Integrand size = 18, antiderivative size = 9 \[ \int \frac {e^x}{1+2 e^x+e^{2 x}} \, dx=-\frac {1}{1+e^x} \]
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Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2320, 32} \[ \int \frac {e^x}{1+2 e^x+e^{2 x}} \, dx=-\frac {1}{e^x+1} \]
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Rule 32
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,e^x\right ) \\ & = -\frac {1}{1+e^x} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {e^x}{1+2 e^x+e^{2 x}} \, dx=-\frac {1}{1+e^x} \]
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Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {1}{1+{\mathrm e}^{x}}\) | \(9\) |
norman | \(-\frac {1}{1+{\mathrm e}^{x}}\) | \(9\) |
risch | \(-\frac {1}{1+{\mathrm e}^{x}}\) | \(9\) |
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none
Time = 0.30 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89 \[ \int \frac {e^x}{1+2 e^x+e^{2 x}} \, dx=-\frac {1}{e^{x} + 1} \]
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Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {e^x}{1+2 e^x+e^{2 x}} \, dx=- \frac {1}{e^{x} + 1} \]
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none
Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89 \[ \int \frac {e^x}{1+2 e^x+e^{2 x}} \, dx=-\frac {1}{e^{x} + 1} \]
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none
Time = 0.32 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89 \[ \int \frac {e^x}{1+2 e^x+e^{2 x}} \, dx=-\frac {1}{e^{x} + 1} \]
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Time = 0.06 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89 \[ \int \frac {e^x}{1+2 e^x+e^{2 x}} \, dx=-\frac {1}{{\mathrm {e}}^x+1} \]
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