I am wondering if I'm overlooking a simple way of solving this, without expanding the squared term.
If I have to first expand, that's fine. I was just thinking I can do it without.
Do it the easiest way that you can (which in this case is probably by expanding), when you are no longer studying this stuff you may as well just use technology to do these things; there is little intrinsic merit in remembering a bag of tricks for systematic integration (because you will never be as good at it as a machine implementing the Risch algorithm).
The only area in which you may be better off on your own is knowing which integrals can be expressed in terms of higher transcendental function after some rearrangement.