If C is the curve given by $\mathbf{r} \left( t \right) = \left( 1 + 2 \sin t \right) \mathbf{i} + \left( 1 + 3 \sin^{2} t \right) \mathbf{j} + \left( 1 + 1 \sin^{3} t \right) \mathbf{k}$, $0 \leq t \leq \frac{\pi}{2}$ and F is the radial vector field $\mathbf{F} \left( x, y, z \right) = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}$, compute the work done by F on a particle moving along C.