def test_hyper_as_trig():
from sympy.simplify.fu import _osborne as o, _osbornei as i, TR12
eq = sinh(x)**2 + cosh(x)**2
t, f = hyper_as_trig(eq)
assert f(fu(t)) == cosh(2 * x)
e, f = hyper_as_trig(tanh(x + y))
assert f(TR12(e)) == (tanh(x) + tanh(y)) / (tanh(x) * tanh(y) + 1)
d = Dummy()
assert o(sinh(x), d) == I * sin(x * d)
assert o(tanh(x), d) == I * tan(x * d)
assert o(coth(x), d) == cot(x * d) / I
assert o(cosh(x), d) == cos(x * d)
assert o(sech(x), d) == sec(x * d)
assert o(csch(x), d) == csc(x * d) / I
for func in (sinh, cosh, tanh, coth, sech, csch):
h = func(pi)
assert i(o(h, d), d) == h
# /!\ the _osborne functions are not meant to work
# in the o(i(trig, d), d) direction so we just check
# that they work as they are supposed to work
assert i(cos(x * y + z), y) == cosh(x + z * I)
assert i(sin(x * y + z), y) == sinh(x + z * I) / I
assert i(tan(x * y + z), y) == tanh(x + z * I) / I
assert i(cot(x * y + z), y) == coth(x + z * I) * I
assert i(sec(x * y + z), y) == sech(x + z * I)
assert i(csc(x * y + z), y) == csch(x + z * I) * I
def test_math_functions():
dort('sin(37)', sympy.sin(37))
dort('cos(38)', sympy.cos(38))
dort('tan(38)', sympy.tan(38))
dort('sec(39)', sympy.sec(39))
dort('csc(40)', sympy.csc(40))
dort('cot(41)', sympy.cot(41))
dort('asin(42)', sympy.asin(42))
dort('acos(43)', sympy.acos(43))
dort('atan(44)', sympy.atan(44))
dort('asec(45)', sympy.asec(45))
dort('acsc(46)', sympy.acsc(46))
dort('acot(47)', sympy.acot(47))
dort('sind(37)', sympy.sin(37 * sympy.pi / sympy.Number(180)))
dort('cosd(38)', sympy.cos(38 * sympy.pi / sympy.Number(180)))
dort('tand(38)', sympy.tan(38 * sympy.pi / sympy.Number(180)))
dort('secd(39)', sympy.sec(39 * sympy.pi / sympy.Number(180)))
dort('cscd(40)', sympy.csc(40 * sympy.pi / sympy.Number(180)))
dort('cotd(41)', sympy.cot(41 * sympy.pi / sympy.Number(180)))
dort('asind(42)', sympy.asin(42) * sympy.Number(180) / sympy.pi)
dort('acosd(43)', sympy.acos(43) * sympy.Number(180) / sympy.pi)
dort('atand(44)', sympy.atan(44) * sympy.Number(180) / sympy.pi)
dort('asecd(45)', sympy.asec(45) * sympy.Number(180) / sympy.pi)
dort('acscd(46)', sympy.acsc(46) * sympy.Number(180) / sympy.pi)
dort('acotd(47)', sympy.acot(47) * sympy.Number(180) / sympy.pi)
dort('sinh(4)', sympy.sinh(4))
dort('cosh(5)', sympy.cosh(5))
dort('tanh(6)', sympy.tanh(6))
dort('asinh(4)', sympy.asinh(4))
dort('acosh(5)', sympy.acosh(5))
dort('atanh(6)', sympy.atanh(6))
dort('int(E)', int(sympy.E))
dort('int(-E)', int(-sympy.E))
def test_periodicity():
x = Symbol('x')
y = Symbol('y')
assert periodicity(sin(2*x), x) == pi
assert periodicity((-2)*tan(4*x), x) == pi/4
assert periodicity(sin(x)**2, x) == 2*pi
assert periodicity(3**tan(3*x), x) == pi/3
assert periodicity(tan(x)*cos(x), x) == 2*pi
assert periodicity(sin(x)**(tan(x)), x) == 2*pi
assert periodicity(tan(x)*sec(x), x) == 2*pi
assert periodicity(sin(2*x)*cos(2*x) - y, x) == pi/2
assert periodicity(tan(x) + cot(x), x) == pi
assert periodicity(sin(x) - cos(2*x), x) == 2*pi
assert periodicity(sin(x) - 1, x) == 2*pi
assert periodicity(sin(4*x) + sin(x)*cos(x), x) == pi
assert periodicity(exp(sin(x)), x) == 2*pi
assert periodicity(log(cot(2*x)) - sin(cos(2*x)), x) == pi
assert periodicity(sin(2*x)*exp(tan(x) - csc(2*x)), x) == pi
assert periodicity(cos(sec(x) - csc(2*x)), x) == 2*pi
assert periodicity(tan(sin(2*x)), x) == pi
assert periodicity(2*tan(x)**2, x) == pi
assert periodicity(sin(x)**2 + cos(x)**2, x) == S.Zero
assert periodicity(tan(x), y) == S.Zero
assert periodicity(exp(x), x) is None
assert periodicity(log(x), x) is None
assert periodicity(exp(x)**sin(x), x) is None
assert periodicity(sin(x)**y, y) is None
def trig_rule(integral):
integrand, symbol = integral
if isinstance(integrand, sympy.sin) or isinstance(integrand, sympy.cos):
arg = integrand.args[0]
if not isinstance(arg, sympy.Symbol):
return # perhaps a substitution can deal with it
if isinstance(integrand, sympy.sin):
func = "sin"
else:
func = "cos"
return TrigRule(func, arg, integrand, symbol)
if integrand == sympy.sec(symbol) ** 2:
return TrigRule("sec**2", symbol, integrand, symbol)
elif integrand == sympy.csc(symbol) ** 2:
return TrigRule("csc**2", symbol, integrand, symbol)
if isinstance(integrand, sympy.tan):
rewritten = sympy.sin(*integrand.args) / sympy.cos(*integrand.args)
elif isinstance(integrand, sympy.cot):
rewritten = sympy.cos(*integrand.args) / sympy.sin(*integrand.args)
elif isinstance(integrand, sympy.sec):
arg = integrand.args[0]
rewritten = (sympy.sec(arg) ** 2 + sympy.tan(arg) * sympy.sec(arg)) / (sympy.sec(arg) + sympy.tan(arg))
elif isinstance(integrand, sympy.csc):
arg = integrand.args[0]
rewritten = (sympy.csc(arg) ** 2 + sympy.cot(arg) * sympy.csc(arg)) / (sympy.csc(arg) + sympy.cot(arg))
else:
return
return RewriteRule(rewritten, integral_steps(rewritten, symbol), integrand, symbol)
def trig_rule(integral):
integrand, symbol = integral
if isinstance(integrand, sympy.sin) or isinstance(integrand, sympy.cos):
arg = integrand.args[0]
if not isinstance(arg, sympy.Symbol):
return # perhaps a substitution can deal with it
if isinstance(integrand, sympy.sin):
func = 'sin'
else:
func = 'cos'
return TrigRule(func, arg, integrand, symbol)
if isinstance(integrand, sympy.tan):
rewritten = sympy.sin(*integrand.args) / sympy.cos(*integrand.args)
elif isinstance(integrand, sympy.cot):
rewritten = sympy.cos(*integrand.args) / sympy.sin(*integrand.args)
elif isinstance(integrand, sympy.sec):
arg = integrand.args[0]
rewritten = ((sympy.sec(arg)**2 + sympy.tan(arg) * sympy.sec(arg)) /
(sympy.sec(arg) + sympy.tan(arg)))
elif isinstance(integrand, sympy.csc):
arg = integrand.args[0]
rewritten = ((sympy.csc(arg)**2 + sympy.cot(arg) * sympy.csc(arg)) /
(sympy.csc(arg) + sympy.cot(arg)))
return RewriteRule(rewritten, integral_steps(rewritten, symbol), integrand,
symbol)
def test_periodicity():
x = Symbol('x')
y = Symbol('y')
assert periodicity(sin(2 * x), x) == pi
assert periodicity((-2) * tan(4 * x), x) == pi / 4
assert periodicity(sin(x)**2, x) == 2 * pi
assert periodicity(3**tan(3 * x), x) == pi / 3
assert periodicity(tan(x) * cos(x), x) == 2 * pi
assert periodicity(sin(x)**(tan(x)), x) == 2 * pi
assert periodicity(tan(x) * sec(x), x) == 2 * pi
assert periodicity(sin(2 * x) * cos(2 * x) - y, x) == pi / 2
assert periodicity(tan(x) + cot(x), x) == pi
assert periodicity(sin(x) - cos(2 * x), x) == 2 * pi
assert periodicity(sin(x) - 1, x) == 2 * pi
assert periodicity(sin(4 * x) + sin(x) * cos(x), x) == pi
assert periodicity(exp(sin(x)), x) == 2 * pi
assert periodicity(log(cot(2 * x)) - sin(cos(2 * x)), x) == pi
assert periodicity(sin(2 * x) * exp(tan(x) - csc(2 * x)), x) == pi
assert periodicity(cos(sec(x) - csc(2 * x)), x) == 2 * pi
assert periodicity(tan(sin(2 * x)), x) == pi
assert periodicity(2 * tan(x)**2, x) == pi
assert periodicity(sin(x)**2 + cos(x)**2, x) == S.Zero
assert periodicity(tan(x), y) == S.Zero
assert periodicity(exp(x), x) is None
assert periodicity(log(x), x) is None
assert periodicity(exp(x)**sin(x), x) is None
assert periodicity(sin(x)**y, y) is None
assert periodicity(x**3 - x**2 + 1, x) is None
assert periodicity(Abs(x), x) is None
assert periodicity(Abs(x**2 - 1), x) is None
def test_hyper_as_trig():
from sympy.simplify.fu import _osborne as o, _osbornei as i, TR12
eq = sinh(x)**2 + cosh(x)**2
t, f = hyper_as_trig(eq)
assert f(fu(t)) == cosh(2*x)
e, f = hyper_as_trig(tanh(x + y))
assert f(TR12(e)) == (tanh(x) + tanh(y))/(tanh(x)*tanh(y) + 1)
d = Dummy()
assert o(sinh(x), d) == I*sin(x*d)
assert o(tanh(x), d) == I*tan(x*d)
assert o(coth(x), d) == cot(x*d)/I
assert o(cosh(x), d) == cos(x*d)
assert o(sech(x), d) == sec(x*d)
assert o(csch(x), d) == csc(x*d)/I
for func in (sinh, cosh, tanh, coth, sech, csch):
h = func(pi)
assert i(o(h, d), d) == h
# /!\ the _osborne functions are not meant to work
# in the o(i(trig, d), d) direction so we just check
# that they work as they are supposed to work
assert i(cos(x*y + z), y) == cosh(x + z*I)
