The phrase "study at ridicule" means to scrutinize or examine closely, often in a critical or mocking way. When it comes to synonyms for this phrase, there are several options that convey similar meanings. For example, one could say "analyze with derision," "scrutinize mockingly," or "inspect with a critical eye." Other phrases that capture this sense of close examination or criticism include "examine scornfully," "scrutinize critically," and "analyze with contempt." Depending on the context in which you're using the phrase, choosing the most appropriate synonym can help you better convey the meaning you intend.