Optimal. Leaf size=33 \[ e^{-x-\frac {\left (e^x+\frac {\log (x)}{x \log \left (x^2\right )}\right )^2}{4 x^4}} x \]
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Rubi [F] time = 27.95, 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 {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{x^6 \log ^3\left (x^2\right )} \, dx\\ &=\frac {1}{2} \int \left (-\frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) (-2+x)}{x^4}-\frac {\exp \left (x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (-2 \log (x)+\log \left (x^2\right )-5 \log (x) \log \left (x^2\right )+x \log (x) \log \left (x^2\right )\right )}{x^5 \log ^2\left (x^2\right )}+\frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (2 \log ^2(x)-\log (x) \log \left (x^2\right )+3 \log ^2(x) \log \left (x^2\right )+2 x^6 \log ^3\left (x^2\right )-2 x^7 \log ^3\left (x^2\right )\right )}{x^6 \log ^3\left (x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) (-2+x)}{x^4} \, dx\right )-\frac {1}{2} \int \frac {\exp \left (x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (-2 \log (x)+\log \left (x^2\right )-5 \log (x) \log \left (x^2\right )+x \log (x) \log \left (x^2\right )\right )}{x^5 \log ^2\left (x^2\right )} \, dx+\frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (2 \log ^2(x)-\log (x) \log \left (x^2\right )+3 \log ^2(x) \log \left (x^2\right )+2 x^6 \log ^3\left (x^2\right )-2 x^7 \log ^3\left (x^2\right )\right )}{x^6 \log ^3\left (x^2\right )} \, dx\\ &=-\left (\frac {1}{2} \int \left (-\frac {2 \exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^4}+\frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^3}\right ) \, dx\right )+\frac {1}{2} \int \left (-2 \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) (-1+x)+\frac {2 \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^3\left (x^2\right )}+\frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x) (-1+3 \log (x))}{x^6 \log ^2\left (x^2\right )}\right ) \, dx-\frac {1}{2} \int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (-2 \log (x)+\log \left (x^2\right )-5 \log (x) \log \left (x^2\right )+x \log (x) \log \left (x^2\right )\right )}{x^5 \log ^2\left (x^2\right )} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^3} \, dx\right )+\frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x) (-1+3 \log (x))}{x^6 \log ^2\left (x^2\right )} \, dx-\frac {1}{2} \int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (\log \left (x^2\right )+\log (x) \left (-2+(-5+x) \log \left (x^2\right )\right )\right )}{x^5 \log ^2\left (x^2\right )} \, dx-\int \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) (-1+x) \, dx+\int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^4} \, dx+\int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^3\left (x^2\right )} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^3} \, dx\right )+\frac {1}{2} \int \left (-\frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^6 \log ^2\left (x^2\right )}+\frac {3 \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^2\left (x^2\right )}\right ) \, dx-\frac {1}{2} \int \left (-\frac {2 \exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^5 \log ^2\left (x^2\right )}+\frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) (1-5 \log (x)+x \log (x))}{x^5 \log \left (x^2\right )}\right ) \, dx+\int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^4} \, dx-\int \left (-\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )+\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) x\right ) \, dx+\int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^3\left (x^2\right )} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^3} \, dx\right )-\frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^6 \log ^2\left (x^2\right )} \, dx-\frac {1}{2} \int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) (1-5 \log (x)+x \log (x))}{x^5 \log \left (x^2\right )} \, dx+\frac {3}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^2\left (x^2\right )} \, dx+\int \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \, dx+\int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^4} \, dx-\int \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) x \, dx+\int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^3\left (x^2\right )} \, dx+\int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^5 \log ^2\left (x^2\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.47, size = 56, normalized size = 1.70 \begin {gather*} e^{-\frac {e^{2 x}}{4 x^4}-x-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}} x^{1-\frac {e^x}{2 x^5 \log \left (x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 29, normalized size = 0.88 \begin {gather*} x e^{\left (-\frac {16 \, x^{7} + 4 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{x} + 1}{16 \, x^{6}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left ({\left (2 \, x^{7} - 2 \, x^{6} + {\left (x^{3} - 2 \, x^{2}\right )} e^{\left (2 \, x\right )}\right )} \log \left (x^{2}\right )^{3} + {\left ({\left (x^{2} - 5 \, x\right )} e^{x} \log \relax (x) + x e^{x}\right )} \log \left (x^{2}\right )^{2} - {\left ({\left (2 \, x e^{x} - 1\right )} \log \relax (x) + 3 \, \log \relax (x)^{2}\right )} \log \left (x^{2}\right ) - 2 \, \log \relax (x)^{2}\right )} e^{\left (-\frac {2 \, x e^{x} \log \left (x^{2}\right ) \log \relax (x) + {\left (4 \, x^{7} + x^{2} e^{\left (2 \, x\right )}\right )} \log \left (x^{2}\right )^{2} + \log \relax (x)^{2}}{4 \, x^{6} \log \left (x^{2}\right )^{2}}\right )}}{2 \, x^{6} \log \left (x^{2}\right )^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.39, size = 549, normalized size = 16.64
method | result | size |
risch | \(x \,{\mathrm e}^{-\frac {-4 \mathrm {csgn}\left (i x^{2}\right )^{6} \pi ^{2} x^{7}+16 \mathrm {csgn}\left (i x^{2}\right )^{5} \mathrm {csgn}\left (i x \right ) \pi ^{2} x^{7}-24 \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i x \right )^{2} \pi ^{2} x^{7}+16 \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x \right )^{3} \pi ^{2} x^{7}-4 \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )^{4} \pi ^{2} x^{7}-\mathrm {csgn}\left (i x^{2}\right )^{6} \pi ^{2} {\mathrm e}^{2 x} x^{2}+4 \mathrm {csgn}\left (i x^{2}\right )^{5} \mathrm {csgn}\left (i x \right ) \pi ^{2} {\mathrm e}^{2 x} x^{2}-6 \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i x \right )^{2} \pi ^{2} {\mathrm e}^{2 x} x^{2}+4 \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x \right )^{3} \pi ^{2} {\mathrm e}^{2 x} x^{2}-\mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )^{4} \pi ^{2} {\mathrm e}^{2 x} x^{2}-32 i \mathrm {csgn}\left (i x^{2}\right )^{3} \pi \ln \relax (x ) x^{7}+16 i \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right ) \pi \ln \relax (x ) {\mathrm e}^{2 x} x^{2}-8 i \mathrm {csgn}\left (i x^{2}\right )^{3} \pi \ln \relax (x ) {\mathrm e}^{2 x} x^{2}+64 i \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right ) \pi \ln \relax (x ) x^{7}+8 i \ln \relax (x ) {\mathrm e}^{x} x \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right ) \pi -4 i \ln \relax (x ) {\mathrm e}^{x} x \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2} \pi +64 x^{7} \ln \relax (x )^{2}-4 i \ln \relax (x ) {\mathrm e}^{x} x \mathrm {csgn}\left (i x^{2}\right )^{3} \pi -8 i \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2} \pi \ln \relax (x ) {\mathrm e}^{2 x} x^{2}-32 i \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2} \pi \ln \relax (x ) x^{7}+16 \ln \relax (x )^{2} {\mathrm e}^{2 x} x^{2}+16 x \,{\mathrm e}^{x} \ln \relax (x )^{2}+4 \ln \relax (x )^{2}}{4 x^{6} \left (4 \ln \relax (x )-i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+2 i \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )^{2}}}\) | \(549\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 28, normalized size = 0.85 \begin {gather*} x e^{\left (-x - \frac {e^{\left (2 \, x\right )}}{4 \, x^{4}} - \frac {e^{x}}{4 \, x^{5}} - \frac {1}{16 \, x^{6}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.13, size = 48, normalized size = 1.45 \begin {gather*} x\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^x\,\ln \relax (x)}{2\,x^5\,\ln \left (x^2\right )}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-\frac {{\ln \relax (x)}^2}{4\,x^6\,{\ln \left (x^2\right )}^2}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{2\,x}}{4\,x^4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.70, size = 48, normalized size = 1.45 \begin {gather*} x e^{- \frac {x e^{x} \log {\relax (x )}^{2} + \left (4 x^{7} + x^{2} e^{2 x}\right ) \log {\relax (x )}^{2} + \frac {\log {\relax (x )}^{2}}{4}}{4 x^{6} \log {\relax (x )}^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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