itab.cpp

   1 /* Table of integrals
   2
   3 The symbol f is just a dummy symbol for creating a list f(A,B,C,C,...) where
   4
   5         A       is the template expression
   6
   7         B       is the result expression
   8
   9         C       is an optional list of conditional expressions
  10 */
  11
  12 #include "stdafx.h"
  13 #include "defs.h"
  14
  15 const char *itab[] = {
  16 // 1
  17         "f(a,a*x)",
  18 // 9 (need a caveat for 7 so we can put 9 after 7)
  19         "f(1/x,log(x))",
  20 // 7
  21         "f(x^a,x^(a+1)/(a+1))",
  22 // 12
  23         "f(exp(a*x),1/a*exp(a*x))",
  24         "f(exp(a*x+b),1/a*exp(a*x+b))",
  25         "f(x*exp(a*x^2),exp(a*x^2)/(2*a))",
  26         "f(x*exp(a*x^2+b),exp(a*x^2+b)/(2*a))",
  27 // 14
  28         "f(log(a*x),x*log(a*x)-x)",
  29 // 15
  30         "f(a^x,a^x/log(a),or(not(number(a)),a>0))",
  31 // 16
  32         "f(1/(a+x^2),1/sqrt(a)*arctan(x/sqrt(a)),or(not(number(a)),a>0))",
  33 // 17
  34         "f(1/(a-x^2),1/sqrt(a)*arctanh(x/sqrt(a)))",
  35 // 19
  36         "f(1/sqrt(a-x^2),arcsin(x/(sqrt(a))))",
  37 // 20
  38         "f(1/sqrt(a+x^2),log(x+sqrt(a+x^2)))",
  39 // 27
  40         "f(1/(a+b*x),1/b*log(a+b*x))",
  41 // 28
  42         "f(1/(a+b*x)^2,-1/(b*(a+b*x)))",
  43 // 29
  44         "f(1/(a+b*x)^3,-1/(2*b)*1/(a+b*x)^2)",
  45 // 30
  46         "f(x/(a+b*x),x/b-a*log(a+b*x)/b/b)",
  47 // 31
  48         "f(x/(a+b*x)^2,1/b^2*(log(a+b*x)+a/(a+b*x)))",
  49 // 33
  50         "f(x^2/(a+b*x),1/b^2*(1/2*(a+b*x)^2-2*a*(a+b*x)+a^2*log(a+b*x)))",
  51 // 34
  52         "f(x^2/(a+b*x)^2,1/b^3*(a+b*x-2*a*log(a+b*x)-a^2/(a+b*x)))",
  53 // 35
  54         "f(x^2/(a+b*x)^3,1/b^3*(log(a+b*x)+2*a/(a+b*x)-1/2*a^2/(a+b*x)^2))",
  55 // 37
  56         "f(1/x*1/(a+b*x),-1/a*log((a+b*x)/x))",
  57 // 38
  58         "f(1/x*1/(a+b*x)^2,1/a*1/(a+b*x)-1/a^2*log((a+b*x)/x))",
  59 // 39
  60         "f(1/x*1/(a+b*x)^3,1/a^3*(1/2*((2*a+b*x)/(a+b*x))^2+log(x/(a+b*x))))",
  61 // 40
  62         "f(1/x^2*1/(a+b*x),-1/(a*x)+b/a^2*log((a+b*x)/x))",
  63 // 41
  64         "f(1/x^3*1/(a+b*x),(2*b*x-a)/(2*a^2*x^2)+b^2/a^3*log(x/(a+b*x)))",
  65 // 42
  66         "f(1/x^2*1/(a+b*x)^2,-(a+2*b*x)/(a^2*x*(a+b*x))+2*b/a^3*log((a+b*x)/x))",
  67 // 60
  68         "f(1/(a+b*x^2),1/sqrt(a*b)*arctan(x*sqrt(a*b)/a),or(not(number(a*b)),a*b>0))",
  69 // 61
  70         "f(1/(a+b*x^2),1/(2*sqrt(-a*b))*log((a+x*sqrt(-a*b))/(a-x*sqrt(-a*b))),or(not(number(a*b)),a*b<0))",
  71 // 62 is the same as 60
