4.7 x^m trig(a+b log(cx^n))^p.nb

1. à u Sin@a + b Log@c xn DDp â x
Rules for integrands of the form xm Trig@a + b Log@c xnDDp
1. à Sin@a + b Log@c xn DDp â x
1: à Sin@a + b Log@c xn DDp â x when b2 n2 Hp + 2L2 + 1 0 ì p ¹ - 1
Note: When p 1 it is better to use rule for Sin@a + b Log@c xn DD.
Rule: If b2 n2 Hp + 2L2 + 1 0 ì p ¹ - 1, then
Program code:
p
n
à Sin@a + b Log@c x DD â x
x Hp + 2L Sin@a + b Log@c xn DDp+2
+
p+1
x Cot@a + b Log@c xn DD Sin@a + b Log@c xn DDp+2
b n Hp + 1L
Int@Sin@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
x*Hp+2L*Sin@a+b*Log@c*x^nDD^Hp+2LHp+1L +
x*Cot@a+b*Log@c*x^nDD*Sin@a+b*Log@c*x^nDD^Hp+2LHb*n*Hp+1LL ;
FreeQ@8a,b,c,n,p<,xD && ZeroQ@b^2*n^2*Hp+2L^2+1D && NonzeroQ@p+1D
Int@Cos@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
x*Hp+2L*Cos@a+b*Log@c*x^nDD^Hp+2LHp+1L x*Tan@a+b*Log@c*x^nDD*Cos@a+b*Log@c*x^nDD^Hp+2LHb*n*Hp+1LL ;
FreeQ@8a,b,c,n,p<,xD && ZeroQ@b^2*n^2*Hp+2L^2+1D && NonzeroQ@p+1D
2. à Sin@a + b Log@c xn DDp â x when p > 0
1: à Sin@a + b Log@c xn DDp â x when p Î Z+ ì b2 n2 p2 + 1 0
Derivation: Algebraic expansion
Basis: If b2 n2 p2 + 1 0 ì p Î Z, then Sin@a + b Log@c xn DDp K 2ã b n p Hc xn L
abnp
Basis: If b2 n2 p2 + 1 0 ì p Î Z, then Cos@a + b Log@c xn DDp K ã
abnp
2
-
Hc xn L
-
1
np
-
ã-a b n p
2bnp
+
ã-a b n p
2
1
np
Note: The above identities need to be formally derived, and possibly the domain of p expanded.
Rule: If p Î Z+ ì b2 n2 p2 + 1 0, then
Hc xn L n p O
1
Hc xn L n p O
1
p
p
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
Program code:
2
p
n
à Sin@a + b Log@c x DD â x à ExpandIntegrandB
ãa b n p
2bnp
Hc xn L
-
ã-a b n p
1
np
2bnp
Hc xn L n p
1
p
, xF â x
Int@Sin@a_.+b_.*Log@c_.*x_^n_.DD^p_.,x_SymbolD :=
Int@ExpandIntegrand@HE^Ha*b*n*pLH2*b*n*pL*Hc*x^nL^H-1Hn*pLL-E^H-a*b*n*pLH2*b*n*pL*Hc*x^nL^H1Hn*pLLL^p,xD,xD ;
FreeQ@8a,b,c,n<,xD && PositiveIntegerQ@pD && ZeroQ@b^2*n^2*p^2+1D
Int@Cos@a_.+b_.*Log@c_.*x_^n_.DD^p_.,x_SymbolD :=
Int@ExpandIntegrand@HE^Ha*b*n*pL2*Hc*x^nL^H-1Hn*pLL-E^H-a*b*n*pL2*Hc*x^nL^H1Hn*pLLL^p,xD,xD ;
FreeQ@8a,b,c,n<,xD && PositiveIntegerQ@pD && ZeroQ@b^2*n^2*p^2+1D
2. à Sin@a + b Log@c xn DDp â x when p > 0 ì b2 n2 p2 + 1 ¹ 0
1: à Sin@a + b Log@c xn DD â x when b2 n2 + 1 ¹ 0
Rule: If b2 n2 + 1 ¹ 0, then
Program code:
n
à Sin@a + b Log@c x DD â x
x Sin@a + b Log@c xn DD
-
b2 n2 + 1
b n x Cos@a + b Log@c xn DD
b2 n2 + 1
Int@Sin@a_.+b_.*Log@c_.*x_^n_.DD,x_SymbolD :=
x*Sin@a+b*Log@c*x^nDDHb^2*n^2+1L b*n*x*Cos@a+b*Log@c*x^nDDHb^2*n^2+1L ;
FreeQ@8a,b,c,n<,xD && NonzeroQ@b^2*n^2+1D
Int@Cos@a_.+b_.*Log@c_.*x_^n_.DD,x_SymbolD :=
x*Cos@a+b*Log@c*x^nDDH1+b^2*n^2L +
b*n*x*Sin@a+b*Log@c*x^nDDHb^2*n^2+1L ;
FreeQ@8a,b,c,n<,xD && NonzeroQ@b^2*n^2+1D
2: à Sin@a + b Log@c xn DDp â x when p > 1 ì b2 n2 p2 + 1 ¹ 0
Rule: If p > 1 ì b2 n2 p2 + 1 ¹ 0, then
-
p
n
à Sin@a + b Log@c x DD â x
+
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
x Sin@a + b Log@c xn DDp
b2 n2 p2 + 1
-
p
n
à Sin@a + b Log@c x DD â x
3
b n p x Cos@a + b Log@c xn DD Sin@a + b Log@c xn DDp-1
b2 n2 p2 + 1
+
b2 n2 p Hp - 1L
b2 n2 p2 + 1
Program code:
p-2
n
âx
à Sin@a + b Log@c x DD
Int@Sin@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
x*Sin@a+b*Log@c*x^nDD^pHb^2*n^2*p^2+1L b*n*p*x*Cos@a+b*Log@c*x^nDD*Sin@a+b*Log@c*x^nDD^Hp-1LHb^2*n^2*p^2+1L +
b^2*n^2*p*Hp-1LHb^2*n^2*p^2+1L*Int@Sin@a+b*Log@c*x^nDD^Hp-2L,xD ;
FreeQ@8a,b,c,n<,xD && RationalQ@pD && p>1 && NonzeroQ@b^2*n^2*p^2+1D
Int@Cos@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
x*Cos@a+b*Log@c*x^nDD^pHb^2*n^2*p^2+1L +
b*n*p*x*Cos@a+b*Log@c*x^nDD^Hp-1L*Sin@a+b*Log@c*x^nDDHb^2*n^2*p^2+1L +
b^2*n^2*p*Hp-1LHb^2*n^2*p^2+1L*Int@Cos@a+b*Log@c*x^nDD^Hp-2L,xD ;
