Integrand size = 152, antiderivative size = 30 \[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=\frac {\log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )}{x} \]
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\[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=\int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {e^{3+\frac {e^3 x}{3}} \log ^2(3 x)}{9 x \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )}-\frac {2 e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x-e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x \log (3 x)+e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x \log (3 x) \log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )+\log ^3(3 x) \log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )}{x^2 \log (3 x) \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )}\right ) \, dx \\ & = -\left (\frac {1}{9} \int \frac {e^{3+\frac {e^3 x}{3}} \log ^2(3 x)}{x \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx\right )-\int \frac {2 e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x-e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x \log (3 x)+e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x \log (3 x) \log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )+\log ^3(3 x) \log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )}{x^2 \log (3 x) \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx \\ & = -\left (\frac {1}{9} \int \frac {e^{3+\frac {e^3 x}{3}} \log ^2(3 x)}{x \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx\right )-\int \left (\frac {(-2+\log (3 x)) \log (3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )}+\frac {2-\log (3 x)+\log (3 x) \log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )}{x^2 \log (3 x)}\right ) \, dx \\ & = -\left (\frac {1}{9} \int \frac {e^{3+\frac {e^3 x}{3}} \log ^2(3 x)}{x \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx\right )-\int \frac {(-2+\log (3 x)) \log (3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx-\int \frac {2-\log (3 x)+\log (3 x) \log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )}{x^2 \log (3 x)} \, dx \\ & = -\left (\frac {1}{9} \int \frac {e^{3+\frac {e^3 x}{3}} \log ^2(3 x)}{x \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx\right )-\int \left (-\frac {2 \log (3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )}+\frac {\log ^2(3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )}\right ) \, dx-\int \left (\frac {2-\log (3 x)}{x^2 \log (3 x)}+\frac {\log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )}{x^2}\right ) \, dx \\ & = -\left (\frac {1}{9} \int \frac {e^{3+\frac {e^3 x}{3}} \log ^2(3 x)}{x \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx\right )+2 \int \frac {\log (3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx-\int \frac {2-\log (3 x)}{x^2 \log (3 x)} \, dx-\int \frac {\log ^2(3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx-\int \frac {\log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )}{x^2} \, dx \\ & = -3 \operatorname {ExpIntegralEi}(-\log (3 x)) (2-\log (3 x))-\frac {1}{9} \int \frac {e^{3+\frac {e^3 x}{3}} \log ^2(3 x)}{x \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx+2 \int \frac {\log (3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx-\int \frac {3 \operatorname {ExpIntegralEi}(-\log (3 x))}{x} \, dx-\int \frac {\log ^2(3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx-\int \frac {\log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )}{x^2} \, dx \\ & = -3 \operatorname {ExpIntegralEi}(-\log (3 x)) (2-\log (3 x))-\frac {1}{9} \int \frac {e^{3+\frac {e^3 x}{3}} \log ^2(3 x)}{x \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx+2 \int \frac {\log (3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx-3 \int \frac {\operatorname {ExpIntegralEi}(-\log (3 x))}{x} \, dx-\int \frac {\log ^2(3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx-\int \frac {\log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )}{x^2} \, dx \\ & = -3 \operatorname {ExpIntegralEi}(-\log (3 x)) (2-\log (3 x))-\frac {1}{9} \int \frac {e^{3+\frac {e^3 x}{3}} \log ^2(3 x)}{x \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx+2 \int \frac {\log (3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx-3 \text {Subst}(\int \operatorname {ExpIntegralEi}(-x) \, dx,x,\log (3 x))-\int \frac {\log ^2(3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx-\int \frac {\log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )}{x^2} \, dx \\ & = -\frac {1}{x}-3 \operatorname {ExpIntegralEi}(-\log (3 x)) (2-\log (3 x))-3 \operatorname {ExpIntegralEi}(-\log (3 x)) \log (3 x)-\frac {1}{9} \int \frac {e^{3+\frac {e^3 x}{3}} \log ^2(3 x)}{x \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx+2 \int \frac {\log (3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx-\int \frac {\log ^2(3 x)}{x^2 \left (e^{\frac {1}{3} e^{\frac {e^3 x}{3}}} x+\log ^2(3 x)\right )} \, dx-\int \frac {\log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )}{x^2} \, dx \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=\frac {\log \left (e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}}+\frac {x}{\log ^2(3 x)}\right )}{x} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 300, normalized size of antiderivative = 10.00
