$\text{ If : } \alpha=\displaystyle \int_{0}^1 (e^{9x+3\tan^{-1}x}) \left( \frac{12+9x^2}{1+x^2}\right)dx$
where $$\tan^{-1}x$$ takes only principal values, then find the value of $\left( \log_e |1+\alpha|-\frac{3\pi}{4} \right)$