To solve the given integral, we need to interpret and analyze the mathematical expression from the provided image.

The integral is:

$I = \int_{-\infty}^{\infty} \frac{1}{x + a \sqrt{\text{atan}(x)} + x^2 + x^2 \cos(x)} \, dx$

To solve this integral, follow these steps:

1. Simplify the Expression: Check if there are any symmetries or substitutions that can simplify the integral. For example, look for odd/even function properties, or potential substitutions like $u = x^2$.

2. Evaluate the Behavior at Infinity: Determine if the integral converges by analyzing the behavior of the integrand as $x \to \infty$ and $x \to -\infty$.

3. Apply Integration Techniques: Use appropriate integration techniques such as integration by parts, partial fractions, or contour integration if it’s a complex function.

Step-by-Step Analysis

1. Behavior at Infinity:

   For $x \to \infty$ or $x \to -\infty$, the dominant term in the denominator is $x^2$. Therefore, the integrand behaves like $\frac{1}{x^2}$ for large $|x|$, which suggests the integral converges since $\int \frac{1}{x^2} dx$ converges.

2. Symmetry and Simplifications:

   Check if the function inside the integral has any symmetry properties. However, in this case, the presence of both $x$ and $x^2$ terms complicates this approach.

3. Integration Technique:

   This specific integral does not lend itself easily to standard techniques like substitution or partial fractions due to the complex expression in the denominator. Numerical integration methods or advanced techniques might be necessary.

Given the complexity of the integrand, it's likely this integral does not have a simple closed-form solution and might require numerical methods for evaluation.