Symbolic Integration¶

class sage.symbolic.integration.integral.DefiniteIntegral[source]¶

Bases: BuiltinFunction

The symbolic function representing a definite integral.

EXAMPLES:

Sage

sage: from sage.symbolic.integration.integral import definite_integral
sage: definite_integral(sin(x),x,0,pi)
2

Python

>>> from sage.all import *
>>> from sage.symbolic.integration.integral import definite_integral
>>> definite_integral(sin(x),x,Integer(0),pi)
2

Sage Live

from sage.symbolic.integration.integral import definite_integral
definite_integral(sin(x),x,0,pi)

class sage.symbolic.integration.integral.IndefiniteIntegral[source]¶

Bases: BuiltinFunction

Class to represent an indefinite integral.

EXAMPLES:

Sage

sage: from sage.symbolic.integration.integral import indefinite_integral
sage: indefinite_integral(log(x), x) #indirect doctest
x*log(x) - x
sage: indefinite_integral(x^2, x)
1/3*x^3
sage: indefinite_integral(4*x*log(x), x)
2*x^2*log(x) - x^2
sage: indefinite_integral(exp(x), 2*x)
2*e^x

Python

>>> from sage.all import *
>>> from sage.symbolic.integration.integral import indefinite_integral
>>> indefinite_integral(log(x), x) #indirect doctest
x*log(x) - x
>>> indefinite_integral(x**Integer(2), x)
1/3*x^3
>>> indefinite_integral(Integer(4)*x*log(x), x)
2*x^2*log(x) - x^2
>>> indefinite_integral(exp(x), Integer(2)*x)
2*e^x

Sage Live

from sage.symbolic.integration.integral import indefinite_integral
indefinite_integral(log(x), x) #indirect doctest
indefinite_integral(x^2, x)
indefinite_integral(4*x*log(x), x)
indefinite_integral(exp(x), 2*x)

sage.symbolic.integration.integral.integral(expression, v=None, a=None, b=None, algorithm=None, hold=False)[source]¶

Return the indefinite integral with respect to the variable \(v\), ignoring the constant of integration. Or, if endpoints \(a\) and \(b\) are specified, returns the definite integral over the interval \([a, b]\).

If self has only one variable, then it returns the integral with respect to that variable.

If definite integration fails, it could be still possible to evaluate the definite integral using indefinite integration with the Newton - Leibniz theorem (however, the user has to ensure that the indefinite integral is continuous on the compact interval \([a,b]\) and this theorem can be applied).

INPUT:

v – a variable or variable name; this can also be a tuple of the variable (optional) and endpoints (i.e., (x,0,1) or (0,1))
a – (optional) lower endpoint of definite integral
b – (optional) upper endpoint of definite integral
algorithm – (default: 'maxima', 'libgiac' and 'sympy') one of
- 'maxima' – use maxima
- 'sympy' – use sympy (also in Sage)
- 'mathematica_free' – use http://integrals.wolfram.com/
- 'fricas' – use FriCAS (the optional fricas spkg has to be installed)
- 'giac' – use Giac
- 'libgiac' – use libgiac

To prevent automatic evaluation, use the hold argument.

See also

To integrate a polynomial over a polytope, use the optional latte_int package sage.geometry.polyhedron.base.Polyhedron_base.integrate().

EXAMPLES:

Sage

sage: x = var('x')
sage: h = sin(x)/(cos(x))^2
sage: h.integral(x)
1/cos(x)

Python

>>> from sage.all import *
>>> x = var('x')
>>> h = sin(x)/(cos(x))**Integer(2)
>>> h.integral(x)
1/cos(x)

Sage Live

x = var('x')
h = sin(x)/(cos(x))^2
h.integral(x)

Sage

sage: f = x^2/(x+1)^3
sage: f.integral(x)
1/2*(4*x + 3)/(x^2 + 2*x + 1) + log(x + 1)

Python

>>> from sage.all import *
>>> f = x**Integer(2)/(x+Integer(1))**Integer(3)
>>> f.integral(x)
1/2*(4*x + 3)/(x^2 + 2*x + 1) + log(x + 1)

Sage Live

f = x^2/(x+1)^3
f.integral(x)

Python

>>> from sage.all import *
>>> f = x**Integer(2)/(x+Integer(1))**Integer(3)
>>> f.integral(x)
1/2*(4*x + 3)/(x^2 + 2*x + 1) + log(x + 1)

Sage Live

f = x^2/(x+1)^3
f.integral(x)

Sage

sage: f = x*cos(x^2)
sage: f.integral(x, 0, sqrt(pi))
0
sage: f.integral(x, a=-pi, b=pi)
0

Python

>>> from sage.all import *
>>> f = x*cos(x**Integer(2))
>>> f.integral(x, Integer(0), sqrt(pi))
0
>>> f.integral(x, a=-pi, b=pi)
0

Sage Live

f = x*cos(x^2)
f.integral(x, 0, sqrt(pi))
f.integral(x, a=-pi, b=pi)

Python

>>> from sage.all import *
>>> f = x*cos(x**Integer(2))
>>> f.integral(x, Integer(0), sqrt(pi))
0
>>> f.integral(x, a=-pi, b=pi)
0

Sage Live

f = x*cos(x^2)
f.integral(x, 0, sqrt(pi))
f.integral(x, a=-pi, b=pi)

Sage

sage: f(x) = sin(x)
sage: f.integral(x, 0, pi/2)
1

Python

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression(sin(x)).function(x)
>>> f.integral(x, Integer(0), pi/Integer(2))
1

Sage Live

f(x) = sin(x)
f.integral(x, 0, pi/2)

Python

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression(sin(x)).function(x)
>>> f.integral(x, Integer(0), pi/Integer(2))
1

Sage Live

f(x) = sin(x)
f.integral(x, 0, pi/2)

The variable is required, but the endpoints are optional:

