Composition of two axis-angle rotations

Please note that I am not referring to Euler angles of the form (α,β,γ). I am referring to the axis-angle representation, in which a unit vector indicates the direction axis of a rotation and a scalar the magnitude of the rotation.

Let $(\hat{n_1},\theta_1)$ refer to the first rotation and $(\hat{n_2},\theta_2)$ refer to the second rotation. What is the value of the first rotation followed by the second rotation, in axis-angle representation?

I understand that the composition of two rotations represented by quaternions $q_1$ and $q_2$ is equal to their product $q_2q_1$. Is there a way to find the composition of axis-angle rotations (without having to convert them to quaternions, multiply them, and convert them back to axis-angle) in a similar manner? Is there a simplified formula for this operation?

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I do not believe there is without passing through some alternate representation (quaternion, matrix, ...). This is one of the known disadvantages of axis-angle compared to the others, while an advantage is the triviality of inversion (simply negate the angle or the axis).

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The quaternion procedure is probably the simplest, easiest to implement, and most computationally economical way to go.

In practice you would likely be doing all of this in a computer anyhow, and computing the product of two quaternions (in the big scheme of things) is not much harder than two real numbers, or two complex numbers. I think the multiplication is more computatationally efficient than multiplying two $3\times 3$ matrices, at least.

Actually, if you sit down and work the quaternion solution, you can probably work out a formula completely in terms of the coordinates of the $n_i$ and the angles $\theta_i$. It would be monstrous, but it would be totally in terms of your data (and maybe inverse trigonometric functions.)

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