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available_examples.txt
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/*
This file is to be run by the EXAMPLE command, and may not
otherwise work.
The following are either acceptable lines to Maxima, or they are
two successive '&' characters
and then followed by the name of the section of examples, and then followed by
a sequence of Maxima forms, e.g.
'&'& topic (without the quotes)
/* This is a comment */
<Maxima form 1>;
<Maxima form 2>;
...
*/
&& additive
declare(f,additive);
f(2*a+3*b);
&& algsys
f1:2*x*(1-l1)-2*(x-1)*l2$
f2:l2-l1$
f3:l1*(1-x**2-y)$
f4:l2*(y-(x-1)**2)$
algsys([f1,f2,f3,f4],[x,y,l1,l2]);
f1:x**2-y**2$
f2:x**2-x+2*y**2-y-1$
algsys([f1,f2],[x,y]);
&& allroots
(2*x+1)^3=13.5*(x^5+1);
allroots(%);
&& antisymmetric
declare(h,antisymmetric);
h(x,z,y);
&& append
append([y+x,0,-3.2],[2.5e20,x]);
&& arrayinfo
b[1,x]:1$
array(f,2,3);
arrayinfo(b);
arrayinfo(f);
&& arrays
a[n]:=n*a[n-1];
a[0]:1$
a[5];
a[n]:=n$
a[6];
a[4];
&& at
atvalue(f(x,y),[x=0,y=1],a^2);
atvalue('diff(f(x,y),x),x=0,y+1);
printprops(all,atvalue);
diff(4*f(x,y)^2-u(x,y)^2,x);
at(%,[x=0,y=1]);
&& atvalue
kill(f,x,a,u,g);
atvalue(f(x,y),[x=0,y=1],a^2)$
atvalue('diff(f(x,y),x),x=0,y+1);
printprops(all,atvalue);
diff(4*f(x,y)^2-u(x,y)^2,x);
at(%,[x=0,y=1]);
&& augcoefmatrix
[2*x-(a-1)*y=5*b,a*x+b*y+c=0]$
augcoefmatrix(%,[x,y]);
&& bezout
bezout(a*y+x^2+1,y^2+x*y+b,x);
expand(determinant(%));
%-expand(resultant(a*y+x^2+1,y^2+x*y+b,x));
&& block
kill(f);
hessian(f):=block([dfxx,dfxy,dfxz,dfyy,dfyz,dfzz],
dfxx:diff(f,x,2),dfxy:diff(f,x,1,y,1),
dfxz:diff(f,x,1,z,1),dfyy:diff(f,y,2),
dfyz:diff(f,y,1,z,1),dfzz:diff(f,z,2),
determinant(matrix([dfxx,dfxy,dfxz],[dfxy,dfyy,dfyz],
[dfxz,dfyz,dfzz])))$
kill(x,y,z);
hessian(x^3-3*a*x*y*z+y^3);
subst(1,z,quotient(%,-54*a^2));
f(x):=block([y,use_fast_arrays:false], local(a), y:4, a[y]:x, display(a[y]))$
y:2$
a[y+2]:0$
f(9);
ff(x):=block([y,a,use_fast_arrays:true], y:4, a[y]:x, display(a[y]))$
ff(10);
a[y+2];
&& bothcoeff
islinear(exp,var):=block([c],
c:bothcoef(rat(exp,var),var),
is(freeof(var,c) and c[1]#0))$
islinear((r^2-(x-r)^2)/x,x);
&& catch
g(l):=catch(map(lambda([x],if x<0 then throw(x) else f(x)),l))$
g([1,2,3,7]);
g([1,2,-3,7]);
&& cf
cf([1,2,-3]+[1,-2,1]);
cfdisrep(%);
cflength:4$
cf(sqrt(3));
cfexpand(%);
ev(%[1,2]/%[2,2],numer);
&& cfdisrep
cf([1,2,-3]+[1,-2,1]);
cfdisrep(%);
&& cfexpand
cflength:4$
cf(sqrt(3));
cfexpand(%);
ev(%[1,2]/%[2,2],numer);
&& changevar
'integrate(%e^(sqrt(a)*sqrt(y)),y,0,4);
changevar(%,y-z^2/a,z,y);
