Thanks for your analysis. I will take more detailed look at it at a later time. My approach was different. Here it is...
Considering the transformation x=1/t, one obtains
Hence, we accordingly obtain
Consequently, we see thatNotes1. It is worth mentioning that the given integral does not depend on parameter 'n'. In my opinion, this is far from obvious.
2. Probably, by suitable simplifications your result can be rewritten as the simple expression above.
3. Thank you for your interest in solving such old problems. They may look complicated, but solving them can be interesting and surprising.