I tried Fricas on the examples from the paper: INTEGRATION IN FINITE TERMS WITH SPECIAL FUNCTIONS: THE LOGARITHMIC INTEGRAL
which was written by CHERRY in 1986.
I just wanted to share my findings in case this is useful.
INPUT
---------------------------------------------------
-- Examples taken from:
-- "INTEGRATION IN FINITE TERMS WITH SPECIAL FUNCTIONS: THE LOGARITHMIC
INTEGRAL"
-- written by G. W. CHERRY
-- published in SIAM J. COMPUT. Vol. 15, No. 1, February 1986
-- Page 1
-- Example 1.1
i := x/log(x)^2
I := integrate(i, x)
Ic := 2*li(x^2) - x^2/log(x)
D(Ic, x)
-- Page 16
-- Example 5.1
i := x^3 / log(x^2-1)
I := integrate(i, x)
Ic := li(x^4-2*x^2+1)/2 + li(x^2-1)/2
D(Ic, x)
-- Example 5.2
i := x^2 / log(x^2-1)
I := integrate(i, x)
-- No closed form
-- Page 17
-- Example 5.3
i := (2*x+3)/(3*log(x)+2*x) * exp(log(x)/2+x) + 1/(x+1)*(exp(log(x)/2+x))^2
I := integrate(i, x)
-- Example 5.3a
i := (2*x+3)/(3*log(x)+2*x) * exp(log(x)/2+x)
I := integrate(i, x)
Ic := li(x*exp(log(x)/2+x))
D(Ic, x)
-- Example 5.3b
i := 1/(x+1)*(exp(log(x)/2+x))^2
I := integrate(i, x)
i := x/(x+1)*exp(2*x)
I := integrate(i, x)
Ic := exp(2*x)/2 - exp(-2)*li(exp(2*x+2))
D(Ic, x)
-- Page 18
-- Example 5.4
i := ((2*x^3-x^2-6*x)/(x^2+3*x+2) + (2*x-3)/(log(x)+1)) * exp(x*log(x)+x)
I := integrate(i, x)
Ic := -3/exp(1)*li(exp(x*log(x)+x+log(x)+1)) +
2/exp(2)*li(exp(x*log(x)+x+2*log(x)+2))
D(Ic, x)
-- Page 19
-- Example 6
i := cos(x)^2 / x^3
I := integrate(i, x)
Ic := -Ci(2*x) + sin(2*x)/(2*x) - cos(2*x)/(4*x^2) - 1/(4*x^2)
D(Ic, x)
-- Appendix
-- Example d1 (same as 1.1)
i := x/log(x)^2
I := integrate(i, x)
-- Example d2
i := 1/(log(x)+3)
I := integrate(i, x)
Ic := exp(-3)*li(exp(3)*x)
D(Ic, x)
-- Example d3
i := x^2 / log(x+1)
I := integrate(i, x)
Ic := li(x^3+3*x^2+3*x+1) - 2*li(x^2+2*x+1) + li(x+1)
D(Ic, x)
-- Example d4
i := (log(x)^2+3)/(log(x)^2+3*log(x)+2)
I := integrate(i, x)
Ic := -7*exp(-2)*li(exp(2)*x)+4*exp(-1)*li(exp(1)*x)+x
D(Ic, x)
-- Example d5, d6, d7
I := li(x*exp(x)/log(x))
i := D(I, x)
I := integrate(i, x)
-- Example d8
i := exp(x)/(x+1)^2
I := integrate(i, x)
Ic := exp(-1)*Ei(x+1) - exp(x)/(x+1)
D(Ic, x)
-- Example d9
i := sin(x)/x
I := integrate(i, x)
-- Example d10 (same as 6)
i := cos(x)^2/x^3
I := integrate(i, x)
-- Example d11
i := (x^2+3)*exp(x)/(x^2+3*x+2)
I := integrate(i, x)
Ic := -7*exp(-2)*Ei(x+2) + 4*exp(-1)*Ei(x+1) + exp(x)
D(Ic, x)
-- Example d12
i := (x^2+1)*exp(x)/(x^2+x+1)
I := integrate(i, x)
---------------------------------------------------
OUTPUT
---------------------------------------------------
(1) -> )read li.input
-- Examples taken from:
-- "INTEGRATION IN FINITE TERMS WITH SPECIAL FUNCTIONS: THE LOGARITHMIC
INTEGRAL"
-- written by G. W. CHERRY
-- published in SIAM J. COMPUT. Vol. 15, No. 1, February 1986
-- Page 1
-- Example 1.1
i := x/log(x)^2
x
(1) -------
2
log(x)
Type: Expression(Integer)
I := integrate(i, x)
2 2
2li(x )log(x) - x
(2) ------------------
log(x)
Type: Union(Expression(Integer),...)
