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{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 
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0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Another example for the ex
ponential case." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 357 "restart
; f := (1+exp(x^2+1)+2*x^2+2*x^2*exp(x^2+1)-7*exp(x^2+1)^2*x+4*exp(x^2
+1)^4*x-6*x*exp\n(x^2+1)^3+6*x^2*exp(x^2+1)^3-2*x^3*exp(x^2+1)+8*exp(x
^2+1)^2*x^2-12*exp(x^2+1)\n^6*x+12*exp(x^2+1)^4*x^2-2*exp(x^2+1)^2+3*e
xp(x^2+1)^4-exp(x^2+1)^3-exp(x^2+1)\n^6-4*x^3-4*exp(x^2+1)^8*x+8*exp(x
^2+1)^6*x^2-16*exp(x^2+1)^2*x^3+8*x^4)/exp(x^\n2+1)^2/(-exp(x^2+1)^2+x
)^2;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"fG*&,L\"\"\"F'-%$expG6#,&*
$)%\"xG\"\"#\"\"\"F'F'F'F'F,F/*&F-F0F(F'F/*&)F(F/F0F.F'!\"(*&)F(\"\"%F
0F.F0F7*&F.F0)F(\"\"$F0!\"'*&F-F0F9F0\"\"'*&)F.F:F0F(F0!\"#*&F3F0F-F0
\"\")*&)F(F=F0F.F0!#7*&F6F0F-F0\"#7*$F3F0F@*$F6F0F:*$F9F0!\"\"*$FDF0FK
*$F?F0!\"%*&)F(FBF0F.F0FN*&FDF0F-F0FB*&F3F0F?F0!#;*$)F.F7F0FBF0*&)F(\"
\"#F0),&FHFKF.F'\"\"#F0!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 96 "theta:=exp(x^2+1); usef:=proc(a) subs(Theta=theta,a) end; usev:=
proc(a) subs(theta=Theta,a) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
&thetaG-%$expG6#,&*$)%\"xG\"\"#\"\"\"\"\"\"F.F." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%usefGR6#%\"aG6\"F(F(-%%subsG6$/%&ThetaG%&thetaG9$F(F
(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%usevGR6#%\"aG6\"F(F(-%%subsG
6$/%&thetaG%&ThetaG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "denom(normal(f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)-%$expG6#,
&*$)%\"xG\"\"#\"\"\"\"\"\"F.F.F,F-),&*$F$F-!\"\"F+F.F,F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sqrfree(usev(%),Theta);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"7$7$%&ThetaG\"\"#7$,&*$)F'F(\"\"\"F$
%\"xG!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A := (c[0]+
c[1]*theta)/(theta^2-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG*&,&
&%\"cG6#\"\"!\"\"\"*&&F(6#F+F+-%$expG6#,&*$)%\"xG\"\"#\"\"\"F+F+F+F+F+
F7,&*$)F/F6F7F+F5!\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
54 "rem(usev(numer(normal(f-diff(A,x)))),Theta^2-x,Theta);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,4*&,.\"\"\"F&*$)%\"xG\"\"#\"\"\"!\"%*$)F)\"
\"$F+\"\"%F)!\"\"*&&%\"cG6#F&F&F)F&F1*&F.F+F3F+F0F&%&ThetaGF&F&F&F&F'!
\"#*$)F)F0F+!\")F-\"\"(*&F.F+&F46#\"\"!F&F0*$)F)\"\"&F+F0F)F8*&F>F+F)F
+F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solve( \{coeffs(%,Th
eta)\}, \{c[0],c[1]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/&%\"cG6#
\"\"!,$*&,**$)%\"xG\"\"$\"\"\"\"\"\"*$)F.\"\"#F0!\"#F.F4!\"\"F1F0F.!\"
\"F6/&F&6#F1,$*&,&F.F1F6F1F0F.F7F6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 74 "A:=subs(%,A); newf:=normal(f-diff(A,x)); sqrfree(usev
(denom(newf)),Theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG*&,&*&,
**$)%\"xG\"\"$\"\"\"\"\"\"*$)F+\"\"#F-!\"#F+F1!\"\"F.F-F+!\"\"F3*&*&,&
F+F.F3F.F.-%$expG6#,&F/F.F.F.F.F-F+F4F3F-,&*$)F8F1F-F.F+F3F4" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%%newfG*&,F*$)%\"xG\"\"&\"\"\"\"\")*&)F)\"
\"%F+)-%$expG6#,&*$)F)\"\"#F+\"\"\"F8F8F/F+!\"%*$F.F+F9*&F.F+F1F8!\"#*
&F.F+)F1F7F+!\")*&F>F+)F)\"\"$F+FB*&FAF+)F1\"\"'F+F/*&FAF+F0F+\"#7*&FA
F+F1F+F7*$FAF+F7*&FAF+)F1FBF+F7*&F6F+FKF+F<*&F>F+F6F+!\"&*&F0F+F6F+F8*
&F)F8F1F+F8F)F8*$FKF+!\"\"*$F>F+FRF+*()F)\"\"#F+,&FSFRF)F8\"\"\")F1\"
\"#F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*$)%\"xG\"\"#\"\"\"!
