Optimal. Leaf size=27 \[ \log ^2\left (\frac {4 e^{-x^2-x^2 \log (x)} x}{5 \log (x)}\right ) \]
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Rubi [F] time = 0.70, 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 {\left (-2+\left (2-6 x^2\right ) \log (x)-4 x^2 \log ^2(x)\right ) \log \left (\frac {4 e^{-x^2-x^2 \log (x)} x}{5 \log (x)}\right )}{x \log (x)} \, 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 {2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x}-6 x \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )-\frac {2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)}-4 x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )\right ) \, dx\\ &=2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx-6 \int x \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx\\ &=-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx+6 \int \frac {x \left (-1-\left (-1+3 x^2\right ) \log (x)-2 x^2 \log ^2(x)\right )}{2 \log (x)} \, dx\\ &=-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx+3 \int \frac {x \left (-1-\left (-1+3 x^2\right ) \log (x)-2 x^2 \log ^2(x)\right )}{\log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx\\ &=-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx+3 \int \left (x-3 x^3-\frac {x}{\log (x)}-2 x^3 \log (x)\right ) \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx\\ &=\frac {3 x^2}{2}-\frac {9 x^4}{4}-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-3 \int \frac {x}{\log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx-6 \int x^3 \log (x) \, dx\\ &=\frac {3 x^2}{2}-\frac {15 x^4}{8}-\frac {3}{2} x^4 \log (x)-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-3 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx\\ &=\frac {3 x^2}{2}-\frac {15 x^4}{8}-3 \text {Ei}(2 \log (x))-\frac {3}{2} x^4 \log (x)-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 27, normalized size = 1.00 \begin {gather*} \log ^2\left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 24, normalized size = 0.89 \begin {gather*} \log \left (\frac {4 \, x e^{\left (-x^{2} \log \relax (x) - x^{2}\right )}}{5 \, \log \relax (x)}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 86, normalized size = 3.19 \begin {gather*} x^{4} - 4 \, x^{2} \log \relax (2) + {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \relax (x)^{2} + 2 \, {\left (x^{4} - x^{2} {\left (2 \, \log \relax (2) + 1\right )}\right )} \log \relax (x) + 4 \, \log \relax (2) \log \relax (x) + 2 \, {\left (x^{2} \log \relax (x) + x^{2} - \log \relax (x)\right )} \log \left (5 \, \log \relax (x)\right ) + \log \left (5 \, \log \relax (x)\right )^{2} - 4 \, \log \relax (2) \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.35, size = 1507, normalized size = 55.81
method | result | size |
risch | \(-4 \ln \relax (2) \ln \left (\ln \relax (x )\right )-4 x^{2} \ln \relax (2) \ln \relax (x )+2 \ln \left (\ln \relax (x )\right ) \ln \relax (5)+2 x^{2} \ln \relax (5)+\ln \left (\ln \relax (x )\right )^{2}+\ln \relax (x )^{2}-x^{4}+4 \ln \relax (2) \ln \relax (x )-x^{4} \ln \relax (x )^{2}-2 \ln \relax (x ) \ln \left (\ln \relax (x )\right )-2 x^{4} \ln \relax (x )-4 x^{2} \ln \relax (2)-i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3}+i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3}-i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3}+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3}+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3}+i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3}+2 x^{2} \ln \relax (5) \ln \relax (x )-2 \ln \relax (5) \ln \relax (x )+i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )+\left (2 x^{2} \ln \relax (x )+2 x^{2}-2 \ln \relax (x )+2 \ln \left (\ln \relax (x )\right )\right ) \ln \left (x^{x^{2}} {\mathrm e}^{x^{2}}\right )-i \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) x^{2} \pi +i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}+i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}-i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}-i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}-i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}-i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}+i \pi \ln \relax (x ) \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}+i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2}-i \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) x^{2} \pi -i \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) x^{2} \pi -i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )-i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \ln \relax (x )-i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \ln \relax (x )+i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )+i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )-i \pi \,x^{2} \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \ln \relax (x )-i \pi \,x^{2} \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \ln \relax (x )-i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3} \ln \relax (x )+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{3} \ln \relax (x )-i \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (i x \right ) x^{2} \pi +i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x \,x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \ln \relax (x )+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (i x^{-x^{2}} {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{-x^{2}} {\mathrm e}^{-x^{2}}}{\ln \relax (x )}\right ) \ln \relax (x )\) | \(1507\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 100, normalized size = 3.70 \begin {gather*} -x^{4} - {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \relax (x)^{2} - 2 \, {\left (x^{4} - x^{2}\right )} \log \relax (x) - 2 \, {\left (x^{2} \log \relax (x) + x^{2} - \log \relax (x) + \log \left (\log \relax (x)\right )\right )} \log \left (\frac {4 \, x e^{\left (-x^{2} \log \relax (x) - x^{2}\right )}}{5 \, \log \relax (x)}\right ) - 2 \, {\left (x^{2} + {\left (x^{2} - 1\right )} \log \relax (x)\right )} \log \left (\log \relax (x)\right ) - \log \left (\log \relax (x)\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.91, size = 23, normalized size = 0.85 \begin {gather*} {\ln \left (\frac {4\,x\,{\mathrm {e}}^{-x^2}}{5\,x^{x^2}\,\ln \relax (x)}\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.57, size = 24, normalized size = 0.89 \begin {gather*} \log {\left (\frac {4 x e^{- x^{2} \log {\relax (x )} - x^{2}}}{5 \log {\relax (x )}} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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