In[]:=
g[x_]:=3Sin[x]^2-(Sin[2x]-Sin[x])^2​​h[x_]:=16Cos[x]Sin[x]^5-10Cos[x]Sin[x]^3-24Sin[x]^5-10Sin[x]^3​​f[x_]:=(1/2(TrigReduce[-h[x]]+Sqrt[TrigReduce[4g[x]^3+h[x]^2]]))^(1/3)​​​​F[x_]:=ArcSin[1/3(Sin[2x]-Sin[x]+f[x]-g[x]/f[x])]
In[]:=
N[Pi^2/8]​​NIntegrate[F[x],{x,0,Pi}]
Out[]=
1.2337
Out[]=
1.2337