∫\ (( x-1))/((1+( x)^2.)\ ^2 d x is equal to

\begin{aligned} & \int \frac{(\log x-1)^2}{\left(1+(\log x)^2\right)^2} d x \\ & =\int\left[\frac{1}{\left(1+(\log x)^2\right)}-\frac{2 \log x}{\left(1+(\log x)^2\right)^2}\right] d x \\ & =\int\left[\frac{e^t}{1+t^2}-\frac{2 t e^t}{\left(1+t^2\right)^2}\right] d t \text{ put } \log x=t \Rightarrow d x=e^t d t \\ & \int e^t\left[\frac{1}{1+t^2}-\frac{2 t}{\left(1+t^2\right)^2}\right] d t \\ & =\frac{e^t}{1+t^2}+c=\frac{x}{1+(\log x)^2}+c \end{aligned}