In [1]:

    

)version


    

    






 
Value = "FriCAS 1.2.3 compiled at Wed, May 07, 2014  3:39:04 PM"

In [2]:

    

x^n


    

    






 
         n
   (3)  x
                                                    Type: Expression(Integer)

In [3]:

    

)set output tex on


    

In [4]:

    

x^n


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{x} \sp {n} 
\leqno(4)
$$

In [8]:

    

integrate(1/(x * (a+b*x)^(1/3)),x)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{-{\log 
\left(
{{{{\root {3} \of {a}} \  {{{\root {3} \of {{{b \  x}+a}}}} \sp 
{2}}}+{{{{\root {3} \of {a}}} \sp {2}} \  {\root {3} \of {{{b \  x}+a}}}}+a}} 
\right)}+{2
\  {\log 
\left(
{{{{{{\root {3} \of {a}}} \sp {2}} \  {\root {3} \of {{{b \  x}+a}}}} -a}} 
\right)}}+{2
\  {\sqrt {3}} \  {\arctan 
\left(
{{{{2 \  {\sqrt {3}} \  {{{\root {3} \of {a}}} \sp {2}} \  {\root {3} \of 
{{{b \  x}+a}}}}+{a \  {\sqrt {3}}}} \over {3 \  a}}} 
\right)}}}
\over {2 \  {\root {3} \of {a}}} 
\leqno(7)
$$

In [9]:

    

series(log(cot(x)),x = %pi/2)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{\log 
\left(
{{{-{2 \  x}+\pi} \over 2}} 
\right)}+{{1
\over 3} \  {{{\left( x -{\pi \over 2} 
\right)}}
\sp {2}}}+{{7 \over {90}} \  {{{\left( x -{\pi \over 2} 
\right)}}
\sp {4}}}+{{{62} \over {2835}} \  {{{\left( x -{\pi \over 2} 
\right)}}
\sp {6}}}+{{{127} \over {18900}} \  {{{\left( x -{\pi \over 2} 
\right)}}
\sp {8}}}+{{{146} \over {66825}} \  {{{\left( x -{\pi \over 2} 
\right)}}
\sp {{10}}}}+{O 
\left(
{{{{\left( x -{\pi \over 2} 
\right)}}
\sp {{11}}}} 
\right)}
\leqno(8)
$$

In [10]:

    

M:=matrix [ [x + %i,0], [1,-2] ]


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
\left[
\begin{array}{cc}
{x+i} & 0 \\ 
1 & -2 
\end{array}
\right]
\leqno(9)
$$

In [11]:

    

inverse(M)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
\left[
\begin{array}{cc}
{1 \over {x+i}} & 0 \\ 
{1 \over {{2 \  x}+{2 \  i}}} & -{1 \over 2} 
\end{array}
\right]
\leqno(10)
$$

In [13]:

    

S := [3*x^3 + y + 1 = 0,y^2 = 4]


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
\left[
{{y+{3 \  {{x} \sp {3}}}+1}=0}, \: {{{y} \sp {2}}=4} 
\right]
\leqno(11)
$$

In [14]:

    

radicalSolve(S)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
\left[
{\left[ {y=2}, \: {x=-1} 
\right]},
\: {\left[ {y=2}, \: {x={{-{\sqrt {-3}}+1} \over 2}} 
\right]},
\: {\left[ {y=2}, \: {x={{{\sqrt {-3}}+1} \over 2}} 
\right]},
\: {\left[ {y=-2}, \: {x={1 \over {\root {3} \of {3}}}} 
\right]},
\: {\left[ {y=-2}, \: {x={{{{\sqrt {-1}} \  {\sqrt {3}}} -1} \over {2 \  
{\root {3} \of {3}}}}} 
\right]},
\: {\left[ {y=-2}, \: {x={{-{{\sqrt {-1}} \  {\sqrt {3}}} -1} \over {2 \  
{\root {3} \of {3}}}}} 
\right]}
\right]
\leqno(12)
$$

In [15]:

    

continuedFraction(6543/210)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{31}+ \zag{1}{6}+ \zag{1}{2}+ \zag{1}{1}+ \zag{1}{3} 
\leqno(13)
$$

In [17]:

    

(3*a^4 + 27*a - 36)::Polynomial PrimeField 7


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{3 \  {{a} \sp {4}}}+{6 \  a}+6 
\leqno(14)
$$

In [19]:

    

[i^2 for i in 1..10]


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
\left[
1, \: 4, \: 9, \: {16}, \: {25}, \: {36}, \: {49}, \: {64}, \: {81}, \: {100} 
\right]
\leqno(15)
$$

In [20]:

    

[i for i in 1..10 | even?(i)]


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
\left[
2, \: 4, \: 6, \: 8, \: {10} 
\right]
\leqno(16)
$$

