Optimal. Leaf size=26 \[ -\frac{2 \log (1-a x)}{a^2}-\frac{2 x}{a}-\frac{x^2}{2} \]
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Rubi [A] time = 0.0287868, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6126, 77} \[ -\frac{2 \log (1-a x)}{a^2}-\frac{2 x}{a}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 77
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x \, dx &=\int \frac{x (1+a x)}{1-a x} \, dx\\ &=\int \left (-\frac{2}{a}-x-\frac{2}{a (-1+a x)}\right ) \, dx\\ &=-\frac{2 x}{a}-\frac{x^2}{2}-\frac{2 \log (1-a x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0107103, size = 26, normalized size = 1. \[ -\frac{2 \log (1-a x)}{a^2}-\frac{2 x}{a}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 24, normalized size = 0.9 \begin{align*} -{\frac{{x}^{2}}{2}}-2\,{\frac{x}{a}}-2\,{\frac{\ln \left ( ax-1 \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963384, size = 35, normalized size = 1.35 \begin{align*} -\frac{a x^{2} + 4 \, x}{2 \, a} - \frac{2 \, \log \left (a x - 1\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61819, size = 61, normalized size = 2.35 \begin{align*} -\frac{a^{2} x^{2} + 4 \, a x + 4 \, \log \left (a x - 1\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.38652, size = 22, normalized size = 0.85 \begin{align*} - \frac{x^{2}}{2} - \frac{2 x}{a} - \frac{2 \log{\left (a x - 1 \right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14283, size = 41, normalized size = 1.58 \begin{align*} -\frac{a^{2} x^{2} + 4 \, a x}{2 \, a^{2}} - \frac{2 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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