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It seems bananas to me that Mathematica and other existing CASes can't solve that first sample equation: it reads like a straightforward integration, taking just a parameter substitution and an inverse trig derivative (which I feel like everyone doing calculus knows by heart?). Maybe I should go do it on paper and I'm missing the hard part?

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The risch algorithm is not trivial to implement.

Sure, but this is Mathematica, the self-proclaimed leading mathematics package!

That surprised me too. I found that Mathematica could solve it if you give it the denominator as 1-u^2 for the eventual substitution. Perhaps it just doesn't try very hard to look for substitutions? I could believe it not giving a lot of "coverage" per time spent compared to more advanced techniques.

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