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Integrate[1/(x^(-2*a) + x^a), x] ==
(x*(-Hypergeometric2F1[1, -a^(-1), (-1 + a)/a, -x^(-a)] - (x^a*((1 + (-1)^(2/3)*x^a)^(-1))^ a^(-1)*Hypergeometric2F1[1 + a^(-1), 1 + a^(-1), 2 + a^(-1), (-1)^(2/3)/ ((-1)^(2/3) + x^(-a))])/ ((1 + a)*(I*Sqrt[3] + (-1)^(1/3)* (-2 + (-1)^(1/3))*x^a)) + (I*(-((x^a*(-(-1 + (-1)^(1/3)*x^a)^(-1))^ (1 + a^(-1))*Hypergeometric2F1[1 + a^(-1), 1 + a^(-1), 2 + a^(-1), ((-1)^(1/3)*x^a)/ (-1 + (-1)^(1/3)*x^a)])/(1 + a)) + ((1 + (-1)^(2/3)*x^a)^(-1))^a^(-1)* Hypergeometric2F1[a^(-1), a^(-1), 1 + a^(-1), (-1)^(2/3)/((-1)^(2/3) + x^(-a))] - (-(-1 + (-1)^(1/3)*x^a)^(-1))^a^(-1)* Hypergeometric2F1[a^(-1), a^(-1), 1 + a^(-1), ((-1)^(1/3)*x^a)/(-1 + (-1)^(1/3)*x^a)]))/ Sqrt[3]))/3

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

                            1   -1 + a    -a
(x (-Hypergeometric2F1[1, -(-), ------, -x  ] - 
                            a     a
 
        a        1        1/a
      (x  (--------------)    
                   2/3  a
           1 + (-1)    x
 
                               1      1      1
         Hypergeometric2F1[1 + -, 1 + -, 2 + -, 
                               a      a      a
 
                 2/3
             (-1)
          -------------]) / 
              2/3    -a
          (-1)    + x
 
       ((1 + a) (I Sqrt[3] + 
 
               1/3           1/3   a
           (-1)    (-2 + (-1)   ) x )) + 
 
              a           1         1 + 1/a
      (I (-((x  (-(---------------))        
                            1/3  a
                   -1 + (-1)    x
 
                                      1      1
                Hypergeometric2F1[1 + -, 1 + -, 
                                      a      a
 
                              1/3  a
                     1    (-1)    x
                 2 + -, ---------------]) / (1 + a))\
                     a           1/3  a
                        -1 + (-1)    x
 
                     1        1/a
            + (--------------)    
                       2/3  a
               1 + (-1)    x
 
                              1  1      1
            Hypergeometric2F1[-, -, 1 + -, 
                              a  a      a
 
                    2/3
                (-1)
             -------------] - 
                 2/3    -a
             (-1)    + x
 
                     1         1/a
           (-(---------------))    
                       1/3  a
              -1 + (-1)    x
 
                              1  1      1
            Hypergeometric2F1[-, -, 1 + -, 
                              a  a      a
 
                   1/3  a
               (-1)    x
             ---------------])) / Sqrt[3])) / 3
                      1/3  a
             -1 + (-1)    x