Optimal. Leaf size=22 \[ -2 x-\frac{e^{-2 x}}{2}+\frac{e^{2 x}}{2} \]
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Rubi [A] time = 0.0177986, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2282, 266, 43} \[ -2 x-\frac{e^{-2 x}}{2}+\frac{e^{2 x}}{2} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (-e^{-x}+e^x\right )^2 \, dx &=\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^3} \, dx,x,e^x\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(1-x)^2}{x^2} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (1+\frac{1}{x^2}-\frac{2}{x}\right ) \, dx,x,e^{2 x}\right )\\ &=-\frac{1}{2} e^{-2 x}+\frac{e^{2 x}}{2}-2 x\\ \end{align*}
Mathematica [A] time = 0.0058902, size = 22, normalized size = 1. \[ -2 x-\frac{e^{-2 x}}{2}+\frac{e^{2 x}}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 19, normalized size = 0.9 \begin{align*}{\frac{ \left ({{\rm e}^{x}} \right ) ^{2}}{2}}-2\,\ln \left ({{\rm e}^{x}} \right ) -{\frac{1}{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 = 0.926443, size = 22, normalized size = 1. \begin{align*} -2 \, x + \frac{1}{2} \, e^{\left (2 \, x\right )} - \frac{1}{2} \, e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86786, size = 58, normalized size = 2.64 \begin{align*} -\frac{1}{2} \,{\left (4 \, x e^{\left (2 \, x\right )} - e^{\left (4 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.101106, size = 17, normalized size = 0.77 \begin{align*} - 2 x + \frac{e^{2 x}}{2} - \frac{e^{- 2 x}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05804, size = 32, normalized size = 1.45 \begin{align*} \frac{1}{2} \,{\left (2 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )} - 2 \, x + \frac{1}{2} \, e^{\left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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