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int = Integrate[1/(x^3 - 1), x];
Map[Framed, int, Infinity]

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When I called Wolfram|Alpha it gave the answer 1/4. You can check that it is correct with an estimate:

f[x_, y_] = (x^2*(1 + Sin[y]) + y^2)/(4 x^2 + 4 y^2);
Plot3D[f[x, y],
 {x, -1, 1}, {y, -1, 1}]
Reduce[RealAbs[f[x, y] - 1/4] <= RealAbs[x]/8, Reals]

While it is true that Wolfram|Alpha can interpret some Mathematica syntax, it is not able to interpret all Mathematica syntax and that presumably means Mathematica syntax is not all being interpreted in exactly the same way by Mathematica and Wolfram|Alpha.

In examples I have seen and tried Wolfram|Alpha does seem to correctly evaluate limits involving a single variable. If you find any counterexample to that then please make that known.

I have seen examples in the past where multivariate limits do not seem to all be understood and supported by Wolfram|Alpha, as you seem to have demonstrated.

Possibly Alpha is running an older version that does not have more recent methods for handling multivariate limits.