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

So,

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 Sqrt[3]Integrate[(Csc[u]-Sin[u]),u] , then with u=ArcSin[x/Sqrt[3]] to have x back, the same solution in achived.