sum(a[k]*x^(N-1-k), k, 0, N - 1);
changevar(%, m = N - 1 - k, m, k);
&& charpoly
a:matrix([3,1],[2,4]);
expand(charpoly(a,lambda));
(programmode:true,solve(%));
matrix([x1],[x2]);
ev(a.%-lambda*%,%th(2)[1]);
%[1,1]=0;
x1^2+x2^2=1;
solve([%th(2),%],[x1,x2]);
&& coeff
coeff(2*a*tan(x)+tan(x)+b=5*tan(x)+3,tan(x));
coeff(y+x*%e^x+1,x,0);
&& combine
combine(a/x+b/x+a/y+b/y);
&& commutative
declare(h,commutative);
h(x,z,y);
&& complex
(sqrt(-4)+sqrt(2.25))^2;
expand(%);
expand(sqrt(2*%i));
&& content
content(2*x*y+4*x^2*y^2,y);
&& defmatch
nonzeroandfreeof(x,e):=is(e#0 and freeof(x,e));
matchdeclare(a,nonzeroandfreeof(x),b,freeof(x));
defmatch(linear,a*x+b,x);
linear(3*z+(y+1)*z+y**2,z);
matchdeclare([a,f],true);
constinterval(l,h):=constantp(h-l)$
matchdeclare(b,constinterval(a))$
matchdeclare(x,atom)$
block(remove(integrate,outative),
defmatch(checklimits,'integrate(f,x,a,b)),
declare(integrate,outative))$
'integrate(sin(t),t,x+%pi,x+2*%pi)$
checklimits(%);
'integrate(sin(t),t,0,x)$
checklimits(%);
remvalue(a,b,f,x)$
&& deftaylor
deftaylor(f(x),x^2+sum(x^i/(2^i*i!^2),i,4,inf));
taylor(%e^sqrt(f(x)),x,0,4);
&& delete
delete(sin(x),x+sin(x)+y);
&& depends
kill(a,x,f,y,t);
depends(a,x);
diff(a.a,x);
depends(f,[x,y],[x,y],t);
diff(f,t);
&& derivdegree
'diff(y,x,2)+'diff(y,z,3)*2+'diff(y,x)*x^2;
derivdegree(%,y,x);
&& desolve
eqn1:'diff(f(x),x)='diff(g(x),x)+sin(x);
eqn2:'diff(g(x),x,2)='diff(f(x),x)-cos(x);
atvalue('diff(g(x),x),x=0,a);
atvalue(f(x),x=0,1);
desolve([eqn1,eqn2],[f(x),g(x)]);
/* verification */
[eqn1,eqn2],%,diff;
&& diff
kill(f,g,h,x,y);
diff(sin(x)+x^3+2*x^2,x);
diff(sin(x)*cos(x),x);
diff(sin(x)*cos(x),x,2);
derivabbrev:true$
diff(exp(f(x)),x,2);
'integrate(f(x,y),y,g(x),h(x));
diff(%,x);
&& display
display(b[1,2]);
&& divide
divide(x+y,x-y,x);
divide(x+y,x-y);
&& do
for a:-3 thru 26 step 7 do ldisplay(a)$
s:0$
for i:1 while i<=10 do s:s+i;
s;
series:1$
term:exp(sin(x))$
for p:1 unless p>7 do
(term:diff(term,x)/p,
series:series+subst(x=0,term)*x^p)$
series;
poly:0$
for i:1 thru 5 do
for j:i step -1 thru 1 do
poly:poly+i*x^j$
poly;
guess:-3.0$
for i thru 10 do (guess:subst(guess,x,0.5*(x+10/x)),
if abs(guess^2-10)<0.00005 then return(guess));
for count:2 next 3*count thru 20
do ldisplay(count)$
x:1000;
thru 10 while x#0 do x:0.5*(x+5/x)$
x;
remvalue(x);
newton(f,guess):=block([numer,y],local(f,df,x,guess),
numer:true,
define(df(x),diff(f(x),x)),
do (y:df(guess), if y=0 then error(
"derivative at",guess,"is zero"),
guess:guess-f(guess)/y,
if abs(f(guess))<5.0e-6 then return(guess)))$
sqr(x):=x^2-5.0$
newton(sqr,1000);
