Evaluate.
\[
\int \frac{e^{11 x}}{\left(e^{11 x}+13\right)^{4}} d x=
\]

Step 1 ：Given the integral problem \(\int \frac{e^{11 x}}{(e^{11 x}+13)^{4}} dx\)

Step 2 ：This integral is of the form \(\int\frac{f'(x)}{(f(x))^n} dx\), which can be solved using the formula \(\int\frac{f'(x)}{(f(x))^n} dx = -\frac{1}{n-1} \cdot \frac{1}{(f(x))^{n-1}} + C\), where C is the constant of integration.

Step 3 ：In this case, we have \(f(x) = e^{11x} + 13\), \(f'(x) = 11e^{11x}\), and \(n = 4\).

Step 4 ：Substituting these values into the formula, we get the integral of the function as \(-\frac{1}{33e^{33x} + 1287e^{22x} + 16731e^{11x} + 72501}\).

Step 5 ：Thus, the final answer is \(\boxed{-\frac{1}{33e^{33x} + 1287e^{22x} + 16731e^{11x} + 72501} + C}\), where C is the constant of integration.