In [21]:

    

[1..3,5,6,8..10]


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
\left[
{1..3}, \: {5..5}, \: {6..6}, \: {8..{10}} 
\right]
\leqno(17)
$$

In [22]:

    

factor 643238070748569023720594412551704344145570763243


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{{{11}} \sp {{13}}} \  {{{13}} \sp {{11}}} \  {{{17}} \sp {7}} \  {{{19}} \sp 
{5}} \  {{{23}} \sp {3}} \  {{{29}} \sp {2}} 
\leqno(18)
$$

In [23]:

    

roman(1992)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
MCMXCII 
\leqno(19)
$$

In [24]:

    

(2/3 + %i)^3


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
-{{46} \over {27}}+{{1 \over 3} \  i} 
\leqno(20)
$$

In [25]:

    

q:=quatern(1,2,3,4)*quatern(5,6,7,8) - quatern(5,6,7,8)*quatern(1,2,3,4)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
-{8 \  i}+{{16} \  j} -{8 \  k} 
\leqno(21)
$$

In [26]:

    

matrix([ [1/(i + j - x) for i in 1..4] for j in 1..4])


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
\left[
\begin{array}{cccc}
-{1 \over {x -2}} & -{1 \over {x -3}} & -{1 \over {x -4}} & -{1 \over {x -5}} \\ 
-{1 \over {x -3}} & -{1 \over {x -4}} & -{1 \over {x -5}} & -{1 \over {x -6}} \\ 
-{1 \over {x -4}} & -{1 \over {x -5}} & -{1 \over {x -6}} & -{1 \over {x -7}} \\ 
-{1 \over {x -5}} & -{1 \over {x -6}} & -{1 \over {x -7}} & -{1 \over {x -8}} 
\end{array}
\right]
\leqno(22)
$$

In [28]:

    

p: UP(x,INT) := (3*x-1)^2 * (2*x + 8)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{{18} \  {{x} \sp {3}}}+{{60} \  {{x} \sp {2}}} -{{46} \  x}+8 
\leqno(23)
$$

In [29]:

    

g := csc(a*x) / csch(b*x)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{\csc 
\left(
{{a \  x}} 
\right)}
\over {\csch 
\left(
{{b \  x}} 
\right)}
\leqno(24)
$$

In [30]:

    

limit(g,x=0)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
b \over a 
\leqno(25)
$$

In [32]:

    

h := (1 + k/x)^x


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{{{x+k} \over x}} \sp {x} 
\leqno(26)
$$

In [33]:

    

limit(h,x=%plusInfinity)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{e} \sp {k} 
\leqno(27)
$$

In [34]:

    

series(sin(a*x),x = 0)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{a \  x} -{{{{a} \sp {3}} \over 6} \  {{x} \sp {3}}}+{{{{a} \sp {5}} \over 
{120}} \  {{x} \sp {5}}} -{{{{a} \sp {7}} \over {5040}} \  {{x} \sp 
{7}}}+{{{{a} \sp {9}} \over {362880}} \  {{x} \sp {9}}} -{{{{a} \sp {{11}}} 
\over {39916800}} \  {{x} \sp {{11}}}}+{O 
\left(
{{{x} \sp {{12}}}} 
\right)}
\leqno(28)
$$

In [35]:

    

series(sin(a*x),x = %pi/4)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{\sin 
\left(
{{{a \  \pi} \over 4}} 
\right)}+{a
\  {\cos 
\left(
{{{a \  \pi} \over 4}} 
\right)}
\  {\left( x -{\pi \over 4} 
\right)}}
-{{{{{a} \sp {2}} \  {\sin 
\left(
{{{a \  \pi} \over 4}} 
\right)}}
\over 2} \  {{{\left( x -{\pi \over 4} 
\right)}}
\sp {2}}} -{{{{{a} \sp {3}} \  {\cos 
\left(
{{{a \  \pi} \over 4}} 
\right)}}
\over 6} \  {{{\left( x -{\pi \over 4} 
\right)}}
\sp {3}}}+{{{{{a} \sp {4}} \  {\sin 
\left(
{{{a \  \pi} \over 4}} 
\right)}}
\over {24}} \  {{{\left( x -{\pi \over 4} 
\right)}}
\sp {4}}}+{{{{{a} \sp {5}} \  {\cos 
\left(
{{{a \  \pi} \over 4}} 
\right)}}
\over {120}} \  {{{\left( x -{\pi \over 4} 
\right)}}
\sp {5}}} -{{{{{a} \sp {6}} \  {\sin 
\left(
{{{a \  \pi} \over 4}} 
\right)}}
\over {720}} \  {{{\left( x -{\pi \over 4} 
\right)}}
\sp {6}}} -{{{{{a} \sp {7}} \  {\cos 
\left(
{{{a \  \pi} \over 4}} 
\right)}}
\over {5040}} \  {{{\left( x -{\pi \over 4} 
\right)}}
\sp {7}}}+{{{{{a} \sp {8}} \  {\sin 
\left(
{{{a \  \pi} \over 4}} 
\right)}}
\over {40320}} \  {{{\left( x -{\pi \over 4} 
\right)}}
\sp {8}}}+{{{{{a} \sp {9}} \  {\cos 
\left(
{{{a \  \pi} \over 4}} 
\right)}}
\over {362880}} \  {{{\left( x -{\pi \over 4} 
\right)}}
\sp {9}}} -{{{{{a} \sp {{10}}} \  {\sin 
\left(
{{{a \  \pi} \over 4}} 
\right)}}
\over {3628800}} \  {{{\left( x -{\pi \over 4} 
\right)}}
\sp {{10}}}}+{O 
\left(
{{{{\left( x -{\pi \over 4} 
\right)}}
\sp {{11}}}} 
\right)}
\leqno(29)
$$

