changed: -integrate((100e^(2x)+100e^(-2x)-199)^0.5,x) \begin{axiom} integrate((100*e^(2*x)+100*e^(-2*x)-199)^(1/2), x) \end{axiom}
This page is set to execute \begin{axiom}... \end{axiom} commands using FriCAS. See also FriCASIntegration.
(1) -> )version
"FriCAS 1.3.12 compiled at Sat 7 Jun 23:54:49 CEST 2025"
Any comments added here use this version of FriCAS.
solve(s=v*t+a*t^2/2,t)
radicalSolve(s=v*t+a*t^2/2,t)
solve(v*t+a*t^2/2-s=0,t)
solve(v*t+a*t/2-s=0,t)
solve(v*t+a*t^2/2-s=0,t)
radicalSolve(p^3 - p + 1/10=0,p)
R1 ==> Record(foo1: String,remLexs: List Integer)
R2 ==> Record(foo2: String,remLexs: List Integer)
r1: R1 := ["a",[1, 2, 3]]
r2: R2 := ["b",[5]]
r1.remLexs
r2.remLexs
)set output tex off
)set output algebra on
guessPRec [1,0, 1, 1, 4, 10, 35, 120, 455, 1792, 7413, 31780, 140833, 641928, 3000361, 14338702, 69902535, 346939792, 1750071307, 8958993507, 46484716684, 244187539270, 1297395375129, 6965930587924]
(13) [ [ f(n): 2 2 (- n - 17 n - 72)f(n + 3) + (4 n + 30 n + 44)f(n + 2) + 2 2 (19 n + 113 n + 150)f(n + 1) + (14 n + 42 n + 28)f(n) = 0 ,f(0) = 1, f(1) = 0, f(2) = 1] ]
guessRec([1,1, 0, 1, - 1, 2, - 1, 5, - 4, 29, - 13, 854, - 685])
2 (14) [[f(n): f(n + 2) + f(n + 1) - f(n) = 0,f(0) = 1, f(1) = 1]]
integrate(exp(x-x^2),x)
1 - 4 2 x - 1 +---+ %e erf(-------)\|%pi 2 (15) --------------------- 2
1+1
(16) 2
2^3
(17) 8
2**3
There are no library operations named ** Use HyperDoc Browse or issue )what op ** to learn if there is any operation containing " ** " in its name.
Cannot find a definition or applicable library operation named ** with argument type(s) PositiveInteger PositiveInteger
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need.
sqrt(2)^2-2
(18) 0
a:=sqrt(2)
+-+ (19) \|2
b:=sqrt(a)
+----+ | +-+ (20) \|\|2
b^2-a
(21) 0
f(a) == { local b=sqrt(a); b^2-a}
Line 1: f(a) == { local b=sqrt(a); b^2-a}
........A........B
Error A: Missing mate.
Error B: syntax error at top level
Error B: Possibly missing a }
3 error(s) parsing
local needs to be on separate statement:
f(a) == { local b; b :=sqrt(a); b^2-a}
f(2)
Compiling function f with type PositiveInteger -> AlgebraicNumber
(23) 0
f(sqrt(2))
Compiling function f with type AlgebraicNumber -> AlgebraicNumber
(24) 0
[q,r]
(25) [q,r]
f(a) == ( local b; b:=sqrt(a); b^2-a)
Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f
f(2)
Compiling function f with type PositiveInteger -> AlgebraicNumber
(27) 0
f(sqrt(2))
Compiling function f with type AlgebraicNumber -> AlgebraicNumber
(28) 0
f(a) == ( local b; b:=sqrt(a); b^2-a)
Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f
f(2)
Compiling function f with type PositiveInteger -> AlgebraicNumber
(30) 0
f(sqrt(2))
Compiling function f with type AlgebraicNumber -> AlgebraicNumber
(31) 0
f(z)
Compiling function f with type Variable(z) -> Expression(Integer)
(32) 0
f(%i)
Compiling function f with type Complex(Integer) -> Expression( Complex(Integer))
(33) 0
f(a) == ( local b; b:=sqrt(a); b^2-a)
Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f
f(2)
Compiling function f with type PositiveInteger -> AlgebraicNumber
(35) 0
c:=f(sqrt(2))
Compiling function f with type AlgebraicNumber -> AlgebraicNumber
(36) 0
)display properties c
Properties of c : Value (has type AlgebraicNumber): 0
f(a) == ( local b; b:=sqrt(a); b^2 = a)
Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f
f(a) == ( local b; b:=sqrt(a); b^2 = a)
1 old definition(s) deleted for function or rule f
f(2)
