Deriving Bloch vector $dr$ from master equation

I am trying to derive the Bloch vector $dr$ for a measurement of a observable in any arbitrary direction $theta$. For context this is the setup and derivation I have for continuous measurement along the $z$ axis.

The equation of continuous measurement on observable X has the following form:

$frac{d rho}{d t}=mathcal{D}[X] rho+sqrt{eta} mathcal{H}[X] rho xi(t)$

$mathcal{D}[X] rho=X rho X^{dagger}-frac{1}{2}left(X^{dagger} X rho+rho X^{dagger} Xright)$

$mathcal{H}[X] rho=X rho+rho X-leftlangle X+X^{dagger}rightrangle rho$

Kappa is the measurement strength.

In Bloch Vector form,

$rho=frac{1}{2}left(I+x sigma_{x}+y sigma_{y}+z sigma_{z}right)$

Then,

$mathcal{D}[X] rho=2 kappaleft(sigma_{z} rho sigma_{z}-rhoright)$

$mathcal{H}[X] rho=sqrt{2 kappa}left(sigma_{z} rho+rho sigma_{z}-2 z rhoright)$

To find $dx$

$frac{d x}{d t}=frac{d T rleft(sigma_{x} rhoright)}{d t}=2 kappaleft(operatorname{Tr}left(sigma_{z} sigma_{x} sigma_{z} rhoright)-xright)+sqrt{2 kappa eta}left(operatorname{Tr}left(left(sigma_{x} sigma_{z}+sigma_{z} sigma_{x}right) rhoright)-2 x zright) xi(t)$

$=-4 kappa x-sqrt{8 kappa eta} x z xi(t)$

$frac{d z}{d t}=frac{d operatorname{Tr}left(sigma_{z} rhoright)}{d t}=2 kappaleft(operatorname{Tr}left(sigma_{z}^{2} rho sigma_{z}right)-operatorname{Tr}left(sigma_{z} rhoright)right)+sqrt{2 kappa eta}left(operatorname{Tr}left(sigma_{z}^{2} rho+sigma_{z} rho sigma_{z}right)-2 z operatorname{Tr}left(sigma_{z} rhoright)right) xi(t)$

$=sqrt{8 kappa eta}left(1-z^{2}right) xi(t)$

Now I am trying to find the $frac{d z}{d t}$ and $frac{d x}{d t}$, in the case that the $mathcal{D}[X]$ and $mathcal{H}[X]$ terms for the same master equation where becomes $X = cos(Theta )sigma _z+sin(Theta )sigma_x$ along measurement angle $theta$ are:

$mathcal{D}[X] rho=X rho X^{dagger}-frac{1}{2}left(X^{dagger} X rho+rho X^{dagger} Xright)$

$= cos^2(Theta )sigma _zrho sigma _z+sin^2(Theta )sigma _xrho sigma _x$

I am confused on what the simplified form of $mathcal{H}[X]rho$ will be as my simplification skills are not very strong

I would also need help in finding $frac{d T rleft(sigma_{x} rhoright)}{d t}$ and $frac{d T rleft(sigma_{z} rhoright)}{d t}$ with the $sin$ and $cos$ terms like the above simplification.