Optimal. Leaf size=16 \[ -\frac{2 \log (1-a x)}{a}-x \]
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Rubi [A] time = 0.0120185, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6125, 43} \[ -\frac{2 \log (1-a x)}{a}-x \]
Antiderivative was successfully verified.
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Rule 6125
Rule 43
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} \, dx &=\int \frac{1+a x}{1-a x} \, dx\\ &=\int \left (-1-\frac{2}{-1+a x}\right ) \, dx\\ &=-x-\frac{2 \log (1-a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0094114, size = 16, normalized size = 1. \[ -\frac{2 \log (1-a x)}{a}-x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 16, normalized size = 1. \begin{align*} -x-2\,{\frac{\ln \left ( ax-1 \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952724, size = 20, normalized size = 1.25 \begin{align*} -x - \frac{2 \, \log \left (a x - 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74374, size = 36, normalized size = 2.25 \begin{align*} -\frac{a x + 2 \, \log \left (a x - 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.187183, size = 12, normalized size = 0.75 \begin{align*} - x - \frac{2 \log{\left (a x - 1 \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17178, size = 22, normalized size = 1.38 \begin{align*} -x - \frac{2 \, \log \left ({\left | a x - 1 \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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