> The expression for the antiderivative of 1/(1 + x^5) > tells me the value is a transcendental number that is > a closed form number in the sense of Timothy Y. Chow. > It also tells me something about the "complexity depth" > (a term I made up just now) of the closed form number.
Oops, I jumped too fast. "Closed form number in the sense of Timothy Y. Chow" is clear, but "transcendental" isn't. Note that algebraic combinations of transcendental numbers may not be transcendental: log_10(2) + log_10(5) = 1 and pi + (-pi) = 0 are examples that showing even the sum of transcendental numbers can be non-transcendental. Indeed, according to the following paper, it wasn't even known in 1949 whether the integral from x=0 to x=1 of 1/(1 + x^3), which equals (ln 2)/3 + (pi)(sqrt 3)/9.
However, if I understand the corollary to the main theorem in this paper correctly, then for each positive integer n, the integral from x=0 to x=1 of 1/(1 + x^n) is transcendental.
More generally, if the following hold
(1) P(x) and Q(x) are polynomials (2) the coefficients of P(x) and Q(x) are real and algebraic (3) P(x) and Q(x) have no common non-constant factors (4) deg(P) < deg(Q) (5) the complex zeros of Q(x) are all distinct (6) the numbers a and b are real and algebraic (7) none of the real zeros of Q(x) belong to the interval [a,b]
then the integral from x=a to x=b of P(x)/Q(x) is either zero or transcendental.