How do I repair drywall around a bathtub?
I don't have experience with it, but there are fiberglass-covered drywall products available. If that is not available, I would recommend using 1/2" James Hardie Tile Backer Board. The smooth side is smooth enough to be painted. As far as taping, I would use a fiber glass mesh tape and a joint compound mix like Sheetrock 90, which hardens better than pre-mix joint compound.

But to really fix the problem permanently, I would pull cut away the bottom 18" of drywall all the way around the tub, replace it with Hardie Tile Backer Board, cover it with Schuter KERDI and finish with tile. Google or Bing to get more information about those products and how to use them.

Tip I picked up from Mike Holmes: don't run the tile backer board over the tub flange that goes up the studs. Stop the tile backer board above the flange, fill with thinset and cover with KERDI. Use KERDI-FIX to adhere an 1/8" fold of the KERDI to the tub deck.

This all assumes this is a tub only with no shower.