\( \mathrm{Ii} \int_{0}^{2} \frac{\ln (1+2 x)}{1+x^{2}}=\left(\tan ...

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\( \mathrm{Ii} \int_{0}^{2} \frac{\ln (1+2 x)}{1+x^{2}}=\left(\tan ^{-1} a\right)(\ln \sqrt{b}) \) where \( a, b \in N \), find the value of \( \left(a^{2}+b^{2}\right) \)
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