Optimal. Leaf size=12 \[ -\frac{\log (1-a x)}{a} \]
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Rubi [A] time = 0.0341307, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {6140, 31} \[ -\frac{\log (1-a x)}{a} \]
Antiderivative was successfully verified.
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Rule 6140
Rule 31
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx &=\int \frac{1}{1-a x} \, dx\\ &=-\frac{\log (1-a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0047174, size = 12, normalized size = 1. \[ -\frac{\log (1-a x)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 12, normalized size = 1. \begin{align*} -{\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 = 0.962576, size = 15, normalized size = 1.25 \begin{align*} -\frac{\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 = 1.68061, size = 23, normalized size = 1.92 \begin{align*} -\frac{\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 = 0.062884, size = 8, normalized size = 0.67 \begin{align*} - \frac{\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 = 1.15186, size = 16, normalized size = 1.33 \begin{align*} -\frac{\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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