Integrand size = 143, antiderivative size = 30 \[ \int \frac {80 e^{-2 e^5+2 x}+40 e^{-e^5+x} \log (5)+5 \log ^2(5)+e^{\frac {5 e^{-e^5+x}+2 e^{-e^5+x} \log (\log (4))}{4 e^{-e^5+x}+\log (5)}} \left (5 e^{-e^5+x} \log (5)+2 e^{-e^5+x} \log (5) \log (\log (4))\right )}{16 e^{-2 e^5+2 x}+8 e^{-e^5+x} \log (5)+\log ^2(5)} \, dx=e^{\frac {5+2 \log (\log (4))}{4+e^{e^5-x} \log (5)}}+5 x \]
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\[ \int \frac {80 e^{-2 e^5+2 x}+40 e^{-e^5+x} \log (5)+5 \log ^2(5)+e^{\frac {5 e^{-e^5+x}+2 e^{-e^5+x} \log (\log (4))}{4 e^{-e^5+x}+\log (5)}} \left (5 e^{-e^5+x} \log (5)+2 e^{-e^5+x} \log (5) \log (\log (4))\right )}{16 e^{-2 e^5+2 x}+8 e^{-e^5+x} \log (5)+\log ^2(5)} \, dx=\int \frac {80 e^{-2 e^5+2 x}+40 e^{-e^5+x} \log (5)+5 \log ^2(5)+\exp \left (\frac {5 e^{-e^5+x}+2 e^{-e^5+x} \log (\log (4))}{4 e^{-e^5+x}+\log (5)}\right ) \left (5 e^{-e^5+x} \log (5)+2 e^{-e^5+x} \log (5) \log (\log (4))\right )}{16 e^{-2 e^5+2 x}+8 e^{-e^5+x} \log (5)+\log ^2(5)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {80 x^2+5 \log ^2(5)+x \log (5) \left (40+e^{\frac {5 x}{4 x+\log (5)}} \log ^{\frac {2 x}{4 x+\log (5)}}(4) (5+2 \log (\log (4)))\right )}{x (4 x+\log (5))^2} \, dx,x,e^{\frac {1}{2} \left (-2 e^5+2 x\right )}\right ) \\ & = \text {Subst}\left (\int \left (\frac {5}{x}+\frac {e^{\frac {5 x}{4 x+\log (5)}} \log ^{\frac {2 x}{4 x+\log (5)}}(4) \log (5) (5+2 \log (\log (4)))}{(4 x+\log (5))^2}\right ) \, dx,x,e^{\frac {1}{2} \left (-2 e^5+2 x\right )}\right ) \\ & = 5 x+(\log (5) (5+2 \log (\log (4)))) \text {Subst}\left (\int \frac {e^{\frac {5 x}{4 x+\log (5)}} \log ^{\frac {2 x}{4 x+\log (5)}}(4)}{(4 x+\log (5))^2} \, dx,x,e^{\frac {1}{2} \left (-2 e^5+2 x\right )}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(30)=60\).
Time = 0.71 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.03 \[ \int \frac {80 e^{-2 e^5+2 x}+40 e^{-e^5+x} \log (5)+5 \log ^2(5)+e^{\frac {5 e^{-e^5+x}+2 e^{-e^5+x} \log (\log (4))}{4 e^{-e^5+x}+\log (5)}} \left (5 e^{-e^5+x} \log (5)+2 e^{-e^5+x} \log (5) \log (\log (4))\right )}{16 e^{-2 e^5+2 x}+8 e^{-e^5+x} \log (5)+\log ^2(5)} \, dx=5 x+5^{-\frac {5 e^{e^5}}{4 \left (4 e^x+e^{e^5} \log (5)\right )}} e^{5/4} \log ^{\frac {2 e^x}{4 e^x+e^{e^5} \log (5)}}(4) \]
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Time = 0.68 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23
method | result | size |
parallelrisch | \(5 x +{\mathrm e}^{\frac {{\mathrm e}^{-{\mathrm e}^{5}+x} \left (5+\ln \left (4 \ln \left (2\right )^{2}\right )\right )}{4 \,{\mathrm e}^{-{\mathrm e}^{5}+x}+\ln \left (5\right )}}\) | \(37\) |
risch | \(5 x +{\mathrm e}^{\frac {{\mathrm e}^{-{\mathrm e}^{5}+x} \left (5+2 \ln \left (2\right )+2 \ln \left (\ln \left (2\right )\right )\right )}{4 \,{\mathrm e}^{-{\mathrm e}^{5}+x}+\ln \left (5\right )}}\) | \(39\) |
