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.

We consider the additive decomposition problem in primitive towers and present an algorithm to decompose a function in a certain kind of primitive tower which we call S-primitive, as a sum of a derivative in the tower and a remainder which is minimal in some sense. Special instances of S-primitive towers include differential fields generated by finitely many logarithmic functions and logarithmic integrals. A function in an S-primitive tower is integrable in the tower if and only if the remainder is equal to zero. The additive decomposition is achieved by viewing our towers not as a traditional chain of extension fields, but rather as a direct sum of certain subrings. Furthermore, we can determine whether or not a function in an S-primitive tower has an elementary integral without the need to deal with differential equations explicitly. We also show that any logarithmic tower can be embedded into a particular extension where we can further decompose the given function. The extension is constructed using only differential field operations without introducing any new constants.

This paper extends the classical Hermite-Ostrogradsky reduction for rational functions to more general functions in primitive extensions of certain types. For an element f in such an extension K , the extended reduction decomposes f as the sum of a derivative in K and another element r such that f has an antiderivative in K if and only if r=0; and f has an elementary antiderivative over K if and only if r is a linear combination of logarithmic derivatives over the constants when K is a logarithmic extension. Moreover, r is minimal in some sense. Additive decompositions may lead to reduction-based creative-telescoping methods for nested logarithmic functions, which are not necessarily D -finite.