Rules for integrands involving inverse secants and cosecants
1. à u Ha + b ArcSec@c xDLn â x
1. à Ha + b ArcSec@c xDLn â x
1: à ArcSec@c xD â x
Ÿ Reference: G&R 2.821.2, CRC 445, A&S 4.4.62
Ÿ Reference: G&R 2.821.1, CRC 446, A&S 4.4.61
Ÿ Derivation: Integration by parts
Ÿ Rule:
à ArcSec@c xD â x ™ x ArcSec@c xD -
Ÿ Program code:
Int@ArcSec@c_.*x_D,x_SymbolD :=
x*ArcSec@c*xD - 1c*Int@1Hx*Sqrt@1-1Hc^2*x^2LDL,xD ;
FreeQ@c,xD
Int@ArcCsc@c_.*x_D,x_SymbolD :=
x*ArcCsc@c*xD + 1c*Int@1Hx*Sqrt@1-1Hc^2*x^2LDL,xD ;
FreeQ@c,xD
1
c
á
1
âx
x
1-
1
c2 x2
Rules for integrands involving inverse secants and cosecants
2
2: à Ha + b ArcSec@c xDLn â x
Ÿ Derivation: Integration by substitution
Ÿ Basis: 1 Š
1
c
Sec@ArcSec@c xDD Tan@ArcSec@c xDD ¶x ArcSec@c xD
Ÿ Rule:
Ÿ Program code:
n
à Ha + b ArcSec@c xDL â x ™
1
c
SubstBà Ha + b xLn Sec@xD Tan@xD â x, x, ArcSec@c xDF
Int@Ha_.+b_.*ArcSec@c_.*x_DL^n_,x_SymbolD :=
1c*Subst@Int@Ha+b*xL^n*Sec@xD*Tan@xD,xD,x,ArcSec@c*xDD ;
FreeQ@8a,b,c,n<,xD
Int@Ha_.+b_.*ArcCsc@c_.*x_DL^n_,x_SymbolD :=
-1c*Subst@Int@Ha+b*xL^n*Csc@xD*Cot@xD,xD,x,ArcCsc@c*xDD ;
FreeQ@8a,b,c,n<,xD
Rules for integrands involving inverse secants and cosecants
3
2. à xm Ha + b ArcSec@c xDLn â x
1. à xm Ha + b ArcSec@c xDL â x
1: à
a + b ArcSec@c xD
âx
x
Ÿ Derivation: Integration by substitution
Ÿ Basis: ArcSec@zD Š ArcCosA 1z E
FA E
1
Ÿ Basis:
x
x
Š - SubstA
F@xD
,
x
Ÿ Rule:
x,
1
E
x
¶x
1
x
a + b ArcCosA c x E
1
Ÿ Program code:
á
a + b ArcSec@c xD
x
âx ™ á
IntAHa_.+b_.*ArcSec@c_.*x_DL‘x_,x_SymbolE :=
-Subst@Int@Ha+b*ArcCos@xcDLx,xD,x,1xD ;
FreeQ@8a,b,c<,xD
IntAHa_.+b_.*ArcCsc@c_.*x_DL‘x_,x_SymbolE :=
-Subst@Int@Ha+b*ArcSin@xcDLx,xD,x,1xD ;
FreeQ@8a,b,c<,xD
x
a + b ArcCosA c E
x
â x ™ - SubstBá
x
1
â x, x,
x
F
Rules for integrands involving inverse secants and cosecants
4
2: à xm Ha + b ArcSec@c xDL â x when m ¹ - 1
Ÿ Reference: CRC 474
Ÿ Reference: CRC 477
Ÿ Derivation: Integration by parts
Ÿ Rule: If m ¹ - 1, then
m
à x Ha + b ArcSec@c xDL â x ™
xm+1 Ha + b ArcSec@c xDL
m+1
b
-
c Hm + 1L
á
xm-1
âx
1-
1
c2 x2
Ÿ Program code:
Int@x_^m_.*Ha_.+b_.*ArcSec@c_.*x_DL,x_SymbolD :=
x^Hm+1L*Ha+b*ArcSec@c*xDLHm+1L bHc*Hm+1LL*Int@x^Hm-1LSqrt@1-1Hc^2*x^2LD,xD ;
FreeQ@8a,b,c,m<,xD && NonzeroQ@m+1D
Int@x_^m_.*Ha_.+b_.*ArcCsc@c_.*x_DL,x_SymbolD :=
x^Hm+1L*Ha+b*ArcCsc@c*xDLHm+1L +
bHc*Hm+1LL*Int@x^Hm-1LSqrt@1-1Hc^2*x^2LD,xD ;
FreeQ@8a,b,c,m<,xD && NonzeroQ@m+1D
2: à xm Ha + b ArcSec@c xDLn â x when m Î Z
Ÿ Derivation: Integration by substitution
Ÿ Basis: If m Î Z, then xm Š
1
cm+1
Sec@ArcSec@c xDDm+1 Tan@ArcSec@c xDD ¶x ArcSec@c xD
Ÿ Rule: If m Î Z, then
Ÿ Program code:
m
n
à x Ha + b ArcSec@c xDL â x ™
1
cm+1
SubstBà Ha + b xLn Sec@xDm+1 Tan@xD â x, x, ArcSec@c xDF
Int@x_^m_.*Ha_.+b_.*ArcSec@c_.*x_DL^n_,x_SymbolD :=
1c^Hm+1L*Subst@Int@Ha+b*xL^n*Sec@xD^Hm+1L*Tan@xD,xD,x,ArcSec@c*xDD ;
FreeQ@8a,b,c,n<,xD && IntegerQ@mD
Rules for integrands involving inverse secants and cosecants
5
Int@x_^m_.*Ha_.+b_.*ArcCsc@c_.*x_DL^n_,x_SymbolD :=
-1c^Hm+1L*Subst@Int@Ha+b*xL^n*Csc@xD^Hm+1L*Cot@xD,xD,x,ArcCsc@c*xDD ;
FreeQ@8a,b,c,n<,xD && IntegerQ@mD
X: à xm Ha + b ArcSec@c xDLn â x
Ÿ Rule:
Ÿ Program code:
m
n
m
n
à x Ha + b ArcSec@c xDL â x ™ à x Ha + b ArcSec@c xDL â x
Int@x_^m_.*Ha_.+b_.*ArcSec@c_.*x_DL^n_.,x_SymbolD :=
Defer@IntD@x^m*Ha+b*ArcSec@c*xDL^n,xD ;
FreeQ@8a,b,c,m,n<,xD
Int@x_^m_.*Ha_.+b_.*ArcCsc@c_.*x_DL^n_.,x_SymbolD :=
Defer@IntD@x^m*Ha+b*ArcCsc@c*xDL^n,xD ;
FreeQ@8a,b,c,m,n<,xD
Rules for integrands involving inverse secants and cosecants
6
3. à Id + e x2 M Ha + b ArcSec@c xDLn â x
p
1: à Id + e x2 M Ha + b ArcSec@c xDL â x when p Î Z+ ë p +
p
1
2
Î Z-
Ÿ Derivation: Integration by parts and piecewise constant extraction
Ÿ Basis: ¶x Ha + b ArcSec@c xDL Š
Ÿ Basis: ¶x
x
bc
c2 x2
c2 x2 -1
Š0
c2 x2
Ÿ Note: If p Î Z+ ë p +
Ÿ Rule: If p Î Z+ ë p +
1
2
1
2
Î Z- , then Ù Id + e x2 Mp â x is expressible as an algebraic function not involving logarithms, inverse trig or inverse hyperbolic functions.
