I want to do some functionality on log-out, if the user directly closed his browser then same functionality want to do, we can not do on page unload because there are more
You can have a button for the "logout" case.
Unfortunately, there is no reliable way to be notified if the user closes their browser. Other than client-side page unload, which you've said you don't want, about the only other option is periodic Ajax-based polling; both are ugly and notoriously unreliable.
In general, server-side timeouts are a better approach.