Sage

sage: y = var('y')
sage: integral(sin(x), x)
-cos(x)
sage: integral(sin(x), y)
y*sin(x)
sage: integral(sin(x), x, pi, 2*pi)
-2
sage: integral(sin(x), y, pi, 2*pi)
pi*sin(x)
sage: integral(sin(x), (x, pi, 2*pi))
-2
sage: integral(sin(x), (y, pi, 2*pi))
pi*sin(x)

Python

>>> from sage.all import *
>>> y = var('y')
>>> integral(sin(x), x)
-cos(x)
>>> integral(sin(x), y)
y*sin(x)
>>> integral(sin(x), x, pi, Integer(2)*pi)
-2
>>> integral(sin(x), y, pi, Integer(2)*pi)
pi*sin(x)
>>> integral(sin(x), (x, pi, Integer(2)*pi))
-2
>>> integral(sin(x), (y, pi, Integer(2)*pi))
pi*sin(x)

Sage Live

y = var('y')
integral(sin(x), x)
integral(sin(x), y)
integral(sin(x), x, pi, 2*pi)
integral(sin(x), y, pi, 2*pi)
integral(sin(x), (x, pi, 2*pi))
integral(sin(x), (y, pi, 2*pi))

Using the hold parameter it is possible to prevent automatic evaluation, which can then be evaluated via simplify():

Sage

sage: integral(x^2, x, 0, 3)
9
sage: a = integral(x^2, x, 0, 3, hold=True) ; a
integrate(x^2, x, 0, 3)
sage: a.simplify()
9

Python

>>> from sage.all import *
>>> integral(x**Integer(2), x, Integer(0), Integer(3))
9
>>> a = integral(x**Integer(2), x, Integer(0), Integer(3), hold=True) ; a
integrate(x^2, x, 0, 3)
>>> a.simplify()
9

Sage Live

integral(x^2, x, 0, 3)
a = integral(x^2, x, 0, 3, hold=True) ; a
a.simplify()

Constraints are sometimes needed:

Sage

sage: var('x, n')
(x, n)
sage: integral(x^n,x)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(n>0)', see `assume?`
for more details)
Is n equal to -1?
sage: assume(n > 0)
sage: integral(x^n,x)
x^(n + 1)/(n + 1)
sage: forget()

Python

>>> from sage.all import *
>>> var('x, n')
(x, n)
>>> integral(x**n,x)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(n>0)', see `assume?`
for more details)
Is n equal to -1?
>>> assume(n > Integer(0))
>>> integral(x**n,x)
x^(n + 1)/(n + 1)
>>> forget()

Sage Live

var('x, n')
integral(x^n,x)
assume(n > 0)
integral(x^n,x)
forget()

Usually the constraints are of sign, but others are possible:

Sage

sage: assume(n==-1)
sage: integral(x^n,x)
log(x)

Python

>>> from sage.all import *
>>> assume(n==-Integer(1))
>>> integral(x**n,x)
log(x)

Sage Live

assume(n==-1)
integral(x^n,x)

Note that an exception is raised when a definite integral is divergent:

Sage

sage: forget() # always remember to forget assumptions you no longer need
sage: integrate(1/x^3,(x,0,1))
Traceback (most recent call last):
...
ValueError: Integral is divergent.
sage: integrate(1/x^3,x,-1,3)
Traceback (most recent call last):
...
ValueError: Integral is divergent.

Python

>>> from sage.all import *
>>> forget() # always remember to forget assumptions you no longer need
>>> integrate(Integer(1)/x**Integer(3),(x,Integer(0),Integer(1)))
Traceback (most recent call last):
...
ValueError: Integral is divergent.
>>> integrate(Integer(1)/x**Integer(3),x,-Integer(1),Integer(3))
Traceback (most recent call last):
...
ValueError: Integral is divergent.

Sage Live

forget() # always remember to forget assumptions you no longer need
integrate(1/x^3,(x,0,1))
integrate(1/x^3,x,-1,3)

But Sage can calculate the convergent improper integral of this function:

Sage

sage: integrate(1/x^3,x,1,infinity)
1/2

Python

>>> from sage.all import *
>>> integrate(Integer(1)/x**Integer(3),x,Integer(1),infinity)
1/2

Sage Live

integrate(1/x^3,x,1,infinity)

The examples in the Maxima documentation:

Sage

sage: var('x, y, z, b')
(x, y, z, b)
sage: integral(sin(x)^3, x)
1/3*cos(x)^3 - cos(x)
sage: integral(x/sqrt(b^2-x^2), b)
x*log(2*b + 2*sqrt(b^2 - x^2))
sage: integral(x/sqrt(b^2-x^2), x)
-sqrt(b^2 - x^2)
sage: integral(cos(x)^2 * exp(x), x, 0, pi)
3/5*e^pi - 3/5
sage: integral(x^2 * exp(-x^2), x, -oo, oo)
1/2*sqrt(pi)

Python

>>> from sage.all import *
>>> var('x, y, z, b')
(x, y, z, b)
>>> integral(sin(x)**Integer(3), x)
1/3*cos(x)^3 - cos(x)
>>> integral(x/sqrt(b**Integer(2)-x**Integer(2)), b)
x*log(2*b + 2*sqrt(b^2 - x^2))
>>> integral(x/sqrt(b**Integer(2)-x**Integer(2)), x)
-sqrt(b^2 - x^2)
>>> integral(cos(x)**Integer(2) * exp(x), x, Integer(0), pi)
3/5*e^pi - 3/5
>>> integral(x**Integer(2) * exp(-x**Integer(2)), x, -oo, oo)
1/2*sqrt(pi)

Sage Live

var('x, y, z, b')
integral(sin(x)^3, x)
integral(x/sqrt(b^2-x^2), b)
integral(x/sqrt(b^2-x^2), x)
integral(cos(x)^2 * exp(x), x, 0, pi)
integral(x^2 * exp(-x^2), x, -oo, oo)

We integrate the same function in both Mathematica and Sage (via Maxima):

