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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&& Y/home/luca/screen-checking/.venv/lib/python3.14/site-packages/sympy/integrals/heurisch.pyrTrT-s8,UFzz!}} ;;;1+++ JJqM$ M#]]]aoooVV*Q** JJqM MXXX j+ +F55###55$$$JJqvvx15577';;< M j2aS88F MVV*Q** Mzdict[str, list[Dummy]]_symbols_cachec \V,p\V4V8d/VP\ RV\V43,44K>VRV# \d.pT\T&L[i;i)z*get vector of symbols local to this modulez%s%iN)r`KeyErrorlenappendr )namenlsymss&& r^_symbolsrh`si%t$ e*q. eFdCJ%778: !9 %$t%s AA+*A+Nc x^RIHp Hp \V4pVP V4'g W,#\ WW#WEVWx4 p \ V \4'gV #.p \V !V 44Fp W!V .RV3R7, p K V 'gV #\\V 44p .pV !V4Fp W!V .RV3R7, pK V Uu.uF qV9gK VNK p pV 'gV #\V 4^8d].pV FApTPVP4UUu.uFwpp\VV4NK upp4KC V !VRV3R7V ,p .pV Fp\ VP!V4WW4VWgV4 p\#VP4UUu.uFwpp\VV4NK upp!p\%VP4UUu.uFwpp\'VV4NK upp!pVf\)VP!V4V4pVP+VV34K \V4^8Xd%\ WW#WEWgV4 X3V^,^,R3.pM VP+\ WW#WEWgV4 R34\-V!# \dEK!i;i \dEKi;iuupiuuppiuuppiuuppi)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 )solvedenomsT)dictexclude)sympy.solvers.solversrjrkr rMheurisch isinstancerrNotImplementedErrorlistr>rcextenditemsrsubsr<r=rrJrdr8)rZr[rewritehintsmappingsretries degree_offsetunnecessary_permutations _try_heurischrjrkresslnsdslns0seqssub_dictkeyvaluepairsexprcondgenerics&&&&&&&&& r^heurisch_wrapperrns:4 A ::a==s 1M+ #a7a8a9a:a;aa?a@aAaBaC\V4pVRJdFVP\\\\ \ \\\\4 'dR#VPV4'g W,#VP'gVPV4wrM\Pp \ \"\$3\&\(\*\,3\./p V'd,V P14FwrVP3W4pK M1V P54Fp VP!V !'gKM Rp\7W4p \9V4pVEeV'Eg\;RV.R7p\;RV.R7p\;RV.R7p\=V 4EFpVP>'Ed.\AV\B4'EdVPD^,PGWV,,4pVeV PIV\CVV,VVV,,,4VV,VVV,,,RVV,, ,\KVV,^,\MVV,VVV,,,4,VV,, 4,, ,4EK1EK4\AV\N4'EdVPD^,PGW^,,4pVe~VV,PP'd3V PI\S\UVV,4V,44M2V PI\W\UVV,)4V,44VPD^,PGW^,,VV,,V,4pVEeVV,PP'dV PI\U\X^, VV,),4\OVV,VV,^,^VV,,, , 4,\S\UVV,4V,VV,^\UVV,4,, ,4,4MVV,PZ'dV PI\U\X^, VV,),4\OVV,VV,^,^VV,,, , 4,\W\UVV,)4V,VV,^\UVV,)4,, , 4,4VPD^,PGV\MV4^,,4pVeVV,PP'd`V PI\S\UVV,4\MV4,^^\UVV,4,, ,44VV,PZ'deV PI\W\UVV,)4\MV4,^^\UVV,)4,, , 44EK;EK>EKAEKDVP\'gEKYVPNP^'gEKxVPNP`^8XgEKVPbPGW^,,V,4pVeVV,PP'dVV,PP'dAV PI\e\UVV,VV,, 4V,44MYVV,PZ'dAV PI\g\UVV,)VV,, 4V,44VPbPGW^,,V, 4pVfEKVV,PP'gEKVV,PP'd^\UVV,V^,,VV,, 4, p\M^\UVV,4,\UVV,V^,,VV,, 4,^VV,,V,,4\UVV,4, o7VVPhS7&V PIS74EKVV,PZ'gEKV PIVV,)^, \UVV,)4,\k\UVV,)4V,\UVV,V^,,VV,, 4, 4,4EK MV \=V4,p \=V 4F%pV \7VPmV4V4,p K' \oR\qV 44o8\s\u\s\w\y\wV S84Uu.uF#q^,PV4^,V3NK% up4!4^,44o>S>UUu/uF wppVVbK pppVfS>R,^,V8XgQhS>P{R4.p\}\r4pS>F)pVwppV\V4,PV4K+ VUu.uF pVV,NK upo>V>3RlpV!4pM T;'g.pV>3R lpVFo>\sS>4o>S>V,o>V Uu.uFpV!VPmV44NK ppVUu.uFpVP4^,NK pp\;QJdV83R lV4F 'dK R M RM!V83R lV44'gKV!V4P!S8!'gK\V83R lV4o=M( V'g\WRVVR 7p V e V V ,#R#VUu.uFp\S=V,4NK upo?V8V?3Rlo:V8V9V:3Rlo9V8V:V<3Rlo'gK\AV"\&4'dR V!^V!V"4^,,&KH\AV"\.4'd'R V!^V!V"4,&R V!^V!V"4, &K\AV"\4'gKRV!V!V"4&K V!V4o7S7P4wp#p$S RV7V8V:V=V@VAVBVC3Rllp2\;QJdRS84F 'dK R M RM !RS844'dS7P\=S84, p3MVS7P4p4V4P\\wS8RS84444PV4P,p3V3'gV2!R4p5V5fV2!4p5MV2!4p5V5e^V5PV4p6\V64P4p6V6P'dV6PV4^,p6V V6,#V^8d$\WWBW5^, VR7p V e V V ,#R#uupiuuppiuupiuupiuupiuupiuuppiuup)iuup)iuup.pi)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. 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