Rules for integrands involving inverse sines and cosines
1. à u Ha + b ArcSin@c + d xDLn â x
1: à Ha + b ArcSin@c + d xDLn â x
Ÿ Derivation: Integration by substitution
Ÿ Rule:
Ÿ Program code:
n
à Ha + b ArcSin@c + d xDL â x ™
1
d
SubstBà Ha + b ArcSin@xDLn â x, x, c + d xF
Int@Ha_.+b_.*ArcSin@c_+d_.*x_DL^n_.,x_SymbolD :=
1d*Subst@Int@Ha+b*ArcSin@xDL^n,xD,x,c+d*xD ;
FreeQ@8a,b,c,d,n<,xD
Int@Ha_.+b_.*ArcCos@c_+d_.*x_DL^n_.,x_SymbolD :=
1d*Subst@Int@Ha+b*ArcCos@xDL^n,xD,x,c+d*xD ;
FreeQ@8a,b,c,d,n<,xD
2: à He + f xLm Ha + b ArcSin@c + d xDLn â x
Ÿ Derivation: Integration by substitution
Ÿ Rule:
Ÿ Program code:
m
n
à He + f xL Ha + b ArcSin@c + d xDL â x ™
1
d
SubstBà
de-cf
fx
+
d
Int@He_.+f_.*x_L^m_.*Ha_.+b_.*ArcSin@c_+d_.*x_DL^n_.,x_SymbolD :=
1d*Subst@Int@HHd*e-c*fLd+f*xdL^m*Ha+b*ArcSin@xDL^n,xD,x,c+d*xD ;
FreeQ@8a,b,c,d,e,f,m,n<,xD
Int@He_.+f_.*x_L^m_.*Ha_.+b_.*ArcCos@c_+d_.*x_DL^n_.,x_SymbolD :=
1d*Subst@Int@HHd*e-c*fLd+f*xdL^m*Ha+b*ArcCos@xDL^n,xD,x,c+d*xD ;
FreeQ@8a,b,c,d,e,f,m,n<,xD
d
m
Ha + b ArcSin@xDLn â x, x, c + d xF
Rules for integrands involving inverse sines and cosines
2
3: à IA + B x + C x2 M Ha + b ArcSin@c + d xDLn â x when B I1 - c2 M + 2 A c d Š 0 ì 2 c C - B d Š 0
p
Ÿ Derivation: Integration by substitution
Ÿ Basis: If B I1 - c2 M + 2 A c d Š 0 ì 2 c C - B d Š 0, then A + B x + C x2 Š - dC2 +
Ÿ Rule: If B I1 - c2 M + 2 A c d Š 0 ì 2 c C - B d Š 0, then
2
n
à IA + B x + C x M Ha + b ArcSin@c + d xDL â x ™
p
Ÿ Program code:
1
d
C
d2
SubstBà -
Hc + d xL2
C x2
C
p
+
d2
d2
Ha + b ArcSin@xDLn â x, x, c + d xF
Int@HA_.+B_.*x_+C_.*x_^2L^p_.*Ha_.+b_.*ArcSin@c_+d_.*x_DL^n_.,x_SymbolD :=
1d*Subst@Int@H-Cd^2+Cd^2*x^2L^p*Ha+b*ArcSin@xDL^n,xD,x,c+d*xD ;
FreeQ@8a,b,c,d,A,B,C,n,p<,xD && ZeroQ@B*H1-c^2L+2*A*c*dD && ZeroQ@2*c*C-B*dD
Int@HA_.+B_.*x_+C_.*x_^2L^p_.*Ha_.+b_.*ArcCos@c_+d_.*x_DL^n_.,x_SymbolD :=
1d*Subst@Int@H-Cd^2+Cd^2*x^2L^p*Ha+b*ArcCos@xDL^n,xD,x,c+d*xD ;
FreeQ@8a,b,c,d,A,B,C,n,p<,xD && ZeroQ@B*H1-c^2L+2*A*c*dD && ZeroQ@2*c*C-B*dD
4: à He + f xLm IA + B x + C x2 M Ha + b ArcSin@c + d xDLn â x when B I1 - c2 M + 2 A c d Š 0 ì 2 c C - B d Š 0
p
Ÿ Derivation: Integration by substitution
Ÿ Basis: If B I1 - c2 M + 2 A c d Š 0 ì 2 c C - B d Š 0, then A + B x + C x2 Š - dC2 +
Ÿ Rule: If B I1 - c2 M + 2 A c d Š 0 ì 2 c C - B d Š 0, then
m
2
n
à He + f xL IA + B x + C x M Ha + b ArcSin@c + d xDL â x ™
p
Ÿ Program code:
1
d
SubstBà
C
d2
Hc + d xL2
de-cf
fx
+
d
m
d
C x2
C
-
+
d2
d2
p
Ha + b ArcSin@xDLn â x, x, c + d xF
Int@He_.+f_.*x_L^m_.*HA_.+B_.*x_+C_.*x_^2L^p_.*Ha_.+b_.*ArcSin@c_+d_.*x_DL^n_.,x_SymbolD :=
1d*Subst@Int@HHd*e-c*fLd+f*xdL^m*H-Cd^2+Cd^2*x^2L^p*Ha+b*ArcSin@xDL^n,xD,x,c+d*xD ;
FreeQ@8a,b,c,d,e,f,A,B,C,m,n,p<,xD && ZeroQ@B*H1-c^2L+2*A*c*dD && ZeroQ@2*c*C-B*dD
Int@He_.+f_.*x_L^m_.*HA_.+B_.*x_+C_.*x_^2L^p_.*Ha_.+b_.*ArcCos@c_+d_.*x_DL^n_.,x_SymbolD :=
1d*Subst@Int@HHd*e-c*fLd+f*xdL^m*H-Cd^2+Cd^2*x^2L^p*Ha+b*ArcCos@xDL^n,xD,x,c+d*xD ;
FreeQ@8a,b,c,d,e,f,A,B,C,m,n,p<,xD && ZeroQ@B*H1-c^2L+2*A*c*dD && ZeroQ@2*c*C-B*dD
