Here is an integral that looks ominous.
Give it a whirl if you so desire.
$\displaystyle \int_{e^{e}}^{e^{e+1}}\left[\frac{1}{ln(x)\cdot ln(ln(x))}+ln(ln(ln(x)))\right]dx$
Mathstud?. Krizarama?. anyone?.
Hmmm...well I have fiddled around with the indefinite integral. I tried a bunch of tricks but to no avail. So I am going to say that this has no closed form answer. But for the definite part, I am not completely done yet, but letting $\displaystyle u=\ln(\ln(x))$ then using differentation under the integral sign is looking promising.
EDIT: Also, based upon human nature I am going to guess that a sub of $\displaystyle u=\ln(x)$ will be in here somewhere, I got this based upon human being's propensity for nice solutions, e.g. look at the limits of integration
$\displaystyle \int_{e^{e}}^{e^{e+1}}\left[\frac{1}{ln(x)\cdot ln(ln(x))}+ln(ln(ln(x)))\right]dx$
Let $\displaystyle f(x)=\ln \ln \ln x$,
then $\displaystyle f'(x) = \frac{1}{x \ln x \ln \ln x}$
So the integral is $\displaystyle \int_{e^{e}}^{e^{e+1}}\left[ xf(x)\right ]'~dx$
It's obvious from here
I am using it less and less, but seriously, hate me or not, you cannot deny that if you really did have an answer and the online integrator said there was no solution you would disagree with it. I admit I did not see the solution till Dystopia said product rule, but nonetheless. Stop making it sound like I cannot do anything, that all my answers are from my calculator, I almost never use it anymore. Your complete and utter faith in my mathematical abilities is flattering Moo.
I had a chance to look at it and what I saw happens to be along wingless lines, it would appear.
Take note that the derivative of $\displaystyle xln(ln(ln(x)))=\text{our integral}$
So, we get:
$\displaystyle \int(\frac{d}{dx}\left[xln(ln(ln(x)))\right])dx$
Which we just end up with:
$\displaystyle (e^{e+1})ln(ln(ln(e^{e+1})))\approx{11.226}$
Now, when $\displaystyle e^{e}$ we get 0, so we are done.
I had this integral written down amongst some papers and it was not finished. I do not know where it came from.