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intg.cc
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// -*- mode:C++ ; compile-command: "g++ -I.. -g -c intg.cc -fno-strict-aliasing -DGIAC_GENERIC_CONSTANTS -DHAVE_CONFIG_H -DIN_GIAC " -*-
#include "giacPCH.h"
// #define LOGINT
/*
* Copyright (C) 2000,2014 B. Parisse, Institut Fourier, 38402 St Martin d'Heres
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
using namespace std;
#include <stdexcept>
#include "vector.h"
#include <cmath>
#include <cstdlib>
#include <limits>
#include "sym2poly.h"
#include "usual.h"
#include "intg.h"
#include "subst.h"
#include "derive.h"
#include "lin.h"
#include "vecteur.h"
#include "gausspol.h"
#include "plot.h"
#include "prog.h"
#include "modpoly.h"
#include "series.h"
#include "tex.h"
#include "ifactor.h"
#include "risch.h"
#include "solve.h"
#include "intgab.h"
#include "moyal.h"
#include "maple.h"
#include "rpn.h"
#include "modpoly.h"
#include "giacintl.h"
#ifdef HAVE_CONFIG_H
#include "config.h"
#endif
#ifdef HAVE_LIBGSL
#include <gsl/gsl_math.h>
#include <gsl/gsl_sf_gamma.h>
#include <gsl/gsl_sf_psi.h>
#include <gsl/gsl_sf_zeta.h>
#include <gsl/gsl_odeiv.h>
#include <gsl/gsl_errno.h>
#endif
#if defined HAVE_LIBBERNMM && !defined BF2GMP_H
#include <bern_rat.h>
#endif
#ifndef NO_NAMESPACE_GIAC
namespace giac {
#endif // ndef NO_NAMESPACE_GIAC
// Left redimension p to degree n, i.e. size n+1
void lrdm(modpoly & p,int n){
int s=int(p.size());
if (n+1>s)
p=mergevecteur(vecteur(n+1-s),p);
}
struct pf1 {
vecteur num;
vecteur den;
vecteur fact;
int mult; // den=cste*fact^mult
pf1():num(0),den(makevecteur(1)),fact(makevecteur(1)),mult(1) {}
pf1(const pf1 & a) : num(a.num), den(a.den), fact(a.fact),mult(a.mult) {}
pf1(const vecteur &n, const vecteur & d, const vecteur & f,int m) : num(n), den(d), fact(f), mult(m) {};
pf1(const polynome & n,const polynome & d,const polynome & f,int m): num(polynome2poly1(n,1)),den(polynome2poly1(d,1)),fact(polynome2poly1(f,1)),mult(m) {}
};
gen complex_subst(const gen & e,const vecteur & substin,const vecteur & substout,GIAC_CONTEXT){
bool save_complex_mode=complex_mode(contextptr);
complex_mode(true,contextptr);
bool save_eval_abs=eval_abs(contextptr);
eval_abs(false,contextptr);
gen res=simplifier(eval(subst(e,substin,substout,false,contextptr),1,contextptr),contextptr);
// eval is used since after subst * are not flattened
complex_mode(save_complex_mode,contextptr);
eval_abs(save_eval_abs,contextptr);
return res;
}
gen complex_subst(const gen & e,const gen & x,const gen & newx,GIAC_CONTEXT){
bool save_complex_mode=complex_mode(contextptr);
complex_mode(true,contextptr);
bool save_eval_abs=eval_abs(contextptr);
eval_abs(false,contextptr);
gen res=subst(e,x,newx,false,contextptr);
eval_abs(save_eval_abs,contextptr);
// avoid rewrite of fractional powers
vecteur v=lop(newx,at_pow);
int i=0;
for (;i<v.size();++i){
gen tmp=v[i];
if (tmp.is_symb_of_sommet(at_pow)){
tmp=tmp._SYMBptr->feuille;
if (tmp.type==_VECT && tmp._VECTptr->size()==2){
tmp=tmp._VECTptr->back();
if (tmp.type==_FRAC && tmp._FRACptr->den.type==_INT_ ){
tmp=tmp._FRACptr->den;
if (tmp.val % 2==1)
break;
}
}
}
}
complex_mode(save_complex_mode,contextptr);
if (i==v.size())
res=eval(res,1,contextptr);
return res;
}
static bool has_nop_var(const vecteur & v){
const_iterateur it=v.begin(),itend=v.end();
for (;it!=itend;++it){
if (contains(*it,at_nop))
return true;
}
return false;
}
static gen nop_inv(const gen & e,GIAC_CONTEXT){
return symbolic(at_nop,gen(symbolic(at_inv,e)));
}
static gen nop_pow(const gen & e,GIAC_CONTEXT){
if ( (e.type!=_VECT) || (e._VECTptr->size()!=2))
return symbolic(at_pow,e);
if ( (e._VECTptr->back().type!=_INT_) || (e._VECTptr->back().val>=0) || ( (e._VECTptr->front().type==_SYMB) && (e._VECTptr->front()._SYMBptr->sommet==at_exp) ) )
