(%i1) (%o1) - sin(x) (%i2) %pi (%o2) --- 4 (%i3) 2 x - 1 2 atan(-------) log(x - x + 1) sqrt(3) log(x + 1) (%o3) --------------- + ------------- - ---------- 6 sqrt(3) 3 (%i4) (%o4) - sin(x) (%i5) 2 x - 1 2 atan(-------) log(x - x + 1) sqrt(3) log(x + 1) (%o5) - --------------- + ------------- + ---------- 6 sqrt(3) 3 (%i6) %pi (%o6) --- 4 (%i7) (%i1) (%i1) clear ; (%o1) clear (%i2) (%i1) e; (%o1) e (%i2) E; (%o2) E (%i3) evalf(E); (%o3) evalf(E) (%i4) evalf(e) ; (%o4) evalf(e) (%i5) (%o5) [x = 3, x = 2] (%i6) 2 x - 1 2 atan(-------) log(x - x + 1) sqrt(3) log(x + 1) (%o6) --------------- + ------------- - ---------- 6 sqrt(3) 3 (%i7) (%o7) - sin(x) (%i8) 2 (%o8) - 3 (%i9) 45 1 19 (%o9) [[x = --, y = --, z = --]] 13 13 13 (%i10) [ 1 2 ] (%o10) [ ] [ 3 4 ] (%i11) [ 1 1 ] (%o11) [ ] [ 1 1 ] (%i12) [ - 2 1 ] [ ] (%o12) [ 3 1 ] [ - - - ] [ 2 2 ] (%i13) (%o13) a d - b c (%i14) 2 2 3 x 3 x ---- - ---- / 2 2 [ %e sin(x) (%o14) y = %e (I ------------- dx + %c) ] x / (%i15) Maxima encountered a Lisp error: Ctrl-C: User break Automatically continuing. To reenable the Lisp debugger set *debugger-hook* to nil. (%i15) float(%e); (%o15) 2.718281828459045 (%i16) float(%pi); (%o16) 3.141592653589793 (%i17) bfloat(%e); (%o17) 2.718281828459045b0 (%i18) tex((x^2 + integrate(log(z),z,1,x))/x); Is x - 1 positive, negative, or zero? positive ; $${{x\,\log x+x^2-x+1}\over{x}}$$ (%o18) false (%i19) assume(x > 1); (%o19) [x > 1] (%i20) tex((x^2 + integrate(log(z),z,1,x))/x); $${{x\,\log x+x^2-x+1}\over{x}}$$ (%o20) false (%i21) g(x)g(x) = (x - r)^2*(x - s); 2 (%o21) g(x) = (x - r) (x - s) (%i22) delta; (%o22) delta (%i23) \delta; (%o23) delta (%i24) L(r,d) = integrate(sqrt(1 + (diff(f(x),x) + d*diff(g(x),x))^2),x,-r,r); r / [ d d 2 (%o24) L(r, d) = I sqrt((d (-- (g(x))) + -- (f(x))) + 1) dx ] dx dx / - r (%i25) simplify(%o24) ; r / [ d d 2 (%o25) simplify(L(r, d) = I sqrt((d (-- (g(x))) + -- (f(x))) + 1) dx) ] dx dx / - r (%i26) new; (%o26) new (%i27) kill(all); (%o0) done (%i1) g(x) := (x - r)^2*(x - s); 2 (%o1) g(x) := (x - r) (x - s) (%i2) L(r,d) := integrate(sqrt(1 + (diff(f(x),x) + d*diff(g(x),x))^2),x,-r,r); 2 (%o2) L(r, d) := integrate(sqrt(1 + (diff(f(x), x) + d diff(g(x), x)) ), x, - r, r) (%i3) assume(r > 0); (%o3) [r > 0] (%i4) assume(d > 0); (%o4) [d > 0] (%i5) L(1,1); 1 / [ d 2 2 (%o5) I sqrt((-- (f(x)) + 2 (x - 1) (x - s) + (x - 1) ) + 1) dx ] dx / - 1 (%i6) describe(taylor); -- Function: taylor (, , , ) -- Function: taylor (, [, , ...], , ) -- Function: taylor (, [, , , 'asymp]) -- Function: taylor (, [, , ...], [, , ...], [, , ...]) -- Function: taylor (, [, , ], [, , ], ...) `taylor (, , , )' expands the expression in a truncated Taylor or Laurent series in the variable around the point , containing terms through `( - )^'. If is of the form `()/()' and `()' has no terms up to degree then `taylor' attempts to expand `()' up to degree `2 '. If there are still no nonzero terms, `taylor' doubles the degree of the expansion of `()' so long as the degree of the expansion is less than or equal to ` 2^taylordepth'. `taylor (, [, , ...], , )' returns a truncated power series of degree in all variables , , ... about the point `(, , ...)'. `taylor (, [, , ], [, , ], ...)' returns a truncated power series in the variables , , ... about the point `(, , ...)', truncated at , , .... `taylor (, [, , ...], [, , ...], [, , ...])' returns a truncated power series in the variables , , ... about the point `(, , ...)', truncated at , , .... `taylor (, [, , , 'asymp])' returns an expansion of in negative powers of ` - '. The highest order term is `( - )^<-n>'. When `maxtayorder' is `true', then during algebraic manipulation of (truncated) Taylor series, `taylor' tries to retain as many terms as are known to be correct. When `psexpand' is `true', an extended rational function expression is displayed fully expanded. The switch `ratexpand' has the same effect. When `psexpand' is `false', a multivariate expression is displayed just as in the rational function package. When `psexpand' is `multi', then terms with the same total degree in the variables are grouped together. See also the `taylor_logexpand' switch for controlling expansion. Examples: (%i1) taylor (sqrt (sin(x) + a*x + 1), x, 0, 3); 2 2 (a + 1) x (a + 2 a + 1) x (%o1)/T/ 1 + --------- - ----------------- 2 8 3 2 3 (3 a + 9 a + 9 a - 1) x + -------------------------- + . . . 48 (%i2) %^2; 3 x (%o2)/T/ 1 + (a + 1) x - -- + . . . 6 (%i3) taylor (sqrt (x + 1), x, 0, 5); 2 3 4 5 x x x 5 x 7 x (%o3)/T/ 1 + - - -- + -- - ---- + ---- + . . . 2 8 16 128 256 (%i4) %^2; (%o4)/T/ 1 + x + . . . (%i5) product ((1 + x^i)^2.5, i, 1, inf)/(1 + x^2); inf /===\ ! ! i 2.5 ! ! (x + 1) ! ! i = 1 (%o5) ----------------- 2 x + 1 (%i6) ev (taylor(%, x, 0, 3), keepfloat); 2 3 (%o6)/T/ 1 + 2.5 x + 3.375 x + 6.5625 x + . . . (%i7) taylor (1/log (x + 1), x, 0, 3); 2 3 1 1 x x 19 x (%o7)/T/ - + - - -- + -- - ----- + . . . x 2 12 24 720 (%i8) taylor (cos(x) - sec(x), x, 0, 5); 4 2 x (%o8)/T/ - x - -- + . . . 6 (%i9) taylor ((cos(x) - sec(x))^3, x, 0, 5); (%o9)/T/ 0 + . . . (%i10) taylor (1/(cos(x) - sec(x))^3, x, 0, 5); 2 4 1 1 11 347 6767 x 15377 x (%o10)/T/ - -- + ---- + ------ - ----- - ------- - -------- 6 4 2 15120 604800 7983360 x 2 x 120 x + . . . (%i11) taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6); 2 2 4 2 4 k x (3 k - 4 k ) x (%o11)/T/ 1 - ----- - ---------------- 2 24 6 4 2 6 (45 k - 60 k + 16 k ) x - -------------------------- + . . . 720 (%i12) taylor ((x + 1)^n, x, 0, 4); 2 2 3 2 3 (n - n) x (n - 3 n + 2 n) x (%o12)/T/ 1 + n x + ----------- + -------------------- 2 6 4 3 2 4 (n - 6 n + 11 n - 6 n) x + ---------------------------- + . . . 24 (%i13) taylor (sin (y + x), x, 0, 3, y, 0, 3); 3 2 y y (%o13)/T/ y - -- + . . . + (1 - -- + . . .) x 6 2 3 2 y y 2 1 y 3 + (- - + -- + . . .) x + (- - + -- + . . .) x + . . . 2 12 6 12 (%i14) taylor (sin (y + x), [x, y], 0, 3); 3 2 2 3 x + 3 y x + 3 y x + y (%o14)/T/ y + x - ------------------------- + . . . 6 (%i15) taylor (1/sin (y + x), x, 0, 3, y, 0, 3); 1 y 1 1 1 2 (%o15)/T/ - + - + . . . + (- -- + - + . . .) x + (-- + . . .) x y 6 2 6 3 y y 1 3 + (- -- + . . .) x + . . . 4 y (%i16) taylor (1/sin (y + x), [x, y], 0, 3); 3 2 2 3 1 x + y 7 x + 21 y x + 21 y x + 7 y (%o16)/T/ ----- + ----- + ------------------------------- + . . . x + y 6 360 There are also some inexact matches for `taylor'. Try `?? taylor' to see them. (%o6) true (%i7) taylor((x-1)^4*(y -1)^4, [x,0,2], [y, 0, 1]); 2 (%o7)/T/ 1 - 4 y + . . . + (- 4 + 16 y + . . .) x + (6 - 24 y + . . .) x + . . . (%i8) taylor(L(r,d), [r,0,2], [d,0,1]); ! ! r ! r 2 d 2 ! / ! / (2 (x - r) (x - s) + (x - r) ) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) ! [ d 2 2 ! [ dx ! (%o8)/T/ I sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) dx! + (I ----------------------------------------------------------------------------- dx! ) ] dx ! ] d 2 2 ! / ! / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) ! - r ! - r dx ! !r = 0 !r = 0 ! r d 2 ! ! ! / (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) ! d ! 