Integrand size = 117, antiderivative size = 28 \[ \int \frac {2 x-3 x^2+x^3-x^4+5 e^{-e^x-x} \left (1-x-e^x x+x^2\right )+\left (-5 e^{-e^x-x}-x+x^2\right ) \log \left (-5 e^{-e^x-x} x-x^2+x^3\right )}{5 e^{-e^x-x} x^2+x^3-x^4} \, dx=x+\frac {\log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )}{x} \]
[Out]
Time = 17.72 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14, number of steps used = 76, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6874, 6820, 1634, 2631, 78} \[ \int \frac {2 x-3 x^2+x^3-x^4+5 e^{-e^x-x} \left (1-x-e^x x+x^2\right )+\left (-5 e^{-e^x-x}-x+x^2\right ) \log \left (-5 e^{-e^x-x} x-x^2+x^3\right )}{5 e^{-e^x-x} x^2+x^3-x^4} \, dx=\frac {\log \left (-\left ((1-x) x^2\right )-5 e^{-x-e^x} x\right )}{x}+x \]
[In]
[Out]
Rule 78
Rule 1634
Rule 2631
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {5 e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {e^{-e^x} \left (5-2 e^{e^x}+3 e^{e^x} x-e^{e^x} x^2+e^{e^x} x^3+e^{e^x} \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )-e^{e^x} x \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )\right )}{(-1+x) x^2}\right ) \, dx \\ & = 5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+\int \frac {e^{-e^x} \left (5-2 e^{e^x}+3 e^{e^x} x-e^{e^x} x^2+e^{e^x} x^3+e^{e^x} \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )-e^{e^x} x \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )\right )}{(-1+x) x^2} \, dx \\ & = 5 \int \frac {e^{-e^x} \left (5+e^{e^x} \left (-1+x+x^2\right )\right )}{(1-x) x^2 \left (5-e^{e^x+x} (-1+x) x\right )} \, dx+\int \frac {e^{-e^x} \left (-5-e^{e^x} \left (-2+3 x-x^2+x^3\right )+e^{e^x} (-1+x) \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )\right )}{(1-x) x^2} \, dx \\ & = 5 \int \left (\frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}-\frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}-\frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx+\int \left (\frac {5 e^{-e^x}}{(-1+x) x^2}+\frac {-2+3 x-x^2+x^3+\log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )-x \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )}{(-1+x) x^2}\right ) \, dx \\ & = 5 \int \frac {e^{-e^x}}{(-1+x) x^2} \, dx+5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+\int \frac {-2+3 x-x^2+x^3+\log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )-x \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )}{(-1+x) x^2} \, dx \\ & = 5 \int \left (\frac {e^{-e^x}}{-1+x}-\frac {e^{-e^x}}{x^2}-\frac {e^{-e^x}}{x}\right ) \, dx-5 \int \frac {e^{-e^x} \left (-5-e^{e^x} \left (-1+x+x^2\right )\right )}{x^2 \left (5-e^{e^x+x} (-1+x) x\right )} \, dx-5 \int \frac {e^{-e^x} \left (-5-e^{e^x} \left (-1+x+x^2\right )\right )}{x \left (5-e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {e^{-e^x} \left (5+e^{e^x} \left (-1+x+x^2\right )\right )}{(1-x) \left (5-e^{e^x+x} (-1+x) x\right )} \, dx+\int \frac {2-3 x+x^2-x^3+(-1+x) \log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )}{(1-x) x^2} \, dx \\ & = 5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx-5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}-\frac {1}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {5 e^{-e^x}}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {1}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx-5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}-\frac {1}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {5 e^{-e^x}}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2}\right ) \, dx+5 \int \left (-\frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {5 e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x^2}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx+\int \left (\frac {-2+3 x-x^2+x^3}{(-1+x) x^2}-\frac {\log \left (-5 e^{-e^x-x} x+(-1+x) x^2\right )}{x^2}\right ) \, dx \\ & = 5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx-2 \left (5 \int \frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx\right )-5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \frac {1}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx+5 \int \frac {x}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \frac {x^2}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+\int \frac {-2+3 x-x^2+x^3}{(-1+x) x^2} \, dx-\int \frac {\log \left (-5 e^{-e^x-x} x+(-1+x) x^2\right )}{x^2} \, dx \\ & = \frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx-2 \left (5 \int \frac {1}{-5+e^{e^x+x} (-1+x) x} \, dx\right )+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-5 \int \frac {x}{-5+e^{e^x+x} (-1+x) x} \, dx-5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}+\frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx+5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}+\frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2}\right ) \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+\int \left (1+\frac {1}{-1+x}+\frac {2}{x^2}-\frac {1}{x}\right ) \, dx-\int \frac {5-\left (5+5 e^x-2 e^{e^x+x}\right ) x-3 e^{e^x+x} x^2}{x^2 \left (5-e^{e^x+x} (-1+x) x\right )} \, dx \\ & = -\frac {2}{x}+x+\log (1-x)-\log (x)+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx-2 \left (5 \int \frac {1}{-5+e^{e^x+x} (-1+x) x} \, dx\right )+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-5 \int \frac {x}{-5+e^{e^x+x} (-1+x) x} \, dx+2 \left (5 \int \frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx\right )+5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-\int \left (\frac {e^{-e^x} \left (5-2 e^{e^x}+3 e^{e^x} x\right )}{(-1+x) x^2}+\frac {5 e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx \\ & = -\frac {2}{x}+x+\log (1-x)-\log (x)+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-\int \frac {e^{-e^x} \left (5-2 e^{e^x}+3 e^{e^x} x\right )}{(-1+x) x^2} \, dx \\ & = -\frac {2}{x}+x+\log (1-x)-\log (x)+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \frac {e^{-e^x} \left (5+e^{e^x} \left (-1+x+x^2\right )\right )}{(1-x) x^2 \left (5-e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-\int \frac {2-5 e^{-e^x}-3 x}{(1-x) x^2} \, dx \\ & = -\frac {2}{x}+x+\log (1-x)-\log (x)+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \left (\frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}-\frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}-\frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-\int \left (\frac {5 e^{-e^x}}{(-1+x) x^2}+\frac {-2+3 x}{(-1+x) x^2}\right ) \, dx \\ & = -\frac {2}{x}+x+\log (1-x)-\log (x)+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{(-1+x) x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \frac {e^{-e^x} \left (5-e^{e^x}+e^{e^x} x+e^{e^x} x^2\right )}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-\int \frac {-2+3 x}{(-1+x) x^2} \, dx \\ & = -\frac {2}{x}+x+\log (1-x)-\log (x)+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}-5 \int \left (\frac {e^{-e^x}}{-1+x}-\frac {e^{-e^x}}{x^2}-\frac {e^{-e^x}}{x}\right ) \, dx+5 \int \frac {e^{-e^x}}{-1+x} \, dx-5 \int \frac {e^{-e^x}}{x^2} \, dx-5 \int \frac {e^{-e^x}}{x} \, dx+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \frac {e^{-e^x} \left (-5-e^{e^x} \left (-1+x+x^2\right )\right )}{x^2 \left (5-e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {e^{-e^x} \left (-5-e^{e^x} \left (-1+x+x^2\right )\right )}{x \left (5-e^{e^x+x} (-1+x) x\right )} \, dx-5 \int \frac {e^{-e^x} \left (5+e^{e^x} \left (-1+x+x^2\right )\right )}{(1-x) \left (5-e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-\int \left (\frac {1}{-1+x}+\frac {2}{x^2}-\frac {1}{x}\right ) \, dx \\ & = x+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}-\frac {1}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {5 e^{-e^x}}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {1}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx+5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}-\frac {1}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {5 e^{-e^x}}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2}\right ) \, dx-5 \int \left (-\frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {5 e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x^2}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx \\ & = x+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+5 \int \frac {1}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+2 \left (5 \int \frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx\right )+2 \left (5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx\right )-5 \int \frac {1}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+5 \int \frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx-5 \int \frac {x}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-5 \int \frac {x^2}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx-25 \int \frac {e^{-e^x}}{x^2 \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx-25 \int \frac {e^{-e^x}}{x \left (-5+e^{e^x+x} (-1+x) x\right )} \, dx+25 \int \frac {e^{-e^x}}{x^2 \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx+25 \int \frac {e^{-e^x}}{x \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx \\ & = x+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+2 \left (5 \int \frac {1}{-5+e^{e^x+x} (-1+x) x} \, dx\right )+5 \int \frac {x}{-5+e^{e^x+x} (-1+x) x} \, dx+2 \left (5 \int \frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )} \, dx\right )-5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}+\frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}\right ) \, dx-5 \int \left (\frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2}+\frac {1}{(-1+x) \left (-5-e^{e^x+x} x+e^{e^x+x} x^2\right )}+\frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2}\right ) \, dx \\ & = x+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x}+2 \left (5 \int \frac {1}{-5+e^{e^x+x} (-1+x) x} \, dx\right )+5 \int \frac {x}{-5+e^{e^x+x} (-1+x) x} \, dx-2 \left (5 \int \frac {1}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx\right )-5 \int \frac {x}{-5-e^{e^x+x} x+e^{e^x+x} x^2} \, dx \\ & = x+\frac {\log \left (-5 e^{-e^x-x} x-(1-x) x^2\right )}{x} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {2 x-3 x^2+x^3-x^4+5 e^{-e^x-x} \left (1-x-e^x x+x^2\right )+\left (-5 e^{-e^x-x}-x+x^2\right ) \log \left (-5 e^{-e^x-x} x-x^2+x^3\right )}{5 e^{-e^x-x} x^2+x^3-x^4} \, dx=1+x+\frac {\log \left (x \left (-5 e^{-e^x-x}+(-1+x) x\right )\right )}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(95\) vs. \(2(30)=60\).
