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mZmZ ddlmZmZmZ dd„ Zdd„ Zdd	d
„Zdd„ Zdd„ ZdS )zAThis module implements tools for integrating rational functions. é    )ÚSÚSymbolÚsymbolsÚIÚlogÚatanÚrootsÚRootSumÚLambdaÚcancelÚDummy)ÚPolyÚ	resultantÚZZc              
   K   s*  t | ƒtur|  ¡ \}}n| \}}t||dddt||ddd}}| |¡\}}}| |¡\}}| |¡ ¡ }|jr?|| S t	|||ƒ\}}	|	 ¡ \}
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dd¡}t|tƒs}t|ƒ}n| ¡ }t||||ƒ}| 
d¡}|du r¹t | ƒturœ|  ¡ }n| \}}| ¡ | ¡ B }||h D ]	}|js¶d} nq­d}tj}|sÞ|D ]\}	}|	 ¡ \}}	|t|t||t|	 ¡ ƒ ƒdd7 }qÀn/|D ],\}	}|	 ¡ \}}	t|	|||ƒ}|durú||7 }qà|t|t||t|	 ¡ ƒ ƒdd7 }qà||7 }|| S )	aa  
    Performs indefinite integration of rational functions.

    Explanation
    ===========

    Given a field :math:`K` and a rational function :math:`f = p/q`,
    where :math:`p` and :math:`q` are polynomials in :math:`K[x]`,
    returns a function :math:`g` such that :math:`f = g'`.

    Examples
    ========

    >>> from sympy.integrals.rationaltools import ratint
    >>> from sympy.abc import x

    >>> ratint(36/(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2), x)
    (12*x + 6)/(x**2 - 1) + 4*log(x - 2) - 4*log(x + 1)

    References
    ==========

    .. [1] M. Bronstein, Symbolic Integration I: Transcendental
       Functions, Second Edition, Springer-Verlag, 2005, pp. 35-70

    See Also
    ========

    sympy.integrals.integrals.Integral.doit
    sympy.integrals.rationaltools.ratint_logpart
    sympy.integrals.rationaltools.ratint_ratpart

    FT)Ú	compositeÚfieldÚsymbolÚtÚrealN)Z	quadratic)ÚtypeÚtupleZas_numer_denomr   r   ÚdivZ	integrateÚas_exprÚis_zeroÚratint_ratpartÚgetÚ
isinstancer   r   Zas_dummyÚratint_logpartÚatomsÚis_extended_realr   ÚZeroÚ	primitiver	   r
   r   Úlog_to_real)ÚfÚxÚflagsÚpÚqÚcoeffZpolyÚresultÚgÚhÚPÚQÚrr   r   ÚLr   r   ÚeltZepsÚ_ÚR© r3   ú?/usr/lib/python3/dist-packages/sympy/integrals/rationaltools.pyÚratint   sf   ""
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
ÿr5   c                    s$  ddl m} t| |ƒ} t||ƒ}| | ¡ ¡\}}}| ¡ ‰| ¡ ‰ ‡fdd„tdˆƒD ƒ}‡ fdd„tdˆ ƒD ƒ}|| }	t||t|	 d}
t||t|	 d}| |
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¡  |¡}
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| 
¡  |ƒ}t|| 
¡  |ƒ}||fS )a«  
    Horowitz-Ostrogradsky algorithm.

    Explanation
    ===========

    Given a field K and polynomials f and g in K[x], such that f and g
    are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
    such that f/g = A' + B and B has square-free denominator.

