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-(((-a^2 + x^2)*(a + Sqrt[a^2 - x^2]*Log[a*x] - Sqrt[a^2 - x^2]*Log[2*a^3*(a + Sqrt[a^2 - x^2])]))/(a^3*Sqrt[(a^2 - x^2)^3]))


Integrate[1/(x*Sqrt[(a^2 - x^2)^3]), x] ==
-(((-a^2 + x^2)*(a + Sqrt[a^2 - x^2]*Log[a*x] - Sqrt[a^2 - x^2]* Log[2*a^3*(a + Sqrt[a^2 - x^2])]))/ (a^3*Sqrt[(a^2 - x^2)^3]))



                  1
Integrate[------------------, x] == 
                   2    2 3
          x Sqrt[(a  - x ) ]

      2    2             2    2
-(((-a  + x ) (a + Sqrt[a  - x ] Log[a x] - 
 
              2    2         3            2    2
        Sqrt[a  - x ] Log[2 a  (a + Sqrt[a  - x ])]))\
 
         3        2    2 3
     / (a  Sqrt[(a  - x ) ]))

Time to compute: 0.15 second

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