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Integrate[1/(x*Sqrt[a^2 + x^2]), x] ==
Log[a*x]/a - Log[2*(a^2 + a*Sqrt[a^2 + x^2])]/a

                 1
Integrate[---------------, x] == 
                  2    2
          x Sqrt[a  + x ]

                   2           2    2
Log[a x]   Log[2 (a  + a Sqrt[a  + x ])]
-------- - -----------------------------
   a                     a