Hmm, I don't know whether with "lucrative" it is actually meant "efficient"?
There is a possibility in Pari/GP to formulate the derivative of the eta() by a divergent sum-procedure
sumalt(), applicable even to formal series. What I do in cases like this, where derivatives of eta at zero (or zeta at zero: Stieltjes-numbers) are involved is the following procedure
default(seriesprecision,64) \\ number of taylor-coefficients of a series
psaeta = sumalt(k=0,(-1)^k/(1+k)^x); \\ gives the formal powerseries for aeta
pcaeta= polcoeffs(psaeta,64)~; \\ extracts the taylor-coefficients
\\ = eta^(k)(0) /k!
\\ now the functional definition for the c_j coefficients
mrb = sum(r=1,63,mrbcj(r)*pcaeta[1+r]) \\ this gives a rough estimate to about 4 digits
\\ using Eulersummation I can improve the number of correct digits dramatically
The terms of the 60 to 64'th partial sums with the Euler-summation are
and the digits up to the last "7" seem to be correct.
Perhaps there is something similar in Math'ca , at least they have manipulations for formal divergent series symbolically...
The Pari/GP-evaluation of the initial formula is very simple btw. and gives an accurate result very easy; I got
mrb = sumalt(k=1,(-1)^k *(exp(log(k)/k)-1))
\\ %2909 = 0.187859642462067120248517934054273230055903094900138786172005