AppellF1 calculation hangs indefinitely

Question

The built-in AppellF1 function seems generally useless. For example,

AppellF1[3/4, 1/2, 1/2, 7/4, (7 + 4 Sqrt[3]), (7 - 4 Sqrt[3])]

hangs indefinitely on my system. For the curious, this particular instance can be naturally encountered in the parameterization of minimal surfaces, for example,

\[Theta] = 38.014773989108068108 \[Pi]/180;
f = Function[z, E^(I \[Theta])/Sqrt[1 - 14 z^4 + z^8]];
g = Function[z, z];
\[Phi]k = Function[z, f[z] { 1 - g[z]^2, I (1 + g[z]^2), 2 g[z]}];
xk = Function[z, 
   Evaluate[Re[(\[Integral]\[Phi]k[w] \[DifferentialD]w)]] /. w -> z];

Many of the examples in the Wolfram documentation seem to suggest that Mathematica can deal competently with this function's asymptotes, but that contradicts my personal experience.

Mathworld mentions the following (nearly) globally convergent double contour integral:

which I believe should work for all complex values that aren't exactly negative real numbers, but even when specifying complex parameters and taking care to avoid any with principal argument exactly equal to pi, Mathematica still almost always hangs.

Plotting the function outside of anything but the most restrictive domains (painstakingly delineated via trial and error over the course of dozens of kernel quits) is obviously out of the question.

Is there any way to get anything resembling reasonable use out of this function, or a general way around using it?