tests/rtest5.mac

   1 /*************** -*- Mode: MACSYMA; Package: MAXIMA -*-  ******************/
   2 /***************************************************************************
   3 ***                                                                    *****
   4 ***     Copyright (c) 1984 by William Schelter,University of Texas     *****
   5 ***     All rights reserved                                            *****
   6 ***************************************************************************/
   7
   8
   9 kill(all);
  10 done$
  11 partition(2*a*x*f(x),x);
  12 [2*a,x*f(x)]$
  13 partition(a+b,x);
  14 [b+a,0]$
  15 2*(a*log(x)+2*a*log(y));
  16 2*(2*a*log(y)+a*log(x))$
  17 logcontract(%);
  18 a*log(x^2*y^4)$
  19 ratsimp(logcontract(log(sqrt(x+1)+sqrt(x))+log(sqrt(x+1)-sqrt(x))));
  20 0$
  21
  22 /* Two examples from Bug report ID: 947808 - logcontract and ratfac */
  23 logexpand:false; /* To prevent simplification of the expected result */
  24 false;
  25 logcontract(2*log(x+1)+1);
  26 log((x+1)^2)+1;
  27 logexpand:true;
  28 true;
  29 logcontract(1000*log(x+1)+log(x));
  30 log(x*(x+1)^1000)$
  31
  32 rootsconmode:false;
  33 false$
  34
  35 rootscontract(x^(1/2)*y^(3/2));
  36 sqrt(x*y^3)$
  37
  38 rootscontract(x^(1/2)*y^(1/4));
  39 sqrt(x)*y^(1/4)$
  40
  41 rootsconmode:true;
  42 true$
  43
  44 expand(rootscontract(x^(1/2)*y^(1/4)) - sqrt(x*sqrt(y)),0,0);
  45 0$
  46
  47 expand(rootscontract(x^(1/2)*y^(1/3)) - sqrt(x)*y^(1/3),0,0);
  48 0$
  49
  50 rootsconmode:all;
  51 all$
  52 expand(rootscontract(x^(1/2)*y^(1/4)) - (x^2*y)^(1/4),0,0);
  53 0$
  54
  55 expand(rootscontract(x^(1/2)*y^(1/3)) -(x^3*y^2)^(1/6),0,0);
  56 0$
  57
  58 rootsconmode:false;
  59 false$
  60 rootscontract(sqrt(sqrt(x+1)+sqrt(x))*sqrt(sqrt(x+1)-sqrt(x)));
  61 1$
  62 rootsconmode:true;
  63 true$
  64
  65 rootscontract(sqrt(sqrt(5)+5)-5^(1/4)*sqrt(sqrt(5)+1));
  66 0$
  67
  68 kill(f,g,h,x,y);
  69 done$
  70 diff(sin(x)+x^3+2*x^2,x);
  71 cos(x)+3*x^2+4*x$
  72 diff(sin(x)*cos(x),x);
  73 cos(x)^2-sin(x)^2$
  74 diff(sin(x)*cos(x),x,2);
  75 -4*cos(x)*sin(x)$
  76 derivabbrev:true;
  77 true$
  78 diff(exp(f(x)),x,2);
  79 %e^f(x)*'diff(f(x),x,2)+%e^f(x)*('diff(f(x),x,1))^2$
  80 'integrate(f(x,y),y,g(x),h(x));
  81 'integrate(f(x,y),y,g(x),h(x))$
  82 diff(%,x);
  83 'integrate('diff(f(x,y),x,1),y,g(x),h(x))
  84  +f(x,h(x))*'diff(h(x),x,1)-f(x,g(x))*'diff(g(x),x,1)$
  85 kill(a,x,f,y,t);
  86 done$
  87
  88 /* -----------------------------------------------------------------------------
  89  * Examples for dependencies
  90  * -------------------------------------------------------------------------- */
  91 depends(a,x);
  92 [a(x)]$
  93 diff(a . a,x);
