Firefox works flawlessly in my case. Only when I restart it I have to click on a checkbox once (or in seldom cases twice) that I am human, but that's it. Current firefox on ubuntu, no addons.
Possible explanations for these different experiences:
1) Browser addons
2) Privacy or other settings
3) cloudflares measures are not as simple and may depend on network parameters (like provider, router settings,...), browser ids and many small details.
@meilodaresh: Maybe time to update or reinstall your ubuntu, not only because of this forum (and stay with LTS versions if you haven't done already)