Prove that
`3 tan^(-1) x= {(tan^(-1) ((3x - x^(3))/(1 - 3x^(2))),"if " -(1)/(sqrt3) lt x lt (1)/(sqrt3)),(pi + tan^(-1) ((3x - x^(3))/(1 - 3x^(2))),"if " x gt (1)/(sqrt3)),(-pi + tan^(-1) ((3x - x^(3))/(1 - 3x^(2))),"if " x lt - (1)/(sqrt3)):}`

Explanation

Let `x = tan theta`, where `theta in (-pi//2, pi//2)`
`:. Tan^(-1). (3 x -x^(3))/(1 - 3x^(2)) = tan^(-1).(3 tan theta - tan^(3) theta)/(1 -3 tan^(2) theta)`
`= tan^(-1)(tan 3 theta), " where " 3 theta in (-3pi//2, 3 pi//2)`
`:. Tan^(-1).(3x - x^(3))/(1 - 3x^(2)) = {(3 theta,"if "-(pi)/(2) lt 3 theta lt (pi)/(2)),(3 theta - pi,"if " (pi)/(2) lt 3 theta lt (3pi)/(2)),(3 theta + pi,"if " -(3pi)/(2) lt 3 theta lt -(pi)/(2)):}`
Now if `-(pi)/(2) lt 3 theta lt (pi)/(2)`
`-(pi)/(2) lt 3 tan^(-1) x lt (pi)/(2)`
`rArr -(pi)/(6) lt tan^(-1) x lt (pi)/(6)`
`rArr -(1)/(sqrt3) lt x lt (1)/(sqrt3)`
Similarly, from `(pi)/(2) lt 3 theta lt (3pi)/(2)`, we get `x gt (1)/(sqrt3)`
And from `-(3pi)/(2) lt 3 theta lt -(pi)/(2)`, we get `x lt -(1)/(sqrt3)`
Thus, tan^(-1).(3x -x^(3))/(1 - 3x^(2)) = {(3 tan^(-1) x,"if " -(1)/(sqrt3) lt x lt (1)/(sqrt3)),(3 tan^(-1) x - pi," if " (1)/(sqrt3) lt x lt oo),(3 tan^(-1) x + pi,"if " -oo lt x lt -(1)/(sqrt3)):}`