Intégration de fonctions de dérivées dans Sympy

Question

Je suis impatient de faire un calcul variationnel en Python avec sympy et j'ai couru vers un problème, et je ne comprends pas très bien l'origine. J'utilise la suite Anaconda, Jupyter Notebook et IPython version 5.3.0, avec python 3.6.

Si je writte

from sympy import Funtion, symbols, integrate 
from IPython.display import display 

t, s = symbols('t s') 
f = Function('f') 
c = Function('c') 

calculer les expressions suivantes (ils seront les éléments de la partie qui ne)

display(f(t, s)) 
display(c(t, s)) 
display(f(c(t, s))) 
display(f(c(t, s), c(t, s).diff(t))) 
display(integrate(f(t, s), t)) 
display(integrate(f(c(t, s)), t)) 
display(integrate(f(c(t, s).diff(t)), t)) 
display(integrate(f(c(t, s),c(s,t)), t)) 

les expressions attendues (latex) apparaissent. Je vois qu'il peut gérer la dérivation et l'intégration de plusieurs variables, mais lorsque je tente de mettre certaines choses togheter, comme dans

display(integrate(f(c(t, s), c(t, s).diff(t)), t)) 

l'erreur suivante est soulevée

Can't calculate 1st derivative wrt Derivative(_x1, t). 

D'autre part,

display(integrate(f(c(t,s), c(t,s).diff(t)), s)) 

affiche la sortie correcte. Des idées comment on pourrait résoudre ce problème?

Nous vous remercions à l'avance

EDIT: .doit() n'a pas aidé.

L'erreur est précédée par

--------------------------------------------------------------------------- 
ValueError        Traceback (most recent call last) 
<ipython-input-21-66abd1a98aff> in <module>() 
    11 display(integrate(f(c(t,s).diff(t)), t)) 
12 display(integrate(f(c(t,s), c(s,t)), t)) 
---> 13 display(integrate(f(c(t,s), c(t,s).diff(t)), t)) 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\integrals.py in integrate(*args, **kwargs) 
1278  if isinstance(integral, Integral): 
1279   return integral.doit(deep=False, meijerg=meijerg, conds=conds, 
-> 1280        risch=risch, manual=manual) 
1281  else: 
1282   return integral 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\integrals.py in doit(self, **hints) 
484      function, xab[0], 
485      meijerg=meijerg1, risch=risch, manual=manual, 
--> 486      conds=conds) 
487     if antideriv is None and meijerg1 is True: 
488      ret = try_meijerg(function, xab) 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\integrals.py in _eval_integral(self, f, x, meijerg, risch, manual, conds) 
885     try: 
886      if conds == 'piecewise': 
--> 887       h = heurisch_wrapper(g, x, hints=[]) 
888      else: 
889       h = heurisch(g, x, hints=[]) 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\heurisch.py in heurisch_wrapper(f, x, rewrite, hints, mappings, retries, degree_offset, unnecessary_permutations) 
128 
129  res = heurisch(f, x, rewrite, hints, mappings, retries, degree_offset, 
--> 130     unnecessary_permutations) 
131  if not isinstance(res, Basic): 
132   return res 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\heurisch.py in heurisch(f, x, rewrite, hints, mappings, retries, degree_offset, unnecessary_permutations) 
672  else: 
673   if retries >= 0: 
--> 674    result = heurisch(f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations) 
675 
676    if result is not None: 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\heurisch.py in heurisch(f, x, rewrite, hints, mappings, retries, degree_offset, unnecessary_permutations) 
672  else: 
673   if retries >= 0: 
--> 674    result = heurisch(f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations) 
675 
676    if result is not None: 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\heurisch.py in heurisch(f, x, rewrite, hints, mappings, retries, degree_offset, unnecessary_permutations) 
672  else: 
673   if retries >= 0: 
--> 674    result = heurisch(f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations) 
675 
676    if result is not None: 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\heurisch.py in heurisch(f, x, rewrite, hints, mappings, retries, degree_offset, unnecessary_permutations) 
451   mapping = list(mapping) 
452   mapping = mapping + unnecessary_permutations 
--> 453   diffs = [ _substitute(dcache.get_diff(g)) for g in terms ] 
454   denoms = [ g.as_numer_denom()[1] for g in diffs ] 
455   if all(h.is_polynomial(*V) for h in denoms) and _substitute(f).is_rational_function(*V): 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\heurisch.py in <listcomp>(.0) 
451   mapping = list(mapping) 
452   mapping = mapping + unnecessary_permutations 
--> 453   diffs = [ _substitute(dcache.get_diff(g)) for g in terms ] 
454   denoms = [ g.as_numer_denom()[1] for g in diffs ] 
455   if all(h.is_polynomial(*V) for h in denoms) and _substitute(f).is_rational_function(*V): 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\heurisch.py in _substitute(expr) 
446 
447  def _substitute(expr): 
--> 448   return expr.subs(mapping) 
449 
450  for mapping in mappings: 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\basic.py in subs(self, *args, **kwargs) 
900    rv = self 
901    for old, new in sequence: 
--> 902     rv = rv._subs(old, new, **kwargs) 
903     if not isinstance(rv, Basic): 
904      break 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\cache.py in wrapper(*args, **kwargs) 
91    def wrapper(*args, **kwargs): 
92     try: 
---> 93      retval = cfunc(*args, **kwargs) 
94     except TypeError: 
95      retval = func(*args, **kwargs) 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\basic.py in _subs(self, old, new, **hints) 
1014   rv = self._eval_subs(old, new) 
1015   if rv is None: 
-> 1016    rv = fallback(self, old, new) 
1017   return rv 
1018 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\basic.py in fallback(self, old, new) 
986     if not hasattr(arg, '_eval_subs'): 
987      continue 
--> 988     arg = arg._subs(old, new, **hints) 
989     if not _aresame(arg, args[i]): 
990      hit = True 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\cache.py in wrapper(*args, **kwargs) 
91    def wrapper(*args, **kwargs): 
92     try: 
---> 93      retval = cfunc(*args, **kwargs) 
94     except TypeError: 
95      retval = func(*args, **kwargs) 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\basic.py in _subs(self, old, new, **hints) 
1012    return new 
1013 
-> 1014   rv = self._eval_subs(old, new) 
1015   if rv is None: 
1016    rv = fallback(self, old, new) 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\function.py in _eval_subs(self, old, new) 
1340      variables = self_vars_front + self_vars 
1341      return Derivative(new, *variables) 
-> 1342   return Derivative(*(x._subs(old, new) for x in self.args)) 
1343 
1344  def _eval_lseries(self, x, logx): 

....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\function.py in __new__(cls, expr, *variables, **assumptions) 
1068     ordinal = 'st' if last_digit == 1 else 'nd' if last_digit == 2 else 'rd' if last_digit == 3 else 'th' 
1069     raise ValueError(filldedent(''' 
-> 1070     Can\'t calculate %s%s derivative wrt %s.''' % (count, ordinal, v))) 
1071 
1072    if all_zero and not count == 0: 

ValueError: 
Can't calculate 1st derivative wrt Derivative(_x1, t). 

Answer 1

J'utiliser Integral pour afficher Intégrales, qui les construit uniquement et ne tenter de les calculer (à moins que votre effectuer une doit()). Les travaux suivants

display(Integral(f(c(t, s), c(t, s).diff(t)), t))