def test_issue_10847(): assert manualintegrate(x**2 / (x**2 - c), x) == c * atan(x / sqrt(-c)) / sqrt(-c) + x rc = Symbol('c', real=True) assert manualintegrate(x**2 / (x**2 - rc), x) == \ rc*Piecewise((atan(x/sqrt(-rc))/sqrt(-rc), -rc > 0), (-acoth(x/sqrt(rc))/sqrt(rc), And(-rc < 0, x**2 > rc)), (-atanh(x/sqrt(rc))/sqrt(rc), And(-rc < 0, x**2 < rc))) + x assert manualintegrate(sqrt(x - y) * log(z / x), x) == \ 4*y**Rational(3, 2)*atan(sqrt(x - y)/sqrt(y))/3 - 4*y*sqrt(x - y)/3 +\ 2*(x - y)**Rational(3, 2)*log(z/x)/3 + 4*(x - y)**Rational(3, 2)/9 ry = Symbol('y', real=True) rz = Symbol('z', real=True) assert manualintegrate(sqrt(x - ry) * log(rz / x), x) == \ 4*ry**2*Piecewise((atan(sqrt(x - ry)/sqrt(ry))/sqrt(ry), ry > 0), (-acoth(sqrt(x - ry)/sqrt(-ry))/sqrt(-ry), And(x - ry > -ry, ry < 0)), (-atanh(sqrt(x - ry)/sqrt(-ry))/sqrt(-ry), And(x - ry < -ry, ry < 0)))/3 \ - 4*ry*sqrt(x - ry)/3 + 2*(x - ry)**Rational(3, 2)*log(rz/x)/3 \ + 4*(x - ry)**Rational(3, 2)/9 assert manualintegrate( sqrt(x) * log(x), x) == 2 * x**Rational(3, 2) * log(x) / 3 - 4 * x**Rational(3, 2) / 9 assert manualintegrate(sqrt(a*x + b) / x, x) == \ 2*b*atan(sqrt(a*x + b)/sqrt(-b))/sqrt(-b) + 2*sqrt(a*x + b) ra = Symbol('a', real=True) rb = Symbol('b', real=True) assert manualintegrate(sqrt(ra*x + rb) / x, x) == \ -2*rb*Piecewise((-atan(sqrt(ra*x + rb)/sqrt(-rb))/sqrt(-rb), -rb > 0), (acoth(sqrt(ra*x + rb)/sqrt(rb))/sqrt(rb), And(-rb < 0, ra*x + rb > rb)), (atanh(sqrt(ra*x + rb)/sqrt(rb))/sqrt(rb), And(-rb < 0, ra*x + rb < rb))) \ + 2*sqrt(ra*x + rb) assert expand(manualintegrate(sqrt(ra*x + rb) / (x + rc), x)) == -2*ra*rc*Piecewise((atan(sqrt(ra*x + rb)/sqrt(ra*rc - rb))/sqrt(ra*rc - rb), \ ra*rc - rb > 0), (-acoth(sqrt(ra*x + rb)/sqrt(-ra*rc + rb))/sqrt(-ra*rc + rb), And(ra*rc - rb < 0, ra*x + rb > -ra*rc + rb)), \ (-atanh(sqrt(ra*x + rb)/sqrt(-ra*rc + rb))/sqrt(-ra*rc + rb), And(ra*rc - rb < 0, ra*x + rb < -ra*rc + rb))) \ + 2*rb*Piecewise((atan(sqrt(ra*x + rb)/sqrt(ra*rc - rb))/sqrt(ra*rc - rb), ra*rc - rb > 0), \ (-acoth(sqrt(ra*x + rb)/sqrt(-ra*rc + rb))/sqrt(-ra*rc + rb), And(ra*rc - rb < 0, ra*x + rb > -ra*rc + rb)), \ (-atanh(sqrt(ra*x + rb)/sqrt(-ra*rc + rb))/sqrt(-ra*rc + rb), And(ra*rc - rb < 0, ra*x + rb < -ra*rc + rb))) + 2*sqrt(ra*x + rb) assert manualintegrate( sqrt(2 * x + 3) / (x + 1), x) == 2 * sqrt(2 * x + 3) - log(sqrt(2 * x + 3) + 1) + log(sqrt(2 * x + 3) - 1) assert manualintegrate( sqrt(2 * x + 3) / 2 * x, x ) == (2 * x + 3)**Rational(5, 2) / 20 - (2 * x + 3)**Rational(3, 2) / 4 assert manualintegrate( x**Rational(3, 2) * log(x), x) == 2 * x**Rational(5, 2) * log(x) / 5 - 4 * x**Rational(5, 2) / 25 assert manualintegrate(x**(-3) * log(x), x) == -log(x) / (2 * x**2) - 1 / (4 * x**2) assert manualintegrate(log(y)/(y**2*(1 - 1/y)), y) == \ log(y)*log(-1 + 1/y) - Integral(log(-1 + 1/y)/y, y)
