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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 P/var/www/auris/envauris/lib/python3.13/site-packages/sympy/integrals/heurisch.pyrUrU-s',UFzz!}} ;;1++ JJqM$ M#]]aooVV*Q** JJqM MXX j+ +F55##55$$JJqvvx15577';;< M j2aS88F MVV*Q** Mzdict[str, list[Dummy]]_symbols_cachec [Un[U5U:a9UR[ SU[U54-55 [U5U:aM9USU$![a /nU[U'Ndf=f)z*get vector of symbols local to this modulez%s%iN)raKeyErrorlenappendr )namenlsymss r__symbolsri`sx%t$ e*q. eFdCJ%778: e*q. !9 %$t%s AA0/A0Nc $SSKJn Jn [U5nUR U5(dX-$[ XX#XEUXx5 n [ U [5(dU $/n [U "U 55Hn X"U /SU4S9- n M U (dU $[[U 55n /nU "U5Hn X"U /SU4S9- nM U Vs/sH oU;dM UPM n nU (dU $[U 5S:aV/nU HBnURUR5VVs/sHunn[UU5PM snn5 MD U "USU4S9U -n /nU Hn[ UR!U5XX4UXgU5 n[#UR5VVs/sHunn[UU5PM snn6n[%UR5VVs/sHunn['UU5PM snn6nUc[)UR!U5U5nUR+UU45 M [U5S:Xa[ XX#XEXgU5 W4USSS4/nO UR+[ XX#XEXgU5 S45 [-U6$![a GMf=f![a GMf=fs snfs snnfs snnfs snnf)aC 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)dictexcluderL)sympy.solvers.solversrkrlr rNheurisch isinstancerrNotImplementedErrorlistr>rdextenditemsrsubsr<r=rrJrer8)r[r\rewritehintsmappingsretries degree_offsetunnecessary_permutations _try_heurischrkrlresslnsdslns0seqssub_dictkeyvaluepairsexprcondgenerics r_heurisch_wrapperrns:4 A ::a==s 1M+ ^?^@^A^B^C[U5nUSLaCUR[[[[ [ [[[[5 (agURU5(dX-$UR(dURU5upO[Rn [ ["[$4[&[([*[,4[.0n U(a+U R15HupUR3X5nM O/U R55Hn UR"U 6(dM O Sn[7X5n [9U5nUGbdU(GdN[;SU/S9n[;SU/S9n[;SU/S9n[=U 5GHnUR>(Ga[AU[B5(aURDSRGXU--5nUbsU RIU[CUUUUU--5UUUUU--SUU- -[KUUS -[MUUUUU--5-UU- 5-- -5 MM[AU[N5(GaURDSRGXS --5nUbhUURP(a*U RI[S[UUU5U-55 O*U RI[W[UUU*5U-55 URDSRGXS --UU--U-5nUGb#UURP(a}U RI[U[XS - UU*-5[OUUUUS -S UU-- - 5-[S[UUU5U-UUS [UUU5-- -5-5 OUURZ(a~U RI[U[XS - UU*-5[OUUUUS -S UU-- - 5-[W[UUU*5U-UUS [UUU*5-- - 5-5 URDSRGU[MU5S --5nUbUURP(aGU RI[S[UUU5[MU5-S S [UUU5-- -55 UURZ(aLU RI[W[UUU*5[MU5-S S [UUU*5-- - 55 GMGMGMGMUR\(dGMURNR^(dGMURNR`S :XdGMURbRGXS --U-5nUbUURP(aUURP(a0U RI[e[UUUUU- 5U-55 ODUURZ(a0U RI[g[UUU*UU- 5U-55 URbRGXS --U- 5nUcGMUURP(dGMUURP(aS [UUUUS --UU- 5- n[MS [UUU5-[UUUUS --UU- 5-S UU-U--5[UUU5- m7UURhT7'U RIT75 GMUURZ(dGMU RIUU*S - [UUU*5-[k[UUU*5U-[UUUUS --UU- 5- 5-5 GM OU [=U5-n [=U 5H!nU [7URmU5U5-n M# [oS [qU 55m8[s[u[s[w[y[wU T85Vs/sHoSRU5S U4PM sn565S 55m>T>VVs0sH unnUU_M