> zt:=exp(I*t);
int(subs(z=zt,1/z)*diff(zt,t),t=0..2*Pi);

> zt:=2*exp(I*t);
int(subs(z=zt,1/z)*diff(zt,t),t=0..2*Pi);

> zt:=1+2*exp(I*t);
int(subs(z=zt,1/z)*diff(zt,t),t=0..2*Pi);

> zt:=1+2*exp(I*t);
int(evalc(subs(z=zt,1/z)*diff(zt,t)),t=0..2*Pi);

> zt:=3+2*exp(I*t);
int(evalc(subs(z=zt,1/z)*diff(zt,t)),t=0..2*Pi);

>