1. The term “highly transcendental” needs care in interpretation-cf. the reprise below. Again Abel, in a paper published in 1826, showed the existence of polynomials R, F such that
$$
\int {\frac{{F dx}}
{{\sqrt R }} = \ln \left( {\frac{{P + \sqrt {RQ} }}
{{R - \sqrt {RQ} }}} \right)}
$$
has solutions for relatively prime polynomials P, Q. Here, R is a polynomial of degree 2n with distinct roots and F is a polynomial of degree n-1, so that the integrand is a differential of the 3rd kind. This is an “exceptional” case where the integral is transcendental but expressible in terms of elementary functions.
2. Integrating a DE means finding a solution by an iterative process. Since there are no derivations of ℚ the methods of calculus break down-one must break the problem into “increments” by some other means, perhaps either by an iteration process that at each stage decreases the “arithmetic complexity” of the 0-cycle, or by analyzing the DE’s (4.15) and (4.20) in the completions of ℚ under all valuations.