Integrand size = 281, antiderivative size = 29 \[ \int \frac {e^x (-4+2 x) \log (-2+x)+\left (-e^x x+e^x (-2+x) \log (-2+x)\right ) \log \left (x^2\right )+e^x \left (2 x^2-x^3\right ) \log ^2\left (x^2\right )+\left (e^x (-2+x) \log (-2+x) \log \left (x^2\right )+e^x \left (-2 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )+\left (e^x \left (2-3 x+x^2\right ) \log (-2+x) \log \left (x^2\right )+e^x \left (2 x^2-3 x^3+x^4\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right ) \log \left (\frac {x}{\log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}\right )}{\left (\left (-2 x^2+x^3\right ) \log (-2+x) \log \left (x^2\right )+\left (-2 x^4+x^5\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )} \, dx=\frac {e^x \log \left (\frac {x}{\log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )}\right )}{x} \]
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Time = 70.71 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 57, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6820, 6874, 2208, 2209, 2228, 2635} \[ \int \frac {e^x (-4+2 x) \log (-2+x)+\left (-e^x x+e^x (-2+x) \log (-2+x)\right ) \log \left (x^2\right )+e^x \left (2 x^2-x^3\right ) \log ^2\left (x^2\right )+\left (e^x (-2+x) \log (-2+x) \log \left (x^2\right )+e^x \left (-2 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )+\left (e^x \left (2-3 x+x^2\right ) \log (-2+x) \log \left (x^2\right )+e^x \left (2 x^2-3 x^3+x^4\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right ) \log \left (\frac {x}{\log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}\right )}{\left (\left (-2 x^2+x^3\right ) \log (-2+x) \log \left (x^2\right )+\left (-2 x^4+x^5\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )} \, dx=\frac {e^x \log \left (\frac {x}{\log \left (\frac {\log (x-2)}{x \log \left (x^2\right )}+x\right )}\right )}{x} \]
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Rule 2208
Rule 2209
Rule 2228
Rule 2635
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \left (-2 (-2+x) \log (-2+x)-(-x+(-2+x) \log (-2+x)) \log \left (x^2\right )+(-2+x) x^2 \log ^2\left (x^2\right )-(-2+x) \log \left (x^2\right ) \left (\log (-2+x)+x^2 \log \left (x^2\right )\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )-\left (2-3 x+x^2\right ) \log \left (x^2\right ) \left (\log (-2+x)+x^2 \log \left (x^2\right )\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right ) \log \left (\frac {x}{\log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )}\right )\right )}{(2-x) x^2 \log \left (x^2\right ) \left (\log (-2+x)+x^2 \log \left (x^2\right )\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )} \, dx \\ & = \int \left (\frac {e^x \left (-4 \log (-2+x)+2 x \log (-2+x)-x \log \left (x^2\right )-2 \log (-2+x) \log \left (x^2\right )+x \log (-2+x) \log \left (x^2\right )+2 x^2 \log ^2\left (x^2\right )-x^3 \log ^2\left (x^2\right )-2 \log (-2+x) \log \left (x^2\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )+x \log (-2+x) \log \left (x^2\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )-2 x^2 \log ^2\left (x^2\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )+x^3 \log ^2\left (x^2\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )\right )}{(-2+x) x^2 \log \left (x^2\right ) \left (\log (-2+x)+x^2 \log \left (x^2\right )\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )}+\frac {e^x (-1+x) \log \left (\frac {x}{\log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )}\right )}{x^2}\right ) \, dx \\ & = \int \frac {e^x \left (-4 \log (-2+x)+2 x \log (-2+x)-x \log \left (x^2\right )-2 \log (-2+x) \log \left (x^2\right )+x \log (-2+x) \log \left (x^2\right )+2 x^2 \log ^2\left (x^2\right )-x^3 \log ^2\left (x^2\right )-2 \log (-2+x) \log \left (x^2\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )+x \log (-2+x) \log \left (x^2\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )-2 x^2 \log ^2\left (x^2\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )+x^3 \log ^2\left (x^2\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )\right )}{(-2+x) x^2 \log \left (x^2\right ) \left (\log (-2+x)+x^2 \log \left (x^2\right )\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )} \, dx+\int \frac {e^x (-1+x) \log \left (\frac {x}{\log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )}\right )}{x^2} \, dx \\ & = \frac {e^x \log \left (\frac {x}{\log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )}\right )}{x}+\int \frac {e^x \left (-x \log \left (x^2\right ) \left (-1+(-2+x) x \log \left (x^2\right ) \left (-1+\log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )\right )\right )-(-2+x) \log (-2+x) \left (2+\log \left (x^2\right ) \left (1+\log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )\right )\right )\right )}{(2-x) x^2 \log \left (x^2\right ) \left (\log (-2+x)+x^2 \log \left (x^2\right )\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )} \, dx-\int \frac {e^x \left (x \log \left (x^2\right ) \left (-1+(-2+x) x \log \left (x^2\right ) \left (-1+\log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )\right )\right )+(-2+x) \log (-2+x) \left (2+\log \left (x^2\right ) \left (1+\log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )\right )\right )\right )}{(-2+x) x^2 \log \left (x^2\right ) \left (\log (-2+x)+x^2 \log \left (x^2\right )\right ) \log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )} \, dx \\ & = \frac {e^x \log \left (\frac {x}{\log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )}\right )}{x} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {e^x (-4+2 x) \log (-2+x)+\left (-e^x x+e^x (-2+x) \log (-2+x)\right ) \log \left (x^2\right )+e^x \left (2 x^2-x^3\right ) \log ^2\left (x^2\right )+\left (e^x (-2+x) \log (-2+x) \log \left (x^2\right )+e^x \left (-2 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )+\left (e^x \left (2-3 x+x^2\right ) \log (-2+x) \log \left (x^2\right )+e^x \left (2 x^2-3 x^3+x^4\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right ) \log \left (\frac {x}{\log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}\right )}{\left (\left (-2 x^2+x^3\right ) \log (-2+x) \log \left (x^2\right )+\left (-2 x^4+x^5\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )} \, dx=\frac {e^x \log \left (\frac {x}{\log \left (x+\frac {\log (-2+x)}{x \log \left (x^2\right )}\right )}\right )}{x} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.78 (sec) , antiderivative size = 22245, normalized size of antiderivative = 767.07
\[\text {output too large to display}\]
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {e^x (-4+2 x) \log (-2+x)+\left (-e^x x+e^x (-2+x) \log (-2+x)\right ) \log \left (x^2\right )+e^x \left (2 x^2-x^3\right ) \log ^2\left (x^2\right )+\left (e^x (-2+x) \log (-2+x) \log \left (x^2\right )+e^x \left (-2 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )+\left (e^x \left (2-3 x+x^2\right ) \log (-2+x) \log \left (x^2\right )+e^x \left (2 x^2-3 x^3+x^4\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right ) \log \left (\frac {x}{\log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}\right )}{\left (\left (-2 x^2+x^3\right ) \log (-2+x) \log \left (x^2\right )+\left (-2 x^4+x^5\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )} \, dx=\frac {e^{x} \log \left (\frac {x}{\log \left (\frac {x^{2} \log \left (x^{2}\right ) + \log \left (x - 2\right )}{x \log \left (x^{2}\right )}\right )}\right )}{x} \]
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Timed out. \[ \int \frac {e^x (-4+2 x) \log (-2+x)+\left (-e^x x+e^x (-2+x) \log (-2+x)\right ) \log \left (x^2\right )+e^x \left (2 x^2-x^3\right ) \log ^2\left (x^2\right )+\left (e^x (-2+x) \log (-2+x) \log \left (x^2\right )+e^x \left (-2 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )+\left (e^x \left (2-3 x+x^2\right ) \log (-2+x) \log \left (x^2\right )+e^x \left (2 x^2-3 x^3+x^4\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right ) \log \left (\frac {x}{\log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}\right )}{\left (\left (-2 x^2+x^3\right ) \log (-2+x) \log \left (x^2\right )+\left (-2 x^4+x^5\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )} \, dx=\text {Timed out} \]
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Time = 0.42 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {e^x (-4+2 x) \log (-2+x)+\left (-e^x x+e^x (-2+x) \log (-2+x)\right ) \log \left (x^2\right )+e^x \left (2 x^2-x^3\right ) \log ^2\left (x^2\right )+\left (e^x (-2+x) \log (-2+x) \log \left (x^2\right )+e^x \left (-2 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )+\left (e^x \left (2-3 x+x^2\right ) \log (-2+x) \log \left (x^2\right )+e^x \left (2 x^2-3 x^3+x^4\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right ) \log \left (\frac {x}{\log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}\right )}{\left (\left (-2 x^2+x^3\right ) \log (-2+x) \log \left (x^2\right )+\left (-2 x^4+x^5\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )} \, dx=\frac {e^{x} \log \left (x\right ) - e^{x} \log \left (-\log \left (2\right ) + \log \left (2 \, x^{2} \log \left (x\right ) + \log \left (x - 2\right )\right ) - \log \left (x\right ) - \log \left (\log \left (x\right )\right )\right )}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2231 vs. \(2 (28) = 56\).
