A team of mathematicians from RUDN University added a new symbolic integration feature to the Sage computerized algebra system. The group established the ideas and methods proposed by the German mathematician Karl Weierstrass in the 1870s. The results were published in the newspaper *Journal of Symbolic Computing*.

The first computer program capable of calculating the integrals of basic functions was developed in the late 1950s. Once created, the developers confirmed that a computer was capable of coping with tasks that required a certain level of “thinking” in addition to performing simple calculations. Symbolic integration, i.e. the integration of letters and abstract symbols instead of numbers, is an example of such a task.

At the same time, scientists realized that neither humans nor computers could determine whether a particular integral could be taken into basic functions (if those humans or computers used the methods studied in a university exam course and gave a finite number of steps). Therefore, mathematicians working on symbolic integrators in the 1960s began to refer to the methods suggested by Liouville in the 1830s. Since then, computer scientists have been using classical scientific heritage.

The calculation of primitive algebraic functions is one of the bottlenecks in the process of integration development. Prior to World War II, the integration of algebraic functions or abelian integrals was one of the most important mathematical topics, but was later forgotten.

“Today’s computer algebra systems are able to meet even the most exotic demands of students in mathematical analysis, but at the same time, many of these systems do not know the integrals of basic functions. Many packages allow only the integration of algebraic or Abelian functions with integrals, but their development stopped 15 years ago, and their functionality leaves much to be desired, ”says Mikhail Malykh, Assistant PhD in Physical and Mathematical Sciences Applied Computer Science and Probability Theory Department, RUDN University.

One of the theories developed by the German mathematician Karl Weierstrass in the 1870s reduces the calculation of an integral of an algebraic function to the discovery of a given set of three known integral types. The initial integral is represented as the sum of the standard integrals (this construction is known as the normal representation of an abelian integral). The RUDN University team has confirmed that this representation is an indicator of whether a particular integral can be calculated in basic functions. To confirm their theory, mathematicians tested them in simple elliptic integrals in 2017 using a software package created by the team. The package helps to calculate the coefficients of the normal shape of an integral. In the future, the team plans to conduct similar studies for a wider range of integrals.

“This work is just a step towards an ambitious goal: we want to express Weierstrass’s theory of abelian integrals and functions using computer algebraic language and implement it in the Sage system, giving researchers free access to researchers around the world,” added Mikhail Malykh of RUDN University. k.

Reference: “On the Symbolic Integration of Algebraic Functions” by MD Malykh, LA Sevastianova, and Y. Yu, September 11, 2020, *Journal of Symbolic Computing*.

DOI: 10.1016 / j.jsc.2020.09.002