f[z_]:=Exp[I a][-1/z+1/z^3);
x=Integrate[Simplify[[1-z^2)f[z]],z]
X=Assuming[a∈Reals&&u∈Reals&&v∈Reals,FullSimplify[Re[ReplaceAll[x,z->Exp[u+I*v]]]]]

Re[(E^(I a)) (-2 Log[(E^(u+I v))]+Sinh[2 u+2 I v])]

FullSimplify[X]

Exit

FullSimplify[{1/2*Cos[a]*Exp[2*u]*Cos[2*v]-1/2*Sin[a]*Exp[2*u]*Sin[2*v]-2*Cos[a]*u+2*Sin[a]*v-1/2*Cos[a]*Exp[-2*u]*Cos[2*v]-1/2*Sin[a]*Exp[-2*u]*Sin[2*v],1/2*Cos[a]*Exp[2*u]*Sin[2*v]-1/2*Cos[a]*Exp[-2*u]*Sin[2*v]+1/2*Sin[a]*Exp[2*u]*Cos[2*v]+1/2*Sin[a]*Exp[-2*u]*Cos[2*v],-2*Cos[a]*Exp[u]*Cos[v]-2*Cos[a]*Exp[-u]*Cos[v]+2*Sin[a]*Exp[u]*Sin[v]-2*Sin[a]*Exp[-u]*Sin[v]}]

Exit

FullSimplify[{Cos[a]*Exp[u]*Cos[v]-1/3*Cos[a]*Exp[3*u]*Cos[3*v]+1/3*Cos[a]*Exp[-3*u]*Cos[3*v]-Cos[a]*Exp[-u]*Cos[v]-Sin[a]*Exp[u]*Sin[v]+1/3*Sin[a]*Exp[3*u]*Sin[3*v]+1/3*Sin[a]*Exp[-3*u]*Sin[3*v]-Sin[a]*Exp[-u]*Sin[v],-1/3*Cos[a]*Exp[3*u]*Sin[3*v]-Cos[a]*Exp[u]*Sin[v]+1/3*Cos[a]*Exp[-3*u]*Sin[3*v]+Cos[a]*Exp[-u]*Sin[v]-1/3*Sin[a]*Exp[3*u]*Cos[3*v]-Sin[a]*Exp[u]*Cos[v]-1/3*Sin[a]*Exp[-3*u]*Cos[3*v]-Sin[a]*Exp[-u]*Cos[v],Cos[a]*Exp[2*u]*Cos[2*v]+Cos[a]*Exp[-2*u]*Cos[2*v]-Sin[a]*Exp[2*u]*Sin[2*v]+Sin[a]*Exp[-2*u]*Sin[2*v]}]