This sort of thing is called a Computer Algebra System (CAS) and there are good free ones.

https://www.sagemath.org/

http://maxima.sourceforge.net/

https://www.sympy.org/en/index.html

It's probably not a good solution for most people, but here's what I do: I have a TI-89 emulator on my android phone. I used one of those calculators for about six years in high school and college, and I'll never lose the muscle memory of the interface. (I also still own the device, which justifies using the emulator ethically, if not strictly legally.) I believe it's actually faster for me than, say, sympy, which I also use in other contexts. Also faster than using pencil and paper, for sufficiently large problems.

If you're not familiar, it uses a CAS called derive which handles algebraic simplification (including trigonometry), differentiation and integration. Oh, and a function called "rref" (reduced row echelon form; familiar to matlab users), which solves a linear system (including symbolically), with absolutely minimal overhead in the "CLI".

Now that I say that, I want an actual CLI tool called rref that works in the same way.

How hard have you looked? You'd be hard pressed to find a computer alebgra system (CAS) that doesn't support algebraic reduction. It's been available for decades at this point. Popular systems offering this can do this include Mathematica, Maple and Sage. Even MATLAB had extensions for symbolic manipulation now.

Here's something worth investigating, IMO:

> PRESS (PRolog Equation Solving System) is a system for solving symbolic, transcendental, non-differential equations. The methods used for solving equations are described, together with the service facilities. The principal technique, meta-level inference, appears to have applications in the broader field of symbolic and algebraic manipulation.

https://dream.inf.ed.ac.uk/software/press/

And: https://github.com/maths/PRESS

> This directory contains a copy of the PRESS ... modified to run on SWI Prolog

Such things exists under the name "Computer Algebra Systems". Notable commercial offerings include Mathematica and Maple, but there is also free software such as Sage and Sympy.