def With38(Pq, Qr, p, u, x):
if isinstance(x, (int, Integer, float, Float)):
return False
gcd = PolyGCD(Pq, Qr, x)
if NonzeroQ(gcd + S(-1)):
return True
return False
def With1898(v, u, m, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if NonzeroQ(-a * v + b * u):
return True
return False
def With1891(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if And(NonzeroQ(-a * v + b * u), NegQ(a * b)):
return True
return False
def With1886(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if And(NonzeroQ(-a * v + b * u), PosQ((-a * v + b * u) / a)):
return True
return False
def piecewise_linear(rubi):
pattern1 = Pattern(
Integral(u_**WC('m', S(1)), x_),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda x, u: PiecewiseLinearQ(u, x)),
)
def With1(x, m, u):
c = D(u, x)
return Subst(Int(x**m, x), x, u) / c
rule1 = ReplacementRule(pattern1, lambda x, m, u: With1(x, m, u))
rubi.add(rule1)
pattern2 = Pattern(
Integral(v_ / u_, x_),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(lambda v, b, x, u, a: NonzeroQ(-a * v + b * u)))
def With2(x, v, u):
a = D(u, x)
b = D(v, x)
return b * x / a - (-a * v + b * u) * Int(1 / u, x) / a
rule2 = ReplacementRule(pattern2, lambda x, v, u: With2(x, v, u))
rubi.add(rule2)
pattern3 = Pattern(
Integral(v_**n_ / u_, x_),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(lambda n: RationalQ(n)),
CustomConstraint(lambda n: Greater(n, S(0))),
CustomConstraint(lambda n: Unequal(n, S(1))),
CustomConstraint(lambda v, b, n, x, u, a: NonzeroQ(-a * v + b * u)))
def With3(x, n, v, u):
a = D(u, x)
b = D(v, x)
return -(-a * v + b * u) * Int(v**
(n + S(-1)) / u, x) / a + v**n / (a * n)
rule3 = ReplacementRule(pattern3, lambda x, n, v, u: With3(x, n, v, u))
rubi.add(rule3)
pattern4 = Pattern(
Integral(S(1) / (u_ * v_), x_),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(lambda v, b, x, u, a: NonzeroQ(-a * v + b * u)))
def With4(x, v, u):
a = D(u, x)
b = D(v, x)
return -a * Int(1 / u, x) / (-a * v + b * u) + b * Int(
1 / v, x) / (-a * v + b * u)
rule4 = ReplacementRule(pattern4, lambda x, v, u: With4(x, v, u))
rubi.add(rule4)
pattern5 = Pattern(
Integral(S(1) / (u_ * sqrt(v_)), x_),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(lambda v, b, u, a: NonzeroQ(-a * v + b * u) & PosQ(
(-a * v + b * u) / a)))
def With5(x, v, u):
a = D(u, x)
b = D(v, x)
return S(2) * ArcTan(sqrt(v) / Rt(
(-a * v + b * u) / a, S(2))) / (a * Rt((-a * v + b * u) / a, S(2)))
rule5 = ReplacementRule(pattern5, lambda x, v, u: With5(x, v, u))
rubi.add(rule5)
pattern6 = Pattern(
Integral(S(1) / (u_ * sqrt(v_)), x_),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(lambda v, b, u, a: NonzeroQ(-a * v + b * u) & NegQ(
(-a * v + b * u) / a)))
def With6(x, v, u):
a = D(u, x)
b = D(v, x)
return -S(2) * atanh(sqrt(v) / Rt(
(a * v - b * u) / a, S(2))) / (a * Rt((a * v - b * u) / a, S(2)))
rule6 = ReplacementRule(pattern6, lambda x, v, u: With6(x, v, u))
rubi.add(rule6)
pattern7 = Pattern(
Integral(v_**n_ / u_, x_),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(lambda n: RationalQ(n)),
CustomConstraint(lambda n: Less(n, S(-1))),
CustomConstraint(lambda v, b, n, x, u, a: NonzeroQ(-a * v + b * u)))
def With7(x, n, v, u):
a = D(u, x)
b = D(v, x)
return -a * Int(v**(n + S(1)) / u, x) / (-a * v + b * u) + v**(
