Define the integral and apply the substitution.

Step 1: Define the integral and apply the substitution.
Let the given integral be $I$.

$$I = \int_{0}^{1} \frac{\log (1+x)}{1+x^{2}} dx$$

We are given the substitution $x = \tan \left(\frac{\pi}{4} - t\right)$.

Step 2: Calculate $dx$ and change the limits of integration.
Differentiate $x$ with respect to $t$:

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