Clear["Global`*"]

Bondar Action Integral

S=Expand[ (∫_t1^t0)((1/2)(((kx -(F/ω)*Sin[ω*t]))^2)+(1/2)(ko^2)+Ip)\[DifferentialD]t]

S=Simplify[S/.{t0 -> (1/ω)(ArcSin[β] + I*Log[α+(Sqrt[(α^2)-1])]),t1-> (1/ω)*ArcSin[β]}]

Take imaginary part and simplify:

ImS = Im[(1/(8 (ω^3)))(8 F kx ω Cos[ArcSin[β]+i Log[α+(Sqrt[-1+(α^2)])]]+2 i (F^2) Log[α+(Sqrt[-1+(α^2)])]+8 i Ip (ω^2) Log[α+(Sqrt[-1+(α^2)])]+4 i (ko^2) (ω^2) Log[α+(Sqrt[-1+(α^2)])]+4 i (kx^2) (ω^2) Log[α+(Sqrt[-1+(α^2)])]-(F^2) Sin[2 ArcSin[β]+2 i Log[α+(Sqrt[-1+(α^2)])]])];

Log[α+(Sqrt[-1+(α^2)])] = ArcCosh[α]

ImS = Im[(1/(8 (ω^3)))(8 F kx ω Cos[ArcSin[β]+i ArcCosh[α]]+2 i (F^2)ArcCosh[α]+8 i Ip (ω^2) ArcCosh[α]+4 i (ko^2) (ω^2) ArcCosh[α]+4 i (kx^2) (ω^2) ArcCosh[α]-(F^2) Sin[2 ArcSin[β]+2 i ArcCosh[α]])];

Cos (x + Iy) = Cos (x) Cosh (y) - i Sin (x) Sinh (y)

ImS = Im[(1/(8 (ω^3)))(-i8 F kx ω Sin[ArcSin[β]]*Sinh[ ArcCosh[α]]+2 i (F^2)ArcCosh[α]+8 i Ip (ω^2) ArcCosh[α]+4 i (ko^2) (ω^2) ArcCosh[α]+4 i (kx^2) (ω^2) ArcCosh[α]-((iF)^2) Cos[2 ArcSin[β]]*Sinh[2  ArcCosh[α]])];

Sinh = (Sqrt[(Cosh^2) - 1])

ImS = Im[(1/(8 (ω^3)))(-i8 F kx ω β(Sqrt[(α^2)-1])+2 i (F^2)ArcCosh[α]+8 i Ip (ω^2) ArcCosh[α]+4 i (ko^2) (ω^2) ArcCosh[α]+4 i (kx^2) (ω^2) ArcCosh[α]-((iF)^2) Cos[2 ArcSin[β]]*Sinh[2  ArcCosh[α]])];

Sinh (2 x) = 2*Sinh (x)*Cosh (x)

ImS = Im[(1/(8 (ω^3)))(-i8 F kx ω β(Sqrt[(α^2)-1])+2 i (F^2)ArcCosh[α]+8 i Ip (ω^2) ArcCosh[α]+4 i (ko^2) (ω^2) ArcCosh[α]+4 i (kx^2) (ω^2) ArcCosh[α]-((iF)^2) Cos[2 ArcSin[β]]*2*(Sqrt[(α^2)-1])*α)];

ImS = (1/(8 (ω^3)))(-8 F kx ω β(Sqrt[(α^2)-1])+2  (F^2)ArcCosh[α]+8  Ip (ω^2) ArcCosh[α]+4  (ko^2) (ω^2) ArcCosh[α]+4  (kx^2) (ω^2) ArcCosh[α]-(F^2) Cos[2 ArcSin[β]]*2*(Sqrt[(α^2)-1])*α)

Cos(2x) = 1-2*(sin^2)(x)

ImS = (1/(8 (ω^3)))(-8 F kx ω β(Sqrt[(α^2)-1])+2  (F^2)ArcCosh[α]+8  Ip (ω^2) ArcCosh[α]+4  (ko^2) (ω^2) ArcCosh[α]+4  (kx^2) (ω^2) ArcCosh[α]-(F^2) (1-2*(β^2))*2*(Sqrt[(α^2)-1])*α)

Simplify[(1/(8 (ω^3)))(-8 F kx ω β(Sqrt[(α^2)-1])+2  (F^2)ArcCosh[α]+8  Ip (ω^2) ArcCosh[α]+4  (ko^2) (ω^2) ArcCosh[α]+4  (kx^2) (ω^2) ArcCosh[α]-(F^2) (1-2*(β^2))*2*(Sqrt[(α^2)-1])*α)]

Probability

Γ = Exp[-2*Im[S]], (ko^2)+(kx^2) = (k^2)

Γ = Exp[-2*(1/(4 (ω^3)))(F (Sqrt[-1+(α^2)]) (F α (-1+2 (β^2))-4 kx β ω)+((F^2)+2 (2 Ip+(k^2)) (ω^2)) ArcCosh[α])];

factor out (Ip/ω)

Expand[(ω/Ip)*((1/(4 (ω^3)))(F (Sqrt[-1+(α^2)]) (F α (-1+2 (β^2))-4 kx β ω)+((F^2)+2 (2 Ip+(k^2)) (ω^2)) ArcCosh[α]))]

γ = ((Sqrt[2*Ip])/F)ω

Γ = Exp[-((2 Ip)/ω)* ( (1 + ((k^2)/(2 Ip))+ (1/(2 (γ^2))) ) ArcCosh(α) - (Sqrt[-1+(α^2)])((((1-2(β^2))α)/(2 (γ^2)))) + (Sqrt[(2/Ip)])kx β)];