EdfddlmZmZddlmZddlmZddlm Z ddl m Z ddl m Z ddlmZmZmZddl mZdd lmZmZmZdd lmZmZdd lmZdd lmZdd lmZddl m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&m'Z'ddl m(Z(m)Z)m*Z*m+Z+m,Z,m-Z-ddl m.Z.m/Z/m0Z0m1Z1m2Z2ddl m3Z3m4Z4m5Z5m6Z6ddl m7Z7m8Z8m9Z9m:Z:ddl;mZ>m?Z?m@Z@ddlAmBZBddlCmDZDmEZEddlFmGZGddlHmIZImJZJddlKmLZLddlMmNZNmOZOddlPmQZQddlRmSZSmTZTmUZUmVZVmWZWmXZXddlYmZZZddl[m\Z\ddl]m^Z^ddl_m`Z`dd lambZbdd!lcmdZdd"ZeiZfd#Zg d-d'ZhGd(d)Zid%ajGd*d+Zk d-d,Zld%S).)DictList) permutations)reduce)Add)Basic)Mul)WildDummySymbol)sympify)RationalpiI)EqNe)S)ordered) iterfreeargs)expsincostancotasinatan)logsinhcoshtanhcothasinh)sqrterferfiliEi)besseljbesselybesselibesselk)hankel1hankel2jnyn)Absreimsignarg)LambertW)floorceiling) Piecewise) Heaviside DiracDelta)collect)AndOr)uniq)quogcdlcm factor_listcancelPolynomialError) itermonomials) root_factors)PolyRing) solve_lin_sys)construct_domain) integratecft}||r |jr|jr||n|js|jr3|jD]}|t||z}||n|j r|t|j |z}|j j s^|j j r6||j td|j jzn:|t|j ||hzz}n|jD]}|t||z}|S)a Returns a set of all functional components of the given expression which includes symbols, function applications and compositions and non-integer powers. Fractional powers are collected with minimal, positive exponents. Examples ======== >>> from sympy import cos, sin >>> from sympy.abc import x >>> from sympy.integrals.heurisch import components >>> components(sin(x)*cos(x)**2, x) {x, sin(x), cos(x)} See Also ======== heurisch )sethas_free is_symbolis_commutativeadd is_Function is_Derivativeargs componentsis_Powbaser is_Integer is_Rationalrq)fxresultgs 8lib/python3.11/site-packages/sympy/integrals/heurisch.pyrUrU,s=,UUFzz!}}+ ; +1+ + JJqMMMM ] +ao +V + +*Q*** JJqMMMM X + j++ +F5# 95$9JJqvx157';';;<<<<j22aS88FV + +*Q*** Mc   t|}n#t$rg}|t|<YnwxYwt||krG|t d|t|fzt||kG|d|S)z*get vector of symbols local to this modulez%s%iN)_symbols_cacheKeyErrorlenappendr )namenlsymss r__symbolsri_s%t$ %%%$t% e**q.; eFdCJJ%7788::: e**q.; !9s ))FNc ddlm} m} t|}||s||zSt ||||||||| } t | ts| Sg} t| | D]&} | | | gd|fz } #t$rY#wxYw| s| Stt| } g| |D]&} | | gd|fz #t$rY#wxYwfd| D} | s| St| dkrJg}| D]3}| d|D4| |d|f| z} g}| D]}t |||||||||| }t!d|D}t#d |D}|#t%|||}|||ft|dkr*t ||||||||| |f|dddfg}n,|t ||||||||| dft)|S) a A wrapper around the heurisch integration algorithm. Explanation =========== This method takes the result from heurisch and checks for poles in the denominator. For each of these poles, the integral is reevaluated, and the final integration result is given in terms of a Piecewise. Examples ======== >>> from sympy import cos, symbols >>> from sympy.integrals.heurisch import heurisch, heurisch_wrapper >>> n, x = symbols('n x') >>> heurisch(cos(n*x), x) sin(n*x)/n >>> heurisch_wrapper(cos(n*x), x) Piecewise((sin(n*x)/n, Ne(n, 