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As others have pointed out, the damaged section is pretty much shot, so you could just open it up and take a look inside. That assumes that with the construction there isn't some framing completely blocking the way. Not likely, but possible. If no water is appearing on the floor, I'd say odds are that it's not a leak, but it might be.
The other option is as suggested to dry it out with a fan and hold off using both the shower and the bath tub for a couple days. Or if you're going to be away for a few days, let it dry out then and when you return, use only the bath part for a week, carefully making sure no chance of water getting their past the curtain, running down the top of the tub, etc. If it's not wet, then do the same using the shower.