It seems the integral is too complex to compute symbolically within a reasonable time frame using the current approach. However, we can proceed with a numerical approximation.

Let's use numerical integration to approximate the value of the given integral:

$\int_{0}^{1} \frac{\log(1 + x^2 + \sqrt{3})}{1 + x} \, dx$

I'll calculate this numerically.The numerical approximation of the integral

$\int_{0}^{1} \frac{\log(1 + x^2 + \sqrt{3})}{1 + x} \, dx$

is approximately $0.7610$.

Would you like any further details or have any questions about this solution?

Here are some additional questions you might want to explore:

1. How is numerical integration performed in general?
2. What are the different methods for numerical integration?
3. How can one verify the accuracy of a numerical integration result?
4. What are some real-world applications of definite integrals?
5. How do symbolic and numerical integration methods compare?

Tip: When dealing with complex integrals, numerical methods can often provide a quick and accurate approximation when symbolic methods are infeasible.