I am at the final part of a problem where I have derived
$t+c=\frac{1}{\sqrt2}\log\left(\frac{x}{2+\sqrt{4-2x^2}}\right)$
where $c$ is a constant, and now I need to express it in explicit form $x(t)$. Of course I can get rid of the logarithm and get an exponent on the left hand side, but beyond that point I have tried lots of different things (including using surds for the fraction which doesn't seem to help) but can't seem to get anywhere.
$\def\inv#1{\frac{1}{#1}}$ $$t+c=\inv{\sqrt{2}}\log\left(\frac{x}{2+\sqrt{4-2x^2}}\right)$$ $$e^{\sqrt 2(t+c)}=\frac{x}{2+\sqrt{4-2x^2}}$$ As a comment suggested, we do $K=e^{\sqrt 2(t+c)}$ $$K(2+\sqrt{4-2x^2})=x$$ $$\frac{x-2K}{K}=\sqrt{4-2x^2}$$ Now we square both sides $$\frac{x^2-4Kx+4K^2}{K^2}=4-2x^2$$