To solve the integral:

$I = \int \frac{e^x (1 + \cos x)}{1 + \sin x} \, dx$

We can attempt to simplify the expression and use substitution. The first step will be to break the integral into two parts. Let's rewrite it as:

$I = \int \frac{e^x}{1 + \sin x} \, dx + \int \frac{e^x \cos x}{1 + \sin x} \, dx$

Step 1: Simplify the first integral

The first integral, $\int \frac{e^x}{1 + \sin x} \, dx$, is challenging, and there's no immediate standard technique that simplifies it easily. However, if we focus on the second part, the form of $\frac{\cos x}{1 + \sin x}$ might suggest a substitution that simplifies the expression.

Step 2: Handle the second part using substitution

Consider the second term:

$\int \frac{e^x \cos x}{1 + \sin x} \, dx$

Let's make the substitution: $u = 1 + \sin x \quad \Rightarrow \quad du = \cos x \, dx$

This transforms the integral into: $\int \frac{e^x}{u} \, du$

Now the integral becomes much simpler: $\int \frac{e^x}{u} \, du = e^x \ln |u| + C = e^x \ln |1 + \sin x| + C$

Step 3: Combine the results

Since the first part of the integral doesn't simplify easily, we leave it in its original form:

$I = \int \frac{e^x}{1 + \sin x} \, dx + e^x \ln |1 + \sin x| + C$

Thus, the solution can be expressed as the sum of two integrals:

$I = \int \frac{e^x}{1 + \sin x} \, dx + e^x \ln |1 + \sin x| + C$

You might need numerical or approximation methods to evaluate the first integral.

Would you like more details on solving this, or any clarifications on any part? Here's a list of related topics you might find useful:

1. Substitution methods for integrals.
2. Techniques for handling integrals involving trigonometric functions.
3. Solving integrals involving exponential functions.
4. Series expansions for approximating integrals.
5. Use of integration tables and computer algebra systems.

Tip: When faced with difficult integrals, try to break them into smaller parts, and look for substitutions that might simplify individual terms.