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?

Simplifying complex expresion (without ComplexExpand or User defined rules)

Many times, I’ve struggled with simplifying complex expressions, such as this extraordinarily simple expression, and Mathematica wont do it:

$  Assumptions = {l>0,a>0,a \[Element]Reals, l \[Element]Reals} Conjugate[(Exp[I a])^(2 l)] //FullSimplify 

Where Mathematica returns the same thing with no simplification. I told it the assumptions which make it very obvious to just change the sign of the exponent, and I don’t understand why it doesn’t work.

I looked here, but it seems like they have to define their own rules to do this kind of thing. I’m also saw here that you can do //ComplexExpand //FullSimplify for this expression, but it doesn’t seem to work in all cases. Why should it be nessecary and why doesn’t FullSimplify do it already? Also, I think ComplexExpand assumes that all variables are real, which was the case in this expression, but isn’t always the case. So how would I do it then?