Optimal. Leaf size=13 \[ \frac{\tanh ^{-1}(a x)^3}{3 a} \]
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Rubi [A] time = 0.0261026, 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} \[ \frac{\tanh ^{-1}(a x)^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 5948
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx &=\frac{\tanh ^{-1}(a x)^3}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0048607, size = 13, normalized size = 1. \[ \frac{\tanh ^{-1}(a x)^3}{3 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 12, normalized size = 0.9 \begin{align*}{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{3\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.975451, size = 171, normalized size = 13.15 \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 )^{2} - \frac{{\left (\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}\right )} \operatorname{artanh}\left (a x\right )}{4 \, a} + \frac{\log \left (a x + 1\right )^{3} - 3 \, \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 3 \, \log \left (a x + 1\right ) \log \left (a x - 1\right )^{2} - \log \left (a x - 1\right )^{3}}{24 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27843, size = 49, normalized size = 3.77 \begin{align*} \frac{\log \left (-\frac{a x + 1}{a x - 1}\right )^{3}}{24 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.23219, size = 10, normalized size = 0.77 \begin{align*} \begin{cases} \frac{\operatorname{atanh}^{3}{\left (a x \right )}}{3 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 = 1.163, size = 30, normalized size = 2.31 \begin{align*} \frac{\log \left (-\frac{a x + 1}{a x - 1}\right )^{3}}{24 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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