Establish the integral formula for arccsc x

Establish that the following integration formula is correct:

$\int \operatorname{arccsc} x \, dx = x \operatorname{arccsc} x + \frac{x}{|x|} \log \left| x \sqrt{x^2-1} \right| + C.$

Proof. We integrate by parts, letting

$\begin{align*} u &= \operatorname{arccsc} x & du &= \frac{-1}{|x| \sqrt{x^2-1}} \, dx \\ dv &= dx & v &= x. \end{align*}$

(Where we established the formula for the derivative of $\operatorname{arccsc} x$ in this exercise, Section 6.22, Exercise #5). Therefore,

$\int \operatorname{arccsc} x \, dx = x \operatorname{arccsc} x + \int \frac{x}{|x| \sqrt{x^2-1}} \, dx.$

In the previous exercise we showed that

$\int \frac{x}{|x| \sqrt{x^2-1}} \, dx = \frac{x}{|x|} \log \left| x + \sqrt{x^2-1} \right| + C.$

Therefore,

$\int \operatorname{arccsc} x \, dx = x \operatorname{arccsc} x + \frac{x}{|x|} \log \left| x + \sqrt{x^2-1} \right| + C.$

Stumbling Robot