# Closed-form expression for a nested integral

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 $t$?

• For $n=2$, it is quite simple. For $n=3$, it is doable but quite nasty. – Claude Leibovici May 19 '18 at 5:44
• @ClaudeLeibovici: So, do you think there isn't any chance to find a closed-form expression (or, at least, a tractable bound)? – Ludwig May 19 '18 at 19:03