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 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 eval(cls, n, m, z=None): if z is not None: n, z, m = n, m, z k = 2 * z / pi if n == S.Zero: return elliptic_f(z, m) elif n == S.One: return elliptic_f(z, m) + (sqrt(1 - m * sin(z) ** 2) * tan(z) - elliptic_e(z, m)) / (1 - m) elif k.is_integer: return k * elliptic_pi(n, m) elif m == S.Zero: return atanh(sqrt(n - 1) * tan(z)) / sqrt(n - 1) elif n == m: return elliptic_f(z, n) - elliptic_pi(1, z, n) + tan(z) / sqrt(1 - n * sin(z) ** 2) elif n in (S.Infinity, S.NegativeInfinity): return S.Zero elif m in (S.Infinity, S.NegativeInfinity): return S.Zero elif z.could_extract_minus_sign(): return -elliptic_pi(n, -z, m) else: if n == S.Zero: return elliptic_k(m) elif n == S.One: return S.ComplexInfinity elif m == S.Zero: return pi / (2 * sqrt(1 - n)) elif m == S.One: return -S.Infinity / sign(n - 1) elif n == m: return elliptic_e(n) / (1 - n) elif n in (S.Infinity, S.NegativeInfinity): return S.Zero elif m in (S.Infinity, S.NegativeInfinity): return S.Zero
def test_branch_cuts(): assert limit(asin(I * x + 2), x, 0) == pi - asin(2) assert limit(asin(I * x + 2), x, 0, '-') == asin(2) assert limit(asin(I * x - 2), x, 0) == -asin(2) assert limit(asin(I * x - 2), x, 0, '-') == -pi + asin(2) assert limit(acos(I * x + 2), x, 0) == -acos(2) assert limit(acos(I * x + 2), x, 0, '-') == acos(2) assert limit(acos(I * x - 2), x, 0) == acos(-2) assert limit(acos(I * x - 2), x, 0, '-') == 2 * pi - acos(-2) assert limit(atan(x + 2 * I), x, 0) == I * atanh(2) assert limit(atan(x + 2 * I), x, 0, '-') == -pi + I * atanh(2) assert limit(atan(x - 2 * I), x, 0) == pi - I * atanh(2) assert limit(atan(x - 2 * I), x, 0, '-') == -I * atanh(2) assert limit(atan(1 / x), x, 0) == pi / 2 assert limit(atan(1 / x), x, 0, '-') == -pi / 2 assert limit(atan(x), x, oo) == pi / 2 assert limit(atan(x), x, -oo) == -pi / 2 assert limit(acot(x + S(1) / 2 * I), x, 0) == pi - I * acoth(S(1) / 2) assert limit(acot(x + S(1) / 2 * I), x, 0, '-') == -I * acoth(S(1) / 2) assert limit(acot(x - S(1) / 2 * I), x, 0) == I * acoth(S(1) / 2) assert limit(acot(x - S(1) / 2 * I), x, 0, '-') == -pi + I * acoth(S(1) / 2) assert limit(acot(x), x, 0) == pi / 2 assert limit(acot(x), x, 0, '-') == -pi / 2 assert limit(asec(I * x + S(1) / 2), x, 0) == asec(S(1) / 2) assert limit(asec(I * x + S(1) / 2), x, 0, '-') == -asec(S(1) / 2) assert limit(asec(I * x - S(1) / 2), x, 0) == 2 * pi - asec(-S(1) / 2) assert limit(asec(I * x - S(1) / 2), x, 0, '-') == asec(-S(1) / 2) assert limit(acsc(I * x + S(1) / 2), x, 0) == acsc(S(1) / 2) assert limit(acsc(I * x + S(1) / 2), x, 0, '-') == pi - acsc(S(1) / 2) assert limit(acsc(I * x - S(1) / 2), x, 0) == -pi + acsc(S(1) / 2) assert limit(acsc(I * x - S(1) / 2), x, 