When it comes to synonyms for the phrase "brain work", there are a multitude of options to choose from. One possibility is using the term "mental exertion" to describe the challenging thinking required for a task. Alternatively, one might use "intellectual labor" to convey that the task is complex and requires mental effort. "Cognitive activity" could also be used to describe the mental process involved in the task. In certain contexts, words like "mental strain", "cerebral endeavor", or "intellectual endeavor" might also be appropriate and convey the same idea. Ultimately, the specific synonym chosen will depend on the context and the level of complexity involved in the task at hand.