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Log[1 + 2*x]*Log[1 - (1 + 2*x)/(1 - 2*a)] + PolyLog[2, (1 + 2*x)/(1 - 2*a)]

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

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

                     1 + 2 x               1 + 2 x
Log[1 + 2 x] Log[1 - -------] + PolyLog[2, -------]
                     1 - 2 a               1 - 2 a
Time to compute: 0.01 second
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