2

As suggested by @cvgmt (assuming here that a>0) res=Integrate[(x^2 + (a/y^2))^(-1/2), {x, 0, 1}, {y, 0, 1}, Assumptions -> a > 0] And a quick check: NIntegrate[(x^2 + (a/y^2))^(-1/2) /. a -> 2, {x, 0, 1}, {y, 0, 1}] 0.3406417035798416` res/. a-> 2. 0.3406417035798416`


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