Integrand size = 89, antiderivative size = 26 \[ \int \frac {-9+e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )+e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{-90+45 x+e^{e^x} (15 x+(-60+30 x) \log (5))+e^{2 e^x} \left (5 x \log (5)+(-10+5 x) \log ^2(5)\right )} \, dx=2-\frac {1}{5} \log \left (-2+x+\frac {x}{3 e^{-e^x}+\log (5)}\right ) \]
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\[ \int \frac {-9+e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )+e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{-90+45 x+e^{e^x} (15 x+(-60+30 x) \log (5))+e^{2 e^x} \left (5 x \log (5)+(-10+5 x) \log ^2(5)\right )} \, dx=\int \frac {-9+e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )+e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{-90+45 x+e^{e^x} (15 x+(-60+30 x) \log (5))+e^{2 e^x} \left (5 x \log (5)+(-10+5 x) \log ^2(5)\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {9-e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )-e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{5 \left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx \\ & = \frac {1}{5} \int \frac {9-e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )-e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx \\ & = \frac {1}{5} \int \left (\frac {9}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}+\frac {3 e^{e^x+x} x}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}+\frac {e^{2 e^x} \log (5) (1+\log (5))}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}+\frac {3 e^{e^x} (1+\log (25))}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}\right ) \, dx \\ & = \frac {3}{5} \int \frac {e^{e^x+x} x}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx+\frac {9}{5} \int \frac {1}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{5} (\log (5) (1+\log (5))) \int \frac {e^{2 e^x}}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{5} (3 (1+\log (25))) \int \frac {e^{e^x}}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx \\ & = \frac {3}{5} \int \left (\frac {e^{e^x+x} \log (5)}{3 \left (3+e^{e^x} \log (5)\right )}+\frac {e^{e^x+x} (x (1+\log (5))-\log (25))}{3 \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}\right ) \, dx+\frac {9}{5} \int \left (\frac {x (1+\log (5))-\log (25)}{3 x \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}+\frac {\log (5)}{x \left (9+e^{e^x} \log (125)\right )}\right ) \, dx+\frac {1}{5} (\log (5) (1+\log (5))) \int \left (\frac {e^{2 e^x} (x (1+\log (5))-\log (25))}{3 x \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}+\frac {e^{2 e^x} \log (5)}{x \left (9+e^{e^x} \log (125)\right )}\right ) \, dx+\frac {1}{5} (3 (1+\log (25))) \int \left (\frac {e^{e^x} (x (1+\log (5))-\log (25))}{3 x \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}+\frac {e^{e^x} \log (5)}{x \left (9+e^{e^x} \log (125)\right )}\right ) \, dx \\ & = \frac {1}{5} \int \frac {e^{e^x+x} (x (1+\log (5))-\log (25))}{6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))} \, dx+\frac {3}{5} \int \frac {x (1+\log (5))-\log (25)}{x \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{5} \log (5) \int \frac {e^{e^x+x}}{3+e^{e^x} \log (5)} \, dx+\frac {1}{5} (9 \log (5)) \int \frac {1}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{15} (\log (5) (1+\log (5))) \int \frac {e^{2 e^x} (x (1+\log (5))-\log (25))}{x \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{5} \left (\log ^2(5) (1+\log (5))\right ) \int \frac {e^{2 e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{5} (1+\log (25)) \int \frac {e^{e^x} (x (1+\log (5))-\log (25))}{x \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{5} (3 \log (5) (1+\log (25))) \int \frac {e^{e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx \\ & = \frac {1}{5} \int \left (\frac {e^{e^x+x} \log (25)}{-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))}+\frac {e^{e^x+x} x (1+\log (5))}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)}\right ) \, dx+\frac {3}{5} \int \left (\frac {\log (25)}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )}+\frac {1+\log (5)}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)}\right ) \, dx+\frac {1}{5} \log (5) \text {Subst}\left (\int \frac {e^x}{3+e^x \log (5)} \, dx,x,e^x\right )+\frac {1}{5} (9 \log (5)) \int \frac {1}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{15} (\log (5) (1+\log (5))) \int \left (\frac {e^{2 e^x} \log (25)}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )}+\frac {e^{2 e^x} (1+\log (5))}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)}\right ) \, dx+\frac {1}{5} \left (\log ^2(5) (1+\log (5))\right ) \int \frac {e^{2 e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{5} (1+\log (25)) \int \left (\frac {e^{e^x} \log (25)}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )}+\frac {e^{e^x} (1+\log (5))}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)}\right ) \, dx+\frac {1}{5} (3 \log (5) (1+\log (25))) \int \frac {e^{e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx \\ & = \frac {1}{5} \log (5) \text {Subst}\left (\int \frac {1}{3+x \log (5)} \, dx,x,e^{e^x}\right )+\frac {1}{5} (9 \log (5)) \int \frac {1}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{5} (1+\log (5)) \int \frac {e^{e^x+x} x}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} (3 (1+\log (5))) \int \frac {1}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} \left (\log ^2(5) (1+\log (5))\right ) \int \frac {e^{2 e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{15} \left (\log (5) (1+\log (5))^2\right ) \int \frac {e^{2 e^x}}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} \log (25) \int \frac {e^{e^x+x}}{-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))} \, dx+\frac {1}{5} (3 \log (25)) \int \frac {1}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{15} (\log (5) (1+\log (5)) \log (25)) \int \frac {e^{2 e^x}}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{5} (3 \log (5) (1+\log (25))) \int \frac {e^{e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{5} ((1+\log (5)) (1+\log (25))) \int \frac {e^{e^x}}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} (\log (25) (1+\log (25))) \int \frac {e^{e^x}}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )} \, dx \\ & = \frac {1}{5} \log \left (3+e^{e^x} \log (5)\right )+\frac {1}{5} (9 \log (5)) \int \frac {1}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{5} (1+\log (5)) \int \frac {e^{e^x+x} x}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} (3 (1+\log (5))) \int \frac {1}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} \left (\log ^2(5) (1+\log (5))\right ) \int \frac {e^{2 e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{15} \left (\log (5) (1+\log (5))^2\right ) \int \frac {e^{2 e^x}}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} \log (25) \int \frac {e^{e^x+x}}{-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))} \, dx+\frac {1}{5} (3 \log (25)) \int \frac {1}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{15} (\log (5) (1+\log (5)) \log (25)) \int \frac {e^{2 e^x}}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{5} (3 \log (5) (1+\log (25))) \int \frac {e^{e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{5} ((1+\log (5)) (1+\log (25))) \int \frac {e^{e^x}}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} (\log (25) (1+\log (25))) \int \frac {e^{e^x}}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )} \, dx \\ \end{align*}
\[ \int \frac {-9+e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )+e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{-90+45 x+e^{e^x} (15 x+(-60+30 x) \log (5))+e^{2 e^x} \left (5 x \log (5)+(-10+5 x) \log ^2(5)\right )} \, dx=\int \frac {-9+e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )+e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{-90+45 x+e^{e^x} (15 x+(-60+30 x) \log (5))+e^{2 e^x} \left (5 x \log (5)+(-10+5 x) \log ^2(5)\right )} \, dx \]
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Time = 0.64 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54
method | result | size |
norman | \(\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{x}} \ln \left (5\right )+3\right )}{5}-\frac {\ln \left (\ln \left (5\right ) {\mathrm e}^{{\mathrm e}^{x}} x -2 \,{\mathrm e}^{{\mathrm e}^{x}} \ln \left (5\right )+x \,{\mathrm e}^{{\mathrm e}^{x}}+3 x -6\right )}{5}\) | \(40\) |
parallelrisch | \(\frac {\ln \left (\frac {{\mathrm e}^{{\mathrm e}^{x}} \ln \left (5\right )+3}{\ln \left (5\right )}\right )}{5}-\frac {\ln \left (\frac {\ln \left (5\right ) {\mathrm e}^{{\mathrm e}^{x}} x -2 \,{\mathrm e}^{{\mathrm e}^{x}} \ln \left (5\right )+x \,{\mathrm e}^{{\mathrm e}^{x}}+3 x -6}{\ln \left (5\right )+1}\right )}{5}\) | \(52\) |
risch | \(-\frac {\ln \left (\left (\ln \left (5\right )+1\right ) x -2 \ln \left (5\right )\right )}{5}-\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{x}}+\frac {-6+3 x}{x \ln \left (5\right )-2 \ln \left (5\right )+x}\right )}{5}+\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{x}}+\frac {3}{\ln \left (5\right )}\right )}{5}\) | \(53\) |
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).