Time = 13.82 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.43
method | result | size |
parallelrisch | \(-\frac {2 x \ln \left (x \right )+2 \ln \left (-{\mathrm e}^{\ln \left (5 \,{\mathrm e}^{-{\mathrm e}^{x}}\right )-x}+x^{2}-x \right ) x -2 x^{2}-2 \ln \left (-x \left (-x^{2}+{\mathrm e}^{\ln \left (5 \,{\mathrm e}^{-{\mathrm e}^{x}}\right )-x}+x \right )\right ) x -2 \ln \left (-x \left (-x^{2}+{\mathrm e}^{\ln \left (5 \,{\mathrm e}^{-{\mathrm e}^{x}}\right )-x}+x \right )\right )}{2 x}\) | \(96\) |
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Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {2 x-3 x^2+x^3-x^4+5 e^{-e^x-x} \left (1-x-e^x x+x^2\right )+\left (-5 e^{-e^x-x}-x+x^2\right ) \log \left (-5 e^{-e^x-x} x-x^2+x^3\right )}{5 e^{-e^x-x} x^2+x^3-x^4} \, dx=\frac {x^{2} + \log \left (x^{3} - x^{2} - x e^{\left (-x - e^{x} + \log \left (5\right )\right )}\right )}{x} \]
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Time = 0.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {2 x-3 x^2+x^3-x^4+5 e^{-e^x-x} \left (1-x-e^x x+x^2\right )+\left (-5 e^{-e^x-x}-x+x^2\right ) \log \left (-5 e^{-e^x-x} x-x^2+x^3\right )}{5 e^{-e^x-x} x^2+x^3-x^4} \, dx=x + \frac {\log {\left (x^{3} - x^{2} - 5 x e^{- x} e^{- e^{x}} \right )}}{x} \]
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Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {2 x-3 x^2+x^3-x^4+5 e^{-e^x-x} \left (1-x-e^x x+x^2\right )+\left (-5 e^{-e^x-x}-x+x^2\right ) \log \left (-5 e^{-e^x-x} x-x^2+x^3\right )}{5 e^{-e^x-x} x^2+x^3-x^4} \, dx=\frac {x^{2} - e^{x} + \log \left ({\left (x^{2} - x\right )} e^{\left (x + e^{x}\right )} - 5\right ) + \log \left (x\right )}{x} \]
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Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {2 x-3 x^2+x^3-x^4+5 e^{-e^x-x} \left (1-x-e^x x+x^2\right )+\left (-5 e^{-e^x-x}-x+x^2\right ) \log \left (-5 e^{-e^x-x} x-x^2+x^3\right )}{5 e^{-e^x-x} x^2+x^3-x^4} \, dx=\frac {x^{2} - e^{x} + \log \left (x^{2} e^{\left (x + e^{x}\right )} - x e^{\left (x + e^{x}\right )} - 5\right ) + \log \left (x\right )}{x} \]
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Time = 9.33 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {2 x-3 x^2+x^3-x^4+5 e^{-e^x-x} \left (1-x-e^x x+x^2\right )+\left (-5 e^{-e^x-x}-x+x^2\right ) \log \left (-5 e^{-e^x-x} x-x^2+x^3\right )}{5 e^{-e^x-x} x^2+x^3-x^4} \, dx=x+\frac {\ln \left (x^3-x^2-5\,x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-{\mathrm {e}}^x}\right )}{x} \]
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