The phrase "be on case" is commonly used to mean "to investigate or pursue something diligently." There are a variety of synonyms that can be used interchangeably for this phrase, such as "be on the trail of," "be hot on the heels of," "be after," "be after the truth," "be on the hunt for," and "be following up on." Each of these phrases conveys a sense of determination and persistence, indicating that someone is committed to finding out the truth or achieving a specific goal. Using these synonyms can help to vary your language and add depth to your writing, allowing your message to be communicated effectively.