Integrand size = 223, antiderivative size = 32 \[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx=e^{-2+e^{5-e^5-\frac {\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} x}+x \]
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\[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx=\int \frac {\exp \left (-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}\right ) \left (\exp \left (\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}\right ) \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+\exp \left (\exp \left (-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}\right ) \left (-2 \exp \left (\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}\right )+x\right )\right ) \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx \]
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Rubi steps Aborted
Time = 2.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.75 \[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx=e^{-2+e^{-\frac {\left (-5+e^5\right ) x}{x+\log (x)}+\frac {\left (5-e^5\right ) \log (x)}{x+\log (x)}} x \left (\frac {x}{\log (x)}\right )^{-\frac {1}{x+\log (x)}}}+x \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 90.03 (sec) , antiderivative size = 155, normalized size of antiderivative = 4.84
method | result | size |
risch | \(x +{\mathrm e}^{\left (x \ln \left (x \right )^{\frac {1}{x +\ln \left (x \right )}} x^{\frac {4}{x +\ln \left (x \right )}} {\mathrm e}^{-\frac {-i \pi \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{3}+i \pi \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (i x \right )+i \pi \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )-i \pi \,\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )+2 x \,{\mathrm e}^{5}-10 x}{2 \left (x +\ln \left (x \right )\right )}}-2 x^{\frac {{\mathrm e}^{5}}{x +\ln \left (x \right )}}\right ) x^{-\frac {{\mathrm e}^{5}}{x +\ln \left (x \right )}}}\) | \(155\) |
parallelrisch | \(\frac {8 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+4 x \ln \left (x \right )+8 x \ln \left (\ln \left (x \right )\right )+8 \ln \left (x \right ) \ln \left (\frac {x}{\ln \left (x \right )}\right )-8 \ln \left (x \right )^{2}+12 x^{2}+12 \ln \left (x \right ) {\mathrm e}^{\left (-2 \,{\mathrm e}^{\frac {\ln \left (\frac {x}{\ln \left (x \right )}\right )+\left ({\mathrm e}^{5}-5\right ) \ln \left (x \right )+x \,{\mathrm e}^{5}-5 x}{x +\ln \left (x \right )}}+x \right ) {\mathrm e}^{-\frac {\ln \left (\frac {x}{\ln \left (x \right )}\right )+\left ({\mathrm e}^{5}-5\right ) \ln \left (x \right )+x \,{\mathrm e}^{5}-5 x}{x +\ln \left (x \right )}}}+12 \,{\mathrm e}^{\left (-2 \,{\mathrm e}^{\frac {\ln \left (\frac {x}{\ln \left (x \right )}\right )+\left ({\mathrm e}^{5}-5\right ) \ln \left (x \right )+x \,{\mathrm e}^{5}-5 x}{x +\ln \left (x \right )}}+x \right ) {\mathrm e}^{-\frac {\ln \left (\frac {x}{\ln \left (x \right )}\right )+\left ({\mathrm e}^{5}-5\right ) \ln \left (x \right )+x \,{\mathrm e}^{5}-5 x}{x +\ln \left (x \right )}}} x +8 \ln \left (\frac {x}{\ln \left (x \right )}\right ) x}{12 x +12 \ln \left (x \right )}\) | \(203\) |
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Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.16 \[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx=x + e^{\left ({\left (x - 2 \, e^{\left (\frac {x e^{5} + {\left (e^{5} - 5\right )} \log \left (x\right ) - 5 \, x + \log \left (\frac {x}{\log \left (x\right )}\right )}{x + \log \left (x\right )}\right )}\right )} e^{\left (-\frac {x e^{5} + {\left (e^{5} - 5\right )} \log \left (x\right ) - 5 \, x + \log \left (\frac {x}{\log \left (x\right )}\right )}{x + \log \left (x\right )}\right )}\right )} \]
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Exception generated. \[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx=\text {Exception raised: TypeError} \]
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none
Time = 1.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx={\left (x e^{2} + e^{\left (x e^{\left (-\frac {\log \left (x\right )}{x + \log \left (x\right )} + \frac {\log \left (\log \left (x\right )\right )}{x + \log \left (x\right )} - e^{5} + 5\right )}\right )}\right )} e^{\left (-2\right )} \]
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\[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx=\int { \frac {{\left ({\left ({\left (2 \, x - 1\right )} \log \left (x\right )^{2} + \log \left (x\right )^{3} + {\left (x + 1\right )} \log \left (x\right ) \log \left (\frac {x}{\log \left (x\right )}\right ) + {\left (x^{2} - x + 1\right )} \log \left (x\right ) + x\right )} e^{\left ({\left (x - 2 \, e^{\left (\frac {x e^{5} + {\left (e^{5} - 5\right )} \log \left (x\right ) - 5 \, x + \log \left (\frac {x}{\log \left (x\right )}\right )}{x + \log \left (x\right )}\right )}\right )} e^{\left (-\frac {x e^{5} + {\left (e^{5} - 5\right )} \log \left (x\right ) - 5 \, x + \log \left (\frac {x}{\log \left (x\right )}\right )}{x + \log \left (x\right )}\right )}\right )} + {\left (x^{2} \log \left (x\right ) + 2 \, x \log \left (x\right )^{2} + \log \left (x\right )^{3}\right )} e^{\left (\frac {x e^{5} + {\left (e^{5} - 5\right )} \log \left (x\right ) - 5 \, x + \log \left (\frac {x}{\log \left (x\right )}\right )}{x + \log \left (x\right )}\right )}\right )} e^{\left (-\frac {x e^{5} + {\left (e^{5} - 5\right )} \log \left (x\right ) - 5 \, x + \log \left (\frac {x}{\log \left (x\right )}\right )}{x + \log \left (x\right )}\right )}}{x^{2} \log \left (x\right ) + 2 \, x \log \left (x\right )^{2} + \log \left (x\right )^{3}} \,d x } \]
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Time = 17.84 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx=x+{\mathrm {e}}^{\frac {x\,x^{\frac {5}{x+\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {5\,x}{x+\ln \left (x\right )}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^5}{x+\ln \left (x\right )}}}{x^{\frac {{\mathrm {e}}^5}{x+\ln \left (x\right )}}\,{\left (\frac {x}{\ln \left (x\right )}\right )}^{\frac {1}{x+\ln \left (x\right )}}}}\,{\mathrm {e}}^{-2} \]
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