If the solution curve y=y(x) of the differential equation (1+y^2)(1+ log_e x) d x+x d y=0, x>0 passes through the point (1,1) and y(e)=(α- tan…

If the solution curve $\displaystyle y=y(x)$ of the differential equation $\displaystyle \left(1+y^2\right)\left(1+\log _{\mathrm{e}} x\right) d x+x$ $\displaystyle d y=0, x>0$ passes through the point $\displaystyle (1,1)$ and $\displaystyle y(e)=\frac{\alpha-\tan \left(\dfrac{3}{2}\right)}{\beta+\tan \left(\dfrac{3}{2}\right)}$, then $\displaystyle \alpha+2 \beta$ is $\displaystyle \_\_\_\_$.