Integrate 1/x^2/(x^2+4)

I tried:

A/x + B/(x+1) + (C*x+D)/(x^2+4)

1 = A(x^3+x^2+4*x+4)+B(x^3+4*x)+(C*x+D)*(x^2+x)

1=(A+B+C)*x^3+(A+C+D)*x^2+(4*(A+B)+D)*x+4*A

let x=0..... --> A=1/4

(4*(A+B)+D)=0--> D=-4*B-1 --> D=-1/5

(A+C+D)=0 --> (1/4)+C+-4*B-1=0 --> C=4*B+3/4 --> C= -1/20

(A+B+C)=0 --> (1/4+B+(4*B+3/4) --> B=-1/5

Then I plugged in:

(1/4)/x + (-1/5)/(x+1) + ((-1/20)*x+(-1/20))/(x^2+4)

I still don't know how to get to the solution of the partial fraction step which is supposed to be:

-1/(4*(x^2+4))+1/(4*x^2)

Help me out, I have posted this before and still didn't get it.