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The Risch Algorithm for transcendental function integration.
The core algorithms for the Risch algorithm are here. The subproblem
algorithms are in the rde.py and prde.py files for the Risch
Differential Equation solver and the parametric problems solvers,
respectively. All important information concerning the differential extension
for an integrand is stored in a DifferentialExtension object, which in the code
is usually called DE. Throughout the code and Inside the DifferentialExtension
object, the conventions/attribute names are that the base domain is QQ and each
differential extension is x, t0, t1, ..., tn-1 = DE.t. DE.x is the variable of
integration (Dx == 1), DE.D is a list of the derivatives of
x, t1, t2, ..., tn-1 = t, DE.T is the list [x, t1, t2, ..., tn-1], DE.t is the
outer-most variable of the differential extension at the given level (the level
can be adjusted using DE.increment_level() and DE.decrement_level()),
k is the field C(x, t0, ..., tn-2), where C is the constant field. The
numerator of a fraction is denoted by a and the denominator by
d. If the fraction is named f, fa == numer(f) and fd == denom(f).
Fractions are returned as tuples (fa, fd). DE.d and DE.t are used to
represent the topmost derivation and extension variable, respectively.
The docstring of a function signifies whether an argument is in k[t], in
which case it will just return a Poly in t, or in k(t), in which case it
will return the fraction (fa, fd). Other variable names probably come
from the names used in Bronstein's book.
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real_roots�default_sort_key��z)�Lambda)�ilcm�oo�I)�Mul)�Pow)�Ne)�S)�Symbol�Dummy)�ordered)�_symbols) �acos�acot�asin�atan�cos�cot�exp�log� Piecewise�sin�tan)�sinh�cosh�tanh�coth)�Integral� integrate)�gcd�cancel�PolynomialError�Poly�reduced�RootSum� DomainError)�numbered_symbols)�
GeneratorType)�reducec s� i }| D ]#}|D ]}t || �}|jr || �||f� n q|tjfg||< qi }|D ]!}ttdd� || D ��� |� }� fdd�|| D �}|||< q,tt|� � �dd� d�S )a
Rewrites a list of expressions as integer multiples of each other.
Explanation
===========
For example, if you have [x, x/2, x**2 + 1, 2*x/3], then you can rewrite
this as [(x/6) * 6, (x/6) * 3, (x**2 + 1) * 1, (x/6) * 4]. This is useful
in the Risch integration algorithm, where we must write exp(x) + exp(x/2)
as (exp(x/2))**2 + exp(x/2), but not as exp(x) + sqrt(exp(x)) (this is
because only the transcendental case is implemented and we therefore cannot
integrate algebraic extensions). The integer multiples returned by this
function for each term are the smallest possible (their content equals 1).
Returns a list of tuples where the first element is the base term and the
second element is a list of `(item, factor)` terms, where `factor` is the
integer multiplicative factor that must multiply the base term to obtain
the original item.
The easiest way to understand this is to look at an example:
>>> from sympy.abc import x
>>> from sympy.integrals.risch import integer_powers
>>> integer_powers([x, x/2, x**2 + 1, 2*x/3])
[(x/6, [(x, 6), (x/2, 3), (2*x/3, 4)]), (x**2 + 1, [(x**2 + 1, 1)])]
We can see how this relates to the example at the beginning of the
docstring. It chose x/6 as the first base term. Then, x can be written as
(x/2) * 2, so we get (0, 2), and so on. Now only element (x**2 + 1)
remains, and there are no other terms that can be written as a rational
multiple of that, so we get that it can be written as (x**2 + 1) * 1.
c S s g | ]
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\}}||� f�qS r3 r3 �r0 r2 �j�� common_denomr3 r4 r5 r r6 c S s | d � � S �Nr )�sort_key)�itemr3 r3 r4 �<lambda>u s z integer_powers.<locals>.<lambda>��key)
r$ � is_Rational�appendr
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A container for all the information relating to a differential extension.
Explanation
===========
The attributes of this object are (see also the docstring of __init__):
- f: The original (Expr) integrand.
- x: The variable of integration.
- T: List of variables in the extension.
- D: List of derivations in the extension; corresponds to the elements of T.
- fa: Poly of the numerator of the integrand.
- fd: Poly of the denominator of the integrand.
- Tfuncs: Lambda() representations of each element of T (except for x).
For back-substitution after integration.
- backsubs: A (possibly empty) list of further substitutions to be made on
the final integral to make it look more like the integrand.
- exts:
- extargs:
- cases: List of string representations of the cases of T.
- t: The top level extension variable, as defined by the current level
(see level below).
- d: The top level extension derivation, as defined by the current
derivation (see level below).
- case: The string representation of the case of self.d.
(Note that self.T and self.D will always contain the complete extension,
regardless of the level. Therefore, you should ALWAYS use DE.t and DE.d
instead of DE.T[-1] and DE.D[-1]. If you want to have a list of the
derivations or variables only up to the current level, use
DE.D[:len(DE.D) + DE.level + 1] and DE.T[:len(DE.T) + DE.level + 1]. Note
that, in particular, the derivation() function does this.)
The following are also attributes, but will probably not be useful other
than in internal use:
- newf: Expr form of fa/fd.
- level: The number (between -1 and -len(self.T)) such that
self.T[self.level] == self.t and self.D[self.level] == self.d.
Use the methods self.increment_level() and self.decrement_level() to change
the current level.
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Tries to build a transcendental extension tower from ``f`` with respect to ``x``.
Explanation
===========
If it is successful, creates a DifferentialExtension object with, among
others, the attributes fa, fd, D, T, Tfuncs, and backsubs such that
fa and fd are Polys in T[-1] with rational coefficients in T[:-1],
fa/fd == f, and D[i] is a Poly in T[i] with rational coefficients in
T[:i] representing the derivative of T[i] for each i from 1 to len(T).
