1)
 

> Sum(1/(k^2+1),k=1..infinity);
> value(");
> expand(");

(a)
> t*BesselJ(2,t): eq1 := "=expand(");

(b)
> t^2*BesselJ(3,t): eq2 := "=expand(");
> rhs(eq2)-4*rhs(eq1) = lhs(eq2)-4*lhs(eq1);
> normal((rhs(")-lhs("))/t=0 );

> expand(");

(a)
> f := ln(x*y) + sin(x)^2 + cos(x)^2;
> expand(f);

(b)
> simplify(f);

(c)
> simplify(f,symbolic);

(a)
> (exp(x)+x)/(exp(2*x)+2*x*exp(x)+x^2);
> simplify(");

(b)
> (x^5+40*x^4+595*x^3+3905*x^2+9680*x+1331)^(1/3);
> simplify(");
> simplify(",symbolic);

> (x^3)^(1/3);
> simplify(",assume=positive);
> simplify("",assume=negative);

(c)
> (x-2)^(3/2)/(x^2-4*x+4)^(1/4);
> simplify(");
> simplify(",symbolic);

(d)
> (sqrt(x)-y)/(x-y^2);
> subs(x=s^2,");
> simplify(",assume=positive);
> subs(s=sqrt(x),");

(e)
> 1/(2+5^(1/3));
> rationalize(");

(f)
> cos(x+y)+sin(x)*sin(y)+2^(x+y);
> expand(",2^(x+y));

(g)
> 2*cos(x)^2-cos(2*x);
> simplify(");

(a)
> Int(sqrt((1+x^2)^3),x);
> value(");
> diff(",x);
> normal(");
> factor(");
> simplify(",symbolic);

(b)
> int( 1/(x^4-4), x );
> diff(",x);
> normal("",expanded);

(c)
> int( sin(3*x)*cos(2*x), x );
> diff(",x);
> subs(5*x=y,2*");
> readlib(trigsubs)(");
> subs( y=5*x, op(")/2 );
> ";

> int( sin(3*x)*cos(2*x), x );
> diff(",x);
> expand( " - sin(3*x)*cos(2*x) );

(a)
> (2^(1/3)+4^(1/3))^3 -
> 6*(2^(1/3)+4^(1/3)) - 6 = 0;
> testeq(");
> expand("");
> simplify(");

(b)
> ln( tan(1/2*x+Pi/4)) -
> arcsinh( tan(x) ) = 0;
> testeq(");

> f := lhs("");
> diff(f,x);
> expand(");
> simplify(",symbolic);
> subs(x=0,f);
> evalf(");

(c)
> tan(x)+tan(y) = sin(x+y)/(cos(x)*cos(y));
> testeq(");
> simplify( lhs("") - expand(rhs("")) );

(d)
> sin(5*x) = 5*sin(x) -20*sin(x)^3 +
> 16*sin(x)^5;
> testeq(");
> expand( lhs("") );
> simplify(", {cos(x)^2+sin(x)^2=1},
> [cos(x),sin(x)] );