Compiling function f with type PositiveInteger -> Equation( AlgebraicNumber)
(39) 2 = 2
f(%i)
Compiling function f with type Complex(Integer) -> Equation( Expression(Complex(Integer)))
(40) %i = %i
f(sqrt(2))
Compiling function f with type AlgebraicNumber -> Equation( AlgebraicNumber)
+-+ +-+ (41) \|2 = \|2
f(2^(1/2))
+-+ +-+ (42) \|2 = \|2
f(2^(1/3))
3+-+ 3+-+ (43) \|2 = \|2
f(a) == ( local b; b:=sqrt(a); b^2-a)
Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f
f(sqrt(2))
Compiling function f with type AlgebraicNumber -> AlgebraicNumber
(45) 0
)set output fortran
--------------------------- The fortran Option ----------------------------
Description: create output in FORTRAN format
)set output fortran is used to tell FriCAS to turn FORTRAN-style output printing on and off,and where to place the output. By default, the destination for the output is the screen but printing is turned off.
Syntax: )set output fortran <arg> where arg can be one of on turn FORTRAN printing on off turn FORTRAN printing off (default state) console send FORTRAN output to screen (default state) fp<.fe> send FORTRAN output to file with file prefix fp and file extension .fe. If not given,.fe defaults to .sfort.
If you wish to send the output to a file,you may need to issue this command twice: once with on and once with the file name. For example, to send FORTRAN output to the file polymer.sfort, issue the two commands
)set output fortran on )set output fortran polymer
The output is placed in the directory from which you invoked FriCAS or the one you set with the )cd system command. The current setting is: Off:CONSOLE f(sqrt(2))
(46) 0
)set output tex
----------------------------- The tex Option ------------------------------
Description: create output in TeX style
)set output tex is used to tell FriCAS to turn TeX-style output printing on and off,and where to place the output. By default, the destination for the output is the screen but printing is turned off.
Syntax: )set output tex <arg> where arg can be one of on turn TeX printing on off turn TeX printing off (default state) console send TeX output to screen (default state) fp<.fe> send TeX output to file with file prefix fp and file extension .fe. If not given,.fe defaults to .stex.
If you wish to send the output to a file,you may need to issue this command twice: once with on and once with the file name. For example, to send TeX output to the file polymer.stex, issue the two commands
)set output tex on )set output tex polymer
The output is placed in the directory from which you invoked FriCAS or the one you set with the )cd system command. The current setting is: Off:CONSOLE f(sqrt(2))
(47) 0
)set output script
--------------------------- The scripts Option ----------------------------
Description: show subscripts,... linearly
The scripts option may be followed by any one of the following:
yes no
The current setting is indicated within the list.
f(sqrt(2))
(48) 0
f(a) == ( local b; b:=sqrt(a); b^2-a)
Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f
f(sqrt(2))
Compiling function f with type AlgebraicNumber -> AlgebraicNumber
(50) 0
)set output fortran on
f(sqrt(2))
R51=0
(51) 0
)set output tex on
f(sqrt(2))
R52=0
(52) 0
)set output script on
--------------------------- The scripts Option ----------------------------
Description: show subscripts,... linearly
The scripts option may be followed by any one of the following:
yes no
The current setting is indicated within the list.
