Optimal. Leaf size=13 \[ \frac{\tanh ^{-1}(a x)^2}{2 a} \]
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Rubi [A] time = 0.0157754, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {5948} \[ \frac{\tanh ^{-1}(a x)^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 5948
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx &=\frac{\tanh ^{-1}(a x)^2}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0042339, size = 13, normalized size = 1. \[ \frac{\tanh ^{-1}(a x)^2}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 12, normalized size = 0.9 \begin{align*}{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.975797, size = 88, normalized size = 6.77 \begin{align*} \frac{1}{2} \,{\left (\frac{\log \left (a x + 1\right )}{a} - \frac{\log \left (a x - 1\right )}{a}\right )} \operatorname{artanh}\left (a x\right ) - \frac{\log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + \log \left (a x - 1\right )^{2}}{8 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87087, size = 47, normalized size = 3.62 \begin{align*} \frac{\log \left (-\frac{a x + 1}{a x - 1}\right )^{2}}{8 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.45779, size = 10, normalized size = 0.77 \begin{align*} \begin{cases} \frac{\operatorname{atanh}^{2}{\left (a x \right )}}{2 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13503, size = 30, normalized size = 2.31 \begin{align*} \frac{\log \left (-\frac{a x + 1}{a x - 1}\right )^{2}}{8 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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