Symbolic Integrals

Symbolics.jl provides a the Integral operator for defining integrals. These can be used with various functionalities in order to represent integro-differential equations and as a representation for symbolic solving of integrals.

Defining Symbolic Integrals

Note that integration domains are defined using DomainSets.jl.

Integral(domain)

I1 = Integral(x in ClosedInterval(1, 5))
I2 = Integral(x in ClosedInterval(1, 5))

@variables a b
I = Integral(x in ClosedInterval(a, b))
@test isequal(I(0), 0)
@test isequal(I(2), 2*(b -a))

Solving Symbolic Integrals

Symbolics.jl currently has the following options for solving integrals symbolically:

| Option | Description | Pros | Cons | | –––– | –––- | –––- | | SymbolicNumericIntegration.jl | Uses numeric integrators with symbolic regression machine learning methods | Can solve some hard integrals very fast and easily | Can be unreliable in easy cases, will give floats (0.5) instead of rational values (1//2) | | SymPy's Integrate | Uses SymPy (through SymPy.jl) to integrate the expression | Reasonably robust and tested solution | Extremely slow |

SymbolicNumericIntegration.jl

SymbolicNumericIntegration.jl is a package for solving symbolic integrals using numerical methods. Specifically, it mixes numerical integration with machine learning symbolic regression in order to discover the basis for the integrals solution, for which it then finds the coefficients and uses differentiation to prove the correctness of the result. While this technique is unusual and new, see the paper for details, it has the advantage of working very differently from the purely symbolic methods, and thus the problems which it finds hard can be entirely different from ones which the purely symbolic methods find hard. That is, while purely symbolic methods have difficulty depending on the complexity of the expression, this method has difficulty depending on the uniqueness of the numerical solution of the expression, and there are many complex expressions which have a very distinct solution behavior and thus can be easy to solve through this method.

One major downside to this method is that, because it works numerically, there can be precision loss in the coefficients. Thus the returned solution may have 0.5 instead of 1//2.

For using SymbolicNumericIntegration.jl, see the SymbolicNumericIntegration.jl documentation for more details. A quick example is:

using Symbolics
using SymbolicNumericIntegration

@variables x a b

integrate(3x^3 + 2x - 5)
# (x^2 + (3 // 4) * (x^4) - (5 // 1) * x, 0, 0)

integrate(exp(a * x), x; symbolic = true)
# (exp(a * x) / a, 0, 0)

integrate(sin(a * x) * cos(b * x), x; symbolic = true, detailed = false)
# (-a * cos(a * x) * cos(b * x) - b * sin(a * x) * sin(b * x)) / (a^2 - (b^2))

SymPy

The function sympy_integrate allows one to call SymPy's integration functionality. While it's generally slow, if you leave your computer on for long enough it can solve some difficult expressions.

sympy_integrate(expr, var)

@variables x
expr = x^2
result = sympy_integrate(expr, x)