The arc length for a full period is \frac{4\sqrt{2}\,\pi^{3/2}}{\Gamma(1/4)^2} + \frac{\Gamma(1/4)^2}{\sqrt{2\pi}}=7.640395578\ldots
The arc length of the sine curve from 0 to x is the above number divided by 2\pi times x, plus a correction that varies periodically in x with period \pi.