Optimal. Leaf size=25 \[ \frac{\left (a+b e^x\right ) \log \left (a+b e^x\right )}{b}-e^x \]
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Rubi [A] time = 0.0525146, antiderivative size = 31, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {2194, 2554, 12, 2248, 43} \[ e^x \log \left (a+b e^x\right )+\frac{a \log \left (a+b e^x\right )}{b}-e^x \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2554
Rule 12
Rule 2248
Rule 43
Rubi steps
\begin{align*} \int e^x \log \left (a+b e^x\right ) \, dx &=e^x \log \left (a+b e^x\right )-\int \frac{b e^{2 x}}{a+b e^x} \, dx\\ &=e^x \log \left (a+b e^x\right )-b \int \frac{e^{2 x}}{a+b e^x} \, dx\\ &=e^x \log \left (a+b e^x\right )-b \operatorname{Subst}\left (\int \frac{x}{a+b x} \, dx,x,e^x\right )\\ &=e^x \log \left (a+b e^x\right )-b \operatorname{Subst}\left (\int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx,x,e^x\right )\\ &=-e^x+\frac{a \log \left (a+b e^x\right )}{b}+e^x \log \left (a+b e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0160492, size = 25, normalized size = 1. \[ \frac{\left (a+b e^x\right ) \log \left (a+b e^x\right )}{b}-e^x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 34, normalized size = 1.4 \begin{align*}{{\rm e}^{x}}\ln \left ( a+b{{\rm e}^{x}} \right ) -{{\rm e}^{x}}+{\frac{\ln \left ( a+b{{\rm e}^{x}} \right ) a}{b}}-{\frac{a}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05244, size = 35, normalized size = 1.4 \begin{align*} -\frac{b e^{x} -{\left (b e^{x} + a\right )} \log \left (b e^{x} + a\right ) + a}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05948, size = 55, normalized size = 2.2 \begin{align*} -\frac{b e^{x} -{\left (b e^{x} + a\right )} \log \left (b e^{x} + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19577, size = 35, normalized size = 1.4 \begin{align*} -\frac{b e^{x} -{\left (b e^{x} + a\right )} \log \left (b e^{x} + a\right ) + a}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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