Let's delve into how to derive the integral:
\[
\int_{0}^{1} \frac{\arctan(x)}{1+x} \, dx
\]

This integral is not straightforward, and while it can be evaluated using advanced techniques like series expansions or recognizing it as a special function integral, I'll outline a method that gives us the result based on known integrals.

### Method: Recognizing a Known Integral

1. **Rewrite the Integral**:
   We aim to evaluate:
   \[
   I = \int_{0}^{1} \frac{\arctan(x)}{1+x} \, dx
   \]
   This integral doesn't simplify easily using elementary methods like substitution or integration by parts, but it's known in mathematical tables that:
   \[
   \int_{0}^{1} \frac{\arctan(x)}{1+x} \, dx = \frac{\pi}{8} \ln(2)
   \]

2. **Understanding the Result**:
   The result \( \frac{\pi}{8} \ln(2) \) comes from deeper analysis involving the properties of the arctangent function and the logarithmic function.

   To briefly outline why this is true:
   - The arctangent function \( \arctan(x)

Learn how to evaluate the integral \( \int_{0}^{1} \frac{\arctan(x)}{1+x} \, dx \). This advanced mathematical problem involves integral calculus and the properties of arctangent and logarithmic functions.

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