Optimal. Leaf size=22 \[ -\frac {e^{x^2}}{2}+\frac {1}{2} e^{x^2} x^2 \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2243, 2240}
\begin {gather*} \frac {1}{2} e^{x^2} x^2-\frac {e^{x^2}}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2240
Rule 2243
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*}
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Mathematica [A]
time = 0.02, size = 14, normalized size = 0.64 \begin {gather*} \frac {1}{2} e^{x^2} \left (-1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 17, normalized size = 0.77
method | result | size |
gosper | \(\frac {\left (x^{2}-1\right ) {\mathrm e}^{x^{2}}}{2}\) | \(12\) |
risch | \(\left (\frac {x^{2}}{2}-\frac {1}{2}\right ) {\mathrm e}^{x^{2}}\) | \(13\) |
meijerg | \(\frac {1}{2}-\frac {\left (-2 x^{2}+2\right ) {\mathrm e}^{x^{2}}}{4}\) | \(16\) |
derivativedivides | \(-\frac {{\mathrm e}^{x^{2}}}{2}+\frac {{\mathrm e}^{x^{2}} x^{2}}{2}\) | \(17\) |
default | \(-\frac {{\mathrm e}^{x^{2}}}{2}+\frac {{\mathrm e}^{x^{2}} x^{2}}{2}\) | \(17\) |
norman | \(-\frac {{\mathrm e}^{x^{2}}}{2}+\frac {{\mathrm e}^{x^{2}} x^{2}}{2}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 11, normalized size = 0.50 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 11, normalized size = 0.50 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 10, normalized size = 0.45 \begin {gather*} \frac {\left (x^{2} - 1\right ) e^{x^{2}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.77, size = 11, normalized size = 0.50 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 11, normalized size = 0.50 \begin {gather*} \frac {{\mathrm {e}}^{x^2}\,\left (x^2-1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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