Z(t)=(2/(1+t))-((1/(1+t))+(1/(-1+3 t))) e^(-(m/(2)) (1+t))+((1/(1+t))+(1/(-1+3 t))) e^(m (-1+t))

d/dt Q_1(t) =
d/dt(log_e((2/(1 + t) - (1/(1 + t) + 1/(-1 + 3 t)) e^(-1/2 m (1 + t)) + (1/(1 + t) + 1/(-1 + 3 t)) e^(m (-1 + t))) (1 + t) s)/(1 + t)) 
= 
(-2 (-1 + 3 t) e^(-1/2 m (1 + t)) (2 t e^(1/2 m (-1 + 3 t)) + (-1 + 3 t) e^(1/2 m (1 + t)) - 2 t) log_e((2 s (t (-2 e^(m (-1 + t)) + 2 e^(-1/2 m (1 + t)) - 3) + 1))/(1 - 3 t)) - 4 (3 t^2 + 1) e^(m (-1 + t)) + 4 (3 t^2 + 1) e^(-1/2 m (1 + t)) + 2 (1 - 3 t) (t (-2 e^(m (-1 + t)) + 2 e^(-1/2 m (1 + t)) - 3) + 1) + 4 m t (-1 + 3 t) (1 + t) e^(m (-1 + t)) + 2 m t (-1 + 3 t) (1 + t) e^(-1/2 m (1 + t)) - 2 (1 - 3 t)^2)/(2 (1 - 3 t) (1 + t)^2 (t (-2 e^(m (-1 + t)) + 2 e^(-1/2 m (1 + t)) - 3) + 1)) 

d/dt Q_2(t) =
d/dt(-log_e(-(2/(1 + t) - (1/(1 + t) + 1/(-1 + 3 t)) e^(-1/2 m (1 + t)) + (1/(1 + t) + 1/(-1 + 3 t)) e^(m (-1 + t))) (1 + t) s + 2)/(1 + t)) 
= 
(log_e((s (t (4 e^(m (-1 + t)) - 4 e^(-1/2 m (1 + t)) + 6) - 2) - 6 t + 2)/(1 - 3 t)) - (s (1 + t) (m t (-1 + 3 t) + e^(1/2 m (-1 + 3 t)) (2 m t (-1 + 3 t) - 2) + 2))/((-1 + 3 t) (2 s t e^(1/2 m (-1 + 3 t)) + (s - 1) (-1 + 3 t) e^(1/2 m (1 + t)) - 2 s t)))/(1 + t)^2

d/dt Q_3(t) =
d/dt((log_e((-1 + 3 t) (s (2/(1 + t) - (1/(1 + t) + 1/(-1 + 3 t)) e^(-1/2 m (1 + t)) + (1/(1 + t) + 1/(-1 + 3 t)) e^(m (-1 + t))) - 2/(1 + t) + (1/(1 + t) + 1/(-1 + 3 t)) e^(1/2 (-m) (1 + t)))) + 2 t m)/(-1 + 3 t)) 
= 
((3 (s ((1/(3 t - 1) + 1/(t + 1)) e^(m (t - 1)) - (1/(3 t - 1) + 1/(t + 1)) e^(-1/2 m (t + 1)) + 2/(t + 1)) + (1/(3 t - 1) + 1/(t + 1)) e^(-1/2 m (t + 1)) - 2/(t + 1)) + (3 t - 1) (s ((-3/(3 t - 1)^2 - 1/(t + 1)^2) e^(m (t - 1)) - (-3/(3 t - 1)^2 - 1/(t + 1)^2) e^(-1/2 m (t + 1)) + m (1/(3 t - 1) + 1/(t + 1)) e^(m (t - 1)) + 1/2 m (1/(3 t - 1) + 1/(t + 1)) e^(-1/2 m (t + 1)) - 2/(t + 1)^2) + (-3/(3 t - 1)^2 - 1/(t + 1)^2) e^(-1/2 m (t + 1)) - 1/2 m (1/(3 t - 1) + 1/(t + 1)) e^(-1/2 m (t + 1)) + 2/(t + 1)^2))/((3 t - 1) (s ((1/(3 t - 1) + 1/(t + 1)) e^(m (t - 1)) - (1/(3 t - 1) + 1/(t + 1)) e^(-1/2 m (t + 1)) + 2/(t + 1)) + (1/(3 t - 1) + 1/(t + 1)) e^(-1/2 m (t + 1)) - 2/(t + 1))) + 2 m)/(3 t - 1) - (3 (log_e((3 t - 1) (s ((1/(3 t - 1) + 1/(t + 1)) e^(m (t - 1)) - (1/(3 t - 1) + 1/(t + 1)) e^(-1/2 m (t + 1)) + 2/(t + 1)) + (1/(3 t - 1) + 1/(t + 1)) e^(-1/2 m (t + 1)) - 2/(t + 1))) + 2 m t))/(3 t - 1)^2

d/dt Q_4(t) =
d/dt((-log_e((2/(1 + t) - (1/(1 + t) + 1/(-1 + 3 t)) e^(-1/2 m (1 + t)) + (1/(1 + t) + 1/(-1 + 3 t)) e^(m (-1 + t))) (1 + t) (1 - s)) + 2 t m)/(1 + t)) 
= 
(log_e(-(2 (s - 1) e^(-1/2 m (1 + t)) (2 t e^(1/2 m (-1 + 3 t)) + (-1 + 3 t) e^(1/2 m (1 + t)) - 2 t))/(-1 + 3 t)) - 2 m t + ((1 + t) (2 m (1 - 3 t)^2 e^(1/2 m (1 + t)) + 5 m t (1 - 3 t) + e^(1/2 m (-1 + 3 t)) (2 m t (-1 + 3 t) + 2) - 2))/((-1 + 3 t) (2 t e^(1/2 m (-1 + 3 t)) + (-1 + 3 t) e^(1/2 m (1 + t)) - 2 t)))/(1 + t)^2
