Optimal. Leaf size=38 \[ x \log \left (a+b e^x\right )-x \log \left (1+\frac {b e^x}{a}\right )-\text {Li}_2\left (-\frac {b e^x}{a}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2318, 2221,
2317, 2438} \begin {gather*} -\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2318
Rule 2438
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 )+\text {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*}
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Mathematica [A]
time = 0.00, size = 38, normalized size = 1.00 \begin {gather*} 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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 28, normalized size = 0.74
method | result | size |
derivativedivides | \(\dilog \left (-\frac {b \,{\mathrm e}^{x}}{a}\right )+\ln \left (a +b \,{\mathrm e}^{x}\right ) \ln \left (-\frac {b \,{\mathrm e}^{x}}{a}\right )\) | \(28\) |
default | \(\dilog \left (-\frac {b \,{\mathrm e}^{x}}{a}\right )+\ln \left (a +b \,{\mathrm e}^{x}\right ) \ln \left (-\frac {b \,{\mathrm e}^{x}}{a}\right )\) | \(28\) |
risch | \(x \ln \left (a +b \,{\mathrm e}^{x}\right )-\ln \left (\frac {a +b \,{\mathrm e}^{x}}{a}\right ) x -\dilog \left (\frac {a +b \,{\mathrm e}^{x}}{a}\right )\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 34, normalized size = 0.89 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 40, normalized size = 1.05 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - b \int \frac {x e^{x}}{a + b e^{x}}\, dx + x \log {\left (a + b e^{x} \right )} \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 = 0.38, size = 35, normalized size = 0.92 \begin {gather*} x\,\ln \left (a+b\,{\mathrm {e}}^x\right )-x\,\ln \left (\frac {b\,{\mathrm {e}}^x}{a}+1\right )-\mathrm {polylog}\left (2,-\frac {b\,{\mathrm {e}}^x}{a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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