Optimal. Leaf size=9 \[ \frac{\log \left (\tanh ^{-1}(a x)\right )}{a} \]
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Rubi [A] time = 0.0275559, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {5946} \[ \frac{\log \left (\tanh ^{-1}(a x)\right )}{a} \]
Antiderivative was successfully verified.
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Rule 5946
Rubi steps
\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)} \, dx &=\frac{\log \left (\tanh ^{-1}(a x)\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0286535, size = 9, normalized size = 1. \[ \frac{\log \left (\tanh ^{-1}(a x)\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 10, normalized size = 1.1 \begin{align*}{\frac{\ln \left ({\it Artanh} \left ( ax \right ) \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.955185, size = 28, normalized size = 3.11 \begin{align*} \frac{\log \left (-\log \left (a x + 1\right ) + \log \left (-a x + 1\right )\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.26805, size = 46, normalized size = 5.11 \begin{align*} \frac{\log \left (\log \left (-\frac{a x + 1}{a x - 1}\right )\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.891838, size = 7, normalized size = 0.78 \begin{align*} \frac{\log{\left (\operatorname{atanh}{\left (a x \right )} \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17304, size = 28, normalized size = 3.11 \begin{align*} \frac{\log \left ({\left | \log \left (-\frac{a x + 1}{a x - 1}\right ) \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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