Tfuncs is a list of Lambda objects for back replacing the functions
after integrating. Lambda() is only used (instead of lambda) to make
them easier to test and debug. Note that Tfuncs corresponds to the
elements of T, except for T[0] == x, but they should be back-substituted
in reverse order. backsubs is a (possibly empty) back-substitution list
that should be applied on the completed integral to make it look more
like the original integrand.
If it is unsuccessful, it raises NotImplementedError.
You can also create an object by manually setting the attributes as a
dictionary to the extension keyword argument. You must include at least
D. Warning, any attribute that is not given will be set to None. The
attributes T, t, d, cases, case, x, and level are set automatically and
do not need to be given. The functions in the Risch Algorithm will NOT
check to see if an attribute is None before using it. This also does not
check to see if the extension is valid (non-algebraic) or even if it is
self-consistent. Therefore, this should only be used for
testing/debugging purposes.
rS zUAt least the key D must be included with the extension flag to DifferentialExtension.Nz8Either both f and x or a manual extension must be given.)r r z,handle_first must be 'log' or 'exp', not %s.)TTc 3 s � | ]}|� � �V qd S �N)�has�r0 r2 �rQ r3 r4 � <genexpr>� s � z1DifferentialExtension.__init__.<locals>.<genexpr>z1Trigonometric extensions are not supported (yet!)TzJCouldn't find an elementary transcendental extension for %s. Try using a z)manual extension with the extension flag.r r ))�
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_rewrite_logs� _exp_part� _log_part�frac_inr\ rT rU )�selfrP rQ � handle_firstra � extension�rewrite_complex�attr�exp_new_extension�log_new_extension� rewritables�
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is_Integerr� r� r3 r4 r> X s
z,Algebraic extensions are not supported (%s).Nc r� rb r� r� r� r3 r4 r> v s )�sympy.integrals.prder� r^ rl r �unionr rW �xreplace�dict� update_setsrp r r r� rA ro rj rv r\ ru rB )rw r� r� r� r� r} r� �ratpows� ratpows_replr2 �old�new�basea�based�A�ans�u�constrL r3 r� r4 rr ' sZ $�
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�||f� q t t |�t
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Rewrite logs for better processing.
c s$ | j d j� j� o| j d j� j� S r; )�argsrq rR rc r� r� r3 r4 r> � s z5DifferentialExtension._rewrite_logs.<locals>.<lambda>c s< | j � j� o | jd jo | jd jj� j� o | jd jj S r; )rc rR r� r� r� rq r r� r� r� r3 r4 r> � s �r r? )r^ rl r r� r r� r� rB r r� rW rD rp r )rw r� r� rl r2 �lbaser� r3 r� r4 rs � s
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z#DifferentialExtension._rewrite_logsc C sx | j s dd� | jD �| _ | js| j d | _dd� t| j| j �D �| _d| _| j | j | _| j| j | _| j| j | _dS )zB
Set attributes that are generated automatically.
c S � g | ]}|j �qS r3 )�genrd r3 r3 r4 r5 � � z5DifferentialExtension._auto_attrs.<locals>.<listcomp>r c S s g | ] \}}t ||��qS r3 )�get_case)r0 r] r\ r3 r3 r4 r5 � � ���N) rR rS rQ �ziprZ r_ r\ r] r[ r� r3 r3 r4 ri � s z!DifferentialExtension._auto_attrsc s� ddl m} d}d}dd� |D �}t|�}|D �]8\}}|jdd� d� t|�j�\} }
|| |
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� �d kr�t �fdd�� D �� ��j���fdd�|D ���_�j�t t t
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�� |�� }t�j��_�j��j� �j�|� �j�d� �j�|j �jdd�t�j�jdd� � �j �r*t d�}nt d�}� jt |t|�!�j"|���g7 _�j��fdd�|D ���_d}q|�rVd S |S )a�
Try to build an exponential extension.
Returns
=======
Returns True if there was a new extension, False if there was no new
extension but it was able to rewrite the given exponentials in terms
of the existing extension, and None if the entire extension building
process should be restarted. If the process fails because there is no
way around an algebraic extension (e.g., exp(log(x)/2)), it will raise
NotImplementedError.
r )�is_log_deriv_k_t_radicalFc S r� r3 �r rd r3 r3 r4 r5 � r� z3DifferentialExtension._exp_part.<locals>.<listcomp>c S s | d S )Nr. r3 r� r3 r3 r4 r> � s z1DifferentialExtension._exp_part.<locals>.<lambda>r? Nr� r. c S s g | ] \}}|| f�qS r3 r3 r7 r3 r3 r4 r5 � r� c S � g | ]\}}|| �qS r3 r3 �r0 r� �powerr3 r3 r4 r5 � s �c s8 i | ]\}}t || �t �| �td d� � D �� �qS )c S r� r3 r3 r� r3 r3 r4 r5 � r� z>DifferentialExtension._exp_part.<locals>.<dictcomp>.<listcomp>)r r
�r0 �exparg�p)r� r� r3 r4 �
<dictcomp>� s
� �z3DifferentialExtension._exp_part.<locals>.<dictcomp>c s g | ]
\}}||� �qS r3 r3 )r0 rI r� )�nr3 r4 r5 r6 c s* i | ]\}}t || �t � | �� �qS r3 r� r� )r� �radr3 r4 r� s
��c � g | ]}|� j ��qS r3 re �r0 rP r� r3 r4 r5 � Tz+Cannot integrate over algebraic extensions.� ��includer ��expandr2 c s i | ] \}}t |�� j| �qS r3 )r r\ r� r� r3 r4 r� * s )#r� r� rN �sortrv r\ r^ r� r r
�lenr� �listr� �reversedrR rV ro �
derivationr& r$ �as_expr�nextr` rB rY rX rS �as_polyra r r r �subsrQ )rw r� r� �
new_extension�restart�expargs�ip�arg�others�arga�argdr� r� �darga�dargd�dargr2 r3 )r� r� r� r� rw r4 rt � sx �
�
� ��
�
$�z DifferentialExtension._exp_partc
C s� ddl m} d}dd� |D �}t|�D ]�}t|| j�\}}|||| �}|dur>|\} }
} t| �|
} | j�t|�| i�| _qt|| j�\}}|tt || j�| � |tt || j�| � }
|d }|
�
� |�
� }t | j �| _| j
�| j� | j�|� | j�d� | j�t|�
� | �j| jdd �� | jr�td
�}ntd
�}| jt|t|�| j|���g7 _| j�t|�| ji�| _d }q|S ) a;
Try to build a logarithmic extension.