Î Z- , let u = Ù Id + e x2 Mp â x, then
2
à Id + e x M Ha + b ArcSec@c xDL â x ™ u Ha + b ArcSec@c xDL - b c à
p
u
c2 x2
c2 x2 - 1
Ÿ Program code:
â x ™ u Ha + b ArcSec@uDL -
bcx
c2 x2
Int@Hd_.+e_.*x_^2L^p_.*Ha_.+b_.*ArcSec@c_.*x_DL,x_SymbolD :=
With@8u=IntHide@Hd+e*x^2L^p,xD<,
Dist@Ha+b*ArcSec@c*xDL,u,xD - b*c*xSqrt@c^2*x^2D*Int@SimplifyIntegrand@uHx*Sqrt@c^2*x^2-1DL,xD,xDD ;
FreeQ@8a,b,c,d,e<,xD && HPositiveIntegerQ@pD ÈÈ NegativeIntegerQ@p+12DL
Int@Hd_.+e_.*x_^2L^p_.*Ha_.+b_.*ArcCsc@c_.*x_DL,x_SymbolD :=
With@8u=IntHide@Hd+e*x^2L^p,xD<,
Dist@Ha+b*ArcCsc@c*xDL,u,xD + b*c*xSqrt@c^2*x^2D*Int@SimplifyIntegrand@uHx*Sqrt@c^2*x^2-1DL,xD,xDD ;
FreeQ@8a,b,c,d,e<,xD && HPositiveIntegerQ@pD ÈÈ NegativeIntegerQ@p+12DL
2: à Id + e x2 M Ha + b ArcSec@c xDLn â x when p Î Z
p
Ÿ Derivation: Integration by substitution
Ÿ Basis: ArcSec@zD Š ArcCosA 1z E
Ÿ Basis: FA 1x E Š - SubstA
Ÿ Rule: If p Î Z, then
F@xD
,
x2
x,
1
E
x
¶x
1
x
2
n
à Id + e x M Ha + b ArcSec@c xDL â x ™ à
p
1
-2 p
d
e+
x
p
1
a + b ArcCosB
x2
cx
F
n
âx
à
u
âx
x
c2 x2 - 1
Rules for integrands involving inverse secants and cosecants
7
Ie + d x2 M Ia + b ArcCosA c EM
p
Ÿ Program code:
™ - SubstBá
x
x2 Hp+1L
Int@Hd_.+e_.*x_^2L^p_.*Ha_.+b_.*ArcSec@c_.*x_DL^n_.,x_SymbolD :=
-Subst@Int@He+d*x^2L^p*Ha+b*ArcCos@xcDL^nx^H2*Hp+1LL,xD,x,1xD ;
FreeQ@8a,b,c,d,e,n<,xD && IntegerQ@pD
Int@Hd_.+e_.*x_^2L^p_.*Ha_.+b_.*ArcCsc@c_.*x_DL^n_.,x_SymbolD :=
-Subst@Int@He+d*x^2L^p*Ha+b*ArcSin@xcDL^nx^H2*Hp+1LL,xD,x,1xD ;
FreeQ@8a,b,c,d,e,n<,xD && IntegerQ@pD
n
1
â x, x,
x
F
Rules for integrands involving inverse secants and cosecants
8
3. à Id + e x2 M Ha + b ArcSec@c xDLn â x when c2 d + e Š 0 í p +
p
1
2
ÎZ
1: à Id + e x2 M Ha + b ArcSec@c xDLn â x when c2 d + e Š 0 í p +
p
1
2
Ÿ Derivation: Piecewise constant extraction and integration by substitution
d+e x2
Ÿ Basis: ¶x
x
e+
Š0
d
x2
Ÿ Basis: ArcSec@zD Š ArcCosA 1z E
Ÿ Basis: FA 1x E Š - SubstA
F@xD
,
x2
Ÿ Basis: If e > 0 ß d < 0, then
Ÿ Rule: If c2 d + e Š 0 í p +
1
E
x
x,
d+e x2
e+
1
2
¶x
Š
1
x
x2
d
x2
Î Z í e > 0 í d < 0, then
2
n
á Id + e x M Ha + b ArcSec@c xDL â x ™
d + e x2
p
x2
™ x
Ÿ Program code:
ÎZ í e>0 í d<0
x
e+
d
x2
à
1
d
e+
1
a + b ArcCosB
Ie + d x2 M Ia + b ArcCosA c EM
x
x2 Hp+1L
p
x2
x
p
SubstBá
-2 p
cx
n
1
â x, x,
x
F
Int@Hd_.+e_.*x_^2L^p_*Ha_.+b_.*ArcSec@c_.*x_DL^n_.,x_SymbolD :=
-Sqrt@x^2Dx*Subst@Int@He+d*x^2L^p*Ha+b*ArcCos@xcDL^nx^H2*Hp+1LL,xD,x,1xD ;
FreeQ@8a,b,c,d,e,n<,xD && ZeroQ@c^2*d+eD && IntegerQ@p+12D && PositiveQ@eD && Negative@dD
Int@Hd_.+e_.*x_^2L^p_*Ha_.+b_.*ArcCsc@c_.*x_DL^n_.,x_SymbolD :=
-Sqrt@x^2Dx*Subst@Int@He+d*x^2L^p*Ha+b*ArcSin@xcDL^nx^H2*Hp+1LL,xD,x,1xD ;
FreeQ@8a,b,c,d,e,n<,xD && ZeroQ@c^2*d+eD && IntegerQ@p+12D && PositiveQ@eD && Negative@dD
F
n
âx
Rules for integrands involving inverse secants and cosecants
9
2: à Id + e x2 M Ha + b ArcSec@c xDLn â x when c2 d + e Š 0 í p +
p
1
2
Ÿ Derivation: Piecewise constant extraction and integration by substitution
