Integrand size = 251, antiderivative size = 31 \[ \int \frac {24 x+16 x^3+e^x \left (12+8 x^2\right )+\left (6 x-40 x^3+e^x \left (3 x-20 x^3\right )+\left (-8 x^3-4 e^x x^3\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right ) \log \left (\log ^2\left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )\right )}{\left (-12 x^3+80 x^5+e^{2 x} \left (-3 x+20 x^3\right )+e^x \left (-12 x^2+80 x^4\right )+\left (4 e^{2 x} x^3+16 e^x x^4+16 x^5\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )} \, dx=\frac {\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (2+\frac {e^x}{x}\right ) x} \]
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\[ \int \frac {24 x+16 x^3+e^x \left (12+8 x^2\right )+\left (6 x-40 x^3+e^x \left (3 x-20 x^3\right )+\left (-8 x^3-4 e^x x^3\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right ) \log \left (\log ^2\left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )\right )}{\left (-12 x^3+80 x^5+e^{2 x} \left (-3 x+20 x^3\right )+e^x \left (-12 x^2+80 x^4\right )+\left (4 e^{2 x} x^3+16 e^x x^4+16 x^5\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )} \, dx=\int \frac {24 x+16 x^3+e^x \left (12+8 x^2\right )+\left (6 x-40 x^3+e^x \left (3 x-20 x^3\right )+\left (-8 x^3-4 e^x x^3\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right ) \log \left (\log ^2\left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )\right )}{\left (-12 x^3+80 x^5+e^{2 x} \left (-3 x+20 x^3\right )+e^x \left (-12 x^2+80 x^4\right )+\left (4 e^{2 x} x^3+16 e^x x^4+16 x^5\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-24 x-16 x^3-4 e^x \left (3+2 x^2\right )+\left (2+e^x\right ) x \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right ) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{x \left (e^x+2 x\right )^2 \left (3-20 x^2-4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )} \, dx \\ & = \int \left (\frac {2 (-1+x) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2}-\frac {-12-8 x^2-3 x \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right ) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )+20 x^3 \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right ) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )+4 x^3 \log (x) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right ) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{x \left (e^x+2 x\right ) \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )}\right ) \, dx \\ & = 2 \int \frac {(-1+x) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2} \, dx-\int \frac {-12-8 x^2-3 x \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right ) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )+20 x^3 \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right ) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )+4 x^3 \log (x) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right ) \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{x \left (e^x+2 x\right ) \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )} \, dx \\ & = 2 \int \left (-\frac {\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2}+\frac {x \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2}\right ) \, dx-\int \frac {-\frac {4 \left (3+2 x^2\right )}{x \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )}+\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{e^x+2 x} \, dx \\ & = -\left (2 \int \frac {\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2} \, dx\right )+2 \int \frac {x \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2} \, dx-\int \left (-\frac {12}{x \left (e^x+2 x\right ) \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )}-\frac {8 x}{\left (e^x+2 x\right ) \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )}+\frac {\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{e^x+2 x}\right ) \, dx \\ & = -\left (2 \int \frac {\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2} \, dx\right )+2 \int \frac {x \log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{\left (e^x+2 x\right )^2} \, dx+8 \int \frac {x}{\left (e^x+2 x\right ) \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )} \, dx+12 \int \frac {1}{x \left (e^x+2 x\right ) \left (-3+20 x^2+4 x^2 \log (x)\right ) \log \left (5-\frac {3}{4 x^2}+\log (x)\right ) \log \left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )} \, dx-\int \frac {\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{e^x+2 x} \, dx \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {24 x+16 x^3+e^x \left (12+8 x^2\right )+\left (6 x-40 x^3+e^x \left (3 x-20 x^3\right )+\left (-8 x^3-4 e^x x^3\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right ) \log \left (\log ^2\left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )\right )}{\left (-12 x^3+80 x^5+e^{2 x} \left (-3 x+20 x^3\right )+e^x \left (-12 x^2+80 x^4\right )+\left (4 e^{2 x} x^3+16 e^x x^4+16 x^5\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )} \, dx=\frac {\log \left (\log ^2\left (\log \left (5-\frac {3}{4 x^2}+\log (x)\right )\right )\right )}{e^x+2 x} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 8.56 (sec) , antiderivative size = 820, normalized size of antiderivative = 26.45