assert i(sin(x*y + z), y) == sinh(x + z*I)/I
assert i(tan(x*y + z), y) == tanh(x + z*I)/I
assert i(cot(x*y + z), y) == coth(x + z*I)*I
assert i(sec(x*y + z), y) == sech(x + z*I)
assert i(csc(x*y + z), y) == csch(x + z*I)*I
def test_trigintegrate_mixed():
assert trigintegrate(sin(x) * sec(x), x) == -log(sin(x)**2 - 1) / 2
assert trigintegrate(sin(x) * csc(x), x) == x
assert trigintegrate(sin(x) * cot(x), x) == sin(x)
assert trigintegrate(cos(x) * sec(x), x) == x
assert trigintegrate(cos(x) * csc(x), x) == log(cos(x)**2 - 1) / 2
assert trigintegrate(cos(x) * tan(x), x) == -cos(x)
assert trigintegrate(cos(x)*cot(x), x) == log(cos(x) - 1)/2 \
- log(cos(x) + 1)/2 + cos(x)
def test_manualintegrate_trigonometry():
assert manualintegrate(sin(x), x) == -cos(x)
assert manualintegrate(tan(x), x) == -log(cos(x))
assert manualintegrate(sec(x), x) == log(sec(x) + tan(x))
assert manualintegrate(csc(x), x) == -log(csc(x) + cot(x))
assert manualintegrate(sin(x) * cos(x), x) in [sin(x) ** 2 / 2, -cos(x)**2 / 2]
assert manualintegrate(-sec(x) * tan(x), x) == -sec(x)
assert manualintegrate(csc(x) * cot(x), x) == -csc(x)
def test_manualintegrate_trigonometry():
assert manualintegrate(sin(x), x) == -cos(x)
assert manualintegrate(tan(x), x) == -log(cos(x))
assert manualintegrate(sec(x), x) == log(sec(x) + tan(x))
assert manualintegrate(csc(x), x) == -log(csc(x) + cot(x))
assert manualintegrate(sin(x) * cos(x), x) in [sin(x) ** 2 / 2, -cos(x)**2 / 2]
assert manualintegrate(-sec(x) * tan(x), x) == -sec(x)
assert manualintegrate(csc(x) * cot(x), x) == -csc(x)
def test_trigintegrate_mixed():
assert trigintegrate(sin(x)*sec(x), x) == -log(sin(x)**2 - 1)/2
assert trigintegrate(sin(x)*csc(x), x) == x
assert trigintegrate(sin(x)*cot(x), x) == sin(x)
assert trigintegrate(cos(x)*sec(x), x) == x
assert trigintegrate(cos(x)*csc(x), x) == log(cos(x)**2 - 1)/2
assert trigintegrate(cos(x)*tan(x), x) == -cos(x)
assert trigintegrate(cos(x)*cot(x), x) == log(cos(x) - 1)/2 \
- log(cos(x) + 1)/2 + cos(x)
def test_periodicity():
x = Symbol('x')
y = Symbol('y')
assert periodicity(sin(2 * x), x) == pi
assert periodicity((-2) * tan(4 * x), x) == pi / 4
assert periodicity(sin(x)**2, x) == 2 * pi
assert periodicity(3**tan(3 * x), x) == pi / 3
assert periodicity(tan(x) * cos(x), x) == 2 * pi
assert periodicity(sin(x)**(tan(x)), x) == 2 * pi
assert periodicity(tan(x) * sec(x), x) == 2 * pi
assert periodicity(sin(2 * x) * cos(2 * x) - y, x) == pi / 2
assert periodicity(tan(x) + cot(x), x) == pi
assert periodicity(sin(x) - cos(2 * x), x) == 2 * pi
assert periodicity(sin(x) - 1, x) == 2 * pi
assert periodicity(sin(4 * x) + sin(x) * cos(x), x) == pi
assert periodicity(exp(sin(x)), x) == 2 * pi
assert periodicity(log(cot(2 * x)) - sin(cos(2 * x)), x) == pi
assert periodicity(sin(2 * x) * exp(tan(x) - csc(2 * x)), x) == pi
assert periodicity(cos(sec(x) - csc(2 * x)), x) == 2 * pi
assert periodicity(tan(sin(2 * x)), x) == pi
assert periodicity(2 * tan(x)**2, x) == pi
assert periodicity(sin(x % 4), x) == 4
assert periodicity(sin(x) % 4, x) == 2 * pi
assert periodicity(tan((3 * x - 2) % 4), x) == 4 / 3
assert periodicity((sqrt(2) * (x + 1) + x) % 3, x) == 3 / (sqrt(2) + 1)
assert periodicity((x**2 + 1) % x, x) == None
assert periodicity(sin(x)**2 + cos(x)**2, x) == S.Zero
assert periodicity(tan(x), y) == S.Zero
assert periodicity(exp(x), x) is None
assert periodicity(log(x), x) is None
assert periodicity(exp(x)**sin(x), x) is None
assert periodicity(sin(x)**y, y) is None
assert periodicity(Abs(sin(Abs(sin(x)))), x) == pi
assert all(
periodicity(Abs(f(x)), x) == pi
for f in (cos, sin, sec, csc, tan, cot))
assert periodicity(Abs(sin(tan(x))), x) == pi
assert periodicity(Abs(sin(sin(x) + tan(x))), x) == 2 * pi
assert periodicity(sin(x) > S.Half, x) is 2 * pi
assert periodicity(x > 2, x) is None
assert periodicity(x**3 - x**2 + 1, x) is None
assert periodicity(Abs(x), x) is None
assert periodicity(Abs(x**2 - 1), x) is None
assert periodicity((x**2 + 4) % 2, x) is None
assert periodicity((E**x) % 3, x) is None
def test_periodicity():
x = Symbol('x')
y = Symbol('y')
assert periodicity(sin(2*x), x) == pi
assert periodicity((-2)*tan(4*x), x) == pi/4
assert periodicity(sin(x)**2, x) == 2*pi
assert periodicity(3**tan(3*x), x) == pi/3
assert periodicity(tan(x)*cos(x), x) == 2*pi
assert periodicity(sin(x)**(tan(x)), x) == 2*pi
assert periodicity(tan(x)*sec(x), x) == 2*pi
assert periodicity(sin(2*x)*cos(2*x) - y, x) == pi/2
assert periodicity(tan(x) + cot(x), x) == pi
assert periodicity(sin(x) - cos(2*x), x) == 2*pi
assert periodicity(sin(x) - 1, x) == 2*pi
assert periodicity(sin(4*x) + sin(x)*cos(x), x) == pi
assert periodicity(exp(sin(x)), x) == 2*pi
assert periodicity(log(cot(2*x)) - sin(cos(2*x)), x) == pi
assert periodicity(sin(2*x)*exp(tan(x) - csc(2*x)), x) == pi
assert periodicity(cos(sec(x) - csc(2*x)), x) == 2*pi
assert periodicity(tan(sin(2*x)), x) == pi
assert periodicity(2*tan(x)**2, x) == pi
assert periodicity(sin(x%4), x) == 4
assert periodicity(sin(x)%4, x) == 2*pi
assert periodicity(tan((3*x-2)%4), x) == 4/3
assert periodicity((sqrt(2)*(x+1)+x) % 3, x) == 3 / (sqrt(2)+1)
assert periodicity((x**2+1) % x, x) == None
assert periodicity(sin(x)**2 + cos(x)**2, x) == S.Zero
assert periodicity(tan(x), y) == S.Zero
assert periodicity(exp(x), x) is None
assert periodicity(log(x), x) is None
assert periodicity(exp(x)**sin(x), x) is None
assert periodicity(sin(x)**y, y) is None
assert periodicity(Abs(sin(Abs(sin(x)))),x) == pi
assert all(periodicity(Abs(f(x)),x) == pi for f in (
cos, sin, sec, csc, tan, cot))
assert periodicity(Abs(sin(tan(x))), x) == pi
assert periodicity(Abs(sin(sin(x) + tan(x))), x) == 2*pi
assert periodicity(sin(x) > S.Half, x) is 2*pi
assert periodicity(x > 2, x) is None
assert periodicity(x**3 - x**2 + 1, x) is None
assert periodicity(Abs(x), x) is None
assert periodicity(Abs(x**2 - 1), x) is None
assert periodicity((x**2 + 4)%2, x) is None
assert periodicity((E**x)%3, x) is None
def test_manualintegrate_trigpowers():
assert manualintegrate(sin(x)**2 * cos(x), x) == sin(x)**3 / 3
assert manualintegrate(sin(x)**2 * cos(x) **2, x) == \
x / 8 - sin(4*x) / 32
assert manualintegrate(sin(x) * cos(x)**3, x) == -cos(x)**4 / 4
assert manualintegrate(sin(x)**3 * cos(x)**2, x) == \
cos(x)**5 / 5 - cos(x)**3 / 3
assert manualintegrate(tan(x)**3 * sec(x), x) == sec(x)**3 / 3 - sec(x)
assert manualintegrate(tan(x) * sec(x)**2, x) == sec(x)**2 / 2
assert manualintegrate(cot(x)**5 * csc(x), x) == \
-csc(x)**5/5 + 2*csc(x)**3/3 - csc(x)
assert manualintegrate(cot(x)**2 * csc(x)**6, x) == \
-cot(x)**7/7 - 2*cot(x)**5/5 - cot(x)**3/3
def test_manualintegrate_trigpowers():
assert manualintegrate(sin(x)**2 * cos(x), x) == sin(x)**3 / 3
assert manualintegrate(sin(x)**2 * cos(x) **2, x) == \
x / 8 - sin(4*x) / 32
assert manualintegrate(sin(x) * cos(x)**3, x) == -cos(x)**4 / 4
assert manualintegrate(sin(x)**3 * cos(x)**2, x) == \
cos(x)**5 / 5 - cos(x)**3 / 3
assert manualintegrate(tan(x)**3 * sec(x), x) == sec(x)**3/3 - sec(x)
assert manualintegrate(tan(x) * sec(x) **2, x) == sec(x)**2/2
assert manualintegrate(cot(x)**5 * csc(x), x) == \
-csc(x)**5/5 + 2*csc(x)**3/3 - csc(x)
assert manualintegrate(cot(x)**2 * csc(x)**6, x) == \
-cot(x)**7/7 - 2*cot(x)**5/5 - cot(x)**3/3
def test_sin_rewrite():
assert sin(x).rewrite(exp) == -I * (exp(I * x) - exp(-I * x)) / 2
assert sin(x).rewrite(tan) == 2 * tan(x / 2) / (1 + tan(x / 2)**2)
assert sin(x).rewrite(cot) == 2 * cot(x / 2) / (1 + cot(x / 2)**2)
assert sin(sinh(x)).rewrite(exp).subs(x,
3).n() == sin(x).rewrite(exp).subs(
x, sinh(3)).n()
assert sin(cosh(x)).rewrite(exp).subs(x,
3).n() == sin(x).rewrite(exp).subs(
x, cosh(3)).n()
assert sin(tanh(x)).rewrite(exp).subs(x,
3).n() == sin(x).rewrite(exp).subs(
x, tanh(3)).n()
assert sin(coth(x)).rewrite(exp).subs(x,
3).n() == sin(x).rewrite(exp).subs(
x, coth(3)).n()
assert sin(sin(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(
x, sin(3)).n()
assert sin(cos(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(
