Optimal. Leaf size=16 \[ \log (2) \log (x)-\text {Li}_2\left (-\frac {e x}{2}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 16, 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 = {2392, 2391} \[ \log (2) \log (x)-\text {PolyLog}\left (2,-\frac {e x}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2392
Rubi steps
\begin {align*} \int \frac {\log (2+e x)}{x} \, dx &=\log (2) \log (x)+\int \frac {\log \left (1+\frac {e x}{2}\right )}{x} \, dx\\ &=\log (2) \log (x)-\text {Li}_2\left (-\frac {e x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \[ \log (2) \log (x)-\text {Li}_2\left (-\frac {e x}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (e x + 2\right )}{x}, x\right ) \]
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 \[ \int \frac {\log \left (e x + 2\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 33, normalized size = 2.06 \[ -\dilog \left (\frac {e x}{2}+1\right )+\left (-\ln \left (\frac {e x}{2}+1\right )+\ln \left (e x +2\right )\right ) \ln \left (-\frac {e x}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 20, normalized size = 1.25 \[ \log \left (e x + 2\right ) \log \left (-\frac {1}{2} \, e x\right ) + {\rm Li}_2\left (\frac {1}{2} \, e x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 18, normalized size = 1.12 \[ {\mathrm {Li}}_{\mathrm {2}}\left (-\frac {e\,x}{2}\right )+\ln \left (e\,x+2\right )\,\ln \left (-\frac {e\,x}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.92, size = 68, normalized size = 4.25 \[ \begin {cases} \log {\relax (2 )} \log {\relax (x )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{2}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (2 )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{2}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (2 )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (2 )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{2}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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