Optimal. Leaf size=22 \[ x+2 e^{-x} x \log \left (e^{12+2 e^2} x\right ) \]
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Rubi [B] time = 0.25, antiderivative size = 54, normalized size of antiderivative = 2.45, number of steps used = 10, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {6742, 2194, 6688, 2176, 2554} \begin {gather*} -4 \left (6+e^2\right ) e^{-x} (1-x)+x+4 \left (6+e^2\right ) e^{-x}-2 e^{-x} (1-x) \log (x)+2 e^{-x} \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rule 2554
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+2 e^{-x}+2 e^{-x} (1-x) \left (12 \left (1+\frac {e^2}{6}\right )+\log (x)\right )\right ) \, dx\\ &=x+2 \int e^{-x} \, dx+2 \int e^{-x} (1-x) \left (12 \left (1+\frac {e^2}{6}\right )+\log (x)\right ) \, dx\\ &=-2 e^{-x}+x+2 \int e^{-x} (1-x) \left (2 \left (6+e^2\right )+\log (x)\right ) \, dx\\ &=-2 e^{-x}+x+2 \int \left (-2 e^{-x} \left (6+e^2\right ) (-1+x)-e^{-x} (-1+x) \log (x)\right ) \, dx\\ &=-2 e^{-x}+x-2 \int e^{-x} (-1+x) \log (x) \, dx-\left (4 \left (6+e^2\right )\right ) \int e^{-x} (-1+x) \, dx\\ &=-2 e^{-x}-4 e^{-x} \left (6+e^2\right ) (1-x)+x+2 e^{-x} \log (x)-2 e^{-x} (1-x) \log (x)-2 \int e^{-x} \, dx-\left (4 \left (6+e^2\right )\right ) \int e^{-x} \, dx\\ &=4 e^{-x} \left (6+e^2\right )-4 e^{-x} \left (6+e^2\right ) (1-x)+x+2 e^{-x} \log (x)-2 e^{-x} (1-x) \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 22, normalized size = 1.00 \begin {gather*} x+2 e^{-x} \left (2 \left (6+e^2\right ) x+x \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 23, normalized size = 1.05 \begin {gather*} {\left (x e^{x} + 2 \, x \log \left (x e^{\left (2 \, e^{2} + 12\right )}\right )\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 27, normalized size = 1.23 \begin {gather*} 2 \, x e^{\left (-x\right )} \log \relax (x) + 24 \, x e^{\left (-x\right )} + 4 \, x e^{\left (-x + 2\right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 20, normalized size = 0.91
method | result | size |
default | \(2 x \,{\mathrm e}^{-x} \ln \left (x \,{\mathrm e}^{2 \,{\mathrm e}^{2}+12}\right )+x\) | \(20\) |
risch | \(2 x \,{\mathrm e}^{-x} \ln \left (x \,{\mathrm e}^{2 \,{\mathrm e}^{2}+12}\right )+x\) | \(20\) |
norman | \(\left ({\mathrm e}^{x} x +2 \ln \left (x \,{\mathrm e}^{2 \,{\mathrm e}^{2}+12}\right ) x \right ) {\mathrm e}^{-x}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 2 \, {\left (x + 1\right )} e^{\left (-x\right )} \log \relax (x) - 2 \, e^{\left (-x\right )} \log \left (x e^{\left (2 \, e^{2} + 12\right )}\right ) + x + 2 \, {\rm Ei}\left (-x\right ) - 2 \, e^{\left (-x\right )} - 2 \, \int \frac {{\left (2 \, x^{2} {\left (e^{2} + 6\right )} + x + 1\right )} e^{\left (-x\right )}}{x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.46, size = 18, normalized size = 0.82 \begin {gather*} x\,{\mathrm {e}}^{-x}\,\left (4\,{\mathrm {e}}^2+{\mathrm {e}}^x+2\,\ln \relax (x)+24\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 19, normalized size = 0.86 \begin {gather*} x + 2 x e^{- x} \log {\left (x e^{12 + 2 e^{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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