return symbolic(at_pow,change_subtype(e,_SEQ__VECT));
return nop_inv(symbolic(at_pow,gen(makevecteur(e._VECTptr->front(),-e._VECTptr->back()),_SEQ__VECT)),contextptr);
}
static gen sin_over_cos(const gen & e,GIAC_CONTEXT){
return rdiv(symb_sin(e),symb_cos(e),contextptr);
}
const gen_op_context invpowtan2_tab[]={nop_inv,nop_pow,sin_over_cos,0};
// remove nop if nop() does not contain x
gen remove_nop(const gen & g,const gen & x,GIAC_CONTEXT){
if (g.type==_VECT){
vecteur res(*g._VECTptr);
iterateur it=res.begin(),itend=res.end();
for (;it!=itend;++it){
*it=remove_nop(*it,x,contextptr);
}
return gen(res,g.subtype);
}
if (g.type!=_SYMB)
return g;
if (g._SYMBptr->sommet!=at_nop)
return symbolic(g._SYMBptr->sommet,remove_nop(g._SYMBptr->feuille,x,contextptr));
if (is_zero(derive(g._SYMBptr->feuille,x,contextptr)))
return g._SYMBptr->feuille;
else
return g;
}
vecteur lvarxwithinv(const gen &e,const gen & x,GIAC_CONTEXT){
gen ee=subst(e,invpowtan_tab,invpowtan2_tab,false,contextptr);
ee=remove_nop(ee,x,contextptr);
vecteur v(lvarx(ee,x));
return v; // to remove nop do a return *(eval(v)._VECTptr);
}
bool is_constant_wrt(const gen & e,const gen & x,GIAC_CONTEXT){
if (e.type==_VECT){
const_iterateur it=e._VECTptr->begin(),itend=e._VECTptr->end();
for (;it!=itend;++it){
if (!is_constant_wrt(*it,x,contextptr))
return false;
}
return true;
}
if (e==x)
return false;
if (e.type!=_SYMB)
return true;
return is_exactly_zero(derive(e,x,contextptr));
}
// return true if e=a*x+b
bool is_linear_wrt(const gen & e,const gen &x,gen & a,gen & b,GIAC_CONTEXT){
a=derive(e,x,contextptr);
if (is_undef(a) || !is_constant_wrt(a,x,contextptr))
return false;
if (x*a==e)
b=0;
else
b=ratnormal(e-a*x,contextptr);
return lvarx(b,x).empty();
}
// return true if e=a*x+b
bool is_quadratic_wrt(const gen & e,const gen &x,gen & a,gen & b,gen & c,GIAC_CONTEXT){
gen tmp=derive(e,x,contextptr);
if (is_undef(tmp) || !is_linear_wrt(tmp,x,a,b,contextptr))
return false;
a=ratnormal(rdiv(a,plus_two,contextptr),contextptr);
c=ratnormal(e-a*x*x-b*x,contextptr);
return true;
}
void decompose_plus(const vecteur & arg,const gen & x,vecteur & non_constant,gen & plus_constant,GIAC_CONTEXT){
non_constant.clear();
plus_constant=zero;
const_iterateur it=arg.begin(),itend=arg.end();
for (;it!=itend;++it){
if (is_constant_wrt(*it,x,contextptr))
plus_constant=plus_constant+(*it);
else
non_constant.push_back(*it);
}
// if (contains(plus_constant,x)) plus_constant=ratnormal(plus_constant,contextptr);
}
void decompose_prod(const vecteur & arg,const gen & x,vecteur & non_constant,gen & prod_constant,bool signcst,GIAC_CONTEXT){
non_constant.clear();
prod_constant=plus_one;
const_iterateur it=arg.begin(),itend=arg.end();
for (;it!=itend;++it){
gen tst=*it;
if (!signcst && it->is_symb_of_sommet(at_sign))
tst=it->_SYMBptr->feuille;
if (is_constant_wrt(tst,x,contextptr))
prod_constant=prod_constant*(*it);
else
non_constant.push_back(*it);
}
// if (contains(prod_constant,x)) prod_constant=ratnormal(prod_constant,contextptr);
}
gen extract_cst(gen & u,const gen & x,GIAC_CONTEXT){
if (!u.is_symb_of_sommet(at_prod) || u._SYMBptr->feuille.type!=_VECT)
return 1;
vecteur non_constant; gen prod_constant=1;
decompose_prod(*u._SYMBptr->feuille._VECTptr,x,non_constant,prod_constant,false,contextptr);
if (non_constant.size()==0)
u=1;
if (non_constant.size()==1)
u=non_constant.front();
if (non_constant.size()>1)
u=symbolic(at_prod,gen(non_constant,_SEQ__VECT));
return prod_constant;
}
// applies linearity of f. + & neg are distributed as well as * with respect
// to terms that are constant w.r.t. x
// e is assumed to be a scalar
gen linear_apply(const gen & e,const gen & x,gen & remains, int intmode,GIAC_CONTEXT, gen (* f)(const gen &,const gen &,gen &,int,const context *)){
if (is_constant_wrt(e,x,contextptr) || (e==x) )
