1% Test cases for definite integration. 2 3int(x/(x+2),x,2,6); 4 5 62*( - log(2) + 2) 7 8 9int(sin x,x,0,pi/2); 10 11 121 13 14 15int(log(x),x,1,5); 16 17 185*log(5) - 4 19 20 21int((1+x**2/p**2)**(1/2),x,0,p); 22 23 24 p*(sqrt(2) + log(sqrt(2) + 1)) 25-------------------------------- 26 2 27 28 29int(x**9+y+y**x+x,x,0,2); 30 31 32 2 33 10*log(y)*y + 522*log(y) + 5*y - 5 34------------------------------------- 35 5*log(y) 36 37 38 39% Collected by Kerry Gaskell, ZIB, 1993/94. 40 41int(x^2*log(1+x),x,0,infinity); 42 43 44 2 45int(x *log(1 + x),x,0,infinity) 46 47 48int(x*e^(-1/2x),x,0,infinity); 49 50 514 52 53 54int(x/4*e^(-1/2x),x,0,infinity); 55 56 571 58 59 60int(sqrt(2)*x^(1/2)*e^(-1/2x),x,0,infinity); 61 62 632*sqrt(pi) 64 65 66int(x^(3/2)*e^(-x),x,0,infinity); 67 68 69 3*sqrt(pi) 70------------ 71 4 72 73 74int(sqrt(pi)*x^(3/2)*e^(-x),x,0,infinity); 75 76 77 3*pi 78------ 79 4 80 81 82int(x*log(1+1/x),x,0,infinity); 83 84 85 1 86int(x*log(1 + ---),x,0,infinity) 87 x 88 89 90int(si(1/x),x,0,infinity); 91 92 93 1 94int(si(---),x,0,infinity) 95 x 96 97 98int(cos(1/x),x,0,infinity); 99 100 101 1 102int(cos(---),x,0,infinity) 103 x 104 105 106int(sin(x^2),x,0,infinity); 107 108 109 sqrt(pi)*sqrt(2) 110------------------ 111 4 112 113 114int(sin(x^(3/2)),x,0,infinity); 115 116 117 2/3 5 118 sqrt(pi)*2 *gamma(---) 119 6 120-------------------------- 121 2 122 3*gamma(---) 123 3 124 125 126int(besselj(2,x),x,0,infinity); 127 128 1291 130 131 132int(besselj(2,y^(5/4)),y,0,infinity); 133 134 135 4/5 7 136 2*2 *gamma(---) 137 5 138------------------- 139 8 140 5*gamma(---) 141 5 142 143 144int(x^(-1)*besselj(2,sqrt(x)),x,0,infinity); 145 146 1471 148 149 150int(bessely(2,x),x,0,infinity); 151 152 153int(bessely(2,x),x,0,infinity) 154 155 156int(x*besseli(2,x),x,0,infinity); 157 158 159int(x*besseli(2,x),x,0,infinity) 160 161 162int(besselk(0,x),x,0,infinity); 163 164 165 pi 166---- 167 2 168 169 170int(x^2*besselk(2,x),x,0,infinity); 171 172 173 3*pi 174------ 175 2 176 177 178int(sinh(x),x,0,infinity); 179 180 181int(sinh(x),x,0,infinity) 182 183 184int(cosh(2*x),x,0,infinity); 185 186 187int(cosh(2*x),x,0,infinity) 188 189 190int(-3*ei(-x),x,0,infinity); 191 192 1933 194 195 196int(x*shi(x),x,0,infinity); 197 198 199int(x*shi(x),x,0,infinity) 200 201 202int(x*fresnel_c(x),x,0,infinity); 203 204 205int(x*fresnel_c(x),x,0,infinity) 206 207 208int(x^3*e^(-2*x),x,0,infinity); 209 210 211 3 212--- 213 8 214 215 216int(x^(-1)*sin(x/3),x,0,infinity); 217 218 219 pi 220---- 221 2 222 223 224int(x^(-1/2)*sin(x),x,0,infinity); 225 226 227 sqrt(pi)*sqrt(2) 228------------------ 229 2 230 231 232int(2*x^(-1/2)*cos(x),x,0,infinity); 233 234 235sqrt(pi)*sqrt(2) 236 237 238int(sin x + cos x,x,0,infinity); 239 240 241int(sin(x) + cos(x),x,0,infinity) 242 243 244int(ei(-x) + sin(x^2),x,0,infinity); 245 246 247 sqrt(pi)*sqrt(2) - 4 248---------------------- 