Is it permissible for a husband to continue marital life with his wife, who disclosed a private secret, and the husband divorced her then took her back, and after a period he intended to separate from her and agreed with her family on that but they recanted, and the husband said to another person, "I am divorced," jokingly?

Cases one and three: The phrasing you used includes an allusion to divorce, so it depends on your intention. If you intended divorce, it occurred; otherwise, it did not.

Case two: Divorce definitely occurred in this case, because you uttered an explicit divorce, which does not require intention.

Your statement: "I am divorced" is not explicit. If it was a joke, nothing results from it, especially if it was a false statement.

In some cases, the reference point is your intention, and you know best what it was, and from it you can determine the number of divorces: - If it was three divorces, your wife is forbidden to you until she marries another husband and then he separates from her. - If it was not three divorces, you have the right to take her back.

Divorce should only be resorted to if its benefit outweighs its harm.