As no-one has come forward I will do my best! I have a vague recollection of someone doing JIRA integration before but I don't know the specifics I'm afraid. I haven't looked at integrating with Google at all.
The general rule is if you can call a SOAP web service we can integrate with it. We can tap into other systems web services via LPM or directly in Service Desk from 7.7.1, and the Event Manager web service in Service Desk can receive web service calls from any other system. We also provide the ability to call Event Manager via command line applications and a flat URL.
I have integrated a lot with LPM and not found anything yet that could not be done. As they improve integration, LPM will be even more useful. LPM works with the outside world through e-mail listeners where parameters are passed, web services, and database listeners. This gives you lots of capabilities above and beyond what Service Desk does. Like Service desk, they have a nice process engine.
The great thing about LPM is that even if you cannot do it directly from the web service call action within LPM. You can always run a script (VB, PS, JS, etc...) that can call the web service itself. There are a lot of options when you throw all the actions LPM has available into the equation.
So while I can't offer up any specific advice for doing what you're asking, if you do attempt this and have any questions please bump this thread and I'll see if I can help.
JIRA only support REST calls according to their website so I'm not totally sure if LPM can handle that or not. I haven't had my hands in LPM too much (aside from automated software deployment from Service Desk and AD stuff). Can LPM do stuff using REST?
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As great as REST is you don't get a formal definition stored for each service like you do with SOAP so its hard for other applications to do the dirty work for you. The request and response syntax could potentially be different with every call to the same URL which makes it impossible to read the response in a structred way, map it to attributes, etc.
Neither Service Desk or LPM can call a REST service directly but they are very easy to call from VB, .NET, etc. if you have tha patience to write a go-between.