Q.Evaluate: $\int \csc(\log x)\left[1-\cot(\log x)\right]dx$

The stem as literally extracted, $\csc(\log x)[1-\cot(\log x)]$, carries no explicit $dx/x$ factor, and integrated purely 'dx' as printed this has no elementary closed form (it is not an e^x[f+f'] pattern like the rest of this group, since there is no e^x factor at all, and log x alone inside csc/cot with a bare dx does not reduce by any standard technique). This strongly suggests a $\dfrac1x$ factor was lost in extracting the source PDF (a known artifact in this source, since $d(\log x)=\dfrac{dx}{x}$ is exactly what would make this integrable by substitution, matching the shape of the surrounding I.13/I.19/III.8 items which are all substitution-in-log-x problems). Solving under that assumption — i.e. treating the intended integral as $\displaystyle\int\csc(\log x)\left[1-\cot(\log x)\right]\dfrac{dx}{x}$ — substitute $t=\log x$, $dt=\dfrac{dx}x$:

$\int\csc t(1-\cot t)\,dt=\int\csc t\,dt-\int\csc t\cot t\,dt$ …

Substitution t = log x (assuming a dropped 1/x factor — see note), standa …

Forcing this into the e^x[f+f'] template used by the rest of group III (there is no e^x factor here at all); also, forgetting the sign …