def test_heurisch_function():
df = diff(f(x), x)
assert heurisch(f(x), x) == None
assert heurisch(f(x)*df, x) == f(x)**2/2
assert heurisch(f(x)**2 * df, x) == f(x)**3/3
assert heurisch(df / f(x), x) == log(f(x))
def test_heurisch_hyperbolic():
assert heurisch(sinh(x), x) == cosh(x)
assert heurisch(cosh(x), x) == sinh(x)
assert heurisch(x*sinh(x), x) == x*cosh(x) - sinh(x)
assert heurisch(x*cosh(x), x) == x*sinh(x) - cosh(x)
assert heurisch(x*asinh(x/2), x) == x**2*asinh(x/2)/2 + asinh(x/2) - x*sqrt(4+x**2)/4
def test_heurisch_radicals():
assert heurisch(1/sqrt(x), x) == 2*sqrt(x)
assert heurisch(1/sqrt(x)**3, x) == -2/sqrt(x)
assert heurisch(sqrt(x)**3, x) == 2*sqrt(x)**5/5
assert heurisch(sin(x)*sqrt(cos(x)), x) == -2*sqrt(cos(x))**3/3
assert heurisch(sin(y*sqrt(x)), x) == 2/y**2*sin(y*sqrt(x)) - \
2*sqrt(x)*cos(y*sqrt(x))/y
def test_heurisch_radicals():
assert heurisch(x**Rational(-1,2), x) == 2*x**Rational(1,2)
assert heurisch(x**Rational(-3,2), x) == -2*x**Rational(-1,2)
assert heurisch(x**Rational(3,2), x) == 2*x**Rational(5,2) / 5
assert heurisch(sin(x)*sqrt(cos(x)), x) == -2*cos(x)**Rational(3,2) / 3
assert heurisch(sin(y*sqrt(x)), x) == 2*y**(-2)*sin(y*x**S.Half) - \
2*x**S.Half*cos(y*x**S.Half)/y
def test_pmint_erf():
skip('takes too much time')
f = exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1)
g = sqrt(pi) / 4 * (-1 / (erf(x) - 1) - log(erf(x) + 1) / 2 +
log(erf(x) - 1) / 2)
assert heurisch(f, x) == g
def test_heurisch_trigonometric():
assert heurisch(sin(x), x) == -cos(x)
assert heurisch(pi*sin(x)+1, x) == x-pi*cos(x)
assert heurisch(cos(x), x) == sin(x)
assert heurisch(tan(x), x) == log(1 + tan(x)**2)/2
assert heurisch(sin(x)*sin(y), x) == -cos(x)*sin(y)
assert heurisch(sin(x)*sin(y), y) == -cos(y)*sin(x)
# gives sin(x) in answer when run via setup.py and cos(x) when run via py.test
assert heurisch(sin(x)*cos(x), x) in [sin(x)**2 / 2, -cos(x)**2 / 2]
assert heurisch(cos(x)/sin(x), x) == log(sin(x))
assert heurisch(x*sin(7*x), x) == sin(7*x) / 49 - x*cos(7*x) / 7
assert heurisch(1/pi/4 * x**2*cos(x), x) == 1/pi/4*(x**2*sin(x) - 2*sin(x) + 2*x*cos(x))
def test_heurisch_trigonometric():
assert heurisch(sin(x), x) == -cos(x)
assert heurisch(pi*sin(x)+1, x) == x-pi*cos(x)
assert heurisch(cos(x), x) == sin(x)
assert heurisch(tan(x), x) == log(1 + tan(x)**2)/2
assert heurisch(sin(x)*sin(y), x) == -cos(x)*sin(y)
assert heurisch(sin(x)*sin(y), y) == -cos(y)*sin(x)
# gives sin(x) in answer when run via setup.py and cos(x) when run via py.test
assert heurisch(sin(x)*cos(x), x) in [sin(x)**2 / 2, -cos(x)**2 / 2]
