Ok I believe that your function is in fact BIBO stable. I have to study for history so I cant write a real proof or anything but here is my basic idea:
Given any bounded function
f(x) there must then be a maximal derivative
f'(tmax) != infinity
Then for all t on f'(t) we have f'(t) <= f'(tmax)
and so f'(t) is then bounded
and y = f'(t) + f(t) is then the sum of two bounded functions
and therfore bounded.
Note that proofs are my worst area in math.
I'm more of a linear algebra guy.
Predicatably enough taking the integral from -infinity to infinity wqould basically involve the exact same logic, this just came to me first.