For a fixed $\gamma > 0$, consider the family of curves parametrized by $\Theta=(\vec a,\vec b)$:
$$\vec v_\Theta(t) = \frac{\vec a + e^{\gamma t}\vec b}{||\vec a + e^{\gamma t}\vec b||_2}$$
I'm interested in their integrals
$$\vec x_\Theta(t) = \int_0^t \vec v(t')\,dt'$$ $$\vec w_\Theta(t) = \int_0^t e^{\gamma t'}\vec v(t')\,dt'$$
I don't need $\vec x_\Theta(t)$ and $\vec w_\Theta(t)$ as functions of $\Theta=(\vec a,\vec b)$. Rather, I'd like to know if there's a nice form for a general element of the set
$$\{\vec x_\Theta(\cdot): \Theta\in \mathbb R^n \times \mathbb R^n\}$$
and similarly for $\vec w_\Theta(\cdot)$. For example, an answer might take the form of another parametric family, parametrized by $\Theta$ or by something more convenient. I suppose there is no loss of generality in assuming we're in two dimensions (spanned by $\vec a$ and $\vec b$), in which case we can say that $\vec v$ is a unit vector whose direction satisfies
$$\tan\theta(t) = \frac{a_y + e^{\gamma t}b_y}{a_x + e^{\gamma t}b_x}$$
Are these well-known types of curves? At the very minimum, is there a faster and higher-precision way to compute the curves than numerical integration?
Project $\vec x$ into directions along $\hat a$ and $\hat b$. Then $$x_a(t)-x_a(0)=\int_0^t\frac 1{\sqrt{a^2+b^2e^{2\lambda t'}}}dt'$$ If either $a$ or $b$ are zero, the integral is trivial. If both are nonzero, use $c=b/a$ and $-2\lambda t'=\tau$ to get an integral $$\int\frac1{\sqrt{1+c^2e^{-\tau}}}d\tau$$ Now use $u=1+c^2e^{-\tau}$ to get an integral of the form $$\int\frac1{(u-1)\sqrt u}du$$ Substitute $v=\sqrt u$ and you get $$\int\frac1{v^2-1}dv$$ You will need to get all the constants and the limits. The last integral is trivial. Similarly for the other direction: $$x_b(t)-x_b(0)=\int_0^t\frac {be^{\lambda t'}}{\sqrt{a^2+b^2e^{2\lambda t'}}}dt'$$ Use $u=\frac bae^{\lambda t'}$ to get an integral of the form $$\int\frac1{\sqrt{1+u^2}}du$$ This is a standard integral