1COMMENT 2 3 THE REDUCE INTEGRATION TEST PACKAGE 4 5 Edited By 6 7 Anthony C. Hearn 8 The RAND Corporation 9 10 11This file is designed to provide a set of representative tests of the 12Reduce integration package. Not all examples go through, even when an 13integral exists, since some of the arguments are outside the domain of 14applicability of the current package. However, future improvements to 15the package will result in more closed-form evaluations in later 16releases. We would appreciate any additional contributions to this test 17file either because they illustrate some feature (good or bad) of the 18current package, or suggest domains which future versions should handle. 19Any suggestions for improved organization of this test file (e.g., in a 20way which corresponds more directly to the organization of a standard 21integration table book such as Gradshteyn and Ryznik) are welcome. 22 23Acknowledgments: 24 25The examples in this file have been contributed by the following. 26Any omissions to this list should be reported to the Editor. 27 28David M. Dahm 29James H. Davenport 30John P. Fitch 31Steven Harrington 32Anthony C. Hearn 33K. Siegfried Koelbig 34Ernst Krupnikov 35Arthur C. Norman 36Herbert Stoyan 37; 38 39 40Comment we first set up a suitable testing functions; 41 42 43fluid '(gcknt!*); 44 45 46 47global '(faillist!* gcnumber!* inittime number!-of!-integrals 48 unintlist!*); 49 50 51 52symbolic operator time; 53 54 55 56symbolic procedure initialize!-integral!-test; 57 begin 58 faillist!* := unintlist!* := nil; 59 number!-of!-integrals := 0; 60 gcnumber!* := gcknt!*; 61 inittime := time() 62 end; 63 64 65initialize!-integral!-test 66 67 68symbolic procedure summarize!-integral!-test; 69 begin scalar totaltime; 70 totaltime := time()-inittime; 71 prin2t 72 " ***** SUMMARY OF INTEGRAL TESTS *****"; 73 terpri(); 74 prin2 "Number of integrals tested: "; 75 prin2t number!-of!-integrals; 76 terpri(); 77 prin2 "Total time taken: "; 78 prin2 totaltime; 79 prin2t " ms"; 80 terpri(); 81 if gcnumber!* 82 then <<prin2 "Number of garbage collections: "; 83 prin2t (gcknt!* - gcnumber!*); 84 terpri()>>; 85 prin2 "Number of incorrect integrals: "; 86 prin2t length faillist!*; 87 terpri(); 88 prin2 "Number of unevaluated integrals: "; 89 prin2t length unintlist!*; 90 terpri(); 91 if faillist!* 92 then <<prin2t "Integrands of incorrect integrals are:"; 93 for each x in reverse faillist!* do mathprint car x>>; 94 if unintlist!* 95 then <<prin2t "Integrands of unevaluated integrals are:"; 96 terpri(); 97 for each x in reverse unintlist!* do mathprint car x>> 98 end; 99 100 101summarize!-integral!-test 102 103 104procedure testint(a,b); 105 begin scalar der,diffce,res,tt; 106 tt:=time(); 107 symbolic (number!-of!-integrals := number!-of!-integrals + 1); 108 res:=int(a,b); 109% write "time for integral: ",time()-tt," ms"; 110 der := df(res,b); 111 diffce := der-a; 112 if diffce neq 0 113 then begin for all x let cot x=cos x/sin x, 114 sec x=1/cos x, 115 sin x**2=1-cos x**2, 116 tan(x/2)=sin x/(1+cos x), 117 tan x=sin x/cos x, 118 tanh x= 119 (e**(x)-e**(-x))/(e**x+e**(-x)), 120 coth x= 1/tanh x; 121 diffce := diffce; 122 for all x clear cot x,sec x,sin x**2,tan x,tan(x/2), 123 tanh x,coth x 124 end; 125 %hopefully, difference appeared non-zero due to absence of 126 %above transformations; 127 if diffce neq 0 128 then <<on combineexpt; diffce := diffce; off combineexpt>>; 129 if diffce neq 0 130 then begin scalar !*reduced; 131 symbolic(!*reduced := t); 132 for all x let cos(2x)= 1-2sin x**2, sin x**2=1-cos x**2; 133 diffce := diffce; 134 for all x clear cos(2x),sin x**2 135 end; 136 if diffce neq 0 137 then <<write 138 " ***** DERIVATIVE OF INTEGRAL NOT EQUAL TO INTEGRAND *****"; 139 symbolic(faillist!* := list(a,b,res,der) . faillist!*)>>; 140 symbolic if smemq('int,res) 141 then unintlist!* := list(a,b,res) . unintlist!*; 142 return res 143 end; 144 145 146testint 147 148 149symbolic initialize!-integral!-test(); 150 151 152 153% References are to Gradshteyn and Ryznik. 154 155testint(1+x+x**2,x); 156 157 158 2 159 x*(2*x + 3*x + 6) 160-------------------- 161 6 162 163testint(x**2*(2*x**2+x)**2,x); 164 165 166 5 2 167 x *(60*x + 70*x + 21) 168------------------------ 169 105 170 171testint(x*(x**2+2*x+1),x); 172 173 174 2 2 175 x *(3*x + 8*x + 6) 176--------------------- 177 12 178 179testint(1/x,x); 180 181 182log(x) 183 % 2.01 #2 184testint((x+1)**3/(x-1)**4,x); 185 186 187 3 2 3 188 3*log(x - 1)*x - 9*log(x - 1)*x + 9*log(x - 1)*x - 3*log(x - 1) - 6*x - 2 189------------------------------------------------------------------------------ 190 3 2 191 3*(x - 3*x + 3*x - 1) 192 193testint(1/(x*(x-1)*(x+1)**2),x); 194 195 196(log(x - 1)*x + log(x - 1) + 3*log(x + 1)*x + 3*log(x + 1) - 4*log(x)*x 197 198 - 4*log(x) + 2*x)/(4*(x + 1)) 199 200testint((a*x+b)/((x-p)*(x-q)),x); 201 202 203 log(p - x)*a*p + log(p - x)*b - log(q - x)*a*q - log(q - x)*b 204--------------------------------------------------------------- 205 p - q 206 207testint(1/(a*x**2+b*x+c),x); 208 209 210 2 2*a*x + b 211 2*sqrt(4*a*c - b )*atan(------------------) 212 2 213 sqrt(4*a*c - b ) 214--------------------------------------------- 215 2 216 4*a*c - b 217 218testint((a*x+b)/(1+x**2),x); 219 220 221 2 222 2*atan(x)*b + log(x + 1)*a 223----------------------------- 224 2 225 226testint(1/(x**2-2*x+3),x); 227 228 229 x - 1 230 sqrt(2)*atan(---------) 231 sqrt(2) 232------------------------- 233 2 234 235 236% Rational function examples from Hardy, Pure Mathematics, p 253 et seq. 237 238testint(1/((x-1)*(x**2+1))**2,x); 239 240 241 3 2 2 3 2 2 242(atan(x)*x - atan(x)*x + atan(x)*x - atan(x) + log(x + 1)*x - log(x + 1)*x 243 244 2 2 3 2 245 + log(x + 1)*x - log(x + 1) - 2*log(x - 1)*x + 2*log(x - 1)*x 246 247 3 3 2 248 - 2*log(x - 1)*x + 2*log(x - 1) - x - 2*x + 1)/(4*(x - x + x - 1)) 249 250testint(x/((x-a)*(x-b)*(x-c)),x); 251 252 253(log(a - x)*a*b - log(a - x)*a*c - log(b - x)*a*b + log(b - x)*b*c 254 255 2 2 2 2 2 2 256 + log(c - x)*a*c - log(c - x)*b*c)/(a *b - a *c - a*b + a*c + b *c - b*c ) 257 258testint(x/((x**2+a**2)*(x**2+b**2)),x); 259 260 261 2 2 2 2 262 - log(a + x ) + log(b + x ) 263-------------------------------- 264 2 2 265 2*(a - b ) 266 267testint(x**2/((x**2+a**2)*(x**2+b**2)),x); 268 269 270 x x 271 atan(---)*a - atan(---)*b 272 a b 273--------------------------- 274 2 2 275 a - b 276 277testint(x/((x-1)*(x**2+1)),x); 278 279 280 2 281 2*atan(x) - log(x + 1) + 2*log(x - 1) 282---------------------------------------- 283 4 284 285testint(x/(1+x**3),x); 286 287 288 2*x - 1 2 289 2*sqrt(3)*atan(---------) + log(x - x + 1) - 2*log(x + 1) 290 sqrt(3) 291------------------------------------------------------------ 292 6 293 294testint(x**3/((x-1)**2*(x**3+1)),x); 295 296 297 2 2 298( - 4*log(x - x + 1)*x + 4*log(x - x + 1) + 9*log(x - 1)*x - 9*log(x - 1) 299 300 - log(x + 1)*x + log(x + 1) - 6*x)/(12*(x - 1)) 301 302testint(1/(1+x**4),x); 303 304 305 sqrt(2) - 2*x sqrt(2) + 2*x 306(sqrt(2)*( - 2*atan(---------------) + 2*atan(---------------) 307 sqrt(2) sqrt(2) 308 309 2 2 310 - log( - sqrt(2)*x + x + 1) + log(sqrt(2)*x + x + 1)))/8 311 312testint(x**2/(1+x**4),x); 313 314 315 sqrt(2) - 2*x sqrt(2) + 2*x 316(sqrt(2)*( - 2*atan(---------------) + 2*atan(---------------) 317 sqrt(2) sqrt(2) 318 319 2 2 320 + log( - sqrt(2)*x + x + 1) - log(sqrt(2)*x + x + 1)))/8 321 322testint(1/(1+x**2+x**4),x); 323 324 325 2*x - 1 2*x + 1 2 326(2*sqrt(3)*atan(---------) + 2*sqrt(3)*atan(---------) - 3*log(x - x + 1) 327 sqrt(3) sqrt(3) 328 329 2 330 + 3*log(x + x + 1))/12 331 332 333% Examples involving a+b*x. 334 335z := a+b*x; 336 337 338z := a + b*x 339 340 341testint(z**p,x); 342 343 344 p 345 (a + b*x) *(a + b*x) 346---------------------- 347 b*(p + 1) 348 349testint(x*z**p,x); 350 351 352 p 2 2 2 2 2 353 (a + b*x) *( - a + a*b*p*x + b *p*x + b *x ) 354------------------------------------------------ 355 2 2 356 b *(p + 3*p + 2) 357 358testint(x**2*z**p,x); 359 360 361 p 362((a + b*x) 363 364 3 2 2 2 2 2 2 3 2 3 3 3 3 3 365 *(2*a - 2*a *b*p*x + a*b *p *x + a*b *p*x + b *p *x + 3*b *p*x + 2*b *x )) 366 367 3 3 2 368/(b *(p + 6*p + 11*p + 6)) 369 370testint(1/z,x); 371 372 373 log(a + b*x) 374-------------- 375 b 376 377testint(1/z**2,x); 378 379 380 x 381------------- 382 a*(a + b*x) 383 384testint(x/z,x); 385 386 387 - log(a + b*x)*a + b*x 388------------------------- 389 2 390 b 391 392testint(x**2/z,x); 393 394 395 2 2 2 396 2*log(a + b*x)*a - 2*a*b*x + b *x 397------------------------------------- 398 3 399 2*b 400 401testint(1/(x*z),x); 402 403 404 - log(a + b*x) + log(x) 405-------------------------- 406 a 407 408testint(1/(x**2*z),x); 409 410 411 log(a + b*x)*b*x - log(x)*b*x - a 412----------------------------------- 413 2 414 a *x 415 416testint(1/(x*z)**2,x); 417 418 419 2 2 2 2 420(2*log(a + b*x)*a*b*x + 2*log(a + b*x)*b *x - 2*log(x)*a*b*x - 2*log(x)*b *x 421 422 2 2 2 3 423 - a + 2*b *x )/(a *x*(a + b*x)) 424 425testint(1/(c**2+x**2),x); 426 427 428 x 429 atan(---) 430 c 431----------- 432 c 433 434testint(1/(c**2-x**2),x); 435 436 437 log( - c - x) - log(c - x) 438---------------------------- 439 2*c 440 441 442% More complicated rational function examples, mostly contributed 443% by David M. Dahm, who also developed the code to integrate them. 444 445testint(1/(2*x**3-1),x); 446 447 448 1/3 449 2/3 2*2 *x + 1 2/3 2 1/3 450(2 *( - 2*sqrt(3)*atan(--------------) - log(2 *x + 2 *x + 1) 451 sqrt(3) 452 453 1/3 454 + 2*log(2 *x - 1)))/12 455 456testint(1/(x**3-2),x); 457 458 459 1/3 460 1/3 2 + 2*x 2/3 1/3 2 461(2 *( - 2*sqrt(3)*atan(--------------) - log(2 + 2 *x + x ) 462 1/3 463 2 *sqrt(3) 464 465 1/3 466 + 2*log( - 2 + x)))/12 467 468testint(1/(a*x**3-b),x); 469 470 471 1/3 1/3 472 1/3 2*a *x + b 2/3 2 1/3 1/3 2/3 473(b *( - 2*sqrt(3)*atan(-----------------) - log(a *x + b *a *x + b ) 474 1/3 475 b *sqrt(3) 476 477 1/3 1/3 1/3 478 + 2*log(a *x - b )))/(6*a *b) 479 480testint(1/(x**4-2),x); 481 482 483 1/4 x 1/4 1/4 484 2 *( - 2*atan(------) - log(2 + x) + log( - 2 + x)) 485 1/4 486 2 487------------------------------------------------------------- 488 8 489 490testint(1/(5*x**4-1),x); 491 492 493 1/4 sqrt(5)*x 1/4 1/4 494 sqrt(5)*5 *( - 2*atan(-----------) + log(5 *x - 1) - log(5 *x + 1)) 495 1/4 496 5 497--------------------------------------------------------------------------- 498 20 499 500testint(1/(3*x**4+7),x); 501 502 503 1/4 504 1/4 sqrt(2)*21 - 2*sqrt(3)*x 505(sqrt(6)*21 *( - 2*atan(-----------------------------) 506 1/4 507 sqrt(2)*21 508 509 1/4 510 sqrt(2)*21 + 2*sqrt(3)*x 511 + 2*atan(-----------------------------) 512 1/4 513 sqrt(2)*21 514 515 1/4 2 516 - log( - sqrt(2)*21 *x + sqrt(7) + sqrt(3)*x ) 517 518 1/4 2 519 + log(sqrt(2)*21 *x + sqrt(7) + sqrt(3)*x )))/168 520 521testint(1/(x**4+3*x**2-1),x); 522 523 524 2*x 525(sqrt(2)*(6*sqrt(sqrt(13) + 3)*sqrt(13)*atan(----------------------------) 526 sqrt(sqrt(13) + 3)*sqrt(2) 527 528 2*x 529 - 26*sqrt(sqrt(13) + 3)*atan(----------------------------) + 3 530 sqrt(sqrt(13) + 3)*sqrt(2) 531 532 *sqrt(sqrt(13) - 3)*sqrt(13)*log( - sqrt(sqrt(13) - 3) + sqrt(2)*x) 533 534 - 3*sqrt(sqrt(13) - 3)*sqrt(13)*log(sqrt(sqrt(13) - 3) + sqrt(2)*x) 535 536 + 13*sqrt(sqrt(13) - 3)*log( - sqrt(sqrt(13) - 3) + sqrt(2)*x) 537 538 - 13*sqrt(sqrt(13) - 3)*log(sqrt(sqrt(13) - 3) + sqrt(2)*x)))/104 539 540testint(1/(x**4-3*x**2-1),x); 541 542 543 2*x 544(sqrt(2)*( - 6*sqrt(sqrt(13) - 3)*sqrt(13)*atan(----------------------------) 545 sqrt(sqrt(13) - 3)*sqrt(2) 546 547 2*x 548 - 26*sqrt(sqrt(13) - 3)*atan(----------------------------) - 3 549 sqrt(sqrt(13) - 3)*sqrt(2) 550 551 *sqrt(sqrt(13) + 3)*sqrt(13)*log( - sqrt(sqrt(13) + 3) + sqrt(2)*x) 552 553 + 3*sqrt(sqrt(13) + 3)*sqrt(13)*log(sqrt(sqrt(13) + 3) + sqrt(2)*x) 554 555 + 13*sqrt(sqrt(13) + 3)*log( - sqrt(sqrt(13) + 3) + sqrt(2)*x) 556 557 - 13*sqrt(sqrt(13) + 3)*log(sqrt(sqrt(13) + 3) + sqrt(2)*x)))/104 558 559testint(1/(x**4-3*x**2+1),x); 560 561 562( - sqrt(5)*log( - sqrt(5) + 2*x - 1) - sqrt(5)*log( - sqrt(5) + 2*x + 1) 563 564 + sqrt(5)*log(sqrt(5) + 2*x - 1) + sqrt(5)*log(sqrt(5) + 2*x + 1) 565 566 + 5*log( - sqrt(5) + 2*x - 1) - 5*log( - sqrt(5) + 2*x + 1) 567 568 + 5*log(sqrt(5) + 2*x - 1) - 5*log(sqrt(5) + 2*x + 