src/sinint.lisp

   1 ;;; -*-  Mode: Lisp; Package: Maxima; Syntax: Common-Lisp; Base: 10 -*- ;;;;
   2 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
   3 ;;;     The data in this file contains enhancements.                   ;;;;;
   4 ;;;                                                                    ;;;;;
   5 ;;;  Copyright (c) 1984,1987 by William Schelter,University of Texas   ;;;;;
   6 ;;;     All rights reserved                                            ;;;;;
   7 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
   8 ;;;     (c) Copyright 1982 Massachusetts Institute of Technology         ;;;
   9 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
  10
  11 (in-package :maxima)
  12
  13 (macsyma-module sinint)
  14
  15 (load-macsyma-macros ratmac)
  16
  17 (defun rootfac (q sinint-var)
  18   (prog (nthdq nthdq1 simproots ans n-loops)
  19      (setq nthdq (pgcd q (pderivative q sinint-var)))
  20      (setq simproots (pquotient q nthdq))
  21      (setq ans (list (pquotient simproots (pgcd nthdq simproots))))
  22      (setq n-loops 0)
  23    amen
  24      (cond
  25        ((= n-loops $factor_max_degree)
  26         (return (list q)))
  27        ((or (pcoefp nthdq) (pointergp sinint-var (car nthdq)))
  28         (return (reverse ans))))
  29      (setq nthdq1 (pgcd (pderivative nthdq sinint-var) nthdq))
  30      (push (pquotient (pgcd nthdq simproots)
  31                       (pgcd nthdq1 simproots))
  32            ans)
  33      (setq nthdq nthdq1)
  34      (incf n-loops)
  35      (go amen)))
  36
  37 (defun aprog (q sinint-var)
  38   (setq q (oldcontent q))
  39   (let ((sinint-rootfactor
  40           (rootfac (cadr q) sinint-var)))
  41     (setq sinint-rootfactor
  42           (cons (ptimes (car q) (car sinint-rootfactor))
  43                 (cdr sinint-rootfactor)))
  44     (let (sinint-pardenom)
  45       (do ((pd (list (car sinint-rootfactor)))
  46            (rf (cdr sinint-rootfactor) (cdr rf))
  47            (n 2 (1+ n)))
  48           ((null rf)
  49            (setq sinint-pardenom (reverse pd)))
  50         (push (pexpt (car rf) n)
  51               pd))
  52       (values sinint-rootfactor sinint-pardenom))))
  53
  54 (defun cprog (top bottom sinint-var sinint-pardenom)
  55   (prog (frpart pardenomc ppdenom thebpg sinint-parnumer sinint-wholepart)
  56      (setq frpart (pdivide top bottom))
  57      (setq sinint-wholepart (car frpart))
  58      (setq frpart (cadr frpart))
  59      (if (= (length sinint-pardenom) 1)
  60          (return (values (list frpart) sinint-wholepart)))
  61      (setq pardenomc (cdr sinint-pardenom))
  62      (setq ppdenom (list (car sinint-pardenom)))
  63    dseq
  64      (if (= (length pardenomc) 1)
  65          (go ok))
  66      (setq ppdenom (cons (ptimes (car ppdenom) (car pardenomc))
  67                          ppdenom))
  68      (setq pardenomc (cdr pardenomc))
  69      (go dseq)
  70    ok
  71      (setq pardenomc (reverse sinint-pardenom))
  72    numc
  73      (setq thebpg (bprog (car pardenomc)
  74                          (car ppdenom)
  75                          sinint-var))
  76      (setq sinint-parnumer
  77            (cons (cdr (ratdivide (ratti frpart (cdr thebpg) t)
  78                                  (car pardenomc)))
  79                  sinint-parnumer))
  80      (setq frpart
  81            (cdr (ratdivide (ratti frpart (car thebpg) t)
  82                            (car ppdenom))))
  83      (setq pardenomc (cdr pardenomc))
  84      (setq ppdenom (cdr ppdenom))
  85      (if (null ppdenom)
  86          (return (values (cons frpart sinint-parnumer)
  87                          sinint-wholepart)))
  88      (go numc)))
  89
  90 (defun polyint (p sinint-var)
  91   (labels
  92       ((polyint1 (p sinint-var)