x, cos(3)).n()
assert sin(tan(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(
x, tan(3)).n()
assert sin(cot(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(
x, cot(3)).n()
assert sin(log(x)).rewrite(Pow) == I * x**-I / 2 - I * x**I / 2
assert sin(x).rewrite(csc) == 1 / csc(x)
def test_trig_functions(self, printer, x):
# Trig functions
assert printer.doprint(sp.acos(x)) == 'acos(x)'
assert printer.doprint(sp.acosh(x)) == 'acosh(x)'
assert printer.doprint(sp.asin(x)) == 'asin(x)'
assert printer.doprint(sp.asinh(x)) == 'asinh(x)'
assert printer.doprint(sp.atan(x)) == 'atan(x)'
assert printer.doprint(sp.atanh(x)) == 'atanh(x)'
assert printer.doprint(sp.ceiling(x)) == 'ceil(x)'
assert printer.doprint(sp.cos(x)) == 'cos(x)'
assert printer.doprint(sp.cosh(x)) == 'cosh(x)'
assert printer.doprint(sp.exp(x)) == 'exp(x)'
assert printer.doprint(sp.factorial(x)) == 'factorial(x)'
assert printer.doprint(sp.floor(x)) == 'floor(x)'
assert printer.doprint(sp.log(x)) == 'log(x)'
assert printer.doprint(sp.sin(x)) == 'sin(x)'
assert printer.doprint(sp.sinh(x)) == 'sinh(x)'
assert printer.doprint(sp.tan(x)) == 'tan(x)'
assert printer.doprint(sp.tanh(x)) == 'tanh(x)'
# extra trig functions
assert printer.doprint(sp.sec(x)) == '1 / cos(x)'
assert printer.doprint(sp.csc(x)) == '1 / sin(x)'
assert printer.doprint(sp.cot(x)) == '1 / tan(x)'
assert printer.doprint(sp.asec(x)) == 'acos(1 / x)'
assert printer.doprint(sp.acsc(x)) == 'asin(1 / x)'
assert printer.doprint(sp.acot(x)) == 'atan(1 / x)'
assert printer.doprint(sp.sech(x)) == '1 / cosh(x)'
assert printer.doprint(sp.csch(x)) == '1 / sinh(x)'
assert printer.doprint(sp.coth(x)) == '1 / tanh(x)'
assert printer.doprint(sp.asech(x)) == 'acosh(1 / x)'
assert printer.doprint(sp.acsch(x)) == 'asinh(1 / x)'
assert printer.doprint(sp.acoth(x)) == 'atanh(1 / x)'
def csc(self, value):
"""
Verilen değeri kullanarak, sumpy.csc döndürür
:param value: değer
:return: sympy.csc
"""
return sp.csc(value_checker(value))
def test_conv7():
x = Symbol("x")
y = Symbol("y")
assert sin(x / 3) == sin(sympy.Symbol("x") / 3)
assert cos(x / 3) == cos(sympy.Symbol("x") / 3)
assert tan(x / 3) == tan(sympy.Symbol("x") / 3)
assert cot(x / 3) == cot(sympy.Symbol("x") / 3)
assert csc(x / 3) == csc(sympy.Symbol("x") / 3)
assert sec(x / 3) == sec(sympy.Symbol("x") / 3)
assert asin(x / 3) == asin(sympy.Symbol("x") / 3)
assert acos(x / 3) == acos(sympy.Symbol("x") / 3)
assert atan(x / 3) == atan(sympy.Symbol("x") / 3)
assert acot(x / 3) == acot(sympy.Symbol("x") / 3)
assert acsc(x / 3) == acsc(sympy.Symbol("x") / 3)
assert asec(x / 3) == asec(sympy.Symbol("x") / 3)
assert sin(x / 3)._sympy_() == sympy.sin(sympy.Symbol("x") / 3)
assert sin(x / 3)._sympy_() != sympy.cos(sympy.Symbol("x") / 3)
assert cos(x / 3)._sympy_() == sympy.cos(sympy.Symbol("x") / 3)
assert tan(x / 3)._sympy_() == sympy.tan(sympy.Symbol("x") / 3)
assert cot(x / 3)._sympy_() == sympy.cot(sympy.Symbol("x") / 3)
assert csc(x / 3)._sympy_() == sympy.csc(sympy.Symbol("x") / 3)
assert sec(x / 3)._sympy_() == sympy.sec(sympy.Symbol("x") / 3)
assert asin(x / 3)._sympy_() == sympy.asin(sympy.Symbol("x") / 3)
assert acos(x / 3)._sympy_() == sympy.acos(sympy.Symbol("x") / 3)
assert atan(x / 3)._sympy_() == sympy.atan(sympy.Symbol("x") / 3)
assert acot(x / 3)._sympy_() == sympy.acot(sympy.Symbol("x") / 3)
assert acsc(x / 3)._sympy_() == sympy.acsc(sympy.Symbol("x") / 3)
assert asec(x / 3)._sympy_() == sympy.asec(sympy.Symbol("x") / 3)
def test_conv7():
x = Symbol("x")
y = Symbol("y")
assert sin(x/3) == sin(sympy.Symbol("x") / 3)
assert cos(x/3) == cos(sympy.Symbol("x") / 3)
assert tan(x/3) == tan(sympy.Symbol("x") / 3)
assert cot(x/3) == cot(sympy.Symbol("x") / 3)
assert csc(x/3) == csc(sympy.Symbol("x") / 3)
assert sec(x/3) == sec(sympy.Symbol("x") / 3)
assert asin(x/3) == asin(sympy.Symbol("x") / 3)
assert acos(x/3) == acos(sympy.Symbol("x") / 3)
assert atan(x/3) == atan(sympy.Symbol("x") / 3)
assert acot(x/3) == acot(sympy.Symbol("x") / 3)
assert acsc(x/3) == acsc(sympy.Symbol("x") / 3)
assert asec(x/3) == asec(sympy.Symbol("x") / 3)
assert sin(x/3)._sympy_() == sympy.sin(sympy.Symbol("x") / 3)
assert sin(x/3)._sympy_() != sympy.cos(sympy.Symbol("x") / 3)
assert cos(x/3)._sympy_() == sympy.cos(sympy.Symbol("x") / 3)
assert tan(x/3)._sympy_() == sympy.tan(sympy.Symbol("x") / 3)
assert cot(x/3)._sympy_() == sympy.cot(sympy.Symbol("x") / 3)
assert csc(x/3)._sympy_() == sympy.csc(sympy.Symbol("x") / 3)
assert sec(x/3)._sympy_() == sympy.sec(sympy.Symbol("x") / 3)
assert asin(x/3)._sympy_() == sympy.asin(sympy.Symbol("x") / 3)
assert acos(x/3)._sympy_() == sympy.acos(sympy.Symbol("x") / 3)
assert atan(x/3)._sympy_() == sympy.atan(sympy.Symbol("x") / 3)
assert acot(x/3)._sympy_() == sympy.acot(sympy.Symbol("x") / 3)
assert acsc(x/3)._sympy_() == sympy.acsc(sympy.Symbol("x") / 3)
assert asec(x/3)._sympy_() == sympy.asec(sympy.Symbol("x") / 3)
def test_find_substitutions():
assert find_substitutions((cot(x)**2 + 1)**2*csc(x)**2*cot(x)**2, x, u) == \
[(cot(x), 1, -u**6 - 2*u**4 - u**2)]
assert find_substitutions(
(sec(x)**2 + tan(x) * sec(x)) / (sec(x) + tan(x)), x,
u) == [(sec(x) + tan(x), 1, 1 / u)]
assert find_substitutions(x * exp(-x**2), x,
u) == [(-x**2, -S.Half, exp(u))]
def eval_trig(func, arg, integrand, symbol):
if func == 'sin':
return -sympy.cos(arg)
elif func == 'cos':
return sympy.sin(arg)
elif func == 'sec*tan':
return sympy.sec(arg)
elif func == 'csc*cot':
return sympy.csc(arg)
def eval_trig(func, arg, integrand, symbol):
if func == 'sin':
return -sympy.cos(arg)
elif func == 'cos':
return sympy.sin(arg)
elif func == 'sec*tan':
return sympy.sec(arg)
elif func == 'csc*cot':
return sympy.csc(arg)
def test_find_substitutions():
assert find_substitutions(
(cot(x) ** 2 + 1) ** 2 * csc(x) ** 2 * cot(x) ** 2, x, u
) == [(cot(x), 1, -(u ** 6) - 2 * u ** 4 - u ** 2)]
assert find_substitutions(
(sec(x) ** 2 + tan(x) * sec(x)) / (sec(x) + tan(x)), x, u
) == [(sec(x) + tan(x), 1, 1 / u)]
assert find_substitutions(x * exp(-(x ** 2)), x, u) == [
(-(x ** 2), Rational(-1, 2), exp(u))
]
def test_inverses():
raises(AttributeError, lambda: sin(x).inverse())
raises(AttributeError, lambda: cos(x).inverse())
assert tan(x).inverse() == atan
assert cot(x).inverse() == acot
raises(AttributeError, lambda: csc(x).inverse())
raises(AttributeError, lambda: sec(x).inverse())
assert asin(x).inverse() == sin
assert acos(x).inverse() == cos
assert atan(x).inverse() == tan
assert acot(x).inverse() == cot
def test_inverses():
raises(AttributeError, lambda: sin(x).inverse())
raises(AttributeError, lambda: cos(x).inverse())
assert tan(x).inverse() == atan
assert cot(x).inverse() == acot
raises(AttributeError, lambda: csc(x).inverse())
raises(AttributeError, lambda: sec(x).inverse())
assert asin(x).inverse() == sin
assert acos(x).inverse() == cos
assert atan(x).inverse() == tan
assert acot(x).inverse() == cot
def manual_diff(f, symbol):
"""Derivative of f in form expected by find_substitutions
SymPy's derivatives for some trig functions (like cot) aren't in a form
that works well with finding substitutions; this replaces the
derivatives for those particular forms with something that works better.
"""
if f.args:
arg = f.args[0]
if isinstance(f, sympy.tan):
return arg.diff(symbol) * sympy.sec(arg)**2
elif isinstance(f, sympy.cot):
return -arg.diff(symbol) * sympy.csc(arg)**2
elif isinstance(f, sympy.sec):
return arg.diff(symbol) * sympy.sec(arg) * sympy.tan(arg)
elif isinstance(f, sympy.csc):
return -arg.diff(symbol) * sympy.csc(arg) * sympy.cot(arg)
elif isinstance(f, sympy.Add):
return sum([manual_diff(arg, symbol) for arg in f.args])
return f.diff(symbol)
def manual_diff(f, symbol):
"""Derivative of f in form expected by find_substitutions
SymPy's derivatives for some trig functions (like cot) aren't in a form
that works well with finding substitutions; this replaces the
derivatives for those particular forms with something that works better.