1))/20 569 570testint(1/(x**4-4*x**2+1),x); 571 572 573 2*x 2*x 574(sqrt(2)*(2*sqrt(3)*atanh(-------------------) + 6*atanh(-------------------) 575 sqrt(6) - sqrt(2) sqrt(6) - sqrt(2) 576 577 - sqrt(6) - sqrt(2) + 2*x 578 - sqrt(3)*log(----------------------------) 579 2 580 581 sqrt(6) + sqrt(2) + 2*x 582 + sqrt(3)*log(-------------------------) 583 2 584 585 - sqrt(6) - sqrt(2) + 2*x 586 + 3*log(----------------------------) 587 2 588 589 sqrt(6) + sqrt(2) + 2*x 590 - 3*log(-------------------------)))/24 591 2 592 593testint(1/(x**4+4*x**2+1),x); 594 595 596 2*x 2*x 597(sqrt(2)*(2*sqrt(3)*atan(-------------------) - 6*atan(-------------------) 598 sqrt(6) + sqrt(2) sqrt(6) + sqrt(2) 599 600 - sqrt(6)*i + sqrt(2)*i + 2*x 601 - sqrt(3)*log(--------------------------------)*i 602 2 603 604 sqrt(6)*i - sqrt(2)*i + 2*x 605 + sqrt(3)*log(-----------------------------)*i 606 2 607 608 - sqrt(6)*i + sqrt(2)*i + 2*x 609 - 3*log(--------------------------------)*i 610 2 611 612 sqrt(6)*i - sqrt(2)*i + 2*x 613 + 3*log(-----------------------------)*i))/24 614 2 615 616testint(1/(x**4+x**2+2),x); 617 618 619 sqrt(2*sqrt(2) - 1) - 2*x 620(2*sqrt(2*sqrt(2) + 1)*sqrt(2)*atan(---------------------------) 621 sqrt(2*sqrt(2) + 1) 622 623 sqrt(2*sqrt(2) - 1) - 2*x 624 - 8*sqrt(2*sqrt(2) + 1)*atan(---------------------------) 625 sqrt(2*sqrt(2) + 1) 626 627 sqrt(2*sqrt(2) - 1) + 2*x 628 - 2*sqrt(2*sqrt(2) + 1)*sqrt(2)*atan(---------------------------) 629 sqrt(2*sqrt(2) + 1) 630 631 sqrt(2*sqrt(2) - 1) + 2*x 632 + 8*sqrt(2*sqrt(2) + 1)*atan(---------------------------) 633 sqrt(2*sqrt(2) + 1) 634 635 2 636 - sqrt(2*sqrt(2) - 1)*sqrt(2)*log( - sqrt(2*sqrt(2) - 1)*x + sqrt(2) + x ) 637 638 2 639 + sqrt(2*sqrt(2) - 1)*sqrt(2)*log(sqrt(2*sqrt(2) - 1)*x + sqrt(2) + x ) 640 641 2 642 - 4*sqrt(2*sqrt(2) - 1)*log( - sqrt(2*sqrt(2) - 1)*x + sqrt(2) + x ) 643 644 2 645 + 4*sqrt(2*sqrt(2) - 1)*log(sqrt(2*sqrt(2) - 1)*x + sqrt(2) + x ))/56 646 647testint(1/(x**4-x**2+2),x); 648 649 650 sqrt(2*sqrt(2) + 1) - 2*x 651( - 2*sqrt(2*sqrt(2) - 1)*sqrt(2)*atan(---------------------------) 652 sqrt(2*sqrt(2) - 1) 653 654 sqrt(2*sqrt(2) + 1) - 2*x 655 - 8*sqrt(2*sqrt(2) - 1)*atan(---------------------------) 656 sqrt(2*sqrt(2) - 1) 657 658 sqrt(2*sqrt(2) + 1) + 2*x 659 + 2*sqrt(2*sqrt(2) - 1)*sqrt(2)*atan(---------------------------) 660 sqrt(2*sqrt(2) - 1) 661 662 sqrt(2*sqrt(2) + 1) + 2*x 663 + 8*sqrt(2*sqrt(2) - 1)*atan(---------------------------) 664 sqrt(2*sqrt(2) - 1) 665 666 2 667 + sqrt(2*sqrt(2) + 1)*sqrt(2)*log( - sqrt(2*sqrt(2) + 1)*x + sqrt(2) + x ) 668 669 2 670 - sqrt(2*sqrt(2) + 1)*sqrt(2)*log(sqrt(2*sqrt(2) + 1)*x + sqrt(2) + x ) 671 672 2 673 - 4*sqrt(2*sqrt(2) + 1)*log( - sqrt(2*sqrt(2) + 1)*x + sqrt(2) + x ) 674 675 2 676 + 4*sqrt(2*sqrt(2) + 1)*log(sqrt(2*sqrt(2) + 1)*x + sqrt(2) + x ))/56 677 678testint(1/(x**6-1),x); 679 680 681 2*x - 1 2*x + 1 2 682( - 2*sqrt(3)*atan(---------) - 2*sqrt(3)*atan(---------) + log(x - x + 1) 683 sqrt(3) sqrt(3) 684 685 2 686 - log(x + x + 1) + 2*log(x - 1) - 2*log(x + 1))/12 687 688testint(1/(x**6-2),x); 689 690 691 1/6 1/6 692 1/6 2 - 2*x 2 + 2*x 693(2 *(2*sqrt(3)*atan(--------------) - 2*sqrt(3)*atan(--------------) 694 1/6 1/6 695 2 *sqrt(3) 2 *sqrt(3) 696 697 1/6 1/6 1/6 1/3 2 698 - 2*log(2 + x) + 2*log( - 2 + x) + log( - 2 *x + 2 + x ) 699 700 1/6 1/3 2 701 - log(2 *x + 2 + x )))/24 702 703testint(1/(x**6+2),x); 704 705 706 1/6 1/6 707 1/6 2 *sqrt(3) - 2*x 2 *sqrt(3) + 2*x 708(2 *( - 2*atan(--------------------) + 2*atan(--------------------) 709 1/6 1/6 710 2 2 711 712 x 1/6 1/3 2 713 + 4*atan(------) - sqrt(3)*log( - 2 *sqrt(3)*x + 2 + x ) 714 1/6 715 2 716 717 1/6 1/3 2 718 + sqrt(3)*log(2 *sqrt(3)*x + 2 + x )))/24 719 720testint(1/(x**8+1),x); 721 722 723 sqrt( - sqrt(2) + 2) - 2*x 724( - 2*sqrt(sqrt(2) + 2)*atan(----------------------------) 725 sqrt(sqrt(2) + 2) 726 727 sqrt( - sqrt(2) + 2) + 2*x 728 + 2*sqrt(sqrt(2) + 2)*atan(----------------------------) 729 sqrt(sqrt(2) + 2) 730 731 sqrt(sqrt(2) + 2) - 2*x 732 - 2*sqrt( - sqrt(2) + 2)*atan(-------------------------) 733 sqrt( - sqrt(2) + 2) 734 735 sqrt(sqrt(2) + 2) + 2*x 736 + 2*sqrt( - sqrt(2) + 2)*atan(-------------------------) 737 sqrt( - sqrt(2) + 2) 738 739 2 740 - sqrt( - sqrt(2) + 2)*log( - sqrt( - sqrt(2) + 2)*x + x + 1) 741 742 2 743 + sqrt( - sqrt(2) + 2)*log(sqrt( - sqrt(2) + 2)*x + x + 1) 744 745 2 746 - sqrt(sqrt(2) + 2)*log( - sqrt(sqrt(2) + 2)*x + x + 1) 747 748 2 749 + sqrt(sqrt(2) + 2)*log(sqrt(sqrt(2) + 2)*x + x + 1))/16 750 751testint(1/(x**8-1),x); 752 753 754 sqrt(2) - 2*x sqrt(2) + 2*x 755(2*sqrt(2)*atan(---------------) - 2*sqrt(2)*atan(---------------) - 4*atan(x) 756 sqrt(2) sqrt(2) 757 758 2 2 759 + sqrt(2)*log( - sqrt(2)*x + x + 1) - sqrt(2)*log(sqrt(2)*x + x + 1) 760 761 + 2*log(x - 1) - 2*log(x + 1))/16 762 763testint(1/(x**8-x**4+1),x); 764 765 766 sqrt(6) + sqrt(2) - 4*x 767( - 2*sqrt( - sqrt(3) + 2)*sqrt(3)*atan(-------------------------) 768 2*sqrt( - sqrt(3) + 2) 769 770 sqrt(6) + sqrt(2) - 4*x 771 - 6*sqrt( - sqrt(3) + 2)*atan(-------------------------) 772 2*sqrt( - sqrt(3) + 2) 773 774 sqrt(6) + sqrt(2) + 4*x 775 + 2*sqrt( - sqrt(3) + 2)*sqrt(3)*atan(-------------------------) 776 2*sqrt( - sqrt(3) + 2) 777 778 sqrt(6) + sqrt(2) + 4*x 779 + 6*sqrt( - sqrt(3) + 2)*atan(-------------------------) 780 2*sqrt( - sqrt(3) + 2) 781 782 2*sqrt( - sqrt(3) + 2) - 4*x 783 - 2*sqrt(6)*atan(------------------------------) 784 sqrt(6) + sqrt(2) 785 786 2*sqrt( - sqrt(3) + 2) + 4*x 787 + 2*sqrt(6)*atan(------------------------------) 788 sqrt(6) + sqrt(2) 789 790 2 791 - sqrt( - sqrt(3) + 2)*sqrt(3)*log( - sqrt( - sqrt(3) + 2)*x + x + 1) 792 793 2 794 + sqrt( - sqrt(3) + 2)*sqrt(3)*log(sqrt( - sqrt(3) + 2)*x + x + 1) 795 796 2 797 - 3*sqrt( - sqrt(3) + 2)*log( - sqrt( - sqrt(3) + 2)*x + x + 1) 798 799 2 800 + 3*sqrt( - sqrt(3) + 2)*log(sqrt( - sqrt(3) + 2)*x + x + 1) 801 802 2 803 - sqrt(6)*x - sqrt(2)*x + 2*x + 2 804 - sqrt(6)*log(-------------------------------------) 805 2 806 807 2 808 sqrt(6)*x + sqrt(2)*x + 2*x + 2 809 + sqrt(6)*log(----------------------------------))/24 810 2 811 812testint(x**7/(x**12+1),x); 813 814 815 sqrt(6) + sqrt(2) - 4*x 816( - sqrt( - sqrt(3) + 2)*sqrt(6)*atan(-------------------------) 817 2*sqrt( - sqrt(3) + 2) 818 819 sqrt(6) + sqrt(2) - 4*x 820 - 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan(-------------------------) 821 2*sqrt( - sqrt(3) + 2) 822 823 sqrt(6) + sqrt(2) + 4*x 824 - sqrt( - sqrt(3) + 2)*sqrt(6)*atan(-------------------------) 825 2*sqrt( - sqrt(3) + 2) 826 827 sqrt(6) + sqrt(2) + 4*x 828 - 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan(-------------------------) 829 2*sqrt( - sqrt(3) + 2) 830 831 2*sqrt( - sqrt(3) + 2) - 4*x 832 + sqrt( - sqrt(3) + 2)*sqrt(6)*atan(------------------------------) 833 sqrt(6) + sqrt(2) 834 835 2*sqrt( - sqrt(3) + 2) - 4*x 836 + 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan(------------------------------) 837 sqrt(6) + sqrt(2) 838 839 2*sqrt( - sqrt(3) + 2) + 4*x 840 + sqrt( - sqrt(3) + 2)*sqrt(6)*atan(------------------------------) 841 sqrt(6) + sqrt(2) 842 843 2*sqrt( - sqrt(3) + 2) + 4*x 844 + 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan(------------------------------) 845 sqrt(6) + sqrt(2) 846 847 2 2 848 + log( - sqrt( - sqrt(3) + 2)*x + x + 1) - 2*log( - sqrt(2)*x + x + 1) 849 850 2 2 851 + log(sqrt( - sqrt(3) + 2)*x + x + 1) - 2*log(sqrt(2)*x + x + 1) 852 853 2 854 - sqrt(6)*x - sqrt(2)*x + 2*x + 2 855 + log(-------------------------------------) 856 2 857 858 2 859 sqrt(6)*x + sqrt(2)*x + 2*x + 2 860 + log(----------------------------------))/24 861 2 862 863 864% Examples involving logarithms. 865 866testint(log x,x); 867 868 869x*(log(x) - 1) 870 871testint(x*log x,x); 872 873 874 2 875 x *(2*log(x) - 1) 876------------------- 877 4 878 879testint(x**2*log x,x); 880 881 882 3 883 x *(3*log(x) - 1) 884------------------- 885 9 886 887testint(x**p*log x,x); 888 889 890 p 891 x *x*(log(x)*p + log(x) - 1) 892------------------------------ 893 2 894 p + 2*p + 1 895 896testint((log x)**2,x); 897 898 899 2 900x*(log(x) - 2*log(x) + 2) 901 902testint(x**9*log x**11,x); 903 904 905 10 11 10 9 906(x *(15625000*log(x) - 17187500*log(x) + 17187500*log(x) 907 908 8 7 6 5 909 - 15468750*log(x) + 12375000*log(x) - 8662500*log(x) + 5197500*log(x) 910 911 4 3 2 912 - 2598750*log(x) + 1039500*log(x) - 311850*log(x) + 62370*log(x) 913 914 - 6237))/156250000 915 916testint(log x**2/x,x); 917 918 919 3 920 log(x) 921--------- 922 3 923 924testint(1/log x,x); 925 926 927ei(log(x)) 928 929testint(1/log(x+1),x); 930 931 932ei(log(x + 1)) 933 934testint(1/(x*log x),x); 935 936 937log(log(x)) 938 939testint(1/(x*log x)**2,x); 940 941 942 - (ei( - log(x))*log(x)*x + 1) 943--------------------------------- 944 log(x)*x 945 946testint((log x)**p/x,x); 947 948 949 p 950 log(x) *log(x) 951---------------- 952 p + 1 953 954testint(log x *(a*x+b),x); 955 956 957 x*(2*log(x)*a*x + 4*log(x)*b - a*x - 4*b) 958------------------------------------------- 959 4 960 961testint((a*x+b)**2*log x,x); 962 963 964 2 2 2 2 2 2 965(x*(6*log(x)*a *x + 18*log(x)*a*b*x + 18*log(x)*b - 2*a *x - 9*a*b*x - 18*b ) 966 967 )/18 968 969testint(log x/(a*x+b)**2,x); 970 971 972 - log(a*x + b)*a*x - log(a*x + b)*b + log(x)*a*x 973--------------------------------------------------- 974 a*b*(a*x + b) 975 976testint(x*log (a*x+b),x); 977 978 979 2 2 2 2 2 980 2*log(a*x + b)*a *x - 2*log(a*x + b)*b - a *x + 2*a*b*x 981------------------------------------------------------------ 982 2 983 4*a 984 985testint(x**2*log(a*x+b),x); 986 987 988 3 3 3 3 3 2 2 2 989 6*log(a*x + b)*a *x + 6*log(a*x + b)*b - 2*a *x + 3*a *b*x - 6*a*b *x 990--------------------------------------------------------------------------- 991 3 992 18*a 993 994testint(log(x**2+a**2),x); 995 996 997 x 2 2 9982*atan(---)*a + log(a + x )*x - 2*x 999 a 1000 1001testint(x*log(x**2+a**2),x); 1002 1003 1004 2 2 2 2 2 2 2 1005 log(a + x )*a + log(a + x )*x - x 1006---------------------------------------- 1007 2 1008 1009testint(x**2*log(x**2+a**2),x); 1010 1011 1012 x 3 2 2 3 2 3 1013 - 6*atan(---)*a + 3*log(a + x )*x + 6*a *x - 2*x 1014 a 1015------------------------------------------------------- 1016 9 1017 1018testint(x**4*log(x**2+a**2),x); 1019 1020 1021 x 5 2 2 5 4 2 3 5 1022 30*atan(---)*a + 15*log(a + x )*x - 30*a *x + 10*a *x - 6*x 1023 a 1024------------------------------------------------------------------ 1025 75 1026 1027testint(log(x**2-a**2),x); 1028 1029 1030 2 2 2 2 1031 - log( - a + x )*a + log( - a + x )*x + 2*log( - a - x)*a - 2*x 1032 1033testint(log(log(log(log(x)))),x); 1034 1035 1036 1 1037 - int(-------------------------------------,x) + log(log(log(log(x))))*x 1038 log(log(log(x)))*log(log(x))*log(x) 1039 1040 1041% Examples involving circular functions. 