Alternative form of ArcSin[Sin[x]]

We can take the Floor and Ceiling from Carl's answer and expand them out:

PiecewiseExpand[
  PowerExpand[ArcSin[Sin[x]], Assumptions -> x ∈ Reals], 
  -π/2 < x < 3π/2
]

Edit

As it turns out, we can just pass the interval into PowerExpand:

PowerExpand[ArcSin[Sin[x]], Assumptions -> -π/2 < x < 3π/2]

@kglr was on the right track with PowerExpand. With the default option Assumptions->Automatic, Mathematica may return a result that is not valid. On the other hand, if you give PowerExpand a non-default assumption, then it will return a result valid given those assumptions. So, for your example:

Assuming[
    x ∈ Reals,
    Simplify @ PowerExpand[ArcSin[Sin[x]], Assumptions -> x ∈ Reals]
]

1/2 (-1)^(Ceiling[1/2 + x/π] + Floor[-(1/2) + x/π] + Floor[1/2 + x/π]) (π + (-1)^( Ceiling[1/2 + x/π] + Floor[1/2 + x/π]) π + 2 x - 2 π Floor[1/2 + x/π])