in particular I'm curious if sympy is really as good as Mathematica. I haven't used Mathematica since doing physics hw as an undergrad but it's symbolic manipulation was amazing most of the time
On the other hand, open source tools are catching up. Part of the SymPy project is SymEngine which is a reconstruction in C++. SymEngine is not as feature-filled as SymPy yet, but it flies. It's much faster than SymPy, and also Mathematica. SymEngine.jl works very well since by using the symbolic types in side of Julia you get both the speed (via function specialization) while getting a lot of free features through generic code. For example, Julia code is generic over number types, so if you call inv on a matrix of SymEngine symbols the built-in Base version will compile a fast version for SymEngine symbols and use it. So you have all of Julia's Base available along with Julia packages (yeah, you can put them in a neural net if you wanted to). So SymEngine.jl is the only thing that works when I need that speed, but I do have to keep Mathematica installed for its special handling of specific equations.
And I hate to say it but... the Mathematica notebook is really pretty. You put the exponents up, the fractions on top of each other...
Mathematica is undergoing a compiler re-vamp, which may change that.
That's a video where they lay out how they want it to work, if you're interested and haven't seen it.
yup. for quick and dirty type setting i preferred it to latex
SymPy is doing nicely, but it's decades behind Mathematica when it comes to symbolic computation.
By the way, if you want to invert a symbolic matrix, chances are that generic Gaussian elimination working with symbolic expressions isn't the best algorithm (it might even give wrong answers if zero testing is done incorrectly), and compiled code isn't going to change that. The state of the art in symbolic linear algebra uses specialized algorithms like evaluation-interpolation, modular computation, and all kinds of low level optimizations.