Should I get another credit card?

I know this isn't exactly frugal-living related, but this seemed like the best place to post. I'll try to keep this as short as possible.

This is my tentative plan: Visa offers an American Airlines card that will give me 75,000 frequent flier miles if I spend $4,000 in the first six months (this is totally doable if I use the card for all my normal purchases, especially considering that Christmas is coming up—I wouldn't need to spend extra to meet this requirement). I could definitely use these points in the near future for (nearly) free flights I know I'll need to take next year. I would then either a) cancel the card when they try to charge me the regular annual fee or b) keep the card and use it occasionally if I get them to waive the annual fee.

My question is, how would opening (and potentially closing in the near future) this card impact my credit score? I know that applying will temporarily lower your score, but I've never been able to get a clear answer about how adding and deleting credit sources impacts the score.

Other info that might be useful: I currently have three credit cards—a Visa I've had for about 10 years that I rarely use and has no benefits but that I'm afraid to close because of how it might affect my credit, a Chase Visa I've had for about 2 years that has good rewards and that I use for most everything, and a Delta-affiliated AmEx that I opened about 6 months ago in order to get the big signup mileage bonus and now only use once or twice a month. I am responsible with these credit cards—I ALWAYS pay in full each month and have never made a late payment. The last time I checked my credit (about 2.5 years ago, back when I only had one credit card), my score was something like 735 or 740 (which, at least in the old system, was considered "excellent"). I don't anticipate anyone needing to check my credit score in the next few months.

Can someone please set me straight on how getting another credit card would impact my credit, positively or negatively? I'd like to reap the benefits of this American Airlines card, but not if it will be detrimental to my credit in the long run.