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