return f(e,x,remains,intmode,contextptr);
// e must be of type _SYMB
if (e.type==_VECT){
vecteur v(*e._VECTptr);
vecteur r(v.size());
for (unsigned i=0;i<v.size();++i){
v[i]=linear_apply(v[i],x,r[i],intmode,contextptr,f);
}
remains=r;
return gen(v,e.subtype);
}
if (e.type!=_SYMB) return gensizeerr(gettext("in linear_apply"));
unary_function_ptr u(e._SYMBptr->sommet);
gen arg(e._SYMBptr->feuille);
gen res;
if (u==at_neg){
res=-linear_apply(arg,x,remains,intmode,contextptr,f);
remains=-remains;
return res;
} // end at_neg
if (u==at_plus){
if (arg.type!=_VECT)
return linear_apply(arg,x,remains,intmode,contextptr,f);
const_iterateur it=arg._VECTptr->begin(),itend=arg._VECTptr->end();
for (gen tmp;it!=itend;++it){
res = res + linear_apply(*it,x,tmp,intmode,contextptr,f);
remains =remains + tmp;
}
return res;
} // end at_plus
if (u==at_prod){
if (arg.type!=_VECT)
return linear_apply(arg,x,remains,intmode,contextptr,f);
// find all constant terms in the product
vecteur non_constant;
gen prod_constant;
decompose_prod(*arg._VECTptr,x,non_constant,prod_constant,false,contextptr);
if (non_constant.empty()) return gensizeerr(gettext("in linear_apply 2")); // otherwise the product would be constant
if (non_constant.size()==1)
res = linear_apply(non_constant.front(),x,remains,intmode,contextptr,f);
else
res = f(symbolic(at_prod,gen(non_constant,_SEQ__VECT)),x,remains,intmode,contextptr);
remains = prod_constant * remains;
return prod_constant * res;
} // end at_prod
return f(e,x,remains,intmode,contextptr);
}
gen lnabs(const gen & x,GIAC_CONTEXT){
bool _lnabs=do_lnabs(contextptr);
if (!complex_mode(contextptr) && _lnabs && !has_i(x))
return ln(abs(x,contextptr),contextptr);
else
return ln(x,contextptr);
}
gen lnabs2(const gen & x,const gen & xvar,GIAC_CONTEXT){
if (xvar.type!=_IDNT)
return lnabs(x,contextptr);
bool _lnabs=do_lnabs(contextptr);
if (!complex_mode(contextptr) && _lnabs && !has_i(x)){
return symbolic(at_ln,symbolic(at_abs,x));
}
else {
if (is_positive(-x,contextptr))
return symbolic(at_ln,-x);
return symbolic(at_ln,x);
}
}
static gen normal_norootof(const gen & g,GIAC_CONTEXT){
gen res=normal(g,contextptr);
if (!lop(res,at_rootof).empty())
res=ratnormal(normalize_sqrt(g,contextptr),contextptr);
return res;
}
// eval N at X=e with e=x*exp(i*dephasage*pi/n) and returns N*ln(X-e)+conj
static gen substconj_(const gen & N,const gen & X,const gen & x,const gen & dephasage_,bool residue_only,GIAC_CONTEXT){
int mode=angle_mode(contextptr);
gen pi=cst_pi;
gen dephasage(dephasage_);
if (mode==1){
dephasage=ratnormal(gen(180)/cst_pi*dephasage,contextptr);
pi=180;
}
if (mode==2){
dephasage=ratnormal(gen(200)/cst_pi*dephasage,contextptr);
pi=200;
}
gen c=cos(dephasage,contextptr);
gen s=sin(dephasage,contextptr);
if (c.is_symb_of_sommet(at_cos) && c._SYMBptr->feuille==dephasage){
gen c2=cos(ratnormal(2*dephasage,contextptr),contextptr);
if (!c2.is_symb_of_sommet(at_cos)){
c=sign(c,contextptr)*sqrt((1+c2)/2,contextptr);
s=sign(s,contextptr)*sqrt((1-c2)/2,contextptr);
}
}
gen e=x*(c+cst_i*s);
gen b=subst(N,X,e,false,contextptr),rb,ib;
reim(b,rb,ib,contextptr);
gen N2=normal_norootof(-2*ib,contextptr); // same
if (residue_only)
return N2*sign(s*x,contextptr);
gen res=normal_norootof(rb,contextptr)*symbolic(at_ln,pow(X,2)+ratnormal(-2*c*x,contextptr)*X+x.squarenorm(contextptr));
gen atanterm=pi/cst_pi*symbolic(at_atan,(X-c*x)/(s*x));
if (X.is_symb_of_sommet(at_tan))
atanterm += pi*sign(s*x,contextptr)*symbolic(at_floor,X._SYMBptr->feuille/pi+plus_one_half);
res=res+N2*atanterm;
return res;
}
static gen substconj(const gen & N,const gen & X,const gen & x,const gen & dephasage,bool residue_only,GIAC_CONTEXT){
if (has_i(N)){
gen Nr,Ni;
reim(N,Nr,Ni,contextptr);
return substconj_(Nr,X,x,dephasage,residue_only,contextptr)+cst_i*substconj_(Ni,X,x,dephasage,residue_only,contextptr);
