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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 $$ \int_0^\infty \exp(-(x/c)^2)\; dx = \sqrt{\pi} c/2 $$