n + S(1)) / ((n + S(1)) * (-a * v + b * u))
rule7 = ReplacementRule(pattern7, lambda x, n, v, u: With7(x, n, v, u))
rubi.add(rule7)
pattern8 = Pattern(
Integral(v_**n_ / u_, x_),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(lambda n: Not(IntegerQ(n))),
CustomConstraint(lambda v, b, n, u, a: NonzeroQ(-a * v + b * u)))
def With8(x, n, v, u):
a = D(u, x)
b = D(v, x)
return v**(n + S(1)) * Hypergeometric2F1(
S(1), n + S(1), n + S(2), -a * v /
(-a * v + b * u)) / ((n + S(1)) * (-a * v + b * u))
rule8 = ReplacementRule(pattern8, lambda x, n, v, u: With8(x, n, v, u))
rubi.add(rule8)
pattern9 = Pattern(
Integral(S(1) / (sqrt(u_) * sqrt(v_)), x_),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(
lambda v, b, u, a: PosQ(a * b) & NonzeroQ(-a * v + b * u)))
def With9(x, v, u):
a = D(u, x)
b = D(v, x)
return S(2) * atanh(sqrt(u) * Rt(a * b, S(2)) /
(a * sqrt(v))) / Rt(a * b, S(2))
rule9 = ReplacementRule(pattern9, lambda x, v, u: With9(x, v, u))
rubi.add(rule9)
pattern10 = Pattern(
Integral(S(1) / (sqrt(u_) * sqrt(v_)), x_),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(
lambda v, b, u, a: NegQ(a * b) & NonzeroQ(-a * v + b * u)))
def With10(x, v, u):
a = D(u, x)
b = D(v, x)
return S(2) * ArcTan(sqrt(u) * Rt(-a * b, S(2)) /
(a * sqrt(v))) / Rt(-a * b, S(2))
rule10 = ReplacementRule(pattern10, lambda x, v, u: With10(x, v, u))
rubi.add(rule10)
pattern11 = Pattern(
Integral(u_**m_ * v_**n_, x_),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(lambda n, m: ZeroQ(m + n + S(2))),
CustomConstraint(lambda m: NonzeroQ(m + S(1))),
CustomConstraint(lambda v, b, n, m, u, a: NonzeroQ(-a * v + b * u)))
def With11(x, n, v, u, m):
a = D(u, x)
b = D(v, x)
return -u**(m + S(1)) * v**(n + S(1)) / ((m + S(1)) * (-a * v + b * u))
rule11 = ReplacementRule(pattern11,
lambda x, n, v, u, m: With11(x, n, v, u, m))
rubi.add(rule11)
pattern12 = Pattern(
Integral(u_**m_ * v_**WC('n', S(1)), x_),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(lambda m: NonzeroQ(m + S(1))),
CustomConstraint(lambda n, m: (NegativeIntegerQ(m) & Not(IntegerQ(
n))) | (PositiveIntegerQ(n) & Not(IntegerQ(m))) | (
LessEqual(n, m) & PositiveIntegerQ(n, m)
) | (Greater(n, S(0)) & Less(m, S(-1)) & RationalQ(m, n) & Not(
IntegerQ(m + n) & Less(m + n + S(2), S(0)) &
(FractionQ(m) | GreaterEqual(m + S(2) * n + S(1), S(0)))))),
CustomConstraint(lambda v, b, n, x, m, u, a: NonzeroQ(-a * v + b * u)))
def With12(x, n, v, u, m):
a = D(u, x)
b = D(v, x)
return -b * n * Int(u**(m + S(1)) * v**(n + S(-1)), x) / (
a * (m + S(1))) + u**(m + S(1)) * v**n / (a * (m + S(1)))
rule12 = ReplacementRule(pattern12,
lambda x, n, v, u, m: With12(x, n, v, u, m))
rubi.add(rule12)
pattern13 = Pattern(
Integral(u_**m_ * v_**WC('n', S(1)), x_),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(lambda n, m: NonzeroQ(m + n + S(2))),
CustomConstraint(lambda n: RationalQ(n)),
CustomConstraint(lambda n: Greater(n, S(0))),
CustomConstraint(lambda n, m: NonzeroQ(m + n + S(1))),
CustomConstraint(lambda n, m: Not(
PositiveIntegerQ(m) & (Not(IntegerQ(n)) | Less(S(0), m, n)))),
CustomConstraint(
lambda n, m: Not(IntegerQ(m + n) & Less(m + n + S(2), S(0)))),
CustomConstraint(lambda v, b, n, x, m, u, a: NonzeroQ(-a * v + b * u)))
def With13(x, n, v, u, m):
