Exact iterates [in iterated maps] For any integer a the n th iterate of x  FractionalPart[a x] can be written as FractionalPart[a n x] , or equivalently 1/2 - ArcTan[Cot[a n π x]]/ π . In the specific case a = 2 the iterates of If[x < 1/2, a x, a (1 - x)] have the form ArcCos[Cos[2 n π x]]/ π .

For rational functions f[x] , Integrate[f[x], {x, 0, 1}] must always be a linear function of Log and ArcTan applied to algebraic numbers ( f[x] = 1/(1 + x 2 ) for example yields π /4 ). Multiple integrals of rational functions can be more complicated, as in Integrate[1/(1 + x 2 + y 2 ), {x, 0, 1}, {y, 0, 1}]  HypergeometricPFQ[{1/2, 1, 1}, {3/2, 3/2}, 1/9]/6 + 1/2 π ArcSinh[1] - Catalan and presumably often cannot be expressed at all in terms of standard mathematical functions.