sigmaNtilde = ((sigma1^2/sigma2^2) (sigma1^2 + 
                sigma2^2))^(1/2)

sigmaXtilde = ((sigmaX^2 (sigma1^2 + sigma2^2))/(sigmaX^2 + 
                sigma1^2 + sigma2^2))^(1/2)

rho0 = (sigmaXtilde D1/(sigmaNtilde (sigmaXtilde^2 - D1) 
                  D2)) (*big bracket*) (
(D2 (sigmaXtilde^2 - D2))^(1/
                  2)
 -  (*sqrt start*) (
(sigmaXtilde^2 (D2/D1) (sigmaNtilde^2 - D1) - 
                      sigmaNtilde^2 D2) ((D2/D1) - 
                      1)
)^(1/
                  2) (*sqrt end*) 
) (*big bracket*)

rate  =  sigmaXtilde^2/(D2 (1 - rho0^2))

FullSimplify[
      ExpandAll[
        D2 ((sigma1^2) + (sigma2^2) + 
              (sigmaX^2)) (1 - ((D1^2) (sigma2^2) (sigmaX^2) ((((Sqrt[D2 (-D2 + ((((sigma1^2) + (sigma2^2)) (sigmaX^2))/((sigma1^2) + (sigma2^2) + (sigmaX^2))))]) - √((-1 + (D2/D1)) (-((D2 (sigma1^2) ((sigma1^2) + (sigma2^2)))/(sigma2^2)) + ((D2 ((sigma1^2) + (sigma2^2)) (-D1 + (((sigma1^2) ((sigma1^2) + (sigma2^2)))/(sigma2^2))) (sigmaX^2))/(D1 ((sigma1^2) + (sigma2^2) + (sigmaX^2))))))))^2))/((D2^2) (sigma1^2) ((sigma1^2) + (sigma2^2) + 
                        (sigmaX^2)) (((-D1 + ((((sigma1^2) + (sigma2^2)) (sigmaX^2))/((sigma1^2) + (sigma2^2) + (sigmaX^2)))))^2)))]]

(*write the rate with the simplified denominator:*)

rate = (((sigma1^2) + (sigma2^2)) (sigmaX^2) 
            sigma1^2) (D2 (((((sigma1^2) + (sigma2^2)) (sigmaX^2) - D1  ((sigma1^2) + (sigma2^2) + (sigmaX^2))))^2))/(D1 ((sigma1^2) + 
                  (sigma2^2) + 
                  (sigmaX^2)) (D2 ((sigma1^2) + 
                        (sigma2^2)) (sigmaX^2) (-D2 (sigma1^2) ((sigma1^2) + (sigma2^2)) + (-D2 (sigma1^2) + (sigma1^4) + 
                              D2 (sigma2^2) + 
                              (sigma1^2) (sigma2^2)) (sigmaX^2)) + 
                  D1 ((D2^2) ((sigma1^2) + (sigma2^2)) ((sigma1^2) + 
                              (sigmaX^2)) ((sigma1^2) + (sigma2^2) + 
                              (sigmaX^2)) - 
                        D2 ((sigma1^2) + 
                              (sigma2^2)) (sigmaX^2) ((sigma1^4) + 
                              2 (sigma2^2) (sigmaX^2) + 
                              (sigma1^2) ((sigma2^2) + (sigmaX^2))) + 
                        2 (sigma2^2) (sigmaX^2) ((sigma1^2) + (sigma2^2) + 
                              (sigmaX^2)) √(((D1 - 
                                        D2) D2 ((((sigma1^2) + (sigma2^2)))^2) (D1 (sigma1^2) + (D1 - 
                                        (sigma1^2)) (sigmaX^2)))/((D1^2) (sigma2^2) ((sigma1^2) + (sigma2^2) + 
                                        (sigmaX^2)))) (Sqrt[((D2 (((sigma1^2) + (sigma2^2)) (sigmaX^2) - D2 ((sigma1^2) + (sigma2^2) + (sigmaX^2))))/((sigma1^2) + (sigma2^2) + (sigmaX^2)))]))))

TeXForm[rate] >> conditional_Kaspi_rate.tex