For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I know that this is a classical problem which results in an incomplete elliptic integral of the second kind: $$\tag{1} l=a E\left(\phi \left|\sqrt{1-\frac{b^2}{a^2}}\right.\right) $$
What I would like to know is whether an algebraic approximation to this equation is known which is applicable with good accuracy in the range $1 < \frac{a}{b} < 3$? I would want to use this approximation to determine $l=f(\phi)$.
Of course, I could do a Taylor series around $a=b$ but that is fairly inaccurate unless you use a significant number of terms. What I would love to have is an equation akin to the beautiful approximation due to Ramanujan (1914) for the complete elliptic integral of the second kind: $$\tag{2} \frac{1}{4} \pi (a+b) \left(\frac{3 (a-b)^2}{(a+b)^2 \left(\sqrt{4-\frac{3 (a-b)^2}{(a+b)^2}}+10\right)}+1\right)$$