catch all unhandled exceptions in ASP.NET Web Api

How do I catch all unhandled exceptions that occur in ASP.NET Web Api so that I can log them?

So far I have tried:

Create and register an ExceptionHandlingAttribute
Implement an Application_Error method in Global.asax.cs
Subscribe to AppDomain.CurrentDomain.UnhandledException
Subscribe to TaskScheduler.UnobservedTaskException

The ExceptionHandlingAttribute successfully handles exceptions that are thrown within controller action methods and action filters, but other exceptions are not handled, for example:

Exceptions thrown when an IQueryable returned by an action method fails to execute
Exceptions thrown by a message handler (i.e. HttpConfiguration.MessageHandlers)
Exceptions thrown when creating a controller instance

Basically, if an exception is going to cause a 500 Internal Server Error to be returned to the client, I want it logged. Implementing Application_Error did this job well in Web Forms and MVC - what can I use in Web Api?

Solution

This is now possible with WebAPI 2.1 (see the What's New):

Create one or more implementations of IExceptionLogger. For example:

public class TraceExceptionLogger : ExceptionLogger
{
    public override void Log(ExceptionLoggerContext context)
    {
        Trace.TraceError(context.ExceptionContext.Exception.ToString());
    }
}

Then register with your application's HttpConfiguration, inside a config callback like so:

config.Services.Add(typeof(IExceptionLogger), new TraceExceptionLogger());

or directly:

GlobalConfiguration.Configuration.Services.Add(typeof(IExceptionLogger), new TraceExceptionLogger());