\[\frac {\ln \left (\ln \left (3 x \right )^{2} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}}{3}}}{3}}+x \right )}{x}-\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (3 x \right )^{2} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}}{3}}}{3}}+x \right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (3 x \right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3 x \right )^{2} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}}{3}}}{3}}+x \right )}{\ln \left (3 x \right )^{2}}\right )-i \pi \,\operatorname {csgn}\left (i \left (\ln \left (3 x \right )^{2} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}}{3}}}{3}}+x \right )\right ) {\operatorname {csgn}\left (\frac {i \left (\ln \left (3 x \right )^{2} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}}{3}}}{3}}+x \right )}{\ln \left (3 x \right )^{2}}\right )}^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (3 x \right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (\ln \left (3 x \right )^{2} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}}{3}}}{3}}+x \right )}{\ln \left (3 x \right )^{2}}\right )}^{2}-i \pi \operatorname {csgn}\left (i \ln \left (3 x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (3 x \right )^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i \ln \left (3 x \right )\right ) \operatorname {csgn}\left (i \ln \left (3 x \right )^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i \ln \left (3 x \right )^{2}\right )^{3}+i \pi {\operatorname {csgn}\left (\frac {i \left (\ln \left (3 x \right )^{2} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}}{3}}}{3}}+x \right )}{\ln \left (3 x \right )^{2}}\right )}^{3}+4 \ln \left (\ln \left (3 x \right )\right )}{2 x}\]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=\frac {\log \left (\frac {{\left (e^{\left (\frac {1}{3} \, x e^{3} - \frac {1}{3} \, e^{\left (\frac {1}{3} \, x e^{3}\right )} + 3\right )} \log \left (3 \, x\right )^{2} + x e^{\left (\frac {1}{3} \, x e^{3} + 3\right )}\right )} e^{\left (-\frac {1}{3} \, x e^{3} - 3\right )}}{\log \left (3 \, x\right )^{2}}\right )}{x} \]
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Time = 4.62 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=\frac {\log {\left (\frac {x + e^{- \frac {e^{\frac {x e^{3}}{3}}}{3}} \log {\left (3 x \right )}^{2}}{\log {\left (3 x \right )}^{2}} \right )}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (23) = 46\).
Time = 0.34 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=-\frac {e^{\left (\frac {1}{3} \, x e^{3}\right )} - 3 \, \log \left (x e^{\left (\frac {1}{3} \, e^{\left (\frac {1}{3} \, x e^{3}\right )}\right )} + \log \left (3\right )^{2} + 2 \, \log \left (3\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right ) + 6 \, \log \left (\log \left (3\right ) + \log \left (x\right )\right )}{3 \, x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (23) = 46\).
Time = 13.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=\frac {\log \left ({\left (e^{\left (\frac {1}{3} \, x e^{3} - \frac {1}{3} \, e^{\left (\frac {1}{3} \, x e^{3}\right )}\right )} \log \left (3 \, x\right )^{2} + x e^{\left (\frac {1}{3} \, x e^{3}\right )}\right )} e^{\left (-\frac {1}{3} \, x e^{3}\right )}\right ) - \log \left (\log \left (3 \, x\right )^{2}\right )}{x} \]
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Timed out. \[ \int \frac {-18 x+9 x \log (3 x)-e^{3-\frac {1}{3} e^{\frac {e^3 x}{3}}+\frac {e^3 x}{3}} x \log ^3(3 x)+\left (-9 x \log (3 x)-9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^3(3 x)\right ) \log \left (\frac {x+e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} \log ^2(3 x)}{\log ^2(3 x)}\right )}{9 x^3 \log (3 x)+9 e^{-\frac {1}{3} e^{\frac {e^3 x}{3}}} x^2 \log ^3(3 x)} \, dx=\int -\frac {18\,x-9\,x\,\ln \left (3\,x\right )+\ln \left (\frac {{\mathrm {e}}^{-\frac {{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^3}{3}}}{3}}\,{\ln \left (3\,x\right )}^2+x}{{\ln \left (3\,x\right )}^2}\right )\,\left (9\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^3}{3}}}{3}}\,{\ln \left (3\,x\right )}^3+9\,x\,\ln \left (3\,x\right )\right )+x\,{\ln \left (3\,x\right )}^3\,{\mathrm {e}}^3\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^3}{3}}}{3}}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^3}{3}}}{9\,x^3\,\ln \left (3\,x\right )+9\,x^2\,{\ln \left (3\,x\right )}^3\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^3}{3}}}{3}}} \,d x \]
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