assert heurisch(cos(x)/sin(x), x) == log(sin(x))
assert heurisch(x*sin(7*x), x) == sin(7*x) / 49 - x*cos(7*x) / 7
assert heurisch(1/pi/4 * x**2*cos(x), x) == 1/pi/4*(x**2*sin(x) - 2*sin(x) + 2*x*cos(x))
def test_pmint_logexp():
skip('takes too much time')
f = (1 + x + x * exp(x)) * (x + log(x) + exp(x) - 1) / (x + log(x) +
exp(x))**2 / x
g = 1 / (x + log(x) + exp(x)) + log(x + log(x) + exp(x))
assert heurisch(f, x) == g
def test_pmint_rat():
f = (x**7 - 24 * x**4 - 4 * x**2 + 8 * x - 8) / (x**8 + 6 * x**6 +
12 * x**4 + 8 * x**2)
g = (4 + 8 * x**2 + 6 * x + 3 * x**3) / (x *
(x**4 + 4 * x**2 + 4)) + log(x)
assert heurisch(f, x) == g
def test_pmint_rat():
skip('takes too much time')
f = (x**7 - 24 * x**4 - 4 * x**2 + 8 * x - 8) / (x**8 + 6 * x**6 +
12 * x**4 + 8 * x**2)
g = (4 + 8 * x**2 + 6 * x + 3 * x**3) / (x *
(x**4 + 4 * x**2 + 4)) + log(x)
assert heurisch(f, x) == g
def test_heurisch_exp():
assert heurisch(exp(x), x) == exp(x)
assert heurisch(exp(-x), x) == -exp(-x)
assert heurisch(exp(17*x), x) == exp(17*x) / 17
assert heurisch(x*exp(x), x) == x*exp(x) - exp(x)
assert heurisch(x*exp(x**2), x) == exp(x**2) / 2
assert heurisch(exp(-x**2), x) is None
assert heurisch(2**x, x) == 2**x/log(2)
assert heurisch(x*2**x, x) == x*2**x/log(2) - 2**x*log(2)**(-2)
def test_heurisch_exp():
assert heurisch(exp(x), x) == exp(x)
assert heurisch(exp(-x), x) == -exp(-x)
assert heurisch(exp(17 * x), x) == exp(17 * x) / 17
assert heurisch(x * exp(x), x) == x * exp(x) - exp(x)
assert heurisch(x * exp(x**2), x) == exp(x**2) / 2
assert heurisch(exp(-x**2), x) is None
assert heurisch(2**x, x) == 2**x / log(2)
assert heurisch(x * 2**x, x) == x * 2**x / log(2) - 2**x * log(2)**(-2)
def test_heurisch_fractions():
assert heurisch(1/x, x) == log(x)
assert heurisch(1/(2 + x), x) == log(x + 2)
assert heurisch(1/(x+sin(y)), x) == log(x+sin(y))
# Up to a constant, where C = 5*pi*I/12, Mathematica gives identical
# result in the first case. The difference is because sympy changes
# signs of expressions without any care.
# XXX ^ ^ ^ is this still correct?
assert heurisch(5*x**5/(2*x**6 - 5), x) in [5*log(2*x**6 - 5) / 12, 5*log(-2*x**6 + 5) / 12]
assert heurisch(5*x**5/(2*x**6 + 5), x) == 5*log(2*x**6 + 5) / 12
assert heurisch(1/x**2, x) == -1/x
assert heurisch(-1/x**5, x) == 1/(4*x**4)
def test_heurisch_fractions():
assert heurisch(1/x, x) == log(x)
assert heurisch(1/(2 + x), x) == log(x + 2)
assert heurisch(1/(x+sin(y)), x) == log(x+sin(y))
