Clear[K, ϕ, θ, ψ, tofF, t, f, F]

f = -K (((ψ (ϕ - θ))/F) + 
              1);

(∫_0^Ff) 
              (F/(ψ (ϕ - θ) + F)) \[DifferentialD]F // 
      Simplify

(∫_0^tf)-K \[DifferentialD]T // Simplify

K = 1.97;
ϕ = 0.518;
θ = 0.318;
 ψ = -9.37;

tofF = ((F  - (ϕ  -  θ) ψ Log[1 + (F/((ϕ  -  θ) ψ))])/(-K))

Plot[{f, tofF}, {F, -1.0, -20.0}];