+ ikRt^RIHt^RIHt^RIHt^RIHt^RI H t H t H t H t ^RIHtHt^RIHt^RIHtHtHtHtHtHtHtR tR tRR ltR tRRltRRlt Rt!Rt"Rt#Rt$Rt%Rt&RRlt'Rt(R #)a Algorithms for solving the Risch differential equation. Given a differential field K of characteristic 0 that is a simple monomial extension of a base field k and f, g in K, the Risch Differential Equation problem is to decide if there exist y in K such that Dy + f*y == g and to find one if there are some. If t is a monomial over k and the coefficients of f and g are in k(t), then y is in k(t), and the outline of the algorithm here is given as: 1. Compute the normal part n of the denominator of y. The problem is then reduced to finding y' in k, where y == y'/n. 2. Compute the special part s of the denominator of y. The problem is then reduced to finding y'' in k[t], where y == y''/(n*s) 3. Bound the degree of y''. 4. Reduce the equation Dy + f*y == g to a similar equation with f, g in k[t]. 5. Find the solutions in k[t] of bounded degree of the reduced equation. See Chapter 6 of "Symbolic Integration I: Transcendental Functions" by Manuel Bronstein. See also the docstring of risch.py. )mul)reduce)oo)Dummy)PolygcdZZcancel)imre)sqrt)gcdex_diophantinefrac_in derivation splitfactorNonElementaryIntegralExceptionDecrementLevelrecognize_log_derivativecXVP'd\#V\W"48Xd.VPV4P 4^,^,#.pTpVP V4p^pVP'd7VP WF34WD,pV^,pVP V4pKH^p\^V4p\V4^8wdXVP4p W^,,p VP V 4pVP'gKTWy^,, pT pKgV#)a9 Computes the order of a at p, with respect to t. Explanation =========== For a, p in k[t], the order of a at p is defined as nu_p(a) = max({n in Z+ such that p**n|a}), where a != 0. If a == 0, nu_p(a) = +oo. To compute the order at a rational function, a/b, use the fact that nu_p(a/b) == nu_p(a) - nu_p(b). ) is_zerorras_polyETremappendlenpop) apt power_listp1r tracks_powernproductfinalproductfs &&& s/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/integrals/rde.pyorder_atr()s yyy DJyy| #A&& J B b AL )))2+, U EE"I A1ajG j/Q  8# EE(O 999 qMAG HcVP'd\#VPV4VPV4, #)z Computes the order of a/d at oo (infinity), with respect to t. For f in k(t), the order or f at oo is defined as deg(d) - deg(a), where f == a/d. )rrdegree)rdrs&&&r' order_at_oor-Ts- yyy 88A;! $$r)NcT;'g \R4p\W4wrE\WDPVP44pVP V4pVP \Wv44p\ VP V4PVP4VPVP4VPVP44wrV\W2P4\W4,, PVP4PVPVP44p \W4p V PPV4'g\^VP4W33#V P4U u.uFq\9gKV ^8gKV NK p p \\ V Uu.uF9p\V\WP4\W4,, V4NK; up\^VP44p\W4pW,VV,, pW,pVP#VRR7wppVVV33#uup iuupi)aP Weak normalization. Explanation =========== Given a derivation D on k[t] and f == a/d in k(t), return q in k[t] such that f - Dq/q is weakly normalized with respect to t. f in k(t) is said to be "weakly normalized" with respect to t if residue_p(f) is not a positive integer for any normal irreducible p in k[t] such that f is in R_p (Definition 6.1.1). If f has an elementary integral, this is equivalent to no logarithm of integral(f) whose argument depends on t has a positive integer coefficient, where the arguments of the logarithms not in k(t) are in k[t]. Returns (q, f - Dq/q) zTinclude)rrrdiffrquor rrr resultantexprhas real_rootsrrrr )rr,DEr/dndsg d_sqf_partd1a1br!iNr#qdqsnsds&&&& r'weak_normalizerrF`s( U3ZA  FB B AJ J* +B aeeBi//5rzz"$$7G "$$ EB T!TT]:b- - -66rtt<FF 244 A Q A 66::a==Q v&&LLN8Nq2g!a%NA8sANAqST!TT]:b+===rBAN Q  A A B qtB B YYr4Y (FB Bx= 9Ns I/0I/9I/?I4 cD\W4wrV\W44wrxVPV4p VPVPVP44P V PV PVP444p WZ,p W,p V P V4^,'d\ hW,p V PVRR7wrW,V\W4,V,, pVPVRR7wppWV3W3V 3#)a Normal part of the denominator. Explanation =========== Given a derivation D on k[t] and f, g in k(t) with f weakly normalized with respect to t, either raise NonElementaryIntegralException, in which case the equation Dy + f*y == g has no solution in k(t), or the quadruplet (a, b, c, h) such that a, h in k[t], b, c in k, and for any solution y in k(t) of Dy + f*y == g, q = y*h in k satisfies a*Dq + b*q == c. This constitutes step 1 in the outline given in the rde.py docstring. Tr0) rrr2rr3divrr r)fafdgagdr8r9r:enesrhrccacdbabds&&&&& r' normal_denomrUs  FB  FB r A rwwrtt}!!!%%rtt "56A A AuuRy||,, B YYr4Y (FB :a$$R' 'B YYr4Y (FB Bx"1 %%r)c  VR8Xd VPpVR8Xd"\VPVP4pMVR8Xd0\VP^,^,VP4pMpVR9dXVP4P V4pVP4P V4p WV \^VP43#\ RV,4h\ WVP4\ W'VP4, p \ W7VP4\ WGVP4, p \^V \^V 4, 4p V 'Eg^RIH p VR8XdVPP \VPVP44p\V4;_uu_4\VP^4)VP^4, VP^4, VP4wpp\WP4wppV !VVVVV4pVeVwpppV^8Xd \V V4p RRR4EM VR8XEdVPP \VP^,^,VP44p\V4;_uu_4\\VP\R44)VP\R44, VP\R44, 4VP4wpp\\!VP\R44)VP\R44, VP\R44, 4VP4wpp\WP4wpp\#\^VP4V,VV4'djV !V\\R4VP4,V,VV,,VV,VVV4pVeVwpppV^8Xd \V V4p RRR4\%^V )W, 4pVV,pW|),pVV,pVVP V4,\WP4V,\'Wu4P V4,V,,pVV,V,P V4p TpVWV3# +'giL;i +'giL;i) a Special part of the denominator. Explanation =========== case is one of {'exp', 'tan', 'primitive'} for the hyperexponential, hypertangent, and primitive cases, respectively. For the hyperexponential (resp. hypertangent) case, given a derivation D on k[t] and a in k[t], b, c, in k with Dt/t in k (resp. Dt/(t**2 + 1) in k, sqrt(-1) not in k), a != 0, and gcd(a, t) == 1 (resp. gcd(a, t**2 + 1) == 1), return the quadruplet (A, B, C, 1/h) such that A, B, C, h in k[t] and for any solution q in k of a*Dq + b*q == c, r = qh in k[t] satisfies A*Dr + B*r == C. For ``case == 'primitive'``, k == k[t], so it returns (a, b, c, 1) in this case. This constitutes step 2 of the outline given in the rde.py docstring. autoexptanz@case must be one of {'exp', 'tan', 'primitive', 'base'}, not %s.parametric_log_derivN) primitivebase)caserrto_fieldr3 ValueErrorr(minprder[r,rrevalr r r rmaxr)rrSrTrQrRr8r_rBCnbncr#r[dcoeffalphaaalphadetaaetadAQmr/betaabetadrApNpnrOs&&&&&&& r' special_denomrvs. v~ww u} rtt   q1bdd # & & KKM  b ! KKM  b !aa''!