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Symbolic evaluation of sums and products

fricas
(1) -> sum(i, i=1..n)
 (1)
Type: Fraction(Polynomial(Integer))
fricas
product(x+k, k=0..n-1)
 (2)
Type: Expression(Integer)

Result should be pi

fricas
integrate((4*sqrt(1/2)-8*x^3-4*sqrt(1/2)*x^4-8*x^5)/(1-x^8), x=0..sqrt(1/2))
 (3)
Type: Union(pole: potentialPole,...)

Use "noPole"

Axioms answer is "potentialPole", which indicates that there might be a pole within the interval of integration. If you are sure, that there is no pole within this interval, use "noPole":

fricas
integrate((4*sqrt(1/2)-8*x^3-4*sqrt(1/2)*x^4-8*x^5)/(1-x^8), x=0..sqrt(1/2), "noPole")
 (4)
Type: Union(f1: OrderedCompletion?(Expression(AlgebraicNumber?)),...)

It's numeric value is roughly

fricas
%::Expression Float
 (5)
Type: Expression(Float)

To check, do a numeric integration:

fricas
romberg(x+->(4*sqrt(1/2)-8*x^3-4*sqrt(1/2)*x^4-8*x^5)/(1-x^8),0.0,sqrt(1/2)::Float,0.1,0.1,6,10)
 (6)
Type: Record(value: Float,error: Float,totalpts: Integer,success: Boolean)

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