d+e x2
Ÿ Basis: ¶x
x
e+
Î Z í Ø He > 0 ì d < 0L
Š0
d
x2
Ÿ Basis: ArcSec@zD Š ArcCosA 1z E
Ÿ Basis: FA 1x E Š - SubstA
F@xD
,
x2
Ÿ Rule: If c2 d + e Š 0 í p +
1
2
x,
1
E
x
¶x
1
x
Î Z í Ø He > 0 ß d < 0L, then
2
n
á Id + e x M Ha + b ArcSec@c xDL â x ™
d + e x2
p
d + e x2
™ x
e+
d
x2
x
e+
d
x2
à
1
-2 p
x
x2
x
x2 Hp+1L
1
a + b ArcCosB
cx
Ie + d x2 M Ia + b ArcCosA c EM
p
SubstBá
p
d
e+
n
1
â x, x,
x
F
n
âx
F
Ÿ Program code:
Int@Hd_.+e_.*x_^2L^p_*Ha_.+b_.*ArcSec@c_.*x_DL^n_.,x_SymbolD :=
-Sqrt@d+e*x^2DHx*Sqrt@e+dx^2DL*Subst@Int@He+d*x^2L^p*Ha+b*ArcCos@xcDL^nx^H2*Hp+1LL,xD,x,1xD ;
FreeQ@8a,b,c,d,e,n<,xD && ZeroQ@c^2*d+eD && IntegerQ@p+12D && Not@PositiveQ@eD && Negative@dDD
Int@Hd_.+e_.*x_^2L^p_*Ha_.+b_.*ArcCsc@c_.*x_DL^n_.,x_SymbolD :=
-Sqrt@d+e*x^2DHx*Sqrt@e+dx^2DL*Subst@Int@He+d*x^2L^p*Ha+b*ArcSin@xcDL^nx^H2*Hp+1LL,xD,x,1xD ;
FreeQ@8a,b,c,d,e,n<,xD && ZeroQ@c^2*d+eD && IntegerQ@p+12D && Not@PositiveQ@eD && Negative@dDD
Rules for integrands involving inverse secants and cosecants
10
X: à Id + e x2 M Ha + b ArcSec@c xDLn â x
p
Ÿ Rule:
2
n
2
n
à Id + e x M Ha + b ArcSec@c xDL â x ™ à Id + e x M Ha + b ArcSec@c xDL â x
p
Ÿ Program code:
Int@Hd_.+e_.*x_^2L^p_.*Ha_.+b_.*ArcSec@c_.*x_DL^n_.,x_SymbolD :=
Defer@IntD@Hd+e*x^2L^p*Ha+b*ArcSec@c*xDL^n,xD ;
FreeQ@8a,b,c,d,e,n,p<,xD
Int@Hd_.+e_.*x_^2L^p_.*Ha_.+b_.*ArcCsc@c_.*x_DL^n_.,x_SymbolD :=
Defer@IntD@Hd+e*x^2L^p*Ha+b*ArcCsc@c*xDL^n,xD ;
FreeQ@8a,b,c,d,e,n,p<,xD
p
Rules for integrands involving inverse secants and cosecants
11
4. à xm Id + e x2 M Ha + b ArcSec@c xDLn â x
p
1. à xm Id + e x2 M Ha + b ArcSec@c xDL â x when
p
Jp Î Z+ í Ø J
m-1
2
Î Z- í m + 2 p + 3 > 0NN ë J
m+1
2
Î Z+ í Ø Hp Î Z- ì m + 2 p + 3 > 0LN ë J
Î Z- í
m+2 p+1
2
1: à x Id + e x2 M Ha + b ArcSec@c xDL â x when p ¹ - 1
p
Ï Z- N
m-1
2
Ÿ Derivation: Integration by parts and piecewise constant extraction
Ÿ Basis: x Id + e x2 M Š ¶x
p
Hd+e x2 Lp+1
2 e Hp+1L
Ÿ Basis: ¶x Ha + b ArcSec@c xDL Š
Ÿ Basis: ¶x
x
bc
c2 x2
c2 x2 -1
Š0
c2 x2
Ÿ Rule: If p ¹ - 1, then
2
á x Id + e x M Ha + b ArcSec@c xDL â x ™
p
Id + e x2 M
p+1
™
Ÿ Program code:
Id + e x2 M
p+1
2 e Hp + 1L
Ha + b ArcSec@c xDL
2 e Hp + 1L
Ha + b ArcSec@c xDL
-
bcx
2 e Hp + 1L
c2 x2
bc
-
2 e Hp + 1L
á
Int@x_*Hd_.+e_.*x_^2L^p_.*Ha_.+b_.*ArcSec@c_.*x_DL,x_SymbolD :=
Hd+e*x^2L^Hp+1L*Ha+b*ArcSec@c*xDLH2*e*Hp+1LL b*c*xH2*e*Hp+1L*Sqrt@c^2*x^2DL*Int@Hd+e*x^2L^Hp+1LHx*Sqrt@c^2*x^2-1DL,xD ;
FreeQ@8a,b,c,d,e,p<,xD && NonzeroQ@p+1D
Int@x_*Hd_.+e_.*x_^2L^p_.*Ha_.+b_.*ArcCsc@c_.*x_DL,x_SymbolD :=
Hd+e*x^2L^Hp+1L*Ha+b*ArcCsc@c*xDLH2*e*Hp+1LL +
b*c*xH2*e*Hp+1L*Sqrt@c^2*x^2DL*Int@Hd+e*x^2L^Hp+1LHx*Sqrt@c^2*x^2-1DL,xD ;
FreeQ@8a,b,c,d,e,p<,xD && NonzeroQ@p+1D
p+1
á
Id + e x2 M
x
Id + e x2 M
c2 x2
p+1
c2 x2 - 1
âx
c2 x2 - 1
âx
Rules for integrands involving inverse secants and cosecants
12
2: à xm Id + e x2 M Ha + b ArcSec@c xDL â x when
p
Jp Î Z+ í Ø J
m-1
2
Î Z- í m + 2 p + 3 > 0NN ë J
m+1
2
