Yes you can - integrate ln(ln x) by parts twice, integrating 1 each time (and differentiating the integrand each time)... see what happens. Pic in a mo.

Edit: sorry for the delay. Just in case a picture helps...

... where

... is lazy integration by parts, with...

... the product rule. Straight lines differentiate downwards (integrate up) with respect to x.

Spoiler:

Hence we can write...

Spoiler:

You may find it helpful to zoom in on the chain rule, inside the product rule...

Spoiler:

... where

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

Yes you can - integrate ln(ln x) by parts twice, integrating 1 each time (and differentiating the integrand each time)... see what happens. Pic in a mo.

Edit: sorry for the delay. Just in case a picture helps...

... where

... is lazy integration by parts, with...

... the product rule. Straight lines differentiate downwards (integrate up) with respect to x.

Spoiler:

Hence we can write...

Spoiler:

You may find it helpful to zoom in on the chain rule, inside the product rule...

Spoiler:

... where

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

Wow!! Where do I start and what do I say. You're simply fantastic Tom. Thank you so much. I kind of figured it out before you posted the pics but they're a visual delight. I loved it when the two integrals involving the messy 1 / (lnx)^2 just cancelled out. However, if we just had the integral of ln(lnx), I don't think it can be done with elementary functions.