Number 361 in my List of Fiendish Integrals:

Prove that:

I tried the Weierstrass substitution but that got very complicated:

... and I can't get Integration by Parts to work. I fleetingly looked at manipulating it into the form:

... but that doesn't look hopeful either.

It's probably all about the substitution, as usual, but I haven't found one that works.

Any takers?