def test_catalan():
n = Symbol('n', integer=True)
m = Symbol('n', integer=True, positive=True)
catalans = [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786]
for i, c in enumerate(catalans):
assert catalan(i) == c
assert catalan(n).rewrite(factorial).subs(n, i) == c
assert catalan(n).rewrite(Product).subs(n, i).doit() == c
assert catalan(x) == catalan(x)
assert catalan(2*x).rewrite(binomial) == binomial(4*x, 2*x)/(2*x + 1)
assert catalan(Rational(1, 2)).rewrite(gamma) == 8/(3*pi)
assert catalan(Rational(1, 2)).rewrite(factorial).rewrite(gamma) ==\
8 / (3 * pi)
assert catalan(3*x).rewrite(gamma) == 4**(
3*x)*gamma(3*x + Rational(1, 2))/(sqrt(pi)*gamma(3*x + 2))
assert catalan(x).rewrite(hyper) == hyper((-x + 1, -x), (2,), 1)
assert catalan(n).rewrite(factorial) == factorial(2*n) / (factorial(n + 1)
* factorial(n))
assert isinstance(catalan(n).rewrite(Product), catalan)
assert isinstance(catalan(m).rewrite(Product), Product)
assert diff(catalan(x), x) == (polygamma(
0, x + Rational(1, 2)) - polygamma(0, x + 2) + log(4))*catalan(x)
assert catalan(x).evalf() == catalan(x)
c = catalan(S.Half).evalf()
assert str(c) == '0.848826363156775'
c = catalan(I).evalf(3)
assert str((re(c), im(c))) == '(0.398, -0.0209)'
def test_euler():
assert euler(0) == 1
assert euler(1) == 0
assert euler(2) == -1
assert euler(3) == 0
assert euler(4) == 5
assert euler(6) == -61
assert euler(8) == 1385
assert euler(20, evaluate=False) != 370371188237525
n = Symbol('n', integer=True)
assert euler(n) != -1
assert euler(n).subs(n, 2) == -1
raises(ValueError, lambda: euler(-2))
raises(ValueError, lambda: euler(-3))
raises(ValueError, lambda: euler(2.3))
assert euler(20).evalf() == 370371188237525.0
assert euler(20, evaluate=False).evalf() == 370371188237525.0
assert euler(n).rewrite(Sum) == euler(n)
n = Symbol('n', integer=True, nonnegative=True)
assert euler(2 * n + 1).rewrite(Sum) == 0
_j = Dummy('j')
_k = Dummy('k')
assert euler(2 * n).rewrite(Sum).dummy_eq(I * Sum(
(-1)**_j * 2**(-_k) * I**(-_k) *
(-2 * _j + _k)**(2 * n + 1) * binomial(_k, _j) / _k, (_j, 0, _k),
(_k, 1, 2 * n + 1)))
def test_catalan():
n = Symbol('n', integer=True)
m = Symbol('n', integer=True, positive=True)
catalans = [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786]
for i, c in enumerate(catalans):
assert catalan(i) == c
assert catalan(n).rewrite(factorial).subs(n, i) == c
assert catalan(n).rewrite(Product).subs(n, i).doit() == c
assert catalan(x) == catalan(x)
assert catalan(2 *
x).rewrite(binomial) == binomial(4 * x, 2 * x) / (2 * x + 1)
assert catalan(Rational(1, 2)).rewrite(gamma) == 8 / (3 * pi)
assert catalan(Rational(1, 2)).rewrite(factorial).rewrite(gamma) ==\
8 / (3 * pi)
assert catalan(3 * x).rewrite(gamma) == 4**(
3 * x) * gamma(3 * x + Rational(1, 2)) / (sqrt(pi) * gamma(3 * x + 2))
assert catalan(x).rewrite(hyper) == hyper((-x + 1, -x), (2, ), 1)
assert catalan(n).rewrite(factorial) == factorial(
2 * n) / (factorial(n + 1) * factorial(n))
assert isinstance(catalan(n).rewrite(Product), catalan)
assert isinstance(catalan(m).rewrite(Product), Product)
assert diff(catalan(x), x) == (polygamma(0, x + Rational(1, 2)) -