1042 1043testint(sin x,x); 1044 1045 1046 - cos(x) 1047 % 2.01 #5 1048testint(cos x,x); 1049 1050 1051sin(x) 1052 % #6 1053testint(tan x,x); 1054 1055 1056 2 1057 log(tan(x) + 1) 1058------------------ 1059 2 1060 % #11 1061testint(1/tan(x),x); 1062 1063 1064 2 1065 - log(tan(x) + 1) + 2*log(tan(x)) 1066------------------------------------- 1067 2 1068 % 2.01 #12 1069testint(1/(1+tan(x))**2,x); 1070 1071 1072 2 2 1073( - log(tan(x) + 1)*tan(x) - log(tan(x) + 1) + 2*log(tan(x) + 1)*tan(x) 1074 1075 + 2*log(tan(x) + 1) + 2*tan(x))/(4*(tan(x) + 1)) 1076 1077testint(1/cos x,x); 1078 1079 1080 x x 1081 - log(tan(---) - 1) + log(tan(---) + 1) 1082 2 2 1083 1084testint(1/sin x,x); 1085 1086 1087 x 1088log(tan(---)) 1089 2 1090 1091testint(sin x**2,x); 1092 1093 1094 - cos(x)*sin(x) + x 1095---------------------- 1096 2 1097 1098testint(x**3*sin(x**2),x); 1099 1100 1101 2 2 2 1102 - cos(x )*x + sin(x ) 1103------------------------- 1104 2 1105 1106testint(sin x**3,x); 1107 1108 1109 2 1110 - cos(x)*sin(x) - 2*cos(x) + 2 1111---------------------------------- 1112 3 1113 1114testint(sin x**p,x); 1115 1116 1117 p 1118int(sin(x) ,x) 1119 1120testint((sin x**2+1)**2*cos x,x); 1121 1122 1123 4 2 1124 sin(x)*(3*sin(x) + 10*sin(x) + 15) 1125-------------------------------------- 1126 15 1127 1128testint(cos x**2,x); 1129 1130 1131 cos(x)*sin(x) + x 1132------------------- 1133 2 1134 1135testint(cos x**3,x); 1136 1137 1138 2 1139 sin(x)*( - sin(x) + 3) 1140------------------------- 1141 3 1142 1143testint(sin(a*x+b),x); 1144 1145 1146 - cos(a*x + b) 1147----------------- 1148 a 1149 1150testint(1/cos x**2,x); 1151 1152 1153 sin(x) 1154-------- 1155 cos(x) 1156 1157testint(sin x*sin(2*x),x); 1158 1159 1160 - 2*cos(2*x)*sin(x) + cos(x)*sin(2*x) 1161---------------------------------------- 1162 3 1163 1164testint(x*sin x,x); 1165 1166 1167 - cos(x)*x + sin(x) 1168 1169testint(x**2*sin x,x); 1170 1171 1172 2 1173 - cos(x)*x + 2*cos(x) + 2*sin(x)*x 1174 1175testint(x*sin x**2,x); 1176 1177 1178 2 2 1179 - 2*cos(x)*sin(x)*x + sin(x) + x - 2 1180----------------------------------------- 1181 4 1182 1183testint(x**2*sin x**2,x); 1184 1185 1186 2 2 3 1187 - 6*cos(x)*sin(x)*x + 3*cos(x)*sin(x) + 6*sin(x) *x + 2*x - 3*x 1188-------------------------------------------------------------------- 1189 12 1190 1191testint(x*sin x**3,x); 1192 1193 1194 2 3 1195 - 3*cos(x)*sin(x) *x - 6*cos(x)*x + sin(x) + 6*sin(x) 1196--------------------------------------------------------- 1197 9 1198 1199testint(x*cos x,x); 1200 1201 1202cos(x) + sin(x)*x 1203 1204testint(x**2*cos x,x); 1205 1206 1207 2 12082*cos(x)*x + sin(x)*x - 2*sin(x) 1209 1210testint(x*cos x**2,x); 1211 1212 1213 2 2 1214 2*cos(x)*sin(x)*x - sin(x) + x + 2 1215-------------------------------------- 1216 4 1217 1218testint(x**2*cos x**2,x); 1219 1220 1221 2 2 3 1222 6*cos(x)*sin(x)*x - 3*cos(x)*sin(x) - 6*sin(x) *x + 2*x + 3*x 1223----------------------------------------------------------------- 1224 12 1225 1226testint(x*cos x**3,x); 1227 1228 1229 2 3 1230 - cos(x)*sin(x) + 7*cos(x) - 3*sin(x) *x + 9*sin(x)*x + 1 1231------------------------------------------------------------- 1232 9 1233 1234testint(sin x/x,x); 1235 1236 1237si(x) 1238 1239testint(cos x/x,x); 1240 1241 1242ci(x) 1243 1244testint(sin x/x**2,x); 1245 1246 1247 ci(x)*x - sin(x) 1248------------------ 1249 x 1250 1251testint(sin x**2/x,x); 1252 1253 1254 - ci(2*x) + log(x) 1255--------------------- 1256 2 1257 1258testint(tan x**3,x); 1259 1260 1261 2 2 1262 - log(tan(x) + 1) + tan(x) 1263------------------------------- 1264 2 1265 1266% z := a+b*x; 1267testint(sin z,x); 1268 1269 1270 - cos(a + b*x) 1271----------------- 1272 b 1273 1274testint(cos z,x); 1275 1276 1277 sin(a + b*x) 1278-------------- 1279 b 1280 1281testint(tan z,x); 1282 1283 1284 2 1285 log(tan(a + b*x) + 1) 1286------------------------ 1287 2*b 1288 1289testint(1/tan z,x); 1290 1291 1292 2 1293 - log(tan(a + b*x) + 1) + 2*log(tan(a + b*x)) 1294------------------------------------------------- 1295 2*b 1296 1297testint(1/sin z,x); 1298 1299 1300 a + b*x 1301 log(tan(---------)) 1302 2 1303--------------------- 1304 b 1305 1306testint(1/cos z,x); 1307 1308 1309 a + b*x a + b*x 1310 - log(tan(---------) - 1) + log(tan(---------) + 1) 1311 2 2 1312------------------------------------------------------ 1313 b 1314 1315testint(sin z**2,x); 1316 1317 1318 - cos(a + b*x)*sin(a + b*x) + b*x 1319------------------------------------ 1320 2*b 1321 1322testint(sin z**3,x); 1323 1324 1325 2 1326 - cos(a + b*x)*sin(a + b*x) - 2*cos(a + b*x) + 2 1327---------------------------------------------------- 1328 3*b 1329 1330testint(cos z**2,x); 1331 1332 1333 cos(a + b*x)*sin(a + b*x) + b*x 1334--------------------------------- 1335 2*b 1336 1337testint(cos z**3,x); 1338 1339 1340 2 1341 sin(a + b*x)*( - sin(a + b*x) + 3) 1342------------------------------------- 1343 3*b 1344 1345testint(1/cos z**2,x); 1346 1347 1348 sin(a + b*x) 1349---------------- 1350 cos(a + b*x)*b 1351 1352testint(1/(1+cos x),x); 1353 1354 1355 x 1356tan(---) 1357 2 1358 1359testint(1/(1-cos x),x); 1360 1361 1362 - 1 1363---------- 1364 x 1365 tan(---) 1366 2 1367 1368testint(1/(1+sin x),x); 1369 1370 1371 x 1372 2*tan(---) 1373 2 1374-------------- 1375 x 1376 tan(---) + 1 1377 2 1378 1379testint(1/(1-sin x),x); 1380 1381 1382 x 1383 - 2*tan(---) 1384 2 1385--------------- 1386 x 1387 tan(---) - 1 1388 2 1389 1390testint(1/(a+b*sin x),x); 1391 1392 1393 x 1394 tan(---)*a + b 1395 2 2 2 1396 2*sqrt(a - b )*atan(----------------) 1397 2 2 1398 sqrt(a - b ) 1399---------------------------------------- 1400 2 2 1401 a - b 1402 1403testint(1/(a+b*sin x+cos x),x); 1404 1405 1406 x x 1407 tan(---)*a - tan(---) + b 1408 2 2 2 2 1409 2*sqrt(a - b - 1)*atan(---------------------------) 1410 2 2 1411 sqrt(a - b - 1) 1412------------------------------------------------------- 1413 2 2 1414 a - b - 1 1415 1416testint(x**2*sin z**2,x); 1417 1418 1419 2 2 1420( - 6*cos(a + b*x)*sin(a + b*x)*b *x + 3*cos(a + b*x)*sin(a + b*x) 1421 1422 2 3 3 3 1423 + 6*sin(a + b*x) *b*x + 9*a + 2*b *x - 3*b*x)/(12*b ) 1424 1425testint(cos x*cos(2*x),x); 1426 1427 1428 - cos(2*x)*sin(x) + 2*cos(x)*sin(2*x) 1429---------------------------------------- 1430 3 1431 1432testint(x**2*cos z**2,x); 1433 1434 1435 2 2 1436(6*cos(a + b*x)*sin(a + b*x)*b *x - 3*cos(a + b*x)*sin(a + b*x) 1437 1438 2 3 3 3 1439 - 6*sin(a + b*x) *b*x + 2*b *x + 3*b*x)/(12*b ) 1440 1441testint(1/tan x**3,x); 1442 1443 1444 2 2 2 1445 log(tan(x) + 1)*tan(x) - 2*log(tan(x))*tan(x) - 1 1446------------------------------------------------------ 1447 2 1448 2*tan(x) 1449 1450testint(x**3*tan(x)**4,x); 1451 1452 1453 2 2 3 3 2 2 1454(48*int(tan(x)*x ,x) - 6*log(tan(x) + 1) + 4*tan(x) *x - 6*tan(x) *x 1455 1456 3 4 2 1457 - 12*tan(x)*x + 12*tan(x)*x + 3*x - 6*x )/12 1458 1459testint(x**3*tan(x)**6,x); 1460 1461 1462 2 2 5 3 4 2 1463( - 276*int(tan(x)*x ,x) + 60*log(tan(x) + 1) + 12*tan(x) *x - 9*tan(x) *x 1464 1465 3 3 3 2 2 2 3 1466 - 20*tan(x) *x + 6*tan(x) *x + 48*tan(x) *x - 3*tan(x) + 60*tan(x)*x 1467 1468 4 2 1469 - 114*tan(x)*x - 15*x + 57*x )/60 1470 1471testint(x*tan(x)**2,x); 1472 1473 1474 2 2 1475 - log(tan(x) + 1) + 2*tan(x)*x - x 1476--------------------------------------- 1477 2 1478 1479testint(sin(2*x)*cos(3*x),x); 1480 1481 1482 2*cos(3*x)*cos(2*x) + 3*sin(3*x)*sin(2*x) 1483------------------------------------------- 1484 5 1485 1486testint(sin x**2*cos x**2,x); 1487 1488 1489 3 1490 2*cos(x)*sin(x) - cos(x)*sin(x) + x 1491-------------------------------------- 1492 8 1493 1494testint(1/(sin x**2*cos x**2),x); 1495 1496 1497 2 1498 2*sin(x) - 1 1499--------------- 1500 cos(x)*sin(x) 1501 1502testint(d**x*sin x,x); 1503 1504 1505 x 1506 d *( - cos(x) + log(d)*sin(x)) 1507-------------------------------- 1508 2 1509 log(d) + 1 1510 1511testint(d**x*cos x,x); 1512 1513 1514 x 1515 d *(cos(x)*log(d) + sin(x)) 1516----------------------------- 1517 2 1518 log(d) + 1 1519 1520testint(x*d**x*sin x,x); 1521 1522 1523 x 2 3 1524(d *( - cos(x)*log(d) *x + 2*cos(x)*log(d) - cos(x)*x + log(d) *sin(x)*x 1525 1526 2 4 2 1527 - log(d) *sin(x) + log(d)*sin(x)*x + sin(x)))/(log(d) + 2*log(d) + 1) 1528 1529testint(x*d**x*cos x,x); 1530 1531 1532 x 3 2 1533(d *(cos(x)*log(d) *x - cos(x)*log(d) + cos(x)*log(d)*x + cos(x) 1534 1535 2 4 2 1536 + log(d) *sin(x)*x - 2*log(d)*sin(x) + sin(x)*x))/(log(d) + 2*log(d) + 1 1537 1538 ) 1539 1540testint(x**2*d**x*sin x,x); 1541 1542 1543 x 4 2 3 2 2 1544(d *( - cos(x)*log(d) *x + 4*cos(x)*log(d) *x - 2*cos(x)*log(d) *x 1545 1546 2 2 1547 - 6*cos(x)*log(d) + 4*cos(x)*log(d)*x - cos(x)*x + 2*cos(x) 1548 1549 5 2 4 3 2 1550 + log(d) *sin(x)*x - 2*log(d) *sin(x)*x + 2*log(d) *sin(x)*x 1551 1552 3 2 1553 + 2*log(d) *sin(x) + log(d)*sin(x)*x - 6*log(d)*sin(x) + 2*sin(x)*x))/( 1554 1555 6 4 2 1556 log(d) + 3*log(d) + 3*log(d) + 1) 1557 1558testint(x**2*d**x*cos x,x); 1559 1560 1561 x 5 2 4 3 2 1562(d *(cos(x)*log(d) *x - 2*cos(x)*log(d) *x + 2*cos(x)*log(d) *x 1563 1564 3 2 1565 + 2*cos(x)*log(d) + cos(x)*log(d)*x - 6*cos(x)*log(d) + 2*cos(x)*x 1566 1567 4 2 3 2 2 1568 + log(d) *sin(x)*x - 4*log(d) *sin(x)*x + 2*log(d) *sin(x)*x 1569 1570 2 2 6 1571 + 6*log(d) *sin(x) - 4*log(d)*sin(x)*x + sin(x)*x - 2*sin(x)))/(log(d) 1572 1573 4 2 1574 + 3*log(d) + 3*log(d) + 1) 1575 1576testint(x**3*d**x*sin x,x); 1577 1578 1579 x 6 3 5 2 4 3 1580(d *( - cos(x)*log(d) *x + 6*cos(x)*log(d) *x - 3*cos(x)*log(d) *x 1581 1582 4 3 2 3 1583 - 18*cos(x)*log(d) *x + 12*cos(x)*log(d) *x + 24*cos(x)*log(d) 1584 1585 2 3 2 2 1586 - 3*cos(x)*log(d) *x - 12*cos(x)*log(d) *x + 6*cos(x)*log(d)*x 1587 1588 3 7 3 1589 - 24*cos(x)*log(d) - cos(x)*x + 6*cos(x)*x + log(d) *sin(x)*x 1590 1591 6 2 5 3 5 1592 - 3*log(d) *sin(x)*x + 3*log(d) *sin(x)*x + 6*log(d) *sin(x)*x 1593 1594 4 2 4 3 3 1595 - 3*log(d) *sin(x)*x - 6*log(d) *sin(x) + 3*log(d) *sin(x)*x 1596 1597 3 2 2 2 1598 - 12*log(d) *sin(x)*x + 3*log(d) *sin(x)*x + 36*log(d) *sin(x) 1599 1600 3 2 1601 + log(d)*sin(x)*x - 18*log(d)*sin(x)*x + 3*sin(x)*x - 6*sin(x)))/( 1602 1603 8 6 4 2 1604 log(d) + 4*log(d) + 6*log(d) + 4*log(d) + 1) 1605 1606testint(x**3*d**x*cos x,x); 1607 1608 1609 x 7 3 6 2 5 3 1610(d *(cos(x)*log(d) *x - 3*cos(x)*log(d) *x + 3*cos(x)*log(d) *x 1611 1612 5 4 2 4 1613 + 6*cos(x)*log(d) *x - 3*cos(x)*log(d) *x - 6*cos(x)*log(d) 1614 1615 3 3 3 2 2 1616 + 3*cos(x)*log(d) *x - 12*cos(x)*log(d) *x + 3*cos(x)*log(d) *x 1617 1618 2 3 2 1619 + 36*cos(x)*log(d) + cos(x)*log(d)*x - 18*cos(x)*log(d)*x + 3*cos(x)*x 1620 1621 6 3 5 2 4 3 1622 - 6*cos(x) + log(d) *sin(x)*x - 6*log(d) *sin(x)*x + 3*log(d) *sin(x)*x 1623 1624 4 3 2 3 1625 + 18*log(d) *sin(x)*x - 12*log(d) *sin(x)*x - 24*log(d) *sin(x) 1626 1627 2 3 2 2 1628 + 3*log(d) *sin(x)*x + 12*log(d) *sin(x)*x - 6*log(d)*sin(x)*x 1629 1630 3 8 6 1631 + 24*log(d)*sin(x) + sin(x)*x - 6*sin(x)*x))/(log(d) + 4*log(d) 1632 1633 4 2 1634 + 6*log(d) + 4*log(d) + 1) 1635 1636testint(sin x*sin(2*x)*sin(3*x),x); 1637 1638 1639( - cos(3*x)*cos(2*x)*cos(x) + 6*cos(3*x)*cos(2*x)*sin(x)*x 1640 1641 + 6*cos(3*x)*cos(x)*sin(2*x)*x - 8*cos(3*x)*sin(2*x)*sin(x) 1642 1643 - 6*cos(2*x)*cos(x)*sin(3*x)*x + 3*cos(2*x)*sin(3*x)*sin(x) 1644 1645 + 6*sin(3*x)*sin(2*x)*sin(x)*x)/24 1646 1647testint(cos x*cos(2*x)*cos(3*x),x); 1648 1649 1650(6*cos(3*x)*cos(2*x)*cos(x)*x + 8*cos(3*x)*cos(2*x)*sin(x) 1651 1652 + 5*cos(3*x)*cos(x)*sin(2*x) - 6*cos(3*x)*sin(2*x)*sin(x)*x 1653 1654 + 6*cos(2*x)*sin(3*x)*sin(x)*x + 6*cos(x)*sin(3*x)*sin(2*x)*x 1655 1656 + 9*sin(3*x)*sin(2*x)*sin(x))/24 1657 1658testint(sin(x*kx)**3*x**2,x); 1659 1660 1661 2 2 2 2 2 2 1662( - 9*cos(kx*x)*sin(kx*x) *kx *x + 2*cos(kx*x)*sin(kx*x) - 18*cos(kx*x)*kx *x 1663 1664 3 3 1665 + 40*cos(kx*x) + 6*sin(kx*x) *kx*x + 36*sin(kx*x)*kx*x + 16)/(27*kx ) 1666 1667testint(x*cos(xi/sin(x))*cos(x)/sin(x)**2,x); 1668 1669 1670 xi 1671 cos(--------)*cos(x)*x 1672 sin(x) 1673int(------------------------,x) 1674 2 1675 sin(x) 1676 1677 1678% Mixed angles and half angles. 1679 1680int(cos(x)/(sin(x)*tan(x/2)),x); 1681 1682 1683 x 1684 - (tan(---)*x + 1) 1685 2 1686--------------------- 1687 x 1688 tan(---) 1689 2 1690 1691 1692% This integral produces a messy result because the code for 1693% converting half angle tans to sin and cos is not effective enough. 1694 1695testint(sin(a*x)/(b+c*sin(a*x))**2,x); 1696 1697 1698 a*x 1699 tan(-----)*b + c 1700 2 2 2 2 1701( - 2*sqrt(b - c )*atan(------------------)*sin(a*x)*c 1702 2 2 1703 sqrt(b - c ) 1704 1705 a*x 1706 tan(-----)*b + c 1707 2 2 2 3 2 1708 - 2*sqrt(b - c )*atan(------------------)*b*c - cos(a*x)*b + cos(a*x)*b*c )/ 1709 2 2 1710 sqrt(b - c ) 1711 1712 4 2 3 5 5 3 2 4 1713(a*(sin(a*x)*b *c - 2*sin(a*x)*b *c + sin(a*x)*c + b - 2*b *c + b*c )) 1714 1715 1716% Examples involving logarithms and circular functions. 1717 1718testint(sin log x,x); 1719 1720 1721 x*( - cos(log(x)) + sin(log(x))) 1722---------------------------------- 1723 2 1724 1725testint(cos log x,x); 1726 1727 1728 x*(cos(log(x)) + sin(log(x))) 1729------------------------------- 1730 2 1731 1732 1733% Examples involving exponentials. 