Let $r>0$ and $|\cdot|$ the Euclidean norm.
We consider $f(y)=\int_{\mathbb{R}^d}\cos(\langle x,y\rangle)e^{-|x|^4+r|x|^2}dx,y\in \mathbb{R}^d.$
How can we prove that $f\in L^1?$
Attempt: I tried computing the integral bit this seems impossible, also tried to write the series expansion of $\cos,$ this lead to no where, since the computation is still difficult