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;;; calcalg2.el --- more algebraic functions for Calc

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;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc.
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;; Author: David Gillespie <daveg@synaptics.com>
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;; Maintainers: D. Goel <deego@gnufans.org>
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;;              Colin Walters <walters@debian.org>
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;; This file is part of GNU Emacs.

;; GNU Emacs is distributed in the hope that it will be useful,
;; but WITHOUT ANY WARRANTY.  No author or distributor
;; accepts responsibility to anyone for the consequences of using it
;; or for whether it serves any particular purpose or works at all,
;; unless he says so in writing.  Refer to the GNU Emacs General Public
;; License for full details.

;; Everyone is granted permission to copy, modify and redistribute
;; GNU Emacs, but only under the conditions described in the
;; GNU Emacs General Public License.   A copy of this license is
;; supposed to have been given to you along with GNU Emacs so you
;; can know your rights and responsibilities.  It should be in a
;; file named COPYING.  Among other things, the copyright notice
;; and this notice must be preserved on all copies.

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;;; Commentary:
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;;; Code:
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;; This file is autoloaded from calc-ext.el.
(require 'calc-ext)

(require 'calc-macs)

(defun calc-Need-calc-alg-2 () nil)


(defun calc-derivative (var num)
  (interactive "sDifferentiate with respect to: \np")
  (calc-slow-wrapper
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   (when (< num 0)
     (error "Order of derivative must be positive"))
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   (let ((func (if (calc-is-hyperbolic) 'calcFunc-tderiv 'calcFunc-deriv))
	 n expr)
     (if (or (equal var "") (equal var "$"))
	 (setq n 2
	       expr (calc-top-n 2)
	       var (calc-top-n 1))
       (setq var (math-read-expr var))
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       (when (eq (car-safe var) 'error)
	 (error "Bad format in expression: %s" (nth 1 var)))
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       (setq n 1
	     expr (calc-top-n 1)))
     (while (>= (setq num (1- num)) 0)
       (setq expr (list func expr var)))
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     (calc-enter-result n "derv" expr))))
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(defun calc-integral (var)
  (interactive "sIntegration variable: ")
  (calc-slow-wrapper
   (if (or (equal var "") (equal var "$"))
       (calc-enter-result 2 "intg" (list 'calcFunc-integ
					 (calc-top-n 2)
					 (calc-top-n 1)))
     (let ((var (math-read-expr var)))
       (if (eq (car-safe var) 'error)
	   (error "Bad format in expression: %s" (nth 1 var)))
       (calc-enter-result 1 "intg" (list 'calcFunc-integ
					 (calc-top-n 1)
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					 var))))))
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(defun calc-num-integral (&optional varname lowname highname)
  (interactive "sIntegration variable: ")
  (calc-tabular-command 'calcFunc-ninteg "Integration" "nint"
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			nil varname lowname highname))
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(defun calc-summation (arg &optional varname lowname highname)
  (interactive "P\nsSummation variable: ")
  (calc-tabular-command 'calcFunc-sum "Summation" "sum"
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			arg varname lowname highname))
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(defun calc-alt-summation (arg &optional varname lowname highname)
  (interactive "P\nsSummation variable: ")
  (calc-tabular-command 'calcFunc-asum "Summation" "asum"
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			arg varname lowname highname))
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(defun calc-product (arg &optional varname lowname highname)
  (interactive "P\nsIndex variable: ")
  (calc-tabular-command 'calcFunc-prod "Index" "prod"
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			arg varname lowname highname))
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(defun calc-tabulate (arg &optional varname lowname highname)
  (interactive "P\nsIndex variable: ")
  (calc-tabular-command 'calcFunc-table "Index" "tabl"
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			arg varname lowname highname))
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(defun calc-tabular-command (func prompt prefix arg varname lowname highname)
  (calc-slow-wrapper
   (let (var (low nil) (high nil) (step nil) stepname stepnum (num 1) expr)
     (if (consp arg)
	 (setq stepnum 1)
       (setq stepnum 0))
     (if (or (equal varname "") (equal varname "$") (null varname))
	 (setq high (calc-top-n (+ stepnum 1))
	       low (calc-top-n (+ stepnum 2))
	       var (calc-top-n (+ stepnum 3))
	       num (+ stepnum 4))
       (setq var (if (stringp varname) (math-read-expr varname) varname))
       (if (eq (car-safe var) 'error)
	   (error "Bad format in expression: %s" (nth 1 var)))
       (or lowname
	   (setq lowname (read-string (concat prompt " variable: " varname
					      ", from: "))))
       (if (or (equal lowname "") (equal lowname "$"))
	   (setq high (calc-top-n (+ stepnum 1))
		 low (calc-top-n (+ stepnum 2))
		 num (+ stepnum 3))
	 (setq low (if (stringp lowname) (math-read-expr lowname) lowname))
	 (if (eq (car-safe low) 'error)
	     (error "Bad format in expression: %s" (nth 1 low)))
	 (or highname
	     (setq highname (read-string (concat prompt " variable: " varname
						 ", from: " lowname
						 ", to: "))))
	 (if (or (equal highname "") (equal highname "$"))
	     (setq high (calc-top-n (+ stepnum 1))
		   num (+ stepnum 2))
	   (setq high (if (stringp highname) (math-read-expr highname)
			highname))
	   (if (eq (car-safe high) 'error)
	       (error "Bad format in expression: %s" (nth 1 high)))
	   (if (consp arg)
	       (progn
		 (setq stepname (read-string (concat prompt " variable: "
						     varname
						     ", from: " lowname
						     ", to: " highname
						     ", step: ")))
		 (if (or (equal stepname "") (equal stepname "$"))
		     (setq step (calc-top-n 1)
			   num 2)
		   (setq step (math-read-expr stepname))
		   (if (eq (car-safe step) 'error)
		       (error "Bad format in expression: %s"
			      (nth 1 step)))))))))
     (or step
	 (if (consp arg)
	     (setq step (calc-top-n 1))
	   (if arg
	       (setq step (prefix-numeric-value arg)))))
     (setq expr (calc-top-n num))
     (calc-enter-result num prefix (append (list func expr var low high)
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					   (and step (list step)))))))
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(defun calc-solve-for (var)
  (interactive "sVariable to solve for: ")
  (calc-slow-wrapper
   (let ((func (if (calc-is-inverse)
		   (if (calc-is-hyperbolic) 'calcFunc-ffinv 'calcFunc-finv)
		 (if (calc-is-hyperbolic) 'calcFunc-fsolve 'calcFunc-solve))))
     (if (or (equal var "") (equal var "$"))
	 (calc-enter-result 2 "solv" (list func
					   (calc-top-n 2)
					   (calc-top-n 1)))
       (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var)
			   (not (string-match "\\[" var)))
		      (math-read-expr (concat "[" var "]"))
		    (math-read-expr var))))
	 (if (eq (car-safe var) 'error)
	     (error "Bad format in expression: %s" (nth 1 var)))
	 (calc-enter-result 1 "solv" (list func
					   (calc-top-n 1)
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					   var)))))))
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(defun calc-poly-roots (var)
  (interactive "sVariable to solve for: ")
  (calc-slow-wrapper
   (if (or (equal var "") (equal var "$"))
       (calc-enter-result 2 "prts" (list 'calcFunc-roots
					 (calc-top-n 2)
					 (calc-top-n 1)))
     (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var)
			 (not (string-match "\\[" var)))
		    (math-read-expr (concat "[" var "]"))
		  (math-read-expr var))))
       (if (eq (car-safe var) 'error)
	   (error "Bad format in expression: %s" (nth 1 var)))
       (calc-enter-result 1 "prts" (list 'calcFunc-roots
					 (calc-top-n 1)
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					 var))))))
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(defun calc-taylor (var nterms)
  (interactive "sTaylor expansion variable: \nNNumber of terms: ")
  (calc-slow-wrapper
   (let ((var (math-read-expr var)))
     (if (eq (car-safe var) 'error)
	 (error "Bad format in expression: %s" (nth 1 var)))
     (calc-enter-result 1 "tylr" (list 'calcFunc-taylor
				       (calc-top-n 1)
				       var
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				       (prefix-numeric-value nterms))))))
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;; The following are global variables used by math-derivative and some 
;; related functions
(defvar math-deriv-var)
(defvar math-deriv-total)
(defvar math-deriv-symb)