249 4 250 251 252int(x^(-1)*(sin (-2*x) + sin(x^2)),x,0,infinity); 253 254 255 - pi 256------- 257 4 258 259 260int(x^(-1)*(cos(x/2) - cos(x/3)),x,0,infinity); 261 262 263 3 264 - log(---) 265 2 266 267 268int(x^(-1)*(cos x - cos(2*x)),x,0,infinity); 269 270 271log(2) 272 273 274int(x^(-1)*(cos(x) - cos(x)),x,0,infinity); 275 276 2770 278 279 280int(2,x,0,infinity); 281 282 283int(2,x,0,infinity) 284 285 286int(cos(x)*si(x),x,0,infinity); 287 288 289int(cos(x)*si(x),x,0,infinity) 290 291 292int(2*cos(x)*e^(-x),x,0,infinity); 293 294 2951 296 297 298int(x/2*cos(x)*e^(-x),x,0,infinity); 299 300 3010 302 303 304int(x^2/4*cos(x)*e^(-2*x),x,0,infinity); 305 306 307 1 308----- 309 125 310 311 312int(1/(2*x)*sin(x)*e^(-3*x),x,0,infinity); 313 314 315 1 316 atan(---) 317 3 318----------- 319 2 320 321 322int(3/x^2*sin(x)*e^(-x),x,0,infinity); 323 324 325 3 - x 326int(----*sin(x)*e ,x,0,infinity) 327 2 328 x 329 330 331int(cos(sqrt(x))*e^(-x),x,0,infinity); 332 333 334 i 1/4 335 sqrt( - pi)*erf(---) + 2*e 336 2 337------------------------------- 338 1/4 339 2*e 340 341 342int(e^(-x)*besselj(2,x),x,0,infinity); 343 344 345 - 2*sqrt(2) + 3 346------------------ 347 sqrt(2) 348 349 350int(cos(x^2)*e^(-x),x,0,infinity); 351 352 353 1 1 1 1 1 354(pi*( - 2*cos(---)*fresnel_s(---) + cos(---) + 2*fresnel_c(---)*sin(---) 355 4 4 4 4 4 356 357 1 358 - sin(---)))/(2*sqrt(pi)*sqrt(2)) 359 4 360 361 362int(erf(x)*e^(-x),x,0,infinity); 363 364 365 1/4 1 366e *( - erf(---) + 1) 367 2 368 369 370int(besseli(2,x)*e^(-x),x,0,infinity); 371 372 373 - 1 1 3742*hypergeometric({------},{},1) + hypergeometric({---},{},1) - 2 375 2 2 376 377 378int(e^(-x^2)*cos(x),x,0,infinity); 379 380 381 sqrt(pi) 382---------- 383 1/4 384 2*e 385 386 387int(x^(-1)*sin(x)*cos(x),x,0,infinity); 388 389 390 pi 391---- 392 4 393 394 395int(x^(-1)*sin(x)*cos(2*x),x,0,infinity); 396 397 3980 399 400 401int(x^(-1)*sin(x)*cos(x/2),x,0,infinity); 402 403 404 pi 405---- 406 2 407 408 409int(e^x,x,0,infinity); 410 411 412 x 413int(e ,x,0,infinity) 414 415 416int(e^(-x^2 - x),x,0,infinity); 417 418 419 1/4 1 420 e *pi*( - erf(---) + 1) 421 2 422--------------------------- 423 2*sqrt(pi) 424 425 426int(e^(-(x+x^2+1)),x,0,infinity); 427 428 429 1/4 1 430 e *pi*( - erf(---) + 1) 431 2 432--------------------------- 433 2*sqrt(pi)*e 434 435 436int(e^(-(x+1/x)^2),x,0,infinity); 437 438 439 sqrt(pi) 440---------- 441 4 442 2*e 443 444 445int(e^(-(x+2))*sin(x),x,0,infinity); 446 447 448 1 449------ 450 2 451 2*e 452 453 454int(-3*x*e^(-1/2x),x,0,infinity); 455 456 457-12 458 459 460int(x*e^(-1/2*x^2),x,0,infinity); 461 462 4631 464 465 466int(x^2*besselj(2,x),x,0,infinity); 467 468 469 2 470int(x *besselj(2,x),x,0,infinity) 471 472 473int(x*besselk(1,x),x,0,infinity); 474 475 476 pi 477---- 478 2 479 480 481int(-3*ei(-x),x,0,infinity); 482 483 4843 485 486 487int(x^3*e^(-2*x^2),x,0,infinity); 