# Up to a constant, where C = 5*pi*I/12, Mathematica gives identical
# result in the first case. The difference is because sympy changes
# signs of expressions without any care.
# XXX ^ ^ ^ is this still correct?
assert heurisch(5*x**5/(2*x**6 - 5), x) in [5*log(2*x**6 - 5) / 12, 5*log(-2*x**6 + 5) / 12]
assert heurisch(5*x**5/(2*x**6 + 5), x) == 5*log(2*x**6 + 5) / 12
assert heurisch(1/x**2, x) == -1/x
assert heurisch(-1/x**5, x) == 1/(4*x**4)
def test_heurisch_hacking():
assert heurisch(sqrt(1 + 7*x**2), x, hints=[]) == \
x*sqrt(1+7*x**2)/2 + sqrt(7)*asinh(sqrt(7)*x)/14
assert heurisch(sqrt(1 - 7*x**2), x, hints=[]) == \
x*sqrt(1-7*x**2)/2 + sqrt(7)*asin(sqrt(7)*x)/14
assert heurisch(1/sqrt(1 + 7*x**2), x, hints=[]) == \
sqrt(7)*asinh(sqrt(7)*x)/7
assert heurisch(1/sqrt(1 - 7*x**2), x, hints=[]) == \
sqrt(7)*asin(sqrt(7)*x)/7
assert heurisch(exp(-7*x**2),x,hints=[]) == \
sqrt(7*pi)*erf(sqrt(7)*x)/14
assert heurisch(1/sqrt(9 - 4*x**2), x, hints=[]) == \
asin(2*x/3)/2
assert heurisch(1/sqrt(9 + 4*x**2), x, hints=[]) == \
asinh(2*x/3)/2
def test_heurisch_hacking():
assert heurisch(sqrt(1 + 7*x**2), x, hints=[]) == \
x*sqrt(1+7*x**2)/2 + sqrt(7)*asinh(sqrt(7)*x)/14
assert heurisch(sqrt(1 - 7*x**2), x, hints=[]) == \
x*sqrt(1-7*x**2)/2 + sqrt(7)*asin(sqrt(7)*x)/14
assert heurisch(1/sqrt(1 + 7*x**2), x, hints=[]) == \
sqrt(7)*asinh(sqrt(7)*x)/7
assert heurisch(1/sqrt(1 - 7*x**2), x, hints=[]) == \
sqrt(7)*asin(sqrt(7)*x)/7
assert heurisch(exp(-7*x**2),x,hints=[]) == \
sqrt(7*pi)*erf(sqrt(7)*x)/14
assert heurisch(1/sqrt(9 - 4*x**2), x, hints=[]) == \
asin(2*x/3)/2
assert heurisch(1/sqrt(9 + 4*x**2), x, hints=[]) == \
asinh(2*x/3)/2
def test_pmint_trig():
skip('takes too much time')
f = (x-tan(x)) / tan(x)**2 + tan(x)
g = (-x - tan(x)*x**2 / 2) / tan(x) + log(1+tan(x)**2) / 2
assert heurisch(f, x) == g
def test_pmint_logexp():
skip('takes too much time')
f = (1+x+x*exp(x))*(x+log(x)+exp(x)-1)/(x+log(x)+exp(x))**2/x
g = 1/(x+log(x)+exp(x)) + log(x + log(x) + exp(x))
assert heurisch(f, x) == g
def _eval_integral(self, f, x):
"""Calculate the anti-derivative to the function f(x).
This is a powerful function that should in theory be able to integrate
everything that can be integrated. If you find something, that it
doesn't, it is easy to implement it.
(1) Simple heuristics (based on pattern matching and integral table):
- most frequently used functions (e.g. polynomials)
- functions non-integrable by any of the following algorithms (e.g.
exp(-x**2))
(2) Integration of rational functions:
(a) using apart() - apart() is full partial fraction decomposition
procedure based on Bronstein-Salvy algorithm. It gives formal
decomposition with no polynomial factorization at all (so it's fast
and gives the most general results). However it needs much better
implementation of RootsOf class (if fact any implementation).
(b) using Trager's algorithm - possibly faster than (a) but needs
implementation :)
(3) Whichever implementation of pmInt (Mateusz, Kirill's or a
combination of both).
- this way we can handle efficiently huge class of elementary and
special functions
(4) Recursive Risch algorithm as described in Bronstein's integration
tutorial.
- this way we can handle those integrable functions for which (3)
fails
(5) Powerful heuristics based mostly on user defined rules.
