Solution of differential equation $x'=\cos(x),$ with condition $x(0)=0$

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Write out the Picard iteration scheme. If possible, find the solution.

$x'=cos(x),x(0)=0$

I did the picard iterations but I don't know how to get the solution from this

$$u_0=0,...,u_3=\int_0^t\cos(\sin(s))ds$$

I know the solution is this

The solution to the IVP is $x(t)=2\tan^{-1}(\tanh(\frac t2))$