I don’t see anywhere in the Bible where any kind of statement must be repeated when baptizing a penitent believer. Most brethren I know say something like: “I baptize you in the name of the Father, Son, and Holy Spirit, for the remission of your sins.” This comes from Jesus’ words as found in Matthew 28:19 in the King James Version.
However, I do not believe this is a statement or credo that must be repeated prior to or during baptism. In fact, this is not even a proper translation of the phrase in question. A more accurate rendering is found in the American Standard Version (1901), which reads, “baptizing them into the name of the Father, and of the Son, and of the Holy Spirit.”
The issue in verse 19 is not authority, but relationship. The authority principle is addressed in verse 18, where Jesus says “all authority has been given to Me in heaven and in earth.” Still, there is no intent of our Lord to institute some kind of creed to be repeated by the baptizer. The Father, Son, and Holy Spirit are one. They do not teach differing doctrines. Jesus pointed to the Father as the source of His message (John 12:49-50). Jesus said the Holy Spirit would bring to the apostles remembrance of all Jesus had taught them (John 14:26) and then some (John 16:12-13).
To be baptized “in the name of Jesus” (Acts 19:5) means to be baptized with the baptism He authorized His disciples to administer, a baptism for remission of sins. Those Ephesians in Acts 19 were baptized again because the baptism they were taught (John’s) looked forward to the coming of Jesus (v 4). The baptism of the Great Commission is a baptism that looks backward to the resurrected Jesus (Acts 2:22-24, 32, 36-28).
Any attempt to make an issue of what is said at baptism is to miss the point of the discussion. So long as a person believes in Jesus’ deity (John 8:24), repents of sins (Luke 13:3, 5), confesses their faith in Christ (Romans 10:10), and understands the purpose of baptism is to receive remission of sins (Acts 2:38; Colossians 2:12), it makes no difference if the baptizer is silent during the process. I make the statement “into the name for remission of sins” simply to teach our children and any non-Christian in the audience why the person is being baptized.