cos(x) / ( 1 + cos(x) ) = [cos(x) / (1 + cos(x) )][ (1 - cos(x))/(1-cos(x))]
= cos(x)(1 - cos(x))/ ( (1+cos(x))(1-cos(x)) )
= cos(x)(1-cos(x) )/ ( 1-cos^2(x) )
= cos(x)(1 - cos(x))/sin^2(x)
= (cos(x) - cos^2(x)) / sin^2(x)
= (cos(x) - (1 - sin^2(x)) / sin^2(x)
= (cos(x) - 1 + sin^2(x) ) / sin^2(x)
= cos(x)/sin^2(x) - 1/sin^2(x) + sin^2(x)/sin^2(x)
= cos(x)sin^2(x) - cosec^2(x) + 1
Thus,
int ( cos(x)/(cos(x) + 1) dx ) = int( cos(x)/sin^2(x) dx ) - int(cosec^2(x) dx) + int(dx)
= -1/sin(x) + cot(x) + x + C ; C = integration constant
= -cosec(x) + cot(x) + x + C