2 d ! 2 [ dx ! d + . . . + (sqrt((-- (f(r))! ) + 1) + sqrt((---- (f(- r))! ) + 1) + (I ------------------------------------------------------------------------ dx! ) d dr ! d- r ! ] d 2 2 ! !r = 0 !r = 0 / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) ! - r dx ! !r = 0 ! 2 ! ! ! ! d ! d ! 2 d ! d ! 2 + . . .) r + (((--- (f(r))! ) sqrt((-- (f(r))! ) + 1) (-- (f(r))! ) (---- (f(- r))! ) 2 ! dr ! dr ! d- r ! dr ! !r = 0 !r = 0 !r = 0 !r = 0 ! ! 2 ! ! 2 ! ! ! d ! d ! 2 d ! d ! 2 d ! + ((------- (f(- r))! ) (-- (f(r))! ) + ------- (f(- r))! ) sqrt((---- (f(- r))! ) + 1) (---- (f(- r))! ) d- r dr ! dr ! d- r dr ! d- r ! d- r ! !r = 0 !r = 0 !r = 0 !r = 0 !r = 0 ! 2 ! ! ! ! ! ! d ! d ! 2 d ! d ! 2 d ! 2 d ! 2 + (--- (f(r))! ) sqrt((-- (f(r))! ) + 1) (-- (f(r))! ))/((2 (-- (f(r))! ) + 2) (---- (f(- r))! ) + 2 (-- (f(r))! ) + 2) 2 ! dr ! dr ! dr ! d- r ! dr ! dr ! !r = 0 !r = 0 !r = 0 !r = 0 !r = 0 !r = 0 r d 2 ! ! / 2 (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) d ! 2 d ! [ dx + (((2 sqrt((-- (f(r))! ) + 1) (-- (f(r))! ) s + (at(I (--------------------------------------------------------- dr ! dr ! ] d 2 2 !r = 0 !r = 0 / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) - r dx 2 d 2 2 2 d (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) ! d (- 2 (x - s) - 2 (x - r)) dx d ! 2 + --------------------------------------------------------- - ----------------------------------------------------------------------------) dx, r = 0)) (-- (f(r))! ) d 2 2 d 2 2 3/2 dr ! sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) ((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) !r = 0 dx dx r d 2 / 2 (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) 2 [ dx d (- 2 (x - s) - 2 (x - r)) + at(I (--------------------------------------------------------- + --------------------------------------------------------- ] d 2 2 d 2 2 / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) - r dx dx 2 d 2 2 d (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) ! dx d ! 2 - ----------------------------------------------------------------------------) dx, r = 0)) (---- (f(- r))! ) d 2 2 3/2 d- r ! ((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) !r = 0 dx ! ! ! ! ! d ! 2 d ! 2 d ! d ! 2 d ! + (6 (-- (f(r))! ) + 6) s sqrt((---- (f(- r))! ) + 1) (---- (f(- r))! ) + 2 sqrt((-- (f(r))! ) + 1) (-- (f(r))! ) s dr ! d- r ! d- r ! dr ! dr ! !r = 0 !r = 0 !r = 0 !r = 0 !r = 0 r d 2 / 2 (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) 2 [ dx d (- 2 (x - s) - 2 (x - r)) + (at(I (--------------------------------------------------------- + --------------------------------------------------------- ] d 2 2 d 2 2 / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) - r dx dx 2 d 2 2 d (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) ! dx d ! 2 - ----------------------------------------------------------------------------) dx, r = 0)) (-- (f(r))! ) d 2 2 3/2 dr ! ((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) !r = 0 dx r d 2 / 2 (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) 2 [ dx d (- 2 (x - s) - 2 (x - r)) + at(I (--------------------------------------------------------- + --------------------------------------------------------- ] d 2 2 d 2 2 / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) - r dx dx 2 d 2 2 d (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) ! ! ! dx d ! 2 d ! 2 d ! 2 - ----------------------------------------------------------------------------) dx, r = 0)) d)/((2 (-- (f(r))! ) + 2) (---- (f(- r))! ) + 2 (-- (f(r))! ) + 2) d 2 2 3/2 dr ! d- r ! dr ! ((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) !r = 0 !r = 0 !r = 0 dx 2 + . . .) r + . . . (%i9) simplify(%o8); r d 2 ! ! / 2 (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) 2 d ! 2 d ! 2 [ dx (%o9) simplify(r ((d ((---- (f(- r))! ) ((-- (f(r))! ) (at(I (--------------------------------------------------------- d- r ! dr ! ] d 2 2 !r = 0 !r = 0 / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) - r dx 2 d 2 2 2 d (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) d (- 2 (x - s) - 2 (x - r)) dx + --------------------------------------------------------- - ----------------------------------------------------------------------------) dx, r = 0)) d 2 2 d 2 2 3/2 sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) ((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) dx dx r d 2 / 2 (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) 2 [ dx d (- 2 (x - s) - 2 (x - r)) + at(I (--------------------------------------------------------- + --------------------------------------------------------- ] d 2 2 d 2 2 / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) - r dx dx 2 d 2 2 d (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) ! ! dx d ! d ! 2 - ----------------------------------------------------------------------------) dx, r = 0) + 2 (-- (f(r))! ) sqrt((-- (f(r))! ) + 1) s) d 2 2 3/2 dr ! dr ! ((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) !r = 0 !r = 0 dx r d 2 ! / 2 (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) 2 d ! 2 [ dx d (- 2 (x - s) - 2 (x - r)) + (-- (f(r))! ) (at(I (--------------------------------------------------------- + --------------------------------------------------------- dr ! ] d 2 2 d 2 2 !r = 0 / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) - r dx dx 2 d 2 2 d (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) dx - ----------------------------------------------------------------------------) dx, r = 0)) d 2 2 3/2 ((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) dx r d 2 / 2 (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) 2 [ dx d (- 2 (x - s) - 2 (x - r)) + at(I (--------------------------------------------------------- + --------------------------------------------------------- ] d 2 2 d 2 2 / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) - r dx dx 2 d 2 2 d (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) ! ! ! dx d ! d ! 2 d ! 2 - ----------------------------------------------------------------------------) dx, r = 0) + (---- (f(- r))! ) sqrt((---- (f(- r))! ) + 1) (6 (-- (f(r))! ) + 6) s d 2 2 3/2 d- r ! d- r ! dr ! ((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) !r = 0 !r = 0 !r = 0 dx ! ! ! ! ! d ! d ! 2 d ! 2 d ! 2 d ! 2 + 2 (-- (f(r))! ) sqrt((-- (f(r))! ) + 1) s))/((---- (f(- r))! ) (2 (-- (f(r))! ) + 2) + 2 (-- (f(r))! ) + 2) dr ! dr ! d- r ! dr ! dr ! !r = 0 !r = 0 !r = 0 !r = 0 !r = 0 ! ! ! ! ! 2 ! ! ! 2 ! d ! 2 d ! d ! 2 d ! d ! d ! 2 d ! + ((---- (f(- r))! ) (-- (f(r))! ) sqrt((-- (f(r))! ) + 1) (--- (f(r))! ) + (-- (f(r))! ) sqrt((-- (f(r))! ) + 1) (--- (f(r))! ) d- r ! dr ! dr ! 2 ! dr ! dr ! 2 ! !r = 0 !r = 0 !r = 0 dr ! !r = 0 !r = 0 dr ! !r = 0 !r = 0 ! ! ! ! 2 ! ! 2 ! d ! d ! 2 d ! d ! 2 d ! + (---- (f(- r))! ) sqrt((---- (f(- r))! ) + 1) ((------- (f(- r))! ) (-- (f(r))! ) + ------- (f(- r))! )) d- r ! d- r ! d- r dr ! dr ! d- r dr ! !r = 0 !r = 0 !r = 0 !r = 0 !r = 0 ! r ! ! ! ! / ! d ! 2 d ! 2 d ! 