Integrand size = 7, antiderivative size = 10 \[ \int \frac {1}{1+e^x} \, dx=x-\log \left (1+e^x\right ) \]
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Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {2320, 36, 29, 31} \[ \int \frac {1}{1+e^x} \, dx=x-\log \left (e^x+1\right ) \]
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Rule 29
Rule 31
Rule 36
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,e^x\right ) \\ & = \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )-\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^x\right ) \\ & = x-\log \left (1+e^x\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+e^x} \, dx=-2 \text {arctanh}\left (1+2 e^x\right ) \]
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Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00
method | result | size |
norman | \(x -\ln \left (1+{\mathrm e}^{x}\right )\) | \(10\) |
risch | \(x -\ln \left (1+{\mathrm e}^{x}\right )\) | \(10\) |
parallelrisch | \(x -\ln \left (1+{\mathrm e}^{x}\right )\) | \(10\) |
derivativedivides | \(-\ln \left (1+{\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x}\right )\) | \(12\) |
default | \(-\ln \left (1+{\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x}\right )\) | \(12\) |
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none
Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {1}{1+e^x} \, dx=x - \log \left (e^{x} + 1\right ) \]
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Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {1}{1+e^x} \, dx=x - \log {\left (e^{x} + 1 \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {1}{1+e^x} \, dx=x - \log \left (e^{x} + 1\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {1}{1+e^x} \, dx=x - \log \left (e^{x} + 1\right ) \]
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Time = 0.05 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {1}{1+e^x} \, dx=x-\ln \left ({\mathrm {e}}^x+1\right ) \]
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