Optimal. Leaf size=13 \[ -\frac{1}{2 \left (e^{2 x}+1\right )} \]
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Rubi [A] time = 0.0124029, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2282, 261} \[ -\frac{1}{2 \left (e^{2 x}+1\right )} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 261
Rubi steps
\begin{align*} \int \frac{1}{\left (e^{-x}+e^x\right )^2} \, dx &=\operatorname{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*}
Mathematica [A] time = 0.0086088, size = 13, normalized size = 1. \[ -\frac{1}{2 e^{2 x}+2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 11, normalized size = 0.9 \begin{align*} -{\frac{1}{2+2\, \left ({{\rm e}^{x}} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965802, size = 14, normalized size = 1.08 \begin{align*} \frac{1}{2 \,{\left (e^{\left (-2 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.79847, size = 27, normalized size = 2.08 \begin{align*} -\frac{1}{2 \,{\left (e^{\left (2 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.075101, size = 10, normalized size = 0.77 \begin{align*} - \frac{1}{2 e^{2 x} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31566, size = 14, normalized size = 1.08 \begin{align*} -\frac{1}{2 \,{\left (e^{\left (2 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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