Sage

sage: _ = var('x, y, z')
sage: f = sin(x^2) + y^z
sage: g = mathematica(f)                            # optional - mathematica
sage: print(g)                                      # optional - mathematica
          z        2
         y  + Sin[x ]
sage: print(g.Integrate(x))                         # optional - mathematica
            z        Pi                2
         x y  + Sqrt[--] FresnelS[Sqrt[--] x]
                     2                 Pi
sage: print(f.integral(x))
x*y^z + 1/16*sqrt(pi)*((I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x)
                        + (I - 1)*sqrt(2)*erf((1/2*I - 1/2)*sqrt(2)*x)
                        - (I - 1)*sqrt(2)*erf(sqrt(-I)*x)
                        + (I + 1)*sqrt(2)*erf((-1)^(1/4)*x))

Python

>>> from sage.all import *
>>> _ = var('x, y, z')
>>> f = sin(x**Integer(2)) + y**z
>>> g = mathematica(f)                            # optional - mathematica
>>> print(g)                                      # optional - mathematica
          z        2
         y  + Sin[x ]
>>> print(g.Integrate(x))                         # optional - mathematica
            z        Pi                2
         x y  + Sqrt[--] FresnelS[Sqrt[--] x]
                     2                 Pi
>>> print(f.integral(x))
x*y^z + 1/16*sqrt(pi)*((I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x)
                        + (I - 1)*sqrt(2)*erf((1/2*I - 1/2)*sqrt(2)*x)
                        - (I - 1)*sqrt(2)*erf(sqrt(-I)*x)
                        + (I + 1)*sqrt(2)*erf((-1)^(1/4)*x))

Sage Live

_ = var('x, y, z')
f = sin(x^2) + y^z
g = mathematica(f)                            # optional - mathematica
print(g)                                      # optional - mathematica
print(g.Integrate(x))                         # optional - mathematica
print(f.integral(x))

Alternatively, just use algorithm=’mathematica_free’ to integrate via Mathematica over the internet (does NOT require a Mathematica license!):

Sage

sage: _ = var('x, y, z')  # optional - internet
sage: f = sin(x^2) + y^z   # optional - internet
sage: f.integrate(x, algorithm='mathematica_free')   # optional - internet
x*y^z + sqrt(1/2)*sqrt(pi)*fresnel_sin(sqrt(2)*x/sqrt(pi))

Python

>>> from sage.all import *
>>> _ = var('x, y, z')  # optional - internet
>>> f = sin(x**Integer(2)) + y**z   # optional - internet
>>> f.integrate(x, algorithm='mathematica_free')   # optional - internet
x*y^z + sqrt(1/2)*sqrt(pi)*fresnel_sin(sqrt(2)*x/sqrt(pi))

Sage Live

_ = var('x, y, z')  # optional - internet
f = sin(x^2) + y^z   # optional - internet
f.integrate(x, algorithm='mathematica_free')   # optional - internet

We can also use Sympy:

Sage

sage: integrate(x*sin(log(x)), x)
-1/5*x^2*(cos(log(x)) - 2*sin(log(x)))
sage: integrate(x*sin(log(x)), x, algorithm='sympy')                            # needs sympy
-1/5*x^2*cos(log(x)) + 2/5*x^2*sin(log(x))
sage: _ = var('y, z')
sage: (x^y - z).integrate(y)
-y*z + x^y/log(x)
sage: (x^y - z).integrate(y, algorithm='sympy')                                 # needs sympy
-y*z + cases(((log(x) != 0, x^y/log(x)), (1, y)))

Python

>>> from sage.all import *
>>> integrate(x*sin(log(x)), x)
-1/5*x^2*(cos(log(x)) - 2*sin(log(x)))
>>> integrate(x*sin(log(x)), x, algorithm='sympy')                            # needs sympy
-1/5*x^2*cos(log(x)) + 2/5*x^2*sin(log(x))
>>> _ = var('y, z')
>>> (x**y - z).integrate(y)
-y*z + x^y/log(x)
>>> (x**y - z).integrate(y, algorithm='sympy')                                 # needs sympy
-y*z + cases(((log(x) != 0, x^y/log(x)), (1, y)))

Sage Live

integrate(x*sin(log(x)), x)
integrate(x*sin(log(x)), x, algorithm='sympy')                            # needs sympy
_ = var('y, z')
(x^y - z).integrate(y)
(x^y - z).integrate(y, algorithm='sympy')                                 # needs sympy

We integrate the above function in Maple now:

Sage

sage: g = maple(f); g.sort()         # optional - maple
y^z+sin(x^2)
sage: g.integrate(x).sort()          # optional - maple
x*y^z+1/2*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*x)

Python

>>> from sage.all import *
>>> g = maple(f); g.sort()         # optional - maple
y^z+sin(x^2)
>>> g.integrate(x).sort()          # optional - maple
x*y^z+1/2*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*x)

Sage Live

g = maple(f); g.sort()         # optional - maple
g.integrate(x).sort()          # optional - maple

We next integrate a function with no closed form integral. Notice that the answer comes back as an expression that contains an integral itself.

Sage

sage: A = integral(1/ ((x-4) * (x^4+x+1)), x); A
integrate(1/((x^4 + x + 1)*(x - 4)), x)

Python

>>> from sage.all import *
>>> A = integral(Integer(1)/ ((x-Integer(4)) * (x**Integer(4)+x+Integer(1))), x); A
integrate(1/((x^4 + x + 1)*(x - 4)), x)

Sage Live

A = integral(1/ ((x-4) * (x^4+x+1)), x); A

Sometimes, in this situation, using the algorithm “maxima” gives instead a partially integrated answer:

Sage

sage: integral(1/(x**7-1),x,algorithm='maxima')
-1/7*integrate((x^5 + 2*x^4 + 3*x^3 + 4*x^2 + 5*x + 6)/(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1), x) + 1/7*log(x - 1)

Python

>>> from sage.all import *
>>> integral(Integer(1)/(x**Integer(7)-Integer(1)),x,algorithm='maxima')
-1/7*integrate((x^5 + 2*x^4 + 3*x^3 + 4*x^2 + 5*x + 6)/(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1), x) + 1/7*log(x - 1)

Sage Live

integral(1/(x**7-1),x,algorithm='maxima')

We now show that floats are not converted to rationals automatically since we by default have keepfloat: true in maxima.

