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 $(F, \, ^\prime)$ 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 $f = g^\prime + r$. 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 $F$.

翻译：超越Liouvillian扩张是一种微分域，可用于模拟多对数函数、超指数函数、三角函数、对数积分及其（嵌套）有理表达式。对于此类具有常数子域$C$的扩张$(F, \, ^\prime)$，我们为$F$中导数的$C$-子空间构造了一个互补子空间$W$，并开发了一种算法：对任意$f \in F$，该算法可计算满足$f = g^\prime + r$的配对$(g,r) \in F \times W$。特别地，$f$是$F$中的导数当且仅当$r=0$。该算法通过计算参数化对数部分来确定$F$上的初等可积性，并为可表示为$F$中元素的函数提供基于约化的望远镜算子构造方法。