Optimal. Leaf size=29 \[ -\frac{x}{a^2}-\frac{\log (1-a x)}{a^3}-\frac{x^2}{2 a} \]
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Rubi [A] time = 0.0906503, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6150, 43} \[ -\frac{x}{a^2}-\frac{\log (1-a x)}{a^3}-\frac{x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^2}{\sqrt{1-a^2 x^2}} \, dx &=\int \frac{x^2}{1-a x} \, dx\\ &=\int \left (-\frac{1}{a^2}-\frac{x}{a}-\frac{1}{a^2 (-1+a x)}\right ) \, dx\\ &=-\frac{x}{a^2}-\frac{x^2}{2 a}-\frac{\log (1-a x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0164728, size = 29, normalized size = 1. \[ -\frac{x}{a^2}-\frac{\log (1-a x)}{a^3}-\frac{x^2}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 27, normalized size = 0.9 \begin{align*} -{\frac{{x}^{2}}{2\,a}}-{\frac{x}{{a}^{2}}}-{\frac{\ln \left ( ax-1 \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.942624, size = 35, normalized size = 1.21 \begin{align*} -\frac{a x^{2} + 2 \, x}{2 \, a^{2}} - \frac{\log \left (a x - 1\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49826, size = 61, normalized size = 2.1 \begin{align*} -\frac{a^{2} x^{2} + 2 \, a x + 2 \, \log \left (a x - 1\right )}{2 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.25169, size = 22, normalized size = 0.76 \begin{align*} - \frac{x^{2}}{2 a} - \frac{x}{a^{2}} - \frac{\log{\left (a x - 1 \right )}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15345, size = 36, normalized size = 1.24 \begin{align*} -\frac{a x^{2} + 2 \, x}{2 \, a^{2}} - \frac{\log \left ({\left | a x - 1 \right |}\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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