488 489 490 1 491--- 492 8 493 494 495int(sqrt(2)/2*x^(-3/2)*sin x,x,0,infinity); 496 497 498sqrt(pi) 499 500 501int(x^(-1)*(sin(-2*x) + sin(x^2)),x,0,infinity); 502 503 504 - pi 505------- 506 4 507 508 509int(x^(-1)*(cos(3*x) - cos(x/2)),x,0,infinity); 510 511 512 - log(6) 513 514 515int(x^(-1)*(sin x - sin(2*x)),x,0,infinity); 516 517 5180 519 520 521int(1/x*sin(x)*e^(-3*x),x,0,infinity); 522 523 524 1 525atan(---) 526 3 527 528 529int(sin(x)*e^(-x),x,0,infinity); 530 531 532 1 533--- 534 2 535 536 537int(x^(-1)*sin(x)*cos(x),x,0,infinity); 538 539 540 pi 541---- 542 4 543 544 545int(e^(1-x)*e^(2-x^2),x,0,infinity); 546 547 548 1/4 3 1 549 e *e *pi*( - erf(---) + 1) 550 2 551------------------------------ 552 2*sqrt(pi) 553 554 555int(e^(-1/2x),x,0,y); 556 557 558 y/2 559 2*(e - 1) 560-------------- 561 y/2 562 e 563 564 565int(si(x),x,0,y); 566 567 568si(y)*y - 1 + cos(y) 569 570 571int(besselj(2,x^(1/4)),x,0,y); 572 573 574 1/4 1/4 575 - 2*((8*sqrt(y) - y)*besselj(0,y ) - besselj(2,y )*y 576 577 1/4 1/4 1/4 1/4 578 + 4*y *(y + 2)*(y - 2)*besselj(1,y )) 579 580 581int(x*besseli(2,x),x,0,y); 582 583 584 - (2*(besseli(0,y) - 1) - besseli(1,y)*y) 585 586 587int(x^(3/2)*e^(-x),x,0,y); 588 589 590 y 591 3*sqrt(pi)*e *erf(sqrt(y)) - 4*sqrt(y)*y - 6*sqrt(y) 592------------------------------------------------------ 593 y 594 4*e 595 596 597int(sinh(x),x,0,y); 598 599 600 y 2 601 (e - 1) 602----------- 603 y 604 2*e 605 606 607int(cosh(2*x),x,0,y); 608 609 610 2*y y y 611 (e + 1)*(e + 1)*(e - 1) 612------------------------------ 613 2*y 614 4*e 615 616 617int(x*shi(x),x,0,y); 618 619 620 y y 2 621 - (e *(e *(y - 1) - 2*shi(y)*y ) + y + 1) 622-------------------------------------------- 623 y 624 4*e 625 626 627int(x^2*e^(-x^2),x,0,y); 628 629 630 2 631 y 632 sqrt(pi)*e *erf(y) - 2*y 633--------------------------- 634 2 635 y 636 4*e 637 638 639int(x^(-1)/2*sin(x),x,0,y); 640 641 642 si(y) 643------- 644 2 645 646 647int(sin x + cos x,x,0,y); 648 649 650sin(y) + 1 - cos(y) 651 652 653int(sin x + sin(-2*x),x,0,y); 654 655 656 - (2*cos(y) - 1 - cos(2*y)) 657------------------------------ 658 2 659 660 661int(sin(n*x),x,0,y); 662 663 664 - (cos(n*y) - 1) 665------------------- 666 n 667 668 669int(heaviside(x-1),x,0,y); 670 671 672(y - 1)*heaviside(y - 1) 673 674 675 676% Tests of transformations defined in defint package. 677 678laplace_transform(1,x); 679 680 681 1 682--- 683 s 684 685 686laplace_transform(x,x); 687 688 689 1 690---- 691 2 692 s 693 694 695laplace_transform(x^a/factorial(a),x); 696 697 698 1 699------ 700 a 701 s *s 702 703 704laplace_transform(x,e^(-a*x),x); 705 706 707 1 708----------------- 709 2 2 710 a + 2*a*s + s 711 712 713laplace_transform(x^k,e^(-a*x),x); 714 715 716 gamma(k + 1) 717------------------------- 718 k k 719 (a + s) *a + (a + s) *s 720 721 722laplace_transform(cosh(a*x),x); 723 724 725 - s 