Is the book using British English? Because in North American and Canadian English, the past participle of "to get" is "gotten". But in most other types of English (e.g. British English), the past participle is "got" instead of "gotten". "Gotten" is only used in phrases such as "ill-gotten gains" instead.
Otherwise, the book might be supporting the use of "got" instead of "gotten" as a past participle because the present perfect form, "have gotten", can only be used in some contexts, but not others, making it a little confusing. For example:
You can say "I have gotten your letter." and "I've got your letter"
But you can only say "I have got to go now or I'll be late." and not "I've gotten to go now or I'll be late."
So I think in order to avoid the confusion, most types of English prefer standardise the verb and use "got" as a past participle instead.