My table of Integrals has the final scenario ##int frac dx x^2+a^2 ## to be a 'standard' integral far too, not only the case the place ##a=one##. But I believe this currently being a Calculus research challenge it is predicted from the OP to figure out how to do it. https://gabletoppak.com/water-in-a-sustainable-box-the-future-of-eco-friendly-hydration/