SymbolicIntegration.jl

SymbolicIntegration.jl solves indefinite integrals using one of the implemented algorithms: Risch method and Rule based method

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.

Usage

julia> using Pkg; Pkg.add("SymbolicIntegration") # installation

julia> using SymbolicIntegration, Symbolics

julia> @variables x
1-element Vector{Num}:
 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 RischMethod will be tried, then RuleBasedMethod:

julia> integrate(sqrt(x))
┌ Warning: NotImplementedError: integrand contains unsupported expression sqrt(x)
└ @ SymbolicIntegration ~/.julia/dev/SymbolicIntegration.jl_official/src/methods/risch/frontend.jl:826

 > RischMethod failed returning ∫(sqrt(x), x) 
 > Trying with RuleBasedMethod...

(2//3)*(x^(3//2))

julia> integrate(abs(x))
┌ Warning: NotImplementedError: integrand contains unsupported expression abs(x)
└ @ SymbolicIntegration ~/.julia/dev/SymbolicIntegration.jl_official/src/methods/risch/frontend.jl:826

 > RischMethod failed returning ∫(abs(x), x) 
 > Trying with RuleBasedMethod...

No rule found for ∫(abs(x), x)

 > RuleBasedMethod failed returning ∫(abs(x), x) 
 > Sorry we cannot integrate this expression :(

Integration Methods

Currently two algorithms 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	❌	❌
more than one symbolic variable in the expression	❌	✅

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.

Documentation

Complete documentation with method selection guidance, algorithm details, and examples is available at: docs.sciml.ai/SymbolicIntegration

Citation

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

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