}
return substconj_(N,X,x,dephasage,residue_only,contextptr);
}
gen surd(const gen & c,int n,GIAC_CONTEXT){
if (is_exactly_zero(c))
return c;
if (n%2 && is_positive(-c,contextptr)){
if (c.type==_FLOAT_)
return -exp(ln(-c,contextptr)/n,contextptr);
return -pow(-c,inv(n,contextptr),contextptr);
}
else {
if (c.type==_FLOAT_)
return exp(ln(c,contextptr)/n,contextptr);
return pow(c,inv(n,contextptr),contextptr);
}
}
gen _surd(const gen & args,GIAC_CONTEXT){
if ( args.type==_STRNG && args.subtype==-1) return args;
if (args.type!=_VECT || args._VECTptr->size()!=2)
return gensizeerr(contextptr);
gen a=args._VECTptr->front(),aa,b=args._VECTptr->back(),c;
if (a.is_symb_of_sommet(at_abs) || a.is_symb_of_sommet(at_exp))
return pow(a,inv(b,contextptr),contextptr);
if (is_equal(a)){
gen a0=a._SYMBptr->feuille[0],a1=a._SYMBptr->feuille[1];
return symbolic(at_equal,makesequence(_surd(makesequence(a0,b),contextptr),_surd(makesequence(a1,b),contextptr)));
}
if (is_undef(a)) return a;
if (is_undef(b)) return b;
if (is_inf(b)){
if (is_inf(a) || is_zero(a))
return undef;
return 1;
}
if (is_zero(b))
return undef;
if (is_inf(a))
return pow(a,inv(b,contextptr),contextptr);
c=_floor(b,contextptr);
if (c.type==_FLOAT_)
c=get_int(c._FLOAT_val);
if (!has_evalf(a,aa,1,contextptr)){
if (c.type==_INT_ && c==b && (c.val %2 ==0 || (a.is_symb_of_sommet(at_pow) && a._SYMBptr->feuille[1].type==_INT_ && a._SYMBptr->feuille[1].val % c.val==0)) )
return pow(a,inv(c,contextptr),contextptr);
return symbolic(at_NTHROOT,gen(makevecteur(b,a),_SEQ__VECT));
}
if (c.type==_INT_ && c==b)
return surd(a,c.val,contextptr);
else
return pow(a,inv(b,contextptr),contextptr);
}
static const char _surd_s []="surd";
static define_unary_function_eval (__surd,&_surd,_surd_s);
define_unary_function_ptr5( at_surd ,alias_at_surd,&__surd,0,true);
static gen makelnatan(const gen & N,const gen & X,const gen & c0,int n,bool residue_only,GIAC_CONTEXT){
gen c(c0),res(0);
if (n%2){
if (is_positive(-c,contextptr))
c=-pow(-c,inv(n,contextptr),contextptr);
else
c=pow(c,inv(n,contextptr),contextptr);
if (!residue_only)
res += subst(N,X,c,false,contextptr)*lnabs(X-c,contextptr);
for (int i=1;i<=n/2;++i)
res += substconj(N,X,c,gen(2*i)/n*cst_pi,residue_only,contextptr);
return res;
}
if (is_positive(c,contextptr) ){
if (n==2)
c=sqrt(c,contextptr);
else
c=pow(c,inv(n,contextptr),contextptr);
if (!residue_only)
res += normal_norootof(subst(N,X,c,false,contextptr),contextptr)*lnabs2(X-c,X,contextptr)+normal_norootof(subst(N,X,-c,false,contextptr),contextptr)*lnabs2(X+c,X,contextptr);
for (int i=1;i<n/2;++i)
res += substconj(N,X,c,gen(2*i)/n*cst_pi,residue_only,contextptr);
}
else {
if (n==2)
c=sqrt(-c,contextptr);
else
c=pow(-c,inv(n,contextptr),contextptr);
for (int i=0;i<n/2;++i)
res += substconj(N,X,c,gen(2*i+1)/n*cst_pi,residue_only,contextptr);
}
return res;
}
gen symb_atan(const polynome & d_,const polynome & a_,const vecteur & l,GIAC_CONTEXT){
polynome d(d_),a(a_);
simplify(d,a);
if (a.coord.empty())
return 0;
gen D=r2e(d,l,contextptr);
if (is_positive(-D*a.coord.front().value,contextptr))
return -symb_atan(r2e(-a,l,contextptr)/D);
return symb_atan(r2e(a,l,contextptr)/D);
}
// im(ln(a+i*b)), a and b polynomials rewritten as sum of atan without denominators
// im(ln(a+i*b)+ln(u-i*v))=im(ln(a*u+b*v)+i(b*u-a*v))
gen ln2sumatan(const polynome & a,const polynome & b,const vecteur & l,GIAC_CONTEXT){
if (a.lexsorted_degree()>b.lexsorted_degree())
return -ln2sumatan(b,a,l,contextptr);
polynome u,v,d;
egcd(a,b,u,v,d);
if (v.coord.empty()){ // a divides b
return symb_atan(a,b,l,contextptr);
}
gen tmp=-ln2sumatan(v,u,l,contextptr);
tmp += symb_atan(d,b*u-a*v,l,contextptr);
return tmp;
}
gen ln2sumatan(const gen & a,const gen & b,const vecteur & l,GIAC_CONTEXT){
