Mon Feb 9 04:20:03 MET 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Indefinite Integrals ---------- -- This integral only makes sense for x real => x |x|/2 integrate(abs(x), x) x ++ (1) | abs(%I)d%I ++ Type: Union(Expression Integer,...) Time: 0.33 (IN) + 0.62 (EV) + 0.13 (OT) + 0.22 (GC) = 1.30 sec -- Calculus on a piecewise defined function a(x) == if x < 0 then -x else x Type: Void Time: 0 sec -- => if x < 0 then -x^2/2 else x^2/2 integrate(a(x), x) Compiling function a with type Variable x -> Polynomial Integer 1 2 (3) - x 2 Type: Polynomial Fraction Integer Time: 0.30 (IN) + 0.02 (EV) + 0.02 (OT) = 0.33 sec )clear properties a Compiled code for a has been cleared. -- This would be very difficult to do by hand -- => 2^(1/3)/6 [1/2 log([x + 2^(1/3)]^2/[x^2 - 2^(1/3) x + 2^(2/3)]) -- + sqrt(3) arctan({[sqrt(3) x]/[2^(4/3) - x] or -- [2 x - 2^(1/3)]/[2^(1/3) sqrt(3)]}) -- [Gradshteyn and Ryzhik 2.126(1)] 1/(x**3 + 2) 1 (4) ------ 3 x + 2 Type: Fraction Polynomial Integer Time: 0.08 (IN) + 0.03 (OT) = 0.12 sec integrate(%, x) (5) +-+ 2 3+-+2 3+-+ +-+ 3+-+ - \|3 log(x \|4 - 2x\|4 + 4) + 2\|3 log(x\|4 + 2) + +-+3+-+ +-+ x\|3 \|4 - \|3 6atan(----------------) 3 / +-+3+-+ 6\|3 \|4 Type: Union(Expression Integer,...) Time: 0.10 (IN) + 0.75 (EV) + 0.07 (OT) + 0.07 (GC) = 0.98 sec D(%, x) 1 (6) ------ 3 x + 2 Type: Expression Integer Time: 0.12 (EV) = 0.12 sec -- This integral is easy if one realizes that 4^x = (2^x)^2 -- => arcsinh(2^x)/log(2) [Robert Israel in sci.math.symbolic] integrate(2**x/sqrt(1 + 4**x), x) >> Error detected within library code: integrate: implementation incomplete (constant residues) initial (7) -> real 15.2 user 3.8 sys 0.3 ------------------------------------------------------------------------------- Mon Feb 9 04:37:53 MET 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Indefinite Integrals ---------- -- => (-9 x^2 + 16 x - 41/5)/(2 x - 1)^(5/2) -- [Gradshteyn and Ryzhik 2.244(8)] integrate((3*x - 5)**2/(2*x - 1)**(7/2), x) 2 - 45x + 80x - 41 (1) ------------------------- 2 +------+ (20x - 20x + 5)\|2x - 1 Type: Union(Expression Integer,...) Time: 0.49 (IN) + 0.67 (EV) + 0.20 (OT) + 0.20 (GC) = 1.55 sec -- => 1/[2 m sqrt(10)] log([-5 + e^(m x) sqrt(10)]/[-5 - e^(m x) sqrt(10)]) -- [Gradshteyn and Ryzhik 2.314] integrate(1/(2*exp(m*x) - 5*exp(-m*x)), x) +--+ m x 2 m x +--+ 2\|10 (%e ) - 20%e + 5\|10 log(---------------------------------) m x 2 2(%e ) - 5 (2) -------------------------------------- +--+ 2m\|10 Type: Union(Expression Integer,...) Time: 0.15 (IN) + 0.75 (EV) + 0.08 (OT) + 0.03 (GC) = 1.02 sec -- => -3/2 x + 1/4 sinh(2 x) + tanh x [Gradshteyn and Ryzhik 2.423(24)] integrate(sinh(x)**4/cosh(x)**2, x) 3 2 sinh(x) + (3cosh(x) + 9)sinh(x) + (- 12x - 8)cosh(x) (3) ------------------------------------------------------ 8cosh(x) Type: Union(Expression Integer,...) Time: 0.13 (IN) + 0.17 (EV) + 0.02 (OT) = 0.32 sec simplify(%) 2 (cosh(x) + 2)sinh(x) + (- 3x - 2)cosh(x) (4) ----------------------------------------- 2cosh(x) Type: Expression Integer Time: 0.03 (IN) + 0.05 (EV) + 0.02 (OT) = 0.10 sec -- This example involves several symbolic parameters -- => 1/sqrt(b^2 - a^2) log([sqrt(b^2 - a^2) tan(x/2) + a + b]/ -- [sqrt(b^2 - a^2) tan(x/2) - a - b]) (a^2 < b^2) -- [Gradshteyn and Ryzhik 2.553(3)] integrate(1/(a + b*cos(x)), x) (5) +-------+ | 2 2 2 2 (- a cos(x) - b)\|b - a + (- b + a )sin(x) log(----------------------------------------------) b cos(x) + a [---------------------------------------------------, +-------+ | 2 2 \|b - a +---------+ | 2 2 sin(x)\|- b + a 2atan(---------------------) (b + a)cos(x) + b + a ----------------------------] +---------+ | 2 2 \|- b + a Type: Union(List Expression Integer,...) Time: 0.12 (IN) + 0.67 (EV) + 0.05 (OT) = 0.83 sec map(simplify, map(f +-> D(f, x), %)) 1 1 (6) [------------,------------] b cos(x) + a b cos(x) + a Type: List Expression Integer Time: 0.32 (EV) + 0.02 (OT) = 0.33 sec -- The integral of 1/(a + 3 cos x + 4 sin x) can have 4 different forms -- depending on the value of a ! [Gradshteyn and Ryzhik 2.558(4)] -- => (a = 3) 1/4 log[3 + 4 tan(x/2)] integrate(1/(3 + 3*cos(x) + 4*sin(x)), x) 4sin(x) + 3cos(x) + 3 log(---------------------) cos(x) + 1 (7) -------------------------- 4 Type: Union(Expression Integer,...) Time: 0.05 (IN) + 0.20 (EV) + 0.02 (OT) = 0.27 sec -- => (a = 4) 1/3 log([tan(x/2) + 1]/[tan(x/2) + 7]) integrate(1/(4 + 3*cos(x) + 4*sin(x)), x) sin(x) + 7cos(x) + 7 sin(x) + cos(x) + 1 - log(--------------------) + log(-------------------) cos(x) + 1 cos(x) + 1 (8) ------------------------------------------------------ 3 Type: Union(Expression Integer,...) Time: 0.27 (EV) + 0.02 (OT) = 0.28 sec -- => (a = 5) -1/[2 + tan(x/2)] integrate(1/(5 + 3*cos(x) + 4*sin(x)), x) - cos(x) - 1 (9) -------------------- sin(x) + 2cos(x) + 2 Type: Union(Expression Integer,...) Time: 0.03 (IN) + 0.08 (EV) + 0.02 (OT) = 0.13 sec -- => (a = 6) 2/sqrt(11) arctan([3 tan(x/2) + 4]/sqrt(11)) integrate(1/(6 + 3*cos(x) + 4*sin(x)), x) +--+ +--+ +--+ 3\|11 sin(x) + 4\|11 cos(x) + 4\|11 2atan(------------------------------------) 11cos(x) + 11 (10) ------------------------------------------- +--+ \|11 Type: Union(Expression Integer,...) Time: 0.22 (EV) + 0.05 (OT) = 0.27 sec -- => x log|x^2 - a^2| - 2 x + a log|(x + a)/(x - a)| -- [Gradshteyn and Ryzhik 2.736(1)] integrate(log(abs(x**2 - a**2)), x) 2 2 (11) x log(abs(x - a )) + a log(x + a) - a log(x - a) - 2x Type: Union(Expression Integer,...) Time: 0.05 (IN) + 0.20 (EV) + 0.02 (OT) = 0.27 sec -- => (a x)/2 + (pi x^2)/4 - 1/2 (x^2 + a^2) arctan(x/a) -- [Gradshteyn and Ryzhik 2.822(4)] or -- (a x)/2 + 1/2 (x^2 + a^2) arccot(x/a) [Gradshteyn and Ryzhik 2.853(2)] integrate(x*acot(x/a), x) 2 2 2a x (x + a )atan(-------) + 2a x 2 2 x - a (12) ----------------------------- 4 Type: Union(Expression Integer,...) Time: 0.07 (IN) + 0.80 (EV) + 0.02 (OT) + 0.03 (GC) = 0.92 sec -- => [sin(5 x) Ci(2 x)]/5 - [Si(7 x) + Si(3 x)]/10 -- [Gradshteyn and Ryzhik 5.31(1)] integrate(cos(5*x)*real(Ei(%i*2*x)), x) >> Error detected within library code: ker2trigs: cannot convert kernel to gaussian function initial (13) -> real 30.1 user 7.6 sys 0.3 ------------------------------------------------------------------------------- Mon Feb 9 04:40:40 MET 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Indefinite Integrals ---------- -- => 1/2 [f(x) - g(x)]/[f(x) + g(x)] [Gradshteyn and Ryzhik 2.02(25)] f:= operator('f); Type: BasicOperator Time: 0.03 (IN) + 0.08 (OT) + 0.03 (GC) = 0.15 sec g:= operator('g); Type: BasicOperator Time: 0.02 (OT) = 0.02 sec integrate((D(f(x), x)*g(x) - f(x)*D(g(x), x))/(f(x)**2 - g(x)**2), x) , , x f(%I)g (%I) - g(%I)f (%I) ++ (3) | ------------------------- d%I ++ 2 2 g(%I) - f(%I) Type: Union(Expression Integer,...) Time: 0.72 (IN) + 0.82 (EV) + 0.20 (OT) + 0.07 (GC) = 1.80 sec )clear properties f g -- ---------- Quit ---------- )quit real 9.1 user 2.5 sys 0.2