def _rewriter(integral):
integrand, symbol = integral
if condition(*integral):
rewritten = rewrite(*integral)
if rewritten != integrand:
substep = integral_steps(rewritten, symbol)
if not isinstance(substep, DontKnowRule) and substep:
return RewriteRule(rewritten, substep, integrand, symbol)
def _proxy_rewriter(criteria):
criteria, integral = criteria
integrand, symbol = integral
args = criteria + list(integral)
if condition(*args):
rewritten = rewrite(*args)
if rewritten != integrand:
return RewriteRule(rewritten, integral_steps(rewritten, symbol), integrand, symbol)
def _rewriter(integral):
integrand, symbol = integral
if condition(*integral):
rewritten = rewrite(*integral)
if rewritten != integrand:
return RewriteRule(
rewritten,
integral_steps(rewritten, symbol),
integrand, symbol)
def _proxy_rewriter(criteria):
criteria, integral = criteria
integrand, symbol = integral
args = criteria + list(integral)
if condition(*args):
rewritten = rewrite(*args)
if rewritten != integrand:
return RewriteRule(rewritten,
integral_steps(rewritten,
symbol), integrand, symbol)
def _rewriter(integral):
integrand, symbol = integral
if condition(*integral):
rewritten = rewrite(*integral)
if rewritten != integrand:
substep = integral_steps(rewritten, symbol)
if not isinstance(substep, DontKnowRule) and substep:
return RewriteRule(
rewritten,
substep,
integrand, symbol)
def integral_steps(integrand, symbol, **options):
"""Returns the steps needed to compute an integral.
This function attempts to mirror what a student would do by hand as
closely as possible.
SymPy Gamma uses this to provide a step-by-step explanation of an
integral. The code it uses to format the results of this function can be
found at
https://github.com/sympy/sympy_gamma/blob/master/app/logic/intsteps.py.
Examples
========
>>> from sympy import exp, sin, cos
>>> from sympy.integrals.manualintegrate import integral_steps
>>> from sympy.abc import x
>>> print(repr(integral_steps(exp(x) / (1 + exp(2 * x)), x))) \
# doctest: +NORMALIZE_WHITESPACE
URule(u_var=_u, u_func=exp(x), constant=1,
substep=ArctanRule(context=1/(_u**2 + 1), symbol=_u),
context=exp(x)/(exp(2*x) + 1), symbol=x)
>>> print(repr(integral_steps(sin(x), x))) \
# doctest: +NORMALIZE_WHITESPACE
TrigRule(func='sin', arg=x, context=sin(x), symbol=x)
>>> print(repr(integral_steps((x**2 + 3)**2 , x))) \
# doctest: +NORMALIZE_WHITESPACE
RewriteRule(rewritten=x**4 + 6*x**2 + 9,
substep=AddRule(substeps=[PowerRule(base=x, exp=4, context=x**4, symbol=x),
ConstantTimesRule(constant=6, other=x**2,
substep=PowerRule(base=x, exp=2, context=x**2, symbol=x),
context=6*x**2, symbol=x),
ConstantRule(constant=9, context=9, symbol=x)],
context=x**4 + 6*x**2 + 9, symbol=x), context=(x**2 + 3)**2, symbol=x)
Returns
=======
rule : namedtuple
The first step; most rules have substeps that must also be
considered. These substeps can be evaluated using ``manualintegrate``
to obtain a result.
"""
cachekey = (integrand, symbol)
if cachekey in _integral_cache:
if _integral_cache[cachekey] is None:
# cyclic integral! null_safe will eliminate that path
return None
else:
return _integral_cache[cachekey]
else:
_integral_cache[cachekey] = None
integral = IntegralInfo(integrand, symbol)
def key(integral):
integrand = integral.integrand
if isinstance(integrand, TrigonometricFunction):
return TrigonometricFunction
elif isinstance(integrand, sympy.Derivative):
return sympy.Derivative
elif symbol not in integrand.free_symbols:
return sympy.Number
else:
for cls in (sympy.Pow, sympy.Symbol, sympy.exp, sympy.log,
sympy.Add, sympy.Mul, sympy.atan, sympy.asin, sympy.acos, sympy.Heaviside):
if isinstance(integrand, cls):
return cls
def integral_is_subclass(*klasses):
def _integral_is_subclass(integral):
k = key(integral)
return k and issubclass(k, klasses)
return _integral_is_subclass
result = do_one(
null_safe(switch(key, {
sympy.Pow: do_one(null_safe(power_rule), null_safe(inverse_trig_rule)),
sympy.Symbol: power_rule,
sympy.exp: exp_rule,
sympy.Add: add_rule,
sympy.Mul: do_one(null_safe(mul_rule), null_safe(trig_product_rule)),
sympy.Derivative: derivative_rule,
TrigonometricFunction: trig_rule,
sympy.Heaviside: heaviside_rule,
sympy.Number: constant_rule
})),
do_one(
null_safe(trig_rule),
null_safe(alternatives(
rewrites_rule,
substitution_rule,
condition(
integral_is_subclass(sympy.Mul, sympy.Pow),
partial_fractions_rule),
condition(
integral_is_subclass(sympy.Mul, sympy.log, sympy.atan, sympy.asin, sympy.acos),
parts_rule),
condition(
integral_is_subclass(sympy.Mul, sympy.Pow),
distribute_expand_rule),
trig_powers_products_rule
)),
null_safe(trig_substitution_rule)
),
fallback_rule)(integral)
del _integral_cache[cachekey]
return result
def integral_steps(integrand, symbol, **options):
"""Returns the steps needed to compute an integral.
