Wilson-Sommerfeld. m=1, hbar=1, V(x)=x^2/2+x^4/10
a  is the energy E

Solve[1-u^2/2-a u^4/10==0,u]

u0:=((Sqrt[-(5/a)+(((Sqrt[5]) (Sqrt[5+8 a]))/a)])/(Sqrt[2]))

f[a_]= 4 Sqrt[2] a Assuming[a>0,Integrate[Sqrt[1-u^2/2-a u^4/10],{u,0, u0}]]

FullSimplify[%]

f[a_]:= ((((-5+(Sqrt[5]) (Sqrt[5+8 a])))^(3/4)) (-40 ((((((a^3) (5+8 a))/(((-5+(Sqrt[5]) (Sqrt[5+8 a])))^3))))^(1/4)) EllipticE[((-5-4 a+(Sqrt[5]) (Sqrt[5+8 a]))/(4 a))]+(5 ((((((5+8 a) (-5+(Sqrt[5]) (Sqrt[5+8 a])))/a)))^(1/4))+8 ((((a^3) (5+8 a) (-5+(Sqrt[5]) (Sqrt[5+8 a]))))^(1/4))+(Sqrt[5]) ((((((((5+8 a))^3) (-5+(Sqrt[5]) (Sqrt[5+8 a])))/a)))^(1/4))) EllipticK[((-5-4 a+(Sqrt[5]) (Sqrt[5+8 a]))/(4 a))]))/(3 (5^(1/4)) (a^(1/4)) (Sqrt[5+8 a-(Sqrt[5]) (Sqrt[5+8 a])]))

g[n_]:= 2 Pi n

For[n=1,n<10,n++,b:=FindRoot[f[a]==g[n],{a,n}];Print["n=",n," ",b]]

gmodified[n_]:= 2 Pi (n-1/2)

For[n=1,n<10,n++,b:=FindRoot[f[a]==gmodified[n],{a,n}];Print["n=",n," ",b]]