I was initially getting four hours screen on time like you so I contacted Samsung on twitter. They told me to put my phone into safe mode and see what apps were missing. Whichever apps were missing were the ones causing the battery drain. They recommended I uninstall any apps I didn't want or need out of those apps. I uninstalled and reinstalled the apps and discovered the source of the drain. In my case it was avg and the weather channel.
I also installed Weyo which is a dark theme onto my phone. I replaced carbon with Twitter which runs rings around it and is able to be stopped in the background unlike the official app. I also gave my phone a black wallpaper as the phone lights up each individual pixel.
I recently installed greenify and Doze and I could not be happier. It has increased my standby time exponentially. I was asleep and just woke up to this.
Try any all or none of these tips and let me know how you get on
Ive actually recently updated to official 5.1.1 and factory reset the phone and have been running it clean (without the apps i usually use like Nova Launcher, Google Keyboard, Lightflow, and leaving the capacitive lights OFF) and i havent seen any improvement. im running the phone stock since 5 days and it hasnt done anything.
The fact that i have to install "Apps" to help battery life shouldn't be here in 2015. My old kitkat phones (Z1c Moto G) have me. 5 hours+ SoT running everything i wanted with 100's of apps.
Todays is a perfect example
left phone with 100% battery before sleeping, wake up at 90% been using the phone for about 10 minutes and at 81% that is ridiculous. ive been looking thru forums and checked instagram, no videos no streaming no camera use nothing "draining" was done since last night