If $l^r\left(x\right)$ means log log log ................. (x), then 'log' being repeated 'r' times, then $\int \frac{1}{xl\left(x\right)l^2\left(x\right)l^3\left(x\right)\mathrm{.......}{l}^r\left(x\right)}dx=$

$l^r\left(x\right)=\mathrm{logloglog}\mathrm{......}\left(x\right) (r times)$

$I=\int \frac{1}{xl\left(x\right)l^2\left(x\right)l^3\left(x\right)\mathrm{.........}{l}^r\left(x\right)}dx$

Put $l^r\left(x\right)=t$  $\Rightarrow \frac{1}{l^{r-1}\left(x\right).l^{r-2}\left(x\right)\mathrm{.....}{l}^2\left(x\right)l\left(x\right).x}dx=dt$

$I=\int \frac{1}{\left[xl\left(x\right)l^2\left(x\right)\mathrm{.......}{l}^{r-1}\left(x\right)\right]l^r\left(x\right)}dx$

$=\int \frac{dt}{t}=\log t+c$

$=\log \left(l^r\left(x\right)\right)+c$

$=l^{r+1}\left(x\right)+c$