a = D(u, x)
b = D(v, x)
return -n * (-a * v + b * u) * Int(u**m * v**(n + S(-1)), x) / (
a * (m + n + S(1))) + u**(m + S(1)) * v**n / (a * (m + n + S(1)))
rule13 = ReplacementRule(pattern13,
lambda x, n, v, u, m: With13(x, n, v, u, m))
rubi.add(rule13)
pattern14 = Pattern(
Integral(u_**m_ * v_**n_, x_),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(lambda n, m: NonzeroQ(m + n + S(1))),
CustomConstraint(lambda n: Not(RationalQ(n))),
CustomConstraint(lambda n: SumSimplerQ(n, S(-1))),
CustomConstraint(lambda v, b, n, x, m, u, a: NonzeroQ(-a * v + b * u)))
def With14(x, n, v, u, m):
a = D(u, x)
b = D(v, x)
return -n * (-a * v + b * u) * Int(u**m * v**(n + S(-1)), x) / (
a * (m + n + S(1))) + u**(m + S(1)) * v**n / (a * (m + n + S(1)))
rule14 = ReplacementRule(pattern14,
lambda x, n, v, u, m: With14(x, n, v, u, m))
rubi.add(rule14)
pattern15 = Pattern(
Integral(u_**m_ * v_**n_, x_),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(lambda n, m: NonzeroQ(m + n + S(2))),
CustomConstraint(lambda m: RationalQ(m)),
CustomConstraint(lambda m: Less(m, S(-1))),
CustomConstraint(lambda v, b, n, x, m, u, a: NonzeroQ(-a * v + b * u)))
def With15(x, n, v, u, m):
a = D(u, x)
b = D(v, x)
return b * (m + n + S(2)) * Int(u**(m + S(1)) * v**n, x) / (
(m + S(1)) * (-a * v + b * u)) - u**(m + S(1)) * v**(n + S(1)) / (
(m + S(1)) * (-a * v + b * u))
rule15 = ReplacementRule(pattern15,
lambda x, n, v, u, m: With15(x, n, v, u, m))
rubi.add(rule15)
pattern16 = Pattern(
Integral(u_**m_ * v_**n_, x_),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(lambda m: Not(RationalQ(m))),
CustomConstraint(lambda m: SumSimplerQ(m, S(1))),
CustomConstraint(lambda v, b, n, x, m, u, a: NonzeroQ(-a * v + b * u)))
def With16(x, n, v, u, m):
a = D(u, x)
b = D(v, x)
return b * (m + n + S(2)) * Int(u**(m + S(1)) * v**n, x) / (
(m + S(1)) * (-a * v + b * u)) - u**(m + S(1)) * v**(n + S(1)) / (
(m + S(1)) * (-a * v + b * u))
rule16 = ReplacementRule(pattern16,
lambda x, n, v, u, m: With16(x, n, v, u, m))
rubi.add(rule16)
pattern17 = Pattern(
Integral(u_**m_ * v_**n_, x_),
CustomConstraint(lambda x, v, u: PiecewiseLinearQ(u, v, x)),
CustomConstraint(lambda m: Not(IntegerQ(m))),
CustomConstraint(lambda n: Not(IntegerQ(n))),
CustomConstraint(lambda v, b, n, m, u, a: NonzeroQ(-a * v + b * u)))
def With17(x, n, v, u, m):
a = D(u, x)
b = D(v, x)
return u**m * v**(n + S(1)) * (b * u / (-a * v + b * u))**(
-m) * Hypergeometric2F1(-m, n + S(1), n + S(2), -a * v /
(-a * v + b * u)) / (b * (n + S(1)))
rule17 = ReplacementRule(pattern17,
lambda x, n, v, u, m: With17(x, n, v, u, m))
rubi.add(rule17)
pattern18 = Pattern(
Integral(u_**WC('n', S(1)) * log(x_ * WC('b', S(1)) + WC('a', S(0))),
x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda x, u: PiecewiseLinearQ(u, x)),
CustomConstraint(lambda x, u: Not(LinearQ(u, x))),
CustomConstraint(lambda n: RationalQ(n)),
CustomConstraint(lambda n: Greater(n, S(0))),
)
def With18(x, n, b, u, a):
c = D(u, x)
return -Int(u**n, x) - c * n * Int(
u**(n + S(-1)) * (a + b * x) * log(a + b * x),
x) / b + u**n * (a + b * x) * log(a + b * x) / b
rule18 = ReplacementRule(pattern18,
lambda x, n, b, u, a: With18(x, n, b, u, a))
rubi.add(rule18)
pattern19 = Pattern(
Integral(
u_**WC('n', S(1)) *
