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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.

Learn how to numerically approximate the integral \(\int_{0}^{1} \frac{\log(1 + x^2 + \sqrt{3})}{1 + x} \, dx\). Discover the method of numerical integration and its application in solving complex integrals. Suitable for undergraduate students and those interested in numerical methods.

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