There's always Mathics, which is an open source implementation of (some) of Mathematica, using SymPy underneath: https://mathics.github.io/
Any thoughts on Mathics? Have heard of it but haven't tried working with it in depth.
1) If you use it, I'd recommend installing it locally (pretty easy with pip install mathics)
2) When doing something complex, it can be a bit slow (it relies on SymPy, which isn't a speed demon itself, plus it has to parse and interpret the Mathematica syntax, which adds overhead)
3) The subset of Mathematica that it supports is pretty much what SymPy does -- so pretty much classic mathematics and graphing rather than the knowledge based stuff like populations of countries and their GDPs or whatever that Mathematica has been adding over the last few versions.
How are good like things like its simplifying algorithm? I've used Sage which uses sympy from what I understand and it didn't quite cut it. Is Mathics just an interface or does it tweak the algorithms?
The Mathematica/Wolfram language support of Mathics is good enough that some of the functionality is implemented in itself rather than a straight SymPy wrapper, but if Sage (a much bigger system than Mathics) can't simplify the expressions you want, probably Mathics can't either.
For regular mortals the cheapest way to get Mathematica is to get Raspberry Pi. Raspbian has Mathematica in its repo (sudo apt install wolfram-engine). The performance is not great but maybe the latest RPi 4 will help.
Wow! Now that's a life hack! :-)
And you can get a cluster of these for a price of one PC. Awesome!
It will only have the performance of one PC though. ;)
If you don't care about the UI and enjoy a command line version (or scripting with your favorite editor), you can know install the Wolfram engine for free: https://www.wolfram.com/engine/
WolframAlpha + student licence is comparable for some tasks, I've found (hard integrals from applied probability theory).