Integrand size = 115, antiderivative size = 25 \[ \int \frac {e^{-x} \left (e^x (1-x) \log (-1+x)+e^x (-1+x) \log (-1+x) \log (x)+4^{4 e^{-x}} \left (e^x (-1+x)+e^x (1-x) \log (x)\right )+\left (e^x x \log (x)+4^{4 e^{-x}} \left (-4 x+4 x^2\right ) \log (4) \log (x)\right ) \log \left (\frac {2 x}{\log (x)}\right )\right )}{\left (-x+x^2\right ) \log (x)} \, dx=\left (-4^{4 e^{-x}}+\log (-1+x)\right ) \log \left (\frac {2 x}{\log (x)}\right ) \]
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\[ \int \frac {e^{-x} \left (e^x (1-x) \log (-1+x)+e^x (-1+x) \log (-1+x) \log (x)+4^{4 e^{-x}} \left (e^x (-1+x)+e^x (1-x) \log (x)\right )+\left (e^x x \log (x)+4^{4 e^{-x}} \left (-4 x+4 x^2\right ) \log (4) \log (x)\right ) \log \left (\frac {2 x}{\log (x)}\right )\right )}{\left (-x+x^2\right ) \log (x)} \, dx=\int \frac {e^{-x} \left (e^x (1-x) \log (-1+x)+e^x (-1+x) \log (-1+x) \log (x)+4^{4 e^{-x}} \left (e^x (-1+x)+e^x (1-x) \log (x)\right )+\left (e^x x \log (x)+4^{4 e^{-x}} \left (-4 x+4 x^2\right ) \log (4) \log (x)\right ) \log \left (\frac {2 x}{\log (x)}\right )\right )}{\left (-x+x^2\right ) \log (x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (e^x (1-x) \log (-1+x)+e^x (-1+x) \log (-1+x) \log (x)+4^{4 e^{-x}} \left (e^x (-1+x)+e^x (1-x) \log (x)\right )+\left (e^x x \log (x)+4^{4 e^{-x}} \left (-4 x+4 x^2\right ) \log (4) \log (x)\right ) \log \left (\frac {2 x}{\log (x)}\right )\right )}{(-1+x) x \log (x)} \, dx \\ & = \int \frac {-256^{e^{-x}}+\frac {256^{e^{-x}}}{\log (x)}+\frac {\log (-1+x) (-1+\log (x))}{\log (x)}+\frac {e^{-x} x \left (e^x+4^{1+4 e^{-x}} (-1+x) \log (4)\right ) \log \left (\frac {2 x}{\log (x)}\right )}{-1+x}}{x} \, dx \\ & = \int \left (4^{e^{-x} \left (4+e^x\right )} e^{-x} \log (4) \log \left (\frac {2 x}{\log (x)}\right )-\frac {256^{e^{-x}}-256^{e^{-x}} x-\log (-1+x)+x \log (-1+x)-256^{e^{-x}} \log (x)+256^{e^{-x}} x \log (x)+\log (-1+x) \log (x)-x \log (-1+x) \log (x)-x \log (x) \log \left (\frac {2 x}{\log (x)}\right )}{(-1+x) x \log (x)}\right ) \, dx \\ & = \log (4) \int 4^{e^{-x} \left (4+e^x\right )} e^{-x} \log \left (\frac {2 x}{\log (x)}\right ) \, dx-\int \frac {256^{e^{-x}}-256^{e^{-x}} x-\log (-1+x)+x \log (-1+x)-256^{e^{-x}} \log (x)+256^{e^{-x}} x \log (x)+\log (-1+x) \log (x)-x \log (-1+x) \log (x)-x \log (x) \log \left (\frac {2 x}{\log (x)}\right )}{(-1+x) x \log (x)} \, dx \\ & = -4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )-\log (4) \int \frac {256^{e^{-x}} (1-\log (x))}{x \log (4) \log (x)} \, dx-\int \frac {256^{e^{-x}} (-1+x)+(-1+x) \log (-1+x) (-1+\log (x))-\log (x) \left (256^{e^{-x}} (-1+x)-x \log \left (\frac {2 x}{\log (x)}\right )\right )}{(1-x) x \log (x)} \, dx \\ & = -4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )-\int \frac {256^{e^{-x}} (1-\log (x))}{x \log (x)} \, dx-\int \left (\frac {256^{e^{-x}} (-1+\log (x))}{x \log (x)}+\frac {-\log (-1+x)+x \log (-1+x)+\log (-1+x) \log (x)-x \log (-1+x) \log (x)-x \log (x) \log \left (\frac {2 x}{\log (x)}\right )}{(-1+x) x \log (x)}\right ) \, dx \\ & = -4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )-\int \left (-\frac {256^{e^{-x}}}{x}+\frac {256^{e^{-x}}}{x \log (x)}\right ) \, dx-\int \frac {256^{e^{-x}} (-1+\log (x))}{x \log (x)} \, dx-\int \frac {-\log (-1+x)+x \log (-1+x)+\log (-1+x) \log (x)-x \log (-1+x) \log (x)-x \log (x) \log \left (\frac {2 x}{\log (x)}\right )}{(-1+x) x \log (x)} \, dx \\ & = -4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )+\int \frac {256^{e^{-x}}}{x} \, dx-\int \left (\frac {256^{e^{-x}}}{x}-\frac {256^{e^{-x}}}{x \log (x)}\right ) \, dx-\int \frac {256^{e^{-x}}}{x \log (x)} \, dx-\int \left (-\frac {\log (-1+x) (-1+\log (x))}{x \log (x)}-\frac {\log \left (\frac {2 x}{\log (x)}\right )}{-1+x}\right ) \, dx \\ & = -4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )+\int \frac {\log (-1+x) (-1+\log (x))}{x \log (x)} \, dx+\int \frac {\log \left (\frac {2 x}{\log (x)}\right )}{-1+x} \, dx \\ & = -4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )+\int \left (\frac {\log (-1+x)}{x}-\frac {\log (-1+x)}{x \log (x)}\right ) \, dx+\int \frac {\log \left (\frac {2 x}{\log (x)}\right )}{-1+x} \, dx \\ & = -4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )+\int \frac {\log (-1+x)}{x} \, dx-\int \frac {\log (-1+x)}{x \log (x)} \, dx+\int \frac {\log \left (\frac {2 x}{\log (x)}\right )}{-1+x} \, dx \\ & = \log (-1+x) \log (x)-4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )-\int \frac {\log (-1+x)}{x \log (x)} \, dx-\int \frac {\log (x)}{-1+x} \, dx+\int \frac {\log \left (\frac {2 x}{\log (x)}\right )}{-1+x} \, dx \\ & = \log (-1+x) \log (x)-4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )+\text {Li}_2(1-x)-\int \frac {\log (-1+x)}{x \log (x)} \, dx+\int \frac {\log \left (\frac {2 x}{\log (x)}\right )}{-1+x} \, dx \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-x} \left (e^x (1-x) \log (-1+x)+e^x (-1+x) \log (-1+x) \log (x)+4^{4 e^{-x}} \left (e^x (-1+x)+e^x (1-x) \log (x)\right )+\left (e^x x \log (x)+4^{4 e^{-x}} \left (-4 x+4 x^2\right ) \log (4) \log (x)\right ) \log \left (\frac {2 x}{\log (x)}\right )\right )}{\left (-x+x^2\right ) \log (x)} \, dx=-\left (\left (256^{e^{-x}}-\log (-1+x)\right ) \log \left (\frac {2 x}{\log (x)}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.76 (sec) , antiderivative size = 261, normalized size of antiderivative = 10.44
\[\left (-\ln \left (-1+x \right )+256^{{\mathrm e}^{-x}}\right ) \ln \left (\ln \left (x \right )\right )+\ln \left (x \right ) \ln \left (-1+x \right )+\frac {i 256^{{\mathrm e}^{-x}} \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )}{2}+\frac {i 256^{{\mathrm e}^{-x}} \pi \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{3}}{2}-\frac {i 256^{{\mathrm e}^{-x}} \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2}}{2}-\frac {i \ln \left (-1+x \right ) \pi \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{3}}{2}-\frac {i 256^{{\mathrm e}^{-x}} \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2}}{2}+\frac {i \ln \left (-1+x \right ) \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2}}{2}+\frac {i \ln \left (-1+x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2}}{2}-\frac {i \ln \left (-1+x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )}{2}+\ln \left (-1+x \right ) \ln \left (2\right )-256^{{\mathrm e}^{-x}} \ln \left (2\right )-256^{{\mathrm e}^{-x}} \ln \left (x \right )\]
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Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x} \left (e^x (1-x) \log (-1+x)+e^x (-1+x) \log (-1+x) \log (x)+4^{4 e^{-x}} \left (e^x (-1+x)+e^x (1-x) \log (x)\right )+\left (e^x x \log (x)+4^{4 e^{-x}} \left (-4 x+4 x^2\right ) \log (4) \log (x)\right ) \log \left (\frac {2 x}{\log (x)}\right )\right )}{\left (-x+x^2\right ) \log (x)} \, dx=-{\left (2^{8 \, e^{\left (-x\right )}} - \log \left (x - 1\right )\right )} \log \left (\frac {2 \, x}{\log \left (x\right )}\right ) \]