%&' ' " "!6 6B " "!6 6B ArC2JA 2. 5=TTXXd244./F##!("''!*RWWQZ)?q )I244!P$VTT2 d(tRH=GAq!Av1I$#U]TTXXd244719bdd34F##!(RWWT"X->,>rwwtBx?P,PQRQWQWX\]_X`Qa,a)bdfdhdh!i&r27748+<*bound_degree..(s,A"$$s'*r]r\)limited_integratezLength of m should be 1!is_log_deriv_k_t_radical_in_fieldNrXrZzScase must be one of {'exp', 'tan', 'primitive', 'other_nonlinear', 'base'}, not %s.)rYother_nonlinear)r_r+rrer rLCas_expr is_Integerrr,Tlevelrrcr}rrarrrr[r3r)rr?cQr8r_ parametricdadbdcalphar#rmrnt1rkrlr}zazdrqrroaar/betarrrsr[deltalams&&&f&& r' bound_degreer s( v~ww "$$B "$$B ,, , YYrtt_ AIIbddO&&(0022 "$$$$&' (E v~ 2BQ' ( a! !A F H-$#, Hsb2,U"T>1C-U"U2 U  U" U  U" UU"UU"" U3 6 V cH\^VP4p\^VP4p\^VP4pVP'dWU^WW3#V^8RJd\hVP V4pVP V4P'g\hVP V4VP V4VP V4r!pVPVP4^8XdDVP4P V4pVP4P V4pWW6V3#\WV4wrV\W4, pV \W4, pW0PVP4,pWvV ,, pW`,pEKa)az Rothstein's Special Polynomial Differential Equation algorithm. Explanation =========== Given a derivation D on k[t], an integer n and ``a``,``b``,``c`` in k[t] with ``a != 0``, either raise NonElementaryIntegralException, in which case the equation a*Dq + b*q == c has no solution of degree at most ``n`` in k[t], or return the tuple (B, C, m, alpha, beta) such that B, C, alpha, beta in k[t], m in ZZ, and any solution q in k[t] of degree at most n of a*Dq + b*q == c must be of the form q == alpha*h + beta, where h in k[t], deg(h) <= m, and Dh + B*h == C. This constitutes step 4 of the outline given in the rde.py docstring. T) rrrrrrr3r+r`r r) rr?rPr#r8zerorrr;r!r/s &&&&& r'spdersE" 244=D BDDME 244=D  9994. . Ed?0 0 EE!HuuQx0 0%%(AEE!HaeeAha 88BDD>Q    #A   #A!D) ) q) Z  1! ! XXbdd^   r)cx\^VP4pVP'EgVPVP4VPVP4, p^Tu;8:dV8:g\h\h\VP VP4P 4VP VP4P 4, VPV,,VPRR7pWF,pV^, pV\Wc4, W,, pEK#V#)a Poly Risch Differential Equation - No cancellation: deg(b) large enough. Explanation =========== Given a derivation D on k[t], ``n`` either an integer or +oo, and ``b``,``c`` in k[t] with ``b != 0`` and either D == d/dt or deg(b) > max(0, deg(D) - 1), either raise NonElementaryIntegralException, in which case the equation ``Dq + b*q == c`` has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with ``deg(q) < n``. Fexpand)rrrr+rrrrr?rPr#r8rBrqrs&&&& r'no_cancel_b_largers Q Aiii HHRTTNQXXbdd^ +A{{0 00 0 244##%aiio&8&8&::2447BBDD  E E 1! !AC ' Hr)cD\^VP4pVP'EgwV^8Xd^pMLVPVP4VPPVP4, ^,p^Tu;8:dV8:g\ h\ hV^8d\VP VP4P4WSPP VP4P4,, VPV,,VPRR7pEM2VPVP4VPVP48wd\ hVPVP4^8XdgW@P VPVP^, ,4VP VPVP^, ,43#\VP VP4P4VP VP4P4, VPRR7pWF,pV^, pV\Wc4, W,, pEKV#)ah Poly Risch Differential Equation - No cancellation: deg(b) small enough. Explanation =========== Given a derivation D on k[t], ``n`` either an integer or +oo, and ``b``,``c`` in k[t] with deg(b) < deg(D) - 1 and either D == d/dt or deg(D) >= 2, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n, or the tuple (h, b0, c0) such that h in k[t], b0, c0, in k, and for any solution q in k[t] of degree at most n of Dq + bq == c, y == q - h is a solution in k of Dy + b0*y == c0. Fr) rrrr+r,rrrrrrrs&&&& r'no_cancel_b_smallrs Q Aiii 6ARTT!22Q6AA{{0 00 0 q5QYYrtt_'')1TT\\"$$-?