\"\"7$7$,&*$)%&ThetaGF(F)\"\"\"F'F*F17$F0F(" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 69 "Pole order = 1.  So then (and only then) we use the res
ultant method." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "a,b:=nume
r(newf),denom(newf): bprime:=diff(b,x):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "factor(resultant( usev(a-bprime*z), usev(b), Theta));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.)%\"xG\"#?\"\"\"),&F%\"\"#!\"\"
\"\"\"F*F'),&F%F*F,F,F*F'),&\"\"$F,%\"zGF,F*F'),(*$)F%F*F'\"\"%F%F*F,F
,F*F'),(F5F*F%!\"#F,F,F*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "\{solve(%,z)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#!\"$" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Only 1 residue,  -3." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "usev(normal(a-bprime*(-3))), usev(b
);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$,H*$)%\"xG\"\"&\"\"\"\"\")*&)F&
\"\"%F()%&ThetaGF,F(!\"%*$F+F(F/*&F+F(F.\"\"\"!\"#*&F+F()F.\"\"#F(F,*&
F5F()F&\"\"$F(F9*&F8F()F.\"\"'F(F,*&F8F(F-F(!#7*&F8F(F.F(F6*$F8F(F6*&F
8F()F.F9F(F6*&)F&F6F(FBF(F3*&F5F(FDF(F,*&F-F(FDF(F2*&F.F(F&F2F2F&F2*$F
BF(!\"\"*$F5F(FI*&F-F(F&F(!\"'*(FDF(,&FJFIF&F2F2F5F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "gcd(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,&*$)%&ThetaG\"\"#\"\"\"!\"\"%\"xG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 16 "A2 := -3*log(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#A2G,$-%#lnG6#,&*$)%&ThetaG\"\"#\"\"\"!\"\"%\"xG\"\"\"!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "normal(newf-diff(A2,x));" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,fo*()%\"xG\"\"$\"\"\")-%$expG6#,&*$
)F'\"\"#F)\"\"\"F2F2\"\"'F))%&ThetaGF1F)!\"%*$)F'F3F)\"\")*&)F'\"\"%F)
)F+F<F)\"#7*&F;F)F+F2F1*&F;F))F+F1F)F(*&F&F)F=F)F2*&F&F))F+F(F)!\"#*&F
;F)F4F)F<*&F4F)F&F)FE*&F4F)F'F2!\"\"*(F;F)F=F)F4F)F<*(F;F)FAF)F4F)F9*&
F0F)F+F)F2*&FAF)F'F)FI*&F'F)FDF)FIF/F2*$F;F)F1*&F=F)F0F)!\"$*&FAF)F&F)
FE*(F;F)F+F)F4F)F1*(FAF)F&F)F4F)FQ*$)F'\"\"&F)F6*(F&F)F=F)F4F)!#7*(F&F
)FDF)F4F)FE*(F0F)FDF)F4F)F1*(FAF)F0F)F4F)FW*(F=F)F0F)F4F)FI*(F+F)F'F)F
4F)FI*&FVF)F4F)!\")*&F;F)FDF)F1*&FVF)F+F)FE*(F&F)F+F)F4F)FE*&F;F)F*F)F
<*&FVF)FAF)Fjn*&F=F)FVF)F6*&FDF)F4F)F2*&FAF)F4F)F2F)**)F'\"\"#F),&*$FA
F)FIF'F2\"\"\")F+\"\"#F),&*$F4F)FIF'F2\"\"\"!\"\"" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 77 "This is wrong: I have both exp(..) and the variabl
e Theta in this expression." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 10 "Try again:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "A2:=usef(A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A
2G,$-%#lnG6#,&*$)-%$expG6#,&*$)%\"xG\"\"#\"\"\"\"\"\"F5F5F3F4!\"\"F2F5
!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "normal(newf-diff(A2
,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,4*$)%\"xG\"\"%\"\"\"\"\")*
&)F'\"\"$F))-%$expG6#,&*$)F'\"\"#F)\"\"\"F6F6F(F)!\"%*$F,F)F7*&F,F)F/F
6!\"#*&)F/F5F)F4F)!\"\"F3F5*&F4F)F/F)F5F/F6F6F6F)*&)F/\"\"#F))F'\"\"#F
)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "newf:=collect(%,t
heta,normal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%newfG,**&)-%$expG6
#,&*$)%\"xG\"\"#\"\"\"\"\"\"F1F1F/F0F.F1!\"%!\"\"F1*&,(*$)F.\"\"$F0F/F
3F1F,!\"#F0*&)F.\"\"#F0F(\"\"\"!\"\"F3*&,*F,F/*$)F.\"\"%F0\"\")F6F2F1F
1F0*&)F.\"\"#F0)F(\"\"#F0F>F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "
I have a term of" }}{PARA 0 "" 0 "" {TEXT -1 10 "  degree 2" }}{PARA 