In [36]:

    

series(n +-> (-1)^((3*n - 4)/6)/factorial(n - 1/3),x=0,4/3..,2)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{{x} \sp {{4 \over 3}}} -{{1 \over 6} \  {{x} \sp {{{10} \over 3}}}}+{O 
\left(
{{{x} \sp {5}}} 
\right)}
\leqno(30)
$$

In [37]:

    

f := taylor(exp(x))


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
1+x+{{1 \over 2} \  {{x} \sp {2}}}+{{1 \over 6} \  {{x} \sp {3}}}+{{1 \over 
{24}} \  {{x} \sp {4}}}+{{1 \over {120}} \  {{x} \sp {5}}}+{{1 \over {720}} \  
{{x} \sp {6}}}+{{1 \over {5040}} \  {{x} \sp {7}}}+{{1 \over {40320}} \  {{x} 
\sp {8}}}+{{1 \over {362880}} \  {{x} \sp {9}}}+{{1 \over {3628800}} \  {{x} 
\sp {{10}}}}+{O 
\left(
{{{x} \sp {{11}}}} 
\right)}
\leqno(31)
$$

In [38]:

    

F := operator 'F; x := operator 'x; y := operator 'y


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
y 
\leqno(32)
$$

In [40]:

    

a := F(x z, y z, z^2) + x y(z+1)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{x 
\left(
{{y 
\left(
{{z+1}} 
\right)}}
\right)}+{F
\left(
{{x 
\left(
{z} 
\right)},
\: {y 
\left(
{z} 
\right)},
\: {{z} \sp {2}}} 
\right)}
\leqno(33)
$$

In [41]:

    

dadz := D(a, z)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{2 \  z \  {{F \sb {{,3}}} 
\left(
{{x 
\left(
{z} 
\right)},
\: {y 
\left(
{z} 
\right)},
\: {{z} \sp {2}}} 
\right)}}+{{{y
\sb {{\ }} \sp {,}} 
\left(
{z} 
\right)}
\  {{F \sb {{,2}}} 
\left(
{{x 
\left(
{z} 
\right)},
\: {y 
\left(
{z} 
\right)},
\: {{z} \sp {2}}} 
\right)}}+{{{x
\sb {{\ }} \sp {,}} 
\left(
{z} 
\right)}
\  {{F \sb {{,1}}} 
\left(
{{x 
\left(
{z} 
\right)},
\: {y 
\left(
{z} 
\right)},
\: {{z} \sp {2}}} 
\right)}}+{{{x
\sb {{\ }} \sp {,}} 
\left(
{{y 
\left(
{{z+1}} 
\right)}}
\right)}
\  {{y \sb {{\ }} \sp {,}} 
\left(
{{z+1}} 
\right)}}
\leqno(34)
$$

In [42]:

    

eval(eval(dadz, 'x, z +-> exp z), 'y, z +-> log(z+1))


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{{{\left( {2 \  {{z} \sp {2}}}+{2 \  z} 
\right)}
\  {{F \sb {{,3}}} 
\left(
{{{e} \sp {z}}, \: {\log 
\left(
{{z+1}} 
\right)},
\: {{z} \sp {2}}} 
\right)}}+{{F
\sb {{,2}}} 
\left(
{{{e} \sp {z}}, \: {\log 
\left(
{{z+1}} 
\right)},
\: {{z} \sp {2}}} 
\right)}+{{\left(
z+1 
\right)}
\  {{e} \sp {z}} \  {{F \sb {{,1}}} 
\left(
{{{e} \sp {z}}, \: {\log 
\left(
{{z+1}} 
\right)},
\: {{z} \sp {2}}} 
\right)}}+z+1}
\over {z+1} 
\leqno(35)
$$