"""
if f.args:
arg = f.args[0]
if isinstance(f, sympy.tan):
return arg.diff(symbol) * sympy.sec(arg)**2
elif isinstance(f, sympy.cot):
return -arg.diff(symbol) * sympy.csc(arg)**2
elif isinstance(f, sympy.sec):
return arg.diff(symbol) * sympy.sec(arg) * sympy.tan(arg)
elif isinstance(f, sympy.csc):
return -arg.diff(symbol) * sympy.csc(arg) * sympy.cot(arg)
elif isinstance(f, sympy.Add):
return sum([manual_diff(arg, symbol) for arg in f.args])
return f.diff(symbol)
def eval_trig(func, arg, integrand, symbol):
if func == "sin":
return -sympy.cos(arg)
elif func == "cos":
return sympy.sin(arg)
elif func == "sec*tan":
return sympy.sec(arg)
elif func == "csc*cot":
return sympy.csc(arg)
elif func == "sec**2":
return sympy.tan(arg)
elif func == "csc**2":
return -sympy.cot(arg)
def trig_powers_products_rule(integral):
integrand, symbol = integral
if any(integrand.has(f) for f in (sympy.sin, sympy.cos)):
pattern, a, b, m, n = sincos_pattern(symbol)
match = integrand.match(pattern)
if match:
a, b, m, n = match.get(a, 0), match.get(b, 0), match.get(
m, 0), match.get(n, 0)
return multiplexer({
sincos_botheven_condition: sincos_botheven,
sincos_sinodd_condition: sincos_sinodd,
sincos_cosodd_condition: sincos_cosodd
})((a, b, m, n, integrand, symbol))
integrand = integrand.subs({1 / sympy.cos(symbol): sympy.sec(symbol)})
if any(integrand.has(f) for f in (sympy.tan, sympy.sec)):
pattern, a, b, m, n = tansec_pattern(symbol)
match = integrand.match(pattern)
if match:
a, b, m, n = match.get(a, 0), match.get(b, 0), match.get(
m, 0), match.get(n, 0)
return multiplexer({
tansec_tanodd_condition: tansec_tanodd,
tansec_seceven_condition: tansec_seceven
})((a, b, m, n, integrand, symbol))
integrand = integrand.subs({
1 / sympy.sin(symbol):
sympy.csc(symbol),
1 / sympy.tan(symbol):
sympy.cot(symbol),
sympy.cos(symbol) / sympy.tan(symbol):
sympy.cot(symbol)
})
if any(integrand.has(f) for f in (sympy.cot, sympy.csc)):
pattern, a, b, m, n = cotcsc_pattern(symbol)
match = integrand.match(pattern)
if match:
a, b, m, n = match.get(a, 0), match.get(b, 0), match.get(
m, 0), match.get(n, 0)
return multiplexer({
cotcsc_cotodd_condition: cotcsc_cotodd,
cotcsc_csceven_condition: cotcsc_csceven
})((a, b, m, n, integrand, symbol))
def test_conv7b():
x = sympy.Symbol("x")
y = sympy.Symbol("y")
assert sympify(sympy.sin(x / 3)) == sin(Symbol("x") / 3)
assert sympify(sympy.sin(x / 3)) != cos(Symbol("x") / 3)
assert sympify(sympy.cos(x / 3)) == cos(Symbol("x") / 3)
assert sympify(sympy.tan(x / 3)) == tan(Symbol("x") / 3)
assert sympify(sympy.cot(x / 3)) == cot(Symbol("x") / 3)
assert sympify(sympy.csc(x / 3)) == csc(Symbol("x") / 3)
assert sympify(sympy.sec(x / 3)) == sec(Symbol("x") / 3)
assert sympify(sympy.asin(x / 3)) == asin(Symbol("x") / 3)
assert sympify(sympy.acos(x / 3)) == acos(Symbol("x") / 3)
assert sympify(sympy.atan(x / 3)) == atan(Symbol("x") / 3)
assert sympify(sympy.acot(x / 3)) == acot(Symbol("x") / 3)
assert sympify(sympy.acsc(x / 3)) == acsc(Symbol("x") / 3)
assert sympify(sympy.asec(x / 3)) == asec(Symbol("x") / 3)
def test_hyper_as_trig():
from sympy.simplify.fu import _osborne, _osbornei
eq = sinh(x)**2 + cosh(x)**2
t, f = hyper_as_trig(eq)
assert f(fu(t)) == cosh(2*x)
assert _osborne(cosh(x)) == cos(x)
assert _osborne(sinh(x)) == I*sin(x)
assert _osborne(tanh(x)) == I*tan(x)
assert _osborne(coth(x)) == cot(x)/I
assert _osbornei(cos(x)) == cosh(x)
assert _osbornei(sin(x)) == sinh(x)/I
assert _osbornei(tan(x)) == tanh(x)/I
assert _osbornei(cot(x)) == coth(x)*I
assert _osbornei(sec(x)) == 1/cosh(x)
assert _osbornei(csc(x)) == I/sinh(x)
def test_conv7b():
x = sympy.Symbol("x")
y = sympy.Symbol("y")
assert sympify(sympy.sin(x/3)) == sin(Symbol("x") / 3)
assert sympify(sympy.sin(x/3)) != cos(Symbol("x") / 3)
assert sympify(sympy.cos(x/3)) == cos(Symbol("x") / 3)
assert sympify(sympy.tan(x/3)) == tan(Symbol("x") / 3)
assert sympify(sympy.cot(x/3)) == cot(Symbol("x") / 3)
assert sympify(sympy.csc(x/3)) == csc(Symbol("x") / 3)
assert sympify(sympy.sec(x/3)) == sec(Symbol("x") / 3)
assert sympify(sympy.asin(x/3)) == asin(Symbol("x") / 3)
assert sympify(sympy.acos(x/3)) == acos(Symbol("x") / 3)
assert sympify(sympy.atan(x/3)) == atan(Symbol("x") / 3)
assert sympify(sympy.acot(x/3)) == acot(Symbol("x") / 3)
assert sympify(sympy.acsc(x/3)) == acsc(Symbol("x") / 3)
assert sympify(sympy.asec(x/3)) == asec(Symbol("x") / 3)
def test_hyper_as_trig():
from sympy.simplify.fu import _osborne, _osbornei
eq = sinh(x)**2 + cosh(x)**2
t, f = hyper_as_trig(eq)
assert f(fu(t)) == cosh(2 * x)
assert _osborne(cosh(x)) == cos(x)
assert _osborne(sinh(x)) == I * sin(x)
assert _osborne(tanh(x)) == I * tan(x)
assert _osborne(coth(x)) == cot(x) / I
assert _osbornei(cos(x)) == cosh(x)
assert _osbornei(sin(x)) == sinh(x) / I
assert _osbornei(tan(x)) == tanh(x) / I
assert _osbornei(cot(x)) == coth(x) * I
assert _osbornei(sec(x)) == 1 / cosh(x)
assert _osbornei(csc(x)) == I / sinh(x)
def trig_powers_products_rule(integral):
integrand, symbol = integral
if any(integrand.has(f) for f in (sympy.sin, sympy.cos)):
pattern, a, b, m, n = sincos_pattern(symbol)
match = integrand.match(pattern)
if match:
a, b, m, n = match.get(a, 0),match.get(b, 0), match.get(m, 0), match.get(n, 0)
return multiplexer({
sincos_botheven_condition: sincos_botheven,
sincos_sinodd_condition: sincos_sinodd,
sincos_cosodd_condition: sincos_cosodd
})((a, b, m, n, integrand, symbol))
integrand = integrand.subs({
1 / sympy.cos(symbol): sympy.sec(symbol)
})
if any(integrand.has(f) for f in (sympy.tan, sympy.sec)):
pattern, a, b, m, n = tansec_pattern(symbol)
match = integrand.match(pattern)
if match:
a, b, m, n = match.get(a, 0),match.get(b, 0), match.get(m, 0), match.get(n, 0)
return multiplexer({
tansec_tanodd_condition: tansec_tanodd,
tansec_seceven_condition: tansec_seceven
})((a, b, m, n, integrand, symbol))
integrand = integrand.subs({
1 / sympy.sin(symbol): sympy.csc(symbol),
1 / sympy.tan(symbol): sympy.cot(symbol),
sympy.cos(symbol) / sympy.tan(symbol): sympy.cot(symbol)
})
if any(integrand.has(f) for f in (sympy.cot, sympy.csc)):
pattern, a, b, m, n = cotcsc_pattern(symbol)
match = integrand.match(pattern)
if match:
a, b, m, n = match.get(a, 0),match.get(b, 0), match.get(m, 0), match.get(n, 0)
return multiplexer({
cotcsc_cotodd_condition: cotcsc_cotodd,
cotcsc_csceven_condition: cotcsc_csceven
})((a, b, m, n, integrand, symbol))
def trig_cotcsc_rule(integral):
integrand, symbol = integral
integrand = integrand.subs({
1 / sympy.sin(symbol): sympy.csc(symbol),
1 / sympy.tan(symbol): sympy.cot(symbol),
sympy.cos(symbol) / sympy.tan(symbol): sympy.cot(symbol)
})
if any(integrand.has(f) for f in (sympy.cot, sympy.csc)):
pattern, a, b, m, n = cotcsc_pattern(symbol)
match = integrand.match(pattern)
if match:
a, b, m, n = match.get(a, 0),match.get(b, 0), match.get(m, 0), match.get(n, 0)
return multiplexer({
cotcsc_cotodd_condition: cotcsc_cotodd,
cotcsc_csceven_condition: cotcsc_csceven
})((a, b, m, n, integrand, symbol))
def trig_cotcsc_rule(integral):
integrand, symbol = integral
integrand = integrand.subs({
1 / sympy.sin(symbol): sympy.csc(symbol),
1 / sympy.tan(symbol): sympy.cot(symbol),
sympy.cos(symbol) / sympy.tan(symbol): sympy.cot(symbol)
})
if any(integrand.has(f) for f in (sympy.cot, sympy.csc)):
pattern, a, b, m, n = cotcsc_pattern(symbol)
match = integrand.match(pattern)
if match:
a, b, m, n = match.get(a, 0),match.get(b, 0), match.get(m, 0), match.get(n, 0)
return multiplexer({
cotcsc_cotodd_condition: cotcsc_cotodd,
cotcsc_csceven_condition: cotcsc_csceven
})((a, b, m, n, integrand, symbol))
def test_manualintegrate_trigonometry():
assert manualintegrate(sin(x), x) == -cos(x)
assert manualintegrate(tan(x), x) == -log(cos(x))
assert manualintegrate(sec(x), x) == log(sec(x) + tan(x))
assert manualintegrate(csc(x), x) == -log(csc(x) + cot(x))
assert manualintegrate(sin(x) * cos(x), x) in [sin(x) ** 2 / 2, -cos(x)**2 / 2]
assert manualintegrate(-sec(x) * tan(x), x) == -sec(x)
assert manualintegrate(csc(x) * cot(x), x) == -csc(x)
assert manualintegrate(sec(x)**2, x) == tan(x)
assert manualintegrate(csc(x)**2, x) == -cot(x)
assert manualintegrate(x * sec(x**2), x) == log(tan(x**2) + sec(x**2))/2
assert manualintegrate(cos(x)*csc(sin(x)), x) == -log(cot(sin(x)) + csc(sin(x)))
assert manualintegrate(cos(3*x)*sec(x), x) == -x + sin(2*x)
assert manualintegrate(sin(3*x)*sec(x), x) == \
-3*log(cos(x)) + 2*log(cos(x)**2) - 2*cos(x)**2
def test_manualintegrate_trigonometry():
assert manualintegrate(sin(x), x) == -cos(x)
assert manualintegrate(tan(x), x) == -log(cos(x))
assert manualintegrate(sec(x), x) == log(sec(x) + tan(x))
assert manualintegrate(csc(x), x) == -log(csc(x) + cot(x))
assert manualintegrate(sin(x) * cos(x), x) in [sin(x) ** 2 / 2, -cos(x)**2 / 2]
assert manualintegrate(-sec(x) * tan(x), x) == -sec(x)
