Integrals of the form , with , , and  polynomials, often return answers in closed form. For example, when  has degree 1 or 2, the integral can always be returned in terms of elementary functions, such as logs, exponentials, trig, or arctrig functions. Thus, you have:

Note that the previous example evaluated the integral in an equation with the inert form on the left-hand side in order to see the integral "prettyprinted."

When y(x) has degree 3 or 4, the integral is called an elliptic integral.  Only sometimes do these return answers in terms of elementary functions.  In these cases, they are called pseudo-elliptic integrals.

However, in most cases, elliptic integrals return answers in terms of nonelementary functions, for example, the Legendre elliptic functions. A small variation of the previous problem gives:

Elliptic integrals appear in a variety of applications (some mentioned in the classical text by Byrd and Friedmann). For example, the length of an arc of a hyperbola  measured from the vertex to any point  is determined by the integral

For example, when  and , the length from the vertex to the point is given by

which is

with the result given in terms of Elliptic functions. Numerically, this is approximated by

The following equations were constructed to show Maple definitions of the Legendre F, E, and Pi functions.

with the variable  in . For , you obtain the complete forms of these integrals.  For example,

For more information, see EllipticE, EllipticF, EllipticK, and EllipticPi.

Integrals of the form  where  is a rational function of  and  and  is a degree 2 polynomial in  and  are elliptic integrals in trigonometric form. Corresponding forms for hyperbolic trig expressions (as above) also exist.  Such integrals  can also be expressed in terms of Legendre's elliptic functions. For example,

and

In both cases you can compare the results numerically:

Because of the possible presence of poles in the integrand, elliptic integrals are often very difficult to compute numerically with standard numerical integration methods. However, the Legendre functions can be numerically evaluated very efficiently using the AGM method. Therefore, reducing to normal form allows for easy numerical evaluation. For example,

which can be (quickly) numerically approximated by

Of course, one can always compute the numerical approximation directly by using

However, having a formula for an answer allows one to also compute to higher digits by using the same form:

which can also be done directly with evalf. However, in this case, the numerical integrator takes considerable time to come up with a result:

 Let

and consider the problem of integrating

and

The classical technique of reducing such integrals to Legendre normal form involves working with the poles of the rational function, converting the rational function to partial fractions, and reducing each "smaller" part. However, the poles of  and  are seen to be

so all the poles involve nested radicals. Numerically, at the current setting of Digits, these are given by:

Converting say  to a complete partial fraction decomposition can be accomplished by

again an expression that involves complicated nested radicals. The elliptic integration algorithm in Maple takes special care to avoid working with unnecessary radicals whenever possible. Integrals are first reduced to ones involving no repeated poles by using a classical reduction technique discovered by Hermite in the last century which introduces no new radicals during the reduction. Thus, you obtain

In addition, whenever possible, the elliptic integration algorithm in Maple works with an implicit rather than with an explicit representation of such roots. After Hermite reduction to eliminate multiple poles, a final answer is reduced to normal form by using the sum over roots functionality available in Maple. Thus, you obtain

You can evaluate this numerically by

If you want to evaluate the integral numerically to higher digits, you would simply do

Sometimes one also uses the roots of the polynomial under the radical in an implicit rather than an explicit form. For example, you obtain (after some waiting and using alias to obtain a simpler form of answer):

Here the answer is expressed in a (complicated) expression involving implicit roots indexed by their proximity to a numerical root. In this case, you can evaluate such an integral numerically to any number of digits by using:

You can also use a reduction procedure to return closed form solutions of Elliptic integrals in the indefinite case. In such cases, one does not return a form with elliptic functions having a k between 0 and 1; instead, an arbitrary complex number is used. For example, you have

a result which is verified by

Similarly, in the case of the trig form of elliptic integrals, you have

which again can be verified by

diff, EllipticE, EllipticF, EllipticK, EllipticPi, evalf, Int, simplify