∫(x + sinx)/(1 + cosx)dx

=∫(x)/(1-cos(x)) dx+∫sin(x)/(1-cos(x))dx

=I_{1 +}I₂

I₁=∫(x)/(2cos²(x/2)) dx

   =0.5∫(x)*sec²(x/2) dx ...........(use byparts, x as differential and sec²(x/2) as integral)

   =0.5[2*x*tan(x/2)  +  ∫(1)*2*tan(x/2) dx]

   =0.5[2*x*tan(x/2)  +  2*(- 2* ln(|cos(x/2)|) )]

   =xtan(x/2)-2 ln(|cos(x/2)|) +c

I₂=∫(2*sin(x/2)*cos(x/2))/(2*cos²(x/2)) dx

   =∫tan(x/2) dx

   =-2ln(|cos(x/2)|) + c