Computing the indefinite integral of a non-rationalized fraction

Question

Compute the indefinite integral $$\int \frac{x-1} {1 + \sqrt{x^2+2x-3}}dx$$

What do I need to do then? Fraction isn't simplified.

Answer 1

Hint:

Use substitution: to eliminate the radical from $$\sqrt{x^2+2x-3}=\sqrt{(x+1)^2-4}$$ set $x+1=2\cosh t,\enspace t\ge 0$, $\mathrm d\mkern1mu x=2\sinh t\,\mathrm d\mkern1mu t$. You'll obtain $$\int\frac{2(\cosh t-1)}{1+2\sinh t}2\sinh t\,\mathrm d\mkern1mu t$$ Then transform into a rational function of $u=\mathrm e^t$, which is standard.