With the substitution and a little computation [...] the integral...
(1)
... becomes the integral of a rational function in z...
Kind regards
if that is the case, then you should have worked your original substitution down to this point ...
let's ignore the constant (16) for a bit and focus effort on the integral of secant cubed. using integration by parts ...
...
...
put the constant (16) back ...
I'll leave you the task of back substituting.