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Relevant wiki: $u$ -Substitution

$\LARGE{I = \int^{\sqrt{e}^{\pi}}_{\sqrt[6]{e}^{\pi}} {\dfrac{\sin{(\log{(\sin{(\log{(x)})})})}\cos{(\log{(x)})}} {x \sin{(\log{(x)})}}} dx}$

We can use a substitution to simplify the integrand. We see this instantly if we set $u = \log{(\sin{(\log{(x)})})}\\$ . This gives that $du = \dfrac{\cos{(\log{(x)})}}{x\sin{(\log{(x)})}}dx.$

This leaves us with $\large{\displaystyle{I = \int^{0}_{-\log{2}} {\sin{(u)}du}} = -\cos{(u)}}$ . Substituting again to have $\large{u}$ in terms of $\large{x}$ , we see that $\displaystyle{\large{I = -\cos{(\log{(\sin{(\log{(x)})})})}}|^{\sqrt{e^{\pi}}}_{\sqrt[6]{e}^{\pi}}}$ So $I = -1+\cos{(\log{(2)})}\\$ Thus, $A=-1$ and $B=\log{(2)}\\$ . $\therefore I+A+B = -1+\cos{(\log{(2)})}-1+\log{(2)} \approx \boxed{-0.54}$ $(2$ decimal places $)$