Parametric, Polar and Vector-Valued Functions · Superchallenge problem

For $t\ge1$ a curve is defined by $x=\displaystyle\int_1^t\frac{\cos u}{u}\,du$, $y=\displaystyle\int_1^t\frac{\sin u}{u}\,du$. Find the length of the curve between the first two points (with $t>1$) at which its tangent line is vertical.