Answer
$$\phi (x,y,z) = \sin (x y z)+K
$$
Work Step by Step
Given $$\mathbf{F}=\langle y z \cos (x y z), x z \cos (x y z), x y \cos (x y z)\rangle$$
We need to find $\phi(x,y,z)$ such that
$$\frac{\partial \phi }{\partial x} = y z \cos (x y z),\ \ \ \ \frac{\partial \phi }{\partial y} = x z \cos (x y z),\ \ \ \ \frac{\partial \phi }{\partial z} =xy \cos (x y z) $$
Then
\begin{align*}
\phi(x,y,z)&= \sin (x y z)+C_1\\
\phi(x,y,z)&= \sin (x y z)+C_1\\
\phi(x,y,z)&= \sin (x y z)+C_1
\end{align*}
Hence, we can choose $$\phi(x,y,z)= \sin (x y z)+K$$
