Complete the square under the root:
#int dx/sqrt(x^2-16x-7) = int dx/sqrt((x-8)^2 -73)#
#int dx/sqrt(x^2-16x-7) = 1/sqrt73 int dx/sqrt(((x-8)/sqrt73)^2 -1)#
Substitute:
#u = (x-8)/sqrt73#
#du = dx/sqrt73#
so:
#int dx/sqrt(x^2-16x-7) = int (du)/sqrt(u^2-1)#
Restrict now to the interval #u in (1,+oo)# and substitute:
#u = cosht#
#du = sinht dt#
with #t in (0,+oo)#
#int dx/sqrt(x^2-16x-7) = int (sinht dt )/sqrt(cosh^2t-1)#
As #cosh^2t-sinh^2t = 1# and #sinht > 0# for #t>0#:
#sqrt(cosh^2t-1) = sinht#
#int dx/sqrt(x^2-16x-7) = int (sinht dt )/sinht = int dt = t+C#
undoing the substitution:
#int dx/sqrt(x^2-16x-7) = "arccosh"( (x-8)/sqrt73)+C#
and using the logarithmic form of the inverse hyperbolic cosine:
#int dx/sqrt(x^2-16x-7) = ln abs ((x-8)/sqrt73 + sqrt(((x-8)/sqrt73)^2-1))+C#
Multiplying the argument of the logarithm by #sqrt73# only changes the constant #C#, so:
#int dx/sqrt(x^2-16x-7) = ln abs ((x-8) + sqrt((x-8)^2-73))+C#
#int dx/sqrt(x^2-16x-7) = ln abs ((x-8) + sqrt(x^2 -16x-7))+C#
and by differentiating we can verify that the solution is valid also for #x-8 < -sqrt73#
