I've tried that already, but it's not conclusive. Whatever I try with integration by parts, I only manage to express the above integral in terms of a similar integral, but with the logarithm squared, or the denominator squared. So it's making things only more difficult instead of simpler.
By the way, I happened to find the exact value of this integral by playing around with WolframAlpha and doing some extra algebra. It reads as follows:
and is in agreement with the numerical computation.
But that doesn't tell me the steps to get to this result…