Optimal. Leaf size=15 \[ \frac{1}{2 \left (1-e^{2 x}\right )} \]
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Rubi [A] time = 0.0139409, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2282, 261} \[ \frac{1}{2 \left (1-e^{2 x}\right )} \]
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 = 0.0112566, size = 11, normalized size = 0.73 \[ \frac{1}{2-2 e^{2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 11, normalized size = 0.7 \begin{align*} -{\frac{1}{2\, \left ({{\rm e}^{x}} \right ) ^{2}-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961202, size = 14, normalized size = 0.93 \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 = 0.889163, size = 27, normalized size = 1.8 \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 = 0.074859, size = 10, normalized size = 0.67 \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 = 1.29891, size = 14, normalized size = 0.93 \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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