FreeQ@8a,b,c,n<,xD && RationalQ@pD && p>1 && NonzeroQ@b^2*n^2*p^2+1D
3: à Sin@a + b Log@c xn DDp â x when p < - 1 ì p ¹ - 2 ì b2 n2 Hp + 2L2 + 1 ¹ 0
Rule: If p < - 1 ì p ¹ - 2 ì b2 n2 Hp + 2L2 + 1 ¹ 0, then
x Cot@a + b Log@c xn DD Sin@a + b Log@c xn DDp+2
Program code:
b n Hp + 1L
p
n
à Sin@a + b Log@c x DD â x
-
x Sin@a + b Log@c xn DDp+2
b2 n2 Hp + 1L Hp + 2L
+
b2 n2 Hp + 2L2 + 1
b2 n2 Hp + 1L Hp + 2L
Int@Sin@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
x*Cot@a+b*Log@c*x^nDD*Sin@a+b*Log@c*x^nDD^Hp+2LHb*n*Hp+1LL x*Sin@a+b*Log@c*x^nDD^Hp+2LHb^2*n^2*Hp+1L*Hp+2LL +
Hb^2*n^2*Hp+2L^2+1LHb^2*n^2*Hp+1L*Hp+2LL*Int@Sin@a+b*Log@c*x^nDD^Hp+2L,xD ;
FreeQ@8a,b,c,n<,xD && RationalQ@pD && p<-1 && p¹-2 && NonzeroQ@b^2*n^2*Hp+2L^2+1D
Int@Cos@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
-x*Tan@a+b*Log@c*x^nDD*Cos@a+b*Log@c*x^nDD^Hp+2LHb*n*Hp+1LL x*Cos@a+b*Log@c*x^nDD^Hp+2LHb^2*n^2*Hp+1L*Hp+2LL +
Hb^2*n^2*Hp+2L^2+1LHb^2*n^2*Hp+1L*Hp+2LL*Int@Cos@a+b*Log@c*x^nDD^Hp+2L,xD ;
FreeQ@8a,b,c,n<,xD && RationalQ@pD && p<-1 && p¹-2 && NonzeroQ@b^2*n^2*Hp+2L^2+1D
p+2
n
âx
à Sin@a + b Log@c x DD
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
4
4: à Sin@a + b Log@c xn DDp â x when p Ï Z+ ì b2 n2 p2 + 1 ¹ 0
Rule: If b2 n2 p2 + 1 ¹ 0, then
x Iä ã-ä a Hc xn L-ä b - ä ãä a Hc xn Lä b M
p
Program code:
H1 - ä b n pL I2 - 2
ã2 ä a
Hc
p
xn L2 ä b M
p
n
à Sin@a + b Log@c x DD â x
1-äbnp
Hypergeometric2F1B- p,
2äbn
1-äbnp
, 1+
2äbn
, ã2 ä a Hc xn L2 ä b F
Int@Sin@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
x*HIHE^HI*aL*Hc*x^nL^HI*bLL-I*E^HI*aL*Hc*x^nL^HI*bLL^pHH1-I*b*n*pL*H2-2*E^H2*I*aL*Hc*x^nL^H2*I*bLL^pL*
Hypergeometric2F1@-p,H1-I*b*n*pLH2*I*b*nL,1+H1-I*b*n*pLH2*I*b*nL,E^H2*I*aL*Hc*x^nL^H2*I*bLD ;
FreeQ@8a,b,c,n,p<,xD && NonzeroQ@b^2*n^2*p^2+1D
Int@Cos@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
x*H1HE^HI*aL*Hc*x^nL^HI*bLL+E^HI*aL*Hc*x^nL^HI*bLL^pHH1-I*b*n*pL*H2+2*E^H2*I*aL*Hc*x^nL^H2*I*bLL^pL*
Hypergeometric2F1@-p,H1-I*b*n*pLH2*I*b*nL,1+H1-I*b*n*pLH2*I*b*nL,-E^H2*I*aL*Hc*x^nL^H2*I*bLD ;
FreeQ@8a,b,c,n,p<,xD && NonzeroQ@b^2*n^2*p^2+1D
2. à xm Sin@a + b Log@c xn DDp â x
1: à xm Sin@a + b Log@c xn DDp â x when b2 n2 Hp + 2L2 + Hm + 1L2 0 ì p ¹ - 1 ì m ¹ - 1
Note: When p 1 it is better to use rule for xm Sin@a + b Log@c xn DD.
Rule: If b2 n2 Hp + 2L2 + Hm + 1L2 0 ì p ¹ - 1 ì m ¹ - 1, then
p
m
n
à x Sin@a + b Log@c x DD â x
Program code:
Hp + 2L xm+1 Sin@a + b Log@c xn DDp+2
Hm + 1L Hp + 1L
+
xm+1 Cot@a + b Log@c xn DD Sin@a + b Log@c xn DDp+2
b n Hp + 1L
Int@x_^m_.*Sin@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
Hp+2L*x^Hm+1L*Sin@a+b*Log@c*x^nDD^Hp+2LHHm+1L*Hp+1LL +
x^Hm+1L*Cot@a+b*Log@c*x^nDD*Sin@a+b*Log@c*x^nDD^Hp+2LHb*n*Hp+1LL ;
FreeQ@8a,b,c,m,n,p<,xD && ZeroQ@b^2*n^2*Hp+2L^2+Hm+1L^2D && NonzeroQ@p+1D && NonzeroQ@m+1D
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
5
Int@x_^m_.*Cos@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
Hp+2L*x^Hm+1L*Cos@a+b*Log@c*x^nDD^Hp+2LHHm+1L*Hp+1LL x^Hm+1L*Tan@a+b*Log@c*x^nDD*Cos@a+b*Log@c*x^nDD^Hp+2LHb*n*Hp+1LL ;
FreeQ@8a,b,c,m,n,p<,xD && ZeroQ@b^2*n^2*Hp+2L^2+Hm+1L^2D && NonzeroQ@p+1D && NonzeroQ@m+1D
2. à xm Sin@a + b Log@c xn DDp â x when p > 0
1: à xm Sin@a + b Log@c xn DDp â x when p Î Z+ ì b2 n2 p2 + Hm + 1L2 0
Derivation: Algebraic expansion
Basis: If b2 n2 p2 + Hm + 1L2 0 ì p Î Z, then Sin@a + b Log@c xn DDp
Basis: If b2 n2 p2 + Hm + 1L2 0 ì p Î Z, then Cos@a + b Log@c xn DDp
1
2p
1
2p
K bm+1
ã
np
Kã
abnp
m+1
abnp
m+1
Hc xn L
-
Hc xn L
-
m+1
np
m+1
np
+ ã-
-
m+1
bnp
abnp
m+1
Note: The above identities need to be formally derived, and possibly the domain of p expanded.