726--------- 727 2 2 728 a - s 729 730 731laplace_transform(1/(2*a^3),sinh(a*x)-sin(a*x),x); 732 733 734 - 1 735--------- 736 4 4 737 a - s 738 739 740laplace_transform(1/(a^2),1-cos(a*x),x); 741 742 743 1 744----------- 745 2 3 746 a *s + s 747 748 749laplace_transform(1/(b^2-a^2),cos(a*x)-cos(b*x),x); 750 751 752 s 753---------------------------- 754 2 2 2 2 2 2 4 755 a *b + a *s + b *s + s 756 757 758laplace_transform(besselj(0,2*sqrt(k*x)),x); 759 760 761 1 762-------- 763 k/s 764 e *s 765 766 767laplace_transform(Heaviside(x-1),x); 768 769 770 1 771------ 772 s 773 e *s 774 775 776laplace_transform(1/x,sin(k*x),x); 777 778 779 k 780atan(---) 781 s 782 783 784laplace_transform(1/(k*sqrt(pi)),e^(-x^2/(4*k^2)),x); 785 786 787 2 2 2 2 788 k *s k *s 789 - e *erf(k*s) + e 790 791 792laplace_transform(1/k,e^(-k^2/(4*x)),x); 793 794 795 besselk(1,sqrt(s)*k) 796---------------------- 797 sqrt(s) 798 799 800laplace_transform(2/(sqrt(pi*x)),besselk(0,2*sqrt(2*k*x)),x); 801 802 803 k/s k 804 e *besselk(0,---) 805 s 806--------------------- 807 sqrt(s) 808 809 810hankel_transform(x,x); 811 812 813 n + 4 814 gamma(-------) 815 2 816------------------- 817 n - 2 2 818 gamma(-------)*s 819 2 820 821 822Y_transform(x,x); 823 824 825 - n + 4 n + 4 826 gamma(----------)*gamma(-------) 827 2 2 828------------------------------------- 829 - n + 3 n - 1 2 830 gamma(----------)*gamma(-------)*s 831 2 2 832 833 834K_transform(x,x); 835 836 837 - n + 4 n + 4 838 gamma(----------)*gamma(-------) 839 2 2 840---------------------------------- 841 2 842 2*s 843 844 845struveh_transform(x,x); 846 847 848 - n - 3 n + 5 849 gamma(----------)*gamma(-------) 850 2 2 851------------------------------------- 852 - n - 2 n - 2 2 853 gamma(----------)*gamma(-------)*s 854 2 2 855 856 857fourier_sin(e^(-x),x); 858 859 860 s 861-------- 862 2 863 s + 1 864 865 866fourier_sin(sqrt(x),e^(-1/2*x),x); 867 868 869 3*atan(2*s) 870 2*sin(-------------)*pi 871 2 872-------------------------------- 873 2 3/4 874 sqrt(pi)*(4*s + 1) *sqrt(2) 875 876 877fourier_sin(1/x,e^(-a*x),x); 878 879 880 s 881atan(---) 882 a 883 884 885fourier_sin(x^k,x); 886 887 888 k/2 - k k 889 4 *gamma(------)*gamma(---)*k 890 2 2 891--------------------------------- 892 k k 893 4*s *2 *gamma( - k)*s 894 895 896fourier_sin(1/(b-a),(e^(-a*x)-e^(-b*x)),x); 897 898 899 a*s + b*s 900---------------------------- 901 2 2 2 2 2 2 4 902 a *b + a *s + b *s + s 903 904 905fourier_sin(besselj(0,a*x),x); 906 907 908 2 2 909 - a + s 910 heaviside(------------) 911 2 912 a 913------------------------- 914 2 2 915 sqrt( - a + s ) 916 917 918fourier_sin(1/sqrt(pi*x),cos(2*sqrt(k*x)),x); 919 920 921 k k 922 sqrt(s)*sqrt(2)*cos(---) - sqrt(s)*sqrt(2)*sin(---) 923 s s 924----------------------------------------------------- 925 2*s 926 927 928fourier_sin(1/(k*sqrt(pi)),e^(-x^2/(4*k^2)),x); 929 930 931 erf(i*k*s)*i 