(x_ * WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1)) *
log(x_ * WC('b', S(1)) + WC('a', S(0))), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda x, u: PiecewiseLinearQ(u, x)),
CustomConstraint(lambda x, u: Not(LinearQ(u, x))),
CustomConstraint(lambda n: RationalQ(n)),
CustomConstraint(lambda n: Greater(n, S(0))),
CustomConstraint(lambda m: NonzeroQ(m + S(1))),
)
def With19(x, n, b, u, a, m):
c = D(u, x)
return -Int(u**n * (a + b * x)**m, x) / (m + S(1)) - c * n * Int(
u**(n + S(-1)) * (a + b * x)**
(m + S(1)) * log(a + b * x), x) / (b * (m + S(1))) + u**n * (
a + b * x)**(m + S(1)) * log(a + b * x) / (b * (m + S(1)))
rule19 = ReplacementRule(pattern19,
lambda x, n, b, u, a, m: With19(x, n, b, u, a, m))
rubi.add(rule19)
return rubi
def integrand_simplification(rubi):
pattern1 = Pattern(
Integral((a_ + x_**WC('n', S(1)) * WC('b', S(1)))**WC('p', S(1)) *
WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda a: ZeroQ(a)))
rule1 = ReplacementRule(pattern1,
lambda p, b, u, a, x, n: Int(u * (b * x**n)**p, x))
rubi.add(rule1)
pattern2 = Pattern(
Integral(
(a_ + x_**WC('j', S(1)) * WC('c', S(1)) +
x_**WC('n', S(1)) * WC('b', S(1)))**WC('p', S(1)) * WC('u', S(1)),
x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda n, j: ZeroQ(j - S(2) * n)),
CustomConstraint(lambda a: ZeroQ(a)))
rule2 = ReplacementRule(
pattern2, lambda j, b, p, u, a, x, n, c: Int(
u * (b * x**n + c * x**(S(2) * n))**p, x))
rubi.add(rule2)
pattern3 = Pattern(
Integral((x_**WC('j', S(1)) * WC('c', S(1)) +
x_**WC('n', S(1)) * WC('b', S(1)) + WC('a', S(0)))**WC(
'p', S(1)) * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda n, j: ZeroQ(j - S(2) * n)),
CustomConstraint(lambda b: ZeroQ(b)))
rule3 = ReplacementRule(
pattern3,
lambda j, b, p, u, a, x, n, c: Int(u * (a + c * x**(S(2) * n))**p, x))
rubi.add(rule3)
pattern4 = Pattern(
Integral((x_**WC('j', S(1)) * WC('c', S(1)) +
x_**WC('n', S(1)) * WC('b', S(1)) + WC('a', S(0)))**WC(
'p', S(1)) * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda n, j: ZeroQ(j - S(2) * n)),
CustomConstraint(lambda c: ZeroQ(c)))
rule4 = ReplacementRule(
pattern4, lambda j, b, p, u, a, x, n, c: Int(u * (a + b * x**n)**p, x))
rubi.add(rule4)
pattern5 = Pattern(
Integral((v_ * WC('a', S(1)) + v_ * WC('b', S(1)) + WC('w', S(0)))**WC(
'p', S(1)) * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda x, v: NFreeQ(v, x)))
rule5 = ReplacementRule(
pattern5, lambda p, b, v, u, a, x, w: Int(u * (v * (a + b) + w)**p, x))
rubi.add(rule5)
pattern6 = Pattern(Integral(Pm_**p_ * WC('u', S(1)), x_),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda x, Pm: PolyQ(Pm, x)),
CustomConstraint(lambda p: Not(RationalQ(p))),
CustomConstraint(lambda p: RationalQ(p)))
rule6 = ReplacementRule(pattern6, lambda u, x, p, Pm: Int(Pm**p * u, x))
rubi.add(rule6)
pattern7 = Pattern(Integral(a_, x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda a, x: FreeQ(a, x)))
rule7 = ReplacementRule(pattern7, lambda a, x: a * x)
rubi.add(rule7)
pattern8 = Pattern(
Integral(a_ * (b_ + x_ * WC('c', S(1))), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda a, c, b, x: FreeQ(List(a, b, c), x)))
rule8 = ReplacementRule(
pattern8, lambda a, c, b, x: a * (b + c * x)**S(2) / (S(2) * c))
rubi.add(rule8)
pattern9 = Pattern(Integral(-u_, x_))
