Optimal. Leaf size=19 \[ -\frac {\log (1-a x)}{a^2}-\frac {x}{a} \]
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Rubi [A] time = 0.06, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 43
Rule 6150
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*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.00 \[ -\frac {\log (1-a x)}{a^2}-\frac {x}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 15, normalized size = 0.79 \[ -\frac {a x + \log \left (a x - 1\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 19, normalized size = 1.00 \[ -\frac {x}{a} - \frac {\log \left ({\left | a x - 1 \right |}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 19, normalized size = 1.00 \[ -\frac {x}{a}-\frac {\ln \left (a x -1\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 18, normalized size = 0.95 \[ -\frac {x}{a} - \frac {\log \left (a x - 1\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.86, size = 15, normalized size = 0.79 \[ -\frac {\ln \left (a\,x-1\right )+a\,x}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 14, normalized size = 0.74 \[ - \frac {x}{a} - \frac {\log {\left (a x - 1 \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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