My desk of Integrals has the final scenario ##int frac dx x^2+a^two ## as a 'normal' integral far too, not just the case exactly where ##a=1##. But I think this remaining a Calculus research issue it is anticipated from the OP to determine how to get it done. Hints got https://gabletoppak.com/a-revolutionary-step-towards-sustainable-water-bottle-packaging/