-B-B-D+DEbddAgMU$Axx~"$$/44xx~"99RTT"((Q,%78IIbdd288a<0133QYYrtt_'')!))BDD/*<*<*>>A E E 1! !AC ' Hr)c\^VP4p\VPVP4P 4)VP PVP4P 4, 4pVP 'dVP'dTpMRpVP'Eg9\WaPVP4VP PVP4, ^,4p^Tu;8:dV8:g\h\h\WsP PVP4P 4,VPVP4P 4,4pVP'dWGV3#V^8d`\VPVP4P 4V, VPV,,VPRR7p MVPVP4VP PVP4^, 8wd\hVPVP4P 4VPVP4P 4, p WI,pV^, pV\W4, W ,, pEKKV#)ai Poly Risch Differential Equation - No cancellation: deg(b) == deg(D) - 1 Explanation =========== Given a derivation D on k[t] with deg(D) >= 2, n either an integer or +oo, and b, c in k[t] with deg(b) == deg(D) - 1, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n, or the tuple (h, m, C) such that h in k[t], m in ZZ, and C in k[t], and for any solution q in k[t] of degree at most n of Dq + b*q == c, y == q - h is a solution in k[t] of degree at most m of Dy + b*y == C. Frr^) rrr rrr,r is_positiverrer+rr) r?rPr#r8rBlcMrqurs &&&& r'no_cancel_equalrs Q A 244##%%bddll244&8&;&;&== >B }}}  iii 88BDD>BDDKK$559 :A{{0 00 0 1TT\\"$$'**,,qyy/A/A/CC D 999!9  q5QYYrtt_'')!+BDD!G3RTT%HAxx~RTT!2Q!6644IIbddO&&(244););)== E E 1! !AC ' Hr)c ^RIHp\V4;_uu_4\WP4wrVV!WVV4pVeVwr(V^8Xd \ R4hRRR4VP 'dV#W!PVP48d\h\^VP4p VP 'EgVPVP4p W*8d\h\V4;_uu_4\VP4VP4wr\XXWV4wrRRR4\X P4XP4, VPV ,,VPRR7pW, p V ^, pWV,\W4,,pEKV # +'giELt;i +'giL;i)a Poly Risch Differential Equation - Cancellation: Primitive case. Explanation =========== Given a derivation D on k[t], n either an integer or +oo, ``b`` in k, and ``c`` in k[t] with Dt in k and ``b != 0``, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. r~Nz7is_deriv_in_field() is required to solve this problem.Fr)rcrrrrNotImplementedErrorrr+rrrrischDErr)r?rPr#r8rrSrTror/rBrqa2aa2dsarEstms&&&& r'cancel_primitiver.sa8   DD! -bb 9 =DAAv)++,,  yyy88BDD>,, Q Aiii HHRTTN 50 0 B  qttvrtt,HCRSr2FB 2::< ,RTT1W4bdd5I  E sUZ( (( HA   0 s;G97G G  G$ c \^RIHpVPP\ VP VP 44P 4p\V4;_uu_4\WSP 4wrg\WP 4wrV!WWgV4p V eV wrp V ^8Xd \R4hRRR4VP'dV#W!PVP 48d\h\ ^VP 4pVP'EggVPVP 4p W,8d\hVP 4p\V4;_uu_4\WP 4wppVX,XV,\ WP 4,,pVV,p\VP4VP 4wpp\VVVVV4wppRRR4\ XP 4XP 4, VP V ,,VP RR7pVV, pV ^, pWV,\VV4,,pEKyV# +'giEL;i +'giL;i)a Poly Risch Differential Equation - Cancellation: Hyperexponential case. Explanation =========== Given a derivation D on k[t], n either an integer or +oo, ``b`` in k, and ``c`` in k[t] with Dt/t in k and ``b != 0``, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. rZNz6is_deriv_in_field() is required to solve this