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Integrate[1/(Sqrt[x^2 - 1] + 4), x] ==
(-4*ArcTanh[(4*x)/(Sqrt[17]*Sqrt[-1 + x^2])])/ Sqrt[17] - (2*Log[Sqrt[17] - x])/Sqrt[17] + (2*Log[Sqrt[17] + x])/Sqrt[17] + Log[2*(x + Sqrt[-1 + x^2])]



                 1
Integrate[----------------, x] == 
                2
          Sqrt[x  - 1] + 4

                    4 x
-4 ArcTanh[----------------------]
                               2
           Sqrt[17] Sqrt[-1 + x ]
---------------------------------- - 
             Sqrt[17]
 
  2 Log[Sqrt[17] - x]   2 Log[Sqrt[17] + x]
  ------------------- + ------------------- + 
       Sqrt[17]              Sqrt[17]
 
                        2
  Log[2 (x + Sqrt[-1 + x ])]

Time to compute: 0.06 second

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