Optimal. Leaf size=26 \[ 2 e^{-x} x^2 \left (e^x+e^{-3+x+x^2} x^2\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 18, normalized size of antiderivative = 0.69, number of steps
used = 9, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {6820, 2258,
2243, 2240} \begin {gather*} 2 x^2+2 e^{x^2-3} x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 2240
Rule 2243
Rule 2258
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (4 x+4 e^{-3+x^2} x^3 \left (2+x^2\right )\right ) \, dx\\ &=2 x^2+4 \int e^{-3+x^2} x^3 \left (2+x^2\right ) \, dx\\ &=2 x^2+4 \int \left (2 e^{-3+x^2} x^3+e^{-3+x^2} x^5\right ) \, dx\\ &=2 x^2+4 \int e^{-3+x^2} x^5 \, dx+8 \int e^{-3+x^2} x^3 \, dx\\ &=2 x^2+4 e^{-3+x^2} x^2+2 e^{-3+x^2} x^4-8 \int e^{-3+x^2} x \, dx-8 \int e^{-3+x^2} x^3 \, dx\\ &=-4 e^{-3+x^2}+2 x^2+2 e^{-3+x^2} x^4+8 \int e^{-3+x^2} x \, dx\\ &=2 x^2+2 e^{-3+x^2} x^4\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.09, size = 18, normalized size = 0.69 \begin {gather*} 2 x^2+2 e^{-3+x^2} x^4 \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs.
\(2(23)=46\).
time = 0.06, size = 54, normalized size = 2.08
method | result | size |
risch | \(2 x^{2}+2 x^{4} {\mathrm e}^{x^{2}-3}\) | \(18\) |
norman | \(\left (2 \,{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x^{2}+x -3} x^{4}\right ) {\mathrm e}^{-x}\) | \(26\) |
default | \(2 x^{2}+8 \,{\mathrm e}^{-3} \left (\frac {x^{2} {\mathrm e}^{x^{2}}}{2}-\frac {{\mathrm e}^{x^{2}}}{2}\right )+4 \,{\mathrm e}^{-3} \left (\frac {x^{4} {\mathrm e}^{x^{2}}}{2}-x^{2} {\mathrm e}^{x^{2}}+{\mathrm e}^{x^{2}}\right )\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 37, normalized size = 1.42 \begin {gather*} 2 \, x^{2} + 2 \, {\left (x^{4} - 2 \, x^{2} + 2\right )} e^{\left (x^{2} - 3\right )} + 4 \, {\left (x^{2} - 1\right )} e^{\left (x^{2} - 3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 24, normalized size = 0.92 \begin {gather*} 2 \, {\left (x^{4} e^{\left (x^{2} + x - 3\right )} + x^{2} e^{x}\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 20, normalized size = 0.77 \begin {gather*} 2 x^{4} e^{- x} e^{x^{2} + x - 3} + 2 x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 37, normalized size = 1.42 \begin {gather*} 2 \, {\left (x^{2} e^{3} + {\left (x^{4} - 2 \, x^{2} + 2\right )} e^{\left (x^{2}\right )} + 2 \, {\left (x^{2} - 1\right )} e^{\left (x^{2}\right )}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.55, size = 17, normalized size = 0.65 \begin {gather*} 2\,x^2+2\,x^4\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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