I don't know that there is still interest in the mathematics of a case of a rocket with a variable proper acceleration, but there's an approach I find convenient.
One needs shared definitions of three things: proper time, denoted by ##\tau##, proper acceleration, denoted by a, and rapidity, denoted by w.
Wiki has articles on
proper time,
proper acceleration, and
rapidity. One needs to understand all three terms to follow the argument.
The key features of rapidities is that, unlike velocities, in special relativity rapiditiy add. Thus one can write
$$w(\tau) = \int dw = \int \frac{dw}{d\tau} d\tau$$
This works with rapidites (and does not work with velocities), because in 1-space, 1-time, special relativity, rapidities add linearly, while velocities add according to the
"relativistic velocity addition law".
The remaining step is to note that ##\frac{dw}{d\tau} = a/c##, the rate of change of rapidity with respect to proper time is proportional to the proper acceleration. This can be conveniently done by considering the instantaneous frame of the rocket, and noting that in that frame dv=dw/c and ##dt=d\tau##. But dv/dt in the instantaneous rest frame of the rocket is just the proper acceleration of the rocket.
So then we can solve the problem of the non-uniformly accelerating rocket by writing
$$w(\tau) = \int \frac{a(\tau)}{c} d\tau$$
This follows from the chain rule, and the additive nature of rapidity.
As a check, we can consider the case where a is constant, where we get ##w = a\tau##, We use the basic conversions of rapidity to velocity and vica-versa (see the wiki article for a discussion or look for a clearer one). As the equations imply, there is a 1:1 correspondence between velocity and rapidity, if we know the velocity, we can compute the rapidity, and vica-versa. These basic conversion equations are:
$$w = \tanh \frac{v}{c} \quad v = c \, \tanh^{-1} w$$
substituting ##a(\tau) = a## into the integral, we find
$$w = \tanh v/c = a \tau / c$$
which matches such references as
http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html for the constant proper acceleration case.