My guess was that the symbolic output of Integrate could then be evaluated to much higher numerical precision in a reasonable time, but I couldn't transform the result using simple Mathematica operations into something tractable.
It's a bit of a challenge to get either the first or second forms you provided to converge if requesting 10 or 15 digits of precision. However, your third form involving the Gamma function evaluates to very high numerical precision quickly. I can get 1,000 digits in under 50 milliseconds, which is "good enough"!
Your usage of NIntegrate tuning options shows a few things:
1) That these days I only use Mathematica as a glorified calculator, evoking a mental image of a 500-ton press being used to crack walnuts.
2) Even the best symbolic CAS in the world isn't magic, and requires hand-holding.
3) With the right knowledge even a very hard nut can be easily cracked!
I wonder why Wolfram Research hasn't added some basic multi-threading capabilities to Mathematica to try different "flags" in parallel, racing the various approaches on each CPU core to see which one wins...
It's a bit of a challenge to get either the first or second forms you provided to converge if requesting 10 or 15 digits of precision. However, your third form involving the Gamma function evaluates to very high numerical precision quickly. I can get 1,000 digits in under 50 milliseconds, which is "good enough"!
Your usage of NIntegrate tuning options shows a few things:
1) That these days I only use Mathematica as a glorified calculator, evoking a mental image of a 500-ton press being used to crack walnuts.
2) Even the best symbolic CAS in the world isn't magic, and requires hand-holding.
3) With the right knowledge even a very hard nut can be easily cracked!
I wonder why Wolfram Research hasn't added some basic multi-threading capabilities to Mathematica to try different "flags" in parallel, racing the various approaches on each CPU core to see which one wins...