The integral $$\displaystyle\int_0^1 \cot^{-1}(1 + x + x^2)\,dx$$ is equal to :

A

$$2\tan^{-1}2 + \frac{1}{2}\log_e\frac{5}{4} + \frac{\pi}{2}$$

B

$$2\tan^{-1}2 + \frac{1}{2}\log_e\frac{5}{4} - \frac{\pi}{2}$$

C

$$2\tan^{-1}2 - \frac{1}{2}\log_e\frac{5}{4} + \frac{\pi}{2}$$

D

$$2\tan^{-1}2 - \frac{1}{2}\log_e\frac{5}{4} - \frac{\pi}{2}$$