  72 // 63
  73         "f(x/(a+b*x^2),1/2*1/b*log(a+b*x^2))",
  74 //64
  75         "f(x^2/(a+b*x^2),x/b-a/b*integral(1/(a+b*x^2),x))",
  76 //65
  77         "f(1/(a+b*x^2)^2,x/(2*a*(a+b*x^2))+1/2*1/a*integral(1/(a+b*x^2),x))",
  78 //66 is covered by 61
  79 //70
  80         "f(1/x*1/(a+b*x^2),1/2*1/a*log(x^2/(a+b*x^2)))",
  81 //71
  82         "f(1/x^2*1/(a+b*x^2),-1/(a*x)-b/a*integral(1/(a+b*x^2),x))",
  83 //74
  84         "f(1/(a+b*x^3),1/3*1/a*(a/b)^(1/3)*(1/2*log(((a/b)^(1/3)+x)^3/(a+b*x^3))+sqrt(3)*arctan((2*x-(a/b)^(1/3))*(a/b)^(-1/3)/sqrt(3))))",
  85 //76
  86         "f(x^2/(a+b*x^3),1/3*1/b*log(a+b*x^3))",
  87 //77
  88         "f(1/(a+b*x^4),1/2*1/a*(a/b/4)^(1/4)*(1/2*log((x^2+2*(a/b/4)^(1/4)*x+2*(a/b/4)^(1/2))/(x^2-2*(a/b/4)^(1/4)*x+2*(a/b/4)^(1/2)))+arctan(2*(a/b/4)^(1/4)*x/(2*(a/b/4)^(1/2)-x^2))),or(not(number(a*b)),a*b>0))",
  89 //78
  90         "f(1/(a+b*x^4),1/2*(-a/b)^(1/4)/a*(1/2*log((x+(-a/b)^(1/4))/(x-(-a/b)^(1/4)))+arctan(x*(-a/b)^(-1/4))),or(not(number(a*b)),a*b<0))",
  91 //79
  92         "f(x/(a+b*x^4),1/2*sqrt(b/a)/b*arctan(x^2*sqrt(b/a)),or(not(number(a*b)),a*b>0))",
  93 //80
  94         "f(x/(a+b*x^4),1/4*sqrt(-b/a)/b*log((x^2-sqrt(-a/b))/(x^2+sqrt(-a/b))),or(not(number(a*b)),a*b<0))",
  95 //81
  96         "f(x^2/(a+b*x^4),1/4*1/b*(a/b/4)^(-1/4)*(1/2*log((x^2-2*(a/b/4)^(1/4)*x+2*sqrt(a/b/4))/(x^2+2*(a/b/4)^(1/4)*x+2*sqrt(a/b/4)))+arctan(2*(a/b/4)^(1/4)*x/(2*sqrt(a/b/4)-x^2))),or(not(number(a*b)),a*b>0))",
  97 //82
  98         "f(x^2/(a+b*x^4),1/4*1/b*(-a/b)^(-1/4)*(log((x-(-a/b)^(1/4))/(x+(-a/b)^(1/4)))+2*arctan(x*(-a/b)^(-1/4))),or(not(number(a*b)),a*b<0))",
  99 //83
 100         "f(x^3/(a+b*x^4),1/4*1/b*log(a+b*x^4))",
 101 //124
 102         "f(sqrt(a+b*x),2/3*1/b*sqrt((a+b*x)^3))",
 103 //125
 104         "f(x*sqrt(a+b*x),-2*(2*a-3*b*x)*sqrt((a+b*x)^3)/15/b^2)",
 105 //126
 106         "f(x^2*sqrt(a+b*x),2*(8*a^2-12*a*b*x+15*b^2*x^2)*sqrt((a+b*x)^3)/105/b^3)",
 107 //128
 108         "f(sqrt(a+b*x)/x,2*sqrt(a+b*x)+a*integral(1/x*1/sqrt(a+b*x),x))",
 109 //129
 110         "f(sqrt(a+b*x)/x^2,-sqrt(a+b*x)/x+b/2*integral(1/x*1/sqrt(a+b*x),x))",
 111 //131
 112         "f(1/sqrt(a+b*x),2*sqrt(a+b*x)/b)",
 113 //132
 114         "f(x/sqrt(a+b*x),-2/3*(2*a-b*x)*sqrt(a+b*x)/b^2)",
 115 //133
 116         "f(x^2/sqrt(a+b*x),2/15*(8*a^2-4*a*b*x+3*b^2*x^2)*sqrt(a+b*x)/b^3)",
 117 //135
 118         "f(1/x*1/sqrt(a+b*x),1/sqrt(a)*log((sqrt(a+b*x)-sqrt(a))/(sqrt(a+b*x)+sqrt(a))),or(not(number(a)),a>0))",
