  We analyze the vibrational resonance in the Duffing oscillator system in the
presence of (i) a gamma distributed time-delayed feedback and (ii) integrative
time-delayed (uniformly distributed time delays over a finite interval)
feedback. Particularly, applying a theoretical procedure we obtain an
expression for the response amplitude $Q$ at the low-frequency of the driving
biharmonic force. For both double-well potential and single-well potential
cases we are able to identify the regions in parameter space where either (i)
two resonances, (ii) a single resonance or (iii) no resonance occur.
Theoretically predicted values of $Q$ and the values of a control parameter at
which resonance occurs are in good agreement with our numerical simulation. The
analysis shows a strong influence of both types of time-delayed feedback on
vibrational resonance.
