Optimal. Leaf size=14 \[ a \log (x)-b \text{PolyLog}(2,-e x) \]
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Rubi [A] time = 0.0183462, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2392, 2391} \[ a \log (x)-b \text{PolyLog}(2,-e x) \]
Antiderivative was successfully verified.
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Rule 2392
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log (1+e x)}{x} \, dx &=a \log (x)+b \int \frac{\log (1+e x)}{x} \, dx\\ &=a \log (x)-b \text{Li}_2(-e x)\\ \end{align*}
Mathematica [A] time = 0.0015947, size = 14, normalized size = 1. \[ a \log (x)-b \text{PolyLog}(2,-e x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 17, normalized size = 1.2 \begin{align*} a\ln \left ( ex \right ) -b{\it dilog} \left ( ex+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3834, size = 35, normalized size = 2.5 \begin{align*}{\left (\log \left (e x + 1\right ) \log \left (-e x\right ) +{\rm Li}_2\left (e x + 1\right )\right )} b + a \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06495, size = 36, normalized size = 2.57 \begin{align*} -b{\rm Li}_2\left (-e x\right ) + a \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.45331, size = 15, normalized size = 1.07 \begin{align*} a \log{\left (x \right )} - b \operatorname{Li}_{2}\left (e x e^{i \pi }\right ) \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{b \log \left (e x + 1\right ) + a}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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