While it is not a back-end integration, it is definitively a work around that meets the need. The element to consider is to have SSO (Single Sign On) for both, the Jive and the Moodle instance. This ensures that users can access both platforms having only logged in once.
Then, on the Jive side, create a document with the Moodle link for each of the courses. Since you provide the name, description and tags in Moodle when setting up a course, you can use the exact information in the Jive document. This ensures that the Moodle course will show when doing a search in Jive.
Then on the Moodle side, you can maintain a closed discussion forum in a course. I recommend adding a link pointing to a course discussion forum in a space dedicated to the Moodle courses. This way, the Jive discussion forum feeds the Knowledge base in Jive.
In terms of reporting, you can build the reports in Moodle and then just add the corresponding Moodle link in a Jive group, space, etc.
In summary, let Moodle handle the formal structured learning, and let Jive drive the informal/social learning and collaboration.
Any update on this whether Jive can be integrated now with Moodle LMS?