Optimal. Leaf size=26 \[ -e^{-x} x^2-2 e^{-x} x-2 e^{-x} \]
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Rubi [A] time = 0.0179771, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2176, 2194} \[ -e^{-x} x^2-2 e^{-x} x-2 e^{-x} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{-x} x^2 \, dx &=-e^{-x} x^2+2 \int e^{-x} x \, dx\\ &=-2 e^{-x} x-e^{-x} x^2+2 \int e^{-x} \, dx\\ &=-2 e^{-x}-2 e^{-x} x-e^{-x} x^2\\ \end{align*}
Mathematica [A] time = 0.0059813, size = 16, normalized size = 0.62 \[ e^{-x} \left (-x^2-2 x-2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 15, normalized size = 0.6 \begin{align*} -{\frac{{x}^{2}+2\,x+2}{{{\rm e}^{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.938155, size = 19, normalized size = 0.73 \begin{align*} -{\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06137, size = 34, normalized size = 1.31 \begin{align*} -{\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.078733, size = 12, normalized size = 0.46 \begin{align*} \left (- x^{2} - 2 x - 2\right ) e^{- x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05395, size = 19, normalized size = 0.73 \begin{align*} -{\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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