0)), (x, True)) See Also ======== heurisch r)solvedenomsT)dictexcludecg|]}|v| Srq).0sslns0s r_ z$heurisch_wrapper..s" . . .!q~ .A . . .r`rLc4g|]\}}t||Srqrrrkeyvalues r_ruz$heurisch_wrapper..s$JJJ:33JJJr`c4g|]\}}t||Srqrwrxs r_ruz$heurisch_wrapper..s$GGG URU^^GGGr`c4g|]\}}t||Srq)rrxs r_ruz$heurisch_wrapper..s$III*#ur#u~~IIIr`)sympy.solvers.solversrlrmr rNheurisch isinstancerrNotImplementedErrorlistr>rdextenditemssubsr<r=rJrer8)r[r\rewritehintsmappingsretries degree_offsetunnecessary_permutations _try_heurischrlrmresslnsdeqssub_dictpairsexprcondgenericrts @r_heurisch_wrapperrmsv:43333333 A ::a==s 1a%7M+] < B BBC&& C32C3c*eZdZdZdZdZdZdZdS) BesselTablez~ Derivatives of Bessel functions of orders n and n-1 in terms of each other. See the docstring of DiffCache. ci|_td|_td|_|dS)Nrgz)tabler rgr _create_table)selfs r___init__zBesselTable.__init__s; ss r`c|j|j|j}}}ttt t fD]L}||dz |||||z|z z |dz ||dz |z|z |||z f||<Mt}||dz |||||z|z z |dz ||dz |z|z |||zf||<t}||dz | ||||z|z z |dz ||dz |z|z |||z f||<ttfD]O}||dz ||dz|||z|z z |dz ||dz |z|z |||z f||<PdS)NrL) rrgrr(r)r,r-r*r+r.r/)trrgrr[s r_rzBesselTable._create_tablesgqsAC!q7GW5 5 5A!A#q Aaa1ggIaK/1aa!Qii)AAaGG35E!HH Aac1II!!Aq'' ! +qS!!AaC))OA%!Q/1a QqsAYYJ11Q771,qS!!AaC))OA%!Q/1ab 5 5A!A#q QqS!!Aq''M!O31aa!Qii)AAaGG35E!HH 5 5r`c||jvrL|j|\}}|j|f|j|fg}||||fSdSN)rrgrr)rr[rgrdiff0diff1repls r_diffszBesselTable.diffssb < 871:LE5S!HqsAh'DJJt$$ejj&6&67 7 8 8r`c||jvSr)r)rr[s r_haszBesselTable.hassAG|r`N)__name__ __module__ __qualname____doc__rrrrrqr`r_rrsZ 555"888 r`rceZdZdZdZdZdS) DiffCacheau Store for derivatives of expressions. Explanation =========== The standard form of the derivative of a Bessel function of order n contains two Bessel functions of orders n-1 and n+1, respectively. Such forms cannot be used in parallel Risch algorithm, because there is a linear recurrence relation between the three functions while the algorithm expects that functions and derivatives are represented in terms of algebraically independent transcendentals. The solution is to take two of the functions, e.g., those of orders n and n-1, and to express the derivatives in terms of the pair. To guarantee that the proper form is used the two derivatives are cached as soon as one is encountered. Derivatives of other functions are also cached at no extra cost. All derivatives are with respect to the same variable `x`. cPi|_||_tstadSdSr)cacher\ _bessel_tabler)rr\s r_rzDiffCache.__init__s1  *'MMMMM * *r`c|j}||vrnt|drt|js+t ||j||<nj|j\}}t |j||\}}| |}||z||<||z|||dz |<||S)NfuncrL) rhasattrrrrrCdiffr\rTrget_diff)rr[rrgrd0d1dzs r_rzDiffCache.get_diffs  : * !V$$ *!!!&)) *affTVnn--E!HH6DAq"((A66FBq!!B"uE!H$&rEE!