0, '-') == -acsc(S(1) / 2) assert limit(log(I * x - 1), x, 0) == I * pi assert limit(log(I * x - 1), x, 0, '-') == -I * pi assert limit(log(-I * x - 1), x, 0) == -I * pi assert limit(log(-I * x - 1), x, 0, '-') == I * pi assert limit(sqrt(I * x - 1), x, 0) == I assert limit(sqrt(I * x - 1), x, 0, '-') == -I assert limit(sqrt(-I * x - 1), x, 0) == -I assert limit(sqrt(-I * x - 1), x, 0, '-') == I assert limit(cbrt(I * x - 1), x, 0) == (-1)**(S(1) / 3) assert limit(cbrt(I * x - 1), x, 0, '-') == -(-1)**(S(2) / 3) assert limit(cbrt(-I * x - 1), x, 0) == -(-1)**(S(2) / 3) assert limit(cbrt(-I * x - 1), x, 0, '-') == (-1)**(S(1) / 3)
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 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 test_P(): assert P(0, z, m) == F(z, m) assert P(1, z, m) == F(z, m) + \ (sqrt(1 - m*sin(z)**2)*tan(z) - E(z, m))/(1 - m) assert P(n, i*pi/2, m) == i*P(n, m) assert P(n, z, 0) == atanh(sqrt(n - 1)*tan(z))/sqrt(n - 1) assert P(n, z, n) == F(z, n) - P(1, z, n) + tan(z)/sqrt(1 - n*sin(z)**2) assert P(oo, z, m) == 0 assert P(-oo, z, m) == 0 assert P(n, z, oo) == 0 assert P(n, z, -oo) == 0 assert P(0, m) == K(m) assert P(1, m) is zoo assert P(n, 0) == pi/(2*sqrt(1 - n)) assert P(2, 1) is -oo assert P(-1, 1) is oo assert P(n, n) == E(n)/(1 - n) assert P(n, -z, m) == -P(n, z, m) ni, mi = Symbol('n', real=False), Symbol('m', real=False) assert P(ni, z, mi).conjugate() == \ P(ni.conjugate(), z.conjugate(), mi.conjugate()) nr, mr = Symbol('n', negative=True), \ Symbol('m', negative=True) assert P(nr, z, mr).conjugate() == P(nr, z.conjugate(), mr) assert P(n, m).conjugate() == P(n.conjugate(), m.conjugate()) assert P(n, z, m).diff(n) == (E(z, m) + (m - n)*F(z, m)/n + (n**2 - m)*P(n, z, m)/n - n*sqrt(1 - m*sin(z)**2)*sin(2*z)/(2*(1 - n*sin(z)**2)))/(2*(m - n)*(n - 1)) assert P(n, z, m).diff(z) == 1/(sqrt(1 - m*sin(z)**2)*(1 - n*sin(z)**2)) assert P(n, z, m).diff(m) == (E(z, m)/(m - 1) + P(n, z, m) - m*sin(2*z)/(2*(m - 1)*sqrt(1 - m*sin(z)**2)))/(2*(n - m)) assert P(n, m).diff(n) == (E(m) + (m - n)*K(m)/n + (n**2 - m)*P(n, m)/n)/(2*(m - n)*(n - 1)) assert P(n, m).diff(m) == (E(m)/(m - 1) + P(n, m))/(2*(n - m)) # These tests fail due to # https://github.com/fredrik-johansson/mpmath/issues/571#issuecomment-777201962 # https://github.com/sympy/sympy/issues/20933#issuecomment-777080385 # # rx, ry = randcplx(), randcplx() # assert td(P(n, rx, ry), n) # assert td(P(rx, z, ry), z) # assert td(P(rx, ry, m), m) assert P(n, z, m).series(z) == z + z**3*(m/6 + n/3) + \ z**5*(3*m**2/40 + m*n/10 - m/30 + n**2/5 - n/15) + O(z**6) assert P(n, z, m).rewrite(Integral).dummy_eq( Integral(1/((1 - n*sin(t)**2)*sqrt(1 - m*sin(t)**2)), (t, 0, z))) assert P(n, m).rewrite(Integral).dummy_eq( Integral(1/((1 - n*sin(t)**2)*sqrt(1 - m*sin(t)**2)), (t, 0, pi/2)))