2 [ d 2 2 ! /((---- (f(- r))! ) (2 (-- (f(r))! ) + 2) + 2 (-- (f(r))! ) + 2)) + I sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) dx! d- r ! dr ! dr ! ] dx ! !r = 0 !r = 0 !r = 0 / ! - r ! !r = 0 ! r 2 d 2 ! / (2 (x - r) (x - s) + (x - r) ) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) ! [ dx ! + d (I ----------------------------------------------------------------------------- dx! ) ] d 2 2 ! / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) ! - r dx ! !r = 0 ! r d 2 ! / (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) ! ! ! [ dx ! d ! 2 d ! 2 + r (d (I ------------------------------------------------------------------------ dx! ) + sqrt((-- (f(r))! ) + 1) + sqrt((---- (f(- r))! ) + 1))) ] d 2 2 ! dr ! d- r ! / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) ! !r = 0 !r = 0 - r dx ! !r = 0 (%i10) f(x) := A*x^2/2 + B*x + C; 2 A x (%o10) f(x) := ---- + B x + C 2 (%i11) L(r,d); r / [ 2 2 (%o11) I sqrt((B + x A + d (2 (x - r) (x - s) + (x - r) )) + 1) dx ] / - r (%i12) simplify(%o8); r d 2 ! ! / 2 (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) 2 d ! 2 d ! 2 [ dx (%o12) simplify(r ((d ((---- (f(- r))! ) ((-- (f(r))! ) (at(I (--------------------------------------------------------- d- r ! dr ! ] d 2 2 !r = 0 !r = 0 / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) - r dx 2 d 2 2 2 d (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) d (- 2 (x - s) - 2 (x - r)) dx + --------------------------------------------------------- - ----------------------------------------------------------------------------) dx, r = 0)) d 2 2 d 2 2 3/2 sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) ((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) dx dx r d 2 / 2 (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) 2 [ dx d (- 2 (x - s) - 2 (x - r)) + at(I (--------------------------------------------------------- + --------------------------------------------------------- ] d 2 2 d 2 2 / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) - r dx dx 2 d 2 2 d (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) ! ! dx d ! d ! 2 - ----------------------------------------------------------------------------) dx, r = 0) + 2 (-- (f(r))! ) sqrt((-- (f(r))! ) + 1) s) d 2 2 3/2 dr ! dr ! ((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) !r = 0 !r = 0 dx r d 2 ! / 2 (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) 2 d ! 2 [ dx d (- 2 (x - s) - 2 (x - r)) + (-- (f(r))! ) (at(I (--------------------------------------------------------- + --------------------------------------------------------- dr ! ] d 2 2 d 2 2 !r = 0 / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) - r dx dx 2 d 2 2 d (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) dx - ----------------------------------------------------------------------------) dx, r = 0)) d 2 2 3/2 ((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) dx r d 2 / 2 (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) 2 [ dx d (- 2 (x - s) - 2 (x - r)) + at(I (--------------------------------------------------------- + --------------------------------------------------------- ] d 2 2 d 2 2 / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) - r dx dx 2 d 2 2 d (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) ! ! ! dx d ! d ! 2 d ! 2 - ----------------------------------------------------------------------------) dx, r = 0) + (---- (f(- r))! ) sqrt((---- (f(- r))! ) + 1) (6 (-- (f(r))! ) + 6) s d 2 2 3/2 d- r ! d- r ! dr ! ((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) !r = 0 !r = 0 !r = 0 dx ! ! ! ! ! d ! d ! 2 d ! 2 d ! 2 d ! 2 + 2 (-- (f(r))! ) sqrt((-- (f(r))! ) + 1) s))/((---- (f(- r))! ) (2 (-- (f(r))! ) + 2) + 2 (-- (f(r))! ) + 2) dr ! dr ! d- r ! dr ! dr ! !r = 0 !r = 0 !r = 0 !r = 0 !r = 0 ! ! ! ! ! 2 ! ! ! 2 ! d ! 2 d ! d ! 2 d ! d ! d ! 2 d ! + ((---- (f(- r))! ) (-- (f(r))! ) sqrt((-- (f(r))! ) + 1) (--- (f(r))! ) + (-- (f(r))! ) sqrt((-- (f(r))! ) + 1) (--- (f(r))! ) d- r ! dr ! dr ! 2 ! dr ! dr ! 2 ! !r = 0 !r = 0 !r = 0 dr ! !r = 0 !r = 0 dr ! !r = 0 !r = 0 ! ! ! ! 2 ! ! 2 ! d ! d ! 2 d ! d ! 2 d ! + (---- (f(- r))! ) sqrt((---- (f(- r))! ) + 1) ((------- (f(- r))! ) (-- (f(r))! ) + ------- (f(- r))! )) d- r ! d- r ! d- r dr ! dr ! d- r dr ! !r = 0 !r = 0 !r = 0 !r = 0 !r = 0 ! r ! ! ! ! / ! d ! 2 d ! 2 d ! 2 [ d 2 2 ! /((---- (f(- r))! ) (2 (-- (f(r))! ) + 2) + 2 (-- (f(r))! ) + 2)) + I sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) dx! d- r ! dr ! dr ! ] dx ! !r = 0 !r = 0 !r = 0 / ! - r ! !r = 0 ! r 2 d 2 ! / (2 (x - r) (x - s) + (x - r) ) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) ! [ dx ! + d (I ----------------------------------------------------------------------------- dx! ) ] d 2 2 ! / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) ! - r dx ! !r = 0 ! r d 2 ! / (- 2 (x - s) - 2 (x - r)) (-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) ! ! ! [ dx ! d ! 2 d ! 2 + r (d (I ------------------------------------------------------------------------ dx! ) + sqrt((-- (f(r))! ) + 1) + sqrt((---- (f(- r))! ) + 1))) ] d 2 2 ! dr ! d- r ! / sqrt((-- (f(x)) + d (2 (x - r) (x - s) + (x - r) )) + 1) ! !r = 0 !r = 0 - r dx ! !r = 0 (%i13) lam(r,d,x) := sqrt(1 + (diff(f(x),x) + d*diff(g(x),x))^2); 2 (%o13) lam(r, d, x) := sqrt(1 + (diff(f(x), x) + d diff(g(x), x)) ) (%i14) tlam(r,d,x) := taylor(lam(r,d,x), [r, 0, 3], [d, 0, 1]); (%o14) tlam(r, d, x) := taylor(lam(r, d, x), [r, 0, 3], [d, 0, 1]) (%i15) taylor(lam(r,d,x), [r, 0, 3], [d, 0, 1]); 2 2 2 (%o15)/T/ sqrt(B + 2 x A B + x A + 1) 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 ((3 sqrt(B + 2 x A B + x A + 1) x - 2 sqrt(B + 2 x A B + x A + 1) s x) B + (3 sqrt(B + 2 x A B + x A + 1) x - 2 sqrt(B + 2 x A B + x A + 1) s x ) A) d + -------------------------------------------------------------------------------------------------------------------------------------------------------------------- + . . . 2 2 2 B + 2 x A B + x A + 1 2 2 2 2 2 2 2 2 2 2 2 2 2 ((4 sqrt(B + 2 x A B + x A + 1) x - 2 sqrt(B + 2 x A B + x A + 1) s) B + (4 sqrt(B + 2 x A B + x A + 1) x - 2 sqrt(B + 2 x A B + x A + 1) s x) A) d + (- ---------------------------------------------------------------------------------------------------------------------------------------------------------------- + . . .) r 2 2 2 B + 2 x A B + x A + 1 2 2 2 2 2 2 (sqrt(B + 2 x A B + x A + 1) B + sqrt(B + 2 x A B + x A + 1) x A) d 2 + (------------------------------------------------------------------------- + . . .) r + . . . 2 2 2 B + 2 x A B + x A + 1 (%i16) taylor(lam(r,d,x), [r, 0, 3], [d, 0, 1], [x, 0, 3]); 2 3 3 2 2 2 2 (sqrt(B + 1) A B) x (sqrt(B + 1) A ) x (sqrt(B + 1) A B) x 2 (%o16)/T/ - ---------------------- + -------------------- + -------------------- + sqrt(B + 1) + . . . 