Optimal. Leaf size=11 \[ \log (x)-2 \log (a x+1) \]
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Rubi [A] time = 0.023803, antiderivative size = 11, 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} \[ \log (x)-2 \log (a x+1) \]
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 = 0.0061687, size = 11, normalized size = 1. \[ \log (x)-2 \log (a x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 12, normalized size = 1.1 \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 = 0.957287, size = 15, normalized size = 1.36 \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 = 1.75575, size = 35, normalized size = 3.18 \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 = 0.153514, size = 10, normalized size = 0.91 \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 [B] time = 1.14313, size = 58, normalized size = 5.27 \begin{align*} a{\left (\frac{\log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{\log \left ({\left | -\frac{1}{a x + 1} + 1 \right |}\right )}{a}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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