nnnUc~T>SSU:XdeT>R{S5/n[}[r5nT>H%nUunnU[U5RU5 M' UVs/sHnUUPM snm>U>4S jnU"5nO U=(d /nU>4SjnUHm>[sT>5m>T>U-m>U Vs/sHnU"URmU55PM nnUVs/sHnUR5S PM nn[U84SjU55(dMyU"U5R"T86(dM[U84SjU5m= O U(d[XSUUS9n U bU U -$gUVs/sHn[T=U-5PM snm?U8U?4Sjm:U8U9U:4Sjm9U8U:U<4Sjm<0n!U Hn"U"R>(dM[AU"[&5(aSU!S U"U"5S --'M>[AU"[.5(aSU!S U"U"5-'SU!S U"U"5- 'Mq[AU"[5(dMSU!U"U"5'M U"U5m7T7R5un#n$T<"T=5n%T<"U$5n&[=[sU&5U%S/-[sU!R555-5n'U%S[U!R15VVs/sHunnU(dMUPM snn6-n(U(U#U$4V)s/sHn)U)R"T86PM n*n)SU*;agU*V)s/sHn)U)R5PM sn)unnnU(U&S-T9"U&S 5-R5mAU;4Sjm;T;"U5U[UU5-n,n+U+S :a.U,S :a([[y[T8U+U,-S - U-555n-O$[[y[T8U+U,-U-555n-[oS[qU-55m@[[U-5VV.s/sHunn.T@UU.-PM sn.n6mB[=5mC[yU'5H;n/[U//T8Q76un0n1TCRIU05 TCRSU155 M= SU7U8U:U=U@UAUBUC4Sjjn2[ST855(aT7R[=T85- n3ORT7R5n4U4R[[wT8ST85555RU4R-n3U3(dU2"S5n5U5cU2"5n5OU2"5n5U5bTU5RU5n6[U65R5n6U6R(aU6RU5S n6U U6-$US:a[XXBX5S - US9n U bU U -$gs snfs snnfs snfs snfs snfs snfs snnfs sn)fs sn)fs sn.nf)aM Compute indefinite integral using heuristic Risch algorithm. Explanation =========== This is a heuristic approach to indefinite integration in finite terms using the extended heuristic (parallel) Risch algorithm, based on Manuel Bronstein's "Poor Man's Integrator". The algorithm supports various classes of functions including transcendental elementary or special functions like Airy, Bessel, Whittaker and Lambert. Note that this algorithm is not a decision procedure. If it isn't able to compute the antiderivative for a given function, then this is not a proof that such a functions does not exist. One should use recursive Risch algorithm in such case. It's an open question if this algorithm can be made a full decision procedure. This is an internal integrator procedure. You should use top level 'integrate' function in most cases, as this procedure needs some preprocessing steps and otherwise may fail. Specification ============= heurisch(f, x, rewrite=False, hints=None) where f : expression x : symbol rewrite -> 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] https://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)rnbcrrLr\c3># [T5H,nUVVs/sHn[U5HoPM M snnv M. gs snnf7fr)rr)ijmappings r__iter_mappings heurisch.._iter_mappingss9!'