Returns
=======
Returns True if there was a new extension and False if there was no new
extension but it was able to rewrite the given logarithms in terms
of the existing extension. Unlike with exponential extensions, there
is no way that a logarithm is not transcendental over and cannot be
rewritten in terms of an already existing extension in a non-algebraic
way, so this function does not ever return None or raise
NotImplementedError.
r r� Fc S s g | ]}|j d �qS )r )r� rd r3 r3 r4 r5 C r� z3DifferentialExtension._log_part.<locals>.<listcomp>Nr� r r� r2 T)r� r� r rv r\ r r^ r� r� r& r� r� r` rR rB rY rX rS r$ r� ra r r rV r r� rQ )rw r� r� r� �logargsr� r� r� r� r� r� r� rL r� r� r� r2 r3 r3 r4 ru 1 s>
� �
$z DifferentialExtension._log_partc C s$ | j | j| j| j| j| j| j| jfS )z�
Returns some of the more important attributes of self.
Explanation
===========
Used for testing and debugging purposes.
The attributes are (fa, fd, D, T, Tfuncs, backsubs,
exts, extargs).
)rT rU rS rR rV rW rX rY r� r3 r3 r4 �_important_attrsg s
�z&DifferentialExtension._important_attrsc s$ � fdd�� j D �}� jjd| S )Nc s* g | ]}t t� |�t�s|t� |�f�qS r3 )r� �getattrr+ )r0 r{ r� r3 r4 r5 } s �z2DifferentialExtension.__repr__.<locals>.<listcomp>z
(dict(%r)))r� � __class__�__name__)rw �rr3 r� r4 �__repr__{ s z DifferentialExtension.__repr__c C s | j jd| j| j| jf S )Nz({fa=%s, fd=%s, D=%s}))r� r� rT rU rS r� r3 r3 r4 �__str__� s �z DifferentialExtension.__str__c C s@ | j jD ]}t| |�t||�}}t|t�s ||ks dS qdS )NFT)r� r� r� r� r+ )rw �otherr{ �d1�d2r3 r3 r4 �__eq__� s �z DifferentialExtension.__eq__c C sp | j | _| j g| _td| j �g| _d| _dg| _dg| _| jr't dt
d�| _ nt d�| _ g | _ g | _
| j| _dS )zD
Reset self to an initial state. Used by __init__.
r. r� Nr\ )�cls)rQ r\ rR r& rS r_ rX rY ra r* r r` rW rV rP r^ r� r3 r3 r4 rk � s
zDifferentialExtension.resetc s � fdd�t | j�D �S )aV
Parameters
==========
extension : str
Represents a valid extension type.
Returns
=======
list: A list of indices of 'exts' where extension of
type 'extension' is present.
Examples
========
>>> from sympy.integrals.risch import DifferentialExtension
>>> from sympy import log, exp
>>> from sympy.abc import x
>>> DE = DifferentialExtension(log(x) + exp(x), x, handle_first='exp')
>>> DE.indices('log')
[2]
>>> DE.indices('exp')
[1]
c s g | ]
\}}|� kr|�qS r3 r3 )r0 r2 �ext�ry r3 r4 r5 � r6 z1DifferentialExtension.indices.<locals>.<listcomp>)� enumeraterX )rw ry r3 r� r4 �indices� s z DifferentialExtension.indicesc C sN | j dkr td��| j d7 _ | j| j | _| j| j | _| j| j | _dS )a+
Increment the level of self.
Explanation
===========
This makes the working differential extension larger. self.level is
given relative to the end of the list (-1, -2, etc.), so we don't need
do worry about it when building the extension.
r� zJThe level of the differential extension cannot be incremented any further.r. N)r_ rg rR r\ rS r] rZ r[ r� r3 r3 r4 �increment_level� s
z%DifferentialExtension.increment_levelc C sV | j t| j� kr
td��| j d8 _ | j| j | _| j| j | _| j| j | _dS )a+
Decrease the level of self.
Explanation
===========
This makes the working differential extension smaller. self.level is
given relative to the end of the list (-1, -2, etc.), so we don't need
do worry about it when building the extension.
zJThe level of the differential extension cannot be decremented any further.r. N) r_ r� rR rg r\ rS r] rZ r[ r� r3 r3 r4 �decrement_level� s z%DifferentialExtension.decrement_level)NNr FNN)r� �
__module__� __qualname__�__doc__r� r� r� rr rs ri rt ru �propertyr� r� r� r� rk r� r� r� r3 r3 r3 r4 rO x s& -
wk"n6
rO c C s6 t | �}|�|�}|| }|�tt||��� t|�S rb )rp � intersection�updater� �filter)�seqrl �func�sr� r3 r3 r4 r� � s
r� c @ s, e Zd ZdZdZdd� Zdd� Zdd� Zd S )
�DecrementLevelzR
A context manager for decrementing the level of a DifferentialExtension.
