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I don't want to re-invent the wheel here, and I know that there are a lot of good math libraries out there for all sorts of things; what I'm wondering is if there's one that generates its answers in LaTex? (Could be any sort of TeX; I'm not really familiar with which ones are used for what)

Well, it could look like anything (just linear algebra, just calculus, just physics or chemistry, etc.) but it would be nice if it were an application with its own GUI that generates copy-and-paste LaTex code from different inputs and a whole set of operations at the user's disposal.

It doesn't have to be that great, though. I wouldn't mind coding this myself, but before I embark, I'd like to see what my options are so that I know where to start.

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Sage, Maxima, Mathematica... – mbork Jan 23 '14 at 7:24
Welcome to TeX.SE. I think the answer depend on the specific type of computation you are thinking of. One example, albeit fairly simple computation, that I know of is Automatically add fractions and reduce the result (if neccessary). There is also the polynom package as illustrated in Polynomial Long Division Using Polynom. – Peter Grill Jan 23 '14 at 7:54
On OS X you get the (highly underrated) Grapher which can produce LaTeX formulas from your input. – Habi Jan 23 '14 at 9:20

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