Time = 0.44 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {-9+e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )+e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{-90+45 x+e^{e^x} (15 x+(-60+30 x) \log (5))+e^{2 e^x} \left (5 x \log (5)+(-10+5 x) \log ^2(5)\right )} \, dx=-\frac {1}{5} \, \log \left ({\left (x - 2\right )} \log \left (5\right ) + x\right ) + \frac {1}{5} \, \log \left (e^{\left (e^{x}\right )} \log \left (5\right ) + 3\right ) - \frac {1}{5} \, \log \left (\frac {{\left ({\left (x - 2\right )} \log \left (5\right ) + x\right )} e^{\left (e^{x}\right )} + 3 \, x - 6}{{\left (x - 2\right )} \log \left (5\right ) + x}\right ) \]
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Exception generated. \[ \int \frac {-9+e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )+e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{-90+45 x+e^{e^x} (15 x+(-60+30 x) \log (5))+e^{2 e^x} \left (5 x \log (5)+(-10+5 x) \log ^2(5)\right )} \, dx=\text {Exception raised: PolynomialError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (22) = 44\).
Time = 0.33 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.62 \[ \int \frac {-9+e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )+e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{-90+45 x+e^{e^x} (15 x+(-60+30 x) \log (5))+e^{2 e^x} \left (5 x \log (5)+(-10+5 x) \log ^2(5)\right )} \, dx=-\frac {1}{5} \, \log \left (x {\left (\log \left (5\right ) + 1\right )} - 2 \, \log \left (5\right )\right ) - \frac {1}{5} \, \log \left (\frac {{\left (x {\left (\log \left (5\right ) + 1\right )} - 2 \, \log \left (5\right )\right )} e^{\left (e^{x}\right )} + 3 \, x - 6}{x {\left (\log \left (5\right ) + 1\right )} - 2 \, \log \left (5\right )}\right ) + \frac {1}{5} \, \log \left (\frac {e^{\left (e^{x}\right )} \log \left (5\right ) + 3}{\log \left (5\right )}\right ) \]
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Time = 0.41 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {-9+e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )+e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{-90+45 x+e^{e^x} (15 x+(-60+30 x) \log (5))+e^{2 e^x} \left (5 x \log (5)+(-10+5 x) \log ^2(5)\right )} \, dx=-\frac {1}{5} \, e^{x} - \frac {1}{5} \, \log \left (x e^{\left (e^{x}\right )} \log \left (5\right ) + x e^{\left (e^{x}\right )} - 2 \, e^{\left (e^{x}\right )} \log \left (5\right ) + 3 \, x - 6\right ) + \frac {1}{5} \, \log \left (e^{\left (e^{x}\right )} \log \left (5\right ) + 3\right ) \]
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Timed out. \[ \int \frac {-9+e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )+e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{-90+45 x+e^{e^x} (15 x+(-60+30 x) \log (5))+e^{2 e^x} \left (5 x \log (5)+(-10+5 x) \log ^2(5)\right )} \, dx=\int -\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (6\,\ln \left (5\right )+3\,x\,{\mathrm {e}}^x+3\right )+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (\ln \left (5\right )+{\ln \left (5\right )}^2\right )+9}{45\,x+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (5\,x\,\ln \left (5\right )+{\ln \left (5\right )}^2\,\left (5\,x-10\right )\right )+{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (15\,x+\ln \left (5\right )\,\left (30\,x-60\right )\right )-90} \,d x \]
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