(defun math-derivative (expr)
  (cond ((equal expr math-deriv-var)
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	 1)
	((or (Math-scalarp expr)
	     (eq (car expr) 'sdev)
	     (and (eq (car expr) 'var)
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		  (or (not math-deriv-total)
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		      (math-const-var expr)
		      (progn
			(math-setup-declarations)
			(memq 'const (nth 1 (or (assq (nth 2 expr)
						      math-decls-cache)
						math-decls-all)))))))
	 0)
	((eq (car expr) '+)
	 (math-add (math-derivative (nth 1 expr))
		   (math-derivative (nth 2 expr))))
	((eq (car expr) '-)
	 (math-sub (math-derivative (nth 1 expr))
		   (math-derivative (nth 2 expr))))
	((memq (car expr) '(calcFunc-eq calcFunc-neq calcFunc-lt
					calcFunc-gt calcFunc-leq calcFunc-geq))
	 (list (car expr)
	       (math-derivative (nth 1 expr))
	       (math-derivative (nth 2 expr))))
	((eq (car expr) 'neg)
	 (math-neg (math-derivative (nth 1 expr))))
	((eq (car expr) '*)
	 (math-add (math-mul (nth 2 expr)
			     (math-derivative (nth 1 expr)))
		   (math-mul (nth 1 expr)
			     (math-derivative (nth 2 expr)))))
	((eq (car expr) '/)
	 (math-sub (math-div (math-derivative (nth 1 expr))
			     (nth 2 expr))
		   (math-div (math-mul (nth 1 expr)
				       (math-derivative (nth 2 expr)))
			     (math-sqr (nth 2 expr)))))
	((eq (car expr) '^)
	 (let ((du (math-derivative (nth 1 expr)))
	       (dv (math-derivative (nth 2 expr))))
	   (or (Math-zerop du)
	       (setq du (math-mul (nth 2 expr)
				  (math-mul (math-normalize
					     (list '^
						   (nth 1 expr)
						   (math-add (nth 2 expr) -1)))
					    du))))
	   (or (Math-zerop dv)
	       (setq dv (math-mul (math-normalize
				   (list 'calcFunc-ln (nth 1 expr)))
				  (math-mul expr dv))))
	   (math-add du dv)))
	((eq (car expr) '%)
	 (math-derivative (nth 1 expr)))   ; a reasonable definition
	((eq (car expr) 'vec)
	 (math-map-vec 'math-derivative expr))
	((and (memq (car expr) '(calcFunc-conj calcFunc-re calcFunc-im))
	      (= (length expr) 2))
	 (list (car expr) (math-derivative (nth 1 expr))))
	((and (memq (car expr) '(calcFunc-subscr calcFunc-mrow calcFunc-mcol))
	      (= (length expr) 3))
	 (let ((d (math-derivative (nth 1 expr))))
	   (if (math-numberp d)
	       0    ; assume x and x_1 are independent vars
	     (list (car expr) d (nth 2 expr)))))
	(t (or (and (symbolp (car expr))
		    (if (= (length expr) 2)
			(let ((handler (get (car expr) 'math-derivative)))
			  (and handler
			       (let ((deriv (math-derivative (nth 1 expr))))
				 (if (Math-zerop deriv)
				     deriv
				   (math-mul (funcall handler (nth 1 expr))
					     deriv)))))
		      (let ((handler (get (car expr) 'math-derivative-n)))
			(and handler
			     (funcall handler expr)))))
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	       (and (not (eq math-deriv-symb 'pre-expand))
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		    (let ((exp (math-expand-formula expr)))
		      (and exp
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			   (or (let ((math-deriv-symb 'pre-expand))
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				 (catch 'math-deriv (math-derivative expr)))
			       (math-derivative exp)))))
	       (if (or (Math-objvecp expr)
		       (eq (car expr) 'var)
		       (not (symbolp (car expr))))
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		   (if math-deriv-symb
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		       (throw 'math-deriv nil)
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		     (list (if math-deriv-total 'calcFunc-tderiv 'calcFunc-deriv)
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			   expr
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			   math-deriv-var))
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		 (let ((accum 0)
		       (arg expr)
		       (n 1)
		       derv)
		   (while (setq arg (cdr arg))
		     (or (Math-zerop (setq derv (math-derivative (car arg))))
			 (let ((func (intern (concat (symbol-name (car expr))
						     "'"
						     (if (> n 1)
							 (int-to-string n)
						       ""))))
			       (prop (cond ((= (length expr) 2)
					    'math-derivative-1)
					   ((= (length expr) 3)
					    'math-derivative-2)
					   ((= (length expr) 4)
					    'math-derivative-3)
					   ((= (length expr) 5)
					    'math-derivative-4)
					   ((= (length expr) 6)
					    'math-derivative-5))))
			   (setq accum
				 (math-add
				  accum
				  (math-mul
				   derv
				   (let ((handler (get func prop)))
				     (or (and prop handler
					      (apply handler (cdr expr)))
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					 (if (and math-deriv-symb
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						  (not (get func
							    'calc-user-defn)))
					     (throw 'math-deriv nil)
					   (cons func (cdr expr))))))))))
		     (setq n (1+ n)))
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		   accum))))))
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(defun calcFunc-deriv (expr math-deriv-var &optional deriv-value math-deriv-symb)
  (let* ((math-deriv-total nil)
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	 (res (catch 'math-deriv (math-derivative expr))))
    (or (eq (car-safe res) 'calcFunc-deriv)
	(null res)
	(setq res (math-normalize res)))
    (and res
	 (if deriv-value
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	     (math-expr-subst res math-deriv-var deriv-value)
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	   res))))
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(defun calcFunc-tderiv (expr math-deriv-var &optional deriv-value math-deriv-symb)
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  (math-setup-declarations)
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  (let* ((math-deriv-total t)
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	 (res (catch 'math-deriv (math-derivative expr))))
    (or (eq (car-safe res) 'calcFunc-tderiv)
	(null res)
	(setq res (math-normalize res)))
    (and res
	 (if deriv-value
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	     (math-expr-subst res math-deriv-var deriv-value)
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	   res))))
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(put 'calcFunc-inv\' 'math-derivative-1
     (function (lambda (u) (math-neg (math-div 1 (math-sqr u))))))

(put 'calcFunc-sqrt\' 'math-derivative-1
     (function (lambda (u) (math-div 1 (math-mul 2 (list 'calcFunc-sqrt u))))))

(put 'calcFunc-deg\' 'math-derivative-1
     (function (lambda (u) (math-div-float '(float 18 1) (math-pi)))))

(put 'calcFunc-rad\' 'math-derivative-1
     (function (lambda (u) (math-pi-over-180))))

(put 'calcFunc-ln\' 'math-derivative-1
     (function (lambda (u) (math-div 1 u))))

(put 'calcFunc-log10\' 'math-derivative-1
     (function (lambda (u)
		 (math-div (math-div 1 (math-normalize '(calcFunc-ln 10)))
			   u))))

(put 'calcFunc-lnp1\' 'math-derivative-1
     (function (lambda (u) (math-div 1 (math-add u 1)))))

