Optimal. Leaf size=33 \[ \log ^2\left (\frac {\log (4 x)}{-2 x^2+\log \left (2-x^2 \left (\frac {e^x}{x}+x\right )\right )}\right ) \]
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Rubi [F] time = 10.10, 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 (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int 2 \left (\frac {1}{x \log (4 x)}+\frac {e^x \left (1+x-4 x^2\right )+x \left (8+3 x-4 x^3\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right ) \, dx\\ &=2 \int \left (\frac {1}{x \log (4 x)}+\frac {e^x \left (1+x-4 x^2\right )+x \left (8+3 x-4 x^3\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right ) \, dx\\ &=2 \int \left (-\frac {\left (-2-2 x-2 x^3+x^4\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}-\frac {\left (-2 x^2-\log (4 x)-x \log (4 x)+4 x^2 \log (4 x)+\log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \log (4 x) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {\left (-2-2 x-2 x^3+x^4\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx\right )-2 \int \frac {\left (-2 x^2-\log (4 x)-x \log (4 x)+4 x^2 \log (4 x)+\log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \log (4 x) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx\\ &=-\left (2 \int \left (-\frac {2 \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}-\frac {2 \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}-\frac {2 x^2 \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}+\frac {x^3 \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}\right ) \, dx\right )-2 \int \left (-\frac {\log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{2 x^2-\log \left (2-e^x x-x^3\right )}-\frac {\log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}+\frac {4 x \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{2 x^2-\log \left (2-e^x x-x^3\right )}-\frac {2 x \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\log (4 x) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}+\frac {\log \left (2-e^x x-x^3\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \log (4 x) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}\right ) \, dx\\ &=2 \int \frac {\log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{2 x^2-\log \left (2-e^x x-x^3\right )} \, dx+2 \int \frac {\log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx-2 \int \frac {x^3 \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx-2 \int \frac {\log \left (2-e^x x-x^3\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \log (4 x) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx+4 \int \frac {\log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx+4 \int \frac {\log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx+4 \int \frac {x^2 \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx+4 \int \frac {x \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\log (4 x) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx-8 \int \frac {x \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{2 x^2-\log \left (2-e^x x-x^3\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.17, size = 30, normalized size = 0.91 \begin {gather*} \log ^2\left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 32, normalized size = 0.97 \begin {gather*} \log \left (-\frac {\log \left (4 \, x\right )}{2 \, x^{2} - \log \left (-x^{3} - x e^{x} + 2\right )}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.89, size = 119, normalized size = 3.61 \begin {gather*} \log \left (2 \, x^{2} - \log \left (-x^{3} - x e^{x} + 2\right )\right )^{2} - 2 \, \log \left (-2 \, x^{2} + \log \left (-x^{3} - x e^{x} + 2\right )\right ) \log \left (-\log \left (4 \, x\right )\right ) + \log \left (-\log \left (4 \, x\right )\right )^{2} - 2 \, \log \left (2 \, x^{2} - \log \left (-x^{3} - x e^{x} + 2\right )\right ) \log \left (\log \left (4 \, x\right )\right ) + 2 \, \log \left (-2 \, x^{2} + \log \left (-x^{3} - x e^{x} + 2\right )\right ) \log \left (\log \left (4 \, x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.35, size = 727, normalized size = 22.03
method | result | size |
risch | \(\ln \left (x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}\right )^{2}-2 \ln \left (x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}\right ) \ln \left (\ln \left (4 x \right )\right )+\ln \left (\ln \left (4 x \right )\right )^{2}+i \pi \ln \left (\ln \left (4 x \right )\right ) \mathrm {csgn}\left (\frac {i}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right ) \mathrm {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{2}+i \pi \ln \left (\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )-2 x^{2}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right ) \mathrm {csgn}\left (i \ln \left (4 x \right )\right ) \mathrm {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )+i \pi \ln \left (\ln \left (4 x \right )\right ) \mathrm {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{3}-2 i \pi \ln \left (\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )-2 x^{2}\right )-2 i \pi \ln \left (\ln \left (4 x \right )\right ) \mathrm {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{2}-i \pi \ln \left (\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )-2 x^{2}\right ) \mathrm {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{3}+2 i \pi \ln \left (\ln \left (4 x \right )\right )+2 i \pi \ln \left (\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )-2 x^{2}\right ) \mathrm {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{2}-i \pi \ln \left (\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )-2 x^{2}\right ) \mathrm {csgn}\left (i \ln \left (4 x \right )\right ) \mathrm {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{2}-i \pi \ln \left (\ln \left (4 x \right )\right ) \mathrm {csgn}\left (\frac {i}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right ) \mathrm {csgn}\left (i \ln \left (4 x \right )\right ) \mathrm {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )-i \pi \ln \left (\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )-2 x^{2}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right ) \mathrm {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{2}+i \pi \ln \left (\ln \left (4 x \right )\right ) \mathrm {csgn}\left (i \ln \left (4 x \right )\right ) \mathrm {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{2}+2 \ln \left (\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )-2 x^{2}\right ) \ln \relax (2)-2 \ln \relax (2) \ln \left (\ln \left (4 x \right )\right )\) | \(727\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 130, normalized size = 3.94 \begin {gather*} -\log \left (-2 \, x^{2} + \log \left (-x^{3} - x e^{x} + 2\right )\right )^{2} - 2 \, {\left (\log \left (-2 \, x^{2} + \log \left (-x^{3} - x e^{x} + 2\right )\right ) - \log \left (2 \, \log \relax (2) + \log \relax (x)\right )\right )} \log \left (-\frac {\log \left (4 \, x\right )}{2 \, x^{2} - \log \left (-x^{3} - x e^{x} + 2\right )}\right ) + 2 \, \log \left (-2 \, x^{2} + \log \left (-x^{3} - x e^{x} + 2\right )\right ) \log \left (2 \, \log \relax (2) + \log \relax (x)\right ) - \log \left (2 \, \log \relax (2) + \log \relax (x)\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.08, size = 29, normalized size = 0.88 \begin {gather*} {\ln \left (\frac {\ln \left (4\,x\right )}{\ln \left (2-x^3-x\,{\mathrm {e}}^x\right )-2\,x^2}\right )}^2 \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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