Integrate a chain function

Question

I have to calculate $\int\operatorname{arccot}(\cot(x))\ dx.$ If I had to find the derivative it would be easy with the chain rule. How can I do this?

Answer 1

Hint: integrating by parts we get $$\int arccot(\cot(x))dx=x arccot(\cot(x))-\int xdx$$ Second hint: $$(arccot(cot(x))'=-\frac{-1-\cot(x)^2}{1+\cot(x)^2}=1$$

Answer 2

Let arccot$(\cot x)=y$

$\implies \cot x=\cot y$

$x=n\pi+y$ where $n$ is an integer

$\int y\ dx=?$