0 "" 0 "" {TEXT -1 10 "  degree 0" }}{PARA 0 "" 0 "" {TEXT -1 11 "  de
gree -1" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT 
-1 11 "  degree -2" }}{PARA 0 "" 0 "" {TEXT -1 9 "in theta." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "We integrate th
e terms seperately. Lets assign each of these terms to some variable:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "for i from ldegree(new
f,theta) to degree(newf,theta) do\n    term[i] := coeff(newf,theta,i) \+
* theta^i\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%termG6#!\"#*&,**
$)%\"xG\"\"#\"\"\"F-*$)F,\"\"%F.\"\")*$)F,\"\"$F.!\"%\"\"\"F7F.*&)F,\"
\"#F.)-%$expG6#,&F*F7F7F7\"\"#F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>&%%termG6#!\"\",$*&,(*$)%\"xG\"\"$\"\"\"\"\"#F'\"\"\"*$)F-F0F/!\"#
F/*&)F-\"\"#F/-%$expG6#,&F2F1F1F1\"\"\"!\"\"F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%%termG6#\"\"!!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>&%%termG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%termG6
#\"\"#,$*&)-%$expG6#,&*$)%\"xGF'\"\"\"\"\"\"F3F3F'F2F1F3!\"%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "The degree 0 term can be integrat
ed with \"int\". That's OK because it contains no theta (so it is in a
 smaller differential field)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "TERM[0] := int(term[0],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&
%%TERMG6#\"\"!,$%\"xG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Now
 the other terms:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 41 "Lets start with the term of degree k = -2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "k := -2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"kG!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "TERM[k] := c(x)*theta^k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%TE
RMG6#!\"#*&-%\"cG6#%\"xG\"\"\"*$)-%$expG6#,&*$)F,\"\"#F-\"\"\"F7F7\"\"
#F-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "ZERO := normal( term[k]
 - diff(TERM[k], x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ZEROG*&,.
*$)%\"xG\"\"#\"\"\"F**$)F)\"\"%F+\"\")*$)F)\"\"$F+!\"%\"\"\"F4*&-%%dif
fG6$-%\"cG6#F)F)F4F(F+!\"\"*&F9F4F1F+F.F+*&)-%$expG6#,&F'F4F4F4\"\"#F+
)F)\"\"#F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ZERO:=ZE
RO*theta^(-k);  # get rid of theta" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%%ZEROG*&,.*$)%\"xG\"\"#\"\"\"F**$)F)\"\"%F+\"\")*$)F)\"\"$F+!\"%\"\"
\"F4*&-%%diffG6$-%\"cG6#F)F)F4F(F+!\"\"*&F9F4F1F+F.F+*$)F)\"\"#F+!\"\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ratsols(ZERO,c(x));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7\",$*&,(\"\"\"F(%\"xG!\"\"*$)F)\"\"
#\"\"\"F-F.F)!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "TER
M[k] := subs(c(x)=%[2],TERM[k]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&
%%TERMG6#!\"#,$*&,(\"\"\"F+%\"xG!\"\"*$)F,\"\"#\"\"\"F0F1*&F,\"\"\")-%
$expG6#,&F.F+F+F+\"\"#F1!\"\"F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
104 "Now the term of degree k = -1   (I'll simply copy the above comma
nds, and just replace k:=-2 by k:=-1). " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "k:=-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG!\"\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "TERM[k] := c(x)*theta^
k; ZERO := normal( term[k] - diff(TERM[k], x) ); ZERO:=ZERO*theta^(-k)
; ratsols(ZERO,c(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%TERMG6#!