parts | \(5 x -\frac {\left (\ln \left (5\right ) \ln \left (4 \ln \left (2\right )^{2}\right )+5 \ln \left (5\right )\right ) \left (-4 \ln \left (5\right ) \ln \left (4 \ln \left (2\right )^{2}\right )-20 \ln \left (5\right )\right ) {\mathrm e}^{\frac {-\frac {\ln \left (5\right ) \ln \left (4 \ln \left (2\right )^{2}\right )}{4}-\frac {5 \ln \left (5\right )}{4}}{4 \,{\mathrm e}^{-{\mathrm e}^{5}+x}+\ln \left (5\right )}+\frac {5}{4}+\frac {\ln \left (4 \ln \left (2\right )^{2}\right )}{4}}}{4 \ln \left (5\right )^{2} \left (4 \ln \left (2\right )^{2}+8 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+4 \ln \left (\ln \left (2\right )\right )^{2}+20 \ln \left (2\right )+20 \ln \left (\ln \left (2\right )\right )+25\right )}\) | \(118\) |
norman | \(\frac {\ln \left (5\right ) {\mathrm e}^{\frac {{\mathrm e}^{-{\mathrm e}^{5}+x} \ln \left (4 \ln \left (2\right )^{2}\right )+5 \,{\mathrm e}^{-{\mathrm e}^{5}+x}}{4 \,{\mathrm e}^{-{\mathrm e}^{5}+x}+\ln \left (5\right )}}+20 x \,{\mathrm e}^{-{\mathrm e}^{5}+x}+5 x \ln \left (5\right )+4 \,{\mathrm e}^{-{\mathrm e}^{5}+x} {\mathrm e}^{\frac {{\mathrm e}^{-{\mathrm e}^{5}+x} \ln \left (4 \ln \left (2\right )^{2}\right )+5 \,{\mathrm e}^{-{\mathrm e}^{5}+x}}{4 \,{\mathrm e}^{-{\mathrm e}^{5}+x}+\ln \left (5\right )}}}{4 \,{\mathrm e}^{-{\mathrm e}^{5}+x}+\ln \left (5\right )}\) | \(126\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1818\) |
default | \(\text {Expression too large to display}\) | \(1818\) |
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Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.50 \[ \int \frac {80 e^{-2 e^5+2 x}+40 e^{-e^5+x} \log (5)+5 \log ^2(5)+e^{\frac {5 e^{-e^5+x}+2 e^{-e^5+x} \log (\log (4))}{4 e^{-e^5+x}+\log (5)}} \left (5 e^{-e^5+x} \log (5)+2 e^{-e^5+x} \log (5) \log (\log (4))\right )}{16 e^{-2 e^5+2 x}+8 e^{-e^5+x} \log (5)+\log ^2(5)} \, dx=5 \, x + e^{\left (\frac {e^{\left (x - e^{5}\right )} \log \left (4 \, \log \left (2\right )^{2}\right ) + 5 \, e^{\left (x - e^{5}\right )}}{4 \, e^{\left (x - e^{5}\right )} + \log \left (5\right )}\right )} \]
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Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {80 e^{-2 e^5+2 x}+40 e^{-e^5+x} \log (5)+5 \log ^2(5)+e^{\frac {5 e^{-e^5+x}+2 e^{-e^5+x} \log (\log (4))}{4 e^{-e^5+x}+\log (5)}} \left (5 e^{-e^5+x} \log (5)+2 e^{-e^5+x} \log (5) \log (\log (4))\right )}{16 e^{-2 e^5+2 x}+8 e^{-e^5+x} \log (5)+\log ^2(5)} \, dx=5 x + e^{\frac {e^{x - e^{5}} \log {\left (4 \log {\left (2 \right )}^{2} \right )} + 5 e^{x - e^{5}}}{4 e^{x - e^{5}} + \log {\left (5 \right )}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (29) = 58\).
