On Sat, Sep 24, 2022 at 08:46:35AM -0700, 'Nasser M. Abbasi' via FriCAS -
computer algebra system wrote:
> " BTW: I am not sure how you maintain "has_known_anti". The following
> 9 have has_known_anti = 0, but belong to well-known class having
> elementary integrals:"
>
> Hello;
>
> I myself do not edit or modify the input files. The Rubi input files are
> maintained by Albert Rich.
> This way there is only only source. If I modify or edit these files, then
> things will get out of sync.
>
> I will forward these integrals you found to Albert Rich and let him know
> that these have known
> anti-derivatives so he can update the files on his end for next release.
> But first I would
> need to know the list of the known antiderivatives for these to send them
> as well.
>
> For example the first integral in your list is integral number 399 in file
> 4_Trig_functions/4.1_Sine/4.1.7-d_trig-^m-a+b-c_sin-^n-^p.txt
>
> And the Rubi input file says
>
> {Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2, x, 0, Unintegrable[Cos[c +
> d*x]^4/(a + b*Sin[c + d*x]^3)^2, x]}
>
> It is the 4th field above which says it is not integrable. Otherwise it
> will have the optimal
> antiderivative there.
>
> Mathematica gives solution but in terms of rootsof
>
> In[2]:= Integrate[Cos[c+d*x]^4/(a+b*Sin[c+d*x]^3)^2,x]
> Out[2]= (1/(18 a b d))(-I RootSum[-I b+3 I b #1^2+8 a #1^3-3 I b #1^4+I b
> #1^6&,(2 b ArcTan[Sin[c+d x]/(Cos[c+d x]-#1)]-I b Log[1-2 Cos[c+d x]
> #1+#1^2]+4 I a ArcTan[Sin[c+d x]/(Cos[c+d x]-#1)] #1+2 a Log[1-2 Cos[c+d x]
> #1+#1^2] #1+12 b ArcTan[Sin[c+d x]/(Cos[c+d x]-#1)] #1^2-6 I b Log[1-2
> Cos[c+d x] #1+#1^2] #1^2-4 I a ArcTan[Sin[c+d x]/(Cos[c+d x]-#1)] #1^3-2 a
> Log[1-2 Cos[c+d x] #1+#1^2] #1^3+2 b ArcTan[Sin[c+d x]/(Cos[c+d x]-#1)]
> #1^4-I b Log[1-2 Cos[c+d x] #1+#1^2] #1^4)/(b #1-4 I a #1^2-2 b #1^3+b
> #1^5)&]+(24 Cos[c+d x] (a+b Sin[c+d x]))/(4 a+3 b Sin[c+d x]-b Sin[3 (c+d
> x)]))
>
> And Rubi does not solve it
>
> In[4]:= Int[Cos[c+d*x]^4/(a+b*Sin[c+d*x]^3)^2,x]
> Out[4]= Int[Cos[c+d x]^4/(a+b Sin[c+d x]^3)^2,x]
Well, that is known weakness of Rubi.
> Maple gives solution is also in terms of rootof
>
<snip>
>
> Looking at Fricas I see it timed out on this:
>
> sqlite> select fricas_anti from main where rowid=37649;
> Timed out
As I wrote large size/timeout on such integrals is artifact of
Sage interface. You can see answer on FriCAS command line.
> But this must be because of the use of the flag set to true.
> When I tried it now without setting the flag:
>
> ii:=integrate(cos(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x);
>
> Fricas gave a very large output also using Root object. But you said
> antiderivative is
> in terms of elementary functions?
Yes, RootSum (and rootof) are considered elementary functions.
And results of integration look exactly as they should.
Here (in all 9 examples) in fact implicit roots are only used
to provide constants.
--
Waldek Hebisch
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