Evaluate the integral. \[ \int_{0}^{\pi / 4} \frac{8+5 \cos ^{2} \theta}{\cos ^{2} \theta} d \theta \]

Step 1 ：Rewrite the integral by dividing each term in the numerator by \(\cos^2 \theta\), which gives us two separate integrals to solve: \(\int_{0}^{\pi / 4} 8 \sec^2 \theta d \theta\) and \(\int_{0}^{\pi / 4} 5 d \theta\).

Step 2 ：The integral of \(\sec^2 \theta\) is \(\tan \theta\) and the integral of a constant is the constant times the variable.

Step 3 ：Evaluate the first integral: \(\int_{0}^{\pi / 4} 8 \sec^2 \theta d \theta = 8 \tan \theta \Big|_0^{\pi/4} = 8\).

Step 4 ：Evaluate the second integral: \(\int_{0}^{\pi / 4} 5 d \theta = 5 \theta \Big|_0^{\pi/4} = \frac{5\pi}{4}\).

Step 5 ：Add the results of the two integrals to get the final answer: \(\frac{5\pi}{4} + 8\).

Step 6 ：\(\boxed{\frac{5\pi}{4} + 8}\) is the final answer.