Î Z+ í Ø Hp Î Z- ì m + 2 p + 3 > 0LN ë J
m+2 p+1
Ÿ Derivation: Integration by parts and piecewise constant extraction
Ÿ Basis: ¶x Ha + b ArcSec@c xDL Š
Ÿ Basis: ¶x
x
2
Î Z- í
m-1
2
Ï Z- N
bc
c2 x2
c2 x2 -1
Š0
c2 x2
Ÿ Note: If Jp Î Z+ í Ø J m-1
Î Z- í m + 2 p + 3 > 0NN ë J m+1
Î Z+ í Ø Hp Î Z- ì m + 2 p + 3 > 0LN ë J
2
2
m+2 p+1
2
Î Z- í
m-1
2
Ï Z- N,
then Ù xm Id + e x2 Mp â x is expressible as an algebraic function not involving logarithms, inverse trig or inverse hyperbolic functions.
Ÿ Rule: If Jp Î Z+ í Ø J m-1
Î Z- í m + 2 p + 3 > 0NN ë J m+1
Î Z+ í Ø Hp Î Z- ì m + 2 p + 3 > 0LN ë J
2
2
u = Ù xm Id + e x2 M â x, then
p
m
2
à x Id + e x M Ha + b ArcSec@c xDL â x ™ u Ha + b ArcSec@c xDL - b c à
u
p
Ÿ Program code:
c2 x2
c2 x2 - 1
m+2 p+1
2
Î Z- í
â x ™ u Ha + b ArcSec@uDL -
m-1
2
bcx
c2 x2
Int@x_^m_.*Hd_.+e_.*x_^2L^p_.*Ha_.+b_.*ArcSec@c_.*x_DL,x_SymbolD :=
With@8u=IntHide@x^m*Hd+e*x^2L^p,xD<,
Dist@Ha+b*ArcSec@c*xDL,u,xD - b*c*xSqrt@c^2*x^2D*Int@SimplifyIntegrand@uHx*Sqrt@c^2*x^2-1DL,xD,xDD ;
FreeQ@8a,b,c,d,e,m,p<,xD && H
PositiveIntegerQ@pD && Not@NegativeIntegerQ@Hm-1L2D && m+2*p+3>0D ÈÈ
PositiveIntegerQ@Hm+1L2D && Not@NegativeIntegerQ@pD && m+2*p+3>0D ÈÈ
NegativeIntegerQ@Hm+2*p+1L2D && Not@NegativeIntegerQ@Hm-1L2DD L
Int@x_^m_.*Hd_.+e_.*x_^2L^p_.*Ha_.+b_.*ArcCsc@c_.*x_DL,x_SymbolD :=
With@8u=IntHide@x^m*Hd+e*x^2L^p,xD<,
Dist@Ha+b*ArcCsc@c*xDL,u,xD + b*c*xSqrt@c^2*x^2D*Int@SimplifyIntegrand@uHx*Sqrt@c^2*x^2-1DL,xD,xDD ;
FreeQ@8a,b,c,d,e,m,p<,xD && H
PositiveIntegerQ@pD && Not@NegativeIntegerQ@Hm-1L2D && m+2*p+3>0D ÈÈ
PositiveIntegerQ@Hm+1L2D && Not@NegativeIntegerQ@pD && m+2*p+3>0D ÈÈ
NegativeIntegerQ@Hm+2*p+1L2D && Not@NegativeIntegerQ@Hm-1L2DD L
Ï Z- N, let
à
u
âx
x
c2 x2 - 1
Rules for integrands involving inverse secants and cosecants
13
2: à xm Id + e x2 M Ha + b ArcSec@c xDLn â x when Hm
p
pL Î Z
Ÿ Derivation: Integration by substitution
Ÿ Basis: ArcSec@zD Š ArcCosA 1z E
Ÿ Basis: FA 1x E Š - SubstA
Ÿ Rule: If Hm
pL Î Z, then
F@xD
,
x2
x,
1
E
x
¶x
1
x
m
2
n
à x Id + e x M Ha + b ArcSec@c xDL â x ™ à
p
1
-m-2 p
Ÿ Program code:
Int@x_^m_.*Hd_.+e_.*x_^2L^p_.*Ha_.+b_.*ArcSec@c_.*x_DL^n_.,x_SymbolD :=
-Subst@Int@He+d*x^2L^p*Ha+b*ArcCos@xcDL^nx^Hm+2*Hp+1LL,xD,x,1xD ;
FreeQ@8a,b,c,d,e,n<,xD && IntegersQ@m,pD
Int@x_^m_.*Hd_.+e_.*x_^2L^p_.*Ha_.+b_.*ArcCsc@c_.*x_DL^n_.,x_SymbolD :=
-Subst@Int@He+d*x^2L^p*Ha+b*ArcSin@xcDL^nx^Hm+2*Hp+1LL,xD,x,1xD ;
FreeQ@8a,b,c,d,e,n<,xD && IntegersQ@m,pD
1
a + b ArcCosB
Ie + d x2 M Ia + b ArcCosA c EM
xm+2 Hp+1L
p
x2
x
p
™ - SubstBá
d
e+
x
cx
n
1
â x, x,
x
F
F
n
âx
Rules for integrands involving inverse secants and cosecants
14
3. à xm Id + e x2 M Ha + b ArcSec@c xDLn â x when c2 d + e Š 0 í m Î Z í p +
p
1
2
ÎZ
1: à xm Id + e x2 M Ha + b ArcSec@c xDLn â x when c2 d + e Š 0 í m Î Z í p +
p
Ÿ Derivation: Piecewise constant extraction and integration by substitution
d+e x2
Ÿ Basis: ¶x
x
e+
ÎZ í e>0 í d<0
Š0
d
x2
Ÿ Basis: ArcSec@zD Š ArcCosA 1z E
