Trig substitution is a powerful tool, but it is just that: a tool. And as always, it is important to use the right tool for the right job. I tried to continue your work with the trig substitution, and it was unworkable as a secant-tangent type trig integral. I then converted it to a sine-cosine integral, and it was no better. Since those are the only 2 options I can see for continuing, I conclude that using trig substitution for this integral is as inappropriate as using a hacksaw to tighten a screw.
But if you use integration by parts, the answer comes out almost immediately.