Compute the following integral.
First, we multiply the numerator and denominator by and do some rearranging to get a friendlier integral,
For the integral on the left we make the substitution , and obtain
Then, for the integral on the right we make the substitution , . This gives us
Therefore for the integral on the right we have
Now, we have to use partial fractions on the integrand,
This gives us the equation
Putting these integrals back into our original expression we have
Using substitution of x=sqrt(3)sin(u) and dx=sqrt(3)cos(u) we can obtain SqrtIntegrate[(Csc[u]-Sin[u]),u] , then with u=ArcSin[x/Sqrt] to have x back, the same solution in achived.