Time = 7.28 (sec) , antiderivative size = 2231, normalized size of antiderivative = 76.93 \[ \int \frac {e^x (-4+2 x) \log (-2+x)+\left (-e^x x+e^x (-2+x) \log (-2+x)\right ) \log \left (x^2\right )+e^x \left (2 x^2-x^3\right ) \log ^2\left (x^2\right )+\left (e^x (-2+x) \log (-2+x) \log \left (x^2\right )+e^x \left (-2 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )+\left (e^x \left (2-3 x+x^2\right ) \log (-2+x) \log \left (x^2\right )+e^x \left (2 x^2-3 x^3+x^4\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right ) \log \left (\frac {x}{\log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}\right )}{\left (\left (-2 x^2+x^3\right ) \log (-2+x) \log \left (x^2\right )+\left (-2 x^4+x^5\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {e^x (-4+2 x) \log (-2+x)+\left (-e^x x+e^x (-2+x) \log (-2+x)\right ) \log \left (x^2\right )+e^x \left (2 x^2-x^3\right ) \log ^2\left (x^2\right )+\left (e^x (-2+x) \log (-2+x) \log \left (x^2\right )+e^x \left (-2 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )+\left (e^x \left (2-3 x+x^2\right ) \log (-2+x) \log \left (x^2\right )+e^x \left (2 x^2-3 x^3+x^4\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right ) \log \left (\frac {x}{\log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}\right )}{\left (\left (-2 x^2+x^3\right ) \log (-2+x) \log \left (x^2\right )+\left (-2 x^4+x^5\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )} \, dx=\int -\frac {\ln \left (\frac {x}{\ln \left (\frac {\ln \left (x-2\right )+x^2\,\ln \left (x^2\right )}{x\,\ln \left (x^2\right )}\right )}\right )\,\ln \left (\frac {\ln \left (x-2\right )+x^2\,\ln \left (x^2\right )}{x\,\ln \left (x^2\right )}\right )\,\left ({\mathrm {e}}^x\,\left (x^4-3\,x^3+2\,x^2\right )\,{\ln \left (x^2\right )}^2+\ln \left (x-2\right )\,{\mathrm {e}}^x\,\left (x^2-3\,x+2\right )\,\ln \left (x^2\right )\right )-\ln \left (\frac {\ln \left (x-2\right )+x^2\,\ln \left (x^2\right )}{x\,\ln \left (x^2\right )}\right )\,\left ({\ln \left (x^2\right )}^2\,{\mathrm {e}}^x\,\left (2\,x^2-x^3\right )-\ln \left (x-2\right )\,\ln \left (x^2\right )\,{\mathrm {e}}^x\,\left (x-2\right )\right )-\ln \left (x^2\right )\,\left (x\,{\mathrm {e}}^x-\ln \left (x-2\right )\,{\mathrm {e}}^x\,\left (x-2\right )\right )+{\ln \left (x^2\right )}^2\,{\mathrm {e}}^x\,\left (2\,x^2-x^3\right )+\ln \left (x-2\right )\,{\mathrm {e}}^x\,\left (2\,x-4\right )}{\ln \left (\frac {\ln \left (x-2\right )+x^2\,\ln \left (x^2\right )}{x\,\ln \left (x^2\right )}\right )\,\left (\left (2\,x^4-x^5\right )\,{\ln \left (x^2\right )}^2+\ln \left (x-2\right )\,\left (2\,x^2-x^3\right )\,\ln \left (x^2\right )\right )} \,d x \]
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