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# sympy.integrals.risch.splitfactor

All Samples(32)  |  Call(26)  |  Derive(0)  |  Import(6)

```from sympy.polys import Poly, lcm, cancel, sqf_list

from sympy.integrals.risch import (gcdex_diophantine, frac_in, derivation,
NonElementaryIntegralException, residue_reduce, splitfactor,
residue_reduce_derivation, DecrementLevel, recognize_log_derivative)
```
```    q == y*h in k<t> satisfies a*Dq + b*q == Sum(ci*Gi, (i, 1, m)).
"""
dn, ds = splitfactor(fd, DE)
Gas, Gds = list(zip(*G))
gd = reduce(lambda i, j: i.lcm(j), Gds, Poly(1, DE.t))
en, es = splitfactor(gd, DE)
```
```    that Dt == a*t + b with for some a, b in k*.
"""
dn, ds = splitfactor(fd, DE)
E = [splitfactor(gd, DE) for _, gd in G]
En, Es = list(zip(*E))
```

```from sympy.polys import Poly, lcm, cancel, sqf_list

from sympy.integrals.risch import (gcdex_diophantine, frac_in, derivation,
NonElementaryIntegralException, residue_reduce, splitfactor,
residue_reduce_derivation, DecrementLevel, recognize_log_derivative)
```
```    q == y*h in k<t> satisfies a*Dq + b*q == Sum(ci*Gi, (i, 1, m)).
"""
dn, ds = splitfactor(fd, DE)
Gas, Gds = list(zip(*G))
gd = reduce(lambda i, j: i.lcm(j), Gds, Poly(1, DE.t))
en, es = splitfactor(gd, DE)
```
```    that Dt == a*t + b with for some a, b in k*.
"""
dn, ds = splitfactor(fd, DE)
E = [splitfactor(gd, DE) for _, gd in G]
En, Es = list(zip(*E))
```

```from sympy.polys import Poly, gcd, ZZ, cancel

from sympy.integrals.risch import (gcdex_diophantine, frac_in, derivation,
splitfactor, NonElementaryIntegralException, DecrementLevel)

```
```    """
z = z or Dummy('z')
dn, ds = splitfactor(d, DE)

# Compute d1, where dn == d1*d2**2*...*dn**n is a square-free
```
```    This constitutes step 1 in the outline given in the rde.py docstring.
"""
dn, ds = splitfactor(fd, DE)
en, es = splitfactor(gd, DE)

```

```from sympy.polys import Poly, gcd, ZZ, cancel

from sympy.integrals.risch import (gcdex_diophantine, frac_in, derivation,
splitfactor, NonElementaryIntegralException, DecrementLevel)

```
```    """
z = z or Dummy('z')
dn, ds = splitfactor(d, DE)

# Compute d1, where dn == d1*d2**2*...*dn**n is a square-free
```
```    This constitutes step 1 in the outline given in the rde.py docstring.
"""
dn, ds = splitfactor(fd, DE)
en, es = splitfactor(gd, DE)

```

```"""Most of these tests come from the examples in Bronstein's book."""
from sympy import (Poly, I, S, Function, log, symbols, exp, tan, sqrt,
Symbol, Lambda, sin, cos, Eq, Piecewise, factor)
from sympy.integrals.risch import (gcdex_diophantine, frac_in, as_poly_1t,
derivation, splitfactor, splitfactor_sqf, canonical_representation,
```
```def test_splitfactor():
p = Poly(4*x**4*t**5 + (-4*x**3 - 4*x**4)*t**4 + (-3*x**2 + 2*x**3)*t**3 +
(2*x + 7*x**2 + 2*x**3)*t**2 + (1 - 4*x - 4*x**2)*t - 1 + 2*x, t, field=True)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - 3/(2*x)*t + 1/(2*x), t)]})
assert splitfactor(p, DE) == (Poly(4*x**4*t**3 + (-8*x**3 - 4*x**4)*t**2 +
(4*x**2 + 8*x**3)*t - 4*x**2, t), Poly(t**2 + 1/x*t + (1 - 2*x)/(4*x**2), t, domain='ZZ(x)'))
assert splitfactor(Poly(x, t), DE) == (Poly(x, t), Poly(1, t))
r = Poly(-4*x**4*z**2 + 4*x**6*z**2 - z*x**3 - 4*x**5*z**3 + 4*x**3*z**3 + x**4 + z*x**5 - x**6, t)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
assert splitfactor(r, DE, coefficientD=True) == \
```
```    assert splitfactor_sqf(r, DE, coefficientD=True) == \
(((Poly(x*z - x**2 - z*x**3 + x**4, t), 1),), ((Poly(-x**2 + 4*x**2*z**2, t), 1),))
assert splitfactor(Poly(0, t), DE) == (Poly(0, t), Poly(1, t))
assert splitfactor_sqf(Poly(0, t), DE) == (((Poly(0, t), 1),), ())

```

```"""Most of these tests come from the examples in Bronstein's book."""
from sympy import (Poly, I, S, Function, log, symbols, exp, tan, sqrt,
Symbol, Lambda, sin, cos, Eq, Piecewise, factor)
from sympy.integrals.risch import (gcdex_diophantine, frac_in, as_poly_1t,
derivation, splitfactor, splitfactor_sqf, canonical_representation,
```
```def test_splitfactor():
p = Poly(4*x**4*t**5 + (-4*x**3 - 4*x**4)*t**4 + (-3*x**2 + 2*x**3)*t**3 +
(2*x + 7*x**2 + 2*x**3)*t**2 + (1 - 4*x - 4*x**2)*t - 1 + 2*x, t, field=True)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - 3/(2*x)*t + 1/(2*x), t)]})
assert splitfactor(p, DE) == (Poly(4*x**4*t**3 + (-8*x**3 - 4*x**4)*t**2 +
(4*x**2 + 8*x**3)*t - 4*x**2, t), Poly(t**2 + 1/x*t + (1 - 2*x)/(4*x**2), t, domain='ZZ(x)'))
assert splitfactor(Poly(x, t), DE) == (Poly(x, t), Poly(1, t))
r = Poly(-4*x**4*z**2 + 4*x**6*z**2 - z*x**3 - 4*x**5*z**3 + 4*x**3*z**3 + x**4 + z*x**5 - x**6, t)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
assert splitfactor(r, DE, coefficientD=True) == \
```
```    assert splitfactor_sqf(r, DE, coefficientD=True) == \
(((Poly(x*z - x**2 - z*x**3 + x**4, t), 1),), ((Poly(-x**2 + 4*x**2*z**2, t), 1),))
assert splitfactor(Poly(0, t), DE) == (Poly(0, t), Poly(1, t))
assert splitfactor_sqf(Poly(0, t), DE) == (((Poly(0, t), 1),), ())

```