hyint: Hybrid (Symbolic-Numeric) Integration Package

hyint is a Python package to computes indefinite integral of univariable expressions with constant coeficients using symbolic-numeric methodolgy. It is built on top of sympy symbolic manipulation ecosystem of Python, but applies numerical methods to solve integral problems.

hyint can solve a large subset of basic standard integrals (polynomials, exponential/logarithmic, trigonometric and hyperbolic, inverse trigonometric and hyperbolic, rational and square root) ( see The Basis of Symbolic-Numeric Integration for a brief introduction to the algorithm. It can even find some integrals not found by the current version of sympy.integrate.

The symbolic part of the algorithm is similar (but not identical) to the Risch-Bronstein's poor man's integrator and generates a list of ansatzes (candidate terms). The numerical part uses sparse regression adopted from the Sparse identification of nonlinear dynamics (SINDy) algorithm to prune down the ansatzes and find the corresponding coefficients.

Prerequisites

hyint requires numpy/scipy and sympy to have be installed.

Installation

Install hyint as

pip install hyint

Tutorial

Basic Usage

The main function exported by hyint is integrate(eq, x). It accepts two arguments, where eq is a univariable expression in `x'. It results either the integral or 0 otherwise.

Some examples:

from sympy import *
import hyint

x = Symbol('x')

In: hyint.integrate(x**3 - x + 1, x)
Out: x**4/4 - x**2/2 + x

In: hyint.integrate(x**2 * sin(2*x), x)
Out: -x**2*cos(2*x)/2 + x*sin(2*x)/2 + cos(2*x)/4

In: hyint.integrate(sqrt(x**2 + x - 1), x)
Out: x*sqrt(x**2 + x - 1)/2 + sqrt(x**2 + x - 1)/4 - 5*log(2*x + 2*sqrt(x**2 + x - 1) + 1)/8

In: hyint.integrate(x/(x**2 + 4), x)
Out: log(x**2 + 4)/2

In: hyint.integrate(x**2*log(x)**2 , x)
Out: x**3*log(x)**2/3 - 2*x**3*log(x)/9 + 0.0740740740740739*x**3

In: hyint.integrate(1 / (x**3 - 2*x + 1), x)
Out: 2.17082039324994*log(x - 1) + 1.34164078649988*log(x + 1/2 + sqrt(5)/2) - 1.17082039324993*log(x**3 - 2*x + 1)

In: hyint.integrate(x*exp(x)*cos(2*x), x)
Out: 2*x*exp(x)*sin(2*x)/5 + x*exp(x)*cos(2*x)/5 - 0.16*exp(x)*sin(2*x) + 0.12*exp(x)*cos(2*x)

In: hyint.integrate(log(log(x))/x, x)
Out: log(x)*log(log(x)) - log(x)

In: hyint.integrate(log(cos(x))*tan(x), x)
Out: -log(cos(x))**2/2

In: hyint.integrate(exp(x + 1)/(x + 1), x)
Out: Ei(x + 1)

In: hyint.integrate(exp(x)/x - exp(x)/x**2 , x)
Out: exp(x)/x

In: hyint.integrate(exp(x**2) , x)
Out: 0.886226925452758*erfi(x)

# sympy.integrate does not solve this example:
In: hyint.integrate(sqrt(1 - sin(x)), x)
Out: 2*cos(x)/sqrt(1 - sin(x))

As an Ansatz Generator

hyint can be used as an standalone integrator (integrate(eq, x)); however, it is also useful as a helper untility for other integration routines by running hints(eq, x), which returns a filtered list of ansatzes. In this role, eq can have symbolic constants in addition to the numerical ones.

The following example shows how it can augment heurisch integrator. heurisch is one of the sympy integrators and is a true symbolic implementation of the the Risch-Bronstein's poor man's algorithm). It has a hints arguments that can accept a list of ansatzes from hyint.

In: from sympy import log
In: from sympy.integrals.heurisch import heurisch
In: from sympy.abc import a, x
In: import hyint   

In: y = log(log(x) + a) / x

In: heurisch(y, x)
Out:    # None is returned, meaning no solution is found

In: hints = hyint.hints(y, x)
In: print(hints)
Out: [log(a + log(x)), log(x)*log(a + log(x)), log(x)]
    
In: heurisch(y, x, hints=hints)
Out: a*log(a + log(x)) + log(x)*log(a + log(x)) - log(x)    

Testing

A test suite of 170 basic integrals can be run as hyint.run_tests().

Citation

hyint is a adopted from and is a rewrite of SymbolicNumericIntegration.jl. Citation: Symbolic-Numeric Integration of Univariate Expressions based on Sparse Regression:

@article{Iravanian2022,
author = {Shahriar Iravanian and Carl Julius Martensen and Alessandro Cheli and Shashi Gowda and Anand Jain and Julia Computing and Yingbo Ma and Chris Rackauckas},
doi = {10.48550/arxiv.2201.12468},
month = {1},
title = {Symbolic-Numeric Integration of Univariate Expressions based on Sparse Regression},
url = {https://arxiv.org/abs/2201.12468v2},
year = {2022},
}