    Examples
    ========

        >>> from sympy.integrals.rationaltools import ratint_ratpart
        >>> from sympy.abc import x, y
        >>> from sympy import Poly
        >>> ratint_ratpart(Poly(1, x, domain='ZZ'),
        ... Poly(x + 1, x, domain='ZZ'), x)
        (0, 1/(x + 1))
        >>> ratint_ratpart(Poly(1, x, domain='EX'),
        ... Poly(x**2 + y**2, x, domain='EX'), x)
        (0, 1/(x**2 + y**2))
        >>> ratint_ratpart(Poly(36, x, domain='ZZ'),
        ... Poly(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2, x, domain='ZZ'), x)
        ((12*x + 6)/(x**2 - 1), 12/(x**2 - x - 2))

    See Also
    ========

    ratint, ratint_logpart
    r   )Úsolvec                    ó    g | ]}t d tˆ | ƒ ƒ‘qS )Úa©r   Ústr©Ú.0Úi)Únr3   r4   Ú
<listcomp>    ó     z"ratint_ratpart.<locals>.<listcomp>c                    r7   )Úbr9   r;   )Úmr3   r4   r?   ¡   r@   )Údomain)Úsympyr6   r   Z	cofactorsÚdiffÚdegreeÚranger   ÚquoÚcoeffsr   Úsubsr   )r#   r*   r$   r6   ÚuÚvr1   ZA_coeffsZB_coeffsZC_coeffsÚAÚBÚHr)   Zrat_partZlog_partr3   )rB   r>   r4   r   v   s$    

.r   Nc                 C   sÂ  t | |ƒt ||ƒ} }|ptdƒ}|| | ¡ t ||ƒ  }}t||dd\}}t ||dd}|s9J d||f ƒ‚i g }}	|D ]}
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 ¡ < q@dd„ }| ¡ \}}|||ƒ |D ]„\}}| ¡ \}}| ¡ |krr|	 ||f¡ qZ|| }t | ¡ |dd	}|jdd
\}}|||ƒ |D ]\}}| 	t | 
|¡| |ƒ¡}qŽ| |¡tjg}}| ¡ dd… D ]}| |j¡}||  |¡}| | ¡ ¡ q²t ttt| ¡ |ƒƒƒ|ƒ}|	 ||f¡ qZ|	S )an  
    Lazard-Rioboo-Trager algorithm.

    Explanation
    ===========

    Given a field K and polynomials f and g in K[x], such that f and g
    are coprime, deg(f) < deg(g) and g is square-free, returns a list
    of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i
    in K[t, x] and q_i in K[t], and::

                           ___    ___
                 d  f   d  \  `   \  `
                 -- - = --  )      )   a log(s_i(a, x))
                 dx g   dx /__,   /__,
                          i=1..n a | q_i(a) = 0

    Examples
    ========

    >>> from sympy.integrals.rationaltools import ratint_logpart
    >>> from sympy.abc import x
    >>> from sympy import Poly
    >>> ratint_logpart(Poly(1, x, domain='ZZ'),
    ... Poly(x**2 + x + 1, x, domain='ZZ'), x)
    [(Poly(x + 3*_t/2 + 1/2, x, domain='QQ[_t]'),
    ...Poly(3*_t**2 + 1, _t, domain='ZZ'))]
    >>> ratint_logpart(Poly(12, x, domain='ZZ'),
    ... Poly(x**2 - x - 2, x, domain='ZZ'), x)
    [(Poly(x - 3*_t/8 - 1/2, x, domain='QQ[_t]'),
    ...Poly(-_t**2 + 16, _t, domain='ZZ'))]

    See Also
    ========

    ratint, ratint_ratpart
    r   T)Z
includePRSF)r   z$BUG: resultant(%s, %s) can't be zeroc                 S   sF   | j r| dk dkr!|d \}}|  |j¡}|| |f|d< d S d S d S )Nr   T)r   Úas_polyÚgens)ÚcZsqfr+   ÚkZc_polyr3   r3   r4   Ú_include_signê   s
   ýz%ratint_logpart.<locals>._include_sign)r   )Úallé   N)r   r   rE   r   rF   Zsqf_listr!   ÚappendZLCrH   ZgcdÚinvertr   ÚOnerI   rP   rQ   Zremr   ÚdictÚlistÚzipZmonoms)r#   r*   r$   r   r8   rA   Úresr2   ZR_maprO   r.   rT   ÚCZres_sqfr'   r=   r1   r+   Zh_lcrR   Zh_lc_sqfÚjÚinvrI   r(   ÚTr3   r3   r4   r   µ   s<   &


r   c           	      C   s–   |   ¡ |  ¡ k r| | } }|  ¡ } | ¡ }|  |¡\}}|jr(dt| ¡ ƒ S | |  ¡\}}}| | ||   |¡}dt| ¡ ƒ }|t||ƒ S )a0  
    Convert complex logarithms to real arctangents.

