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

Optimal. Leaf size=61 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \tanh ^{-1}(a x)},x\right )}{a}-\frac{a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\text{Chi}\left (2 \tanh ^{-1}(a x)\right )-\frac{1}{a x \tanh ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.341975, 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 = {} \[ \int \frac{1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.96287, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.162, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( -{a}^{2}{x}^{2}+1 \right ) ^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2}{{\left (a^{3} x^{3} - a x\right )} \log \left (a x + 1\right ) -{\left (a^{3} x^{3} - a x\right )} \log \left (-a x + 1\right )} - \int -\frac{2 \,{\left (3 \, a^{2} x^{2} - 1\right )}}{{\left (a^{5} x^{6} - 2 \, a^{3} x^{4} + a x^{2}\right )} \log \left (a x + 1\right ) -{\left (a^{5} x^{6} - 2 \, a^{3} x^{4} + a x^{2}\right )} \log \left (-a x + 1\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/((a^3*x^3 - a*x)*log(a*x + 1) - (a^3*x^3 - a*x)*log(-a*x + 1)) - integrate(-2*(3*a^2*x^2 - 1)/((a^5*x^6 - 2*
a^3*x^4 + a*x^2)*log(a*x + 1) - (a^5*x^6 - 2*a^3*x^4 + a*x^2)*log(-a*x + 1)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{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)^2/arctanh(a*x)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (a x - 1\right )^{2} \left (a x + 1\right )^{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)**2/atanh(a*x)**2,x)

[Out]

Integral(1/(x*(a*x - 1)**2*(a*x + 1)**2*atanh(a*x)**2), 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 )}^{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)^2/arctanh(a*x)^2,x, algorithm="giac")

[Out]

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