Music transfer from iPhone to new computer?

I recently got a new PC because my last one is broken, but when I sync my iPhone 4 only the few amount of songs I've purchased from iTunes are displayed. How can I get the songs that I've downloaded elsewhere, which are present on my phone, onto iTunes on my computer? I tried one transfer software but the free trial only allowed me to transfer 3 songs, so if you know of any efficient ways to transfer that are harmless and free that would be of much help.

I know a nice app called Wireless Transfer App on the App Store. Maybe it can give you some help.

I would like to share one reliable software program with you. It is named iStonsoft iPhone to computer transfer.

Here is a detailed description of how to transfer:

You can refer to iTransfer, a powerful tool to help you transfer music from iPhone to computer freely and easily. Of course, there are no limitation of numbers in the trial version. Therefore, you can get all your songs back again.

This iPhone to PC Transfer just cost $19.00, buy one and use forever. It can help transfer muisc, movies, photos, sms, contacts, ibooks, whether free or purchased to computer or iTunes, i think it's worthy to invest.

http://www.transfer-iphone-sms.com/transfer-iphone-to-pc.html

There's a really good article on Apple discussions here providing instructions on how to sync your iPhone to a new computer…
https://discussions.apple.com/...s/DOC-3141