assert manualintegrate(csc(x) * cot(x), x) == -csc(x)
assert manualintegrate(sec(x)**2, x) == tan(x)
assert manualintegrate(csc(x)**2, x) == -cot(x)
assert manualintegrate(x * sec(x**2), x) == log(tan(x**2) + sec(x**2))/2
assert manualintegrate(cos(x)*csc(sin(x)), x) == -log(cot(sin(x)) + csc(sin(x)))
assert manualintegrate(cos(3*x)*sec(x), x) == -x + sin(2*x)
assert manualintegrate(sin(3*x)*sec(x), x) == \
-3*log(cos(x)) + 2*log(cos(x)**2) - 2*cos(x)**2
def trig_cotcsc_rule(integral):
integrand, symbol = integral
integrand = integrand.subs({
1 / sympy.sin(symbol): sympy.csc(symbol),
1 / sympy.tan(symbol): sympy.cot(symbol),
sympy.cos(symbol) / sympy.tan(symbol): sympy.cot(symbol)
})
if any(integrand.has(f) for f in (sympy.cot, sympy.csc)):
pattern, a, b, m, n = cotcsc_pattern(symbol)
match = integrand.match(pattern)
if not match:
return
return multiplexer({
cotcsc_cotodd_condition: cotcsc_cotodd,
cotcsc_csceven_condition: cotcsc_csceven
})(tuple(
[match.get(i, ZERO) for i in (a, b, m, n)] +
[integrand, symbol]))
def trig_cotcsc_rule(integral):
integrand, symbol = integral
integrand = integrand.subs({
1 / sympy.sin(symbol): sympy.csc(symbol),
1 / sympy.tan(symbol): sympy.cot(symbol),
sympy.cos(symbol) / sympy.tan(symbol): sympy.cot(symbol)
})
if any(integrand.has(f) for f in (sympy.cot, sympy.csc)):
pattern, a, b, m, n = cotcsc_pattern(symbol)
match = integrand.match(pattern)
if not match:
return
return multiplexer({
cotcsc_cotodd_condition: cotcsc_cotodd,
cotcsc_csceven_condition: cotcsc_csceven
})(tuple(
[match.get(i, ZERO) for i in (a, b, m, n)] +
[integrand, symbol]))
def test_sin_rewrite():
assert sin(x).rewrite(exp) == -I*(exp(I*x) - exp(-I*x))/2
assert sin(x).rewrite(tan) == 2*tan(x/2)/(1 + tan(x/2)**2)
assert sin(x).rewrite(cot) == 2*cot(x/2)/(1 + cot(x/2)**2)
assert sin(sinh(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sinh(3)).n()
assert sin(cosh(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cosh(3)).n()
assert sin(tanh(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tanh(3)).n()
assert sin(coth(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, coth(3)).n()
assert sin(sin(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sin(3)).n()
assert sin(cos(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cos(3)).n()
assert sin(tan(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tan(3)).n()
assert sin(cot(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cot(3)).n()
assert sin(log(x)).rewrite(Pow) == I*x**-I / 2 - I*x**I /2
assert sin(x).rewrite(csc) == 1/csc(x)
def trig_product_rule(integral):
integrand, symbol = integral
sectan = sympy.sec(symbol) * sympy.tan(symbol)
q = integrand / sectan
if symbol not in q.free_symbols:
rule = TrigRule('sec*tan', symbol, sectan, symbol)
if q != 1:
rule = ConstantTimesRule(q, sectan, rule, integrand, symbol)
return rule
csccot = -sympy.csc(symbol) * sympy.cot(symbol)
q = integrand / csccot
if symbol not in q.free_symbols:
rule = TrigRule('csc*cot', symbol, csccot, symbol)
if q != 1:
rule = ConstantTimesRule(q, csccot, rule, integrand, symbol)
return rule
def trig_product_rule(integral):
integrand, symbol = integral
sectan = sympy.sec(symbol) * sympy.tan(symbol)
q = integrand / sectan
if symbol not in q.free_symbols:
rule = TrigRule('sec*tan', symbol, sectan, symbol)
if q != 1 and rule:
rule = ConstantTimesRule(q, sectan, rule, integrand, symbol)
return rule
csccot = -sympy.csc(symbol) * sympy.cot(symbol)
q = integrand / csccot
if symbol not in q.free_symbols:
rule = TrigRule('csc*cot', symbol, csccot, symbol)
if q != 1 and rule:
rule = ConstantTimesRule(q, csccot, rule, integrand, symbol)
return rule
def test_TR1():
assert TR1(2*csc(x) + sec(x)) == 1/cos(x) + 2/sin(x)
sympy.tan(a*symbol) ** m ))
tansec_tanodd_condition = uncurry(lambda a, b, m, n, i, s: m.is_odd)
tansec_tanodd = trig_rewriter(
lambda a, b, m, n, i, symbol: ( (sympy.sec(a*symbol)**2 - 1) ** ((m - 1) / 2) *
sympy.tan(a*symbol) *
sympy.sec(b*symbol) ** n ))
tan_tansquared_condition = uncurry(lambda a, b, m, n, i, s: m == 2 and n == 0)
tan_tansquared = trig_rewriter(
lambda a, b, m, n, i, symbol: ( sympy.sec(a*symbol)**2 - 1))
cotcsc_csceven_condition = uncurry(lambda a, b, m, n, i, s: n.is_even and n >= 4)
cotcsc_csceven = trig_rewriter(
lambda a, b, m, n, i, symbol: ( (1 + sympy.cot(b*symbol)**2) ** (n/2 - 1) *
sympy.csc(b*symbol)**2 *
sympy.cot(a*symbol) ** m ))
cotcsc_cotodd_condition = uncurry(lambda a, b, m, n, i, s: m.is_odd)
cotcsc_cotodd = trig_rewriter(
lambda a, b, m, n, i, symbol: ( (sympy.csc(a*symbol)**2 - 1) ** ((m - 1) / 2) *
sympy.cot(a*symbol) *
sympy.csc(b*symbol) ** n ))
def trig_sincos_rule(integral):
integrand, symbol = integral
if any(integrand.has(f) for f in (sympy.sin, sympy.cos)):
pattern, a, b, m, n = sincos_pattern(symbol)
match = integrand.match(pattern)
def cotcsc_pattern(symbol):
a, b, m, n = make_wilds(symbol)
pattern = sympy.cot(a * symbol)**m * sympy.csc(b * symbol)**n
return pattern, a, b, m, n
def test_invert_real():
x = Dummy(real=True)
n = Symbol('n')
minus_n = Intersection(Interval(-oo, 0), FiniteSet(-n))
plus_n = Intersection(Interval(0, oo), FiniteSet(n))
assert solveset(abs(x) - n, x, S.Reals) == Union(minus_n, plus_n)
n = Symbol('n', real=True)
assert invert_real(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_real(x*3, y, x) == (x, FiniteSet(y / 3))
assert invert_real(exp(x), y, x) == (x, FiniteSet(log(y)))
assert invert_real(exp(3*x), y, x) == (x, FiniteSet(log(y) / 3))
assert invert_real(exp(x + 3), y, x) == (x, FiniteSet(log(y) - 3))
assert invert_real(exp(x) + 3, y, x) == (x, FiniteSet(log(y - 3)))
assert invert_real(exp(x)*3, y, x) == (x, FiniteSet(log(y / 3)))
assert invert_real(log(x), y, x) == (x, FiniteSet(exp(y)))
assert invert_real(log(3*x), y, x) == (x, FiniteSet(exp(y) / 3))
assert invert_real(log(x + 3), y, x) == (x, FiniteSet(exp(y) - 3))
minus_y = Intersection(Interval(-oo, 0), FiniteSet(-y))
plus_y = Intersection(Interval(0, oo), FiniteSet(y))
assert invert_real(Abs(x), y, x) == (x, Union(minus_y, plus_y))
assert invert_real(2**x, y, x) == (x, FiniteSet(log(y)/log(2)))
assert invert_real(2**exp(x), y, x) == (x, FiniteSet(log(log(y)/log(2))))
assert invert_real(x**2, y, x) == (x, FiniteSet(sqrt(y), -sqrt(y)))
assert invert_real(x**Rational(1, 2), y, x) == (x, FiniteSet(y**2))
raises(ValueError, lambda: invert_real(x, x, x))
raises(ValueError, lambda: invert_real(x**pi, y, x))
raises(ValueError, lambda: invert_real(S.One, y, x))
assert invert_real(x**31 + x, y, x) == (x**31 + x, FiniteSet(y))
y_1 = Intersection(Interval(-1, oo), FiniteSet(y - 1))
y_2 = Intersection(Interval(-oo, -1), FiniteSet(-y - 1))
assert invert_real(Abs(x**31 + x + 1), y, x) == (x**31 + x,
Union(y_1, y_2))
assert invert_real(sin(x), y, x) == \
(x, imageset(Lambda(n, n*pi + (-1)**n*asin(y)), S.Integers))
assert invert_real(sin(exp(x)), y, x) == \
(x, imageset(Lambda(n, log((-1)**n*asin(y) + n*pi)), S.Integers))
assert invert_real(csc(x), y, x) == \
(x, imageset(Lambda(n, n*pi + (-1)**n*acsc(y)), S.Integers))
assert invert_real(csc(exp(x)), y, x) == \
(x, imageset(Lambda(n, log((-1)**n*acsc(y) + n*pi)), S.Integers))
assert invert_real(cos(x), y, x) == \
(x, Union(imageset(Lambda(n, 2*n*pi + acos(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - acos(y)), S.Integers)))
assert invert_real(cos(exp(x)), y, x) == \
(x, Union(imageset(Lambda(n, log(2*n*pi + acos(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - acos(y))), S.Integers)))
assert invert_real(sec(x), y, x) == \
(x, Union(imageset(Lambda(n, 2*n*pi + asec(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - asec(y)), S.Integers)))
assert invert_real(sec(exp(x)), y, x) == \
(x, Union(imageset(Lambda(n, log(2*n*pi + asec(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - asec(y))), S.Integers)))
assert invert_real(tan(x), y, x) == \
(x, imageset(Lambda(n, n*pi + atan(y)), S.Integers))
assert invert_real(tan(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + atan(y))), S.Integers))
assert invert_real(cot(x), y, x) == \
(x, imageset(Lambda(n, n*pi + acot(y)), S.Integers))
assert invert_real(cot(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + acot(y))), S.Integers))
assert invert_real(tan(tan(x)), y, x) == \
(tan(x), imageset(Lambda(n, n*pi + atan(y)), S.Integers))
x = Symbol('x', positive=True)
assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1/pi)))
# Test for ``set_h`` containing information about the domain
n = Dummy('n')
x = Symbol('x')
h1 = Intersection(Interval(-3, oo), FiniteSet(a + b - 3),
imageset(Lambda(n, -n + a - 3), Interval(-oo, 0)))
h2 = Intersection(Interval(-oo, -3), FiniteSet(-a + b - 3),
imageset(Lambda(n, n - a - 3), Interval(0, oo)))
h3 = Intersection(Interval(-3, oo), FiniteSet(a - b - 3),
imageset(Lambda(n, -n + a - 3), Interval(0, oo)))
h4 = Intersection(Interval(-oo, -3), FiniteSet(-a - b - 3),
