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Symbolic evaluation of sums and products

fricas
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
numeric %
There are 4 exposed and 0 unexposed library operations named numeric
having 1 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op numeric
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named
numeric with argument type(s)
Union(f1: OrderedCompletion(Expression(AlgebraicNumber)),f2: List(OrderedCompletion(Expression(AlgebraicNumber))),fail: failed,pole: potentialPole)
Perhaps you should use "@" to indicate the required return type,
or "\$" to specify which version of the function you need.

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) (5)
Type: Record(value: Float,error: Float,totalpts: Integer,success: Boolean)

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