Ic := 2*li(x^2) - x^2/log(x)
2 2
2li(x )log(x) - x
(3) ------------------
log(x)
Type: Expression(Integer)
D(Ic, x)
2 2
(- 2x log(x) + x)log(x ) + 4x log(x)
(4) -------------------------------------
2 2
log(x) log(x )
Type: Expression(Integer)
-- Page 16
-- Example 5.1
i := x^3 / log(x^2-1)
3
x
(5) -----------
2
log(x - 1)
Type: Expression(Integer)
I := integrate(i, x)
4 2 2
li(x - 2x + 1) + li(x - 1)
(6) -----------------------------
2
Type: Union(Expression(Integer),...)
Ic := li(x^4-2*x^2+1)/2 + li(x^2-1)/2
4 2 2
li(x - 2x + 1) + li(x - 1)
(7) -----------------------------
2
Type: Expression(Integer)
D(Ic, x)
4 2 3 2
x log(x - 2x + 1) + (2x - 2x)log(x - 1)
(8) -------------------------------------------
2 4 2
log(x - 1)log(x - 2x + 1)
Type: Expression(Integer)
-- Example 5.2
i := x^2 / log(x^2-1)
2
x
(9) -----------
2
log(x - 1)
Type: Expression(Integer)
I := integrate(i, x)
x 2
++ %A
(10) | ------------ d%A
++ 2
log(%A - 1)
Type: Union(Expression(Integer),...)
-- No closed form
-- Page 17
-- Example 5.3
i := (2*x+3)/(3*log(x)+2*x) * exp(log(x)/2+x) + 1/(x+1)*(exp(log(x)/2+x))^2
log(x) + 2x 2 log(x) + 2x
----------- -----------
2 2 2
(3log(x) + 2x)(%e ) + (2x + 5x + 3)%e
(11) ------------------------------------------------------------
2
(3x + 3)log(x) + 2x + 2x
Type: Expression(Integer)
I := integrate(i, x)
(12)
INTSIGN
,
x
,
log(%A) + 2%A 2 log(%A) + 2%A
------------- -------------
2 2 2
(3log(%A) + 2%A)(%e ) + (2%A + 5%A + 3)%e
-------------------------------------------------------------------- d%A
2
(3%A + 3)log(%A) + 2%A + 2%A
Type: Union(Expression(Integer),...)
-- Example 5.3a
i := (2*x+3)/(3*log(x)+2*x) * exp(log(x)/2+x)
log(x) + 2x
-----------
2
(2x + 3)%e
(13) ---------------------
3log(x) + 2x
Type: Expression(Integer)
I := integrate(i, x)
log(%A) + 2%A
-------------
x 2
++ (2%A + 3)%e
(14) | ------------------------ d%A
++ 3log(%A) + 2%A
Type: Union(Expression(Integer),...)
Ic := li(x*exp(log(x)/2+x))
log(x) + 2x
-----------
2
(15) li(x %e )
Type: Expression(Integer)
D(Ic, x)
log(x) + 2x
-----------
2
(2x + 3)%e
(16) ---------------------
log(x) + 2x
-----------
2
2log(x %e )
Type: Expression(Integer)
-- Example 5.3b
i := 1/(x+1)*(exp(log(x)/2+x))^2
log(x) + 2x 2
-----------
2
(%e )
(17) ----------------
x + 1
Type: Expression(Integer)
I := integrate(i, x)
log(%A) + 2%A 2
-------------
x 2
++ (%e )
(18) | ------------------ d%A
++ %A + 1
Type: Union(Expression(Integer),...)
i := x/(x+1)*exp(2*x)
2x
x %e
(19) ------
x + 1
Type: Expression(Integer)
I := integrate(i, x)
x 2%A
++ %A %e
(20) | -------- d%A
++ %A + 1
Type: Union(Expression(Integer),...)
