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I am trying to evaluate the following integral

$$\int^\infty_{-\infty}\frac{\log(t^2-a^2)}{(b^2+t^2)^2}dt$$ with $a,b\in \mathbb{R}^{+}$

I know that similar integrals do appear in other posts but somehow I am stuck and can not see how to evaluate this. If I try to use contour integration methods, should I make a branch cut between $-a$ and $a$ and close the contour in the upper half plane ? Or is there another method without using residue calculus ? Any help would be appreciated.

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    What happens when $|t| < a$ ? Have you chosen a branch or have you excluded that with a $\log|t^2-a^2|$ instead? – Ninad Munshi Jul 18 '21 at 01:07
  • Why you are closing this question ? I was just curious about improper integrals involving logarithm. How this question do violate any rules ? –  Jul 18 '21 at 17:58

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