The problem not only persists with your iphone 5,but will all apple devices which have updated to ios 6.1.3
Tips for saving battery life
1)minimise brightness and turn on auto brightness
2)turn off location services
3)turn off bluetooth and wifi
4)turn off 3G and try t use 2G or edge
5)close all running apps
6)charge iphone only when the battery reaches 20%,this wil help preserve your battery life
Something may have gone amiss in the upgrade to iOS 6.1.3 on your device.
To correct this follow these steps:
- Backup your device using iTunes or iCloud.
- Use iTunes to completely restore your device as new.
- Setup your device and monitor battery usage to see if it is normal. If not, you may have a hardware problem - if so, contact Apple support.
- Restore your device from your backup above.
- Check battery usage to confirm that it remains normal.
It's hard to grasp because the vast majority of iPhone users are having no battery problem at all with iOS 6.1.3. If you are having problems then there is something special about your iPhone. Take some action to solve this problem or determine if it is a hardware problem. Don't count on any software fix.