Optimal. Leaf size=12 \[ \log (x)-2 \log (1-a x) \]
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Rubi [A] time = 0.0253327, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6126, 72} \[ \log (x)-2 \log (1-a x) \]
Antiderivative was successfully verified.
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Rule 6126
Rule 72
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)}}{x} \, dx &=\int \frac{1+a x}{x (1-a x)} \, dx\\ &=\int \left (\frac{1}{x}-\frac{2 a}{-1+a x}\right ) \, dx\\ &=\log (x)-2 \log (1-a x)\\ \end{align*}
Mathematica [A] time = 0.0065252, size = 12, normalized size = 1. \[ \log (x)-2 \log (1-a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 12, normalized size = 1. \begin{align*} \ln \left ( x \right ) -2\,\ln \left ( ax-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.951546, size = 15, normalized size = 1.25 \begin{align*} -2 \, \log \left (a x - 1\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76841, size = 35, normalized size = 2.92 \begin{align*} -2 \, \log \left (a x - 1\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.2531, size = 10, normalized size = 0.83 \begin{align*} \log{\left (x \right )} - 2 \log{\left (x - \frac{1}{a} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20251, size = 18, normalized size = 1.5 \begin{align*} -2 \, \log \left ({\left | a x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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