Sage

sage: integral(e^(-x^2),(x, 0, 0.1))
0.05623145800914245*sqrt(pi)

Python

>>> from sage.all import *
>>> integral(e**(-x**Integer(2)),(x, Integer(0), RealNumber('0.1')))
0.05623145800914245*sqrt(pi)

Sage Live

integral(e^(-x^2),(x, 0, 0.1))

An example of an integral that fricas can integrate:

Sage

sage: f(x) = sqrt(x+sqrt(1+x^2))/x
sage: integrate(f(x), x, algorithm='fricas')      # optional - fricas
2*sqrt(x + sqrt(x^2 + 1)) - 2*arctan(sqrt(x + sqrt(x^2 + 1)))
 - log(sqrt(x + sqrt(x^2 + 1)) + 1) + log(sqrt(x + sqrt(x^2 + 1)) - 1)

Python

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression(sqrt(x+sqrt(Integer(1)+x**Integer(2)))/x).function(x)
>>> integrate(f(x), x, algorithm='fricas')      # optional - fricas
2*sqrt(x + sqrt(x^2 + 1)) - 2*arctan(sqrt(x + sqrt(x^2 + 1)))
 - log(sqrt(x + sqrt(x^2 + 1)) + 1) + log(sqrt(x + sqrt(x^2 + 1)) - 1)

Sage Live

f(x) = sqrt(x+sqrt(1+x^2))/x
integrate(f(x), x, algorithm='fricas')      # optional - fricas

where the default integrator obtains another answer:

Sage

sage: integrate(f(x), x)  # long time
1/8*sqrt(x)*gamma(1/4)*gamma(-1/4)^2*hypergeometric((-1/4, -1/4, 1/4),
                                                    (1/2, 3/4),
                                                    -1/x^2)/(pi*gamma(3/4))

Python

>>> from sage.all import *
>>> integrate(f(x), x)  # long time
1/8*sqrt(x)*gamma(1/4)*gamma(-1/4)^2*hypergeometric((-1/4, -1/4, 1/4),
                                                    (1/2, 3/4),
                                                    -1/x^2)/(pi*gamma(3/4))

Sage Live

integrate(f(x), x)  # long time

The following definite integral is not found by maxima:

Sage

sage: f(x) = (x^4 - 3*x^2 + 6) / (x^6 - 5*x^4 + 5*x^2 + 4)
sage: integrate(f(x), x, 1, 2, algorithm='maxima')
integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4), x, 1, 2)

Python

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression((x**Integer(4) - Integer(3)*x**Integer(2) + Integer(6)) / (x**Integer(6) - Integer(5)*x**Integer(4) + Integer(5)*x**Integer(2) + Integer(4))).function(x)
>>> integrate(f(x), x, Integer(1), Integer(2), algorithm='maxima')
integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4), x, 1, 2)

Sage Live

f(x) = (x^4 - 3*x^2 + 6) / (x^6 - 5*x^4 + 5*x^2 + 4)
integrate(f(x), x, 1, 2, algorithm='maxima')

but is nevertheless computed:

Sage

sage: integrate(f(x), x, 1, 2)
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)

Python

>>> from sage.all import *
>>> integrate(f(x), x, Integer(1), Integer(2))
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)

Sage Live

integrate(f(x), x, 1, 2)

Both fricas and sympy give the correct result:

Sage

sage: integrate(f(x), x, 1, 2, algorithm='fricas')  # optional - fricas
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)
sage: integrate(f(x), x, 1, 2, algorithm='sympy')                               # needs sympy
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)

Python

>>> from sage.all import *
>>> integrate(f(x), x, Integer(1), Integer(2), algorithm='fricas')  # optional - fricas
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)
>>> integrate(f(x), x, Integer(1), Integer(2), algorithm='sympy')                               # needs sympy
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)

Sage Live

integrate(f(x), x, 1, 2, algorithm='fricas')  # optional - fricas
integrate(f(x), x, 1, 2, algorithm='sympy')                               # needs sympy

Using Giac to integrate the absolute value of a trigonometric expression:

Sage

sage: integrate(abs(cos(x)), x, 0, 2*pi, algorithm='giac')
4
sage: result = integrate(abs(cos(x)), x, 0, 2*pi)
...
sage: result
4

Python

>>> from sage.all import *
>>> integrate(abs(cos(x)), x, Integer(0), Integer(2)*pi, algorithm='giac')
4
>>> result = integrate(abs(cos(x)), x, Integer(0), Integer(2)*pi)
...
>>> result
4

Sage Live

integrate(abs(cos(x)), x, 0, 2*pi, algorithm='giac')
result = integrate(abs(cos(x)), x, 0, 2*pi)
result

ALIASES: integral() and integrate() are the same.

EXAMPLES:

Here is an example where we have to use assume:

Sage

sage: a,b = var('a,b')
sage: integrate(1/(x^3 *(a+b*x)^(1/3)), x)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(a>0)', see `assume?`
for more details)
Is a positive or negative?

Python

>>> from sage.all import *
>>> a,b = var('a,b')
>>> integrate(Integer(1)/(x**Integer(3) *(a+b*x)**(Integer(1)/Integer(3))), x)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(a>0)', see `assume?`
for more details)
Is a positive or negative?