for f in [log, rho, atan] do ldisp(f(1.0))$
ev(concat(e,linenum-1),numer);
&& dotscrules
declare(l,scalar,[m1,m2,m3],nonscalar);
expand((1-l*m1).(1-l*m2).(1-l*m3));
%,dotscrules;
rat(%,l);
&& dpart
dpart(x+y/z^2,1,2,1);
expand((b+a)^4);
(b+a)^2*(y+x)^2;
expand(%);
%th(3)/%;
factor(%);
dpart(%th(2),2,4);
part(%th(3),2,4);
&& echelon
matrix([2,1-a,-5*b],[a,b,c]);
echelon(%);
&& eliminate
exp1:2*x^2+y*x+z;
exp2:3*x+5*y-z-1;
exp3:z^2+x-y^2+5;
eliminate([exp3,exp2,exp1],[y,z]);
&& entermatrix
entermatrix(2,1);
&& equations
x+1=y^2;
x-1=2*y+1$
%th(2)+%;
%th(3)/y;
1/%;
&& ev
kill(y,x,w);
sin(x)+cos(y)+(w+1)^2+'diff(sin(w),w);
ev(%,sin,expand,diff,x=2,y=1);
ev(x+y,x:a+y,y:2);
'diff(y^2+x*y+x^2,x,2,y,1);
ev(%,diff);
2*x-3*y=3$
-3*x+2*y=-4$
solve([%th(2),%]);
ev(%th(3),%);
x+1/x>gamma(1/2);
ev(%,numer,x=1/2);
ev(%,pred);
&& evaluation
diff(x*f(x),x);
f(x):=sin(x)$
ev(%th(2),diff);
x;
x:3$
x;
'x;
f(x):=x^2;
'f(2);
ev(%,f);
'(f(2));
''%;
sum(i!,i,1,4);
'sum(i!,i,1,4);
remvalue(x);
'integrate(f(x),x,a,b);
for i thru 5 do s:s+i^2;
s;
ev(%,s:0);
ev(%th(2));
'sum(g(i),i,0,n);
z*%e^z;
ev(%,z:x^2);
subst(x^2,z,%th(3));
a:%;
a+1;
kill(a,y);
a;
/* declare(integrate,noun)$ */
integrate(y^2,y);
''integrate(y^2,y);
f(y):=diff(y*log(y),y,2);
f(y):=''(diff(y*log(y),y,2));
''(concat(c,linenum-1));
(x+y)^3$
diff(%,x);
y:x^2+1$
''(concat(c,linenum-2));
&& evenfun
declare(g,evenfun);
g(-x);
&& exp
ev(%e^x*sin(x)^2,exponentialize);
kill(x);
integrate(%,x);
ev(%,demoivre);
ans:ev(%,ratexpand);
ev(%,x:1,numer)-ev(%,x:0,numer);
integrate(%e^x*sin(x)^2,x);
trigreduce(%);
%-ans;
ev(sin(x),%emode);
&& expand
(1/(x+y)^4-3/(y+z)^3)^2;
expand(%,2,0);
expand(a.(b+c.(d+e)+f));
expand((x+1)^3);
(x+1)^7;
expand(%);
expand(%th(2),7,7);
ev(a*(b+c)+a*(b+c)^2,expop:1);
&& factcomb
(n+1)^2*n!^2;
factcomb(%);
&& factor
factor(2^63-1);
factor(z^2*(x+2*y)-4*x-8*y);
x^2*y^2+2*x*y^2+y^2-x^2-2*x-1;
block([dontfactor:[x]],factor(%/36/(y^2+2*y+1)));
factor(%e^(3*x)+1);
factor(x^4+1,a^2-2);
factor(x^3+x^2*y^2-x*z^2-y^2*z^2);
(x+2)/(x+3)/(x+b)/(x+c)^2;
ratsimp(%);
partfrac(%,x);
map('factor,%);
ratsimp((x^5-1)/(x-1));
subst(a,x,%);
factor(%th(2),%);
factor(x^12+1);
factor(x^99+1);
&& factorsum
ev((x+1)*((u+v)^2+a*(w+z)^2),expand);
factorsum(%);
&& featurep
declare(j,even)$
featurep(j,integer);
&& freeof
freeof(y,sin(x+2*y));
freeof(cos(y),"*",sin(y)+cos(x));
&& fullmap
fullmap(g,a+b*c);
map(g,a+b*c);
&& fullmapl
fullmapl("+",[3,[4,5]],[[a,1],[0,-1.5]]);
&& funcsolve
funcsolve((n+1)*f(n)-(n+3)*f(n+1)/(n+1)=(n-1)/(n+2),f(n));
&& functions
kill(x,y,f,g,h);
f(x):=x^2+y;
