Optimal. Leaf size=32 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \tanh ^{-1}(a x)},x\right )}{a}-\frac{1}{a x \tanh ^{-1}(a x)} \]
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Rubi [A] time = 0.0777517, 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 = {} \[ \int \frac{1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \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{\int \frac{1}{x^2 \tanh ^{-1}(a x)} \, dx}{a}\\ \end{align*}
Mathematica [A] time = 0.140771, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.118, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( -{a}^{2}{x}^{2}+1 \right ) \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2}{a x \log \left (a x + 1\right ) - a x \log \left (-a x + 1\right )} + 2 \, \int -\frac{1}{a x^{2} \log \left (a x + 1\right ) - a x^{2} \log \left (-a x + 1\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{1}{{\left (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.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{a^{2} x^{3} \operatorname{atanh}^{2}{\left (a x \right )} - x \operatorname{atanh}^{2}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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 )} x \operatorname{artanh}\left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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