∫((x+sinx)/(1+cosx))dx
=∫[(x+sinx)/2cos²(x/2)]dx
=∫(x+sinx)d(tan(x/2))
=(x+sinx)*tan(x/2)-∫tan(x/2)d(x+sinx)
=xtan(x/2)+sinx*tan(x/2)-∫tan(x/2)(1+cosx)dx
=xtan(x/2)+2sin(x/2)cos(x/2)*tan(x/2)-∫tan(x/2)*2cos²(x/2)dx
=xtan(x/2)+2sin²(x/2)-∫sin(x)dx
=xtan(x/2)+2sin²(x/2)-cos(x)+C
=xtan(x/2)+C1