Known conversion between Euler angle sequences?

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Is there a simple close form formula for converting angles in one Euler angle sequence to another?

For example if one knows the Tait–Bryan angles (pitch, yaw, roll or XYZ) can one easily find the Euler angles $\alpha,\beta, \gamma$ corresponding to the sequence $ZX'Z''$?

I realize that I could write down the explicit matrix form for both, but I suspect that would result in some rather intractable equations. $$\text{Tait–Bryan angles}\quad\quad\quad\quad\quad\text{Euler angles}$$ $\quad\quad\quad\quad\quad\quad$ enter image description here $\quad$ enter image description here