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(2*E^x*(-1 + x) - x^2 + RootSum[1 + #1^4 & , x*Log[1 - E^x/#1] + PolyLog[2, E^x/#1] & ])/2


Integrate[x*(E^x + Tanh[2*x]), x] ==
(2*E^x*(-1 + x) - x^2 + RootSum[1 + #1^4 & , x*Log[1 - E^x/#1] + PolyLog[2, E^x/#1] & ])/2



              x
Integrate[x (E  + Tanh[2 x]), x] == 

    x             2
(2 E  (-1 + x) - x  + 
 
                  4
    RootSum[1 + #1  & , 
 
                x                x
               E                E
     x Log[1 - --] + PolyLog[2, --] & ]) / 2
               #1               #1

Time to compute: 0.07 second

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