def replacement1883(v, x, u):
a = D(u, x)
b = D(v, x)
rubi.append(1883)
return -Dist(
(-a * v + b * u) / a, Int(S(1) / u, x), x) + Simp(b * x / a, x)
def replacement1890(u, v, x):
a = D(u, x)
b = D(v, x)
return Simp(
-S(2) * atanh(sqrt(v) / Rt(-(-a * v + b * u) / a, S(2))) /
(a * Rt(-(-a * v + b * u) / a, S(2))), x)
def replacement1885(v, x, u):
a = D(u, x)
b = D(v, x)
rubi.append(1885)
return -Dist(a / (-a * v + b * u), Int(S(1) / u, x), x) + Dist(
b / (-a * v + b * u), Int(S(1) / v, x), x)
def replacement1892(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1892)
return -Simp(
u**(m + S(1)) * v**(n + S(1)) / ((m + S(1)) * (-a * v + b * u)), x)
def replacement1896(m, n, u, v, x):
a = D(u, x)
b = D(v, x)
return -Dist(b * n / (a *
(m + S(1))), Int(u**(m + S(1)) * v**(n + S(-1)), x),
x) + Simp(u**(m + S(1)) * v**n / (a * (m + S(1))), x)
def replacement1894(u, v, x):
a = D(u, x)
b = D(v, x)
return Simp(
S(2) * ArcTan(sqrt(u) * Rt(-a * b, S(2)) / (a * sqrt(v))) /
Rt(-a * b, S(2)), x)
def replacement1893(u, v, x):
a = D(u, x)
b = D(v, x)
return Simp(
S(2) * atanh(sqrt(u) * Rt(a * b, S(2)) / (a * sqrt(v))) /
Rt(a * b, S(2)), x)
def replacement1889(u, v, x):
a = D(u, x)
b = D(v, x)
return Simp(
S(2) * ArcTan(sqrt(v) / Rt((-a * v + b * u) / a, S(2))) / (a * Rt(
(-a * v + b * u) / a, S(2))), x)
def replacement1901(m, n, u, v, x):
a = D(u, x)
b = D(v, x)
return Simp(
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))), x)
def replacement1898(m, n, u, v, x):
a = D(u, x)
b = D(v, x)
return -Dist(n * (-a * v + b * u) /
(a *
(m + n + S(1))), Int(u**m * v**(n + S(-1)), x), x) + Simp(
u**(m + S(1)) * v**n / (a * (m + n + S(1))), x)
def replacement1892(n, u, v, x):
a = D(u, x)
b = D(v, x)
return Simp(
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)), x)
def replacement1884(v, x, n, u):
a = D(u, x)
b = D(v, x)
rubi.append(1884)
return -Dist(
(-a * v + b * u) / a, Int(v**(n + S(-1)) / u, x), x) + Simp(
v**n / (a * n), x)
def replacement1886(v, x, u):
a = D(u, x)
b = D(v, x)
rubi.append(1886)
return Simp(
S(2) * ArcTan(sqrt(v) / Rt((-a * v + b * u) / a, S(2))) / (a * Rt(
(-a * v + b * u) / a, S(2))), x)
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 replacement1895(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1895)
return -Dist(n * (-a * v + b * u) /
(a * (m + n + S(1))), Int(u**m * v**(n + S(-1)), x),
x) + Simp(u**(m + S(1)) * v**n / (a * (m + n + S(1))), x)
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 replacement1893(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1893)
return -Dist(b * n /
(a * (m + S(1))), Int(u**(m + S(1)) * v**(n + S(-1)), x),
x) + Simp(u**(m + S(1)) * v**n / (a * (m + S(1))), x)
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 replacement1888(v, x, n, u):
a = D(u, x)
b = D(v, x)
rubi.append(1888)
return -Dist(a /
(-a * v + b * u), Int(v**(n + S(1)) / u, x), x) + Simp(
v**(n + S(1)) / ((n + S(1)) * (-a * v + b * u)), x)
def replacement1890(v, x, u):
a = D(u, x)
b = D(v, x)
rubi.append(1890)
return Simp(
S(2) * atanh(sqrt(u) * Rt(a * b, S(2)) / (a * sqrt(v))) /
Rt(a * b, S(2)), x)
def replacement1887(v, x, u):
a = D(u, x)
b = D(v, x)
rubi.append(1887)
return Simp(
-S(2) * atanh(sqrt(v) / Rt(-(-a * v + b * u) / a, S(2))) /
(a * Rt(-(-a * v + b * u) / a, S(2))), x)
def replacement1891(v, x, u):
a = D(u, x)
b = D(v, x)
rubi.append(1891)
return Simp(
S(2) * ArcTan(sqrt(u) * Rt(-a * b, S(2)) / (a * sqrt(v))) /
Rt(-a * b, S(2)), x)
def replacement1898(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1898)
return Simp(
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))), x)
def replacement1900(m, n, u, v, x):
a = D(u, x)
b = D(v, x)
return Dist(b * (m + n + S(2)) /
((m + S(1)) *
(-a * v + b * u)), Int(u**(m + S(1)) * v**n, x), x) - Simp(
u**(m + S(1)) * v**(n + S(1)) / ((m + S(1)) *
(-a * v + b * u)), x)
def replacement1897(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1897)
return Dist(b * (m + n + S(2)) / (
(m + S(1)) *
(-a * v + b * u)), Int(u**(m + S(1)) * v**n, x), x) - Simp(
u**(m + S(1)) * v**(n + S(1)) / ((m + S(1)) *
(-a * v + b * u)), x)
def replacement1889(v, x, n, u):
a = D(u, x)
b = D(v, x)
rubi.append(1889)
return Simp(
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)), x)
def With1899(u, b, a, n, x):
c = D(u, x)
rubi.append(1899)
return -Dist(c * n / b,
Int(u**(n + S(-1)) *
(a + b * x) * log(a + b * x), x), x) - Int(
u**n, x) + Simp(
u**n * (a + b * x) * log(a + b * x) / b, x)
def With1903(a, b, m, n, u, x):
c = D(u, x)
return -Dist(
c * n / (b * (m + S(1))),
Int(u**(n + S(-1)) *
(a + b * x)**(m + S(1)) * log(a + b * x), x), x) - Dist(
S(1) / (m + S(1)), Int(u**n * (a + b * x)**m, x), x) + Simp(
u**n * (a + b * x)**(m + S(1)) * log(a + b * x) /
(b * (m + S(1))), x)
def With34(p, Pm, b, n2, c, a, n, Qm, x):
if isinstance(x, (int, Integer, float, Float)):
return False
m = Expon(Pm, x)
if And(
Equal(Expon(Qm, x), m + S(-1)),
ZeroQ(-Qm * m * Coeff(Pm, x, m) +
Coeff(Qm, x, m + S(-1)) * D(Pm, x))):
return True
return False
def With1900(u, m, b, a, n, x):
c = D(u, x)
rubi.append(1900)
return -Dist(
c * n / (b * (m + S(1))),
Int(u**(n + S(-1)) *
(a + b * x)**(m + S(1)) * log(a + b * x), x), x) - Dist(
S(1) /
(m + S(1)), Int(u**n * (a + b * x)**m, x), x) + Simp(
u**n * (a + b * x)**(m + S(1)) * log(a + b * x) /
(b * (m + S(1))), x)
