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Integrate[x*Sec[x]^5, x] ==
((72*I)*PolyLog[2, (-I)*E^(I*x)] + Sec[x]^4*(30 - 70*Cos[x] + 40*Cos[2*x] - 18*Cos[3*x] + 10*Cos[4*x] + 27*x*Log[1 - I*E^(I*x)] + 36*x*Cos[2*x]* Log[1 - I*E^(I*x)] + 9*x*Cos[4*x]* Log[1 - I*E^(I*x)] - 27*x*Log[1 + I*E^(I*x)] - 36*x*Cos[2*x]*Log[1 + I*E^(I*x)] - 9*x*Cos[4*x]*Log[1 + I*E^(I*x)] - (72*I)*Cos[x]^4*PolyLog[2, I*E^(I*x)] + 66*x*Sin[x] + 18*x*Sin[3*x]))/192



                  5
Integrate[x Sec[x] , x] == 

                       I x
((72 I) PolyLog[2, -I E   ] + 
 
          4
    Sec[x]  (30 - 70 Cos[x] + 40 Cos[2 x] - 
 
       18 Cos[3 x] + 10 Cos[4 x] + 
 
                       I x
       27 x Log[1 - I E   ] + 
 
                                I x
       36 x Cos[2 x] Log[1 - I E   ] + 
 
                               I x
       9 x Cos[4 x] Log[1 - I E   ] - 
 
                       I x
       27 x Log[1 + I E   ] - 
 
                                I x
       36 x Cos[2 x] Log[1 + I E   ] - 
 
                               I x
       9 x Cos[4 x] Log[1 + I E   ] - 
 
                    4               I x
       (72 I) Cos[x]  PolyLog[2, I E   ] + 
 
       66 x Sin[x] + 18 x Sin[3 x])) / 192

Time to compute: 1.52 second

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