a
    ҶDf*                     @   s   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.     )Lambda)I)S)DummySymbolsymbols)log)atan)roots)cancel)RootSum)Poly	resultantZZc              
   K   s8  t | tr| \}}n|  \}}t||dddt||ddd }}||\}}}||\}}|| }|jr||| S t	|||\}}	|	 \}
}t|
|}
t||}|
|\}}||||  7 }|js0|
dd}t |tst|}n| }t||||}|
d}|du rxt | trH| \}}| | B }n|  }||h D ]}|jsZd} qxqZd}tj}|s|D ]:\}	}|	 \}}	|t|t||t|	  dd7 }qnb|D ]\\}	}|	 \}}	t|	|||}|dur ||7 }n$|t|t||t|	  dd7 }q||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)	compositefieldsymboltrealN)Z	quadratic)
isinstancetupleZas_numer_denomr   r   divZ	integrateas_expris_zeroratint_ratpartgetr   r   Zas_dummyratint_logpartatomsis_extended_realr   Zero	primitiver   r   r   log_to_real)fxflagspqcoeffZpolyresultghPQrr   r   Lr   r   eltZeps_R r2   j/nfs/NAS7/SABIOD/METHODE/ermites/ermites_venv/lib/python3.9/site-packages/sympy/integrals/rationaltools.pyratint   sb    "

"








r4   c                    s$  ddl m} t| |} t||}|| \}}}| |  fddtdD } fddtd D }|| }	t||t|	 d}
t||t|	 d}| |
 |  |
| | |  ||  }||	 |	}|

 |}
|
 |}t|
|
  |}t||
  |}||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   )solvec                    s    g | ]}t d t |  qS )ar   str.0i)nr2   r3   
<listcomp>       z"ratint_ratpart.<locals>.<listcomp>c                    s    g | ]}t d t |  qS )br7   r9   )mr2   r3   r=      r>   )domain)Zsympy.solvers.solversr5   r   Z	cofactorsdiffdegreeranger   quocoeffsr   subsr   )r"   r)   r#   r5   uvr0   ZA_coeffsZB_coeffsZC_coeffsABHr(   Zrat_partZlog_partr2   )r@   r<   r3   r   |   s$     

.r   Nc                 C   s  t | |t || } }|p td}|| | t ||   }}t||dd\}}t ||dd}|srJ d||f i g  }}	|D ]}
|
||
 < qdd }| \}}||| |D ]\}}| \}}| |kr|	||f q|| }t | |dd	}|jdd
\}}||| |D ]$\}}|	t |
|| |}q||tjg }}| dd D ].}||j}|| |}||  qht ttt| ||}|	||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   as_polygens)cZsqfr*   kZc_polyr2   r2   r3   _include_sign   s    z%ratint_logpart.<locals>._include_sign)r   )all   N)r   r   rB   r   rC   Zsqf_listr    appendZLCrE   gcdinvertr   OnerF   rM   rN   remr   dictlistzipZmonoms)r"   r)   r#   r   r6   r?   resr1   ZR_maprL   r-   rQ   CZres_sqfr&   r;   r0   r*   Zh_lcrO   Zh_lc_sqfjinvrF   r'   Tr2   r2   r3   r      s<    &


r   c           	      C   s   |   |  k r| |  } }|  } | }| |\}}|jrPdt|  S ||  \}}}| | ||  |}dt|  }|t|| 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
       N)	rC   Zto_fieldr   r   r	   r   ZgcdexrE   log_to_atan)	r"   r)   r%   r&   sr   r*   rH   rJ   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
t	j|
tt	j }}tt||||}t|dd}t|| krd	S t	j}| D ]*}t|||i|}t|dd}t|| kr2 d	S g }|D ]N}||vr:| |vr:|jsf| rt||  n|js:|| q:|D ]}|||||i}|jd
ddkrqt|||||i|}t|||||i|}|d |d   }||t| |t||  7 }qqt|dd}t|| krDd	S | D ]"}||t|  || 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   )collectzu,v)clsF)evaluater1   )filterNT)Zchopra   )Zsympy.simplify.radsimprd   r   r   r   rG   r   expandr   r   rW   r   r   r   r
   lenZcount_rootskeysZis_negativeZcould_extract_minus_signrT   r   Zevalfr   rb   )r*   r&   r#   r   rd   rH   rI   rL   r,   ZH_mapZQ_mapr6   r?   rO   dr1   ZR_ur(   Zr_ur]   ZR_vZ
R_v_pairedZr_vDrJ   rK   ZABZR_qr-   r2   r2   r3   r!   G  sN    !  $ r!   )N)__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   <module>   s   m?
[1