��DEc C s
|| _ d S rb r� )rw r� r3 r3 r4 r� � s zDecrementLevel.__init__c C � | j �� d S rb )r� r� r� r3 r3 r4 � __enter__ � zDecrementLevel.__enter__c C r� rb )r� r� )rw �exc_type� exc_value� tracebackr3 r3 r4 �__exit__ r� zDecrementLevel.__exit__N)r� r� r� r� r� r� r� r� r3 r3 r3 r4 r� � s r� c @ � e Zd ZdZdS )� NonElementaryIntegralExceptionz�
Exception used by subroutines within the Risch algorithm to indicate to one
another that the function being integrated does not have an elementary
integral in the given differential field.
N�r� r� r� r� r3 r3 r3 r4 r� s r� c C sX | � |�\}}||�|�9 }|r |�� |�� kr |�|�\}}||| �|�}||fS )a
Extended Euclidean Algorithm, Diophantine version.
Explanation
===========
Given ``a``, ``b`` in K[x] and ``c`` in (a, b), the ideal generated by ``a`` and
``b``, return (s, t) such that s*a + t*b == c and either s == 0 or s.degree()
< b.degree().
)�
half_gcdex�exquo�degree�div)rJ �b�cr� �gr1 r\ r3 r3 r4 �gcdex_diophantine s r F�r$ c K s� t | �tu r| \}}|�� |�� } | �� �� \}}|j|fi |��|j|fi |��}}|r8|j|dd�\}}|du s@|du rHtd| |f ��||fS )ar
Returns the tuple (fa, fd), where fa and fd are Polys in t.
Explanation
===========
This is a common idiom in the Risch Algorithm functions, so we abstract
it out here. ``f`` should be a basic expression, a Poly, or a tuple (fa, fd),
where fa and fd are either basic expressions or Polys, and f == fa/fd.
**kwargs are applied to Poly.
Tr� Nz(Could not turn %s into a fraction in %s.)�type�tupler� r/ r� r$ rg )rP r\ r$ �kwargsrT rU r3 r3 r4 rv + s &rv c
C s4 t | |dd�\}}|jstd| ||f ��|�|�}|�d|d �}|�� |�� }|| }z |�� �|�}W n tyG } zt| ��d} ~ ww t j
t |j j �g|j
�R d|ji�}d| k rdtk ron n |t || |�9 }|�||�}|�|�r�|t d|�|� |dd �9 }|j||dd �|j||dd � }
|
S )
a
(Hackish) way to convert an element ``p`` of K[t, 1/t] to K[t, z].
In other words, ``z == 1/t`` will be a dummy variable that Poly can handle
better.
See issue 5131.
Examples
========
>>> from sympy import random_poly
>>> from sympy.integrals.risch import as_poly_1t
>>> from sympy.abc import x, z
>>> p1 = random_poly(x, 10, -10, 10)
>>> p2 = random_poly(x, 10, -10, 10)
>>> p = p1 + p2.subs(x, 1/x)
>>> as_poly_1t(p, x, z).as_expr().subs(z, 1/x) == p
True
Tr z$%s is not an element of K[%s, 1/%s].r r. N�domainFr� )rv � is_monomialr% r �slice�to_fieldr r) ro r& � from_listr� �rep�gensr
r �replace�nthr� ) r� r\ r �pa�pdr] �
one_t_partr� �t_part�er� r3 r3 r4 �
as_poly_1tC s2
���
�r c C s |rd}nt d|j�}|j}|r |jt|j� kr|S |�� |jdt|j�|j d � }|jdt|j�|j d � }t||�D ]1\}} | �| �}
|
du sQ|rU| � � }
|rc||� � |
�
| � 7 }qB||� � |
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| �� � �|�7 }qB|rzt |�}|r�|� � |S )aC
Computes Dp.
Explanation
===========
Given the derivation D with D = d/dx and p is a polynomial in t over
K(x), return Dp.
If coefficientD is True, it computes the derivation kD
(kappaD), which is defined as kD(sum(ai*Xi**i, (i, 0, n))) ==
sum(Dai*Xi**i, (i, 1, n)) (Definition 3.2.2, page 80). X in this case is
T[-1], so coefficientD computes the derivative just with respect to T[:-1],
with T[-1] treated as a constant.
If ``basic=True``, the returns a Basic expression. Elements of D can still be
instances of Poly.
r Nr. )
r& r\ r_ r� rR r� rS r� r� r� �diffr$ r� ) r� r� � coefficientD�basicr� r\ rS rR r] �v�pvr3 r3 r4 r� y s,
"r� c C sd | j �|�s
| jr dS dS | �t||��jrdS | �td|d |��jr'dS | �|�dkr0dS dS ) z�
Returns the type of the derivation d.
Returns one of {'exp', 'tan', 'base', 'primitive', 'other_linear',
'other_nonlinear'}.
r� � primitiver r. r� r �other_nonlinear� other_linear)�exprrc �is_one�remr& �is_zeror )r] r\ r3 r3 r4 r� � s r� Nc C s dd� |j d|j� D �}|r|�|� td|j| �� d�}t| ||d�}| jr,| |fS | j� |j�sK| j
|� � |j
|� ��
|j�}| � |�}||fS |js�| � |��
� } | � | �|j���
� }
| � |
�}|�|j�dkrq| |fS t| � |�||d�} | d | d | fS | |fS )a
Splitting factorization.
Explanation
===========
Given a derivation D on k[t] and ``p`` in k[t], return (p_n, p_s) in
k[t] x k[t] such that p = p_n*p_s, p_s is special, and each square
factor of p_n is normal.
