Integrals of the form d t, where P (t) is a polynomial of the third or fourth degree and R is a rational function, have the simplest algebraic integrands that can lead to nonelementary1 integrals. Equivalent integrals occur in trigonometric and other forms, in pure and applied mathematics. Such integrals are known as elliptic integrals because a special example of this type arose in the rectification of the arc of an ellipse.
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