6 4 2 4 2 2 2 B + 6 B + 6 B + 2 2 B + 4 B + 2 B + 1 2 2 3 2 2 2 2 2 2 2 2 3 (2 sqrt(B + 1) s B) x (3 sqrt(B + 1) B + 3 sqrt(B + 1) B - 2 sqrt(B + 1) s A) x (3 sqrt(B + 1) A B + 3 sqrt(B + 1) s A B + 3 sqrt(B + 1) A) x + (- ---------------------- + -------------------------------------------------------------- + ------------------------------------------------------------------- + . . .) d 2 4 2 6 4 2 B + 1 B + 2 B + 1 B + 3 B + 3 B + 1 2 2 3 2 2 2 2 2 2 2 2 2 sqrt(B + 1) s B (4 sqrt(B + 1) B + 4 sqrt(B + 1) B - 2 sqrt(B + 1) s A) x (4 sqrt(B + 1) A B + 3 sqrt(B + 1) s A B + 4 sqrt(B + 1) A) x + . . . + ((------------------ - ------------------------------------------------------------- - ------------------------------------------------------------------- 2 4 2 6 4 2 B + 1 B + 2 B + 1 B + 3 B + 3 B + 1 2 2 3 2 3 2 2 2 2 3 3 (6 sqrt(B + 1) A B + 4 sqrt(B + 1) s A B + 6 sqrt(B + 1) A B - sqrt(B + 1) s A ) x + -------------------------------------------------------------------------------------------- + . . .) d + . . .) r 8 6 4 2 B + 4 B + 6 B + 4 B + 1 2 2 2 2 2 2 3 2 2 3 3 sqrt(B + 1) B (sqrt(B + 1) A) x (3 sqrt(B + 1) A B) x (4 sqrt(B + 1) A B - sqrt(B + 1) A ) x 2 + ((-------------- + ------------------ - ------------------------ + ------------------------------------------- + . . .) d + . . .) r + . . . 2 4 2 6 4 2 8 6 4 2 B + 1 B + 2 B + 1 2 B + 6 B + 6 B + 2 2 B + 8 B + 12 B + 8 B + 2 (%i17) tlam(r,d,x) := %o16; (%o17) tlam(r, d, x) := %o16 (%i18) integrate(tlam(r,d,x), x, -r, r); 2 8 2 2 7 6 2 2 5 3 2 3 4 4 (%o18) (sqrt(B + 1) (24 r B + (12 r A + 24 d r s) B + 96 r B + (36 r A + 72 d r s) B + (4 r A + (8 d r s - 2 d r ) A + 144 r) B 4 3 5 4 2 2 2 3 5 6 3 3 2 3 4 2 + (- 3 r A + (24 d r - 6 d r s) A + 36 r A + 72 d r s) B + ((24 d r s + 12 d r ) A + 8 r A + (16 d r s - 4 d r ) A + 96 r) B 4 3 5 4 2 2 2 5 6 3 3 2 3 4 8 6 4 2 + (- 3 r A + (24 d r - 6 d r s) A + 12 r A + 24 d r s) B + (- 6 d r s - 3 d r ) A + 4 r A + (8 d r s - 2 d r ) A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 2 8 2 2 3 7 6 2 2 3 5 3 2 3 4 4 + (sqrt(B + 1) (24 r B + (- 12 r A + 72 d r s + 96 d r ) B + 96 r B + (- 36 r A + 216 d r s + 288 d r ) B + (4 r A + (- 40 d r s - 62 d r ) A + 144 r) B 4 3 4 5 2 2 2 3 3 5 6 3 3 2 3 4 2 + (3 r A + (- 42 d r s - 48 d r ) A - 36 r A + 216 d r s + 288 d r ) B + ((- 24 d r s - 12 d r ) A + 8 r A + (- 80 d r s - 124 d r ) A + 96 r) B 4 3 4 5 2 2 2 3 5 6 3 3 2 3 4 + (3 r A + (- 42 d r s - 48 d r ) A - 12 r A + 72 d r s + 96 d r ) B + (6 d r s + 3 d r ) A + 4 r A + (- 40 d r s - 62 d r ) A + 24 r)) 8 6 4 2 /(24 B + 96 B + 144 B + 96 B + 24) (%i19) help(subs); (%o19) type `describe(topic);' or `example(topic);' or `? topic' (%i20) describe(subs); No exact match found for topic `subs'. Try `?? subs' (inexact match) instead. (%o20) false (%i21) describe(subst); -- Function: subst (, , ) Substitutes for in . must be an atom or a complete subexpression of . For example, `x+y+z' is a complete subexpression of `2*(x+y+z)/w' while `x+y' is not. When does not have these characteristics, one may sometimes use `substpart' or `ratsubst' (see below). Alternatively, if is of the form `e/f' then one could use `subst (a*f, e, c)' while if is of the form `e^(1/f)' then one could use `subst (a^f, e, c)'. The `subst' command also discerns the `x^y' in `x^-y' so that `subst (a, sqrt(x), 1/sqrt(x))' yields `1/a'. and may also be operators of an expression enclosed in double-quotes `"' or they may be function names. If one wishes to substitute for the independent variable in derivative forms then the `at' function (see below) should be used. `subst' is an alias for `substitute'. `subst (, )' or `subst ([, ..., ], )' are other permissible forms. The are equations indicating substitutions to be made. For each equation, the right side will be substituted for the left in the expression . `exptsubst' if `true' permits substitutions like `y' for `%e^x' in `%e^(a*x)' to take place. When `opsubst' is `false', `subst' will not attempt to substitute into the operator of an expression. E.g. `(opsubst: false, subst (x^2, r, r+r[0]))' will work. Examples: (%i1) subst (a, x+y, x + (x+y)^2 + y); 2 (%o1) y + x + a (%i2) subst (-%i, %i, a + b*%i); (%o2) a - %i b For further examples, do `example (subst)'. There are also some inexact matches for `subst'. Try `?? subst' to see them. (%o21) true (%i22) subst(0,d,diff(%o18,d)); 2 2 3 7 2 3 5 3 4 4 4 5 2 2 3 3 (%o22) (sqrt(B + 1) ((72 r s + 96 r ) B + (216 r s + 288 r ) B + (- 40 r s - 62 r ) A B + ((- 42 r s - 48 r ) A + 216 r s + 288 r ) B 5 6 3 3 4 2 4 5 2 2 3 5 6 3 3 4 + ((- 24 r s - 12 r ) A + (- 80 r s - 124 r ) A) B + ((- 42 r s - 48 r ) A + 72 r s + 96 r ) B + (6 r s + 3 r ) A + (- 40 r s - 62 r ) A)) 8 6 4 2 2 2 7 2 5 3 4 4 5 4 2 2 3 /(24 B + 96 B + 144 B + 96 B + 24) + (sqrt(B + 1) (24 r s B + 72 r s B + (8 r s - 2 r ) A B + ((24 r - 6 r s) A + 72 r s) B 5 6 3 3 4 2 5 4 2 2 5 6 3 3 4 8 6 4 2 + ((24 r s + 12 r ) A + (16 r s - 4 r ) A) B + ((24 r - 6 r s) A + 24 r s) B + (- 6 r s - 3 r ) A + (8 r s - 2 r ) A))/(24 B + 96 B + 144 B + 96 B + 24) (%i23) describe(solve) ; -- Function: solve (, ) -- Function: solve () -- Function: solve ([, ..., ], [, ..., ]) Solves the algebraic equation for the variable and returns a list of solution equations in . If is not an equation, the equation ` = 0' is assumed in its place. may be a function (e.g. `f(x)'), or other non-atomic expression except a sum or product. may be omitted if contains only one variable. may be a rational expression, and may contain trigonometric functions, exponentials, etc. The following method is used: Let be the expression and be the variable. If is linear in then it is trivially solved for . Otherwise if is of the form `A*X^N + B' then the result is `(-B/A)^1/N)' times the `N''th roots of unity. If is not linear in then the gcd of the exponents of in (say ) is divided into the exponents and the multiplicity of the roots is multiplied by . Then `solve' is called again on the result. If factors then `solve' is called on each of the factors. Finally `solve' will use the quadratic, cubic, or quartic formulas where necessary. In the case where is a polynomial in some function of the variable to be solved for, say `F(X)', then it is first solved for `F(X)' (call the result ), then the equation `F(X)=C' can be solved for provided the inverse of the function is known. `breakup' if `false' will cause `solve' to express the solutions of cubic or quartic equations as single expressions rather than as made up of several common subexpressions which is the default. `multiplicities' - will be set to a list of the multiplicities of the individual solutions returned by `solve', `realroots', or `allroots'. Try `apropos (solve)' for the switches which affect `solve'. `describe' may then by used on the individual switch names if their purpose is not clear. `solve ([, ..., ], [, ..., ])' solves a system of simultaneous (linear or non-linear) polynomial equations by calling `linsolve' or `algsys' and returns a list of the solution lists in the variables. In the case of `linsolve' this list would contain a single list of solutions. It takes two lists as arguments. The first list represents the equations to be solved; the second list is a list of the unknowns to be determined. If the total number of variables in the equations is equal to the number of equations, the second