Optimal. Leaf size=9 \[ \log \left (\frac{\log (x)}{x}+1\right ) \]
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Rubi [A] time = 0.0718203, antiderivative size = 9, normalized size of antiderivative = 1., 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 = {6712, 31} \[ \log \left (\frac{\log (x)}{x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 6712
Rule 31
Rubi steps
\begin{align*} \int \frac{1-\log (x)}{x (x+\log (x))} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\frac{\log (x)}{x}\right )\\ &=\log \left (1+\frac{\log (x)}{x}\right )\\ \end{align*}
Mathematica [A] time = 0.051494, size = 10, normalized size = 1.11 \[ \log (x+\log (x))-\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 11, normalized size = 1.2 \begin{align*} -\ln \left ( x \right ) +\ln \left ( x+\ln \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19336, size = 14, normalized size = 1.56 \begin{align*} \log \left (x + \log \left (x\right )\right ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15737, size = 35, normalized size = 3.89 \begin{align*} \log \left (x + \log \left (x\right )\right ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.114662, size = 8, normalized size = 0.89 \begin{align*} - \log{\left (x \right )} + \log{\left (x + \log{\left (x \right )} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3485, size = 19, normalized size = 2.11 \begin{align*} -\log \left (x\right ) + \log \left (-x - \log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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