Rules for integrands involving inverse sines and cosines
3
2. à Ia + b ArcSinAc + d x2 EM â x when c2 Š 1
n
1. à Ia + b ArcSinAc + d x2 EM â x when c2 Š 1 ì n > 0
n
1. à
a + b ArcSinAc + d x2 E â x when c2 Š 1
a + b ArcSinAc + d x2 E â x when c2 Š 1
1: à
Ÿ Derivation: Integration by parts
Ÿ Rule: If c2 Š 1, then
a + b ArcSinAc + d x2 E â x ™ x
à
a + b ArcSinAc + d x2 E - b d á
™ x
Π x ICosA 2 b E + c SinA 2 b EM FresnelCB
a
c
b
a
c
Πb
1
a + b ArcSinAc + d x2 E F
Ÿ Program code:
- 2 c d x2 - d2 x4
a + b ArcSinAc + d x2 E -
ICosA 2 ArcSinAc + d x2 EE - c SinA 2 ArcSinAc + d x2 EEM
1
x2
Π x ICosA 2 b E - c SinA 2 b EM FresnelSB
a
+
a + b ArcSinAc + d x2 E
c
b
a
1
Ÿ Rule:
a + b ArcCosAc + d x2 E â x when c2 Š 1
1: à
a + b ArcCosA1 + d x2 E â x
á
a + b ArcSinAc + d x2 E F
ICosA 2 ArcSinAc + d x2 EE - c SinA 2 ArcSinAc + d x2 EEM
Int@Sqrt@a_.+b_.*ArcSin@c_+d_.*x_^2DD,x_SymbolD :=
x*Sqrt@a+b*ArcSin@c+d*x^2DD Sqrt@PiD*x*HCos@aH2*bLD+c*Sin@aH2*bLDL*FresnelC@Sqrt@cHPi*bLD*Sqrt@a+b*ArcSin@c+d*x^2DDD
HSqrt@cbD*HCos@ArcSin@c+d*x^2D2D-c*Sin@ArcSin@c+d*x^2D2DLL +
Sqrt@PiD*x*HCos@aH2*bLD-c*Sin@aH2*bLDL*FresnelS@Sqrt@cHPi*bLD*Sqrt@a+b*ArcSin@c+d*x^2DDD
HSqrt@cbD*HCos@ArcSin@c+d*x^2D2D-c*Sin@ArcSin@c+d*x^2D2DLL ;
FreeQ@8a,b,c,d<,xD && ZeroQ@c^2-1D
2. à
c
Πb
âx
a + b ArcCosA1 + d x2 E â x ™
+
1
Rules for integrands involving inverse sines and cosines
4
2
-
a + b ArcCosA1 + d x2 E SinA 2 ArcCosA1 + d x2 EE
2
1
+
dx
1
a
-
2
Π SinB
2b
1
b
dx
1
a
2
Π CosB
2b
1
b
dx
F SinB
1
F SinB
1
2
2
ArcCosA1 + d x2 EF FresnelCB
ArcCosA1 + d x2 EF FresnelSB
1
Πb
1
Πb
a + b ArcCosA1 + d x2 E F +
a + b ArcCosA1 + d x2 E F
Ÿ Program code:
Int@Sqrt@a_.+b_.*ArcCos@1+d_.*x_^2DD,x_SymbolD :=
-2*Sqrt@a+b*ArcCos@1+d*x^2DD*Sin@ArcCos@1+d*x^2D2D^2Hd*xL 2*Sqrt@PiD*Sin@aH2*bLD*Sin@ArcCos@1+d*x^2D2D*FresnelC@Sqrt@1HPi*bLD*Sqrt@a+b*ArcCos@1+d*x^2DDDHSqrt@1bD*d*xL +
2*Sqrt@PiD*Cos@aH2*bLD*Sin@ArcCos@1+d*x^2D2D*FresnelS@Sqrt@1HPi*bLD*Sqrt@a+b*ArcCos@1+d*x^2DDDHSqrt@1bD*d*xL ;
FreeQ@8a,b,d<,xD
Rules for integrands involving inverse sines and cosines
Ÿ Rule:
2: à
5
a + b ArcCosA- 1 + d x2 E â x
á
a + b ArcCosA- 1 + d x2 E â x ™
a + b ArcCosA- 1 + d x2 E CosA 2 ArcCosA- 1 + d x2 EE
2
2
1
-
dx
1
a
2
Π CosB
2b
1
b
dx
1
F CosB
a
2
Π SinB
2b
1
b
dx
ArcCosA- 1 + d x2 EF FresnelCB
1
2
F CosB
1
2
ArcCosA- 1 + d x2 EF FresnelSB
1
Πb
1
Πb
a + b ArcCosA- 1 + d x2 E F a + b ArcCosA- 1 + d x2 E F
Ÿ Program code:
Int@Sqrt@a_.+b_.*ArcCos@-1+d_.*x_^2DD,x_SymbolD :=
2*Sqrt@a+b*ArcCos@-1+d*x^2DD*Cos@H12L*ArcCos@-1+d*x^2DD^2Hd*xL 2*Sqrt@PiD*Cos@aH2*bLD*Cos@ArcCos@-1+d*x^2D2D*FresnelC@Sqrt@1HPi*bLD*Sqrt@a+b*ArcCos@-1+d*x^2DDDHSqrt@1bD*d*xL 2*Sqrt@PiD*Sin@aH2*bLD*Cos@ArcCos@-1+d*x^2D2D*FresnelS@Sqrt@1HPi*bLD*Sqrt@a+b*ArcCos@-1+d*x^2DDDHSqrt@1bD*d*xL ;
FreeQ@8a,b,d<,xD
2: à Ia + b ArcSinAc + d x2 EM â x when c2 Š 1 ì n > 1
n
Ÿ Derivation: Integration by parts twice
Ÿ Basis: If c2 Š 1, then ¶x Ia + b ArcSinAc + d x2 EM Š
n
Ÿ Basis:
x2
-d x2 H2 c+d x2 L
Š - ¶x
Ÿ Rule: If c2 Š 1 ì n > 1, then
2 b d n x Ha+b ArcSin@c+d x2 DLn-1
-2 c d x2 -d2 x4
