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API Reference

Main Functions

SymbolicIntegration.integrate — Function
integrate(f, x, method::AbstractIntegrationMethod=RischMethod(); kwargs...)

Compute the symbolic integral of expression f with respect to variable x using the specified integration method.

Arguments

  • f: Symbolic expression to integrate (Symbolics.Num)
  • x: Integration variable (Symbolics.Num)
  • method: Integration method to use (AbstractIntegrationMethod, default: RischMethod())

Keyword Arguments

  • Method-specific keyword arguments are passed to the method implementation

Returns

  • Symbolic expression representing the antiderivative (Symbolics.Num)

Examples

using SymbolicIntegration, Symbolics
@variables x

# Using default Risch method
integrate(x^2, x)  # (1//3)*(x^3)

# Explicit method with options
integrate(1/(x^2 + 1), x, RischMethod(use_algebraic_closure=true))  # atan(x)

# Method configuration
risch = RischMethod(use_algebraic_closure=false, catch_errors=true)
integrate(exp(x), x, risch)  # exp(x)
source

Integration Methods

Available Methods

SymbolicIntegration.RischMethod — Type
RischMethod <: AbstractIntegrationMethod

Risch algorithm for symbolic integration of elementary functions.

Fields

  • use_algebraic_closure::Bool: Whether to use algebraic closure for complex roots (default: true)
  • catch_errors::Bool: Whether to catch and handle algorithm errors gracefully (default: true)
source

Method Traits

Algorithm Overview

SymbolicIntegration.jl implements the complete symbolic integration algorithms from Manuel Bronstein's book "Symbolic Integration I: Transcendental Functions".

Supported Function Classes

  • Polynomial functions: ∫xⁿ dx
  • Rational functions: ∫P(x)/Q(x) dx using Rothstein-Trager method
  • Exponential functions: ∫exp(f(x)) dx using Risch algorithm
  • Logarithmic functions: ∫log(f(x)) dx using integration by parts
  • Trigonometric functions: Transformed to exponential form

Algorithm Components

The package includes implementations of:

  • Hermite reduction for rational functions
  • Rothstein-Trager method for logarithmic parts
  • Risch algorithm for transcendental functions
  • Differential field tower construction
  • Complex root finding for arctangent terms

Internal Structure

The package is organized into several algorithm modules:

  • rational_functions.jl: Rational function integration algorithms
  • transcendental_functions.jl: Risch algorithm implementation
  • differential_fields.jl: Differential field operations
  • complex_fields.jl: Complex number field handling
  • frontend.jl: User interface and expression conversion

Error Handling

The package defines custom exception types:

  • NotImplementedError: For unsupported function types
  • AlgorithmFailedError: When no elementary antiderivative exists
  • AlgebraicNumbersInvolved: When algebraic numbers complicate the result

Index