Optimal. Leaf size=19 \[ \frac{\log \left (a+b e^{c+d x}\right )}{b d} \]
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Rubi [A] time = 0.0359485, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2246, 31} \[ \frac{\log \left (a+b e^{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 2246
Rule 31
Rubi steps
\begin{align*} \int \frac{e^{c+d x}}{a+b e^{c+d x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac{\log \left (a+b e^{c+d x}\right )}{b d}\\ \end{align*}
Mathematica [A] time = 0.0063149, size = 19, normalized size = 1. \[ \frac{\log \left (a+b e^{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 19, normalized size = 1. \begin{align*}{\frac{\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10066, size = 24, normalized size = 1.26 \begin{align*} \frac{\log \left (b e^{\left (d x + c\right )} + a\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47187, size = 41, normalized size = 2.16 \begin{align*} \frac{\log \left (b e^{\left (d x + c\right )} + a\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.218728, size = 14, normalized size = 0.74 \begin{align*} \frac{\log{\left (\frac{a}{b} + e^{c + d x} \right )}}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28083, size = 26, normalized size = 1.37 \begin{align*} \frac{\log \left ({\left | b e^{\left (d x + c\right )} + a \right |}\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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