(put 'calcFunc-log\' 'math-derivative-2
     (function (lambda (x b)
		 (and (not (Math-zerop b))
		      (let ((lnv (math-normalize
				  (list 'calcFunc-ln b))))
			(math-div 1 (math-mul lnv x)))))))

(put 'calcFunc-log\'2 'math-derivative-2
     (function (lambda (x b)
		 (let ((lnv (list 'calcFunc-ln b)))
		   (math-neg (math-div (list 'calcFunc-log x b)
				       (math-mul lnv b)))))))

(put 'calcFunc-exp\' 'math-derivative-1
     (function (lambda (u) (math-normalize (list 'calcFunc-exp u)))))

(put 'calcFunc-expm1\' 'math-derivative-1
     (function (lambda (u) (math-normalize (list 'calcFunc-expm1 u)))))

(put 'calcFunc-sin\' 'math-derivative-1
     (function (lambda (u) (math-to-radians-2 (math-normalize
					       (list 'calcFunc-cos u))))))

(put 'calcFunc-cos\' 'math-derivative-1
     (function (lambda (u) (math-neg (math-to-radians-2
				      (math-normalize
				       (list 'calcFunc-sin u)))))))

(put 'calcFunc-tan\' 'math-derivative-1
     (function (lambda (u) (math-to-radians-2
			    (math-div 1 (math-sqr
					 (math-normalize
					  (list 'calcFunc-cos u))))))))

(put 'calcFunc-arcsin\' 'math-derivative-1
     (function (lambda (u)
		 (math-from-radians-2
		  (math-div 1 (math-normalize
			       (list 'calcFunc-sqrt
				     (math-sub 1 (math-sqr u)))))))))

(put 'calcFunc-arccos\' 'math-derivative-1
     (function (lambda (u)
		 (math-from-radians-2
		  (math-div -1 (math-normalize
				(list 'calcFunc-sqrt
				      (math-sub 1 (math-sqr u)))))))))

(put 'calcFunc-arctan\' 'math-derivative-1
     (function (lambda (u) (math-from-radians-2
			    (math-div 1 (math-add 1 (math-sqr u)))))))

(put 'calcFunc-sinh\' 'math-derivative-1
     (function (lambda (u) (math-normalize (list 'calcFunc-cosh u)))))

(put 'calcFunc-cosh\' 'math-derivative-1
     (function (lambda (u) (math-normalize (list 'calcFunc-sinh u)))))

(put 'calcFunc-tanh\' 'math-derivative-1
     (function (lambda (u) (math-div 1 (math-sqr
					(math-normalize
					 (list 'calcFunc-cosh u)))))))

(put 'calcFunc-arcsinh\' 'math-derivative-1
     (function (lambda (u)
		 (math-div 1 (math-normalize
			      (list 'calcFunc-sqrt
				    (math-add (math-sqr u) 1)))))))

(put 'calcFunc-arccosh\' 'math-derivative-1
     (function (lambda (u)
		  (math-div 1 (math-normalize
			       (list 'calcFunc-sqrt
				     (math-add (math-sqr u) -1)))))))

(put 'calcFunc-arctanh\' 'math-derivative-1
     (function (lambda (u) (math-div 1 (math-sub 1 (math-sqr u))))))

(put 'calcFunc-bern\'2 'math-derivative-2
     (function (lambda (n x)
		 (math-mul n (list 'calcFunc-bern (math-add n -1) x)))))

(put 'calcFunc-euler\'2 'math-derivative-2
     (function (lambda (n x)
		 (math-mul n (list 'calcFunc-euler (math-add n -1) x)))))

(put 'calcFunc-gammag\'2 'math-derivative-2
     (function (lambda (a x) (math-deriv-gamma a x 1))))

(put 'calcFunc-gammaG\'2 'math-derivative-2
     (function (lambda (a x) (math-deriv-gamma a x -1))))

(put 'calcFunc-gammaP\'2 'math-derivative-2
     (function (lambda (a x) (math-deriv-gamma a x
					       (math-div
						1 (math-normalize
						   (list 'calcFunc-gamma
							 a)))))))

(put 'calcFunc-gammaQ\'2 'math-derivative-2
     (function (lambda (a x) (math-deriv-gamma a x
					       (math-div
						-1 (math-normalize
						    (list 'calcFunc-gamma
							  a)))))))

(defun math-deriv-gamma (a x scale)
  (math-mul scale
	    (math-mul (math-pow x (math-add a -1))
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		      (list 'calcFunc-exp (math-neg x)))))
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(put 'calcFunc-betaB\' 'math-derivative-3
     (function (lambda (x a b) (math-deriv-beta x a b 1))))

(put 'calcFunc-betaI\' 'math-derivative-3
     (function (lambda (x a b) (math-deriv-beta x a b
						(math-div
						 1 (list 'calcFunc-beta
							 a b))))))

(defun math-deriv-beta (x a b scale)
  (math-mul (math-mul (math-pow x (math-add a -1))
		      (math-pow (math-sub 1 x) (math-add b -1)))
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	    scale))
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(put 'calcFunc-erf\' 'math-derivative-1
     (function (lambda (x) (math-div 2
				     (math-mul (list 'calcFunc-exp
						     (math-sqr x))
					       (if calc-symbolic-mode
						   '(calcFunc-sqrt
						     (var pi var-pi))
						 (math-sqrt-pi)))))))

(put 'calcFunc-erfc\' 'math-derivative-1
     (function (lambda (x) (math-div -2
				     (math-mul (list 'calcFunc-exp
						     (math-sqr x))
					       (if calc-symbolic-mode
						   '(calcFunc-sqrt
						     (var pi var-pi))
						 (math-sqrt-pi)))))))

(put 'calcFunc-besJ\'2 'math-derivative-2
     (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besJ
						       (math-add v -1)
						       z)
						 (list 'calcFunc-besJ
						       (math-add v 1)
						       z))
				       2))))

(put 'calcFunc-besY\'2 'math-derivative-2
     (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besY
						       (math-add v -1)
						       z)
						 (list 'calcFunc-besY
						       (math-add v 1)
						       z))
				       2))))

(put 'calcFunc-sum 'math-derivative-n
     (function
      (lambda (expr)
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	(if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var)
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	    (throw 'math-deriv nil)
	  (cons 'calcFunc-sum
		(cons (math-derivative (nth 1 expr))
		      (cdr (cdr expr))))))))

(put 'calcFunc-prod 'math-derivative-n
     (function
      (lambda (expr)
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	(if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var)
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	    (throw 'math-deriv nil)
	  (math-mul expr
		    (cons 'calcFunc-sum
			  (cons (math-div (math-derivative (nth 1 expr))
					  (nth 1 expr))
				(cdr (cdr expr)))))))))

(put 'calcFunc-integ 'math-derivative-n
     (function
      (lambda (expr)
	(if (= (length expr) 3)
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	    (if (equal (nth 2 expr) math-deriv-var)
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		(nth 1 expr)
	      (math-normalize
	       (list 'calcFunc-integ
		     (math-derivative (nth 1 expr))
		     (nth 2 expr))))
	  (if (= (length expr) 5)
	      (let ((lower (math-expr-subst (nth 1 expr) (nth 2 expr)
					    (nth 3 expr)))
		    (upper (math-expr-subst (nth 1 expr) (nth 2 expr)
					    (nth 4 expr))))
		(math-add (math-sub (math-mul upper
					      (math-derivative (nth 4 expr)))
				    (math-mul lower
					      (math-derivative (nth 3 expr))))
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			  (if (equal (nth 2 expr) math-deriv-var)
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			      0
			    (math-normalize
			     (list 'calcFunc-integ
				   (math-derivative (nth 1 expr)) (nth 2 expr)
				   (nth 3 expr) (nth 4 expr)))))))))))