Rule: If p Î Z+ ì b2 n2 p2 + Hm + 1L2 0, then
Program code:
p
m
n
à x Sin@a + b Log@c x DD â x
1
2p
à ExpandIntegrandBx
m
m+1
abnp
ã
bnp
m+1
Hc xn L
-
m+1
m+1
np
bnp
Int@x_^m_.*Sin@a_.+b_.*Log@c_.*x_^n_.DD^p_.,x_SymbolD :=
12^p*Int@ExpandIntegrand@x^m*HHm+1LHb*n*pL*E^Ha*b*n*pHm+1LL*Hc*x^nL^H-Hm+1LHn*pLL Hm+1LHb*n*pL*E^H-a*b*n*pHm+1LL*Hc*x^nL^HHm+1LHn*pLLL^p,xD,xD ;
FreeQ@8a,b,c,m,n<,xD && PositiveIntegerQ@pD && ZeroQ@b^2*n^2*p^2+Hm+1L^2D
Int@x_^m_.*Cos@a_.+b_.*Log@c_.*x_^n_.DD^p_.,x_SymbolD :=
12^p*Int@ExpandIntegrand@x^m*HE^Ha*b*n*pHm+1LL*Hc*x^nL^H-Hm+1LHn*pLL E^H-a*b*n*pHm+1LL*Hc*x^nL^HHm+1LHn*pLLL^p,xD,xD ;
FreeQ@8a,b,c,m,n<,xD && PositiveIntegerQ@pD && ZeroQ@b^2*n^2*p^2+Hm+1L^2D
2. à xm Sin@a + b Log@c xn DDp â x when p > 0 ì b2 n2 p2 + Hm + 1L2 ¹ 0
1: à xm Sin@a + b Log@c xn DD â x when b2 n2 + Hm + 1L2 ¹ 0
Rule: If b2 n2 + Hm + 1L2 ¹ 0, then
ã-
abnp
m+1
Hc xn L n p O
Hc xn L n p O
m+1
ã-
abnp
m+1
m+1
p
p
Hc xn L n p
m+1
p
, xF â x
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
Program code:
6
m
n
à x Sin@a + b Log@c x DD â x
Hm + 1L xm+1 Sin@a + b Log@c xn DD
b2 n2 + Hm + 1L2
-
b n xm+1 Cos@a + b Log@c xn DD
b2 n2 + Hm + 1L2
Int@x_^m_.*Sin@a_.+b_.*Log@c_.*x_^n_.DD,x_SymbolD :=
Hm+1L*x^Hm+1L*Sin@a+b*Log@c*x^nDDHb^2*n^2+Hm+1L^2L b*n*x^Hm+1L*Cos@a+b*Log@c*x^nDDHb^2*n^2+Hm+1L^2L ;
FreeQ@8a,b,c,m,n<,xD && NonzeroQ@b^2*n^2+Hm+1L^2D
Int@x_^m_.*Cos@a_.+b_.*Log@c_.*x_^n_.DD,x_SymbolD :=
Hm+1L*x^Hm+1L*Cos@a+b*Log@c*x^nDDHb^2*n^2+Hm+1L^2L +
b*n*x^Hm+1L*Sin@a+b*Log@c*x^nDDHb^2*n^2+Hm+1L^2L ;
FreeQ@8a,b,c,m,n<,xD && NonzeroQ@b^2*n^2+Hm+1L^2D
2: à xm Sin@a + b Log@c xn DDp â x when p > 1 ì b2 n2 p2 + Hm + 1L2 ¹ 0
Rule: If p > 1 ì b2 n2 p2 + Hm + 1L2 ¹ 0, then
Hm + 1L xm+1 Sin@a + b Log@c xn DDp
b2 n2 p2 + Hm + 1L2
Program code:
-
p
m
n
à x Sin@a + b Log@c x DD â x
b n p xm+1 Cos@a + b Log@c xn DD Sin@a + b Log@c xn DDp-1
b2 n2 p2 + Hm + 1L2
+
b2 n2 p Hp - 1L
b2 n2 p2 + Hm + 1L2
Int@x_^m_.*Sin@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
Hm+1L*x^Hm+1L*Sin@a+b*Log@c*x^nDD^pHb^2*n^2*p^2+Hm+1L^2L b*n*p*x^Hm+1L*Cos@a+b*Log@c*x^nDD*Sin@a+b*Log@c*x^nDD^Hp-1LHb^2*n^2*p^2+Hm+1L^2L +
b^2*n^2*p*Hp-1LHb^2*n^2*p^2+Hm+1L^2L*Int@x^m*Sin@a+b*Log@c*x^nDD^Hp-2L,xD ;
FreeQ@8a,b,c,m,n<,xD && RationalQ@pD && p>1 && NonzeroQ@b^2*n^2*p^2+Hm+1L^2D
Int@x_^m_.*Cos@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
Hm+1L*x^Hm+1L*Cos@a+b*Log@c*x^nDD^pHb^2*n^2*p^2+Hm+1L^2L +
b*n*p*x^Hm+1L*Sin@a+b*Log@c*x^nDD*Cos@a+b*Log@c*x^nDD^Hp-1LHb^2*n^2*p^2+Hm+1L^2L +
b^2*n^2*p*Hp-1LHb^2*n^2*p^2+Hm+1L^2L*Int@x^m*Cos@a+b*Log@c*x^nDD^Hp-2L,xD ;
FreeQ@8a,b,c,m,n<,xD && RationalQ@pD && p>1 && NonzeroQ@b^2*n^2*p^2+Hm+1L^2D
p-2
m
n
âx
à x Sin@a + b Log@c x DD
