I presume you're interested in $c \to \infty$.  Without absolute value you get
$\dfrac{\sqrt{\pi}}{2} c \exp(-c^2/4)$, which is not **quite** of order $\exp(-c^2/4)$.  With absolute value, a lower bound is 
$$ \int_0^\infty \exp(-(x/c)^2) \cos^2 x\; dx = \sqrt{\pi} c (1 + \exp(-c^2))/4$$
so it is not even $O(1)$.  An upper bound is
$$ \int_0^\infty \exp(-(x/c)^2)\; dx = \sqrt{\pi} c/2 $$