Optimal. Leaf size=12 \[ \frac {1}{2} e^{x^2} x^2 \]
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Rubi [A]
time = 0.03, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2258, 2240,
2243} \begin {gather*} \frac {1}{2} e^{x^2} x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 2240
Rule 2243
Rule 2258
Rubi steps
\begin {align*} \int e^{x^2} x \left (1+x^2\right ) \, dx &=\int \left (e^{x^2} x+e^{x^2} x^3\right ) \, dx\\ &=\int e^{x^2} x \, dx+\int e^{x^2} x^3 \, dx\\ &=\frac {e^{x^2}}{2}+\frac {1}{2} e^{x^2} x^2-\int e^{x^2} x \, dx\\ &=\frac {1}{2} e^{x^2} x^2\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 12, normalized size = 1.00 \begin {gather*} \frac {1}{2} e^{x^2} x^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 10, normalized size = 0.83
method | result | size |
gosper | \(\frac {{\mathrm e}^{x^{2}} x^{2}}{2}\) | \(10\) |
derivativedivides | \(\frac {{\mathrm e}^{x^{2}} x^{2}}{2}\) | \(10\) |
default | \(\frac {{\mathrm e}^{x^{2}} x^{2}}{2}\) | \(10\) |
norman | \(\frac {{\mathrm e}^{x^{2}} x^{2}}{2}\) | \(10\) |
risch | \(\frac {{\mathrm e}^{x^{2}} x^{2}}{2}\) | \(10\) |
meijerg | \(-\frac {\left (-2 x^{2}+2\right ) {\mathrm e}^{x^{2}}}{4}+\frac {{\mathrm e}^{x^{2}}}{2}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.33, size = 18, normalized size = 1.50 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} + \frac {1}{2} \, e^{\left (x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.99, size = 9, normalized size = 0.75 \begin {gather*} \frac {1}{2} \, x^{2} 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 = 8, normalized size = 0.67 \begin {gather*} \frac {x^{2} e^{x^{2}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 18, normalized size = 1.50 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} + \frac {1}{2} \, e^{\left (x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 9, normalized size = 0.75 \begin {gather*} \frac {x^2\,{\mathrm {e}}^{x^2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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