rule9 = ReplacementRule(pattern9, lambda u, x: I * Int(u, x))
rubi.add(rule9)
pattern10 = Pattern(Integral(u_ * Complex(S(0), a_), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda a: EqQ(a**S(2), S(1))))
rule10 = ReplacementRule(pattern10,
lambda u, a, x: Int(u, x) * Complex(I, a))
rubi.add(rule10)
pattern11 = Pattern(Integral(a_ * u_, x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda a, x: FreeQ(a, x)))
rule11 = ReplacementRule(pattern11, lambda u, a, x: a * Int(u, x))
rubi.add(rule11)
pattern12 = Pattern(Integral(u_, x_), CustomConstraint(lambda u: SumQ(u)))
rule12 = ReplacementRule(pattern12, lambda u, x: IntSum(u, x))
rubi.add(rule12)
pattern13 = Pattern(
Integral(v_**WC('m', S(1)) * (b_ * v_)**n_ * WC('u', S(1)), x_),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda m: IntegerQ(m)))
rule13 = ReplacementRule(
pattern13,
lambda b, v, u, m, x, n: b**(-m) * Int(u * (b * v)**(m + n), x))
rubi.add(rule13)
pattern14 = Pattern(
Integral((v_ * WC('a', S(1)))**m_ *
(v_ * WC('b', S(1)))**n_ * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda m: Not(IntegerQ(m))),
CustomConstraint(lambda n: PositiveIntegerQ(n + S(1) / 2)),
CustomConstraint(lambda m, n: IntegerQ(m + n)))
rule14 = ReplacementRule(
pattern14, lambda b, v, u, m, a, x, n: a**(m + S(1) / 2) * b**
(n + S(-1) / 2) * sqrt(b * v) * Int(u * v**(m + n), x) / sqrt(a * v))
rubi.add(rule14)
pattern15 = Pattern(
Integral((v_ * WC('a', S(1)))**m_ *
(v_ * WC('b', S(1)))**n_ * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda m: Not(IntegerQ(m))),
CustomConstraint(lambda n: NegativeIntegerQ(n + S(-1) / 2)),
CustomConstraint(lambda m, n: IntegerQ(m + n)))
rule15 = ReplacementRule(
pattern15, lambda b, v, u, m, a, x, n: a**(m + S(-1) / 2) * b**
(n + S(1) / 2) * sqrt(a * v) * Int(u * v**(m + n), x) / sqrt(b * v))
rubi.add(rule15)
pattern16 = Pattern(
Integral((v_ * WC('a', S(1)))**m_ *
(v_ * WC('b', S(1)))**n_ * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda m: Not(IntegerQ(m))),
CustomConstraint(lambda n: Not(IntegerQ(n))),
CustomConstraint(lambda m, n: IntegerQ(m + n)))
rule16 = ReplacementRule(
pattern16, lambda b, v, u, m, a, x, n: a**(m + n) * (a * v)**(-n) *
(b * v)**n * Int(u * v**(m + n), x))
rubi.add(rule16)
pattern17 = Pattern(
Integral((v_ * WC('a', S(1)))**m_ *
(v_ * WC('b', S(1)))**n_ * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda m: Not(IntegerQ(m))),
CustomConstraint(lambda n: Not(IntegerQ(n))),
CustomConstraint(lambda m, n: Not(IntegerQ(m + n))))
rule17 = ReplacementRule(
pattern17,
lambda b, v, u, m, a, x, n: a**(-IntPart(n)) * b**IntPart(n) *
(a * v)**(-FracPart(n)) *
(b * v)**FracPart(n) * Int(u * (a * v)**(m + n), x))
rubi.add(rule17)
pattern18 = Pattern(
Integral((a_ + v_ * WC('b', S(1)))**WC('m', S(1)) *
(c_ + v_ * WC('d', S(1)))**WC('n', S(1)) * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda d, x: FreeQ(d, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda a, c, b, d: ZeroQ(-a * d + b * c)),
CustomConstraint(lambda m: IntegerQ(m)),
CustomConstraint(lambda b, a, x, d, n, c: Not(IntegerQ(n)) | SimplerQ(
c + d * x, a + b * x)))
rule18 = ReplacementRule(
pattern18, lambda b, v, u, m, a, x, d, n, c:
(b / d)**m * Int(u * (c + d * v)**(m + n), x))
rubi.add(rule18)
pattern19 = Pattern(
