Optimal. Leaf size=12 \[ \frac {\log \left (a+b e^x\right )}{b} \]
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Rubi [A]
time = 0.01, 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 = {2278, 31}
\begin {gather*} \frac {\log \left (a+b e^x\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2278
Rubi steps
\begin {align*} \int \frac {e^x}{a+b e^x} \, dx &=\text {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 \begin {gather*} \frac {\log \left (a+b e^x\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 12, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {\ln \left (a +b \,{\mathrm e}^{x}\right )}{b}\) | \(12\) |
default | \(\frac {\ln \left (a +b \,{\mathrm e}^{x}\right )}{b}\) | \(12\) |
norman | \(\frac {\ln \left (a +b \,{\mathrm e}^{x}\right )}{b}\) | \(12\) |
risch | \(\frac {\ln \left ({\mathrm e}^{x}+\frac {a}{b}\right )}{b}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 11, normalized size = 0.92 \begin {gather*} \frac {\log \left (b e^{x} + a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 11, normalized size = 0.92 \begin {gather*} \frac {\log \left (b e^{x} + a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 8, normalized size = 0.67 \begin {gather*} \frac {\log {\left (\frac {a}{b} + e^{x} \right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.07, size = 12, normalized size = 1.00 \begin {gather*} \frac {\log \left ({\left | b e^{x} + a \right |}\right )}{b} \end {gather*}
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 \begin {gather*} \frac {\ln \left (a+b\,{\mathrm {e}}^x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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