How to solve this rational integral?

Question

I am trying to find a solution for this integral, but can't find any form of analytically solvable rational integrals that can be related to this one. The integral is:

$$\int {\frac{-b(1+a)\,dx}{(x+\sqrt{x^2-1+a^2 })\sqrt{x^2-1+a^2} }}\,,$$

but since $a,b$ are constants, I guess it can be simplified to

$$\int \frac{dx}{(x+\sqrt{x^2+m})\sqrt{x^2+m} } \,,$$

where $m=a^2-1\,$. I have tried some variable changes with no result, but I may be that I'm a little rusty. Also tried expressing it like this

$$\int \frac{dx}{x\sqrt{x^2+m}+x^2+m } \,,$$

but still cannot relate it to typical rational integrals forms. Any help will be appreciated.

Answer 1

Hint: Substitute $u=\dfrac{1}{\sqrt{x^2+m}+x}$ into your original equation (the one you mention second)

You should get $F(x)=-\dfrac{1}{\sqrt{x^2+m}+x}+C$

Answer 2

In the second representation, one can multiply numerator and denominator by $\sqrt{x^2+m}-x$, and obtain the integral under the form $$I=\frac{1}{m}\int \frac{\sqrt{x^2+m}-x}{\sqrt{x^2+m}}\,dx$$. It can be splitted and it remains to integrate $\frac{x}{\sqrt{x^2+m}}$ and $1$. $$I=\frac{1}{m}(x-\sqrt{x^2+m})$$