1/*************** -*- Mode: MACSYMA; Package: MAXIMA -*- ******************/ 2/*************************************************************************** 3*** ***** 4*** Copyright (c) 1984 by William Schelter,University of Texas ***** 5*** All rights reserved ***** 6***************************************************************************/ 7 8 9kill(functions,arrays,values); 10done$ 11ev(sin(x),exponentialize); 12-%i*(%e^(%i*x)-%e^-(%i*x))/2$ 13taylor(sin(x)/x,x,0,4); 14''(taylor(1-x^2/6+x^4/120,x,0,4))$ 15ev(cos(x)^2-sin(x)^2,sin(x)^2 = 1-cos(x)^2); 162*cos(x)^2-1$ 17(sqrt(-4)+sqrt(2.25))^2; 18(2*%i+1.5)^2$ 19expand(%); 206.0*%i-1.75$ 21expand(sqrt(2*%i)); 22(-1)^(1/4)*sqrt(2)$ 23kill(all); 24done$ 25sin(x)+cos(y)+(w+1)^2+'diff(sin(w),w); 26cos(y)+sin(x)+'diff(sin(w),w,1)+(w+1)^2$ 27ev(%,sin,expand,diff,x = 2,y = 1); 28cos(w)+w^2+2*w+cos(1)+sin(2)+1; 29ev(x+y,x:a+y,y:2); 30y+a+2$ 31'diff(y^2+x*y+x^2,x,2,y,1); 32'diff(y^2+x*y+x^2,x,2,y,1)$ 33ev(%,diff); 340$ 35exp:2*x-3*y = 3; 362*x-3*y = 3$ 37-3*x+2*y = -4; 382*y-3*x = -4$ 39solve([exp,%]); 40[[y = -1/5,x = 6/5]]$ 41ev(exp,%); 423 = 3$ 43x+1/x > gamma(1/2); 44x+1/x > sqrt(%pi)$ 45ev(%,numer,x = 1/2); 462.5 > 1.772453850905516; 47/* tops 20 :2.5 > 1.77245384$ */ 48ev(%,pred); 49true$ 50zeroequiv(sin(2*x)-2*sin(x)*cos(x),x); 51true$ 52zeroequiv(%e^x+x,x); 53false$ 54zeroequiv(log(a*b)-log(a)-log(b),a); 55dontknow$ 56(1/(x+y)^4-3/(y+z)^3)^2; 57(1/(y+x)^4-3/(z+y)^3)^2$ 58expand(%,2,0); 59-6/((y+x)^4*(z+y)^3)+9/(z+y)^6+1/(y+x)^8$ 60expand(a . (b+c . (d+e)+f)); 61a . f+a . c . e+a . c . d+a . b$ 62expand((x+1)^3); 63x^3+3*x^2+3*x+1$ 64exp:(x+1)^7; 65(x+1)^7$ 66expand(%); 67x^7+7*x^6+21*x^5+35*x^4+35*x^3+21*x^2+7*x+1$ 68expand(exp,7,7); 69x^7+7*x^6+21*x^5+35*x^4+35*x^3+21*x^2+7*x+1$ 70ev(a*(b+c)+a*(b+c)^2,expop:1); 71a*(c+b)^2+a*c+a*b$ 72ratexpand((2*x-3*y)^3); 73-27*y^3+54*x*y^2-36*x^2*y+8*x^3$ 74exp:(x-1)/(x+1)^2+1/(x-1); 75(x-1)/(x+1)^2+1/(x-1)$ 76expand(%); 77x/(x^2+2*x+1)-1/(x^2+2*x+1)+1/(x-1)$ 78ratexpand(exp); 792*x^2/(x^3+x^2-x-1)+2/(x^3+x^2-x-1)$ 80sin(x/(x^2+x)) = %e^((log(x)+1)^2-log(x)^2); 81sin(x/(x^2+x)) = %e^((log(x)+1)^2-log(x)^2)$ 82ratsimp(%); 83sin(1/(x+1)) = %e*x^2$ 84b*(a/b-x)+b*x+a; 85b*x+b*(a/b-x)+a$ 86ratsimp(%); 872*a$ 88((x-1)^(3/2)-(x+1)*sqrt(x-1))/sqrt(x-1)/sqrt(x+1); 89((x-1)^(3/2)-sqrt(x-1)*(x+1))/(sqrt(x-1)*sqrt(x+1))$ 90ratsimp(%); 91-2/sqrt(x+1)$ 92ev(x^(a+1/a),ratsimpexpons); 93x^((a^2+1)/a)$ 94(log(x^2+x)-log(x))^a/log(x+1)^(a/2); 95(log(x^2+x)-log(x))^a/log(x+1)^(a/2)$ 96radcan(%); 97log(x+1)^(a/2)$ 98log(a^(2*x)+2*a^x+1)/log(a^x+1); 