`int(cotx+cot^3x)/(1+cot^3x)dx`

Explanation

`int(cotx+cot^3x)/(1+cot^3x)dx`
`=int(cotx)(cosec^2x))/(1+cot^3x)dx`
taking `y=cotx`
`=>dy=-cosec^2xdx`
putting these values we get,
`inty/(1+y^3)dy`
`=-inty/((1+y)(y^2-y+1))`
`=-[int(y+1)/(3(y^2-y+1))-int1/((y+1)^3)]`
`=-int [(2y-1+3)/(6(y^2-y+1))-1/3ln (y+1)]`
`=-[1/6 int(2y-1)/(y^2-y+1)+1/6 int 3/(y^2-y+1)-1/3 ln(y+1)]`
`=-[1/6 ln(y^2-y+1)+1/2int1/((y-1/2)^2+(sqrt3/2)^2)-1/3 ln (y+1)]`
`=-1/6 ln(y^2-y+1)-1/sqrt3 tan^-1((2y-1)/sqrt3)+1/3ln (y+1)]+c`
where `y=cotx`