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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}
(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)

\label{eq2}-{\log \left({{\sqrt{{{x}^{2}}+ 1}}- x}\right)}(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"

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))))

Coherent type inference --gdr, Thu, 21 Feb 2008 20:30:01 -0800 reply
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,'Tue)
(20) [Mon,Tue]
Type: Tuple(OrderedVariableList([Mon,Tue]))

fricas
a := 'x :: OutputForm
(21) x
fricas
b := 'y :: OutputForm
(22) y
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
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;
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),basis: List(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 )
Type: UnivariatePuiseuxSeries?(Expression(Integer),x,0)
fricas
1.2*d
21 (10) O(x )
Type: UnivariatePuiseuxSeries?(Expression(Float),x,0.0)

quicksort average --grinder, Wed, 26 Nov 2008 13:29:45 -0800 reply
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)))

simple integration --kylekanos, Tue, 09 Dec 2008 09:55:35 -0800 reply
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))))

A hard to compute integral --PAP, Tue, 28 Sep 2010 17:55:14 -0700 reply
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)




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