How to backup music from my iPhone to my iTunes?

Basically my computer broke and now I lost all my data, I had the latest version of iOS 7 and the latest iTunes update installed, but not all my music is from iTunes, and I asked the Genius Bar and didn't get a straight response. So is there any programs or anything to transfer my music?

Plug in your iPhone into your computer. Open iTunes. Click iPhone. Go onto summary. Click back up now

Can you hook up the drive to another computer and mount it using a live CD Linux? AFAIK, Debian and Ubuntu both support read/write on HFS+.

Just plug inyour iPhone to the computer, then open iTunes, click backup to copy them. However, if this is not aviable, you can use AnyMP4 iPhone Transfer. It's a transfer tool I have used, i can transfer my files from iPhone to pc and vice verse. Also, just like you said, it can back up iPhone music to iTunes for later use.
You can downliad the trail version to have a try.

Follow the steps:

1. Find and install an iPhone Manage, connect your iPhone with your pc.
2. With the iPhone Manager, Backup all your music from your iPhone to your pc.
3. Finally, transfer your music from pc to your iTunes.

You can try this free iPhone Manager: Syncios

Refer to this tutorial:
http://www.syncios.com/how-to-transfer-music-from-iphone-4s-to-computer.html