Optimal. Leaf size=13 \[ -\frac{1}{2 a \tanh ^{-1}(a x)^2} \]
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Rubi [A] time = 0.0257797, antiderivative size = 13, 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 = {5948} \[ -\frac{1}{2 a \tanh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 5948
Rubi steps
\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3} \, dx &=-\frac{1}{2 a \tanh ^{-1}(a x)^2}\\ \end{align*}
Mathematica [A] time = 0.005255, size = 13, normalized size = 1. \[ -\frac{1}{2 a \tanh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 12, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.973626, size = 57, normalized size = 4.38 \begin{align*} -\frac{2}{a \log \left (a x + 1\right )^{2} - 2 \, a \log \left (a x + 1\right ) \log \left (-a x + 1\right ) + a \log \left (-a x + 1\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24365, size = 49, normalized size = 3.77 \begin{align*} -\frac{2}{a \log \left (-\frac{a x + 1}{a x - 1}\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.4995, size = 12, normalized size = 0.92 \begin{align*} - \frac{1}{2 a \operatorname{atanh}^{2}{\left (a x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19174, size = 30, normalized size = 2.31 \begin{align*} -\frac{2}{a \log \left (-\frac{a x + 1}{a x - 1}\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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