Optimal. Leaf size=22 \[ -1+5 e^{-6-4 e^2+x+\frac {x^2}{2}}+x \]
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Rubi [A]
time = 0.21, antiderivative size = 24, normalized size of antiderivative = 1.09, number of steps
used = 3, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6820, 2268}
\begin {gather*} 5 e^{\frac {x^2}{2}+x-2 \left (3+2 e^2\right )}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 2268
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+5 e^{-2 \left (3+2 e^2\right )+x+\frac {x^2}{2}} (1+x)\right ) \, dx\\ &=x+5 \int e^{-2 \left (3+2 e^2\right )+x+\frac {x^2}{2}} (1+x) \, dx\\ &=5 e^{-2 \left (3+2 e^2\right )+x+\frac {x^2}{2}}+x\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 21, normalized size = 0.95 \begin {gather*} 5 e^{-6-4 e^2+x+\frac {x^2}{2}}+x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 18, normalized size = 0.82
method | result | size |
default | \(x +5 \,{\mathrm e}^{-4 \,{\mathrm e}^{2}+\frac {x^{2}}{2}+x -6}\) | \(18\) |
risch | \(x +5 \,{\mathrm e}^{-4 \,{\mathrm e}^{2}+\frac {x^{2}}{2}+x -6}\) | \(18\) |
norman | \(\left (5+x \,{\mathrm e}^{4 \,{\mathrm e}^{2}-\frac {x^{2}}{2}-x +6}\right ) {\mathrm e}^{-4 \,{\mathrm e}^{2}+\frac {x^{2}}{2}+x -6}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.54, size = 88, normalized size = 4.00 \begin {gather*} -\frac {5}{2} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {2} x + \frac {1}{2} i \, \sqrt {2}\right ) e^{\left (-4 \, e^{2} - \frac {13}{2}\right )} - \frac {5}{2} \, \sqrt {2} {\left (\frac {\sqrt {\pi } {\left (x + 1\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-{\left (x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x + 1\right )}^{2}}} - \sqrt {2} e^{\left (\frac {1}{2} \, {\left (x + 1\right )}^{2}\right )}\right )} e^{\left (-4 \, e^{2} - \frac {13}{2}\right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 33, normalized size = 1.50 \begin {gather*} {\left (x e^{\left (-\frac {1}{2} \, x^{2} - x + 4 \, e^{2} + 6\right )} + 5\right )} e^{\left (\frac {1}{2} \, x^{2} + x - 4 \, e^{2} - 6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 17, normalized size = 0.77 \begin {gather*} x + 5 e^{\frac {x^{2}}{2} + x - 4 e^{2} - 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 17, normalized size = 0.77 \begin {gather*} x + 5 \, e^{\left (\frac {1}{2} \, x^{2} + x - 4 \, e^{2} - 6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 17, normalized size = 0.77 \begin {gather*} x+5\,{\mathrm {e}}^{\frac {x^2}{2}+x-4\,{\mathrm {e}}^2-6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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