f(2);
ev(f(2),y:7);
f(x):=sin(x)^2+1;
f(x+1);
g(y,z):=f(z)+3*y;
ev(g(2*y+z,-0.5),y:7);
h(n):=sum(i*x^i,i,0,n);
functions;
t[n](x):=ratexpand(2*x*t[n-1](x)-t[n-2](x));
t[0](x):=1$
t[1](x):=x$
t[4](y);
g[n](x):=sum(ev(x),i,n,n+2);
h(n,x):=sum(ev(x),i,n,n+2);
g[2](i^2);
h(2,i^2);
p[n](x):=ratsimp(1/(2^n*n!)*diff((x^2-1)^n,x,n));
q(n,x):=ratsimp(1/(2^n*n!)*diff((x^2-1)^n,x,n));
p[2];
p[2](y+1);
q(2,y);
p[2](5);
f[i,j](x,y):=x^i+y^j;
g(fun,a,b):=print(fun," applied to ",a," and ",b," is ",fun(a,b))$
g(f[2,1],sin(%pi),2*c);
&& genmatrix
h[i,j]:=1/(i+j-1)$
genmatrix(h,3,3);
&& get
put(%e,transcendental,type);
put(%pi,transcendental,type)$
put(%i,algebraic,type)$
typeof(x):=block([q], if numberp(x)
then return(algebraic),
if not atom(x)
then return(maplist(typeof,x)),
q:get(x,type), if q=false then
error("not numeric") else q)$
errcatch(typeof(2*%e+x*%pi));
typeof(2*%e+%pi);
&& gfactor
gfactor(x^4-1);
&& gradef
depends(y,x);
kill(f,g,j);
gradef(f(x,y),x^2,g(x,y));
diff(f(x,y),x);
gradef(j(n,z),'diff(j(n,z),n),
j(n-1,z)-n/z*j(n,z))$
ratsimp(diff(j(2,x),x,2));
&& horner
poly:1.0e-20*x^2-5.5*x+5.2e20;
errcatch(ev(%,x=1.0e20));
horner(poly,x),keepfloat;
ev(%,x=1.0e20);
&& if
fib[n]:=if n=1 or n=2 then 1 else fib[n-1]+fib[n-2];
fib[1]+fib[2];
fib[3];
fib[5];
eta(mu,nu):=if mu=nu then mu else if mu>nu then mu-nu else mu+nu;
eta(5,6);
eta(eta(7,7),eta(1,2));
if not 5>=2 and 6<=5 or 4+1>3 then a else b;
&& ilt
'integrate(sinh(a*x)*f(t-x),x,0,t)+b*f(t)=t^2;
laplace(%,t,s);
linsolve([%],['laplace(f(t),t,s)]);
ilt(ev(%[1]),s,t);
&& inpart
x+y+w*z;
inpart(%,3,2);
'limit(f(x)^g(x+1),x,0,minus);
inpart(%,1,2);
&& integrate
test(f):=block([u],u:integrate(f,x),ratsimp(f-diff(u,x)));
test(sin(x));
test(1/(1+x));
test(1/(1+x^2));
integrate(sin(x)^3,x);
kill(q)$
integrate(%e^x/(%e^x+2),x);
integrate(1/(x*log(x)),x);
integrate(sin(2*x+3),x);
integrate(%e^x*erf(x),x);
integrate(x/(x^3+1),x);
diff(%,x);
ratsimp(%);
integrate(x^(5/4)/(x+1)^(5/2),x,0,inf);
gradef(q(x),sin(x^2));
diff(log(q(r(x))),x);
integrate(%,x);
&& is
is(x^2>=2*x-1);
assume(a>1);
is(log(log(a+1)+1)>0 and a^2+1>2*a);
&& isolate
(a+b)^4*(1+x*(2*x+(c+d)^2));
isolate(%,x);
ratexpand(%)$
ev(%);
(a+b)*(x+a+b)^2*%e^(x^2+a*x+b);
isolate(%,x),exptisolate:true;
&& lambda
lambda([x,y,z],x^2+y^2+z^2);
%(1,2,a);
"+"(1,2,a);
&& laplace
laplace(%e^(2*t+a)*sin(t)*t,t,s);
&& lassociative
declare(g,lassociative);
g(g(a,b),g(c,d));
g(g(a,b),g(c,d))-g(a,g(b,g(c,d)));
&& let
matchdeclare([a,a1,a2],true);
oneless(x,y):=is(x=y-1)$
let(a1*a2!,a1!,oneless,a2,a1);
let(a1!/a1,(a1-1)!),letrat;
