(f, g, x, t=None)
return H
Lazard-Rioboo-Trager algorithm.

    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
Lazard-Rioboo-Trager algorithm.
Lazard - Rioboo - Trager algorithm .
def ratint_logpart(f, g, x, t=None):
    """
    Lazard-Rioboo-Trager algorithm.

    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
    """
    f, g = Poly(f, x), Poly(g, x)

    t = t or Dummy('t')
    a, b = g, f - g.diff()*Poly(t, x)

    res, R = resultant(a, b, includePRS=True)
    res = Poly(res, t, composite=False)

    assert res, "BUG: resultant(%s, %s) can't be zero" % (a, b)

    R_map, H = {}, []

    for r in R:
        R_map[r.degree()] = r

    def _include_sign(c, sqf):
        if (c < 0) == True:
            h, k = sqf[0]
            sqf[0] = h*c, k

    C, res_sqf = res.sqf_list()
    _include_sign(C, res_sqf)

    for q, i in res_sqf:
        _, q = q.primitive()

        if g.degree() == i:
            H.append((g, q))
        else:
            h = R_map[i]
            h_lc = Poly(h.LC(), t, field=True)

            c, h_lc_sqf = h_lc.sqf_list(all=True)
            _include_sign(c, h_lc_sqf)

            for a, j in h_lc_sqf:
                h = h.quo(Poly(a.gcd(q)**j, x))

            inv, coeffs = h_lc.invert(q), [S(1)]

            for coeff in h.coeffs()[1:]:
                T = (inv*coeff).rem(q)
                coeffs.append(T.as_expr())

            h = Poly(dict(list(zip(h.monoms(), coeffs))), x)

            H.append((h, q))

    return H
def ratint_logpart ( f , g , x , t = None ) : f , g = Poly ( f , x ) , Poly ( g , x ) t = t or Dummy ( 't' ) a , b = g , f - g . diff ( ) * Poly ( t , x ) res , R = resultant ( a , b , includePRS = True ) res = Poly ( res , t , composite = False ) assert res , "BUG: resultant(%s, %s) can't be zero" % ( a , b ) R_map , H = { } , [ ] for r in R : R_map [ r . degree ( ) ] = r def _include_sign ( c , sqf ) : if ( c < 0 ) == True : h , k = sqf [ 0 ] sqf [ 0 ] = h * c , k C , res_sqf = res . sqf_list ( ) _include_sign ( C , res_sqf ) for q , i in res_sqf : _ , q = q . primitive ( ) if g . degree ( ) == i : H . append ( ( g , q ) ) else : h = R_map [ i ] h_lc = Poly ( h . LC ( ) , t , field = True ) c , h_lc_sqf = h_lc . sqf_list ( all = True ) _include_sign ( c , h_lc_sqf ) for a , j in h_lc_sqf : h = h . quo ( Poly ( a . gcd ( q ) ** j , x ) ) inv , coeffs = h_lc . invert ( q ) , [ S ( 1 ) ] for coeff in h . coeffs ( ) [ 1 : ] : T = ( inv * coeff ) . rem ( q ) coeffs . append ( T . as_expr ( ) ) h = Poly ( dict ( list ( zip ( h . monoms ( ) , coeffs ) ) ) , x ) H . append ( ( h , q ) ) return H