Calculating line integral I'm working on this problem:

Calculate integral
  \begin{equation}
\int_C\frac{z\arctan(z)}{\sqrt{1+z^2}}\,dz + (y-z^3)\,dx - (2x+z^3)\,dy,
\end{equation}
  where the contour $C$ is defined by equations
  $$ 
\sqrt{1-x^2-y^2}=z, \quad 4x^2+9y^2 = 1.
$$

Seems to me that I know the solution, but I have  feeling that I could lost something. Would you help me to clarify this.
First, it is easy to parametrize the contour: $x=\frac{1}{2}\cos\varphi$, $y=\frac{1}{3}\sin\varphi$, $z=\sqrt{1-\frac{\cos^2\varphi}{4}-\frac{\sin^2\varphi}{9}}$ and $\varphi$ goes from $0$ to $2\pi$. So we will have the integral $\int_0^{2\pi}F(\varphi)\,d\varphi$, where the function $F(\varphi)$ is quite complicated.
But I thought about another method. The contour is symmetric and it would provide some simplifications:
When variable $z$ goes up and down (on the contour) the value of the function $\frac{z\arctan(z)}{\sqrt{1+z^2}}$ are same in such up-and-down points. So I can write
$$
\int_C\dots = \int_C + (y-z^3)\,dx - (2x+z^3)\,dy.
$$ 
The same I can conclude for variable $x$ and function $z^3$ and for variable $y$ and function $z^3$. So
$$
\int_C\dots = \int_C  y\,dx - (2x+z^3)\,dy = \int_C y\,dx - 2x\,dy.
$$ 
After that it is much easier to compute the integral using parameterization.
$$
\int_C y\,dx - 2x\,dy = \int_0^{2\pi}-\frac{1}{6}\sin^2\varphi -2\frac{1}{6}\cos^2\varphi \, d\varphi = -\frac{1}{6}\int_0^{2\pi}1+\cos^2\varphi\,d\varphi = -\frac{\pi}{2}
$$
So the answer is $-\frac{\pi}{2}$.

It's absolutely fine to exploit the symmetries in the given problem. But we need a clear cut argument. Observing that a "variable goes up and down" doesn't suffice.
Relevant are the following symmetries in the parametrization of $C$:
$$x(\phi+\pi)=-x(\phi),\quad y(\phi+\pi)=-y(\phi),\quad z(\phi)=z(-\phi)=z(\phi+\pi)\ .$$
This implies
$$\dot x(\phi+\pi)=-\dot x(\phi),\quad \dot y(\phi+\pi)=-\dot y(\phi),\quad \dot z(-\phi)=-\dot z(\phi)\ .$$
When computing $W:=\int_C \bigl(P\ dx+Q\ dy+ R\ dz\bigr)$ for $P$, $Q$, $R$ as in your question we therefore immediately get
$$\eqalign{\int_C P\ dx&=\int_0^{2\pi}(y(\phi)-z^3(\phi))\dot x(\phi)\ d\phi=\int_0^{2\pi}y(\phi)\ \dot x(\phi)\ d\phi=-{1\over6}\int_0^{2\pi}\sin^2\phi\ d\phi=-{\pi\over6},\cr
\int_C Q\ dy&=\int_0^{2\pi}(-2x(\phi)-z^3(\phi))\dot y(\phi)\ d\phi=-2
\int_0^{2\pi}x(\phi)\ \dot y(\phi)\ d\phi=\ldots=-{\pi\over3},\cr
\int_C R\ dz&=\int_{-\pi}^\pi \tilde R(\phi)\dot z(\phi)\ d\phi=0\cr}$$
(where we have used that $\tilde R(-\phi)=\tilde R(\phi)$). It follows that $W=-{\pi\over2}$.
