Integrand size = 128, antiderivative size = 26 \[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=5+\log ^2\left (e^{6+\left (3+x-\log \left (16 e^{-2 x}\right )\right )^2}+x\right ) \]
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\[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=\int \frac {\left (2+\exp \left (15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right ) \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (\exp \left (15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{\exp \left (15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (12 \left (3+x-\log \left (16 e^{-2 x}\right )\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )-\frac {2^{25+8 x} \left (e^{-2 x}\right )^{2 x} \left (-1+18 x+6 x^2-6 x \log \left (16 e^{-2 x}\right )\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+2^{24+8 x} \left (e^{-2 x}\right )^{2 x} x}\right ) \, dx \\ & = 12 \int \left (3+x-\log \left (16 e^{-2 x}\right )\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right ) \, dx-\int \frac {2^{25+8 x} \left (e^{-2 x}\right )^{2 x} \left (-1+18 x+6 x^2-6 x \log \left (16 e^{-2 x}\right )\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+2^{24+8 x} \left (e^{-2 x}\right )^{2 x} x} \, dx \\ & = 12 \int \left (3 \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )+x \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )-\log \left (16 e^{-2 x}\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )\right ) \, dx-\int \left (-\frac {2^{25+8 x} \left (e^{-2 x}\right )^{2 x} \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x}+\frac {9\ 2^{26+8 x} \left (e^{-2 x}\right )^{2 x} x \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x}+\frac {3\ 2^{26+8 x} \left (e^{-2 x}\right )^{2 x} x^2 \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x}-\frac {3\ 2^{26+8 x} \left (e^{-2 x}\right )^{2 x} x \log \left (16 e^{-2 x}\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x}\right ) \, dx \\ & = -\left (3 \int \frac {2^{26+8 x} \left (e^{-2 x}\right )^{2 x} x^2 \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x} \, dx\right )+3 \int \frac {2^{26+8 x} \left (e^{-2 x}\right )^{2 x} x \log \left (16 e^{-2 x}\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x} \, dx-9 \int \frac {2^{26+8 x} \left (e^{-2 x}\right )^{2 x} x \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x} \, dx+12 \int x \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right ) \, dx-12 \int \log \left (16 e^{-2 x}\right ) \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right ) \, dx+36 \int \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right ) \, dx+\int \frac {2^{25+8 x} \left (e^{-2 x}\right )^{2 x} \log \left (\exp \left (15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )\right )+x\right )}{e^{15+18 x+x^2+\log ^2\left (16 e^{-2 x}\right )}+256^{3+x} \left (e^{-2 x}\right )^{2 x} x} \, dx \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=\log ^2\left (e^{15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right ) \]
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Time = 0.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42
\[\ln \left ({\mathrm e}^{\ln \left (16 \,{\mathrm e}^{-2 x}\right )^{2}+\left (-2 x -6\right ) \ln \left (16 \,{\mathrm e}^{-2 x}\right )+x^{2}+6 x +15}+x \right )^{2}\]
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Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=\log \left (x + e^{\left (9 \, x^{2} - 24 \, {\left (x + 1\right )} \log \left (2\right ) + 16 \, \log \left (2\right )^{2} + 18 \, x + 15\right )}\right )^{2} \]
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Timed out. \[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=\int { \frac {2 \, {\left (6 \, {\left (x - \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + 3\right )} e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )} + 1\right )} \log \left (x + e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )}\right )}{x + e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )}} \,d x } \]
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\[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=\int { \frac {2 \, {\left (6 \, {\left (x - \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + 3\right )} e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )} + 1\right )} \log \left (x + e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )}\right )}{x + e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )}} \,d x } \]
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Time = 16.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 4.54 \[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=576\,{\ln \left (2\right )}^2\,x^2-48\,\ln \left (2\right )\,x\,\ln \left (16777216\,2^{24\,x}\,x+{\mathrm {e}}^{18\,x}\,{\mathrm {e}}^{15}\,{\mathrm {e}}^{16\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{9\,x^2}\right )+1152\,{\ln \left (2\right )}^2\,x+{\ln \left (16777216\,2^{24\,x}\,x+{\mathrm {e}}^{18\,x}\,{\mathrm {e}}^{15}\,{\mathrm {e}}^{16\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{9\,x^2}\right )}^2-48\,\ln \left (2\right )\,\ln \left (16777216\,2^{24\,x}\,x+{\mathrm {e}}^{18\,x}\,{\mathrm {e}}^{15}\,{\mathrm {e}}^{16\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{9\,x^2}\right ) \]
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