1734 1735testint(e**x,x); 1736 1737 1738 x 1739e 1740 % 2.01 #3 1741testint(a**x,x); 1742 1743 1744 x 1745 a 1746-------- 1747 log(a) 1748 % 2.01 #4 1749testint(e**(a*x),x); 1750 1751 1752 a*x 1753 e 1754------ 1755 a 1756 1757testint(e**(a*x)/x,x); 1758 1759 1760ei(a*x) 1761 1762testint(1/(a+b*e**(m*x)),x); 1763 1764 1765 m*x 1766 - log(e *b + a) + m*x 1767-------------------------- 1768 a*m 1769 1770testint(e**(2*x)/(1+e**x),x); 1771 1772 1773 x x 1774e - log(e + 1) 1775 1776testint(e**(2*x)*e**(a*x),x); 1777 1778 1779 a*x + 2*x 1780 e 1781------------ 1782 a + 2 1783 1784testint(1/(a*e**(m*x)+b*e**(-m*x)),x); 1785 1786 1787 m*x 1788 e *a 1789 sqrt(b)*sqrt(a)*atan(-----------------) 1790 sqrt(b)*sqrt(a) 1791----------------------------------------- 1792 a*b*m 1793 1794testint(x*e**(a*x),x); 1795 1796 1797 a*x 1798 e *(a*x - 1) 1799---------------- 1800 2 1801 a 1802 1803testint(x**20*e**x,x); 1804 1805 1806 x 20 19 18 17 16 15 14 1807e *(x - 20*x + 380*x - 6840*x + 116280*x - 1860480*x + 27907200*x 1808 1809 13 12 11 10 1810 - 390700800*x + 5079110400*x - 60949324800*x + 670442572800*x 1811 1812 9 8 7 1813 - 6704425728000*x + 60339831552000*x - 482718652416000*x 1814 1815 6 5 4 1816 + 3379030566912000*x - 20274183401472000*x + 101370917007360000*x 1817 1818 3 2 1819 - 405483668029440000*x + 1216451004088320000*x - 2432902008176640000*x 1820 1821 + 2432902008176640000) 1822 1823testint(a**x/b**x,x); 1824 1825 1826 x 1827 a 1828---------------------- 1829 x 1830 b *(log(a) - log(b)) 1831 1832testint(a**x*b**x,x); 1833 1834 1835 x x 1836 b *a 1837----------------- 1838 log(a) + log(b) 1839 1840testint(a**x/x**2,x); 1841 1842 1843 x 1844 ei(log(a)*x)*log(a)*x - a 1845---------------------------- 1846 x 1847 1848testint(x*a**x/(1+b*x)**2,x); 1849 1850 1851 x 1852 a *x 1853int(-----------------------------------------------------------,x)*(log(a) - b) 1854 2 2 3 2 2 1855 log(a)*b *x + 2*log(a)*b*x + log(a) - b *x - 2*b *x - b 1856 1857testint(x*e**(a*x)/(1+a*x)**2,x); 1858 1859 1860 a*x 1861 e 1862-------------- 1863 2 1864 a *(a*x + 1) 1865 1866testint(x*k**(x**2),x); 1867 1868 1869 2 1870 x 1871 k 1872---------- 1873 2*log(k) 1874 1875testint(e**(x**2),x); 1876 1877 1878 - sqrt(pi)*erf(i*x)*i 1879------------------------ 1880 2 1881 1882testint(x*e**(x**2),x); 1883 1884 1885 2 1886 x 1887 e 1888----- 1889 2 1890 1891testint((x+1)*e**(1/x)/x**4,x); 1892 1893 1894 1/x 2 1895 e *( - x + x - 1) 1896---------------------- 1897 2 1898 x 1899 1900testint((2*x**3+x)*(e**(x**2))**2*e**(1-x*e**(x**2))/(1-x*e**(x**2))**2, 1901 x); 1902 1903 1904 - e 1905-------------------- 1906 2 1907 x 2 1908 e *x x 1909 e *(e *x - 1) 1910 1911testint(e**(e**(e**(e**x))),x); 1912 1913 1914 x 1915 e 1916 e 1917 e 1918int(e ,x) 1919 1920 1921% Examples involving exponentials and logarithms. 1922 1923testint(e**x*log x,x); 1924 1925 1926 x 1927 - ei(x) + e *log(x) 1928 1929testint(x*e**x*log x,x); 1930 1931 1932 x x x 1933ei(x) + e *log(x)*x - e *log(x) - e 1934 1935testint(e**(2*x)*log(e**x),x); 1936 1937 1938 2*x 1939 e *(2*x - 1) 1940---------------- 1941 4 1942 1943 1944% Examples involving square roots. 1945 1946testint(sqrt(2)*x**2 + 2*x,x); 1947 1948 1949 2 1950 x *(sqrt(2)*x + 3) 1951-------------------- 1952 3 1953 1954testint(log x/sqrt(a*x+b),x); 1955 1956 1957(2*(sqrt(a*x + b)*log(x) - 2*sqrt(a*x + b) 1958 1959 - 2*sqrt(b)*log(sqrt(a*x + b) - sqrt(b)) + sqrt(b)*log(x)))/a 1960 1961u:=sqrt(a+b*x); 1962 1963 1964u := sqrt(a + b*x) 1965 1966v:=sqrt(c+d*x); 1967 1968 1969v := sqrt(c + d*x) 1970 1971testint(u*v,x); 1972 1973 1974 2 2 1975(sqrt(c + d*x)*sqrt(a + b*x)*a*b*d + sqrt(c + d*x)*sqrt(a + b*x)*b *c*d 1976 1977 2 2 1978 + 2*sqrt(c + d*x)*sqrt(a + b*x)*b *d *x 1979 1980 sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x) 2 2 1981 - sqrt(d)*sqrt(b)*log(-----------------------------------------------)*a *d + 1982 sqrt(a*d - b*c) 1983 1984 sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x) 1985 2*sqrt(d)*sqrt(b)*log(-----------------------------------------------)*a*b*c*d 1986 sqrt(a*d - b*c) 1987 1988 sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x) 2 2 1989 - sqrt(d)*sqrt(b)*log(-----------------------------------------------)*b *c )/ 1990 sqrt(a*d - b*c) 1991 1992 2 2 1993(4*b *d ) 1994 1995testint(u,x); 1996 1997 1998 2*sqrt(a + b*x)*(a + b*x) 1999--------------------------- 2000 3*b 2001 2002testint(x*u,x); 2003 2004 2005 2 2 2 2006 2*sqrt(a + b*x)*( - 2*a + a*b*x + 3*b *x ) 2007--------------------------------------------- 2008 2 2009 15*b 2010 2011testint(x**2*u,x); 2012 2013 2014 3 2 2 2 3 3 2015 2*sqrt(a + b*x)*(8*a - 4*a *b*x + 3*a*b *x + 15*b *x ) 2016---------------------------------------------------------- 2017 3 2018 105*b 2019 2020testint(u/x,x); 2021 2022 20232*sqrt(a + b*x) + sqrt(a)*log(sqrt(a + b*x) - sqrt(a)) 2024 2025 - sqrt(a)*log(sqrt(a + b*x) + sqrt(a)) 2026 2027testint(u/x**2,x); 2028 2029 2030( - 2*sqrt(a + b*x)*a + sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*b*x 2031 2032 - sqrt(a)*log(sqrt(a + b*x) + sqrt(a))*b*x)/(2*a*x) 2033 2034testint(1/u,x); 2035 2036 2037 2*sqrt(a + b*x) 2038----------------- 2039 b 2040 2041testint(x/u,x); 2042 2043 2044 2*sqrt(a + b*x)*( - 2*a + b*x) 2045-------------------------------- 2046 2 2047 3*b 2048 2049testint(x**2/u,x); 2050 2051 2052 2 2 2 2053 2*sqrt(a + b*x)*(8*a - 4*a*b*x + 3*b *x ) 2054-------------------------------------------- 2055 3 2056 15*b 2057 2058testint(1/(x*u),x); 2059 2060 2061 sqrt(a)*(log(sqrt(a + b*x) - sqrt(a)) - log(sqrt(a + b*x) + sqrt(a))) 2062----------------------------------------------------------------------- 2063 a 2064 2065testint(1/(x**2*u),x); 2066 2067 2068( - 2*sqrt(a + b*x)*a - sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*b*x 2069 2070 2 2071 + sqrt(a)*log(sqrt(a + b*x) + sqrt(a))*b*x)/(2*a *x) 2072 2073testint(u**p,x); 2074 2075 2076 p/2 2077 2*(a + b*x) *(a + b*x) 2078-------------------------- 2079 b*(p + 2) 2080 2081testint(x*u**p,x); 2082 2083 2084 p/2 2 2 2 2 2 2085 2*(a + b*x) *( - 2*a + a*b*p*x + b *p*x + 2*b *x ) 2086-------------------------------------------------------- 2087 2 2 2088 b *(p + 6*p + 8) 2089 2090testint(atan((-sqrt(2)+2*x)/sqrt(2)),x); 2091 2092 2093 sqrt(2) - 2*x sqrt(2) - 2*x 2094(2*sqrt(2)*atan(---------------) - 4*atan(---------------)*x 2095 sqrt(2) sqrt(2) 2096 2097 2 2098 - sqrt(2)*log( - sqrt(2)*x + x + 1))/4 2099 2100testint(1/sqrt(x**2-1),x); 2101 2102 2103 2 2104log(sqrt(x - 1) + x) 2105 2106testint(sqrt(x+1)*sqrt x,x); 2107 2108 2109 2*sqrt(x)*sqrt(x + 1)*x + sqrt(x)*sqrt(x + 1) - log(sqrt(x + 1) + sqrt(x)) 2110---------------------------------------------------------------------------- 2111 4 2112 2113 2114testint(sin(sqrt x),x); 2115 2116 21172*( - sqrt(x)*cos(sqrt(x)) + sin(sqrt(x))) 2118 2119testint(x*(1-x^2)^(-9/4),x); 2120 2121 2122 2 1/4 2123 - 2*( - x + 1) 2124---------------------------- 2125 2 2 2126 5*sqrt( - x + 1)*(x - 1) 2127 2128testint(x/sqrt(1-x^4),x); 2129 2130 2131 2 2132 asin(x ) 2133---------- 2134 2 2135 2136testint(1/(x*sqrt(1+x^4)),x); 2137 2138 2139 4 2 4 2 2140 log(sqrt(x + 1) + x - 1) - log(sqrt(x + 1) + x + 1) 2141--------------------------------------------------------- 2142 2 2143 2144testint(x/sqrt(1+x^2+x^4),x); 2145 2146 2147 4 2 2 2148 2*sqrt(x + x + 1) + 2*x + 1 2149 log(--------------------------------) 2150 sqrt(3) 2151--------------------------------------- 2152 2 2153 2154testint(1/(x*sqrt(x^2-1-x^4)),x); 2155 2156 2157 1 2158int(------------------------,x) 2159 4 2 2160 sqrt( - x + x - 1)*x 2161 2162 2163% Examples from James Davenport's thesis: 2164 2165testint(1/sqrt(x**2-1)+10/sqrt(x**2-4),x); 2166 2167 2168 2 2169 2 sqrt(x - 4) + x 2170log(sqrt(x - 1) + x) + 10*log(------------------) 2171 2 2172 % p. 173 2173testint(sqrt(x+sqrt(x**2+a**2))/x,x); 2174 2175 2176 2 2 2 2 2177sqrt(a)*atan((sqrt(a)*sqrt(sqrt(a + x ) + x)*sqrt(a + x ) 2178 2179 2 2 2180 - sqrt(a)*sqrt(sqrt(a + x ) + x)*a 2181 2182 2 2 2 2183 - sqrt(a)*sqrt(sqrt(a + x ) + x)*x)/(2*a )) 2184 2185 2 2 2 2 2186 + 2*sqrt(sqrt(a + x ) + x) + sqrt(a)*log(sqrt(sqrt(a + x ) + x) - sqrt(a)) 2187 2188 2 2 2189 - sqrt(a)*log(sqrt(sqrt(a + x ) + x) + sqrt(a)) 2190 2191 2192% Examples generated by differentiating various functions. 2193 2194testint(df(sqrt(1+x**2)/(1-x),x),x); 2195 2196 2197 2 2198 - sqrt(x + 1) 2199----------------- 2200 x - 1 2201 2202testint(df(log(x+sqrt(1+x**2)),x),x); 2203 2204 2205 2 2206log(sqrt(x + 1) + x) 2207 2208testint(df(sqrt(x)+sqrt(x+1)+sqrt(x+2),x),x); 2209 2210 2211sqrt(x + 2) + sqrt(x + 1) + sqrt(x) 2212 2213testint(df(sqrt(x**5-2*x+1)-sqrt(x**3+1),x),x); 2214 2215 2216 5 3 2217sqrt(x - 2*x + 1) - sqrt(x + 1) 2218 2219 2220% Another such example from James Davenport's thesis (p. 146). 2221% It contains a point of order 3, which is found by use of Mazur's 2222% bound on the torsion of elliptic curves over the rationals; 2223 2224testint(df(log(1+sqrt(x**3+1)),x),x); 2225 2226 2227 3 3 2228 - log(sqrt(x + 1) - 1) + log(sqrt(x + 1) + 1) + 3*log(x) 2229------------------------------------------------------------- 2230 2 2231 2232 2233% Examples quoted by Joel Moses: 2234 2235testint(1/sqrt(2*h*r**2-alpha**2),r); 2236 2237 2238 2 2 2239 sqrt( - alpha + 2*h*r ) + sqrt(h)*sqrt(2)*r 2240 sqrt(h)*sqrt(2)*log(----------------------------------------------) 2241 alpha 2242--------------------------------------------------------------------- 2243 2*h 2244 2245testint(1/(r*sqrt(2*h*r**2-alpha**2-epsilon**2)),r); 2246 2247 2248 2 2 2249(2*sqrt(alpha + epsilon ) 2250 2251 2 2 2 2252 sqrt( - alpha - epsilon + 2*h*r ) + sqrt(h)*sqrt(2)*r 2 2253 *atan(---------------------------------------------------------))/(alpha 2254 2 2 2255 sqrt(alpha + epsilon ) 2256 2257 2 2258 + epsilon ) 2259 2260testint(1/(r*sqrt(2*h*r**2-alpha**2-2*k*r)),r); 2261 2262 2263 2 2 2264 sqrt(h)*sqrt( - alpha + 2*h*r - 2*k*r)*sqrt(2) + 2*h*r 2265 2*atan(----------------------------------------------------------) 2266 sqrt(h)*sqrt(2)*alpha 2267-------------------------------------------------------------------- 2268 alpha 2269 2270testint(1/(r*sqrt(2*h*r**2-alpha**2-epsilon**2-2*k*r)),r); 2271 2272 2273 2 2 2274(2*sqrt(alpha + epsilon ) 2275 2276 2 2 2 2277 sqrt(h)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r)*sqrt(2) + 2*h*r 2278 *atan(---------------------------------------------------------------------))/( 2279 2 2 2280 sqrt(h)*sqrt(alpha + epsilon )*sqrt(2) 2281 2282 2 2 2283 alpha + epsilon ) 2284 2285testint(r/sqrt(2*e*r**2-alpha**2),r); 2286 2287 2288 2 2 2289 sqrt( - alpha + 2*e*r ) 2290-------------------------- 2291 2*e 2292 2293testint(r/sqrt(2*e*r**2-alpha**2-epsilon**2),r); 2294 2295 2296 2 2 2 2297 sqrt( - alpha + 2*e*r - epsilon ) 2298------------------------------------- 2299 2*e 2300 2301testint(r/sqrt(2*e*r**2-alpha**2-2*k*r**4),r); 2302 2303 2304 2 2305 e*i - 2*i*k*r 2306 sqrt(k)*sqrt(2)*asinh(--------------------------)*i 2307 2 2 2308 sqrt( - 2*alpha *k + e ) 2309----------------------------------------------------- 2310 4*k 2311 2312testint(r/sqrt(2*e*r**2-alpha**2-2*k*r),r); 2313 2314 2315 2 2 2316(2*sqrt( - alpha + 2*e*r - 2*k*r)*e + sqrt(e)*sqrt(2) 2317 2318 2 2 2319 sqrt(e)*sqrt( - alpha + 2*e*r - 2*k*r)*sqrt(2) + 2*e*r - k 2 2320 *log(--------------------------------------------------------------)*k)/(4*e ) 2321 2 2 2322 sqrt(2*alpha *e + k ) 2323 2324testint(1/(r*sqrt(2*h*r**2-alpha**2-2*k*r**4)),r); 2325 2326 2327 2 2 2328(i*(256*sqrt(k)*sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h )*sqrt(2)*atan( 2329 2330 2 2 4 2 2 2 2331 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2332 ------------------------------------------------------------------------) 2333 2 2 2334 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2335 2336 15 7 2 2 2337 *alpha *k + 9856*sqrt(k)*sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2338 2339 *sqrt(2)*atan( 2340 2341 2 2 4 2 2 2 2342 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2343 ------------------------------------------------------------------------) 2344 2 2 2345 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2346 2347 13 2 6 2348 *alpha *h *k + 40768*sqrt(k) 2349 2350 2 2 2351 *sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h )*sqrt(2)*atan( 2352 2353 2 2 4 2 2 2 2354 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2355 ------------------------------------------------------------------------) 2356 2 2 2357 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2358 2359 11 4 5 2360 *alpha *h *k + 32032*sqrt(k) 2361 2362 2 2 2363 *sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h )*sqrt(2)*atan( 2364 2365 2 2 4 2 2 2 2366 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2367 ------------------------------------------------------------------------) 2368 2 2 2369 