Ÿ Basis: FA 1x E Š - SubstA
F@xD
,
x2
Ÿ Basis: If e > 0 ß d < 0, then
1
E
x
x,
d+e x2
e+
¶x
1
x
x2
Š
d
x2
Ÿ Rule: If c2 d + e Š 0 í m Î Z í p +
1
2
Î Z í e > 0 í d < 0, then
m
2
n
á x Id + e x M Ha + b ArcSec@c xDL â x ™
p
x2
™ x
Ÿ Program code:
1
2
d + e x2
x
e+
d
x2
à
1
d
e+
Ie + d x2 M Ia + b ArcCosA c EM
x
xm+2 Hp+1L
p
1
a + b ArcCosB
x2
x
p
SubstBá
-m-2 p
cx
n
1
â x, x,
x
F
n
âx
F
Int@x_^m_.*Hd_.+e_.*x_^2L^p_*Ha_.+b_.*ArcSec@c_.*x_DL^n_.,x_SymbolD :=
-Sqrt@x^2Dx*Subst@Int@He+d*x^2L^p*Ha+b*ArcCos@xcDL^nx^Hm+2*Hp+1LL,xD,x,1xD ;
FreeQ@8a,b,c,d,e,n<,xD && ZeroQ@c^2*d+eD && IntegerQ@mD && IntegerQ@p+12D && PositiveQ@eD && Negative@dD
Int@x_^m_.*Hd_.+e_.*x_^2L^p_*Ha_.+b_.*ArcCsc@c_.*x_DL^n_.,x_SymbolD :=
-Sqrt@x^2Dx*Subst@Int@He+d*x^2L^p*Ha+b*ArcSin@xcDL^nx^Hm+2*Hp+1LL,xD,x,1xD ;
FreeQ@8a,b,c,d,e,n<,xD && ZeroQ@c^2*d+eD && IntegerQ@mD && IntegerQ@p+12D && PositiveQ@eD && Negative@dD
Rules for integrands involving inverse secants and cosecants
15
2: à xm Id + e x2 M Ha + b ArcSec@c xDLn â x when c2 d + e Š 0 í m Î Z í p +
p
1
2
Ÿ Derivation: Piecewise constant extraction and integration by substitution
d+e x2
Ÿ Basis: ¶x
x
e+
Î Z í Ø He > 0 ì d < 0L
Š0
d
x2
Ÿ Basis: ArcSec@zD Š ArcCosA 1z E
Ÿ Basis: FA 1x E Š - SubstA
F@xD
,
x2
x,
1
E
x
¶x
Ÿ Rule: If c2 d + e Š 0 í m Î Z í p +
1
2
1
x
Î Z í Ø He > 0 ß d < 0L, then
m
2
n
á x Id + e x M Ha + b ArcSec@c xDL â x ™
p
d + e x2
™ x
e+
d
x2
d + e x2
x
e+
d
x2
à
1
-m-2 p
1
a + b ArcCosB
Ie + d x2 M Ia + b ArcCosA c EM
x
xm+2 Hp+1L
p
x2
x
p
SubstBá
d
e+
cx
n
1
â x, x,
x
F
n
âx
F
Ÿ Program code:
Int@x_^m_.*Hd_.+e_.*x_^2L^p_*Ha_.+b_.*ArcSec@c_.*x_DL^n_.,x_SymbolD :=
-Sqrt@d+e*x^2DHx*Sqrt@e+dx^2DL*Subst@Int@He+d*x^2L^p*Ha+b*ArcCos@xcDL^nx^Hm+2*Hp+1LL,xD,x,1xD ;
FreeQ@8a,b,c,d,e,n<,xD && ZeroQ@c^2*d+eD && IntegerQ@mD && IntegerQ@p+12D && Not@PositiveQ@eD && Negative@dDD
Int@x_^m_.*Hd_.+e_.*x_^2L^p_*Ha_.+b_.*ArcCsc@c_.*x_DL^n_.,x_SymbolD :=
-Sqrt@d+e*x^2DHx*Sqrt@e+dx^2DL*Subst@Int@He+d*x^2L^p*Ha+b*ArcSin@xcDL^nx^Hm+2*Hp+1LL,xD,x,1xD ;
FreeQ@8a,b,c,d,e,n<,xD && ZeroQ@c^2*d+eD && IntegerQ@mD && IntegerQ@p+12D && Not@PositiveQ@eD && Negative@dDD
Rules for integrands involving inverse secants and cosecants
16
X: à xm Id + e x2 M Ha + b ArcSec@c xDLn â x
p
Ÿ Rule:
m
2
n
m
2
n
à x Id + e x M Ha + b ArcSec@c xDL â x ™ à x Id + e x M Ha + b ArcSec@c xDL â x
p
Ÿ Program code:
p
Int@x_^m_.*Hd_.+e_.*x_^2L^p_.*Ha_.+b_.*ArcSec@c_.*x_DL^n_.,x_SymbolD :=
Defer@IntD@x^m*Hd+e*x^2L^p*Ha+b*ArcSec@c*xDL^n,xD ;
FreeQ@8a,b,c,d,e,m,n,p<,xD
Int@x_^m_.*Hd_.+e_.*x_^2L^p_.*Ha_.+b_.*ArcCsc@c_.*x_DL^n_.,x_SymbolD :=
Defer@IntD@x^m*Hd+e*x^2L^p*Ha+b*ArcCsc@c*xDL^n,xD ;
FreeQ@8a,b,c,d,e,m,n,p<,xD
2. à u ArcSec@a + b xDn â x
1. à ArcSec@a + b xDn â x
1: à ArcSec@a + b xD â x
Ÿ Reference: G&R 2.821.2, CRC 445, A&S 4.4.62
Ÿ Reference: G&R 2.821.1, CRC 446, A&S 4.4.61
Ÿ Derivation: Integration by parts
Ÿ Rule:
à ArcSec@a + b xD â x ™
Ÿ Program code:
Int@ArcSec@a_+b_.*x_D,x_SymbolD :=
