Integrand size = 16, antiderivative size = 28 \[ \int \frac {1}{a+b e^{-c-d x}} \, dx=\frac {x}{a}+\frac {\log \left (a+b e^{-c-d x}\right )}{a d} \]
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Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 36, 29, 31} \[ \int \frac {1}{a+b e^{-c-d x}} \, dx=\frac {\log \left (a+b e^{-c-d x}\right )}{a d}+\frac {x}{a} \]
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Rule 29
Rule 31
Rule 36
Rule 2320
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {1}{a+b e^{-c-d x}} \, dx=\frac {\log \left (b+a e^{c+d x}\right )}{a d} \]
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Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
method | result | size |
parallelrisch | \(\frac {d x +\ln \left (a +b \,{\mathrm e}^{-d x -c}\right )}{a d}\) | \(26\) |
norman | \(\frac {x}{a}+\frac {\ln \left (a +b \,{\mathrm e}^{-d x -c}\right )}{a d}\) | \(28\) |
risch | \(\frac {x}{a}+\frac {c}{a d}+\frac {\ln \left ({\mathrm e}^{-d x -c}+\frac {a}{b}\right )}{a d}\) | \(38\) |
derivativedivides | \(-\frac {\ln \left ({\mathrm e}^{-d x -c}\right )}{d a}+\frac {\ln \left (a +b \,{\mathrm e}^{-d x -c}\right )}{a d}\) | \(41\) |
default | \(-\frac {\ln \left ({\mathrm e}^{-d x -c}\right )}{d a}+\frac {\ln \left (a +b \,{\mathrm e}^{-d x -c}\right )}{a d}\) | \(41\) |
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Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {1}{a+b e^{-c-d x}} \, dx=\frac {d x + \log \left (b e^{\left (-d x - c\right )} + a\right )}{a d} \]
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Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {1}{a+b e^{-c-d x}} \, dx=\frac {x}{a} + \frac {\log {\left (\frac {a}{b} + e^{- c - d x} \right )}}{a d} \]
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Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {1}{a+b e^{-c-d x}} \, dx=\frac {d x + c}{a d} + \frac {\log \left (b e^{\left (-d x - c\right )} + a\right )}{a d} \]
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Time = 0.40 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {1}{a+b e^{-c-d x}} \, dx=\frac {\frac {d x + c}{a} + \frac {\log \left ({\left | b e^{\left (-d x - c\right )} + a \right |}\right )}{a}}{d} \]
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Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {1}{a+b e^{-c-d x}} \, dx=\frac {\ln \left (a+b\,{\mathrm {e}}^{-c}\,{\mathrm {e}}^{-d\,x}\right )+d\,x}{a\,d} \]
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