  93          (cond ((or (null p) (equal p 0))
  94                 (cons 0 1))
  95                ((atom p)
  96                 (list sinint-var 1 p))
  97                ((not (numberp (car p)))
  98                 (if (pointergp sinint-var (car p))
  99                     (list sinint-var 1 p)
 100                     (polyint1 (cdr p) sinint-var)))
 101                (t
 102                 (ratplus (polyint2 p sinint-var)
 103                          (polyint1 (cddr p) sinint-var)))))
 104
 105        (polyint2 (p sinint-var)
 106          (cons (list sinint-var
 107                      (1+ (car p))
 108                      (cadr p))
 109                (1+ (car p)))))
 110     (ratqu (polyint1 (ratnumerator p) sinint-var)
 111            (ratdenominator p))))
 112
 113
 114 (defun dprog (ratarg sinint-ratform sinint-var)
 115   (prog (klth kx arootf deriv thebpg thetop thebot prod1 prod2 ans
 116          sinint-logptdx sinint-parnumer sinint-pardenom sinint-rootfactor sinint-wholepart)
 117      (setq ans (cons 0 1))
 118      (if (or (pcoefp (cdr ratarg)) (pointergp sinint-var (cadr ratarg)))
 119          (return (values (disrep (polyint ratarg sinint-var) sinint-ratform)
 120                          sinint-logptdx)))
 121      (multiple-value-setq (sinint-rootfactor sinint-pardenom)
 122        (aprog (ratdenominator ratarg) sinint-var))
 123      (multiple-value-setq (sinint-parnumer sinint-wholepart)
 124        (cprog (ratnumerator ratarg)
 125               (ratdenominator ratarg)
 126               sinint-var
 127               sinint-pardenom))
 128      (setq sinint-rootfactor (reverse sinint-rootfactor))
 129      (setq sinint-parnumer (reverse sinint-parnumer))
 130      (setq klth (length sinint-rootfactor))
 131    intg
 132      (if (= klth 1) (go simp))
 133      (setq arootf (car sinint-rootfactor))
 134      (if (zerop (pdegree arootf sinint-var))
 135          (go reset))
 136      (setq deriv (pderivative arootf sinint-var))
 137      (setq thebpg (bprog arootf deriv sinint-var))
 138      (setq kx (1- klth))
 139      (setq thetop (car sinint-parnumer))
 140    iter
 141      (setq prod1 (ratti thetop (car thebpg) t))
 142      (setq prod2 (ratti thetop (cdr thebpg) t))
 143      (setq thebot (pexpt arootf kx))
 144      (setq ans
 145            (ratplus ans (ratqu (ratminus prod2)
 146                                (ratti kx thebot t))))
 147      (setq thetop
 148            (ratplus prod1
 149                     (ratqu (ratreduce (pderivative (car prod2) sinint-var)
 150                                       (cdr prod2))
 151                            kx)))
 152      (setq thetop (cdr (ratdivide thetop thebot)))
 153      (cond ((= kx 1)
 154             (setq sinint-logptdx (cons (ratqu thetop arootf)
 155                                        sinint-logptdx))
 156             (go reset)))
 157      (setq kx (1- kx))
 158      (go iter)
 159    reset
 160      (setq sinint-rootfactor (cdr sinint-rootfactor))
 161      (setq sinint-parnumer (cdr sinint-parnumer))
 162      (decf klth)
 163      (go intg)
 164    simp
 165      (push (ratqu (car sinint-parnumer) (car sinint-rootfactor))
 166            sinint-logptdx)
 167      (if (equal ans 0)
 168          (return (values (disrep (polyint sinint-wholepart sinint-var) sinint-ratform)
 169                          sinint-logptdx)))
 170      (setq thetop
 171            (cadr (pdivide (ratnumerator ans) (ratdenominator ans))))
 172      (return (values (list '(mplus)
 173                            (disrep (polyint sinint-wholepart sinint-var) sinint-ratform)
 174                            (disrep (ratqu thetop (ratdenominator ans)) sinint-ratform))
 175                      sinint-logptdx))))
 176
 177 (defun logmabs (x)
 178   (list '(%log) (if $logabs