- handle complicated, rarely used cases
"""
# if it is a poly(x) then let the polynomial integrate itself (fast)
#
# It is important to make this check first, otherwise the other code
# will return a sympy expression instead of a Polynomial.
#
# see Polynomial for details.
if isinstance(f, Poly):
return f.integrate(x)
# Piecewise antiderivatives need to call special integrate.
if f.func is Piecewise:
return f._eval_integral(x)
# let's cut it short if `f` does not depend on `x`
if not f.has(x):
return f*x
# try to convert to poly(x) and then integrate if successful (fast)
poly = f.as_poly(x)
if poly is not None:
return poly.integrate(x).as_basic()
# since Integral(f=g1+g2+...) == Integral(g1) + Integral(g2) + ...
# we are going to handle Add terms separately,
# if `f` is not Add -- we only have one term
parts = []
for g in make_list(f, Add):
coeff, g = g.as_independent(x)
# g(x) = const
if g is S.One:
parts.append(coeff * x)
continue
# c
# g(x) = (a*x+b)
if g.is_Pow and not g.exp.has(x):
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
M = g.base.match(a*x + b)
if M is not None:
if g.exp == -1:
h = C.log(g.base)
else:
h = g.base**(g.exp+1) / (g.exp+1)
parts.append(coeff * h / M[a])
continue
# poly(x)
# g(x) = -------
# poly(x)
if g.is_rational_function(x):
parts.append(coeff * ratint(g, x))
continue
# g(x) = Mul(trig)
h = trigintegrate(g, x)
if h is not None:
parts.append(coeff * h)
continue
# g(x) has at least a DiracDelta term
h = deltaintegrate(g,x)
if h is not None:
parts.append(coeff * h)
continue
# fall back to the more general algorithm
h = heurisch(g, x, hints=[])
if h is not None:
parts.append(coeff * h)
else:
return None
return C.Add(*parts)
def test_heurisch_symbolic_coeffs_1130():
assert heurisch(1/(x**2+y), x) in [I/sqrt(y)*log(x + sqrt(-y))/2 - \
I/sqrt(y)*log(x - sqrt(-y))/2, I*log(x + I*sqrt(y)) / \
(2*sqrt(y)) - I*log(x - I*sqrt(y))/(2*sqrt(y))]
def test_heurisch_special():
assert heurisch(erf(x), x) == x*erf(x) + exp(-x**2)/sqrt(pi)
assert heurisch(exp(-x**2)*erf(x), x) == sqrt(pi)*erf(x)**2 / 4
def _eval_integral(self, f, x, meijerg=None):
"""Calculate the anti-derivative to the function f(x).
This is a powerful function that should in theory be able to integrate
everything that can be integrated. If you find something, that it
doesn't, it is easy to implement it.
(1) Simple heuristics (based on pattern matching and integral table):
- most frequently used functions (e.g. polynomials)
- functions non-integrable by any of the following algorithms (e.g.
exp(-x**2))
(2) Integration of rational functions:
(a) using apart() - apart() is full partial fraction decomposition
procedure based on Bronstein-Salvy algorithm. It gives formal
decomposition with no polynomial factorization at all (so it's fast
and gives the most general results). However it needs much better
implementation of RootsOf class (if fact any implementation).
(b) using Trager's algorithm - possibly faster than (a) but needs
implementation :)
(3) Whichever implementation of pmInt (Mateusz, Kirill's or a
combination of both).
- this way we can handle efficiently huge class of elementary and
special functions
(4) Recursive Risch algorithm as described in Bronstein's integration
tutorial.
- this way we can handle those integrable functions for which (3)
fails
(5) Powerful heuristics based mostly on user defined rules.
- handle complicated, rarely used cases
"""
# if it is a poly(x) then let the polynomial integrate itself (fast)
#
# It is important to make this check first, otherwise the other code
# will return a sympy expression instead of a Polynomial.
#
# see Polynomial for details.
if isinstance(f, Poly) and not meijerg:
return f.integrate(x)
# Piecewise antiderivatives need to call special integrate.
if f.func is Piecewise:
return f._eval_integral(x)
# let's cut it short if `f` does not depend on `x`
if not f.has(x):
return f*x
# try to convert to poly(x) and then integrate if successful (fast)
poly = f.as_poly(x)
if poly is not None and not meijerg:
return poly.integrate().as_expr()
