For any problem, where g[x] is independent of y and does not contain y and g[x] is "nice" and does not go to infinity, etc., is it true that
Integrate[g[x],{y,a,b}]==g[x]*(b-a)
If that is the case then can
Integrate[Integrate[f[s]^(1/n), {s, 0, y}], {t, 0, y1}] +
Integrate[Integrate[f[s]^(1/n), {s, 0, y1}], {t, y1, y2}] +
Integrate[Integrate[(-f[s])^(1/n), {s, y, y1}], {t, y1, y2}] +
Integrate[Integrate[f[s]^(1/n), {s, 0, y1}], {t, y2, 1}] +
Integrate[Integrate[(-f[s])^(1/n), {s, y2, y1}], {t, y2, 1}] +
Integrate[Integrate[f[s]^(1/n), {s, y2, y}], {t, y2, 1}]
be simplified to
Integrate[f[s]^(1/n), {s, 0, y}] y1 -
Integrate[f[s]^(1/n), {s, 0, y1}] (y1 - y2) -
Integrate[(-f[s])^(1/n), {s, y, y1}] (y1 - y2) +
Integrate[f[s]^(1/n), {s, 0, y1}] (1 - y2) +
Integrate[(-f[s])^(1/n), {s, y2, y1}] (1 - y2) +
Integrate[f[s]^(1/n), {s, y2, y}] (1 - y2)
If this is correct then this might be a somewhat easier problem.
Integrating a fractional exponent of a quadratic equation seems to be a difficult problem for Mathematica.
Is anything known about the value of n or is it completely unknown? If more information were available about that then this might help.
Is there anything else that is known about the problem that might make it easier to solve?