SymbolicIntegration.jl

SymbolicIntegration.jl solves indefinite integrals (find primitives of functions) using one of the implemented algorithms: Risch method and Rule based method

Usage

julia> using SymbolicIntegration, Symbolics

julia> @variables x

julia> integrate(exp(2x) + 2x^2 + sin(x))
(1//2)*exp(2x) + (2//3)*(x^3) - cos(x)

The first argument is the expression to integrate, second argument is the variable of integration. If the variable is not specified, it will be guessed from the expression. The +c is omitted :)

Method Selection

You can explicitly choose a integration method like this:

risch = RischMethod(use_algebraic_closure=true, catch_errors=false)
integrate(f, x, risch)

or like this:

rbm = RuleBasedMethod(verbose=true, use_gamma=false)
integrate(f, x, rbm)

If no method is specified, first RuleBasedMethod will be tried, then RischMethod:

julia> integrate(abs(x);verbose=true)
No rule found for ∫(abs(x), x)
 > RuleBasedMethod(use_gamma=false, verbose=false) failed, returning ∫(abs(x), x) 
 > Trying with RischMethod(use_algebraic_closure=false, catch_errors=true)...

┌ Warning: NotImplementedError: integrand contains unsupported expression abs(x)
└ @ SymbolicIntegration ~/.julia/dev/SymbolicIntegration.jl_official/src/methods/risch/frontend.jl:821

 > RischMethod(use_algebraic_closure=false, catch_errors=true) failed, returning ∫(abs(x), x) 
 > Sorry we cannot integrate this expression :(

Documentation

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.

Integration Methods

Currently two methods are implemented: Risch algorithm and Rule based integration.

feature	Risch	Rule based
rational functions	✅	✅
non integers powers	❌	✅
exponential functions	✅	✅
logarithms	✅	✅
trigonometric functions	?	sometimes
hyperbolic functions	✅	sometimes
Nonelementary integrals	❌	most of them
Special functions	❌	❌
multiple symbols	❌	✅

More info about them in the methods documentation

Risch Method

Complete symbolic integration using the Risch algorithm from Manuel Bronstein's "Symbolic Integration I: Transcendental Functions".

RuleBasedMethod

This method uses a large number of integration rules that specify how to integrate various mathematical expressions. There are now more than 3400 rules impelmented.

Citation

If you use SymbolicIntegration.jl in your research, please cite:

@software{SymbolicIntegration.jl,
  author = {Mattia Micheletta Merlin, Harald Hofstätter and Chris Rackauckas},
  title = {SymbolicIntegration.jl: Symbolic Integration for Julia},
  url = {https://github.com/JuliaSymbolics/SymbolicIntegration.jl},
  year = {2023-2025}
}