If anyone searching for a solution to this problem comes across this page, here's how its done.
INT sec(^2)x dx
You have to use a substitution. Let u = tan x. We know tan x = sin x / cos x and using the quotient rule, du/dx = 1/sec^2 x. So rearranging, du = sec(^2)x * dx
Substitute and you have INT du
which is just u + c.
Substitute u = tan x and you have tan x + c