Optimal. Leaf size=20 \[ \log (e x) \log (-1+e x)+\text {Li}_2(1-e x) \]
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Rubi [A]
time = 0.01, 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 = {2441, 2352}
\begin {gather*} \text {PolyLog}(2,1-e x)+\log (e x) \log (e x-1) \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2441
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 \begin {gather*} \log (e x) \log (-1+e x)+\text {Li}_2(1-e x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 17, normalized size = 0.85
method | result | size |
derivativedivides | \(\dilog \left (e x \right )+\ln \left (e x \right ) \ln \left (e x -1\right )\) | \(17\) |
default | \(\dilog \left (e x \right )+\ln \left (e x \right ) \ln \left (e x -1\right )\) | \(17\) |
risch | \(\dilog \left (e x \right )+\ln \left (e x \right ) \ln \left (e x -1\right )\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 22, normalized size = 1.10 \begin {gather*} \log \left (x e - 1\right ) \log \left (x e\right ) + {\rm Li}_2\left (-x e + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.83, size = 60, normalized size = 3.00 \begin {gather*} \begin {cases} - \operatorname {Li}_{2}\left (e x\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\i \pi \log {\left (x \right )} - \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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} {\mathrm {Li}}_{\mathrm {2}}\left (e\,x\right )+\ln \left (e\,x-1\right )\,\ln \left (e\,x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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