(put 'calcFunc-if 'math-derivative-n
     (function
      (lambda (expr)
	(and (= (length expr) 4)
	     (list 'calcFunc-if (nth 1 expr)
		   (math-derivative (nth 2 expr))
		   (math-derivative (nth 3 expr)))))))

(put 'calcFunc-subscr 'math-derivative-n
     (function
      (lambda (expr)
	(and (= (length expr) 3)
	     (list 'calcFunc-subscr (nth 1 expr)
		   (math-derivative (nth 2 expr)))))))


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(defvar math-integ-var '(var X ---))
(defvar math-integ-var-2 '(var Y ---))
(defvar math-integ-vars (list 'f math-integ-var math-integ-var-2))
(defvar math-integ-var-list (list math-integ-var))
(defvar math-integ-var-list-list (list math-integ-var-list))
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;; math-integ-depth is a local variable for math-try-integral, but is used
;; by math-integral and math-tracing-integral
;; which are called (directly or indirectly) by math-try-integral.
(defvar math-integ-depth)
;; math-integ-level is a local variable for math-try-integral, but is used
;; by math-integral, math-do-integral, math-tracing-integral, 
;; math-sub-integration, math-integrate-by-parts and 
;; math-integrate-by-substitution, which are called (directly or 
;; indirectly) by math-try-integral.
(defvar math-integ-level)
;; math-integral-limit is a local variable for calcFunc-integ, but is
;; used by math-tracing-integral, math-sub-integration and 
;; math-try-integration. 
(defvar math-integral-limit)

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(defmacro math-tracing-integral (&rest parts)
  (list 'and
	'trace-buffer
	(list 'save-excursion
	      '(set-buffer trace-buffer)
	      '(goto-char (point-max))
	      (list 'and
		    '(bolp)
		    '(insert (make-string (- math-integral-limit
					     math-integ-level) 32)
			     (format "%2d " math-integ-depth)
			     (make-string math-integ-level 32)))
	      ;;(list 'condition-case 'err
		    (cons 'insert parts)
		;;    '(error (insert (prin1-to-string err))))
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	      '(sit-for 0))))
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;;; The following wrapper caches results and avoids infinite recursion.
;;; Each cache entry is: ( A B )          Integral of A is B;
;;;			 ( A N )          Integral of A failed at level N;
;;;			 ( A busy )	  Currently working on integral of A;
;;;			 ( A parts )	  Currently working, integ-by-parts;
;;;			 ( A parts2 )	  Currently working, integ-by-parts;
;;;			 ( A cancelled )  Ignore this cache entry;
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;;;			 ( A [B] )        Same result as for math-cur-record = B.

;; math-cur-record is a local variable for math-try-integral, but is used
;; by math-integral, math-replace-integral-parts and math-integrate-by-parts
;; which are called (directly or indirectly) by math-try-integral, as well as
;; by calc-dump-integral-cache
(defvar math-cur-record)
;; math-enable-subst and math-any-substs are local variables for
;; calcFunc-integ, but are used by math-integral and math-try-integral.
(defvar math-enable-subst)
(defvar math-any-substs)

;; math-integ-msg is a local variable for math-try-integral, but is
;; used (both locally and non-locally) by math-integral.
(defvar math-integ-msg)

(defvar math-integral-cache nil)
(defvar math-integral-cache-state nil)

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(defun math-integral (expr &optional simplify same-as-above)
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  (let* ((simp math-cur-record)
	 (math-cur-record (assoc expr math-integral-cache))
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	 (math-integ-depth (1+ math-integ-depth))
	 (val 'cancelled))
    (math-tracing-integral "Integrating "
			   (math-format-value expr 1000)
			   "...\n")
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    (and math-cur-record
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	 (progn
	   (math-tracing-integral "Found "
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				  (math-format-value (nth 1 math-cur-record) 1000))
	   (and (consp (nth 1 math-cur-record))
		(math-replace-integral-parts math-cur-record))
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	   (math-tracing-integral " => "
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				  (math-format-value (nth 1 math-cur-record) 1000)
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				  "\n")))
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    (or (and math-cur-record
	     (not (eq (nth 1 math-cur-record) 'cancelled))
	     (or (not (integerp (nth 1 math-cur-record)))
		 (>= (nth 1 math-cur-record) math-integ-level)))
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	(and (math-integral-contains-parts expr)
	     (progn
	       (setq val nil)
	       t))
	(unwind-protect
	    (progn
	      (let (math-integ-msg)
		(if (eq calc-display-working-message 'lots)
		    (progn
		      (calc-set-command-flag 'clear-message)
		      (setq math-integ-msg (format
					    "Working... Integrating %s"
					    (math-format-flat-expr expr 0)))
		      (message math-integ-msg)))
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		(if math-cur-record
		    (setcar (cdr math-cur-record)
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			    (if same-as-above (vector simp) 'busy))
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		  (setq math-cur-record
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			(list expr (if same-as-above (vector simp) 'busy))
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			math-integral-cache (cons math-cur-record
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						  math-integral-cache)))
		(if (eq simplify 'yes)
		    (progn
		      (math-tracing-integral "Simplifying...")
		      (setq simp (math-simplify expr))
		      (setq val (if (equal simp expr)
				    (progn
				      (math-tracing-integral " no change\n")
				      (math-do-integral expr))
				  (math-tracing-integral " simplified\n")
				  (math-integral simp 'no t))))
		  (or (setq val (math-do-integral expr))
		      (eq simplify 'no)
		      (let ((simp (math-simplify expr)))
			(or (equal simp expr)
			    (progn
			      (math-tracing-integral "Trying again after "
						     "simplification...\n")
			      (setq val (math-integral simp 'no t))))))))
	      (if (eq calc-display-working-message 'lots)
		  (message math-integ-msg)))
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	  (setcar (cdr math-cur-record) (or val
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				       (if (or math-enable-subst
					       (not math-any-substs))
					   math-integ-level
					 'cancelled)))))
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    (setq val math-cur-record)
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    (while (vectorp (nth 1 val))
      (setq val (aref (nth 1 val) 0)))
    (setq val (if (memq (nth 1 val) '(parts parts2))
		  (progn
		    (setcar (cdr val) 'parts2)
		    (list 'var 'PARTS val))
		(and (consp (nth 1 val))
		     (nth 1 val))))
    (math-tracing-integral "Integral of "
			   (math-format-value expr 1000)
			   "  is  "
			   (math-format-value val 1000)
			   "\n")
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    val))
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(defun math-integral-contains-parts (expr)
  (if (Math-primp expr)
      (and (eq (car-safe expr) 'var)
	   (eq (nth 1 expr) 'PARTS)
	   (listp (nth 2 expr)))
    (while (and (setq expr (cdr expr))
		(not (math-integral-contains-parts (car expr)))))
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    expr))
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(defun math-replace-integral-parts (expr)
  (or (Math-primp expr)
      (while (setq expr (cdr expr))
	(and (consp (car expr))
	     (if (eq (car (car expr)) 'var)
		 (and (eq (nth 1 (car expr)) 'PARTS)
		      (consp (nth 2 (car expr)))
		      (if (listp (nth 1 (nth 2 (car expr))))
			  (progn
			    (setcar expr (nth 1 (nth 2 (car expr))))
			    (math-replace-integral-parts (cons 'foo expr)))
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			(setcar (cdr math-cur-record) 'cancelled)))
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	       (math-replace-integral-parts (car expr)))))))
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(defvar math-linear-subst-tried t
  "Non-nil means that a linear substitution has been tried.")