 119 //136
 120         "f(1/x*1/sqrt(a+b*x),2/sqrt(-a)*arctan(sqrt(-(a+b*x)/a)),or(not(number(a)),a<0))",
 121 //137
 122         "f(1/x^2*1/sqrt(a+b*x),-sqrt(a+b*x)/a/x-1/2*b/a*integral(1/x*1/sqrt(a+b*x),x))",
 123 //156
 124         "f(sqrt(x^2+a),1/2*(x*sqrt(x^2+a)+a*log(x+sqrt(x^2+a))))",
 125 //157
 126         "f(1/sqrt(x^2+a),log(x+sqrt(x^2+a)))",
 127 //158
 128         "f(1/x*1/sqrt(x^2+a),arcsec(x/sqrt(-a))/sqrt(-a),or(not(number(a)),a<0))",
 129 //159
 130         "f(1/x*1/sqrt(x^2+a),-1/sqrt(a)*log((sqrt(a)+sqrt(x^2+a))/x),or(not(number(a)),a>0))",
 131 //160
 132         "f(sqrt(x^2+a)/x,sqrt(x^2+a)-sqrt(a)*log((sqrt(a)+sqrt(x^2+a))/x),or(not(number(a)),a>0))",
 133 //161
 134         "f(sqrt(x^2+a)/x,sqrt(x^2+a)-sqrt(-a)*arcsec(x/sqrt(-a)),or(not(number(a)),a<0))",
 135 //162
 136         "f(x/sqrt(x^2+a),sqrt(x^2+a))",
 137 //163
 138         "f(x*sqrt(x^2+a),1/3*sqrt((x^2+a)^3))",
 139 //164 need an unexpanded version?
 140         "f(sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/4*(x*sqrt((x^2+a^(1/3))^3)+3/2*a^(1/3)*x*sqrt(x^2+a^(1/3))+3/2*a^(2/3)*log(x+sqrt(x^2+a^(1/3)))))",
 141         // match doesn't work for the following
 142         "f(sqrt(-a+x^6-3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/4*(x*sqrt((x^2-a^(1/3))^3)-3/2*a^(1/3)*x*sqrt(x^2-a^(1/3))+3/2*a^(2/3)*log(x+sqrt(x^2-a^(1/3)))))",
 143 //165
 144         "f(1/sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),x/a^(1/3)/sqrt(x^2+a^(1/3)))",
 145 //166
 146         "f(x/sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),-1/sqrt(x^2+a^(1/3)))",
 147 //167
 148         "f(x*sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/5*sqrt((x^2+a^(1/3))^5))",
 149 //168
 150         "f(x^2*sqrt(x^2+a),1/4*x*sqrt((x^2+a)^3)-1/8*a*x*sqrt(x^2+a)-1/8*a^2*log(x+sqrt(x^2+a)))",
 151 //169
 152         "f(x^3*sqrt(x^2+a),(1/5*x^2-2/15*a)*sqrt((x^2+a)^3),and(number(a),a>0))",
 153 //170
 154         "f(x^3*sqrt(x^2+a),sqrt((x^2+a)^5)/5-a*sqrt((x^2+a)^3)/3,and(number(a),a<0))",
 155 //171
 156         "f(x^2/sqrt(x^2+a),1/2*x*sqrt(x^2+a)-1/2*a*log(x+sqrt(x^2+a)))",
 157 //172
 158         "f(x^3/sqrt(x^2+a),1/3*sqrt((x^2+a)^3)-a*sqrt(x^2+a))",
 159 //173
 160         "f(1/x^2*1/sqrt(x^2+a),-sqrt(x^2+a)/a/x)",
 161 //174
 162         "f(1/x^3*1/sqrt(x^2+a),-1/2*sqrt(x^2+a)/a/x^2+1/2*log((sqrt(a)+sqrt(x^2+a))/x)/a^(3/2),or(not(number(a)),a>0))",
 163 //175
 164         "f(1/x^3*1/sqrt(x^2-a),1/2*sqrt(x^2-a)/a/x^2+1/2*1/(a^(3/2))*arcsec(x/(a^(1/2))),or(not(number(a)),a>0))",
