Symbolics.jl is a fast and modern Computer Algebra System (CAS) for a fast and modern programming language (Julia). The goal is to have a high-performance and parallelized symbolic algebra system that is directly extendable in the same language as the users.

To install Symbolics.jl, use the Julia package manager:

julia> using Pkg
julia> Pkg.add("Symbolics")

For information on using the package, see the stable documentation. Use the in-development documentation for the version of the documentation which contains the unreleased features.

• SymbolicUtils.jl: This is a rule-rewriting system that is the core of Symbolics.jl. Symbolics.jl builds off of SymbolicUtils.jl to extend it to a whole symbolic algebra system, complete with support for differentation, solving symbolic systems of equations, etc. If you're looking for the barebones to build a new CAS for specific algebras, SymbolicUtils.jl is that foundation. Otherwise, Symbolics.jl is for you.
• ModelingToolkit.jl: This is a symbolic-numeric modeling system for the SciML ecosystem. It heavily uses Symbolics.jl for its representation of symbolic equations along with tools like differentiation, and adds the representation of common modeling systems like ODEs, SDEs, and more.

julia> using Symbolics

julia> @variables t x y
julia> D = Differential(t)

julia> z = t + t^2
julia> D(z) # symbolic representation of derivative(t + t^2, t)
Differential(t)(t + t^2)

julia> expand_derivatives(D(z))
1 + 2t

julia> Symbolics.jacobian([x + x*y, x^2 + y],[x, y])
2×2 Matrix{Num}:
 1 + y  x
    2x  1

julia> B = simplify.([t^2 + t + t^2  2t + 4t
                  x + y + y + 2t  x^2 - x^2 + y^2])
2×2 Matrix{Num}:
  t + 2(t^2)   6t
 x + 2t + 2y  y^2

julia> simplify.(substitute.(B, (Dict(x => y^2),)))
2×2 Matrix{Num}:
    t + 2(t^2)   6t
 2t + y^2 + 2y  y^2

julia> substitute.(B, (Dict(x => 2.0, y => 3.0, t => 4.0),))
2×2 Matrix{Num}:
 36.0  24.0
 16.0   9.0

If you use Symbolics.jl, please cite this paper

@article{gowda2021high,
  title={High-performance symbolic-numerics via multiple dispatch},
  author={Gowda, Shashi and Ma, Yingbo and Cheli, Alessandro and Gwozdz, Maja and Shah, Viral B and Edelman, Alan and Rackauckas, Christopher},
  journal={arXiv preprint arXiv:2105.03949},
  year={2021}
}