Sage Live

a,b = var('a,b')
integrate(1/(x^3 *(a+b*x)^(1/3)), x)

So we just assume that \(a>0\) and the integral works:

Sage

sage: assume(a>0)
sage: integrate(1/(x^3 *(a+b*x)^(1/3)), x)
2/9*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(7/3)
 - 1/9*b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3)
 + 2/9*b^2*log((b*x + a)^(1/3) - a^(1/3))/a^(7/3) + 1/6*(4*(b*x + a)^(5/3)*b^2
 - 7*(b*x + a)^(2/3)*a*b^2)/((b*x + a)^2*a^2 - 2*(b*x + a)*a^3 + a^4)

Python

>>> from sage.all import *
>>> assume(a>Integer(0))
>>> integrate(Integer(1)/(x**Integer(3) *(a+b*x)**(Integer(1)/Integer(3))), x)
2/9*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(7/3)
 - 1/9*b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3)
 + 2/9*b^2*log((b*x + a)^(1/3) - a^(1/3))/a^(7/3) + 1/6*(4*(b*x + a)^(5/3)*b^2
 - 7*(b*x + a)^(2/3)*a*b^2)/((b*x + a)^2*a^2 - 2*(b*x + a)*a^3 + a^4)

Sage Live

assume(a>0)
integrate(1/(x^3 *(a+b*x)^(1/3)), x)

sage.symbolic.integration.integral.integrate(expression, v=None, a=None, b=None, algorithm=None, hold=False)[source]¶

If self has only one variable, then it returns the integral with respect to that variable.

INPUT:

v – a variable or variable name; this can also be a tuple of the variable (optional) and endpoints (i.e., (x,0,1) or (0,1))
a – (optional) lower endpoint of definite integral
b – (optional) upper endpoint of definite integral
algorithm – (default: 'maxima', 'libgiac' and 'sympy') one of
- 'maxima' – use maxima
- 'sympy' – use sympy (also in Sage)
- 'mathematica_free' – use http://integrals.wolfram.com/
- 'fricas' – use FriCAS (the optional fricas spkg has to be installed)
- 'giac' – use Giac
- 'libgiac' – use libgiac

To prevent automatic evaluation, use the hold argument.

See also

To integrate a polynomial over a polytope, use the optional latte_int package sage.geometry.polyhedron.base.Polyhedron_base.integrate().

EXAMPLES:

Sage

sage: x = var('x')
sage: h = sin(x)/(cos(x))^2
sage: h.integral(x)
1/cos(x)

Python

>>> from sage.all import *
>>> x = var('x')
>>> h = sin(x)/(cos(x))**Integer(2)
>>> h.integral(x)
1/cos(x)

Sage Live

x = var('x')
h = sin(x)/(cos(x))^2
h.integral(x)

Sage

sage: f = x^2/(x+1)^3
sage: f.integral(x)
1/2*(4*x + 3)/(x^2 + 2*x + 1) + log(x + 1)

Python

>>> from sage.all import *
>>> f = x**Integer(2)/(x+Integer(1))**Integer(3)
>>> f.integral(x)
1/2*(4*x + 3)/(x^2 + 2*x + 1) + log(x + 1)

Sage Live

f = x^2/(x+1)^3
f.integral(x)

Python

>>> from sage.all import *
>>> f = x**Integer(2)/(x+Integer(1))**Integer(3)
>>> f.integral(x)
1/2*(4*x + 3)/(x^2 + 2*x + 1) + log(x + 1)

Sage Live

f = x^2/(x+1)^3
f.integral(x)

Sage

sage: f = x*cos(x^2)
sage: f.integral(x, 0, sqrt(pi))
0
sage: f.integral(x, a=-pi, b=pi)
0

Python

>>> from sage.all import *
>>> f = x*cos(x**Integer(2))
>>> f.integral(x, Integer(0), sqrt(pi))
0
>>> f.integral(x, a=-pi, b=pi)
0

Sage Live

f = x*cos(x^2)
f.integral(x, 0, sqrt(pi))
f.integral(x, a=-pi, b=pi)

Python

>>> from sage.all import *
>>> f = x*cos(x**Integer(2))
>>> f.integral(x, Integer(0), sqrt(pi))
0
>>> f.integral(x, a=-pi, b=pi)
0

Sage Live

f = x*cos(x^2)
f.integral(x, 0, sqrt(pi))
f.integral(x, a=-pi, b=pi)

Sage

sage: f(x) = sin(x)
sage: f.integral(x, 0, pi/2)
1

Python

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression(sin(x)).function(x)
>>> f.integral(x, Integer(0), pi/Integer(2))
1

Sage Live

f(x) = sin(x)
f.integral(x, 0, pi/2)

Python

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression(sin(x)).function(x)
>>> f.integral(x, Integer(0), pi/Integer(2))
1

Sage Live

f(x) = sin(x)
f.integral(x, 0, pi/2)

The variable is required, but the endpoints are optional:

Sage

sage: y = var('y')
sage: integral(sin(x), x)
-cos(x)
sage: integral(sin(x), y)
y*sin(x)
sage: integral(sin(x), x, pi, 2*pi)
-2
sage: integral(sin(x), y, pi, 2*pi)
pi*sin(x)
sage: integral(sin(x), (x, pi, 2*pi))
-2
sage: integral(sin(x), (y, pi, 2*pi))
pi*sin(x)

Python

>>> from sage.all import *
>>> y = var('y')
>>> integral(sin(x), x)
-cos(x)
>>> integral(sin(x), y)
y*sin(x)
>>> integral(sin(x), x, pi, Integer(2)*pi)
-2
>>> integral(sin(x), y, pi, Integer(2)*pi)
pi*sin(x)
>>> integral(sin(x), (x, pi, Integer(2)*pi))
-2
>>> integral(sin(x), (y, pi, Integer(2)*pi))
pi*sin(x)

Sage Live

y = var('y')
integral(sin(x), x)
integral(sin(x), y)
integral(sin(x), x, pi, 2*pi)
integral(sin(x), y, pi, 2*pi)
integral(sin(x), (x, pi, 2*pi))
integral(sin(x), (y, pi, 2*pi))