 165 //176+
 166         "f(x^2*sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/6*x*sqrt((x^2+a^(1/3))^5)-1/24*a^(1/3)*x*sqrt((x^2+a^(1/3))^3)-1/16*a^(2/3)*x*sqrt(x^2+a^(1/3))-1/16*a*log(x+sqrt(x^2+a^(1/3))),or(not(number(a)),a>0))",
 167 //176-
 168         "f(x^2*sqrt(-a-3*a^(1/3)*x^4+3*a^(2/3)*x^2+x^6),1/6*x*sqrt((x^2-a^(1/3))^5)+1/24*a^(1/3)*x*sqrt((x^2-a^(1/3))^3)-1/16*a^(2/3)*x*sqrt(x^2-a^(1/3))+1/16*a*log(x+sqrt(x^2-a^(1/3))),or(not(number(a)),a>0))",
 169 //177+
 170         "f(x^3*sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/7*sqrt((x^2+a^(1/3))^7)-1/5*a^(1/3)*sqrt((x^2+a^(1/3))^5),or(not(number(a)),a>0))",
 171 //177-
 172         "f(x^3*sqrt(-a-3*a^(1/3)*x^4+3*a^(2/3)*x^2+x^6),1/7*sqrt((x^2-a^(1/3))^7)+1/5*a^(1/3)*sqrt((x^2-a^(1/3))^5),or(not(number(a)),a>0))",
 173 //196
 174         "f(1/(x-a)/sqrt(x^2-a^2),-sqrt(x^2-a^2)/a/(x-a))",
 175 //197
 176         "f(1/(x+a)/sqrt(x^2-a^2),sqrt(x^2-a^2)/a/(x+a))",
 177 //200+
 178         "f(sqrt(a-x^2),1/2*(x*sqrt(a-x^2)+a*arcsin(x/sqrt(abs(a)))))",
 179 //201           (seems to be handled somewhere else)
 180 //202
 181         "f(1/x*1/sqrt(a-x^2),-1/sqrt(a)*log((sqrt(a)+sqrt(a-x^2))/x),or(not(number(a)),a>0))",
 182 //203
 183         "f(sqrt(a-x^2)/x,sqrt(a-x^2)-sqrt(a)*log((sqrt(a)+sqrt(a-x^2))/x),or(not(number(a)),a>0))",
 184 //204
 185         "f(x/sqrt(a-x^2),-sqrt(a-x^2))",
 186 //205
 187         "f(x*sqrt(a-x^2),-1/3*sqrt((a-x^2)^3))",
 188 //210
 189         "f(x^2*sqrt(a-x^2),-x/4*sqrt((a-x^2)^3)+1/8*a*(x*sqrt(a-x^2)+a*arcsin(x/sqrt(a))),or(not(number(a)),a>0))",
 190 //211
 191         "f(x^3*sqrt(a-x^2),(-1/5*x^2-2/15*a)*sqrt((a-x^2)^3),or(not(number(a)),a>0))",
 192 //214
 193         "f(x^2/sqrt(a-x^2),-x/2*sqrt(a-x^2)+a/2*arcsin(x/sqrt(a)),or(not(number(a)),a>0))",
 194 //215
 195         "f(1/x^2*1/sqrt(a-x^2),-sqrt(a-x^2)/a/x,or(not(number(a)),a>0))",
 196 //216
 197         "f(sqrt(a-x^2)/x^2,-sqrt(a-x^2)/x-arcsin(x/sqrt(a)),or(not(number(a)),a>0))",
 198 //217
 199         "f(sqrt(a-x^2)/x^3,-1/2*sqrt(a-x^2)/x^2+1/2*log((sqrt(a)+sqrt(a-x^2))/x)/sqrt(a),or(not(number(a)),a>0))",
 200 //218
 201         "f(sqrt(a-x^2)/x^4,-1/3*sqrt((a-x^2)^3)/a/x^3,or(not(number(a)),a>0))",
 202 // 273
 203         "f(sqrt(a*x^2+b),x*sqrt(a*x^2+b)/2+b*log(x*sqrt(a)+sqrt(a*x^2+b))/2/sqrt(a),and(number(a),a>0))",
 204 // 274
 205         "f(sqrt(a*x^2+b),x*sqrt(a*x^2+b)/2+b*arcsin(x*sqrt(-a/b))/2/sqrt(-a),and(number(a),a<0))",
 206 // 290
 207         "f(sin(a*x),-cos(a*x)/a)",
