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Abstract

We describe the development of a term-rewriting system for indenite integration; it is also called a rule-based evaluation system. The development is separated into modules, and we describe the module for a wide class of integrands containing the tangent function.
f(x)
F(x)
F(x) = f(x)
Æ
2 tan x dx =1
2ln 22 tan x22 tan x
arctan 2 tan x1
+1
2ln 22 tan x+ 22 tan x
arctan 2 tan x+ 1,
2 tan x dx =
[x]
2[x]212[x]
+2 1 + 2[x]
+12[x] + [x]
1 + 2[x] + [x] ,
= ln cos x+ ln(1 + 2 tan xtan x)
+ arctan 1 + tan x
2 tan x,
=arctan 2 tan x
1 + tan x+ arctanh 2 tan x
1tan x,
= arctan 1 + tan x
2 tan x+ arctanh 1tan x
2 tan x.
f F
F=f
dx
x=ln x ,
ln |x|,
(nπ, nπ +π/2) nZ
F=f
2 tan x x =+π/2
x=
x= (n+
1/2)π
tan x=±1
x=+π/2
x2+ 2
x43x2+ 4 dx = arctan x
2x2,
= arctan 2x+7+ arctan 2x7.
Æ
2 tan x dx =tan xcos xarccos (cos xsin x)
cos xsin x
ln cos x+2tan xcos x+ sin x.
x→ −x
f(ax +b)dx
y=ax +b y
(m+n+ 1)
m+n+ 1 ̸= 0
T1= tan(c+dx)m m 1
m1
m+n+ 1 = 0
C= 0
tanm(c+dx)(a+btan(c+dx))n
(A+Btan(c+dx) + Ctan2(c+dx)) ,
a, b, c, d, A, B, C Cm, n R
m n
B=C= 0
a2+b2= 0
u= tan(c+dx)
u
u= tan((c+dx)/2)
tan(1 + i+x)
412 tan(1 + i+x) + 9 tan(1 + i+x)2
(2 3 tan(1 + i+x))3/2dx .
m= 1 n=3/2n≤ −1
Ab2abB +
a2C= 0
1
13 tan(1 + i+x)(26 + 39 tan(1 + i+x))
23 tan(1 + i+x)dx .
tan(1 + i+x)23 tan(1 + i+x)dx .
dn T1Tn
2dx Tn
2dn Tn1
2T3(b, a, 0) dx
3 + 2 tan(1 + i+x)
23 tan(1 + i+x)dx + 223 tan(1 + i+x).
A2+B2̸= 0 a2+b2̸= 0
A+Btan(c+dx)
a+btan(c+dx)dx
1
2(ABi)1 + itan(c+dx)
a+btan(c+dx)dx+
1
2(A+Bi)1itan(c+dx)
a+btan(c+dx)dx
A2+B2= 0 bA +aB ̸= 0
A+Btan(c+dx)
a+btan(c+dx)dx
2Barctanh a+btan(c+dx)
a+bA
B
da+bA
B
+ 223 tan(1 + i+x)
23iarctanh 23 tan(1 + i+x)
23i
2 + 3iarctanh 23 tan(1 + i+x)
2+3i
(sin(c+dx))1
csc(c+dx) tan(c+dx)1
cot(c+dx)
T1= tan(c+dx), T2=a+btan(c+dx),
T3(A, B, C ) = A+Btan(c+dx) + Ctan2(c+dx).
