+\text{Chi}\left (\tanh ^{-1}(a x)\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.417318, 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 = {} \[ \int \frac{1}{x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 5.89887, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.287, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}} \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)^2,x)

[Out]

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

________________________________________________________________________________________

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 )^{2}}\,{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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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 )^{2}}, 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)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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}^{2}{\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)**2,x)

[Out]

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

________________________________________________________________________________________

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 )^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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