 208 // 291
 209         "f(cos(a*x),sin(a*x)/a)",
 210 // 292
 211         "f(tan(a*x),-log(cos(a*x))/a)",
 212 // 293
 213         "f(1/tan(a*x),log(sin(a*x))/a)",
 214 // 294
 215         "f(1/cos(a*x),log(tan(pi/4+a*x/2))/a)",
 216 // 295
 217         "f(1/sin(a*x),log(tan(a*x/2))/a)",
 218 // 296
 219         "f(sin(a*x)^2,x/2-sin(2*a*x)/(4*a))",
 220 // 297
 221         "f(sin(a*x)^3,-cos(a*x)*(sin(a*x)^2+2)/(3*a))",
 222 // 298
 223         "f(sin(a*x)^4,3/8*x-sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a))",
 224 // 302
 225         "f(cos(a*x)^2,x/2+sin(2*a*x)/(4*a))",
 226 // 303
 227         "f(cos(a*x)^3,sin(a*x)*(cos(a*x)^2+2)/(3*a))",
 228 // 304
 229         "f(cos(a*x)^4,3/8*x+sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a))",
 230 // 308
 231         "f(1/sin(a*x)^2,-1/(a*tan(a*x)))",
 232 // 312
 233         "f(1/cos(a*x)^2,tan(a*x)/a)",
 234 // 318
 235         "f(sin(a*x)*cos(a*x),sin(a*x)^2/(2*a))",
 236 // 320
 237         "f(sin(a*x)^2*cos(a*x)^2,-sin(4*a*x)/(32*a)+x/8)",
 238 // 326
 239         "f(sin(a*x)/cos(a*x)^2,1/(a*cos(a*x)))",
 240 // 327
 241         "f(sin(a*x)^2/cos(a*x),(log(tan(pi/4+a*x/2))-sin(a*x))/a)",
 242 // 328
 243         "f(cos(a*x)/sin(a*x)^2,-1/(a*sin(a*x)))",
 244 // 329
 245         "f(1/(sin(a*x)*cos(a*x)),log(tan(a*x))/a)",
 246 // 330
 247         "f(1/(sin(a*x)*cos(a*x)^2),(1/cos(a*x)+log(tan(a*x/2)))/a)",
 248 // 331
 249         "f(1/(sin(a*x)^2*cos(a*x)),(log(tan(pi/4+a*x/2))-1/sin(a*x))/a)",
 250 // 333
 251         "f(1/(sin(a*x)^2*cos(a*x)^2),-2/(a*tan(2*a*x)))",
 252 // 335
 253         "f(sin(a+b*x),-cos(a+b*x)/b)",
 254 // 336
 255         "f(cos(a+b*x),sin(a+b*x)/b)",
 256 // 337+ (with the addition of b)
 257         "f(1/(b+b*sin(a*x)),-tan(pi/4-a*x/2)/a/b)",
 258 // 337- (with the addition of b)
 259         "f(1/(b-b*sin(a*x)),tan(pi/4+a*x/2)/a/b)",
 260 // 338 (with the addition of b)
 261         "f(1/(b+b*cos(a*x)),tan(a*x/2)/a/b)",
 262 // 339 (with the addition of b)
 263         "f(1/(b-b*cos(a*x)),-1/tan(a*x/2)/a/b)",
 264 // 340
 265         "f(1/(a+b*sin(x)),1/sqrt(b^2-a^2)*log((a*tan(x/2)+b-sqrt(b^2-a^2))/(a*tan(x/2)+b+sqrt(b^2-a^2))),b^2-a^2)", // check that b^2-a^2 is not zero
 266 // 341
 267         "f(1/(a+b*cos(x)),1/sqrt(b^2-a^2)*log((sqrt(b^2-a^2)*tan(x/2)+a+b)/(sqrt(b^2-a^2)*tan(x/2)-a-b)),b^2-a^2)", // check that b^2-a^2 is not zero
 268 // 389
 269         "f(x*sin(a*x),sin(a*x)/a^2-x*cos(a*x)/a)",
 270 // 390