A, B, C C
d(m+ 1) Tm
1Tn
2T3(A, B, C )dx =
AT m+1
1Tn
2+dTm+1
1Tn1
2T3(ˆ
A, ˆ
B, ˆ
C)dx ,
ˆ
A=aB(m+ 1) Abn ,
ˆ
B= (bB aA +aC)(m+ 1) ,
ˆ
C=bC(m+ 1) Ab(m+n+ 1) .
d(m+n+ 1) Tm
1Tn
2T3(A, B, C )dx =
CT m+1
1Tn
2+dTm
1Tn1
2T3(ˆ
A, ˆ
B, ˆ
C)dx ,
ˆ
A=Aa(m+n+ 1) C(m+ 1)a ,
ˆ
B= (aB +bA bC)(m+n+ 1) ,
ˆ
C=aCn +bB(m+n+ 1) .
bd(n+ 1) a2+b2Tm
1Tn
2T3dx =
Ab2abB +a2CTm
1Tn+1
2+dTm1
1Tn+1
2ˆ
T3dx ,
ˆ
A=Ab2abB +a2Cm ,
ˆ
B=b(bB +aA aC)(n+ 1) ,
ˆ
C= (m+n+ 1)(aB Ab)bma2C+ (n+ 1)b2C .
ad(n+ 1) a2+b2Tm
1Tn
2T3dx =
Ab2abB +a2CTm+1
1Tn+1
2+dTm
1Tn+1
2ˆ
T3dx
ˆ
A=Aa2(n+ 1) + b2(m+n+ 2)
a(bB aC)(m+ 1) ,ˆ
B=a(aB bA +bC)(n+ 1),
ˆ
C=Ab2abB +a2C(m+n+ 2) .
bd(m+n+ 1) Tm
1Tn
2T3(A, B, C )dx =
CT m
1Tn+1
2dTm1
1Tn
2ˆ
T3dx ,
ˆ
A=aCm , ˆ
B=b(CA)(m+n+ 1) ,
ˆ
C=aCm bB(m+n+ 1) .
ad(m+ 1) Tm
1Tn
2T3(A, B, C )dx =
AT m+1
1Tn+1
2+dTm+1
1Tn
2ˆ
T3dx ,
ˆ
A=aB(m+ 1) Ab(m+n+ 2) ,
ˆ
B=a(AC)(m+ 1) ,ˆ
C=Ab(m+n+ 2) .
a2+b2= 0
T3
d(m+ 1) Tm
1Tn
2T3(A, B, 0) dx =
AaT m+1
1Tn1
2dTm+1
1Tn1
2T3(ˆ
A, ˆ
B, 0) dx ,
ˆ
A=Ab(n1) (Ab +Ba)(m+ 1) ,
ˆ
B=Aa(m+n)Bb(m+ 1) .
d(m+n)Tm
1Tn
2T3(A, B, 0) dx =
BbT m+1
1Tn1
2+dTm
1Tn1
2T3(ˆ
A, ˆ
B, 0) dx ,
ˆ
A=Aa(n+m)Bb(m+ 1) ,
ˆ
B=Ba(n1) + (Ab +Ba)(m+n).
2a2nd Tm
1Tn
2T3(A, B, 0) dx =
BbT m
1Tn
2+dTm1
1Tn+1
2T3(ˆ
A, ˆ
B, 0) dx ,
ˆ
A= (Ab Ba)m , ˆ
B=Bb(mn) + Aa(m+n).
2a2nd Tm
1Tn
2T3(A, B, 0) dx =
a(aA +bB)Tm+1
1Tn
2+dTm
1Tn+1
2ˆ
T3(ˆ
A, ˆ
B, 0) dx ,
ˆ
A=bB(m+ 1) + aA(m+ 2n+ 1) ,
ˆ
B= (aB Ab)(m+n+ 1) .
ad(m+n)Tm
1Tn
2T3(A, B, 0) dx =
aBT m
1Tn
2+dTm1
1Tn
2ˆ
T3(ˆ
A, ˆ
B, 0) dx ,
ˆ
A=aBm , ˆ
B=Aam + (Aa Bb)n .
ad(m+ 1) Tm
1Tn
2T3(A, B, 0) dx =
aAT m+1
1Tn
2+dTm+1
1Tn
2ˆ
T3(ˆ
A, ˆ
B, 0) dx ,
ˆ
A=Abn Ba(m+ 1) ,ˆ
B=Aa(m+n+ 1) .
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