How to integrate $\int \frac{\sin^2 x \cos^2 x}{(\sin^3 x+\cos^3 x)^2} \, dx$?

Question

How to find

$$\int \frac{\sin^2 x \cos^2 x}{(\sin^3 x+\cos^3 x)^2} \, dx.$$

I have tried but not been able to find a solution.

Thank you.

Answer 1

Hint:

$$\dfrac{\sin^2x\cos^2x}{(\sin^3x+\cos^3x)^2}=\dfrac{\sin^2x\cos^2x}{\cos^6x(\tan^3x+1)^2}=\dfrac{\tan^2x\sec^2x}{(\tan^3x+1)^2}$$

Set $\tan^3x+1=u$