I have this function : \begin{equation}\tag{1} a(\eta) = \sqrt{\sin{2 \eta}}, \end{equation} and this time variable : \begin{equation}\tag{2} t(\eta) = \int_0^\eta a(\eta') \, d\eta'. \end{equation} This integral is very difficult to express in an analytical way, because of the square-root.
I would like to know the function $a$ parametrized as a power series of $t$. How can I achieve this ?
When $\eta$ is very small, I could get \begin{equation}\tag{3} a(t) \approx (3 \, t)^{\frac{1}{3}}. \end{equation} When $\eta$ isn't so small, I'm expecting something like \begin{equation}\tag{4} a(t) = (3 \, t)^{\frac{1}{3}} f(t), \end{equation} where $f(t)$ could be (?) Taylor expanded. How to find this function ?