According to Cambridge Dictionary, "include" means
to contain something as a part of something else, or to make something part of something else
while "contain" means
to have something inside or include something as a part
The distinction between 'includes' and 'contains' is a bit more subtle. 'Contains' is used when there is something else that is doing the containing. The box is a container, and contains the hats. You would not, generally, say that the box includes the hats. Inside the box is a set of objects, and that set (which 'contents' refers to) includes (and consists of) the hats.
Common combat sports include mixed martial arts, boxing ...
Is it still natural if I substitute "contain" for "include"?
Common combat sports contain mixed martial arts, boxing ...