When I attempt to solve this integral using Wolfram Alpha, it says that no integral exists in standard mathematical form. Why is it so? It is possible to integrate this?
The bottom limit is 0 and the upper limit is 23260.
2026-04-03 04:00:45.1775188845
How to solve this integral relating to time taken on an ellipse?
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The Risch algorithm provides a formal way to check whether an integral is elementarily solvable. It would return a negative result for this integral, in spite of its (usually) long running time.
However, there is a simpler, if less rigorous, way to see this: the preponderance of square-rooted terms that cannot be simplified, because they are added to other terms with different square roots, or none at all. Even the simpler $$\int\sqrt{\frac1{\sqrt x+1}}\,dx$$ is not elementarily solvable.