Optimal. Leaf size=30 \[ 4 \left (4+\left (1+\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )^2\right ) \]
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Rubi [F] time = 4.01, 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 {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )+\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (32-32 x+16 \log \left (e^{-x} x\right )-16 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16 \log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (1+\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right ) \left (-2 (-1+x)-\log \left (e^{-x} x\right ) \left (-1+\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right )} \, dx\\ &=16 \int \frac {\log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (1+\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right ) \left (-2 (-1+x)-\log \left (e^{-x} x\right ) \left (-1+\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right )} \, dx\\ &=16 \int \left (\frac {\log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (2-2 x+\log \left (e^{-x} x\right )-\log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right )}-\frac {\log ^{-1+\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (-2+2 x-\log \left (e^{-x} x\right )+\log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right )}\right ) \, dx\\ &=16 \int \frac {\log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (2-2 x+\log \left (e^{-x} x\right )-\log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right )} \, dx-16 \int \frac {\log ^{-1+\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (-2+2 x-\log \left (e^{-x} x\right )+\log \left (e^{-x} x\right ) \log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )\right )}{x^2 \log \left (e^{-x} x\right )} \, dx\\ &=16 \int \left (\frac {\left (2-2 x+\log \left (e^{-x} x\right )\right ) \log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right )}-\frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )}{x^2}\right ) \, dx-16 \int \left (\frac {\left (-2+2 x-\log \left (e^{-x} x\right )\right ) \log ^{-1+\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right )}+\frac {\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )}{x^2}\right ) \, dx\\ &=16 \int \frac {\left (2-2 x+\log \left (e^{-x} x\right )\right ) \log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right )} \, dx-16 \int \frac {\left (-2+2 x-\log \left (e^{-x} x\right )\right ) \log ^{-1+\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right )} \, dx-16 \int \frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )}{x^2} \, dx-16 \int \frac {\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )}{x^2} \, dx\\ &=16 \int \left (\frac {\log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2}-\frac {2 (-1+x) \log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right )}\right ) \, dx-16 \int \left (-\frac {\log ^{-1+\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2}+\frac {2 (-1+x) \log ^{-1+\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right )}\right ) \, dx-16 \int \frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )}{x^2} \, dx-16 \int \frac {\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )}{x^2} \, dx\\ &=16 \int \frac {\log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2} \, dx+16 \int \frac {\log ^{-1+\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2} \, dx-16 \int \frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )}{x^2} \, dx-16 \int \frac {\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )}{x^2} \, dx-32 \int \frac {(-1+x) \log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right )} \, dx-32 \int \frac {(-1+x) \log ^{-1+\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right )} \, dx\\ &=16 \int \frac {\log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2} \, dx+16 \int \frac {\log ^{-1+\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2} \, dx-16 \int \frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )}{x^2} \, dx-16 \int \frac {\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )}{x^2} \, dx-32 \int \left (-\frac {\log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right )}+\frac {\log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x \log \left (e^{-x} x\right )}\right ) \, dx-32 \int \left (-\frac {\log ^{-1+\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right )}+\frac {\log ^{-1+\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x \log \left (e^{-x} x\right )}\right ) \, dx\\ &=16 \int \frac {\log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2} \, dx+16 \int \frac {\log ^{-1+\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2} \, dx-16 \int \frac {\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )}{x^2} \, dx-16 \int \frac {\log ^{\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \log \left (\log \left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right )}{x^2} \, dx+32 \int \frac {\log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right )} \, dx-32 \int \frac {\log ^{-1+\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x \log \left (e^{-x} x\right )} \, dx+32 \int \frac {\log ^{-1+\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x^2 \log \left (e^{-x} x\right )} \, dx-32 \int \frac {\log ^{-1+\frac {4}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )}{x \log \left (e^{-x} x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 48, normalized size = 1.60 \begin {gather*} 4 \log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right ) \left (2+\log ^{\frac {2}{x}}\left (\frac {1}{9} x \log ^2\left (e^{-x} x\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 43, normalized size = 1.43 \begin {gather*} 4 \, \log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )^{\frac {4}{x}} + 8 \, \log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )^{\frac {2}{x}} \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 {16 \, {\left ({\left (\log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right ) \log \left (x e^{\left (-x\right )}\right ) \log \left (\log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )\right ) + 2 \, x - \log \left (x e^{\left (-x\right )}\right ) - 2\right )} \log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )^{\frac {4}{x}} + {\left (\log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right ) \log \left (x e^{\left (-x\right )}\right ) \log \left (\log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )\right ) + 2 \, x - \log \left (x e^{\left (-x\right )}\right ) - 2\right )} \log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right )^{\frac {2}{x}}\right )}}{x^{2} \log \left (\frac {1}{9} \, x \log \left (x e^{\left (-x\right )}\right )^{2}\right ) \log \left (x e^{\left (-x\right )}\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (-16 \ln \left (x \,{\mathrm e}^{-x}\right ) \ln \left (\frac {x \ln \left (x \,{\mathrm e}^{-x}\right )^{2}}{9}\right ) \ln \left (\ln \left (\frac {x \ln \left (x \,{\mathrm e}^{-x}\right )^{2}}{9}\right )\right )+16 \ln \left (x \,{\mathrm e}^{-x}\right )-32 x +32\right ) {\mathrm e}^{\frac {4 \ln \left (\ln \left (\frac {x \ln \left (x \,{\mathrm e}^{-x}\right )^{2}}{9}\right )\right )}{x}}+\left (-16 \ln \left (x \,{\mathrm e}^{-x}\right ) \ln \left (\frac {x \ln \left (x \,{\mathrm e}^{-x}\right )^{2}}{9}\right ) \ln \left (\ln \left (\frac {x \ln \left (x \,{\mathrm e}^{-x}\right )^{2}}{9}\right )\right )+16 \ln \left (x \,{\mathrm e}^{-x}\right )-32 x +32\right ) {\mathrm e}^{\frac {2 \ln \left (\ln \left (\frac {x \ln \left (x \,{\mathrm e}^{-x}\right )^{2}}{9}\right )\right )}{x}}}{x^{2} \ln \left (x \,{\mathrm e}^{-x}\right ) \ln \left (\frac {x \ln \left (x \,{\mathrm e}^{-x}\right )^{2}}{9}\right )}\, dx\]
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*} 4 \, {\left (-2 \, \log \relax (3) + \log \relax (x) + 2 \, \log \left (-x + \log \relax (x)\right )\right )}^{\frac {4}{x}} + 8 \, {\left (-2 \, \log \relax (3) + \log \relax (x) + 2 \, \log \left (-x + \log \relax (x)\right )\right )}^{\frac {2}{x}} + 16 \, \int -\frac {2 \, {\left ({\left (3 \, x - \log \relax (x) - 2\right )} \log \left (x - \log \relax (x)\right ) - {\left (3 \, x - \log \relax (x) - 2\right )} \log \left (-x + \log \relax (x)\right )\right )} {\left (-2 \, \log \relax (3) + \log \relax (x) + 2 \, \log \left (-x + \log \relax (x)\right )\right )}^{\frac {4}{x}}}{4 \, x^{3} \log \relax (3)^{2} - x^{2} \log \relax (x)^{3} + {\left (x^{3} + 4 \, x^{2} \log \relax (3)\right )} \log \relax (x)^{2} - 2 \, {\left (2 \, x^{3} \log \relax (3) + x^{2} \log \relax (x)^{2} - {\left (x^{3} + 2 \, x^{2} \log \relax (3)\right )} \log \relax (x)\right )} \log \left (x - \log \relax (x)\right ) - 4 \, {\left (x^{3} \log \relax (3) + x^{2} \log \relax (3)^{2}\right )} \log \relax (x) - 2 \, {\left (2 \, x^{3} \log \relax (3) + x^{2} \log \relax (x)^{2} - 2 \, {\left (x^{3} - x^{2} \log \relax (x)\right )} \log \left (x - \log \relax (x)\right ) - {\left (x^{3} + 2 \, x^{2} \log \relax (3)\right )} \log \relax (x)\right )} \log \left (-x + \log \relax (x)\right )}\,{d x} + 16 \, \int -\frac {2 \, {\left ({\left (3 \, x - \log \relax (x) - 2\right )} \log \left (x - \log \relax (x)\right ) - {\left (3 \, x - \log \relax (x) - 2\right )} \log \left (-x + \log \relax (x)\right )\right )} {\left (-2 \, \log \relax (3) + \log \relax (x) + 2 \, \log \left (-x + \log \relax (x)\right )\right )}^{\frac {2}{x}}}{4 \, x^{3} \log \relax (3)^{2} - x^{2} \log \relax (x)^{3} + {\left (x^{3} + 4 \, x^{2} \log \relax (3)\right )} \log \relax (x)^{2} - 2 \, {\left (2 \, x^{3} \log \relax (3) + x^{2} \log \relax (x)^{2} - {\left (x^{3} + 2 \, x^{2} \log \relax (3)\right )} \log \relax (x)\right )} \log \left (x - \log \relax (x)\right ) - 4 \, {\left (x^{3} \log \relax (3) + x^{2} \log \relax (3)^{2}\right )} \log \relax (x) - 2 \, {\left (2 \, x^{3} \log \relax (3) + x^{2} \log \relax (x)^{2} - 2 \, {\left (x^{3} - x^{2} \log \relax (x)\right )} \log \left (x - \log \relax (x)\right ) - {\left (x^{3} + 2 \, x^{2} \log \relax (3)\right )} \log \relax (x)\right )} \log \left (-x + \log \relax (x)\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.65, size = 58, normalized size = 1.93 \begin {gather*} 4\,{\ln \left (\frac {x^3}{9}-\frac {2\,x^2\,\ln \relax (x)}{9}+\frac {x\,{\ln \relax (x)}^2}{9}\right )}^{2/x}\,\left ({\ln \left (\frac {x^3}{9}-\frac {2\,x^2\,\ln \relax (x)}{9}+\frac {x\,{\ln \relax (x)}^2}{9}\right )}^{2/x}+2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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