The given integral is:

$\int \frac{e^{2x} - e^x + 1}{(e^x \sin x + \cos x)(e^x \cos x - \sin x)} \, dx$

Step-by-step solution:

1. Expression Simplification: The denominator is a product of two terms: $(e^x \sin x + \cos x)(e^x \cos x - \sin x)$. To simplify, let's multiply these two terms out: $(e^x \sin x + \cos x)(e^x \cos x - \sin x)$ Using the distributive property: $e^x \sin x \cdot e^x \cos x - e^x \sin x \cdot \sin x + \cos x \cdot e^x \cos x - \cos x \cdot \sin x$ Simplify each term: $e^{2x} \sin x \cos x - e^x \sin^2 x + e^x \cos^2 x - \cos x \sin x$

   We can group the terms involving $e^x$: $e^{2x} \sin x \cos x + e^x (\cos^2 x - \sin^2 x) - \cos x \sin x$ Now, notice that $\cos^2 x - \sin^2 x = \cos(2x)$, so this becomes: $e^{2x} \sin x \cos x + e^x \cos(2x) - \cos x \sin x$

2. Analyzing the Numerator: The numerator is $e^{2x} - e^x + 1$.

3. Factorization: Unfortunately, it does not appear that a simple factorization or substitution will immediately simplify this integral further. However, this problem might suggest using a trigonometric identity, or alternatively, we could attempt a substitution involving both $e^x$ and trigonometric functions, though the complexity suggests it may require more advanced techniques.

Let me know if you'd like to explore an advanced technique such as substitution or partial fractions further.

Would you like to continue with a substitution approach or discuss a different technique for this problem?

Here are some related questions for practice:

1. How do we simplify products of trigonometric and exponential functions?
2. What strategies work best for integrating complex rational expressions?
3. How can we use trigonometric identities to simplify integrals?
4. How does substitution help in solving integrals involving exponentials?
5. When should we consider numerical methods for solving integrals?

Tip: Always consider looking for patterns or symmetries in the integrand to simplify the process.