imageset(Lambda(n, n - a - 3), Interval(-oo, 0)))
assert invert_real(Abs(Abs(x + 3) - a) - b, 0, x) == (x, Union(h1, h2, h3, h4))
(n / 2 - 1) * sympy.sec(b * symbol)**2 * sympy.tan(a * symbol)**m))
tansec_tanodd_condition = uncurry(lambda a, b, m, n, i, s: m.is_odd)
tansec_tanodd = trig_rewriter(
lambda a, b, m, n, i, symbol: ((sympy.sec(a * symbol)**2 - 1)**(
(m - 1) / 2) * sympy.tan(a * symbol) * sympy.sec(b * symbol)**n))
tan_tansquared_condition = uncurry(lambda a, b, m, n, i, s: m == 2 and n == 0)
tan_tansquared = trig_rewriter(lambda a, b, m, n, i, symbol:
(sympy.sec(a * symbol)**2 - 1))
cotcsc_csceven_condition = uncurry(
lambda a, b, m, n, i, s: n.is_even and n >= 4)
cotcsc_csceven = trig_rewriter(lambda a, b, m, n, i, symbol: (
(1 + sympy.cot(b * symbol)**2)**
(n / 2 - 1) * sympy.csc(b * symbol)**2 * sympy.cot(a * symbol)**m))
cotcsc_cotodd_condition = uncurry(lambda a, b, m, n, i, s: m.is_odd)
cotcsc_cotodd = trig_rewriter(
lambda a, b, m, n, i, symbol: ((sympy.csc(a * symbol)**2 - 1)**(
(m - 1) / 2) * sympy.cot(a * symbol) * sympy.csc(b * symbol)**n))
def trig_sincos_rule(integral):
integrand, symbol = integral
if any(integrand.has(f) for f in (sympy.sin, sympy.cos)):
pattern, a, b, m, n = sincos_pattern(symbol)
match = integrand.match(pattern)
if not match:
return
def test_periodicity():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z', real=True)
assert periodicity(sin(2*x), x) == pi
assert periodicity((-2)*tan(4*x), x) == pi/4
assert periodicity(sin(x)**2, x) == 2*pi
assert periodicity(3**tan(3*x), x) == pi/3
assert periodicity(tan(x)*cos(x), x) == 2*pi
assert periodicity(sin(x)**(tan(x)), x) == 2*pi
assert periodicity(tan(x)*sec(x), x) == 2*pi
assert periodicity(sin(2*x)*cos(2*x) - y, x) == pi/2
assert periodicity(tan(x) + cot(x), x) == pi
assert periodicity(sin(x) - cos(2*x), x) == 2*pi
assert periodicity(sin(x) - 1, x) == 2*pi
assert periodicity(sin(4*x) + sin(x)*cos(x), x) == pi
assert periodicity(exp(sin(x)), x) == 2*pi
assert periodicity(log(cot(2*x)) - sin(cos(2*x)), x) == pi
assert periodicity(sin(2*x)*exp(tan(x) - csc(2*x)), x) == pi
assert periodicity(cos(sec(x) - csc(2*x)), x) == 2*pi
assert periodicity(tan(sin(2*x)), x) == pi
assert periodicity(2*tan(x)**2, x) == pi
assert periodicity(sin(x%4), x) == 4
assert periodicity(sin(x)%4, x) == 2*pi
assert periodicity(tan((3*x-2)%4), x) == S(4)/3
assert periodicity((sqrt(2)*(x+1)+x) % 3, x) == 3 / (sqrt(2)+1)
assert periodicity((x**2+1) % x, x) == None
assert periodicity(sin(re(x)), x) == 2*pi
assert periodicity(sin(x)**2 + cos(x)**2, x) == S.Zero
assert periodicity(tan(x), y) == S.Zero
assert periodicity(sin(x) + I*cos(x), x) == 2*pi
assert periodicity(x - sin(2*y), y) == pi
assert periodicity(exp(x), x) is None
assert periodicity(exp(I*x), x) == 2*pi
assert periodicity(exp(I*z), z) == 2*pi
assert periodicity(exp(z), z) is None
assert periodicity(exp(log(sin(z) + I*cos(2*z)), evaluate=False), z) == 2*pi
assert periodicity(exp(log(sin(2*z) + I*cos(z)), evaluate=False), z) == 2*pi
assert periodicity(exp(sin(z)), z) == 2*pi
assert periodicity(exp(2*I*z), z) == pi
assert periodicity(exp(z + I*sin(z)), z) is None
assert periodicity(exp(cos(z/2) + sin(z)), z) == 4*pi
assert periodicity(log(x), x) is None
assert periodicity(exp(x)**sin(x), x) is None
assert periodicity(sin(x)**y, y) is None
assert periodicity(Abs(sin(Abs(sin(x)))), x) == pi
assert all(periodicity(Abs(f(x)), x) == pi for f in (
cos, sin, sec, csc, tan, cot))
assert periodicity(Abs(sin(tan(x))), x) == pi
assert periodicity(Abs(sin(sin(x) + tan(x))), x) == 2*pi
assert periodicity(sin(x) > S.Half, x) is 2*pi
assert periodicity(x > 2, x) is None
assert periodicity(x**3 - x**2 + 1, x) is None
assert periodicity(Abs(x), x) is None
assert periodicity(Abs(x**2 - 1), x) is None
assert periodicity((x**2 + 4)%2, x) is None
assert periodicity((E**x)%3, x) is None
def test_invert_real():
x = Symbol('x', real=True)
y = Symbol('y')
n = Symbol('n')
def ireal(x, s=S.Reals):
return Intersection(s, x)
minus_n = Intersection(Interval(-oo, 0), FiniteSet(-n))
plus_n = Intersection(Interval(0, oo), FiniteSet(n))
assert solveset(abs(x) - n, x, S.Reals) == Union(minus_n, plus_n)
assert invert_real(exp(x), y, x) == (x, ireal(FiniteSet(log(y))))
y = Symbol('y', positive=True)
n = Symbol('n', real=True)
assert invert_real(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_real(x * 3, y, x) == (x, FiniteSet(y / 3))
assert invert_real(exp(x), y, x) == (x, FiniteSet(log(y)))
assert invert_real(exp(3 * x), y, x) == (x, FiniteSet(log(y) / 3))
assert invert_real(exp(x + 3), y, x) == (x, FiniteSet(log(y) - 3))
assert invert_real(exp(x) + 3, y, x) == (x, ireal(FiniteSet(log(y - 3))))
assert invert_real(exp(x) * 3, y, x) == (x, FiniteSet(log(y / 3)))
assert invert_real(log(x), y, x) == (x, FiniteSet(exp(y)))
assert invert_real(log(3 * x), y, x) == (x, FiniteSet(exp(y) / 3))
assert invert_real(log(x + 3), y, x) == (x, FiniteSet(exp(y) - 3))
minus_y = Intersection(Interval(-oo, 0), FiniteSet(-y))
plus_y = Intersection(Interval(0, oo), FiniteSet(y))
assert invert_real(Abs(x), y, x) == (x, Union(minus_y, plus_y))
assert invert_real(2**x, y, x) == (x, FiniteSet(log(y) / log(2)))
assert invert_real(2**exp(x), y,
x) == (x, ireal(FiniteSet(log(log(y) / log(2)))))
assert invert_real(x**2, y, x) == (x, FiniteSet(sqrt(y), -sqrt(y)))
assert invert_real(x**Rational(1, 2), y, x) == (x, FiniteSet(y**2))
raises(ValueError, lambda: invert_real(x, x, x))
raises(ValueError, lambda: invert_real(x**pi, y, x))
raises(ValueError, lambda: invert_real(S.One, y, x))
assert invert_real(x**31 + x, y, x) == (x**31 + x, FiniteSet(y))
y_1 = Intersection(Interval(-1, oo), FiniteSet(y - 1))
y_2 = Intersection(Interval(-oo, -1), FiniteSet(-y - 1))
assert invert_real(Abs(x**31 + x + 1), y,
x) == (x**31 + x, Union(y_1, y_2))
assert invert_real(sin(x), y, x) == \
(x, imageset(Lambda(n, n*pi + (-1)**n*asin(y)), S.Integers))
assert invert_real(sin(exp(x)), y, x) == \
(x, imageset(Lambda(n, log((-1)**n*asin(y) + n*pi)), S.Integers))
assert invert_real(csc(x), y, x) == \
(x, imageset(Lambda(n, n*pi + (-1)**n*acsc(y)), S.Integers))
assert invert_real(csc(exp(x)), y, x) == \
(x, imageset(Lambda(n, log((-1)**n*acsc(y) + n*pi)), S.Integers))
assert invert_real(cos(x), y, x) == \
(x, Union(imageset(Lambda(n, 2*n*pi + acos(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - acos(y)), S.Integers)))
assert invert_real(cos(exp(x)), y, x) == \
(x, Union(imageset(Lambda(n, log(2*n*pi + acos(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - acos(y))), S.Integers)))
assert invert_real(sec(x), y, x) == \
(x, Union(imageset(Lambda(n, 2*n*pi + asec(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - asec(y)), S.Integers)))
assert invert_real(sec(exp(x)), y, x) == \
(x, Union(imageset(Lambda(n, log(2*n*pi + asec(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - asec(y))), S.Integers)))
assert invert_real(tan(x), y, x) == \
(x, imageset(Lambda(n, n*pi + atan(y)), S.Integers))
assert invert_real(tan(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + atan(y))), S.Integers))
assert invert_real(cot(x), y, x) == \
(x, imageset(Lambda(n, n*pi + acot(y)), S.Integers))
assert invert_real(cot(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + acot(y))), S.Integers))
assert invert_real(tan(tan(x)), y, x) == \
(tan(x), imageset(Lambda(n, n*pi + atan(y)), S.Integers))
x = Symbol('x', positive=True)
assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1 / pi)))
# Test for ``set_h`` containing information about the domain
n = Dummy('n')
x = Symbol('x')
h1 = Intersection(Interval(-oo, -3), FiniteSet(-a + b - 3),
imageset(Lambda(n, n - a - 3), Interval(0, oo)))
h2 = Intersection(Interval(-3, oo), FiniteSet(a - b - 3),
imageset(Lambda(n, -n + a - 3), Interval(0, oo)))
assert invert_real(Abs(Abs(x + 3) - a) - b, 0, x) == (x, Union(h1, h2))
def test_csc():
assert csc(x).diff(x) == -cot(x)*csc(x)
class TestAllGood(object):
# These latex strings should parse to the corresponding SymPy expression
GOOD_PAIRS = [
("0", Rational(0)),
("1", Rational(1)),
("-3.14", Rational(-314, 100)),
("5-3", _Add(5, _Mul(-1, 3))),
("(-7.13)(1.5)", _Mul(Rational('-7.13'), Rational('1.5'))),
("\\left(-7.13\\right)\\left(1.5\\right)", _Mul(Rational('-7.13'), Rational('1.5'))),
("x", x),
("2x", 2 * x),
("x^2", x**2),
("x^{3 + 1}", x**_Add(3, 1)),
("x^{\\left\\{3 + 1\\right\\}}", x**_Add(3, 1)),
("-3y + 2x", _Add(_Mul(2, x), Mul(-1, 3, y, evaluate=False))),
("-c", -c),
("a \\cdot b", a * b),
("a / b", a / b),
("a \\div b", a / b),
("a + b", a + b),
("a + b - a", Add(a, b, _Mul(-1, a), evaluate=False)),
("a^2 + b^2 = c^2", Eq(a**2 + b**2, c**2)),
("a^2 + b^2 != 2c^2", Ne(a**2 + b**2, 2 * c**2)),
("a\\mod b", Mod(a, b)),
("\\sin \\theta", sin(theta)),
("\\sin(\\theta)", sin(theta)),
("\\sin\\left(\\theta\\right)", sin(theta)),