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
7
3: à xm Sin@a + b Log@c xn DDp â x when p < - 1 ì p ¹ - 2 ì b2 n2 Hp + 2L2 + Hm + 1L2 ¹ 0
Rule: If p < - 1 ì p ¹ - 2 ì b2 n2 Hp + 2L2 + Hm + 1L2 ¹ 0, then
xm+1 Cot@a + b Log@c xn DD Sin@a + b Log@c xn DDp+2
Program code:
b n Hp + 1L
-
p
m
n
à x Sin@a + b Log@c x DD â x
Hm + 1L xm+1 Sin@a + b Log@c xn DDp+2
b2 n2 Hp + 1L Hp + 2L
+
b2 n2 Hp + 2L2 + Hm + 1L2
b2 n2 Hp + 1L Hp + 2L
p+2
m
n
âx
à x Sin@a + b Log@c x DD
Int@x_^m_.*Sin@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
x^Hm+1L*Cot@a+b*Log@c*x^nDD*Sin@a+b*Log@c*x^nDD^Hp+2LHb*n*Hp+1LL Hm+1L*x^Hm+1L*Sin@a+b*Log@c*x^nDD^Hp+2LHb^2*n^2*Hp+1L*Hp+2LL +
Hb^2*n^2*Hp+2L^2+Hm+1L^2LHb^2*n^2*Hp+1L*Hp+2LL*Int@x^m*Sin@a+b*Log@c*x^nDD^Hp+2L,xD ;
FreeQ@8a,b,c,m,n<,xD && RationalQ@pD && p<-1 && p¹-2 && NonzeroQ@b^2*n^2*Hp+2L^2+Hm+1L^2D
Int@x_^m_.*Cos@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
-x^Hm+1L*Tan@a+b*Log@c*x^nDD*Cos@a+b*Log@c*x^nDD^Hp+2LHb*n*Hp+1LL Hm+1L*x^Hm+1L*Cos@a+b*Log@c*x^nDD^Hp+2LHb^2*n^2*Hp+1L*Hp+2LL +
Hb^2*n^2*Hp+2L^2+Hm+1L^2LHb^2*n^2*Hp+1L*Hp+2LL*Int@x^m*Cos@a+b*Log@c*x^nDD^Hp+2L,xD ;
FreeQ@8a,b,c,m,n<,xD && RationalQ@pD && p<-1 && p¹-2 && NonzeroQ@b^2*n^2*Hp+2L^2+Hm+1L^2D
4: à xm Sin@a + b Log@c xn DDp â x when p Ï Z+ ì b2 n2 p2 + Hm + 1L2 ¹ 0
Rule: If b2 n2 p2 + Hm + 1L2 ¹ 0, then
xm+1 Iä ã-ä a Hc xn L-ä b - ä ãä a Hc xn Lä b M
p
Program code:
Hm + 1 - ä b n pL I2 - 2 ã2 ä a Hc
p
xn L2 ä b M
p
m
n
à x Sin@a + b Log@c x DD â x
m+1-äbnp
Hypergeometric2F1B- p,
2äbn
m+1-äbnp
, 1+
2äbn
, ã2 ä a Hc xn L2 ä b F
Int@x_^m_.*Sin@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
x^Hm+1L*HI*E^H-I*aL*Hc*x^nL^H-I*bL-I*E^HI*aL*Hc*x^nL^HI*bLL^pHHm+1-I*b*n*pL*H2-2*E^H2*I*aL*Hc*x^nL^H2*I*bLL^pL*
Hypergeometric2F1@-p,Hm+1-I*b*n*pLH2*I*b*nL,1+Hm+1-I*b*n*pLH2*I*b*nL,E^H2*I*aL*Hc*x^nL^H2*I*bLD ;
FreeQ@8a,b,c,m,n,p<,xD && NonzeroQ@b^2*n^2*p^2+Hm+1L^2D
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
8
Int@x_^m_.*Cos@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
x^Hm+1L*HE^H-I*aL*Hc*x^nL^H-I*bL+E^HI*aL*Hc*x^nL^HI*bLL^pHHm+1-I*b*n*pL*H2+2*E^H2*I*aL*Hc*x^nL^H2*I*bLL^pL*
Hypergeometric2F1@-p,Hm+1-I*b*n*pLH2*I*b*nL,1+Hm+1-I*b*n*pLH2*I*b*nL,-E^H2*I*aL*Hc*x^nL^H2*I*bLD ;
FreeQ@8a,b,c,m,n,p<,xD && NonzeroQ@b^2*n^2*p^2+Hm+1L^2D
2. à u Sec@a + b Log@c xn DDp â x
1. à Sec@a + b Log@c xn DDp â x
1. à Sec@a + b Log@c xn DDp â x when b2 n2 Hp - 2L2 + 1 0
1: à Sec@a + b Log@c xn DD â x when b2 n2 + 1 0
Derivation: Algebraic expansion
Basis: If b2 n2 + 1 0, then Sec@a + b Log@c xn DD 2 ãa b n
Hc xn L1n
ã2 a b n +Hc xn L2n
Basis: If b2 n2 + 1 0, then Csc@a + b Log@c xn DD 2 b n ãa b n
Note: The above identities need to be formally derived.
