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 $-\frac{\pi}{2}$.

Since I don't want to do these messy calculation by myself, I throw this to SAGE with natural parametrization. Here's the code.
x, y, z, t = var("x, y, z, t")

x = (1/2)*cos(t)
y = (1/3)*sin(t)
z = sqrt(1 - x^2 - y^2)

dx = x.derivative(t)
dy = y.derivative(t)
dz = z.derivative(t)

fx = y - z^3
fy = -(2*x + z^3)
fz = (z*arctan(z))/sqrt(1 + z^2)

df = fx*dx + fy*dy + fz*dz
print N(integral(df(t), (t, 0, 2*pi)))

It returns $-1.5707963267948961 \approx -\pi/2$ so it seems your calculation is right.