polygamma(0, x + 2) + log(4)) * catalan(x)
assert catalan(x).evalf() == catalan(x)
c = catalan(S.Half).evalf()
assert str(c) == '0.848826363156775'
c = catalan(I).evalf(3)
assert str((re(c), im(c))) == '(0.398, -0.0209)'
def test_catalan():
assert catalan(1) == 1
assert catalan(2) == 2
assert catalan(3) == 5
assert catalan(4) == 14
assert catalan(x) == catalan(x)
assert catalan(2*x).rewrite(binomial) == binomial(4*x, 2*x)/(2*x + 1)
assert catalan(Rational(1, 2)).rewrite(gamma) == 8/(3*pi)
assert catalan(3*x).rewrite(gamma) == 4**(
3*x)*gamma(3*x + Rational(1, 2))/(sqrt(pi)*gamma(3*x + 2))
assert catalan(x).rewrite(hyper) == hyper((-x + 1, -x), (2,), 1)
assert diff(catalan(x), x) == (polygamma(
0, x + Rational(1, 2)) - polygamma(0, x + 2) + log(4))*catalan(x)
c = catalan(0.5).evalf()
assert str(c) == '0.848826363156775'
def test_catalan():
assert catalan(1) == 1
assert catalan(2) == 2
assert catalan(3) == 5
assert catalan(4) == 14
assert catalan(x) == catalan(x)
assert catalan(2 *
x).rewrite(binomial) == binomial(4 * x, 2 * x) / (2 * x + 1)
assert catalan(Rational(1, 2)).rewrite(gamma) == 8 / (3 * pi)
assert catalan(3 * x).rewrite(gamma) == 4**(
3 * x) * gamma(3 * x + Rational(1, 2)) / (sqrt(pi) * gamma(3 * x + 2))
assert catalan(x).rewrite(hyper) == hyper((-x + 1, -x), (2, ), 1)
assert diff(catalan(x), x) == (polygamma(0, x + Rational(1, 2)) -
polygamma(0, x + 2) + log(4)) * catalan(x)
c = catalan(0.5).evalf()
assert str(c) == '0.848826363156775'
def test_catalan():
n = Symbol("n", integer=True)
m = Symbol("m", integer=True, positive=True)
k = Symbol("k", integer=True, nonnegative=True)
p = Symbol("p", nonnegative=True)
catalans = [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786]
for i, c in enumerate(catalans):
assert catalan(i) == c
assert catalan(n).rewrite(factorial).subs(n, i) == c
assert catalan(n).rewrite(Product).subs(n, i).doit() == c
assert unchanged(catalan, x)
assert catalan(2 *
x).rewrite(binomial) == binomial(4 * x, 2 * x) / (2 * x + 1)
assert catalan(S.Half).rewrite(gamma) == 8 / (3 * pi)
assert catalan(S.Half).rewrite(factorial).rewrite(gamma) == 8 / (3 * pi)
assert catalan(3 * x).rewrite(gamma) == 4**(
3 * x) * gamma(3 * x + S.Half) / (sqrt(pi) * gamma(3 * x + 2))
assert catalan(x).rewrite(hyper) == hyper((-x + 1, -x), (2, ), 1)
assert catalan(n).rewrite(factorial) == factorial(
2 * n) / (factorial(n + 1) * factorial(n))
assert isinstance(catalan(n).rewrite(Product), catalan)
assert isinstance(catalan(m).rewrite(Product), Product)
assert diff(catalan(x), x) == (polygamma(0, x + S.Half) -
polygamma(0, x + 2) + log(4)) * catalan(x)
assert catalan(x).evalf() == catalan(x)
c = catalan(S.Half).evalf()
assert str(c) == "0.848826363156775"
c = catalan(I).evalf(3)
assert str((re(c), im(c))) == "(0.398, -0.0209)"
# Assumptions
assert catalan(p).is_positive is True
assert catalan(k).is_integer is True
assert catalan(m + 3).is_composite is True
def test_Function_change_name():
assert mcode(abs(x)) == "abs(x)"
assert mcode(ceiling(x)) == "ceil(x)"
assert mcode(arg(x)) == "angle(x)"
assert mcode(im(x)) == "imag(x)"
assert mcode(re(x)) == "real(x)"
assert mcode(conjugate(x)) == "conj(x)"
