AFAIK, if your usage stays within the free quota of Content Safety (both analysis and the logs/metrics they provide internally), you would not pay extra. If you exceed free quotas, you’ll pay for analysis (text/image) per request/record. If you use additional Azure services for monitoring/logging (e.g. exporting logs to Azure Monitor / Log Analytics, long-term retention, alerting etc.), those will have their own costs.
If the above response helps answer your question, remember to "Accept Answer" so that others in the community facing similar issues can easily find the solution. Your contribution is highly appreciated.
hth
Marcin