Extrema(s) of Trigonometric Functions!

First of all lets use Cauchy-Swartz inequality: $(\sin A + \sin B + \sin C)(\frac{1}{A}+\frac{1}{B}+\frac{1}{C})\ge(\sqrt{\frac{\sin A}{A}}+\sqrt{\frac{\sin B}{B}}+\sqrt{\frac{\sin C}{C}})^2$

Note that, we can now analyse the funcion that apears on the right side of the inequality using Jensen inequality! Define: $f(x)=\sqrt{\frac{\sin x}{x}}$

Hence, from the cited inequality one can write (also noting that the second derived of this funcion is negative for the angles inside the triangle, which are the ones that matters in that case):

$\frac{\sqrt{\frac{\sin A}{A}}+\sqrt{\frac{\sin B}{B}}+\sqrt{\frac{\sin C}{C}}}{3}\ge\sqrt{\frac{\sin (\frac{A+B+C}{3})}{\frac{A+B+C}{3}}}=\sqrt{\frac{3\sqrt{3}}{2\pi}}$

After all, joining, the two inequalities:

$(\sin A + \sin B + \sin C)(\frac{1}{A}+\frac{1}{B}+\frac{1}{C})\ge\frac{\sqrt{3^7}}{2\pi}$

Then. $A=3, B=7,C=2,D=1$ which gives the sum of $\boxed{13}$