Definite integral with mutually inverse limits

Question

Compute the integral $$ \int_{1/2}^{2}\left(1+x-\frac{1}{x}\right)e^{x+\frac{1}{x}}dx. $$ The only idea that I have is the substitute $t=1/x$. Then we obtain the following: $$ \int_{1/2}^{2}\left(\frac{t+1-t^{2}}{t^3}\right)e^{t+\frac{1}{t}}dt. $$