Rules for integrands of the form Hg Sin@e + f xDLp Ha + b Sec@e + f xDLm
1: à Hg Sin@e + f xDLp Ha + b Sec@e + f xDLm â x when m Î Z
Ÿ Derivation: Algebraic normalization
Ÿ Basis: If m Î Z, then Ha + b Sec@zDLm Š
Ÿ Rule: If m Î Z, then
Hb+a Cos@zDLm
Cos@zDm
p
m
à Hg Sin@e + f xDL Ha + b Sec@e + f xDL â x ™ à
Ÿ Program code:
Hg Sin@e + f xDLp Hb + a Cos@e + f xDLm
Cos@e + f xDm
âx
Int@Hg_.*cos@e_.+f_.*x_DL^p_.*Ha_+b_.*csc@e_.+f_.*x_DL^m_.,x_SymbolD :=
Int@Hg*Cos@e+f*xDL^p*Hb+a*Sin@e+f*xDL^mSin@e+f*xD^m,xD ;
FreeQ@8a,b,e,f,g,p<,xD && IntegerQ@mD
2. à Sin@e + f xDp Ha + b Sec@e + f xDLm â x when
p-1
2
ÎZ
1: à Sin@e + f xDp Ha + b Sec@e + f xDLm â x when
p-1
2
Ÿ Derivation: Integration by substitution
Ÿ Basis: If
Ÿ Rule: If
p-1
2
p-1
2
ÎZ í
a2
-
b2
Š 0, then Sin@e + f
Î Z í a2 - b2 Š 0, then
Ÿ Program code:
Î Z í a2 - b2 Š 0
xDp
Š
1
f bp-1
p
m
á Sin@e + f xD Ha + b Sec@e + f xDL â x ™
SubstB
1
f bp-1
H-a+b xL
SubstBá
p-1
2
Ha+b xL
xp+1
p-1
2
, x, Sec@e + f xDF ¶x Sec@e + f xD
H- a + b xL
p-1
2
Ha + b xLm+
p-1
2
â x, x, Sec@e + f xDF
xp+1
Int@cos@e_.+f_.*x_D^p_.*Ha_+b_.*csc@e_.+f_.*x_DL^m_,x_SymbolD :=
-1Hf*b^Hp-1LL*Subst@Int@H-a+b*xL^HHp-1L2L*Ha+b*xL^Hm+Hp-1L2Lx^Hp+1L,xD,x,Csc@e+f*xDD ;
FreeQ@8a,b,e,f,m<,xD && IntegerQ@Hp-1L2D && ZeroQ@a^2-b^2D
Rules for integrands of the form (g sin(e+f x))^p (a+b sec(e+f x))^m
2: à Sin@e + f xDp Ha + b Sec@e + f xDLm â x when
2
Î Z í a2 - b2 ¹ 0
p-1
2
Ÿ Derivation: Integration by substitution
Ÿ Basis: If
Ÿ Rule: If
p-1
2
p-1
2
Î Z, then Sin@e + f
xDp
Š
1
f
SubstB
Î Z í a2 - b2 ¹ 0, then
H-1+xL
p-1
2
H1+xL
xp+1
á Sin@e + f xD Ha + b Sec@e + f xDL â x ™
p
Ÿ Program code:
m
1
f
p-1
2
, x, Sec@e + f xDF ¶x Sec@e + f xD
SubstBá
H- 1 + xL
p-1
2
H1 + xL
xp+1
p-1
2
Ha + b xLm
â x, x, Sec@e + f xDF
Int@cos@e_.+f_.*x_D^p_.*Ha_+b_.*csc@e_.+f_.*x_DL^m_,x_SymbolD :=
-1f*Subst@Int@H-1+xL^HHp-1L2L*H1+xL^HHp-1L2L*Ha+b*xL^mx^Hp+1L,xD,x,Csc@e+f*xDD ;
FreeQ@8a,b,e,f,m<,xD && IntegerQ@Hp-1L2D && NonzeroQ@a^2-b^2D
3: à
Ha + b sec@e + f xDLm
âx
Sin@e + f xD2
Ÿ Derivation: Integration by parts
Ÿ Basis: Ù
Ÿ Basis: -
1
Sin@e+f xD2
Cot@e+f xD
f
Ÿ Rule:
Ÿ Program code:
à
âx Š -
Cot@e+f xD
f
¶x Ha + b Sec@e + f xDLm Š - b m Sec@e + f xD Ha + b Sec@e + f xDLm-1
Ha + b sec@e + f xDLm
Sin@e + f xD2
âx ™ -
Cot@e + f xD Ha + b Sec@e + f xDLm
f
IntAHa_+b_.*csc@e_.+f_.*x_DL^m_‘cos@e_.+f_.*x_D^2,x_SymbolE :=
+ b m à Sec@e + f xD Ha + b Sec@e + f xDLm-1 â x
