Integrand size = 10, antiderivative size = 25 \[ \int \frac {\log (-2+e x)}{x} \, dx=\log \left (\frac {e x}{2}\right ) \log (-2+e x)+\operatorname {PolyLog}\left (2,1-\frac {e x}{2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2441, 2352} \[ \int \frac {\log (-2+e x)}{x} \, dx=\operatorname {PolyLog}\left (2,1-\frac {e x}{2}\right )+\log \left (\frac {e x}{2}\right ) \log (e x-2) \]
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Rule 2352
Rule 2441
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\log (-2+e x)}{x} \, dx=\log \left (\frac {e x}{2}\right ) \log (-2+e x)+\operatorname {PolyLog}\left (2,\frac {1}{2} (2-e x)\right ) \]
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Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\operatorname {dilog}\left (\frac {e x}{2}\right )+\ln \left (\frac {e x}{2}\right ) \ln \left (e x -2\right )\) | \(19\) |
default | \(\operatorname {dilog}\left (\frac {e x}{2}\right )+\ln \left (\frac {e x}{2}\right ) \ln \left (e x -2\right )\) | \(19\) |
risch | \(\operatorname {dilog}\left (\frac {e x}{2}\right )+\ln \left (\frac {e x}{2}\right ) \ln \left (e x -2\right )\) | \(19\) |
parts | \(\ln \left (e x -2\right ) \ln \left (x \right )-e \left (\frac {\left (\ln \left (x \right )-\ln \left (\frac {e x}{2}\right )\right ) \ln \left (1-\frac {e x}{2}\right )}{e}-\frac {\operatorname {dilog}\left (\frac {e x}{2}\right )}{e}\right )\) | \(46\) |
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\[ \int \frac {\log (-2+e x)}{x} \, dx=\int { \frac {\log \left (e x - 2\right )}{x} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.97 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.08 \[ \int \frac {\log (-2+e x)}{x} \, dx=\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x}{2}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (2 \right )} \log {\left (x \right )} + 3 i \pi \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x}{2}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (2 \right )} \log {\left (\frac {1}{x} \right )} - 3 i \pi \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x}{2}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (2 \right )} - 3 i \pi {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (2 \right )} + 3 i \pi {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} - \operatorname {Li}_{2}\left (\frac {e x}{2}\right ) & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {\log (-2+e x)}{x} \, dx=\log \left (e x - 2\right ) \log \left (\frac {1}{2} \, e x\right ) + {\rm Li}_2\left (-\frac {1}{2} \, e x + 1\right ) \]
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\[ \int \frac {\log (-2+e x)}{x} \, dx=\int { \frac {\log \left (e x - 2\right )}{x} \,d x } \]
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Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {\log (-2+e x)}{x} \, dx={\mathrm {Li}}_{\mathrm {2}}\left (\frac {e\,x}{2}\right )+\ln \left (e\,x-2\right )\,\ln \left (\frac {e\,x}{2}\right ) \]
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