Link to actual problem [9184] \[ \boxed {y^{\prime }-f_{1} \left (y-\ln \left (\sin \left (x \right )\right )+\ln \left (\cos \left (x \right )+1\right )\right )=\frac {1}{\sin \left (x \right )}} \]
type detected by program
{"unknown"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {8 \cos \left (x \right )+6+2 \cos \left (2 x \right )}{\sin \left (3 x \right )+5 \sin \left (x \right )+4 \sin \left (2 x \right )}\right ] \\ \\ \end{align*}