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2370 2371 9 6 4 2372 *alpha *h *k - 6864*sqrt(k) 2373 2374 2 2 2375 *sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h )*sqrt(2)*atan( 2376 2377 2 2 4 2 2 2 2378 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2379 ------------------------------------------------------------------------) 2380 2 2 2381 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2382 2383 7 8 3 2384 *alpha *h *k - 8008*sqrt(k) 2385 2386 2 2 2387 *sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h )*sqrt(2)*atan( 2388 2389 2 2 4 2 2 2 2390 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2391 ------------------------------------------------------------------------) 2392 2 2 2393 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2394 2395 5 10 2 2396 *alpha *h *k - 1092*sqrt(k) 2397 2398 2 2 2399 *sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h )*sqrt(2)*atan( 2400 2401 2 2 4 2 2 2 2402 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2403 ------------------------------------------------------------------------) 2404 2 2 2405 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2406 2407 3 12 2 2 2408 *alpha *h *k - 26*sqrt(k)*sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2409 2410 *sqrt(2)*atan( 2411 2412 2 2 4 2 2 2 2413 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2414 ------------------------------------------------------------------------) 2415 2 2 2416 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2417 2418 14 2 2 2419 *alpha*h - 3328*sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h )*atan( 2420 2421 2 2 4 2 2 2 2422 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2423 ------------------------------------------------------------------------) 2424 2 2 2425 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2426 2427 14 7 2 2 2428 *alpha *h*k - 34944*sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h )*atan 2429 2430 2 2 4 2 2 2 2431 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2432 (------------------------------------------------------------------------) 2433 2 2 2434 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2435 2436 12 3 6 2 2 2437 *alpha *h *k - 64064*sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h )* 2438 2439 2 2 4 2 2 2 2440 atan((sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2441 2442 2 2 10 5 5 2443 )/sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ))*alpha *h *k - 2444 2445 2 2 2446 13728*sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h )*atan( 2447 2448 2 2 4 2 2 2 2449 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2450 ------------------------------------------------------------------------) 2451 2 2 2452 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2453 2454 8 7 4 2 2 2455 *alpha *h *k + 16016*sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h )*atan 2456 2457 2 2 4 2 2 2 2458 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2459 (------------------------------------------------------------------------) 2460 2 2 2461 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2462 2463 6 9 3 2 2 2464 *alpha *h *k + 5096*sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h )*atan( 2465 2466 2 2 4 2 2 2 2467 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2468 ------------------------------------------------------------------------) 2469 2 2 2470 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2471 2472 4 11 2 2 2 2473 *alpha *h *k + 308*sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h )*atan( 2474 2475 2 2 4 2 2 2 2476 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2477 ------------------------------------------------------------------------) 2478 2 2 2479 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2480 2481 2 13 2 2 2482 *alpha *h *k + 2*sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h )*atan( 2483 2484 2 2 4 2 2 2 2485 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i - h*i + 2*i *k*r 2486 ------------------------------------------------------------------------) 2487 2 2 2488 sqrt(2*sqrt(k)*sqrt(2)*alpha*h - 2*alpha *k - h ) 2489 2490 15 2 2 2491 *h - 128*sqrt(k)*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*sqrt(2) 2492 2493 2 2 4 2494 *log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2495 2496 2 2 2 2497 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 2498 2499 15 7 2 2 2500 *alpha *k - 6720*sqrt(k)*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) 2501 2502 2 2 4 2503 *sqrt(2)*log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2504 2505 2 2 2506 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h 2507 2508 2 13 2 6 2509 - 2*k*r )*alpha *h *k - 43680*sqrt(k) 2510 2511 2 2 2512 *sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*sqrt(2)*log( 2513 2514 2 2 4 2515 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2516 2517 2 2 2 2518 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 2519 2520 11 4 5 2521 *alpha *h *k - 80080*sqrt(k) 2522 2523 2 2 2524 *sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*sqrt(2)*log( 2525 2526 2 2 4 2527 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2528 2529 2 2 2 9 2530 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2531 2532 6 4 2 2 2533 *h *k - 51480*sqrt(k)*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) 2534 2535 2 2 4 2536 *sqrt(2)*log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2537 2538 2 2 2539 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h 2540 2541 2 7 8 3 2542 - 2*k*r )*alpha *h *k - 12012*sqrt(k) 2543 2544 2 2 2545 *sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*sqrt(2)*log( 2546 2547 2 2 4 2548 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2549 2550 2 2 2 5 2551 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2552 2553 10 2 2 2 2554 *h *k - 910*sqrt(k)*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) 2555 2556 2 2 4 2557 *sqrt(2)*log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2558 2559 2 2 2560 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h 2561 2562 2 3 12 2563 - 2*k*r )*alpha *h *k - 15*sqrt(k) 2564 2565 2 2 2566 *sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*sqrt(2)*log( 2567 2568 2 2 4 2569 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2570 2571 2 2 2 2572 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2573 2574 14 2 2 2575 *h + 128*sqrt(k)*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*sqrt(2) 2576 2577 2 2 4 2578 *log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2579 2580 2 2 2 2581 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 2582 2583 15 7 2 2 2584 *alpha *k + 6720*sqrt(k)*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) 2585 2586 2 2 4 2587 *sqrt(2)*log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2588 2589 2 2 2590 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h 2591 2592 2 13 2 6 2593 - 2*k*r )*alpha *h *k + 43680*sqrt(k) 2594 2595 2 2 2596 *sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*sqrt(2)*log( 2597 2598 2 2 4 2599 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2600 2601 2 2 2 2602 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 2603 2604 11 4 5 2605 *alpha *h *k + 80080*sqrt(k) 2606 2607 2 2 2608 *sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*sqrt(2)*log( 2609 2610 2 2 4 2611 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2612 2613 2 2 2 9 2614 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2615 2616 6 4 2 2 2617 *h *k + 51480*sqrt(k)*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) 2618 2619 2 2 4 2620 *sqrt(2)*log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2621 2622 2 2 2623 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h 2624 2625 2 7 8 3 2626 - 2*k*r )*alpha *h *k + 12012*sqrt(k) 2627 2628 2 2 2629 *sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*sqrt(2)*log( 2630 2631 2 2 4 2632 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2633 2634 2 2 2 5 2635 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2636 2637 10 2 2 2 2638 *h *k + 910*sqrt(k)*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) 2639 2640 2 2 4 2641 *sqrt(2)*log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2642 2643 2 2 2644 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h 2645 2646 2 3 12 2647 - 2*k*r )*alpha *h *k + 15*sqrt(k) 2648 2649 2 2 2650 *sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*sqrt(2)*log( 2651 2652 2 2 4 2653 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2654 2655 2 2 2 2656 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2657 2658 14 2 2 2659 *h + 1920*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log( 2660 2661 2 2 4 2662 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2663 2664 2 2 2 2665 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 2666 2667 14 7 2 2 2668 *alpha *h*k + 29120*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log( 2669 2670 2 2 4 2671 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2672 2673 2 2 2 2674 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 2675 2676 12 3 6 2 2 2677 *alpha *h *k + 96096*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log 2678 2679 2 2 4 2680 (sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2681 2682 2 2 2 10 2683 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2684 2685 5 5 2 2 2686 *h *k + 102960*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log( 2687 2688 2 2 4 2689 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2690 2691 2 2 2 8 2692 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2693 2694 7 4 2 2 2695 *h *k + 40040*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log( 2696 2697 2 2 4 2698 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2699 2700 2 2 2 6 2701 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2702 2703 9 3 2 2 2704 *h *k + 5460*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log( 2705 2706 2 2 4 2707 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2708 2709 2 2 2 4 2710 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2711 2712 11 2 2 2 2713 *h *k + 210*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log( 2714 2715 2 2 4 2716 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2717 2718 2 2 2 2 2719 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2720 2721 13 2 2 2722 *h *k + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log( 2723 2724 2 2 4 2725 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2726 2727 2 2 2 15 2728 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*h - 2729 2730 2 2 2731 1920*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log( 2732 2733 2 2 4 2734 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2735 2736 2 2 2 2737 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 2738 2739 14 7 2 2 2740 *alpha *h*k - 29120*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log( 2741 2742 2 2 4 2743 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2744 2745 2 2 2 2746 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 2747 2748 12 3 6 2 2 2749 *alpha *h *k - 96096*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log 2750 2751 2 2 4 2752 (sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2753 2754 2 2 2 10 2755 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2756 2757 5 5 2 2 2758 *h *k - 102960*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log( 2759 2760 2 2 4 2761 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2762 2763 2 2 2 8 2764 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2765 2766 7 4 2 2 2767 *h *k - 40040*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log( 2768 2769 2 2 4 2770 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2771 2772 2 2 2 6 2773 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2774 2775 9 3 2 2 2776 *h *k - 5460*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log( 2777 2778 2 2 4 2779 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2780 2781 2 2 2 4 2782 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2783 2784 11 2 2 2 2785 *h *k - 210*sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log( 2786 2787 2 2 4 2788 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2789 2790 2 2 2 2 2791 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2792 