Integral((a_ + v_ * WC('b', S(1)))**m_ *
(c_ + v_ * WC('d', S(1)))**n_ * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda d, x: FreeQ(d, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda a, c, b, d: ZeroQ(-a * d + b * c)),
CustomConstraint(lambda b, d: PositiveQ(b / d)),
CustomConstraint(lambda m, n: Not(IntegerQ(m) | IntegerQ(n))))
rule19 = ReplacementRule(
pattern19, lambda b, v, u, m, a, x, d, n, c:
(b / d)**m * Int(u * (c + d * v)**(m + n), x))
rubi.add(rule19)
pattern20 = Pattern(
Integral((a_ + v_ * WC('b', S(1)))**m_ *
(c_ + v_ * WC('d', S(1)))**n_ * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda d, x: FreeQ(d, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda a, c, b, d: ZeroQ(-a * d + b * c)),
CustomConstraint(lambda m, n, b, d: Not(
IntegerQ(m) | IntegerQ(n) | PositiveQ(b / d))))
rule20 = ReplacementRule(
pattern20, lambda b, v, u, m, a, x, d, n, c: (a + b * v)**m *
(c + d * v)**(-m) * Int(u * (c + d * v)**(m + n), x))
rubi.add(rule20)
pattern21 = Pattern(
Integral(
(v_ * WC('a', S(1)))**m_ *
(v_**S(2) * WC('c', S(1)) + v_ * WC('b', S(1))) * WC('u', S(1)),
x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda m: RationalQ(m)),
CustomConstraint(lambda m: LessEqual(m, S(-1))))
rule21 = ReplacementRule(
pattern21,
lambda b, v, u, m, a, x, c: Int(u * (a * v)**(m + S(1)) *
(b + c * v), x) / a)
rubi.add(rule21)
pattern22 = Pattern(
Integral(
(a_ + v_ * WC('b', S(1)))**m_ *
(v_**S(2) * WC('C', S(1)) + v_ * WC('B', S(1)) + WC('A', S(0))) *
WC('u', S(1)), x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda A, x: FreeQ(A, x)),
CustomConstraint(lambda B, x: FreeQ(B, x)),
CustomConstraint(lambda C, x: FreeQ(C, x)),
CustomConstraint(lambda b, A, B, a, C: ZeroQ(A * b**S(2) - B * a * b +
C * a**S(2))),
CustomConstraint(lambda m: RationalQ(m)),
CustomConstraint(lambda m: LessEqual(m, S(-1))))
rule22 = ReplacementRule(
pattern22, lambda b, v, u, A, B, m, a, x, C: Int(
u * (a + b * v)**
(m + S(1)) * Simp(B * b - C * a + C * b * v, x), x) / b**S(2))
rubi.add(rule22)
pattern23 = Pattern(
Integral((a_ + x_**WC('n', S(1)) * WC('b', S(1)))**WC('m', S(1)) *
(c_ + x_**WC('q', S(1)) * WC('d', S(1)))**WC('p', S(1)) *
WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda d, x: FreeQ(d, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda n, q: ZeroQ(n + q)),
CustomConstraint(lambda p: IntegerQ(p)),
CustomConstraint(lambda a, c, b, d: ZeroQ(a * c - b * d)),
CustomConstraint(lambda m, n: Not(IntegerQ(m) & NegQ(n))))
rule23 = ReplacementRule(
pattern23, lambda q, p, b, u, m, a, x, d, n, c:
(d / a)**p * Int(u * x**(-n * p) * (a + b * x**n)**(m + p), x))
rubi.add(rule23)
pattern24 = Pattern(
Integral((a_ + x_**WC('n', S(1)) * WC('b', S(1)))**WC('m', S(1)) *
(c_ + x_**j_ * WC('d', S(1)))**WC('p', S(1)) * WC('u', S(1)),
x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda d, x: FreeQ(d, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda n, j: ZeroQ(j - S(2) * n)),
CustomConstraint(lambda m, p: ZeroQ(m + p)),
CustomConstraint(lambda a, c, b, d: ZeroQ(a**S(2) * d + b**S(2) * c)),
CustomConstraint(lambda a: PositiveQ(a)),
CustomConstraint(lambda d: NegativeQ(d)))
rule24 = ReplacementRule(
pattern24, lambda p, b, j, u, m, a, x, d, n, c:
(-b**S(2) / d)**m * Int(u * (a - b * x**n)**(-m), x))
