I found this integral in a handbook of integrals: $$\int \frac{1}{\sqrt[4]{1+x^4}} dx$$
I already have evaluated this integral by trigonometric substitutions and my answer agrees with that of the book. Interestingly, the integral is expressible by using the natural logarithm and arc tangent functions. This motivated the idea that maybe $$\frac{1}{\sqrt{1+x^4}} $$
can be parametrized and turns into a rational function. Just like what we do with sine and cosine functions when we use Euler/Wierstrauss substitutions.
So, is it possible to find $f(t)$, $g(t)$ and $h(t)$ in the field of rational functions such that $x=f(t)$ and $y=g(t)$ and $dx=h(t)dt$?