argument-list may be omitted. For linear systems if the given equations are not compatible, the message `inconsistent' will be displayed (see the `solve_inconsistent_error' switch); if no unique solution exists, then `singular' will be displayed. When `programmode' is `false', `solve' displays solutions with intermediate expression (`%t') labels, and returns the list of labels. When `globalsolve' is `true' and the problem is to solve two or more linear equations, each solved-for variable is bound to its value in the solution of the equations. Examples: (%i1) solve (asin (cos (3*x))*(f(x) - 1), x); SOLVE is using arc-trig functions to get a solution. Some solutions will be lost. %pi (%o1) [x = ---, f(x) = 1] 6 (%i2) ev (solve (5^f(x) = 125, f(x)), solveradcan); log(125) (%o2) [f(x) = --------] log(5) (%i3) [4*x^2 - y^2 = 12, x*y - x = 2]; 2 2 (%o3) [4 x - y = 12, x y - x = 2] (%i4) solve (%, [x, y]); (%o4) [[x = 2, y = 2], [x = .5202594388652008 %i - .1331240357358706, y = .0767837852378778 - 3.608003221870287 %i], [x = - .5202594388652008 %i - .1331240357358706, y = 3.608003221870287 %i + .0767837852378778], [x = - 1.733751846381093, y = - .1535675710019696]] (%i5) solve (1 + a*x + x^3, x); 3 sqrt(3) %i 1 sqrt(4 a + 27) 1 1/3 (%o5) [x = (- ---------- - -) (--------------- - -) 2 2 6 sqrt(3) 2 sqrt(3) %i 1 (---------- - -) a 2 2 - --------------------------, x = 3 sqrt(4 a + 27) 1 1/3 3 (--------------- - -) 6 sqrt(3) 2 3 sqrt(3) %i 1 sqrt(4 a + 27) 1 1/3 (---------- - -) (--------------- - -) 2 2 6 sqrt(3) 2 sqrt(3) %i 1 (- ---------- - -) a 2 2 - --------------------------, x = 3 sqrt(4 a + 27) 1 1/3 3 (--------------- - -) 6 sqrt(3) 2 3 sqrt(4 a + 27) 1 1/3 a (--------------- - -) - --------------------------] 6 sqrt(3) 2 3 sqrt(4 a + 27) 1 1/3 3 (--------------- - -) 6 sqrt(3) 2 (%i6) solve (x^3 - 1); sqrt(3) %i - 1 sqrt(3) %i + 1 (%o6) [x = --------------, x = - --------------, x = 1] 2 2 (%i7) solve (x^6 - 1); sqrt(3) %i + 1 sqrt(3) %i - 1 (%o7) [x = --------------, x = --------------, x = - 1, 2 2 sqrt(3) %i + 1 sqrt(3) %i - 1 x = - --------------, x = - --------------, x = 1] 2 2 (%i8) ev (x^6 - 1, %[1]); 6 (sqrt(3) %i + 1) (%o8) ----------------- - 1 64 (%i9) expand (%); (%o9) 0 (%i10) x^2 - 1; 2 (%o10) x - 1 (%i11) solve (%, x); (%o11) [x = - 1, x = 1] (%i12) ev (%th(2), %[1]); (%o12) 0 There are also some inexact matches for `solve'. Try `?? solve' to see them. (%o23) true (%i24) solve(%o22,s); 5 3 2 2 3 2 2 12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A (%o24) [s = - -----------------------------------------------------------] 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A (%i25) s := op(1,%o24); Improper function definition: s -- an error. To debug this try debugmode(true); (%i26) op(%o24,1); Maxima encountered a Lisp error: APPLY: too many arguments given to # Automatically continuing. To reenable the Lisp debugger set *debugger-hook* to nil. (%i27) op(%o24); (%o27) [ (%i28) describe(op); -- Function: op () Returns the main operator of the expression . `op ()' is equivalent to `part (, 0)'. `op' returns a string if the main operator is a built-in or user-defined prefix, binary or n-ary infix, postfix, matchfix, or nofix operator. Otherwise, if is a subscripted function expression, `op' returns the subscripted function; in this case the return value is not an atom. Otherwise, is an array function or ordinary function expression, and `op' returns a symbol. `op' observes the value of the global flag `inflag'. `op' evaluates it argument. See also `args'. Examples: (%i1) stringdisp: true$ (%i2) op (a * b * c); (%o2) "*" (%i3) op (a * b + c); (%o3) "+" (%i4) op ('sin (a + b)); (%o4) sin (%i5) op (a!); (%o5) "!" (%i6) op (-a); (%o6) "-" (%i7) op ([a, b, c]); (%o7) "[" (%i8) op ('(if a > b then c else d)); (%o8) "if" (%i9) op ('foo (a)); (%o9) foo (%i10) prefix (foo); (%o10) "foo" (%i11) op (foo a); (%o11) "foo" (%i12) op (F [x, y] (a, b, c)); (%o12) F x, y (%i13) op (G [u, v, w]); (%o13) G There are also some inexact matches for `op'. Try `?? op' to see them. (%o28) true (%i29) args(%o24); 5 3 2 2 3 2 2 12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A (%o29) [s = - -----------------------------------------------------------] 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A (%i30) args(%o24)[1]; 5 3 2 2 3 2 2 12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A (%o30) s = - ----------------------------------------------------------- 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A (%i31) args(args(%o24)); 5 3 2 2 3 2 2 12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A (%o31) [s = - -----------------------------------------------------------] 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A (%i32) args(args(%o24))[1]; 5 3 2 2 3 2 2 12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A (%o32) s = - ----------------------------------------------------------- 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A (%i33) args(%o24[1]); 5 3 2 2 3 2 2 12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A (%o33) [s, - -----------------------------------------------------------] 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A (%i34) %o24[1][1]; 5 3 2 2 3 2 2 12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A (%o34) (s = - -----------------------------------------------------------) 5 3 2 2 2 1 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A (%i35) %o24; 5 3 2 2 3 2 2 12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A (%o35) [s = - -----------------------------------------------------------] 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A (%i36) describe(ev) ; -- Function: ev (, , ..., ) Evaluates the expression in the environment specified by the arguments , ..., . The arguments are switches (Boolean flags), assignments, equations, and functions. `ev' returns the result (another expression) of the evaluation. The evaluation is carried out in steps, as follows. 1. First the environment is set up by scanning the arguments which may be any or all of the following. * `simp' causes to be simplified regardless of the setting of the switch `simp' which inhibits simplification if `false'. * `noeval' supresses the evaluation phase of `ev' (see step (4) below). This is useful in conjunction with the other switches and in causing to be resimplified without being reevaluated. * `nouns' causes the evaluation of noun forms (typically unevaluated functions such as `'integrate' or `'diff') in . * `expand' causes expansion. * `expand (, )' causes expansion, setting the values of `maxposex' and `maxnegex' to and respectively. * `detout' causes any matrix inverses computed in to have their determinant kept outside of the inverse rather than dividing through each element. * `diff' causes all differentiations indicated in to be performed. * `derivlist (, , , ...)' causes only differentiations with respect to the indicated variables. * `float' causes non-integral rational numbers to be converted to floating point. * `numer' causes some mathematical functions (including exponentiation) with numerical arguments to be evaluated in floating point. It causes variables in which have been given numervals to be replaced by their values. It also sets the `float' switch on. * `pred' causes predicates (expressions which evaluate to `true' or `false') to be evaluated. * `eval' causes an extra post-evaluation of to occur. (See step (5) below.) `eval' may occur multiple times. For each instance of `eval', the expression is evaluated again. * `A' where `A' is an atom declared to be an evaluation flag (see `evflag') causes `A' to be bound to `true' during the evaluation of . * `V: expression' (or alternately `V=expression') causes `V' to be bound to the value of `expression' during the evaluation of . Note that if `V' is a Maxima option, then `expression' is used for its value during the evaluation of . If more than one argument to `ev' is of this type then the binding is done in parallel. If `V' is a non-atomic expression then a substitution rather than a binding is performed. * `F' where `F', a function name, has been declared to be an evaluation function (see `evfun') causes `F' to be applied to . * Any other function names (e.g., `sum') cause evaluation of occurrences of those names in as though they were verbs. * In addition a function occurring in (say `F(x)') may be defined locally for the purpose of this evaluation of by giving `F(x) := expression' as an argument to `ev'. * If an atom not mentioned above or a subscripted variable or subscripted expression was given as an argument, it is evaluated and if the result is an equation or assignment then the indicated binding or substitution is performed. If the result is a list then the members of the list are treated as if they were additional arguments given to `ev'. This permits a list of equations to be given (e.g. `[X=1, Y=A**2]') or a list of names of equations (e.g., `[%t1, %t2]' where `%t1' and `%t2' are equations) such as that returned by `solve'. The arguments of `ev' may be given in any order with the exception of substitution equations which are handled in sequence, left to right, and evaluation functions which are composed, e.g., `ev (, ratsimp, realpart)' is handled as `realpart (ratsimp ())'. The `simp', `numer', `float', and `pred' switches may also be set locally in a block, or globally in Maxima so that they will remain in effect until being reset. If is a canonical rational expression (CRE), then the expression returned by `ev' is also a CRE, provided the `numer' and `float' switches are not both `true'. 2. During step (1), a list is made of the non-subscripted variables appearing on the left side of equations in the arguments or in the value of some arguments if the value is an equation. The variables (subscripted variables which do not have associated array functions as well as non-subscripted variables) in the expression are replaced by their global values, except for those appearing in this list. Usually, is just a label or `%' (as in `%i2' in the example below), so this step simply retrieves the expression named by the label, so that `ev' may work on it. 3. If any substitutions are indicated by the arguments, they are carried out now. 4. The resulting expression is then re-evaluated (unless one of the arguments was `noeval') and simplified according to the arguments. Note that any function calls in will be carried out after the variables in it are evaluated and that `ev(F(x))' thus may behave like `F(ev(x))'. 5. For each instance of `eval' in the arguments, steps (3) and (4) are repeated. Examples (%i1) sin(x) + cos(y) + (w+1)^2 + 'diff (sin(w), w); d 2 (%o1) cos(y) + sin(x) + -- (sin(w)) + (w + 1) dw (%i2) ev (%, sin, expand, diff, x=2, y=1); 2 (%o2) cos(w) + w + 2 w + cos(1) + 1.909297426825682 An alternate top level syntax has been provided for `ev', whereby one may just type in its arguments, without the `ev()'. That is, one may write simply , , ..., This is not permitted as part of another expression, e.g., in functions, blocks, etc. Notice the parallel binding process in the following example. (%i3) programmode: false; (%o3) false (%i4) x+y, x: a+y, y: 2; (%o4) y + a + 2 (%i5) 2*x - 3*y = 3$ (%i6) -3*x + 2*y = -4$ (%i7) solve ([%o5, %o6]); Solution 1 (%t7) y = - - 5 6 (%t8) x = - 5 (%o8) [[%t7, %t8]] (%i8) %o6, %o8; (%o8) - 4 = - 4 (%i9) x + 1/x > gamma (1/2); 1 (%o9) x + - > sqrt(%pi) x (%i10) %, numer, x=1/2; (%o10) 2.5 > 1.772453850905516 (%i11) %, pred; (%o11) true There are also some inexact matches for `ev'. Try `?? ev' to see them. (%o36) true (%i37) %i18; (%o37) integrate(tlam(r, d, x), x, - r, r) (%i38) ev(%o18,%o24[1]); 5 5 3 2 2 3 2 2 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (%o38) (sqrt(B + 1) (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 2 8 6 4 2 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) (%i39) simplify(%); 5 5 3 2 2 3 2 2 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (%o39) simplify((sqrt(B + 1) (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 2 8 6 4 2 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24)) (%i40) taylor(%,[r,0,3],[d,0,1]); 5 5 3 2 2 3 2 2 d 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (%o40)/T/ simplify(0) + (at(-- (simplify((sqrt(B + 1) (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) dr 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 3 2 8 6 4 2 d 2 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24))), r = 0) + (at(----- (simplify((sqrt(B + 1) dd dr 5 5 3 2 2 3 2 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 3 2 8 6 4 2 d 2 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24))), r = 0)) d + . . .) r + ((at(--- (simplify((sqrt(B + 1) 2 dr 5 5 3 2 2 3 2 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 3 2 8 6 4 2 d 2 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24))), r = 0))/2 + ((at(------ (simplify((sqrt(B + 1) 2 dd dr 5 5 3 2 2 3 2 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 3 2 8 6 4 2 2 d 2 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24))), r = 0)) d)/2 + . . .) r + ((at(--- (simplify((sqrt(B + 1) 3 dr 5 5 3 2 2 3 2 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 3 2 8 6 4 2 d 2 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24))), r = 0))/6 + ((at(------ (simplify((sqrt(B + 1) 3 dd dr 5 5 3 2 2 3 2 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 2 8 6 4 2 3 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24))), r = 0)) d)/6 + . . .) r + . . . (%i41) simplify(%); 3 5 5 3 2 2 3 2 2 3 d 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (%o41) simplify(r ((at(--- (simplify((sqrt(B + 1) (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) 3 5 3 2 2 2 dr 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 3 2 8 6 4 2 d 2 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24))), r = 0))/6 + (d (at(------ (simplify((sqrt(B + 1) 3 dd dr 5 5 3 2 2 3 2 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 3 2 8 6 4 2 2 d 2 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24))), r = 0)))/6) + r ((at(--- (simplify((sqrt(B + 1) 2 dr 5 5 3 2 2 3 2 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 3 2 8 6 4 2 d 2 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24))), r = 0))/2 + (d (at(------ (simplify((sqrt(B + 1) 2 dd dr 5 5 3 2 2 3 2 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 2 8 6 4 2 d 2 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24))), r = 0)))/2) + r (at(-- (simplify((sqrt(B + 1) dr 5 5 3 2 2 3 2 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 3 2 8 6 4 2 d 2 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24))), r = 0) + d (at(----- (simplify((sqrt(B + 1) dd dr 5 5 3 2 2 3 2 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 2 8 6 4 2 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24))), r = 0))) + simplify(0)) (%i42) factor(%); ! ! 