\[\text {Expression too large to display}\]
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Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {24 x+16 x^3+e^x \left (12+8 x^2\right )+\left (6 x-40 x^3+e^x \left (3 x-20 x^3\right )+\left (-8 x^3-4 e^x x^3\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right ) \log \left (\log ^2\left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )\right )}{\left (-12 x^3+80 x^5+e^{2 x} \left (-3 x+20 x^3\right )+e^x \left (-12 x^2+80 x^4\right )+\left (4 e^{2 x} x^3+16 e^x x^4+16 x^5\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )} \, dx=\frac {\log \left (\log \left (\log \left (\frac {4 \, x^{2} \log \left (x\right ) + 20 \, x^{2} - 3}{4 \, x^{2}}\right )\right )^{2}\right )}{2 \, x + e^{x}} \]
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Timed out. \[ \int \frac {24 x+16 x^3+e^x \left (12+8 x^2\right )+\left (6 x-40 x^3+e^x \left (3 x-20 x^3\right )+\left (-8 x^3-4 e^x x^3\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right ) \log \left (\log ^2\left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )\right )}{\left (-12 x^3+80 x^5+e^{2 x} \left (-3 x+20 x^3\right )+e^x \left (-12 x^2+80 x^4\right )+\left (4 e^{2 x} x^3+16 e^x x^4+16 x^5\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )} \, dx=\text {Timed out} \]
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Time = 0.42 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {24 x+16 x^3+e^x \left (12+8 x^2\right )+\left (6 x-40 x^3+e^x \left (3 x-20 x^3\right )+\left (-8 x^3-4 e^x x^3\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right ) \log \left (\log ^2\left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )\right )}{\left (-12 x^3+80 x^5+e^{2 x} \left (-3 x+20 x^3\right )+e^x \left (-12 x^2+80 x^4\right )+\left (4 e^{2 x} x^3+16 e^x x^4+16 x^5\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )} \, dx=\frac {2 \, \log \left (\log \left (-2 \, \log \left (2\right ) + \log \left (4 \, x^{2} \log \left (x\right ) + 20 \, x^{2} - 3\right ) - 2 \, \log \left (x\right )\right )\right )}{2 \, x + e^{x}} \]
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Time = 2.53 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {24 x+16 x^3+e^x \left (12+8 x^2\right )+\left (6 x-40 x^3+e^x \left (3 x-20 x^3\right )+\left (-8 x^3-4 e^x x^3\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right ) \log \left (\log ^2\left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )\right )}{\left (-12 x^3+80 x^5+e^{2 x} \left (-3 x+20 x^3\right )+e^x \left (-12 x^2+80 x^4\right )+\left (4 e^{2 x} x^3+16 e^x x^4+16 x^5\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )} \, dx=\frac {\log \left (\log \left (-2 \, \log \left (2\right ) + \log \left (4 \, x^{2} \log \left (x\right ) + 20 \, x^{2} - 3\right ) - 2 \, \log \left (x\right )\right )^{2}\right )}{2 \, x + e^{x}} \]
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Timed out. \[ \int \frac {24 x+16 x^3+e^x \left (12+8 x^2\right )+\left (6 x-40 x^3+e^x \left (3 x-20 x^3\right )+\left (-8 x^3-4 e^x x^3\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right ) \log \left (\log ^2\left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )\right )}{\left (-12 x^3+80 x^5+e^{2 x} \left (-3 x+20 x^3\right )+e^x \left (-12 x^2+80 x^4\right )+\left (4 e^{2 x} x^3+16 e^x x^4+16 x^5\right ) \log (x)\right ) \log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right ) \log \left (\log \left (\frac {-3+20 x^2+4 x^2 \log (x)}{4 x^2}\right )\right )} \, dx=-\int \frac {24\,x+{\mathrm {e}}^x\,\left (8\,x^2+12\right )+16\,x^3+\ln \left (\frac {x^2\,\ln \left (x\right )+5\,x^2-\frac {3}{4}}{x^2}\right )\,\ln \left ({\ln \left (\ln \left (\frac {x^2\,\ln \left (x\right )+5\,x^2-\frac {3}{4}}{x^2}\right )\right )}^2\right )\,\ln \left (\ln \left (\frac {x^2\,\ln \left (x\right )+5\,x^2-\frac {3}{4}}{x^2}\right )\right )\,\left (6\,x+{\mathrm {e}}^x\,\left (3\,x-20\,x^3\right )-40\,x^3-\ln \left (x\right )\,\left (4\,x^3\,{\mathrm {e}}^x+8\,x^3\right )\right )}{\ln \left (\frac {x^2\,\ln \left (x\right )+5\,x^2-\frac {3}{4}}{x^2}\right )\,\ln \left (\ln \left (\frac {x^2\,\ln \left (x\right )+5\,x^2-\frac {3}{4}}{x^2}\right )\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (3\,x-20\,x^3\right )+{\mathrm {e}}^x\,\left (12\,x^2-80\,x^4\right )-\ln \left (x\right )\,\left (16\,x^4\,{\mathrm {e}}^x+4\,x^3\,{\mathrm {e}}^{2\,x}+16\,x^5\right )+12\,x^3-80\,x^5\right )} \,d x \]
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