Integral of (cot4x)^5 (csc4x)^7
ah! good one! Looks easy, but there's some algebra and trig identities involved. Rather than slog through it here, I refer you to
http://www.wolframalpha.com/input/?i=integral+%28cot4x%29^5+%28csc4x%29^7
and hit the "step by step solution" button.