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 = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 77
Rule 6126
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*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.00 \[ -\frac {2 \log (1-a x)}{a^2}-\frac {2 x}{a}-\frac {x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 25, normalized size = 0.96 \[ -\frac {a^{2} x^{2} + 4 \, a x + 4 \, \log \left (a x - 1\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 30, normalized size = 1.15 \[ -\frac {a^{2} x^{2} + 4 \, a x}{2 \, a^{2}} - \frac {2 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 24, normalized size = 0.92 \[ -\frac {x^{2}}{2}-\frac {2 x}{a}-\frac {2 \ln \left (a x -1\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 26, normalized size = 1.00 \[ -\frac {a x^{2} + 4 \, x}{2 \, a} - \frac {2 \, \log \left (a x - 1\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 23, normalized size = 0.88 \[ -\frac {2\,\ln \left (a\,x-1\right )}{a^2}-\frac {2\,x}{a}-\frac {x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 22, normalized size = 0.85 \[ - \frac {x^{2}}{2} - \frac {2 x}{a} - \frac {2 \log {\left (a x - 1 \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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