&&1a.. !Qxr`N)rrrrrrrqr`r_rrs<,***r`rc ./0123456789:;<t|}|durL|tttt t tttt rdS| s|zS|j s| \} }n tj} t t"t$ft&t(t*t,ft.i} |r1| D]\} } || | }n%| D]} |j| rnd}t7|} t95|4|st;dg}t;dg}t;dg}t=| D]}|jrOtA|tBr|j"d#||zz}|| $tC||||zz||||zzd||z ztK||d ztM||||zzz||z zz ztA|tNr]|j"d#|d zz}|||j(r9| $tStU||zn9| $tWtU|| z|j"d#|d zz|zz|z}|q||j(r| $tUtXd z || ztO||||d zd ||zz z ztStU||z||d tU||zz zzn||j-r| $tUtXd z || ztO||||d zd ||zz z ztWtU|| z||d tU|| zz z z|j"d#|tMd zz}|||j(ra| $tStU||tMzd d tU||zz z||j-rc| $tWtU|| tMzd d tU|| zz z Z|j.ru|j'j/rh|j'j0d krW|j1#|d zz|z}|||j(r||j(rB| $tetU||||z znO||j-rB| $tgtU|| ||z z|j1#|d zz|z }|a||j(rS||j(rd tU||d zz||z z }tMd tU||ztU||d zz||z zd ||zzztU||z .|5j4.<| $.H||j-r| $|| d z tU|| ztktU|| ztU||d zz||z z zn| t=|z} t=| D](}| t756|z} )tod tq| /tstutstwtyfd tw| /Dd 7d7D}|:7ddksJ7=dg}t}7}n|pg}7fd4|D]t7ts777|z745fd| D}d|D}t/fd|Dr(4|j@/rt/fd|6n u|st|d||}|| |zSdS6fd|D8/8fd1/01fd0/13fd3i}| D]}|jrtA|t&rd|d 4|d zz<3tA|t.r#d|d 4|z<d|d 4|z <ktA|trd|4|<4|..D\}}36}3|}t=ts||dgzts|z}|dtd|Dz}/fd|||fD} d| vrdSd| D\}}}||dz0|d zF:2fd22||t||z}"}!|!d kr:|"d kr4ttyt/|!|"zd z |z}#n0ttyt/|!|"z|z}#todtq|#9t9fdt|#D;t= force rewrite 'f' in terms of 'tan' and 'tanh' hints -> a list of functions that may appear in anti-derivate - hints = None --> no suggestions at all - hints = [ ] --> try to figure out - hints = [f1, ..., fn] --> we know better Examples ======== >>> from sympy import tan >>> from sympy.integrals.heurisch import heurisch >>> from sympy.abc import x, y >>> heurisch(y*tan(x), x) y*log(tan(x)**2 + 1)/2 See Manuel Bronstein's "Poor Man's Integrator": References ========== .. [1] http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.html For more information on the implemented algorithm refer to: .. [2] K. Geddes, L. Stefanus, On the Risch-Norman Integration Method and its Implementation in Maple, Proceedings of ISSAC'89, ACM Press, 212-217. .. [3] J. H. Davenport, On the Parallel Risch Algorithm (I), Proceedings of EUROCAM'82, LNCS 144, Springer, 144-157. .. [4] J. H. Davenport, On the Parallel Risch Algorithm (III): Use of Tangents, SIGSAM Bulletin 16 (1982), 3-6. .. [5] J. H. Davenport, B. M. Trager, On the Parallel Risch Algorithm (II), ACM Transactions on Mathematical Software 11 (1985), 356-362. See Also ======== sympy.integrals.integrals.Integral.doit sympy.integrals.integrals.Integral sympy.integrals.heurisch.components TNa)robcrrLr\cVg|]%}|dd|f&S)rrL)as_independent)rrrr\s r_ruzheurisch..s5???A!A$  a  #Q '???r`ci|]\}}|| Srqrqrrkvs r_ zheurisch..s,,,DAq1a,,,r`c.|Sr)r)rmappings r_ _substitutezheurisch.._substitutesyy!!!r`cLg|] }|!Srq)r)rrr^rdcaches r_ruzheurisch..s/BBBa++fooa0011BBBr`cBg|]}|dS)rL)as_numer_denom)rrr^s r_ruzheurisch..s)999Q1##%%a(999r`c3,K|]}|jVdSr) is_polynomial)rrhVs r_ zheurisch..s,33qq"333333r`c t||gRSr)rA)prZrs r_zheurisch..sAq 1 r`)rrrc4g|]}t|zSrq)rC)rrr^denoms r_ruzheurisch..s# / / /1veAg / / /r`cLtfdtDS)NcFg|]\}}||zSrq)r)rrrrrs r_ruz1heurisch.._derivation..s+@@@1a!