Integrand size = 180, antiderivative size = 34 \[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=\frac {\log \left (x^2\right )}{\log \left (5+\frac {1}{5} \left (2-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \]
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\[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=\int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\frac {8 \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )}{x}+\frac {e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \left (5+2 x^3\right ) \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (-5+x^3\right )^2}}{4 \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx \\ & = \frac {1}{4} \int \frac {\frac {8 \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )}{x}+\frac {e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \left (5+2 x^3\right ) \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (-5+x^3\right )^2}}{\log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx \\ & = \frac {1}{4} \int \left (\frac {8}{x \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )}+\frac {e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \left (5+2 x^3\right ) \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (-5+x^3\right )^2 \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )}\right ) \, dx \\ & = \frac {1}{4} \int \frac {e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \left (5+2 x^3\right ) \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (-5+x^3\right )^2 \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx+2 \int \frac {1}{x \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx \\ & = \frac {1}{4} \int \left (\frac {15 e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (-5+x^3\right )^2 \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )}+\frac {2 e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (-5+x^3\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )}\right ) \, dx+2 \int \frac {1}{x \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx \\ & = \frac {1}{2} \int \frac {e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (-5+x^3\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx+2 \int \frac {1}{x \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx+\frac {15}{4} \int \frac {e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (-5+x^3\right )^2 \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx \\ & = \frac {1}{2} \int \left (-\frac {e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \log \left (x^2\right )}{3\ 5^{2/3} \left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (\sqrt [3]{5}-x\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )}-\frac {e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \log \left (x^2\right )}{3\ 5^{2/3} \left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (\sqrt [3]{5}+\sqrt [3]{-1} x\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )}-\frac {e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \log \left (x^2\right )}{3\ 5^{2/3} \left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (\sqrt [3]{5}-(-1)^{2/3} x\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )}\right ) \, dx+2 \int \frac {1}{x \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx+\frac {15}{4} \int \frac {e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (-5+x^3\right )^2 \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx \\ & = 2 \int \frac {1}{x \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx+\frac {15}{4} \int \frac {e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (-5+x^3\right )^2 \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx-\frac {\int \frac {e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (\sqrt [3]{5}-x\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx}{6\ 5^{2/3}}-\frac {\int \frac {e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (\sqrt [3]{5}+\sqrt [3]{-1} x\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx}{6\ 5^{2/3}}-\frac {\int \frac {e^{e^{\frac {x}{4 \left (-5+x^3\right )}}+\frac {x}{4 \left (-5+x^3\right )}} \log \left (x^2\right )}{\left (-27+e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right ) \left (\sqrt [3]{5}-(-1)^{2/3} x\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \, dx}{6\ 5^{2/3}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=\frac {\log \left (x^2\right )}{\log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{4 \left (-5+x^3\right )}}}\right )\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.51 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.24
\[\frac {4 \ln \left (x \right )-i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}}{2 \ln \left (-\frac {{\mathrm e}^{{\mathrm e}^{\frac {x}{4 x^{3}-20}}}}{5}+\frac {27}{5}\right )}\]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.91 \[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=\frac {\log \left (x^{2}\right )}{\log \left (-\frac {1}{5} \, {\left (e^{\left (\frac {4 \, {\left (x^{3} - 5\right )} e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )} + x}{4 \, {\left (x^{3} - 5\right )}}\right )} - 27 \, e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} e^{\left (-\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} \]
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Time = 10.87 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=\frac {\log {\left (x^{2} \right )}}{\log {\left (\frac {27}{5} - \frac {e^{e^{\frac {x}{4 x^{3} - 20}}}}{5} \right )}} \]
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Time = 0.51 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=-\frac {2 \, \log \left (x\right )}{\log \left (5\right ) - \log \left (-e^{\left (e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} + 27\right )} \]
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\[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=\int { -\frac {{\left (2 \, x^{4} + 5 \, x\right )} e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}} + e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} \log \left (x^{2}\right ) - 8 \, {\left (27 \, x^{6} - 270 \, x^{3} - {\left (x^{6} - 10 \, x^{3} + 25\right )} e^{\left (e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} + 675\right )} \log \left (-\frac {1}{5} \, e^{\left (e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} + \frac {27}{5}\right )}{4 \, {\left (27 \, x^{7} - 270 \, x^{4} - {\left (x^{7} - 10 \, x^{4} + 25 \, x\right )} e^{\left (e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} + 675 \, x\right )} \log \left (-\frac {1}{5} \, e^{\left (e^{\left (\frac {x}{4 \, {\left (x^{3} - 5\right )}}\right )}\right )} + \frac {27}{5}\right )^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (-5400+2160 x^3-216 x^6+e^{e^{\frac {x}{-20+4 x^3}}} \left (200-80 x^3+8 x^6\right )\right ) \log \left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )+e^{e^{\frac {x}{-20+4 x^3}}+\frac {x}{-20+4 x^3}} \left (5 x+2 x^4\right ) \log \left (x^2\right )}{\left (-2700 x+1080 x^4-108 x^7+e^{e^{\frac {x}{-20+4 x^3}}} \left (100 x-40 x^4+4 x^7\right )\right ) \log ^2\left (\frac {1}{5} \left (27-e^{e^{\frac {x}{-20+4 x^3}}}\right )\right )} \, dx=\int -\frac {\ln \left (\frac {27}{5}-\frac {{\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{4\,x^3-20}}}}{5}\right )\,\left ({\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{4\,x^3-20}}}\,\left (8\,x^6-80\,x^3+200\right )+2160\,x^3-216\,x^6-5400\right )+\ln \left (x^2\right )\,{\mathrm {e}}^{\frac {x}{4\,x^3-20}}\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{4\,x^3-20}}}\,\left (2\,x^4+5\,x\right )}{{\ln \left (\frac {27}{5}-\frac {{\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{4\,x^3-20}}}}{5}\right )}^2\,\left (2700\,x-{\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{4\,x^3-20}}}\,\left (4\,x^7-40\,x^4+100\,x\right )-1080\,x^4+108\,x^7\right )} \,d x \]
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