#int x^2/(3-x^2) color(red)(dx)#
#= int ((- 3 + x^2) + 3)/(3-x^2) dx#
#= int -1 + (3)/(3-x^2) dx#
#= -x + color(blue)( int (3)/(3-x^2) dx) star#
For the blue bit, we will use the hyperbolic identity:
#1 - tanh^2 y = sech^2 y#
So we let #x = sqrt 3 tanh y implies dx =sqrt 3 sech^2 y dy# so the blue part of #star# becomes:
# int (3)/(3-3 tanh^2 y) sqrt 3 sech^2 y \ dy#
#= int (3)/(3-3 tanh^2 y) sqrt 3 sech^2 y \ dy#
#= int 1/sech^2 y sqrt 3 sech^2 y \ dy#
#= sqrt 3 int dy#
#= sqrt 3 tanh^(-1) (x/ (sqrt 3)) + C#
Feeding this into #star#
#implies - x + sqrt 3 tanh^(-1) (x/ (sqrt 3)) + C#