&&))m@@@r`)rzip)rrnumerss`r_ _derivationzheurisch.._derivations,@@@@FA@@@AAr`cFD]}||s|tjurm||\}}|t |||zcS|Sr)rrZeroas_poly primitiver@ras_expr)ryrrZr _deflationrs r_rzheurisch.._deflations A AA5588 {1~~QV+ Ayy||--//1!z!}}SAFF1II%6%6%>%>%@%@@@@@ Ar`c D]I}||s |tjur||\}}|}t | ||}t|t |||||} |}|| dkr|d||dzfcS t||z }|d|dz|z|d|dzfcSKtj |fS)NrrL) rrrrrrr@r?rdegreerCOne) rrrrZrrsc_splitq_splitrr _splitters r_rzheurisch.._splitter!sW H HA5588 {1~~QV+ Hyy||--//1IIKK;;q>>1--3q!&&))Q//33#)A,,99Q<<&&((A-8#AJGAJ7777#)F1q5MM22 71:-a/GAJ1FGGGG H"qzr`Fcg|] \}}|| Srqrqrs r_ruzheurisch..Ns!@@@$!QQ@A@@@r`c$g|] }|j Srq)r)rrrrs r_ruzheurisch..Os!3331A333r`c6g|]}|Srq) total_degree)rrrs r_ruzheurisch..Us"444Q  444r`cR|jrp|jjrb|jjdkrR|jjdkr|jj|jjzdz St |jj|jjzSdS|js'|jr tfd|jDSdS)NrLrc&g|] }|Srqrq)rrr _exponents r_ruz/heurisch.._exponent..cs!777!1777r`) rVrrYrZrabsis_AtomrTmax)r^rs r_rzheurisch.._exponentYs 8 u  QUW\ 57Q;257QUW,q00quw0111q qv 7777qv77788 81r`Ac,g|]\}}||zSrqrq)rrimonomial poly_coeffss r_ruzheurisch..ps6/// Ax#1~h.///r`c  t}t}|dkrt%}nt}t}t%D]W}|tt|z}t|D]&}tt|||}||z}Xgg} } t|D]}t |t d} | t tj} | r| tj tj} | t s| t r| | | f| ||r| \} } | | f|vrs| | | f| r| } | | | z| | zz| t| | z n | | t | zz|t!dt#|}t!dt#|}t%t't)t||D]I\}}|j r:"|| |t-|zJt%t't)t||D]<\}}|j r-"|| ||z=$#z t/| zt/| z} |!z z }|d}t"tztfd |t3d \}}n#t4$rYdSwxYwt7"|}t7|} ||}n#t:$rt4wxYwt=||d }|dS| | tCt)"tjgt#"zS) NQ)filterF)evaluateBCrc|js|jrdS|vrdS|js|dS|js|js|jr$tt|j dStr) rXrYrNrQis_Addis_MulrVrmaprTrD)r find_non_symsnon_symssymss r_rz3heurisch.._integrate..find_non_symss &$"2 & &"T]D) & T""""" &  &t{ &S 2233333&%r`T)field)_raw)"rMrrrFr;rgetrrrrrQremovepopcould_extract_minus_signrrirdreversedrrrerrrrIrDrG from_expr ValueErrorrHcoeffsxreplacern)&ratansr irreduciblessetVpolyzVrrslog_part atan_partmrr\rrrr candidater raw_numerground_ coeff_ringringnumersolutionrrrFrrrr poly_denom poly_part reducibless& @@@r_ _integratezheurisch.._integrate{s C< z??LLq66D55L ++  C T 2 2333 ALq???@@A A%L ")L)) * *Da%000Aa  A *EE!%((5588quuQxx 1a&!!!##D))) *99;;DAqA2w% * a!W%%%--//A  1qs+++ $qs))$$$$  QqS))) * S#l++ , , S#e** % % S)>)>%B%B C CDD / /GD!tx| /""1%%%CII ...S%;%; < <== + +GD!tx| +""1%%%  T***j(3>9COK I&&. .$$&&q) ;#a&&(55 & & & & & & & ? M) $ $ $)>>>IFAA   44  k622 :&& "NN9--EE " " "! ! " %HHH  C4%%h//88Sqvhs;/?/?&?