 179                     (simplify (list '(mabs) x))
 180                     x)))
 181
 182 (defun npask (npask-exp)
 183   (cond ((freeof '$%i npask-exp)
 184          (learn `((mnotequal) ,npask-exp 0)
 185                 t)
 186          (asksign npask-exp))
 187         (t '$positive)))
 188
 189 (defmvar $integrate_use_rootsof nil
 190   "Use the rootsof form for integrals when denominator does not factor"
 191   :setting-list (nil t))
 192
 193 (defun integrate-use-rootsof (f q variable sinint-ratform)
 194   (let ((dummy (make-param))
 195         (qprime (disrep (pderivative q (p-var q)) sinint-ratform))
 196         (ff (disrep f sinint-ratform))
 197         (qq (disrep q sinint-ratform)))
 198     ;; This basically does a partial fraction expansion and integrates
 199     ;; the result.  Let r be one (simple) root of the denominator
 200     ;; polynomial q.  Then the partial fraction expansion is
 201     ;;
 202     ;;   f(x)/q(x) = A/(x-r) + similar terms.
 203     ;;
 204     ;; Then
 205     ;;
 206     ;;   f(x) = A*q(x)/(x-r) + others
 207     ;;
 208     ;; Take the limit as x -> r.
 209     ;;
 210     ;;   f(r) = A*limit(q(x)/(x-r),x,r) + others
 211     ;;        = A*at(diff(q(x),r), [x=r])
 212     ;;
 213     ;; Hence, A = f(r)/at(diff(q(x),x),[x=r])
 214     ;;
 215     ;; Then it follows that the integral is
 216     ;;
 217     ;;    A*log(x-r)
 218     ;;
 219     ;; Note that we don't express the polynomial in terms of the
 220     ;; variable of integration, but in our dummy variable instead.
 221     ;; Using the variable of integration results in a wrong answer
 222     ;; when a substitution was done previously, since when the
 223     ;; substitution is finally undone, that modifies the polynomial.
 224     `((%lsum) ((mtimes)
 225                ,(div* (subst dummy variable ff)
 226                       (subst dummy variable qprime))
 227                ((%log) ,(sub* variable dummy)))
 228       ,dummy
 229       (($rootsof) ,(subst dummy variable qq) ,dummy))))
 230
 231 (defun eprog (p sinint-ratform sinint-var sinint-switch1)
 232   (prog (p1e p2e a1e a2e a3e discrim repart sign ncc dcc allcc xx deg
 233          sinint-parnumer sinint-pardenom sinint-wholepart)
 234      (if (or (equal p 0) (equal (car p) 0))
 235          (return 0))
 236      (setq p1e (ratnumerator p)
 237            p2e (ratdenominator p))
 238      (cond ((or sinint-switch1
 239                 (and (not (atom p2e))
 240                      (eq (car (setq xx (cadr (oldcontent p2e))))
 241                          sinint-var)
 242                      (member (setq deg (pdegree xx sinint-var)) '(5 6) :test #'equal)
 243                      (zerocoefl xx deg sinint-var)
 244                      (or (equal deg 5)
 245                          (not (pminusp (car (last xx)))))))
 246             (go efac)))
 247      (setq a1e (intfactor p2e))
 248      (if (> (length a1e) 1)
 249          (go e40))
 250    efac
 251      (setq ncc (oldcontent p1e))
 252      (setq p1e (cadr ncc))
 253      (setq dcc (oldcontent p2e))
 254      (setq p2e (cadr dcc))
 255      (setq allcc (ratqu (car ncc) (car dcc)))
 256      (setq deg (pdegree p2e sinint-var))
 257      (setq a1e (pderivative p2e sinint-var))
 258      (setq a2e (ratqu (polcoef p1e (pdegree p1e sinint-var) sinint-var)
 259                       (polcoef a1e (pdegree a1e sinint-var) sinint-var)))
 260      (cond ((equal (ratti a2e a1e t) (cons p1e 1))
 261             (return (list '(mtimes)
 262                           (disrep (ratti allcc a2e t) sinint-ratform )
 263                           (logmabs (disrep p2e sinint-ratform))))))
 264      (cond ((equal deg 1) (go e10))
 265            ((equal deg 2) (go e20))
 266            ((and (equal deg 3) (equal (polcoef p2e 2 sinint-var) 0)