assert mcode(chebyshevt(y, x)) == "chebyshevT(y, x)"
assert mcode(chebyshevu(y, x)) == "chebyshevU(y, x)"
assert mcode(laguerre(x, y)) == "laguerreL(x, y)"
assert mcode(Chi(x)) == "coshint(x)"
assert mcode(Shi(x)) == "sinhint(x)"
assert mcode(Ci(x)) == "cosint(x)"
assert mcode(Si(x)) == "sinint(x)"
assert mcode(li(x)) == "logint(x)"
assert mcode(loggamma(x)) == "gammaln(x)"
assert mcode(polygamma(x, y)) == "psi(x, y)"
assert mcode(RisingFactorial(x, y)) == "pochhammer(x, y)"
assert mcode(DiracDelta(x)) == "dirac(x)"
assert mcode(DiracDelta(x, 3)) == "dirac(3, x)"
assert mcode(Heaviside(x)) == "heaviside(x, 1/2)"
assert mcode(Heaviside(x, y)) == "heaviside(x, y)"
assert mcode(binomial(x, y)) == "bincoeff(x, y)"
assert mcode(Mod(x, y)) == "mod(x, y)"
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)
1/2*sin(x)**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 = 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 n.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_ = (n > m) # NB: careful here, one of the
m_ = not (n > m) # conditions *must* be true
res = S.Zero
if n_:
# 2k 2 k i 2i
# C = (1-S ) = sum(i, (-) * B(k,i) * S )
for i in range(0,m/2+1):
res += (-1)**i * binomial(m/2,i) * Sin_2k_integrate(n/2+i, x)
elif m_:
# 2k 2 k i 2i
# S = (1 -C ) = sum(i, (-) * B(k,i) * C )
for i in range(0,n/2+1):
res += (-1)**i * binomial(n/2,i) * Cos_2k_integrate(m/2+i, x)
return res.subs(x, a*x) / a
def rsolve_poly(coeffs, f, n, **hints):
"""
Given linear recurrence operator `\operatorname{L}` of order
`k` with polynomial coefficients and inhomogeneous equation
`\operatorname{L} y = f`, where `f` is a polynomial, we seek for
all polynomial solutions over field `K` of characteristic zero.
The algorithm performs two basic steps:
(1) Compute degree `N` of the general polynomial solution.
(2) Find all polynomials of degree `N` or less
of `\operatorname{L} y = f`.
There are two methods for computing the polynomial solutions.
If the degree bound is relatively small, i.e. it's smaller than
or equal to the order of the recurrence, then naive method of
undetermined coefficients is being used. This gives system
of algebraic equations with `N+1` unknowns.
In the other case, the algorithm performs transformation of the
initial equation to an equivalent one, for which the system of
algebraic equations has only `r` indeterminates. This method is
quite sophisticated (in comparison with the naive one) and was
invented together by Abramov, Bronstein and Petkovsek.
It is possible to generalize the algorithm implemented here to
the case of linear q-difference and differential equations.
Lets say that we would like to compute `m`-th Bernoulli polynomial
up to a constant. For this we can use `b(n+1) - b(n) = m n^{m-1}`
recurrence, which has solution `b(n) = B_m + C`. For example:
>>> from sympy import Symbol, rsolve_poly
>>> n = Symbol('n', integer=True)
>>> rsolve_poly([-1, 1], 4*n**3, n)
C0 + n**4 - 2*n**3 + n**2
References
==========
.. [1] S. A. Abramov, M. Bronstein and M. Petkovsek, On polynomial
solutions of linear operator equations, in: T. Levelt, ed.,
Proc. ISSAC '95, ACM Press, New York, 1995, 290-296.
