U
    —9%eª*  ã                   @   sº   d Z ddlmZ ddlmZ ddlmZ ddlmZm	Z	m
Z
 ddlmZ ddlmZ ddlmZ dd	lmZ dd
l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Lambda)ÚI)ÚS)ÚDummyÚSymbolÚsymbols)Úlog)Úatan)Úroots)Úcancel)ÚRootSum)ÚPolyÚ	resultantÚZZc              
   K   s8  t | tƒr| \}}n|  ¡ \}}t||dddt||ddd }}| |¡\}}}| |¡\}}| |¡ ¡ }|jr||| S t	|||ƒ\}}	|	 ¡ \}
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t||ƒ}|
 |¡\}}||| |¡ ¡  7 }|js0| 
dd¡}t |tƒsøt|ƒ}n| ¡ }t||||ƒ}| 
d¡}|dkrxt | tƒrH| \}}| ¡ | ¡ B }n|  ¡ }||h D ]}|jsZd} qxqZd}tj}|sÆ|D ]:\}	}|	 ¡ \}}	|t|t||t|	 ¡ ƒ ƒdd7 }qˆnb|D ]\\}	}|	 ¡ \}}	t|	|||ƒ}|dk	r ||7 }n$|t|t||t|	 ¡ ƒ ƒdd7 }qÊ||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
isinstanceÚtupleZas_numer_denomr   r   ÚdivZ	integrateÚas_exprÚis_zeroÚratint_ratpartÚgetr   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© r2   ú\/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/integrals/rationaltools.pyÚratint   sj    "

"


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
  ÿ

  ÿr4   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}| |
 ¡ |  |
| ¡ |  |¡  ||  }|| 	¡ |	ƒ}|
 
¡  |¡}
| 
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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                    s    g | ]}t d tˆ | ƒ ƒ‘qS )Úa©r   Ústr©Ú.0Úi)Únr2   r3   Ú
<listcomp>¦   s     z"ratint_ratpart.<locals>.<listcomp>c                    s    g | ]}t d tˆ | ƒ ƒ‘qS )Úbr7   r9   )Úmr2   r3   r=   §   s     )Údomain)Zsympy.solvers.solversr5   r   Z	cofactorsÚdiffÚdegreeÚranger   ÚquoÚcoeffsr   Úsubsr   )r"   r)   r#   r5   ÚuÚvr0   ZA_coeffsZB_coeffsZC_coeffsÚAÚBÚHr(   Zrat_partZlog_partr2   )r?   r<   r3   r   |   s$     

.r   Nc                 C   sÈ  t | |ƒt ||ƒ } }|p tdƒ}|| | ¡ t ||ƒ   }}t||dd\}}t ||dd}|srtd||f ƒ‚i g  }}	|D ]}
|
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 ¡ < q€dd„ }| ¡ \}}|||ƒ |D ]\}}| ¡ \}}| ¡ |kræ|	 ||f¡ q´|| }t | 	¡ |dd	}|jdd
\}}|||ƒ |D ]$\}}| 
t | |¡| |ƒ¡}q| |¡tjg }}| ¡ dd… D ].}| |j¡}||  |¡}| | ¡ ¡ qht ttt| ¡ |ƒƒƒ|ƒ}|	 ||f¡ q´|	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) cannot be zeroc                 S   s>   | j r:| dk dkr:|d \}}|  |j¡}|| |f|d< d S )Nr   T)r   Úas_polyÚgens)ÚcZsqfr*   ÚkZc_polyr2   r2   r3   Ú_include_signð   s    z%ratint_logpart.<locals>._include_sign)r   )Úallé   N)r   r   rA   r   ÚAssertionErrorrB   Zsqf_listr    ÚappendZLCrD   ÚgcdÚinvertr   ÚOnerE   rL   rM   Úremr   ÚdictÚlistÚzipZmonoms)r"   r)   r#   r   r6   r>   Úresr1   ZR_maprK   r-   rP   ÚCZres_sqfr&   r;   r0   r*   Zh_lcrN   Zh_lc_sqfÚjÚinvrE   r'   ÚTr2   r2   r3   r   »   s<    &


r   c           	      C   sš   |   ¡ |  ¡ k r| |  } }|  ¡ } | ¡ }|  |¡\}}|jrPdt| ¡ ƒ S | |  ¡\}}}| | ||   |¡}dt| ¡ ƒ }|t||ƒ S d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   N)	rB   Zto_fieldr   r   r	   r   ZgcdexrD   Úlog_to_atan)	r"   r)   r%   r&   Úsr   r*   rG   rI   r2   r2   r3   rb     s    rb   c              	   C   st  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¡ }}|
 t	j
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 tt	j¡ }}tt|||ƒ|ƒ}t|dd}t|ƒ| ¡ kræd	S t	j}| ¡ D ]*}t| ||i¡|ƒ}t|dd}t|ƒ| ¡ kr2 d	S g }|D ]N}||kr:| |kr:|jsf| ¡ rt| | ¡ n|js:| |¡ q:|D ]}| ||||i¡}|jd
ddkrºqŽt| ||||i¡|ƒ}t| ||||i¡|ƒ}|d |d   ¡ }||t|ƒ |t||ƒ  7 }qŽqôt|dd}t|ƒ| ¡ krDd	S | ¡ D ]"}||t|  ¡  ||¡ƒ 7 }qL|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)Úevaluater1   )ÚfilterNT)Zchopra   )Zsympy.simplify.radsimprd   r   r   r   rF   r   Úexpandr   r   rW   r   r   r   r
   ÚlenZcount_rootsÚkeysZis_negativeZcould_extract_minus_signrT   r   Zevalfr   rb   )r*   r&   r#   r   rd   rG   rH   rK   r,   ZH_mapZQ_mapr6   r>   rN   Údr1   ZR_ur(   Zr_ur]   ZR_vZ
R_v_pairedZr_vÚDrI   rJ   ZABZR_qr-   r2   r2   r3   r!   G  sN    !  $ r!   )N)Ú__doc__Zsympy.core.functionr   Zsympy.core.numbersr   Zsympy.core.singletonr   Zsympy.core.symbolr   r   r   Z&sympy.functions.elementary.exponentialr   Z(sympy.functions.elementary.trigonometricr	   Zsympy.polys.polyrootsr
   Zsympy.polys.polytoolsr   Zsympy.polys.rootoftoolsr   Zsympy.polysr   r   r   r4   r   r   rb   r!   r2   r2   r2   r3   Ú<module>   s   m?
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