Optimal. Leaf size=25 \[ \text{PolyLog}\left (2,1-\frac{e x}{2}\right )+\log \left (\frac{e x}{2}\right ) \log (e x-2) \]
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Rubi [A] time = 0.0197499, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2394, 2315} \[ \text{PolyLog}\left (2,1-\frac{e x}{2}\right )+\log \left (\frac{e x}{2}\right ) \log (e x-2) \]
Antiderivative was successfully verified.
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Rule 2394
Rule 2315
Rubi steps
\begin{align*} \int \frac{\log (-2+e x)}{x} \, dx &=\log \left (\frac{e x}{2}\right ) \log (-2+e x)-e \int \frac{\log \left (\frac{e x}{2}\right )}{-2+e x} \, dx\\ &=\log \left (\frac{e x}{2}\right ) \log (-2+e x)+\text{Li}_2\left (1-\frac{e x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0019247, size = 27, normalized size = 1.08 \[ \text{PolyLog}\left (2,\frac{1}{2} (2-e x)\right )+\log \left (\frac{e x}{2}\right ) \log (e x-2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 19, normalized size = 0.8 \begin{align*}{\it dilog} \left ({\frac{ex}{2}} \right ) +\ln \left ({\frac{ex}{2}} \right ) \ln \left ( ex-2 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2387, size = 27, normalized size = 1.08 \begin{align*} \log \left (e x - 2\right ) \log \left (\frac{1}{2} \, e x\right ) +{\rm Li}_2\left (-\frac{1}{2} \, e x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (e x - 2\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.99814, size = 88, normalized size = 3.52 \begin{align*} \begin{cases} \log{\left (2 \right )} \log{\left (x \right )} + 3 i \pi \log{\left (x \right )} - \operatorname{Li}_{2}\left (\frac{e x}{2}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (2 \right )} \log{\left (\frac{1}{x} \right )} - 3 i \pi \log{\left (\frac{1}{x} \right )} - \operatorname{Li}_{2}\left (\frac{e x}{2}\right ) & \text{for}\: \frac{1}{\left |{x}\right |} < 1 \\-{G_{2, 2}^{2, 0}\left (\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle |{x} \right )} \log{\left (2 \right )} - 3 i \pi{G_{2, 2}^{2, 0}\left (\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle |{x} \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} \log{\left (2 \right )} + 3 i \pi{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} - \operatorname{Li}_{2}\left (\frac{e x}{2}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (e x - 2\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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