This function attempts to mirror what a student would do by hand as
closely as possible.
SymPy Gamma uses this to provide a step-by-step explanation of an
integral. The code it uses to format the results of this function can be
found at
https://github.com/sympy/sympy_gamma/blob/master/app/logic/intsteps.py.
Examples
========
>>> from sympy import exp, sin, cos
>>> from sympy.integrals.manualintegrate import integral_steps
>>> from sympy.abc import x
>>> print(repr(integral_steps(exp(x) / (1 + exp(2 * x)), x))) \
# doctest: +NORMALIZE_WHITESPACE
URule(u_var=_u, u_func=exp(x), constant=1,
substep=PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=1, c=1, context=1/(_u**2 + 1), symbol=_u), True),
(ArccothRule(a=1, b=1, c=1, context=1/(_u**2 + 1), symbol=_u), False),
(ArctanhRule(a=1, b=1, c=1, context=1/(_u**2 + 1), symbol=_u), False)],
context=1/(_u**2 + 1), symbol=_u), context=exp(x)/(exp(2*x) + 1), symbol=x)
>>> print(repr(integral_steps(sin(x), x))) \
# doctest: +NORMALIZE_WHITESPACE
TrigRule(func='sin', arg=x, context=sin(x), symbol=x)
>>> print(repr(integral_steps((x**2 + 3)**2 , x))) \
# doctest: +NORMALIZE_WHITESPACE
RewriteRule(rewritten=x**4 + 6*x**2 + 9,
substep=AddRule(substeps=[PowerRule(base=x, exp=4, context=x**4, symbol=x),
ConstantTimesRule(constant=6, other=x**2,
substep=PowerRule(base=x, exp=2, context=x**2, symbol=x),
context=6*x**2, symbol=x),
ConstantRule(constant=9, context=9, symbol=x)],
context=x**4 + 6*x**2 + 9, symbol=x), context=(x**2 + 3)**2, symbol=x)
Returns
=======
rule : namedtuple
The first step; most rules have substeps that must also be
considered. These substeps can be evaluated using ``manualintegrate``
to obtain a result.
"""
cachekey = (integrand, symbol)
if cachekey in _integral_cache:
if _integral_cache[cachekey] is None:
# cyclic integral! null_safe will eliminate that path
return None
else:
return _integral_cache[cachekey]
else:
_integral_cache[cachekey] = None
integral = IntegralInfo(integrand, symbol)
def key(integral):
integrand = integral.integrand
if isinstance(integrand, TrigonometricFunction):
return TrigonometricFunction
elif isinstance(integrand, sympy.Derivative):
return sympy.Derivative
elif symbol not in integrand.free_symbols:
return sympy.Number
else:
for cls in (sympy.Pow, sympy.Symbol, sympy.exp, sympy.log,
sympy.Add, sympy.Mul, sympy.atan, sympy.asin,
sympy.acos, sympy.Heaviside):
if isinstance(integrand, cls):
return cls
def integral_is_subclass(*klasses):
def _integral_is_subclass(integral):
k = key(integral)
return k and issubclass(k, klasses)
return _integral_is_subclass
result = do_one(
null_safe(switch(key, {
sympy.Pow: do_one(null_safe(power_rule), null_safe(inverse_trig_rule), \
null_safe(quadratic_denom_rule)),
sympy.Symbol: power_rule,
sympy.exp: exp_rule,
sympy.Add: add_rule,
sympy.Mul: do_one(null_safe(mul_rule), null_safe(trig_product_rule), \
null_safe(heaviside_rule), null_safe(quadratic_denom_rule), \
null_safe(root_mul_rule)),
sympy.Derivative: derivative_rule,
TrigonometricFunction: trig_rule,
sympy.Heaviside: heaviside_rule,
sympy.Number: constant_rule
})),
do_one(
null_safe(trig_rule),
null_safe(alternatives(
rewrites_rule,
substitution_rule,
condition(
integral_is_subclass(sympy.Mul, sympy.Pow),
partial_fractions_rule),
condition(
integral_is_subclass(sympy.Mul, sympy.Pow),
cancel_rule),
condition(
integral_is_subclass(sympy.Mul, sympy.log, sympy.atan, sympy.asin, sympy.acos),
parts_rule),
condition(
integral_is_subclass(sympy.Mul, sympy.Pow),
distribute_expand_rule),
trig_powers_products_rule,
trig_expand_rule
)),
null_safe(trig_substitution_rule)
),
fallback_rule)(integral)
del _integral_cache[cachekey]
return result
def test_condition():
rl = condition(lambda x: x % 2 == 0, posdec)
assert rl(5) == 5
assert rl(4) == 3
def integral_steps(integrand, symbol, **options):
"""Returns the steps needed to compute an integral.
