Integrand size = 93, antiderivative size = 27 \[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=-e^{e^{2 x \left (-\frac {x}{1+e}+\log (x)\right )} x \log ^2(\log (2))} \]
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\[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=\int \frac {\exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\log ^2(\log (2)) \int \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \, dx}{1+e} \\ & = \frac {\log ^2(\log (2)) \int \left (-\exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right )-2 \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) x+4 \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) x^2-\exp \left (1+\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) (1+2 x)-2 \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) (1+e) x \log (x)\right ) \, dx}{1+e} \\ & = -\left (\left (2 \log ^2(\log (2))\right ) \int \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) x \log (x) \, dx\right )-\frac {\log ^2(\log (2)) \int \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) \, dx}{1+e}-\frac {\log ^2(\log (2)) \int \exp \left (1+\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) (1+2 x) \, dx}{1+e}-\frac {\left (2 \log ^2(\log (2))\right ) \int \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) x \, dx}{1+e}+\frac {\left (4 \log ^2(\log (2))\right ) \int \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) x^2 \, dx}{1+e} \\ & = -\left (\left (2 \log ^2(\log (2))\right ) \int \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) x \log (x) \, dx\right )-\frac {\log ^2(\log (2)) \int \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) \, dx}{1+e}-\frac {\log ^2(\log (2)) \int \left (\exp \left (1+\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right )+2 \exp \left (1+\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) x\right ) \, dx}{1+e}-\frac {\left (2 \log ^2(\log (2))\right ) \int \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) x \, dx}{1+e}+\frac {\left (4 \log ^2(\log (2))\right ) \int \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) x^2 \, dx}{1+e} \\ & = -\left (\left (2 \log ^2(\log (2))\right ) \int \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) x \log (x) \, dx\right )-\frac {\log ^2(\log (2)) \int \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) \, dx}{1+e}-\frac {\log ^2(\log (2)) \int \exp \left (1+\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) \, dx}{1+e}-\frac {\left (2 \log ^2(\log (2))\right ) \int \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) x \, dx}{1+e}-\frac {\left (2 \log ^2(\log (2))\right ) \int \exp \left (1+\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) x \, dx}{1+e}+\frac {\left (4 \log ^2(\log (2))\right ) \int \exp \left (\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) x^2 \, dx}{1+e} \\ \end{align*}
Time = 5.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=-e^{e^{-\frac {2 x^2}{1+e}} x^{1+2 x} \log ^2(\log (2))} \]
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Time = 2.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19
method | result | size |
risch | \(-{\mathrm e}^{x \ln \left (\ln \left (2\right )\right )^{2} {\mathrm e}^{\frac {2 x \left ({\mathrm e} \ln \left (x \right )+\ln \left (x \right )-x \right )}{1+{\mathrm e}}}}\) | \(32\) |
parallelrisch | \(\frac {-{\mathrm e}^{x \ln \left (\ln \left (2\right )\right )^{2} {\mathrm e}^{\frac {2 x \left ({\mathrm e} \ln \left (x \right )+\ln \left (x \right )-x \right )}{1+{\mathrm e}}}} {\mathrm e}-{\mathrm e}^{x \ln \left (\ln \left (2\right )\right )^{2} {\mathrm e}^{\frac {2 x \left ({\mathrm e} \ln \left (x \right )+\ln \left (x \right )-x \right )}{1+{\mathrm e}}}}}{1+{\mathrm e}}\) | \(75\) |
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.11 \[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=-e^{\left (\frac {{\left (x e + x\right )} e^{\left (-\frac {2 \, {\left (x^{2} - {\left (x e + x\right )} \log \left (x\right )\right )}}{e + 1}\right )} \log \left (\log \left (2\right )\right )^{2} - 2 \, x^{2} + 2 \, {\left (x e + x\right )} \log \left (x\right )}{e + 1} + \frac {2 \, {\left (x^{2} - {\left (x e + x\right )} \log \left (x\right )\right )}}{e + 1}\right )} \]
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Time = 1.68 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=- e^{x e^{\frac {2 \left (- x^{2} + \left (x + e x\right ) \log {\left (x \right )}\right )}{1 + e}} \log {\left (\log {\left (2 \right )} \right )}^{2}} \]
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Time = 0.49 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=-e^{\left (x e^{\left (\frac {2 \, x e \log \left (x\right )}{e + 1} - \frac {2 \, x^{2}}{e + 1} + \frac {2 \, x \log \left (x\right )}{e + 1}\right )} \log \left (\log \left (2\right )\right )^{2}\right )} \]
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\[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=\int { \frac {{\left (4 \, x^{2} - {\left (2 \, x + 1\right )} e - 2 \, {\left (x e + x\right )} \log \left (x\right ) - 2 \, x - 1\right )} e^{\left (x e^{\left (-\frac {2 \, {\left (x^{2} - {\left (x e + x\right )} \log \left (x\right )\right )}}{e + 1}\right )} \log \left (\log \left (2\right )\right )^{2} - \frac {2 \, {\left (x^{2} - {\left (x e + x\right )} \log \left (x\right )\right )}}{e + 1}\right )} \log \left (\log \left (2\right )\right )^{2}}{e + 1} \,d x } \]
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Time = 8.81 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=-{\mathrm {e}}^{x\,x^{\frac {2\,x\,\mathrm {e}}{\mathrm {e}+1}}\,x^{\frac {2\,x}{\mathrm {e}+1}}\,{\mathrm {e}}^{-\frac {2\,x^2}{\mathrm {e}+1}}\,{\ln \left (\ln \left (2\right )\right )}^2} \]
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