List of integrals of irrational functions

The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, see lists of integrals. Throughout this article the constant of integration is omitted for brevity.

Integrals involving r = a2 + x2Edit


Integrals involving s = x2a2Edit

Assume x2 > a2 (for x2 < a2, see next section):


Here  , where the positive value of   is to be taken.


Integrals involving u = a2x2Edit


Integrals involving R = ax2 + bx + cEdit

Assume (ax2 + bx + c) cannot be reduced to the following expression (px + q)2 for some p and q.


Integrals involving S = ax + bEdit



  • Peirce, Benjamin Osgood (1929) [1899]. "Chap. 3". A Short Table of Integrals (3rd revised ed.). Boston: Ginn and Co. pp. 16–30.
  • Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables 1972, Dover: New York. (See chapter 3.)
  • Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. LCCN 2014010276. (Several previous editions as well.)