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from sympy.core import cacheit, Dummy, Ne, Integer, Rational, S, Wildfrom sympy.functions import binomial, sin, cos, Piecewisefrom .integrals import integrate
# TODO sin(a*x)*cos(b*x) -> sin((a+b)x) + sin((a-b)x) ?
# creating, each time, Wild's and sin/cos/Mul is expensive. Also, our match &# subs are very slow when not cached, and if we create Wild each time, we# effectively block caching.## so we cache the pattern
# need to use a function instead of lamda since hash of lambda changes on# each call to _pat_sincosdef _integer_instance(n): return isinstance(n, Integer)
@cacheitdef _pat_sincos(x): a = Wild('a', exclude=[x]) n, m = [Wild(s, exclude=[x], properties=[_integer_instance]) for s in 'nm'] pat = sin(a*x)**n * cos(a*x)**m return pat, a, n, m
_u = Dummy('u')
def trigintegrate(f, x, conds='piecewise'): """
Integrate f = Mul(trig) over x.
Examples ========
>>> from sympy import sin, cos, tan, sec >>> from sympy.integrals.trigonometry import trigintegrate >>> from sympy.abc import x
>>> trigintegrate(sin(x)*cos(x), x) sin(x)**2/2
>>> trigintegrate(sin(x)**2, x) x/2 - sin(x)*cos(x)/2
>>> trigintegrate(tan(x)*sec(x), x) 1/cos(x)
>>> trigintegrate(sin(x)*tan(x), x) -log(sin(x) - 1)/2 + log(sin(x) + 1)/2 - sin(x)
References ==========
.. [1] http://en.wikibooks.org/wiki/Calculus/Integration_techniques
See Also ========
sympy.integrals.integrals.Integral.doit sympy.integrals.integrals.Integral """
pat, a, n, m = _pat_sincos(x)
f = f.rewrite('sincos') M = f.match(pat)
if M is None: return
n, m = M[n], M[m] if n.is_zero and m.is_zero: return x zz = x if n.is_zero else S.Zero
a = M[a]
if n.is_odd or m.is_odd: u = _u n_, m_ = n.is_odd, m.is_odd
# take smallest n or m -- to choose simplest substitution if n_ and m_:
# Make sure to choose the positive one # otherwise an incorrect integral can occur. if n < 0 and m > 0: m_ = True n_ = False elif m < 0 and n > 0: n_ = True m_ = False # Both are negative so choose the smallest n or m # in absolute value for simplest substitution. elif (n < 0 and m < 0): n_ = n > m m_ = not (n > m)
# Both n and m are odd and positive else: n_ = (n < m) # NB: careful here, one of the m_ = not (n < m) # conditions *must* be true
# n m u=C (n-1)/2 m # S(x) * C(x) dx --> -(1-u^2) * u du if n_: ff = -(1 - u**2)**((n - 1)/2) * u**m uu = cos(a*x)
# n m u=S n (m-1)/2 # S(x) * C(x) dx --> u * (1-u^2) du elif m_: ff = u**n * (1 - u**2)**((m - 1)/2) uu = sin(a*x)
fi = integrate(ff, u) # XXX cyclic deps fx = fi.subs(u, uu) if conds == 'piecewise': return Piecewise((fx / a, Ne(a, 0)), (zz, True)) return fx / a
# n & m are both even # # 2k 2m 2l 2l # we transform S (x) * C (x) into terms with only S (x) or C (x) # # example: # 100 4 100 2 2 100 4 2 # S (x) * C (x) = S (x) * (1-S (x)) = S (x) * (1 + S (x) - 2*S (x)) # # 104 102 100 # = S (x) - 2*S (x) + S (x) # 2k # then S is integrated with recursive formula
# take largest n or m -- to choose simplest substitution n_ = (abs(n) > abs(m)) m_ = (abs(m) > abs(n)) res = S.Zero
if n_: # 2k 2 k i 2i # C = (1 - S ) = sum(i, (-) * B(k, i) * S ) if m > 0: for i in range(0, m//2 + 1): res += (S.NegativeOne**i * binomial(m//2, i) * _sin_pow_integrate(n + 2*i, x))
elif m == 0: res = _sin_pow_integrate(n, x) else:
# m < 0 , |n| > |m| # / # | # | m n # | cos (x) sin (x) dx = # | # | #/ # / # | # -1 m+1 n-1 n - 1 | m+2 n-2 # ________ cos (x) sin (x) + _______ | cos (x) sin (x) dx # | # m + 1 m + 1 | # /
res = (Rational(-1, m + 1) * cos(x)**(m + 1) * sin(x)**(n - 1) + Rational(n - 1, m + 1) * trigintegrate(cos(x)**(m + 2)*sin(x)**(n - 2), x))
elif m_: # 2k 2 k i 2i # S = (1 - C ) = sum(i, (-) * B(k, i) * C ) if n > 0:
# / / # | | # | m n | -m n # | cos (x)*sin (x) dx or | cos (x) * sin (x) dx # | | # / / # # |m| > |n| ; m, n >0 ; m, n belong to Z - {0} # n 2 # sin (x) term is expanded here in terms of cos (x), # and then integrated. #
for i in range(0, n//2 + 1): res += (S.NegativeOne**i * binomial(n//2, i) * _cos_pow_integrate(m + 2*i, x))
elif n == 0:
# / # | # | 1 # | _ _ _ # | m # | cos (x) # / #
res = _cos_pow_integrate(m, x) else:
# n < 0 , |m| > |n| # / # | # | m n # | cos (x) sin (x) dx = # | # | #/ # / # | # 1 m-1 n+1 m - 1 | m-2 n+2 # _______ cos (x) sin (x) + _______ | cos (x) sin (x) dx # | # n + 1 n + 1 | # /
res = (Rational(1, n + 1) * cos(x)**(m - 1)*sin(x)**(n + 1) + Rational(m - 1, n + 1) * trigintegrate(cos(x)**(m - 2)*sin(x)**(n + 2), x))
else: if m == n: ##Substitute sin(2x)/2 for sin(x)cos(x) and then Integrate. res = integrate((sin(2*x)*S.Half)**m, x) elif (m == -n): if n < 0: # Same as the scheme described above. # the function argument to integrate in the end will # be 1, this cannot be integrated by trigintegrate. # Hence use sympy.integrals.integrate. res = (Rational(1, n + 1) * cos(x)**(m - 1) * sin(x)**(n + 1) + Rational(m - 1, n + 1) * integrate(cos(x)**(m - 2) * sin(x)**(n + 2), x)) else: res = (Rational(-1, m + 1) * cos(x)**(m + 1) * sin(x)**(n - 1) + Rational(n - 1, m + 1) * integrate(cos(x)**(m + 2)*sin(x)**(n - 2), x)) if conds == 'piecewise': return Piecewise((res.subs(x, a*x) / a, Ne(a, 0)), (zz, True)) return res.subs(x, a*x) / a
def _sin_pow_integrate(n, x): if n > 0: if n == 1: #Recursion break return -cos(x)
# n > 0 # / / # | | # | n -1 n-1 n - 1 | n-2 # | sin (x) dx = ______ cos (x) sin (x) + _______ | sin (x) dx # | | # | n n | #/ / # #
return (Rational(-1, n) * cos(x) * sin(x)**(n - 1) + Rational(n - 1, n) * _sin_pow_integrate(n - 2, x))
if n < 0: if n == -1: ##Make sure this does not come back here again. ##Recursion breaks here or at n==0. return trigintegrate(1/sin(x), x)
# n < 0 # / / # | | # | n 1 n+1 n + 2 | n+2 # | sin (x) dx = _______ cos (x) sin (x) + _______ | sin (x) dx # | | # | n + 1 n + 1 | #/ / #
return (Rational(1, n + 1) * cos(x) * sin(x)**(n + 1) + Rational(n + 2, n + 1) * _sin_pow_integrate(n + 2, x))
else: #n == 0 #Recursion break. return x
def _cos_pow_integrate(n, x): if n > 0: if n == 1: #Recursion break. return sin(x)
# n > 0 # / / # | | # | n 1 n-1 n - 1 | n-2 # | sin (x) dx = ______ sin (x) cos (x) + _______ | cos (x) dx # | | # | n n | #/ / #
return (Rational(1, n) * sin(x) * cos(x)**(n - 1) + Rational(n - 1, n) * _cos_pow_integrate(n - 2, x))
if n < 0: if n == -1: ##Recursion break return trigintegrate(1/cos(x), x)
# n < 0 # / / # | | # | n -1 n+1 n + 2 | n+2 # | cos (x) dx = _______ sin (x) cos (x) + _______ | cos (x) dx # | | # | n + 1 n + 1 | #/ / #
return (Rational(-1, n + 1) * sin(x) * cos(x)**(n + 1) + Rational(n + 2, n + 1) * _cos_pow_integrate(n + 2, x)) else: # n == 0 #Recursion Break. return x
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