# since Integral(f=g1+g2+...) == Integral(g1) + Integral(g2) + ...
# we are going to handle Add terms separately,
# if `f` is not Add -- we only have one term
parts = []
args = Add.make_args(f)
for g in args:
coeff, g = g.as_independent(x)
# g(x) = const
if g is S.One and not meijerg:
parts.append(coeff*x)
continue
# g(x) = expr + O(x**n)
order_term = g.getO()
if order_term is not None:
h = self._eval_integral(g.removeO(), x)
if h is not None:
h_order_expr = self._eval_integral(order_term.expr, x)
if h_order_expr is not None:
h_order_term = order_term.func(h_order_expr, *order_term.variables)
parts.append(coeff*(h + h_order_term))
continue
# NOTE: if there is O(x**n) and we fail to integrate then there is
# no point in trying other methods because they will fail anyway.
return None
# c
# g(x) = (a*x+b)
if g.is_Pow and not g.exp.has(x) and not meijerg:
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
M = g.base.match(a*x + b)
if M is not None:
if g.exp == -1:
h = C.log(g.base)
else:
h = g.base**(g.exp + 1) / (g.exp + 1)
parts.append(coeff * h / M[a])
continue
# poly(x)
# g(x) = -------
# poly(x)
if g.is_rational_function(x) and not meijerg:
parts.append(coeff * ratint(g, x))
continue
if not meijerg:
# g(x) = Mul(trig)
h = trigintegrate(g, x)
if h is not None:
parts.append(coeff * h)
continue
# g(x) has at least a DiracDelta term
h = deltaintegrate(g, x)
if h is not None:
parts.append(coeff * h)
continue
if not meijerg:
# fall back to the more general algorithm
try:
h = heurisch(g, x, hints=[])
except PolynomialError:
# XXX: this exception means there is a bug in the
# implementation of heuristic Risch integration
# algorithm.
h = None
else:
h = None
if meijerg is not False and h is None:
# rewrite using G functions
h = meijerint_indefinite(g, x)
if h is not None:
parts.append(coeff * h)
continue
# if we failed maybe it was because we had
# a product that could have been expanded,
# so let's try an expansion of the whole
# thing before giving up; we don't try this
# out the outset because there are things
# that cannot be solved unless they are
# NOT expanded e.g., x**x*(1+log(x)). There
# should probably be a checker somewhere in this
# routine to look for such cases and try to do
# collection on the expressions if they are already
# in an expanded form
if not h and len(args) == 1:
f = f.expand(mul=True, deep=False)
if f.is_Add:
return self._eval_integral(f, x, meijerg)
if h is not None:
parts.append(coeff * h)
else:
return None
return Add(*parts)
def test_atom_bug():
from sympy import meijerg
from sympy.integrals.risch import heurisch
assert heurisch(meijerg([], [], [1], [], x), x) is None
def test_heurisch_symbolic_coeffs_1130():
assert heurisch(1/(x**2+y), x) in [I*y**(-S.Half)*log(x + (-y)**S.Half)/2 - \
I*y**(-S.Half)*log(x - (-y)**S.Half)/2, I*log(x + I*y**Rational(1,2)) / \
(2*y**Rational(1,2)) - I*log(x - I*y**Rational(1,2))/(2*y**Rational(1,2))]
def test_heurisch_symbolic_coeffs():
assert heurisch(1 / (x + y), x) == log(x + y)
assert heurisch(1 / (x + sqrt(2)), x) == log(x + sqrt(2))
assert simplify(diff(heurisch(log(x + y + z), y), y)) == log(x + y + z)
def test_heurisch_special():
assert heurisch(erf(x), x) == x * erf(x) + exp(-x**2) / sqrt(pi)
assert heurisch(exp(-x**2) * erf(x), x) == sqrt(pi) * erf(x)**2 / 4
