Optimal. Leaf size=31 \[ 3 x+\frac{e^{-2 x}}{2}-\frac{3 e^{2 x}}{2}+\frac{e^{4 x}}{4} \]
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Rubi [A] time = 0.0366155, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2282, 266, 43} \[ 3 x+\frac{e^{-2 x}}{2}-\frac{3 e^{2 x}}{2}+\frac{e^{4 x}}{4} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 266
Rule 43
Rubi steps
\begin{align*} \int e^x \left (-e^{-x}+e^x\right )^3 \, dx &=\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^3}{x^3} \, dx,x,e^x\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(-1+x)^3}{x^2} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-3-\frac{1}{x^2}+\frac{3}{x}+x\right ) \, dx,x,e^{2 x}\right )\\ &=\frac{e^{-2 x}}{2}-\frac{3 e^{2 x}}{2}+\frac{e^{4 x}}{4}+3 x\\ \end{align*}
Mathematica [A] time = 0.0126826, size = 29, normalized size = 0.94 \[ \frac{1}{2} \left (6 x+e^{-2 x}-3 e^{2 x}+\frac{e^{4 x}}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 25, normalized size = 0.8 \begin{align*}{\frac{ \left ({{\rm e}^{x}} \right ) ^{4}}{4}}-{\frac{3\, \left ({{\rm e}^{x}} \right ) ^{2}}{2}}+3\,\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 = 0.968184, size = 32, normalized size = 1.03 \begin{align*} -\frac{1}{4} \,{\left (6 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (4 \, x\right )} + 3 \, 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 = 0.856181, size = 74, normalized size = 2.39 \begin{align*} \frac{1}{4} \,{\left (12 \, x e^{\left (2 \, x\right )} + e^{\left (6 \, x\right )} - 6 \, 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 = 0.133643, size = 26, normalized size = 0.84 \begin{align*} 3 x + \frac{e^{4 x}}{4} - \frac{3 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 = 1.20512, size = 41, normalized size = 1.32 \begin{align*} -\frac{1}{2} \,{\left (3 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )} + 3 \, x + \frac{1}{4} \, e^{\left (4 \, x\right )} - \frac{3}{2} \, e^{\left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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