 271         "f(x^2*sin(a*x),2*x*sin(a*x)/a^2-(a^2*x^2-2)*cos(a*x)/a^3)",
 272 // 393
 273         "f(x*cos(a*x),cos(a*x)/a^2+x*sin(a*x)/a)",
 274 // 394
 275         "f(x^2*cos(a*x),2*x*cos(a*x)/a^2+(a^2*x^2-2)*sin(a*x)/a^3)",
 276 // 441
 277         "f(arcsin(a*x),x*arcsin(a*x)+sqrt(1-a^2*x^2)/a)",
 278 // 442
 279         "f(arccos(a*x),x*arccos(a*x)+sqrt(1-a^2*x^2)/a)",
 280 // 443
 281         "f(arctan(a*x),x*arctan(a*x)-1/2*log(1+a^2*x^2)/a)",
 282 // 485 (with addition of a)
 283         "f(log(a*x),x*log(a*x)-x)",
 284 // 486 (with addition of a)
 285         "f(x*log(a*x),x^2*log(a*x)/2-x^2/4)",
 286 // 487 (with addition of a)
 287         "f(x^2*log(a*x),x^3*log(a*x)/3-1/9*x^3)",
 288 // 489
 289         "f(log(x)^2,x*log(x)^2-2*x*log(x)+2*x)",
 290 // 493 (with addition of a)
 291         "f(1/x*1/(a+log(x)),log(a+log(x)))",
 292 // 499
 293         "f(log(a*x+b),(a*x+b)*log(a*x+b)/a-x)",
 294 // 500
 295         "f(log(a*x+b)/x^2,a/b*log(x)-(a*x+b)*log(a*x+b)/b/x)",
 296 // 554
 297         "f(sinh(x),cosh(x))",
 298 // 555
 299         "f(cosh(x),sinh(x))",
 300 // 556
 301         "f(tanh(x),log(cosh(x)))",
 302 // 560
 303         "f(x*sinh(x),x*cosh(x)-sinh(x))",
 304 // 562
 305         "f(x*cosh(x),x*sinh(x)-cosh(x))",
 306 // 566
 307         "f(sinh(x)^2,sinh(2*x)/4-x/2)",
 308 // 569
 309         "f(tanh(x)^2,x-tanh(x))",
 310 // 572
 311         "f(cosh(x)^2,sinh(2*x)/4+x/2)",
 312 // ?
 313         "f(x^3*exp(a*x^2),exp(a*x^2)*(x^2/a-1/(a^2))/2)",
 314 // ?
 315         "f(x^3*exp(a*x^2+b),exp(a*x^2)*exp(b)*(x^2/a-1/(a^2))/2)",
 316 // ?
 317         "f(exp(a*x^2),-i*sqrt(pi)*erf(i*sqrt(a)*x)/sqrt(a)/2)",
 318 // ?
 319         "f(erf(a*x),x*erf(a*x)+exp(-a^2*x^2)/a/sqrt(pi))",
 320
 321 // these are needed for the surface integral in the manual
 322
 323         "f(x^2*(1-x^2)^(3/2),(x*sqrt(1-x^2)*(-8*x^4+14*x^2-3)+3*arcsin(x))/48)",
 324         "f(x^2*(1-x^2)^(5/2),(x*sqrt(1-x^2)*(48*x^6-136*x^4+118*x^2-15)+15*arcsin(x))/384)",
 325         "f(x^4*(1-x^2)^(3/2),(-x*sqrt(1-x^2)*(16*x^6-24*x^4+2*x^2+3)+3*arcsin(x))/128)",
 326
 327         "f(x*exp(a*x),exp(a*x)*(a*x-1)/(a^2))",
 328         "f(x*exp(a*x+b),exp(a*x+b)*(a*x-1)/(a^2))",
 329
 330         "f(x^2*exp(a*x),exp(a*x)*(a^2*x^2-2*a*x+2)/(a^3))",
 331         "f(x^2*exp(a*x+b),exp(a*x+b)*(a^2*x^2-2*a*x+2)/(a^3))",
 332
 333         "f(x^3*exp(a*x),exp(a*x)*x^3/a-3/a*integral(x^2*exp(a*x),x))",
 334         "f(x^3*exp(a*x+b),exp(a*x+b)*x^3/a-3/a*integral(x^2*exp(a*x+b),x))",
 335
 336         NULL,
 337 };