def test_heurisch_mixed():
assert heurisch(sin(x) * exp(x),
x) == exp(x) * sin(x) / 2 - exp(x) * cos(x) / 2
def test_pmint_lambertw():
g = (x**2 + (LambertW(x) * x)**2 - LambertW(x) * x**2) / (x * LambertW(x))
assert simplify(heurisch(LambertW(x), x) - g) == 0
def test_heurisch_hyperbolic():
assert heurisch(sinh(x), x) == cosh(x)
assert heurisch(cosh(x), x) == sinh(x)
assert heurisch(x*sinh(x), x) == x*cosh(x) - sinh(x)
assert heurisch(x*cosh(x), x) == x*sinh(x) - cosh(x)
def test_heurisch_symbolic_coeffs_1130():
assert heurisch(1/(x**2+y), x) in [I/sqrt(y)*log(x + sqrt(-y))/2 - \
I/sqrt(y)*log(x - sqrt(-y))/2, I*log(x + I*sqrt(y)) / \
(2*sqrt(y)) - I*log(x - I*sqrt(y))/(2*sqrt(y))]
def test_pmint_rat():
skip('takes too much time')
f = (x**7-24*x**4-4*x**2+8*x-8) / (x**8+6*x**6+12*x**4+8*x**2)
g = (4 + 8*x**2 + 6*x + 3*x**3) / (x*(x**4 + 4*x**2 + 4)) + log(x)
assert heurisch(f, x) == g
def test_heurisch_trigonometric():
assert heurisch(sin(x), x) == -cos(x)
assert heurisch(pi * sin(x) + 1, x) == x - pi * cos(x)
assert heurisch(cos(x), x) == sin(x)
assert heurisch(tan(x), x) in [
log(1 + tan(x)**2) / 2,
log(tan(x) + I) + I * x,
log(tan(x) - I) - I * x,
]
assert heurisch(sin(x) * sin(y), x) == -cos(x) * sin(y)
assert heurisch(sin(x) * sin(y), y) == -cos(y) * sin(x)
# gives sin(x) in answer when run via setup.py and cos(x) when run via py.test
assert heurisch(sin(x) * cos(x), x) in [sin(x)**2 / 2, -cos(x)**2 / 2]
assert heurisch(cos(x) / sin(x), x) == log(sin(x))
assert heurisch(x * sin(7 * x), x) == sin(7 * x) / 49 - x * cos(7 * x) / 7
assert heurisch(
1 / pi / 4 * x**2 * cos(x),
x) == 1 / pi / 4 * (x**2 * sin(x) - 2 * sin(x) + 2 * x * cos(x))
assert heurisch(acos(x/4) * asin(x/4), x) == 2*x - ((16-x**2)**Rational(1,2))*asin(x/4) \
+ ((16 - x**2)**Rational(1,2))*acos(x/4) + x*asin(x/4)*acos(x/4)
def test_pmint_logexp():
f = (1 + x + x * exp(x)) * (x + log(x) + exp(x) - 1) / (x + log(x) +
exp(x))**2 / x
g = 1 / (x + log(x) + exp(x)) + log(x + log(x) + exp(x))
assert heurisch(f, x) == g
def _eval_integral(self, f, x):
"""Calculate the anti-derivative to the function f(x).
This is a powerful function that should in theory be able to integrate
everything that can be integrated. If you find something, that it
doesn't, it is easy to implement it.
(1) Simple heuristics (based on pattern matching and integral table):
- most frequently used functions (e.g. polynomials)
- functions non-integrable by any of the following algorithms (e.g.
exp(-x**2))
(2) Integration of rational functions:
(a) using apart() - apart() is full partial fraction decomposition
procedure based on Bronstein-Salvy algorithm. It gives formal
decomposition with no polynomial factorization at all (so it's fast
and gives the most general results). However it needs much better
implementation of RootsOf class (if fact any implementation).
(b) using Trager's algorithm - possibly faster than (a) but needs
implementation :)
(3) Whichever implementation of pmInt (Mateusz, Kirill's or a
combination of both).
- this way we can handle efficiently huge class of elementary and
special functions
(4) Recursive Risch algorithm as described in Bronstein's integration
tutorial.
- this way we can handle those integrable functions for which (3)
fails
(5) Powerful heuristics based mostly on user defined rules.
