Rules for integrands of the form Hg Tan@e + f xDLp Ha + b Tan@e + f xDLm Hc + d Tan@e + f xDLn
X: à Hg Tan@e + f xDLp Ha + b Tan@e + f xDLm Hc + d Tan@e + f xDLn â x
Ÿ Rule:
p
m
n
p
m
n
à Hg Tan@e + f xDL Ha + b Tan@e + f xDL Hc + d Tan@e + f xDL â x ™ à Hg Tan@e + f xDL Ha + b Tan@e + f xDL Hc + d Tan@e + f xDL â x
Ÿ Program code:
Int@Hg_.*tan@e_.+f_.*x_DL^p_.*Ha_+b_.*tan@e_.+f_.*x_DL^m_*Hc_+d_.*tan@e_.+f_.*x_DL^n_,x_SymbolD :=
Defer@IntD@Hg*Tan@e+f*xDL^p*Ha+b*Tan@e+f*xDL^m*Hc+d*Tan@e+f*xDL^n,xD ;
FreeQ@8a,b,c,d,e,f,g,m,n,p<,xD
Rules for integrands of the form Hg Tan@e + f xDq Lp Ha + b Tan@e + f xDLm Hc + d Tan@e + f xDLn
1: à Hg Cot@e + f xDLp Ha + b Tan@e + f xDLm Hc + d Tan@e + f xDLn â x when p Ï Z ì m Î Z ì n Î Z
Ÿ Derivation: Algebraic normalization
Ÿ Basis: If m Î Z ß n Î Z, then Ha + b Tan@zDLm Hc + d Tan@zDLn Š
Ÿ Rule: If p Ï Z ì m Î Z ì n Î Z, then
gm+n Hb+a Cot@zDLm Hd+c Cot@zDLn
Hg Cot@zDLm+n
p
m
n
m+n
p-m-n
Hb + a Cot@e + f xDLm Hd + c Cot@e + f xDLn â x
à Hg Cot@e + f xDL Ha + b Tan@e + f xDL Hc + d Tan@e + f xDL â x ™ g
à Hg Cot@e + f xDL
Ÿ Program code:
IntAIg_.‘tan@e_.+f_.*x_DM^p_*Ha_.+b_.*tan@e_.+f_.*x_DL^m_.*Hc_+d_.*tan@e_.+f_.*x_DL^n_.,x_SymbolE :=
g^Hm+nL*Int@Hg*Cot@e+f*xDL^Hp-m-nL*Hb+a*Cot@e+f*xDL^m*Hd+c*Cot@e+f*xDL^n,xD ;
FreeQ@8a,b,c,d,e,f,g,p<,xD && Not@IntegerQ@pDD && IntegerQ@mD && IntegerQ@nD
IntAIg_.‘cot@e_.+f_.*x_DM^p_*Ha_.+b_.*cot@e_.+f_.*x_DL^m_.*Hc_+d_.*cot@e_.+f_.*x_DL^n_.,x_SymbolE :=
g^Hm+nL*Int@Hg*Tan@e+f*xDL^Hp-m-nL*Hb+a*Tan@e+f*xDL^m*Hd+c*Tan@e+f*xDL^n,xD ;
FreeQ@8a,b,c,d,e,f,g,p<,xD && Not@IntegerQ@pDD && IntegerQ@mD && IntegerQ@nD
Rules for integrands of the form (a+b tan(e+f x)^m (c+d tan(e+f x))^n (g tan[e+f x])^p
2
2: à Hg Tan@e + f xDq Lp Ha + b Tan@e + f xDLm Hc + d Tan@e + f xDLn â x when p Ï Z ì Ø Hm Î Z ß n Î ZL
Ÿ Derivation: Piecewise constant extraction
Ÿ Basis: ¶x
Ig Tan@e+f xDq M
p
Hg Tan@e+f xDLp q
Š0
Ÿ Rule: If p Ï Z ì Ø Hm Î Z ß n Î ZL, then
q p
m
n
