Optimal. Leaf size=13 \[ 2 \log (a x+1)-\log (x) \]
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Rubi [A] time = 0.04, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6167, 6126, 72} \[ 2 \log (a x+1)-\log (x) \]
Antiderivative was successfully verified.
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Rule 72
Rule 6126
Rule 6167
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*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ 2 \log (a x+1)-\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 13, 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 [A] time = 0.12, size = 15, normalized size = 1.15 \[ 2 \, \log \left ({\left | a x + 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 14, normalized size = 1.08 \[ -\ln \relax (x )+2 \ln \left (a x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 13, 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 = 0.04, size = 14, normalized size = 1.08 \[ 2\,\ln \left (-3\,a\,x-3\right )-\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 10, normalized size = 0.77 \[ - \log {\relax (x )} + 2 \log {\left (x + \frac {1}{a} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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