How to transfer my iphones itunes library to me new PC?

I've tried several online tutorials and they all fail for one reason or another. Many years ago I was really good with computers but I apparently don't know anything anymore. I have an iphone with all my songs that was synced with a mac laptop but the laptop died, and now I've gone back to PC. The only option itunes gives is to wipe the phone and sync with my (new, non-existent) library. How on Earth do I transfer this music from iphone 5 to itunes installed on a PC? The PC is an authorized device in itunes. I tried Sharepod but it quit after 50 songs.

I've had no problems with this new PC until I tried this with Apple software. I swear, they keep things locked up tighter than a gnat's a$$… This shouldn't be this hard.

Transferring purchases method help to this.

If your iPhone, iPad, or iPod has your purchased content from the iTunes Store, you can transfer the content to a computer that is authorized for your Apple ID.

This option only works for content you purchased from the iTunes Store (as opposed to content from other sources, such as CDs you may have imported). For content you did not purchase from the iTunes Store, synchronization occurs in one direction only. This means your content can be copied from a computer to your iOS device, but can't be copied from your device to a computer.

Open iTunes.
From the Store menu, choose Authorize This Computer.
Enter your Apple ID and password. Then, click the Authorize button.
Connect your device your computer. If it is synced with another iTunes library, you may be asked if you want to "Erase and Sync" or "Transfer Purchases."
If that alert appears, click Transfer Purchases.
Important: Clicking Erase and Sync will delete any existing content currently on your device. The content would then be replaced with the content that is on the computer.
If you were not offered the option to transfer purchases:
After connecting your device to the computer, its icon appears in iTunes. Click the icon.
From the File > Devices menu, choose "Transfer Purchases from 'Your device name.'"

If the songs are bought through iTunes (You payed them through apple) then just sign in to the iTunes on the PC, everything should be able to sync.
If else, Get the demo version of this, plug in your phone and your done http://www.wideanglesoftware.com

Paid content can be shared on another computer. Just sign in to the iTunes on the PC, everything should be able to sync. For content you did not purchase from the iTunes Store, third-party program like iPhone Transfer is your best solution.