1load_package defint; 2 3 4 5% used to do a car/cdr of nil 6int(xxx,x,-infinity, infinity); 7 8 9int(xxx,x, - infinity,infinity) 10 11 12df(erf(x),x); 13 14 15 2*sqrt(pi) 16------------ 17 2 18 x 19 e *pi 20 21 22int(ws,x,-infinity, infinity); 23 24 252 26 27 28int(exp(-x^2)/sqrt(pi),x,-infinity, infinity); 29 30 311 32 33 34exp(-x^2)/sqrt(pi); 35 36 37 1 38-------------- 39 2 40 x 41 sqrt(pi)*e 42 43 44int(ws,x,-infinity, infinity); 45 46 471 48 49 50int(exp(-x^2)/sqrt(pi)/y,x,-infinity, infinity); 51 52 53 1 54--- 55 y 56 57 58exp(-x^2-x)/sqrt(pi); 59 60 61 1 62------------------ 63 2 64 x + x 65 sqrt(pi)*e 66 67 68int(ws,x,-infinity, infinity); 69 70 71 1/4 1 72 e *( - erf(---) + 1) 73 2 74------------------------ 75 2 76 77 78erf(x+1/2); 79 80 81 2*x + 1 82erf(---------) 83 2 84 85 86df(ws,x); 87 88 89 2*sqrt(pi) 90------------------------ 91 2 92 (4*x + 4*x + 1)/4 93 e *pi 94 95 96% doesn't work 97int(ws,x,-infinity, infinity); 98 99 100 x 101 2*sqrt(pi)*e 1 102int(------------------,x,0,infinity) + 1 - erf(---) 103 2 2 104 (4*x + 1)/4 105 e *pi 106 107 108e^(1/4)*erf(x+1/2); 109 110 111 1/4 2*x + 1 112e *erf(---------) 113 2 114 115 116df(ws,x); 117 118 119 1/4 120 2*sqrt(pi)*e 121------------------ 122 2 123 (2*x + 1) /4 124 e *pi 125 126 127% nor does this 128int(ws,x,-infinity, infinity); 129 130 131 1/4 132 2*sqrt(pi)*e 133int(------------------,x, - infinity,infinity) 134 2 135 (2*x + 1) /4 136 e *pi 137 138 139erf(sqrt(log(2))*x); 140 141 142erf(sqrt(log(2))*x) 143 144 145df(ws,x); 146 147 148 2*sqrt(pi)*sqrt(log(2)) 149------------------------- 150 2 151 x 152 2 *pi 153 154 155% or this 156int(ws,x,-infinity, infinity); 157 158 1592 160 161 162int(exp(-5*x^2),x,-infinity, infinity); 163 164 165 sqrt(pi) 166---------- 167 sqrt(5) 168 169 170int(exp(-sqrt(2)*x^2),x,-infinity, infinity); 171 172 173 sqrt(pi) 174---------- 175 1/4 176 2 177 178 179% the following is not recognized as exponential 180int(2^(-x^2),x,-infinity, infinity); 181 182 183 sqrt(pi) 184-------------- 185 sqrt(log(2)) 186 187 188% standard simplification turns the following integrand into 2^(-x^2) 189int(exp(-log(2)*x^2),x,-infinity, infinity); 190 191 192 sqrt(pi) 193-------------- 194 sqrt(log(2)) 195 196 197int(exp(-a*x^2),x,-infinity, infinity); 198 199 200 2 201int(exp( - a*x ),x, - infinity,infinity) 202 203 204int(exp(-abs(a)*x^2),x,-infinity, infinity); 205 206 207 2 208int(exp( - abs(a)*x ),x, - infinity,infinity) 209 210 211int(e^(-abs(a)^2*x^2),x,-infinity,infinity); 212 213 214 2 2 215 - abs(a) *x 216int(e ,x, - infinity,infinity) 217 218 219int(e^(-abs(a^2)*x^2),x,-infinity,infinity); 220 221 222 2 2 223 - abs(a )*x 224int(e ,x, - infinity,infinity) 225 226 227% not simplified since a may be complex 228int(exp(-a^2*x^2),x,-infinity, infinity); 229 230 231 2 2 232int(exp( - a *x ),x, - infinity,infinity) 233 234 235% but works if a is real 236let impart(a)=0; 237 238 239int(exp(-a^2*x^2),x,-infinity, infinity); 240 241 242 sqrt(pi) 243---------- 244 a 245 246 247end; 248 249Tested on x86_64-pc-windows CSL 250Time (counter 1): 63 ms plus GC time: 47 ms 251 252End of Lisp run after 0.06+0.09 seconds 253real 0.33 254user 0.01 255sys 0.03 256