//return symb_atan(b/a);
gen A=e2r(a,l,contextptr),An,Ad;
gen B=e2r(b,l,contextptr),Bn,Bd;
fxnd(A,An,Ad);
fxnd(B,Bn,Bd);
An=Bd*An;
Bn=Ad*Bn;
if (An.type==_POLY && Bn.type==_POLY)
return ln2sumatan(*An._POLYptr,*Bn._POLYptr,l,contextptr);
if (Bn.type!=_POLY)
return -symb_atan(a/b);
return symb_atan(b/a);
}
static bool integrate_rothstein_trager(const polynome & num,const vecteur & v,const vecteur & l,const gen & X,gen & res,int intmode,GIAC_CONTEXT){
// Improve: csolve for resultant(num-t*v',v)
// Example a:=diff(atan((x^2-2x)/(x-1))); b:=int(a);
// v=[1,-4,5,-2,1], roots for resultant +/-i/2
// sum t*ln(gcd(n-t*d',d))
// if t is complex and v real
// t*ln()+conjugate=re(t)*ln(|gcd|^2)-im(t)*atan(im(gcd)/re(gcd))
gen N=r2e(num,l,contextptr);
vecteur Nv(lvar(N));
if (1 || Nv==vecteur(1,X)){ // [commented: do it for univariate only]
gen D=r2e(poly12polynome(v,1),l,contextptr),resadd;
#if 0
gen Dprime=r2e(poly12polynome(derivative(v),1),l,contextptr);
gen Dc=1;
#else
gen Dc=_content(makesequence(D,X),contextptr);
D=_quo(makesequence(D,Dc,X),contextptr);
gen Dprime=derive(D,X,contextptr);
#endif
int Ddeg=v.size()-1;
gen tres(identificateur("tresultant"));
gen R=_resultant(makesequence(N-tres*Dprime,D,X),contextptr);
gen Rprime=derive(R,tres,contextptr);
R=_quo(makesequence(R,gcd(R,Rprime,contextptr),tres),contextptr);
gen Rdeg=_degree(makesequence(R,tres),contextptr);
if (Rdeg.type==_INT_ && Rdeg.val==Ddeg){
// it's easier to extract the roots of D
gen racines=solve(D,X,1,contextptr);
if (!has_i(racines) && racines.type==_VECT && racines._VECTptr->size()==Ddeg){
// apply sum_racines N/D'(racine)*log(x-racine)
gen ND=N/Dprime;
for (int i=0;i<Ddeg;++i){
gen racine=racines[i];
//if (has_op(normal(racine,contextptr),*at_rootof)) return false;
gen coeff=subst(ND,X,racine,false,contextptr);
resadd += coeff*symb_ln(X-racine);
}
res += resadd/Dc;
return true;
}
} // end Rdeg==Ddeg
gen Rt=solve(R,tres,1,contextptr); // _cSolve(makesequence(R,tres),contextptr);
if (Rdeg.type==_INT_ && Rt.type==_VECT && Rt._VECTptr->size()==Rdeg.val){
vecteur w=*Rt._VECTptr;
bool reel=vect_is_real(v,contextptr);
if (!has_num_coeff(w)){
for (size_t wi=0;wi<w.size();++wi){
gen racine=w[wi],racinen=normal(racine,contextptr);
if (has_op(racinen,*at_rootof))
return false;
gen G;
#ifndef NO_STDEXCEPT
try {
#endif
G=_numer(gcd(N-racine*Dprime,D,contextptr),contextptr);
#ifndef NO_STDEXCEPT
}
catch (std::runtime_error & err){
return false;
}
#endif
if (reel){
gen racr,raci;
reim(racine,racr,raci,contextptr);
if (is_zero(raci,contextptr))
resadd += racine*symb_ln(symb_abs(G));
else {
// search conjugate
size_t wj=wi+1; gen cwi=conj(w[wi],contextptr);
for (;wj<w.size();++wj){
if (is_zero(ratnormal(cwi-w[wj],contextptr)))
break;
}
if (wj<w.size()){
gen gcdr,gcdi;
reim(G,gcdr,gcdi,contextptr);
resadd += racr*symb_ln(gcdr*gcdr+gcdi*gcdi)-2*raci*ln2sumatan(gcdr,gcdi,l,contextptr);
w.erase(w.begin()+wj);
}
else
resadd += racine*symb_ln(G);
}
}
else {
resadd += racine*symb_ln(G);
}
}
res += resadd/Dc;
return true;
}
}
}
return false;
}
// integration of cyclotomic-type denominators
static bool integrate_deno_length_2(const polynome & num,const vecteur & v,const vecteur & l,const vecteur & lprime,gen & res,bool residue_only,int intmode,GIAC_CONTEXT){
if (v.size()<2)
return false;
const_iterateur it=v.begin()+1,itend=v.end()-1;
for (;it!=itend;++it){
if (!is_zero(*it))
break;
}
int n=int(v.size())-1,d=int(it-v.begin()),deg;
gen X=l.front();
if (X.type==_VECT)
X=X._VECTptr->front();
gen a=r2e(v.front(),lprime,contextptr);
gen b=r2e(v.back(),lprime,contextptr);
// check for deno of type a*x^2n + A*x^n + b
// FIXME: improve some simplifications of sin/cos(asin()/k) and remove test d==2
if (d==2 && 2*d==n){
++it;
for (;it!=itend;++it){
if (!is_zero(*it))
break;
}
if (it==itend){ // ok!
