Let $\omega_i$, $i=1,\dots,n$ be positive real numbers. Consider the following nested integral $$ \int_0^{t} \cos(\omega_1t_1)\left(\int_0^{t_1} \cos(\omega_2t_2)\cdots \left(\int_0^{t_{n-1}} \cos(\omega_nt_n)\, \mathrm{d}t_n\right)\cdots\mathrm{d}t_2\right) \mathrm{d}t_1 $$
Q. Does there exist a closed-form expression for the latter integral? If not, is it possible to say that the latter integral is bounded in $t$?