Define the curve $\phi$ by $\phi(t):=(f(t)\cos(t),f(t)\sin(t))$, where $f$ be a strictly increasing infinitly many differentiable function . Find an explicit formula for the length of $\phi$ between $\phi(a)$ and $\phi(b)$.
By using the definition of the length of a curve, you could reduce the problem to compute the integral $$\int_a^b \sqrt{f'(x)^2 + f(x)^2} dx $$ But now tricks by change of variables and integration by parts seems not to be helpful. Probably this integral cannot be computed explicitly in terms of antiderivatives.
Indeed $$\int_a^b \sqrt{f'(x)^2 + f(x)^2} dx $$ was meant as a solution.