Can the Risch Algorithm Determine Integrability of Functions Like ∫x^x dx?

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if dy/dx = x^x , find y ?

or

find
∫x^x dx

?
 
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HallsofIvy said:
That cannot be integrated in terms of elementary functions.

Is there a way enable us to know that a function F can be intergrated in terms of elementary functions or not ?