def test_inverses(): x = Symbol('x') assert sinh(x).inverse() == asinh raises(AttributeError, lambda: cosh(x).inverse()) assert tanh(x).inverse() == atanh assert coth(x).inverse() == acoth assert asinh(x).inverse() == sinh assert acosh(x).inverse() == cosh assert atanh(x).inverse() == tanh assert acoth(x).inverse() == coth assert asech(x).inverse() == sech assert acsch(x).inverse() == csch
def test_hyperbolic(): assert sinh(x).nseries(x, n=6) == x + x**3 / 6 + x**5 / 120 + O(x**6) assert cosh(x).nseries(x, n=5) == 1 + x**2 / 2 + x**4 / 24 + O(x**5) assert tanh(x).nseries(x, n=6) == x - x**3 / 3 + 2 * x**5 / 15 + O(x**6) assert coth(x).nseries(x, n=6) == \ 1/x - x**3/45 + x/3 + 2*x**5/945 + O(x**6) assert asinh(x).nseries(x, n=6) == x - x**3 / 6 + 3 * x**5 / 40 + O(x**6) assert acosh(x).nseries(x, n=6) == \ pi*I/2 - I*x - 3*I*x**5/40 - I*x**3/6 + O(x**6) assert atanh(x).nseries(x, n=6) == x + x**3 / 3 + x**5 / 5 + O(x**6) assert acoth(x).nseries( x, n=6) == x + x**3 / 3 + x**5 / 5 + pi * I / 2 + O(x**6)
def test_simplifications(): x = Symbol('x') assert sinh(asinh(x)) == x assert sinh(acosh(x)) == sqrt(x - 1) * sqrt(x + 1) assert sinh(atanh(x)) == x / sqrt(1 - x**2) assert sinh(acoth(x)) == 1 / (sqrt(x - 1) * sqrt(x + 1)) assert cosh(asinh(x)) == sqrt(1 + x**2) assert cosh(acosh(x)) == x assert cosh(atanh(x)) == 1 / sqrt(1 - x**2) assert cosh(acoth(x)) == x / (sqrt(x - 1) * sqrt(x + 1)) assert tanh(asinh(x)) == x / sqrt(1 + x**2) assert tanh(acosh(x)) == sqrt(x - 1) * sqrt(x + 1) / x assert tanh(atanh(x)) == x assert tanh(acoth(x)) == 1 / x assert coth(asinh(x)) == sqrt(1 + x**2) / x assert coth(acosh(x)) == x / (sqrt(x - 1) * sqrt(x + 1)) assert coth(atanh(x)) == 1 / x assert coth(acoth(x)) == x assert csch(asinh(x)) == 1 / x assert csch(acosh(x)) == 1 / (sqrt(x - 1) * sqrt(x + 1)) assert csch(atanh(x)) == sqrt(1 - x**2) / x assert csch(acoth(x)) == sqrt(x - 1) * sqrt(x + 1) assert sech(asinh(x)) == 1 / sqrt(1 + x**2) assert sech(acosh(x)) == 1 / x assert sech(atanh(x)) == sqrt(1 - x**2) assert sech(acoth(x)) == sqrt(x - 1) * sqrt(x + 1) / x
def test_hyperexpand_bases(): assert hyperexpand(hyper([2], [a], z)) == \ a + z**(-a + 1)*(-a**2 + 3*a + z*(a - 1) - 2)*exp(z)* \ lowergamma(a - 1, z) - 1 # TODO [a+1, aRational(-1, 2)], [2*a] assert hyperexpand(hyper([1, 2], [3], z)) == -2/z - 2*log(-z + 1)/z**2 assert hyperexpand(hyper([S.Half, 2], [Rational(3, 2)], z)) == \ -1/(2*z - 2) + atanh(sqrt(z))/sqrt(z)/2 assert hyperexpand(hyper([S.Half, S.Half], [Rational(5, 2)], z)) == \ (-3*z + 3)/4/(z*sqrt(-z + 1)) \ + (6*z - 3)*asin(sqrt(z))/(4*z**Rational(3, 2)) assert hyperexpand(hyper([1, 2], [Rational(3, 2)], z)) == -1/(2*z - 2) \ - asin(sqrt(z))/(sqrt(z)*(2*z - 2)*sqrt(-z + 1)) assert hyperexpand(hyper([Rational(-1, 2) - 1, 1, 2], [S.Half, 3], z)) == \ sqrt(z)*(z*Rational(6, 