Optimal. Leaf size=9 \[ \log \left (1+\frac {\log (x)}{x}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6844, 31}
\begin {gather*} \log \left (\frac {\log (x)}{x}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 6844
Rubi steps
\begin {align*} \int \frac {1-\log (x)}{x (x+\log (x))} \, dx &=\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\frac {\log (x)}{x}\right )\\ &=\log \left (1+\frac {\log (x)}{x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 10, normalized size = 1.11 \begin {gather*} -\log (x)+\log (x+\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 11, normalized size = 1.22
method | result | size |
default | \(-\ln \left (x \right )+\ln \left (x +\ln \left (x \right )\right )\) | \(11\) |
norman | \(-\ln \left (x \right )+\ln \left (x +\ln \left (x \right )\right )\) | \(11\) |
risch | \(-\ln \left (x \right )+\ln \left (x +\ln \left (x \right )\right )\) | \(11\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 10, normalized size = 1.11 \begin {gather*} \log \left (x + \log \left (x\right )\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 10, normalized size = 1.11 \begin {gather*} \log \left (x + \log \left (x\right )\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 8, normalized size = 0.89 \begin {gather*} - \log {\left (x \right )} + \log {\left (x + \log {\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.59, size = 14, normalized size = 1.56 \begin {gather*} -\log \left (x\right ) + \log \left (-x - \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 10, normalized size = 1.11 \begin {gather*} \ln \left (x+\ln \left (x\right )\right )-\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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