@@AACC Cs7 P P%$P% QQ1c3@K|]}t|tVdSr)rr rrrs r_rzheurisch..s, , ,Q:a , , , , , ,r`c32K|]}tVdSr)r r's r_rzheurisch..s&,@,@UWW,@,@,@,@,@,@r`r)rrrrrr)Tr rr0r1r2r3r9r:r6r7r4rNrrrrrrrrrrr!r rrkeysrUrr rMrRrr&rTmatchrQr'rr is_positiver%r#r$r is_negativerVrYrZrWr"rrrrrirdrr rrr rallis_rational_functionrr~r5rr rrtuplerEr enumeraterB free_symbolsas_dummyrrnrrCexpand)=r[r\rrrrrrrindep rewritables candidatesruletermsrrrr^MdF rev_mappingrrmr]specialtermPru_splitv_splitpolysrspolifiedrrmonomsrcoefffactorsfactmulr% more_freeFdr  antiderivr!rrrrrrrrrrrr"r#r$s= ` @@@@@@@@@@@@@@@r_r~r~'sf  A D  55b"dIz5'3 O O  F ::a==s 8##A&&qq c3 tTDK  + 1 1 3 3 , , J *d++AA ,&**,,  Jquj!  G q!  E q\\F C B S1#&&&AS1#&&&AS1#&&&AZZ: W: W=9W!!R(("WF1IOOAadF33~!IIq"QqT!QqT'\*:*:ad1ad7lbQRSTQUg=VWY[\]^[_`a[acfghijgklmopqroslsgsctctZtuvwxuyZyWzWz=z*z'{}}} $As++WF1IOOAadF33> t/> % $tAaDzz!|*<*< = = = = % #dAaD5kk!m*<*< = = =F1IOOAadFQqSL1,<==U t/U % $r!tadU|*<*Q/Q*R*R+S!T!T!T!T!"1!1U % $r!tadU|*<*R.R*S*S+T!U!U!UF1IOOAc!ffaiK88W t/V % $tAaDzz#a&&/@1aQqT lCS/S*T*T U U U t/W % #dAaD5kk#a&&.@1aaPQdU mCT.T*U*U V V VXWu(WQUW\WFLL1a4!44DQqT-=D t/D % %QqT!A$Y0A*B*B C C C C!"1!1D % $tQqTE!A$J/?/?/A*B*B C C CFLL1a4!44WQqT-=W t/W%&tAaDAI!,<'='=%=$'$qt** T!A$q!t)ad:J5K5K(KaPQRSPTfUVh(V$W$WX\]^_`]aXbXb$b24 Q % ! !"1!1W % 1Q4%'$!u++*=+/adU A d1Q419qQRtCS>T>T0T+U+U+V!W!W!Ws: Wz SZZ E ZZ33 FOOA..222 c%jj!!A8Dg????UA???'A'A"BCCCDFGGHHG,,G,,,KBr{1~""""$+KKOO#4 ((#;#Ar """""w--44BBBBB5BBB99%999 3333F333 3 3 8[ A8[]^8_ 5555v>>E E $aD)ACCCF $V|#t / / / / / / /FBBBBBB       0G22   2$$$ 249KK--q0011D$'' 216KK---.16KK---..D(++ 2-1 D))* AA    DAqiGillG W '!*.gllnn1E1EE F FE S@@'--//@@@AAA3333Aq 333H xt54(444GAq!gaj.::gaj#9#99BBDDJ      9Q<<SAYYqA1uIQIw}QA M0IJJKKLLw}QA 0EFFGGHH3F ,,K////$V,,///0IJ!!$T.A...wu  ! !ID# NN4  !hChChChChChChChChChChChChCT , ,! , , ,,,-NSVV+ ZZ\\KKS,@,@a,@,@,@%A%A B BR_-  :c??  $!z||H:<<MM+.. 9%%,,..   7!0033A6IY a< $aXwe]dgh]hD\]]]F $V|#tr`)FNNrjrNN)mtypingrtDictr itertoolsr functoolsrsympy.core.addrsympy.core.basicrsympy.core.mulr sympy.core.symbolr r r r sympy.core.numbersrrrsympy.core.relationalrrsympy.core.singletonrsympy.core.sortingrsympy.core.traversalrsympy.functionsrrrrrrrrrrr r!r"r#r$r%r&r'r(r)r*r+r,r-r.r/$sympy.functions.elementary.complexesr0r1r2r3r4&sympy.functions.elementary.exponentialr5#sympy.functions.elementary.integersr6r7$sympy.functions.elementary.piecewiser8'sympy.functions.special.delta_functionsr9r:sympy.simplify.radsimpr;sympy.logic.boolalgr<r=sympy.utilities.iterablesr> sympy.polysr?r@rArBrCrDsympy.polys.monomialsrEsympy.polys.polyrootsrFsympy.polys.ringsrGsympy.polys.solversrHsympy.polys.constructorrIsympy.integrals.integralsrJrUrbrirrrrr~rqr`r_rhs&&&&&&&&""""""""""""1111111111$$$$$$..........((((((((""""""&&&&&&------??????????????????>>>>>>>>>>>>>>>>33333333333333>>>>>>>>>>>>444444444444GGGGGGGGGGGGGG;;;;;;>>>>>>>>::::::IIIIIIII******''''''''******KKKKKKKKKKKKKKKK//////......&&&&&&------444444//////,,,^   NO?C#'^^^^@&&&&&&&&P ........`FG7;\\\\\\r`