Optimal. Leaf size=26 \[ \frac {e^{-2 x^2} x}{4 (2-x)^2 \log \left (\log \left (x^2\right )\right )} \]
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Rubi [F] time = 1.54, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{-2 x^2} \left (4-2 x+\left (-2-x+8 x^2-4 x^3\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right )}{\left (-32+48 x-24 x^2+4 x^3\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {e^{-2 x^2}}{2 (-2+x)^2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}+\frac {e^{-2 x^2} \left (-2-x+8 x^2-4 x^3\right )}{4 (-2+x)^3 \log \left (\log \left (x^2\right )\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^{-2 x^2} \left (-2-x+8 x^2-4 x^3\right )}{(-2+x)^3 \log \left (\log \left (x^2\right )\right )} \, dx-\frac {1}{2} \int \frac {e^{-2 x^2}}{(-2+x)^2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx\\ &=\frac {1}{4} \int \left (-\frac {4 e^{-2 x^2}}{\log \left (\log \left (x^2\right )\right )}-\frac {4 e^{-2 x^2}}{(-2+x)^3 \log \left (\log \left (x^2\right )\right )}-\frac {17 e^{-2 x^2}}{(-2+x)^2 \log \left (\log \left (x^2\right )\right )}-\frac {16 e^{-2 x^2}}{(-2+x) \log \left (\log \left (x^2\right )\right )}\right ) \, dx-\frac {1}{2} \int \frac {e^{-2 x^2}}{(-2+x)^2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{-2 x^2}}{(-2+x)^2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx\right )-4 \int \frac {e^{-2 x^2}}{(-2+x) \log \left (\log \left (x^2\right )\right )} \, dx-\frac {17}{4} \int \frac {e^{-2 x^2}}{(-2+x)^2 \log \left (\log \left (x^2\right )\right )} \, dx-\int \frac {e^{-2 x^2}}{\log \left (\log \left (x^2\right )\right )} \, dx-\int \frac {e^{-2 x^2}}{(-2+x)^3 \log \left (\log \left (x^2\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.24, size = 24, normalized size = 0.92 \begin {gather*} \frac {e^{-2 x^2} x}{4 (-2+x)^2 \log \left (\log \left (x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 26, normalized size = 1.00 \begin {gather*} \frac {x e^{\left (-2 \, x^{2}\right )}}{4 \, {\left (x^{2} - 4 \, x + 4\right )} \log \left (\log \left (x^{2}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 36, normalized size = 1.38 \begin {gather*} \frac {x e^{\left (-2 \, x^{2}\right )}}{4 \, {\left (x^{2} \log \left (\log \left (x^{2}\right )\right ) - 4 \, x \log \left (\log \left (x^{2}\right )\right ) + 4 \, \log \left (\log \left (x^{2}\right )\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.13, size = 56, normalized size = 2.15
method | result | size |
risch | \(\frac {x \,{\mathrm e}^{-2 x^{2}}}{4 \left (x^{2}-4 x +4\right ) \ln \left (2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 39, normalized size = 1.50 \begin {gather*} \frac {x e^{\left (-2 \, x^{2}\right )}}{4 \, {\left (x^{2} \log \relax (2) - 4 \, x \log \relax (2) + {\left (x^{2} - 4 \, x + 4\right )} \log \left (\log \relax (x)\right ) + 4 \, \log \relax (2)\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.81, size = 21, normalized size = 0.81 \begin {gather*} \frac {x\,{\mathrm {e}}^{-2\,x^2}}{4\,\ln \left (\ln \left (x^2\right )\right )\,{\left (x-2\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 37, normalized size = 1.42 \begin {gather*} \frac {x e^{- 2 x^{2}}}{4 x^{2} \log {\left (\log {\left (x^{2} \right )} \right )} - 16 x \log {\left (\log {\left (x^{2} \right )} \right )} + 16 \log {\left (\log {\left (x^{2} \right )} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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