2793 13 2 2 2794 *h *k - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h )*log( 2795 2796 2 2 4 2797 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2798 2799 2 2 2 15 2800 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*h + 2801 2802 2 2 4 2803 1792*sqrt(k)*sqrt(2)*log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2804 2805 2 2 2 2806 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 2807 2808 15 7 2809 *alpha *h*k + 22400*sqrt(k)*sqrt(2)*log( 2810 2811 2 2 4 2812 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2813 2814 2 2 2 2815 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 2816 2817 13 3 6 2818 *alpha *h *k + 52416*sqrt(k)*sqrt(2)*log( 2819 2820 2 2 4 2821 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2822 2823 2 2 2 2824 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 2825 2826 11 5 5 2827 *alpha *h *k + 22880*sqrt(k)*sqrt(2)*log( 2828 2829 2 2 4 2830 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2831 2832 2 2 2 9 2833 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2834 2835 7 4 2836 *h *k - 11440*sqrt(k)*sqrt(2)*log( 2837 2838 2 2 4 2839 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2840 2841 2 2 2 7 2842 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2843 2844 9 3 2845 *h *k - 6552*sqrt(k)*sqrt(2)*log( 2846 2847 2 2 4 2848 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2849 2850 2 2 2 5 2851 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2852 2853 11 2 2854 *h *k - 700*sqrt(k)*sqrt(2)*log( 2855 2856 2 2 4 2857 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2858 2859 2 2 2 3 2860 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2861 2862 13 2863 *h *k - 14*sqrt(k)*sqrt(2)*log( 2864 2865 2 2 4 2866 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2867 2868 2 2 2 2869 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2870 2871 15 2872 *h + 1792*sqrt(k)*sqrt(2)*log( 2873 2874 2 2 4 2875 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2876 2877 2 2 2 2878 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 2879 2880 15 7 2881 *alpha *h*k + 22400*sqrt(k)*sqrt(2)*log( 2882 2883 2 2 4 2884 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2885 2886 2 2 2 2887 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 2888 2889 13 3 6 2890 *alpha *h *k + 52416*sqrt(k)*sqrt(2)*log( 2891 2892 2 2 4 2893 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2894 2895 2 2 2 2896 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 2897 2898 11 5 5 2899 *alpha *h *k + 22880*sqrt(k)*sqrt(2)*log( 2900 2901 2 2 4 2902 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2903 2904 2 2 2 9 2905 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2906 2907 7 4 2908 *h *k - 11440*sqrt(k)*sqrt(2)*log( 2909 2910 2 2 4 2911 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2912 2913 2 2 2 7 2914 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2915 2916 9 3 2917 *h *k - 6552*sqrt(k)*sqrt(2)*log( 2918 2919 2 2 4 2920 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2921 2922 2 2 2 5 2923 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2924 2925 11 2 2926 *h *k - 700*sqrt(k)*sqrt(2)*log( 2927 2928 2 2 4 2929 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2930 2931 2 2 2 3 2932 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2933 2934 13 2935 *h *k - 14*sqrt(k)*sqrt(2)*log( 2936 2937 2 2 4 2938 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 2939 2940 2 2 2 2941 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 2942 2943 15 2944 *h - 1792*sqrt(k)*sqrt(2)*log( 2945 2946 2 2 4 2947 2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 2948 2949 2 2 4 2 2950 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 2951 2952 2 2 4 15 7 2953 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha *h*k - 22400 2954 2955 2 2 4 2956 *sqrt(k)*sqrt(2)*log(2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 2957 2958 2 2 4 2 2959 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 2960 2961 2 2 4 13 3 6 2962 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha *h *k - 52416 2963 2964 2 2 4 2965 *sqrt(k)*sqrt(2)*log(2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 2966 2967 2 2 4 2 2968 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 2969 2970 2 2 4 11 5 5 2971 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha *h *k - 22880 2972 2973 2 2 4 2974 *sqrt(k)*sqrt(2)*log(2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 2975 2976 2 2 4 2 2977 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 2978 2979 2 2 4 9 7 4 2980 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha *h *k + 11440 2981 2982 2 2 4 2983 *sqrt(k)*sqrt(2)*log(2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 2984 2985 2 2 4 2 2986 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 2987 2988 2 2 4 7 9 3 2989 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha *h *k + 6552 2990 2991 2 2 4 2992 *sqrt(k)*sqrt(2)*log(2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 2993 2994 2 2 4 2 2995 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 2996 2997 2 2 4 5 11 2 2998 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha *h *k + 700 2999 3000 2 2 4 3001 *sqrt(k)*sqrt(2)*log(2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 3002 3003 2 2 4 2 3004 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 3005 3006 2 2 4 3 13 3007 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha *h *k + 14 3008 3009 2 2 4 3010 *sqrt(k)*sqrt(2)*log(2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 3011 3012 2 2 4 2 3013 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 3014 3015 2 2 4 15 3016 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha*h - 256*log( 3017 3018 2 2 4 3019 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3020 3021 2 2 2 3022 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 3023 3024 16 8 2 2 4 3025 *alpha *k - 11520*log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3026 3027 2 2 2 3028 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 3029 3030 14 2 7 3031 *alpha *h *k - 58240*log( 3032 3033 2 2 4 3034 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3035 3036 2 2 2 3037 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 3038 3039 12 4 6 3040 *alpha *h *k - 64064*log( 3041 3042 2 2 4 3043 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3044 3045 2 2 2 3046 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 3047 3048 10 6 5 3049 *alpha *h *k + 16016*log( 3050 3051 2 2 4 3052 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3053 3054 2 2 2 6 3055 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 3056 3057 10 3 2 2 4 3058 *h *k + 3640*log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3059 3060 2 2 2 4 3061 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 3062 3063 12 2 2 2 4 3064 *h *k + 180*log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3065 3066 2 2 2 2 3067 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 3068 3069 14 2 2 4 3070 *h *k + log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3071 3072 2 2 3073 - sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h 3074 3075 2 16 3076 - 2*k*r )*h - 256*log( 3077 3078 2 2 4 3079 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3080 3081 2 2 2 3082 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 3083 3084 16 8 2 2 4 3085 *alpha *k - 11520*log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3086 3087 2 2 2 3088 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 3089 3090 14 2 7 3091 *alpha *h *k - 58240*log( 3092 3093 2 2 4 3094 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3095 3096 2 2 2 3097 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 3098 3099 12 4 6 3100 *alpha *h *k - 64064*log( 3101 3102 2 2 4 3103 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3104 3105 2 2 2 3106 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r ) 3107 3108 10 6 5 3109 *alpha *h *k + 16016*log( 3110 3111 2 2 4 3112 sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3113 3114 2 2 2 6 3115 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 3116 3117 10 3 2 2 4 3118 *h *k + 3640*log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3119 3120 2 2 2 4 3121 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 3122 3123 12 2 2 2 4 3124 *h *k + 180*log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3125 3126 2 2 2 2 3127 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h - 2*k*r )*alpha 3128 3129 14 2 2 4 3130 *h *k + log(sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i 3131 3132 2 2 3133 + sqrt(2*sqrt(k)*sqrt(2)*alpha*h + 2*alpha *k + h ) + h 3134 3135 2 16 3136 - 2*k*r )*h + 256*log( 3137 3138 2 2 4 3139 2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 3140 3141 2 2 4 2 3142 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 3143 3144 2 2 4 16 8 3145 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha *k + 11520*log 3146 3147 2 2 4 3148 (2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 3149 3150 2 2 4 2 3151 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 3152 3153 2 2 4 14 2 7 3154 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha *h *k + 58240* 3155 3156 2 2 4 3157 log(2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 3158 3159 2 2 4 2 3160 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 3161 3162 2 2 4 12 4 6 3163 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha *h *k + 64064 3164 3165 2 2 4 3166 *log(2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 3167 3168 2 2 4 2 3169 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 3170 3171 2 2 4 10 6 5 3172 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha *h *k - 3173 3174 2 2 4 3175 16016*log(2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 3176 3177 2 2 4 2 3178 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 3179 3180 2 2 4 6 10 3 3181 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha *h *k - 3182 3183 2 2 4 3184 3640*log(2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 3185 3186 2 2 4 2 3187 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 3188 3189 2 2 4 4 12 2 3190 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha *h *k - 3191 3192 2 2 4 3193 180*log(2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 3194 3195 2 2 4 2 3196 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 3197 3198 2 2 4 2 14 3199 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*alpha *h *k - 3200 3201 2 2 4 3202 log(2*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*h*i 3203 3204 2 2 4 2 3205 - 4*sqrt(k)*sqrt( - alpha + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 3206 3207 2 2 4 16 3208 + 2*sqrt(k)*sqrt(2)*alpha*h - 8*h*k*r + 8*k *r )*h ))/(4*alpha*( 3209 3210 15 7 13 3 6 3211 1792*sqrt(k)*sqrt(2)*alpha *h*k + 22400*sqrt(k)*sqrt(2)*alpha *h *k 3212 3213 11 5 5 3214 + 52416*sqrt(k)*sqrt(2)*alpha *h *k 3215 3216 9 7 4 7 9 3 3217 + 22880*sqrt(k)*sqrt(2)*alpha *h *k - 11440*sqrt(k)*sqrt(2)*alpha *h *k 3218 3219 5 11 2 3 13 3220 - 6552*sqrt(k)*sqrt(2)*alpha *h *k - 700*sqrt(k)*sqrt(2)*alpha *h *k 3221 3222 15 16 8 14 2 7 3223 - 14*sqrt(k)*sqrt(2)*alpha*h - 256*alpha *k - 11520*alpha *h *k 3224 3225 12 4 6 10 6 5 6 10 3 3226 - 58240*alpha *h *k - 64064*alpha *h *k + 16016*alpha *h *k 3227 3228 4 12 2 2 14 16 3229 + 3640*alpha *h *k + 180*alpha *h *k + h )) 3230 3231testint(1/(r*sqrt(2*h*r**2-alpha**2-epsilon**2-2*k*r**4)),r); 3232 3233 3234 2 2 2 3235(i*( - 2*sqrt(2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h - 2*alpha *k 3236 3237 2 2 2 2 3238 - 2*epsilon *k - h )*sqrt(alpha + epsilon )*atan(( 3239 3240 2 2 2 4 2 3241 sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i - h*i 3242 3243 2 2 2 2 3244 + 2*i *k*r )/sqrt(2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h 3245 3246 2 2 2 3247 - 2*alpha *k - 2*epsilon *k - h ))*h - 2*sqrt(k)*sqrt( 3248 3249 2 2 2 2 3250 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h - 2*alpha *k - 2*epsilon *k 3251 3252 2 3253 - h )*sqrt(2)*atan(( 3254 3255 2 2 2 4 2 3256 sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i - h*i 3257 3258 2 2 2 2 3259 + 2*i *k*r )/sqrt(2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h 3260 3261 2 2 2 2 3262 - 2*alpha *k - 2*epsilon *k - h ))*alpha - 2*sqrt(k)*sqrt( 