Optimal. Leaf size=25 \[ \frac{2 \log (a x+1)}{a^2}-\frac{2 x}{a}+\frac{x^2}{2} \]
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Rubi [A] time = 0.0323883, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6167, 6126, 77} \[ \frac{2 \log (a x+1)}{a^2}-\frac{2 x}{a}+\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6126
Rule 77
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} x \, dx &=-\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 = 0.012387, size = 25, normalized size = 1. \[ \frac{2 \log (a x+1)}{a^2}-\frac{2 x}{a}+\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 24, normalized size = 1. \begin{align*} -2\,{\frac{x}{a}}+{\frac{{x}^{2}}{2}}+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 = 1.05236, size = 35, normalized size = 1.4 \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 = 1.84129, size = 59, normalized size = 2.36 \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 = 0.260727, size = 20, normalized size = 0.8 \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 = 1.11769, size = 41, normalized size = 1.64 \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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