7) - Rational(6, 5))*atanh(sqrt(z)) \ + (-30*z**2 + 32*z - 6)/35/z - 6*log(-z + 1)/(35*z**2) assert hyperexpand(hyper([1 + S.Half, 1, 1], [2, 2], z)) == \ -4*log(sqrt(-z + 1)/2 + S.Half)/z # TODO hyperexpand(hyper([a], [2*a + 1], z)) # TODO [S.Half, a], [Rational(3, 2), a+1] assert hyperexpand(hyper([2], [b, 1], z)) == \ z**(-b/2 + S.Half)*besseli(b - 1, 2*sqrt(z))*gamma(b) \ + z**(-b/2 + 1)*besseli(b, 2*sqrt(z))*gamma(b)
def test_derivs(): x = Symbol('x') assert coth(x).diff(x) == -sinh(x)**(-2) assert sinh(x).diff(x) == cosh(x) assert cosh(x).diff(x) == sinh(x) assert tanh(x).diff(x) == -tanh(x)**2 + 1 assert csch(x).diff(x) == -coth(x) * csch(x) assert sech(x).diff(x) == -tanh(x) * sech(x) assert acoth(x).diff(x) == 1 / (-x**2 + 1) assert asinh(x).diff(x) == 1 / sqrt(x**2 + 1) assert acosh(x).diff(x) == 1 / sqrt(x**2 - 1) assert atanh(x).diff(x) == 1 / (-x**2 + 1) assert asech(x).diff(x) == -1 / (x * sqrt(1 - x**2)) assert acsch(x).diff(x) == -1 / (x**2 * sqrt(1 + x**(-2)))
def eval(cls, n, m, z=None): if z is not None: n, z, m = n, m, z if n.is_zero: return elliptic_f(z, m) elif n is S.One: return (elliptic_f(z, m) + (sqrt(1 - m*sin(z)**2)*tan(z) - elliptic_e(z, m))/(1 - m)) k = 2*z/pi if k.is_integer: return k*elliptic_pi(n, m) elif m.is_zero: return atanh(sqrt(n - 1)*tan(z))/sqrt(n - 1) elif n == m: return (elliptic_f(z, n) - elliptic_pi(1, z, n) + tan(z)/sqrt(1 - n*sin(z)**2)) elif n in (S.Infinity, S.NegativeInfinity): return S.Zero elif m in (S.Infinity, S.NegativeInfinity): return S.Zero elif z.could_extract_minus_sign(): return -elliptic_pi(n, -z, m) if n.is_zero: return elliptic_f(z, m) if m.is_extended_real and m.is_infinite or \ n.is_extended_real and n.is_infinite: return S.Zero else: if n.is_zero: return elliptic_k(m) elif n is S.One: return S.ComplexInfinity elif m.is_zero: return pi/(2*sqrt(1 - n)) elif m == S.One: return S.NegativeInfinity/sign(n - 1) elif n == m: return elliptic_e(n)/(1 - n) elif n in (S.Infinity, S.NegativeInfinity): return S.Zero elif m in (S.Infinity, S.NegativeInfinity): return S.Zero if n.is_zero: return elliptic_k(m) if m.is_extended_real and m.is_infinite or \ n.is_extended_real and n.is_infinite: return S.Zero
def eval(cls, *args): if len(args) == 3: n, z, m = args k = 2 * z / pi if n == S.Zero: return elliptic_f(z, m) elif n == S.One: return (elliptic_f(z, m) + (sqrt(1 - m * sin(z)**2) * tan(z) - elliptic_e(z, m)) / (1 - m)) elif k.is_integer: return k * elliptic_pi(n, m) elif m == S.Zero: return atanh(sqrt(n - 1) * tan(z)) / sqrt(n - 1) elif n == m: return (elliptic_f(z, n) - elliptic_pi(1, z, n) + tan(z) / sqrt(1 - n * sin(z)**2)) elif n in (S.Infinity, S.NegativeInfinity): return S.Zero elif m in (S.Infinity, S.NegativeInfinity): return S.Zero elif z.could_extract_minus_sign(): return -elliptic_pi(n, -z, m) else: n, m = args if n == S.Zero: return elliptic_k(m) elif n == S.One: return S.ComplexInfinity elif m == S.Zero: return pi / (2 * sqrt(1 - n)) elif m == S.One: return -S.Infinity / sign(n - 1) elif n == m: return elliptic_e(n) / (1 - n) elif n in (S.Infinity, S.NegativeInfinity): return S.Zero elif m in (S.Infinity, S.NegativeInfinity): return S.Zero