à Hg Tan@e + f xD L Ha + b Tan@e + f xDL Hc + d Tan@e + f xDL â x ™
Ÿ Program code:
Hg Tan@e + f xDq Lp
Hg Tan@e + f xDLp q
à Hg Tan@e + f xDL
pq
Ha + b Tan@e + f xDLm Hc + d Tan@e + f xDLn â x
Int@Hg_.*tan@e_.+f_.*x_D^q_L^p_*Ha_.+b_.*tan@e_.+f_.*x_DL^m_.*Hc_+d_.*tan@e_.+f_.*x_DL^n_.,x_SymbolD :=
Hg*Tan@e+f*xD^qL^pHg*Tan@e+f*xDL^Hp*qL*Int@Hg*Tan@e+f*xDL^Hp*qL*Ha+b*Tan@e+f*xDL^m*Hc+d*Tan@e+f*xDL^n,xD ;
FreeQ@8a,b,c,d,e,f,g,m,n,p,q<,xD && Not@IntegerQ@pDD && Not@IntegerQ@mD && IntegerQ@nDD
Int@Hg_.*cot@e_.+f_.*x_D^q_L^p_*Ha_.+b_.*cot@e_.+f_.*x_DL^m_.*Hc_+d_.*cot@e_.+f_.*x_DL^n_.,x_SymbolD :=
Hg*Cot@e+f*xD^qL^pHg*Cot@e+f*xDL^Hp*qL*Int@Hg*Cot@e+f*xDL^Hp*qL*Ha+b*Cot@e+f*xDL^m*Hc+d*Cot@e+f*xDL^n,xD ;
FreeQ@8a,b,c,d,e,f,g,m,n,p,q<,xD && Not@IntegerQ@pDD && Not@IntegerQ@mD && IntegerQ@nDD
Rules for integrands of the form Hg Tan@e + f xDLp Ha + b Tan@e + f xDLm Hc + d Cot@e + f xDLn
1: à Hg Tan@e + f xDLp Ha + b Tan@e + f xDLm Hc + d Cot@e + f xDLn â x when n Î Z
Ÿ Derivation: Algebraic normalization
Ÿ Basis: c + d Cot@zD Š
d+c Tan@zD
Tan@zD
Ÿ Rule: If n Î Z, then
p
m
n
n
p-n
Ha + b Tan@e + f xDLm Hd + c Tan@e + f xDLn â x
à Hg Tan@e + f xDL Ha + b Tan@e + f xDL Hc + d Cot@e + f xDL â x ™ g à Hg Tan@e + f xDL
Ÿ Program code:
Int@Hg_.*tan@e_.+f_.*x_DL^p_.*Ha_+b_.*tan@e_.+f_.*x_DL^m_.*Hc_+d_.*cot@e_.+f_.*x_DL^n_.,x_SymbolD :=
g^n*Int@Hg*Tan@e+f*xDL^Hp-nL*Ha+b*Tan@e+f*xDL^m*Hd+c*Tan@e+f*xDL^n,xD ;
FreeQ@8a,b,c,d,e,f,g,m,p<,xD && IntegerQ@nD
Rules for integrands of the form (a+b tan(e+f x)^m (c+d tan(e+f x))^n (g tan[e+f x])^p
3
2. à Hg Tan@e + f xDLp Ha + b Tan@e + f xDLm Hc + d Cot@e + f xDLn â x when n Ï Z
1. à Hg Tan@e + f xDLp Ha + b Tan@e + f xDLm Hc + d Cot@e + f xDLn â x when n Ï Z ß m Î Z
1: à Tan@e + f xDp Ha + b Tan@e + f xDLm Hc + d Cot@e + f xDLn â x when n Ï Z ì m Î Z ì p Î Z
Ÿ Derivation: Algebraic normalization
Ÿ Basis: a + b Tan@zD Š
b+a Cot@zD
Cot@zD
Ÿ Rule: If n Ï Z ì m Î Z ì p Î Z, then
Ÿ Program code:
p
m
n