Time = 0.32 (sec) , antiderivative size = 279, normalized size of antiderivative = 9.30 \[ \int \frac {80 e^{-2 e^5+2 x}+40 e^{-e^5+x} \log (5)+5 \log ^2(5)+e^{\frac {5 e^{-e^5+x}+2 e^{-e^5+x} \log (\log (4))}{4 e^{-e^5+x}+\log (5)}} \left (5 e^{-e^5+x} \log (5)+2 e^{-e^5+x} \log (5) \log (\log (4))\right )}{16 e^{-2 e^5+2 x}+8 e^{-e^5+x} \log (5)+\log ^2(5)} \, dx=5 \, {\left (\frac {1}{4 \, e^{\left (x - e^{5}\right )} \log \left (5\right ) + \log \left (5\right )^{2}} + \frac {x - e^{5}}{\log \left (5\right )^{2}} - \frac {\log \left (4 \, e^{\left (x - e^{5}\right )} + \log \left (5\right )\right )}{\log \left (5\right )^{2}}\right )} \log \left (5\right )^{2} + \frac {\sqrt {2} e^{\left (-\frac {\log \left (5\right ) \log \left (2\right )}{2 \, {\left (4 \, e^{\left (x - e^{5}\right )} + \log \left (5\right )\right )}} - \frac {\log \left (5\right ) \log \left (\log \left (2\right )\right )}{2 \, {\left (4 \, e^{\left (x - e^{5}\right )} + \log \left (5\right )\right )}} - \frac {5 \, \log \left (5\right )}{4 \, {\left (4 \, e^{\left (x - e^{5}\right )} + \log \left (5\right )\right )}} + \frac {5}{4}\right )} \log \left (5\right ) \sqrt {\log \left (2\right )} \log \left (4 \, \log \left (2\right )^{2}\right )}{{\left (2 \, \log \left (\log \left (2\right )\right ) + 5\right )} \log \left (5\right ) + 2 \, \log \left (5\right ) \log \left (2\right )} + \frac {5 \, \sqrt {2} e^{\left (-\frac {\log \left (5\right ) \log \left (2\right )}{2 \, {\left (4 \, e^{\left (x - e^{5}\right )} + \log \left (5\right )\right )}} - \frac {\log \left (5\right ) \log \left (\log \left (2\right )\right )}{2 \, {\left (4 \, e^{\left (x - e^{5}\right )} + \log \left (5\right )\right )}} - \frac {5 \, \log \left (5\right )}{4 \, {\left (4 \, e^{\left (x - e^{5}\right )} + \log \left (5\right )\right )}} + \frac {5}{4}\right )} \log \left (5\right ) \sqrt {\log \left (2\right )}}{{\left (2 \, \log \left (\log \left (2\right )\right ) + 5\right )} \log \left (5\right ) + 2 \, \log \left (5\right ) \log \left (2\right )} - \frac {5 \, \log \left (5\right )}{4 \, e^{\left (x - e^{5}\right )} + \log \left (5\right )} + 5 \, \log \left (4 \, e^{\left (x - e^{5}\right )} + \log \left (5\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (29) = 58\).
Time = 0.44 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.67 \[ \int \frac {80 e^{-2 e^5+2 x}+40 e^{-e^5+x} \log (5)+5 \log ^2(5)+e^{\frac {5 e^{-e^5+x}+2 e^{-e^5+x} \log (\log (4))}{4 e^{-e^5+x}+\log (5)}} \left (5 e^{-e^5+x} \log (5)+2 e^{-e^5+x} \log (5) \log (\log (4))\right )}{16 e^{-2 e^5+2 x}+8 e^{-e^5+x} \log (5)+\log ^2(5)} \, dx={\left (5 \, x e^{x} + 5 \, e^{x} \log \left (4 \, e^{\left (x - e^{5}\right )} + \log \left (5\right )\right ) - 5 \, e^{x} \log \left (-4 \, e^{\left (x - e^{5}\right )} - \log \left (5\right )\right ) + e^{\left (\frac {4 \, x e^{\left (x - e^{5}\right )} + x \log \left (5\right ) + 2 \, e^{\left (x - e^{5}\right )} \log \left (2\right ) + 2 \, e^{\left (x - e^{5}\right )} \log \left (\log \left (2\right )\right ) + 5 \, e^{\left (x - e^{5}\right )}}{4 \, e^{\left (x - e^{5}\right )} + \log \left (5\right )}\right )}\right )} e^{\left (-x\right )} \]
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Time = 0.53 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \frac {80 e^{-2 e^5+2 x}+40 e^{-e^5+x} \log (5)+5 \log ^2(5)+e^{\frac {5 e^{-e^5+x}+2 e^{-e^5+x} \log (\log (4))}{4 e^{-e^5+x}+\log (5)}} \left (5 e^{-e^5+x} \log (5)+2 e^{-e^5+x} \log (5) \log (\log (4))\right )}{16 e^{-2 e^5+2 x}+8 e^{-e^5+x} \log (5)+\log ^2(5)} \, dx=5\,x+2^{\frac {2\,{\mathrm {e}}^{x-{\mathrm {e}}^5}}{4\,{\mathrm {e}}^{x-{\mathrm {e}}^5}+\ln \left (5\right )}}\,{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{-{\mathrm {e}}^5}\,{\mathrm {e}}^x}{\ln \left (5\right )+4\,{\mathrm {e}}^{-{\mathrm {e}}^5}\,{\mathrm {e}}^x}}\,{\ln \left (2\right )}^{\frac {2\,{\mathrm {e}}^{x-{\mathrm {e}}^5}}{4\,{\mathrm {e}}^{x-{\mathrm {e}}^5}+\ln \left (5\right )}} \]
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