Your value on is not among the valid choices.
f(sqrt(2))
R53=0
(53) 0
f(a) == ( local b; b:=sqrt(a); b^2-a)
Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f
f(sqrt(2))
Compiling function f with type AlgebraicNumber -> AlgebraicNumber
R55=0
(55) 0
)set output fortran console
f(sqrt(2))
R56=0
(56) 0
)set output tex console
f(sqrt(2))
R57=0
(57) 0
)set output script console
--------------------------- The scripts Option ----------------------------
Description: show subscripts,... linearly
The scripts option may be followed by any one of the following:
yes no
The current setting is indicated within the list.
Your value console is not among the valid choices.
f(sqrt(2))
R58=0
(58) 0
f(a) == ( local b; b:=sqrt(a); b^2-a)
Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f
f(sqrt(2))
Compiling function f with type AlgebraicNumber -> AlgebraicNumber
R60=0
(60) 0
)set output fortran on
)set output fortran console
f(sqrt(2))
R61=0
(61) 0
)set output tex on
)set output tex console
f(sqrt(2))
R62=0
(62) 0
)set output script on
--------------------------- The scripts Option ----------------------------
Description: show subscripts,... linearly
The scripts option may be followed by any one of the following:
yes no
The current setting is indicated within the list.
Your value on is not among the valid choices.
)set output script console
--------------------------- The scripts Option ----------------------------
Description: show subscripts,... linearly
The scripts option may be followed by any one of the following:
yes no
The current setting is indicated within the list.
Your value console is not among the valid choices.
f(sqrt(2))
R63=0
(63) 0
f(a) == ( local b; b:=sqrt(a); b^2-a)
Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f
f(sqrt(2))
Compiling function f with type AlgebraicNumber -> AlgebraicNumber
R65=0
(65) 0
)set output fortran on
f(sqrt(2))
R66=0
(66) 0
)set output fortran off
)set output tex on
f(sqrt(2))
(67) 0
)set output tex off
)set output script on
--------------------------- The scripts Option ----------------------------
Description: show subscripts,... linearly
The scripts option may be followed by any one of the following:
yes no
The current setting is indicated within the list.
Your value on is not among the valid choices.
f(sqrt(2))
(68) 0
)set output script off
--------------------------- The scripts Option ----------------------------
Description: show subscripts,... linearly
The scripts option may be followed by any one of the following:
yes no
The current setting is indicated within the list.
Your value off is not among the valid choices.
((1+2)/3)::SquareMatrix(3,Integer)
+1 0 0+ | | (69) |0 1 0| | | +0 0 1+
((1+2)/3)::SquareMatrix(3,Integer)
+1 0 0+ | | (70) |0 1 0| | | +0 0 1+
integrate((x+sin(x))^(1/2),x)
>> Error detected within library code: integrate: implementation incomplete (has polynomial part)
integrate((x+sin(x))^(1/2),x)
>> Error detected within library code: integrate: implementation incomplete (has polynomial part)
""; "Notice how the CAS computes the second (displayed) integral,while saying 'proved non-elementary' about the first one, "; "despite the fact that the two are equal for any x in [-1..-1/sqrt(2)]."; "My guess is that the system does not apply transformations that force additional restrictions on the function domain..."; "Notice also how transforming the integrand into a form that the CAS can find an elementary antiderivative for requires"; "quite some discipline in keeping track of the integrand domain, as well as using some not-so-obvious transfromations..."; t:=sqrt(1-x*x); nume:=(acos(x)*t)^(-1); deno:=log(2*acos(x)/%pi); transformedDeno:=log(2+asin(2*x*t)/%pi); failedIntegral:=integrate(nume/transformedDeno, x); integrate(nume/deno, x)
2 acos(x) (71) - log(log(---------)) %pi
integrate((x+sin(x))^(1/3),x)
>> Error detected within library code: integrate: implementation incomplete (has polynomial part)
t:=x+log(x); nume:=(x^2+x+1)*sqrt(t)+(3*x+1)*log(x)+3*x*x+x; deno:=x*t*sqrt(t)+x*x*t; check:=integrate(nume/deno,x); differentiate(2*sqrt(t)+2*log(x+sqrt(t)), x)
2 +----------+ 2 (x + 2 x + 1)\|log(x) + x + (3 x + 1)log(x) + 3 x + x (72) -------------------------------------------------------- 2 +----------+ 2 3 (x log(x) + x )\|log(x) + x + x log(x) + x
No lie, compare:
nume - numer(differentiate(2*sqrt(t)+2*log(x+sqrt(t)),x))
+----------+ (73) - x\|log(x) + x
Still, shame that FriCAS can not do corrected example...
