I am trying to find a transformation that will convert the region bounded by y=0 and y=(1+x^6)^1/6 to a region that is a rectangle.
If you're trying to transform the region bound by
$\displaystyle y = 0, y = (1+x^6)^{1/6}, x = -1 \; \text{and}\; x = 1$
to say a rectangle, try the transformation
$\displaystyle u = x,\;\; v = \dfrac{y}{(1+x^6)^{1/6}}$.