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Topic: why do not CAS see that int( f'(x) ,x) = f(x) ?
Replies: 19   Last Post: Nov 18, 2017 12:46 AM

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 Nasser Abbasi Posts: 6,677 Registered: 2/7/05
why do not CAS see that int( f'(x) ,x) = f(x) ?
Posted: Nov 10, 2017 10:24 PM

This question came up in Mathematica forum.

https://mathematica.stackexchange.com/questions/159688/differentiating-and-integrating-simple-functions-like-fx-log-sinx-log-c

I tried on Maple also. The question is basically saying,
when don't CAS (Mathematica in the question) "see" right away
that

int( f'(x),x ) = f(x)

It seems to me what happens is that Maple and Mathematica
first evaluate f'(x), and then spend long time trying to
integrate the result, coming up with very complicated
anti-derivative.

I run FullSimplify[] on the anti-derivative trying
to see if will give back the original function f(x)
and its been running for hours.

Only Fricas is smart enough to do this. (unless I made
mistake in the input)Below is what I tried

Mathematica:
=============
ClearAll[f,x];
f[x_]:=Log[Sin[x]]Log[Cos[x]];
sol=Integrate[f'[x],x]
... VERY complicated result ....

Maple
=======
restart;
f:=x->log(sin(x))*log(cos(x));
int(diff(f(x),x),x);
... VERY complicated result, like Mathematica...

FriCAS
=======
http://axiom-wiki.newsynthesis.org/FriCASIntegration#bottom

\begin{axiom}
setSimplifyDenomsFlag(true)
f:=log(sin(x))*log(cos(x))
integrate(D(f,x),x)
\end{axiom}

log(sin(x))*log(cos(x))

So my question is: Why M and Maple do not see right away that
int(f'(x),x) = f(x) ? isn't this what FTOC says?

Is it just an issue of parsing? I.e. they will first evaluate
f'(x) and then try to integrate the result, and they just
need to add extra code to detect that the integrand is a derivative
of some function and do the short-circuit FTOC to return that