Optimal. Leaf size=23 \[ \frac {4 e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2}}{x^2 \log (x)} \]
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Rubi [F] time = 2.60, 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^2-4 x \log \left (\frac {1}{x^2}\right )+2 \log ^2\left (\frac {1}{x^2}\right )} \left (-4+\left (-8+32 x+16 x^2+(-32-16 x) \log \left (\frac {1}{x^2}\right )\right ) \log (x)\right )}{x^3 \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2} \left (-4+\left (-8+32 x+16 x^2+(-32-16 x) \log \left (\frac {1}{x^2}\right )\right ) \log (x)\right )}{x^3 \log ^2(x)} \, dx\\ &=\int \left (-\frac {4 e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2}}{x^3 \log ^2(x)}+\frac {8 e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2} \left (-1+4 x+2 x^2-4 \log \left (\frac {1}{x^2}\right )-2 x \log \left (\frac {1}{x^2}\right )\right )}{x^3 \log (x)}\right ) \, dx\\ &=-\left (4 \int \frac {e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2}}{x^3 \log ^2(x)} \, dx\right )+8 \int \frac {e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2} \left (-1+4 x+2 x^2-4 \log \left (\frac {1}{x^2}\right )-2 x \log \left (\frac {1}{x^2}\right )\right )}{x^3 \log (x)} \, dx\\ &=-\left (4 \int \frac {e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2}}{x^3 \log ^2(x)} \, dx\right )+8 \int \left (-\frac {e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2}}{x^3 \log (x)}+\frac {4 e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2}}{x^2 \log (x)}+\frac {2 e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2}}{x \log (x)}-\frac {4 e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2} \log \left (\frac {1}{x^2}\right )}{x^3 \log (x)}-\frac {2 e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2} \log \left (\frac {1}{x^2}\right )}{x^2 \log (x)}\right ) \, dx\\ &=-\left (4 \int \frac {e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2}}{x^3 \log ^2(x)} \, dx\right )-8 \int \frac {e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2}}{x^3 \log (x)} \, dx+16 \int \frac {e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2}}{x \log (x)} \, dx-16 \int \frac {e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2} \log \left (\frac {1}{x^2}\right )}{x^2 \log (x)} \, dx+32 \int \frac {e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2}}{x^2 \log (x)} \, dx-32 \int \frac {e^{2 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2} \log \left (\frac {1}{x^2}\right )}{x^3 \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.59, size = 29, normalized size = 1.26 \begin {gather*} \frac {4 e^{2 \left (x^2+\log ^2\left (\frac {1}{x^2}\right )\right )} \left (\frac {1}{x^2}\right )^{1-4 x}}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 33, normalized size = 1.43 \begin {gather*} -\frac {8 \, e^{\left (2 \, x^{2} - 4 \, x \log \left (\frac {1}{x^{2}}\right ) + 2 \, \log \left (\frac {1}{x^{2}}\right )^{2}\right )}}{x^{2} \log \left (\frac {1}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 27, normalized size = 1.17 \begin {gather*} \frac {4 \, e^{\left (2 \, x^{2} + 8 \, x \log \relax (x) + 8 \, \log \relax (x)^{2}\right )}}{x^{2} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 72, normalized size = 3.13
method | result | size |
risch | \(\frac {4 \,{\mathrm e}^{\frac {\left (-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+4 \ln \relax (x )+2 x \right )^{2}}{2}}}{x^{2} \ln \relax (x )}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 27, normalized size = 1.17 \begin {gather*} \frac {4 \, e^{\left (2 \, x^{2} + 8 \, x \log \relax (x) + 8 \, \log \relax (x)^{2}\right )}}{x^{2} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.09, size = 29, normalized size = 1.26 \begin {gather*} \frac {4\,{\mathrm {e}}^{2\,{\ln \left (\frac {1}{x^2}\right )}^2}\,{\mathrm {e}}^{2\,x^2}\,{\left (x^8\right )}^x}{x^2\,\ln \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 27, normalized size = 1.17 \begin {gather*} \frac {4 e^{2 x^{2} + 8 x \log {\relax (x )} + 8 \log {\relax (x )}^{2}}}{x^{2} \log {\relax (x )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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