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SandBox Risch

Edit detail for SandBox Risch revision 3 of 4

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Editor: Shoenemann Time: 2021/06/06 17:34:41 GMT+0
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By the way, let's try to modify this integral by changing a sign, thus allowing maybe a trigonometric substitution


\begin{axiom}
integrate(sin(x)/(sqrt(1-x^2)),x)
\end{axiom}

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

integrate(x^2,x)

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

(1)

Type: Polynomial(Fraction(Integer))

fricas

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

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

(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)

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

(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)

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

(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