Optimal. Leaf size=14 \[ 2 \log (1-a x)-\log (x) \]
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Rubi [A] time = 0.0396086, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6167, 6126, 72} \[ 2 \log (1-a x)-\log (x) \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6126
Rule 72
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{x} \, dx &=-\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 = 0.008856, size = 14, normalized size = 1. \[ 2 \log (1-a x)-\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 14, 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 = 1.00709, size = 18, normalized size = 1.29 \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 = 1.79277, size = 34, normalized size = 2.43 \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 = 0.179207, size = 10, normalized size = 0.71 \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 = 1.16421, size = 20, normalized size = 1.43 \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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