## ICMS 2016 Session 8: Symbolic Integration

ICMS 2016: Home, Sessions

### Aim and Scope

The symbolic evaluation of integrals is a classic topic in computer algebra. This refers to indefinite integrals, i.e., finding the antiderivative of a given function, as well as to definite integrals. In the past decades tremendous progress has been made in this field, and the capabilities of today's computer algebra systems are truly impressive. As these capabilities are particularly based on table-lookup, there is still a considerable interest in algorithmic approaches to symbolic integration, which also nowadays is a very active field of research. In this session we want to discuss recent developments concerning the algorithmic evaluation of integrals, either in closed form or using some implicit representation, e.g., in terms of a differential equation satisfied by a given parametric integral.

### Publications

• A short abstract will appear on the permanent conference web page (see below) as soon as accepted.

• An extended abstract will appear on the permanent conference web page (see below) as soon as accepted.
It will also appear on the proceedings that will be distributed during the meeting.

• A journal special issue consisting of full papers may be organized immediately after the meeting (will be discussed during the meeting).

### Submission Guidelines

• If you would like to give a talk at ICMS, you need to submit first a short abstract and then later an extended abstract (see the guidelines for the details). Please send your abstract by e-mail to Christoph Koutschan.

• After the meeting, the submission guideline for a journal special issue will be communicated to you by the session organizers.

• ### Session Opening and Overview

Christoph Koutschan (Austrian Academy of Sciences, Austria)

Abstract:
The symbolic evaluation of integrals is a classic topic in computer algebra. This refers to indefinite integrals, i.e., finding the antiderivative of a given function, as well as to definite integrals. In the past decades tremendous progress has been made in this field, and the capabilities of today's computer algebra systems are truly impressive. As these capabilities are particularly based on table-lookup, there is still a considerable interest in algorithmic approaches to symbolic integration, which also nowadays is a very active field of research. In this session we want to discuss recent developments concerning the algorithmic evaluation of integrals, either in closed form or using some implicit representation, e.g., in terms of a differential equation satisfied by a given parametric integral.

• ### Computer algebra tools for integrals

Clemens Raab (Austrian Academy of Sciences, Austria)

Abstract:
In this talk we give a brief overview of some techniques and algorithms for symbolic computation of integrals that have been established over the past decades. Also, software implementing these methods will be shown.
On the one hand algorithmic methods for finding a closed form for indefinite integrals will be discussed. Mainly reduction techniques of some kind play a role here. Their principle is to reduce step by step the integrand to an integrand that is in some sense simpler, obtaining part of the antiderivative in each step. Most notably, in 1969 Risch gave an algorithm for computing indefinite integrals of elementary functions. Apart from reduction algorithms also the Risch-Norman algorithm is a powerful tool, which proceeds by solving for undetermined coefficients in a suitable ansatz for the antiderivative.
Computation of (definite) parameter integrals, on the other hand, does not necessarily require knowledge of an explicit antiderivative of the integrand. Instead one may rely on techniques like differentiation under the integral in order to derive a differential equation that is satisfied by the parameter integral. Especially since the work of Zeilberger in 1990, computer algebra algorithms have been developed for this and similar purposes.

• ### A Discussion of the Practical Issues of Computing Integrals in Maple

Austin Roche (Maplesoft)
John May (Maplesoft)

Abstract:
We'll give an overview of the general strategy for symbolic integration in Maple. We'll discuss the theoretical algorithms we use and how we combine them into a user-level meta-heuristic. We'll give some statistics on the performance of various parts of our method based on our internal test suite. Finally, we'll describe some of our recent improvements and the challenges inherent in creating and evolving practical mathematical software.

• ### Recent Developments in the RUBI Integration Project

David J. Jeffrey (The University of Western Ontario, London, Canada)
Albert D. Rich (The University of Western Ontario, London, Canada)

Abstract:
RUBI (RUle Based Integrator) is a long-term project that demonstrates the feasibility and desirability of organizing mathematical knowledge as a rule-based decision tree. The "proof-of-concept" is provided by the development and implementation of a system of rules for finding optimal integrals (anti-derivatives or primitives) for a large class of integrands. We give an overview of the project and its current status. We discuss different ways of implementing the project in different systems.

• ### Complexity of Integration, Special Values, and Recent Developments

James H. Davenport (University of Bath, UK)

Abstract:
Two questions often come up when the author discusses integration: what is the complexity of the integration process, and for what special values of parameters is an unintegrable function actually integrable. These questions have not been much considered in the formal literature, and where they have been, there is one recent development indicating that the question is more delicate than had been supposed.

• ### Integration in terms of exponential integrals and incomplete gamma functions

Waldek Hebisch (University of Wroclaw, Poland)

Abstract:
We present a new decision procedure for integration in terms of exponential integrals (Ei) and incomplete gamma functions. Our procedure extends classical elementary integration procedure given by Risch and improved among others by Rothstein, Trager and Bronstein. First we give Lioville style theorem which describes possible form of integral. Compared to elementary integration new terms can appear when integrating functions which contain exponential or logarithm as top transcendental. Main effort goes into integration of functions containing exponential as top transcendental. Following Cherry and Knowles we give conditions which allow to find finite number of potential Ei and gamma terms and then solve parametric Risch differential equation to find coefficients and finish integration. For gamma terms we base our conditions on presence of multiple factors. This can be efficiently tested using resultants. We will show examples of difficult cases and how our algorithm handles them.
Compared to Cherry and Knowles our method is general and handles algebraic extensions. It is also more practical. In particular, when finding potential Ei and gamma terms in worst case our algorithm needs to solve bivariate system of polynomial equations, while method of Cherry needs multivariate systems. Large part of our method is implemented in computer algebra system FriCAS.

• ### The Method of Brackets

Lin Jiu (Tulane University, New Orleans, USA)

Abstract:
The method of brackets, developed by Ivan Gonzalez, has its origin on the evaluation of definite integrals arising from the Schwinger parametrization of Feynman diagrams. It especially computes the definite integral $\int_0^\infty f(x)\,dx$, based on the Ramanujan's Master Theorem $$\int_{0}^{\infty}x^{s-1}\left\{ f\left(0\right)-\frac{x}{1!}f\left(1\right)+\frac{x^{2}}{2!}f\left(2\right)-\cdots\right\} dx=f\left(-s\right)\Gamma\left(s\right).$$ Besides introducing basic rules and examples, current achievement and future work, such as packages, modifications and new ideas, will also be presented.