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 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 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_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 test_leading_term():
x = Symbol('x')
assert cosh(x).as_leading_term(x) == 1
assert coth(x).as_leading_term(x) == 1 / x
assert acosh(x).as_leading_term(x) == I * pi / 2
assert acoth(x).as_leading_term(x) == I * pi / 2
for func in [sinh, tanh, asinh, atanh]:
assert func(x).as_leading_term(x) == x
for func in [sinh, cosh, tanh, coth, asinh, acosh, atanh, acoth]:
for arg in (1 / x, S.Half):
eq = func(arg)
assert eq.as_leading_term(x) == eq
for func in [csch, sech]:
eq = func(S.Half)
assert eq.as_leading_term(x) == eq
def test_messy():
from sympy.functions.elementary.complexes import re
from sympy.functions.elementary.hyperbolic import (acosh, acoth)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (asin, atan)
from sympy.functions.special.bessel import besselj
from sympy.functions.special.error_functions import (Chi, E1, Shi, Si)
from sympy.integrals.transforms import (fourier_transform,
laplace_transform)
assert laplace_transform(Si(x), x, s) == ((-atan(s) + pi / 2) / s, 0, True)
assert laplace_transform(Shi(x), x, s) == (acoth(s) / s, -oo, s**2 > 1)
# where should the logs be simplified?
assert laplace_transform(Chi(x), x,
s) == ((log(s**(-2)) - log(1 - 1 / s**2)) /
(2 * s), -oo, s**2 > 1)
# TODO maybe simplify the inequalities? when the simplification
# allows for generators instead of symbols this will work
assert laplace_transform(besselj(a, x), x, s)[1:] == \
(0, (re(a) > -2) & (re(a) > -1))
# NOTE s < 0 can be done, but argument reduction is not good enough yet
ans = fourier_transform(besselj(1, x) / x, x, s, noconds=False)
assert tuple([ans[0].factor(deep=True).expand(), ans[1]]) == \
(Piecewise((0, (s > 1/(2*pi)) | (s < -1/(2*pi))),
(2*sqrt(-4*pi**2*s**2 + 1), True)), s > 0)
# TODO FT(besselj(0,x)) - conditions are messy (but for acceptable reasons)
# - folding could be better
assert integrate(E1(x)*besselj(0, x), (x, 0, oo), meijerg=True) == \
log(1 + sqrt(2))
assert integrate(E1(x)*besselj(1, x), (x, 0, oo), meijerg=True) == \
log(S.Half + sqrt(2)/2)
assert integrate(1/x/sqrt(1 - x**2), x, meijerg=True) == \
Piecewise((-acosh(1/x), abs(x**(-2)) > 1), (I*asin(1/x), True))
def test_sympy__functions__elementary__hyperbolic__acoth():
from sympy.functions.elementary.hyperbolic import acoth
assert _test_args(acoth(2))
def test_sympy__functions__elementary__hyperbolic__acoth():
from sympy.functions.elementary.hyperbolic import acoth
assert _test_args(acoth(2))
def test_acoth_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: acoth(x).fdiff(2))
def test_acoth_series():
x = Symbol('x')
assert acoth(x).series(x, 0, 10) == \
I*pi/2 + x + x**3/3 + x**5/5 + x**7/7 + x**9/9 + O(x**10)
def test_acoth_rewrite():
x = Symbol('x')
assert acoth(x).rewrite(log) == (log(1 + 1 / x) - log(1 - 1 / x)) / 2
def test_acoth():
x = Symbol('x')
#at specific points
assert acoth(0) == I * pi / 2
assert acoth(I) == -I * pi / 4
assert acoth(-I) == I * pi / 4
assert acoth(1) is oo
assert acoth(-1) is -oo
assert acoth(nan) is nan
# at infinites
assert acoth(oo) == 0
assert acoth(-oo) == 0
assert acoth(I * oo) == 0
assert acoth(-I * oo) == 0
assert acoth(zoo) == 0
#properties
assert acoth(-x) == -acoth(x)
assert acoth(I / sqrt(3)) == -I * pi / 3
assert acoth(-I / sqrt(3)) == I * pi / 3
assert acoth(I * sqrt(3)) == -I * pi / 6
assert acoth(-I * sqrt(3)) == I * pi / 6
assert acoth(I * (1 + sqrt(2))) == -pi * I / 8
assert acoth(-I * (sqrt(2) + 1)) == pi * I / 8
assert acoth(I * (1 - sqrt(2))) == pi * I * Rational(3, 8)
assert acoth(I * (sqrt(2) - 1)) == pi * I * Rational(-3, 8)
assert acoth(I * sqrt(5 + 2 * sqrt(5))) == -I * pi / 10
assert acoth(-I * sqrt(5 + 2 * sqrt(5))) == I * pi / 10
assert acoth(I * (2 + sqrt(3))) == -pi * I / 12
assert acoth(-I * (2 + sqrt(3))) == pi * I / 12
assert acoth(I * (2 - sqrt(3))) == pi * I * Rational(-5, 12)
assert acoth(I * (sqrt(3) - 2)) == pi * I * Rational(5, 12)
# Symmetry
assert acoth(Rational(-1, 2)) == -acoth(S.Half)
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))
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)))