3263 3264 2 2 2 2 3265 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h - 2*alpha *k - 2*epsilon *k 3266 3267 2 3268 - h )*sqrt(2)*atan(( 3269 3270 2 2 2 4 2 3271 sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i - h*i 3272 3273 2 2 2 2 3274 + 2*i *k*r )/sqrt(2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h 3275 3276 2 2 2 2 3277 - 2*alpha *k - 2*epsilon *k - h ))*epsilon - sqrt( 3278 3279 2 2 2 2 3280 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k + 2*epsilon *k 3281 3282 2 2 2 3283 + h )*sqrt(alpha + epsilon )*log( 3284 3285 2 2 2 4 3286 sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i - sqrt( 3287 3288 2 2 2 3289 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k 3290 3291 2 2 2 3292 + 2*epsilon *k + h ) + h - 2*k*r )*h + sqrt( 3293 3294 2 2 2 2 3295 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k + 2*epsilon *k 3296 3297 2 2 2 3298 + h )*sqrt(alpha + epsilon )*log( 3299 3300 2 2 2 4 3301 sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i + sqrt( 3302 3303 2 2 2 3304 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k 3305 3306 2 2 2 2 2 3307 + 2*epsilon *k + h ) + h - 2*k*r )*h + 2*sqrt(alpha + epsilon )*log( 3308 3309 2 2 2 4 3310 sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i - sqrt( 3311 3312 2 2 2 3313 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k 3314 3315 2 2 2 2 3316 + 2*epsilon *k + h ) + h - 2*k*r )*alpha *k + 2 3317 3318 2 2 3319 *sqrt(alpha + epsilon )*log( 3320 3321 2 2 2 4 3322 sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i - sqrt( 3323 3324 2 2 2 3325 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k 3326 3327 2 2 2 2 3328 + 2*epsilon *k + h ) + h - 2*k*r )*epsilon *k - 3329 3330 2 2 3331 sqrt(alpha + epsilon )*log( 3332 3333 2 2 2 4 3334 sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i - sqrt( 3335 3336 2 2 2 3337 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k 3338 3339 2 2 2 2 2 2 3340 + 2*epsilon *k + h ) + h - 2*k*r )*h + 2*sqrt(alpha + epsilon )*log 3341 3342 2 2 2 4 3343 (sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i + sqrt( 3344 3345 2 2 2 2 3346 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k + 2*epsilon *k 3347 3348 2 2 2 2 2 3349 + h ) + h - 2*k*r )*alpha *k + 2*sqrt(alpha + epsilon )*log( 3350 3351 2 2 2 4 3352 sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i + sqrt( 3353 3354 2 2 2 3355 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k 3356 3357 2 2 2 2 3358 + 2*epsilon *k + h ) + h - 2*k*r )*epsilon *k - 3359 3360 2 2 3361 sqrt(alpha + epsilon )*log( 3362 3363 2 2 2 4 3364 sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i + sqrt( 3365 3366 2 2 2 3367 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k 3368 3369 2 2 2 2 2 2 3370 + 2*epsilon *k + h ) + h - 2*k*r )*h - 2*sqrt(alpha + epsilon )*log 3371 3372 2 2 3373 (2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h 3374 3375 2 2 2 4 3376 + 2*sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*h*i 3377 3378 2 2 2 4 2 3379 - 4*sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 3380 3381 2 2 4 2 2 2 3382 - 8*h*k*r + 8*k *r )*alpha *k - 2*sqrt(alpha + epsilon )*log( 3383 3384 2 2 3385 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h 3386 3387 2 2 2 4 3388 + 2*sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*h*i 3389 3390 2 2 2 4 2 3391 - 4*sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 3392 3393 2 2 4 2 2 2 3394 - 8*h*k*r + 8*k *r )*epsilon *k + sqrt(alpha + epsilon )*log( 3395 3396 2 2 3397 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h 3398 3399 2 2 2 4 3400 + 2*sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*h*i 3401 3402 2 2 2 4 2 3403 - 4*sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i*k*r 3404 3405 2 2 4 2 3406 - 8*h*k*r + 8*k *r )*h + sqrt(k)*sqrt( 3407 3408 2 2 2 2 3409 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k + 2*epsilon *k 3410 3411 2 3412 + h )*sqrt(2)*log( 3413 3414 2 2 2 4 3415 sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i - sqrt( 3416 3417 2 2 2 3418 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k 3419 3420 2 2 2 2 3421 + 2*epsilon *k + h ) + h - 2*k*r )*alpha + sqrt(k)*sqrt( 3422 3423 2 2 2 2 3424 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k + 2*epsilon *k 3425 3426 2 3427 + h )*sqrt(2)*log( 3428 3429 2 2 2 4 3430 sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i - sqrt( 3431 3432 2 2 2 3433 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k 3434 3435 2 2 2 2 3436 + 2*epsilon *k + h ) + h - 2*k*r )*epsilon - sqrt(k)*sqrt( 3437 3438 2 2 2 2 3439 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k + 2*epsilon *k 3440 3441 2 3442 + h )*sqrt(2)*log( 3443 3444 2 2 2 4 3445 sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i + sqrt( 3446 3447 2 2 2 3448 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k 3449 3450 2 2 2 2 3451 + 2*epsilon *k + h ) + h - 2*k*r )*alpha - sqrt(k)*sqrt( 3452 3453 2 2 2 2 3454 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k + 2*epsilon *k 3455 3456 2 3457 + h )*sqrt(2)*log( 3458 3459 2 2 2 4 3460 sqrt(k)*sqrt( - alpha - epsilon + 2*h*r - 2*k*r )*sqrt(2)*i + sqrt( 3461 3462 2 2 2 3463 2*sqrt(k)*sqrt(alpha + epsilon )*sqrt(2)*h + 2*alpha *k 3464 3465 2 2 2 2 3466 + 2*epsilon *k + h ) + h - 2*k*r )*epsilon ))/(4 3467 3468 4 2 2 2 2 4 2 2 3469 *(2*alpha *k + 4*alpha *epsilon *k - alpha *h + 2*epsilon *k - epsilon *h )) 3470 3471 3472 3473Comment many of these integrals used to require Steve Harrington's 3474 code to evaluate. They originated in Novosibirsk as examples 3475 of using Analytik. There are still a few examples that could 3476 be evaluated using better heuristics; 3477 3478 3479testint(a*sin(3*x+5)**2*cos(3*x+5),x); 3480 3481 3482 3 3483 sin(3*x + 5) *a 3484----------------- 3485 9 3486 3487testint(log(x**2)/x**3,x); 3488 3489 3490 2 3491 - (log(x ) + 1) 3492------------------ 3493 2 3494 2*x 3495 3496testint(x*sin(x+a),x); 3497 3498 3499 - cos(a + x)*x + sin(a + x) 3500 3501testint((log(x)*(1-x)-1)/(e**x*log(x)**2),x); 3502 3503 3504 x 3505----------- 3506 x 3507 e *log(x) 3508 3509testint(x**3*(a*x**2+b)**(-1),x); 3510 3511 3512 2 2 3513 - log(a*x + b)*b + a*x 3514--------------------------- 3515 2 3516 2*a 3517 3518testint(x**(1/2)*(x+1)**(-7/2),x); 3519 3520 3521 2 2 3522(2*( - 2*sqrt(x + 1)*x - 4*sqrt(x + 1)*x - 2*sqrt(x + 1) + 2*sqrt(x)*x 3523 3524 2 3525 + 5*sqrt(x)*x))/(15*sqrt(x + 1)*(x + 2*x + 1)) 3526 3527testint(x**(-1)*(x+1)**(-1),x); 3528 3529 3530 - log(x + 1) + log(x) 3531 3532testint(x**(-1/2)*(2*x-1)**(-1),x); 3533 3534 3535 sqrt(2)*(log(2*sqrt(x) - sqrt(2)) - log(2*sqrt(x) + sqrt(2))) 3536--------------------------------------------------------------- 3537 2 3538 3539testint((x**2+1)*x**(1/2),x); 3540 3541 3542 2 3543 2*sqrt(x)*x*(3*x + 7) 3544------------------------ 3545 21 3546 3547testint(x**(-1)*(x-a)**(1/3),x); 3548 3549 3550 1/6 1/6 3551 2*( - a + x) - a *sqrt(3) 3552( - 2*sqrt(3)*atan(--------------------------------)*a 3553 1/6 3554 a 3555 3556 1/6 1/6 3557 2*( - a + x) + a *sqrt(3) 2/3 1/3 3558 + 2*sqrt(3)*atan(--------------------------------)*a + 6*a *( - a + x) 3559 1/6 3560 a 3561 3562 1/3 1/3 3563 - 2*log(( - a + x) + a )*a 3564 3565 1/6 1/6 1/3 1/3 3566 + log( - a *( - a + x) *sqrt(3) + ( - a + x) + a )*a 3567 3568 1/6 1/6 1/3 1/3 2/3 3569 + log(a *( - a + x) *sqrt(3) + ( - a + x) + a )*a)/(2*a ) 3570 3571testint(x*sinh(x),x); 3572 3573 3574cosh(x)*x - sinh(x) 3575 3576testint(x*cosh(x),x); 3577 3578 3579 - cosh(x) + sinh(x)*x 3580 3581testint(sinh(2*x)/cosh(2*x),x); 3582 3583 3584 log(cosh(2*x)) 3585---------------- 3586 2 3587 3588testint((i*eps*sinh x-1)/(eps*i*cosh x+i*a-x),x); 3589 3590 3591log(cosh(x)*eps*i + a*i - x) 3592 3593testint(sin(2*x+3)*cos(x)**2,x); 3594 3595 3596 2 3597( - 4*cos(2*x + 3)*cos(x)*sin(x)*x + 2*cos(2*x + 3)*sin(x) - 3*cos(2*x + 3) 3598 3599 2 3600 - 4*sin(2*x + 3)*sin(x) *x + 2*sin(2*x + 3)*x + 3)/8 3601 3602testint(x*atan(x),x); 3603 3604 3605 2 3606 atan(x)*x + atan(x) - x 3607-------------------------- 3608 2 3609 3610testint(x*acot(x),x); 3611 3612 3613 2 3614 acot(x)*x + acot(x) + x 3615-------------------------- 3616 2 3617 3618testint(x*log(x**2+a),x); 3619 3620 3621 2 2 2 2 3622 log(a + x )*a + log(a + x )*x - x 3623------------------------------------- 3624 2 3625 3626testint(sin(x+a)*cos(x),x); 3627 3628 3629 - cos(a + x)*cos(x) - cos(a + x)*sin(x)*x + cos(x)*sin(a + x)*x 3630------------------------------------------------------------------ 3631 2 3632 3633testint(cos(x+a)*sin(x),x); 3634 3635 3636 - cos(a + x)*cos(x) + cos(a + x)*sin(x)*x - cos(x)*sin(a + x)*x 3637------------------------------------------------------------------ 3638 2 3639 3640testint((1+sin(x))**(1/2),x); 3641 3642 3643int(sqrt(sin(x) + 1),x) 3644 3645testint((1-sin(x))**(1/2),x); 3646 3647 3648int(sqrt( - sin(x) + 1),x) 3649 3650testint((1+cos(x))**(1/2),x); 3651 3652 3653int(sqrt(cos(x) + 1),x) 3654 3655testint((1-cos(x))**(1/2),x); 3656 3657 3658int(sqrt( - cos(x) + 1),x) 3659 3660testint(1/(x**(1/2)-(x-1)**(1/2)),x); 3661 3662 3663 2*(sqrt(x - 1)*x - sqrt(x - 1) + sqrt(x)*x) 3664--------------------------------------------- 3665 3 3666 3667testint(1/(1-(x+1)**(1/2)),x); 3668 3669 3670 - 2*(sqrt(x + 1) + log(sqrt(x + 1) - 1)) 3671 3672testint(x/(x**4+36)**(1/2),x); 3673 3674 3675 4 2 3676 sqrt(x + 36) + x 3677 log(--------------------) 3678 6 3679--------------------------- 3680 2 3681 3682testint(1/(x**(1/3)+x**(1/2)),x); 3683 3684 3685 1/6 1/3 1/6 36866*x - 3*x + 2*sqrt(x) - 6*log(x + 1) 3687 3688testint(log(2+3*x**2),x); 3689 3690 3691 3*x 2 3692 2*sqrt(6)*atan(---------) + 3*log(3*x + 2)*x - 6*x 3693 sqrt(6) 3694----------------------------------------------------- 3695 3 3696 3697testint(cot(x),x); 3698 3699 3700 x 2 x 3701 - log(tan(---) + 1) + log(tan(---)) 3702 2 2 3703 3704testint(cot x**4,x); 3705 3706 3707 3 3708 - cot(x) + 3*cot(x) + 3*x 3709----------------------------- 3710 3 3711 3712testint(tanh(x),x); 3713 3714 3715 2*x 3716log(e + 1) - x 3717 3718testint(coth(x),x); 3719 3720 3721 x x 3722log(e - 1) + log(e + 1) - x 3723 3724testint(b**x,x); 3725 3726 3727 x 3728 b 3729-------- 3730 log(b) 3731 3732testint((x**4+x**(-4)+2)**(1/2),x); 3733 3734 3735 4 3736 x - 3 3737-------- 3738 3*x 3739 3740testint((2*x+1)/(3*x+2),x); 3741 3742 3743 - log(3*x + 2) + 6*x 3744----------------------- 3745 9 3746 3747testint(x*log(x+(x**2+1)**(1/2)),x); 3748 3749 3750 2 2 2 2 3751 - sqrt(x + 1)*x + 2*log(sqrt(x + 1) + x)*x + log(sqrt(x + 1) + x) 3752------------------------------------------------------------------------ 3753 4 3754 3755testint(x*(e**x*sin(x)+1)**2,x); 3756 3757 3758 2*x 2*x x x 3759( - 2*e *cos(x)*sin(x)*x + e *cos(x)*sin(x) - 8*e *cos(x)*x + 8*e *cos(x) 3760 3761 2*x 2 2*x 2*x x 2 3762 + 2*e *sin(x) *x + e *x - e + 8*e *sin(x)*x + 4*x )/8 3763 3764testint(x*e**x*cos(x),x); 3765 3766 3767 x 3768 e *(cos(x)*x + sin(x)*x - sin(x)) 3769----------------------------------- 3770 2 3771 3772 3773Comment the following set came from Herbert Stoyan; 3774 3775 3776testint(1/(x-3)**4,x); 3777 3778 3779 - 1 3780--------------------------- 3781 3 2 3782 3*(x - 9*x + 27*x - 27) 3783 3784testint(x/(x**3-1),x); 3785 3786 3787 2*x + 1 2 3788 2*sqrt(3)*atan(---------) - log(x + x + 1) + 2*log(x - 1) 3789 sqrt(3) 3790------------------------------------------------------------ 3791 6 3792 3793testint(x/(x**4-1),x); 3794 3795 3796 2 3797 - log(x + 1) + log(x - 1) + log(x + 1) 3798------------------------------------------ 3799 4 3800 3801testint(log(x)*(x**3+1)/(x**4+2),x); 3802 3803 3804 log(x) log(x) 2 3805 - 4*int(----------,x) + 2*int(--------,x) + log(x) 3806 5 4 3807 x + 2*x x + 2 3808------------------------------------------------------ 3809 2 3810 3811testint(log(x)+log(x+1)+log(x+2),x); 3812 3813 3814log(x + 2)*x + 2*log(x + 2) + log(x + 1)*x + log(x + 1) + log(x)*x - 3*x 3815 3816testint(1/(x**3+5),x); 3817 3818 3819 1/3 3820 1/3 5 - 2*x 2/3 1/3 2 3821(5 *( - 2*sqrt(3)*atan(--------------) - log(5 - 5 *x + x ) 3822 1/3 3823 sqrt(3)*5 3824 3825 1/3 3826 + 2*log(5 + x)))/30 3827 3828testint(1/sqrt(1+x**2),x); 3829 3830 3831 2 3832log(sqrt(x + 1) + x) 3833 3834testint(sqrt(x**2+3),x); 3835 3836 3837 2 3838 2 sqrt(x + 3) + x 3839 sqrt(x + 3)*x + 3*log(------------------) 