Why won't iTunes download songs to my computer?

I bought a few songs on my iphone and when I tried to download the songs to my computer it wouldn't work.

At first it showed the songs in my library but when I tried to download them it said my computer wasn't authorized. I authorized the computer and the songs disappeared.

In the itunes store it shows the songs as purchased but they're not there. I've tried a few methods that I've read about to fix the problem but none of them have worked.

Added (1). I believe on of the methods had something to do with the songs being 'hidden' and I had to look at my apple I.D. Info or something but all I know is it didn't work.

1. Open up Itunes and Click on Itunes Store
2. Once you're in the Itunes Store look to right side of the screen and you should see the word "purchased". Click purchased and you'll be sent to a purchase screen.
3. Click on "All Songs" and then it will show all the songs you've downloaded. From there you just hit the button that looks like a cloud with an arrow on it, and it will download all the songs all over again.
Or you can use a 3rd party software like this iPhone transfer, it can help you copy songs, videos and photos from iPhone back to computer.

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