- handle complicated, rarely used cases
"""
# if it is a poly(x) then let the polynomial integrate itself (fast)
#
# It is important to make this check first, otherwise the other code
# will return a sympy expression instead of a Polynomial.
#
# see Polynomial for details.
if isinstance(f, Poly):
return f.integrate(x)
# Piecewise antiderivatives need to call special integrate.
if f.func is Piecewise:
return f._eval_integral(x)
# let's cut it short if `f` does not depend on `x`
if not f.has(x):
return f*x
# try to convert to poly(x) and then integrate if successful (fast)
poly = f.as_poly(x)
if poly is not None:
return poly.integrate().as_basic()
# since Integral(f=g1+g2+...) == Integral(g1) + Integral(g2) + ...
# we are going to handle Add terms separately,
# if `f` is not Add -- we only have one term
parts = []
args = Add.make_args(f)
for g in args:
coeff, g = g.as_independent(x)
# g(x) = const
if g is S.One:
parts.append(coeff*x)
continue
# c
# g(x) = (a*x+b)
if g.is_Pow and not g.exp.has(x):
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
M = g.base.match(a*x + b)
if M is not None:
if g.exp == -1:
h = C.log(g.base)
else:
h = g.base**(g.exp + 1) / (g.exp + 1)
parts.append(coeff * h / M[a])
continue
# poly(x)
# g(x) = -------
# poly(x)
if g.is_rational_function(x):
parts.append(coeff * ratint(g, x))
continue
# g(x) = Mul(trig)
h = trigintegrate(g, x)
if h is not None:
parts.append(coeff * h)
continue
# g(x) has at least a DiracDelta term
h = deltaintegrate(g, x)
if h is not None:
parts.append(coeff * h)
continue
# fall back to the more general algorithm
h = heurisch(g, x, hints=[])
# if we failed maybe it was because we had
# a product that could have been expanded,
# so let's try an expansion of the whole
# thing before giving up; we don't try this
# out the outset because there are things
# that cannot be solved unless they are
# NOT expanded e.g., x**x*(1+log(x)). There
# should probably be a checker somewhere in this
# routine to look for such cases and try to do
# collection on the expressions if they are already
# in an expanded form
if not h and len(args) == 1:
f = f.expand(mul=True, deep=False)
if f.is_Add:
return self._eval_integral(f, x)
if h is not None:
parts.append(coeff * h)
else:
return None
return Add(*parts)
def test_pmint_rat():
f = (x**7-24*x**4-4*x**2+8*x-8) / (x**8+6*x**6+12*x**4+8*x**2)
g = (4 + 8*x**2 + 6*x + 3*x**3) / (x*(x**4 + 4*x**2 + 4)) + log(x)
assert heurisch(f, x) == g
def test_pmint_erf():
f = exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1)
g = sqrt(pi) / 4 * (-1 / (erf(x) - 1) - log(erf(x) + 1) / 2 +
log(erf(x) - 1) / 2)
assert heurisch(f, x) == g
def test_pmint_logexp():
f = (1+x+x*exp(x))*(x+log(x)+exp(x)-1)/(x+log(x)+exp(x))**2/x
g = 1/(x+log(x)+exp(x)) + log(x + log(x) + exp(x))
assert heurisch(f, x) == g
def test_heurisch_symbolic_coeffs():
assert heurisch(1/(x+y), x) == log(x+y)
assert heurisch(1/(x+sqrt(2)), x) == log(x+sqrt(2))
assert simplify(diff(heurisch(log(x+y+z), y), y)) == log(x+y+z)
def test_pmint_lambertw():
g = (x**2 + (LambertW(x)*x)**2 - LambertW(x)*x**2)/(x*LambertW(x))
assert simplify(heurisch(LambertW(x), x) - g) == 0
def test_issue510():
assert heurisch(1/(x * (1 + log(x)**2)), x) == I*log(log(x) + I)/2 - \
