Arc length up to a vertical tangent

Let $(x(t),y(t))$, $1\le t\le\pi$, be the curve defined by

$$x(t)=\int_1^t\frac{\cos z}{z^2}\,dz\quad\text{and}\quad y(t)=\int_1^t\frac{\sin z}{z^2}\,dz.$$

Let $L$ be the length of the arc of this curve from the origin to the point $P$ on the curve at which the tangent is perpendicular to the $x$-axis. Then $L$ equals