Ha+b*xL*ArcSec@a+b*xDb Int@1HHa+b*xL*Sqrt@1-1Ha+b*xL^2DL,xD ;
FreeQ@8a,b<,xD
Ha + b xL ArcSec@a + b xD
b
-á
1
âx
Ha + b xL
1-
1
Ha+b xL2
Rules for integrands involving inverse secants and cosecants
17
Int@ArcCsc@a_+b_.*x_D,x_SymbolD :=
Ha+b*xL*ArcCsc@a+b*xDb +
Int@1HHa+b*xL*Sqrt@1-1Ha+b*xL^2DL,xD ;
FreeQ@8a,b<,xD
2: à ArcSec@a + b xDn â x
Ÿ Derivation: Integration by substitution
Ÿ Basis: ArcSec@a + b xDn Š
1
b
ArcSec@a + b xDn Sec@ArcSec@a + b xDD Tan@ArcSec@a + b xDD ¶x ArcSec@a + b xD
Ÿ Rule:
n
à ArcSec@a + b xD â x ™
Ÿ Program code:
1
b
SubstBà xn Sec@xD Tan@xD â x, x, ArcSec@a + b xDF
Int@ArcSec@a_+b_.*x_D^n_,x_SymbolD :=
1b*Subst@Int@x^n*Sec@xD*Tan@xD,xD,x,ArcSec@a+b*xDD ;
FreeQ@8a,b,n<,xD
Int@ArcCsc@a_+b_.*x_D^n_,x_SymbolD :=
-1b*Subst@Int@x^n*Csc@xD*Cot@xD,xD,x,ArcCsc@a+b*xDD ;
FreeQ@8a,b,n<,xD
2. à xm ArcSec@a + b xDn â x
1. à xm ArcSec@a + b xD â x
Ÿ Rule:
1: à
ArcSec@a + b xD
âx
x
ArcSec@a + b xD LogB1 -
J1 -
1 - a2 N ãä ArcSec@a+b xD
a
á
ArcSec@a + b xD
âx ™
x
F + ArcSec@a + b xD LogB1 -
J1 +
1 - a2 N ãä ArcSec@a+b xD
a
+
F - ArcSec@a + b xD LogA1 + ã2 ä ArcSec@a+b xD E -
Rules for integrands involving inverse secants and cosecants
ä PolyLogB2,
J1 -
18
1 - a2 N ãä ArcSec@a+b xD
a
à
F - ä PolyLogB2,
ArcCsc@a + b xD
x
ArcCsc@a + b xD LogB1 -
ArcCsc@a + b xD LogB1 -
ä PolyLogB2,
ä J1 -
ä J1 +
1 - a2 N ã-ä ArcCsc@a+b xD
a
Ÿ Program code:
J1 +
1 - a2 N ãä ArcSec@a+b xD
a
F+
1
2
ä PolyLogA2, - ã2 ä ArcSec@a+b xD E
â x ™ ä ArcCsc@a + b xD2 +
ä J1 -
1 - a2 N ã-ä ArcCsc@a+b xD
1 - a2 N ã-ä ArcCsc@a+b xD
a
F + ä PolyLogB2,
ä J1 +
a
F+
F - ArcCsc@a + b xD LogA1 - ã2 ä ArcCsc@a+b xD E +
IntAArcSec@a_+b_.*x_D‘x_,x_SymbolE :=
ArcSec@a+b*xD*Log@1-H1-Sqrt@1-a^2DL*E^HI*ArcSec@a+b*xDLaD +
ArcSec@a+b*xD*Log@1-H1+Sqrt@1-a^2DL*E^HI*ArcSec@a+b*xDLaD ArcSec@a+b*xD*Log@1+E^H2*I*ArcSec@a+b*xDLD I*PolyLog@2,H1-Sqrt@1-a^2DL*E^HI*ArcSec@a+b*xDLaD I*PolyLog@2,H1+Sqrt@1-a^2DL*E^HI*ArcSec@a+b*xDLaD +
12*I*PolyLog@2,-E^H2*I*ArcSec@a+b*xDLD ;
FreeQ@8a,b<,xD
IntAArcCsc@a_+b_.*x_D‘x_,x_SymbolE :=
I*ArcCsc@a+b*xD^2 +
ArcCsc@a+b*xD*Log@1-I*H1-Sqrt@1-a^2DLHE^HI*ArcCsc@a+b*xDL*aLD +
ArcCsc@a+b*xD*Log@1-I*H1+Sqrt@1-a^2DLHE^HI*ArcCsc@a+b*xDL*aLD ArcCsc@a+b*xD*Log@1-E^H2*I*ArcCsc@a+b*xDLD +
I*PolyLog@2,I*H1-Sqrt@1-a^2DLHE^HI*ArcCsc@a+b*xDL*aLD +
I*PolyLog@2,I*H1+Sqrt@1-a^2DLHE^HI*ArcCsc@a+b*xDL*aLD +
12*I*PolyLog@2,E^H2*I*ArcCsc@a+b*xDLD ;
FreeQ@8a,b<,xD
1 - a2 N ã-ä ArcCsc@a+b xD
a
F+
1
2
ä PolyLogA2, ã2 ä ArcCsc@a+b xD E
Rules for integrands involving inverse secants and cosecants
19
2: à xm ArcSec@a + b xD â x when m ¹ - 1
Ÿ Derivation: Integration by parts and substitution
Ÿ Basis: xm Š - ¶x
H-aLm+1 -bm+1 xm+1
bm+1 Hm+1L
Ÿ Basis: If m Î Z, then
Ÿ Rule: If m Î
Z+ ,
then
HH-aLm+1 -bm+1 xm+1 L
Ha+b xL2
1
fA a+b
EŠx
m
à x ArcSec@a + b xD â x ™ -
J-
a
a+b x
N
m+1
bJ
-J11
a+b x
N
a
a+b x
m+1
N
1
fA a+b
E ¶x
x
1
a+b x
IH- aLm+1 - bm+1 xm+1 M ArcSec@a + b xD
™ -
bm+1 Hm + 1L
1
+
