Hi,
this is the question which can be solved by the method integration by parts. Integration by parts is technique which perform indefinite & definite integration by expanding the differential of a product of functions and expressing the original integral in terms of a known integral. It is particularly used in the functions having algebraic, trigonometry, logarithmic expressions etc. Now, for your question I gave you a hint to start the question,
Let us suppose, sec^-1(1+z)=t & 1/((3+z)(sqrt(3+z^2)-1))dz = dt
thus we have to do the integration of , t*dt
Now find the integration of t separately. Such that,
(1+z)=sint
d(1+z)=costdt
I=(integration of)(sinttdt)
I=(integration of)(tsintdt)
Now you can solve the remaining steps easily I think by integrating & putting the values.