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trigonometry.py
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trigonometry.py
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# -*- coding: utf-8 -*-
import sympy
from sympy.core import Symbol, Wild, S
from sympy.core.numbers import Rational
from sympy.functions import sin, cos, binomial
from sympy.core.cache import cacheit
# TODO add support for tan^m(x) * sec^n(x)
# 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
@cacheit
def _pat_sincos(x):
a, n, m = [Wild(s, exclude=[x]) for s in 'anm']
pat = sin(a*x)**n * cos(a*x)**m
return pat, a,n,m
_u = Symbol('u', dummy=True)
def trigintegrate(f, x):
"""Integrate f = Mul(trig) over x
>>> from sympy import Symbol, sin, cos
>>> from sympy.integrals.trigonometry import trigintegrate
>>> x = Symbol('x')
>>> trigintegrate(sin(x)*cos(x), x)
sin(x)**2/2
>>> trigintegrate(sin(x)**2, x)
x/2 - cos(x)*sin(x)/2
http://en.wikibooks.org/wiki/Calculus/Further_integration_techniques
"""
pat, a,n,m = _pat_sincos(x)
##m - cos
##n - sin
M = f.match(pat)
if M is None:
return
n, m = M[n], M[m] # should always be there
if n is S.Zero and m is S.Zero:
return x
a = M[a]
if n.is_integer and m.is_integer:
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_:
n_ = n_ and (n < m) # NB: careful here, one of the
m_ = m_ and 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= sympy.integrals.integrate(ff, u) # XXX cyclic deps
fx= fi.subs(u, uu)
return fx / a
# n & m are even
else:
# 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 += (-1)**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 -1 m+1 n-1 n - 1 | m+2 n-2
# | cos (x) sin (x) dx = ________ 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 interms of cos (x), and then integrated.
for i in range(0,n/2+1):
res += (-1)**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 1 m-1 n+1 m - 1 | m-2 n+2
# | cos (x) sin (x) dx = _______ 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=sympy.integrals.integrate((Rational(1,2)*sin(2*x))**m,x)
elif (m == -n):
if n < 0:
##Same as the scheme described above.
res= Rational(1,(n+1))*cos(x)**(m-1)*sin(x)**(n+1) + Rational(m-1,n+1)*sympy.integrals.integrate(cos(x)**(m-2)*sin(x)**(n+2),x) ##the function argument to integrate in the end will be 1 , this cannot be integrated by trigintegrate. Hence use sympy.integrals.integrate.
else:
res=Rational(-1,m+1)*cos(x)**(m+1)*sin(x)**(n-1) + Rational(n-1,m+1)*sympy.integrals.integrate(cos(x)**(m+2)*sin(x)**(n-2),x)
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