I would like to compute the length of the arc of $f(x) = a x + b \sin(x)$ (let's say from $0$ to $\alpha < 2\pi$.).
The traditional method of computing it as the integral $\int_0^\alpha \sqrt{1+[f'(x)]^2} \mathrm{d} x$ does not seem to be helpful as the integral is horrible.
So, I thought there must be a trick to express the arc in another form using which the integral has a known form (or ideally a closed form but I doubt.)
Thanks in advance,