Foolish invertible g(x) !!

Calculus Level 4

\[\large \int _{ \sqrt [ 7 ]{ 1-{ x }^{ 4 } } }^{ g\left( x \right) }{ { e }^{ \ln (f (\sin ^{ 2 }{ t } ) \cos ^{ 2 }{ t ) } } } dt =\sqrt [ 4 ]{ 1-{ x }^{ 7 } } -\sqrt [ 7 ]{ 1-{ x }^{ 4 } } \]

For the equation given above, where \(g \left( x \right) =\sqrt [ 4 ]{ 1-{ x }^{ 7 } } \) is an invertible function, find \(f \left(\frac 12\right)\cdot f \left(\frac 13\right)\).