Optimal. Leaf size=25 \[ -2+x+\frac {1}{5} x^2 \left (3-e^{2 x^2}-\log (x)\right ) \]
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Rubi [A]
time = 0.08, antiderivative size = 32, normalized size of antiderivative = 1.28, number of steps
used = 9, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {12, 1607, 2258,
2240, 2243, 2341} \begin {gather*} -\frac {1}{5} e^{2 x^2} x^2+\frac {3 x^2}{5}-\frac {1}{5} x^2 \log (x)+x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1607
Rule 2240
Rule 2243
Rule 2258
Rule 2341
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \left (5+5 x+e^{2 x^2} \left (-2 x-4 x^3\right )-2 x \log (x)\right ) \, dx\\ &=x+\frac {x^2}{2}+\frac {1}{5} \int e^{2 x^2} \left (-2 x-4 x^3\right ) \, dx-\frac {2}{5} \int x \log (x) \, dx\\ &=x+\frac {3 x^2}{5}-\frac {1}{5} x^2 \log (x)+\frac {1}{5} \int e^{2 x^2} x \left (-2-4 x^2\right ) \, dx\\ &=x+\frac {3 x^2}{5}-\frac {1}{5} x^2 \log (x)+\frac {1}{5} \int \left (-2 e^{2 x^2} x-4 e^{2 x^2} x^3\right ) \, dx\\ &=x+\frac {3 x^2}{5}-\frac {1}{5} x^2 \log (x)-\frac {2}{5} \int e^{2 x^2} x \, dx-\frac {4}{5} \int e^{2 x^2} x^3 \, dx\\ &=-\frac {1}{10} e^{2 x^2}+x+\frac {3 x^2}{5}-\frac {1}{5} e^{2 x^2} x^2-\frac {1}{5} x^2 \log (x)+\frac {2}{5} \int e^{2 x^2} x \, dx\\ &=x+\frac {3 x^2}{5}-\frac {1}{5} e^{2 x^2} x^2-\frac {1}{5} x^2 \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.12, size = 32, normalized size = 1.28 \begin {gather*} x+\frac {3 x^2}{5}-\frac {1}{5} e^{2 x^2} x^2-\frac {1}{5} x^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 26, normalized size = 1.04
method | result | size |
default | \(x +\frac {3 x^{2}}{5}-\frac {x^{2} \ln \left (x \right )}{5}-\frac {{\mathrm e}^{2 x^{2}} x^{2}}{5}\) | \(26\) |
norman | \(x +\frac {3 x^{2}}{5}-\frac {x^{2} \ln \left (x \right )}{5}-\frac {{\mathrm e}^{2 x^{2}} x^{2}}{5}\) | \(26\) |
risch | \(x +\frac {3 x^{2}}{5}-\frac {x^{2} \ln \left (x \right )}{5}-\frac {{\mathrm e}^{2 x^{2}} x^{2}}{5}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 25, normalized size = 1.00 \begin {gather*} -\frac {1}{5} \, x^{2} e^{\left (2 \, x^{2}\right )} - \frac {1}{5} \, x^{2} \log \left (x\right ) + \frac {3}{5} \, x^{2} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 25, normalized size = 1.00 \begin {gather*} -\frac {1}{5} \, x^{2} e^{\left (2 \, x^{2}\right )} - \frac {1}{5} \, x^{2} \log \left (x\right ) + \frac {3}{5} \, x^{2} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 27, normalized size = 1.08 \begin {gather*} - \frac {x^{2} e^{2 x^{2}}}{5} - \frac {x^{2} \log {\left (x \right )}}{5} + \frac {3 x^{2}}{5} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (18) = 36\).
time = 0.39, size = 37, normalized size = 1.48 \begin {gather*} -\frac {1}{5} \, x^{2} \log \left (x\right ) + \frac {3}{5} \, x^{2} - \frac {1}{10} \, {\left (2 \, x^{2} - 1\right )} e^{\left (2 \, x^{2}\right )} + x - \frac {1}{10} \, e^{\left (2 \, x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.55, size = 22, normalized size = 0.88 \begin {gather*} \frac {x\,\left (3\,x-x\,{\mathrm {e}}^{2\,x^2}-x\,\ln \left (x\right )+5\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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