Optimal. Leaf size=24 \[ \frac {e^{-c} \log \left (a+b e^{c+d x}\right )}{b d} \]
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Rubi [A]
time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2279, 2278, 31}
\begin {gather*} \frac {e^{-c} \log \left (a+b e^{c+d x}\right )}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2278
Rule 2279
Rubi steps
\begin {align*} \int \frac {e^{d x}}{a+b e^{c+d x}} \, dx &=e^{-c} \int \frac {e^{c+d x}}{a+b e^{c+d x}} \, dx\\ &=\frac {e^{-c} \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac {e^{-c} \log \left (a+b e^{c+d x}\right )}{b d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 1.00 \begin {gather*} \frac {e^{-c} \log \left (a+b e^{c+d x}\right )}{b d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 23, normalized size = 0.96
method | result | size |
default | \(\frac {\ln \left (a +b \,{\mathrm e}^{d x} {\mathrm e}^{c}\right ) {\mathrm e}^{-c}}{d b}\) | \(23\) |
norman | \(\frac {\ln \left (a +b \,{\mathrm e}^{d x} {\mathrm e}^{c}\right ) {\mathrm e}^{-c}}{d b}\) | \(23\) |
risch | \(\frac {{\mathrm e}^{-c} \ln \left ({\mathrm e}^{d x}+\frac {{\mathrm e}^{-c} a}{b}\right )}{b d}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 22, normalized size = 0.92 \begin {gather*} \frac {e^{\left (-c\right )} \log \left (b e^{\left (d x + c\right )} + a\right )}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 22, normalized size = 0.92 \begin {gather*} \frac {e^{\left (-c\right )} \log \left (b e^{\left (d x + c\right )} + a\right )}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 19, normalized size = 0.79 \begin {gather*} \frac {e^{- c} \log {\left (\frac {a e^{- c}}{b} + e^{d x} \right )}}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.55, size = 23, normalized size = 0.96 \begin {gather*} \frac {e^{\left (-c\right )} \log \left ({\left | b e^{\left (d x + c\right )} + a \right |}\right )}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 22, normalized size = 0.92 \begin {gather*} \frac {\ln \left (a+b\,{\mathrm {e}}^{c+d\,x}\right )\,{\mathrm {e}}^{-c}}{b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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