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Let us integrate functions as explained in: IV_ (https://math.stackexchange.com/users/292527/iv), Does someone know an online-applet for the Risch-algorithm?, URL (version: 2020-08-20): https://math.stackexchange.com/q/3237569

fricas
integrate(x^2,x) (1)
Type: Polynomial(Fraction(Integer))

fricas
integrate(exp(x-x^2),x) (2)
Type: Union(Expression(Integer),...)

Cool that was easy. Now our test function is the one that appear in this math-stackexchange question: https://math.stackexchange.com/questions/4159030/the-integral-int-0x-frac-sint-sqrt1t2-textdt

fricas
integrate(sin(x)/(sqrt(1+x^2)),x) (3)
Type: Union(Expression(Integer),...)

I guess, the Risch algorithm did not find any elementary function here. I guess, this means that this function cannot be solved analytically.

By the way, let's try to modify this integral by changing a sign, thus allowing maybe a trigonometric substitution

fricas
integrate(sin(x)/(sqrt(1-x^2)),x) (4)
Type: Union(Expression(Integer),...)

Some documentation and examples of the integrate() function in FriCAS is here: http://fricas-wiki.math.uni.wroc.pl/FriCASIntegration and here: http://fricas-wiki.math.uni.wroc.pl/FriCASSpecialIntegration

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