Integrand size = 128, antiderivative size = 25 \[ \int \frac {3 e^{\log ^8(4)} \log (x)+e^{\log ^8(4)} (-3-8 x) \log \left (\frac {3+8 x}{x}\right )+3 e^{\log ^8(4)} \log (\log (2))}{\left (3 x+8 x^2\right ) \log ^2(x) \log ^2\left (\frac {3+8 x}{x}\right )+\left (6 x+16 x^2\right ) \log (x) \log ^2\left (\frac {3+8 x}{x}\right ) \log (\log (2))+\left (3 x+8 x^2\right ) \log ^2\left (\frac {3+8 x}{x}\right ) \log ^2(\log (2))} \, dx=\frac {e^{\log ^8(4)}}{\log \left (8+\frac {3}{x}\right ) (\log (x)+\log (\log (2)))} \]
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\[ \int \frac {3 e^{\log ^8(4)} \log (x)+e^{\log ^8(4)} (-3-8 x) \log \left (\frac {3+8 x}{x}\right )+3 e^{\log ^8(4)} \log (\log (2))}{\left (3 x+8 x^2\right ) \log ^2(x) \log ^2\left (\frac {3+8 x}{x}\right )+\left (6 x+16 x^2\right ) \log (x) \log ^2\left (\frac {3+8 x}{x}\right ) \log (\log (2))+\left (3 x+8 x^2\right ) \log ^2\left (\frac {3+8 x}{x}\right ) \log ^2(\log (2))} \, dx=\int \frac {3 e^{\log ^8(4)} \log (x)+e^{\log ^8(4)} (-3-8 x) \log \left (\frac {3+8 x}{x}\right )+3 e^{\log ^8(4)} \log (\log (2))}{\left (3 x+8 x^2\right ) \log ^2(x) \log ^2\left (\frac {3+8 x}{x}\right )+\left (6 x+16 x^2\right ) \log (x) \log ^2\left (\frac {3+8 x}{x}\right ) \log (\log (2))+\left (3 x+8 x^2\right ) \log ^2\left (\frac {3+8 x}{x}\right ) \log ^2(\log (2))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{\log ^8(4)} \left (-\left ((3+8 x) \log \left (8+\frac {3}{x}\right )\right )+3 \log (x \log (2))\right )}{x (3+8 x) \log ^2\left (8+\frac {3}{x}\right ) \log ^2(x \log (2))} \, dx \\ & = e^{\log ^8(4)} \int \frac {-\left ((3+8 x) \log \left (8+\frac {3}{x}\right )\right )+3 \log (x \log (2))}{x (3+8 x) \log ^2\left (8+\frac {3}{x}\right ) \log ^2(x \log (2))} \, dx \\ & = e^{\log ^8(4)} \int \left (-\frac {1}{x \log \left (8+\frac {3}{x}\right ) \log ^2(x \log (2))}+\frac {3}{x (3+8 x) \log ^2\left (8+\frac {3}{x}\right ) \log (x \log (2))}\right ) \, dx \\ & = -\left (e^{\log ^8(4)} \int \frac {1}{x \log \left (8+\frac {3}{x}\right ) \log ^2(x \log (2))} \, dx\right )+\left (3 e^{\log ^8(4)}\right ) \int \frac {1}{x (3+8 x) \log ^2\left (8+\frac {3}{x}\right ) \log (x \log (2))} \, dx \\ & = -\left (e^{\log ^8(4)} \int \frac {1}{x \log \left (8+\frac {3}{x}\right ) \log ^2(x \log (2))} \, dx\right )+\left (3 e^{\log ^8(4)}\right ) \int \left (\frac {1}{3 x \log ^2\left (8+\frac {3}{x}\right ) \log (x \log (2))}-\frac {8}{3 (3+8 x) \log ^2\left (8+\frac {3}{x}\right ) \log (x \log (2))}\right ) \, dx \\ & = -\left (e^{\log ^8(4)} \int \frac {1}{x \log \left (8+\frac {3}{x}\right ) \log ^2(x \log (2))} \, dx\right )+e^{\log ^8(4)} \int \frac {1}{x \log ^2\left (8+\frac {3}{x}\right ) \log (x \log (2))} \, dx-\left (8 e^{\log ^8(4)}\right ) \int \frac {1}{(3+8 x) \log ^2\left (8+\frac {3}{x}\right ) \log (x \log (2))} \, dx \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {3 e^{\log ^8(4)} \log (x)+e^{\log ^8(4)} (-3-8 x) \log \left (\frac {3+8 x}{x}\right )+3 e^{\log ^8(4)} \log (\log (2))}{\left (3 x+8 x^2\right ) \log ^2(x) \log ^2\left (\frac {3+8 x}{x}\right )+\left (6 x+16 x^2\right ) \log (x) \log ^2\left (\frac {3+8 x}{x}\right ) \log (\log (2))+\left (3 x+8 x^2\right ) \log ^2\left (\frac {3+8 x}{x}\right ) \log ^2(\log (2))} \, dx=\frac {e^{\log ^8(4)}}{\log \left (8+\frac {3}{x}\right ) \log (x \log (2))} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 660.22 (sec) , antiderivative size = 125, normalized size of antiderivative = 5.00
\[\frac {2 \,{\mathrm e}^{256 \ln \left (2\right )^{8}}}{\left (\ln \left (x \right )+\ln \left (\ln \left (2\right )\right )\right ) \left (6 \ln \left (2\right )-2 \ln \left (x \right )+2 \ln \left (x +\frac {3}{8}\right )-i \pi \,\operatorname {csgn}\left (i \left (x +\frac {3}{8}\right )\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (x +\frac {3}{8}\right )}{x}\right )+i \pi \,\operatorname {csgn}\left (i \left (x +\frac {3}{8}\right )\right ) \operatorname {csgn}\left (\frac {i \left (x +\frac {3}{8}\right )}{x}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (x +\frac {3}{8}\right )}{x}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (x +\frac {3}{8}\right )}{x}\right )^{3}\right )}\]
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Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {3 e^{\log ^8(4)} \log (x)+e^{\log ^8(4)} (-3-8 x) \log \left (\frac {3+8 x}{x}\right )+3 e^{\log ^8(4)} \log (\log (2))}{\left (3 x+8 x^2\right ) \log ^2(x) \log ^2\left (\frac {3+8 x}{x}\right )+\left (6 x+16 x^2\right ) \log (x) \log ^2\left (\frac {3+8 x}{x}\right ) \log (\log (2))+\left (3 x+8 x^2\right ) \log ^2\left (\frac {3+8 x}{x}\right ) \log ^2(\log (2))} \, dx=\frac {e^{\left (256 \, \log \left (2\right )^{8}\right )}}{\log \left (x\right ) \log \left (\frac {8 \, x + 3}{x}\right ) + \log \left (\frac {8 \, x + 3}{x}\right ) \log \left (\log \left (2\right )\right )} \]
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Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {3 e^{\log ^8(4)} \log (x)+e^{\log ^8(4)} (-3-8 x) \log \left (\frac {3+8 x}{x}\right )+3 e^{\log ^8(4)} \log (\log (2))}{\left (3 x+8 x^2\right ) \log ^2(x) \log ^2\left (\frac {3+8 x}{x}\right )+\left (6 x+16 x^2\right ) \log (x) \log ^2\left (\frac {3+8 x}{x}\right ) \log (\log (2))+\left (3 x+8 x^2\right ) \log ^2\left (\frac {3+8 x}{x}\right ) \log ^2(\log (2))} \, dx=\frac {e^{256 \log {\left (2 \right )}^{8}}}{\left (\log {\left (x \right )} + \log {\left (\log {\left (2 \right )} \right )}\right ) \log {\left (\frac {8 x + 3}{x} \right )}} \]
