In[]:=
R={{(1-d)/2*Cos[g*t/2]^2+(1+d)/2*Sin[g*t/2]^2,(Cos[w*t/2]-I*Sin[w*t/2])^2*I*Sin[g*t/2]*Cos[g*t/2]*(-d)},{(Cos[w*t/2]+I*Sin[w*t/2])^2*I*Sin[g*t/2]*Cos[g*t/2]*d,(1-d)/2*Sin[g*t/2]^2+(1+d)/2*Cos[g*t/2]^2}};
In[]:=
MatrixForm[FullSimplify[R]]
Out[]//MatrixForm=
1
2
(1-dCos[gt])
-
1
2
d
-tw

Sin[gt]
1
2
d
tw

Sin[gt]
1
2
(1+dCos[gt])
In[]:=
A=FullSimplify[MatrixPower[R,1/2]]
Out[]=

1-d
+
1+d
+
1-d
-
1+d
Cos[gt]
2
2
,

1-d
-
1+d
Sin[gt](Cos[tw]+Sin[tw])
2
2
,-

1-d
-
1+d

tw

Sin[gt]
2
2
,
1-d
+
1+d
+-
1-d
+
1+d
Cos[gt]
2
2

In[]:=
H={{w/2,g/2*Cos[w*t]-g/2*I*Sin[w*t]},{g/2*Cos[w*t]+g/2*I*Sin[w*t],-w/2}}
Out[]=

w
2
,
1
2
gCos[tw]-
1
2
gSin[tw],
1
2
gCos[tw]+
1
2
gSin[tw],-
w
2

In[]:=
G=FullSimplify[FullSimplify[A].H-H.FullSimplify[A]]
Out[]=


1-d
-
1+d
gSin[gt]
2
2
,

1-d
-
1+d

-tw

(gCos[gt]-wSin[gt])
2
2
,-

1-d
-
1+d

tw

(gCos[gt]+wSin[gt])
2
2
,-

1-d
-
1+d
gSin[gt]
2
2

In[]:=
k=-1/2*Tr[FullSimplify[G.G]]
Out[]=
-
1
8
-1+
1-
2
d
(2
2
g
+
2
w
-
2
w
Cos[2gt])
In[]:=
K={{Sqrt[(1-d)/2],0},{0,Sqrt[(1+d)/2]}}
Out[]=

1-d
2
,0,0,
1+d
2

In[]:=
o=ArcCos[Tr[K.FullSimplify[A]]]
Out[]=
ArcCos
1
4
1-d

1-d
+
1+d
+
1-d
-
1+d
Cos[gt]+
1
4
1+d

1-d
+
1+d
+-
1-d
+
1+d
Cos[gt]
In[]:=
1/x*IntegrateSqrt-18-1+
1-
2
d
(2
2
g
+
2
w
-
2
w
Cos[2g*t]),{t,0,x}
Out[]=
$Aborted
In[]:=
1/x*Integrate[Sqrt[k],{t,0,x}]​​x/(Integrate[Sqrt[k],{t,0,x}])*o
Out[]=
$Aborted