Wed Feb 11 00:57:32 MST 1998 aquarius% reduce REDUCE 3.6, 15-Jul-95, patched to 15 Apr 96 ... 1: % ----------[ R e d u c e ]---------- % ---------- Initialization ---------- on time; Time: 0 ms % ---------- Indefinite Integrals ---------- % This integral only makes sense for x real => x |x|/2 int(abs(x), x); abs(x)*x ---------- 2 Time: 180 ms % Calculus on a piecewise defined function procedure a(x); if x < 0 then -x else x$ Time: 10 ms % => if x < 0 then -x^2/2 else x^2/2 int(a(x), x); ***** x invalid as number Cont? (Y or N) ?y Time: 0 ms clear a; Time: 0 ms % 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 -------- 3 x + 2 Time: 0 ms int(ws, x); 1/3 1/3 2 - 2*x 2/3 1/3 2 (2 *( - 2*sqrt(3)*atan(--------------) - log(2 - 2 *x + x ) 1/3 2 *sqrt(3) 1/3 + 2*log(2 + x)))/12 Time: 70 ms df(ws, x); 1 -------- 3 x + 2 Time: 10 ms % This integral is easy if one realizes that 4^x = (2^x)^2 % => arcsinh(2^x)/log(2) [Robert Israel in sci.math.symbolic] int(2^x/sqrt(1 + 4^x), x); x x sqrt(4 + 1)*2 int(-----------------,x) x 4 + 1 Time: 70 ms % => (-9 x^2 + 16 x - 41/5)/(2 x - 1)^(5/2) % [Gradshteyn and Ryzhik 2.244(8)] int((3*x - 5)^2/(2*x - 1)^(7/2), x); 2 - 45*x + 80*x - 41 ---------------------------------- 2 5*sqrt(2*x - 1)*(4*x - 4*x + 1) Time: 30 ms % => 1/[2 m sqrt(10)] log([-5 + e^(m x) sqrt(10)]/[-5 - e^(m x) sqrt(10)]) % [Gradshteyn and Ryzhik 2.314] int(1/(2*exp(m*x) - 5*exp(-m*x)), x); m*x m*x sqrt(10)*(log(2*e - sqrt(10)) - log(2*e + sqrt(10))) ------------------------------------------------------------ 20*m Time: 70 ms on combinelogs; Time: 0 ms ws; m*x 2*e - sqrt(10) sqrt(10)*log(-------------------) m*x 2*e + sqrt(10) ----------------------------------- 20*m Time: 60 ms off combinelogs; Time: 0 ms % => -3/2 x + 1/4 sinh(2 x) + tanh x [Gradshteyn and Ryzhik 2.423(24)] int(sinh(x)^4/cosh(x)^2, x); 6*x 4*x 4*x 2*x e - 12*e *x + 18*e - 12*e *x - 1 -------------------------------------------- 2*x 2*x 8*e *(e + 1) Time: 60 ms % 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)] load_package(assist)$ Time: 110 ms int(1/(a + b*cos(x)), x); x x tan(---)*a - tan(---)*b 2 2 2 2 2*sqrt(a - b )*atan(-------------------------) 2 2 sqrt(a - b ) ------------------------------------------------- 2 2 a - b Time: 200 ms plus GC time: 20 ms df(ws, x); x 2 2 x 2 2 2 2 x 2 3 x 2 2 (tan(---) *a - tan(---) *b + a - b )/(tan(---) *a - tan(---) *a *b 2 2 2 2 x 2 2 x 2 3 3 2 2 3 - tan(---) *a*b + tan(---) *b + a + a *b - a*b - b ) 2 2 Time: 30 ms trigreduce(ws); 1 -------------- cos(x)*b + a Time: 40 ms % 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)] int(1/(3 + 3*cos(x) + 4*sin(x)), x); x log(4*tan(---) + 3) 2 --------------------- 4 Time: 70 ms % => (a = 4) 1/3 log([tan(x/2) + 1]/[tan(x/2) + 7]) int(1/(4 + 3*cos(x) + 4*sin(x)), x); x x - log(tan(---) + 7) + log(tan(---) + 1) 2 2 ------------------------------------------ 3 Time: 70 ms on combinelogs; Time: 0 ms ws; x tan(---) + 1 2 log(--------------) x tan(---) + 7 2 --------------------- 3 Time: 0 ms off combinelogs; Time: 0 ms % => (a = 5) -1/[2 + tan(x/2)] int(1/(5 + 3*cos(x) + 4*sin(x)), x); x tan(---) 2 ------------------ x 2*(tan(---) + 2) 2 Time: 80 ms % => (a = 6) 2/sqrt(11) arctan([3 tan(x/2) + 4]/sqrt(11)) int(1/(6 + 3*cos(x) + 4*sin(x)), x); x 3*tan(---) + 4 2 2*sqrt(11)*atan(----------------) sqrt(11) ----------------------------------- 11 Time: 90 ms % => x log|x^2 - a^2| - 2 x + a log|(x + a)/(x - a)| % [Gradshteyn and Ryzhik 2.736(1)] int(log(abs(x^2 - a^2)), x); 2 2 2 2 - log(abs(a - x ))*a + log(abs(a - x ))*x + 2*log( - a - x)*a - 2*x Time: 180 ms % => (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)] int(x*acot(x/a), x); x 2 x 2 acot(---)*a + acot(---)*x + a*x a a ----------------------------------- 2 Time: 50 ms % => [sin(5 x) Ci(2 x)]/5 - [Si(7 x) + Si(3 x)]/10 % [Gradshteyn and Ryzhik 5.31(1)] int(cos(5*x)*Ci(2*x), x); int(ci(2*x)*cos(5*x),x) Time: 130 ms plus GC time: 40 ms % => 1/2 [f(x) - g(x)]/[f(x) + g(x)] [Gradshteyn and Ryzhik 2.02(25)] operator f, g; Time: 0 ms int((df(f(x), x)*g(x) - f(x)*df(g(x), x))/(f(x)^2 - g(x)^2), x); df(f(x),x)*g(x) df(g(x),x)*f(x) int(-----------------,x) - int(-----------------,x) 2 2 2 2 f(x) - g(x) f(x) - g(x) Time: 100 ms % ---------- Quit ---------- quit; Quitting real 11.14 user 1.86 sys 1.35