Optimal. Leaf size=31 \[ \frac{\log \left (\log \left (e^x\right )\right )}{x-\log \left (e^x\right )}-\frac{\log (x)}{x-\log \left (e^x\right )} \]
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Rubi [A] time = 0.0157068, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2160, 2157, 29} \[ \frac{\log \left (\log \left (e^x\right )\right )}{x-\log \left (e^x\right )}-\frac{\log (x)}{x-\log \left (e^x\right )} \]
Antiderivative was successfully verified.
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Rule 2160
Rule 2157
Rule 29
Rubi steps
\begin{align*} \int \frac{1}{x \log \left (e^x\right )} \, dx &=-\frac{\int \frac{1}{x} \, dx}{x-\log \left (e^x\right )}+\frac{\int \frac{1}{\log \left (e^x\right )} \, dx}{x-\log \left (e^x\right )}\\ &=-\frac{\log (x)}{x-\log \left (e^x\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\log \left (e^x\right )\right )}{x-\log \left (e^x\right )}\\ &=-\frac{\log (x)}{x-\log \left (e^x\right )}+\frac{\log \left (\log \left (e^x\right )\right )}{x-\log \left (e^x\right )}\\ \end{align*}
Mathematica [A] time = 0.0083698, size = 21, normalized size = 0.68 \[ \frac{\log \left (\log \left (e^x\right )\right )-\log (x)}{x-\log \left (e^x\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 29, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( \ln \left ({{\rm e}^{x}} \right ) \right ) }{\ln \left ({{\rm e}^{x}} \right ) -x}}+{\frac{\ln \left ( x \right ) }{\ln \left ({{\rm e}^{x}} \right ) -x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00705, size = 7, normalized size = 0.23 \begin{align*} -\frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88555, size = 8, normalized size = 0.26 \begin{align*} -\frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.071003, size = 3, normalized size = 0.1 \begin{align*} - \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27182, size = 7, normalized size = 0.23 \begin{align*} -\frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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