How to play music from my iPhone 4S through my computer?

I have an iPhone 4S, all up to date with software, and would like to have the option to play music directly from my iPhone through the iTunes (Newest Version) on my computer. This issue is that my iTunes account is on a different computer from the one I would be using, so none of the songs I have from my iPhone are on my computer. Is there a way to play this music, controlled from my iPhone, through the iTunes on my laptop and through my speakers? I have been trying to find a way to do this but no one seems to have the same question I have been trying to find an answer for.

I usually just hook my phone to the computer as if I'm going to be charging it and open up the iTunes app double click on (Whatever your phones name is) and you should see more options show under the phone tab. Click music under the phone Tab and it should show you all the music on that phone.

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