letsimp(n*m!*(n-1)!/m),letrat;
let(sin(a)^2,1-cos(a)^2);
sin(x)^4;
letsimp(%);
&& letrules
matchdeclare([a,a1,a2],true);
oneless(x,y):=is(x=y-1)$
let(a1*a2!,a1!,oneless,a2,a1);
let(a1!/a1,(a1-1)!),letrat;
letsimp(n*m!*(n-1)!/m),letrat;
let(sin(a)^2,1-cos(a)^2);
sin(x)^4;
letsimp(%);
&& limit
limit(x*log(x),x,0,plus);
limit((1+x)^(1/x),x,0);
limit(%e^x/x,x,inf);
limit(sin(1/x),x,0);
&& linear
declare(f,linear);
f(2*a+3*b);
f(2*x+y,x);
&& linsolve
x+z=y$
2*a*x-y=2*a^2$
y-2*z=2$
linsolve([%th(3),%th(2),%],[x,y,z]),globalsolve;
&& listofvars
listofvars(f(x[1]+y)/g^(2+a));
&& lists
[x^2,y/3,-2];
%[1]*x;
[a,%th(2),%];
&& logcontract
2*(a*log(x) + 2*a*log(y));
logcontract(%);
logcontract(log(sqrt(x+1)+sqrt(x)) + log(sqrt(x+1)-sqrt(x)));
&& map
map(f,x+a*y+b*z);
map(lambda([u],partfrac(u,x)),x+1/(x^3+4*x^2+5*x+2));
map(ratsimp, x/(x^2+x)+(y^2+y)/y);
map("=",[a,b],[-0.5,3]);
&& matchdeclare
matchdeclare(a,true)$
tellsimp(sin(a)^2,1-cos(a)^2)$
sin(y)^2;
kill(rules);
nonzeroandfreeof(x,e):=is(e#0 and freeof(x,e));
matchdeclare(a,nonzeroandfreeof(x),b,freeof(x));
defmatch(linear,a*x+b,x);
linear(3*z+(y+1)*z+y**2,z);
matchdeclare([a,f],true);
constinterval(l,h):=constantp(h-l)$
matchdeclare(b,constinterval(a))$
matchdeclare(x,atom)$
block(remove(integrate,outative),
defmatch(checklimits,'integrate(f,x,a,b)),
declare(integrate,outative))$
'integrate(sin(t),t,x+%pi,x+2*%pi)$
checklimits(%);
'integrate(sin(t),t,0,x)$
checklimits(%);
&& matrices
m:matrix([a,0],[b,1]);
m^2;
m.m;
m[1,1]*m;
%-%th(2)+1;
m^^-1;
[x,y].m;
matrix([a,b,c],[d,e,f],[g,h,i]);
%^^2;
&& minfactorial
n!/(n+1)!;
minfactorial(%);
&& multiplicative
declare(f,multiplicative);
f(2*a*b);
&& multthru
x/(x-y)^2-1/(x-y)-f(x)/(x-y)^3;
multthru((x-y)^3,%);
ratexpand(%);
((a+b)^10*s^2+2*a*b*s+(a*b)^2)/(a*b*s^2);
multthru(%);
multthru(a.(b+c.(d+e)+f));
&& nary
declare(j,nary);
j(j(a,b),j(c,d));
&& nounify
'limit(f(x)^g(x+1),x,0,minus);
is(inpart(%,0)=nounify(limit));
&& nroots
x^10-2*x^4+1/2;
nroots(%,-6,9.1);
&& numfactor
gamma(7/2);
numfactor(%);
&& nusum
nusum(n*n!,n,0,n);
nusum(n^4*4^n/binomial(2*n,n),n,0,n);
unsum(%,n);
unsum(prod(i^2,i,1,n),n);
nusum(%,n,1,n);
&& oddfun
declare(f,oddfun);
f(-x);
&& ode2
x^2*'diff(y,x) + 3*x*y = sin(x)/x;
soln1:ode2(%,y,x);
ic1(soln1,x=%pi,y=0);
'diff(y,x,2) + y*'diff(y,x)^3 = 0;
soln2:ode2(%,y,x);
ratsimp(ic2(soln2,x=0,y=0,'diff(y,x)=2));
bc2(soln2,x=0,y=1,x=1,y=3);
&& optimize
diff(exp(x^2+y)/(x+y),x,2);
optimize(%);
&& ordergreat
a^2+b*x;
ordergreat(a);
a^2+b*x;
%-%th(3);
unorder();
&& orderless
y^2+b*x;
orderless(y);
y^2+b*x;
%-%th(3);
unorder();
