Indefinite integral $ \int\frac{ \cos 2x}{\sin^4 x+\cos^4 x}\,dx$

Question

Can someone help me with this integral:

$$\int\frac{\cos 2x}{\sin^4 x+\cos^4 x}\,dx$$

I don't understand why $\sin^4 x+\cos^4 x$ is not equal to $1$. Unfortunately, Wolfram doesn't show step by step solution. Thank you for your help.

Answer 1

HINT:

$$\sin^4x+\cos^4x=(\sin^2x+\cos^2x)^2-2\sin^2x\cos^2x=1-\dfrac{(\sin2x)^2}2$$

Let $\sin2x=u$

Answer 2

The reason why is that $$\left(\sin^2(x)+\cos^2(x)\right)^2=\sin^4(x)+\cos^4(x)+2\sin^2(x)\cos^2(x)=1^2=1$$ So, $$\sin^4(x)+\cos^4(x)=1-2\sin^2(x)\cos^2(x)=1-\frac 12 \sin^2(2x)$$ So $$I=\int \frac{\cos (2x)}{\sin^4 (x)+\cos^4 (x)}dx=\int \frac{\cos (2x)}{1-\frac 12 \sin^2(2x)}dx$$

Now, there is a beautiful thing here. I let you the pleasure of finding it and finishing the problem.

I am sure that you can take it from here.

Answer 3

Here are two substitutions that fit this integral well:

1) Since $$ \frac{\cos 2x}{\cos^4x+\sin^4x}=\frac{1}{\cos^2x}\frac{1-\tan^2x}{1+\tan^4x} $$ we let $u=\tan x$. This will transform the integral into $$ \int \frac{1-u^2}{1+u^4}\,du. $$ 2) Next, in the spirit of @juantheron, let $v=u+1/u$. This will transform the integral into $$ \int \frac{1}{2-v^2}\,dv=\frac{1}{\sqrt{2}}\text{artanh}\frac{v}{\sqrt{2}}+C. $$ I leave it to you to do the substitutions back to the $x$ variable.