1. The problem is to find the primitive (antiderivative) of the function $$f(x) = \frac{\cos^2(2x)}{5 - x \sin(x)}$$. 2. The primitive or antiderivative of a function $f(x)$ is a function $F(x)$ such that $$F'(x) = f(x).$$ 3. This function is a quotient of two functions: numerator $\cos^2(2x)$ and denominator $5 - x \sin(x)$. 4. There is no straightforward elementary antiderivative for this function because it involves a composite trigonometric function squared in the numerator and a product inside the denominator. 5. Typically, such integrals require advanced techniques or numerical methods. 6. We can attempt substitution or integration by parts, but no simple substitution simplifies the integral directly. 7. Therefore, the integral $$\int \frac{\cos^2(2x)}{5 - x \sin(x)} \, dx$$ does not have a simple closed-form expression in elementary functions. 8. For practical purposes, numerical integration methods or computer algebra systems are recommended to evaluate this integral over specific intervals. Final answer: The primitive of $$\frac{\cos^2(2x)}{5 - x \sin(x)}$$ cannot be expressed in elementary functions.