3 4 2 2 2 ! 2 4 2 2 2 ! 3 d r (6 B + 12 B + r A + 6) ! 2 d r (6 B + 12 B + r A + 6) ! (%o42) simplify((r (--- (simplify(----------------------------))! ) + 3 r (--- (simplify(----------------------------))! ) 3 2 3/2 ! 2 2 3/2 ! dr 3 (B + 1) ! dr 3 (B + 1) ! !r = 0 !r = 0 ! ! 4 2 2 2 ! 4 4 2 2 2 ! d r (6 B + 12 B + r A + 6) ! 3 d r (6 B + 12 B + r A + 6) ! + 6 r (-- (simplify(----------------------------))! ) + d r (------ (simplify(----------------------------))! ) dr 2 3/2 ! 3 2 3/2 ! 3 (B + 1) ! dd dr 3 (B + 1) ! !r = 0 !r = 0 ! ! 3 4 2 2 2 ! 2 4 2 2 2 ! 2 d r (6 B + 12 B + r A + 6) ! d r (6 B + 12 B + r A + 6) ! + 3 d r (------ (simplify(----------------------------))! ) + 6 d r (----- (simplify(----------------------------))! ) + 6 simplify(0))/6) 2 2 3/2 ! dd dr 2 3/2 ! dd dr 3 (B + 1) ! 3 (B + 1) ! !r = 0 !r = 0 (%i43) %o18; 2 8 2 2 7 6 2 2 5 3 2 3 4 4 (%o43) (sqrt(B + 1) (24 r B + (12 r A + 24 d r s) B + 96 r B + (36 r A + 72 d r s) B + (4 r A + (8 d r s - 2 d r ) A + 144 r) B 4 3 5 4 2 2 2 3 5 6 3 3 2 3 4 2 + (- 3 r A + (24 d r - 6 d r s) A + 36 r A + 72 d r s) B + ((24 d r s + 12 d r ) A + 8 r A + (16 d r s - 4 d r ) A + 96 r) B 4 3 5 4 2 2 2 5 6 3 3 2 3 4 8 6 4 2 + (- 3 r A + (24 d r - 6 d r s) A + 12 r A + 24 d r s) B + (- 6 d r s - 3 d r ) A + 4 r A + (8 d r s - 2 d r ) A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 2 8 2 2 3 7 6 2 2 3 5 3 2 3 4 4 + (sqrt(B + 1) (24 r B + (- 12 r A + 72 d r s + 96 d r ) B + 96 r B + (- 36 r A + 216 d r s + 288 d r ) B + (4 r A + (- 40 d r s - 62 d r ) A + 144 r) B 4 3 4 5 2 2 2 3 3 5 6 3 3 2 3 4 2 + (3 r A + (- 42 d r s - 48 d r ) A - 36 r A + 216 d r s + 288 d r ) B + ((- 24 d r s - 12 d r ) A + 8 r A + (- 80 d r s - 124 d r ) A + 96 r) B 4 3 4 5 2 2 2 3 5 6 3 3 2 3 4 + (3 r A + (- 42 d r s - 48 d r ) A - 12 r A + 72 d r s + 96 d r ) B + (6 d r s + 3 d r ) A + 4 r A + (- 40 d r s - 62 d r ) A + 24 r)) 8 6 4 2 /(24 B + 96 B + 144 B + 96 B + 24) (%i44) %o24; 5 3 2 2 3 2 2 12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A (%o44) [s = - -----------------------------------------------------------] 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A (%i45) ev(%o18,%o24[1]); 5 5 3 2 2 3 2 2 2 8 6 2 3 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 (%o45) (sqrt(B + 1) (24 r B + 96 r B + B (A (--------------------------------------------------------------------- - 12 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 80 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (--------------------------------------------------------------------- - 124 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (--------------------------------------------------------------------- - 48 d r ) - --------------------------------------------------------------------- + 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 3 3 2 42 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 - 12 r A + 96 d r ) + B (A (--------------------------------------------------------------------- - 48 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 3 - ---------------------------------------------------------------------- + 3 r A - 36 r A + 288 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + B (A (--------------------------------------------------------------------- - 62 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 40 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (3 d r - --------------------------------------------------------------------) + A (--------------------------------------------------------------------- - 62 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 7 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 + B (- --------------------------------------------------------------------- - 12 r A + 96 d r ) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 5 216 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 2 3 3 2 8 6 4 2 + B (- ---------------------------------------------------------------------- - 36 r A + 288 d r ) + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 2 8 6 2 3 6 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + (sqrt(B + 1) (24 r B + 96 r B + B (A (12 d r - ---------------------------------------------------------------------) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 16 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 + A (- --------------------------------------------------------------------- - 4 d r ) + 8 r A + 96 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- - 3 r A 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 4 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 2 3 2 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + 12 r A) + B (A (-------------------------------------------------------------------- + 24 d r ) - --------------------------------------------------------------------- 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 5 3 2 2 3 2 2 4 3 2 4 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 3 2 - 3 r A + 36 r A) + B (A (- -------------------------------------------------------------------- - 2 d r ) + 4 r A + 144 r) 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 5 5 3 2 2 3 2 2 3 5 3 2 2 3 2 2 3 6 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 6 8 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 4 + A (-------------------------------------------------------------------- - 3 d r ) + A (- -------------------------------------------------------------------- - 2 d r ) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 2 5 3 2 2 3 2 2 2 5 3 2 2 3 2 2 7 2 24 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) 5 2 72 d r (12 r B + 24 r B - 8 r A B + (12 r - 3 r A ) B - 8 r A) + B (12 r A - ---------------------------------------------------------------------) + B (36 r A - ---------------------------------------------------------------------) 5 3 2 2 2 5 3 2 2 2 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 12 B + 24 B - 4 r A B + (12 - 6 r A ) B - 4 r A 3 2 8 6 4 2 + 4 r A + 24 r))/(24 B + 96 B + 144 B + 96 B + 24) (%i46) factor(%); 4 2 2 2 r (6 B + 12 B + r A + 6) (%o46) ---------------------------- 2 3/2 3 (B + 1) (%i47) factor(%o24[1]); 5 3 2 2 2 r (12 B + 24 B - 8 r A B - 3 r A B + 12 B - 8 r A) (%o47) s = - ------------------------------------------------------- 