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 = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 43
Rule 266
Rule 2282
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.41, size = 25, normalized size = 0.81 \[ \frac {1}{4} \, {\left (12 \, x e^{\left (2 \, x\right )} + e^{\left (6 \, x\right )} - 6 \, e^{\left (4 \, x\right )} + 2\right )} e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 30, normalized size = 0.97 \[ -\frac {1}{2} \, {\left (3 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )} + 3 \, x + \frac {1}{4} \, e^{\left (4 \, x\right )} - \frac {3}{2} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 25, normalized size = 0.81 \[ \frac {{\mathrm e}^{-2 x}}{2}-\frac {3 \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{4 x}}{4}+3 \ln \left ({\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 24, normalized size = 0.77 \[ -\frac {1}{4} \, {\left (6 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (4 \, x\right )} + 3 \, x + \frac {1}{2} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.59, size = 22, normalized size = 0.71 \[ 3\,x+\frac {{\mathrm {e}}^{-2\,x}}{2}-\frac {3\,{\mathrm {e}}^{2\,x}}{2}+\frac {{\mathrm {e}}^{4\,x}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 26, normalized size = 0.84 \[ 3 x + \frac {e^{4 x}}{4} - \frac {3 e^{2 x}}{2} + \frac {e^{- 2 x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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