$\int _0^1\int _0^{\left(1-x^n\right)^{1/n}}\left(-x^n-y^n+1\right)^{1/n}dydx$ Let $n>0$. How does one integrate
$$\int _0^1\int _0^{\left(1-x^n\right)^{1/n}}\left(-x^n-y^n+1\right)^{1/n}dydx$$
?
This integral represents the volume enclosed by $$x>0,y>0,z>0,x^n+y^n+z^n<1$$. 
By substitution of $y=t\left(1-x^n\right)^{1/n}$ we get
$$\int _0^1\left(1-x^n\right)^{1/n}\int _0^1\left(-x^n-\left(t\left(1-x^n\right)^{1/n}\right)^n+1\right)^{1/n}dtdx$$
$$=\int _0^1\left(1-x^n\right)^{1/n}\int _0^1\left((1-x^n)(1-t^n)\right)^{1/n}dtdx$$
$$=\int _0^1\left(1-x^n\right)^{2/n}dx\int _0^1\left(1-t^n\right)^{1/n}dt$$
Some help from Mathematica
$$=\frac{\Gamma \left(1+\frac{1}{n}\right)^2}{\Gamma \left(\frac{n+2}{n}\right)}\int _0^1\left(1-x^n\right)^{2/n}dx$$
I am not sure how to do the rest.
Judging by the pattern, it looks like the final answer is
$$\frac{\Gamma \left(1+\frac{1}{n}\right)^2}{\Gamma \left(\frac{n+2}{n}\right)}
\frac{2 \Gamma \left(1+\frac{1}{n}\right) \Gamma \left(\frac{2}{n}\right)}{3 \Gamma \left(\frac{3}{n}\right)}$$
but I don't see how to actually prove it.
 A: Maple calclulates it for concrete rational values of the parameter $n$. For example,
VectorCalculus:-int((-x^3-y^3+1)^(1/3), [x, y] = Region(0 .. 1, 0 .. (-x^3+1)^(1/3)))

$$  {\frac {4\,\sqrt [3]{2}{\pi }^{5/2}}{81\,\Gamma  \left( 2/3 \right) 
\Gamma  \left( 5/6 \right) }},$$
VectorCalculus:-int((-x^4-y^4+1)^(1/4), [x, y] = Region(0 .. 1, 0 .. (-x^4+1)^(1/4)))

$$1/6\,{\frac {{\pi }^{3/2}{\it EllipticK} \left( i \right) }{ \left( 
\Gamma  \left( 3/4 \right)  \right) ^{2}}},$$
and
VectorCalculus:-int((-x^(3/2)-y^(3/2)+1)^(2/3), [x, y] = Region(0 .. 1, 0 .. (1-x^(3/2))^(2/3)))

$${\frac {4\,\sqrt {\pi }\sqrt {3}{2}^{2/3}\Gamma  \left( 2/3 \right) 
\Gamma  \left( 5/6 \right) }{81}}.$$
I don't see any general formula for the integral under consideration.
A: Final steps thanks to Lucian starting from
$$\frac{\Gamma \left(1+\frac{1}{n}\right)^2}{\Gamma \left(\frac{n+2}{n}\right)}\int _0^1\left(1-x^n\right)^{2/n}dx$$
Let $x=s^{1/n}$ and we get
$$\frac{\Gamma \left(1+\frac{1}{n}\right)^2}{n\Gamma \left(\frac{n+2}{n}\right)}\int _0^1\left(1-s\right)^{2/n}s^{1/n-1}ds$$
$$=\frac{\Gamma \left(1+\frac{1}{n}\right)^2}{n\Gamma \left(\frac{n+2}{n}\right)}\frac{\Gamma \left(\frac{1}{n}\right) \Gamma \left(\frac{n+2}{n}\right)}{\Gamma \left(\frac{n+3}{n}\right)}$$
$$=\frac{\Gamma \left(1+\frac{1}{n}\right)^3}{\Gamma \left(\frac{n+3}{n}\right)}$$
