Why doesn't Mathematica ComplexExpand integrals?

Consider this code example:

ComplexExpand[Re[Integrate[f[t], {t, a, b}]]]

Mathematica gives me the result as

Re[Integrate[f[t], {t, a, b}]]

which is obviously not helpful and not what should happen in my understanding. If all variables are real – and that’s what ComplexExpand assumes according to the documentation – then Re can be dropped from the expression. The same happens for Im. This seems to confuse FullSimplify which leaves me with a long expression that could be shorted. Why is this the case? How can I get Integrate (and Re) to evaluate properly to simply

Integrate[f[t], {t, a, b}]