Indefinite Integration

\(\log (x+\sqrt{x-1})+\sin ^{-1}\left(\sqrt{\frac{x-1}{x}}\right)+C\)

\(\log (x+\sqrt{x-1})-\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2 \sqrt{x-1}+1}{\sqrt{3}}\right)+C\)

\(\log (x+\sqrt{x-1})+C\)

\(\log (x-1+\sqrt{x-1})+\frac{1}{\sqrt{3}} \log \left|\frac{x-2-\sqrt{3}}{x-2+\sqrt{3}}\right|+C\)