Ic := exp(2*x)/2 - exp(-2)*li(exp(2*x+2))
2 2x
%e %e - 2Ei(2x + 2)
(21) ---------------------
2
2%e
Type: Expression(Integer)
D(Ic, x)
2x + 2 2 2x
- %e + (x + 1)%e %e
(22) ---------------------------
2
(x + 1)%e
Type: Expression(Integer)
-- Page 18
-- Example 5.4
i := ((2*x^3-x^2-6*x)/(x^2+3*x+2) + (2*x-3)/(log(x)+1)) * exp(x*log(x)+x)
3 2 3 2 x log(x) + x
((2x - x - 6x)log(x) + 4x + 2x - 11x - 6)%e
(23) -----------------------------------------------------------
2 2
(x + 3x + 2)log(x) + x + 3x + 2
Type: Expression(Integer)
I := integrate(i, x)
(24)
INTSIGN
,
x
,
3 2 3 2 %A log(%A) + %A
((2%A - %A - 6%A)log(%A) + 4%A + 2%A - 11%A - 6)%e
--------------------------------------------------------------------- d%A
2 2
(%A + 3%A + 2)log(%A) + %A + 3%A + 2
Type: Union(Expression(Integer),...)
Ic := -3/exp(1)*li(exp(x*log(x)+x+log(x)+1)) +
2/exp(2)*li(exp(x*log(x)+x+2*log(x)+2))
2
2%e Ei((x + 2)log(x) + x + 2) - 3%e Ei((x + 1)log(x) + x + 1)
(25) -------------------------------------------------------------
2
%e %e
Type: Expression(Integer)
D(Ic, x)
(26)
2 2 (x + 2)log(x) + x + 2
((2x + 2x)%e log(x) + (4x + 8x + 4)%e)%e
+
2 2 2 2 (x + 1)log(x) + x + 1
((- 3x - 6x)%e log(x) + (- 6x - 15x - 6)%e )%e
/
3 2 2 3 2 2
(x + 3x + 2x)%e %e log(x) + (x + 3x + 2x)%e %e
Type: Expression(Integer)
-- Page 19
-- Example 6
i := cos(x)^2 / x^3
2
cos(x)
(27) -------
3
x
Type: Expression(Integer)
I := integrate(i, x)
x 2
++ cos(%A)
(28) | -------- d%A
++ 3
%A
Type: Union(Expression(Integer),...)
Ic := -Ci(2*x) + sin(2*x)/(2*x) - cos(2*x)/(4*x^2) - 1/(4*x^2)
2
2x sin(2x) - cos(2x) - 4x Ci(2x) - 1
(29) ------------------------------------
2
4x
Type: Expression(Integer)
D(Ic, x)
cos(2x) + 1
(30) -----------
3
2x
Type: Expression(Integer)
-- Appendix
-- Example d1 (same as 1.1)
i := x/log(x)^2
x
(31) -------
2
log(x)
Type: Expression(Integer)
I := integrate(i, x)
2 2
2li(x )log(x) - x
(32) ------------------
log(x)
Type: Union(Expression(Integer),...)
-- Example d2
i := 1/(log(x)+3)
1
(33) ----------
log(x) + 3
Type: Expression(Integer)
I := integrate(i, x)
3
li(x %e )
(34) ---------
3
%e
Type: Union(Expression(Integer),...)
Ic := exp(-3)*li(exp(3)*x)
3
li(x %e )
(35) ---------
3
%e
Type: Expression(Integer)
D(Ic, x)
1
(36) ----------
3
log(x %e )
Type: Expression(Integer)
-- Example d3
i := x^2 / log(x+1)
2
x
(37) ----------
log(x + 1)
Type: Expression(Integer)
I := integrate(i, x)
3 2 2
(38) li(x + 3x + 3x + 1) - 2li(x + 2x + 1) + li(x + 1)
Type: Union(Expression(Integer),...)
