`I= int cosh^(2) (8x + 5) dx`.

Answer

To solve the integral \( I = \int \cosh^2(8x + 5) \, dx \), we can use the identity for hyperbolic functions and the method of integration.

### Step-by-Step Solution:

1. **Use the Hyperbolic Identity**:
We know that:
\[ \cosh^2(x) = \frac{1 + \cosh(2x)}{2} \]
Therefore, we can rewrite the integral:
\[ I = \int \cosh^2(8x + 5) \, dx = \int \frac{1 + \cosh(2(8x + 5))}{2} \, dx \]

2. **Simplify the Integral**:
Expanding the integral gives:
\[ I = \frac{1}{2} \int (1 + \cosh(16x + 10)) \, dx \]
This can be separated into two integrals:
\[ I = \frac{1}{2} \left( \int 1 \, dx + \int \cosh(16x + 10) \, dx \right) \]

3. **Integrate Each Part**:
- The first integral is straightforward:
\[ \int 1 \, dx = x \]
- For the second integral, we can use the fact that:
\[ \int \cosh(kx) \, dx = \frac{1}{k} \sinh(kx) \]
Here, \( k = 16 \):
\[ \int \cosh(16x + 10) \, dx = \frac{1}{16} \sinh(16x + 10) \]

4. **Combine the Results**:
Putting it all together, we have:
\[ I = \frac{1}{2} \left( x + \frac{1}{16} \sinh(16x + 10) \right) + C \]
Simplifying this gives:
\[ I = \frac{x}{2} + \frac{1}{32} \sinh(16x + 10) + C \]

### Final Answer:
\[ I = \frac{x}{2} + \frac{1}{32} \sinh(16x + 10) + C \]