SymbolicIntegration.jl

This package provides Julia implementations of symbolic integration algorithms.

The front-end (i.e., the user interface) requires SymbolicUtils.jl. The actual integration algorithms are implemented in a generic way using AbstractAlgebra.jl. Some algorithms require Nemo.jl for calculations with algebraic numbers.

SymbolicIntegration.jl 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. Currently, SymbolicIntegration.jl can integrate rational functions and integrands involving transcendental elementary functions like exp, log, sin, etc. As in the book, integrands involving algebraic functions like sqrt and non-integer powers are not treated.

Note that SymbolicIntegration.jl is still in an early stage of development and should not be expected to run stably in all situations. It comes with absolutely no warranty whatsoever.

Installation

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

Usage

julia> using SymbolicIntegration, SymbolicUtils

julia> @syms x
(x,)

julia> f = (x^3 + x^2 + x + 2)//(x^4 + 3*x^2 + 2)
(2 + x + x^2 + x^3) / (2 + x^4 + 3(x^2))

julia> integrate(f, x)
(1//2)*log(2 + x^2) + atan(x)

julia> f = 1/(x*log(x))
1 / (x*log(x))

julia> integrate(f, x)
log(log(x))

julia> f = 1/(1+2*cos(x))
1 / (1 + 2cos(x))

julia> g = integrate(f, x)
log(-4 - sqrt(16//3)*tan((1//2)*x))*sqrt(1//3) - log(sqrt(16//3)*tan((1//2)*x) - 4)*sqrt(1//3)

Tests

Some test problems taken from the Rubi Integration Test-suites are provided by test_integrate_rational.jl and test_stewart.jl, which produce the output test_integrate_rational.out and test_stewart.out, respectively.