.. [2] M. Petkovsek, Hypergeometric solutions of linear recurrences
with polynomial coefficients, J. Symbolic Computation,
14 (1992), 243-264.
.. [3] M. Petkovsek, H. S. Wilf, D. Zeilberger, A = B, 1996.
"""
f = sympify(f)
if not f.is_polynomial(n):
return None
homogeneous = f.is_zero
r = len(coeffs) - 1
coeffs = [ Poly(coeff, n) for coeff in coeffs ]
polys = [ Poly(0, n) ] * (r + 1)
terms = [ (S.Zero, S.NegativeInfinity) ] *(r + 1)
for i in xrange(0, r + 1):
for j in xrange(i, r + 1):
polys[i] += coeffs[j]*binomial(j, i)
if not polys[i].is_zero:
(exp,), coeff = polys[i].LT()
terms[i] = (coeff, exp)
d = b = terms[0][1]
for i in xrange(1, r + 1):
if terms[i][1] > d:
d = terms[i][1]
if terms[i][1] - i > b:
b = terms[i][1] - i
d, b = int(d), int(b)
x = Dummy('x')
degree_poly = S.Zero
for i in xrange(0, r + 1):
if terms[i][1] - i == b:
degree_poly += terms[i][0]*FallingFactorial(x, i)
nni_roots = roots(degree_poly, x, filter='Z',
predicate=lambda r: r >= 0).keys()
if nni_roots:
N = [max(nni_roots)]
else:
N = []
if homogeneous:
N += [-b - 1]
else:
N += [f.as_poly(n).degree() - b, -b - 1]
N = int(max(N))
if N < 0:
if homogeneous:
if hints.get('symbols', False):
return (S.Zero, [])
else:
return S.Zero
else:
return None
if N <= r:
C = []
y = E = S.Zero
for i in xrange(0, N + 1):
C.append(Symbol('C' + str(i)))
y += C[i] * n**i
for i in xrange(0, r + 1):
E += coeffs[i].as_expr()*y.subs(n, n + i)
solutions = solve_undetermined_coeffs(E - f, C, n)
if solutions is not None:
C = [ c for c in C if (c not in solutions) ]
result = y.subs(solutions)
else:
return None # TBD
else:
A = r
U = N + A + b + 1
nni_roots = roots(polys[r], filter='Z',
predicate=lambda r: r >= 0).keys()
if nni_roots != []:
a = max(nni_roots) + 1
else:
a = S.Zero
def _zero_vector(k):
return [S.Zero] * k
def _one_vector(k):
return [S.One] * k
def _delta(p, k):
B = S.One
D = p.subs(n, a + k)
for i in xrange(1, k + 1):
B *= -Rational(k - i + 1, i)
D += B * p.subs(n, a + k - i)
return D
alpha = {}
for i in xrange(-A, d + 1):
I = _one_vector(d + 1)
for k in xrange(1, d + 1):
I[k] = I[k - 1] * (x + i - k + 1)/k
alpha[i] = S.Zero
for j in xrange(0, A + 1):
for k in xrange(0, d + 1):
B = binomial(k, i + j)
D = _delta(polys[j].as_expr(), k)
alpha[i] += I[k]*B*D
V = Matrix(U, A, lambda i, j: int(i == j))
if homogeneous:
for i in xrange(A, U):
v = _zero_vector(A)
for k in xrange(1, A + b + 1):
if i - k < 0:
break
B = alpha[k - A].subs(x, i - k)
for j in xrange(0, A):
v[j] += B * V[i - k, j]
denom = alpha[-A].subs(x, i)
for j in xrange(0, A):
V[i, j] = -v[j] / denom
else:
G = _zero_vector(U)
for i in xrange(A, U):
v = _zero_vector(A)
g = S.Zero
for k in xrange(1, A + b + 1):
if i - k < 0:
break
B = alpha[k - A].subs(x, i - k)
for j in xrange(0, A):
v[j] += B * V[i - k, j]
g += B * G[i - k]
denom = alpha[-A].subs(x, i)
for j in xrange(0, A):
V[i, j] = -v[j] / denom
G[i] = (_delta(f, i - A) - g) / denom
P, Q = _one_vector(U), _zero_vector(A)
for i in xrange(1, U):
P[i] = (P[i - 1] * (n - a - i + 1)/i).expand()
for i in xrange(0, A):
Q[i] = Add(*[ (v*p).expand() for v, p in zip(V[:, i], P) ])
if not homogeneous:
h = Add(*[ (g*p).expand() for g, p in zip(G, P) ])
C = [ Symbol('C' + str(i)) for i in xrange(0, A) ]
g = lambda i: Add(*[ c*_delta(q, i) for c, q in zip(C, Q) ])
if homogeneous:
E = [ g(i) for i in xrange(N + 1, U) ]
else:
E = [ g(i) + _delta(h, i) for i in xrange(N + 1, U) ]
