How to sync my iPhone library with my new mac?

I just got a new mac and I already have a large library on my phone. I have had two other computers authorized to my phone, my last one I was unable to put my library onto it that i had from my first computer but I could download and sync from that computer, when I click sync this one It doesn't add the songs I've added(I already de-authorized all the other computers and authorized this one).

Get your mac on water an bashing it against your iphone! XD

The music you have on your iMac will be synced to your phone…
The music on your phone will be replace by the music on your iMac's iTunes…

this is what you have to do:
Get a flash drive and put all the music from your old computer onto it
Download all the music from the flash onto your iMac
Open all the music with iTunes then sync…

That is the only way.
You can't transfer music from your phone to your iMac… Only from your iMac to your phone

p.s. You can authorize up to 5 computers…