-2 c d x2 -d2 x4
d2 x
à Ia + b ArcSinAc + d x EM â x ™ x Ia + b ArcSinAc + d x EM - 2 b d n á
2
n
2
n
x2 Ia + b ArcSinAc + d x2 EM
n-1
- 2 c d x2 - d2 x4
âx
Ia + b
ArcSinAc
Rules for integrands involving inverse
and
cosines + d
à sines
™ x Ia + b ArcSinAc + d x2 EM +
n
x2 EM â x ™ x Ia + b ArcSinAc + d x2 EM - 2 b d n á
n
2bn
n
- 2 c d x2 - d2 x4 Ia + b ArcSinAc + d x2 EM
n-1
dx
Ÿ Program code:
Int@Ha_.+b_.*ArcSin@c_+d_.*x_^2DL^n_,x_SymbolD :=
x*Ha+b*ArcSin@c+d*x^2DL^n +
2*b*n*Sqrt@-2*c*d*x^2-d^2*x^4D*Ha+b*ArcSin@c+d*x^2DL^Hn-1LHd*xL 4*b^2*n*Hn-1L*Int@Ha+b*ArcSin@c+d*x^2DL^Hn-2L,xD ;
FreeQ@8a,b,c,d<,xD && ZeroQ@c^2-1D && RationalQ@nD && n>1
Int@Ha_.+b_.*ArcCos@c_+d_.*x_^2DL^n_,x_SymbolD :=
x*Ha+b*ArcCos@c+d*x^2DL^n 2*b*n*Sqrt@-2*c*d*x^2-d^2*x^4D*Ha+b*ArcCos@c+d*x^2DL^Hn-1LHd*xL 4*b^2*n*Hn-1L*Int@Ha+b*ArcCos@c+d*x^2DL^Hn-2L,xD ;
FreeQ@8a,b,c,d<,xD && ZeroQ@c^2-1D && RationalQ@nD && n>1
6
âx
- 2 c d x2 - d2 x4
- 4 b2 n Hn - 1L à Ia + b ArcSinAc + d x2 EM
n-2
âx
Rules for integrands involving inverse sines and cosines
7
2. à Ia + b ArcSinAc + d x2 EM â x when c2 Š 1 ì n < 0
n
1. à
1
a + b ArcSinAc + d x2 E
1: à
â x when c2 Š 1
1
a + b ArcSinAc + d x2 E
â x when c2 Š 1
Ÿ Rule: If c2 Š 1, then
1
a + b ArcSinAc + d x2 E
x Ic CosA 2 b E - SinA 2 b EM CosIntegralA 2 b Ia + b ArcSinAc + d x2 EME
a
-
á
a
c
2 b ICosA 2 ArcSinAc + d x2 EE - c SinA 2 ArcSinAc + d x2 EEM
1
1
Ÿ Program code:
-
âx ™
x Ic CosA 2 b E + SinA 2 b EM SinIntegralA 2 b Ia + b ArcSinAc + d x2 EME
a
a
c
2 b ICosA 2 ArcSinAc + d x2 EE - c SinA 2 ArcSinAc + d x2 EEM
1
IntA1‘Ha_.+b_.*ArcSin@c_+d_.*x_^2DL,x_SymbolE :=
-x*Hc*Cos@aH2*bLD-Sin@aH2*bLDL*CosIntegral@HcH2*bLL*Ha+b*ArcSin@c+d*x^2DLD
H2*b*HCos@ArcSin@c+d*x^2D2D-c*Sin@ArcSin@c+d*x^2D2DLL x*Hc*Cos@aH2*bLD+Sin@aH2*bLDL*SinIntegral@HcH2*bLL*Ha+b*ArcSin@c+d*x^2DLD
H2*b*HCos@ArcSin@c+d*x^2D2D-c*Sin@ArcSin@c+d*x^2D2DLL ;
FreeQ@8a,b,c,d<,xD && ZeroQ@c^2-1D
1
Rules for integrands involving inverse sines and cosines
2. à
Ÿ Rule:
1
a + b ArcCosAc + d x2 E
1: à
8
â x when c2 Š 1
1
a + b ArcCosA1 + d x2 E
âx
á
1
a + b ArcCosA1 + d x2 E
x CosA 2 b E CosIntegralA 2 b Ia + b ArcCosA1 + d x2 EME
a
1
2 b
+
- d x2
âx ™
x SinA 2 b E SinIntegralA 2 b Ia + b ArcCosA1 + d x2 EME
a
1
- d x2
2 b
Ÿ Program code:
IntA1‘Ha_.+b_.*ArcCos@1+d_.*x_^2DL,x_SymbolE :=
x*Cos@aH2*bLD*CosIntegral@Ha+b*ArcCos@1+d*x^2DLH2*bLDHSqrt@2D*b*Sqrt@-d*x^2DL +
x*Sin@aH2*bLD*SinIntegral@Ha+b*ArcCos@1+d*x^2DLH2*bLDHSqrt@2D*b*Sqrt@-d*x^2DL ;
FreeQ@8a,b,d<,xD
Ÿ Rule:
2: à
1
a + b ArcCosA- 1 + d x2 E
âx
á
1
a + b ArcCosA- 1 + d x2 E
x SinA 2 b E CosIntegralA 2 b Ia + b ArcCosA- 1 + d x2 EME
a
1
2 b
d x2
x CosA 2 b E SinIntegralA 2 b Ia + b ArcCosA- 1 + d x2 EME
a
-
âx ™
1
2 b
Ÿ Program code:
IntA1‘Ha_.+b_.*ArcCos@-1+d_.*x_^2DL,x_SymbolE :=
x*Sin@aH2*bLD*CosIntegral@Ha+b*ArcCos@-1+d*x^2DLH2*bLDHSqrt@2D*b*Sqrt@d*x^2DL x*Cos@aH2*bLD*SinIntegral@Ha+b*ArcCos@-1+d*x^2DLH2*bLDHSqrt@2D*b*Sqrt@d*x^2DL ;
FreeQ@8a,b,d<,xD
d x2
Rules for integrands involving inverse sines and cosines
2. á
1
a + b ArcSinAc + d