Page. 100
c S � g | ]}d | �qS r- r3 �r0 rQ r3 r3 r4 r5 � � z splitfactor.<locals>.<listcomp>Nr. )r
)r r )rR r_ rB r& r\ �
get_domainr� r' r$ rc r� r# r r r r � splitfactor) r� r� r r �kinvrC �Dpr� r� �hr �q_splitr3 r3 r4 r, � s(
r, c
C s� dd� |j d|j� D �|j d|j� }|r|g}g }g }| �� }| jr,| dffdfS |D ]>\} }
| j|� �t| |||d�j|� ��|j�} t| |j�} t| |j�} | � | �} | j
sb|� | |
f� | j
sl|� | |
f� q.t |�t |�fS )aO
Splitting Square-free Factorization.
Explanation
===========
Given a derivation D on k[t] and ``p`` in k[t], returns (N1, ..., Nm)
and (S1, ..., Sm) in k[t]^m such that p =
(N1*N2**2*...*Nm**m)*(S1*S2**2*...*Sm**m) is a splitting
factorization of ``p`` and the Ni and Si are square-free and coprime.
c S r( r- r3 r) r3 r3 r4 r5 � r* z#splitfactor_sqf.<locals>.<listcomp>Nr. r3 )r r )
rR r_ �sqf_list_includer' r� r# r� r\ r& r r% rB r )
r� r� r r r �kkinvr
�N�p_sqf�pir2 �Si�Nir3 r3 r4 �splitfactor_sqf� s2 * ���
�r8 c
C s� t d|�� |j�}| �|�|�|�} }| �|�\}}t||�\}}t|�|j�|�|j�|�|j��\}} |�|j�| �|j�}} |||f| |ffS )ak
Canonical Representation.
Explanation
===========
Given a derivation D on k[t] and f = a/d in k(t), return (f_p, f_s,
f_n) in k[t] x k(t) x k(t) such that f = f_p + f_s + f_n is the
canonical representation of f (f_p is a polynomial, f_s is reduced
(has a special denominator), and f_n is simple (has a normal
denominator).
r. )r& �LCr\ �mulr r, r r� )
rJ r] r� �l�qr� �dn�dsr r r3 r3 r4 �canonical_representation s (r? c C s@ t d|�� |j�}| �|�|�|�} }t| ||�\}}}|\} }t d|�� |j�}| �|�|�|�} }t d|j�}t d|j�}t||�} t|| ��|j�}
|�|
�\} } |
� |j�dkr�t|
|�}
t|
|
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�|�\}} | �|
�}|�|
�\}} t
|�|j� |�|j�| �|j��\}}|�|j�|�|j�}}t||��|j�}| �|�\}} |�|j�|�|��|j� } ||
|| }||
}|j |dd�\}}|}
|
� |j�dks`| }| �|�\}} |j |dd�\}}| j |dd�\} }||d ||d |d }|d }|j |dd�\}}||f| |f||ffS )z�
Hermite Reduction - Mack's Linear Version.
Given a derivation D on k(t) and f = a/d in k(t), returns g, h, r in
k(t) such that f = Dg + h + r, h is simple, and r is reduced.
r. r Tr� ) r& r9 r\ r: r? r� r# r� r r r r$ )rJ r] r� r; �fp�fs�fn�ga�gd�dd�dmr> r� �ddm�dm2�dms�ds_ddm� ds_ddm_dmr r �db�ds_dmsr< �rra�rrdr3 r3 r4 �hermite_reduce, sF
* � rP c C s� t d|j�}| �|j�|j�|j�krW| �|j�|j�|j� d }t |j| |j��t | �|j��� ||j�� |j��}||7 }| t||� } | �|j�|j�|j�ks|| fS )z�
Polynomial Reduction.
Explanation
===========
Given a derivation D on k(t) and p in k[t] where t is a nonlinear
monomial over k, return q, r in k[t] such that p = Dq + r, and
deg(r) < deg_t(Dt).
r r. )r& r\ r r] r: r� r9 r� )r� r� r< �m�q0r3 r3 r4 �polynomial_reduced s " ���rS c & C s |� � dkrdS td|�}|�dt� td|j�}td|j�}|�|| �}| |tt| |j� } }
t||�} t||td|j��\} }
t| |td|j��\}}
|}g g g }}}t d|�D ]"}t||�}|�
� |d }|� |� |� t||d || �� q_t d|id�}t d|�D ]�}tt|| |j�||d | }|
}t
||f�d t
||f�d }}|�|�}t d|d �D ] }|�|| || �}q�|
t| |dd��|j� | t|
|dd��|j� |
t| |dd��|j� | t|
|dd��|j� } t|d |j�|
d } | | } }
|�t||��\} } t| |j�\} }!| � |j�|!� |j� dk�r�t||j�| d| ||| }"|"}#|� |#� |"|! �| �}#t| �}$t|$�D ]*}%|t|j|% || |j� t|#�|%�|j� }|t|j|% || |j� }�q\q�|||fS ) a�
Contribution of ``F`` to the full partial fraction decomposition of A/D.