gen c=b;
b=r2e(v[d],lprime,contextptr);
gen delta=b*b-4*a*c;
if (is_zero(delta)) // if (is_positive(-delta,contextptr))
return false;
if ( (intmode &2)==0)
gprintf(step_ratfrac,gettext("Integration of a rational fraction with denominator %gen\nroots are obtained by solving the 2nd order equation %gen=0 then extracting nth-roots"),makevecteur(a*symb_pow(vx_var,2*n)+b*symb_pow(vx_var,n)+c,a*symb_pow(vx_var,2)+b*vx_var+c),contextptr);
// int(num/(a*X^2d+b*X^d+c),X) =
// sum(x=rootof(deno),num*x/(+/-d*sqrt(delta))*ln(X-x))
gen sqrtdelta=sqrt(delta,contextptr);
gen c1=(-b-sqrtdelta)/2/a;
gen c2=(-b+sqrtdelta)/2/a;
gen N=r2e(num,l,contextptr)*X/d/sqrtdelta;
if (is_zero(im(a,contextptr)) && is_zero(im(b,contextptr)) && is_zero(im(c,contextptr))){
if (!is_positive(-delta,contextptr)){
res += makelnatan(N/c2,X,c2,d,residue_only,contextptr);
res -= makelnatan(N/c1,X,c1,d,residue_only,contextptr);
return true;
}
else {
gen module=sqrt(c/a,contextptr);
gen argument=acos(normal(-b/a/2/module,contextptr),contextptr);
// roots are module^(1/d)*exp(i*argument/d)*exp(2*i*pi*k/d)
// for k=0..d-1 and conjugates
gen moduled=pow(c/a,inv(n,contextptr),contextptr);
for (int i=0;i<d;++i)
res += substconj_(N/c2,X,moduled,(argument+2*i*cst_pi)/d,residue_only,contextptr);
}
return true;
}
if (residue_only)
return true;
gen c1s=surd(c1,d,contextptr);
gen c2s=surd(c2,d,contextptr);
for (int i=0;i<d;++i){
gen x=c1s*exp((2*i*cst_i*cst_pi)/d,contextptr);
res -= normal(subst(N,X,x,false,contextptr)/c1,contextptr)*ln(X-x,contextptr);
x=c2s*exp((2*i*cst_i*cst_pi)/d,contextptr);
res += normal(subst(N,X,x,false,contextptr)/c2,contextptr)*ln(X-x,contextptr);
}
return true;
}
} // end if d==2 and n==2d
gen c=normal(-b/a,contextptr);
if (n%d)
return residue_only?false:integrate_rothstein_trager(num,v,l,X,res,intmode,contextptr);
if (d!=n){
// rescale and check cyclotomic
gen tw=v.back()/pow(*it/v.front(),n/d);
if (tw.type!=_INT_ && tw.type!=_POLY)
return residue_only?false:integrate_rothstein_trager(num,v,l,X,res,intmode,contextptr);
tw=r2e(v/v.front(),lprime,contextptr);
if (tw.type!=_VECT)
return residue_only?false:integrate_rothstein_trager(num,v,l,X,res,intmode,contextptr);
vecteur w=*tw._VECTptr;
vecteur w_copy=w;
c=pow(r2e(*it/v.front(),lprime,contextptr),inv(d,contextptr),contextptr);
iterateur jt=w.begin()+1,jtend=w.end();
for (int k=1;jt!=jtend;++jt,++k){
*jt=normal(*jt * pow(c,-k),contextptr);
if (jt->type!=_INT_ && jt->type!=_POLY)
break;
}
deg=is_cyclotomic(w,epsilon(contextptr));
if (!deg){
w=w_copy;
c=pow(r2e(-*it/v.front(),lprime,contextptr),inv(d,contextptr),contextptr);
jt=w.begin()+1,jtend=w.end();
for (int k=1;jt!=jtend;++jt,++k){
*jt=normal(*jt * pow(c,-k),contextptr);
if (jt->type!=_INT_ && jt->type!=_POLY)
break;
}
deg=is_cyclotomic(w,epsilon(contextptr));
}
if (!deg)
return residue_only?false:integrate_rothstein_trager(num,v,l,X,res,intmode,contextptr);
if ( (intmode &2)==0)
gprintf(step_cyclotomic,gettext("Integrate rational fraction with denominator a cyclotomic polynomial, roots are primitive roots of %gen=0"),makevecteur(a*symb_pow(vx_var,deg)+b),contextptr);
// int(num/(a*X^n+b),X)=sum(x=rootof(-b/a),num*x/(-n*b)*ln(X-x))
vecteur vprime=derivative(v),V,Vprime,d;
egcd(v,vprime,0,V,Vprime,d);
if (d.size()!=1)
return residue_only?false:integrate_rothstein_trager(num,v,l,X,res,intmode,contextptr);
gen dd=d.front();
// 1/vprime=Vprime/d
gen N=normal(_quorem(makesequence(r2e(num,l,contextptr)*horner(r2e(Vprime,lprime,contextptr),X),horner(r2e(v,lprime,contextptr),X),X),contextptr)[1]/r2e(dd,lprime,contextptr),contextptr);
if (complex_mode(contextptr) && !residue_only){
for (int i=1;i<deg;++i){
if (gcd(i,deg)!=1)
continue;
gen x=c*exp((2*i*cst_i*cst_pi)/deg,contextptr);
res += normal(subst(N,X,x,false,contextptr),contextptr)*ln(X-x,contextptr);
}
}
else {
for (int i=1;i<=deg/2;++i){
if (gcd(i,deg)!=1)
continue;
res += substconj(N,X,c,2*i*cst_pi/deg,residue_only,contextptr);
}
}
return true;
} // if (d!=n)
else {
if ( (intmode &2)==0)