1
1: á
x2 E
a + b ArcSinAc + d
Ÿ Rule: If c2 Š 1, then
a
-
Π x CosB
2b
F - c SinB
a
2b
9
â x when c2 Š 1
â x when c2 Š 1
x2 E
F FresnelCB
á
1
Π
bc
1
a + b ArcSinAc + d x2 E F “
Π x ICosA 2 b E + c SinA 2 b EM FresnelSB
a
âx ™
a + b ArcSinAc + d x2 E
a
1
bc
2
1
bc
CosB
Π
ArcSinAc + d x2 EF - c SinB
a + b ArcSinAc + d x2 E F
b c ICosA 2 ArcSinAc + d x2 EE - c SinA 2 ArcSinAc + d x2 EEM
1
Ÿ Program code:
1
IntA1‘Sqrt@a_.+b_.*ArcSin@c_+d_.*x_^2DD,x_SymbolE :=
-Sqrt@PiD*x*HCos@aH2*bLD-c*Sin@aH2*bLDL*FresnelC@1HSqrt@b*cD*Sqrt@PiDL*Sqrt@a+b*ArcSin@c+d*x^2DDD
HSqrt@b*cD*HCos@ArcSin@c+d*x^2D2D-c*Sin@ArcSin@c+d*x^2D2DLL Sqrt@PiD*x*HCos@aH2*bLD+c*Sin@aH2*bLDL*FresnelS@H1HSqrt@b*cD*Sqrt@PiDLL*Sqrt@a+b*ArcSin@c+d*x^2DDD
HSqrt@b*cD*HCos@ArcSin@c+d*x^2D2D-c*Sin@ArcSin@c+d*x^2D2DLL ;
FreeQ@8a,b,c,d<,xD && ZeroQ@c^2-1D
2. á
1
a + b ArcCosAc + d
1: á
Ÿ Rule:
1
x2 E
a + b ArcCosA1 + d
â x when c2 Š 1
x2 E
âx
á
1
a + b ArcCosA1 + d
x2 E
âx ™
-
1
2
ArcSinAc + d x2 EF
-
Rules for integrands involving inverse sines and cosines
10
Π
1
-
2
a
CosB
dx
b
2b
Π
1
2
a
SinB
dx
b
2b
Ÿ Program code:
F SinB
1
F SinB
1
2
ArcCosA1 + d x2 EF FresnelCB
2
ArcCosA1 + d x2 EF FresnelSB
1
Πb
1
Πb
a + b ArcCosA1 + d x2 E F a + b ArcCosA1 + d x2 E F
IntA1‘Sqrt@a_.+b_.*ArcCos@1+d_.*x_^2DD,x_SymbolE :=
-2*Sqrt@PibD*Cos@aH2*bLD*Sin@ArcCos@1+d*x^2D2D*FresnelC@Sqrt@1HPi*bLD*Sqrt@a+b*ArcCos@1+d*x^2DDDHd*xL 2*Sqrt@PibD*Sin@aH2*bLD*Sin@ArcCos@1+d*x^2D2D*FresnelS@Sqrt@1HPi*bLD*Sqrt@a+b*ArcCos@1+d*x^2DDDHd*xL ;
FreeQ@8a,b,d<,xD
2: á
Ÿ Rule:
1
a + b ArcCosA- 1 + d x2 E
âx
á
Π
1
2
a
SinB
dx
b
2b
Π
1
2
dx
a
CosB
b
Ÿ Program code:
F CosB
2b
1
2
F CosB
1
2
1
a + b ArcCosA- 1 + d x2 E
âx ™
ArcCosA- 1 + d x2 EF FresnelCB
ArcCosA- 1 + d x2 EF FresnelSB
1
Πb
1
Πb
a + b ArcCosA- 1 + d x2 E F a + b ArcCosA- 1 + d x2 E F
IntA1‘Sqrt@a_.+b_.*ArcCos@-1+d_.*x_^2DD,x_SymbolE :=
2*Sqrt@PibD*Sin@aH2*bLD*Cos@ArcCos@-1+d*x^2D2D*FresnelC@Sqrt@1HPi*bLD*Sqrt@a+b*ArcCos@-1+d*x^2DDDHd*xL 2*Sqrt@PibD*Cos@aH2*bLD*Cos@ArcCos@-1+d*x^2D2D*FresnelS@Sqrt@1HPi*bLD*Sqrt@a+b*ArcCos@-1+d*x^2DDDHd*xL ;
FreeQ@8a,b,d<,xD
3. à Ia + b ArcSinAc + d x2 EM â x when c2 Š 1 ì n < - 1
n
Rules for integrands involving inverse sines and cosines
1. á
11
1
Ia + b ArcSinAc + d x2 EM
32
1: á
1
Ia + b ArcSinAc + d
Ÿ Derivation: Integration by parts
32
x2 EM
â x when c2 Š 1
bdx
Ÿ Basis: If c2 Š 1, then -
-2 c d x2 -d2 x4 Ha+b ArcSin@c+d x2 DL32
Ÿ Rule: If c2 Š 1, then
á
â x when c2 Š 1
Š ¶x
1
a+b ArcSin@c+d x2 D
- 2 c d x2 - d2 x4
1
Ia + b ArcSinAc + d x2 EM
âx ™ -
a + b ArcSinAc + d x2 E
32
bdx
d
b
á
x2
- 2 c d x2 - d2 x4
a + b ArcSinAc + d x2 E
âx
- 2 c d x2 - d2 x4
™ bdx
c
a
32
Π x CosB
b
c
2b
a
32
F + c SinB
Π x CosB
b
Ÿ Program code:
2b
a
2b
F - c SinB
a
F FresnelCB
2b
F FresnelSB
c
Πb
c
Πb
a + b ArcSinAc + d x2 E
a + b ArcSinAc + d x2 E F “
a + b ArcSinAc + d x2 E F “
-
1
CosB
2
ArcSinAc + d x2 EF - c SinB
1
CosB
2
1
2
ArcSinAc + d x2 EF - c SinB
ArcSinAc + d x2 EF +
1
2
IntA1‘Ha_.+b_.*ArcSin@c_+d_.*x_^2DL^H32L,x_SymbolE :=
