iPhone 4 battery problem

I just got a refurbished iPhone 4 from AT&T, and right off the bat, when I plugged it in and tried to charge it, and it took abnormally long to charge, but I didn't think anything of it because it is a new, out of the box, phone and thought maybe it needed a good charge. As soon as it had enough battery to hold a charge I unplugged it and used it, and didn't let it fully charge. Every so often it would just turn off and then I would try to turn it back on and it would show the completely dead battery, with a little charging symbol below it, so naturally I plugged it in and then it would reboot and show that it had above half battery. This happens every once in awhile when I am taking calls and playing Templerun when my phone is at about 64% battery. Should I just ask Apple for another phone or just fix it myself? I fix iPhones, but just never came across this problem. Anything helps, thanks!