A numerical calculation on Mathematica shows that
I think it is very possible that I1=πe24 and I2=πe, but I cannot figure them out. Is there any possible way to prove these identities?
You made a very nice observation! Often it is important to make a good guess than just to solve a prescribed problem. So it is surprising that you made a correct guess, especially considering the complexity of the formula.
I found a solution to the second integral in here, and you can also find a solution to the first integral at the link of this site.