Optimal. Leaf size=26 \[ -\frac{1}{2} e^{-x^2} x^2-\frac{e^{-x^2}}{2} \]
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Rubi [A] time = 0.0199832, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2212, 2209} \[ -\frac{1}{2} e^{-x^2} x^2-\frac{e^{-x^2}}{2} \]
Antiderivative was successfully verified.
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Rule 2212
Rule 2209
Rubi steps
\begin{align*} \int e^{-x^2} x^3 \, dx &=-\frac{1}{2} e^{-x^2} x^2+\int e^{-x^2} x \, dx\\ &=-\frac{e^{-x^2}}{2}-\frac{1}{2} e^{-x^2} x^2\\ \end{align*}
Mathematica [A] time = 0.0023307, size = 16, normalized size = 0.62 \[ -\frac{1}{2} e^{-x^2} \left (x^2+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 14, normalized size = 0.5 \begin{align*} -{\frac{{x}^{2}+1}{2\,{{\rm e}^{{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.942501, size = 18, normalized size = 0.69 \begin{align*} -\frac{1}{2} \,{\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.505699, size = 34, normalized size = 1.31 \begin{align*} -\frac{1}{2} \,{\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.083491, size = 12, normalized size = 0.46 \begin{align*} \frac{\left (- x^{2} - 1\right ) e^{- x^{2}}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08975, size = 18, normalized size = 0.69 \begin{align*} -\frac{1}{2} \,{\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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