932-------------- 933 2 2 934 k *s 935 e 936 937 938fourier_cos(e^(-1/2x),x); 939 940 941 2 942---------- 943 2 944 4*s + 1 945 946 947fourier_cos(x,e^(-x),x); 948 949 950 2 951 - s + 1 952--------------- 953 4 2 954 s + 2*s + 1 955 956 957fourier_cos(x,e^(-1/2*x^2),x); 958 959 960 2 961 i*s s /2 962 sqrt(pi)*erf(---------)*i*s + e *sqrt(2) 963 sqrt(2) 964--------------------------------------------- 965 2 966 s /2 967 e *sqrt(2) 968 969 970fourier_cos(2*x^2,e^(-1/2x),x); 971 972 973 2 974 - 384*s + 32 975--------------------------- 976 6 4 2 977 64*s + 48*s + 12*s + 1 978 979 980fourier_cos(x,e^(-a*x),x); 981 982 983 2 2 984 a - s 985------------------- 986 4 2 2 4 987 a + 2*a *s + s 988 989 990fourier_cos(x^n,e^(-a*x),x); 991 992 993 s s 994 cos(atan(---)*n + atan(---))*gamma(n + 1) 995 a a 996------------------------------------------- 997 2 2 (n + 1)/2 998 (a + s ) 999 1000 1001fourier_cos(1/x,sin(k*x),x); 1002 1003 1004 2 2 1005 sign(k - s )*pi + pi 1006----------------------- 1007 4 1008 1009 1010fourier_cos(1/sqrt(pi*x),cos(2*sqrt(k*x)),x); 1011 1012 1013 k k 1014 sqrt(s)*sqrt(2)*cos(---) + sqrt(s)*sqrt(2)*sin(---) 1015 s s 1016----------------------------------------------------- 1017 2*s 1018 1019 1020fourier_cos(1/(k*sqrt(pi)),e^(-x^2/(4*k^2)),x); 1021 1022 1023 1 1024-------- 1025 2 2 1026 k *s 1027 e 1028 1029 1030fourier_cos(1/(pi*x),sin(2*k*sqrt(x)),x); 1031 1032 1033 2 2 1034 k k 1035intfc(----) + intfs(----) 1036 s s 1037 1038 1039fourier_cos(1/(sqrt(pi*x)),e^(-2*k*sqrt(x)),x); 1040 1041 1042 2 2 2 1043 k k k 1044( - 2*sqrt(s)*cos(----)*fresnel_s(----) + sqrt(s)*cos(----) 1045 s s s 1046 1047 2 2 2 1048 k k k 1049 + 2*sqrt(s)*fresnel_c(----)*sin(----) - sqrt(s)*sin(----))/(sqrt(2)*s) 1050 s s s 1051 1052 1053laplace_transform(x^n/factorial(n)*e^(-a*x),x); 1054 1055 1056 1 1057------------------------- 1058 n n 1059 (a + s) *a + (a + s) *s 1060 1061 1062laplace_transform(1/(2*a^2)*(cosh(a*x)-cos(a*x)),x); 1063 1064 1065 - s 1066--------- 1067 4 4 1068 a - s 1069 1070 1071laplace_transform(k*a^k/x*besselj(k,a*x),x); 1072 1073 1074 2*k 1075 a 1076---------------------- 1077 2 2 k 1078 (sqrt(a + s ) + s) 1079 1080 1081fourier_sin(1/x*e^(-3*x),x); 1082 1083 1084 s 1085atan(---) 1086 3 1087 1088 1089fourier_sin(1/(pi*x)*sin(2*k*sqrt(x)),x); 1090 1091 1092 2 2 1093 k k 1094intfc(----) - intfs(----) 1095 s s 1096 1097 1098fourier_cos(x^n*e^(-a*x),x); 1099 1100 1101 s s 1102 cos(atan(---)*n + atan(---))*gamma(n + 1) 1103 a a 1104------------------------------------------- 1105 2 2 (n + 1)/2 1106 (a + s ) 1107 1108 1109fourier_cos(1/(k*sqrt(pi))*e^(-x^2/(4*k^2)),x); 1110 1111 1112 1 1113-------- 1114 2 2 1115 k *s 1116 e 1117 1118 1119end; 1120 1121Tested on x86_64-pc-windows CSL 1122Time (counter 1): 1622 ms plus GC time: 112 ms 1123 1124End of Lisp run after 1.62+0.18 seconds 1125real 2.04 1126user 0.01 1127sys 0.04 1128