Integrate 1+sinx/sinx(1+cosx)
Numerator can be written as (cos(x/2) + sin(x/2) )2
Denominator can be written as 2sin(x/2).cos(x/2){ 2cos2(x/2) }
Now split to get ,
1/4 { cosecx + tan(x/2).sec2(x/2) + 2sec2(x/2) }
which can be integrated as
I = 1/4 { log(cosecx - cotx) + tan2(x/2) + 4tan(x/2) } + C