Optimal. Leaf size=34 \[ -\frac{2 x}{a^2}-\frac{2 \log (1-a x)}{a^3}-\frac{x^2}{a}-\frac{x^3}{3} \]
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Rubi [A] time = 0.0339557, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6126, 77} \[ -\frac{2 x}{a^2}-\frac{2 \log (1-a x)}{a^3}-\frac{x^2}{a}-\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 77
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x^2 \, dx &=\int \frac{x^2 (1+a x)}{1-a x} \, dx\\ &=\int \left (-\frac{2}{a^2}-\frac{2 x}{a}-x^2-\frac{2}{a^2 (-1+a x)}\right ) \, dx\\ &=-\frac{2 x}{a^2}-\frac{x^2}{a}-\frac{x^3}{3}-\frac{2 \log (1-a x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0134466, size = 34, normalized size = 1. \[ -\frac{2 x}{a^2}-\frac{2 \log (1-a x)}{a^3}-\frac{x^2}{a}-\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 32, normalized size = 0.9 \begin{align*} -{\frac{{x}^{3}}{3}}-{\frac{{x}^{2}}{a}}-2\,{\frac{x}{{a}^{2}}}-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 = 0.960327, size = 46, normalized size = 1.35 \begin{align*} -\frac{a^{2} x^{3} + 3 \, a x^{2} + 6 \, x}{3 \, a^{2}} - \frac{2 \, \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 = 1.68955, size = 77, normalized size = 2.26 \begin{align*} -\frac{a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x + 6 \, \log \left (a x - 1\right )}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.38483, size = 29, normalized size = 0.85 \begin{align*} - \frac{x^{3}}{3} - \frac{x^{2}}{a} - \frac{2 x}{a^{2}} - \frac{2 \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 = 1.17959, size = 51, normalized size = 1.5 \begin{align*} -\frac{a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x}{3 \, a^{3}} - \frac{2 \, \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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