My table of Integrals has the final situation ##int frac dx x^two+a^2 ## as a 'regular' integral too, not merely the case the place ##a=one##. But I do think this staying a Calculus research dilemma it is anticipated in the OP to determine how to make it happen. Hints got https://gabletoppak.com/why-i-switched-to-water-in-a-sustainable-box/