Optimal. Leaf size=22 \[ 5+e^x-2 e^{-2-x} x \left (e^x+x^4\right ) \]
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Rubi [A] time = 0.22, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {12, 6742, 2194, 2176} \begin {gather*} -2 e^{-x-2} x^5-\frac {2 x}{e^2}+e^x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=2 \int e^{-2-x} \left (e^x \left (-1+\frac {e^{2+x}}{2}\right )-5 x^4+x^5\right ) \, dx\\ &=2 \int \left (\frac {-2+e^{2+x}}{2 e^2}-5 e^{-2-x} x^4+e^{-2-x} x^5\right ) \, dx\\ &=2 \int e^{-2-x} x^5 \, dx-10 \int e^{-2-x} x^4 \, dx+\frac {\int \left (-2+e^{2+x}\right ) \, dx}{e^2}\\ &=-\frac {2 x}{e^2}+10 e^{-2-x} x^4-2 e^{-2-x} x^5+10 \int e^{-2-x} x^4 \, dx-40 \int e^{-2-x} x^3 \, dx+\frac {\int e^{2+x} \, dx}{e^2}\\ &=e^x-\frac {2 x}{e^2}+40 e^{-2-x} x^3-2 e^{-2-x} x^5+40 \int e^{-2-x} x^3 \, dx-120 \int e^{-2-x} x^2 \, dx\\ &=e^x-\frac {2 x}{e^2}+120 e^{-2-x} x^2-2 e^{-2-x} x^5+120 \int e^{-2-x} x^2 \, dx-240 \int e^{-2-x} x \, dx\\ &=e^x-\frac {2 x}{e^2}+240 e^{-2-x} x-2 e^{-2-x} x^5-240 \int e^{-2-x} \, dx+240 \int e^{-2-x} x \, dx\\ &=240 e^{-2-x}+e^x-\frac {2 x}{e^2}-2 e^{-2-x} x^5+240 \int e^{-2-x} \, dx\\ &=e^x-\frac {2 x}{e^2}-2 e^{-2-x} x^5\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 1.00 \begin {gather*} e^x-\frac {2 x}{e^2}-2 e^{-2-x} x^5 \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 43, normalized size = 1.95 \begin {gather*} -{\left (x^{5} e^{\left (-2 \, x + 2 \, \log \relax (2) - 4\right )} + {\left (x e^{\left (-x + \log \relax (2) - 2\right )} - 1\right )} e^{\left (\log \relax (2) - 2\right )}\right )} e^{\left (x - \log \relax (2) + 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 23, normalized size = 1.05 \begin {gather*} -{\left (2 \, x^{5} e^{\left (-x\right )} + 2 \, x - e^{\left (x + 2\right )}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 20, normalized size = 0.91
method | result | size |
risch | \(-2 x \,{\mathrm e}^{-2}+{\mathrm e}^{x}-2 x^{5} {\mathrm e}^{-x -2}\) | \(20\) |
norman | \(\left ({\mathrm e}^{2 x}-2 \,{\mathrm e}^{-2} x^{5}-2 x \,{\mathrm e}^{-2} {\mathrm e}^{x}\right ) {\mathrm e}^{-x}\) | \(29\) |
default | \(-2 x \,{\mathrm e}^{-2}-10 \,{\mathrm e}^{-2} \left (-x^{4} {\mathrm e}^{-x}-4 x^{3} {\mathrm e}^{-x}-12 x^{2} {\mathrm e}^{-x}-24 x \,{\mathrm e}^{-x}-24 \,{\mathrm e}^{-x}\right )+2 \,{\mathrm e}^{-2} \left (-{\mathrm e}^{-x} x^{5}-5 x^{4} {\mathrm e}^{-x}-20 x^{3} {\mathrm e}^{-x}-60 x^{2} {\mathrm e}^{-x}-120 x \,{\mathrm e}^{-x}-120 \,{\mathrm e}^{-x}\right )+{\mathrm e}^{x}\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 65, normalized size = 2.95 \begin {gather*} -2 \, x e^{\left (-2\right )} - 2 \, {\left (x^{5} + 5 \, x^{4} + 20 \, x^{3} + 60 \, x^{2} + 120 \, x + 120\right )} e^{\left (-x - 2\right )} + 10 \, {\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x - 2\right )} + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 19, normalized size = 0.86 \begin {gather*} {\mathrm {e}}^x-2\,x\,{\mathrm {e}}^{-2}-2\,x^5\,{\mathrm {e}}^{-x-2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 24, normalized size = 1.09 \begin {gather*} - \frac {2 x}{e^{2}} + \frac {- 2 x^{5} e^{- x} + e^{2} e^{x}}{e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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