Optimal. Leaf size=26 \[ \log ^2(x) \left (x-\log \left (\log \left (2 \left (-e^{-x}+4 x\right )\right )\right )\right )^2 \]
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Rubi [F] time = 19.00, 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 x^2-8 e^x x^2\right ) \log ^2(x)+\left (\left (-2 x^2+8 e^x x^3\right ) \log (x)+\left (-2 x^2+8 e^x x^3\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )+\left (\left (2 x+8 e^x x\right ) \log ^2(x)+\left (\left (4 x-16 e^x x^2\right ) \log (x)+\left (2 x-8 e^x x^2\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right ) \log \left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )+\left (-2+8 e^x x\right ) \log (x) \log \left (e^{-x} \left (-2+8 e^x x\right )\right ) \log ^2\left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )}{\left (-x+4 e^x x^2\right ) \log \left (e^{-x} \left (-2+8 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 {2 \log (x) \left (x \log (x) \left (-1-4 e^x+\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )\right )+\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right ) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )\right ) \left (-x+\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \left (1-4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx\\ &=2 \int \frac {\log (x) \left (x \log (x) \left (-1-4 e^x+\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )\right )+\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right ) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )\right ) \left (-x+\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \left (1-4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx\\ &=2 \int \left (-\frac {(1+x) \log ^2(x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )}+\frac {\log (x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right ) \left (-\log (x)+x \log \left (-2 e^{-x}+8 x\right )+x \log (x) \log \left (-2 e^{-x}+8 x\right )-\log \left (-2 e^{-x}+8 x\right ) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )}\right ) \, dx\\ &=-\left (2 \int \frac {(1+x) \log ^2(x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx\right )+2 \int \frac {\log (x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right ) \left (-\log (x)+x \log \left (-2 e^{-x}+8 x\right )+x \log (x) \log \left (-2 e^{-x}+8 x\right )-\log \left (-2 e^{-x}+8 x\right ) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )} \, dx\\ &=-\left (2 \int \left (\frac {\log ^2(x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )}+\frac {\log ^2(x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )}\right ) \, dx\right )+2 \int \left (\frac {\log (x) \left (-\log (x)+x \log \left (-2 e^{-x}+8 x\right )+x \log (x) \log \left (-2 e^{-x}+8 x\right )\right )}{\log \left (-2 e^{-x}+8 x\right )}-\frac {\log (x) \left (-\log (x)+2 x \log \left (-2 e^{-x}+8 x\right )+x \log (x) \log \left (-2 e^{-x}+8 x\right )\right ) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )}+\frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x}\right ) \, dx\\ &=2 \int \frac {\log (x) \left (-\log (x)+x \log \left (-2 e^{-x}+8 x\right )+x \log (x) \log \left (-2 e^{-x}+8 x\right )\right )}{\log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {\log ^2(x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {\log ^2(x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {\log (x) \left (-\log (x)+2 x \log \left (-2 e^{-x}+8 x\right )+x \log (x) \log \left (-2 e^{-x}+8 x\right )\right ) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x} \, dx\\ &=2 \int \log (x) \left (x+\log (x) \left (x-\frac {1}{\log \left (-2 e^{-x}+8 x\right )}\right )\right ) \, dx+2 \int \frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x} \, dx-2 \int \left (2 \log (x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )+\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )-\frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )}\right ) \, dx-2 \int \left (\frac {x \log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )}-\frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )}\right ) \, dx-2 \int \left (\frac {\log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )}-\frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )}\right ) \, dx\\ &=2 \int \left (x \log (x) (1+\log (x))-\frac {\log ^2(x)}{\log \left (-2 e^{-x}+8 x\right )}\right ) \, dx-2 \int \frac {\log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {x \log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x} \, dx-4 \int \log (x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx\\ &=2 \int x \log (x) (1+\log (x)) \, dx-2 \int \frac {\log ^2(x)}{\log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {\log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {x \log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x} \, dx-4 \int \log (x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx\\ &=-\frac {1}{2} x^2 \log (x)+x^2 \log (x) (1+\log (x))-2 \int \frac {1}{4} x (1+2 \log (x)) \, dx-2 \int \frac {\log ^2(x)}{\log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {\log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {x \log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x} \, dx-4 \int \log (x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx\\ &=-\frac {1}{2} x^2 \log (x)+x^2 \log (x) (1+\log (x))-\frac {1}{2} \int x (1+2 \log (x)) \, dx-2 \int \frac {\log ^2(x)}{\log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {\log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {x \log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x} \, dx-4 \int \log (x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx\\ &=-x^2 \log (x)+x^2 \log (x) (1+\log (x))-2 \int \frac {\log ^2(x)}{\log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {\log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {x \log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x} \, dx-4 \int \log (x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.18, size = 57, normalized size = 2.19 \begin {gather*} 2 \left (\frac {1}{2} x^2 \log ^2(x)-x \log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )+\frac {1}{2} \log ^2(x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 53, normalized size = 2.04 \begin {gather*} x^{2} \log \relax (x)^{2} - 2 \, x \log \relax (x)^{2} \log \left (\log \left (2 \, {\left (4 \, x e^{x} - 1\right )} e^{\left (-x\right )}\right )\right ) + \log \relax (x)^{2} \log \left (\log \left (2 \, {\left (4 \, x e^{x} - 1\right )} e^{\left (-x\right )}\right )\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 53, normalized size = 2.04 \begin {gather*} x^{2} \log \relax (x)^{2} - 2 \, x \log \relax (x)^{2} \log \left (-x + \log \relax (2) + \log \left (4 \, x e^{x} - 1\right )\right ) + \log \relax (x)^{2} \log \left (-x + \log \relax (2) + \log \left (4 \, x e^{x} - 1\right )\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.25, size = 200, normalized size = 7.69
method | result | size |
risch | \(x^{2} \ln \relax (x )^{2}-2 x \ln \relax (x )^{2} \ln \left (3 \ln \relax (2)-\ln \left ({\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x} x -\frac {1}{4}\right )-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right )+\mathrm {csgn}\left (i {\mathrm e}^{-x}\right )\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right )+\mathrm {csgn}\left (i \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right )\right )}{2}\right )+\ln \relax (x )^{2} \ln \left (3 \ln \relax (2)-\ln \left ({\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x} x -\frac {1}{4}\right )-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right )+\mathrm {csgn}\left (i {\mathrm e}^{-x}\right )\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right )+\mathrm {csgn}\left (i \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right )\right )}{2}\right )^{2}\) | \(200\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 53, normalized size = 2.04 \begin {gather*} x^{2} \log \relax (x)^{2} - 2 \, x \log \relax (x)^{2} \log \left (-x + \log \relax (2) + \log \left (4 \, x e^{x} - 1\right )\right ) + \log \relax (x)^{2} \log \left (-x + \log \relax (2) + \log \left (4 \, x e^{x} - 1\right )\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\ln \left ({\mathrm {e}}^{-x}\,\left (8\,x\,{\mathrm {e}}^x-2\right )\right )\,\left (\left (8\,x^3\,{\mathrm {e}}^x-2\,x^2\right )\,{\ln \relax (x)}^2+\left (8\,x^3\,{\mathrm {e}}^x-2\,x^2\right )\,\ln \relax (x)\right )+\ln \left (\ln \left ({\mathrm {e}}^{-x}\,\left (8\,x\,{\mathrm {e}}^x-2\right )\right )\right )\,\left (\ln \left ({\mathrm {e}}^{-x}\,\left (8\,x\,{\mathrm {e}}^x-2\right )\right )\,\left (\left (2\,x-8\,x^2\,{\mathrm {e}}^x\right )\,{\ln \relax (x)}^2+\left (4\,x-16\,x^2\,{\mathrm {e}}^x\right )\,\ln \relax (x)\right )+{\ln \relax (x)}^2\,\left (2\,x+8\,x\,{\mathrm {e}}^x\right )\right )-{\ln \relax (x)}^2\,\left (8\,x^2\,{\mathrm {e}}^x+2\,x^2\right )+\ln \left ({\mathrm {e}}^{-x}\,\left (8\,x\,{\mathrm {e}}^x-2\right )\right )\,{\ln \left (\ln \left ({\mathrm {e}}^{-x}\,\left (8\,x\,{\mathrm {e}}^x-2\right )\right )\right )}^2\,\ln \relax (x)\,\left (8\,x\,{\mathrm {e}}^x-2\right )}{\ln \left ({\mathrm {e}}^{-x}\,\left (8\,x\,{\mathrm {e}}^x-2\right )\right )\,\left (x-4\,x^2\,{\mathrm {e}}^x\right )} \,d x \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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