In [43]:

    

eval(eval(a, 'x, z +-> exp z), 'y, z +-> log(z+1))


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{F 
\left(
{{{e} \sp {z}}, \: {\log 
\left(
{{z+1}} 
\right)},
\: {{z} \sp {2}}} 
\right)}+z+2
\leqno(36)
$$

In [44]:

    

D(%, z)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
{{{\left( {2 \  {{z} \sp {2}}}+{2 \  z} 
\right)}
\  {{F \sb {{,3}}} 
\left(
{{{e} \sp {z}}, \: {\log 
\left(
{{z+1}} 
\right)},
\: {{z} \sp {2}}} 
\right)}}+{{F
\sb {{,2}}} 
\left(
{{{e} \sp {z}}, \: {\log 
\left(
{{z+1}} 
\right)},
\: {{z} \sp {2}}} 
\right)}+{{\left(
z+1 
\right)}
\  {{e} \sp {z}} \  {{F \sb {{,1}}} 
\left(
{{{e} \sp {z}}, \: {\log 
\left(
{{z+1}} 
\right)},
\: {{z} \sp {2}}} 
\right)}}+z+1}
\over {z+1} 
\leqno(37)
$$

In [50]:

    

integrate(1/(u^2 + a),u)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
\left[
{{\log 
\left(
{{{{{\left( -{x 
\left(
{{y 
\left(
{{z+1}} 
\right)}}
\right)}
-{F 
\left(
{{x 
\left(
{z} 
\right)},
\: {y 
\left(
{z} 
\right)},
\: {{z} \sp {2}}} 
\right)}+{{u}
\sp {2}} 
\right)}
\  {\sqrt {{-{x 
\left(
{{y 
\left(
{{z+1}} 
\right)}}
\right)}
-{F 
\left(
{{x 
\left(
{z} 
\right)},
\: {y 
\left(
{z} 
\right)},
\: {{z} \sp {2}}} 
\right)}}}}}+{2
\  u \  {x 
\left(
{{y 
\left(
{{z+1}} 
\right)}}
\right)}}+{2
\  u \  {F 
\left(
{{x 
\left(
{z} 
\right)},
\: {y 
\left(
{z} 
\right)},
\: {{z} \sp {2}}} 
\right)}}}
\over {{x 
\left(
{{y 
\left(
{{z+1}} 
\right)}}
\right)}+{F
\left(
{{x 
\left(
{z} 
\right)},
\: {y 
\left(
{z} 
\right)},
\: {{z} \sp {2}}} 
\right)}+{{u}
\sp {2}}}}} 
\right)}
\over {2 \  {\sqrt {{-{x 
\left(
{{y 
\left(
{{z+1}} 
\right)}}
\right)}
-{F 
\left(
{{x 
\left(
{z} 
\right)},
\: {y 
\left(
{z} 
\right)},
\: {{z} \sp {2}}} 
\right)}}}}}},
\: {{\arctan 
\left(
{{{u \  {\sqrt {{{x 
\left(
{{y 
\left(
{{z+1}} 
\right)}}
\right)}+{F
\left(
{{x 
\left(
{z} 
\right)},
\: {y 
\left(
{z} 
\right)},
\: {{z} \sp {2}}} 
\right)}}}}}
\over {{x 
\left(
{{y 
\left(
{{z+1}} 
\right)}}
\right)}+{F
\left(
{{x 
\left(
{z} 
\right)},
\: {y 
\left(
{z} 
\right)},
\: {{z} \sp {2}}} 
\right)}}}}
\right)}
\over {\sqrt {{{x 
\left(
{{y 
\left(
{{z+1}} 
\right)}}
\right)}+{F
\left(
{{x 
\left(
{z} 
\right)},
\: {y 
\left(
{z} 
\right)},
\: {{z} \sp {2}}} 
\right)}}}}}
\right]
\leqno(38)
$$

In [52]:

    

integrate(log(1 + sqrt(a*u + b)) / u,u)


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
\int \sp{\displaystyle u} {{{\log 
\left(
{{{\sqrt {{{ \%A \  {x 
\left(
{{y 
\left(
{{z+1}} 
\right)}}
\right)}}+{
 \%A \  {F 
\left(
{{x 
\left(
{z} 
\right)},
\: {y 
\left(
{z} 
\right)},
\: {{z} \sp {2}}} 
\right)}}+b}}}+1}}
\right)}
\over \%A} \  {d \%A}} 
\leqno(39)
$$

In [53]:

    

y := operator 'y


    

    






$\def\sp{^}\def\sb{_}\def\leqno(#1){}$$$
y 
\leqno(40)
$$

In [ ]:

    

)quit


    

    




-- Bye. Kernel shutdown [lib.site-packages.ipython-3**.egg]

In [ ]: