This is an impossible integral to complete without resorting to calculators and evaluating at explicit bounds.
http://www.wolframalpha.com/input/?i=integral+of+1%2F%28ln%28lnx%29%29
Let's see how far we can go, though...
int (ln(lnx))^(-1)dx
Let:
u = lnx
du = 1/xdx
xdu = dx -> e^udu = dx
= int (lnu)^(-1)e^udu
= inte^u/(lnu)du
st - int t ds
Let:
s = e^u
ds = e^udu
dt = 1/lnudu
t = ?
Detour. We need to know the integral for 1/lnx. Let:
q = 1/lnu
dq = -lnu/udu
dr = du
r = u
qr - int r dq
= u/lnu + int lnudu
= u/lnu + u ln|u| - u = t
Back to s and t:
= e^u (u/lnu + u ln|u| - u) - int e^u(u/lnu + u lnu - u)du
= (ue^u)/lnu + ue^u ln|u| - u e^u - int (ue^u)/lnudu - int ue^u lnudu + intu e^udu
The only integral we can do with real functions or standard functions is int u e^udu. I'm running out of variables.
op - int p do
Let:
o = u
do = du
dp = e^udu
p = e^u
ue^u - int e^udu
= ue^u - e^u
So now we get:
= (ue^u)/lnu + ue^u ln|u| cancel(- u e^u + ue^u) - e^u - int (ue^u)/lnudu - int ue^u lnudu
= (ue^u)/lnu + ue^u ln|u| - e^u - int (ue^u)/lnudu - int ue^u lnudu
= e^u[u/lnu + u ln|u| - 1] - int (ue^u)/lnudu - int ue^u lnudu
= e^(lnx)[(lnx)/ln(lnx) + (lnx) ln|lnx| - 1] - int ((lnx)e^(lnx))/(xln(lnx))dx - int 1/x(lnx)e^(lnx) ln(lnx)dx
= color(blue)(x[(lnx)/ln(lnx) + (lnx) ln|lnx| - 1] - int (lnx)/(ln(lnx))dx - int (lnx) ln(lnx)dx)