\"\"*&-%\"cG6#%\"xG\"\"\"-%$expG6#,&*$)F,\"\"#F-\"\"\"F5F5!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ZEROG*&,,*$)%\"xG\"\"$\"\"\"!\"#\"
\"\"F-*$)F)\"\"#F+F0*&-%%diffG6$-%\"cG6#F)F)F-F/F+!\"\"*&F5F-F(F+F0F+*
&)F)\"\"#F+-%$expG6#,&F.F-F-F-\"\"\"!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%ZEROG*&,,*$)%\"xG\"\"$\"\"\"!\"#\"\"\"F-*$)F)\"\"#F+
F0*&-%%diffG6$-%\"cG6#F)F)F-F/F+!\"\"*&F5F-F(F+F0F+*$)F)\"\"#F+!\"\"" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7\"*&,&%\"xG\"\"\"!\"\"F(\"\"\"F'!
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "TERM[k] := subs(c(x
)=%[2],TERM[k]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%TERMG6#!\"\"*&
,&%\"xG\"\"\"F'F+\"\"\"*&F*\"\"\"-%$expG6#,&*$)F*\"\"#F,F+F+F+\"\"\"!
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "The degree 0 term is alr
eady done  (we only need \"int\" there, and don't need ratsols because
 there is no theta when the degree is 0)." }}{PARA 0 "" 0 "" {TEXT -1 
18 "The degree 1 term:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "te
rm[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 28 "So that's easy to integrate:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 11 "TERM[1]:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>&%%TERMG6#\"\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "now th
e degree 2 term:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "k:=2;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"#" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 113 "TERM[k] := c(x)*theta^k; ZERO := normal( term[k
] - diff(TERM[k], x) ); ZERO:=ZERO*theta^(-k); ratsols(ZERO,c(x));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%TERMG6#\"\"#*&-%\"cG6#%\"xG\"\"\")
-%$expG6#,&*$)F,F'\"\"\"F-F-F-F'F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%%ZEROG,(*&)-%$expG6#,&*$)%\"xG\"\"#\"\"\"\"\"\"F1F1F/F0F.F1!\"%*&-%%
diffG6$-%\"cG6#F.F.F1F'F0!\"\"*(F7F1F'F0F.F0F2" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%ZEROG*&,(*&)-%$expG6#,&*$)%\"xG\"\"#\"\"\"\"\"\"F2F2
F0F1F/F2!\"%*&-%%diffG6$-%\"cG6#F/F/F2F(F1!\"\"*(F8F2F(F1F/F1F3F1*$)F)
\"\"#F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7\"!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "TERM[k] := subs(c(x)=%[2],TERM[k]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%TERMG6#\"\"#,$*$)-%$expG6#,&*$
)%\"xGF'\"\"\"\"\"\"F3F3F'F2!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
35 "Take the sum of all of these terms:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "A3 := add(TERM[i],i=-2..2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A3G,**&,(\"\"\"F(%\"xG!\"\"*$)F)\"\"#\"\"\"F-F.*&F)
\"\"\")-%$expG6#,&F+F(F(F(\"\"#F.!\"\"F**&,&F)F(F*F(F.*&F)\"\"\"F2\"\"
\"F7F(F)F**$)F2F-F.F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "test:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "normal(diff(A3,x)-newf);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 13 "final result:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "F := A + A2 + A3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,.*&,&*
&,**$)%\"xG\"\"$\"\"\"\"\"\"*$)F,\"\"#F.!\"#F,F2!\"\"F/F.F,!\"\"F4*&*&
,&F,F/F4F/F/-%$expG6#,&F0F/F/F/F/F.F,F5F4F.,&*$)F9F2F.F/F,F4F5F/-%#lnG
6#,&F>F4F,F/!\"$*&,(F/F/F,F4F0F2F.*&F,\"\"\")F9\"\"#F.F5F4*&F8F.*&F,\"
\"\"F9\"\"\"F5F/F,F4F>F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 
"normal( diff(F,x)-f );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Remark:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "In the above worksheet we had t
he following:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 48 "  1)  the residues that we found were constants." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "  2)  the
 term of degree=0  (which was called TERM[0]) could be integrated" }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 130 "  3) Ea
ch time we applied \"ratsols\", it turned out that a \"rational soluti
on\" exists, and so we could find c(x) and hence TERM[k]." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "If any of the abo
ve fails  (which is very likely if you take a random function f)  then
 there is no elementary antiderivative." }}{PARA 0 "" 0 "" {TEXT -1 
263 "In the above example, 1) 2) and 3) work fine. But this is because
 I started with some function F, then took f:=diff(F,x) and then I put
 that in the worksheet. So that way, I know in advance that an element
ary integral exists, and hence that 1),2),3) will succeed." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}}{MARK "53 12 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 
}