MathAction SandBox Risch

Topics

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

Let's start with simple functions

fricas

(1) -> integrate(x^2,x)

$\label{eq1}{\frac{1}{3}}\ {{x}^{3}}$

Type: Polynomial(Fraction(Integer))

fricas

integrate(exp(x-x^2),x)

$\label{eq2}\frac{{{e}^{\frac{1}{4}}}\ {\erf \left({\frac{{2 \ x}- 1}{2}}\right)}\ {\sqrt{\pi}}}{2}$

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)

$\label{eq3}\int^{ \displaystyle x}{{\frac{\sin \left({\%A}\right)}{\sqrt{{{\%A}^{2}}+ 1}}}\ {d \%A}}$

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)

$\label{eq4}\int^{ \displaystyle x}{{\frac{\sin \left({\%A}\right)}{\sqrt{-{{\%A}^{2}}+ 1}}}\ {d \%A}}$

Type: Union(Expression(Integer),...)

Some documentation and examples of the integrate() function in FriCAS is here: FriCASIntegration and here: FriCASSpecialIntegration