gprintf(step_nthroot,gettext("Integrate rational fraction with denominator %gen=0\nroots are deduced from nth-roots of unity"),makevecteur(a*symb_pow(vx_var,n)+b),contextptr);
// int(num/(a*X^n+b),X)=sum(x=rootof(-b/a),num*x/(-n*b)*ln(X-x))
gen N=r2e(num,l,contextptr)*X/(r2e(-n*b,l,contextptr));
if (complex_mode(contextptr) && !residue_only){
c=pow(c,inv(n,contextptr),contextptr);
for (int i=0;i<n;++i){
gen x=c*exp((2*i*cst_i*cst_pi)/n,contextptr);
res += normal(subst(N,X,x,false,contextptr),contextptr)*ln(X-x,contextptr);
}
return true;
}
res += makelnatan(N,X,c,n,residue_only,contextptr);
return true;
}
}
// tests if v is symmetric or antisymmetric
// if it is, compute res such that res[t+-1/t]=v/t^[deg(v)/2]
static int is_symmetric(const vecteur & v,vecteur & res,bool sym){
if (v.empty())
return 0;
int n=int(v.size());
vecteur w;
if (n%2)
w=v;
else {
if (!is_zero(v[n-1]))
return 0;
w=vecteur(v.begin(),v.end()-1);
--n;
}
vecteur w1(w);
reverse(w1.begin(),w1.end());
if (!sym){
for (int i=1;i<n;i+=2){
w1[i] = -w1[i];
}
}
int rescoeff=0;
if (w==w1)
rescoeff=1;
if (w==-w1)
rescoeff=-1;
if (!rescoeff)
return 0;
// if antisym, n/2 is the number of power of (t^2-1), check if it is odd
if (!sym && (n/2)%2)
rescoeff = -rescoeff;
vecteur test(makevecteur(1,0,sym?1:-1)),q,r;
res.clear();
for (n/=2;n>0;n--){
DivRem(w,powmod(test,n,0,0),0,q,r);
if (q.size()>1)
return 0;
w=r.empty()?r:vecteur(r.begin(),r.end()-1);
res.push_back(q.empty()?0:q.front());
}
if (w.empty())
res.push_back(0); // was return 0;
else
res.push_back(w.front());
return rescoeff;
}
// n/d(x) -> newn/newd(t) with x=a/t,
// if dx is true multiplies by dx/dt=-a/t^2
static void xtoinvx(const gen & a,const modpoly & n,const modpoly & d,modpoly & newn, modpoly & newd,bool dx){
int ns=int(n.size()); int nd=int(d.size());
newn=vecteur(ns); newd=vecteur(nd);
gen ad(1);
for (int i=ns-1;i>=0;--i){
newn[ns-1-i]=ad*n[i];
ad = ad*a;
}
ad=1;
for (int i=nd-1;i>=0;--i){
newd[nd-1-i]=ad*d[i];
ad = ad*a;
}
if (dx){
newn=operator_times(-a,newn,0);
ns+=2;
}
trim(newn,0);
trim(newd,0);
for (;ns>nd;--ns){
newd.push_back(0);
}
for (;nd>ns;--nd){
newn.push_back(0);
}
}
static gen integrate_rational(const gen & e, const gen & x, gen & remains_to_integrate,gen & xvar,int intmode,GIAC_CONTEXT);
static void solve_aPprime_plus_P(const gen & anum,const gen & aden,const vecteur & Q,vecteur & R,gen & Pden){
// a P+P'=Q, a=anum/aden, on cherche P sous la forme R/Pden
// On a (k+1)p_(k+1)+ anum/aden*p_k=q_k
// Donc p_k=aden/anum*(q_k-(k+1)p_(k+1))
// n=deg[Q], on a donc Pden=anum^(n+1), puis on cherche R=P*anum^(n+1)
// on multiplie donc Q par anum^(n+1) S=Q*anum^(n+1)/a
// on a aR+R'=aS
// on a donc par ordre decr. r_(n+1)=0
// r_k= s_k - (k+1)*r_(k+1)/a
// avec des divisions sans creation de denominateurs
// par ex. P'+3P=x^2+5x+7 -> r_2=9, r_1=39, r_0=50, a=3, n=2, a^n=9
R.clear();
if (Q.empty()){
Pden=plus_one;
return;
}
int n=int(Q.size())-1;
R.reserve(n+1);
Pden=pow(anum,n);
vecteur S;
multvecteur(Pden*aden,Q,S);
Pden=Pden*anum;
const_iterateur it=S.begin(),itend=S.end();
R.push_back(*it);
++it;
for (int k=n-1;it!=itend;++it,--k){
R.push_back(*it-rdiv(gen(k+1)*R.back()*aden,anum,context0));
}
// should simplify R with Pden
}
static gen integrate_linearizable(const gen & e,const gen & gen_x,gen & remains_to_integrate,int intmode,bool do_risch,GIAC_CONTEXT){
// exp linearization
vecteur vexp;
gen res;
const identificateur & id_x=*gen_x._IDNTptr;
lin(e,vexp,contextptr); // vexp = coeff, arg of exponential
if ( (intmode &2)==0 ){
gen tmp=unlin(vexp,contextptr);
if (vexp.size()>2 || !is_zero(ratnormal(tmp-e,contextptr)))
gprintf(step_linearizable,gettext("Integrate linearizable expression %gen -> %gen"),makevecteur(e,tmp),contextptr);
}
const_iterateur it=vexp.begin(),itend=vexp.end();
for (;it!=itend;){
// trig linearization
vecteur vtrig;
gen coeff=*it;
++it; // it -> on the arg of the exp that must be linear
gen rex2,rea,reb,reaxb=*it;
++it;
if (!is_quadratic_wrt(reaxb,gen_x,rex2,rea,reb,contextptr)){
// IMPROVE using int(exp(-x^a))=1/a*igamma(1/a,x^a)
