We know that the integral of \(\frac{1}{1+x^2}\) is \(\tan^{-1}(x)\). To solve the given integral, we use substitution:
\[
\frac{1}{1 + x^2} \text{is the standard form, whose integral is} \tan^{-1}(x).
\]
Thus, when we perform the integration, we get:
\[
\int \frac{1}{1 + x^2} \, dx = \tan^{-1}(x) + C.
\]
For the given options, option (3) simplifies to this result. Hence, the final solution is
\[
\frac{2}{\sqrt{3}} \tan^{-1} \left(2x + 1 + \sqrt{3}\right).
\]
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