Having lived entirely in math land for several years now, I've found Mathematica to be an indispensable tool for performing those mathemagics. But some people (me) cannot afford a Mathematica license so I went looking for an open source alternative. The software I found is called Sage and is an adequate python based symbolic manipulation, calculus, etc. total package; however, it does require a rather long build process...
Sage: Open Source Mathematics Software
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