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)))
def test_atanh_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: atanh(x).fdiff(2))
def test_atanh_series(): x = Symbol('x') assert atanh(x).series(x, 0, 10) == \ x + x**3/3 + x**5/5 + x**7/7 + x**9/9 + O(x**10)
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))
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))
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 test_manualintegrate_inversetrig(): # atan assert manualintegrate(exp(x) / (1 + exp(2 * x)), x) == atan(exp(x)) assert manualintegrate(1 / (4 + 9 * x**2), x) == atan(3 * x / 2) / 6 assert manualintegrate(1 / (16 + 16 * x**2), x) == atan(x) / 16 assert manualintegrate(1 / (4 + x**2), x) == atan(x / 2) / 2 assert manualintegrate(1 / (1 + 4 * x**2), x) == atan(2 * x) / 2 ra = Symbol('a', real=True) rb = Symbol('b', real=True) assert manualintegrate(1/(ra + rb*x**2), x) == \ Piecewise((atan(x/sqrt(ra/rb))/(rb*sqrt(ra/rb)), ra/rb > 0), (-acoth(x/sqrt(-ra/rb))/(rb*sqrt(-ra/rb)), And(ra/rb < 0, x**2 > -ra/rb)), (-atanh(x/sqrt(-ra/rb))/(rb*sqrt(-ra/rb)), And(ra/rb < 0, x**2 < -ra/rb))) assert manualintegrate(1/(4 + rb*x**2), x) == \ Piecewise((atan(x/(2*sqrt(1/rb)))/(2*rb*sqrt(1/rb)), 4/rb > 0), (-acoth(x/(2*sqrt(-1/rb)))/(2*rb*sqrt(-1/rb)), And(4/rb < 0, x**2 > -4/rb)), (-atanh(x/(2*sqrt(-1/rb)))/(2*rb*sqrt(-1/rb)), And(4/rb < 0, x**2 < -4/rb))) assert manualintegrate(1/(ra + 4*x**2), x) == \ Piecewise((atan(2*x/sqrt(ra))/(2*sqrt(ra)), ra/4 > 0), (-acoth(2*x/sqrt(-ra))/(2*sqrt(-ra)), And(ra/4 < 0, x**2 > -ra/4)), (-atanh(2*x/sqrt(-ra))/(2*sqrt(-ra)), And(ra/4 < 0, x**2 < -ra/4))) assert manualintegrate(1 / (4 + 4 * x**2), x) == atan(x) / 4 assert manualintegrate(1 / (a + b * x**2), x) == atan(x / sqrt(a / b)) / (b * sqrt(a / b)) # asin assert manualintegrate(1 / sqrt(1 - x**2), x) == asin(x) assert manualintegrate(1 / sqrt(4 - 4 * x**2), x) == asin(x) / 2 assert manualintegrate(3 / sqrt(1 - 9 * x**2), x) == asin(3 * x) assert manualintegrate(1 / sqrt(4 - 9 * x**2), x) == asin(x * Rational(3, 2)) / 3 # asinh assert manualintegrate(1/sqrt(x**2 + 1), x) == \ asinh(x) assert manualintegrate(1/sqrt(x**2 + 4), x) == \ asinh(x/2) assert manualintegrate(1/sqrt(4*x**2 + 4), x) == \ asinh(x)/2 assert manualintegrate(1/sqrt(4*x**2 + 1), x) == \ asinh(2*x)/2 assert manualintegrate(1/sqrt(ra*x**2 + 1), x) == \ Piecewise((asin(x*sqrt(-ra))/sqrt(-ra), ra < 0), (asinh(sqrt(ra)*x)/sqrt(ra), ra > 0)) assert