Optimal. Leaf size=35 \[ e^{\left (-x^2+(2-x) x^2-\frac {x}{2 x-\frac {\log (x)}{x}}\right )^2} \]
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Rubi [F] time = 80.25, 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 {x^4-4 x^6+4 x^7+4 x^8-8 x^9+4 x^{10}+\left (2 x^4-2 x^5-4 x^6+8 x^7-4 x^8\right ) \log (x)+\left (x^4-2 x^5+x^6\right ) \log ^2(x)}{4 x^4-4 x^2 \log (x)+\log ^2(x)}\right ) \left (-2 x^3+4 x^5-4 x^6+16 x^7-24 x^8-32 x^9+80 x^{10}-48 x^{11}+\left (2 x^3+2 x^4-24 x^5+32 x^6+48 x^7-120 x^8+72 x^9\right ) \log (x)+\left (8 x^3-10 x^4-24 x^5+60 x^6-36 x^7\right ) \log ^2(x)+\left (4 x^3-10 x^4+6 x^5\right ) \log ^3(x)\right )}{-8 x^6+12 x^4 \log (x)-6 x^2 \log ^2(x)+\log ^3(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 {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) \left (2 x^3-4 x^5+4 x^6-16 x^7+24 x^8+32 x^9-80 x^{10}+48 x^{11}-\left (2 x^3+2 x^4-24 x^5+32 x^6+48 x^7-120 x^8+72 x^9\right ) \log (x)-\left (8 x^3-10 x^4-24 x^5+60 x^6-36 x^7\right ) \log ^2(x)-\left (4 x^3-10 x^4+6 x^5\right ) \log ^3(x)\right )}{\left (2 x^2-\log (x)\right )^3} \, dx\\ &=\int \left (2 \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3 \left (2-5 x+3 x^2\right )-\frac {2 \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3 \left (-1+4 x^2\right )}{\left (2 x^2-\log (x)\right )^3}-\frac {2 \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3 \left (-1-x-4 x^2+4 x^3\right )}{\left (2 x^2-\log (x)\right )^2}+\frac {2 \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3 (-4+5 x)}{2 x^2-\log (x)}\right ) \, dx\\ &=2 \int \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3 \left (2-5 x+3 x^2\right ) \, dx-2 \int \frac {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3 \left (-1+4 x^2\right )}{\left (2 x^2-\log (x)\right )^3} \, dx-2 \int \frac {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3 \left (-1-x-4 x^2+4 x^3\right )}{\left (2 x^2-\log (x)\right )^2} \, dx+2 \int \frac {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3 (-4+5 x)}{2 x^2-\log (x)} \, dx\\ &=2 \int \left (2 \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3-5 \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^4+3 \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^5\right ) \, dx-2 \int \left (-\frac {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3}{\left (2 x^2-\log (x)\right )^3}+\frac {4 \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^5}{\left (2 x^2-\log (x)\right )^3}\right ) \, dx-2 \int \left (-\frac {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3}{\left (2 x^2-\log (x)\right )^2}-\frac {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^4}{\left (2 x^2-\log (x)\right )^2}-\frac {4 \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^5}{\left (2 x^2-\log (x)\right )^2}+\frac {4 \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^6}{\left (2 x^2-\log (x)\right )^2}\right ) \, dx+2 \int \left (-\frac {4 \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3}{2 x^2-\log (x)}+\frac {5 \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^4}{2 x^2-\log (x)}\right ) \, dx\\ &=2 \int \frac {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3}{\left (2 x^2-\log (x)\right )^3} \, dx+2 \int \frac {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3}{\left (2 x^2-\log (x)\right )^2} \, dx+2 \int \frac {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^4}{\left (2 x^2-\log (x)\right )^2} \, dx+4 \int \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3 \, dx+6 \int \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^5 \, dx-8 \int \frac {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^5}{\left (2 x^2-\log (x)\right )^3} \, dx+8 \int \frac {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^5}{\left (2 x^2-\log (x)\right )^2} \, dx-8 \int \frac {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^6}{\left (2 x^2-\log (x)\right )^2} \, dx-8 \int \frac {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^3}{2 x^2-\log (x)} \, dx-10 \int \exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^4 \, dx+10 \int \frac {\exp \left (\frac {x^4 \left (1-2 x^2+2 x^3+\log (x)-x \log (x)\right )^2}{\left (2 x^2-\log (x)\right )^2}\right ) x^4}{2 x^2-\log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.40, size = 77, normalized size = 2.20 \begin {gather*} e^{\frac {x^4 \left (\left (1-2 x^2+2 x^3\right )^2+(-1+x)^2 \log ^2(x)\right )}{\left (-2 x^2+\log (x)\right )^2}} x^{-\frac {2 x^4 \left (-1+x+2 x^2-4 x^3+2 x^4\right )}{\left (-2 x^2+\log (x)\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 95, normalized size = 2.71 \begin {gather*} e^{\left (\frac {4 \, x^{10} - 8 \, x^{9} + 4 \, x^{8} + 4 \, x^{7} - 4 \, x^{6} + x^{4} + {\left (x^{6} - 2 \, x^{5} + x^{4}\right )} \log \relax (x)^{2} - 2 \, {\left (2 \, x^{8} - 4 \, x^{7} + 2 \, x^{6} + x^{5} - x^{4}\right )} \log \relax (x)}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.65, size = 357, normalized size = 10.20 \begin {gather*} e^{\left (\frac {4 \, x^{10}}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} - \frac {8 \, x^{9}}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} - \frac {4 \, x^{8} \log \relax (x)}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} + \frac {4 \, x^{8}}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} + \frac {8 \, x^{7} \log \relax (x)}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} + \frac {x^{6} \log \relax (x)^{2}}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} + \frac {4 \, x^{7}}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} - \frac {4 \, x^{6} \log \relax (x)}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} - \frac {2 \, x^{5} \log \relax (x)^{2}}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} - \frac {4 \, x^{6}}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} - \frac {2 \, x^{5} \log \relax (x)}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} + \frac {x^{4} \log \relax (x)^{2}}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} + \frac {2 \, x^{4} \log \relax (x)}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} + \frac {x^{4}}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 38, normalized size = 1.09
method | result | size |
risch | \({\mathrm e}^{\frac {x^{4} \left (-2 x^{3}+x \ln \relax (x )+2 x^{2}-\ln \relax (x )-1\right )^{2}}{\left (-2 x^{2}+\ln \relax (x )\right )^{2}}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 2 \, \int \frac {{\left (24 \, x^{11} - 40 \, x^{10} + 16 \, x^{9} + 12 \, x^{8} - 8 \, x^{7} + 2 \, x^{6} - 2 \, x^{5} - {\left (3 \, x^{5} - 5 \, x^{4} + 2 \, x^{3}\right )} \log \relax (x)^{3} + x^{3} + {\left (18 \, x^{7} - 30 \, x^{6} + 12 \, x^{5} + 5 \, x^{4} - 4 \, x^{3}\right )} \log \relax (x)^{2} - {\left (36 \, x^{9} - 60 \, x^{8} + 24 \, x^{7} + 16 \, x^{6} - 12 \, x^{5} + x^{4} + x^{3}\right )} \log \relax (x)\right )} e^{\left (\frac {4 \, x^{10} - 8 \, x^{9} + 4 \, x^{8} + 4 \, x^{7} - 4 \, x^{6} + x^{4} + {\left (x^{6} - 2 \, x^{5} + x^{4}\right )} \log \relax (x)^{2} - 2 \, {\left (2 \, x^{8} - 4 \, x^{7} + 2 \, x^{6} + x^{5} - x^{4}\right )} \log \relax (x)}{4 \, x^{4} - 4 \, x^{2} \log \relax (x) + \log \relax (x)^{2}}\right )}}{8 \, x^{6} - 12 \, x^{4} \log \relax (x) + 6 \, x^{2} \log \relax (x)^{2} - \log \relax (x)^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.83, size = 284, normalized size = 8.11 \begin {gather*} \frac {{\mathrm {e}}^{\frac {x^4\,{\ln \relax (x)}^2}{4\,x^4-4\,x^2\,\ln \relax (x)+{\ln \relax (x)}^2}}\,{\mathrm {e}}^{\frac {x^6\,{\ln \relax (x)}^2}{4\,x^4-4\,x^2\,\ln \relax (x)+{\ln \relax (x)}^2}}\,{\mathrm {e}}^{-\frac {2\,x^5\,{\ln \relax (x)}^2}{4\,x^4-4\,x^2\,\ln \relax (x)+{\ln \relax (x)}^2}}\,{\mathrm {e}}^{\frac {x^4}{4\,x^4-4\,x^2\,\ln \relax (x)+{\ln \relax (x)}^2}}\,{\mathrm {e}}^{-\frac {4\,x^6}{4\,x^4-4\,x^2\,\ln \relax (x)+{\ln \relax (x)}^2}}\,{\mathrm {e}}^{\frac {4\,x^7}{4\,x^4-4\,x^2\,\ln \relax (x)+{\ln \relax (x)}^2}}\,{\mathrm {e}}^{\frac {4\,x^8}{4\,x^4-4\,x^2\,\ln \relax (x)+{\ln \relax (x)}^2}}\,{\mathrm {e}}^{\frac {4\,x^{10}}{4\,x^4-4\,x^2\,\ln \relax (x)+{\ln \relax (x)}^2}}\,{\mathrm {e}}^{-\frac {8\,x^9}{4\,x^4-4\,x^2\,\ln \relax (x)+{\ln \relax (x)}^2}}}{x^{\frac {2\,\left (2\,x^8-4\,x^7+2\,x^6+x^5-x^4\right )}{4\,x^4-4\,x^2\,\ln \relax (x)+{\ln \relax (x)}^2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.76, size = 94, normalized size = 2.69 \begin {gather*} e^{\frac {4 x^{10} - 8 x^{9} + 4 x^{8} + 4 x^{7} - 4 x^{6} + x^{4} + \left (x^{6} - 2 x^{5} + x^{4}\right ) \log {\relax (x )}^{2} + \left (- 4 x^{8} + 8 x^{7} - 4 x^{6} - 2 x^{5} + 2 x^{4}\right ) \log {\relax (x )}}{4 x^{4} - 4 x^{2} \log {\relax (x )} + \log {\relax (x )}^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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