3840 sqrt(3) 3841-------------------------------------------- 3842 2 3843 3844testint(x/(x+1)**2,x); 3845 3846 3847 log(x + 1)*x + log(x + 1) - x 3848------------------------------- 3849 x + 1 3850 3851 3852COMMENT The following integrals were used among others as a test of 3853 Moses' SIN program; 3854 3855 3856testint(asin x,x); 3857 3858 3859 2 3860asin(x)*x + sqrt( - x + 1) 3861 3862testint(x**2*asin x,x); 3863 3864 3865 3 2 2 2 3866 3*asin(x)*x + sqrt( - x + 1)*x + 2*sqrt( - x + 1) 3867------------------------------------------------------- 3868 9 3869 3870testint(sec x**2/(1+sec x**2-3*tan x),x); 3871 3872 3873 x x 3874log( - sqrt(5) + 2*tan(---) + 1) - log( - sqrt(2) + tan(---) + 1) 3875 2 2 3876 3877 x x 3878 + log(sqrt(5) + 2*tan(---) + 1) - log(sqrt(2) + tan(---) + 1) 3879 2 2 3880 3881testint(1/sec x**2,x); 3882 3883 3884 cos(x)*sin(x) + x 3885------------------- 3886 2 3887 3888testint((5*x**2-3*x-2)/(x**2*(x-2)),x); 3889 3890 3891 3*log(x - 2)*x + 2*log(x)*x - 1 3892--------------------------------- 3893 x 3894 3895testint(1/(4*x**2+9)**(1/2),x); 3896 3897 3898 2 3899 sqrt(4*x + 9) + 2*x 3900 log(----------------------) 3901 3 3902----------------------------- 3903 2 3904 3905testint((x**2+4)**(-1/2),x); 3906 3907 3908 2 3909 sqrt(x + 4) + x 3910log(------------------) 3911 2 3912 3913testint(1/(9*x**2-12*x+10),x); 3914 3915 3916 3*x - 2 3917 sqrt(6)*atan(---------) 3918 sqrt(6) 3919------------------------- 3920 18 3921 3922testint(1/(x**8-2*x**7+2*x**6-2*x**5+x**4),x); 3923 3924 3925 2 4 2 3 4 3 3926(3*log(x + 1)*x - 3*log(x + 1)*x - 30*log(x - 1)*x + 30*log(x - 1)*x 3927 3928 4 3 4 2 3 3929 + 24*log(x)*x - 24*log(x)*x - 30*x + 12*x + 8*x + 4)/(12*x *(x - 1)) 3930 3931testint((a*x**3+b*x**2+c*x+d)/((x+1)*x*(x-3)),x); 3932 3933 3934(27*log(x - 3)*a + 9*log(x - 3)*b + 3*log(x - 3)*c + log(x - 3)*d 3935 3936 - 3*log(x + 1)*a + 3*log(x + 1)*b - 3*log(x + 1)*c + 3*log(x + 1)*d 3937 3938 - 4*log(x)*d + 12*a*x)/12 3939 3940testint(1/(2-log(x**2+1))**5,x); 3941 3942 3943 2 5 2 4 2 3 2 2 3944 - int(1/(log(x + 1) - 10*log(x + 1) + 40*log(x + 1) - 80*log(x + 1) 3945 3946 2 3947 + 80*log(x + 1) - 32),x) 3948 3949 3950% The next integral appeared in Risch's 1968 paper. 3951 3952testint(2*x*e**(x**2)*log(x)+e**(x**2)/x+(log(x)-2)/(log(x)**2+x)**2+ 3953 ((2/x)*log(x)+(1/x)+1)/(log(x)**2+x),x); 3954 3955 3956 2 2 3957 x 3 x 2 2 2 3958(e *log(x) + e *log(x)*x + log(log(x) + x)*log(x) + log(log(x) + x)*x 3959 3960 2 3961 - log(x))/(log(x) + x) 3962 3963 3964% The following integral would not evaluate in REDUCE 3.3. 3965 3966testint(exp(x*ze+x/2)*sin(pi*ze)**4*x**4,ze); 3967 3968 3969 (2*x*ze + x)/2 3 3 3 3970(e *x *( - 16*cos(pi*ze)*sin(pi*ze) *pi *x 3971 3972 3 3 3 3973 - 4*cos(pi*ze)*sin(pi*ze) *pi*x - 24*cos(pi*ze)*sin(pi*ze)*pi *x 3974 3975 4 2 2 4 4 2 2 2 4 3976 + 4*sin(pi*ze) *pi *x + sin(pi*ze) *x + 12*sin(pi*ze) *pi *x + 24*pi ))/ 3977 3978 4 2 2 4 3979(64*pi + 20*pi *x + x ) 3980 3981 3982% This one evaluates: 3983 3984testint(erf(x),x); 3985 3986 3987 2 3988 x 3989 e *erf(x)*pi*x + sqrt(pi) 3990---------------------------- 3991 2 3992 x 3993 e *pi 3994 3995 3996% So why not this one? 3997 3998testint(erf(x+a),x); 3999 4000 4001int(erf(a + x),x) 4002 4003 4004 4005Comment some interesting integrals of algebraic functions; 4006 4007 4008% The Chebyshev integral. 4009 4010testint((2*x**6+4*x**5+7*x**4-3*x**3-x*x-8*x-8)/ 4011 ((2*x**2-1)**2*sqrt(x**4+4*x**3+2*x**2+1)),x); 4012 4013 4014 4 3 2 4 3 2 4015(2*sqrt(x + 4*x + 2*x + 1)*x + sqrt(x + 4*x + 2*x + 1) 4016 4017 2 2 2 4018 - 4*log(x + 4*x + 2)*x + 2*log(x + 4*x + 2) + 10*log( 4019 4020 4 3 2 4 3 2 4021 - 16*sqrt(x + 4*x + 2*x + 1)*sqrt(2) - 13*sqrt(x + 4*x + 2*x + 1) 4022 4023 2 2 2 4024 + 28*sqrt(2)*x + 32*sqrt(2)*x - 6*sqrt(2) + 35*x + 26*x - 11)*x - 5*log( 4025 4026 4 3 2 4 3 2 4027 - 16*sqrt(x + 4*x + 2*x + 1)*sqrt(2) - 13*sqrt(x + 4*x + 2*x + 1) 4028 4029 2 2 4030 + 28*sqrt(2)*x + 32*sqrt(2)*x - 6*sqrt(2) + 35*x + 26*x - 11) + 4*log( 4031 4032 4 3 2 4 3 2 4033 - 2*sqrt(x + 4*x + 2*x + 1)*sqrt(2) - sqrt(x + 4*x + 2*x + 1) 4034 4035 2 2 2 4036 + 4*sqrt(2)*x + 16*sqrt(2)*x + 10*sqrt(2) - 5*x - 20*x - 9)*x - 2*log( 4037 4038 4 3 2 4 3 2 4039 - 2*sqrt(x + 4*x + 2*x + 1)*sqrt(2) - sqrt(x + 4*x + 2*x + 1) 4040 4041 2 2 4042 + 4*sqrt(2)*x + 16*sqrt(2)*x + 10*sqrt(2) - 5*x - 20*x - 9) 4043 4044 2 4045 - 2*log(sqrt(2) + 2*x)*x + log(sqrt(2) + 2*x) + 2*log( 4046 4047 4 3 2 4 3 2 4048 sqrt(x + 4*x + 2*x + 1)*sqrt(2) + 4*sqrt(x + 4*x + 2*x + 1) 4049 4050 2 4051 - 2*sqrt(2)*x - 4*sqrt(2) - x - 2)*x - log( 4052 4053 4 3 2 4 3 2 4054 sqrt(x + 4*x + 2*x + 1)*sqrt(2) + 4*sqrt(x + 4*x + 2*x + 1) 4055 4056 - 2*sqrt(2)*x - 4*sqrt(2) - x - 2) + 2*log( 4057 4058 4 3 2 4059 25*sqrt(x + 4*x + 2*x + 1)*sqrt(2)*x 4060 4061 4 3 2 4 3 2 4062 + 6*sqrt(x + 4*x + 2*x + 1)*sqrt(2) + 44*sqrt(x + 4*x + 2*x + 1)*x 4063 4064 4 3 2 3 2 4065 + 38*sqrt(x + 4*x + 2*x + 1) - 7*sqrt(2)*x - 36*sqrt(2)*x 4066 4067 3 2 2 4068 - 21*sqrt(2)*x - 10*sqrt(2) - 28*x - 130*x - 84*x + 2)*x - log( 4069 4070 4 3 2 4071 25*sqrt(x + 4*x + 2*x + 1)*sqrt(2)*x 4072 4073 4 3 2 4 3 2 4074 + 6*sqrt(x + 4*x + 2*x + 1)*sqrt(2) + 44*sqrt(x + 4*x + 2*x + 1)*x 4075 4076 4 3 2 3 2 4077 + 38*sqrt(x + 4*x + 2*x + 1) - 7*sqrt(2)*x - 36*sqrt(2)*x 4078 4079 3 2 4080 - 21*sqrt(2)*x - 10*sqrt(2) - 28*x - 130*x - 84*x + 2) 4081 4082 2 2 2 4083 - 10*log(2*x - 1)*x + 5*log(2*x - 1) 4084 4085 2 2 4086 - 4*log(2*sqrt(2)*x + 4*sqrt(2) - x - 4*x - 6)*x 4087 4088 2 2 4089 + 2*log(2*sqrt(2)*x + 4*sqrt(2) - x - 4*x - 6))/(2*(2*x - 1)) 4090 4091 4092% This integral came from Dr. G.S. Joyce of Imperial College London. 4093 4094testint((1+2*y)*sqrt(1-5*y-5*y**2)/(y*(1+y)*(2+y)*sqrt(1-y-y**2)),y); 4095 4096 4097 2 2 4098( - 4*log(y - 2*y + 1) - log(y + 4*y + 4) + 2*log( 4099 4100 2 2 4101 - 3*sqrt( - 5*y - 5*y + 1)*sqrt( - y - y + 1)*i 4102 4103 2 2 4104 + 4*sqrt( - 5*y - 5*y + 1)*sqrt( - y - y + 1) 4105 4106 2 2 4107 - 15*sqrt( - y - y + 1)*i*y + 3*sqrt( - y - y + 1)*i 4108 4109 2 2 4110 - 5*sqrt( - y - y + 1)*y - 4*sqrt( - y - y + 1) 4111 4112 2 2 4113 - sqrt( - 5*y - 5*y + 1)*i*y - 3*sqrt( - 5*y - 5*y + 1)*i 4114 4115 2 2 2 4116 - 7*sqrt( - 5*y - 5*y + 1)*y + 4*sqrt( - 5*y - 5*y + 1) - 5*i*y + 11*i*y 4117 4118 2 4119 + 3*i - 10*y + 2*y - 4) + log( 4120 4121 2 2 2 4122 - 2*sqrt( - 5*y - 5*y + 1)*sqrt( - y - y + 1)*i - 5*sqrt( - y - y + 1)*y 4123 4124 2 2 4125 - 4*sqrt( - y - y + 1) + 3*sqrt( - 5*y - 5*y + 1)*y 4126 4127 2 2 4128 + 4*sqrt( - 5*y - 5*y + 1) - 5*i*y - 6*i*y + 2*i) 4129 4130 2 2 4131 + 2*log(sqrt( - y - y + 1) - sqrt( - 5*y - 5*y + 1) - 4*i*y - 4*i) 4132 4133 2 2 4134 + log(sqrt( - y - y + 1)*i - sqrt( - 5*y - 5*y + 1)*i + 2*y) + 2*log( 4135 4136 2 2 4137 6*sqrt( - 5*y - 5*y + 1)*sqrt( - y - y + 1)*i 4138 4139 2 2 4140 + 8*sqrt( - 5*y - 5*y + 1)*sqrt( - y - y + 1) 4141 4142 2 2 4143 - 20*sqrt( - y - y + 1)*i*y - 4*sqrt( - y - y + 1)*i 4144 4145 2 2 4146 + 15*sqrt( - y - y + 1)*y + 3*sqrt( - y - y + 1) 4147 4148 2 2 4149 - 12*sqrt( - 5*y - 5*y + 1)*i*y + 4*sqrt( - 5*y - 5*y + 1)*i 4150 4151 2 2 2 4152 + 9*sqrt( - 5*y - 5*y + 1)*y - 3*sqrt( - 5*y - 5*y + 1) - 10*i*y 4153 4154 2 4155 - 22*i*y + 14*i - 30*y + 4*y + 2) + 2*log( 4156 4157 2 2 2 4158 6*sqrt( - 5*y - 5*y + 1)*sqrt( - y - y + 1) - 15*sqrt( - y - y + 1)*i*y 4159 4160 2 2 4161 - 3*sqrt( - y - y + 1)*i - 9*sqrt( - 5*y - 5*y + 1)*i*y 4162 4163 2 2 4164 + 3*sqrt( - 5*y - 5*y + 1)*i - 10*y - 22*y + 14) 4165 4166 2 2 4167 + 9*log(3*sqrt( - y - y + 1)*i + sqrt( - 5*y - 5*y + 1)*i - 2*y - 4) 4168 4169 - 2*log(y - 1) - 9*log(y + 2) - 2*log(y + 1) - 3*log(y))/2 4170 4171 4172% This one has a simple result. 4173 4174testint(x*(sqrt(x**2-1)*x**2-4*sqrt(x**2-1)+sqrt(x**2-4)*x**2 4175 -sqrt(x**2-4))/((1+sqrt(x**2-4)+sqrt(x**2-1))*(x**4-5*x**2+4)),x); 4176 4177 4178 2 2 4179log(sqrt(x - 4) + sqrt(x - 1) + 1) 4180 4181 4182% This used to reveal bugs in the integrator which have been fixed. 4183% Since it takes a long time and doesn't have a closed form result, 4184% it has been commented out. 4185 4186% testint(sqrt(-4*sqrt(2)+9)*x-sqrt(x**4+2*x**2+4*x+1)*sqrt(2),x); 4187 4188 4189Comment here is an example of using the integrator with pattern 4190 matching; 4191 4192 4193for all m,n let int(k1**m*log(k1)**n/(p**2-k1**2),k1)=foo(m,n), 4194 int(k1*log(k1)**n/(p**2-k1**2),k1)=foo(1,n), 4195 int(k1**m*log(k1)/(p**2-k1**2),k1)=foo(m,1), 4196 int(k1*log(k1)/(p**2-k1**2),k1)=foo(1,1), 4197 int(log(k1)**n/(k1*(p**2-k1**2)),k1)=foo(-1,n); 4198 4199 4200 4201int(k1**2*log(k1)/(p**2-k1**2),k1); 4202 4203 4204*** foo declared operator 4205 4206foo(2,1) 4207 4208 4209COMMENT It is interesting to see how much of this one can be done; 4210 4211 4212let f1s= (12*log(s/mc**2)*s**2*pi**2*mc**3*(-8*s-12*mc**2+3*mc) 4213 + pi**2*(12*s**4*mc+3*s**4+176*s**3*mc**3-24*s**3*mc**2 4214 -144*s**2*mc**5-48*s*mc**7+24*s*mc**6+4*mc**9-3*mc**8)) 4215 /(384*e**(s/y)*s**2); 4216 4217 4218 4219int(f1s,s); 4220 4221 4222 2 s/y - s 9 s/y - s 8 4223(pi *( - 4*e *ei(------)*mc *s + 3*e *ei(------)*mc *s 4224 y y 4225 4226 s/y - s 7 s/y - s 6 4227 - 48*e *ei(------)*mc *s*y + 24*e *ei(------)*mc *s*y 4228 y y 4229 4230 s/y - s 5 2 s/y - s 4 2 4231 - 144*e *ei(------)*mc *s*y + 36*e *ei(------)*mc *s*y 4232 y y 4233 4234 s/y - s 3 3 s 5 2 4235 - 96*e *ei(------)*mc *s*y + 144*log(-----)*mc *s*y 4236 y 2 4237 mc 4238 4239 s 4 2 s 3 2 2 4240 - 36*log(-----)*mc *s*y + 96*log(-----)*mc *s *y 4241 2 2 4242 mc mc 4243 4244 s 3 3 9 8 5 2 4245 + 96*log(-----)*mc *s*y - 4*mc *y + 3*mc *y + 144*mc *s*y 4246 2 4247 mc 4248 4249 3 2 2 3 3 2 2 2 2 3 3 2 4250 - 176*mc *s *y - 80*mc *s*y + 24*mc *s *y + 24*mc *s*y - 12*mc*s *y 4251 4252 2 3 4 3 2 2 3 4 s/y 4253 - 24*mc*s *y - 24*mc*s*y - 3*s *y - 6*s *y - 6*s*y ))/(384*e *s*y) 4254 4255 4256factor ei,log; 4257 4258 4259 4260ws; 4261 4262 4263 s/y - s 3 2 4264(e *ei(------)*mc *pi *s 4265 y 4266 4267 6 5 4 3 2 2 2 3 4268 *( - 4*mc + 3*mc - 48*mc *y + 24*mc *y - 144*mc *y + 36*mc*y - 96*y ) 4269 4270 s 3 2 2 2 2 9 4271 + 12*log(-----)*mc *pi *s*y *(12*mc - 3*mc + 8*s + 8*y) + pi *y*( - 4*mc 4272 2 4273 mc 4274 4275 8 5 3 2 3 2 2 2 4276 + 3*mc + 144*mc *s*y - 176*mc *s *y - 80*mc *s*y + 24*mc *s *y 4277 4278 2 2 3 2 2 3 3 2 2 4279 + 24*mc *s*y - 12*mc*s *y - 24*mc*s *y - 24*mc*s*y - 3*s *y - 6*s *y 4280 4281 3 s/y 4282 - 6*s*y ))/(384*e *s*y) 4283 4284 4285Comment the following integrals reveal deficiencies in the current 4286integrator; 4287 4288 4289%high degree denominator; 4290%testint(1/(2-log(x**2+1))**5,x); 4291 4292%this example should evaluate; 4293testint(sin(2*x)/cos(x),x); 4294 4295 4296 sin(2*x) 4297int(----------,x) 4298 cos(x) 4299 4300 4301%this example, which appeared in Tobey's thesis, needs factorization 4302%over algebraic fields. It currently gives an ugly answer and so has 4303%been suppressed; 4304 4305% testint((7*x**13+10*x**8+4*x**7-7*x**6-4*x**3-4*x**2+3*x+3)/ 4306% (x**14-2*x**8-2*x**7-2*x**4-4*x**3-x**2+2*x+1),x); 4307 4308symbolic summarize!-integral!-test(); 4309 4310 ***** SUMMARY OF INTEGRAL TESTS ***** 4311 4312Number of integrals tested: 281 4313 4314 4315Number of incorrect integrals: 0 4316 4317Number of unevaluated integrals: 16 4318 4319Integrands of unevaluated integrals are: 4320 4321 4322log(log(log(log(x)))) 4323 4324 4325 p 4326sin(x) 4327 4328 4329 4 3 4330tan(x) *x 4331 4332 4333 6 3 4334tan(x) *x 4335 4336 4337 xi 4338 cos(--------)*cos(x)*x 4339 sin(x) 4340------------------------ 4341 2 4342 sin(x) 4343 4344 4345 x 4346 a *x 4347------------------- 4348 2 2 4349 b *x + 2*b*x + 1 4350 4351 4352 x 4353 e 4354 e 4355 e 4356e 4357 4358 4359 1 4360------------------------ 4361 4 2 4362 sqrt( - x + x - 1)*x 4363 4364 4365sqrt(sin(x) + 1) 4366 4367 4368sqrt( - sin(x) + 1) 4369 4370 4371sqrt(cos(x) + 1) 4372 4373 4374sqrt( - cos(x) + 1) 4375 4376 4377 3 4378 log(x)*x + log(x) 4379-------------------- 4380 4 4381 x + 2 4382 4383 4384 2 5 2 4 2 3 2 2 4385( - 1)/(log(x + 1) - 10*log(x + 1) + 40*log(x + 1) - 80*log(x + 1) 4386 4387 2 4388 + 80*log(x + 1) - 32) 4389 4390 4391erf(a + x) 4392 4393 4394 sin(2*x) 4395---------- 4396 cos(x) 4397 4398 4399 4400end; 4401 4402Tested on x86_64-pc-windows CSL 4403Time (counter 1): 671 ms plus GC time: 15 ms 4404 4405End of Lisp run after 0.67+0.07 seconds 4406real 1.00 4407user 0.01 4408sys 0.07 4409