This function attempts to mirror what a student would do by hand as
closely as possible.
Returns
=======
rule : namedtuple
The first step; most rules have substeps that must also be
considered. These substeps can be evaluated using `manualintegrate`
to obtain a result.
"""
cachekey = (integrand, symbol)
if cachekey in _integral_cache:
if _integral_cache[cachekey] is None:
# cyclic integral! null_safe will eliminate that path
return None
else:
return _integral_cache[cachekey]
else:
_integral_cache[cachekey] = None
integral = IntegralInfo(integrand, symbol)
def key(integral):
integrand = integral.integrand
if isinstance(integrand, TrigonometricFunction):
return TrigonometricFunction
elif symbol not in integrand.free_symbols:
return 'constant'
else:
for cls in (sympy.Pow, sympy.Symbol, sympy.exp,
sympy.Add, sympy.Mul):
if isinstance(integrand, cls):
return cls
result = do_one(
null_safe(switch(key, {
sympy.Pow: do_one(null_safe(power_rule), null_safe(arctan_rule)),
sympy.Symbol: power_rule,
sympy.exp: exp_rule,
sympy.Add: add_rule,
sympy.Mul: do_one(null_safe(mul_rule), null_safe(trig_product_rule)),
TrigonometricFunction: trig_rule,
'constant': constant_rule
})),
null_safe(
alternatives(
substitution_rule,
parts_rule,
condition(lambda integral: key(integral) == sympy.Mul,
partial_fractions_rule),
condition(lambda integral: key(integral) in (sympy.Mul, sympy.Pow),
distribute_expand_rule),
trig_powers_products_rule
)
),
fallback_rule)(integral)
_integral_cache[cachekey] = result
return result
def test_condition():
rl = condition(lambda x: x%2 == 0, posdec)
assert rl(5) == 5
assert rl(4) == 3
def integral_steps(integrand, symbol, **options):
"""Returns the steps needed to compute an integral.
This function attempts to mirror what a student would do by hand as
closely as possible.
Returns
=======
rule : namedtuple
The first step; most rules have substeps that must also be
considered. These substeps can be evaluated using `manualintegrate`
to obtain a result.
"""
cachekey = (integrand, symbol)
if cachekey in _integral_cache:
if _integral_cache[cachekey] is None:
# cyclic integral! null_safe will eliminate that path
return None
else:
return _integral_cache[cachekey]
else:
_integral_cache[cachekey] = None
integral = IntegralInfo(integrand, symbol)
def key(integral):
integrand = integral.integrand
if isinstance(integrand, TrigonometricFunction):
return TrigonometricFunction
elif isinstance(integrand, sympy.Derivative):
return sympy.Derivative
elif symbol not in integrand.free_symbols:
return 'constant'
else:
for cls in (sympy.Pow, sympy.Symbol, sympy.exp, sympy.Add,
sympy.Mul):
if isinstance(integrand, cls):
return cls
result = do_one(
null_safe(
switch(
key, {
sympy.Pow:
do_one(null_safe(power_rule), null_safe(arctan_rule)),
sympy.Symbol:
power_rule,
sympy.exp:
exp_rule,
sympy.Add:
add_rule,
sympy.Mul:
do_one(null_safe(mul_rule), null_safe(trig_product_rule)),
sympy.Derivative:
derivative_rule,
TrigonometricFunction:
trig_rule,
'constant':
constant_rule
})),
null_safe(
alternatives(
substitution_rule, parts_rule,
condition(lambda integral: key(integral) == sympy.Mul,
partial_fractions_rule),
condition(
lambda integral: key(integral) in (sympy.Mul, sympy.Pow),
distribute_expand_rule), trig_powers_products_rule)),
fallback_rule)(integral)
_integral_cache[cachekey] = result
return result