I don't think it is about the OS. I used to have this problem with my 4s. I went to Apple store and the man from Genius Bar asked me to restore the system and then restore from my backup, I tried but it doesn't work. Then I restored the system and didn't restore from my backup, problem solved.
The guy from Genius Bar told me it is because some apps have problems and crashes all the time which makes my battery die so fast, I deleted those apps but it didn't work.
I think the only way to solve it is to restore your OS and give up your backup and set up as a new phone.