def trigintegrate(f, x):
"""Integrate f = Mul(trig) over x
>>> from sympy import Symbol, sin, cos, tan, sec, csc, cot
>>> 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)
http://en.wikibooks.org/wiki/Calculus/Further_integration_techniques
See Also
========
sympy.integrals.integrals.Integral.doit
sympy.integrals.integrals.Integral
"""
pat, a,n,m = _pat_sincos(x)
pat1, s,t,q,r = _pat_gen(x)
M_ = f.match(pat1)
if M_ is None:
return
q = M_[q]
r = M_[r]
###
### f = function1(written in terms of sincos) X function2(written in terms of sincos)
###
if q is not S.Zero and r is not S.Zero:
s = M_[s]
t = M_[t]
if s.args is not () and t.args is not () \
and Trig_Check(s) and Trig_Check(t):
f = s._eval_rewrite_as_sincos(s.args[0])**q * t._eval_rewrite_as_sincos(t.args[0])**r
if q is S.Zero and r is S.Zero:
return x
if q is S.Zero and r is not S.Zero:
t = M_[t]
if t.args is not () and Trig_Check(t):
f = t._eval_rewrite_as_sincos(t.args[0])**r
if r is S.Zero and q is not S.Zero:
s = M_[s]
if s.args is not () and Trig_Check(s):
f = s._eval_rewrite_as_sincos(s.args[0])**q
M= f.match(pat) # matching the rewritten function with the sincos pattern
if M is None:
return
n, m = M[n], M[m]
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 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 trigintegrate(f, x):
"""Integrate f = Mul(trig) over x
>>> from sympy import Symbol, sin, cos
>>> 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 - 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