-Sqrt@-2*c*d*x^2-d^2*x^4DHb*d*x*Sqrt@a+b*ArcSin@c+d*x^2DDL HcbL^H32L*Sqrt@PiD*x*HCos@aH2*bLD+c*Sin@aH2*bLDL*FresnelC@Sqrt@cHPi*bLD*Sqrt@a+b*ArcSin@c+d*x^2DDD
HCos@H12L*ArcSin@c+d*x^2DD-c*Sin@ArcSin@c+d*x^2D2DL +
HcbL^H32L*Sqrt@PiD*x*HCos@aH2*bLD-c*Sin@aH2*bLDL*FresnelS@Sqrt@cHPi*bLD*Sqrt@a+b*ArcSin@c+d*x^2DDD
HCos@H12L*ArcSin@c+d*x^2DD-c*Sin@ArcSin@c+d*x^2D2DL ;
FreeQ@8a,b,c,d<,xD && ZeroQ@c^2-1D
ArcSinAc + d x2 EF
Rules for integrands involving inverse sines and cosines
2. á
12
1
Ia + b ArcCosAc + d x2 EM
32
1: á
â x when c2 Š 1
1
âx
Ia + b ArcCosA1 + d x2 EM
32
Ÿ Derivation: Integration by parts
bdx
Ÿ Basis:
-2 d
x2 -d2
Ÿ Rule:
á
x4
Ha+b ArcCos@1+d
1
Š ¶x
x2 DL32
a+b ArcCos@1+d x2 D
- 2 d x2 - d2 x4
1
Ia + b ArcCosA1 + d x2 EM
âx ™
a + b ArcCosA1 + d x2 E
32
bdx
d
+
b
á
x2
- 2 d x2 - d2 x4
a + b ArcCosA1 + d x2 E
âx
- 2 d x2 - d2 x4
™
bdx
1
1
32
dx
b
1
1
Ÿ Program code:
2b
32
F SinB
a
Π CosB
2
dx
a
Π SinB
2
b
2b
1
2
F SinB
a + b ArcCosA1 + d x2 E
ArcCosA1 + d x2 EF FresnelCB
1
2
ArcCosA1 + d x2 EF FresnelSB
-
1
Πb
1
Πb
a + b ArcCosA1 + d x2 E F +
a + b ArcCosA1 + d x2 E F
IntA1‘Ha_.+b_.*ArcCos@1+d_.*x_^2DL^H32L,x_SymbolE :=
Sqrt@-2*d*x^2-d^2*x^4DHb*d*x*Sqrt@a+b*ArcCos@1+d*x^2DDL 2*H1bL^H32L*Sqrt@PiD*Sin@aH2*bLD*Sin@ArcCos@1+d*x^2D2D*FresnelC@Sqrt@1HPi*bLD*Sqrt@a+b*ArcCos@1+d*x^2DDDHd*xL +
2*H1bL^H32L*Sqrt@PiD*Cos@aH2*bLD*Sin@ArcCos@1+d*x^2D2D*FresnelS@Sqrt@1HPi*bLD*Sqrt@a+b*ArcCos@1+d*x^2DDDHd*xL ;
FreeQ@8a,b,d<,xD
Rules for integrands involving inverse sines and cosines
2: á
13
1
âx
Ia + b ArcCosA- 1 + d x2 EM
32
Ÿ Derivation: Integration by parts
Ÿ Basis:
Ÿ Rule:
bdx
á
1
Š ¶x
2 d x2 -d2 x4 Ha+b ArcCos@-1+d x2 DL32
a+b ArcCos@-1+d x2 D
2 d x2 - d2 x4
1
Ia + b ArcCosA- 1 + d x2 EM
âx ™
32
a + b ArcCosA- 1 + d
bdx
d
x2 E
+
b
á
x2
2d
x2
-
d2
x4
a + b ArcCosA- 1 + d
x2 E
âx
2 d x2 - d2 x4
™
bdx
1
1
32
dx
b
1
1
Ÿ Program code:
2b
32
F CosB
a
Π SinB
2
dx
a
Π CosB
2
b
2b
1
2
F CosB
1
2
a + b ArcCosA- 1 + d
x2 E
ArcCosA- 1 + d x2 EF FresnelCB
ArcCosA- 1 + d x2 EF FresnelSB
-
1
Πb
1
Πb
a + b ArcCosA- 1 + d x2 E F a + b ArcCosA- 1 + d x2 E F
IntA1‘Ha_.+b_.*ArcCos@-1+d_.*x_^2DL^H32L,x_SymbolE :=
Sqrt@2*d*x^2-d^2*x^4DHb*d*x*Sqrt@a+b*ArcCos@-1+d*x^2DDL 2*H1bL^H32L*Sqrt@PiD*Cos@aH2*bLD*Cos@ArcCos@-1+d*x^2D2D*FresnelC@Sqrt@1HPi*bLD*Sqrt@a+b*ArcCos@-1+d*x^2DDDHd*xL 2*H1bL^H32L*Sqrt@PiD*Sin@aH2*bLD*Cos@ArcCos@-1+d*x^2D2D*FresnelS@Sqrt@1HPi*bLD*Sqrt@a+b*ArcCos@-1+d*x^2DDDHd*xL ;
FreeQ@8a,b,d<,xD
Rules for integrands involving inverse sines and cosines
2. á
14
1
Ia + b ArcSinAc + d x2 EM
2
1: á
â x when c2 Š 1
1
Ia + b ArcSinAc + d x2 EM
2
â x when c2 Š 1
Ÿ Derivation: Integration by parts
2bdx
Ÿ Basis: If c2 Š 1, then -2 c d
Ÿ Rule: If
c2
x2 -d2
x4
Š 1, then
á
Ha+b ArcSin@c+d
x2 DL2
Š ¶x
- 2 c d x2 - d2 x4
1
Ia + b ArcSinAc + d x2 EM
âx ™ -
2
d
2 b d x Ia + b ArcSinAc + d x2 EM
2 b d x Ia + b ArcSinAc + d x2 EM
-
2b
á
x2
- 2 c d x2 - d2 x4 Ia + b ArcSinAc + d x2 EM
x ICosA 2 b E + c SinA 2 b EM CosIntegralA 2 b Ia + b ArcSinAc + d x2 EME
a
- 2 c d x2 - d2 x4
™ -
1
a+b ArcSin@c+d x2 D
a
c
