Integrand size = 15, antiderivative size = 12 \[ \int \frac {1+e^x}{1-e^x} \, dx=x-2 \log \left (1-e^x\right ) \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2320, 78} \[ \int \frac {1+e^x}{1-e^x} \, dx=x-2 \log \left (1-e^x\right ) \]
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Rule 78
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1+x}{(1-x) x} \, dx,x,e^x\right ) \\ & = \text {Subst}\left (\int \left (-\frac {2}{-1+x}+\frac {1}{x}\right ) \, dx,x,e^x\right ) \\ & = x-2 \log \left (1-e^x\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08 \[ \int \frac {1+e^x}{1-e^x} \, dx=\log \left (e^x\right )-2 \log \left (-1+e^x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83
method | result | size |
norman | \(x -2 \ln \left (-1+{\mathrm e}^{x}\right )\) | \(10\) |
risch | \(x -2 \ln \left (-1+{\mathrm e}^{x}\right )\) | \(10\) |
parallelrisch | \(x -2 \ln \left (-1+{\mathrm e}^{x}\right )\) | \(10\) |
derivativedivides | \(-2 \ln \left (-1+{\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x}\right )\) | \(12\) |
default | \(-2 \ln \left (-1+{\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x}\right )\) | \(12\) |
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none
Time = 0.26 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \frac {1+e^x}{1-e^x} \, dx=x - 2 \, \log \left (e^{x} - 1\right ) \]
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Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {1+e^x}{1-e^x} \, dx=x - 2 \log {\left (e^{x} - 1 \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \frac {1+e^x}{1-e^x} \, dx=x - 2 \, \log \left (e^{x} - 1\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {1+e^x}{1-e^x} \, dx=x - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \frac {1+e^x}{1-e^x} \, dx=x-2\,\ln \left ({\mathrm {e}}^x-1\right ) \]
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