Optimal. Leaf size=36 \[ 6 x-\frac{e^{-4 x}}{4}+2 e^{-2 x}-2 e^{2 x}+\frac{e^{4 x}}{4} \]
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Rubi [A] time = 0.0255405, antiderivative size = 36, 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} \[ 6 x-\frac{e^{-4 x}}{4}+2 e^{-2 x}-2 e^{2 x}+\frac{e^{4 x}}{4} \]
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 )^4 \, dx &=\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^4}{x^5} \, dx,x,e^x\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(1-x)^4}{x^3} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-4+\frac{1}{x^3}-\frac{4}{x^2}+\frac{6}{x}+x\right ) \, dx,x,e^{2 x}\right )\\ &=-\frac{1}{4} e^{-4 x}+2 e^{-2 x}-2 e^{2 x}+\frac{e^{4 x}}{4}+6 x\\ \end{align*}
Mathematica [A] time = 0.0209597, size = 34, normalized size = 0.94 \[ \frac{1}{4} \left (24 x-e^{-4 x}+8 e^{-2 x}-8 e^{2 x}+e^{4 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 31, normalized size = 0.9 \begin{align*}{\frac{ \left ({{\rm e}^{x}} \right ) ^{4}}{4}}-2\, \left ({{\rm e}^{x}} \right ) ^{2}+6\,\ln \left ({{\rm e}^{x}} \right ) -{\frac{1}{4\, \left ({{\rm e}^{x}} \right ) ^{4}}}+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 = 0.929387, size = 38, normalized size = 1.06 \begin{align*} 6 \, x + \frac{1}{4} \, e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 2 \, e^{\left (-2 \, x\right )} - \frac{1}{4} \, e^{\left (-4 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80387, size = 90, normalized size = 2.5 \begin{align*} \frac{1}{4} \,{\left (24 \, x e^{\left (4 \, x\right )} + e^{\left (8 \, x\right )} - 8 \, e^{\left (6 \, x\right )} + 8 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-4 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.137358, size = 31, normalized size = 0.86 \begin{align*} 6 x + \frac{e^{4 x}}{4} - 2 e^{2 x} + 2 e^{- 2 x} - \frac{e^{- 4 x}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06391, size = 49, normalized size = 1.36 \begin{align*} -\frac{1}{4} \,{\left (18 \, e^{\left (4 \, x\right )} - 8 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-4 \, x\right )} + 6 \, x + \frac{1}{4} \, e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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