Optimal. Leaf size=38 \[ -\text{PolyLog}\left (2,-\frac{b e^x}{a}\right )+x \log \left (a+b e^x\right )-x \log \left (\frac{b e^x}{a}+1\right ) \]
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Rubi [A] time = 0.0482474, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2280, 2190, 2279, 2391} \[ -\text{PolyLog}\left (2,-\frac{b e^x}{a}\right )+x \log \left (a+b e^x\right )-x \log \left (\frac{b e^x}{a}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2280
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \log \left (a+b e^x\right ) \, dx &=x \log \left (a+b e^x\right )-b \int \frac{e^x x}{a+b e^x} \, dx\\ &=x \log \left (a+b e^x\right )-x \log \left (1+\frac{b e^x}{a}\right )+\int \log \left (1+\frac{b e^x}{a}\right ) \, dx\\ &=x \log \left (a+b e^x\right )-x \log \left (1+\frac{b e^x}{a}\right )+\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a}\right )}{x} \, dx,x,e^x\right )\\ &=x \log \left (a+b e^x\right )-x \log \left (1+\frac{b e^x}{a}\right )-\text{Li}_2\left (-\frac{b e^x}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.0022612, size = 38, normalized size = 1. \[ -\text{PolyLog}\left (2,-\frac{b e^x}{a}\right )+x \log \left (a+b e^x\right )-x \log \left (\frac{b e^x}{a}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 28, normalized size = 0.7 \begin{align*}{\it dilog} \left ( -{\frac{b{{\rm e}^{x}}}{a}} \right ) +\ln \left ( a+b{{\rm e}^{x}} \right ) \ln \left ( -{\frac{b{{\rm e}^{x}}}{a}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08078, size = 46, normalized size = 1.21 \begin{align*} \log \left (b e^{x} + a\right ) \log \left (-\frac{b e^{x} + a}{a} + 1\right ) +{\rm Li}_2\left (\frac{b e^{x} + a}{a}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09071, size = 93, normalized size = 2.45 \begin{align*} x \log \left (b e^{x} + a\right ) - x \log \left (\frac{b e^{x} + a}{a}\right ) -{\rm Li}_2\left (-\frac{b e^{x} + a}{a} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - b \int \frac{x e^{x}}{a + b e^{x}}\, dx + x \log{\left (a + b e^{x} \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 \log \left (b e^{x} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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