Given a rocket with known initial position, linear velocity, linear acceleration, orientation, angular velocity, and angular acceleration, and knowing that the linear acceleration will always be in the direction of the rocket's orientation, how would I find its position after time t?
I came up with the following equations: $$ \theta(t)=\theta+\omega t+\frac{\alpha t^2}{2} $$ $$ \vec{a(t)}=a*\hat{\theta(t)} $$ $$ \vec{v(t)}=\vec{v}+\int_0^t\vec{a(x)}dx $$ $$ \vec{r(t)}=\vec{r}+\int_0^t\vec{v(x)}dx $$ However, calculating the Cartesian coordinates of the rocket proves to be extremely messy. Plugging the calculus into Wolfram Alpha yields an expression with Fresnel integrals when solving for x or y from taking the sine or cosine of the quadratic term in the equation, which I have neither the experience nor the desire to deal with.
For calculating something like this, is there a simpler approach that I am missing? Am I doing something wrong?