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;; The variable math-has-rules is a local variable for math-try-integral,
;; but is used by math-do-integral, which is called (non-directly) by
;; math-try-integral.
(defvar math-has-rules)

;; math-old-integ is a local variable for math-do-integral, but is
;; used by math-sub-integration.
(defvar math-old-integ)

;; The variables math-t1, math-t2 and math-t3 are local to 
;; math-do-integral, math-try-solve-for and math-decompose-poly, but
;; are used by functions they call (directly or indirectly); 
;; math-do-integral calls math-do-integral-methods;
;; math-try-solve-for calls math-try-solve-prod, 
;; math-solve-find-root-term and math-solve-find-root-in-prod;
;; math-decompose-poly calls math-solve-poly-funny-powers and
;; math-solve-crunch-poly.
(defvar math-t1)
(defvar math-t2)
(defvar math-t3)

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(defun math-do-integral (expr)
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  (let ((math-linear-subst-tried nil)
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        math-t1 math-t2)
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    (or (cond ((not (math-expr-contains expr math-integ-var))
	       (math-mul expr math-integ-var))
	      ((equal expr math-integ-var)
	       (math-div (math-sqr expr) 2))
	      ((eq (car expr) '+)
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	       (and (setq math-t1 (math-integral (nth 1 expr)))
		    (setq math-t2 (math-integral (nth 2 expr)))
		    (math-add math-t1 math-t2)))
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	      ((eq (car expr) '-)
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	       (and (setq math-t1 (math-integral (nth 1 expr)))
		    (setq math-t2 (math-integral (nth 2 expr)))
		    (math-sub math-t1 math-t2)))
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	      ((eq (car expr) 'neg)
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	       (and (setq math-t1 (math-integral (nth 1 expr)))
		    (math-neg math-t1)))
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	      ((eq (car expr) '*)
	       (cond ((not (math-expr-contains (nth 1 expr) math-integ-var))
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		      (and (setq math-t1 (math-integral (nth 2 expr)))
			   (math-mul (nth 1 expr) math-t1)))
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		     ((not (math-expr-contains (nth 2 expr) math-integ-var))
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		      (and (setq math-t1 (math-integral (nth 1 expr)))
			   (math-mul math-t1 (nth 2 expr))))
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		     ((memq (car-safe (nth 1 expr)) '(+ -))
		      (math-integral (list (car (nth 1 expr))
					   (math-mul (nth 1 (nth 1 expr))
						     (nth 2 expr))
					   (math-mul (nth 2 (nth 1 expr))
						     (nth 2 expr)))
				     'yes t))
		     ((memq (car-safe (nth 2 expr)) '(+ -))
		      (math-integral (list (car (nth 2 expr))
					   (math-mul (nth 1 (nth 2 expr))
						     (nth 1 expr))
					   (math-mul (nth 2 (nth 2 expr))
						     (nth 1 expr)))
				     'yes t))))
	      ((eq (car expr) '/)
	       (cond ((and (not (math-expr-contains (nth 1 expr)
						    math-integ-var))
			   (not (math-equal-int (nth 1 expr) 1)))
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		      (and (setq math-t1 (math-integral (math-div 1 (nth 2 expr))))
			   (math-mul (nth 1 expr) math-t1)))
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		     ((not (math-expr-contains (nth 2 expr) math-integ-var))
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		      (and (setq math-t1 (math-integral (nth 1 expr)))
			   (math-div math-t1 (nth 2 expr))))
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		     ((and (eq (car-safe (nth 1 expr)) '*)
			   (not (math-expr-contains (nth 1 (nth 1 expr))
						    math-integ-var)))
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		      (and (setq math-t1 (math-integral
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				     (math-div (nth 2 (nth 1 expr))
					       (nth 2 expr))))
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			   (math-mul math-t1 (nth 1 (nth 1 expr)))))
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		     ((and (eq (car-safe (nth 1 expr)) '*)
			   (not (math-expr-contains (nth 2 (nth 1 expr))
						    math-integ-var)))
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		      (and (setq math-t1 (math-integral
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				     (math-div (nth 1 (nth 1 expr))
					       (nth 2 expr))))
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			   (math-mul math-t1 (nth 2 (nth 1 expr)))))
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		     ((and (eq (car-safe (nth 2 expr)) '*)
			   (not (math-expr-contains (nth 1 (nth 2 expr))
						    math-integ-var)))
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		      (and (setq math-t1 (math-integral
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				     (math-div (nth 1 expr)
					       (nth 2 (nth 2 expr)))))
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			   (math-div math-t1 (nth 1 (nth 2 expr)))))
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		     ((and (eq (car-safe (nth 2 expr)) '*)
			   (not (math-expr-contains (nth 2 (nth 2 expr))
						    math-integ-var)))
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		      (and (setq math-t1 (math-integral
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				     (math-div (nth 1 expr)
					       (nth 1 (nth 2 expr)))))
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			   (math-div math-t1 (nth 2 (nth 2 expr)))))
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		     ((eq (car-safe (nth 2 expr)) 'calcFunc-exp)
		      (math-integral
		       (math-mul (nth 1 expr)
				 (list 'calcFunc-exp
				       (math-neg (nth 1 (nth 2 expr)))))))))
	      ((eq (car expr) '^)
	       (cond ((not (math-expr-contains (nth 1 expr) math-integ-var))
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		      (or (and (setq math-t1 (math-is-polynomial (nth 2 expr)
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							    math-integ-var 1))
			       (math-div expr
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					 (math-mul (nth 1 math-t1)
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						   (math-normalize
						    (list 'calcFunc-ln
							  (nth 1 expr))))))
			  (math-integral
			   (list 'calcFunc-exp
				 (math-mul (nth 2 expr)
					   (math-normalize
					    (list 'calcFunc-ln
						  (nth 1 expr)))))
			   'yes t)))
		     ((not (math-expr-contains (nth 2 expr) math-integ-var))
		      (if (and (integerp (nth 2 expr)) (< (nth 2 expr) 0))
			  (math-integral
			   (list '/ 1 (math-pow (nth 1 expr) (- (nth 2 expr))))
			   nil t)
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			(or (and (setq math-t1 (math-is-polynomial (nth 1 expr)
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							      math-integ-var
							      1))
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				 (setq math-t2 (math-add (nth 2 expr) 1))
				 (math-div (math-pow (nth 1 expr) math-t2)
					   (math-mul math-t2 (nth 1 math-t1))))
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			    (and (Math-negp (nth 2 expr))
				 (math-integral
				  (math-div 1
					    (math-pow (nth 1 expr)
						      (math-neg
						       (nth 2 expr))))
				  nil t))
			    nil))))))

	;; Integral of a polynomial.
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	(and (setq math-t1 (math-is-polynomial expr math-integ-var 20))
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	     (let ((accum 0)
		   (n 1))
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	       (while math-t1
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		 (if (setq accum (math-add accum
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					   (math-div (math-mul (car math-t1)
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							       (math-pow
								math-integ-var
								n))
						     n))
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			   math-t1 (cdr math-t1))
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		     (setq n (1+ n))))
	       accum))