nume:= 3*(x+exp(x))^(1/3)+(2*x*x+3*x)*exp(x)+5*x*x; deno:= x*(x+exp(x))^(1/3); "integrate(nume/deno,x)" (q:=nume/deno, differentiate(3*x*(x+exp(x))^(2/3)+3*log(x), x))
(74) +-------+ 3| x 2 x 2 3 \|%e + x + (2 x + 3 x)%e + 5 x [-------------------------------------,+-------+ 3| x x\|%e + x +-------+ 3| x 2 x 2 3 \|%e + x + (2 x + 3 x)%e + 5 x -------------------------------------] +-------+ 3| x x\|%e + x
)clear value t
integrate(sqrt(A-2*t*t/((1-t*t)^2))*2/(1+t*t),t)
(75) [ +---------+ \|- 4 A - 2 * log +------------------------+ | 4 2 3 +---------+ |A t + (- 2 A - 2)t + A 4 (2 t - 2 t)\|- 4 A - 2 |------------------------ + A t | 4 2 \| t - 2 t + 1 + 2 (- 6 A - 4)t + A / 4 2 t + 2 t + 1 + +-+ +-+ t\|2 2 \|2 atan(-----------------------------------) +------------------------+ | 4 2 2 |A t + (- 2 A - 2)t + A (t - 1) |------------------------ | 4 2 \| t - 2 t + 1 / 4 ,
+-------+ +-------+ t\|4 A + 2 - \|4 A + 2 atan(-----------------------------------) +------------------------+ | 4 2 2 |A t + (- 2 A - 2)t + A (t - 1) |------------------------ | 4 2 \| t - 2 t + 1 + +-+ +-+ t\|2 \|2 atan(-----------------------------------) +------------------------+ | 4 2 2 |A t + (- 2 A - 2)t + A (t - 1) |------------------------ | 4 2 \| t - 2 t + 1 / 2 ]
integrate(x/sqrt(2*%pi)*exp(-1/2*log(x)^2),x=0..%plusInfinity)
2 +-+ +---+ %e \|2 \|%pi (76) ------------- +-----+ \|2 %pi
guess([3,6, 9, 12, 15, 18, 21, 24, 27, 30])
(77) [3 n + 3]
integrate(5*x^2/sqrt(x^6+x),x)
+------+ 2 | 6 5 (78) log(2 x \|x + x + 2 x + 1)
integrate(x/((x^2+1)*sqrt(x^3-x)),x)
(79) +------+ 2 | 3 4 3 2 (4 x + 8 x - 4)\|x - x + x + 8 x + 2 x - 8 x + 1 log(------------------------------------------------------) 4 2 x + 2 x + 1 + +------+ 2 | 3 4 2 (2 x - 4 x - 2)\|x - x + x - 6 x + 1 2 atan(-----------------------------------------) +------+ 2 | 3 3 (2 x + 4 x - 2)\|x - x + 4 x - 4 x / 8
integrate((100*e^(2*x)+100*e^(-2*x)-199)^(1/2),x)
(80) - 600 * weierstrassZeta 9601 ---- 7500 ,1074401 - ------- 3375000 , 2 x log(e) 9601 1074401 300 %e - 199 weierstrassPInverse(----, - -------, ----------------------) 7500 3375000 300 + 2 x log(e) 9601 1074401 300 %e - 199 - 199 weierstrassPInverse(----, - -------, ----------------------) 7500 3375000 300 + +--------------------------------------------+ | 2 x log(e) 2 2 x log(e) |100 (%e ) - 199 %e + 100 - 30 |-------------------------------------------- | 2 x log(e) \| %e / 30 log(e)