Explanation
===========
Given a field K of characteristic 0 and ``A``,``D``,``F`` in K[x] with D monic,
nonzero, coprime with A, and ``F`` the factor of multiplicity n in the square-
free factorization of D, return the principal parts of the Laurent series of
A/D at all the zeros of ``F``.
r r r. rS r� r� T)r )r r �insertr r& r\ �quor� r �ranger� rB rO r$ r� r� r r# rv r& r r� �eval)&rJ r] �Fr� r� �Z�delta_a�delta_d�E�ha�hd�dF�B�G�C�F_store�V� DE_D_list�H_listr8 r �DE_new�zEha�zEhd�Pa�Pd�Qr2 �Dha�Dhd�Ff�Fr�F_stara�F_stard�QBC�H�alphas�alphar3 r3 r4 �laurent_seriesz s`
$"
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$
0"�
rw c C s� d}| j |dd�\} }| �|�\}}t||d|d�\}}d} |D ]%\}
} t|||
| |�\} }
}t||d ��� }||urAd} |S | d } q |S )a�
Compute the squarefree factorization of the denominator of f
and for each Di the polynomial H in K[x] (see Theorem 2.7.1), using the
LaurentSeries algorithm. Write Di = GiEi where Gj = gcd(Hn, Di) and
gcd(Ei,Hn) = 1. Since the residues of f at the roots of Gj are all 0, and
the residue of f at a root alpha of Ei is Hi(a) != 0, f is the derivative of a
rational function if and only if Ei = 1 for each i, which is equivalent to
Di | H[-1] for each i.
Tr� �r r r. r� F)r$ r r8 rw r# r� )rJ r] r� r �flagr< r� �Np�Spr8 r� r2 rZ r[ rt r r3 r3 r4 �recognize_derivative� s
�r| c C s� |pt d�}| j|dd�\} }| �|�\}} t||j�}t||�}| || }|j|dd�\}} t||�}t||d|d�\}
} | D ]\} }
t| � |��} t
dd� | D ��rX dS qAdS ) a�
There exists a v in K(x)* such that f = dv/v
where f a rational function if and only if f can be written as f = A/D
where D is squarefree,deg(A) < deg(D), gcd(A, D) = 1,
and all the roots of the Rothstein-Trager resultant are integers. In that case,
any of the Rothstein-Trager, Lazard-Rioboo-Trager or Czichowski algorithm
produces u in K(x) such that du/dx = uf.
r Tr� ��
includePRSrx c s s � | ]}|j V qd S rb )r� )r0 r8 r3 r3 r4 rf � s � z+recognize_log_derivative.<locals>.<genexpr>F) r r$ r r& r\ r� � resultantr8 r r� rn )rJ r] r� r r� �pz�Ddr< r� �Rrz r{ r� r2 r3 r3 r4 �recognize_log_derivative� s
�r� Tc s� �pt d��| j|dd�\} }| �� �d|�� �|�� �d|�� �} }dd� � jd� j� D �� jd� j� }| jrBg dfS | �|�\}} t �� j
�}t |� �}| || } |� � j
�|� � j
�krp|j
| dd�\}
} n | j
|dd�\}
} i g } }
| D ]}|| |� � < q�t |
��}
t|
� d�d �\}}|D ]�\}}||� � j
�kr�t |���� }|
�||f� q�| �|�}|du r�q�t |�� j
��� � j
dd
�}|jdd �}|D ] \} }t |� j
dd
��t t| || g|�R � � j
��}q�t |���� }|�rJt |�� j
��� � j
dd d
�}|j�dd
��|�tjg}}|�� dd� D ]}t||�|j� |g�d }|�|� � � �q"t t t t |� � |���� j
�}|
�||f� q�t!� �fdd�|D ��}|
|fS )aE
Lazard-Rioboo-Rothstein-Trager resultant reduction.
Explanation
===========
Given a derivation ``D`` on k(t) and f in k(t) simple, return g
elementary over k(t) and a Boolean b in {True, False} such that f -
Dg in k[t] if b == True or f + h and f + h - Dg do not have an
elementary integral over k(t) for any h in k<t> (reduced) if b ==
False.
Returns (G, b), where G is a tuple of tuples of the form (s_i, S_i),
such that g = Add(*[RootSum(s_i, lambda z: z*log(S_i(z, t))) for
S_i, s_i in G]). f - Dg is the remaining integral, which is elementary
only if b == True, and hence the integral of f is elementary only if
b == True.
f - Dg is not calculated in this function because that would require
explicitly calculating the RootSum. Use residue_reduce_derivation().
r Tr� r. c S r( r- r3 r) r3 r3 r4 r5 r* z"residue_reduce.<locals>.<listcomp>Nr} rx ��field)�allF)r� r� c s( g | ]\}}t |�� ��� j�� �qS r3 )r$ r� rc r\ )r0 r2 r1 �r� r r3 r4 r5 E s ( )"r r$ r �
mul_groundr9 rR r_ r' r r& r\ r� r r r8 �monicrB �getr� r1 r r# �invertr
rC �coeffsr' r r� r� r� r� �monomsr� )rJ r] r� r r� r2 r� r� r� r< r� r� �R_maprt r2 rz r{ r� r/ �h_lc�h_lc_sqfr8 �invr� �coeff�Lr r3 r� r4 �residue_reduce� sT .*
&� r� c sJ t d��ttt� j�t� fdd�� jD �����t���fdd�| D ��S )zR
Converts the tuple returned by residue_reduce() into a Basic expression.
r2 c r� r3 re r� r� r3 r4 r5 P r� z+residue_reduce_to_basic.<locals>.<listcomp>c
3 sJ � | ] }t |d ���t� � t|d �� ���� i���� ��V qdS �r r. N)r( r� r r r� r� �r0 rJ )r2 r� r r3 r4 rf R s � $���z*residue_reduce_to_basic.<locals>.<genexpr>)r r� r� r� rR rV �sum�rt r� r r3 )r� r2 r� r r4 �residue_reduce_to_basicJ s
(�r� c s& t d��tt� ��fdd�| D ���S )z�
Computes the derivation of an expression returned by residue_reduce().
