SymbolicIntegration.jl

A unified interface for symbolic integration methods in Julia

SymbolicIntegration.jl provides a flexible, extensible framework for symbolic integration with multiple algorithm choices. The package uses method dispatch to allow users to select the most appropriate integration algorithm for their specific needs.

Key Features

• 🎯 Multiple Integration Methods: Extensible method dispatch system
• ⚡ Exact Symbolic Results: Guaranteed correct symbolic integration
• 🔢 Complex Root Handling: Produces exact arctangent terms
• ⚙️ Configurable Algorithms: Method-specific options and behavior
• 🏗️ Professional Interface: SciML-style method selection

Integration Methods

RischMethod (Default)

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

Capabilities:

• ✅ Rational functions: Complete integration with Rothstein-Trager method
• ✅ Transcendental functions: Exponential, logarithmic using differential field towers
• ✅ Complex roots: Exact arctangent terms for complex polynomial roots
• ✅ Integration by parts: Logarithmic function integration
• ✅ Trigonometric functions: Via transformation to exponential form

Function Classes:

• Polynomial functions: ∫x^n dx, ∫(ax^2 + bx + c) dx
• Rational functions: ∫P(x)/Q(x) dx → logarithmic and arctangent terms
• Exponential functions: ∫exp(f(x)) dx, ∫x*exp(x) dx
• Logarithmic functions: ∫log(x) dx, ∫1/(x*log(x)) dx
• Trigonometric functions: ∫sin(x) dx, ∫cos(x) dx, ∫tan(x) dx

The framework is designed to support additional integration methods as they are developed.

Installation

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

Usage

Basic Integration

using SymbolicIntegration, Symbolics
@variables x

# Default method (RischMethod) - most cases
integrate(x^2, x)                    # (1//3)*(x^3)
integrate(1/x, x)                    # log(x)
integrate(exp(x), x)                 # exp(x)
integrate(1/(x^2 + 1), x)           # atan(x)

Method Selection

# Explicit method choice
integrate(f, x, RischMethod())

# Method with configuration
risch = RischMethod(use_algebraic_closure=true, catch_errors=false)
integrate(f, x, risch)

Complex Examples

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

# Integration by parts
integrate(log(x), x)  # -x + x*log(x)

# Nested transcendental functions
integrate(1/(x*log(x)), x)  # log(log(x))

Method Framework

SymbolicIntegration.jl uses a modern method dispatch system similar to other SciML packages:

Current Methods

• RischMethod: Complete symbolic integration (default)

Method Configuration

# Research configuration (strict, complete)
RischMethod(use_algebraic_closure=true, catch_errors=false)

# Production configuration (robust, graceful)  
RischMethod(use_algebraic_closure=true, catch_errors=true)

# Performance configuration (faster, simpler)
RischMethod(use_algebraic_closure=false, catch_errors=true)

Extensibility

The framework is designed for easy extension with additional integration methods. The abstract type AbstractIntegrationMethod provides the foundation for implementing new algorithms.

Documentation

Complete documentation with method selection guidance, algorithm details, and examples is available at: https://symbolicintegration.juliasymbolics.org

Citation

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

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