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 = 0.07, 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 = {2247, 2246, 31} \[ \frac {e^{-c} \log \left (a+b e^{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2246
Rule 2247
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} \operatorname {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 = 0.01, size = 24, normalized size = 1.00 \[ \frac {e^{-c} \log \left (a+b e^{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 22, normalized size = 0.92 \[ \frac {e^{\left (-c\right )} \log \left (b e^{\left (d x + c\right )} + a\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 23, normalized size = 0.96 \[ \frac {e^{\left (-c\right )} \log \left ({\left | b e^{\left (d x + c\right )} + a \right |}\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 23, normalized size = 0.96 \[ \frac {{\mathrm e}^{-c} \ln \left (b \,{\mathrm e}^{c} {\mathrm e}^{d x}+a \right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 22, normalized size = 0.92 \[ \frac {e^{\left (-c\right )} \log \left (b e^{\left (d x + c\right )} + a\right )}{b d} \]
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 \[ \frac {\ln \left (a+b\,{\mathrm {e}}^{c+d\,x}\right )\,{\mathrm {e}}^{-c}}{b\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 19, normalized size = 0.79 \[ \frac {e^{- c} \log {\left (\frac {a e^{- c}}{b} + e^{d x} \right )}}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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