4 b2 ICosA 2 ArcSinAc + d x2 EE - c SinA 2 ArcSinAc + d x2 EEM
1
1
x ICosA 2 b E - c SinA 2 b EM SinIntegralA 2 b Ia + b ArcSinAc + d x2 EME
a
a
c
4 b2 ICosA 2 ArcSinAc + d x2 EE - c SinA 2 ArcSinAc + d x2 EEM
1
Ÿ Program code:
1
IntA1‘Ha_.+b_.*ArcSin@c_+d_.*x_^2DL^2,x_SymbolE :=
-Sqrt@-2*c*d*x^2-d^2*x^4DH2*b*d*x*Ha+b*ArcSin@c+d*x^2DLL x*HCos@aH2*bLD+c*Sin@aH2*bLDL*CosIntegral@HcH2*bLL*Ha+b*ArcSin@c+d*x^2DLD
H4*b^2*HCos@ArcSin@c+d*x^2D2D-c*Sin@ArcSin@c+d*x^2D2DLL +
x*HCos@aH2*bLD-c*Sin@aH2*bLDL*SinIntegral@HcH2*bLL*Ha+b*ArcSin@c+d*x^2DLD
H4*b^2*HCos@ArcSin@c+d*x^2D2D-c*Sin@ArcSin@c+d*x^2D2DLL ;
FreeQ@8a,b,c,d<,xD && ZeroQ@c^2-1D
2. á
Ÿ Rule:
1
Ia + b ArcCosAc + d x2 EM
2
1: á
â x when c2 Š 1
1
Ia + b ArcCosA1 + d x2 EM
2
âx
+
âx
Rules for integrands involving inverse sines and cosines
15
á
2 b d x Ia + b ArcCosA1 + d x2 EM
+
âx ™
Ia + b ArcCosA1 + d x2 EM
2
x SinA 2 b E CosIntegralA 2 b Ia + b ArcCosA1 + d x2 EME
a
- 2 d x2 - d2 x4
1
2
1
2 b2
x CosA 2 b E SinIntegralA 2 b Ia + b ArcCosA1 + d x2 EME
a
-
- d x2
1
2 b2
2
- d x2
Ÿ Program code:
IntA1‘Ha_.+b_.*ArcCos@1+d_.*x_^2DL^2,x_SymbolE :=
Sqrt@-2*d*x^2-d^2*x^4DH2*b*d*x*Ha+b*ArcCos@1+d*x^2DLL +
x*Sin@aH2*bLD*CosIntegral@Ha+b*ArcCos@1+d*x^2DLH2*bLDH2*Sqrt@2D*b^2*Sqrt@H-dL*x^2DL x*Cos@aH2*bLD*SinIntegral@Ha+b*ArcCos@1+d*x^2DLH2*bLDH2*Sqrt@2D*b^2*Sqrt@H-dL*x^2DL ;
FreeQ@8a,b,d<,xD
2: á
Ÿ Rule:
1
Ia + b ArcCosA- 1 + d x2 EM
-
á
1
Ia + b ArcCosA- 1 + d x2 EM
âx ™
2
x CosA 2 b E CosIntegralA 2 b Ia + b ArcCosA- 1 + d x2 EME
a
2 d x2 - d2 x4
2 b d x Ia + b ArcCosA- 1 + d x2 EM
âx
2
2
1
2 b2
d x2
x SinA 2 b E SinIntegralA 2 b Ia + b ArcCosA- 1 + d x2 EME
a
-
Ÿ Program code:
IntA1‘Ha_.+b_.*ArcCos@-1+d_.*x_^2DL^2,x_SymbolE :=
Sqrt@2*d*x^2-d^2*x^4DH2*b*d*x*Ha+b*ArcCos@-1+d*x^2DLL x*Cos@aH2*bLD*CosIntegral@Ha+b*ArcCos@-1+d*x^2DLH2*bLDH2*Sqrt@2D*b^2*Sqrt@d*x^2DL x*Sin@aH2*bLD*SinIntegral@Ha+b*ArcCos@-1+d*x^2DLH2*bLDH2*Sqrt@2D*b^2*Sqrt@d*x^2DL ;
FreeQ@8a,b,d<,xD
3: à Ia + b ArcSinAc + d x2 EM â x when c2 Š 1 ì n < - 1 ì n ¹ - 2
n
Ÿ Derivation: Inverted integration by parts twice
Ÿ Rule: If c2 Š 1 ì n < - 1 ì n ¹ - 2, then
1
2
2 b2
d x2
Rules for integrands involving inverse sines and cosines
16
2
á Ia + b ArcSinAc + d x EM â x ™
n
x Ia + b ArcSinAc + d x2 EM
- 2 c d x2 - d2 x4 Ia + b ArcSinAc + d x2 EM
n+2
4
b2
Ÿ Program code:
Hn + 1L Hn + 2L
n+1
2 b d Hn + 1L x
+
4
b2
1
Hn + 1L Hn + 2L
2
à Ia + b ArcSinAc + d x EM
Int@Ha_.+b_.*ArcSin@c_+d_.*x_^2DL^n_,x_SymbolD :=
x*Ha+b*ArcSin@c+d*x^2DL^Hn+2LH4*b^2*Hn+1L*Hn+2LL +
Sqrt@-2*c*d*x^2-d^2*x^4D*Ha+b*ArcSin@c+d*x^2DL^Hn+1LH2*b*d*Hn+1L*xL 1H4*b^2*Hn+1L*Hn+2LL*Int@Ha+b*ArcSin@c+d*x^2DL^Hn+2L,xD ;
FreeQ@8a,b,c,d<,xD && ZeroQ@c^2-1D && RationalQ@nD && n<-1 && n¹-2
Int@Ha_.+b_.*ArcCos@c_+d_.*x_^2DL^n_,x_SymbolD :=
x*Ha+b*ArcCos@c+d*x^2DL^Hn+2LH4*b^2*Hn+1L*Hn+2LL Sqrt@-2*c*d*x^2-d^2*x^4D*Ha+b*ArcCos@c+d*x^2DL^Hn+1LH2*b*d*Hn+1L*xL 1H4*b^2*Hn+1L*Hn+2LL*Int@Ha+b*ArcCos@c+d*x^2DL^Hn+2L,xD ;
FreeQ@8a,b,c,d<,xD && ZeroQ@c^2-1D && RationalQ@nD && n<-1 && n¹-2
3: à
ArcSin@a xp Dn
â x when n Î Z+
