Complete Reduction for Derivatives in a Transcendental Liouvillian Extension

Transcendental Liouvillian extensions are differential fields, in which one can model poly-logarithmic, hyperexponential, and trigonometric functions, logarithmic integrals, and their (nested) rational expressions. For such an extension $(F,{}^{\mathrm{\prime }})$ with the subfield $C$ of constants, we construct a complementary subspace $W$ for the $C$-subspace of derivatives in $F$, and develop an algorithm that, for every $f\in F$, computes a pair $(g,r)\in F\times W$ such that $f=g^{\mathrm{\prime }}+r$. Moreover, $f$ is a derivative in $F$ if and only if $r=0$. The algorithm enables us to determine elementary integrability over $F$ by computing parametric logarithmic parts, and leads to a reduction-based approach to constructing telescopers for functions that can be represented by elements in $F$.