A complete reduction ϕ for derivatives in a differential field is a linear operator on the field over its constant subfield. The reduction enables us to decompose an element f as the sum of a derivative and the remainder ϕ(f). A direct application of ϕ is that f is in-field integrable if and only if ϕ(f) = 0.

We present two evaluation-based algorithms: one for computing logarithmic parts and the other for determining complete logarithmic parts in transcendental function integration. Empirical results illustrate that the new algorithms are markedly faster than those based respectively on resultants, the contraction of ideals, subresultants and Gröbner bases. They may be used to accelerate Risch’s algorithm for transcendental integrands, and help us to compute elementary integrals over logarithmic towers efficiently.