Yesterday I was interested in the so-called Sophie Germain's Identity, see this section of Wikipedia. I was interested in it about composite numbers. After I was thinking about different consequences of this identity. Playing I prosoposed calculate the easy $$\int_0^1\int_0^1\frac{x^4+4y^4}{x^2-2xy+2y^2}dxdy,$$ and after I was interested in integrals $$\int_0^1\int_0^1 f\left(\frac{x^4+4y^4}{x^2-2xy+2y^2}\right)dxdy,$$ for different functions than $f(z)=z$. For example $f(z)=\log z$, that can be solved in closed-form.
A different example was $f(z)=\cos z$, using a CAS I believe that it can not solve in closed-form.
Question. Calculate in terms of series, special functions (integral functions) or well-known constants $$\int_0^1\int_0^1\cos\left((x+y)^2+y^2\right)dxdy.\tag{1}$$ Many thanks.
I am thinking how get an approximation of $(1)$ using calculus, thus avoid provide me hints to this purpose.