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Integrate[Log[a + Sqrt[b + c*x^2]], x] ==
-x - ((a^2 - b)*ArcTan[(Sqrt[c]*x)/Sqrt[-a^2 + b]])/ (Sqrt[-a^2 + b]*Sqrt[c]) + ((a^3 - a*b)*ArcTan[(a*Sqrt[c]*x)/(Sqrt[-a^2 + b]* Sqrt[b + c*x^2])])/(a*Sqrt[-a^2 + b]*Sqrt[c]) + x*Log[a + Sqrt[b + c*x^2]] + (a*Log[2*(c*x + Sqrt[c]*Sqrt[b + c*x^2])])/Sqrt[c]



                              2
Integrate[Log[a + Sqrt[b + c x ]], x] == 

       2               Sqrt[c] x
     (a  - b) ArcTan[-------------]
                            2
                     Sqrt[-a  + b]
-x - ------------------------------ + 
                2
         Sqrt[-a  + b] Sqrt[c]
 
    3                       a Sqrt[c] x
  (a  - a b) ArcTan[----------------------------]
                           2                  2
                    Sqrt[-a  + b] Sqrt[b + c x ]
  ----------------------------------------------- + 
                       2
              a Sqrt[-a  + b] Sqrt[c]
 
                        2
  x Log[a + Sqrt[b + c x ]] + 
 
                                     2
  a Log[2 (c x + Sqrt[c] Sqrt[b + c x ])]
  ---------------------------------------
                  Sqrt[c]

Time to compute: 0.27 second

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