5 3 2 2 2 2 (6 B + 12 B - 2 r A B - 3 r A B + 6 B - 2 r A) (%i48) V := integrate(%pi*(f(x) + d*g(x))^2,x,-r,r); Improper function definition: V -- an error. To debug this try debugmode(true); (%i49) V(r,d) := integrate(%pi*(f(x) + d*ev(g(x),%o24[1]))^2,x,-r,r); 2 (%o49) V(r, d) := integrate(%pi (f(x) + d ev(g(x), %o24 )) , x, - r, r) 1 (%i50) V(r,d); 10 8 2 7 3 2 6 2 5 3 2 4 4 3 2 3 (%o50) %pi (((15120 r B + 60480 r B - 10080 r A B + (90720 r - 15120 r A ) B - 30240 r A B + (60480 r - 28560 r A ) B + (5040 r A - 30240 r A) B 5 4 3 2 2 4 3 2 3 2 2 + (3780 r A - 11760 r A + 15120 r) B + (5040 r A - 10080 r A) B + 1680 r A ) C 2 11 3 4 10 2 9 3 4 8 4 2 5 2 7 + (15120 r B + (5040 r A + 12600 d r ) B + 60480 r B + (10080 r A + 50400 d r ) B + (- 18480 r A - 11760 d r A + 90720 r ) B 5 3 6 2 4 6 4 2 5 2 5 5 3 6 2 3 4 4 + (- 5040 r A - 10080 d r A + 75600 d r ) B + (- 38640 r A - 35280 d r A + 60480 r ) B + (- 4480 r A - 17640 d r A - 10080 r A + 50400 d r ) B 6 4 7 3 4 2 5 2 3 7 5 8 4 5 3 6 2 3 4 2 + (5460 r A + 5040 d r A - 21840 r A - 35280 d r A + 15120 r ) B + (1260 r A + 1890 d r A + 1120 r A - 5040 d r A - 5040 r A + 12600 d r ) B 6 4 7 3 4 2 5 5 3 6 2 3 12 4 5 11 + (1680 r A + 5040 d r A - 1680 r A - 11760 d r A) B + 560 r A + 2520 d r A ) C + 5040 r B + (3780 r A + 3528 d r ) B 5 2 6 2 7 3 10 4 5 9 5 2 6 2 7 3 8 + (756 r A + 756 d r A + 4176 d r + 20160 r ) B + (11760 r A + 14112 d r ) B + (- 4536 r A - 168 d r A + 16704 d r + 30240 r ) B 6 3 7 2 4 2 8 5 7 7 4 8 3 2 9 5 2 6 2 7 3 + (- 4284 r A - 3570 d r A + (12600 r - 5136 d r ) A + 21168 d r ) B + (- 756 r A - 630 d r A + (- 2412 d r - 12544 r ) A - 5040 d r A + 25056 d r + 20160 r ) 6 6 3 7 2 4 2 8 5 5 7 4 8 3 2 9 5 2 6 2 7 3 B + (- 6972 r A - 7140 d r A + (5040 r - 15408 d r ) A + 14112 d r ) B + (1092 r A + 266 d r A + (- 3240 d r - 8456 r ) A - 6552 d r A + 16704 d r + 5040 r ) 4 8 5 9 4 2 10 6 3 7 2 4 2 8 5 3 B + (1197 r A + 861 d r A + (1476 d r - 2772 r ) A - 3570 d r A + (420 r - 15408 d r ) A + 3528 d r ) B 9 6 10 5 2 11 7 4 8 3 2 9 5 2 6 2 7 2 + (189 r A + 126 d r A + (351 d r + 672 r ) A + 1036 d r A + (756 d r - 1204 r ) A - 2436 d r A + 4176 d r ) B 8 5 9 4 2 10 6 3 2 8 7 4 8 3 2 9 2 + (252 r A + 294 d r A + (1476 d r - 84 r ) A - 5136 d r A) B + 84 r A + 140 d r A + 1584 d r A ) 10 8 7 2 2 6 5 2 2 4 3 3 3 4 4 2 2 2 /(15120 B + 60480 B - 10080 r A B + (90720 - 15120 r A ) B - 30240 r A B + (60480 - 28560 r A ) B + (5040 r A - 30240 r A) B + (3780 r A - 11760 r A + 15120) B 3 3 2 2 10 8 2 7 3 2 6 2 5 3 2 4 + (5040 r A - 10080 r A) B + 1680 r A ) + ((15120 r B + 60480 r B - 10080 r A B + (90720 r - 15120 r A ) B - 30240 r A B + (60480 r - 28560 r A ) B 4 3 2 3 5 4 3 2 2 4 3 2 3 2 2 + (5040 r A - 30240 r A) B + (3780 r A - 11760 r A + 15120 r) B + (5040 r A - 10080 r A) B + 1680 r A ) C 2 11 3 4 10 2 9 3 4 8 4 2 5 2 7 + (- 15120 r B + (5040 r A + 27720 d r ) B - 60480 r B + (30240 r A + 110880 d r ) B + (11760 r A - 42000 d r A - 90720 r ) B 5 3 6 2 3 4 6 4 2 5 2 5 5 3 6 2 3 4 4 + (- 5040 r A - 10080 d r A + 60480 r A + 166320 d r ) B + (18480 r A - 126000 d r A - 60480 r ) B + (- 14560 r A - 9240 d r A + 50400 r A + 110880 d r ) B 6 4 7 3 4 2 5 2 3 7 5 8 4 5 3 6 2 3 4 2 + (- 2100 r A + 15120 d r A + 1680 r A - 126000 d r A - 15120 r ) B + (1260 r A - 1890 d r A - 8960 r A + 11760 d r A + 15120 r A + 27720 d r ) B 6 4 7 3 4 2 5 5 3 6 2 3 12 4 5 11 + (1680 r A + 15120 d r A - 5040 r A - 42000 d r A) B + 560 r A + 10920 d r A ) C + 5040 r B + (- 3780 r A - 11592 d r ) B 5 2 6 2 7 3 10 4 5 9 5 2 6 2 7 3 8 + (756 r A + 3276 d r A + 14256 d r + 20160 r ) B + (- 18480 r A - 46368 d r ) B + (504 r A + 35112 d r A + 57024 d r + 30240 r ) B 6 3 7 2 2 8 4 5 7 7 4 8 3 2 9 5 2 6 2 7 3 6 + (3276 r A - 6510 d r A + (- 28656 d r - 32760 r ) A - 69552 d r ) B + (- 756 r A + 630 d r A + (108 d r + 2576 r ) A + 85680 d r A + 85536 d r + 20160 r ) B 6 3 7 2 2 8 4 5 5 7 4 8 3 2 9 5 2 6 2 7 + (7308 r A - 26460 d r A + (- 85968 d r - 25200 r ) A - 46368 d r ) B + (- 1428 r A - 4074 d r A + (18600 d r + 6664 r ) A + 79128 d r A + 57024 d r 3 4 8 5 9 4 6 2 10 3 7 2 2 8 4 5 3 + 5040 r ) B + (- 693 r A + 4851 d r A + (3108 r - 6084 d r ) A - 33390 d r A + (- 85968 d r - 7140 r ) A - 11592 d r ) B 9 6 10 5 2 11 7 4 8 3 2 9 5 2 6 2 7 2 + (189 r A - 1134 d r A + (2241 d r - 1848 r ) A - 2604 d r A + (36876 d r + 3836 r ) A + 25284 d r A + 14256 d r ) B 8 5 9 4 2 10 6 3 7 2 2 8 7 4 8 3 2 9 2 + (252 r A + 2394 d r A + (- 6084 d r - 924 r ) A - 13440 d r A - 28656 d r A) B + 84 r A + 2100 d r A + 18384 d r A ) 10 8 7 2 2 6 5 2 2 4 3 3 3 4 4 2 2 2 /(15120 B + 60480 B - 10080 r A B + (90720 - 15120 r A ) B - 30240 r A B + (60480 - 28560 r A ) B + (5040 r A - 30240 r A) B + (3780 r A - 11760 r A + 15120) B 3 3 2 2 + (5040 r A - 10080 r A) B + 1680 r A )) (%i51) L; (%o51) L (%i52) factor(%o50); 10 2 8 2 7 2 2 2 6 2 6 2 5 2 2 2 4 2 4 2 3 3 3 2 (%o52) (%pi r (15120 B C + 60480 B C - 10080 r A B C - 15120 r A B C + 90720 B C - 30240 r A B C - 28560 r A B C + 60480 B C + 5040 r A B C 3 2 4 4 2 2 2 2 2 2 2 2 3 3 2 2 2 2 2 2 10 3 10 2 8 - 30240 r A B C + 3780 r A B C - 11760 r A B C + 15120 B C + 5040 r A B C - 10080 r A B C + 1680 r A C + 5040 r A B C + 20160 d r B C + 20160 r A B C 3 8 3 2 7 4 7 4 3 6 5 2 6 2 6 3 6 3 2 5 4 5 + 80640 d r B C - 3360 r A B C - 26880 d r A B C - 5040 r A B C - 10080 d r A B C + 30240 r A B C + 120960 d r B C - 10080 r A B C - 80640 d r A B C 4 3 4 5 2 4 2 4 3 4 5 4 3 6 3 3 3 2 3 4 3 6 5 2 - 9520 r A B C - 13440 d r A B C + 20160 r A B C + 80640 d r B C + 1680 r A B C + 10080 d r A B C - 10080 r A B C - 80640 d r A B C + 1260 r A B C 4 3 2 5 2 2 2 2 3 2 5 4 6 3 3 2 4 4 3 5 2 - 3920 r A B C + 3360 d r A B C + 5040 r A B C + 20160 d r B C + 1680 r A B C + 10080 d r A B C - 3360 r A B C - 26880 d r A B C + 560 r A C + 6720 d r A C 2 12 4 11 4 2 10 5 10 2 6 10 2 10 3 9 4 9 4 2 8 5 8 + 5040 r B - 4032 d r B + 756 r A B + 2016 d r A B + 9216 d r B + 20160 r B - 3360 r A B - 16128 d r B - 2016 r A B + 17472 d r A B 2 6 8 2 8 5 3 7 6 2 7 2 7 7 3 7 4 7 6 4 6 2 8 2 6 4 2 6 + 36864 d r B + 30240 r B - 504 r A B - 5040 d r A B - 16896 d r A B - 10080 r A B - 24192 d r B - 756 r A B - 1152 d r A B - 4984 r A B 5 6 2 6 6 2 6 5 3 5 6 2 5 2 7 5 3 5 4 5 6 4 4 7 3 4 + 40320 d r A B + 55296 d r B + 20160 r B + 168 r A B - 16800 d r A B - 50688 d r A B - 10080 r A B - 16128 d r B - 168 r A B - 1904 d r A B 2 8 2 4 4 2 4 5 4 2 6 4 2 4 7 5 3 8 4 3 2 9 3 3 5 3 3 6 2 3 + 7680 d r A B - 896 r A B + 36288 d r A B + 36864 d r B + 5040 r B + 252 r A B + 2856 d r A B - 2304 d r A B + 168 r A B - 18480 d r A B 2 7 3 3 3 4 3 8 6 2 9 5 2 2 10 4 2 6 4 2 7 3 2 2 8 2 2 4 2 2 - 50688 d r A B - 3360 r A B - 4032 d r B + 189 r A B - 504 d r A B + 1296 d r A B - 588 r A B - 784 d r A B + 18816 d r A B + 1316 r A B 5 2 2 6 2 7 5 8 4 2 9 3 5 3 6 2 2 7 6 4 7 3 2 8 2 + 11424 d r A B + 9216 d r B + 252 r A B + 1344 d r A B - 2304 d r A B - 504 r A B - 6720 d r A B - 16896 d r A B + 84 r A + 1120 d r A + 9984 d r A )) 5 3 2 2 2 2 /(210 (6 B + 12 B - 2 r A B - 3 r A B + 6 B - 2 r A) ) (%i53) kill(all); (%o0) done (%i1)