Optimal. Leaf size=31 \[ -x+\frac{e^{-2 x}}{2}+\frac{e^{2 x}}{2}+\frac{e^{4 x}}{4} \]
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Rubi [A] time = 0.0493072, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2282, 14} \[ -x+\frac{e^{-2 x}}{2}+\frac{e^{2 x}}{2}+\frac{e^{4 x}}{4} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 14
Rubi steps
\begin{align*} \int e^x \left (-e^{-x}+e^x\right ) \left (e^{-x}+e^x\right )^2 \, dx &=\operatorname{Subst}\left (\int \frac{-1-\frac{1}{x^2}+x^2+x^4}{x} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{1}{x^3}-\frac{1}{x}+x+x^3\right ) \, dx,x,e^x\right )\\ &=\frac{e^{-2 x}}{2}+\frac{e^{2 x}}{2}+\frac{e^{4 x}}{4}-x\\ \end{align*}
Mathematica [A] time = 0.0082503, size = 31, normalized size = 1. \[ -x+\frac{e^{-2 x}}{2}+\frac{e^{2 x}}{2}+\frac{e^{4 x}}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 23, normalized size = 0.7 \begin{align*} -x+{\frac{ \left ({{\rm e}^{x}} \right ) ^{2}}{2}}+{\frac{ \left ({{\rm e}^{x}} \right ) ^{4}}{4}}+{\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 = 0.97658, size = 32, normalized size = 1.03 \begin{align*} \frac{1}{4} \,{\left (2 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (4 \, x\right )} - 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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Fricas [A] time = 0.67376, size = 74, normalized size = 2.39 \begin{align*} -\frac{1}{4} \,{\left (4 \, x e^{\left (2 \, x\right )} - e^{\left (6 \, x\right )} - 2 \, e^{\left (4 \, x\right )} - 2\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 = 0.117177, size = 22, normalized size = 0.71 \begin{align*} - x + \frac{e^{4 x}}{4} + \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 = 1.2568, size = 38, normalized size = 1.23 \begin{align*} \frac{1}{2} \,{\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )} - x + \frac{1}{4} \, e^{\left (4 \, 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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