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Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {3 e^{\log ^8(4)} \log (x)+e^{\log ^8(4)} (-3-8 x) \log \left (\frac {3+8 x}{x}\right )+3 e^{\log ^8(4)} \log (\log (2))}{\left (3 x+8 x^2\right ) \log ^2(x) \log ^2\left (\frac {3+8 x}{x}\right )+\left (6 x+16 x^2\right ) \log (x) \log ^2\left (\frac {3+8 x}{x}\right ) \log (\log (2))+\left (3 x+8 x^2\right ) \log ^2\left (\frac {3+8 x}{x}\right ) \log ^2(\log (2))} \, dx=\frac {e^{\left (256 \, \log \left (2\right )^{8}\right )}}{{\left (\log \left (x\right ) + \log \left (\log \left (2\right )\right )\right )} \log \left (8 \, x + 3\right ) - \log \left (x\right )^{2} - \log \left (x\right ) \log \left (\log \left (2\right )\right )} \]
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Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int \frac {3 e^{\log ^8(4)} \log (x)+e^{\log ^8(4)} (-3-8 x) \log \left (\frac {3+8 x}{x}\right )+3 e^{\log ^8(4)} \log (\log (2))}{\left (3 x+8 x^2\right ) \log ^2(x) \log ^2\left (\frac {3+8 x}{x}\right )+\left (6 x+16 x^2\right ) \log (x) \log ^2\left (\frac {3+8 x}{x}\right ) \log (\log (2))+\left (3 x+8 x^2\right ) \log ^2\left (\frac {3+8 x}{x}\right ) \log ^2(\log (2))} \, dx=\frac {e^{\left (256 \, \log \left (2\right )^{8}\right )}}{\log \left (8 \, x + 3\right ) \log \left (x\right ) - \log \left (x\right )^{2} + \log \left (8 \, x + 3\right ) \log \left (\log \left (2\right )\right ) - \log \left (x\right ) \log \left (\log \left (2\right )\right )} \]
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Time = 10.13 (sec) , antiderivative size = 181, normalized size of antiderivative = 7.24 \[ \int \frac {3 e^{\log ^8(4)} \log (x)+e^{\log ^8(4)} (-3-8 x) \log \left (\frac {3+8 x}{x}\right )+3 e^{\log ^8(4)} \log (\log (2))}{\left (3 x+8 x^2\right ) \log ^2(x) \log ^2\left (\frac {3+8 x}{x}\right )+\left (6 x+16 x^2\right ) \log (x) \log ^2\left (\frac {3+8 x}{x}\right ) \log (\log (2))+\left (3 x+8 x^2\right ) \log ^2\left (\frac {3+8 x}{x}\right ) \log ^2(\log (2))} \, dx=\frac {\frac {4\,x\,\left ({\mathrm {e}}^{256\,{\ln \left (2\right )}^8}-{\mathrm {e}}^{256\,{\ln \left (2\right )}^8}\,\ln \left (\ln \left (2\right )\right )\right )}{3}-\frac {4\,x\,{\mathrm {e}}^{256\,{\ln \left (2\right )}^8}\,\ln \left (x\right )}{3}}{\ln \left (\ln \left (2\right )\right )+\ln \left (x\right )}+\frac {4\,x\,{\mathrm {e}}^{256\,{\ln \left (2\right )}^8}}{3}+\frac {{\mathrm {e}}^{256\,{\ln \left (2\right )}^8}+\frac {8\,x\,{\mathrm {e}}^{256\,{\ln \left (2\right )}^8}}{3}-\frac {4\,x\,{\mathrm {e}}^{256\,{\ln \left (2\right )}^8}\,\ln \left (\ln \left (2\right )\right )}{3}-\frac {4\,x\,{\mathrm {e}}^{256\,{\ln \left (2\right )}^8}\,\ln \left (x\right )}{3}}{{\ln \left (x\right )}^2+2\,\ln \left (\ln \left (2\right )\right )\,\ln \left (x\right )+{\ln \left (\ln \left (2\right )\right )}^2}+\frac {\frac {{\mathrm {e}}^{256\,{\ln \left (2\right )}^8}}{\ln \left (\ln \left (2\right )\right )+\ln \left (x\right )}-\frac {{\mathrm {e}}^{256\,{\ln \left (2\right )}^8}\,\ln \left (\frac {8\,x+3}{x}\right )\,\left (8\,x+3\right )}{3\,{\left (\ln \left (\ln \left (2\right )\right )+\ln \left (x\right )\right )}^2}}{\ln \left (\frac {8\,x+3}{x}\right )} \]
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