How to Integrate Even Powers of Secants with Tangents
You can integrate even powers of secants with tangents. If you wanted to integrate tan^{m} x sec^{n} x when n is even — for example, tan^{8} x sec^{6} x — you would follow these steps:

Peel off a sec^{2} x and place it next to the dx:

Use the trig identity 1 + tan^{2 }x = sec^{2} x to express the remaining secant factors in terms of tangents:

Use the variable substitution u = tan x and du = sec^{2}x dx:
At this point, the integral is a polynomial, and you can evaluate it.