I'd never bother to play a game that demands captchas: they are really a terrible practice that should be avoided.
Your game seems to have deeper problems than that anyway: ideally a game should give no advantage to a stupid bot, so that it would be pointless to use a stupid bot in the first place. If you cannot achieve that, that's a design problem that will have no real solution, only more-or-less valid workarounds.
With "stupid bot" I mean bots that do not take any meaningful decision and instead just "farm" (that's what happens here). Smart bots (such as aim-bots or chess playing bots) are a completely different matter.
Still, assuming you are comfortable with the idea of making a design-flawed game, there is still room for improvements.
Accept the fact that you can't stop a really determined bot and instead focus on the only thing you can do: make it useless to use a bot. If people have no reason to use a bot... they won't use them (and if they do anyway it won't matter).
A possible solution is to allow one login per week instead of one per day. If people will forget to login for more then a week, they are unlikely to be really interested in playing anyway, so they won't be using bots to keep gaining credits. If, on the other hand, there is someone so screwed that makes a bot that logs once a week and then comes back three months later to attack random people, well, you have just found someone so determined that would have cracked you anyway, whatever system you chose (unless you chose a non-flawed system, of course).
ps: don't make the mistake of putting more effort into working-around design flaws then actually solving them in the first place!