 267                  (equal (polcoef p2e 1 sinint-var) 0))
 268             (return (e3prog p1e p2e allcc sinint-ratform sinint-var sinint-switch1)))
 269            ((and (member deg '(4 5 6) :test #'equal)
 270                  (zerocoefl p2e deg sinint-var))
 271             (return (enprog p1e p2e allcc deg sinint-ratform sinint-var))))
 272      (cond ((and $integrate_use_rootsof
 273                  (equal (car (psqfr p2e)) p2e))
 274             (return (list '(mtimes)
 275                           (disrep allcc sinint-ratform)
 276                           (integrate-use-rootsof p1e p2e
 277                                                  (car (last varlist))
 278                                                  sinint-ratform)))))
 279      (return (list '(mtimes)
 280                    (disrep allcc sinint-ratform)
 281                    (list '(%integrate)
 282                          (list '(mquotient)
 283                                (disrep p1e  sinint-ratform)
 284                                (disrep p2e sinint-ratform))
 285                          (car (last varlist)))))
 286    e10
 287      (return (list '(mtimes)
 288                    (disrep (ratti allcc
 289                                   (ratqu (polcoef p1e (pdegree p1e sinint-var) sinint-var)
 290                                          (polcoef p2e 1 sinint-var))
 291                                   t)
 292                            sinint-ratform)
 293                    (logmabs (disrep p2e sinint-ratform))))
 294    e20
 295      (setq discrim
 296            (ratdifference (cons (pexpt (polcoef p2e 1 sinint-var) 2)
 297                                 1)
 298                           (ratti 4
 299                                  (ratti (polcoef p2e 2 sinint-var)
 300                                         (polcoef p2e 0 sinint-var)
 301                                         t)
 302                                  t)))
 303      (setq a2e (ratti (polcoef p2e (pdegree p2e sinint-var) sinint-var) 2 t))
 304      (setq xx (simplify (disrep discrim sinint-ratform)))
 305      (when (equal ($imagpart xx) 0)
 306        (setq sign (npask xx))
 307        (cond ((eq sign '$negative) (go e30))
 308              ((eq sign '$zero) (go zip))))
 309      (setq a1e (ratsqrt discrim sinint-ratform))
 310      (setq a3e (logmabs
 311                 (list '(mquotient)
 312                       (list '(mplus)
 313                             (list '(mtimes)
 314                                   (disrep a2e sinint-ratform)
 315                                   (disrep (list sinint-var 1 1) sinint-ratform))
 316                             (disrep (polcoef p2e 1 sinint-var) sinint-ratform)
 317                             (list '(mminus) a1e))
 318                       (list '(mplus)
 319                             (list '(mtimes)
 320                                   (disrep a2e sinint-ratform)
 321                                   (disrep (list sinint-var 1 1) sinint-ratform))
 322                             (disrep (polcoef p2e 1 sinint-var) sinint-ratform)
 323                             a1e))))
 324      (cond ((zerop (pdegree p1e sinint-var))
 325             (return (list '(mtimes)
 326                           (disrep allcc sinint-ratform)
 327                           (list '(mquotient)
 328                                 (disrep (polcoef p1e 0 sinint-var) sinint-ratform)
 329                                 a1e)
 330                           a3e))))
 331      (return
 332        (list
 333         '(mplus)
 334         (list '(mtimes)
 335               (disrep (ratti allcc
 336                              (ratqu (polcoef p1e (pdegree p1e sinint-var) sinint-var) a2e)
 337                              t)
 338                       sinint-ratform)