Hc xn L1n
xn L2n
ã2 a b n -Hc
Rule: If b2 n2 + 1 0, then
Program code:
n
abn
à Sec@a + b Log@c x DD â x 2 ã
à
Int@Sec@a_.+b_.*Log@c_.*x_^n_.DD,x_SymbolD :=
2*E^Ha*b*nL*Int@Hc*x^nL^H1nLHE^H2*a*b*nL+Hc*x^nL^H2nLL,xD ;
FreeQ@8a,b,c,n<,xD && ZeroQ@b^2*n^2+1D
Int@Csc@a_.+b_.*Log@c_.*x_^n_.DD,x_SymbolD :=
2*b*n*E^Ha*b*nL*Int@Hc*x^nL^H1nLHE^H2*a*b*nL-Hc*x^nL^H2nLL,xD ;
FreeQ@8a,b,c,n<,xD && ZeroQ@b^2*n^2+1D
2: à Sec@a + b Log@c xn DDp â x when b2 n2 Hp - 2L2 + 1 0 ì p ¹ 1
Rule: If b2 n2 Hp - 2L2 + 1 0 ì p ¹ 1, then
Hc xn L1n
ã2 a b n + Hc xn L2n
âx
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
Program code:
p
n
à Sec@a + b Log@c x DD â x
9
Hp - 2L x Sec@a + b Log@c xn DDp-2
+
p-1
x Tan@a + b Log@c xn DD Sec@a + b Log@c xn DDp-2
b n Hp - 1L
Int@Sec@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
Hp-2L*x*Sec@a+b*Log@c*x^nDD^Hp-2LHp-1L +
x*Tan@a+b*Log@c*x^nDD*Sec@a+b*Log@c*x^nDD^Hp-2LHb*n*Hp-1LL ;
FreeQ@8a,b,c,n,p<,xD && ZeroQ@b^2*n^2*Hp-2L^2+1D && NonzeroQ@p-1D
Int@Csc@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
Hp-2L*x*Csc@a+b*Log@c*x^nDD^Hp-2LHp-1L x*Cot@a+b*Log@c*x^nDD*Csc@a+b*Log@c*x^nDD^Hp-2LHb*n*Hp-1LL ;
FreeQ@8a,b,c,n,p<,xD && ZeroQ@b^2*n^2*Hp-2L^2+1D && NonzeroQ@p-1D
2: à Sec@a + b Log@c xn DDp â x when p > 1 ì p ¹ 2 ì b2 n2 Hp - 2L2 + 1 ¹ 0
Rule: If p > 1 ì p ¹ 2 ì b2 n2 Hp - 2L2 + 1 ¹ 0, then
x Tan@a + b Log@c xn DD Sec@a + b Log@c xn DDp-2
Program code:
b n Hp - 1L
p
n
à Sec@a + b Log@c x DD â x
-
x Sec@a + b Log@c xn DDp-2
b2 n2 Hp - 1L Hp - 2L
+
b2 n2 Hp - 2L2 + 1
b2 n2 Hp - 1L Hp - 2L
Int@Sec@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
x*Tan@a+b*Log@c*x^nDD*Sec@a+b*Log@c*x^nDD^Hp-2LHb*n*Hp-1LL x*Sec@a+b*Log@c*x^nDD^Hp-2LHb^2*n^2*Hp-1L*Hp-2LL +
Hb^2*n^2*Hp-2L^2+1LHb^2*n^2*Hp-1L*Hp-2LL*Int@Sec@a+b*Log@c*x^nDD^Hp-2L,xD ;
FreeQ@8a,b,c,n<,xD && RationalQ@pD && p>1 && p¹2 && NonzeroQ@b^2*n^2*Hp-2L^2+1D
Int@Csc@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
-x*Cot@a+b*Log@c*x^nDD*Csc@a+b*Log@c*x^nDD^Hp-2LHb*n*Hp-1LL x*Csc@a+b*Log@c*x^nDD^Hp-2LHb^2*n^2*Hp-1L*Hp-2LL +
Hb^2*n^2*Hp-2L^2+1LHb^2*n^2*Hp-1L*Hp-2LL*Int@Csc@a+b*Log@c*x^nDD^Hp-2L,xD ;
FreeQ@8a,b,c,n<,xD && RationalQ@pD && p>1 && p¹2 && NonzeroQ@b^2*n^2*Hp-2L^2+1D
p-2
n
âx
à Sec@a + b Log@c x DD
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
10
3: à Sec@a + b Log@c xn DDp â x when p < - 1 ì b2 n2 p2 + 1 ¹ 0
Rule: If p < - 1 ì b2 n2 p2 + 1 ¹ 0, then
-
p
n
à Sec@a + b Log@c x DD â x
b n p x Sin@a + b Log@c xn DD Sec@a + b Log@c xn DDp+1
b2 n2 p2 + 1
+
x Sec@a + b Log@c xn DDp
b2 n2 p2 + 1
+
b2 n2 p Hp + 1L
b2 n2 p2 + 1
Program code:
p+2
n
âx
à Sec@a + b Log@c x DD
Int@Sec@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
-b*n*p*x*Sin@a+b*Log@c*x^nDD*Sec@a+b*Log@c*x^nDD^Hp+1LHb^2*n^2*p^2+1L +
x*Sec@a+b*Log@c*x^nDD^pHb^2*n^2*p^2+1L +
b^2*n^2*p*Hp+1LHb^2*n^2*p^2+1L*Int@Sec@a+b*Log@c*x^nDD^Hp+2L,xD ;
FreeQ@8a,b,c,n<,xD && RationalQ@pD && p<-1 && NonzeroQ@b^2*n^2*p^2+1D
Int@Csc@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
b*n*p*x*Cos@a+b*Log@c*x^nDD*Csc@a+b*Log@c*x^nDD^Hp+1LHb^2*n^2*p^2+1L +
x*Csc@a+b*Log@c*x^nDD^pHb^2*n^2*p^2+1L +
b^2*n^2*p*Hp+1LHb^2*n^2*p^2+1L*Int@Csc@a+b*Log@c*x^nDD^Hp+2L,xD ;
