Optimal. Leaf size=28 \[ e^{\frac {x^2}{2 \left (3-e^{3 e^x}\right )}}-x^2 \]
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Rubi [A]
time = 1.69, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 92, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6873, 12, 6874,
6838} \begin {gather*} e^{\frac {x^2}{2 \left (3-e^{3 e^x}\right )}}-x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6838
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-36 x+24 e^{3 e^x} x-4 e^{6 e^x} x+e^{-\frac {x^2}{-6+2 e^{3 e^x}}} \left (6 x+e^{3 e^x} \left (-2 x+3 e^x x^2\right )\right )}{2 \left (3-e^{3 e^x}\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {-36 x+24 e^{3 e^x} x-4 e^{6 e^x} x+e^{-\frac {x^2}{-6+2 e^{3 e^x}}} \left (6 x+e^{3 e^x} \left (-2 x+3 e^x x^2\right )\right )}{\left (3-e^{3 e^x}\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-4 x+\frac {e^{-\frac {x^2}{2 \left (-3+e^{3 e^x}\right )}} x \left (6-2 e^{3 e^x}+3 e^{3 e^x+x} x\right )}{\left (3-e^{3 e^x}\right )^2}\right ) \, dx\\ &=-x^2+\frac {1}{2} \int \frac {e^{-\frac {x^2}{2 \left (-3+e^{3 e^x}\right )}} x \left (6-2 e^{3 e^x}+3 e^{3 e^x+x} x\right )}{\left (3-e^{3 e^x}\right )^2} \, dx\\ &=e^{\frac {x^2}{2 \left (3-e^{3 e^x}\right )}}-x^2\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.64, size = 32, normalized size = 1.14 \begin {gather*} \frac {1}{2} \left (2 e^{-\frac {x^2}{2 \left (-3+e^{3 e^x}\right )}}-2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 22, normalized size = 0.79
method | result | size |
risch | \(-x^{2}+{\mathrm e}^{-\frac {x^{2}}{2 \left ({\mathrm e}^{3 \,{\mathrm e}^{x}}-3\right )}}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 38, normalized size = 1.36 \begin {gather*} -{\left (x^{2} e^{\left (\frac {x^{2}}{2 \, {\left (e^{\left (3 \, e^{x}\right )} - 3\right )}}\right )} - 1\right )} e^{\left (-\frac {x^{2}}{2 \, {\left (e^{\left (3 \, e^{x}\right )} - 3\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 21, normalized size = 0.75 \begin {gather*} -x^{2} + e^{\left (-\frac {x^{2}}{2 \, {\left (e^{\left (3 \, e^{x}\right )} - 3\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 19, normalized size = 0.68 \begin {gather*} - x^{2} + e^{- \frac {x^{2}}{2 e^{3 e^{x}} - 6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.62, size = 23, normalized size = 0.82 \begin {gather*} {\mathrm {e}}^{-\frac {x^2}{2\,{\mathrm {e}}^{3\,{\mathrm {e}}^x}-6}}-x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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