Mathematica can do nothing this expectation in general: 

[![enter image description here][1]][1]

So, it is highly unlikely that this expectation can be expressed in terms of elementary, or even special, functions. 

However, we have this: 

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/RCJOv.png
  [2]: https://i.sstatic.net/s1z4e.png