Ic := li(x^3+3*x^2+3*x+1) - 2*li(x^2+2*x+1) + li(x+1)
3 2 2
(39) li(x + 3x + 3x + 1) - 2li(x + 2x + 1) + li(x + 1)
Type: Expression(Integer)
D(Ic, x)
(40)
2 3 2
(log(x + 2x + 1) + (- 4x - 4)log(x + 1))log(x + 3x + 3x + 1)
+
2 2
(3x + 6x + 3)log(x + 1)log(x + 2x + 1)
/
2 3 2
log(x + 1)log(x + 2x + 1)log(x + 3x + 3x + 1)
Type: Expression(Integer)
-- Example d4
i := (log(x)^2+3)/(log(x)^2+3*log(x)+2)
2
log(x) + 3
(41) ---------------------
2
log(x) + 3log(x) + 2
Type: Expression(Integer)
I := integrate(i, x)
2 2 2
- 7%e li(x %e ) + 4%e li(x %e) + x %e %e
(42) -----------------------------------------
2
%e %e
Type: Union(Expression(Integer),...)
Ic := -7*exp(-2)*li(exp(2)*x)+4*exp(-1)*li(exp(1)*x)+x
2 2 2
- 7%e li(x %e ) + 4%e li(x %e) + x %e %e
(43) -----------------------------------------
2
%e %e
Type: Expression(Integer)
D(Ic, x)
2
(log(x %e) + 4)log(x %e ) - 7log(x %e)
(44) --------------------------------------
2
log(x %e)log(x %e )
Type: Expression(Integer)
-- Example d5, d6, d7
I := li(x*exp(x)/log(x))
x
x %e
(45) li(------)
log(x)
Type: Expression(Integer)
i := D(I, x)
x x
(x + 1)%e log(x) - %e
(46) ----------------------
x
2 x %e
log(x) log(------)
log(x)
Type: Expression(Integer)
I := integrate(i, x)
x
x %e
(47) li(------)
log(x)
Type: Union(Expression(Integer),...)
-- Example d8
i := exp(x)/(x+1)^2
x
%e
(48) -----------
2
x + 2x + 1
Type: Expression(Integer)
I := integrate(i, x)
x %A
++ %e
(49) | ------------- d%A
++ 2
%A + 2%A + 1
Type: Union(Expression(Integer),...)
Ic := exp(-1)*Ei(x+1) - exp(x)/(x+1)
x
- %e %e + (x + 1)Ei(x + 1)
(50) ---------------------------
(x + 1)%e
Type: Expression(Integer)
D(Ic, x)
x + 1 x
(x + 1)%e - x %e %e
(51) -------------------------
2
(x + 2x + 1)%e
Type: Expression(Integer)
-- Example d9
i := sin(x)/x
sin(x)
(52) ------
x
Type: Expression(Integer)
I := integrate(i, x)
(53) Si(x)
Type: Union(Expression(Integer),...)
-- Example d10 (same as 6)
i := cos(x)^2/x^3
2
cos(x)
(54) -------
3
x
Type: Expression(Integer)
I := integrate(i, x)
x 2
++ cos(%A)
(55) | -------- d%A
++ 3
%A
Type: Union(Expression(Integer),...)
-- Example d11
i := (x^2+3)*exp(x)/(x^2+3*x+2)
2 x
(x + 3)%e
(56) -----------
2
x + 3x + 2
Type: Expression(Integer)
I := integrate(i, x)
x 2 %A
++ (%A + 3)%e
(57) | ------------- d%A
++ 2
%A + 3%A + 2
Type: Union(Expression(Integer),...)
Ic := -7*exp(-2)*Ei(x+2) + 4*exp(-1)*Ei(x+1) + exp(x)
2 x 2
%e %e %e - 7%e Ei(x + 2) + 4%e Ei(x + 1)
(58) -----------------------------------------
2
%e %e
Type: Expression(Integer)
D(Ic, x)
x + 2 2 x + 1 2 2 x
(- 7x - 7)%e %e + (4x + 8)%e %e + (x + 3x + 2)%e %e %e
(59) ------------------------------------------------------------------
2 2
(x + 3x + 2)%e %e
Type: Expression(Integer)
-- Example d12
i := (x^2+1)*exp(x)/(x^2+x+1)
2 x
(x + 1)%e
(60) -----------
2
x + x + 1
Type: Expression(Integer)
I := integrate(i, x)
x 2 %A
++ (%A + 1)%e
(61) | ------------- d%A
++ 2
%A + %A + 1
Type: Union(Expression(Integer),...)
---------------------------------------------------
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li.input
Description: Binary data
li.output
Description: Binary data
