An integral of the form:
A(x) + B(x)sqrt(R(x))
∫ ------------------------ dx
C(x) + D(x)sqrt(R(x))
or
A(x)
∫ -------------- dx
B(x)sqrt(S(x))
where A(x), B(x), C(x), and D(x) are rational functions of x, and R(x) is a polynomial of degree 3 or 4. Equivalently, it is an integral of a rational function of x and w, R(w, x):
∫ R(w, x) dx
where w2 is a cubic or quartic polynomial in x, and R(w, x) contains at least one odd power of w, and w2 has no repeated factors (i.e. all of its zeros are of multiplicity one).
Elliptic integrals can be thought of as generalizations of the inverse trigonometric functions. For example, to compute the arc length of a circle can be obtained by use of the inverse trigonometric functions (without the use of pi), and the arc length of an ellipse can be obtained in a similar way with elliptic integrals.
Elliptic integrals have many applications in applied mathematics and mathematical physics. While the motion of a pendulum as a function of time can be described by trigonometric functions when the displacements are small, the full solution for its motion requires the use of elliptic integrals. Many problems in electromagnetics and gravitation require the use of elliptic integrals as well.
By inverting elliptic integrals one can obtain very useful generalizations of the trigonometric functions which are called elliptic functions, which fall into two classes, the Weierstrass elliptic functions and the Jacobi elliptic functions.
Any elliptic integral may be written in terms of one of the three elliptic integral standard forms, and numerical methods involving Landen's transformation and the arithmetic geometric mean have been developed that are able to compute values.