FreeQ@8a,b,c,n<,xD && RationalQ@pD && p<-1 && NonzeroQ@b^2*n^2*p^2+1D
4: à Sec@a + b Log@c xn DDp â x when p Ï Z- ì b2 n2 p2 + 1 ¹ 0
Rule: If b2 n2 p2 + 1 ¹ 0, then
x I2 + 2 ã2 ä a Hc xn L2 ä b M
p
1+äbnp
Program code:
1+
ãä a Hc xn Lä b
ã2 ä a
Hc
p
n
à Sec@a + b Log@c x DD â x
xn L2 ä b
p
1+äbnp
Hypergeometric2F1Bp,
1+äbnp
, 1+
2äbn
2äbn
, - ã2 ä a Hc xn L2 ä b F
Int@Sec@a_.+b_.*Log@c_.*x_^n_.DD^p_.,x_SymbolD :=
x*H2+2*E^H2*I*aL*Hc*x^nL^H2*I*bLL^pH1+I*b*n*pL*
HE^HI*aL*Hc*x^nL^HI*bLH1+E^H2*I*aL*Hc*x^nL^H2*I*bLLL^p*
Hypergeometric2F1@p,H1+I*b*n*pLH2*I*b*nL,1+H1+I*b*n*pLH2*I*b*nL,-E^H2*I*aL*Hc*x^nL^H2*I*bLD ;
FreeQ@8a,b,c,n,p<,xD && NonzeroQ@b^2*n^2*p^2+1D
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
11
Int@Csc@a_.+b_.*Log@c_.*x_^n_.DD^p_.,x_SymbolD :=
x*H2-2*E^H2*I*aL*Hc*x^nL^H2*I*bLL^pH1+I*b*n*pL*
H-I*E^HI*aL*Hc*x^nL^HI*bLH1-E^H2*I*aL*Hc*x^nL^H2*I*bLLL^p*
Hypergeometric2F1@p,H1+I*b*n*pLH2*I*b*nL,1+H1+I*b*n*pLH2*I*b*nL,E^H2*I*aL*Hc*x^nL^H2*I*bLD ;
FreeQ@8a,b,c,n,p<,xD && NonzeroQ@b^2*n^2*p^2+1D
2. à xm Sec@a + b Log@c xn DDp â x
1. à xm Sec@a + b Log@c xn DDp â x when b2 n2 Hp - 2L2 + Hm + 1L2 0
1: à xm Sec@a + b Log@c xn DD â x when b2 n2 + Hm + 1L2 0
Derivation: Algebraic expansion
Basis: If b2 n2 + Hm + 1L2 0, then Sec@a + b Log@c xn DD 2 ã m+1
abn
Basis: If b2 n2 + Hm + 1L2 0, then Csc@a + b Log@c xn DD
2bn
m+1
Program code:
2abn
ã
m+1
+Hc
abn
ã m+1
m+1
n
xn L
m+1
2 Im+1M
n
Hc xn L
2abn
ã
Note: The above identities need to be formally derived.
Rule: If b2 n2 + Hm + 1L2 0, then
Hc xn L
-Hc
m+1
n
xn L
2 Im+1M
á x Sec@a + b Log@c x DD â x 2 ã m+1 á
n
abn
m
n
xm Hc xn L
2abn
ã
m+1
m+1
n
+ Hc xn L
2 Hm+1L
âx
n
Int@x_^m_.*Sec@a_.+b_.*Log@c_.*x_^n_.DD,x_SymbolD :=
2*E^Ha*b*nHm+1LL*Int@x^m*Hc*x^nL^HHm+1LnLHE^H2*a*b*nHm+1LL+Hc*x^nL^H2*Hm+1LnLL,xD ;
FreeQ@8a,b,c,m,n<,xD && ZeroQ@b^2*n^2+Hm+1L^2D
Int@x_^m_.*Csc@a_.+b_.*Log@c_.*x_^n_.DD,x_SymbolD :=
2*b*nHm+1L*E^Ha*b*nHm+1LL*Int@x^m*Hc*x^nL^HHm+1LnLHE^H2*a*b*nHm+1LL-Hc*x^nL^H2*Hm+1LnLL,xD ;
FreeQ@8a,b,c,m,n<,xD && ZeroQ@b^2*n^2+Hm+1L^2D
2: à xm Sec@a + b Log@c xn DDp â x when b2 n2 Hp - 2L2 + Hm + 1L2 0 ì m ¹ - 1 ì p ¹ 1
Rule: If b2 n2 Hp - 2L2 + Hm + 1L2 0 ì m ¹ - 1 ì p ¹ 1, then
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
p
m
n
à x Sec@a + b Log@c x DD â x
12
Hp - 2L xm+1 Sec@a + b Log@c xn DDp-2
Hm + 1L Hp - 1L
Program code:
+
xm+1 Tan@a + b Log@c xn DD Sec@a + b Log@c xn DDp-2
b n Hp - 1L
Int@x_^m_.*Sec@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
Hp-2L*x^Hm+1L*Sec@a+b*Log@c*x^nDD^Hp-2LHHm+1L*Hp-1LL +
x^Hm+1L*Tan@a+b*Log@c*x^nDD*Sec@a+b*Log@c*x^nDD^Hp-2LHb*n*Hp-1LL ;
FreeQ@8a,b,c,m,n,p<,xD && ZeroQ@b^2*n^2*Hp-2L^2+Hm+1L^2D && NonzeroQ@m+1D && NonzeroQ@p-1D
Int@x_^m_.*Csc@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
Hp-2L*x^Hm+1L*Csc@a+b*Log@c*x^nDD^Hp-2LHHm+1L*Hp-1LL x^Hm+1L*Cot@a+b*Log@c*x^nDD*Csc@a+b*Log@c*x^nDD^Hp-2LHb*n*Hp-1LL ;
FreeQ@8a,b,c,m,n,p<,xD && ZeroQ@b^2*n^2*Hp-2L^2+Hm+1L^2D && NonzeroQ@m+1D && NonzeroQ@p-1D
2: à xm Sec@a + b Log@c xn DDp â x when p > 1 ì p ¹ 2 ì b2 n2 Hp - 2L2 + Hm + 1L2 ¹ 0
Rule: If p > 1 ì p ¹ 2 ì b2 n2 Hp - 2L2 + Hm + 1L2 ¹ 0, then