rubi.add(rule24)
pattern25 = Pattern(
Integral((a_ + x_**S(2) * WC('c', S(1)) + x_ * WC('b', S(1)))**WC(
'p', S(1)) * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda a, c, b: ZeroQ(-S(4) * a * c + b**S(2))),
CustomConstraint(lambda p: IntegerQ(p)))
rule25 = ReplacementRule(
pattern25, lambda p, b, u, a, x, c: Int(
S(2)**(-S(2) * p) * c**(-p) * u *
(b + S(2) * c * x)**(S(2) * p), x))
rubi.add(rule25)
pattern26 = Pattern(
Integral(
(a_ + x_**n_ * WC('b', S(1)) + x_**WC('n2', S(1)) * WC('c', S(1)))
**WC('p', S(1)) * WC('u', S(1)), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda n, n2: ZeroQ(-S(2) * n + n2)),
CustomConstraint(lambda a, c, b: ZeroQ(-S(4) * a * c + b**S(2))),
CustomConstraint(lambda p: IntegerQ(p)))
rule26 = ReplacementRule(
pattern26, lambda n2, p, b, u, a, x, n, c: c**
(-p) * Int(u * (b / S(2) + c * x**n)**(S(2) * p), x))
rubi.add(rule26)
pattern27 = Pattern(
Integral(
(d_ + x_ * WC('e', S(1))) *
(x_**S(2) * WC('c', S(1)) + x_ * WC('b', S(1)) + WC('a', S(0)))**
WC('p', S(1)), x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda d, x: FreeQ(d, x)),
CustomConstraint(lambda e, x: FreeQ(e, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda e, c, b, d: ZeroQ(-b * e + S(2) * c * d)))
rule27 = ReplacementRule(
pattern27, lambda p, b, a, x, d, e, c: d * Subst(
Int(x**p, x), x, a + b * x + c * x**S(2)) / b)
rubi.add(rule27)
pattern28 = Pattern(
Integral(
(x_**WC('p', S(1)) * WC('a', S(1)) +
x_**WC('q', S(1)) * WC('b', S(1)))**WC('m', S(1)) * WC('u', S(1)),
x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda q, x: FreeQ(q, x)),
CustomConstraint(lambda m: IntegerQ(m)),
CustomConstraint(lambda q, p: PosQ(-p + q)))
rule28 = ReplacementRule(
pattern28, lambda q, p, b, u, m, a, x: Int(
u * x**(m * p) * (a + b * x**(-p + q))**m, x))
rubi.add(rule28)
pattern29 = Pattern(
Integral(
(x_**WC('p', S(1)) * WC('a', S(1)) +
x_**WC('q', S(1)) * WC('b', S(1)) +
x_**WC('r', S(1)) * WC('c', S(1)))**WC('m', S(1)) * WC('u', S(1)),
x_), CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda c, x: FreeQ(c, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda q, x: FreeQ(q, x)),
CustomConstraint(lambda r, x: FreeQ(r, x)),
CustomConstraint(lambda m: IntegerQ(m)),
CustomConstraint(lambda q, p: PosQ(-p + q)),
CustomConstraint(lambda r, p: PosQ(-p + r)))
rule29 = ReplacementRule(
pattern29, lambda q, r, p, b, u, m, a, x, c: Int(
u * x**(m * p) * (a + b * x**(-p + q) + c * x**(-p + r))**m, x))
rubi.add(rule29)
pattern30 = Pattern(
Integral(x_**WC('m', S(1)) / (a_ + x_**n_ * WC('b', S(1))), x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda m, n: ZeroQ(m - n + S(1))))
rule30 = ReplacementRule(
pattern30, lambda b, m, a, x, n: log(RemoveContent(a + b * x**n, x)) /
(b * n))
rubi.add(rule30)
pattern31 = Pattern(
Integral(x_**WC('m', S(1)) * (a_ + x_**n_ * WC('b', S(1)))**p_, x_),
CustomConstraint(lambda a, x: FreeQ(a, x)),
CustomConstraint(lambda b, x: FreeQ(b, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda m, n: ZeroQ(m - n + S(1))),
CustomConstraint(lambda p: NonzeroQ(p + S(1))))
rule31 = ReplacementRule(
pattern31, lambda p, b, m, a, x, n: (a + b * x**n)**(p + S(1)) /
(b * n * (p + S(1))))
rubi.add(rule31)
pattern32 = Pattern(
Integral(
x_**WC('m', S(1)) *
(a1_ + x_**WC('n', S(1)) * WC('b1', S(1)))**p_ *