vecteur lv=lvarxwithinv(makevecteur(reaxb,coeff),gen_x,contextptr);
if (lv.size()==1){
gen C=_coeff(makesequence(reaxb,gen_x),contextptr);
if (C.type==_VECT && C._VECTptr->size()>2){
vecteur Cv=*C._VECTptr;
int n=int(Cv.size())-1;
gen c=Cv[0];
gen a=-Cv[1]/(n*c);
// must be c*(x-a)^n
if (C==_coeff(makesequence(c*pow(gen_x-a,n,contextptr),gen_x),contextptr) && ((n%2) || is_positive(-c,contextptr))){
// c=surd(c,n,contextptr);
C=_coeff(makesequence(coeff,gen_x),contextptr);
C=_ptayl(makesequence(C,a,gen_x),contextptr);
if (C.type==_VECT){
c=-c;
gen ca=surd(c,n,contextptr);
Cv=*C._VECTptr;
int m=int(Cv.size())-1;
gen ires=0;
// 1/n*igamma(1/n+b/n,c*x^n)'=x^b*exp(-c*x^n)*c^(b+1)/n
for (int b=0;b<=m;++b){
ires += Cv[m-b]*_lower_incomplete_gamma(makesequence(gen(b+1)/gen(n),c*pow(gen_x-a,n)),contextptr)/pow(ca,b+1,contextptr);
}
if (n%2==0){
ires=ires*abs(gen_x,contextptr)/gen_x; // sign(gen_x,contextptr);
}
ires=ires/n;
res += ires;
continue;
}
}
}
}
remains_to_integrate = remains_to_integrate + coeff*exp(reaxb,contextptr);
continue;
}
if (!is_zero(rex2)){
if (1
//&&is_zero(im(rex2,contextptr))
//&& is_positive(-rex2,contextptr)
){
const vecteur & vx2=lvarxpow(coeff,gen_x);
if ( vx2.size()>1 || (!vx2.empty() && vx2.front()!=gen_x) ){
remains_to_integrate = remains_to_integrate + coeff*exp(reaxb,contextptr);
continue;
}
// int(exp(rex2*x^2+rea*x+reb)*P(x),x)
gen decal=rea/rex2/2;
gen cst=normal(reb-rex2*decal*decal,contextptr);
// exp(cst)*int(exp(rex2*(x+decal)^2)*P(x),x)
coeff=quotesubst(coeff,gen_x,gen_x-decal,contextptr);
// exp(cst)*subst(int(exp(rex2*x^2)*coeff(x),x),x,x+decal)
vecteur les_var(1,gen_x); // insure x is the main var
lvar(makevecteur(coeff,rex2),les_var);
int les_vars=int(les_var.size());
gen in_coeff,in_coeffnum,in_coeffden,ina;
in_coeff=e2r(coeff,les_var,contextptr);
ina=e2r(rex2,vecteur(les_var.begin()+1,les_var.end()),contextptr);
fxnd(in_coeff,in_coeffnum,in_coeffden);
vecteur in_coeffnumv;
if (in_coeffnum.type==_POLY)
in_coeffnumv=polynome2poly1(*in_coeffnum._POLYptr,1);
else
in_coeffnumv.push_back(in_coeffnum);
// now find int(exp(ina*x^2)*P(x)), coeffs of P are in in_coeffnumv
int vs=int(in_coeffnumv.size())-1;
vecteur vres(vs+1);
// integration by part to decrease vs
for (int i=vs;i>=1;--i){
// i is the degree of the term to integrate
gen tmp=in_coeffnumv[vs-i]/ina/2;
vres[vs-(i-1)]=tmp;
if (i>1)
in_coeffnumv[vs-(i-2)] -= (i-1)*tmp;
}
gen vresden;
lcmdeno(vres,vresden,contextptr); // lcmdeno_converted?
gen ppart=subst(r2e(poly12polynome(vres,1,les_vars),les_var,contextptr),gen_x,gen_x+decal,false,contextptr)/r2e(vresden,vecteur(les_var.begin()+1,les_var.end()),contextptr)*exp(reaxb,contextptr);
// add erf part from the last coeff vres[vs]
gen a=-rex2; // r2e(-ina,les_var,contextptr);
gen sqrta=sqrt_noabs(a,contextptr);
gen erfpart=r2e(in_coeffnumv[vs],cdr_VECT(les_var),contextptr)*symbolic(at_sqrt,cst_pi)/sqrta*exp(cst,contextptr)/2*_erf(sqrta*(gen_x+decal),contextptr);
res += (ppart + erfpart)/r2e(in_coeffden,les_var,contextptr);
continue;
}
remains_to_integrate = remains_to_integrate + coeff*exp(reaxb,contextptr);
continue;
}
gen reai=im(rea,contextptr),rebi=im(reb,contextptr);
if (!is_zero(reai) || !is_zero(rebi)){
gen reaxbi=reai*gen_x+rebi;
coeff=coeff*(cos(reaxbi,contextptr)+cst_i*sin(reaxbi,contextptr));
rea=re(rea,contextptr);
reb=re(reb,contextptr);
reaxb=rea*gen_x+reb;
}
tlin(coeff,vtrig,contextptr); // vtrig = coeff , sin/cos(arg)/1
if ( (intmode &2)==0 ){
gen tmp=tunlin(vtrig,contextptr);
if (vtrig.size()>2 || !is_zero(ratnormal(tmp-coeff,contextptr)))
gprintf(step_triglinearizable,gettext("Integrate trigonometric linearizable expression %gen -> %gen"),makevecteur(coeff,tmp),contextptr);
}
const_iterateur jt=vtrig.begin(),jtend=vtrig.end();
for (;jt!=jtend;){
// now check that each arg is linear and coeff polynomial
coeff=*jt;
++jt;
gen ima,imb,imaxb=*jt;
++jt;
if (is_constant_wrt(imaxb,gen_x,contextptr)){
coeff = coeff*imaxb;
imaxb=1;
}
int trig_type=0; // 0 for 1, 1 for sin, 2 for cos