Optimal. Leaf size=12 \[ \frac{\log \left (a+b e^x\right )}{b} \]
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Rubi [A] time = 0.0197353, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2246, 31} \[ \frac{\log \left (a+b e^x\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2246
Rule 31
Rubi steps
\begin{align*} \int \frac{e^x}{a+b e^x} \, dx &=\operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,e^x\right )\\ &=\frac{\log \left (a+b e^x\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0049704, size = 12, normalized size = 1. \[ \frac{\log \left (a+b e^x\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 12, normalized size = 1. \begin{align*}{\frac{\ln \left ( a+b{{\rm e}^{x}} \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05088, size = 15, normalized size = 1.25 \begin{align*} \frac{\log \left (b e^{x} + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43886, size = 24, normalized size = 2. \begin{align*} \frac{\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 [A] time = 0.124032, size = 8, normalized size = 0.67 \begin{align*} \frac{\log{\left (\frac{a}{b} + e^{x} \right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20525, size = 16, normalized size = 1.33 \begin{align*} \frac{\log \left ({\left | b e^{x} + a \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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