# How to integrate $\int \frac{1}{\sin^4(x)\cos^4(x)}\,\mathrm dx$?

How can I integrate $$\int \frac{1}{\sin^4(x)\cos^4(x)}\,\mathrm dx.$$

So I know that for this one we have to use a trigonometric identity or a substitution. Integration by parts is probably not going to help. Can someone please point out what should I do to evaluate this integral?

Thanks!

• The Maple command $$Student[Calculus1]:-IntTutor(1/(sin(x)^4*cos(x)^4), x)$$ finds it step by step with explanations. See here for info. – user64494 Sep 27 '13 at 17:23

For the last let, $\cot(2x) = u$
Hint: Substitute $t=\tan x$.
Then $\frac{dx}{\sin^4 x \cos^4 x} = \frac{(1+t^2)^3}{t^4}dt$.
All integrals of rational functions of $sin(x)$ and $cos(x)$ can be solved by rationally parameterizing the unit circle ( see http://mathnow.wordpress.com/2009/11/06/a-rational-parameterization-of-the-unit-circle/ for example). This will convert the integral into a rational integral, which are all solvable by partial fraction decomposition. Try this out for the infamous $\int sec^3(x)dx$ for instance.