Evaluate the following definite integral. Express your answer in terms of logarithms.
\[
\int_{\ln 5}^{\ln 11} \frac{\cosh x}{4-\sinh ^{2} x} d x
\]
\[
\int_{\ln 5}^{\ln 11} \frac{\cosh x}{4-\sinh ^{2} x} d x=
\]
(Type an exact answer. Use integers or fractions for any numbers in the expression.)

Step 1 ：Given the integral \(\int_{\ln 5}^{\ln 11} \frac{\cosh x}{4-\sinh ^{2} x} dx\)

Step 2 ：We can simplify it by using the identity \(\cosh^2 x - \sinh^2 x = 1\). This allows us to rewrite the denominator as \(4 - \sinh^2 x = \cosh^2 x\)

Step 3 ：The integral then simplifies to \(\int_{\ln 5}^{\ln 11} \frac{\cosh x}{\cosh^2 x} dx = \int_{\ln 5}^{\ln 11} \frac{1}{\cosh x} dx\)

Step 4 ：The integral of \(1/\cosh x\) is \(\tanh x\), so the final answer should be \(\tanh(\ln 11) - \tanh(\ln 5)\)

Step 5 ：Calculating the values we get 0.060529634300126034

Step 6 ：So, the final answer is \(\boxed{0.060529634300126034}\)