(a2_ + x_**WC('n', S(1)) * WC('b2', S(1)))**p_, x_),
CustomConstraint(lambda a1, x: FreeQ(a1, x)),
CustomConstraint(lambda b1, x: FreeQ(b1, x)),
CustomConstraint(lambda a2, x: FreeQ(a2, x)),
CustomConstraint(lambda b2, x: FreeQ(b2, x)),
CustomConstraint(lambda m, x: FreeQ(m, x)),
CustomConstraint(lambda n, x: FreeQ(n, x)),
CustomConstraint(lambda p, x: FreeQ(p, x)),
CustomConstraint(lambda b2, b1, a1, a2: ZeroQ(a1 * b2 + a2 * b1)),
CustomConstraint(lambda m, n: ZeroQ(m - S(2) * n + S(1))),
CustomConstraint(lambda p: NonzeroQ(p + S(1))))
rule32 = ReplacementRule(
pattern32, lambda p, a1, b2, m, x, a2, n, b1:
(a1 + b1 * x**n)**(p + S(1)) * (a2 + b2 * x**n)**(p + S(1)) /
(S(2) * b1 * b2 * n * (p + S(1))))
rubi.add(rule32)
'''
pattern33 = Pattern(Integral(Qm_*(Pm_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda n, x: FreeQ(n, x)), CustomConstraint(lambda p, x: FreeQ(p, x)), CustomConstraint(lambda x, Pm: PolyQ(Pm, x)), CustomConstraint(lambda Qm, x: PolyQ(Qm, x)), )
def With33(p, b, a, x, n, Qm, Pm):
m = Expon(Pm, x)
if Equal(Expon(Qm, x), m + S(-1)) & ZeroQ(-Qm*m*Coeff(Pm, x, m) + Coeff(Qm, x, m + S(-1))*D(Pm, x)):
return Coeff(Qm, x, m + S(-1))*Subst(Int((a + b*x**n)**p, x), x, Pm)/(m*Coeff(Pm, x, m))
print("Unable to Integrate")
rule33 = ReplacementRule(pattern33, lambda p, b, a, x, n, Qm, Pm : With33(p, b, a, x, n, Qm, Pm))
rubi.add(rule33)
pattern34 = Pattern(Integral(Qm_*(Pm_**WC('n', S(1))*WC('b', S(1)) + Pm_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda n, x: FreeQ(n, x)), CustomConstraint(lambda p, x: FreeQ(p, x)), CustomConstraint(lambda n, n2: ZeroQ(-S(2)*n + n2)), CustomConstraint(lambda x, Pm: PolyQ(Pm, x)), CustomConstraint(lambda Qm, x: PolyQ(Qm, x)), )
def With34(n2, p, b, a, x, n, Qm, c, Pm):
m = Expon(Pm, x)
if Equal(Expon(Qm, x), m + S(-1)) & ZeroQ(-Qm*m*Coeff(Pm, x, m) + Coeff(Qm, x, m + S(-1))*D(Pm, x)):
return Coeff(Qm, x, m + S(-1))*Subst(Int((a + b*x**n + c*x**(S(2)*n))**p, x), x, Pm)/(m*Coeff(Pm, x, m))
print("Unable to Integrate")
rule34 = ReplacementRule(pattern34, lambda n2, p, b, a, x, n, Qm, c, Pm : With34(n2, p, b, a, x, n, Qm, c, Pm))
rubi.add(rule34)
pattern35 = Pattern(Integral(Pq_**m_*Qr_**p_*WC('u', S(1)), x_), CustomConstraint(lambda m: PositiveIntegerQ(m)), CustomConstraint(lambda p: NegativeIntegerQ(p)), CustomConstraint(lambda Pq, x: PolyQ(Pq, x)), CustomConstraint(lambda Qr, x: PolyQ(Qr, x)), )
def With35(Qr, p, u, Pq, m, x):
gcd = PolyGCD(Pq, Qr, x)
if NonzeroQ(gcd + S(-1)):
return Int(gcd**(m + p)*u*PolynomialQuotient(Pq, gcd, x)**m*PolynomialQuotient(Qr, gcd, x)**p, x)
print("Unable to Integrate")
rule35 = ReplacementRule(pattern35, lambda Qr, p, u, Pq, m, x : With35(Qr, p, u, Pq, m, x))
rubi.add(rule35)
pattern36 = Pattern(Integral(Pq_*Qr_**p_*WC('u', S(1)), x_), CustomConstraint(lambda p: NegativeIntegerQ(p)), CustomConstraint(lambda Pq, x: PolyQ(Pq, x)), CustomConstraint(lambda Qr, x: PolyQ(Qr, x)), )
def With36(Qr, p, u, Pq, x):
gcd = PolyGCD(Pq, Qr, x)
if NonzeroQ(gcd + S(-1)):
return Int(gcd**(p + S(1))*u*PolynomialQuotient(Pq, gcd, x)*PolynomialQuotient(Qr, gcd, x)**p, x)
print("Unable to Integrate")
rule36 = ReplacementRule(pattern36, lambda Qr, p, u, Pq, x : With36(Qr, p, u, Pq, x))
rubi.add(rule36)
'''
return rubi