Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2942710.22363/2658-4670-2021-29-4-337-346Research ArticleCalculation of integrals in MathPartnerMalaschonokGennadi I.<p>Doctor of Physical and Mathematical Sciences, Professor, Department of Informatics</p>malaschonok@gmail.comhttps://orcid.org/0000-0002-9698-6374SeliverstovAlexandr V.<p>Candidate of Physical and Mathematical Sciences, Leading researcher</p>slvstv@iitp.ruhttps://orcid.org/0000-0003-4746-6396National University of Kyiv-Mohyla AcademyInstitute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)1211202129433734612112021Copyright © 2021, Malaschonok G.I., Seliverstov A.V.2021<p style="text-align: justify;">We present the possibilities provided by the MathPartner service of calculating definite and indefinite integrals. MathPartner contains software implementation of the Risch algorithm and provides users with the ability to compute antiderivatives for elementary functions. Certain integrals, including improper integrals, can be calculated using numerical algorithms. In this case, every user has the ability to indicate the required accuracy with which he needs to know the numerical value of the integral. We highlight special functions allowing us to calculate complete elliptic integrals. These include functions for calculating the arithmetic-geometric mean and the geometric-harmonic mean, which allow us to calculate the complete elliptic integrals of the first kind. The set also includes the modified arithmetic-geometric mean, proposed by Semjon Adlaj, which allows us to calculate the complete elliptic integrals of the second kind as well as the circumference of an ellipse. The Lagutinski algorithm is of particular interest. For given differentiation in the field of bivariate rational functions, one can decide whether there exists a rational integral. The algorithm is based on calculating the Lagutinski determinant. This year we are celebrating 150th anniversary of Mikhail Lagutinski.</p>computer algebra systemMathPartnerintegralarithmetic-geometric meanmodified arithmetic-geometric meanLagutinski determinantMathPartnerсистема компьютерной алгебрыинтеграларифметико-геометрическое среднеемодифицированное арифметикогеометрическое среднееопределитель Лагутинского[G. I. Malaschonok, “Application of the MathPartner service in education,” Computer Tools in Education, no. 3, pp. 29-37, 2017, in Russian.][G. I. Malaschonok, “MathPartner computer algebra,” Programming and Computer Software, vol. 43, pp. 112-118, 2017. DOI: 10.1134/ S0361768817020086.][G. I. Malaschonok and I. A. Borisov, “About MathPartner web service,” Tambov University Reports. 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