SymbolicIntegration.jl

SymbolicIntegration.jl lets you solve indefinite integrals (finds primitives) in Julia Symbolics.jl. It does so using two symbolic integration algorithms: Risch algorithm and Rule based algorithm.

Installation

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

Quick Start

using SymbolicIntegration, Symbolics

@variables x a

# Basic polynomial integration
integrate(x^2, x)  # Returns (1//3)*(x^3)

# Rational function integration
integrate(1/(x^2 + 1), x)  # Returns atan(x)
f = (x^3 + x^2 + x + 2)/(x^4 + 3*x^2 + 2)
integrate(f, x)  # Returns (1//2)*log(2 + x^2) + atan(x)

# Transcendental functions
integrate(exp(x), x)    # Returns exp(x)
integrate(log(x), x)    # Returns -x + x*log(x)

# Method selection and configuration
integrate(f, x, RischMethod())  # Explicit method choice
integrate(f, x, RischMethod(use_algebraic_closure=true))  # With options

Integration Methods

SymbolicIntegration.jl provides two integration algorithms: Risch method and Rule based method.

Default Behavior: When no method is explicitly specified, integrate() will first attempt the Risch method. If the Risch method fails (e.g., due to unsupported expressions like sqrt(x) or abs(x)), it will automatically fall back to the Rule based method. This ensures maximum coverage while prioritizing the theoretically complete Risch algorithm when applicable.

Here is a quick table to see what each method can integrate:

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	❌	✅

→ See complete methods documentation

Example: Automatic Fallback Behavior

When no method is specified, the integration will try Risch first, then fall back to Rule based if needed:

# This uses sqrt which is not supported by Risch, so it falls back to RuleBasedMethod
integrate(sqrt(x))
# ┌ Warning: NotImplementedError: integrand contains unsupported expression sqrt(x)
# └ @ SymbolicIntegration
#
#  > RischMethod failed returning ∫(sqrt(x), x)
#  > Trying with RuleBasedMethod...
#
# (2//3)*(x^(3//2))

You can also explicitly specify which method to use:

integrate(sqrt(x), x, RuleBasedMethod())  # Skip Risch, use Rule based directly
integrate(x^2 + 1, x, RischMethod())      # Use only Risch

RischMethod

This method is based on the algorithms from the book:

Manuel Bronstein, Symbolic Integration I: Transcentental Functions, 2nd ed, Springer 2005,

for which a pretty complete set of reference implementations is provided. As in the book, integrands involving algebraic functions like sqrt and non-integer powers are not treated.

integrate(x^2 + 1, x, RischMethod(use_algebraic_closure=false, catch_errors=true))

use_algebraic_closure does what?
catch_errors does what?

→ See detailed Risch documentation

RuleBased

This method uses a large number of integration rules that specify how to integrate various mathematical expressions.

integrate(x^2 + 1, x, RuleBasedMethod(verbose=true, use_gamma=false))

verbose specifies whether to print or not the integration rules applied (default true)
use_gamma specifies whether to use rules with the gamma function in the result, or not (default false)

→ See detailed Rule based documentation

Contributing

We welcome contributions! Please see the contributing page and the Symbolics.jl contributing guidelines.

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}
}