99log(a^(2*x)+2*a^x+1)/log(a^x+1)$ 100radcan(%); 1012$ 102(%e^x-1)/(%e^(x/2)+1); 103(%e^x-1)/(%e^(x/2)+1)$ 104radcan(%); 105%e^(x/2)-1$ 106kill(all); 107done$ 108combine(a/x+b/x+a/y+b/y); 109(b+a)/y+(b+a)/x$ 110x/(x-y)^2-1/(x-y)-f(x)/(x-y)^3; 111-1/(x-y)+x/(x-y)^2-f(x)/(x-y)^3$ 112multthru((x-y)^3,%); 113-(x-y)^2+x*(x-y)-f(x)$ 114ratexpand(%); 115-y^2+x*y-f(x)$ 116((a+b)^10*s^2+2*a*b*s+(a*b)^2)/(a*b*s^2); 117((b+a)^10*s^2+2*a*b*s+a^2*b^2)/(a*b*s^2)$ 118multthru(%); 1192/s+a*b/s^2+(b+a)^10/(a*b)$ 120multthru(a . (b+c . (d+e)+f)); 121a . f+a . c . (e+d)+a . b$ 122((x+2)^20-2*y)/(x+y)^20+(x+y)^-19-x/(x+y)^20; 1231/(y+x)^19+((x+2)^20-2*y)/(y+x)^20-x/(y+x)^20$ 124xthru(%); 125((x+2)^20-y)/(y+x)^20$ 1262/(x+2)-2/(x+1)+1/(x+1)^2; 1272/(x+2)-2/(x+1)+1/(x+1)^2$ 128ratsimp(%); 129-x/(x^3+4*x^2+5*x+2)$ 130partfrac(%,x); 1312/(x+2)-2/(x+1)+1/(x+1)^2$ 132sort(args(factor(2^63-1))); 133''(sort([7^2,73,127,337,92737,649657]))$ 134factor(z^2*(x+2*y)-4*x-8*y); 135(2*y+x)*(z-2)*(z+2)$ 136x^2*y^2+2*x*y^2+y^2-x^2-2*x-1; 137x^2*y^2+2*x*y^2+y^2-x^2-2*x-1$ 138block([dontfactor:[x]],factor(%/36/(y^2+2*y+1))); 139(x^2+2*x+1)*(y-1)/(36*(y+1))$ 140factor(%e^(3*x)+1); 141(%e^x+1)*(%e^(2*x)-%e^x+1)$ 142factor(x^4+1,a^2-2); 143(x^2-a*x+1)*(x^2+a*x+1)$ 144factor(x^3+x^2*y^2-x*z^2-y^2*z^2); 145-((y^2+x)*(z-x)*(z+x))$ 146(x+2)/(x+3)/(x+b)/(x+c)^2; 147(x+2)/((x+3)*(x+b)*(x+c)^2)$ 148ratsimp(%); 149(x+2)/(x^4+(2*c+b+3)*x^3+(c^2+(2*b+6)*c+3*b)*x^2+((b+3)*c^2+6*b*c)*x+3*b*c^2)$ 150partfrac(%,x); 151-((c^2-4*c-b+6)/((c^4+(-2*b-6)*c^3+(b^2+12*b+9)*c^2+(-6*b^2-18*b)*c+9*b^2) 152 *(x+c))) 153 -((c-2)/((c^2+(-b-3)*c+3*b)*(x+c)^2)) 154 +(b-2)/(((b-3)*c^2+(6*b-2*b^2)*c+b^3-3*b^2)*(x+b)) 155 -(1/(((b-3)*c^2+(18-6*b)*c+9*b-27)*(x+3)))$ 156map('factor,%); 157-((c^2-4*c-b+6)/((c-3)^2*(c-b)^2*(x+c)))-((c-2)/((c-3)*(c-b)*(x+c)^2)) 158 +(b-2)/((b-3)*(c-b)^2*(x+b)) 159 -(1/((b-3)*(c-3)^2*(x+3)))$ 160exp:ratsimp((x^5-1)/(x-1)); 161x^4+x^3+x^2+x+1$ 162subst(a,x,%); 163a^4+a^3+a^2+a+1$ 164factor(exp,%); 165(x-a)*(x-a^2)*(x-a^3)*(x+a^3+a^2+a+1)$ 166factor(x^12+1); 167(x^4+1)*(x^8-x^4+1)$ 168factor(x^99+1); 169(x+1)*(x^2-x+1)*(x^6-x^3+1)*(x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1) 170 *(x^20+x^19-x^17-x^16+x^14+x^13-x^11-x^10-x^9+x^7+x^6-x^4-x^3+x+1) 171 *(x^60+x^57-x^51-x^48+x^42+x^39-x^33-x^30-x^27+x^21+x^18-x^12-x^9+x^3+1)$ 172ev((x+1)*((u+v)^2+a*(w+z)^2),expand); 173a*x*z^2+a*z^2+2*a*w*x*z+2*a*w*z+a*w^2*x+v^2*x+2*u*v*x+u^2*x+a*w^2+v^2+2*u*v 174 +u^2$ 175factorsum(%); 176(x+1)*(a*(z+w)^2+(v+u)^2)$ 177sqfr(4*x^4+4*x^3-3*x^2-4*x-1); 