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This question already has an answer here:

I am trying to evaluate $$\int_0^\infty \frac{\arctan(x)\log(x^2+1)}{x(x^2+1)}$$ I have tried differentiating under the integral sign, with different parameters, as well as contour integration, but I have not succeeded with neither of them. Wolfram Alpha gives a numerical value $\approx 0.754694$, but I suspect that there is a more exact answer.

I would be very glad to get some hints on how to solve it.

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marked as duplicate by Namaste integration Oct 19 '18 at 20:46

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    $\begingroup$ This is solved here. See the middle of the accepted answer there "For the integral let $F(a) =\ldots$" $\endgroup$ – Lee David Chung Lin Oct 19 '18 at 19:04
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    $\begingroup$ Exact value: $\frac{1}{2} \ln ^2(2) \pi$ $\endgroup$ – Mariusz Iwaniuk Oct 19 '18 at 19:28