("\\sin^{-1} a", asin(a)),
("\\sin a \\cos b", _Mul(sin(a), cos(b))),
("\\sin \\cos \\theta", sin(cos(theta))),
("\\sin(\\cos \\theta)", sin(cos(theta))),
("\\arcsin(a)", asin(a)),
("\\arccos(a)", acos(a)),
("\\arctan(a)", atan(a)),
("\\sinh(a)", sinh(a)),
("\\cosh(a)", cosh(a)),
("\\tanh(a)", tanh(a)),
("\\sinh^{-1}(a)", asinh(a)),
("\\cosh^{-1}(a)", acosh(a)),
("\\tanh^{-1}(a)", atanh(a)),
("\\arcsinh(a)", asinh(a)),
("\\arccosh(a)", acosh(a)),
("\\arctanh(a)", atanh(a)),
("\\arsinh(a)", asinh(a)),
("\\arcosh(a)", acosh(a)),
("\\artanh(a)", atanh(a)),
("\\operatorname{arcsinh}(a)", asinh(a)),
("\\operatorname{arccosh}(a)", acosh(a)),
("\\operatorname{arctanh}(a)", atanh(a)),
("\\operatorname{arsinh}(a)", asinh(a)),
("\\operatorname{arcosh}(a)", acosh(a)),
("\\operatorname{artanh}(a)", atanh(a)),
("\\operatorname{gcd}(a, b)", UnevaluatedExpr(gcd(a, b))),
("\\operatorname{lcm}(a, b)", UnevaluatedExpr(lcm(a, b))),
("\\operatorname{gcd}(a,b)", UnevaluatedExpr(gcd(a, b))),
("\\operatorname{lcm}(a,b)", UnevaluatedExpr(lcm(a, b))),
("\\operatorname{floor}(a)", floor(a)),
("\\operatorname{ceil}(b)", ceiling(b)),
("\\cos^2(x)", cos(x)**2),
("\\cos(x)^2", cos(x)**2),
("\\gcd(a, b)", UnevaluatedExpr(gcd(a, b))),
("\\lcm(a, b)", UnevaluatedExpr(lcm(a, b))),
("\\gcd(a,b)", UnevaluatedExpr(gcd(a, b))),
("\\lcm(a,b)", UnevaluatedExpr(lcm(a, b))),
("\\floor(a)", floor(a)),
("\\ceil(b)", ceiling(b)),
("\\max(a, b)", Max(a, b)),
("\\min(a, b)", Min(a, b)),
("\\frac{a}{b}", a / b),
("\\frac{a + b}{c}", _Mul(a + b, _Pow(c, -1))),
("\\frac{7}{3}", Rational(7, 3)),
("(\\csc x)(\\sec y)", csc(x) * sec(y)),
("\\lim_{x \\to 3} a", Limit(a, x, 3)),
("\\lim_{x \\rightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\Rightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\longrightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\Longrightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\to 3^{+}} a", Limit(a, x, 3, dir='+')),
("\\lim_{x \\to 3^{-}} a", Limit(a, x, 3, dir='-')),
("\\infty", oo),
("\\infty\\%", oo),
("\\$\\infty", oo),
("-\\infty", -oo),
("-\\infty\\%", -oo),
("-\\$\\infty", -oo),
("\\lim_{x \\to \\infty} \\frac{1}{x}", Limit(_Mul(1, _Pow(x, -1)), x, oo)),
("\\frac{d}{dx} x", Derivative(x, x)),
("\\frac{d}{dt} x", Derivative(x, t)),
# ("f(x)", f(x)),
# ("f(x, y)", f(x, y)),
# ("f(x, y, z)", f(x, y, z)),
# ("\\frac{d f(x)}{dx}", Derivative(f(x), x)),
# ("\\frac{d\\theta(x)}{dx}", Derivative(theta(x), x)),
("|x|", _Abs(x)),
("\\left|x\\right|", _Abs(x)),
("||x||", _Abs(_Abs(x))),
("|x||y|", _Abs(x) * _Abs(y)),
("||x||y||", _Abs(_Abs(x) * _Abs(y))),
("\\lfloor x\\rfloor", floor(x)),
("\\lceil y\\rceil", ceiling(y)),
("\\pi^{|xy|}", pi**_Abs(x * y)),
("\\frac{\\pi}{3}", _Mul(pi, _Pow(3, -1))),
("\\sin{\\frac{\\pi}{2}}", sin(_Mul(pi, _Pow(2, -1)), evaluate=False)),
("a+bI", a + I * b),
("e^{I\\pi}", Integer(-1)),
("\\int x dx", Integral(x, x)),
("\\int x d\\theta", Integral(x, theta)),
("\\int (x^2 - y)dx", Integral(x**2 - y, x)),
("\\int x + a dx", Integral(_Add(x, a), x)),
("\\int da", Integral(1, a)),
("\\int_0^7 dx", Integral(1, (x, 0, 7))),
("\\int_a^b x dx", Integral(x, (x, a, b))),
("\\int^b_a x dx", Integral(x, (x, a, b))),
("\\int_{a}^b x dx", Integral(x, (x, a, b))),
("\\int^{b}_a x dx", Integral(x, (x, a, b))),
("\\int_{a}^{b} x dx", Integral(x, (x, a, b))),
("\\int_{ }^{}x dx", Integral(x, x)),
("\\int^{ }_{ }x dx", Integral(x, x)),
("\\int^{b}_{a} x dx", Integral(x, (x, a, b))),
# ("\\int_{f(a)}^{f(b)} f(z) dz", Integral(f(z), (z, f(a), f(b)))),
("\\int (x+a)", Integral(_Add(x, a), x)),
("\\int a + b + c dx", Integral(Add(a, b, c, evaluate=False), x)),
("\\int \\frac{dz}{z}", Integral(Pow(z, -1), z)),
("\\int \\frac{3 dz}{z}", Integral(3 * Pow(z, -1), z)),
("\\int \\frac{1}{x} dx", Integral(Pow(x, -1), x)),
("\\int \\frac{1}{a} + \\frac{1}{b} dx", Integral(_Add(_Pow(a, -1), Pow(b, -1)), x)),
("\\int \\frac{3 \\cdot d\\theta}{\\theta}", Integral(3 * _Pow(theta, -1), theta)),
("\\int \\frac{1}{x} + 1 dx", Integral(_Add(_Pow(x, -1), 1), x)),
("x_0", Symbol('x_0', real=True, positive=True)),
("x_{1}", Symbol('x_1', real=True, positive=True)),
("x_a", Symbol('x_a', real=True, positive=True)),
("x_{b}", Symbol('x_b', real=True, positive=True)),
("h_\\theta", Symbol('h_{\\theta}', real=True, positive=True)),
("h_\\theta ", Symbol('h_{\\theta}', real=True, positive=True)),
("h_{\\theta}", Symbol('h_{\\theta}', real=True, positive=True)),
# ("h_{\\theta}(x_0, x_1)", Symbol('h_{theta}', real=True)(Symbol('x_{0}', real=True), Symbol('x_{1}', real=True))),
("x!", _factorial(x)),
("100!", _factorial(100)),
("\\theta!", _factorial(theta)),
("(x + 1)!", _factorial(_Add(x, 1))),
("\\left(x + 1\\right)!", _factorial(_Add(x, 1))),
("(x!)!", _factorial(_factorial(x))),
("x!!!", _factorial(_factorial(_factorial(x)))),
("5!7!", _Mul(_factorial(5), _factorial(7))),
("\\sqrt{x}", sqrt(x)),
("\\sqrt{x + b}", sqrt(_Add(x, b))),
("\\sqrt[3]{\\sin x}", root(sin(x), 3)),
("\\sqrt[y]{\\sin x}", root(sin(x), y)),
("\\sqrt[\\theta]{\\sin x}", root(sin(x), theta)),
("x < y", StrictLessThan(x, y)),
("x \\leq y", LessThan(x, y)),
("x > y", StrictGreaterThan(x, y)),
("x \\geq y", GreaterThan(x, y)),
("\\sum_{k = 1}^{3} c", Sum(c, (k, 1, 3))),
("\\sum_{k = 1}^3 c", Sum(c, (k, 1, 3))),
("\\sum^{3}_{k = 1} c", Sum(c, (k, 1, 3))),
("\\sum^3_{k = 1} c", Sum(c, (k, 1, 3))),
("\\sum_{k = 1}^{10} k^2", Sum(k**2, (k, 1, 10))),
("\\sum_{n = 0}^{\\infty} \\frac{1}{n!}", Sum(_Pow(_factorial(n), -1), (n, 0, oo))),
("\\prod_{a = b}^{c} x", Product(x, (a, b, c))),
("\\prod_{a = b}^c x", Product(x, (a, b, c))),
("\\prod^{c}_{a = b} x", Product(x, (a, b, c))),
("\\prod^c_{a = b} x", Product(x, (a, b, c))),
("\\ln x", _log(x, E)),
("\\ln xy", _log(x * y, E)),
("\\log x", _log(x, 10)),
("\\log xy", _log(x * y, 10)),
# ("\\log_2 x", _log(x, 2)),
("\\log_{2} x", _log(x, 2)),
# ("\\log_a x", _log(x, a)),
("\\log_{a} x", _log(x, a)),
("\\log_{11} x", _log(x, 11)),
("\\log_{a^2} x", _log(x, _Pow(a, 2))),
("[x]", x),
("[a + b]", _Add(a, b)),
("\\frac{d}{dx} [ \\tan x ]", Derivative(tan(x), x)),
("2\\overline{x}", 2 * Symbol('xbar', real=True, positive=True)),
("2\\overline{x}_n", 2 * Symbol('xbar_n', real=True, positive=True)),
("\\frac{x}{\\overline{x}_n}", x / Symbol('xbar_n', real=True, positive=True)),
("\\frac{\\sin(x)}{\\overline{x}_n}", sin(x) / Symbol('xbar_n', real=True, positive=True)),
("2\\bar{x}", 2 * Symbol('xbar', real=True, positive=True)),
("2\\bar{x}_n", 2 * Symbol('xbar_n', real=True, positive=True)),
("\\sin\\left(\\theta\\right) \\cdot4", sin(theta) * 4),
("\\ln\\left(\\theta\\right)", _log(theta, E)),
("\\ln\\left(x-\\theta\\right)", _log(x - theta, E)),
("\\ln\\left(\\left(x-\\theta\\right)\\right)", _log(x - theta, E)),
("\\ln\\left(\\left[x-\\theta\\right]\\right)", _log(x - theta, E)),
("\\ln\\left(\\left\\{x-\\theta\\right\\}\\right)", _log(x - theta, E)),
("\\ln\\left(\\left|x-\\theta\\right|\\right)", _log(_Abs(x - theta), E)),
("\\frac{1}{2}xy(x+y)", Mul(Rational(1, 2), x, y, (x + y), evaluate=False)),
("\\frac{1}{2}\\theta(x+y)", Mul(Rational(1, 2), theta, (x + y), evaluate=False)),
("1-f(x)", 1 - f * x),
("\\begin{matrix}1&2\\\\3&4\\end{matrix}", Matrix([[1, 2], [3, 4]])),
("\\begin{matrix}x&x^2\\\\\\sqrt{x}&x\\end{matrix}", Matrix([[x, x**2], [_Pow(x, S.Half), x]])),
("\\begin{matrix}\\sqrt{x}\\\\\\sin(\\theta)\\end{matrix}", Matrix([_Pow(x, S.Half), sin(theta)])),
("\\begin{pmatrix}1&2\\\\3&4\\end{pmatrix}", Matrix([[1, 2], [3, 4]])),
("\\begin{bmatrix}1&2\\\\3&4\\end{bmatrix}", Matrix([[1, 2], [3, 4]])),
# scientific notation
("2.5\\times 10^2", Rational(250)),
("1,500\\times 10^{-1}", Rational(150)),
# e notation
("2.5E2", Rational(250)),
("1,500E-1", Rational(150)),
# multiplication without cmd
("2x2y", Mul(2, x, 2, y, evaluate=False)),
("2x2", Mul(2, x, 2, evaluate=False)),
("x2", x * 2),
# lin alg processing
("\\theta\\begin{matrix}1&2\\\\3&4\\end{matrix}", MatMul(theta, Matrix([[1, 2], [3, 4]]), evaluate=False)),
("\\theta\\begin{matrix}1\\\\3\\end{matrix} - \\begin{matrix}-1\\\\2\\end{matrix}", MatAdd(MatMul(theta, Matrix([[1], [3]]), evaluate=False), MatMul(-1, Matrix([[-1], [2]]), evaluate=False), evaluate=False)),
("\\theta\\begin{matrix}1&0\\\\0&1\\end{matrix}*\\begin{matrix}3\\\\-2\\end{matrix}", MatMul(theta, Matrix([[1, 0], [0, 1]]), Matrix([3, -2]), evaluate=False)),
("\\frac{1}{9}\\theta\\begin{matrix}1&2\\\\3&4\\end{matrix}", MatMul(Rational(1, 9), theta, Matrix([[1, 2], [3, 4]]), evaluate=False)),
("\\begin{pmatrix}1\\\\2\\\\3\\end{pmatrix},\\begin{pmatrix}4\\\\3\\\\1\\end{pmatrix}", [Matrix([1, 2, 3]), Matrix([4, 3, 1])]),
("\\begin{pmatrix}1\\\\2\\\\3\\end{pmatrix};\\begin{pmatrix}4\\\\3\\\\1\\end{pmatrix}", [Matrix([1, 2, 3]), Matrix([4, 3, 1])]),
("\\left\\{\\begin{pmatrix}1\\\\2\\\\3\\end{pmatrix},\\begin{pmatrix}4\\\\3\\\\1\\end{pmatrix}\\right\\}", [Matrix([1, 2, 3]), Matrix([4, 3, 1])]),
