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Integrate[Log[Sec[x] + Tan[x]], x] ==
(I*Pi*x - Pi*Log[1 - I*E^(I*x)] - 2*x*Log[1 - I*E^(I*x)] - Pi*Log[1 + I*E^(I*x)] + 2*x*Log[1 + I*E^(I*x)] + Pi*Log[-Cos[(Pi + 2*x)/4]] + Pi*Log[Sin[(Pi + 2*x)/4]] + 2*x*Log[Sec[x] + Tan[x]] - (2*I)*PolyLog[2, (-I)*E^(I*x)] + (2*I)*PolyLog[2, I*E^(I*x)])/2



Integrate[Log[Sec[x] + Tan[x]], x] == 

                        I x                   I x
(I Pi x - Pi Log[1 - I E   ] - 2 x Log[1 - I E   ] - 
 
                  I x                   I x
    Pi Log[1 + I E   ] + 2 x Log[1 + I E   ] + 
 
                Pi + 2 x                Pi + 2 x
    Pi Log[-Cos[--------]] + Pi Log[Sin[--------]] + 
                   4                       4
 
    2 x Log[Sec[x] + Tan[x]] - 
 
                         I x
    (2 I) PolyLog[2, -I E   ] + 
 
                        I x
    (2 I) PolyLog[2, I E   ]) / 2

Time to compute: 0.18 second

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