if E != []:
solutions = solve(E, *C)
if not solutions:
if homogeneous:
if hints.get('symbols', False):
return (S.Zero, [])
else:
return S.Zero
else:
return None
else:
solutions = {}
if homogeneous:
result = S.Zero
else:
result = h
for c, q in list(zip(C, Q)):
if c in solutions:
s = solutions[c]*q
C.remove(c)
else:
s = c*q
result += s.expand()
if hints.get('symbols', False):
return (result, C)
else:
return result
def test_euler_failing():
# depends on dummy variables being implemented https://github.com/sympy/sympy/issues/5665
assert euler(2 * n).rewrite(Sum) == I * Sum(
Sum((-1)**_j * 2**(-_k) * I**(-_k) *
(-2 * _j + _k)**(2 * n + 1) * binomial(_k, _j) / _k, (_j, 0, _k)),
(_k, 1, 2 * n + 1))
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
def trigintegrate(f, x, conds='piecewise'):
"""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/Integration_techniques
See Also
========
sympy.integrals.integrals.Integral.doit
sympy.integrals.integrals.Integral
"""
from sympy.integrals.integrals import integrate
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 += ((-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
# | 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 += ((-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
# | 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 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 test_euler_failing():
# depends on dummy variables being implemented https://github.com/sympy/sympy/issues/5665
assert euler(2*n).rewrite(Sum) == I*Sum(Sum((-1)**_j*2**(-_k)*I**(-_k)*(-2*_j + _k)**(2*n + 1)*binomial(_k, _j)/_k, (_j, 0, _k)), (_k, 1, 2*n + 1))
def trigintegrate(f, x, conds='piecewise'):
"""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/Integration_techniques
See Also
========
sympy.integrals.integrals.Integral.doit
sympy.integrals.integrals.Integral
"""
from sympy.integrals.integrals import integrate
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 S.Zero and m is S.Zero:
return x
zz = x if n is S.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_:
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 = integrate(ff, u) # XXX cyclic deps
fx = fi.subs(u, uu)
if conds == 'piecewise':
return Piecewise((zz, Eq(a, 0)), (fx / a, 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 += ((-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
# | 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 += ((-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
# | 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((Rational(1, 2)*sin(2*x))**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((zz, Eq(a, 0)), (res.subs(x, a*x) / a, True))
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)
1/2*sin(x)**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 = 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 n.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_ = (n > m) # NB: careful here, one of the
m_ = not (n > m) # conditions *must* be true
res = S.Zero
if n_:
# 2k 2 k i 2i
# C = (1-S ) = sum(i, (-) * B(k,i) * S )
for i in range(0, m / 2 + 1):
res += (-1)**i * binomial(m / 2, i) * Sin_2k_integrate(
n / 2 + i, x)
elif m_:
# 2k 2 k i 2i
# S = (1 -C ) = sum(i, (-) * B(k,i) * C )
for i in range(0, n / 2 + 1):
res += (-1)**i * binomial(n / 2, i) * Cos_2k_integrate(
m / 2 + i, 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
