try using x=2cos(u). then you can separate the 4 in the numerator i.e 4-x^2=4(1-cos(u)^2)=4sin(u)^2 which is easy to find the square root of. the rest is pretty straightforward.
The second partial fraction seems relatively simple, all you need to do is let 1/x(x+1)=(A/x)+ (B/(x+1)), then multiplying through by x and then x+1 gives
1=A(x+1) + B(x)
then solve by substituting appropriate values for x, i.e let x=0 and then let x=-1 to find A and B respectively. you will then end up with two separate fractions to integrate with only one function of x in each denominator.