Using the hold parameter it is possible to prevent automatic evaluation, which can then be evaluated via simplify():

Sage

sage: integral(x^2, x, 0, 3)
9
sage: a = integral(x^2, x, 0, 3, hold=True) ; a
integrate(x^2, x, 0, 3)
sage: a.simplify()
9

Python

>>> from sage.all import *
>>> integral(x**Integer(2), x, Integer(0), Integer(3))
9
>>> a = integral(x**Integer(2), x, Integer(0), Integer(3), hold=True) ; a
integrate(x^2, x, 0, 3)
>>> a.simplify()
9

Sage Live

integral(x^2, x, 0, 3)
a = integral(x^2, x, 0, 3, hold=True) ; a
a.simplify()

Constraints are sometimes needed:

Sage

sage: var('x, n')
(x, n)
sage: integral(x^n,x)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(n>0)', see `assume?`
for more details)
Is n equal to -1?
sage: assume(n > 0)
sage: integral(x^n,x)
x^(n + 1)/(n + 1)
sage: forget()

Python

>>> from sage.all import *
>>> var('x, n')
(x, n)
>>> integral(x**n,x)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(n>0)', see `assume?`
for more details)
Is n equal to -1?
>>> assume(n > Integer(0))
>>> integral(x**n,x)
x^(n + 1)/(n + 1)
>>> forget()

Sage Live

var('x, n')
integral(x^n,x)
assume(n > 0)
integral(x^n,x)
forget()

Usually the constraints are of sign, but others are possible:

Sage

sage: assume(n==-1)
sage: integral(x^n,x)
log(x)

Python

>>> from sage.all import *
>>> assume(n==-Integer(1))
>>> integral(x**n,x)
log(x)

Sage Live

assume(n==-1)
integral(x^n,x)

Note that an exception is raised when a definite integral is divergent:

Sage

sage: forget() # always remember to forget assumptions you no longer need
sage: integrate(1/x^3,(x,0,1))
Traceback (most recent call last):
...
ValueError: Integral is divergent.
sage: integrate(1/x^3,x,-1,3)
Traceback (most recent call last):
...
ValueError: Integral is divergent.

Python

>>> from sage.all import *
>>> forget() # always remember to forget assumptions you no longer need
>>> integrate(Integer(1)/x**Integer(3),(x,Integer(0),Integer(1)))
Traceback (most recent call last):
...
ValueError: Integral is divergent.
>>> integrate(Integer(1)/x**Integer(3),x,-Integer(1),Integer(3))
Traceback (most recent call last):
...
ValueError: Integral is divergent.

Sage Live

forget() # always remember to forget assumptions you no longer need
integrate(1/x^3,(x,0,1))
integrate(1/x^3,x,-1,3)

But Sage can calculate the convergent improper integral of this function:

Sage

sage: integrate(1/x^3,x,1,infinity)
1/2

Python

>>> from sage.all import *
>>> integrate(Integer(1)/x**Integer(3),x,Integer(1),infinity)
1/2

Sage Live

integrate(1/x^3,x,1,infinity)

The examples in the Maxima documentation:

Sage

sage: var('x, y, z, b')
(x, y, z, b)
sage: integral(sin(x)^3, x)
1/3*cos(x)^3 - cos(x)
sage: integral(x/sqrt(b^2-x^2), b)
x*log(2*b + 2*sqrt(b^2 - x^2))
sage: integral(x/sqrt(b^2-x^2), x)
-sqrt(b^2 - x^2)
sage: integral(cos(x)^2 * exp(x), x, 0, pi)
3/5*e^pi - 3/5
sage: integral(x^2 * exp(-x^2), x, -oo, oo)
1/2*sqrt(pi)

Python

>>> from sage.all import *
>>> var('x, y, z, b')
(x, y, z, b)
>>> integral(sin(x)**Integer(3), x)
1/3*cos(x)^3 - cos(x)
>>> integral(x/sqrt(b**Integer(2)-x**Integer(2)), b)
x*log(2*b + 2*sqrt(b^2 - x^2))
>>> integral(x/sqrt(b**Integer(2)-x**Integer(2)), x)
-sqrt(b^2 - x^2)
>>> integral(cos(x)**Integer(2) * exp(x), x, Integer(0), pi)
3/5*e^pi - 3/5
>>> integral(x**Integer(2) * exp(-x**Integer(2)), x, -oo, oo)
1/2*sqrt(pi)

Sage Live

var('x, y, z, b')
integral(sin(x)^3, x)
integral(x/sqrt(b^2-x^2), b)
integral(x/sqrt(b^2-x^2), x)
integral(cos(x)^2 * exp(x), x, 0, pi)
integral(x^2 * exp(-x^2), x, -oo, oo)

We integrate the same function in both Mathematica and Sage (via Maxima):

Sage

sage: _ = var('x, y, z')
sage: f = sin(x^2) + y^z
sage: g = mathematica(f)                            # optional - mathematica
sage: print(g)                                      # optional - mathematica
          z        2
         y  + Sin[x ]
sage: print(g.Integrate(x))                         # optional - mathematica
            z        Pi                2
         x y  + Sqrt[--] FresnelS[Sqrt[--] x]
                     2                 Pi
sage: print(f.integral(x))
x*y^z + 1/16*sqrt(pi)*((I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x)
                        + (I - 1)*sqrt(2)*erf((1/2*I - 1/2)*sqrt(2)*x)
                        - (I - 1)*sqrt(2)*erf(sqrt(-I)*x)
                        + (I + 1)*sqrt(2)*erf((-1)^(1/4)*x))

