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(x^(1 + p)*((1 + p)*Cos[b*Log[x]] + b*Sin[b*Log[x]]))/(b^2 + (1 + p)^2)

Integrate[x^p*Cos[b*Log[x]], x] ==
(x^(1 + p)*((1 + p)*Cos[b*Log[x]] + b*Sin[b*Log[x]]))/ (b^2 + (1 + p)^2) 

           p
Integrate[x  Cos[b Log[x]], x] == 

 1 + p
x      ((1 + p) Cos[b Log[x]] + b Sin[b Log[x]])
------------------------------------------------
                  2          2
                 b  + (1 + p)
Time to compute: 0.07 second
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