("\\left\\{\\begin{pmatrix}1\\\\2\\\\3\\end{pmatrix},\\begin{pmatrix}4\\\\3\\\\1\\end{pmatrix},\\begin{pmatrix}1\\\\1\\\\1\\end{pmatrix}\\right\\}", [Matrix([1, 2, 3]), Matrix([4, 3, 1]), Matrix([1, 1, 1])]),
("\\left\\{\\begin{pmatrix}1\\\\2\\\\3\\end{pmatrix}\\right\\}", Matrix([1, 2, 3])),
("\\left{\\begin{pmatrix}1\\\\2\\\\3\\end{pmatrix}\\right}", Matrix([1, 2, 3])),
("{\\begin{pmatrix}1\\\\2\\\\3\\end{pmatrix}}", Matrix([1, 2, 3])),
# us dollars
("\\$1,000.00", Rational(1000)),
("\\$543.21", Rational(54321, 100)),
("\\$0.009", Rational(9, 1000)),
# percentages
("100\\%", Rational(1)),
("1.5\\%", Rational(15, 1000)),
("0.05\\%", Rational(5, 10000)),
# empty set
("\\emptyset", S.EmptySet),
# divide by zero
("\\frac{1}{0}", _Pow(0, -1)),
("1+\\frac{5}{0}", _Add(1, _Mul(5, _Pow(0, -1)))),
# adjacent single char sub sup
("4^26^2", _Mul(_Pow(4, 2), _Pow(6, 2))),
("x_22^2", _Mul(Symbol('x_2', real=True, positive=True), _Pow(2, 2)))
]
def test_good_pair(self, s, eq):
assert_equal(s, eq)
def cotcsc_pattern(symbol):
a, b, m, n = make_wilds(symbol)
pattern = sympy.cot(a*symbol)**m * sympy.csc(b*symbol)**n
return pattern, a, b, m, n
)
tansec_tanodd_condition = uncurry(lambda a, b, m, n, i, s: m.is_odd)
tansec_tanodd = trig_rewriter(
lambda a, b, m, n, i, symbol: (
(sympy.sec(a * symbol) ** 2 - 1) ** ((m - 1) / 2) * sympy.tan(a * symbol) * sympy.sec(b * symbol) ** n
)
)
tan_tansquared_condition = uncurry(lambda a, b, m, n, i, s: m == 2 and n == 0)
tan_tansquared = trig_rewriter(lambda a, b, m, n, i, symbol: (sympy.sec(a * symbol) ** 2 - 1))
cotcsc_csceven_condition = uncurry(lambda a, b, m, n, i, s: n.is_even and n >= 4)
cotcsc_csceven = trig_rewriter(
lambda a, b, m, n, i, symbol: (
(1 + sympy.cot(b * symbol) ** 2) ** (n / 2 - 1) * sympy.csc(b * symbol) ** 2 * sympy.cot(a * symbol) ** m
)
)
cotcsc_cotodd_condition = uncurry(lambda a, b, m, n, i, s: m.is_odd)
cotcsc_cotodd = trig_rewriter(
lambda a, b, m, n, i, symbol: (
(sympy.csc(a * symbol) ** 2 - 1) ** ((m - 1) / 2) * sympy.cot(a * symbol) * sympy.csc(b * symbol) ** n
)
)
def trig_sincos_rule(integral):
integrand, symbol = integral
if any(integrand.has(f) for f in (sympy.sin, sympy.cos)):
def test_find_substitutions():
assert find_substitutions((cot(x)**2 + 1)**2*csc(x)**2*cot(x)**2, x, u) == \
[(cot(x), 1, -u**6 - 2*u**4 - u**2)]
assert find_substitutions((sec(x)**2 + tan(x) * sec(x)) / (sec(x) + tan(x)),
x, u) == [(sec(x) + tan(x), 1, 1/u)]
assert find_substitutions(x * exp(-x**2), x, u) == [(-x**2, -S.Half, exp(u))]
def test_csc():
x = symbols('x', real=True)
z = symbols('z')
# https://github.com/sympy/sympy/issues/6707
cosecant = csc('x')
alternate = 1/sin('x')
assert cosecant.equals(alternate) == True
assert alternate.equals(cosecant) == True
assert csc.nargs == FiniteSet(1)
assert csc(0) == zoo
assert csc(pi) == zoo
assert csc(pi/2) == 1
assert csc(-pi/2) == -1
assert csc(pi/6) == 2
assert csc(pi/3) == 2*sqrt(3)/3
assert csc(5*pi/2) == 1
assert csc(9*pi/7) == -csc(2*pi/7)
assert csc(3*pi/4) == sqrt(2) # issue 8421
assert csc(I) == -I/sinh(1)
assert csc(x*I) == -I/sinh(x)
assert csc(-x) == -csc(x)
assert csc(acsc(x)) == x
assert csc(x).rewrite(exp) == 2*I/(exp(I*x) - exp(-I*x))
assert csc(x).rewrite(sin) == 1/sin(x)
assert csc(x).rewrite(cos) == csc(x)
assert csc(x).rewrite(tan) == (tan(x/2)**2 + 1)/(2*tan(x/2))
assert csc(x).rewrite(cot) == (cot(x/2)**2 + 1)/(2*cot(x/2))
assert csc(z).conjugate() == csc(conjugate(z))
assert (csc(z).as_real_imag() ==
(sin(re(z))*cosh(im(z))/(sin(re(z))**2*cosh(im(z))**2 +
cos(re(z))**2*sinh(im(z))**2),
-cos(re(z))*sinh(im(z))/(sin(re(z))**2*cosh(im(z))**2 +
cos(re(z))**2*sinh(im(z))**2)))
assert csc(x).expand(trig=True) == 1/sin(x)
assert csc(2*x).expand(trig=True) == 1/(2*sin(x)*cos(x))
assert csc(x).is_real == True
assert csc(z).is_real == None
assert csc(a).is_algebraic is None
assert csc(na).is_algebraic is False
assert csc(x).as_leading_term() == csc(x)
assert csc(0).is_finite == False
assert csc(x).is_finite == None
assert csc(pi/2).is_finite == True
assert series(csc(x), x, x0=pi/2, n=6) == \
1 + (x - pi/2)**2/2 + 5*(x - pi/2)**4/24 + O((x - pi/2)**6, (x, pi/2))
assert series(csc(x), x, x0=0, n=6) == \
1/x + x/6 + 7*x**3/360 + 31*x**5/15120 + O(x**6)
assert csc(x).diff(x) == -cot(x)*csc(x)
assert csc(x).taylor_term(2, x) == 0
assert csc(x).taylor_term(3, x) == 7*x**3/360
assert csc(x).taylor_term(5, x) == 31*x**5/15120
def test_csc_rewrite_failing():
# Move these 2 tests to test_csc() once bugs fixed
# sin(x).rewrite(pow) raises RuntimeError: maximum recursion depth
# https://github.com/sympy/sympy/issues/7171
assert csc(x).rewrite(pow) == csc(x)
assert csc(x).rewrite(sqrt) == csc(x)
def test_periodicity():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z', real=True)
assert periodicity(sin(2*x), x) == pi
assert periodicity((-2)*tan(4*x), x) == pi/4
assert periodicity(sin(x)**2, x) == 2*pi
assert periodicity(3**tan(3*x), x) == pi/3
assert periodicity(tan(x)*cos(x), x) == 2*pi
assert periodicity(sin(x)**(tan(x)), x) == 2*pi
assert periodicity(tan(x)*sec(x), x) == 2*pi
assert periodicity(sin(2*x)*cos(2*x) - y, x) == pi/2
assert periodicity(tan(x) + cot(x), x) == pi
assert periodicity(sin(x) - cos(2*x), x) == 2*pi
assert periodicity(sin(x) - 1, x) == 2*pi
assert periodicity(sin(4*x) + sin(x)*cos(x), x) == pi
assert periodicity(exp(sin(x)), x) == 2*pi
assert periodicity(log(cot(2*x)) - sin(cos(2*x)), x) == pi
assert periodicity(sin(2*x)*exp(tan(x) - csc(2*x)), x) == pi
assert periodicity(cos(sec(x) - csc(2*x)), x) == 2*pi
assert periodicity(tan(sin(2*x)), x) == pi
assert periodicity(2*tan(x)**2, x) == pi
assert periodicity(sin(x%4), x) == 4
assert periodicity(sin(x)%4, x) == 2*pi
assert periodicity(tan((3*x-2)%4), x) == Rational(4, 3)
assert periodicity((sqrt(2)*(x+1)+x) % 3, x) == 3 / (sqrt(2)+1)
assert periodicity((x**2+1) % x, x) is None
assert periodicity(sin(re(x)), x) == 2*pi
assert periodicity(sin(x)**2 + cos(x)**2, x) is S.Zero
assert periodicity(tan(x), y) is S.Zero
assert periodicity(sin(x) + I*cos(x), x) == 2*pi
assert periodicity(x - sin(2*y), y) == pi
assert periodicity(exp(x), x) is None
assert periodicity(exp(I*x), x) == 2*pi
assert periodicity(exp(I*z), z) == 2*pi
assert periodicity(exp(z), z) is None
assert periodicity(exp(log(sin(z) + I*cos(2*z)), evaluate=False), z) == 2*pi
assert periodicity(exp(log(sin(2*z) + I*cos(z)), evaluate=False), z) == 2*pi
assert periodicity(exp(sin(z)), z) == 2*pi
assert periodicity(exp(2*I*z), z) == pi
assert periodicity(exp(z + I*sin(z)), z) is None
assert periodicity(exp(cos(z/2) + sin(z)), z) == 4*pi
assert periodicity(log(x), x) is None
assert periodicity(exp(x)**sin(x), x) is None
assert periodicity(sin(x)**y, y) is None
assert periodicity(Abs(sin(Abs(sin(x)))), x) == pi
assert all(periodicity(Abs(f(x)), x) == pi for f in (
cos, sin, sec, csc, tan, cot))
assert periodicity(Abs(sin(tan(x))), x) == pi
assert periodicity(Abs(sin(sin(x) + tan(x))), x) == 2*pi
assert periodicity(sin(x) > S.Half, x) == 2*pi
assert periodicity(x > 2, x) is None
assert periodicity(x**3 - x**2 + 1, x) is None
assert periodicity(Abs(x), x) is None
assert periodicity(Abs(x**2 - 1), x) is None
assert periodicity((x**2 + 4)%2, x) is None
assert periodicity((E**x)%3, x) is None
assert periodicity(sin(expint(1, x))/expint(1, x), x) is None
GOOD_PAIRS = [("0", 0), ("1", 1), ("-3.14", _Mul(-1, 3.14)),
("(-7.13)(1.5)", _Mul(_Mul(-1, 7.13), 1.5)), ("x", x),
("2x", 2 * x), ("x^2", x**2), ("x^{3 + 1}", x**_Add(3, 1)),
("-c", -c), ("a \\cdot b", a * b), ("a / b", a / b),
("a \\div b", a / b), ("a + b", a + b),
("a + b - a", _Add(a + b, -a)),
("a^2 + b^2 = c^2", Eq(a**2 + b**2, c**2)),
("\\sin \\theta", sin(theta)), ("\\sin(\\theta)", sin(theta)),
("\\sin^{-1} a", asin(a)),
("\\sin a \\cos b", _Mul(sin(a), cos(b))),
("\\sin \\cos \\theta", sin(cos(theta))),
("\\sin(\\cos \\theta)", sin(cos(theta))),
("\\frac{a}{b}", a / b),
("\\frac{a + b}{c}", _Mul(a + b, _Pow(c, -1))),
("\\frac{7}{3}", _Mul(7, _Pow(3, -1))),
("(\\csc x)(\\sec y)", csc(x) * sec(y)),
("\\lim_{x \\to 3} a", Limit(a, x, 3)),
("\\lim_{x \\rightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\Rightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\longrightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\Longrightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\to 3^{+}} a", Limit(a, x, 3, dir='+')),
("\\lim_{x \\to 3^{-}} a", Limit(a, x, 3, dir='-')),
("\\infty", oo),
("\\lim_{x \\to \\infty} \\frac{1}{x}",
Limit(_Mul(1, _Pow(x, -1)), x, oo)),
("\\frac{d}{dx} x", Derivative(x, x)),
("\\frac{d}{dt} x", Derivative(x, t)), ("f(x)", f(x)),
("f(x, y)", f(x, y)), ("f(x, y, z)", f(x, y, z)),
("\\frac{d f(x)}{dx}", Derivative(f(x), x)),
("\\frac{d\\theta(x)}{dx}", Derivative(Function('theta')(x), x)),
