def fricas_integrator(expression, v, a=None, b=None, noPole=True):
"""
Integration using FriCAS
EXAMPLES::
sage: from sage.symbolic.integration.external import fricas_integrator # optional - fricas
sage: fricas_integrator(sin(x), x) # optional - fricas
-cos(x)
sage: fricas_integrator(cos(x), x) # optional - fricas
sin(x)
sage: fricas_integrator(1/(x^2-2), x, 0, 1) # optional - fricas
1/4*sqrt(2)*(log(3*sqrt(2) - 4) - log(sqrt(2)))
sage: fricas_integrator(1/(x^2+6), x, -oo, oo) # optional - fricas
1/6*sqrt(6)*pi
TESTS:
Check that :trac:`25220` is fixed::
sage: integral(sqrt(1-cos(x)), x, 0, 2*pi, algorithm="fricas") # optional - fricas
4*sqrt(2)
Check that in case of failure one gets unevaluated integral::
sage: integral(cos(ln(cos(x))), x, 0, pi/8, algorithm='fricas') # optional - fricas
integrate(cos(log(cos(x))), x, 0, 1/8*pi)
sage: integral(cos(ln(cos(x))), x, algorithm='fricas') # optional - fricas
integral(cos(log(cos(x))), x)
"""
if not isinstance(expression, Expression):
expression = SR(expression)
from sage.interfaces.fricas import fricas
ex = fricas(expression)
if a is None:
result = ex.integrate(v)
else:
seg = fricas.equation(v, fricas.segment(a, b))
if noPole:
result = ex.integrate(seg, '"noPole"')
else:
result = ex.integrate(seg)
result = result.sage()
if result == "failed":
return expression.integrate(v, a, b, hold=True)
if result == "potentialPole":
raise ValueError("The integrand has a potential pole"
" in the integration interval")
return result
def fricas_integrator(expression, v, a=None, b=None, noPole=True):
"""
Integration using FriCAS
EXAMPLES::
sage: from sage.symbolic.integration.external import fricas_integrator # optional - fricas
sage: fricas_integrator(sin(x), x) # optional - fricas
-cos(x)
sage: fricas_integrator(cos(x), x) # optional - fricas
sin(x)
sage: fricas_integrator(1/(x^2-2), x, 0, 1) # optional - fricas
-1/8*sqrt(2)*(log(2) - log(-24*sqrt(2) + 34))
sage: fricas_integrator(1/(x^2+6), x, -oo, oo) # optional - fricas
1/6*sqrt(6)*pi
TESTS:
Check that :trac:`25220` is fixed::
sage: integral(sqrt(1-cos(x)), x, 0, 2*pi, algorithm="fricas") # optional - fricas
4*sqrt(2)
Check that in case of failure one gets unevaluated integral::
sage: integral(cos(ln(cos(x))), x, 0, pi/8, algorithm='fricas') # optional - fricas
integrate(cos(log(cos(x))), x, 0, 1/8*pi)
sage: integral(cos(ln(cos(x))), x, algorithm='fricas') # optional - fricas
integral(cos(log(cos(x))), x)
Check that :trac:`28641` is fixed::
sage: integrate(sqrt(2)*x^2 + 2*x, x, algorithm="fricas") # optional - fricas
1/3*sqrt(2)*x^3 + x^2
sage: integrate(sqrt(2), x, algorithm="fricas") # optional - fricas
sqrt(2)*x
sage: integrate(1, x, algorithm="fricas") # optional - fricas
x
Check that :trac:`28630` is fixed::
sage: f = polylog(3, x)
sage: f.integral(x, algorithm='fricas') # optional - fricas
-x*dilog(x) - (x - 1)*log(-x + 1) + x*polylog(3, x) + x
Check that :trac:`29043` is fixed::
sage: var("a c d"); f = (I*a*tan(d*x + c) + a)*sec(d*x + c)^10
(a, c, d)
sage: ii = integrate(f, x, algorithm="fricas") # optional - fricas
sage: 1/315*(64512*I*a*e^(10*I*d*x + 10*I*c) + 53760*I*a*e^(8*I*d*x + 8*I*c) + 30720*I*a*e^(6*I*d*x + 6*I*c) + 11520*I*a*e^(4*I*d*x + 4*I*c) + 2560*I*a*e^(2*I*d*x + 2*I*c) + 256*I*a)/(d*e^(20*I*d*x + 20*I*c) + 10*d*e^(18*I*d*x + 18*I*c) + 45*d*e^(16*I*d*x + 16*I*c) + 120*d*e^(14*I*d*x + 14*I*c) + 210*d*e^(12*I*d*x + 12*I*c) + 252*d*e^(10*I*d*x + 10*I*c) + 210*d*e^(8*I*d*x + 8*I*c) + 120*d*e^(6*I*d*x + 6*I*c) + 45*d*e^(4*I*d*x + 4*I*c) + 10*d*e^(2*I*d*x + 2*I*c) + d) - ii # optional - fricas
0
"""
if not isinstance(expression, Expression):
expression = SR(expression)
from sage.interfaces.fricas import fricas
e_fricas = fricas(expression)
v_fricas = fricas(v)
if a is None:
result = e_fricas.integrate(v_fricas)
else:
seg = fricas.equation(v_fricas, fricas.segment(a, b))
if noPole:
result = e_fricas.integrate(seg, '"noPole"')
else:
result = e_fricas.integrate(seg)
result = result.sage()
if result == "failed":
result = expression.integrate(v, a, b, hold=True)
elif result == "potentialPole":
raise ValueError("The integrand has a potential pole"
" in the integration interval")
return result
