Optimal. Leaf size=11 \[ \log (x)-2 \log (a x+1) \]
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Rubi [A] time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 72
Rule 6126
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*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 1.00 \[ \log (x)-2 \log (a x+1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 11, normalized size = 1.00 \[ -2 \, \log \left (a x + 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 43, normalized size = 3.91 \[ a {\left (\frac {\log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {\log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right )}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 12, normalized size = 1.09 \[ \ln \relax (x )-2 \ln \left (a x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 11, normalized size = 1.00 \[ -2 \, \log \left (a x + 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 12, normalized size = 1.09 \[ \ln \relax (x)-2\,\ln \left (3\,a\,x+3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 10, normalized size = 0.91 \[ \log {\relax (x )} - 2 \log {\left (x + \frac {1}{a} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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