Python

>>> from sage.all import *
>>> _ = var('x, y, z')
>>> f = sin(x**Integer(2)) + y**z
>>> g = mathematica(f)                            # optional - mathematica
>>> print(g)                                      # optional - mathematica
          z        2
         y  + Sin[x ]
>>> print(g.Integrate(x))                         # optional - mathematica
            z        Pi                2
         x y  + Sqrt[--] FresnelS[Sqrt[--] x]
                     2                 Pi
>>> print(f.integral(x))
x*y^z + 1/16*sqrt(pi)*((I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x)
                        + (I - 1)*sqrt(2)*erf((1/2*I - 1/2)*sqrt(2)*x)
                        - (I - 1)*sqrt(2)*erf(sqrt(-I)*x)
                        + (I + 1)*sqrt(2)*erf((-1)^(1/4)*x))

Sage Live

_ = var('x, y, z')
f = sin(x^2) + y^z
g = mathematica(f)                            # optional - mathematica
print(g)                                      # optional - mathematica
print(g.Integrate(x))                         # optional - mathematica
print(f.integral(x))

Alternatively, just use algorithm=’mathematica_free’ to integrate via Mathematica over the internet (does NOT require a Mathematica license!):

Sage

sage: _ = var('x, y, z')  # optional - internet
sage: f = sin(x^2) + y^z   # optional - internet
sage: f.integrate(x, algorithm='mathematica_free')   # optional - internet
x*y^z + sqrt(1/2)*sqrt(pi)*fresnel_sin(sqrt(2)*x/sqrt(pi))

Python

>>> from sage.all import *
>>> _ = var('x, y, z')  # optional - internet
>>> f = sin(x**Integer(2)) + y**z   # optional - internet
>>> f.integrate(x, algorithm='mathematica_free')   # optional - internet
x*y^z + sqrt(1/2)*sqrt(pi)*fresnel_sin(sqrt(2)*x/sqrt(pi))

Sage Live

_ = var('x, y, z')  # optional - internet
f = sin(x^2) + y^z   # optional - internet
f.integrate(x, algorithm='mathematica_free')   # optional - internet

We can also use Sympy:

Sage

sage: integrate(x*sin(log(x)), x)
-1/5*x^2*(cos(log(x)) - 2*sin(log(x)))
sage: integrate(x*sin(log(x)), x, algorithm='sympy')                            # needs sympy
-1/5*x^2*cos(log(x)) + 2/5*x^2*sin(log(x))
sage: _ = var('y, z')
sage: (x^y - z).integrate(y)
-y*z + x^y/log(x)
sage: (x^y - z).integrate(y, algorithm='sympy')                                 # needs sympy
-y*z + cases(((log(x) != 0, x^y/log(x)), (1, y)))

Python

>>> from sage.all import *
>>> integrate(x*sin(log(x)), x)
-1/5*x^2*(cos(log(x)) - 2*sin(log(x)))
>>> integrate(x*sin(log(x)), x, algorithm='sympy')                            # needs sympy
-1/5*x^2*cos(log(x)) + 2/5*x^2*sin(log(x))
>>> _ = var('y, z')
>>> (x**y - z).integrate(y)
-y*z + x^y/log(x)
>>> (x**y - z).integrate(y, algorithm='sympy')                                 # needs sympy
-y*z + cases(((log(x) != 0, x^y/log(x)), (1, y)))

Sage Live

integrate(x*sin(log(x)), x)
integrate(x*sin(log(x)), x, algorithm='sympy')                            # needs sympy
_ = var('y, z')
(x^y - z).integrate(y)
(x^y - z).integrate(y, algorithm='sympy')                                 # needs sympy

We integrate the above function in Maple now:

Sage

sage: g = maple(f); g.sort()         # optional - maple
y^z+sin(x^2)
sage: g.integrate(x).sort()          # optional - maple
x*y^z+1/2*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*x)

Python

>>> from sage.all import *
>>> g = maple(f); g.sort()         # optional - maple
y^z+sin(x^2)
>>> g.integrate(x).sort()          # optional - maple
x*y^z+1/2*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*x)

Sage Live

g = maple(f); g.sort()         # optional - maple
g.integrate(x).sort()          # optional - maple

We next integrate a function with no closed form integral. Notice that the answer comes back as an expression that contains an integral itself.

Sage

sage: A = integral(1/ ((x-4) * (x^4+x+1)), x); A
integrate(1/((x^4 + x + 1)*(x - 4)), x)

Python

>>> from sage.all import *
>>> A = integral(Integer(1)/ ((x-Integer(4)) * (x**Integer(4)+x+Integer(1))), x); A
integrate(1/((x^4 + x + 1)*(x - 4)), x)

Sage Live

A = integral(1/ ((x-4) * (x^4+x+1)), x); A

Sometimes, in this situation, using the algorithm “maxima” gives instead a partially integrated answer:

Sage

sage: integral(1/(x**7-1),x,algorithm='maxima')
-1/7*integrate((x^5 + 2*x^4 + 3*x^3 + 4*x^2 + 5*x + 6)/(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1), x) + 1/7*log(x - 1)

Python

>>> from sage.all import *
>>> integral(Integer(1)/(x**Integer(7)-Integer(1)),x,algorithm='maxima')
-1/7*integrate((x^5 + 2*x^4 + 3*x^3 + 4*x^2 + 5*x + 6)/(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1), x) + 1/7*log(x - 1)

Sage Live

integral(1/(x**7-1),x,algorithm='maxima')

We now show that floats are not converted to rationals automatically since we by default have keepfloat: true in maxima.