 339               (logmabs (disrep p2e sinint-ratform)))
 340         (list
 341          '(mtimes)
 342          (list
 343           '(mquotient)
 344           (disrep (ratti allcc
 345                          (ratqu (eprogratd a2e p1e p2e sinint-var) a2e)
 346                          t)
 347                   sinint-ratform)
 348           a1e)
 349          a3e)))
 350    e30
 351      (setq a1e (ratsqrt (ratminus discrim) sinint-ratform))
 352      (setq
 353       repart
 354       (ratqu (cond ((zerop (pdegree p1e sinint-var))
 355                     (ratti a2e (polcoef p1e 0 sinint-var) t))
 356                    (t
 357                     (eprogratd a2e p1e p2e sinint-var)))
 358              (polcoef p2e (pdegree p2e sinint-var) sinint-var)))
 359      (setq a3e (cond ((equal 0 (car repart))
 360                       0)
 361                      (t
 362                       `((mtimes) ((mquotient)
 363                                   ,(disrep (ratti allcc repart t) sinint-ratform)
 364                                   ,a1e)
 365                         ((%atan)
 366                          ((mquotient)
 367                           ,(disrep (pderivative p2e sinint-var) sinint-ratform)
 368                           ,a1e))))))
 369      (if (zerop (pdegree p1e sinint-var))
 370          (return a3e))
 371      (return (list '(mplus)
 372                    (list '(mtimes)
 373                          (disrep (ratti allcc
 374                                         (ratqu (polcoef p1e (pdegree p1e sinint-var) sinint-var) a2e)
 375                                         t)
 376                                  sinint-ratform)
 377                          (logmabs (disrep p2e sinint-ratform)))
 378                    a3e))
 379    zip
 380      (setq
 381          p2e
 382          (ratqu
 383           (psimp
 384            (p-var p2e)
 385            (pcoefadd 2
 386                      (pexpt (ptimes 2 (polcoef p2e 2 sinint-var)) 2)
 387                      (pcoefadd 1
 388                                (ptimes 4 (ptimes (polcoef p2e 2 sinint-var)
 389                                                  (polcoef p2e 1 sinint-var)))
 390                                (pcoefadd 0 (pexpt (polcoef p2e 1 sinint-var) 2) ()))))
 391           (ptimes 4 (polcoef p2e 2 sinint-var))))
 392      (return (fprog (ratti allcc (ratqu p1e p2e) t)
 393                     sinint-ratform
 394                     sinint-var))
 395    e40
 396      (setq sinint-parnumer nil
 397            sinint-pardenom a1e
 398            sinint-switch1 t)
 399      (multiple-value-setq (sinint-parnumer sinint-wholepart)
 400        (cprog p1e p2e sinint-var sinint-pardenom))
 401      (setq a2e
 402            (mapcar #'(lambda (j k)
 403                        (eprog (ratqu j k) sinint-ratform sinint-var sinint-switch1))
 404                    sinint-parnumer sinint-pardenom))
 405      (setq sinint-switch1 nil)
 406      (return (cons '(mplus) a2e))))
 407
 408 (defun e3prog (num denom cont sinint-ratform sinint-var sinint-switch1)
 409   (prog (a b c d e r ratr var* x)
 410      (setq a (polcoef num 2 sinint-var)
 411            b (polcoef num 1 sinint-var)
 412            c (polcoef num 0 sinint-var)
 413            d (polcoef denom 3 sinint-var)
 414            e (polcoef denom 0 sinint-var))
 415      (setq r (cond ((eq (npask (simplify (disrep (ratqu e d) sinint-ratform)))
 416                         '$negative)
 417                     (simpnrt (disrep (ratqu (ratti -1 e t) d) sinint-ratform) 3))
 418                    (t
 419                     (neg (simpnrt (disrep (ratqu e d) sinint-ratform) 3)))))
 420      (setq var* (list sinint-var 1 1))
 421      (newvar r)
 422      (orderpointer varlist)
 423      (setq x (ratf r))
 424      (setq sinint-ratform (car x) ratr (cdr x))
