Step 1 :Evaluate \(\int_{4}^{4} \frac{\cos (x) \ln \left(1+x^{2}\right)+x^{2.8}}{1+e^{1.65 x-63.1}} d x\)
Step 2 :The integral is from 4 to 4, which means the lower limit and the upper limit of the integral are the same.
Step 3 :The value of a definite integral with the same lower and upper limit is always zero, regardless of the function being integrated.
Step 4 :Therefore, the answer to this integral is 0.
Step 5 :Final Answer: \(\boxed{0}\)