Tan@e+f*xD*Ha+b*Csc@e+f*xDL^mf + b*m*Int@Csc@e+f*xD*Ha+b*Csc@e+f*xDL^Hm-1L,xD ;
FreeQ@8a,b,e,f,m<,xD
Rules for integrands of the form (g sin(e+f x))^p (a+b sec(e+f x))^m
3
4: à Hg Sin@e + f xDLp Ha + b Sec@e + f xDLm â x when a2 - b2 Š 0 ê H2 m
pL Î Z
Ÿ Derivation: Piecewise constant extraction
Ÿ Basis: ¶x
Cos@e+f xDm Ha+b Sec@e+f xDLm
Hb+a Cos@e+f xDLm
Ÿ Rule: If a2 - b2 Š 0 ê H2 m
Š0
pL Î Z, then
p
m
à Hg Sin@e + f xDL Ha + b Sec@e + f xDL â x ™
Cos@e + f xDm Ha + b Sec@e + f xDLm
Ÿ Program code:
Hb + a Cos@e + f xDLm
à
Hg Sin@e + f xDLp Hb + a Cos@e + f xDLm
Cos@e + f xDm
Int@Hg_.*cos@e_.+f_.*x_DL^p_.*Ha_+b_.*csc@e_.+f_.*x_DL^m_,x_SymbolD :=
Sin@e+f*xD^FracPart@mD*Ha+b*Csc@e+f*xDL^FracPart@mDHb+a*Sin@e+f*xDL^FracPart@mD*
Int@Hg*Cos@e+f*xDL^p*Hb+a*Sin@e+f*xDL^mSin@e+f*xD^m,xD ;
FreeQ@8a,b,e,f,g,m,p<,xD && HZeroQ@a^2-b^2D ÈÈ IntegersQ@2*m,pDL
X: à Hg Sin@e + f xDLp Ha + b Sec@e + f xDLm â x
Ÿ Rule:
Ÿ Program code:
p
m
p
m
à Hg Sin@e + f xDL Ha + b Sec@e + f xDL â x ™ à Hg Sin@e + f xDL Ha + b Sec@e + f xDL â x
Int@Hg_.*cos@e_.+f_.*x_DL^p_.*Ha_+b_.*csc@e_.+f_.*x_DL^m_.,x_SymbolD :=
Defer@IntD@Hg*Cos@e+f*xDL^p*Ha+b*Csc@e+f*xDL^m,xD ;
FreeQ@8a,b,e,f,g,m,p<,xD
Rules for integrands of the form Hg Csc@e + f xDLp Ha + b Sec@e + f xDLm
x: à Hg Csc@e + f xDLp Ha + b Sec@e + f xDLm â x when p Ï Z ì m Î Z
Ÿ Derivation: Algebraic normalization
Ÿ Basis: If m Î Z, then Ha + b Sec@zDLm Š
Ÿ Rule: If p Ï Z ì m Î Z, then
Hb+a Cos@zDLm
Cos@zDm
âx
Rules for integrands of the form (g sin(e+f x))^p (a+b sec(e+f x))^m
p
m
à Hg Csc@e + f xDL Ha + b Sec@e + f xDL â x ™ à
Ÿ Program code:
4
Hg Csc@e + f xDLp Hb + a Cos@e + f xDLm
Cos@e + f xDm
âx
H* Int@Hg_.*sec@e_.+f_.*x_DL^p_*Ha_+b_.*csc@e_.+f_.*x_DL^m_.,x_SymbolD :=
Int@Hg*Sec@e+f*xDL^p*Hb+a*Sin@e+f*xDL^mSin@e+f*xD^m,xD ;
FreeQ@8a,b,e,f,g,p<,xD && Not@IntegerQ@pDD && IntegerQ@mD *L
1: à Hg Csc@e + f xDLp Ha + b Sec@e + f xDLm â x when p Ï Z
Ÿ Derivation: Piecewise constant extraction
Ÿ Basis: ¶x HHg Csc@e + f xDLp Hg Sin@e + f xDLp L Š 0
Ÿ Rule: If p Ï Z, then
Ÿ Program code:
p
m
p
p
à Hg Csc@e + f xDL Ha + b Sec@e + f xDL â x ™ Hg Csc@e + f xDL Hg Sin@e + f xDL à
Int@Hg_.*sec@e_.+f_.*x_DL^p_*Ha_+b_.*csc@e_.+f_.*x_DL^m_.,x_SymbolD :=
Hg*Sec@e+f*xDL^p*Hg*Cos@e+f*xDL^p*Int@Ha+b*Csc@e+f*xDL^mHg*Cos@e+f*xDL^p,xD ;
FreeQ@8a,b,e,f,g,m,p<,xD && Not@IntegerQ@pDD
Ha + b Sec@e + f xDLm
Hg Sin@e + f xDLp
âx