bm Hm + 1L
IH- aLm+1 - bm+1 xm+1 M ArcSec@a + b xD
bm+1
Ÿ Program code:
m+1
Hm + 1L
1
bm+1
á
IH- aLm+1 - bm+1 xm+1 M
Ha + b xL2
Hm + 1L
SubstBà
1-
âx
1
Ha+b xL2
H- a xLm+1 - H1 - a xLm+1
xm+1
Int@x_^m_.*ArcSec@a_+b_.*x_D,x_SymbolD :=
-HH-aL^Hm+1L-b^Hm+1L*x^Hm+1LL*ArcSec@a+b*xDHb^Hm+1L*Hm+1LL 1Hb^Hm+1L*Hm+1LL*Subst@Int@HHH-aL*xL^Hm+1L-H1-a*xL^Hm+1LLHx^Hm+1L*Sqrt@1-x^2DL,xD,x,1Ha+b*xLD ;
FreeQ@8a,b,m<,xD && IntegerQ@mD && NonzeroQ@m+1D
Int@x_^m_.*ArcCsc@a_+b_.*x_D,x_SymbolD :=
-HH-aL^Hm+1L-b^Hm+1L*x^Hm+1LL*ArcCsc@a+b*xDHb^Hm+1L*Hm+1LL +
1Hb^Hm+1L*Hm+1LL*Subst@Int@HHH-aL*xL^Hm+1L-H1-a*xL^Hm+1LLHx^Hm+1L*Sqrt@1-x^2DL,xD,x,1Ha+b*xLD ;
FreeQ@8a,b,m<,xD && IntegerQ@mD && NonzeroQ@m+1D
1 - x2
1
â x, x,
a+bx
F
Rules for integrands involving inverse secants and cosecants
20
2: à xm ArcSec@a + b xDn â x when m Î Z+
Ÿ Derivation: Integration by substitution
Ÿ Basis: If m Î Z, then xm ArcSec@a + b xDn Š
1
bm+1
Ÿ Rule: If m Î Z+ , then
ArcSec@a + b xDn H- a + Sec@ArcSec@a + b xDDLm Sec@ArcSec@a + b xDD Tan@ArcSec@a + b xDD ¶x ArcSec@a + b xD
m
n
à x ArcSec@a + b xD â x ™
Ÿ Program code:
1
bm+1
SubstBà xn H- a + Sec@xDLm Sec@xD Tan@xD â x, x, ArcSec@a + b xDF
Int@x_^m_.*ArcSec@a_+b_.*x_D^n_,x_SymbolD :=
1b^Hm+1L*Subst@Int@x^n*H-a+Sec@xDL^m*Sec@xD*Tan@xD,xD,x,ArcSec@a+b*xDD ;
FreeQ@8a,b,n<,xD && PositiveIntegerQ@mD
Int@x_^m_.*ArcCsc@a_+b_.*x_D^n_,x_SymbolD :=
-1b^Hm+1L*Subst@Int@x^n*H-a+Csc@xDL^m*Csc@xD*Cot@xD,xD,x,ArcCsc@a+b*xDD ;
FreeQ@8a,b,n<,xD && PositiveIntegerQ@mD
3: à u ArcSecB
c
a + b xn
F âx
m
Ÿ Derivation: Algebraic simplification
Ÿ Basis: ArcSec@zD Š ArcCosA 1z E
Ÿ Rule:
Ÿ Program code:
à u ArcSecB
m
a b xn m
F â x ™ à u ArcCosB +
F âx
a + b xn
c
c
c
IntAu_.*ArcSecAc_.‘Ha_.+b_.*x_^n_.LE^m_.,x_SymbolE :=
Int@u*ArcCos@ac+b*x^ncD^m,xD ;
FreeQ@8a,b,c,n,m<,xD
IntAu_.*ArcCscAc_.‘Ha_.+b_.*x_^n_.LE^m_.,x_SymbolE :=
Int@u*ArcSin@ac+b*x^ncD^m,xD ;
FreeQ@8a,b,c,n,m<,xD
Rules for integrands involving inverse secants and cosecants
21
4: à u fc ArcSec@a+b xD â x
n
Ÿ Derivation: Integration by substitution
Ÿ Basis: F@x, ArcSec@a + b xDD Š
1
b
SubstAFA- ab +
Sec@xD
,
b
xE Sec@xD Tan@xD, x, ArcSec@a + b xDE ¶x ArcSec@a + b xD
Ÿ Rule: If n Î Z+ , then
àuf
c ArcSec@a+b xDn
Ÿ Program code:
1
âx ™
b
SubstBà SubstBu, x, -
a
Sec@xD
+
b
b
F fc x Sec@xD Tan@xD â x, x, ArcSec@a + b xDF
n
Int@u_.*f_^Hc_.*ArcSec@a_.+b_.*x_D^n_.L,x_SymbolD :=
1b*Subst@Int@ReplaceAll@u,x®-ab+Sec@xDbD*f^Hc*x^nL*Sec@xD*Tan@xD,xD,x,ArcSec@a+b*xDD ;
FreeQ@8a,b,c,f<,xD && PositiveIntegerQ@nD
Int@u_.*f_^Hc_.*ArcCsc@a_.+b_.*x_D^n_.L,x_SymbolD :=
-1b*Subst@Int@ReplaceAll@u,x®-ab+Csc@xDbD*f^Hc*x^nL*Csc@xD*Cot@xD,xD,x,ArcCsc@a+b*xDD ;
FreeQ@8a,b,c,f<,xD && PositiveIntegerQ@nD
Rules for integrands involving inverse secants and cosecants
22
5. à v Ha + b ArcSec@uDL â x when u is free of inverse functions
1: à ArcSec@uD â x when u is free of inverse functions
Ÿ Derivation: Integration by parts and piecewise constant extraction
