Integrand size = 203, antiderivative size = 32 \[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx=e^{x+\frac {\log \left (\frac {x^4}{\log ^2(x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \]
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\[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx=\int \frac {\exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x \log (x) \log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2} \, dx \\ & = \int \left (\exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right )+\frac {\exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) \left (\log (3)+x^2 \log (9) \log (x)+2 x^2 \log ^2(x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )}{x \log (x) \log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}-\frac {2 \exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) (-1+2 \log (x))}{x \log (x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}\right ) \, dx \\ & = -\left (2 \int \frac {\exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) (-1+2 \log (x))}{x \log (x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )} \, dx\right )+\int \exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) \, dx+\int \frac {\exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) \left (\log (3)+x^2 \log (9) \log (x)+2 x^2 \log ^2(x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )}{x \log (x) \log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2} \, dx \\ & = -\left (2 \int \left (\frac {2 \exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right )}{x \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}-\frac {\exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right )}{x \log (x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )}\right ) \, dx\right )+\int \exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) \, dx+\int \left (\frac {\exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) x \log (9) \log \left (\frac {x^4}{\log ^2(x)}\right )}{\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}+\frac {\exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) \log (3) \log \left (\frac {x^4}{\log ^2(x)}\right )}{x \log (x) \log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}+\frac {2 \exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) x \log (x) \log \left (\frac {x^4}{\log ^2(x)}\right )}{\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2}\right ) \, dx \\ & = 2 \int \frac {\exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) x \log (x) \log \left (\frac {x^4}{\log ^2(x)}\right )}{\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2} \, dx+2 \int \frac {\exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right )}{x \log (x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )} \, dx-4 \int \frac {\exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right )}{x \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )} \, dx+\log (3) \int \frac {\exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )}{x \log (x) \log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2} \, dx+\log (9) \int \frac {\exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) x \log \left (\frac {x^4}{\log ^2(x)}\right )}{\log (3 x) \left (x^2-\log \left (\frac {\log (3 x)}{\log (x)}\right )\right )^2} \, dx+\int \exp \left (\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}\right ) \, dx \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx=e^x \left (\frac {x^4}{\log ^2(x)}\right )^{\frac {1}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.19 (sec) , antiderivative size = 622, normalized size of antiderivative = 19.44
\[\text {Expression too large to display}\]
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Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.56 \[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx=e^{\left (\frac {x^{3} - x \log \left (\frac {\log \left (3\right ) + \log \left (x\right )}{\log \left (x\right )}\right ) - \log \left (\frac {x^{4}}{\log \left (x\right )^{2}}\right )}{x^{2} - \log \left (\frac {\log \left (3\right ) + \log \left (x\right )}{\log \left (x\right )}\right )}\right )} \]
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Time = 24.63 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx=e^{\frac {- x^{3} + x \log {\left (\frac {\log {\left (x \right )} + \log {\left (3 \right )}}{\log {\left (x \right )}} \right )} + \log {\left (\frac {x^{4}}{\log {\left (x \right )}^{2}} \right )}}{- x^{2} + \log {\left (\frac {\log {\left (x \right )} + \log {\left (3 \right )}}{\log {\left (x \right )}} \right )}}} \]
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\[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx=\int { \frac {{\left (x \log \left (3 \, x\right ) \log \left (x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right )^{2} - 2 \, {\left ({\left (x^{3} - 2\right )} \log \left (x\right ) + 1\right )} \log \left (3 \, x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right ) + {\left ({\left (2 \, x^{2} \log \left (x\right ) + 1\right )} \log \left (3 \, x\right ) - \log \left (x\right )\right )} \log \left (\frac {x^{4}}{\log \left (x\right )^{2}}\right ) + {\left (2 \, x^{2} + {\left (x^{5} - 4 \, x^{2}\right )} \log \left (x\right )\right )} \log \left (3 \, x\right )\right )} e^{\left (\frac {x^{3} - x \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right ) - \log \left (\frac {x^{4}}{\log \left (x\right )^{2}}\right )}{x^{2} - \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right )}\right )}}{x^{5} \log \left (3 \, x\right ) \log \left (x\right ) - 2 \, x^{3} \log \left (3 \, x\right ) \log \left (x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right ) + x \log \left (3 \, x\right ) \log \left (x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right )^{2}} \,d x } \]
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\[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx=\int { \frac {{\left (x \log \left (3 \, x\right ) \log \left (x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right )^{2} - 2 \, {\left ({\left (x^{3} - 2\right )} \log \left (x\right ) + 1\right )} \log \left (3 \, x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right ) + {\left ({\left (2 \, x^{2} \log \left (x\right ) + 1\right )} \log \left (3 \, x\right ) - \log \left (x\right )\right )} \log \left (\frac {x^{4}}{\log \left (x\right )^{2}}\right ) + {\left (2 \, x^{2} + {\left (x^{5} - 4 \, x^{2}\right )} \log \left (x\right )\right )} \log \left (3 \, x\right )\right )} e^{\left (\frac {x^{3} - x \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right ) - \log \left (\frac {x^{4}}{\log \left (x\right )^{2}}\right )}{x^{2} - \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right )}\right )}}{x^{5} \log \left (3 \, x\right ) \log \left (x\right ) - 2 \, x^{3} \log \left (3 \, x\right ) \log \left (x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right ) + x \log \left (3 \, x\right ) \log \left (x\right ) \log \left (\frac {\log \left (3 \, x\right )}{\log \left (x\right )}\right )^{2}} \,d x } \]
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Time = 15.16 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.12 \[ \int \frac {e^{\frac {-x^3+\log \left (\frac {x^4}{\log ^2(x)}\right )+x \log \left (\frac {\log (3 x)}{\log (x)}\right )}{-x^2+\log \left (\frac {\log (3 x)}{\log (x)}\right )}} \left (\left (2 x^2+\left (-4 x^2+x^5\right ) \log (x)\right ) \log (3 x)+\left (-\log (x)+\left (1+2 x^2 \log (x)\right ) \log (3 x)\right ) \log \left (\frac {x^4}{\log ^2(x)}\right )+\left (-2+\left (4-2 x^3\right ) \log (x)\right ) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )\right )}{x^5 \log (x) \log (3 x)-2 x^3 \log (x) \log (3 x) \log \left (\frac {\log (3 x)}{\log (x)}\right )+x \log (x) \log (3 x) \log ^2\left (\frac {\log (3 x)}{\log (x)}\right )} \, dx={\mathrm {e}}^{-\frac {x^3}{\ln \left (\frac {\ln \left (3\,x\right )}{\ln \left (x\right )}\right )-x^2}}\,{\left (\frac {1}{{\ln \left (x\right )}^2}\right )}^{\frac {1}{\ln \left (\frac {\ln \left (3\,x\right )}{\ln \left (x\right )}\right )-x^2}}\,{\left (x^4\right )}^{\frac {1}{\ln \left (\frac {\ln \left (3\,x\right )}{\ln \left (x\right )}\right )-x^2}}\,{\left (\frac {\ln \left (3\,x\right )}{\ln \left (x\right )}\right )}^{\frac {x}{\ln \left (\frac {\ln \left (3\,x\right )}{\ln \left (x\right )}\right )-x^2}} \]
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