What's wrong with my approach to solving this integral via trigonometric substitution?

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(I'm sorry, I've spent the last hour trying to figure out Latex, but nothing's working. I'm just going to paste my work as I've written it in pencil.)

I'm trying to solve this equation: Integral from problem set

And while certain online solvers give me a certain trigonometric solution, I've tried a different one that I think should work. My own work's below: solution, part 1 solution, part 2

That said, I seem to not be getting the right answer. This is what integral-calculator.com gives me: calculator's solution

And having graphed both in Desmos, both equations look quite different when graphed for a = b = 1: desmos graph

Could someone help me figure out what I'm missing here? Thank you!