Ÿ Basis: ¶x ArcSec@f@xDD Š
¶x f@xD
f@xD2
Ÿ Basis: ¶x
f@xD
f@xD2 -1
Š0
f@xD2
Ÿ Rule: If u is free of inverse functions, then
à ArcSec@uD â x ™ x ArcSec@uD - à
Ÿ Program code:
x ¶x u
u
â x ™ x ArcSec@uD -
u2
u2 - 1
u2
à
x ¶x u
âx
u
u2 - 1
Int@ArcSec@u_D,x_SymbolD :=
x*ArcSec@uD uSqrt@u^2D*Int@SimplifyIntegrand@x*D@u,xDHu*Sqrt@u^2-1DL,xD,xD ;
InverseFunctionFreeQ@u,xD && Not@FunctionOfExponentialQ@u,xDD
Int@ArcCsc@u_D,x_SymbolD :=
x*ArcCsc@uD +
uSqrt@u^2D*Int@SimplifyIntegrand@x*D@u,xDHu*Sqrt@u^2-1DL,xD,xD ;
InverseFunctionFreeQ@u,xD && Not@FunctionOfExponentialQ@u,xDD
2: à Hc + d xLm Ha + b ArcSec@uDL â x when m ¹ - 1 ì u is free of inverse functions
Ÿ Derivation: Integration by parts and piecewise constant extraction
Ÿ Basis: ¶x Ha + b ArcSec@f@xDDL Š
Ÿ Basis: ¶x
f@xD
b ¶x f@xD
f@xD2
f@xD2 -1
Š0
f@xD2
Ÿ Rule: If m ¹ - 1 ì u is free of inverse functions, then
m
à Hc + d xL Ha + b ArcSec@uDL â x ™
Hc + d xLm+1 Ha + b ArcSec@uDL
d Hm + 1L
b
-
d Hm + 1L
à
Hc + d xLm+1 ¶x u
u2
u2 - 1
âx
Rules for integrands involving inverse secants and cosecants
™
23
Hc + d xLm+1 Ha + b ArcSec@uDL
d Hm + 1L
Ÿ Program code:
bu
-
d Hm + 1L
u2
à
Hc + d xLm+1 ¶x u
u
âx
u2 - 1
Int@Hc_.+d_.*x_L^m_.*Ha_.+b_.*ArcSec@u_DL,x_SymbolD :=
Hc+d*xL^Hm+1L*Ha+b*ArcSec@uDLHd*Hm+1LL b*uHd*Hm+1L*Sqrt@u^2DL*Int@SimplifyIntegrand@Hc+d*xL^Hm+1L*D@u,xDHu*Sqrt@u^2-1DL,xD,xD ;
FreeQ@8a,b,c,d,m<,xD && NonzeroQ@m+1D && InverseFunctionFreeQ@u,xD && Not@FunctionOfQ@Hc+d*xL^Hm+1L,u,xDD && Not@FunctionOfExponentialQ
Int@Hc_.+d_.*x_L^m_.*Ha_.+b_.*ArcCsc@u_DL,x_SymbolD :=
Hc+d*xL^Hm+1L*Ha+b*ArcCsc@uDLHd*Hm+1LL +
b*uHd*Hm+1L*Sqrt@u^2DL*Int@SimplifyIntegrand@Hc+d*xL^Hm+1L*D@u,xDHu*Sqrt@u^2-1DL,xD,xD ;
FreeQ@8a,b,c,d,m<,xD && NonzeroQ@m+1D && InverseFunctionFreeQ@u,xD && Not@FunctionOfQ@Hc+d*xL^Hm+1L,u,xDD && Not@FunctionOfExponentialQ
3: à v Ha + b ArcSec@uDL â x when u and à v â x are free of inverse functions
Ÿ Derivation: Integration by parts and piecewise constant extraction
Ÿ Basis: ¶x Ha + b ArcSec@f@xDDL Š
Ÿ Basis: ¶x
f@xD
b ¶x f@xD
f@xD2
f@xD2 -1
Š0
f@xD2
Ÿ Rule: If u is free of inverse functions, let w Š Ù v âx, if w is free of inverse functions, then
à v Ha + b ArcSec@uDL â x ™ w Ha + b ArcSec@uDL - b à
Ÿ Program code:
w ¶x u
u2
u2 - 1
â x ™ w Ha + b ArcSec@uDL -
bu
u2
Int@v_*Ha_.+b_.*ArcSec@u_DL,x_SymbolD :=
With@8w=IntHide@v,xD<,
Dist@Ha+b*ArcSec@uDL,w,xD - b*uSqrt@u^2D*Int@SimplifyIntegrand@w*D@u,xDHu*Sqrt@u^2-1DL,xD,xD ;
InverseFunctionFreeQ@w,xDD ;
FreeQ@8a,b<,xD && InverseFunctionFreeQ@u,xD && Not@MatchQ@v, Hc_.+d_.*xL^m_. ; FreeQ@8c,d,m<,xDDD
à
w ¶x u
âx
u
u2 - 1
Rules for integrands involving inverse secants and cosecants
Int@v_*Ha_.+b_.*ArcCsc@u_DL,x_SymbolD :=
With@8w=IntHide@v,xD<,
Dist@Ha+b*ArcCsc@uDL,w,xD + b*uSqrt@u^2D*Int@SimplifyIntegrand@w*D@u,xDHu*Sqrt@u^2-1DL,xD,xD ;
InverseFunctionFreeQ@w,xDD ;
FreeQ@8a,b<,xD && InverseFunctionFreeQ@u,xD && Not@MatchQ@v, Hc_.+d_.*xL^m_. ; FreeQ@8c,d,m<,xDDD
24