Integrand size = 7, antiderivative size = 8 \[ \int \frac {1}{2+e^x} \, dx=-\text {arctanh}\left (1+e^x\right ) \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 2.00, 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} \[ \int \frac {1}{2+e^x} \, dx=\frac {x}{2}-\frac {1}{2} \log \left (e^x+2\right ) \]
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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 (2+x)} \, dx,x,e^x\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{2+x} \, dx,x,e^x\right ) \\ & = \frac {x}{2}-\frac {1}{2} \log \left (2+e^x\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2+e^x} \, dx=-\text {arctanh}\left (1+e^x\right ) \]
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Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.50
method | result | size |
norman | \(\frac {x}{2}-\frac {\ln \left ({\mathrm e}^{x}+2\right )}{2}\) | \(12\) |
risch | \(\frac {x}{2}-\frac {\ln \left ({\mathrm e}^{x}+2\right )}{2}\) | \(12\) |
parallelrisch | \(\frac {x}{2}-\frac {\ln \left ({\mathrm e}^{x}+2\right )}{2}\) | \(12\) |
derivativedivides | \(\frac {\ln \left ({\mathrm e}^{x}\right )}{2}-\frac {\ln \left ({\mathrm e}^{x}+2\right )}{2}\) | \(14\) |
default | \(\frac {\ln \left ({\mathrm e}^{x}\right )}{2}-\frac {\ln \left ({\mathrm e}^{x}+2\right )}{2}\) | \(14\) |
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Time = 0.25 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.38 \[ \int \frac {1}{2+e^x} \, dx=\frac {1}{2} \, x - \frac {1}{2} \, \log \left (e^{x} + 2\right ) \]
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Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.25 \[ \int \frac {1}{2+e^x} \, dx=\frac {x}{2} - \frac {\log {\left (e^{x} + 2 \right )}}{2} \]
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Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.38 \[ \int \frac {1}{2+e^x} \, dx=\frac {1}{2} \, x - \frac {1}{2} \, \log \left (e^{x} + 2\right ) \]
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Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.38 \[ \int \frac {1}{2+e^x} \, dx=\frac {1}{2} \, x - \frac {1}{2} \, \log \left (e^{x} + 2\right ) \]
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Time = 0.05 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.38 \[ \int \frac {1}{2+e^x} \, dx=\frac {x}{2}-\frac {\ln \left ({\mathrm {e}}^x+2\right )}{2} \]
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