Sine and Cosine

Using a couple of fairly standard integrals, $\begin{aligned} S(u) & = \; \int_0^\infty \sin(x^u)\,dx \; = \; \frac{1}{u} \int_0^\infty y^{\frac{1}{u} - 1}\sin y\,dy \; = \; \frac{\Gamma(\tfrac{1}{u})}{u} \sin \tfrac{\pi}{2u} \\ C(u) & = \; \int_0^\infty \cos(x^u)\,dx \; = \; \frac{1}{u} \int_0^\infty y^{\frac{1}{u} - 1}\cos y\,dy \; = \; \frac{\Gamma(\tfrac{1}{u})}{u} \cos \tfrac{\pi}{2u} \end{aligned}$ valid for $u > 1$ . Thus $\frac{S(u)}{C(u)} = \tan\tfrac{\pi}{2u}$ has derivative $-\frac{\pi}{2u^2} \sec^2\tfrac{\pi}{2u} \; = \; -\frac{\pi}{u^2(1 + \cos \tfrac{\pi}{u})}$ for $u > 1$ . When $u = \tfrac13\pi$ this derivative equals $-\frac{9}{\pi(1 + \cos3)}$ making the answer $\boxed{27}$ .