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Calculate the integral $$\int\dfrac{\cos(2x)}{\sin^4(x)+\cos^4(x)}dx$$

My try: $$I=\int\dfrac{\cos(2x)}{1-2\sin^2(x)\cos^2(x)}dx=\int\dfrac{\cos(2x)}{1-\frac{\sin^2(2x)}{2}}dx,$$ using that the denominator is $$(\sin^2x+\cos^2x)^2-2\sin^2x\cos^2x=1-2\sin^2x\cos^2x=1-\dfrac{\sin^2(2x)}{2}$$I don't know what to do from here. Thanks!

SAQ
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  • cosine function on the numerator is the derivative of sine function in the denominator. So you can use a new variable u=sin(2x) – Mehdi Jan 31 '24 at 21:59

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