Optimal. Leaf size=36 \[ -e^{-e^{-e^{\frac {1}{2} (-2-x) x (x+\log (3))}+x}+2 x}+2 x \]
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Rubi [A] time = 1.60, antiderivative size = 52, normalized size of antiderivative = 1.44, number of steps used = 3, number of rules used = 2, integrand size = 137, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {12, 6706} \begin {gather*} 2 x-\exp \left (2 x-\exp \left (x-3^{\frac {1}{2} \left (-x^2-2 x\right )} e^{\frac {1}{2} \left (-x^3-2 x^2\right )}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6706
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \left (4+\exp \left (-\exp \left (-\exp \left (\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )\right )+x\right )+2 x\right ) \left (-4+\exp \left (-\exp \left (\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )\right )+x\right ) \left (2+\exp \left (\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )\right ) \left (4 x+3 x^2+(2+2 x) \log (3)\right )\right )\right )\right ) \, dx\\ &=2 x+\frac {1}{2} \int \exp \left (-\exp \left (-\exp \left (\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )\right )+x\right )+2 x\right ) \left (-4+\exp \left (-\exp \left (\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )\right )+x\right ) \left (2+\exp \left (\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )\right ) \left (4 x+3 x^2+(2+2 x) \log (3)\right )\right )\right ) \, dx\\ &=-\exp \left (-\exp \left (-3^{\frac {1}{2} \left (-2 x-x^2\right )} e^{\frac {1}{2} \left (-2 x^2-x^3\right )}+x\right )+2 x\right )+2 x\\ \end {aligned} \end {gather*}
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Mathematica [F] time = 7.93, size = 139, normalized size = 3.86 \begin {gather*} \frac {1}{2} \int \left (4+e^{-e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x}+2 x} \left (-4+e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x} \left (2+e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )} \left (4 x+3 x^2+(2+2 x) \log (3)\right )\right )\right )\right ) \, dx \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 41, normalized size = 1.14 \begin {gather*} 2 \, x - e^{\left (2 \, x - e^{\left (x - e^{\left (-\frac {1}{2} \, x^{3} - x^{2} - \frac {1}{2} \, {\left (x^{2} + 2 \, x\right )} \log \relax (3)\right )}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 42, normalized size = 1.17 \begin {gather*} 2 \, x - e^{\left (2 \, x - e^{\left (x - e^{\left (-\frac {1}{2} \, x^{3} - \frac {1}{2} \, x^{2} \log \relax (3) - x^{2} - x \log \relax (3)\right )}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 36, normalized size = 1.00
method | result | size |
risch | \(2 x -{\mathrm e}^{-{\mathrm e}^{-3^{-\frac {x \left (2+x \right )}{2}} {\mathrm e}^{-\frac {x^{2} \left (2+x \right )}{2}}+x}+2 x}\) | \(36\) |
default | \(2 x -{\mathrm e}^{-{\mathrm e}^{-{\mathrm e}^{\frac {\left (-x^{2}-2 x \right ) \ln \relax (3)}{2}-\frac {x^{3}}{2}-x^{2}}+x}+2 x}\) | \(44\) |
norman | \(2 x -{\mathrm e}^{-{\mathrm e}^{-{\mathrm e}^{\frac {\left (-x^{2}-2 x \right ) \ln \relax (3)}{2}-\frac {x^{3}}{2}-x^{2}}+x}+2 x}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 42, normalized size = 1.17 \begin {gather*} 2 \, x - e^{\left (2 \, x - e^{\left (x - e^{\left (-\frac {1}{2} \, x^{3} - \frac {1}{2} \, x^{2} \log \relax (3) - x^{2} - x \log \relax (3)\right )}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.96, size = 42, normalized size = 1.17 \begin {gather*} 2\,x-{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-{\mathrm {e}}^{-\frac {{\mathrm {e}}^{-x^2}}{3^x\,\sqrt {{\mathrm {e}}^{x^3}}\,\sqrt {3^{x^2}}}}\,{\mathrm {e}}^x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.04, size = 32, normalized size = 0.89 \begin {gather*} 2 x - e^{2 x - e^{x - e^{- \frac {x^{3}}{2} - x^{2} + \left (- \frac {x^{2}}{2} - x\right ) \log {\relax (3 )}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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