	;; Try looking it up!
	(cond ((= (length expr) 2)
	       (and (symbolp (car expr))
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		    (setq math-t1 (get (car expr) 'math-integral))
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		    (progn
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		      (while (and math-t1
				  (not (setq math-t2 (funcall (car math-t1)
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							 (nth 1 expr)))))
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			(setq math-t1 (cdr math-t1)))
		      (and math-t2 (math-normalize math-t2)))))
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	      ((= (length expr) 3)
	       (and (symbolp (car expr))
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		    (setq math-t1 (get (car expr) 'math-integral-2))
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		    (progn
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		      (while (and math-t1
				  (not (setq math-t2 (funcall (car math-t1)
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							 (nth 1 expr)
							 (nth 2 expr)))))
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			(setq math-t1 (cdr math-t1)))
		      (and math-t2 (math-normalize math-t2))))))
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	;; Integral of a rational function.
	(and (math-ratpoly-p expr math-integ-var)
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	     (setq math-t1 (calcFunc-apart expr math-integ-var))
	     (not (equal math-t1 expr))
	     (math-integral math-t1))
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	;; Try user-defined integration rules.
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	(and math-has-rules
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	     (let ((math-old-integ (symbol-function 'calcFunc-integ))
		   (input (list 'calcFunc-integtry expr math-integ-var))
		   res part)
	       (unwind-protect
		   (progn
		     (fset 'calcFunc-integ 'math-sub-integration)
		     (setq res (math-rewrite input
					     '(var IntegRules var-IntegRules)
					     1))
		     (fset 'calcFunc-integ math-old-integ)
		     (and (not (equal res input))
			  (if (setq part (math-expr-calls
					  res '(calcFunc-integsubst)))
			      (and (memq (length part) '(3 4 5))
				   (let ((parts (mapcar
						 (function
						  (lambda (x)
						    (math-expr-subst
						     x (nth 2 part)
						     math-integ-var)))
						 (cdr part))))
				     (math-integrate-by-substitution
				      expr (car parts) t
				      (or (nth 2 parts)
					  (list 'calcFunc-integfailed
						math-integ-var))
				      (nth 3 parts))))
			    (if (not (math-expr-calls res
						      '(calcFunc-integtry
							calcFunc-integfailed)))
				res))))
		 (fset 'calcFunc-integ math-old-integ))))

	;; See if the function is a symbolic derivative.
	(and (string-match "'" (symbol-name (car expr)))
	     (let ((name (symbol-name (car expr)))
		   (p expr) (n 0) (which nil) (bad nil))
	       (while (setq n (1+ n) p (cdr p))
		 (if (equal (car p) math-integ-var)
		     (if which (setq bad t) (setq which n))
		   (if (math-expr-contains (car p) math-integ-var)
		       (setq bad t))))
	       (and which (not bad)
		    (let ((prime (if (= which 1) "'" (format "'%d" which))))
		      (and (string-match (concat prime "\\('['0-9]*\\|$\\)")
					 name)
			   (cons (intern
				  (concat
				   (substring name 0 (match-beginning 0))
				   (substring name (+ (match-beginning 0)
						      (length prime)))))
				 (cdr expr)))))))

	;; Try transformation methods (parts, substitutions).
	(and (> math-integ-level 0)
	     (math-do-integral-methods expr))

	;; Try expanding the function's definition.
	(let ((res (math-expand-formula expr)))
	  (and res
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	       (math-integral res))))))
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(defun math-sub-integration (expr &rest rest)
  (or (if (or (not rest)
	      (and (< math-integ-level math-integral-limit)
		   (eq (car rest) math-integ-var)))
	  (math-integral expr)
	(let ((res (apply math-old-integ expr rest)))
	  (and (or (= math-integ-level math-integral-limit)
		   (not (math-expr-calls res 'calcFunc-integ)))
	       res)))
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      (list 'calcFunc-integfailed expr)))
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;; math-so-far is a local variable for math-do-integral-methods, but
;; is used by math-integ-try-linear-substitutions and 
;; math-integ-try-substitutions.
(defvar math-so-far)

;; math-integ-expr is a local variable for math-do-integral-methods,
;; but is used by math-integ-try-linear-substitutions and 
;; math-integ-try-substitutions.
(defvar math-integ-expr)

(defun math-do-integral-methods (math-integ-expr)
  (let ((math-so-far math-integ-var-list-list)
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	rat-in)

    ;; Integration by substitution, for various likely sub-expressions.
    ;; (In first pass, we look only for sub-exprs that are linear in X.)
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    (or (math-integ-try-linear-substitutions math-integ-expr)
        (math-integ-try-substitutions math-integ-expr)
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	;; If function has sines and cosines, try tan(x/2) substitution.
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	(and (let ((p (setq rat-in (math-expr-rational-in math-integ-expr))))
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	       (while (and p
			   (memq (car (car p)) '(calcFunc-sin
						 calcFunc-cos
						 calcFunc-tan))
			   (equal (nth 1 (car p)) math-integ-var))
		 (setq p (cdr p)))
	       (null p))
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	     (or (and (math-integ-parts-easy math-integ-expr)
		      (math-integ-try-parts math-integ-expr t))
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		 (math-integrate-by-good-substitution
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		  math-integ-expr (list 'calcFunc-tan (math-div math-integ-var 2)))))
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	;; If function has sinh and cosh, try tanh(x/2) substitution.
	(and (let ((p rat-in))
	       (while (and p
			   (memq (car (car p)) '(calcFunc-sinh
						 calcFunc-cosh
						 calcFunc-tanh
						 calcFunc-exp))
			   (equal (nth 1 (car p)) math-integ-var))
		 (setq p (cdr p)))
	       (null p))
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	     (or (and (math-integ-parts-easy math-integ-expr)
		      (math-integ-try-parts math-integ-expr t))
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		 (math-integrate-by-good-substitution
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		  math-integ-expr (list 'calcFunc-tanh (math-div math-integ-var 2)))))
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	;; If function has square roots, try sin, tan, or sec substitution.
	(and (let ((p rat-in))
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	       (setq math-t1 nil)
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	       (while (and p
			   (or (equal (car p) math-integ-var)
			       (and (eq (car (car p)) 'calcFunc-sqrt)
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				    (setq math-t1 (math-is-polynomial
					      (nth 1 (setq math-t2 (car p)))
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					      math-integ-var 2)))))
		 (setq p (cdr p)))
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	       (and (null p) math-t1))
	     (if (cdr (cdr math-t1))
		 (if (math-guess-if-neg (nth 2 math-t1))
		     (let* ((c (math-sqrt (math-neg (nth 2 math-t1))))
			    (d (math-div (nth 1 math-t1) (math-mul -2 c)))
			    (a (math-sqrt (math-add (car math-t1) (math-sqr d)))))
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		       (math-integrate-by-good-substitution
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			math-integ-expr (list 'calcFunc-arcsin
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				   (math-div-thru
				    (math-add (math-mul c math-integ-var) d)
				    a))))
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		   (let* ((c (math-sqrt (nth 2 math-t1)))
			  (d (math-div (nth 1 math-t1) (math-mul 2 c)))
			  (aa (math-sub (car math-t1) (math-sqr d))))
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		     (if (and nil (not (and (eq d 0) (eq c 1))))
			 (math-integrate-by-good-substitution
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			  math-integ-expr (math-add (math-mul c math-integ-var) d))
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		       (if (math-guess-if-neg aa)
			   (math-integrate-by-good-substitution
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			    math-integ-expr (list 'calcFunc-arccosh
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				       (math-div-thru
					(math-add (math-mul c math-integ-var)
						  d)
					(math-sqrt (math-neg aa)))))
			 (math-integrate-by-good-substitution
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			  math-integ-expr (list 'calcFunc-arcsinh
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				     (math-div-thru
				      (math-add (math-mul c math-integ-var)
						d)
				      (math-sqrt aa))))))))
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	       (math-integrate-by-good-substitution math-integ-expr math-t2)) )
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	;; Try integration by parts.
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	(math-integ-try-parts math-integ-expr)
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	;; Give up.
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	nil)))
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(defun math-integ-parts-easy (expr)
  (cond ((Math-primp expr) t)
	((memq (car expr) '(+ - *))
	 (and (math-integ-parts-easy (nth 1 expr))
	      (math-integ-parts-easy (nth 2 expr))))
	((eq (car expr) '/)
	 (and (math-integ-parts-easy (nth 1 expr))
	      (math-atomic-factorp (nth 2 expr))))
	((eq (car expr) '^)
	 (and (natnump (nth 2 expr))
	      (math-integ-parts-easy (nth 1 expr))))
	((eq (car expr) 'neg)
	 (math-integ-parts-easy (nth 1 expr)))
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	(t t)))
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;; math-prev-parts-v is local to calcFunc-integ (as well as
;; math-integrate-by-parts), but is used by math-integ-try-parts.
(defvar math-prev-parts-v)

