Optimal. Leaf size=22 \[ 6 e^x \left (-1-e^{3-x-x^2} x\right ) \]
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Rubi [A]
time = 0.13, antiderivative size = 18, normalized size of antiderivative = 0.82, number of steps
used = 8, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {6820, 2225,
2258, 2236, 2243} \begin {gather*} -6 e^{3-x^2} x-6 e^x \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 2236
Rule 2243
Rule 2258
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-6 e^x+6 e^{3-x^2} \left (-1+2 x^2\right )\right ) \, dx\\ &=-\left (6 \int e^x \, dx\right )+6 \int e^{3-x^2} \left (-1+2 x^2\right ) \, dx\\ &=-6 e^x+6 \int \left (-e^{3-x^2}+2 e^{3-x^2} x^2\right ) \, dx\\ &=-6 e^x-6 \int e^{3-x^2} \, dx+12 \int e^{3-x^2} x^2 \, dx\\ &=-6 e^x-6 e^{3-x^2} x-3 e^3 \sqrt {\pi } \text {erf}(x)+6 \int e^{3-x^2} \, dx\\ &=-6 e^x-6 e^{3-x^2} x\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.18, size = 18, normalized size = 0.82 \begin {gather*} -6 e^x-6 e^{3-x^2} x \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.15, size = 40, normalized size = 1.82
method | result | size |
risch | \(-6 \,{\mathrm e}^{x}-6 x \,{\mathrm e}^{-x^{2}+3}\) | \(17\) |
norman | \(\left (-6 \,{\mathrm e}^{x} x -6 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}+x -3}\right ) {\mathrm e}^{-x^{2}-x +3}\) | \(28\) |
default | \(-3 \,{\mathrm e}^{3} \sqrt {\pi }\, \erf \left (x \right )+12 \,{\mathrm e}^{3} \left (-\frac {{\mathrm e}^{-x^{2}} x}{2}+\frac {\sqrt {\pi }\, \erf \left (x \right )}{4}\right )-6 \,{\mathrm e}^{x}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 16, normalized size = 0.73 \begin {gather*} -6 \, x e^{\left (-x^{2} + 3\right )} - 6 \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 29, normalized size = 1.32 \begin {gather*} -6 \, {\left (x e^{\left (2 \, x\right )} + e^{\left (x^{2} + 3 \, x - 3\right )}\right )} e^{\left (-x^{2} - 2 \, x + 3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 20, normalized size = 0.91 \begin {gather*} - 6 x e^{x} e^{- x^{2} - x + 3} - 6 e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 16, normalized size = 0.73 \begin {gather*} -6 \, x e^{\left (-x^{2} + 3\right )} - 6 \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 16, normalized size = 0.73 \begin {gather*} -6\,{\mathrm {e}}^x-6\,x\,{\mathrm {e}}^3\,{\mathrm {e}}^{-x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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