Optimal. Leaf size=20 \[ \text {Li}_2(1-e x)+\log (e x) \log (e x-1) \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2394, 2315} \[ \text {PolyLog}(2,1-e x)+\log (e x) \log (e x-1) \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2394
Rubi steps
\begin {align*} \int \frac {\log (-1+e x)}{x} \, dx &=\log (e x) \log (-1+e x)-e \int \frac {\log (e x)}{-1+e x} \, dx\\ &=\log (e x) \log (-1+e x)+\text {Li}_2(1-e x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 20, normalized size = 1.00 \[ \text {Li}_2(1-e x)+\log (e x) \log (e x-1) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (e x - 1\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (e x - 1\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 17, normalized size = 0.85 \[ \ln \left (e x \right ) \ln \left (e x -1\right )+\dilog \left (e x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 19, normalized size = 0.95 \[ \log \left (e x - 1\right ) \log \left (e x\right ) + {\rm Li}_2\left (-e x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 16, normalized size = 0.80 \[ {\mathrm {Li}}_{\mathrm {2}}\left (e\,x\right )+\ln \left (e\,x-1\right )\,\ln \left (e\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.36, size = 48, normalized size = 2.40 \[ \begin {cases} i \pi \log {\relax (x )} - \operatorname {Li}_{2}\left (e x\right ) & \text {for}\: \left |{x}\right | < 1 \\- i \pi \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (e x\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- i \pi {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} + i \pi {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} - \operatorname {Li}_{2}\left (e x\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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