What we are trying to find is:
\[ \int \operatorname{arccot}(x) \ dx \]
We are going to use integration by parts:
\[ \int u \ dv = uv - \int v \ du\]
Let's define \(u\) and \(v\):
\[\begin{gathered} u = \operatorname{arccot}(x), du= - \frac{dx}{x^2 +1} \\ v = x , dv = dx \end{gathered}\]
This means:
\[ \int \operatorname{arccot}(x) \ dx = x(\operatorname{arccot}(x)) - \int - \frac{x}{x^2 +1} \ dx\]
The derivative of \(\ln(x^2 +1)\) is \(\frac{1}{x^2 +1} *2x\), this means:
\[ \int \operatorname{arccot}(x) \ dx = x(\operatorname{arccot}(x)) + \frac{\ln(x^2 +1)}{2} + C\]