Optimal. Leaf size=21 \[ \log (x) (a+b \log (3))-b \text{PolyLog}\left (2,-\frac{e x}{3}\right ) \]
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Rubi [A] time = 0.0208121, antiderivative size = 21, 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} \[ \log (x) (a+b \log (3))-b \text{PolyLog}\left (2,-\frac{e x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 2392
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log (3+e x)}{x} \, dx &=(a+b \log (3)) \log (x)+b \int \frac{\log \left (1+\frac{e x}{3}\right )}{x} \, dx\\ &=(a+b \log (3)) \log (x)-b \text{Li}_2\left (-\frac{e x}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0019859, size = 22, normalized size = 1.05 \[ -b \text{PolyLog}\left (2,-\frac{e x}{3}\right )+a \log (x)+b \log (3) \log (x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.06, size = 46, normalized size = 2.2 \begin{align*} a\ln \left ( ex \right ) +\ln \left ( ex+3 \right ) \ln \left ( -{\frac{ex}{3}} \right ) b-\ln \left ({\frac{ex}{3}}+1 \right ) \ln \left ( -{\frac{ex}{3}} \right ) b-{\it dilog} \left ({\frac{ex}{3}}+1 \right ) b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52034, size = 36, normalized size = 1.71 \begin{align*}{\left (\log \left (e x + 3\right ) \log \left (-\frac{1}{3} \, e x\right ) +{\rm Li}_2\left (\frac{1}{3} \, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (e x + 3\right ) + a}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.91196, size = 75, normalized size = 3.57 \begin{align*} a \log{\left (x \right )} + b \left (\begin{cases} \log{\left (3 \right )} \log{\left (x \right )} - \operatorname{Li}_{2}\left (\frac{e x e^{i \pi }}{3}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (3 \right )} \log{\left (\frac{1}{x} \right )} - \operatorname{Li}_{2}\left (\frac{e x e^{i \pi }}{3}\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 (3 \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} \log{\left (3 \right )} - \operatorname{Li}_{2}\left (\frac{e x e^{i \pi }}{3}\right ) & \text{otherwise} \end{cases}\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 + 3\right ) + a}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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