Why does this not integrate

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why is this integral performed:

forget()
var('n')
assume(n>0)
integrate(1/sqrt(1+x^2*n),x,1,2)

i.e. I get:

-arcsinh(sqrt(n))/sqrt(n) + arcsinh(2*sqrt(n))/sqrt(n)

while this is not

forget()
var('n')
assume(n>0)
integrate(1/sqrt(1+x^2/n),x,1,2)

asked Nov 28 '12

Mark gravatar image Mark
56 3
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Somehow the simply_full() works. It probably forces the integration to happen.

sage: integrate(1/sqrt(1+x^2/n),x,1,2).simplify_full()
-(arcsinh(1/sqrt(n)) - arcsinh(2/sqrt(n)))*sqrt(n)
link

posted Nov 28 '12

ppurka gravatar image ppurka
1896 16 32

Weird. It turns out that simplify_rational and simplify_radical do it, but not some of the others, and not just simplify (just passing to Maxima). But in Maxima itself, radcan and fullratsimp don't seem to have this effect; somehow the "nounform" is evaluated... weird.

kcrisman (Nov 28 '12)

ehem, guys, for a newbs 1st-in-a-life-time three-liner typed into sage this is a little over my head. If this nobrainer integral results in a bug already, is your point that I should use integrate() only in conjunction with simplify_full()? Or is this a 'rare' case? Or am I doing something wrong?

Mark (Nov 28 '12)

It should be considered a bug in my opinion. If the simplify process can make the integration happen, then the integration should have been performed in the first place, without the need for simplify.

ppurka (Nov 29 '12)

Correct, and it's a Maxima bug, as far as I can tell, which I've reported upstream. This should definitely be a 'rare' case, since I've never heard of this happening before (not unevaluated integrals, which are legion, but this simplify behavior).

kcrisman (Nov 29 '12)

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Asked: Nov 28 '12

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