Optimal. Leaf size=19 \[ -\frac{\log (1-a x)}{a^2}-\frac{x}{a} \]
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Rubi [A] time = 0.0622626, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6150, 43} \[ -\frac{\log (1-a x)}{a^2}-\frac{x}{a} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x}{\sqrt{1-a^2 x^2}} \, dx &=\int \frac{x}{1-a x} \, dx\\ &=\int \left (-\frac{1}{a}-\frac{1}{a (-1+a x)}\right ) \, dx\\ &=-\frac{x}{a}-\frac{\log (1-a x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0129541, size = 19, normalized size = 1. \[ -\frac{\log (1-a x)}{a^2}-\frac{x}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 19, normalized size = 1. \begin{align*} -{\frac{x}{a}}-{\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 = 0.944947, size = 24, normalized size = 1.26 \begin{align*} -\frac{x}{a} - \frac{\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 = 1.71153, size = 36, normalized size = 1.89 \begin{align*} -\frac{a x + \log \left (a x - 1\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.247795, size = 14, normalized size = 0.74 \begin{align*} - \frac{x}{a} - \frac{\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 = 1.17948, size = 26, normalized size = 1.37 \begin{align*} -\frac{x}{a} - \frac{\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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