Double substitution when integrating. I need to integrate
$$f(x) = \cos(\sin x)$$
my first thought was substituting so that $u = \cos(x)$, but that doesn't seem to do the trick. Is there any way to do a double substitution on this?
Any ways on how to proceed would be appreciated.

Although the indefinite integral $($or anti-derivative$)$ cannot be expressed in terms of elementary functions, its definite counterpart does have a closed form in terms of the special Bessel function:
$$\int_0^\tfrac\pi2\cos(\sin x)~dx~=~\int_0^\tfrac\pi2\cos(\cos x)~dx~=~\frac\pi2J_0(1).$$
Also, $$\int_0^\tfrac\pi2\sin(\sin x)~dx~=~\int_0^\tfrac\pi2\sin(\cos x)~dx~=~\frac\pi2H_0(1),$$
where H represents the special Struve function.
