If ∫_0^1 4 cot^-1(1-2 x+4 x^2) d x= a tan^-1(2)- b log_e (5), where a, b ∈ N, then (2 a + b ) is equal to ____.

If $\displaystyle \int_0^1 4 \cot ^{-1}\left(1-2 x+4 x^2\right) \mathrm{d} x=\mathrm{a} \tan ^{-1}(2)-\mathrm{b} \log _{\mathrm{e}}(5)$, where $\displaystyle \mathrm{a}, \mathrm{b} \in \mathbf{N}$, then $\displaystyle (2 \mathrm{a}+\mathrm{b})$ is equal to $\displaystyle \_\_\_\_$.