xm+1 Tan@a + b Log@c xn DD Sec@a + b Log@c xn DDp-2
Program code:
b n Hp - 1L
-
p
m
n
à x Sec@a + b Log@c x DD â x
Hm + 1L xm+1 Sec@a + b Log@c xn DDp-2
b2 n2 Hp - 1L Hp - 2L
+
b2 n2 Hp - 2L2 + Hm + 1L2
b2 n2 Hp - 1L Hp - 2L
Int@x_^m_.*Sec@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
x^Hm+1L*Tan@a+b*Log@c*x^nDD*Sec@a+b*Log@c*x^nDD^Hp-2LHb*n*Hp-1LL Hm+1L*x^Hm+1L*Sec@a+b*Log@c*x^nDD^Hp-2LHb^2*n^2*Hp-1L*Hp-2LL +
Hb^2*n^2*Hp-2L^2+Hm+1L^2LHb^2*n^2*Hp-1L*Hp-2LL*Int@x^m*Sec@a+b*Log@c*x^nDD^Hp-2L,xD ;
FreeQ@8a,b,c,m,n<,xD && RationalQ@pD && p>1 && p¹2 && NonzeroQ@b^2*n^2*Hp-2L^2+Hm+1L^2D
Int@x_^m_.*Csc@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
-x^Hm+1L*Cot@a+b*Log@c*x^nDD*Csc@a+b*Log@c*x^nDD^Hp-2LHb*n*Hp-1LL Hm+1L*x^Hm+1L*Csc@a+b*Log@c*x^nDD^Hp-2LHb^2*n^2*Hp-1L*Hp-2LL +
Hb^2*n^2*Hp-2L^2+Hm+1L^2LHb^2*n^2*Hp-1L*Hp-2LL*Int@x^m*Csc@a+b*Log@c*x^nDD^Hp-2L,xD ;
FreeQ@8a,b,c,m,n<,xD && RationalQ@pD && p>1 && p¹2 && NonzeroQ@b^2*n^2*Hp-2L^2+Hm+1L^2D
p-2
m
n
âx
à x Sec@a + b Log@c x DD
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
13
3: à xm Sec@a + b Log@c xn DDp â x when p < - 1 ì b2 n2 p2 + Hm + 1L2 ¹ 0
Rule: If p < - 1 ì b2 n2 p2 + Hm + 1L2 ¹ 0, then
-
p
m
n
à x Sec@a + b Log@c x DD â x
b n p xm+1 Sin@a + b Log@c xn DD Sec@a + b Log@c xn DDp+1
Program code:
b2 n2 p2 + Hm + 1L2
+
Hm + 1L xm+1 Sec@a + b Log@c xn DDp
b2 n2 p2 + Hm + 1L2
+
b2 n2 p Hp + 1L
b2 n2 p2 + Hm + 1L2
p+2
m
n
âx
à x Sec@a + b Log@c x DD
Int@x_^m_.*Sec@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
-b*n*p*x^Hm+1L*Sin@a+b*Log@c*x^nDD*Sec@a+b*Log@c*x^nDD^Hp+1LHb^2*n^2*p^2+Hm+1L^2L +
Hm+1L*x^Hm+1L*Sec@a+b*Log@c*x^nDD^pHb^2*n^2*p^2+Hm+1L^2L +
b^2*n^2*p*Hp+1LHb^2*n^2*p^2+Hm+1L^2L*Int@x^m*Sec@a+b*Log@c*x^nDD^Hp+2L,xD ;
FreeQ@8a,b,c,m,n<,xD && RationalQ@pD && p<-1 && NonzeroQ@b^2*n^2*p^2+Hm+1L^2D
Int@x_^m_.*Csc@a_.+b_.*Log@c_.*x_^n_.DD^p_,x_SymbolD :=
b*n*p*x^Hm+1L*Cos@a+b*Log@c*x^nDD*Csc@a+b*Log@c*x^nDD^Hp+1LHb^2*n^2*p^2+Hm+1L^2L +
Hm+1L*x^Hm+1L*Csc@a+b*Log@c*x^nDD^pHb^2*n^2*p^2+Hm+1L^2L +
b^2*n^2*p*Hp+1LHb^2*n^2*p^2+Hm+1L^2L*Int@x^m*Csc@a+b*Log@c*x^nDD^Hp+2L,xD ;
FreeQ@8a,b,c,m,n<,xD && RationalQ@pD && p<-1 && NonzeroQ@b^2*n^2*p^2+Hm+1L^2D
4: à xm Sec@a + b Log@c xn DDp â x when p Ï Z- ì b2 n2 p2 + Hm + 1L2 ¹ 0
Rule: If b2 n2 p2 + Hm + 1L2 ¹ 0, then
xm+1 I2 + 2 ã2 ä a Hc xn L2 ä b M
p
m+1+äbnp
Program code:
1+
ãä a Hc xn Lä b
ã2 ä a
Hc
p
m
n
à x Sec@a + b Log@c x DD â x
xn L2 ä b
p
m+1+äbnp
Hypergeometric2F1Bp,
2äbn
m+1+äbnp
, 1+
2äbn
, - ã2 ä a Hc xn L2 ä b F
Int@x_^m_.*Sec@a_.+b_.*Log@c_.*x_^n_.DD^p_.,x_SymbolD :=
x^Hm+1L*H2+2*E^H2*I*aL*Hc*x^nL^H2*I*bLL^pHm+1+I*b*n*pL*
HE^HI*aL*Hc*x^nL^HI*bLH1+E^H2*I*aL*Hc*x^nL^H2*I*bLLL^p*
Hypergeometric2F1@p,Hm+1+I*b*n*pLH2*I*b*nL,1+Hm+1+I*b*n*pLH2*I*b*nL,-E^H2*I*aL*Hc*x^nL^H2*I*bLD ;
FreeQ@8a,b,c,m,n,p<,xD && NonzeroQ@b^2*n^2*p^2+Hm+1L^2D
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
14
Int@x_^m_.*Csc@a_.+b_.*Log@c_.*x_^n_.DD^p_.,x_SymbolD :=
x^Hm+1L*H2-2*E^H2*I*aL*Hc*x^nL^H2*I*bLL^pHm+1+I*b*n*pL*
H-I*E^HI*aL*Hc*x^nL^HI*bLH1-E^H2*I*aL*Hc*x^nL^H2*I*bLLL^p*
Hypergeometric2F1@p,Hm+1+I*b*n*pLH2*I*b*nL,1+Hm+1+I*b*n*pLH2*I*b*nL,E^H2*I*aL*Hc*x^nL^H2*I*bLD ;