;; math-good-parts is local to calcFunc-integ (as well as
;; math-integ-try-parts), but is used by math-integrate-by-parts.
(defvar math-good-parts)


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(defun math-integ-try-parts (expr &optional math-good-parts)
  ;; Integration by parts:
  ;;   integ(f(x) g(x),x) = f(x) h(x) - integ(h(x) f'(x),x)
  ;;     where h(x) = integ(g(x),x).
  (or (let ((exp (calcFunc-expand expr)))
	(and (not (equal exp expr))
	     (math-integral exp)))
      (and (eq (car expr) '*)
	   (let ((first-bad (or (math-polynomial-p (nth 1 expr)
						   math-integ-var)
				(equal (nth 2 expr) math-prev-parts-v))))
	     (or (and first-bad   ; so try this one first
		      (math-integrate-by-parts (nth 1 expr) (nth 2 expr)))
		 (math-integrate-by-parts (nth 2 expr) (nth 1 expr))
		 (and (not first-bad)
		      (math-integrate-by-parts (nth 1 expr) (nth 2 expr))))))
      (and (eq (car expr) '/)
	   (math-expr-contains (nth 1 expr) math-integ-var)
	   (let ((recip (math-div 1 (nth 2 expr))))
	     (or (math-integrate-by-parts (nth 1 expr) recip)
		 (math-integrate-by-parts recip (nth 1 expr)))))
      (and (eq (car expr) '^)
	   (math-integrate-by-parts (math-pow (nth 1 expr)
					      (math-sub (nth 2 expr) 1))
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				    (nth 1 expr)))))
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(defun math-integrate-by-parts (u vprime)
  (let ((math-integ-level (if (or math-good-parts
				  (math-polynomial-p u math-integ-var))
			      math-integ-level
			    (1- math-integ-level)))
	(math-doing-parts t)
	v temp)
    (and (>= math-integ-level 0)
	 (unwind-protect
	     (progn
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	       (setcar (cdr math-cur-record) 'parts)
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	       (math-tracing-integral "Integrating by parts, u = "
				      (math-format-value u 1000)
				      ", v' = "
				      (math-format-value vprime 1000)
				      "\n")
	       (and (setq v (math-integral vprime))
		    (setq temp (calcFunc-deriv u math-integ-var nil t))
		    (setq temp (let ((math-prev-parts-v v))
				 (math-integral (math-mul v temp) 'yes)))
		    (setq temp (math-sub (math-mul u v) temp))
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		    (if (eq (nth 1 math-cur-record) 'parts)
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			(calcFunc-expand temp)
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		      (setq v (list 'var 'PARTS math-cur-record)
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			    temp (let (calc-next-why)
				   (math-solve-for (math-sub v temp) 0 v nil)))
		      (and temp (not (integerp temp))
			   (math-simplify-extended temp)))))
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	   (setcar (cdr math-cur-record) 'busy)))))
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;;; This tries two different formulations, hoping the algebraic simplifier
;;; will be strong enough to handle at least one.
(defun math-integrate-by-substitution (expr u &optional user uinv uinvprime)
  (and (> math-integ-level 0)
       (let ((math-integ-level (max (- math-integ-level 2) 0)))
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	 (math-integrate-by-good-substitution expr u user uinv uinvprime))))
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(defun math-integrate-by-good-substitution (expr u &optional user
						 uinv uinvprime)
  (let ((math-living-dangerously t)
	deriv temp)
    (and (setq uinv (if uinv
			(math-expr-subst uinv math-integ-var
					 math-integ-var-2)
		      (let (calc-next-why)
			(math-solve-for u
					math-integ-var-2
					math-integ-var nil))))
	 (progn
	   (math-tracing-integral "Integrating by substitution, u = "
				  (math-format-value u 1000)
				  "\n")
	   (or (and (setq deriv (calcFunc-deriv u
						math-integ-var nil
						(not user)))
		    (setq temp (math-integral (math-expr-subst
					       (math-expr-subst
						(math-expr-subst
						 (math-div expr deriv)
						 u
						 math-integ-var-2)
						math-integ-var
						uinv)
					       math-integ-var-2
					       math-integ-var)
					      'yes)))
	       (and (setq deriv (or uinvprime
				    (calcFunc-deriv uinv
						    math-integ-var-2
						    math-integ-var
						    (not user))))
		    (setq temp (math-integral (math-mul
					       (math-expr-subst
						(math-expr-subst
						 (math-expr-subst
						  expr
						  u
						  math-integ-var-2)
						 math-integ-var
						 uinv)
						math-integ-var-2
						math-integ-var)
					       deriv)
					      'yes)))))
	 (math-simplify-extended
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	  (math-expr-subst temp math-integ-var u)))))
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;;; Look for substitutions of the form u = a x + b.
(defun math-integ-try-linear-substitutions (sub-expr)
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  (setq math-linear-subst-tried t)
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  (and (not (Math-primp sub-expr))
       (or (and (not (memq (car sub-expr) '(+ - * / neg)))
		(not (and (eq (car sub-expr) '^)
			  (integerp (nth 2 sub-expr))))
		(math-expr-contains sub-expr math-integ-var)
		(let ((res nil))
		  (while (and (setq sub-expr (cdr sub-expr))
			      (or (not (math-linear-in (car sub-expr)
						       math-integ-var))
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				  (assoc (car sub-expr) math-so-far)
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				  (progn
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				    (setq math-so-far (cons (list (car sub-expr))
						       math-so-far))
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				    (not (setq res
					       (math-integrate-by-substitution
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						math-integ-expr (car sub-expr))))))))
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		  res))
	   (let ((res nil))
	     (while (and (setq sub-expr (cdr sub-expr))
			 (not (setq res (math-integ-try-linear-substitutions
					 (car sub-expr))))))
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	     res))))
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;;; Recursively try different substitutions based on various sub-expressions.
(defun math-integ-try-substitutions (sub-expr &optional allow-rat)
  (and (not (Math-primp sub-expr))
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       (not (assoc sub-expr math-so-far))
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       (math-expr-contains sub-expr math-integ-var)
       (or (and (if (and (not (memq (car sub-expr) '(+ - * / neg)))
			 (not (and (eq (car sub-expr) '^)
				   (integerp (nth 2 sub-expr)))))
		    (setq allow-rat t)
		  (prog1 allow-rat (setq allow-rat nil)))
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		(not (eq sub-expr math-integ-expr))
		(or (math-integrate-by-substitution math-integ-expr sub-expr)
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		    (and (eq (car sub-expr) '^)
			 (integerp (nth 2 sub-expr))
			 (< (nth 2 sub-expr) 0)
			 (math-integ-try-substitutions
			  (math-pow (nth 1 sub-expr) (- (nth 2 sub-expr)))
			  t))))
	   (let ((res nil))
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	     (setq math-so-far (cons (list sub-expr) math-so-far))
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	     (while (and (setq sub-expr (cdr sub-expr))
			 (not (setq res (math-integ-try-substitutions
					 (car sub-expr) allow-rat)))))
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	     res))))
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;; The variable math-expr-parts is local to math-expr-rational-in,
;; but is used by math-expr-rational-in-rec