à Tan@e + f xD Ha + b Tan@e + f xDL Hc + d Cot@e + f xDL â x ™ à
Hb + a Cot@e + f xDLm Hc + d Cot@e + f xDLn
Cot@e + f xDm+p
âx
Int@tan@e_.+f_.*x_D^p_.*Ha_+b_.*tan@e_.+f_.*x_DL^m_.*Hc_+d_.*cot@e_.+f_.*x_DL^n_,x_SymbolD :=
Int@Hb+a*Cot@e+f*xDL^m*Hc+d*Cot@e+f*xDL^nCot@e+f*xD^Hm+pL,xD ;
FreeQ@8a,b,c,d,e,f,n<,xD && Not@IntegerQ@nDD && IntegerQ@mD && IntegerQ@pD
2: à Hg Tan@e + f xDLp Ha + b Tan@e + f xDLm Hc + d Cot@e + f xDLn â x when n Ï Z ì m Î Z ì p Ï Z
Ÿ Derivation: Algebraic normalization and piecewise constant extraction
Ÿ Basis: a + b Tan@zD Š
b+a Cot@zD
Cot@zD
Ÿ Basis: ¶x HCot@e + f xDp Hg Tan@e + f xDLp L Š 0
Ÿ Rule: If n Ï Z ì m Î Z ì p Ï Z, then
p
m
n
p
p
à Hg Tan@e + f xDL Ha + b Tan@e + f xDL Hc + d Cot@e + f xDL â x ™ Cot@e + f xD Hg Tan@e + f xDL à
Hb + a Cot@e + f xDLm Hc + d Cot@e + f xDLn
Cot@e + f xDm+p
Ÿ Program code:
Int@Hg_.*tan@e_.+f_.*x_DL^p_*Ha_+b_.*tan@e_.+f_.*x_DL^m_.*Hc_+d_.*cot@e_.+f_.*x_DL^n_,x_SymbolD :=
Cot@e+f*xD^p*Hg*Tan@e+f*xDL^p*Int@Hb+a*Cot@e+f*xDL^m*Hc+d*Cot@e+f*xDL^nCot@e+f*xD^Hm+pL,xD ;
FreeQ@8a,b,c,d,e,f,g,n,p<,xD && Not@IntegerQ@nDD && IntegerQ@mD && Not@IntegerQ@pDD
âx
Rules for integrands of the form (a+b tan(e+f x)^m (c+d tan(e+f x))^n (g tan[e+f x])^p
4
2: à Hg Tan@e + f xDLp Ha + b Tan@e + f xDLm Hc + d Cot@e + f xDLn â x when n Ï Z ß m Ï Z
Ÿ Derivation: Piecewise constant extraction
Ÿ Basis: ¶x
Hc+d Cot@e+f xDLn Hg Tan@e+f xDLn
Hd+c Tan@e+f xDLn
Ÿ Rule: If n Ï Z ß m Ï Z, then
Š0
p
m
n
à Hg Tan@e + f xDL Ha + b Tan@e + f xDL Hc + d Cot@e + f xDL â x ™
Hg Tan@e + f xDLn Hc + d Cot@e + f xDLn
Ÿ Program code:
Hd + c Tan@e + f xDLn
à Hg Tan@e + f xDL
p-n
Ha + b Tan@e + f xDLm Hd + c Tan@e + f xDLn â x
Int@Hg_.*tan@e_.+f_.*x_DL^p_.*Ha_+b_.*tan@e_.+f_.*x_DL^m_*Hc_+d_.*cot@e_.+f_.*x_DL^n_,x_SymbolD :=
Hg*Tan@e+f*xDL^n*Hc+d*Cot@e+f*xDL^nHd+c*Tan@e+f*xDL^n*Int@Hg*Tan@e+f*xDL^Hp-nL*Ha+b*Tan@e+f*xDL^m*Hd+c*Tan@e+f*xDL^n,xD ;
FreeQ@8a,b,c,d,e,f,g,m,n,p<,xD && Not@IntegerQ@nDD && Not@IntegerQ@mDD