About restricting variables in an integrand, and also changing the look of an integrands.

Question

So, in the last step of, many, integrands, Wolfram chooses to restrict the $x$-values, even if I didn't specify which values $x$ can take on.

Take for example:

$$\int\frac{dx}{x(x^2-1)^{3/2}} = -\cot(\sec^{-1}(x))-\sec^{-1}(x)+C$$

Checking with Wolfram gives a different option, an option that, in accordance to their step-by-step solver comes from the fact that they restrict the $x$-values. I'm not sure how they restrict it, but it apparently becomes: $$\tan^{-1}\left(\dfrac{1}{\sqrt{x^2-1}}\right) - \dfrac{1}{\sqrt{x^2-1}}$$

Why do they restrict $x$? Is this the usual thing to do? And if so, why?

I also want to ask how to change the expression from LHS to RHS by restricting the $x$-values. $$-\cot(\sec^{-1}(x))-\sec^{-1}(x) = \tan^{-1}\left(\dfrac{1}{\sqrt{x^2-1}}\right) - \dfrac{1}{\sqrt{x^2-1}}$$

Answer 1

• If you are surprised to see that the same indefinite integral $\big($anti-derivative$\big)$ can be expressed in two or more apparently different ways, see for example Gudermannian function.
• To show that the two functions are indeed identical, use $x=\sec t$, and $\sec^2t-1=\tan^2t$.