Integrand size = 11, antiderivative size = 13 \[ \int \frac {1}{\left (e^{-x}+e^x\right )^2} \, dx=-\frac {1}{2 \left (1+e^{2 x}\right )} \]
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Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2320, 267} \[ \int \frac {1}{\left (e^{-x}+e^x\right )^2} \, dx=-\frac {1}{2 \left (e^{2 x}+1\right )} \]
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Rule 267
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x}{\left (1+x^2\right )^2} \, dx,x,e^x\right ) \\ & = -\frac {1}{2 \left (1+e^{2 x}\right )} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (e^{-x}+e^x\right )^2} \, dx=-\frac {1}{2+2 e^{2 x}} \]
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Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {1}{2 \left (1+{\mathrm e}^{2 x}\right )}\) | \(11\) |
norman | \(-\frac {1}{2 \left (1+{\mathrm e}^{2 x}\right )}\) | \(11\) |
risch | \(-\frac {1}{2 \left (1+{\mathrm e}^{2 x}\right )}\) | \(11\) |
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Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (e^{-x}+e^x\right )^2} \, dx=-\frac {1}{2 \, {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
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Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (e^{-x}+e^x\right )^2} \, dx=- \frac {1}{2 e^{2 x} + 2} \]
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Time = 0.19 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (e^{-x}+e^x\right )^2} \, dx=\frac {1}{2 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (e^{-x}+e^x\right )^2} \, dx=-\frac {1}{2 \, {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
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Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (e^{-x}+e^x\right )^2} \, dx=-\frac {1}{2\,\left ({\mathrm {e}}^{2\,x}+1\right )} \]
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