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(defun math-expr-rational-in (expr)
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  (let ((math-expr-parts nil))
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    (math-expr-rational-in-rec expr)
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    (mapcar 'car math-expr-parts)))
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(defun math-expr-rational-in-rec (expr)
  (cond ((Math-primp expr)
	 (and (equal expr math-integ-var)
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	      (not (assoc expr math-expr-parts))
	      (setq math-expr-parts (cons (list expr) math-expr-parts))))
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	((or (memq (car expr) '(+ - * / neg))
	     (and (eq (car expr) '^) (integerp (nth 2 expr))))
	 (math-expr-rational-in-rec (nth 1 expr))
	 (and (nth 2 expr) (math-expr-rational-in-rec (nth 2 expr))))
	((and (eq (car expr) '^)
	      (eq (math-quarter-integer (nth 2 expr)) 2))
	 (math-expr-rational-in-rec (list 'calcFunc-sqrt (nth 1 expr))))
	(t
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	 (and (not (assoc expr math-expr-parts))
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	      (math-expr-contains expr math-integ-var)
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	      (setq math-expr-parts (cons (list expr) math-expr-parts))))))
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(defun math-expr-calls (expr funcs &optional arg-contains)
  (if (consp expr)
      (if (or (memq (car expr) funcs)
	      (and (eq (car expr) '^) (eq (car funcs) 'calcFunc-sqrt)
		   (eq (math-quarter-integer (nth 2 expr)) 2)))
	  (and (or (not arg-contains)
		   (math-expr-contains expr arg-contains))
	       expr)
	(and (not (Math-primp expr))
	     (let ((res nil))
	       (while (and (setq expr (cdr expr))
			   (not (setq res (math-expr-calls
					   (car expr) funcs arg-contains)))))
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	       res)))))
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(defun math-fix-const-terms (expr except-vars)
  (cond ((not (math-expr-depends expr except-vars)) 0)
	((Math-primp expr) expr)
	((eq (car expr) '+)
	 (math-add (math-fix-const-terms (nth 1 expr) except-vars)
		   (math-fix-const-terms (nth 2 expr) except-vars)))
	((eq (car expr) '-)
	 (math-sub (math-fix-const-terms (nth 1 expr) except-vars)
		   (math-fix-const-terms (nth 2 expr) except-vars)))
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	(t expr)))
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;; Command for debugging the Calculator's symbolic integrator.
(defun calc-dump-integral-cache (&optional arg)
  (interactive "P")
  (let ((buf (current-buffer)))
    (unwind-protect
	(let ((p math-integral-cache)
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	      math-cur-record)
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	  (display-buffer (get-buffer-create "*Integral Cache*"))
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	  (set-buffer (get-buffer "*Integral Cache*"))
	  (erase-buffer)
	  (while p
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	    (setq math-cur-record (car p))
	    (or arg (math-replace-integral-parts math-cur-record))
	    (insert (math-format-flat-expr (car math-cur-record) 0)
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		    " --> "
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		    (if (symbolp (nth 1 math-cur-record))
			(concat "(" (symbol-name (nth 1 math-cur-record)) ")")
		      (math-format-flat-expr (nth 1 math-cur-record) 0))
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		    "\n")
	    (setq p (cdr p)))
	  (goto-char (point-min)))
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      (set-buffer buf))))
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;; The variable math-max-integral-limit is local to calcFunc-integ,
;; but is used by math-try-integral.
(defvar math-max-integral-limit)

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(defun math-try-integral (expr)
  (let ((math-integ-level math-integral-limit)
	(math-integ-depth 0)
	(math-integ-msg "Working...done")
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	(math-cur-record nil)   ; a technicality
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	(math-integrating t)
	(calc-prefer-frac t)
	(calc-symbolic-mode t)
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	(math-has-rules (calc-has-rules 'var-IntegRules)))
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    (or (math-integral expr 'yes)
	(and math-any-substs
	     (setq math-enable-subst t)
	     (math-integral expr 'yes))
	(and (> math-max-integral-limit math-integral-limit)
	     (setq math-integral-limit math-max-integral-limit
		   math-integ-level math-integral-limit)
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	     (math-integral expr 'yes)))))
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(defvar var-IntegLimit nil)

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(defun calcFunc-integ (expr var &optional low high)
  (cond
   ;; Do these even if the parts turn out not to be integrable.
   ((eq (car-safe expr) '+)
    (math-add (calcFunc-integ (nth 1 expr) var low high)
	      (calcFunc-integ (nth 2 expr) var low high)))
   ((eq (car-safe expr) '-)
    (math-sub (calcFunc-integ (nth 1 expr) var low high)
	      (calcFunc-integ (nth 2 expr) var low high)))
   ((eq (car-safe expr) 'neg)
    (math-neg (calcFunc-integ (nth 1 expr) var low high)))
   ((and (eq (car-safe expr) '*)
	 (not (math-expr-contains (nth 1 expr) var)))
    (math-mul (nth 1 expr) (calcFunc-integ (nth 2 expr) var low high)))
   ((and (eq (car-safe expr) '*)
	 (not (math-expr-contains (nth 2 expr) var)))
    (math-mul (calcFunc-integ (nth 1 expr) var low high) (nth 2 expr)))
   ((and (eq (car-safe expr) '/)
	 (not (math-expr-contains (nth 1 expr) var))
	 (not (math-equal-int (nth 1 expr) 1)))
    (math-mul (nth 1 expr)
	      (calcFunc-integ (math-div 1 (nth 2 expr)) var low high)))
   ((and (eq (car-safe expr) '/)
	 (not (math-expr-contains (nth 2 expr) var)))
    (math-div (calcFunc-integ (nth 1 expr) var low high) (nth 2 expr)))
   ((and (eq (car-safe expr) '/)
	 (eq (car-safe (nth 1 expr)) '*)
	 (not (math-expr-contains (nth 1 (nth 1 expr)) var)))
    (math-mul (nth 1 (nth 1 expr))
	      (calcFunc-integ (math-div (nth 2 (nth 1 expr)) (nth 2 expr))
			      var low high)))
   ((and (eq (car-safe expr) '/)
	 (eq (car-safe (nth 1 expr)) '*)
	 (not (math-expr-contains (nth 2 (nth 1 expr)) var)))
    (math-mul (nth 2 (nth 1 expr))
	      (calcFunc-integ (math-div (nth 1 (nth 1 expr)) (nth 2 expr))
			      var low high)))
   ((and (eq (car-safe expr) '/)
	 (eq (car-safe (nth 2 expr)) '*)
	 (not (math-expr-contains (nth 1 (nth 2 expr)) var)))
    (math-div (calcFunc-integ (math-div (nth 1 expr) (nth 2 (nth 2 expr)))
			      var low high)
	      (nth 1 (nth 2 expr))))
   ((and (eq (car-safe expr) '/)
	 (eq (car-safe (nth 2 expr)) '*)
	 (not (math-expr-contains (nth 2 (nth 2 expr)) var)))
    (math-div (calcFunc-integ (math-div (nth 1 expr) (nth 1 (nth 2 expr)))
			      var low high)
	      (nth 2 (nth 2 expr))))
   ((eq (car-safe expr) 'vec)
    (cons 'vec (mapcar (function (lambda (x) (calcFunc-integ x var low high)))
		       (cdr expr))))
   (t
    (let ((state (list calc-angle-mode
		       ;;calc-symbolic-mode
		       ;;calc-prefer-frac
		       calc-internal-prec
		       (calc-var-value 'var-IntegRules)
		       (calc-var-value 'var-IntegSimpRules))))
      (or (equal state math-integral-cache-state)
	  (setq math-integral-cache-state state
		math-integral-cache nil)))
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    (let* ((math-max-integral-limit (or (and (natnump var-IntegLimit)
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