    Explanation
    ===========

    Given a real field K and polynomials f and g in K[x], with g != 0,
    returns a sum h of arctangents of polynomials in K[x], such that:

                   dh   d         f + I g
                   -- = -- I log( ------- )
                   dx   dx        f - I g

    Examples
    ========

        >>> from sympy.integrals.rationaltools import log_to_atan
        >>> from sympy.abc import x
        >>> from sympy import Poly, sqrt, S
        >>> log_to_atan(Poly(x, x, domain='ZZ'), Poly(1, x, domain='ZZ'))
        2*atan(x)
        >>> log_to_atan(Poly(x + S(1)/2, x, domain='QQ'),
        ... Poly(sqrt(3)/2, x, domain='EX'))
        2*atan(2*sqrt(3)*x/3 + sqrt(3)/3)

    See Also
    ========

    log_to_real
    é   )	rF   Zto_fieldr   r   r   r   ZgcdexrH   Úlog_to_atan)	r#   r*   r&   r'   Úsr   r+   rK   rM   r3   r3   r4   rc     s   rc   c              	   C   s^  ddl m} tdtd\}}|  ¡  ||t|  i¡ ¡ }| ¡  ||t|  i¡ ¡ }||tdd}	||tdd}
|	 t	j
t	j¡|	 tt	j¡}}|
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 tt	j¡}}tt|||ƒ|ƒ}t|dd}t|ƒ| ¡ krsd	S t	j}| ¡ D ]‹}t| ||i¡|ƒ}t|dd}t|ƒ| ¡ kr— d	S g }|D ]!}||vr¼| |vr¼|js­| ¡ r´| | ¡ q›|js¼| |¡ q›|D ]E}| ||||i¡}|jd
ddkrÓq¿t| ||||i¡|ƒ}t| ||||i¡|ƒ}|d |d   ¡ }||t|ƒ |t||ƒ  7 }q¿qzt|dd}t|ƒ| ¡ krd	S | ¡ D ]}||t|  ¡  ||¡ƒ 7 }q|S )aw  
    Convert complex logarithms to real functions.

    Explanation
    ===========

    Given real field K and polynomials h in K[t,x] and q in K[t],
    returns real function f such that:
                          ___
                  df   d  \  `
                  -- = --  )  a log(h(a, x))
                  dx   dx /__,
                         a | q(a) = 0

    Examples
    ========

        >>> from sympy.integrals.rationaltools import log_to_real
        >>> from sympy.abc import x, y
        >>> from sympy import Poly, S
        >>> log_to_real(Poly(x + 3*y/2 + S(1)/2, x, domain='QQ[y]'),
        ... Poly(3*y**2 + 1, y, domain='ZZ'), x, y)
        2*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/3
        >>> log_to_real(Poly(x**2 - 1, x, domain='ZZ'),
        ... Poly(-2*y + 1, y, domain='ZZ'), x, y)
        log(x**2 - 1)/2

    See Also
    ========

    log_to_atan
    r   )Úcollectzu,v)ÚclsF)Zevaluater2   )ÚfilterNT)Zchoprb   )rD   re   r   r   r   rJ   r   Úexpandr   r   rY   r    r   r   r   ÚlenZcount_rootsÚkeysZis_negativeZcould_extract_minus_signrW   r   Zevalfr   rc   )r+   r'   r$   r   re   rK   rL   rO   r-   ZH_mapZQ_mapr8   rA   rR   Údr2   ZR_ur)   Zr_ur^   ZR_vZ
R_v_pairedZr_vÚDrM   rN   ZABZR_qr.   r3   r3   r4   r"   A  sR   !  
€ ô r"   )N)Ú__doc__rD   r   r   r   r   r   r   r   r	   r
   r   r   Zsympy.polysr   r   r   r5   r   r   rc   r"   r3   r3   r3   r4   Ú<module>   s    4n
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