FreeQ@8a,b,c,m,n,p<,xD && NonzeroQ@b^2*n^2*p^2+Hm+1L^2D
3. à u Sin@a xn Log@b xDp D Log@b xDp â x
1. à Sin@a xn Log@b xDp D Log@b xDp â x
1: à Sin@a x Log@b xDp D Log@b xDp â x when p > 0
Rule: If p > 0, then
Program code:
p
p
à Sin@a x Log@b xD D Log@b xD â x -
Cos@a x Log@b xDp D
Int@Sin@a_.*x_*Log@b_.*x_D^p_.D*Log@b_.*x_D^p_.,x_SymbolD :=
-Cos@a*x*Log@b*xD^pDa p*Int@Sin@a*x*Log@b*xD^pD*Log@b*xD^Hp-1L,xD ;
FreeQ@8a,b<,xD && RationalQ@pD && p>0
Int@Cos@a_.*x_*Log@b_.*x_D^p_.D*Log@b_.*x_D^p_.,x_SymbolD :=
Sin@a*x*Log@b*xD^pDa p*Int@Cos@a*x*Log@b*xD^pD*Log@b*xD^Hp-1L,xD ;
FreeQ@8a,b<,xD && RationalQ@pD && p>0
a
- p à Sin@a x Log@b xDp D Log@b xDp-1 â x
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
15
2: à Sin@a xn Log@b xDp D Log@b xDp â x when p > 0
Rule: If p > 0, then
-
Cos@a xn Log@b xDp D
a n xn-1
Program code:
p
n
n
p
p
à Sin@a x Log@b xD D Log@b xD â x
n
p
p-1
âx à Sin@a x Log@b xD D Log@b xD
n-1
an
à
Cos@a xn Log@b xDp D
âx
xn
Int@Sin@a_.*x_^n_*Log@b_.*x_D^p_.D*Log@b_.*x_D^p_.,x_SymbolD :=
-Cos@a*x^n*Log@b*xD^pDHa*n*x^Hn-1LL pn*Int@Sin@a*x^n*Log@b*xD^pD*Log@b*xD^Hp-1L,xD Hn-1LHa*nL*Int@Cos@a*x^n*Log@b*xD^pDx^n,xD ;
FreeQ@8a,b<,xD && RationalQ@n,pD && p>0
Int@Cos@a_.*x_^n_*Log@b_.*x_D^p_.D*Log@b_.*x_D^p_.,x_SymbolD :=
Sin@a*x^n*Log@b*xD^pDHa*n*x^Hn-1LL pn*Int@Cos@a*x^n*Log@b*xD^pD*Log@b*xD^Hp-1L,xD +
Hn-1LHa*nL*Int@Sin@a*x^n*Log@b*xD^pDx^n,xD ;
FreeQ@8a,b<,xD && RationalQ@n,pD && p>0
2. à xm Sin@a xn Log@b xDp D Log@b xDp â x
1: à xn-1 Sin@a xn Log@b xDp D Log@b xDp â x when p > 0
Rule: If p > 0, then
Program code:
n-1
Sin@a xn Log@b xDp D Log@b xDp â x àx
Cos@a xn Log@b xDp D
p
-
an
Int@x_^m_.*Sin@a_.*x_^n_.*Log@b_.*x_D^p_.D*Log@b_.*x_D^p_.,x_SymbolD :=
-Cos@a*x^n*Log@b*xD^pDHa*nL pn*Int@x^m*Sin@a*x^n*Log@b*xD^pD*Log@b*xD^Hp-1L,xD ;
FreeQ@8a,b,m,n<,xD && ZeroQ@m-n+1D && RationalQ@pD && p>0
n
àx
n-1
Sin@a xn Log@b xDp D Log@b xDp-1 â x
Rules for integrands of the form x^m trig(a+b log(c x^n))^p
16
Int@x_^m_.*Cos@a_.*x_^n_.*Log@b_.*x_D^p_.D*Log@b_.*x_D^p_.,x_SymbolD :=
Sin@a*x^n*Log@b*xD^pDHa*nL pn*Int@x^m*Cos@a*x^n*Log@b*xD^pD*Log@b*xD^Hp-1L,xD ;
FreeQ@8a,b,m,n<,xD && ZeroQ@m-n+1D && RationalQ@pD && p>0
2: à xm Sin@a xn Log@b xDp D Log@b xDp â x when p > 0 ì m - n + 1 ¹ 0
Rule: If p > 0 ì m - n + 1 ¹ 0, then
-
xm-n+1 Cos@a xn Log@b xDp D
an
Program code:
p
n
m
n
p
p
à x Sin@a x Log@b xD D Log@b xD â x
m
n
p
p-1
âx +
à x Sin@a x Log@b xD D Log@b xD
Int@x_^m_.*Sin@a_.*x_^n_.*Log@b_.*x_D^p_.D*Log@b_.*x_D^p_.,x_SymbolD :=
-x^Hm-n+1L*Cos@a*x^n*Log@b*xD^pDHa*nL pn*Int@x^m*Sin@a*x^n*Log@b*xD^pD*Log@b*xD^Hp-1L,xD +
Hm-n+1LHa*nL*Int@x^Hm-nL*Cos@a*x^n*Log@b*xD^pD,xD ;
FreeQ@8a,b,m,n<,xD && RationalQ@pD && p>0 && NonzeroQ@m-n+1D
Int@x_^m_*Cos@a_.*x_^n_.*Log@b_.*x_D^p_.D*Log@b_.*x_D^p_.,x_SymbolD :=
x^Hm-n+1L*Sin@a*x^n*Log@b*xD^pDHa*nL pn*Int@x^m*Cos@a*x^n*Log@b*xD^pD*Log@b*xD^Hp-1L,xD Hm-n+1LHa*nL*Int@x^Hm-nL*Sin@a*x^n*Log@b*xD^pD,xD ;
FreeQ@8a,b,m,n<,xD && RationalQ@pD && p>0 && NonzeroQ@m-n+1D
m-n+1
an
àx
m-n
Cos@a xn Log@b xDp D â x

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4.7 x^m trig(a+b log(cx^n))^p.nb

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