In general, this is a rational function in t, so this returns an
as_expr() result.
r2 c
3 sX � | ]'}t |d ���t��t|d � ��� ���� |d �� ���� ��V qdS r� )r( r� r r� r� r� r� �r� r2 r r3 r4 rf _ s � �� ��z,residue_reduce_derivation.<locals>.<genexpr>)r r
r� r� r3 r� r4 �residue_reduce_derivationV s
�r� c C s� ddl m} td|j�}td|j�}| j�|j�s || dfS | j�|j�s+|| dfS t|j|j|j d �\}}t
|��B | � � }t||j�\}} z||| ||fg|�}
|
du r[t �|
\\} } }
W n t yw || df Y W d � S w W d � n1 s�w Y | �
|j�}|
d �|j�t|j|d |d |j� | �� | �� �|j�t|j| |j� }| t||� } || }q )aH
Integration of primitive polynomials.
Explanation
===========
Given a primitive monomial t over k, and ``p`` in k[t], return q in k[t],
r in k, and a bool b in {True, False} such that r = p - Dq is in k if b is
True, or r = p - Dq does not have an elementary integral over k(t) if b is
False.
r )�limited_integrateTr. NF)r� r� r& r\ r$ rc rv r] rR r_ r� r9 r� r r� r� r� )r� r� r� �Zeror< �Dta�DtbrJ �aa�ad�rv�ba�bdr rQ rR r3 r3 r4 � integrate_primitive_polynomialc s<
�
� �
,(��r� c s� |pt d�}ttt� j�t� fdd�� jD ����}t| |� �\}}}t|d |d � |d�\}} | s�t| � � |� � |d t
|d � � |d t
|d � � � � |d d � � t |� |� �}
t t|
��
|�� j�}
|d � � |d � � �
|�t|� |� |
| fS t|d � � |d � � t |� |� |d � � |d � � �} | �� j�} t| � �\} }
} |d � � |d � � | � � �
|�t|� |� }
| s�t t|
� � ��
|�� j�}
nt|
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Integration of primitive functions.
Explanation
===========
Given a primitive monomial t over k and f in k(t), return g elementary over
k(t), i in k(t), and b in {True, False} such that i = f - Dg is in k if b
is True or i = f - Dg does not have an elementary integral over k(t) if b
is False.
This function returns a Basic expression for the first argument. If b is
True, the second argument is Basic expression in k to recursively integrate.
If b is False, the second argument is an unevaluated Integral, which has
been proven to be nonelementary.
r c r� r3 re r� r� r3 r4 r5 � r� z'integrate_primitive.<locals>.<listcomp>r r. r r� )r r� r� r� rR rV rP r� r$ r� r� r� �NonElementaryIntegralr� rQ r� r� r\ r� )rJ r] r� r r� �g1r/ r� �g2r r2 r� r< �retr3 r� r4 �integrate_primitive� s@ ( $��
�
�� �� $
�
r� c
C s� ddl m} |j}|j�t|j|j��}td|j�}td|j�}d}| jr*|||fS t|��� t| � |� | � |�d �D ]�} | sBq=| dk rR| j
|dd�� | �}
n
| j
|dd�� | �}
t |
|jdd�\} } | j
| dd�\} } t| |�| }
t |
|jdd�\}}z |||t| |j�t| |j�|�\}}t ||f|dd �\}}W n ty� d}Y q=w || |t|| � | }||9 }q=W d
� n1 s�w Y |||fS ) aG
Integration of hyperexponential polynomials.
Explanation
===========
Given a hyperexponential monomial t over k and ``p`` in k[t, 1/t], return q in
k[t, 1/t] and a bool b in {True, False} such that p - Dq in k if b is True,
or p - Dq does not have an elementary integral over k(t) if b is False.
r )�rischDEr. TFr� r� r� r N)�sympy.integrals.rder� r\ r] r r& r' r� rV r r� r rv r$ r� )r� r� r r� �t1�dtt�qa�qdr r2 rJ r� r� �iDt�iDta�iDtd�va�vdr3 r3 r4 �%integrate_hyperexponential_polynomial� s<
$ �
� �
r� � piecewisec sv |pt d�}ttt� j�t� fdd�� jD ����}t| |� �\}}}t|d |d � |d�\} }
|
s�t| � � |� � |d t
|d � � |d t
|d � � � � |d d � � t | � |� �} t t| �
|��� j�} |d � � |d � � �
|�t| � |� | |
fS t|d � � |d � � t | � |� |d � � |d � � �} t| � j|�}
t|
� |�\}}}
|
�dd�} |d � � |d � � �
|�t| � |� }|� � �
|�}|� � �
|�}|dk�r
� j|jv�r
|t|| t|d�ft| | �
� jd��
|�� j�d f�7 }n||| 7 }|
�s6| |t
|� � |t
|� � � � |d � � } t t| ��
|�� j�} || |
fS )
ap
Integration of hyperexponential functions.
Explanation
===========
Given a hyperexponential monomial t over k and f in k(t), return g
elementary over k(t), i in k(t), and a bool b in {True, False} such that
i = f - Dg is in k if b is True or i = f - Dg does not have an elementary
integral over k(t) if b is False.
This function returns a Basic expression for the first argument. If b is
True, the second argument is Basic expression in k to recursively integrate.
If b is False, the second argument is an unevaluated Integral, which has
been proven to be nonelementary.
r c r� r3 re r� r� r3 r4 r5
r� z.integrate_hyperexponential.<locals>.<listcomp>r r. r r� r� T)r r� r� r� rR rV rP r� r$ r� r� r� r� r� rQ r� r r\ r� r � free_symbolsr r r" )rJ r] r� r �condsr� r� r/ r� r� r r2 r� �ppr� r� r� �qas�qdsr3 r� r4 �integrate_hyperexponential� sV ( $��
�
�� ��
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r� c C sR t | |�\}}|j�t|jd d |j��}t|�d�d|�� |j�}||fS )ah
Integration of hypertangent polynomials.