178(2*x+1)^2*(x^2-1)$ 179gfactor(x^4-1); 180(x-1)*(x+1)*(x-%i)*(x+%i)$ 181factor(y^20-7*y^19+11*y^18-8*y^17+59*y^16-461*y^15+1694*y^14+198*y^13-3541* 182 y^12+2818*y^11-33327*y^10-11272*y^9-56656*y^8-12672*y^7+433664*y^6+472064*y^5 183 +241664*y^4+131072*y^3+720896*y^2+1835008*y+1048576); 184(y^8-y^7-11*y^6-19*y^5+101*y^4+76*y^3-176*y^2+64*y+256) 185 *(y^12-6*y^11+16*y^10-39*y^9-19*y^8-75*y^7-315*y^6+300*y^5-304*y^4 186 +2496*y^3+4096*y^2+6144*y+4096)$ 187 188/* sf bug 1650226 */ 189factor(156*x^7+4808*x^6-182041*x^5-1266489*x^4+43104271*x^3+29839285*x^2-2542327662*x+7826952672); 190(x-16)*(x+14)*(x+49)*(2*x-11)*(2*x+21)*(3*x-49)*(13*x-63)$ 191 192/* sf bug 681191 */ 193gfactor(x^4+y^4); 194(y^2-%i*x^2)*(y^2+%i*x^2)$ 195 196/* sf bug 696804 */ 197factor(expand((35*x*y^2+19*x^2+25) * (35*x*y^2+25*x^2+19))); 198(35*x*y^2+19*x^2+25) * (35*x*y^2+25*x^2+19)$ 199 200/* bug reported on mailing list circa 2007-11-27; bignum in exponent */ 201 202is (errcatch (rat (x^(2^128))) = []); 203false; 204 205/* [ 1943756 ] ezgcd() never returns. */ 206ezgcd(n^2+(-2*d+c+b+2)*n+d^2+(-c-b-2)*d+c+b+1, -n^2+(2*d-c-b-2)*n-d^2+(c+b+2)*d+(-b-1)*c-b-1); 207[1,n^2+(-2*d+c+b+2)*n+d^2+(-c-b-2)*d+c+b+1, 208 -n^2+(2*d-c-b-2)*n-d^2+(c+b+2)*d+(-b-1)*c-b-1]; 209 210/* mailing list 2015-12-10: "problem with poly_pseudo_divide" */ 211 212(kill (p, q, a, b, c), 213 p : 50688*y[4]^2*y[5]^10+198144*y[4]^4*y[5]^8+292680*y[4]^6*y[5]^6 214 +193464*y[4]^8*y[5]^4+48240*y[4]^10*y[5]^2, 215 q : (-768*y[5]^7)-2016*y[4]^2*y[5]^5-1788*y[4]^4*y[5]^3 216 -540*y[4]^6*y[5], 217 [a, b, c] : ezgcd(p, q)); 218[12*y[5]^3+12*y[4]^2*y[5], 219 4224*y[4]^2*y[5]^7+12288*y[4]^4*y[5]^5+12102*y[4]^6*y[5]^3 220 +4020*y[4]^8*y[5], 221 (-64*y[5]^4)-104*y[4]^2*y[5]^2-45*y[4]^4]; 222 223is (equal (p, a*b)); 224true; 225 226is (equal (q, a*c)); 227true; 228 229/* bug reported in mailing list thread "when might resimplify be needed?" 230 * on 2019-03-16 231 */ 2323*(2/3^(3/2))^(2/3); 2332^(2/3); 234 235/* [ 1980715 ] wrong radcan-simplification */ 236radcan(3^(1/6)*9^(1/3)); 2373^(5/6); 238 239radcan(81^(1/5)*9^(1/3)); 2403*3^(7/15); 241 242/* SF #2905 Assigning variable twice yields different results */ 243 244block ([var1, var5, var6, diff_expr3], 245 var1 : D^5*Some, 246 var5 : sqrt(var1)*z[D]*r^D, 247 var6 : -sqrt(var1)*r^D/D, 248 diff_expr3 : diff(var5,r), 249 var6*diff_expr3, 250 [var1, var5, var6, diff_expr3, %%]); 251[D^5*Some, 252 r^D*sqrt(Some*D^5)*z[D], 253 -sqrt(Some*D^5)*r^D/D, 254 D*r^(D - 1)*sqrt(Some*D^5)*z[D], 255 -r^(2*D - 1)*Some*D^5*z[D]]; 256