Optimal. Leaf size=25 \[ x \left (-x+4 e^{-x} \left (2+3 e^3\right ) (-2+2 x)\right ) \]
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Rubi [A] time = 0.22, antiderivative size = 46, normalized size of antiderivative = 1.84, number of steps used = 16, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6742, 2194, 2176, 2196} \begin {gather*} 24 e^{3-x} x^2+16 e^{-x} x^2-x^2-24 e^{3-x} x-16 e^{-x} x \end {gather*}
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rule 2196
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-16 e^{-x}-2 x+48 e^{-x} x-16 e^{-x} x^2-24 e^{3-x} \left (1-3 x+x^2\right )\right ) \, dx\\ &=-x^2-16 \int e^{-x} \, dx-16 \int e^{-x} x^2 \, dx-24 \int e^{3-x} \left (1-3 x+x^2\right ) \, dx+48 \int e^{-x} x \, dx\\ &=16 e^{-x}-48 e^{-x} x-x^2+16 e^{-x} x^2-24 \int \left (e^{3-x}-3 e^{3-x} x+e^{3-x} x^2\right ) \, dx-32 \int e^{-x} x \, dx+48 \int e^{-x} \, dx\\ &=-32 e^{-x}-16 e^{-x} x-x^2+16 e^{-x} x^2-24 \int e^{3-x} \, dx-24 \int e^{3-x} x^2 \, dx-32 \int e^{-x} \, dx+72 \int e^{3-x} x \, dx\\ &=24 e^{3-x}-72 e^{3-x} x-16 e^{-x} x-x^2+24 e^{3-x} x^2+16 e^{-x} x^2-48 \int e^{3-x} x \, dx+72 \int e^{3-x} \, dx\\ &=-48 e^{3-x}-24 e^{3-x} x-16 e^{-x} x-x^2+24 e^{3-x} x^2+16 e^{-x} x^2-48 \int e^{3-x} \, dx\\ &=-24 e^{3-x} x-16 e^{-x} x-x^2+24 e^{3-x} x^2+16 e^{-x} x^2\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 27, normalized size = 1.08 \begin {gather*} -e^{-x} x \left (-16 (-1+x)-24 e^3 (-1+x)+e^x x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 32, normalized size = 1.28 \begin {gather*} -{\left (x^{2} e^{x} - 16 \, x^{2} - 24 \, {\left (x^{2} - x\right )} e^{3} + 16 \, x\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 34, normalized size = 1.36 \begin {gather*} -x^{2} + 16 \, {\left (x^{2} - x\right )} e^{\left (-x\right )} + 24 \, {\left (x^{2} - x\right )} e^{\left (-x + 3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 32, normalized size = 1.28
method | result | size |
norman | \(\left (\left (-24 \,{\mathrm e}^{3}-16\right ) x +\left (24 \,{\mathrm e}^{3}+16\right ) x^{2}-{\mathrm e}^{x} x^{2}\right ) {\mathrm e}^{-x}\) | \(32\) |
risch | \(-x^{2}+\left (24 x^{2} {\mathrm e}^{3}-24 x \,{\mathrm e}^{3}+16 x^{2}-16 x \right ) {\mathrm e}^{-x}\) | \(33\) |
default | \(-x^{2}-16 x \,{\mathrm e}^{-x}+16 x^{2} {\mathrm e}^{-x}+24 \,{\mathrm e}^{-x} {\mathrm e}^{3}+72 \,{\mathrm e}^{3} \left (-x \,{\mathrm e}^{-x}-{\mathrm e}^{-x}\right )-24 \,{\mathrm e}^{3} \left (-x^{2} {\mathrm e}^{-x}-2 x \,{\mathrm e}^{-x}-2 \,{\mathrm e}^{-x}\right )\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 78, normalized size = 3.12 \begin {gather*} -x^{2} + 24 \, {\left (x^{2} e^{3} + 2 \, x e^{3} + 2 \, e^{3}\right )} e^{\left (-x\right )} + 16 \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} - 72 \, {\left (x e^{3} + e^{3}\right )} e^{\left (-x\right )} - 48 \, {\left (x + 1\right )} e^{\left (-x\right )} + 16 \, e^{\left (-x\right )} + 24 \, e^{\left (-x + 3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.36, size = 33, normalized size = 1.32 \begin {gather*} x^2\,{\mathrm {e}}^{-x}\,\left (24\,{\mathrm {e}}^3+16\right )-x\,{\mathrm {e}}^{-x}\,\left (24\,{\mathrm {e}}^3+16\right )-x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 29, normalized size = 1.16 \begin {gather*} - x^{2} + \left (16 x^{2} + 24 x^{2} e^{3} - 24 x e^{3} - 16 x\right ) e^{- x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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