Can I include the completion of Udacity and Coursera classes I have attended in an academic CV?

No. Specific coursework (whether formal or informal, online or in-person) does not belong in an academic CV.

If the online course is relevant to your current field of work/study it would help. For example, a graduate student of science could mention a course on scientific computing but a course taken on criminal law would be irrelevant. Also online courses taken can only be shown under professional development and not under academic qualifications since these are non-credit courses.

Yes. The certified courses that connect to skills that doesn't directly apply to your mainstream degree deserves to be in your academic CV. It will show, to an extent, an interdisciplinary qualification that you possess. There are many real life instances where this has helped a lot, especially in industrial placements, career change, and interdisciplinary research.

There are some courses from MOOC you ought not to add in your CV. These include those that overlaps the courses you've already completed as per your educational degrees, minor introductory courses, and those that wouldn't prove a significant impact in your career point of view.