Calculating real rate after taxation - is there a better way?

First I need a sanity check for my calculations.

Say I have this scenario:

I want to calculate the real rate (Y) of this investment.

First I need to get the gross value (G):

G = x * (1 + r)^t = 100.00 * (1 + 13%)¹⁰ = $ 339.46

Then the I get the net value after taxation (N)

N = G – (G – x) * T = 339.46 – (339.46 – 100.00) * 15% = $ 303.54

Then I bring this to present value with the inflation rate, which will be the real value (R):

R = N * (1 + i)^-t = 303.54 * (1 + 5%)^-10 = $ 186.35

To get the real rate (Y) I need to use the rate formula:

Y = (R / x)^1/t – 1 = (186.35 / 100.00)^1/10 – 1 = 6,42%

So in this investment my money really grows 6,42% per year. Is that right? Is there a better way to calculate this? this seems very laborious.

In Excel I’m using this monstrosity:

=rate(10;;-100;-pv(5%;10;;fv(13%;10;;-100)-(fv(13%;10;;-100)-100)*15%))

(13% - 5%) * (1-15%) = 8% * 85% = 6.8%

((((1+13%)^10-1)*85%+1)^(1/10)/(1+5%) -1 = 6.42%