Optimal. Leaf size=12 \[ \frac {\log \left (a+b e^x\right )}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 31
Rule 2246
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*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ \frac {\log \left (a+b e^x\right )}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 11, normalized size = 0.92 \[ \frac {\log \left (b e^{x} + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 12, normalized size = 1.00 \[ \frac {\log \left ({\left | b e^{x} + a \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 12, normalized size = 1.00 \[ \frac {\ln \left (b \,{\mathrm e}^{x}+a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 11, normalized size = 0.92 \[ \frac {\log \left (b e^{x} + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 11, normalized size = 0.92 \[ \frac {\ln \left (a+b\,{\mathrm {e}}^x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 8, normalized size = 0.67 \[ \frac {\log {\left (\frac {a}{b} + e^{x} \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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