  94 'diff(a,x,1) . a+a . 'diff(a,x,1)$
  95 depends(f,[x,y],[x,y],t);
  96 [f(x,y),x(t),y(t)]$
  97 diff(f,t);
  98 'diff(f,y,1)*'diff(y,t,1)+'diff(f,x,1)*'diff(x,t,1)$
  99 depends(y,x);
 100 [y(x)]$
 101 kill(f,g,j);
 102 done$
 103
 104 /* More examples with subscribted variables */
 105
 106 depends(g,f,f,[x[1],x[2],x[3]],x,t);
 107 [g(f),f(x[1],x[2],x[3]),x(t)]$
 108 diff(g,f);
 109 'diff(g,f,1)$
 110 diff(f,x[1]);
 111 'diff(f,x[1],1)$
 112 diff(f,x[1],2);
 113 'diff(f,x[1],2)$
 114 diff(x[1],t);
 115 'diff(x[1],t,1)$
 116 diff(f,x[1],1,x[2],1,x[3],1);
 117 'diff(f,x[1],1,x[2],1,x[3],1)$
 118 diff(f,t);
 119 'diff(x[3],t,1)*'diff(f,x[3],1)+'diff(x[2],t,1)*'diff(f,x[2],1)
 120                                +'diff(x[1],t,1)*'diff(f,x[1],1)$
 121 diff(g,t);
 122 ('diff(x[3],t,1)*'diff(f,x[3],1)+'diff(x[2],t,1)*'diff(f,x[2],1)
 123                                 +'diff(x[1],t,1)*'diff(f,x[1],1))
 124  *'diff(g,f,1)$
 125 kill(g,f,x);
 126 done;
 127
 128 /* The subscript is not differentiated, it is a constant */
 129
 130 depends(f,x[a],x,t,t,a);
 131 [f(x[a]),x(t),t(a)]$
 132 diff(f,x[a]);
 133 'diff(f,x[a],1)$
 134 diff(f,t);
 135 'diff(x[a],t,1)*'diff(f,x[a],1)$
 136 diff(f,a);
 137 'diff(x[a],t,1)*'diff(f,x[a],1)*'diff(t,a,1)$
 138 kill(f,x,t);
 139 done;
 140
 141 /* Maxima has not a problem with circular defined dependencies */
 142
 143 depends(f,x,x,t,t,f);
 144 [f(x),x(t),t(f)]$
 145 diff(f,x);
 146 'diff(f,x,1)$
 147 diff(f,t);
 148 'diff(f,x,1)*'diff(x,t,1)$
 149 diff(t,f);
 150 'diff(t,f,1)$
 151 diff(t,x);
 152 'diff(f,x,1)*'diff(t,f,1)$
 153 kill(f,x,t);
 154 done;
 155 /* -------------------------------------------------------------------------- */
 156
 157 gradef(f(x,y),x^2,g(x,y));
 158 f(x,y)$
 159 diff(f(x,y),x);
 160 g(x,y)*'diff(y,x,1)+x^2$
 161 gradef(j(n,z),'diff(j(n,z),n),j(n-1,z)-n/z*j(n,z));
 162 j(n,z)$
 163 ratsimp(diff(j(2,x),x,2));
 164 (j(0,x)*x^2-3*j(1,x)*x+6*j(2,x))/x^2$
 165 test(f):=block([u],u:integrate(f,x),ratsimp(f-diff(u,x)));
 166 test(f):=block([u],u:integrate(f,x),ratsimp(f-diff(u,x)))$
 167 test(sin(x));
 168 0$
 169 test(1/(1+x));
 170 0$
 171 test(1/(1+x^2));
 172 0$
 173 integrate(sin(x)^3,x);
 174 cos(x)^3/3-cos(x)$
 175 kill(q);
 176 done$
 177 integrate(%e^x/(%e^x+2),x);
 178 log(%e^x+2)$
 179 integrate(1/(x*log(x)),x);
 180 log(log(x))$
 181 integrate(sin(2*x+3),x);
 182 -cos(2*x+3)/2$
 183
 184 /* Correct simplification after change to risch.lisp revision 1.17 */
 185
 186 integrate(%e^x*erf(x),x);
 187 %e^x*erf(x)-%e^(1/4)*erf(x-1/2)$
 188
 189 integrate(x/(x^3+1),x);
 190 log(x^2-x+1)/6+atan((2*x-1)/sqrt(3))/sqrt(3)-log(x+1)/3$
 191 diff(%,x);
 192 2/(3*((2*x-1)^2/3+1))+(2*x-1)/(6*(x^2-x+1))-1/(3*(x+1))$
 193 ratsimp(%);
 194 x/(x^3+1)$