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]:




$\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]:




$\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]




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