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last edited 2 years ago by TheDoctor |
Edit detail for SandBoxFriCAS2 revision 3 of 8
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Editor: TheDoctor
Time: 2022/01/31 23:21:14 GMT+0 |
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added:
From TheDoctor Mon Jan 31 23:21:14 +0000 2022
From: TheDoctor
Date: Mon, 31 Jan 2022 23:21:14 +0000
Subject:
Message-ID: <20220131232114+0000@fricas-wiki.math.uni.wroc.pl>
\begin{axiom}
integrate(1/(1+exp(b*x))),x=-m..%plusInfinity
\end{axiom}
Try FriCAS? calculations here. For example, here is a simple FriCAS? command:
\begin{axiom}
integrate(1/(a+z^3), z=0..1,"noPole")
\end{axiom}
fricas
integrate(1/(a+z^3),z=0..1, "noPole")
![\label{eq1}{\left(
\begin{array}{@{}l}
\displaystyle
-{{\sqrt{3}}\ {\log{\left({{3 \ {{a}^{2}}\ {{\root{3}\of{{a}^{2}}}^{2}}}+{{\left(-{2 \ {{a}^{3}}}+{{a}^{2}}\right)}\ {\root{3}\of{{a}^{2}}}}+{{a}^{4}}-{2 \ {{a}^{3}}}}\right)}}}+{2 \ {\sqrt{3}}\ {\log \left({{{\root{3}\of{{a}^{2}}}^{2}}+{2 \ a \ {\root{3}\of{{a}^{2}}}}+{{a}^{2}}}\right)}}+
\
\
\displaystyle
{{12}\ {\arctan \left({{{2 \ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}-{a \ {\sqrt{3}}}}\over{3 \ a}}\right)}}+{2 \ \pi}](images/4322038355343098202-16.0px.png "\label{eq1}{\left(
\begin{array}{@{}l}
\displaystyle
-{{\sqrt{3}}\ {\log{\left({{3 \ {{a}^{2}}\ {{\root{3}\of{{a}^{2}}}^{2}}}+{{\left(-{2 \ {{a}^{3}}}+{{a}^{2}}\right)}\ {\root{3}\of{{a}^{2}}}}+{{a}^{4}}-{2 \ {{a}^{3}}}}\right)}}}+{2 \ {\sqrt{3}}\ {\log \left({{{\root{3}\of{{a}^{2}}}^{2}}+{2 \ a \ {\root{3}\of{{a}^{2}}}}+{{a}^{2}}}\right)}}+
\
\
\displaystyle
{{12}\ {\arctan \left({{{2 \ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}-{a \ {\sqrt{3}}}}\over{3 \ a}}\right)}}+{2 \ \pi}") | (1) |
Type: Union(f1: OrderedCompletion?(Expression(Integer)),
Remember to type \begin{axiom} before each group of commands and \end{axiom} after the commands.
fricas
integrate(1/sqrt(1+x^2),x)
 | (2) |
Type: Union(Expression(Integer),
fricas
)set output algebra on
fricas
)set output tex off
F1:=integrate(cos(t)*sqrt(cos(2*t)),t)
(3) +-+ 4 +-+ 2 +-+ +------------+ +-+ 32\|2 cos(t) - 48\|2 cos(t) + 17\|2 | 2 - \|2 atan(--------------------------------------) + 8sin(t)\|2cos(t) - 1 +------------+ 2 | 2 (32cos(t) - 24)sin(t)\|2cos(t) - 1 --------------------------------------------------------------------------- 16
Type: Union(Expression(Integer),
fricas
draw(F1,t=-%pi/4..%pi/4)
fricas
Compiling function %BD with type DoubleFloat -> DoubleFloat Graph data being transmitted to the viewport manager... FriCAS2D data being transmitted to the viewport manager...
(4) TwoDimensionalViewport: "FriCAS2D"
Type: TwoDimensionalViewport?
fricas
integrate(cos(t)*sqrt(cos(2*t)),t=-%pi/4..%pi/4)
(5) "potentialPole"
Type: Union(pole: potentialPole,
fricas
integrate(abs(x),x=0..1)
(6) "potentialPole"
Type: Union(pole: potentialPole,
fricas
integrate(abs(x),x=0..1, "noPole")
(7) "failed"
Type: Union(fail: failed,
fricas
integrate(abs(x),x=-1..1)
(8) "potentialPole"
Type: Union(pole: potentialPole,
fricas
integrate(abs(x),x=-1..1, "noPole")
(9) "failed"
Type: Union(fail: failed,
fricas
integrate(sqrt(x^2),x=0..1)
1 (10) - 2
Type: Union(f1: OrderedCompletion?(Expression(Integer)),
fricas
integrate(sqrt(x^2),x=0..1, "noPole")
1 (11) - 2
Type: Union(f1: OrderedCompletion?(Expression(Integer)),
fricas
integrate(sqrt(x^2),x=-1..1)
(12) "potentialPole"
Type: Union(pole: potentialPole,
fricas
integrate(sqrt(x^2),x=-1..1, "noPole")
(13) 0
Type: Union(f1: OrderedCompletion?(Expression(Integer)),
fricas
)version
Value = "FriCAS 1.3.1 compiled at Thu Feb 16 03:24:08 UTC 2017"
... --schweika, Mon, 31 Dec 2007 07:13:59 -0800 reply
fricas
solve(x*b -3*a*b + a*x - 9*b*b-4*b*x = a*a - 9*a*b ,x)
(14) [x = - 3b + a]
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
solve(-1*(a+3*b)^2 - 3*b*x -a*x = 0,x)
(15) [x = - 3b - a]
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
solve(-1*(a+3*b)^2 - 3*b*x +a*x = 0,x)
2 2 - 9b - 6a b - a (16) [x = -----------------] 3b - a
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
solve(-1*(a-3*b)^2 - 3*b*x +a*x = 0,x)
(17) [x = - 3b + a]
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
solve((a-3*b)*(x-a+3*b) = 0,x)
(18) [x = - 3b + a]
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
a : (INT,INT) := (2, 3)
The constructor Tuple takes 1 argument and you have given 2 .
fricas
(1,2)
(19) [1,2]
Type: Tuple(PositiveInteger?)
fricas
('Mon,
(20) [Mon,
Type: Tuple(OrderedVariableList?([Mon,
fricas
a := 'x :: OutputForm
(21) x
Type: OutputForm?
fricas
b := 'y :: OutputForm
(22) y
Type: OutputForm?
fricas
a and b
Argument number 1 to "and" must be a Boolean.
fricas
)clear all
All user variables and function definitions have been cleared.
fricas
y := operator 'y
(1) y
Type: BasicOperator?
fricas
deq := D(y(x),x, 2) + D(y(x), x) + y(x) + cos(y(x)) = 0
,, , (2) y (x) + y (x) + cos(y(x)) + y(x) = 0
Type: Equation(Expression(Integer))
fricas
solve(deq,y, x)
>> Error detected within library code: parseLODE: not a linear ordinary differential equation
fricas
y := operator 'y;
Type: BasicOperator?
fricas
deq := D(y(x),x, 2) + D(y(x), x) + y(x) + 1 = 0;
Type: Equation(Expression(Integer))
fricas
solve(deq,y, x);
Type: Union(Record(particular: Expression(Integer),
SandBoxMaybe?
fricas
factor(1-x^900)
(6) - 2 2 2 4 3 2 (x - 1)(x + 1)(x - x + 1)(x + 1)(x + x + 1)(x - x + x - x + 1) * 4 2 4 3 2 6 3 6 3 (x - x + 1)(x + x + x + x + 1)(x - x + 1)(x + x + 1) * 8 7 5 4 3 8 6 4 2 (x - x + x - x + x - x + 1)(x - x + x - x + 1) * 8 7 5 4 3 12 6 (x + x - x - x - x + x + 1)(x - x + 1) * 16 14 10 8 6 2 20 15 10 5 (x + x - x - x - x + x + 1)(x - x + x - x + 1) * 20 15 10 5 24 21 15 12 9 3 (x + x + x + x + 1)(x - x + x - x + x - x + 1) * 24 21 15 12 9 3 (x + x - x - x - x + x + 1) * 40 35 25 20 15 5 40 30 20 10 (x - x + x - x + x - x + 1)(x - x + x - x + 1) * 40 35 25 20 15 5 (x + x - x - x - x + x + 1) * 48 42 30 24 18 6 (x + x - x - x - x + x + 1) * 80 70 50 40 30 10 (x + x - x - x - x + x + 1) * 120 105 75 60 45 15 (x - x + x - x + x - x + 1) * 120 105 75 60 45 15 (x + x - x - x - x + x + 1) * 240 210 150 120 90 30 (x + x - x - x - x + x + 1)
Type: Factored(Polynomial(Integer))
fricas
simplify((a+b+2*sqrt(a)*sqrt(b))/(sqrt(a)+sqrt(b)))
+-+ +-+ 2\|a \|b + b + a (7) ----------------- +-+ +-+ \|b + \|a
Type: Expression(Integer)
fricas
simplify ( sin(a)^2+sin(%pi/2-a)^2 )
%pi - 2a 2 2 (8) - cos(--------) - cos(a) + 2 2
Type: Expression(Integer)
fricas
d:=sin(series(asin(x)))-x
21 (9) O(x )
fricas
1.2*d
21 (10) O(x )
fricas
solve( n*log(n)/log(m) = 1/4*n*log(n/4)/log(m) + 3/4*n*log(3/4*n)/log(m) + 15/4*n + 8,m)
3n n 4n log(n) - 3n log(--) - n log(-) 4 4 --------------------------------- 15n + 32 (11) [m = %e ]
Type: List(Equation(Expression(Integer)))
fricas
solve( log(n)/log(m) = 1/4*log(n/4)/log(m) + 3/4*log(3/4*n)/log(m) + 15/4*n + 8,m)
3n n 4log(n) - 3log(--) - log(-) 4 4 --------------------------- 15n + 32 (12) [m = %e ]
Type: List(Equation(Expression(Integer)))
fricas
solve( log(n)/log(m) = log(n/4)/log(m) + log(3/4*n)/log(m) + 15/4*n + 8,m)
3n n 4log(n) - 4log(--) - 4log(-) 4 4 ---------------------------- 15n + 32 (13) [m = %e ]
Type: List(Equation(Expression(Integer)))
fricas
integrate((P*cos(x))/(2*e)*cos(x)*sin(x),x=0..%pi, "noPole")
P (14) -- 3e
Type: Union(f1: OrderedCompletion?(Expression(Integer)),
Bessel Integral
fricas
integrate(exp(z*cos(t)),t=0..2*%pi)
(15) "failed"
Type: Union(fail: failed,
fricas
integrate(x/sqrt(x^4+10*x^2-96*x-71),x)
(16) log +--------------------+ 6 4 3 2 | 4 2 8 (x + 15x - 80x + 27x - 528x + 781)\|x + 10x - 96x - 71 + x + 6 5 4 3 2 20x - 128x + 54x - 1408x + 3124x + 10001 / 8
Type: Union(Expression(Integer),
fricas
integrate(1/x,x=1..2)
(17) log(2)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),
fricas
integrate(a/x^3,x)
a (18) - --- 2 2x
Type: Union(Expression(Integer),
fricas
solve(a^2+b^2=c^2,c)
2 2 2 (19) [c - b - a = 0]
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
integrate(1/sqrt(tan(x)),x)
(20) - +-+ 4\|2 * atan 1 ----------------------------------------------------------------- +------------------------------------------+ | +------+ | +-+ |sin(x) |- 2\|2 cos(x) |------ + 2sin(x) + 2cos(x) +------+ | \|cos(x) +-+ |sin(x) |------------------------------------------ + \|2 |------ - 1 \| cos(x) \|cos(x) + - +-+ 4\|2 * 1 atan(---------------------------------------------------------------) +----------------------------------------+ | +------+ | +-+ |sin(x) |2\|2 cos(x) |------ + 2sin(x) + 2cos(x) +------+ | \|cos(x) +-+ |sin(x) |---------------------------------------- + \|2 |------ + 1 \| cos(x) \|cos(x) + +------+ +-+ |sin(x) 2\|2 cos(x) |------ + 2sin(x) + 2cos(x) +-+ \|cos(x) \|2 log(----------------------------------------) cos(x) + +------+ +-+ |sin(x) - 2\|2 cos(x) |------ + 2sin(x) + 2cos(x) +-+ \|cos(x) - \|2 log(------------------------------------------) cos(x) / 4
Type: Union(Expression(Integer),
fricas
a :=simplify(integrate(1/sqrt(tan(x)),x))
(21) - +-+ 4\|2 * atan 1 ----------------------------------------------------------------- +------------------------------------------+ | +------+ | +-+ |sin(x) |- 2\|2 cos(x) |------ + 2sin(x) + 2cos(x) +------+ | \|cos(x) +-+ |sin(x) |------------------------------------------ + \|2 |------ - 1 \| cos(x) \|cos(x) + - +-+ 4\|2 * 1 atan(---------------------------------------------------------------) +----------------------------------------+ | +------+ | +-+ |sin(x) |2\|2 cos(x) |------ + 2sin(x) + 2cos(x) +------+ | \|cos(x) +-+ |sin(x) |---------------------------------------- + \|2 |------ + 1 \| cos(x) \|cos(x) + +------+ +-+ |sin(x) 2\|2 cos(x) |------ + 2sin(x) + 2cos(x) +-+ \|cos(x) \|2 log(----------------------------------------) cos(x) + +------+ +-+ |sin(x) - 2\|2 cos(x) |------ + 2sin(x) + 2cos(x) +-+ \|cos(x) - \|2 log(------------------------------------------) cos(x) / 4
Type: Expression(Integer)
fricas
b:=simplify(differentiate(a,x))
(22) +------+ 3 4 2 |sin(x) (48cos(x) sin(x) + 39cos(x) - 64cos(x) ) |------ \|cos(x) * +------------------------------------------+ | +------+ | +-+ |sin(x) |- 2\|2 cos(x) |------ + 2sin(x) + 2cos(x) | \|cos(x) |------------------------------------------ \| cos(x) + +------+ 3 4 2 |sin(x) (- 64cos(x) sin(x) - 29cos(x) + 64cos(x) ) |------ \|cos(x) + +-+ 3 +-+ +-+ 4 +-+ 2 (45\|2 cos(x) - 64\|2 cos(x))sin(x) - 32\|2 cos(x) + 32\|2 cos(x) * +----------------------------------------+ | +------+ | +-+ |sin(x) |2\|2 cos(x) |------ + 2sin(x) + 2cos(x) | \|cos(x) |---------------------------------------- \| cos(x) + +------+ 3 4 2 |sin(x) (64cos(x) sin(x) + 29cos(x) - 64cos(x) ) |------ \|cos(x) + +-+ 3 +-+ +-+ 4 +-+ 2 (45\|2 cos(x) - 64\|2 cos(x))sin(x) - 32\|2 cos(x) + 32\|2 cos(x) * +------------------------------------------+ | +------+ | +-+ |sin(x) |- 2\|2 cos(x) |------ + 2sin(x) + 2cos(x) | \|cos(x) |------------------------------------------ \| cos(x) + +------+ 3 4 2 |sin(x) ((14cos(x) - 128cos(x))sin(x) - 47cos(x) + 96cos(x) ) |------ \|cos(x) / 3 4 2 ((39cos(x) - 64cos(x))sin(x) - 48cos(x) + 48cos(x) ) * +------------------------------------------+ | +------+ | +-+ |sin(x) |- 2\|2 cos(x) |------ + 2sin(x) + 2cos(x) | \|cos(x) |------------------------------------------ \| cos(x) + +-+ 3 +-+ +-+ 4 (45\|2 cos(x) - 64\|2 cos(x))sin(x) - 32\|2 cos(x) + +-+ 2 32\|2 cos(x) * +------+ |sin(x) |------ \|cos(x) + 3 4 2 (- 29cos(x) + 64cos(x))sin(x) + 64cos(x) - 64cos(x) * +----------------------------------------+ | +------+ | +-+ |sin(x) |2\|2 cos(x) |------ + 2sin(x) + 2cos(x) | \|cos(x) |---------------------------------------- \| cos(x) + +-+ 3 +-+ +-+ 4 (45\|2 cos(x) - 64\|2 cos(x))sin(x) - 32\|2 cos(x) + +-+ 2 32\|2 cos(x) * +------+ |sin(x) |------ \|cos(x) + 3 4 2 (29cos(x) - 64cos(x))sin(x) - 64cos(x) + 64cos(x) * +------------------------------------------+ | +------+ | +-+ |sin(x) |- 2\|2 cos(x) |------ + 2sin(x) + 2cos(x) | \|cos(x) |------------------------------------------ \| cos(x) + 3 4 2 (- 47cos(x) + 96cos(x))sin(x) - 14cos(x) + 142cos(x) - 128
Type: Expression(Integer)
fricas
normalize(b - 1/sqrt(tan(x)))
(23) 0
Type: Expression(Integer)
fricas
integrate(1/(1+exp(b*x)),x=-m..%plusInfinity
Line 1: integrate(1/(1+exp(b*x)),x=-m..%plusInfinity .........A.....................B Error A: Missing mate. Error B: syntax error at top level Error B: Possibly missing a ) 3 error(s) parsing
fricas
integrate(1/(1+exp(b*x))),x=-m..%plusInfinity
There are 4 exposed and 2 unexposed library operations named integrate having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse,or issue )display op integrate to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named integrate with argument type(s) Expression(Integer)
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need.