manualintegrate(1/sqrt(ra + x**2), x) == \ Piecewise((asinh(x*sqrt(1/ra)), ra > 0), (acosh(x*sqrt(-1/ra)), ra < 0)) # acosh assert manualintegrate(1/sqrt(x**2 - 1), x) == \ acosh(x) assert manualintegrate(1/sqrt(x**2 - 4), x) == \ acosh(x/2) assert manualintegrate(1/sqrt(4*x**2 - 4), x) == \ acosh(x)/2 assert manualintegrate(1/sqrt(9*x**2 - 1), x) == \ acosh(3*x)/3 assert manualintegrate(1/sqrt(ra*x**2 - 4), x) == \ Piecewise((acosh(sqrt(ra)*x/2)/sqrt(ra), ra > 0)) assert manualintegrate(1/sqrt(-ra + 4*x**2), x) == \ Piecewise((asinh(2*x*sqrt(-1/ra))/2, -ra > 0), (acosh(2*x*sqrt(1/ra))/2, -ra < 0)) # From https://www.wikiwand.com/en/List_of_integrals_of_inverse_trigonometric_functions # asin assert manualintegrate(asin(x), x) == x * asin(x) + sqrt(1 - x**2) assert manualintegrate(asin(a * x), x) == Piecewise( ((a * x * asin(a * x) + sqrt(-a**2 * x**2 + 1)) / a, Ne(a, 0)), (0, True)) assert manualintegrate(x * asin(a * x), x) == -a * Integral( x**2 / sqrt(-a**2 * x**2 + 1), x) / 2 + x**2 * asin(a * x) / 2 # acos assert manualintegrate(acos(x), x) == x * acos(x) - sqrt(1 - x**2) assert manualintegrate(acos(a * x), x) == Piecewise( ((a * x * acos(a * x) - sqrt(-a**2 * x**2 + 1)) / a, Ne(a, 0)), (pi * x / 2, True)) assert manualintegrate(x * acos(a * x), x) == a * Integral( x**2 / sqrt(-a**2 * x**2 + 1), x) / 2 + x**2 * acos(a * x) / 2 # atan assert manualintegrate(atan(x), x) == x * atan(x) - log(x**2 + 1) / 2 assert manualintegrate(atan(a * x), x) == Piecewise( ((a * x * atan(a * x) - log(a**2 * x**2 + 1) / 2) / a, Ne(a, 0)), (0, True)) assert manualintegrate( x * atan(a * x), x) == -a * (x / a**2 - atan(x / sqrt(a**(-2))) / (a**4 * sqrt(a**(-2)))) / 2 + x**2 * atan(a * x) / 2 # acsc assert manualintegrate( acsc(x), x) == x * acsc(x) + Integral(1 / (x * sqrt(1 - 1 / x**2)), x) assert manualintegrate( acsc(a * x), x) == x * acsc(a * x) + Integral(1 / (x * sqrt(1 - 1 / (a**2 * x**2))), x) / a assert manualintegrate(x * acsc(a * x), x) == x**2 * acsc(a * x) / 2 + Integral( 1 / sqrt(1 - 1 / (a**2 * x**2)), x) / (2 * a) # asec assert manualintegrate( asec(x), x) == x * asec(x) - Integral(1 / (x * sqrt(1 - 1 / x**2)), x) assert manualintegrate( asec(a * x), x) == x * asec(a * x) - Integral(1 / (x * sqrt(1 - 1 / (a**2 * x**2))), x) / a assert manualintegrate(x * asec(a * x), x) == x**2 * asec(a * x) / 2 - Integral( 1 / sqrt(1 - 1 / (a**2 * x**2)), x) / (2 * a) # acot assert manualintegrate(acot(x), x) == x * acot(x) + log(x**2 + 1) / 2 assert manualintegrate(acot(a * x), x) == Piecewise( ((a * x * acot(a * x) + log(a**2 * x**2 + 1) / 2) / a, Ne(a, 0)), (pi * x / 2, True)) assert manualintegrate( x * acot(a * x), x) == a * (x / a**2 - atan(x / sqrt(a**(-2))) / (a**4 * sqrt(a**(-2)))) / 2 + x**2 * acot(a * x) / 2 # piecewise assert manualintegrate(1/sqrt(ra-rb*x**2), x) == \ Piecewise((asin(x*sqrt(rb/ra))/sqrt(rb), And(-rb < 0, ra > 0)), (asinh(x*sqrt(-rb/ra))/sqrt(-rb), And(-rb > 0, ra > 0)), (acosh(x*sqrt(rb/ra))/sqrt(-rb), And(-rb > 0, ra < 0))) assert manualintegrate(1/sqrt(ra + rb*x**2), x) == \ Piecewise((asin(x*sqrt(-rb/ra))/sqrt(-rb), And(ra > 0, rb < 0)), (asinh(x*sqrt(rb/ra))/sqrt(rb), And(ra > 0, rb > 0)), (acosh(x*sqrt(-rb/ra))/sqrt(rb), And(ra < 0, rb > 0)))
