I was playing around with asymptotics and integral formulas of the digamma function and exponential integral when I stumbled upon this one:
Working with it a bit, I ended up with the following integral:
And I was wondering how to evaluate this. By letting $u\mapsto u+1$,
By letting $u\mapsto e^u$,
Perhaps we can apply IBP, but that looks to be messy and involves limits. Likewise, the bounds and integrand don't look like they are very inviteful of a series expansion.
Preferably, I'd like to solve this integral without the use of the digamma function.