&& outative
declare(f,outative);
f(2*a);
&& part
x+y/z^2;
part(%,1,2,2);
remvalue(x);
'integrate(f(x),x,a,b)+x;
part(%,1,1);
x^2+2*x=y^2;
%+1;
lhs(%);
part(%th(2),2);
part(%,1);
27*y^3+54*x*y^2+36*x^2*y+y+8*x^3+x+1;
part(%,2,[1,3]);
sqrt(piece/54);
&& partfrac
2/(x+2)-2/(x+1)+1/(x+1)^2;
ratsimp(%);
partfrac(%,x);
&& partition
partition(2*a*x*f(x),x);
partition(a+b,x);
&& pickapart
integrate(1/(x^3+2),x)$
pickapart(%,1);
&& poissimp
pfeformat:true$
poissimp(sin(x)^2);
(2*a^2-b)*cos(x+2*y)-(a*b+5)*sin(u-4*x);
poisexpt(%,2)$
printpois(%);
poisint(%th(2),y)$
poissimp(%);
poissimp(sin(x)^5+cos(x)^5);
pfeformat:false$
&& polarform
rectform(sin(2*%i+x));
polarform(%);
rectform(log(3+4*%i));
polarform(%);
rectform((2+3.5*%i)^0.25),numer;
polarform(%);
&& poly_discriminant
factor(poly_discriminant((x-a)*(x-b)*(x-c),x));
&& posfun
declare(f,posfun);
is(f(x)>0);
&& powerseries
powerseries(log(sin(x)/x),x,0);
&& printprops
gradef(r,x,x/r)$
gradef(r,y,y/r)$
printprops(r,atomgrad);
propvars(atomgrad);
&& product
product(x+i*(i+1)/2,i,1,4);
&& properties
properties(cons);
assume(var1>0);
properties(var1);
var2:2$
properties(var2);
&& propvars
gradef(r,x,x/r)$
gradef(r,y,y/r)$
printprops(r,atomgrad);
propvars(atomgrad);
&& qunit
qunit(17);
expand(%*(sqrt(17)-4));
&& radcan
(log(x^2+x)-log(x))^a/log(x+1)^(a/2);
radcan(%);
log(a^(2*x)+2*a^x+1)/log(a^x+1);
radcan(%);
(%e^x-1)/(%e^(x/2)+1);
radcan(%);
&& rank
matrix([2,1-a,-5*b],[a,b,c]);
rank(%);
&& rassociative
declare(g,rassociative);
g(g(a,b),g(c,d));
g(g(a,b),g(c,d))-g(a,g(b,g(c,d)));
&& rat
rat(x^2);
diff(f(%),x);
((x-2*y)^4/(x^2-4*y^2)^2+1)*(y+a)*(2*y+x)/(4*y^2+x^2);
rat(%,y,a,x);
(x+3)^20;
rat(%);
diff(%,x);
factor(%);
&& ratcoeff
a*x+b*x+5$
ratcoef(%,a+b);
&& ratdiff
(4*x^3+10*x-11)/(x^5+5);
polymod(%),modulus:3;
ratdiff(%th(2),x);
&& ratexpand
ratexpand((2*x-3*y)^3);
(x-1)/(x+1)^2+1/(x-1);
expand(%);
ratexpand(%th(2));
&& ratsimp
sin(x/(x^2+x))=%e^((log(x)+1)^2-log(x)^2);
ratsimp(%);
b*(a/b-x)+b*x+a;
ratsimp(%);
((x-1)^(3/2)-(x+1)*sqrt(x-1))/sqrt(x-1)/sqrt(x+1);
ratsimp(%);
x^(a+1/a),ratsimpexpons;
&& ratsubst
ratsubst(a,x*y^2,x^4*y^8+x^4*y^3);
1 + cos(x) + cos(x)^2 + cos(x)^3 + cos(x)^4;
ratsubst(1-sin(x)^2,cos(x)^2,%);
ratsubst(1-cos(x)^2,sin(x)^2,sin(x)^4);
&& ratweight
ratweight(a,1,b,1);
rat(a+b+1);
%^2;
ev(%th(2)^2,ratwtlvl:1);
&& realpart
(%i*v+u)/(f+%i*e)+%e^(%i*alpha);
realpart(%);
&& realroots
realroots(x^5-x-1,5.0e-6);
%[1],float;
x^5-x-1,%;
&& residue
residue(s/(s^2+a^2),s,a*%i);
residue(sin(a*x)/x^4,x,0);
&& resultant
resultant(a*y+x^2+1,y^2+x*y+b,x);
&& reveal