Explanation
===========
Given a differential field k such that sqrt(-1) is not in k, a
hypertangent monomial t over k, and p in k[t], return q in k[t] and
c in k such that p - Dq - c*D(t**2 + 1)/(t**1 + 1) is in k and p -
Dq does not have an elementary integral over k(t) if Dc != 0.
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r� c sD |pt d�}ttt� j�t� fdd�� jD ����}t| |� �\}}}t|d |d � |d�\}} | sJ|d �� |d �� � |�t
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Integration of nonlinear monomials with no specials.
Explanation
===========
Given a nonlinear monomial t over k such that Sirr ({p in k[t] | p is
special, monic, and irreducible}) is empty, and f in k(t), returns g
elementary over k(t) and a Boolean b in {True, False} such that f - Dg is
in k if b == True, or f - Dg does not have an elementary integral over k(t)
if b == False.
This function is applicable to all nonlinear extensions, but in the case
where it returns b == False, it will only have proven that the integral of
f - Dg is nonelementary if Sirr is empty.
This function returns a Basic expression.
r c r� r3 re r� r� r3 r4 r5 b r� z3integrate_nonlinear_no_specials.<locals>.<listcomp>r r. r FT)r r� r� r� rR rV rP r� r� r� r� r$ r� r� r\ rS r$ rc )rJ r] r� r r� r� r/ r� r� r r� �q1�q2r� r3 r� r4 � integrate_nonlinear_no_specialsK s6 (
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Represents a nonelementary Integral.
Explanation
===========
If the result of integrate() is an instance of this class, it is
guaranteed to be nonelementary. Note that integrate() by default will try
to find any closed-form solution, even in terms of special functions which
may themselves not be elementary. To make integrate() only give
elementary solutions, or, in the cases where it can prove the integral to
be nonelementary, instances of this class, use integrate(risch=True).
In this case, integrate() may raise NotImplementedError if it cannot make
such a determination.
integrate() uses the deterministic Risch algorithm to integrate elementary
functions or prove that they have no elementary integral. In some cases,
this algorithm can split an integral into an elementary and nonelementary
part, so that the result of integrate will be the sum of an elementary
expression and a NonElementaryIntegral.
Examples
========
>>> from sympy import integrate, exp, log, Integral
>>> from sympy.abc import x
>>> a = integrate(exp(-x**2), x, risch=True)
>>> print(a)
Integral(exp(-x**2), x)
>>> type(a)
<class 'sympy.integrals.risch.NonElementaryIntegral'>
>>> expr = (2*log(x)**2 - log(x) - x**2)/(log(x)**3 - x**2*log(x))
>>> b = integrate(expr, x, risch=True)
>>> print(b)
-log(-x + log(x))/2 + log(x + log(x))/2 + Integral(1/log(x), x)
>>> type(b.atoms(Integral).pop())
<class 'sympy.integrals.risch.NonElementaryIntegral'>
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The Risch Integration Algorithm.
Explanation
===========
Only transcendental functions are supported. Currently, only exponentials
and logarithms are supported, but support for trigonometric functions is
forthcoming.
If this function returns an unevaluated Integral in the result, it means
that it has proven that integral to be nonelementary. Any errors will
result in raising NotImplementedError. The unevaluated Integral will be
an instance of NonElementaryIntegral, a subclass of Integral.
handle_first may be either 'exp' or 'log'. This changes the order in
which the extension is built, and may result in a different (but
equivalent) solution (for an example of this, see issue 5109). It is also
possible that the integral may be computed with one but not the other,
because not all cases have been implemented yet. It defaults to 'log' so
that the outer extension is exponential when possible, because more of the
exponential case has been implemented.
If ``separate_integral`` is ``True``, the result is returned as a tuple (ans, i),
where the integral is ans + i, ans is elementary, and i is either a
NonElementaryIntegral or 0. This useful if you want to try further
integrating the NonElementaryIntegral part using other algorithms to
possibly get a solution in terms of special functions. It is False by
default.
Examples
========
>>> from sympy.integrals.risch import risch_integrate
>>> from sympy import exp, log, pprint
>>> from sympy.abc import x
First, we try integrating exp(-x**2). Except for a constant factor of
2/sqrt(pi), this is the famous error function.
>>> pprint(risch_integrate(exp(-x**2), x))
/
|
| 2
| -x
| e dx
|
/
The unevaluated Integral in the result means that risch_integrate() has
proven that exp(-x**2) does not have an elementary anti-derivative.
In many cases, risch_integrate() can split out the elementary
anti-derivative part from the nonelementary anti-derivative part.
For example,
>>> pprint(risch_integrate((2*log(x)**2 - log(x) - x**2)/(log(x)**3 -
... x**2*log(x)), x))
/
|
log(-x + log(x)) log(x + log(x)) | 1
- ---------------- + --------------- + | ------ dx
2 2 | log(x)
|
/
This means that it has proven that the integral of 1/log(x) is
nonelementary. This function is also known as the logarithmic integral,
and is often denoted as Li(x).
risch_integrate() currently only accepts purely transcendental functions
with exponentials and logarithms, though note that this can include
nested exponentials and logarithms, as well as exponentials with bases
other than E.
>>> pprint(risch_integrate(exp(x)*exp(exp(x)), x))
/ x\
\e /
e
>>> pprint(risch_integrate(exp(exp(x)), x))
/
|
| / x\
| \e /
| e dx
|
/
>>> pprint(risch_integrate(x*x**x*log(x) + x**x + x*x**x, x))
x
x*x
>>> pprint(risch_integrate(x**x, x))
/
|
| x
| x dx
|
/
>>> pprint(risch_integrate(-1/(x*log(x)*log(log(x))**2), x))
1
-----------
log(log(x))
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