 425      (return
 426        (simplify
 427         (list '(mplus)
 428               (list '(mtimes)
 429                     (disrep (ratqu (r* cont (r+ (r* a ratr ratr) (r* b ratr) c))
 430                                    (r* ratr ratr 3 d))
 431                             sinint-ratform)
 432                     (logmabs (disrep (ratpl (ratti -1 ratr t) var*) sinint-ratform)))
 433               (eprog (r* cont (ratqu (r+ (r* (r+ (r* 2 a ratr ratr)
 434                                                  (r* -1 b ratr)
 435                                                  (r* -1 c))
 436                                              var*)
 437                                          (r+ (ratqu (r* -1 a e) d)
 438                                              (r* b ratr ratr)
 439                                              (r* -1 2 c ratr)))
 440                                      (r* 3 d ratr ratr
 441                                          (r+ (ratti var* var* t)
 442                                              (ratti ratr var* t)
 443                                              (ratti ratr ratr t)))))
 444                      sinint-ratform
 445                      sinint-var
 446                      sinint-switch1)
 447               )))))
 448
 449 (defun eprogratd (a2e p1e p2e sinint-var)
 450   (ratdifference (ratti a2e
 451                         (polcoef p1e (1- (pdegree p1e sinint-var)) sinint-var)
 452                         t)
 453                  (ratti (polcoef p2e (1- (pdegree p2e sinint-var)) sinint-var)
 454                         (polcoef p1e (pdegree p1e sinint-var) sinint-var)
 455                         t)))
 456
 457 (defun enprog (num denom cont deg sinint-ratform sinint-var)
 458   ;; Denominator is (A*VAR^4+B) =
 459   ;;   if B<0 then (SQRT(A)*VAR^2 - SQRT(-B)) (SQRT(A)*VAR^2 + SQRT(-B))
 460   ;;         else
 461   ;;    (SQRT(A)*VAR^2 - SQRT(2)*A^(1/4)*B^(1/4)*VAR + SQRT(B)) *
 462   ;;    (SQRT(A)*VAR^2 + SQRT(2)*A^(1/4)*B^(1/4)*VAR + SQRT(B))
 463   ;; or (A*VAR^5+B) =
 464   ;;    (1/4) * (A^(1/5)*VAR + B^(1/5)) *
 465   ;;    (2*A^(2/5)*VAR^2 + (-SQRT(5)-1)*A^(1/5)*B^(1/5)*VAR + 2*B^(2/5)) *
 466   ;;    (2*A^(2/5)*VAR^2 + (+SQRT(5)-1)*A^(1/5)*B^(1/5)*VAR + 2*B^(2/5))
 467   ;; or (A*VAR^6+B) =
 468   ;;   if B<0 then (SQRT(A)*VAR^3 - SQRT(-B)) (SQRT(A)*VAR^3 + SQRT(-B))
 469   ;;         else
 470   ;;    (A^(1/3)*VAR^2 + B^(1/3)) *
 471   ;;    (A^(1/3)*VAR^2 - SQRT(3)*A^(1/6)*B^(1/6)*VAR + B^(1/3)) *
 472   ;;    (A^(1/3)*VAR^2 + SQRT(3)*A^(1/6)*B^(1/6)*VAR + B^(1/3))
 473   (prog (($expop 0) ($expon 0) a b term disvar $algebraic)
 474      #+nil
 475      (setq $expop 0 $expon 0)
 476      (setq a (simplify (disrep (polcoef denom deg sinint-var) sinint-ratform))
 477            b (simplify (disrep (polcoef denom 0 sinint-var) sinint-ratform))
 478            disvar (simplify (get sinint-var 'disrep))
 479            num (simplify (disrep num sinint-ratform))
 480            cont (simplify (disrep cont sinint-ratform)))
 481      (cond ((= deg 4)
 482             (if (eq '$neg ($asksign b))
 483                 (setq denom
 484                       (mul2 (add2 (mul2 (power a '((rat simp) 1 2)) (power disvar 2))
 485                                   (power (mul -1 b) '((rat simp) 1 2)))
 486                             (add2 (mul2 (power a '((rat simp) 1 2)) (power disvar 2))
 487                                   (mul -1 (power (mul -1 b) '((rat simp) 1 2))))))
 488                 (progn
 489                   (setq denom (add2 (mul2 (power a '((rat simp) 1 2)) (power disvar 2))
 490                                     (power b '((rat simp) 1 2)))
 491                         term (muln (list (power 2 '((rat simp) 1 2))
 492                                          (power a '((rat simp) 1 4))