*"#8!QWQZqZq!88+8sAAAc&>URT5$r)rv)rrs r_ _substituteheurisch.._substitute syy!!r`c3@># UHoR"T6v M g7fr) is_polynomial).0hVs r_ heurisch..s3Fq"Fsc>[X/TQ76$r)rA)prZrs r_heurisch..s A 1 r`)rwrxr|c >[[TT5VVs/sHupXRU5-PM snn6$s snnfr)rzipr)rrvrnumerss r_ _derivationheurisch.._derivations2FA@a&&)m@AA@s: c$>THnURU5(dMT"U5[RLdM6URU5R 5up#T"U5[ X3R U55R5-s $ U$r)rrZeroas_poly primitiver@ras_expr)ryrrZr _deflationrs r_rheurisch.._deflation!sqA55881~QVV+yy|--/!!}SFF1I%6%>%>%@@@ r`c @>TGHnURU5(dMT "U5[RLdM7URU5R 5up#UR 5n[ UT "U5U5n[U[ X3RU5U5U5nT "U5nURU5R5S:XaUSX6S-4s $T "[X5- 55nUSUS-U-USUS-4s $ [RU4$)NrrL) rrrrrrr@r?rdegreerCOne) rrrrZrrc_splitq_splitrr _splitters r_rheurisch.._splitter,sA55881~QVV+yy|--/IIK;q>1-3q&&)Q/3#A,99Q<&&(A-#AJAJ77#F15M2 71:-a/GAJ1FGG'*qzr`Fc>UR(aURR(aURRS:waURRS:a0URRURR-S- $[ URRURR-5$gUR (d0UR(a[U4SjUR55$g)NrLrc34># UH nT"U5v M g7frr)rr _exponents r_r.heurisch.._exponent..ns4Vy||Vs) rVrrYrZrabsis_AtomrTmax)r^rs r_rheurisch.._exponentds 88uu  QUUWW\5577Q;5577QUUWW,q00quuww011qvv4QVV44 4r`Ac3*# UH upUv M g7frr)rfactmuls r_rrs894$sc >^^^[5n[5nUS:Xa [T%5nOh[T5n[5n[T%5HDnU[[U55-n[U5Hn[[XWUS95nX8-n MB MF //p[U5Hn[ U[ SS9n U R [ [R5n U (dM?U R [R[R5n U R[ 5(dU R[ 5(aMURX45 URU5 M U(aUR5upX*4U;a`URX*45 U R5(aU *n URX-X--5 UR[X- 55 OURU [ U --5 U(aM[!S[#U55n[!S[#U55n[%['[)[U5U555HJunnUR"T6(dMT"R+U5 U R+U[-U5-5 ML [%['[)[U5U555HAunnUR"T6(dMT"R+U5 U R+UU-5 MC T$T#- [/U 6-[/U 6-nTT "U5T!- - nUR15Sn[T"5[T5-m[5mUUU4SjmT"U5 [3TS S 9unn[7T"U5n[7TU5nUR9U5n[=UR?5USS 9nUcgURAU5RA[C[)T"[R/[#T"5-555$![4a gf=f![:a [4ef=f) NQ)filterF)evaluateBCrcV>UR(dUR(agUT;agUR"T6(dTRU5 gUR(d"UR (dUR (a [[TUR55 g[er) rXrYrNrQis_Addis_MulrVrsmaprTrD)r find_non_symsnon_symssymss r_r3heurisch.._integrate..find_non_symsse$"2"2]]D) T" t{{S 23&%r`T)field)_raw)"rMrrrFr;rgetrrrrrQremovepopcould_extract_minus_signrrirdreversedrsrrerras_numer_denomrIrDrG from_expr ValueErrorrHcoeffsxreplacerm)&ratansr irreduciblessetVpolyzVrrlog_part atan_partmrr\rrrr candidater raw_numerground_ coeff_ringringnumersolutionrrrFrrdenom poly_coeffs poly_denom poly_part reducibless& @@@r_ _integrateheurisch.._integrates C<z?Lq6D5L +C T 233 AL?@A %L%,!")L)Da%0Aa AqEE!%%(5588quuQxx 1&!##D)*99;DA2w% aW%--//A  qs+ $qs)$  QqS)e S#l+ , S#e* % S)>%B CDGD!xx||""1%CI .E  S%; <=GD!xx||""1%  T*>j(3>9COK I&. .$$&q) ;#a&(5 & ? ) $)>IFAk62 :& "NN9-E!%H  %%h/88Sqvvhs;/?&?@AC C!   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