x
Ÿ Derivation: Integration by substitution
Ÿ Basis:
ArcSin@a xp Dn
x
Š
Ÿ Rule: If n Î Z+ , then
Ÿ Program code:
1
p
ArcSin@a xp Dn Cot@ArcSin@a xp DD ¶x ArcSin@a xp D
à
ArcSin@a xp Dn
1
âx ™
x
IntAArcSin@a_.*x_^p_D^n_.‘x_,x_SymbolE :=
1p*Subst@Int@x^n*Cot@xD,xD,x,ArcSin@a*x^pDD ;
FreeQ@8a,p<,xD && PositiveIntegerQ@nD
IntAArcCos@a_.*x_^p_D^n_.‘x_,x_SymbolE :=
-1p*Subst@Int@x^n*Tan@xD,xD,x,ArcCos@a*x^pDD ;
FreeQ@8a,p<,xD && PositiveIntegerQ@nD
p
SubstBà xn Cot@xD â x, x, ArcSin@a xp DF
n+2
âx
Rules for integrands involving inverse sines and cosines
4: à u ArcSinB
c
a+b
17
F âx
m
xn
Ÿ Derivation: Algebraic simplification
Ÿ Basis: ArcSin@zD Š ArcCscA 1z E
Ÿ Rule:
Ÿ Program code:
à u ArcSinB
m
a b xn m
F â x ™ à u ArcCscB +
F âx
a + b xn
c
c
c
IntAu_.*ArcSinAc_.‘Ha_.+b_.*x_^n_.LE^m_.,x_SymbolE :=
Int@u*ArcCsc@ac+b*x^ncD^m,xD ;
FreeQ@8a,b,c,n,m<,xD
IntAu_.*ArcCosAc_.‘Ha_.+b_.*x_^n_.LE^m_.,x_SymbolE :=
Int@u*ArcSec@ac+b*x^ncD^m,xD ;
FreeQ@8a,b,c,n,m<,xD
Rules for integrands involving inverse sines and cosines
1 + b x2 F
18
n
5: á
ArcSinB
âx
1 + b x2
Ÿ Derivation: Piecewise constant extraction and integration by substitution
Ÿ Basis: ¶x
-b x2
x
x ArcSinB
Ÿ Basis:
-b
x2
Š0
1+b x2 F
1+b
x2
1 + b x2 F ¶x
n
Š
1
b
SubstB
ArcSin@xDn
, x,
1-x2
Ÿ Rule:
1 + b x2 F
1 + b x2
n
á
ArcSinB
- b x2
bx
Ÿ Program code:
âx ™
1 + b x2
™
1 + b x2 F
n
- b x2
SubstBà
x
á
x ArcSinB
- b x2
ArcSin@xDn
â x, x,
1-
x2
IntAArcSin@Sqrt@1+b_.*x_^2DD^n_.‘Sqrt@1+b_.*x_^2D,x_SymbolE :=
Sqrt@-b*x^2DHb*xL*Subst@Int@ArcSin@xD^nSqrt@1-x^2D,xD,x,Sqrt@1+b*x^2DD ;
FreeQ@8b,n<,xD
IntAArcCos@Sqrt@1+b_.*x_^2DD^n_.‘Sqrt@1+b_.*x_^2D,x_SymbolE :=
Sqrt@-b*x^2DHb*xL*Subst@Int@ArcCos@xD^nSqrt@1-x^2D,xD,x,Sqrt@1+b*x^2DD ;
FreeQ@8b,n<,xD
1 + b x2
1 + b x2 F
âx
Rules for integrands involving inverse sines and cosines
19
6: à u fc ArcSin@a+b xD â x when n Î Z+
n
Ÿ Derivation: Integration by substitution
Ÿ Basis: F@x, ArcSin@a + b xDD Š
1
b
SubstAFA- ab +
Sin@xD
,
b
xE Cos@xD, x, ArcSin@a + b xDE ¶x ArcSin@a + b xD
Ÿ Rule: If n Î Z+ , then
àuf
c ArcSin@a+b xDn
Ÿ Program code:
1
âx ™
b
SubstBà SubstBu, x, -
a
Sin@xD
+
b
b
F fc x Cos@xD â x, x, ArcSin@a + b xDF
n
Int@u_.*f_^Hc_.*ArcSin@a_.+b_.*x_D^n_.L,x_SymbolD :=
1b*Subst@Int@ReplaceAll@u,x®-ab+Sin@xDbD*f^Hc*x^nL*Cos@xD,xD,x,ArcSin@a+b*xDD ;
FreeQ@8a,b,c,f<,xD && PositiveIntegerQ@nD
Int@u_.*f_^Hc_.*ArcCos@a_.+b_.*x_D^n_.L,x_SymbolD :=
-1b*Subst@Int@ReplaceAll@u,x®-ab+Cos@xDbD*f^Hc*x^nL*Sin@xD,xD,x,ArcCos@a+b*xDD ;
FreeQ@8a,b,c,f<,xD && PositiveIntegerQ@nD
7. à v Ha + b ArcSin@uDL â x when u is free of inverse functions
1. à v Ha + b ArcSin@uDL â x when u is free of inverse functions
1: à ArcSinBa x2 + b
c + d x2 F â x when b2 c Š 1
Ÿ Derivation: Integration by parts and piecewise constant extraction
Ÿ Basis: If b2 c Š 1, then 1 - Ja x2 + b
Ÿ Basis: ¶x
x
-x2
b2 d+a2 x2 +2 a b
Jb2
d+a2
x2 +2
ab
c + d x2 N Š - x2 Jb2 d + a2 x2 + 2 a b
2
c+d x2
c+d
x2
N
Š0
c + d x2 N
Ÿ Note: The resulting integrand is of the form x FAx2 E which can be integrated by substitution.
