The expression $\cos x + \cos 3x + \cos 7x + \cos 9x$ can be written in the equivalent form \[a \cos bx \cos cx \cos dx\]for some positive integers $a,$ $b,$ $c,$ and $d.$ Find $a + b + c + d.$
a + b + c + d = 2 + 5 + 8 + 9 = 24.
