Optimal. Leaf size=26 \[ \frac {x}{a}+\frac {\log \left (a+b e^{c-d x}\right )}{a d} \]
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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2320, 36, 29,
31} \begin {gather*} \frac {\log \left (a+b e^{c-d x}\right )}{a d}+\frac {x}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2320
Rubi steps
\begin {align*} \int \frac {1}{a+b e^{c-d x}} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,e^{c-d x}\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{c-d x}\right )}{a d}+\frac {b \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,e^{c-d x}\right )}{a d}\\ &=\frac {x}{a}+\frac {\log \left (a+b e^{c-d x}\right )}{a d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 23, normalized size = 0.88 \begin {gather*} \frac {\log \left (d \left (b e^c+a e^{d x}\right )\right )}{a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 36, normalized size = 1.38
method | result | size |
norman | \(\frac {x}{a}+\frac {\ln \left (a +b \,{\mathrm e}^{-d x +c}\right )}{a d}\) | \(26\) |
derivativedivides | \(-\frac {\frac {\ln \left ({\mathrm e}^{-d x +c}\right )}{a}-\frac {\ln \left (a +b \,{\mathrm e}^{-d x +c}\right )}{a}}{d}\) | \(36\) |
default | \(-\frac {\frac {\ln \left ({\mathrm e}^{-d x +c}\right )}{a}-\frac {\ln \left (a +b \,{\mathrm e}^{-d x +c}\right )}{a}}{d}\) | \(36\) |
risch | \(\frac {x}{a}-\frac {c}{a d}+\frac {\ln \left ({\mathrm e}^{-d x +c}+\frac {a}{b}\right )}{a d}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 34, normalized size = 1.31 \begin {gather*} \frac {d x - c}{a d} + \frac {\log \left (b e^{\left (-d x + c\right )} + a\right )}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 23, normalized size = 0.88 \begin {gather*} \frac {d x + \log \left (b e^{\left (-d x + c\right )} + a\right )}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 17, normalized size = 0.65 \begin {gather*} \frac {x}{a} + \frac {\log {\left (\frac {a}{b} + e^{c - d x} \right )}}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.05, size = 33, normalized size = 1.27 \begin {gather*} \frac {\frac {d x - c}{a} + \frac {\log \left ({\left | b e^{\left (-d x + c\right )} + a \right |}\right )}{a}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.47, size = 23, normalized size = 0.88 \begin {gather*} \frac {\ln \left (a+b\,{\mathrm {e}}^{-d\,x}\,{\mathrm {e}}^c\right )+d\,x}{a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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