Another Improper Integral Problem

Evaluate $\displaystyle\int_1^\infty\frac{dx}{e^{x+1}+e^{3-x}}$. (Source: Putnam)

Admittedly, this problem took me an embarrasingly long time to solve (3 hours). Nonetheless, I had fun playing with this particular integral.

For this particular problem, it actually turns out that a symmetric nature is extremely helpful in successfully integrating. To achieve this, we will let $u=x-1$.

We can then factor $e^2$ out of the denominator:

To get rid of the $e^{-u}$, we multiply the integrand by $\dfrac{e^u}{e^u}$:

Now we can make the the substitution $v=e^u$!

We know $\displaystyle\lim_{x\to\infty}\arctan{x}=\frac{\pi}{2}$; therefore, our answer is