Stochastic Processes – Inequality of Doobs Martingale

I am confused on how to calculate the expectation of an integral on this issue.

Use the inequality of Doob martingale to estimate

$ mathbb {E} sup {0 leq s leq t} mid int_ {0} ^ {s} cos (u) dB (u) mid ^ 2 quad $ (1)

or $ B (t) $ is a one-dimensional Brownian motion.

I understand that (1) is $ leq 4 mathbb {E} mid int_ {0} ^ {s} cos (u) dB (u) mid ^ 2 $ but I do not know where to go from here. I would like any help!