You can give relevant assumptions to a symbolic integration. However, your region is very simple to describe: it is the whole plane except for a segment:
myRegion =
Simplify[(Im[R Cos[x/2]^2] + Re[R Sin[x]] != Im[R Sin[x/2]^2] ||
Re[R Cos[x/2]^2] < Im[R Sin[x]] + Re[R Sin[x/2]^2] ||
1 + Im[R Sin[x]] + Re[R Sin[x/2]^2] < Re[R Cos[x/2]^2]),
Element[x | R, Reals]];
Reduce[myRegion && 0 <= x < 2 Pi && R >= 0, {x, R}, Reals]