# Indefinite Integration of a trig function

How do you integrate $$1/(\sin2x(\tan^5x+\cot^5x))$$ with respect to $$x$$?
I tried writing tan and cot in terms of sin and cos but when I take $$\mathrm{LCM}$$ I get powers of $$10$$ for $$\sin$$ and $$\cos$$ in the denominator. I can't think of how I would simplify this without the use of binomial expansion but that might make things more complicated.

Is there another method I could use to approach this problem?

$$\displaystyle \int \frac{dx}{\sin2x\cdot(tan^5x+cot^5x)} = \frac12 \int \frac{\cot x \,dx}{\cos^2x\cdot(\tan^5x+\cot^5x)}$$
$$\displaystyle \tan x = t, \frac{dx}{\cos^2x}=dt$$