def test_sympy__functions__elementary__hyperbolic__atanh(): from sympy.functions.elementary.hyperbolic import atanh assert _test_args(atanh(2))
def test_atanh(): x = Symbol('x') #at specific points assert atanh(0) == 0 assert atanh(I) == I * pi / 4 assert atanh(-I) == -I * pi / 4 assert atanh(1) is oo assert atanh(-1) is -oo assert atanh(nan) is nan # at infinites assert atanh(oo) == -I * pi / 2 assert atanh(-oo) == I * pi / 2 assert atanh(I * oo) == I * pi / 2 assert atanh(-I * oo) == -I * pi / 2 assert atanh(zoo) == I * AccumBounds(-pi / 2, pi / 2) #properties assert atanh(-x) == -atanh(x) assert atanh(I / sqrt(3)) == I * pi / 6 assert atanh(-I / sqrt(3)) == -I * pi / 6 assert atanh(I * sqrt(3)) == I * pi / 3 assert atanh(-I * sqrt(3)) == -I * pi / 3 assert atanh(I * (1 + sqrt(2))) == pi * I * Rational(3, 8) assert atanh(I * (sqrt(2) - 1)) == pi * I / 8 assert atanh(I * (1 - sqrt(2))) == -pi * I / 8 assert atanh(-I * (1 + sqrt(2))) == pi * I * Rational(-3, 8) assert atanh(I * sqrt(5 + 2 * sqrt(5))) == I * pi * Rational(2, 5) assert atanh(-I * sqrt(5 + 2 * sqrt(5))) == I * pi * Rational(-2, 5) assert atanh(I * (2 - sqrt(3))) == pi * I / 12 assert atanh(I * (sqrt(3) - 2)) == -pi * I / 12 assert atanh(oo) == -I * pi / 2 # Symmetry assert atanh(Rational(-1, 2)) == -atanh(S.Half) # inverse composition assert unchanged(atanh, tanh(Symbol('v1'))) assert atanh(tanh(-5, evaluate=False)) == -5 assert atanh(tanh(0, evaluate=False)) == 0 assert atanh(tanh(7, evaluate=False)) == 7 assert atanh(tanh(I, evaluate=False)) == I assert atanh(tanh(-I, evaluate=False)) == -I assert atanh(tanh(-11 * I, evaluate=False)) == -11 * I + 4 * I * pi assert atanh(tanh(3 + I)) == 3 + I assert atanh(tanh(4 + 5 * I)) == 4 - 2 * I * pi + 5 * I assert atanh(tanh(pi / 2)) == pi / 2 assert atanh(tanh(pi)) == pi assert atanh(tanh(-3 + 7 * I)) == -3 - 2 * I * pi + 7 * I assert atanh(tanh(9 - I * 2 / 3)) == 9 - I * 2 / 3 assert atanh(tanh(-32 - 123 * I)) == -32 - 123 * I + 39 * I * pi
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)))
def test_atanh_rewrite(): x = Symbol('x') assert atanh(x).rewrite(log) == (log(1 + x) - log(1 - x)) / 2
def test_manualintegrate_rational(): assert manualintegrate(1 / (4 - x**2), x) == Piecewise( (acoth(x / 2) / 2, x**2 > 4), (atanh(x / 2) / 2, x**2 < 4)) assert manualintegrate(1 / (-1 + x**2), x) == Piecewise( (-acoth(x), x**2 > 1), (-atanh(x), x**2 < 1))