problem.Fr)rcr[r,r3rrrrrrrr+rrrr)r?rPr#r8r[etarmrnrSrTrorrqr/rBr>a1aa1drrrrErs&&&& r' cancel_expr`s+ $$((4bdd# $ , , .C   S$$' DD! R 8 =GA!Av)+*++  yyy88BDD>,, Q Aiii HHRTTN 50 0 YY[ B  r44(HCd(T#Xd1ddm33Cd(Cqttvrtt,HCS#sC4FB 2::< ,RTT1W4bdd5I S E sUZR( (( HS   8 s!AJ)B J J  J+ cdVP'gVPR8XgUVPVP4\ ^VP PVP4^, 48d$V'd^RIHpV!WW#4#\WW#4#VP'gLVPVP4VP PVP4^, 8EdVPR8Xg+VP PVP4^8dV'd^RIH pV!WW#4#\WW#4p\V\4'dV#Vwrp \V4;_uu_4V PVP4V PVP4rV f \R4hV f \R4h\!WW#4PVP4p RRR4W,#VP PVP4^8EdEVPVP4VP PVP4^, 8XdW PVP4P#4)VP PVP4P#4, 8dVPVP4P#4P$'g \'R4hV'd \)R4h\+WW#4p\V\4'dV#Vwrp \!W W4p W,#VP'd \)R 4hVPR 8Xd V'd \)R 4h\-WW#4#VPR 8Xd V'd \)R 4h\/WW#4#\)RVP,4h +'giTX ,#;i)z Solve a Polynomial Risch Differential Equation with degree bound ``n``. This constitutes step 4 of the outline given in the rde.py docstring. For parametric=False, cQ is c, a Poly; for parametric=True, cQ is Q == [q1, ..., qm], a list of Polys. r])prde_no_cancel_b_large)prde_no_cancel_b_smallNzb0 should be a non-Null valuezc0 should be a non-Null valuezResult should be a numberz0prde_no_cancel_b_equal() is not yet implemented.zWRemaining cases for Poly (P)RDE are not yet implemented (is_deriv_in_field() required).rXzIParametric RDE cancellation hyperexponential case is not yet implemented.r\zBParametric RDE cancellation primitive case is not yet implemented.zBOther Poly (P)RDE cancellation cases are not yet implemented (%s).)rr_r+rrer,rcrrrr isinstancerrrrasolve_poly_rder is_number TypeErrorrrrr)r?rr#r8rrrRrOb0c0yrqrgs&&&&& r'rrs- 999"''V+ HHRTTNSBDDKK$5$9: :  4)!7 7 .. )))qxx~ BDD(9A(== WW "$$++bdd"3q"8  4)!7 7 aQ + a  HIA2##BDD)2::bdd+;B:$%DEE;$%DEE"21199"$$? $5L RTT a AHHRTTNbddkk"$$6G!6K$K 244##%%bddll244&8&;&;&== =yy!!#---78 8 %'   A1 ) a  HGA!qQ+A5L 999%'BC Cww%-/HII!!//K'-/ABB'q55*+:<>GG+DEEe$#q5Ls A9P P/ cR\WV4wpwr\WW#V4wpwrxwrp \WgWW4wrr\WW4p\ WVVV4wrpppVP'dTpM \WVV4pVV,V,W,3# \d \ pL`i;i)a Solve a Risch Differential Equation: Dy + f*y == g. Explanation =========== See the outline in the docstring of rde.py for more information about the procedure used. Either raise NonElementaryIntegralException, in which case there is no solution y in the given differential field, or return y in k(t) satisfying Dy + f*y == g, or raise NotImplementedError, in which case, the algorithms necessary to solve the given Risch Differential Equation have not yet been implemented. ) rFrUrvrrrrrr)rIrJrKrLr8_rrSrTrQrRhnrorfrghsr#rqrrrs&&&&& r'rrs"""-KAx ,RRR @Ax"2rr6KA!  q % aB/A!UDyyy  1B ' !GdNBE ""   s BB&%B&ry)rW)rWF)F))__doc__operatorr functoolsr sympy.corersympy.core.symbolr sympy.polysrrrr $sympy.functions.elementary.complexesr r (sympy.functions.elementary.miscellaneousr sympy.integrals.rischr rrrrrrr(r-rFrUrvrrrrrrrrrr)r'rs.#--99[[[ ( V %0f"&JQht n-^ <+ ^, ^/ d9 xZz'#r)