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Timed out. \[ \int \frac {e^{-x} \left (e^x (1-x) \log (-1+x)+e^x (-1+x) \log (-1+x) \log (x)+4^{4 e^{-x}} \left (e^x (-1+x)+e^x (1-x) \log (x)\right )+\left (e^x x \log (x)+4^{4 e^{-x}} \left (-4 x+4 x^2\right ) \log (4) \log (x)\right ) \log \left (\frac {2 x}{\log (x)}\right )\right )}{\left (-x+x^2\right ) \log (x)} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{-x} \left (e^x (1-x) \log (-1+x)+e^x (-1+x) \log (-1+x) \log (x)+4^{4 e^{-x}} \left (e^x (-1+x)+e^x (1-x) \log (x)\right )+\left (e^x x \log (x)+4^{4 e^{-x}} \left (-4 x+4 x^2\right ) \log (4) \log (x)\right ) \log \left (\frac {2 x}{\log (x)}\right )\right )}{\left (-x+x^2\right ) \log (x)} \, dx=\int { \frac {{\left ({\left (x - 1\right )} e^{x} \log \left (x - 1\right ) \log \left (x\right ) - {\left (x - 1\right )} e^{x} \log \left (x - 1\right ) - {\left ({\left (x - 1\right )} e^{x} \log \left (x\right ) - {\left (x - 1\right )} e^{x}\right )} 2^{8 \, e^{\left (-x\right )}} + {\left (8 \, {\left (x^{2} - x\right )} 2^{8 \, e^{\left (-x\right )}} \log \left (2\right ) \log \left (x\right ) + x e^{x} \log \left (x\right )\right )} \log \left (\frac {2 \, x}{\log \left (x\right )}\right )\right )} e^{\left (-x\right )}}{{\left (x^{2} - x\right )} \log \left (x\right )} \,d x } \]
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\[ \int \frac {e^{-x} \left (e^x (1-x) \log (-1+x)+e^x (-1+x) \log (-1+x) \log (x)+4^{4 e^{-x}} \left (e^x (-1+x)+e^x (1-x) \log (x)\right )+\left (e^x x \log (x)+4^{4 e^{-x}} \left (-4 x+4 x^2\right ) \log (4) \log (x)\right ) \log \left (\frac {2 x}{\log (x)}\right )\right )}{\left (-x+x^2\right ) \log (x)} \, dx=\int { \frac {{\left ({\left (x - 1\right )} e^{x} \log \left (x - 1\right ) \log \left (x\right ) - {\left (x - 1\right )} e^{x} \log \left (x - 1\right ) - {\left ({\left (x - 1\right )} e^{x} \log \left (x\right ) - {\left (x - 1\right )} e^{x}\right )} 2^{8 \, e^{\left (-x\right )}} + {\left (8 \, {\left (x^{2} - x\right )} 2^{8 \, e^{\left (-x\right )}} \log \left (2\right ) \log \left (x\right ) + x e^{x} \log \left (x\right )\right )} \log \left (\frac {2 \, x}{\log \left (x\right )}\right )\right )} e^{\left (-x\right )}}{{\left (x^{2} - x\right )} \log \left (x\right )} \,d x } \]
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Timed out. \[ \int \frac {e^{-x} \left (e^x (1-x) \log (-1+x)+e^x (-1+x) \log (-1+x) \log (x)+4^{4 e^{-x}} \left (e^x (-1+x)+e^x (1-x) \log (x)\right )+\left (e^x x \log (x)+4^{4 e^{-x}} \left (-4 x+4 x^2\right ) \log (4) \log (x)\right ) \log \left (\frac {2 x}{\log (x)}\right )\right )}{\left (-x+x^2\right ) \log (x)} \, dx=\int -\frac {{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{8\,{\mathrm {e}}^{-x}\,\ln \left (2\right )}\,\left ({\mathrm {e}}^x\,\left (x-1\right )-{\mathrm {e}}^x\,\ln \left (x\right )\,\left (x-1\right )\right )+\ln \left (\frac {2\,x}{\ln \left (x\right )}\right )\,\left (x\,{\mathrm {e}}^x\,\ln \left (x\right )-2\,{\mathrm {e}}^{8\,{\mathrm {e}}^{-x}\,\ln \left (2\right )}\,\ln \left (2\right )\,\ln \left (x\right )\,\left (4\,x-4\,x^2\right )\right )-\ln \left (x-1\right )\,{\mathrm {e}}^x\,\left (x-1\right )+\ln \left (x-1\right )\,{\mathrm {e}}^x\,\ln \left (x\right )\,\left (x-1\right )\right )}{\ln \left (x\right )\,\left (x-x^2\right )} \,d x \]
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