Optimal. Leaf size=20 \[ a \log (x)-a \log (1-a x)-\frac{1}{x} \]
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Rubi [A] time = 0.0874084, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6150, 44} \[ a \log (x)-a \log (1-a x)-\frac{1}{x} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 44
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^2 \sqrt{1-a^2 x^2}} \, dx &=\int \frac{1}{x^2 (1-a x)} \, dx\\ &=\int \left (\frac{1}{x^2}+\frac{a}{x}-\frac{a^2}{-1+a x}\right ) \, dx\\ &=-\frac{1}{x}+a \log (x)-a \log (1-a x)\\ \end{align*}
Mathematica [A] time = 0.012582, size = 20, normalized size = 1. \[ a \log (x)-a \log (1-a x)-\frac{1}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 20, normalized size = 1. \begin{align*} -{x}^{-1}+a\ln \left ( x \right ) -a\ln \left ( ax-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.942688, size = 26, normalized size = 1.3 \begin{align*} -a \log \left (a x - 1\right ) + a \log \left (x\right ) - \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7383, size = 54, normalized size = 2.7 \begin{align*} -\frac{a x \log \left (a x - 1\right ) - a x \log \left (x\right ) + 1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.137859, size = 15, normalized size = 0.75 \begin{align*} - a \left (- \log{\left (x \right )} + \log{\left (x - \frac{1}{a} \right )}\right ) - \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15307, size = 28, normalized size = 1.4 \begin{align*} -a \log \left ({\left | a x - 1 \right |}\right ) + a \log \left ({\left | x \right |}\right ) - \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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