Ÿ Rule: If b2 c Š 1, then
Rules for integrands involving inverse sines and cosines
20
c + d x F â x ™ x ArcSinBa x + b
à ArcSinBa x + b
2
2
2
™ x ArcSinBa x + b
2
c+dx F-
Ÿ Program code:
2
x
c+dx F-á
b2 d + a2 x2 + 2 a b
- x2 Jb2 d + a2 x2 + 2 a b
2
c + d x2
c + d x2 N
c+d
á
x2
x2 Jb d + 2 a
- x2
c+d
x2
Jb2
d+
a2
c + d x2 N
x2
x Jb d + 2 a
b2
d+
a2
+2ab
x2
Int@ArcCos@a_.*x_^2+b_.*Sqrt@c_+d_.*x_^2DD,x_SymbolD :=
x*ArcCos@a*x^2+b*Sqrt@c+d*x^2DD +
x*Sqrt@b^2*d+a^2*x^2+2*a*b*Sqrt@c+d*x^2DDSqrt@H-x^2L*Hb^2*d+a^2*x^2+2*a*b*Sqrt@c+d*x^2DLD*
Int@x*Hb*d+2*a*Sqrt@c+d*x^2DLHSqrt@c+d*x^2D*Sqrt@b^2*d+a^2*x^2+2*a*b*Sqrt@c+d*x^2DDL,xD ;
FreeQ@8a,b,c,d<,xD && EqQ@b^2*c,1D
Ÿ Derivation: Integration by parts
Ÿ Rule: If u is free of inverse functions, then
Ÿ Program code:
à ArcSin@uD â x ™ x ArcSin@uD - à
Int@ArcSin@u_D,x_SymbolD :=
x*ArcSin@uD Int@SimplifyIntegrand@x*D@u,xDSqrt@1-u^2D,xD,xD ;
InverseFunctionFreeQ@u,xD && Not@FunctionOfExponentialQ@u,xDD
x ¶x u
âx
1 - u2
c+d
c + d x2 N
+2ab
Int@ArcSin@a_.*x_^2+b_.*Sqrt@c_+d_.*x_^2DD,x_SymbolD :=
x*ArcSin@a*x^2+b*Sqrt@c+d*x^2DD x*Sqrt@b^2*d+a^2*x^2+2*a*b*Sqrt@c+d*x^2DDSqrt@H-x^2L*Hb^2*d+a^2*x^2+2*a*b*Sqrt@c+d*x^2DLD*
Int@x*Hb*d+2*a*Sqrt@c+d*x^2DLHSqrt@c+d*x^2D*Sqrt@b^2*d +a^2*x^2+2*a*b*Sqrt@c+d*x^2DDL,xD ;
FreeQ@8a,b,c,d<,xD && EqQ@b^2*c,1D
2: à ArcSin@uD â x when u is free of inverse functions
âx
x2
N
âx
c+d
x2
Rules for integrands involving inverse sines and cosines
21
Int@ArcCos@u_D,x_SymbolD :=
x*ArcCos@uD +
Int@SimplifyIntegrand@x*D@u,xDSqrt@1-u^2D,xD,xD ;
InverseFunctionFreeQ@u,xD && Not@FunctionOfExponentialQ@u,xDD
2: à Hc + d xLm Ha + b ArcSin@uDL â x when m ¹ - 1 ì u is free of inverse functions
Ÿ Derivation: Integration by parts
Ÿ Rule: If m ¹ - 1 ì u is free of inverse functions, then
Ÿ Program code:
m
à Hc + d xL Ha + b ArcSin@uDL â x ™
Hc + d xLm+1 Ha + b ArcSin@uDL
d Hm + 1L
b
-
d Hm + 1L
à
Hc + d xLm+1 ¶x u
âx
1 - u2
Int@Hc_.+d_.*x_L^m_.*Ha_.+b_.*ArcSin@u_DL,x_SymbolD :=
Hc+d*xL^Hm+1L*Ha+b*ArcSin@uDLHd*Hm+1LL bHd*Hm+1LL*Int@SimplifyIntegrand@Hc+d*xL^Hm+1L*D@u,xDSqrt@1-u^2D,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_.*ArcCos@u_DL,x_SymbolD :=
Hc+d*xL^Hm+1L*Ha+b*ArcCos@uDLHd*Hm+1LL +
bHd*Hm+1LL*Int@SimplifyIntegrand@Hc+d*xL^Hm+1L*D@u,xDSqrt@1-u^2D,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
Rules for integrands involving inverse sines and cosines
22
3: à v Ha + b ArcSin@uDL â x when u and à v â x are free of inverse functions
Ÿ Derivation: Integration by parts
Ÿ Rule: If u is free of inverse functions, let w Š Ù v âx, if w is free of inverse functions, then
Ÿ Program code:
à v Ha + b ArcSin@uDL â x ™ w Ha + b ArcSin@uDL - b à
w ¶x u
âx
1 - u2
Int@v_*Ha_.+b_.*ArcSin@u_DL,x_SymbolD :=
With@8w=IntHide@v,xD<,
Dist@Ha+b*ArcSin@uDL,w,xD b*Int@SimplifyIntegrand@w*D@u,xDSqrt@1-u^2D,xD,xD ;
InverseFunctionFreeQ@w,xDD ;
FreeQ@8a,b<,xD && InverseFunctionFreeQ@u,xD && Not@MatchQ@v, Hc_.+d_.*xL^m_. ; FreeQ@8c,d,m<,xDDD
Int@v_*Ha_.+b_.*ArcCos@u_DL,x_SymbolD :=
With@8w=IntHide@v,xD<,
Dist@Ha+b*ArcCos@uDL,w,xD +
b*Int@SimplifyIntegrand@w*D@u,xDSqrt@1-u^2D,xD,xD ;
InverseFunctionFreeQ@w,xDD ;
FreeQ@8a,b<,xD && InverseFunctionFreeQ@u,xD && Not@MatchQ@v, Hc_.+d_.*xL^m_. ; FreeQ@8c,d,m<,xDDD