Symbolic integration — the dust settles?

  • A. C. Norman
  • J. H. Davenport
10. Integration
Part of the Lecture Notes in Computer Science book series (LNCS, volume 72)


By the end of the 1960s it had been shown that a computer could find indefinite integrals with a competence exceeding that of typical undergraduates. This practical advance was backed up by algorithmic interpretations of a number of classical results on integration, and by some significant mathematical extensions to these same results. At that time it would have been possible to claim that all the major barriers in the way of a complete system for automated analysis had been breached. In this paper we survey the work that has grown out of the above-mentioned early results, showing where the development has been smooth and where it has spurred work in seemingly unrelated fields.


Structure Theorem Algebraic Function Integration Algorithm Finite Term Symbolic Integration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. (Caffo et al).
    Caffo, M., Remidi, E. and Turrini, S. An algorithm for the analytic evaluation of a class of integrals These Proceedings.Google Scholar
  2. (Caviness and Fateman, 1976).
    Caviness, B. F. and Fateman, R. J. 1976 Simplification of Radical Expressions. Proc SYMSAC 76, pp. 329–38.Google Scholar
  3. (Coates, 1970).
    Coates, J. Construction of Rational Functions on a Curve. Proc. Cam. Phil. Soc. 68(1970) pp. 105–23.Google Scholar
  4. (Davenport, 1979a).
    Davenport, J. H. The Computerisation of Algebraic Geometry, These Proceedings.Google Scholar
  5. (Davenport, 1979b).
    Davenport, J. H. Algorithms for the Integration of Algebraic Functions, These Proceedings.Google Scholar
  6. (Epstein, 1975).
    Epstein, H. I. Algorithms for Elementary Transcendental Function Arithmetic. Ph.D. Thesis, U. of Wisconsin, Madison, 1975.Google Scholar
  7. (Epstein, 1979).
    Epstein, H. I. A natural structure theorem for complex fields To appear in SIAM J. Computing.Google Scholar
  8. (Fitch, 1978).
    Fitch, J. P. private communicationGoogle Scholar
  9. (Harrington, 1979).
    Harrington, S. J. A new Symbolic Integration System in Reduce. Computer Journal 22(1979) 2. See also Utah Computational Physics Report 57, University of Utah, Nov. 1977 (revised May 1978).Google Scholar
  10. (Manove et al, 1968).
    Manove, M., Bloom, S. and Engelman, C. Rational functions in MATHLAB Proc IFIP conference on symbolic manipulation languages, Pisa 1968.Google Scholar
  11. (Moses, 1967).
    Moses, J. Symbolic Integration. Project MAC report 47, M.I.T., 1967.Google Scholar
  12. (Moses, 1969).
    Moses, J. The integration of a class of special functions with the Risch algorithm Memo MAC-M-421, 1969.Google Scholar
  13. (Moses, 1971).
    Moses, J. Symbolic Integration, the stormy decade. Communications ACM 14(1971) pp. 548–60.Google Scholar
  14. (Ng, 1974).
    Ng, E. W. Symbolic Integration of a class of Algebraic Functions. NASA Technical Memorandum 33-713 (1974).Google Scholar
  15. (Norman and Moore, 1977).
    Norman, A. C. and Moore, P. M. A. Implementing the New Risch Integration Algorithm. Proc. 4th. Int. Colloquium on Advanced Computing Methods in Theoretical Physics, Marseilles, 1977.Google Scholar
  16. (Risch, 1969).
    Risch, R. H. The Problem of Integration in Finite Terms. Trans. A.M.S. 139(1969) pp. 167–89(MR 38 #5759).Google Scholar
  17. (Risch, 1970).
    Risch, R. H. The Solution of the Problem of Integration in Finite Terms. Bulletin A.M.S. 76(1970) pp. 605–8.Google Scholar
  18. (Risch, 1974).
    Risch, R. H. A Generalization and geometric Interpretation of Liouville's Theorem on Integration in Finite Terms. IBM Research Report RC 4834 (6 May 1974).Google Scholar
  19. (Rosenlicht, 1976).
    Rosenlicht, M. On Liouville's Theory of Elementary functions. Pacific J. Math 65(1976), pp. 485–92.Google Scholar
  20. (Rothstein and Caviness, 1979).
    Rothstein, M., and Caviness, B. F. A Structure Theorem for Exponential and Primitive Functions. To appear in SIAM J. Computing.Google Scholar
  21. (Schmitt, 1979).
    Schmitt P., Substitution methods for the automatic symbolic solution of differential equations of first order and first degree. These Proceedings.Google Scholar
  22. (Trager, 1976).
    Trager, B. M. 1976 Algebraic Factoring and Rational Function Integration. Proc. SYMSAC 76, pp. 219–26.Google Scholar
  23. (Trager, 1978).
    Trager, B. M. IBM Yorktown Heights Integration Workshop, 28–9 Aug. 1978.Google Scholar
  24. (Trager, 1979).
    Trager, B. M. Integration of Single Radical Extensions. These Proceedings.Google Scholar
  25. (Wang, 1978).
    Wang, P. S. An Improved Multivariate Polynomial Factoring Algorithm. Math Comp 32(1978) pp. 1215–31.Google Scholar
  26. (Yun, 1973).
    Yun, D. Y. Y. The Hensel Lemma in Algebraic Manipulation. M.I.T. Thesis MAC TR-138, 1973.Google Scholar
  27. (Yun, 1976).
    Yun, D. Y. Y. 1976 Algebraic Algorithms using p-adic techniques. Proc. SYMSAC 76, pp. 248–59.Google Scholar
  28. (Yun, 1977).
    Yun, D. Y. Y. Fast Algorithm for rational Function Integration. IBM Research Report RC 6563 (6 Jan. 1977).Google Scholar
  29. (Zippel, 1977).
    Zippel, R. E. B. Radical Simplification Made Easy, Proc 1977 MACSYMA Users Conference (NASA CP-2012), pp. 361–7.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1979

Authors and Affiliations

  • A. C. Norman
    • 1
  • J. H. Davenport
    • 1
  1. 1.University of Cambridge Computer LaboratoryUK

Personalised recommendations