 493                                          (power b '((rat simp) 1 4))
 494                                          disvar)
 495                                    t))
 496                   (setq denom (mul2 (add2 denom term) (sub denom term))))))
 497            ((= deg 5)
 498             (setq term (mul3 (power a '((rat simp) 1 5))
 499                              (power b '((rat simp) 1 5))
 500                              disvar))
 501             (setq denom (add2 (mul3 2 (power a '((rat simp) 2 5))
 502                                     (power disvar 2))
 503                               (sub (mul2 2 (power b '((rat simp) 2 5))) term)))
 504             (setq term (mul2 (power 5 '((rat simp) 1 2)) term))
 505             (setq denom (muln (list '((rat simp) 1 4)
 506                                     (add2 (mul2 (power a '((rat simp) 1 5)) disvar)
 507                                           (power b '((rat simp) 1 5)))
 508                                     (add2 denom term) (sub denom term))
 509                               t)))
 510            (t
 511             (if (eq '$neg ($asksign b))
 512                 (setq denom
 513                       (mul2 (add2 (mul2 (power a '((rat simp) 1 2)) (power disvar 3))
 514                                   (power (mul -1 b) '((rat simp) 1 2)))
 515                             (add2 (mul2 (power a '((rat simp) 1 2)) (power disvar 3))
 516                                   (mul -1 (power (mul -1 b) '((rat simp) 1 2))))))
 517                 (progn
 518                   (setq denom (add2 (mul2 (power a '((rat simp) 1 3)) (power disvar 2))
 519                                     (power b '((rat simp) 1 3)))
 520                         term (muln (list (power 3 '((rat simp) 1 2))
 521                                          (power a '((rat simp) 1 6))
 522                                          (power b '((rat simp) 1 6))
 523                                          disvar)
 524                                    t))
 525                   (setq denom (mul3 denom (add2 denom term) (sub denom term))))
 526                 )))
 527      ;;Needs $ALGEBRAIC NIL so next call to RATF will preserve factorization.
 528      (return (mul2 cont (ratint (div num denom) disvar)))))
 529
 530 (defun zerocoefl (e n sinint-var)
 531   (do ((i 1 (1+ i)))
 532       ((= i n) t)
 533     (if (not (equal (polcoef e i sinint-var) 0))
 534         (return nil))))
 535
 536 (defun ratsqrt (a sinint-ratform)
 537   (let (varlist)
 538     (simpnrt (disrep a sinint-ratform) 2)))
 539
 540 (defun fprog (rat* sinint-ratform sinint-var)
 541   (multiple-value-bind (dprog-ret sinint-logptdx)
 542       (dprog rat* sinint-ratform sinint-var)
 543     (addn (cons dprog-ret
 544                 (mapcar #'(lambda (p)
 545                             (eprog p sinint-ratform sinint-var nil))
 546                         sinint-logptdx))
 547           nil)))
 548
 549 (defun ratint (sinint-exp sinint-var)
 550   (prog (genvar *checkfactors* varlist ratarg $keepfloat)
 551      (setq varlist (list sinint-var))
 552      (setq ratarg (ratf sinint-exp))
 553      (let ((sinint-ratform (car ratarg))
 554            (sinint-var (caadr (ratf sinint-var))))
 555        (return (fprog (cdr ratarg) sinint-ratform sinint-var)))))
 556
 557 (defun intfactor (l)
 558   (labels ((everysecond (a)
 559              (if a (cons (if (numberp (car a))
 560                              (pexpt (car a) (cadr a))
 561                              (car a))
 562                          (everysecond (cddr a))))))
 563
 564     (prog ($factorflag a b)
 565        (setq a (oldcontent l)
 566              b (everysecond (pfactor (cadr a))))
 567        (return (if (equal (car a) 1)
 568                    b
 569                    (cons (car a) b))))))