Sage

sage: integral(e^(-x^2),(x, 0, 0.1))
0.05623145800914245*sqrt(pi)

Python

>>> from sage.all import *
>>> integral(e**(-x**Integer(2)),(x, Integer(0), RealNumber('0.1')))
0.05623145800914245*sqrt(pi)

Sage Live

integral(e^(-x^2),(x, 0, 0.1))

An example of an integral that fricas can integrate:

Sage

sage: f(x) = sqrt(x+sqrt(1+x^2))/x
sage: integrate(f(x), x, algorithm='fricas')      # optional - fricas
2*sqrt(x + sqrt(x^2 + 1)) - 2*arctan(sqrt(x + sqrt(x^2 + 1)))
 - log(sqrt(x + sqrt(x^2 + 1)) + 1) + log(sqrt(x + sqrt(x^2 + 1)) - 1)

Python

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression(sqrt(x+sqrt(Integer(1)+x**Integer(2)))/x).function(x)
>>> integrate(f(x), x, algorithm='fricas')      # optional - fricas
2*sqrt(x + sqrt(x^2 + 1)) - 2*arctan(sqrt(x + sqrt(x^2 + 1)))
 - log(sqrt(x + sqrt(x^2 + 1)) + 1) + log(sqrt(x + sqrt(x^2 + 1)) - 1)

Sage Live

f(x) = sqrt(x+sqrt(1+x^2))/x
integrate(f(x), x, algorithm='fricas')      # optional - fricas

where the default integrator obtains another answer:

Sage

sage: integrate(f(x), x)  # long time
1/8*sqrt(x)*gamma(1/4)*gamma(-1/4)^2*hypergeometric((-1/4, -1/4, 1/4),
                                                    (1/2, 3/4),
                                                    -1/x^2)/(pi*gamma(3/4))

Python

>>> from sage.all import *
>>> integrate(f(x), x)  # long time
1/8*sqrt(x)*gamma(1/4)*gamma(-1/4)^2*hypergeometric((-1/4, -1/4, 1/4),
                                                    (1/2, 3/4),
                                                    -1/x^2)/(pi*gamma(3/4))

Sage Live

integrate(f(x), x)  # long time

The following definite integral is not found by maxima:

Sage

sage: f(x) = (x^4 - 3*x^2 + 6) / (x^6 - 5*x^4 + 5*x^2 + 4)
sage: integrate(f(x), x, 1, 2, algorithm='maxima')
integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4), x, 1, 2)

Python

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression((x**Integer(4) - Integer(3)*x**Integer(2) + Integer(6)) / (x**Integer(6) - Integer(5)*x**Integer(4) + Integer(5)*x**Integer(2) + Integer(4))).function(x)
>>> integrate(f(x), x, Integer(1), Integer(2), algorithm='maxima')
integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4), x, 1, 2)

Sage Live

f(x) = (x^4 - 3*x^2 + 6) / (x^6 - 5*x^4 + 5*x^2 + 4)
integrate(f(x), x, 1, 2, algorithm='maxima')

but is nevertheless computed:

Sage

sage: integrate(f(x), x, 1, 2)
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)

Python

>>> from sage.all import *
>>> integrate(f(x), x, Integer(1), Integer(2))
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)

Sage Live

integrate(f(x), x, 1, 2)

Both fricas and sympy give the correct result:

Sage

sage: integrate(f(x), x, 1, 2, algorithm='fricas')  # optional - fricas
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)
sage: integrate(f(x), x, 1, 2, algorithm='sympy')                               # needs sympy
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)

Python

>>> from sage.all import *
>>> integrate(f(x), x, Integer(1), Integer(2), algorithm='fricas')  # optional - fricas
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)
>>> integrate(f(x), x, Integer(1), Integer(2), algorithm='sympy')                               # needs sympy
-1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)

Sage Live

integrate(f(x), x, 1, 2, algorithm='fricas')  # optional - fricas
integrate(f(x), x, 1, 2, algorithm='sympy')                               # needs sympy

Using Giac to integrate the absolute value of a trigonometric expression:

Sage

sage: integrate(abs(cos(x)), x, 0, 2*pi, algorithm='giac')
4
sage: result = integrate(abs(cos(x)), x, 0, 2*pi)
...
sage: result
4

Python

>>> from sage.all import *
>>> integrate(abs(cos(x)), x, Integer(0), Integer(2)*pi, algorithm='giac')
4
>>> result = integrate(abs(cos(x)), x, Integer(0), Integer(2)*pi)
...
>>> result
4

Sage Live

integrate(abs(cos(x)), x, 0, 2*pi, algorithm='giac')
result = integrate(abs(cos(x)), x, 0, 2*pi)
result

ALIASES: integral() and integrate() are the same.

EXAMPLES:

Here is an example where we have to use assume:

Sage

sage: a,b = var('a,b')
sage: integrate(1/(x^3 *(a+b*x)^(1/3)), x)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(a>0)', see `assume?`
for more details)
Is a positive or negative?

Python

>>> from sage.all import *
>>> a,b = var('a,b')
>>> integrate(Integer(1)/(x**Integer(3) *(a+b*x)**(Integer(1)/Integer(3))), x)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(a>0)', see `assume?`
for more details)
Is a positive or negative?

Sage Live

a,b = var('a,b')
integrate(1/(x^3 *(a+b*x)^(1/3)), x)

So we just assume that \(a>0\) and the integral works:

Sage

sage: assume(a>0)
sage: integrate(1/(x^3 *(a+b*x)^(1/3)), x)
2/9*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(7/3)
 - 1/9*b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3)
 + 2/9*b^2*log((b*x + a)^(1/3) - a^(1/3))/a^(7/3) + 1/6*(4*(b*x + a)^(5/3)*b^2
 - 7*(b*x + a)^(2/3)*a*b^2)/((b*x + a)^2*a^2 - 2*(b*x + a)*a^3 + a^4)

Python

>>> from sage.all import *
>>> assume(a>Integer(0))
>>> integrate(Integer(1)/(x**Integer(3) *(a+b*x)**(Integer(1)/Integer(3))), x)
2/9*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(7/3)
 - 1/9*b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3)
 + 2/9*b^2*log((b*x + a)^(1/3) - a^(1/3))/a^(7/3) + 1/6*(4*(b*x + a)^(5/3)*b^2
 - 7*(b*x + a)^(2/3)*a*b^2)/((b*x + a)^2*a^2 - 2*(b*x + a)*a^3 + a^4)

Sage Live

assume(a>0)
integrate(1/(x^3 *(a+b*x)^(1/3)), x)