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WolframAlpha imposes a limit on the amount of computational time it allows for each computation. If you have WolframAlpha Pro, you can do computations with a higher limit on computational time but I often find that the computation times out even so.

Is there any way to get unlimited computational time, or failing that, at least more than what Pro gives you?

I'm fine if it entails using client-side computational resources, i.e. it uses my computer to do some of the computational work. The only thing is that I'd prefer not to buy Mathematica.

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As a WolframAlpha Pro member you get access to an extended time, however there is not an option to receive unlimited time. The browser will not utilize your computers resources in order to compute things unless you use their native Mathematica software.

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No. My understanding is that a query issued to wolframalpha.com requires a Mathematica kernel to do its computation, therefore it would make no sense for Wolfram|Alpha to allow any Mathematica kernel session as a computational resource to be allocated for an average user for unlimited time, especially for free.

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