Suggest iRule to redirect website but not change the URL in browser

From another devcentral post:

The ProxyPass iRule is a good option if you're running v10 through v11.3. It will seamlessly replace all internal and external references based on pattern matching.

The new rewrite profile in v11.4 replaces the above ProxyPass functionality.

You're looking to rewrite the host and path but not necessarily redirect. If you redirect, it will change the URL in the browser, so we call what you are looking for as a reverse proxy rewrite functionality.