I have developed a differential equation for the variation of a star's semi-major axis with respect to its eccentricity. It is as follows: $$\frac{dy}{dx}=\frac{12}{19}\frac{y\left(1+\left(\frac{73}{24}x^2\right)+\left(\frac{37}{26}x^4\right)\right)}{x\left(1+\left(\frac{121}{304}x^2\right)\right)}$$
Where $y$ is the semi-major axis and $x$ is the eccentricity. The 3-D plots of this equation can be found here
And this is the solution to the above DE: here
My question is this: There is a well defined symmetry for the above equation from the plot. Is it possible to express this in terms of other special function (which have different symmetries). The decay time of stars can be found by solving the following integral: $$T(a_{0},e_{0})=\frac{12(c_{0}^4)}{19\gamma}\int_{0}^{e_0}{\frac{e^{29/19}[1+(121/304)e^2]^{1181/2299}}{(1-e^2)^{3/2}}}de\tag1$$ Where $$\gamma=\frac{64G^3}{5c^5}m_{1}m_{2}(m_{1}+m_{2})$$ For $e_{0}$ close to $1$ the equation becomes: $$T(a_{0},e_{0})\approx\frac{768}{425}T_{f}a_{0}(1-e_{0}^2)^{7/2}\tag2$$ Where $$T_{f}=\frac{a_{0}^4}{4\gamma}$$
I used Appell's hypergeometric functions to solve integral (1), but is there any way in which I can express the solutions in terms of special functions with simpler symmetries, which would make analysis easier.