Problem: Solve integral:
Approaches I've tried:
Solving by parts led me nowhere but to more confusion (maybe I've taken the wrong parts?):
Trying to replace $1+x$ or $1-x$ with another variable led me to the same integral but with different variable names.
I've found this interesting integral on one of the math forums, but I just can't wrap my head around the solution to it. I will really appreciate any help.