Integrand size = 134, antiderivative size = 27 \[ \int \frac {e^x \left (-5+e^3\right )+5 x+5 x^2+e^3 \left (-x-x^2\right )+e^{\left (-e^x+x+x^2\right )^x} \left (e^x-x-x^2+\left (-e^x+x+x^2\right )^x \left (x^2-e^x x^2+2 x^3+\left (-e^x x+x^2+x^3\right ) \log \left (-e^x+x+x^2\right )\right )\right )}{e^x x^2-x^3-x^4} \, dx=\frac {5-e^3-e^{\left (-e^x+x+x^2\right )^x}}{x} \]
[Out]
\[ \int \frac {e^x \left (-5+e^3\right )+5 x+5 x^2+e^3 \left (-x-x^2\right )+e^{\left (-e^x+x+x^2\right )^x} \left (e^x-x-x^2+\left (-e^x+x+x^2\right )^x \left (x^2-e^x x^2+2 x^3+\left (-e^x x+x^2+x^3\right ) \log \left (-e^x+x+x^2\right )\right )\right )}{e^x x^2-x^3-x^4} \, dx=\int \frac {e^x \left (-5+e^3\right )+5 x+5 x^2+e^3 \left (-x-x^2\right )+e^{\left (-e^x+x+x^2\right )^x} \left (e^x-x-x^2+\left (-e^x+x+x^2\right )^x \left (x^2-e^x x^2+2 x^3+\left (-e^x x+x^2+x^3\right ) \log \left (-e^x+x+x^2\right )\right )\right )}{e^x x^2-x^3-x^4} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {5}{e^x-x-x^2}-\frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2}+\frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )}+\frac {e^x \left (-5+e^3\right )}{x^2 \left (e^x-x-x^2\right )}-\frac {5}{x \left (-e^x+x+x^2\right )}+\frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )}+\frac {e^3 (1+x)}{x \left (-e^x+x+x^2\right )}-\frac {e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \left (x-e^x x+2 x^2-e^x \log \left (-e^x+x+x^2\right )+x \log \left (-e^x+x+x^2\right )+x^2 \log \left (-e^x+x+x^2\right )\right )}{x}\right ) \, dx \\ & = 5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+e^3 \int \frac {1+x}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \left (x-e^x x+2 x^2-e^x \log \left (-e^x+x+x^2\right )+x \log \left (-e^x+x+x^2\right )+x^2 \log \left (-e^x+x+x^2\right )\right )}{x} \, dx \\ & = 5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+e^3 \int \left (-\frac {1}{e^x-x-x^2}+\frac {1}{x \left (-e^x+x+x^2\right )}\right ) \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \left (x \left (1-e^x+2 x\right )+\left (-e^x+x+x^2\right ) \log \left (-e^x+x+x^2\right )\right )}{x} \, dx \\ & = 5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx-e^3 \int \frac {1}{e^x-x-x^2} \, dx+e^3 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int \left (e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}+2 e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x}+e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \log \left (-e^x+x+x^2\right )+e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \log \left (-e^x+x+x^2\right )-\frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \left (x+\log \left (-e^x+x+x^2\right )\right )}{x}\right ) \, dx \\ & = -\left (2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right )+5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx-e^3 \int \frac {1}{e^x-x-x^2} \, dx+e^3 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx-\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \log \left (-e^x+x+x^2\right ) \, dx-\int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \log \left (-e^x+x+x^2\right ) \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \left (x+\log \left (-e^x+x+x^2\right )\right )}{x} \, dx \\ & = -\left (2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right )+5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx-e^3 \int \frac {1}{e^x-x-x^2} \, dx+e^3 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int \left (e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}+\frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \log \left (-e^x+x+x^2\right )}{x}\right ) \, dx+\int \frac {\left (-1+e^x-2 x\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{e^x-x (1+x)} \, dx+\int \frac {\left (-1+e^x-2 x\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{e^x-x (1+x)} \, dx \\ & = -\left (2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right )+5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx-e^3 \int \frac {1}{e^x-x-x^2} \, dx+e^3 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \log \left (-e^x+x+x^2\right )}{x} \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx-\frac {\left (-1-x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2}\right ) \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx-\frac {\left (-1-x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2}\right ) \, dx \\ & = -\left (2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right )+5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx-e^3 \int \frac {1}{e^x-x-x^2} \, dx+e^3 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\log \left (-e^x+x+x^2\right ) \int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx\right ) \, dx-\int \frac {\left (-1-x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx-\int \frac {\left (-1+e^x-2 x\right ) \int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx}{e^x-x (1+x)} \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right ) \, dx-\int \frac {\left (-1-x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx \\ & = -\left (2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right )+5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx-e^3 \int \frac {1}{e^x-x-x^2} \, dx+e^3 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\log \left (-e^x+x+x^2\right ) \int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx\right ) \, dx-\int \left (\frac {\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{e^x-x-x^2}-\frac {x \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2}+\frac {x^2 \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2}\right ) \, dx-\int \left (\int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx-\frac {\left (-1-x+x^2\right ) \int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx}{-e^x+x+x^2}\right ) \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right ) \, dx-\int \left (\frac {\int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{e^x-x-x^2}-\frac {x \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2}+\frac {x^2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2}\right ) \, dx \\ & = -\left (2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right )+5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx-e^3 \int \frac {1}{e^x-x-x^2} \, dx+e^3 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\log \left (-e^x+x+x^2\right ) \int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx\right ) \, dx-\int \frac {\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{e^x-x-x^2} \, dx+\int \frac {x \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx-\int \frac {x^2 \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx-\int \left (\int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx\right ) \, dx+\int \frac {\left (-1-x+x^2\right ) \int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx}{-e^x+x+x^2} \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right ) \, dx-\int \frac {\int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{e^x-x-x^2} \, dx+\int \frac {x \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx-\int \frac {x^2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx \\ & = -\left (2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right )+5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx-e^3 \int \frac {1}{e^x-x-x^2} \, dx+e^3 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\log \left (-e^x+x+x^2\right ) \int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx\right ) \, dx-\int \frac {\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{e^x-x-x^2} \, dx+\int \frac {x \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx-\int \frac {x^2 \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx-\int \left (\int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx\right ) \, dx+\int \left (\frac {\int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx}{e^x-x-x^2}-\frac {x \int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx}{-e^x+x+x^2}+\frac {x^2 \int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx}{-e^x+x+x^2}\right ) \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right ) \, dx-\int \frac {\int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{e^x-x-x^2} \, dx+\int \frac {x \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx-\int \frac {x^2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx \\ & = -\left (2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right )+5 \int \frac {1}{e^x-x-x^2} \, dx-5 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx-e^3 \int \frac {1}{e^x-x-x^2} \, dx+e^3 \int \frac {1}{x \left (-e^x+x+x^2\right )} \, dx+\left (-5+e^3\right ) \int \frac {e^x}{x^2 \left (e^x-x-x^2\right )} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\log \left (-e^x+x+x^2\right ) \int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx-\log \left (-e^x+x+x^2\right ) \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx-\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{e^x-x-x^2} \, dx+\int \frac {e^{x+\left (-e^x+x+x^2\right )^x}}{x^2 \left (e^x-x-x^2\right )} \, dx+\int \frac {e^{\left (-e^x+x+x^2\right )^x}}{x \left (-e^x+x+x^2\right )} \, dx-\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx\right ) \, dx-\int \frac {\int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{e^x-x-x^2} \, dx+\int \frac {x \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx-\int \frac {x^2 \int e^{\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx-\int \left (\int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx\right ) \, dx+\int \frac {\int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx}{e^x-x-x^2} \, dx-\int \frac {x \int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx}{-e^x+x+x^2} \, dx+\int \frac {x^2 \int \frac {e^{x+\left (-e^x+x+x^2\right )^x} \left (-e^x+x+x^2\right )^{-1+x}}{x} \, dx}{-e^x+x+x^2} \, dx+\int \left (\int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx\right ) \, dx-\int \frac {\int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{e^x-x-x^2} \, dx+\int \frac {x \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx-\int \frac {x^2 \int e^{\left (-e^x+x+x^2\right )^x} x \left (-e^x+x+x^2\right )^{-1+x} \, dx}{-e^x+x+x^2} \, dx \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {e^x \left (-5+e^3\right )+5 x+5 x^2+e^3 \left (-x-x^2\right )+e^{\left (-e^x+x+x^2\right )^x} \left (e^x-x-x^2+\left (-e^x+x+x^2\right )^x \left (x^2-e^x x^2+2 x^3+\left (-e^x x+x^2+x^3\right ) \log \left (-e^x+x+x^2\right )\right )\right )}{e^x x^2-x^3-x^4} \, dx=-\frac {-5+e^3+e^{\left (-e^x+x+x^2\right )^x}}{x} \]
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Time = 138.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
parallelrisch | \(-\frac {-5+{\mathrm e}^{3}+{\mathrm e}^{{\mathrm e}^{x \ln \left (-{\mathrm e}^{x}+x^{2}+x \right )}}}{x}\) | \(24\) |
risch | \(-\frac {{\mathrm e}^{3}}{x}+\frac {5}{x}-\frac {{\mathrm e}^{\left (-{\mathrm e}^{x}+x^{2}+x \right )^{x}}}{x}\) | \(31\) |
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Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {e^x \left (-5+e^3\right )+5 x+5 x^2+e^3 \left (-x-x^2\right )+e^{\left (-e^x+x+x^2\right )^x} \left (e^x-x-x^2+\left (-e^x+x+x^2\right )^x \left (x^2-e^x x^2+2 x^3+\left (-e^x x+x^2+x^3\right ) \log \left (-e^x+x+x^2\right )\right )\right )}{e^x x^2-x^3-x^4} \, dx=-\frac {e^{3} + e^{\left ({\left (x^{2} + x - e^{x}\right )}^{x}\right )} - 5}{x} \]
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Timed out. \[ \int \frac {e^x \left (-5+e^3\right )+5 x+5 x^2+e^3 \left (-x-x^2\right )+e^{\left (-e^x+x+x^2\right )^x} \left (e^x-x-x^2+\left (-e^x+x+x^2\right )^x \left (x^2-e^x x^2+2 x^3+\left (-e^x x+x^2+x^3\right ) \log \left (-e^x+x+x^2\right )\right )\right )}{e^x x^2-x^3-x^4} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {e^x \left (-5+e^3\right )+5 x+5 x^2+e^3 \left (-x-x^2\right )+e^{\left (-e^x+x+x^2\right )^x} \left (e^x-x-x^2+\left (-e^x+x+x^2\right )^x \left (x^2-e^x x^2+2 x^3+\left (-e^x x+x^2+x^3\right ) \log \left (-e^x+x+x^2\right )\right )\right )}{e^x x^2-x^3-x^4} \, dx=-\frac {e^{3} + e^{\left ({\left (x^{2} + x - e^{x}\right )}^{x}\right )} - 5}{x} \]
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\[ \int \frac {e^x \left (-5+e^3\right )+5 x+5 x^2+e^3 \left (-x-x^2\right )+e^{\left (-e^x+x+x^2\right )^x} \left (e^x-x-x^2+\left (-e^x+x+x^2\right )^x \left (x^2-e^x x^2+2 x^3+\left (-e^x x+x^2+x^3\right ) \log \left (-e^x+x+x^2\right )\right )\right )}{e^x x^2-x^3-x^4} \, dx=\int { -\frac {5 \, x^{2} - {\left (x^{2} + x\right )} e^{3} + {\left ({\left (2 \, x^{3} - x^{2} e^{x} + x^{2} + {\left (x^{3} + x^{2} - x e^{x}\right )} \log \left (x^{2} + x - e^{x}\right )\right )} {\left (x^{2} + x - e^{x}\right )}^{x} - x^{2} - x + e^{x}\right )} e^{\left ({\left (x^{2} + x - e^{x}\right )}^{x}\right )} + {\left (e^{3} - 5\right )} e^{x} + 5 \, x}{x^{4} + x^{3} - x^{2} e^{x}} \,d x } \]
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Time = 14.68 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {e^x \left (-5+e^3\right )+5 x+5 x^2+e^3 \left (-x-x^2\right )+e^{\left (-e^x+x+x^2\right )^x} \left (e^x-x-x^2+\left (-e^x+x+x^2\right )^x \left (x^2-e^x x^2+2 x^3+\left (-e^x x+x^2+x^3\right ) \log \left (-e^x+x+x^2\right )\right )\right )}{e^x x^2-x^3-x^4} \, dx=-\frac {{\mathrm {e}}^{{\left (x-{\mathrm {e}}^x+x^2\right )}^x}+{\mathrm {e}}^3-5}{x} \]
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