I*log(log(x) - I)/2
def test_pmint_erf():
skip('takes too much time')
f = exp(-x**2)*erf(x)/(erf(x)**3-erf(x)**2-erf(x)+1)
g = sqrt(pi)/4 * (-1/(erf(x)-1) - log(erf(x)+1)/2 + log(erf(x)-1)/2)
assert heurisch(f, x) == g
def test_issue510():
assert heurisch(1/(x * (1 + log(x)**2)), x) == I*log(log(x) + I)/2 - \
I*log(log(x) - I)/2
def test_heurisch_hyperbolic():
assert heurisch(sinh(x), x) == cosh(x)
assert heurisch(cosh(x), x) == sinh(x)
assert heurisch(x*sinh(x), x) == x*cosh(x) - sinh(x)
assert heurisch(x*cosh(x), x) == x*sinh(x) - cosh(x)
def test_pmint_trig():
f = (x-tan(x)) / tan(x)**2 + tan(x)
g = (-x - tan(x)*x**2 / 2) / tan(x) + log(1+tan(x)**2) / 2
assert heurisch(f, x) == g
def test_pmint_trig():
f = (x - tan(x)) / tan(x)**2 + tan(x)
g = (-x - tan(x) * x**2 / 2) / tan(x) + log(1 + tan(x)**2) / 2
assert heurisch(f, x) == g
def test_pmint_erf():
f = exp(-x**2)*erf(x)/(erf(x)**3-erf(x)**2-erf(x)+1)
g = sqrt(pi)/4 * (-1/(erf(x)-1) - log(erf(x)+1)/2 + log(erf(x)-1)/2)
assert heurisch(f, x) == g
def test_heurisch_symbolic_coeffs_1130():
assert heurisch(1/(x**2+y), x) == I*y**(-S.Half)*log(x + (-y)**S.Half)/2 - \
I*y**(-S.Half)*log(x - (-y)**S.Half)/2
def test_heurisch_polynomials():
assert heurisch(1, x) == x
assert heurisch(x, x) == x**2/2
assert heurisch(x**17, x) == x**18/18
def test_heurisch_polynomials():
assert heurisch(1, x) == x
assert heurisch(x, x) == x**2 / 2
assert heurisch(x**17, x) == x**18 / 18
def test_heurisch_log():
assert heurisch(log(x), x) == x*log(x) - x
assert heurisch(log(3*x), x) == -x + x*log(3) + x*log(x)
assert heurisch(log(x**2), x) in [x*log(x**2) - 2*x, 2*x*log(x) - 2*x]
def test_heurisch_log():
assert heurisch(log(x), x) == x * log(x) - x
assert heurisch(log(3 * x), x) == -x + x * log(3) + x * log(x)
assert heurisch(log(x**2),
x) in [x * log(x**2) - 2 * x, 2 * x * log(x) - 2 * x]
def test_heurisch_trigonometric():
assert heurisch(sin(x), x) == -cos(x)
assert heurisch(pi*sin(x)+1, x) == x-pi*cos(x)
assert heurisch(cos(x), x) == sin(x)
assert heurisch(tan(x), x) in [
log(1 + tan(x)**2)/2,
log(tan(x) + I) + I*x,
log(tan(x) - I) - I*x,
]
assert heurisch(sin(x)*sin(y), x) == -cos(x)*sin(y)
assert heurisch(sin(x)*sin(y), y) == -cos(y)*sin(x)
# gives sin(x) in answer when run via setup.py and cos(x) when run via py.test
assert heurisch(sin(x)*cos(x), x) in [sin(x)**2 / 2, -cos(x)**2 / 2]
assert heurisch(cos(x)/sin(x), x) == log(sin(x))
assert heurisch(x*sin(7*x), x) == sin(7*x) / 49 - x*cos(7*x) / 7
assert heurisch(1/pi/4 * x**2*cos(x), x) == 1/pi/4*(x**2*sin(x) - 2*sin(x) + 2*x*cos(x))
assert heurisch(acos(x/4) * asin(x/4), x) == 2*x - (sqrt(16-x**2))*asin(x/4) \
+ (sqrt(16 - x**2))*acos(x/4) + x*asin(x/4)*acos(x/4)
def test_heurisch_symbolic_coeffs_1130():
assert heurisch(1/(x**2+y), x) in [I*y**(-S.Half)*log(x + (-y)**S.Half)/2 - \
I*y**(-S.Half)*log(x - (-y)**S.Half)/2, I*log(x + I*y**Rational(1,2)) / \
(2*y**Rational(1,2)) - I*log(x - I*y**Rational(1,2))/(2*y**Rational(1,2))]
def test_heurisch_mixed():
assert heurisch(sin(x)*exp(x), x) == exp(x)*sin(x)/2 - exp(x)*cos(x)/2
def test_pmint_trig():
skip('takes too much time')
f = (x - tan(x)) / tan(x)**2 + tan(x)
g = (-x - tan(x) * x**2 / 2) / tan(x) + log(1 + tan(x)**2) / 2
assert heurisch(f, x) == g
