This study explores prompt engineering for automated white-box integration testing of RESTful APIs using Large Language Models (LLMs). Four versions of prompts were designed and tested across three OpenAI models (GPT-3.5 Turbo, GPT-4 Turbo, and GPT-4o) to assess their impact on code coverage, token consumption, execution time, and financial cost. The results indicate that different prompt versions, especially with more advanced models, achieved up to 90% coverage, although at higher costs. Additionally, combining test sets from different models increased coverage, reaching 96% in some cases. We also compared the results with EvoMaster, a specialized tool for generating tests for REST APIs, where LLM-generated tests achieved comparable or higher coverage in the benchmark projects. Despite higher execution costs, LLMs demonstrated superior adaptability and flexibility in test generation.