Take the integral:  integral csc(x+1) csc(x+2) dx Write csc(x+1) csc(x+2) as 2/(cos(1)-cos(2 x+3)):   =   integral 2/(cos(1)-cos(2 x+3)) dx Factor out constants:   =  2 integral 1/(cos(1)-cos(2 x+3)) dx For the integrand 1/(cos(1)-cos(2 x+3)), substitute u = 2 x+3 and  du = 2  dx:   =   integral 1/(cos(1)-cos(u)) du For the integrand 1/(cos(1)-cos(u)), substitute s = tan(u/2) and  ds = 1/2 sec^2(u/2)  du. Then transform the integrand using the substitutions sin(u) = (2 s)/(s^2+1), cos(u) = (1-s^2)/(s^2+1) and  du = (2  ds)/(s^2+1):   =   integral 2/((s^2+1) (cos(1)-(1-s^2)/(s^2+1))) ds Simplify the integrand 2/((s^2+1) (cos(1)-(1-s^2)/(s^2+1))) to get 2/(s^2+s^2 cos(1)-1+cos(1)):   =   integral 2/(s^2+s^2 cos(1)-1+cos(1)) ds Factor out constants:   =  2 integral 1/(s^2+s^2 cos(1)-1+cos(1)) ds Factor cos(1)-1 from the denominator:   =  2 integral 1/((cos(1)-1) ((s^2 (1+cos(1)))/(cos(1)-1)+1)) ds Factor out constants:   =  2/(cos(1)-1) integral 1/((s^2 (1+cos(1)))/(cos(1)-1)+1) ds For the integrand 1/((s^2 (1+cos(1)))/(cos(1)-1)+1), substitute p = s cot(1/2) and  dp = cot(1/2)  ds:   =  (2 tan(1/2))/(cos(1)-1) integral 1/(1-p^2) dp The integral of 1/(1-p^2) is tanh^(-1)(p):   =  (2 tan(1/2) tanh^(-1)(p))/(cos(1)-1)+constant Substitute back for p = s cot(1/2):   =  csc(1/2) sec(1/2) (-tanh^(-1)(s cot(1/2)))+constant Substitute back for s = tan(u/2):   =  csc(1/2) sec(1/2) (-tanh^(-1)(cot(1/2) tan(u/2)))+constant Substitute back for u = 2 x+3:   =  csc(1/2) sec(1/2) (-tanh^(-1)(cot(1/2) tan(x+3/2)))+constant Which is equivalent for restricted x values to: Answer: |    |   =  csc(1) (log(sin(x+1))-log(sin(x+2)))+constant